Handbook of Pharmacokinetics and Toxicokinetics [2 ed.] 1032197056, 9781032197050

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Handbook of Pharmacokinetics and Toxicokinetics This fully revised and expanded volume is an effort to blend the common approaches to pharmacokinetics and toxicokinetics. It integrates the principles held in common by both felds through a logical and systematic approach, which includes mathematical descriptions of physical and physiological processes employed in the approaches to pharmacokinetics and toxicokinetics modeling. It emphasizes general principles and concepts and related, isolated applications and case study observations. The systematic compilation of mathematical concepts and methodologies allows readers to decide on relevant concepts and approaches for their research, scientifc or regulatory decisions, or for offering advanced courses/workshops and seminars. Features: ◾ Comprehensive handbook on principles and applications of PK/TK appealing to a diverse audience including scientists and students. ◾ An excellent text fully revised and fully updated for anyone interested in the theoretical and practical pharmacokinetics. ◾ The systematic compilation of mathematical concepts and methodologies allows readers to decide on relevant concepts and approaches for their research. ◾ Incorporates research relevant to SDGs and of interest to industrial and regulatory environmental scientists involved in chemical contamination research and regulatory decision making related to soil, water, and ocean. ◾ Includes sections on applications and case studies. Dr Mehdi Boroujerdi earned his PhD in pharmaceutics and pharmacokinetics from the University of North Carolina at Chapel Hill in 1978. He completed his post-doctoral training at the National Institutes of Health, National Institute of Environmental Health Sciences at Research Triangle Park. He served as professor of pharmaceutics/pharmacokinetics with tenure at Northeastern University, Boston, MA (1982–2002); professor of pharmaceutics and pharmacokinetics at the MCPHS University, School of Pharmacy, Boston, MA (2002–2005); professor of pharmaceutical sciences with tenure at the Albany College of Pharmacy and Health Sciences, Albany, New York (2005–2015); and professor of pharmaceutical sciences with tenure at the College of Health Sciences, University of Massachusetts, Lowell, MA (2015–2017). Dr Boroujerdi has also served as Dean of the School of Pharmacy at Bouvé College of Health Sciences at Northeastern University (1988–1999); as Dean of the School of Pharmacy-Boston at MCPHS University (2002–2005); as Dean of Pharmacy and Vice President for Academic Affairs, Provost, at the Albany College of Pharmacy and Health Sciences (2006–2012); and Founding Dean of the School of Pharmacy and Pharmaceutical Sciences at the College of Health Sciences, University of Massachusetts at Lowell (2015–2017). He also served as the Dean of Research and Graduate Studies at MCPHS (2003–2005), and Director of Graduate Programs in Biomedical Sciences at Northeastern University (1988–1999). Dr Mehdi Boroujerdi has 112 peer-reviewed publications and is the sole author of two books Pharmacokinetics, Principles and Applications (McGraw Hill, 2002) and Pharmacokinetics and Toxicokinetics (CRC Publications, 2015). He has trained many graduate and undergraduate students through his research programs focused on pharmacokinetics and toxicodynamics of anticancer drugs, carcinogenesis, and effux proteins. He also served as consultant to fve pharmaceutical companies. As a professor he taught graduate courses in advanced pharmacokinetics and biopharmaceutics, drug metabolism, advanced pharmaceutics, and drug delivery systems. His teaching of undergraduate courses included drug discovery and development, biopharmaceutics and pharmacokinetics, physical pharmacy and pharmaceutics, and pharmacokinetics in disease states.

Handbook of Pharmacokinetics and Toxicokinetics Second Edition

Mehdi Boroujerdi

Second edition published 2023 by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN and by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 © 2023 Mehdi Boroujerdi First edition published by CRC Press 2015 CRC Press is an imprint of Informa UK Limited The right of Mehdi Boroujerdi to be identifed as author of this work has been asserted in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. This book contains information obtained from authentic and highly regarded sources. While all reasonable efforts have been made to publish reliable data and information, neither the author[s] nor the publisher can accept any legal responsibility or liability for any errors or omissions that may be made. The publishers wish to make clear that any views or opinions expressed in this book by individual editors, authors or contributors are personal to them and do not necessarily refect the views/opinions of the publishers. The information or guidance contained in this book is intended for use by medical, scientifc or health-care professionals and is provided strictly as a supplement to the medical or other professional’s own judgement, their knowledge of the patient’s medical history, relevant manufacturer’s instructions and the appropriate best practice guidelines. Because of the rapid advances in medical science, any information or advice on dosages, procedures or diagnoses should be independently verifed. The reader is strongly urged to consult the relevant national drug formulary and the drug companies’ and device or material manufacturers’ printed instructions, and their websites, before administering or utilizing any of the drugs, devices or materials mentioned in this book. This book does not indicate whether a particular treatment is appropriate or suitable for a particular individual. Ultimately it is the sole responsibility of the medical professional to make his or her own professional judgements, so as to advise and treat patients appropriately. The authors and publishers have also attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identifcation and explanation without intent to infringe. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library

Library of Congress Cataloging-in-Publication Data Names: Boroujerdi, Mehdi, author. Title: Handbook of pharmacokinetics and toxicokinetics / Mehdi Boroujerdi. Other titles: Pharmacokinetics and toxicokinetics Description: Second edition. | Abingdon, Oxon ; Boca Raton, FL : CRC Press, 2023. | Revised edition of: Pharmacokinetics and toxicokinetics / Mehdi Boroujerdi. [2015]. | Includes bibliographical references and index. | Identifers: LCCN 2022054962 | ISBN 9781032197050 (hbk) | ISBN 9781032197470 (pbk) | ISBN 9781003260660 (ebk) Subjects: LCSH: Pharmacokinetics. | Drugs--Toxicology. Classifcation: LCC RM301.5 .B658 2023 | DDC 615.7--dc23/eng/20221214 LC record available at https://lccn.loc.gov/2022054962 ISBN: 9781032197050 (hbk) ISBN: 9781032197470 (pbk) ISBN: 9781003260660 (ebk) DOI: 10.1201/9781003260660 Typeset in Warnock Pro by Deanta Global Publishing Services, Chennai, India

I dedicate this book with much love to my sons Mazy and Bob. –Mehdi Boroujerdi

Table of Contents Preface

xxvii

1 Pharmacokinetics and Toxicokinetics

1

1 1 Introduction

1

1 2 Pharmacokinetics and Pharmacodynamics

1

1 2 1 Clinical Pharmacokinetics/Pharmacodynamics

3

1 2 2 PK/PD Modeling and Pharmacometrics

3

1 2 3 Population PK and PK/PD Modeling

4

1 2 3 1 Infuences of Genetics and Genomics on PK/PD and TK/TD

5

1 2 3 2 Biomarkers

7

1 3 Toxicokinetics and Toxicodynamics 1 3 1 TK/TD Modeling, Population Toxicokinetics, and Toxicogenetics

9 9

1 4 Basic Concepts and Assumptions of PK and TK

10

1 5 Introduction to the Routes of Administration

12

References

13

2 PK/TK Considerations of Auricular (Otic) – Buccal/Sublingual, and Ocular/ Ophthalmic Routes of Administration

19

2 1 Auricular or OTIC Route of Administration

19

2 1 1 Overview

19

2 1 2 Blood-Labyrinth-Barrier and Auricular Absorption, Distribution, Metabolism, and Excretion

20

2 1 2 1 Syndromes and the Sites of Absorption

20

2 1 2 2 Auricular Distribution, Metabolism, and Excretion 2 1 3 Auricular Rate Equations and PK/TK Models 2 2 Buccal and Sublingual Routes of Administration

21 22 24

2 2 1 Overview

24

2 2 2 Buccal and Sublingual ADME and Related Rate Equations

25

2 2 3 Saliva

27

2 3 Ocular/Ophthalmic Routes of Administration 2 3 1 Overview

29 29

2 3 2 The Blood Aqueous Barrier

29

2 3 3 The Blood-Retinal Barrier

30

2 3 3 1 BRB Effux Transporters

31

2 3 3 2 BRB Infux Transporters

31

2 3 4 Kinetics of BRB Infux Permeability Clearance – Small Water-Soluble Compounds Given Systemically 2 3 5 Recommended Ocular Routes for Drug Administration 2 3 5 1 Conjunctival Route of Administration

32 33 33

2 3 5 2 Subconjunctival Route of Administration

34

2 3 5 3 Intracameral Route of Administration

34 vii

TABLE OF CONTENTS

2 3 5 4 Intravitreal Route of Administration

34

2 3 5 5 Intracorneal Route of Administration

37

2 3 5 6 Retrobulbar, Peribulbar, and Sub-Tenon Routes of Administration

41

References

42

3 PK-TK Considerations of Nasal, Pulmonary and Oral Routes of Administration 3 1 Nasal Route of Administration/Exposure

50 50

3 1 1 Vestibule, Atrium, Valves, and Turbines

50

3 1 2 Mucosal Epithelium

50

3 1 3 Olfactory Epithelium

52

3 1 4 Nasal ADME of Xenobiotics

53

3 1 5 Nasal Rate Equations – PK/TK Models

54

3 1 5 1 A Nose-to-Systemic Circulation PK/TK Model

54

3 1 5 2 An Inclusive Nose-to-Brain PK/TK Model

56

3 2 Pulmonary Route of Administration/Exposure

58

3 2 1 Overview

58

3 2 2 Morphological Differences of Airways Among Species

59

3 2 3 Pulmonary Microbiome

59

3 2 4 ADME of Xenobiotics in the Pulmonary Tract

60

3 2 4 1 Pulmonary Absorption, Deposition, and Clearance

60

3 2 4 2 Transport Proteins of Pulmonary Tract

60

3 2 4 3 Respiratory Tract Metabolic Enzymes – Lung Metabolism of Xenobiotics

61

3 2 4 4 Pulmonary Deposition and Disposition of Particles

62

3 2 4 5 Pulmonary Absorption of Gases and Vapors

63

3 2 4 6 Relevant Pulmonary Kinetic Parameters

66

3 2 4 7 Role of the Lungs in PK/TK of Xenobiotics: Pulmonary First-Pass Metabolism 3 2 4 8 Pulmonary Rate Equations 3 3 Gastrointestinal (Oral) Route of Administration or Exposure 3 3 1 Overview

67 67 73 73

3 3 2 Physiologic and Dynamic Attributes of the GI Tract Infuencing Xenobiotic Absorption 3 3 2 1 Regional pH of GI Tract and pH-Partition Theory

74

3 3 2 2 Absorptive Surface Area

77

3 3 2 3 Gastric Emptying and Gastric Accommodation

78

3 3 2 4 Intestinal Motility: Small Intestinal Transit Time

80

3 3 2 5 Role of Bile Salts

81

3 3 2 6 Hepatic First-Pass Metabolism (Pre-systemic Hepatic Extraction)

81

3 3 2 7 Gastrointestinal Metabolism – Role of CYP450 Isozymes

84

3 3 2 8 GI Tract Infux and Effux Transport Proteins

86

3 3 2 9 Role of Intestinal Microbiotas References viii

74

89 90

TABLE OF CONTENTS

4 PK/TK Considerations of Intra-Arterial, Intramuscular, Intraperitoneal, Intravenous, and Subcutaneous Routes of Administration 4 1 Intra-Arterial Route of Administration

50 107

4 1 1 Overview

107

4 1 2 Intra-Arterial PK/TK Remarks

107

4 2 Intramuscular Route of Administration

107

4 2 1 Overview

107

4 2 2 ADME of Intramuscular Route of Administration

108

4 2 2 1 Rate Equations of Intramuscularly Injected Xenobiotics 4 3 Intraperitoneal Route of Administration 4 3 1 Overview 4 3 1 1 Applications of the IP Route of Administration 4 3 2 Kinetics of Intraperitoneal Transport of Xenobiotics 4 4 Intravenous Route of Administration 4 4 1 Overview

109 112 112 113 114 119 119

4 4 1 1 Intravenous Injection Drawbacks

120

4 4 1 2 Bolus Injection, Continuous Infusion, Intermittent Infusion

120

4 4 2 Intravenous PK/TK Analysis

120

4 5 Subcutaneous Route of Administration

121

4 5 1 Overview

121

4 5 2 Rate Equations of Subcutaneously Injected Xenobiotics

122

4 5 2 1 Subcutaneous Diffusion Rate-Limited Model

122

4 5 2 2 Subcutaneous Dissolution Rate-Limited Model

123

4 5 2 3 Subcutaneous Capacity-Limited Model

123

4 5 2 4 Subcutaneous Models Based on Diffusion Equations

124

4 5 2 5 Other PK Models for Subcutaneous Insulin

125

References

126

5 PK/TK Considerations of Transdermal, Intradermal, and Intraepidermal Routes of Administration

132

5 1 Transdermal Route of Administration

132

5 1 1 Overview

132

5 1 2 Stratum Corneum

132

5 1 3 Epidermis

134

5 1 4 Dermis

135

5 1 4 1 Dermis Cells

135

5 1 4 2 Dermis Appendages

135

5 1 5 Transdermal Absorption, Metabolism, and Disposition

136

5 1 5 1 Transdermal Absorption

136

5 1 5 2 Cutaneous Metabolism of Xenobiotics

138

5 1 5 3 Skin Transport Proteins

139

5 1 6 Mathematical Interpretations of Transdermal Absorption of Xenobiotics

139 ix

TABLE OF CONTENTS

5 1 6 1 Diffusion Models

140

5 1 6 2 Skin-Perm Model

141

5 1 6 3 One-Layered Diffusion Model

142

5 1 6 4 Two-Layered Diffusion Model

143

5 1 6 5 Compartmental Analysis

145

5 1 6 6 Diffusion–Diffusion Model and Statistical Moments for Percutaneous Absorption 5 1 6 7 Physiological Modeling of Percutaneous Absorption of Xenobiotics

153 155

5 1 6 8 Six-Compartment Intradermal Disposition Kinetics of Xenobiotics with Contralateral Compartments 5 2 Intradermal Route of Administration 5 2 1 Overview

156 158 158

5 2 2 PK/TK Parameters and Constants of Drug Absorption from Intradermal Space to Blood 5 3 Intraepidermal Route of Administration 5 3 1 Overview References

159 159 160

6 PK/TK Considerations of Rectal, Vaginal, and Intraovarian Routes of Administration 6 1 Rectal Route of Administration

167 167

6 1 1 Overview

167

6 1 2 Pharmacokinetic Considerations of the Rectal Route of Administration

167

6 2 Vaginal Route of Administration 6 2 1 Overview

169 169

6 2 2 Vaginal Microbiota

170

6 2 3 Pharmacokinetic Considerations of the Vaginal Route of Administration

170

6 3 Intraovarian Route of Administration 6 3 1 Overview References

173 173 173

7 PK/TK Considerations of Absorption Mechanisms and Rate Equations

177

7 1 Introduction

177

7 2 Passive Diffusion

177

7 2 1 Transcellular and Paracellular Diffusion 7 2 1 1 Transcellular and Paracellular Transport Rate Equations 7 2 2 Partition Coeffcient

177 178 180

7 2 2 1 CLOGPcoeff

181

7 2 2 2 MLOGPcoeff

182

7 2 3 Distribution Coeffcient

183

7 2 4 Diffusion Coeffcient

184

7 2 5 Permeation and Permeability Constant

185

7 2 5 1 Estimation of Apparent Permeability Constant Using Caco-2 Cells 7 3 Carrier-Mediated Transcellular Diffusion x

158

187 188

TABLE OF CONTENTS

7 4 Transcellular Diffusion Subjected to P-Glycoprotein Effux

189

7 4 1 Overview

189

7 4 2 Pgp Structure and Function

189

7 4 3 Pgp Computational Equations

192

7 5 Active Transport

194

7 6 Endocytosis and Pinocytosis

196

7 7 Solvent Drag, Osmosis, and Two-Pore Theory

196

7 8 Ion-Pair Absorption

198

References

200

8 PK – TK Considerations of Distribution Mechanisms and Rate Equations

210

8 1 Introduction

210

8 2 Factors Infuencing the Distribution of Xenobiotics in the Body

210

8 2 1 Infuence of Total Body Water on Xenobiotic Distribution

210

8 2 2 Effect of Blood Flow and Organ/Tissue Perfusion on Xenobiotic Distribution

211

8 2 2 1 Perfusion-Limited Distribution and Permeability-Limited Distribution (Transcapillary Exchange of Xenobiotics) 8 2 3 Effect of Binding to Plasma Proteins on Xenobiotic Distribution 8 2 3 1 Estimation of Protein-Binding Parameters

213 216 217

8 2 4 Infuence of Physicochemical Characteristics of Xenobiotics on Their Distribution

221

8 2 5 Infuence of Extent of Penetration Through the Physiological Barriers, and Parallel Removal Processes on Xenobiotic Distribution 8 2 6 Physiological Barriers

221 222

8 2 6 1 Blood–Brain Barrier

222

8 2 6 2 Blood–Lymph Barrier

227

8 2 6 3 Placental Barrier

227

8 2 6 4 Blood–Testis Barrier

228

8 2 6 5 Blood–Aqueous Humor Barrier (BAB) – also Read Chapter 2, Section 232

229

8 2 7 Effect of Body Weight and Composition on Xenobiotic Distribution 8 2 7 1 Ideal Body Weight (IBW in kg)

229 229

8 2 7 2 Body Surface Area (BSA in m )

229

8 2 7 3 Body Mass Index (BMI in kg/m2)

230

8 2 7 4 Lean Body Mass (LBM in kg)

230

2

8 2 8 Impact of Disease States on Xenobiotic Distribution

230

8 2 8 1 Congestive Heart Failure (CHF)

230

8 2 8 2 Chronic Renal Failure (CRF)

230

8 2 8 3 Hepatic Diseases

231

8 2 8 4 Cystic Fibrosis (CF)

231

8 2 8 5 Other Conditions

231

8 3 Applications and Case Studies

231

References

231 xi

TABLE OF CONTENTS

9 PK/TK Considerations of Xenobiotic Metabolism Mechanisms and Rate Equations

237

9 2 Liver

237

9 3 Metabolic Pathways

239

9 3 1 Phase I Metabolism

240

9 3 1 1 Flavin-Containing Monooxygenases

240

9 3 1 2 Flavin-Containing Amine Oxidoreductases

240

9 3 1 3 Epoxide Hydrolases

241

9 3 1 4 Cytochrome P450

241

9 3 1 5 Alcohol Dehydrogenase

245

9 3 1 6 Diamine Oxidase (Histaminase)

246

9 3 1 7 Aldehyde Dehydrogenases

246

9 3 1 8 Xanthine Oxidase

247

9 3 1 9 Carboxylesterases

247

9 3 1 10 Peptidase (Protease/Proteinase)

247

9 3 2 Phase II Metabolism: Conjugation

248

9 3 2 1 Glucuronidation

248

9 3 2 2 Sulfation

250

9 3 2 3 Methylation

251

9 3 2 4 Acetylation (Acylation)

252

9 3 2 5 Glutathione Conjugation

253

9 3 2 6 Amino Acid Conjugation

254

9 3 3 In Vitro Systems for Xenobiotics Metabolism Study 9 3 3 1 Subcellular Fractions

255 255

9 3 3 2 Cellular Fractions – Hepatocytes

257

9 3 3 3 Organ Fractions (Precision Cut Liver Slices)

258

9 3 3 4 In-Situ and Ex-Vivo Liver Perfusion Techniques

258

9 3 3 5 Antibodies Against CYP Proteins

260

9 3 3 6 bDNA Probes

260

9 3 3 7 Pure and Recombinant Enzymes

260

9 3 3 8 Cell Lines

260

9 3 4 In Vivo Samples for Xenobiotic Metabolism Study

261

9 3 4 1 Serum and Plasma Samples

261

9 3 4 2 Urine Samples

262

9 3 4 3 Bile Samples

262

9 3 4 4 Portal Vein Cannulation

263

9 4 Kinetics of In Vitro Metabolism

xii

237

9 1 Introduction

263

9 4 1 Michaelis–Menten Kinetics

263

9 4 2 In Vitro Intrinsic Metabolic Clearance

267

9 4 3 The Catalytic Effciency and Turnover Number

267

9 4 4 Estimation of the Michaelis–Menten Parameters

267

TABLE OF CONTENTS

9 4 4 1 Lineweaver–Burk Plot or Double Reciprocal Plot

267

9 4 4 2 Hanes–Woolfe Plot

268

9 4 4 3 Eadie–Hofstee Plot

269

9 4 4 4 Direct Linear Plot

269

9 4 4 5 Hill Plot 9 4 5 Assimilation of Intrinsic Clearance in Hepatic Clearance Using Liver Models

270 271

9 4 5 1 The Well-Stirred Model (Venous Equilibration Model)

272

9 4 5 2 The Parallel-Tube Model (Undistributed Sinusoidal Model)

273

9 4 5 3 The Dispersion Model

274

9 4 5 4 Physiological PK/TK Organ Model for the Liver

274

9 4 5 5 Zonal Liver Model

276

9 4 6 Inhibition of Xenobiotic Metabolism 9 4 6 1 Classifcations of Metabolic Inhibition 9 4 7 Induction of Xenobiotic Metabolism

276 277 285

9 5 Applications and Case Studies

286

References

286

10 PK – TK Considerations of Renal Function and Elimination of Xenobiotics - Estimation of Parameters and Constants

309

10 1 Introduction

309

10 2 Glomerular Filtration

309

10 3 Tubular Reabsorption and Secretion

309

10 4 Loop of Henle, Distal Tubule, and Collecting Ducts

311

10 5 Estimation of GFR

312

10 5 1 Exogenous Markers of GFR

312

10 5 1 1 Radioisotope-Labeled Compounds

312

10 5 1 2 Inulin

312

10 5 1 3 Iohexol

313

10 5 2 Endogenous Markers of GFR (GFR Biomarkers)

313

10 5 2 1 Creatinine Clearance

313

10 5 2 2 Cystatin C

317

10 6 PK/TK Analysis of Urinary Data

318

10 6 1 PK/TK Analysis of Urinary Excretion of Unchanged Xenobiotic – Intravenous Bolus Injection

320

10 6 1 1 Rate Plot – Intravenous Bolus Dose

321

10 6 1 2 ARE Plot aka Sigma-Minus Plot – Intravenous Bolus Dose

323

10 6 2 PK/TK Analysis of Urinary Elimination of Xenobiotic Metabolites Following Intravenous Bolus Injection

325

10 6 2 1 Amount of Metabolite Remaining to be Eliminated from the Body Following IV Bolus Dose Administration

326

10 6 2 2 Urinary Elimination Rate of Metabolite and Estimation of Metabolic Rate Constant

327 xiii

TABLE OF CONTENTS

10 6 3 PK/TK Analysis of Urinary Excretion of Unchanged Xenobiotic Following Zero-Order Intravenous Infusion

327

10 6 3 1 Urinary Excretion Rate of Unchanged Xenobiotic During Zero-Order Intravenous Infusion and After Attaining the Steady-State Level

327

10 6 3 2 Cumulative Amount of Urinary Excretion of Unchanged Xenobiotic During Zero-Order Intravenous Infusion

329

10 6 4 PK/TK Analysis of Urinary Excretion of Unchanged Xenobiotic Following First-Order Absorption from an Extravascular Route of Administration

329

10 6 4 1 Urinary Excretion Rate of Unchanged Xenobiotic Following FirstOrder Absorption into the Systemic Circulation and Estimation of Absorption Rate Constant

329

10 6 4 2 Amount of Xenobiotic Remaining to be Excreted Unchanged in the Urine Following the First-Order Absorption into the Systemic Circulation from an Extravascular Route of Administration

331

10 6 5 PK/TK Analysis of Urinary Excretion of Unchanged Xenobiotics that Follow the Two-Compartment Model Subsequent to Intravenous Bolus Injection

333

10 6 5 1 Urinary Excretion Rate of Unchanged Xenobiotic Following Intravenous Bolus Injection – Two-Compartment Model

333

10 6 5 2 Amount of Xenobiotic Remaining to be Excreted Unchanged in the Urine Following an Intravenous Bolus Injection – TwoCompartment Model

334

10 6 6 General Equations of PK/TK Multicompartment Analysis of Urinary Excretion Data – First-Order Absorption and Intravenous Infusion

336

10 6 7 PK/TK Analysis of Urinary Excretion Data Using Principles of NonCompartmental Analysis

337

10 7 Renal Metabolism

338

10 8 Renal Mechanistic Models

338

10 9 Estimation of PK/TK Parameters and Constants of Xenobiotics Elimination When Using Renal Replacement Therapy – Dialysis

340

10 9 1 Overview

340

10 9 2 Hemodialysis

341

10 9 3 Peritoneal Dialysis

341

10 9 4 Composition of Dialysate

341

10 9 5 Dialysis Clearance

342

10 9 6 Effects of Dialysis on PK/TK Parameters and Constants

343

10 10 Applications and Case Studies

346

References

346

11 Elimination Rates and Clearances (Excretion + Metabolism)

xiv

352

11 1 Introduction

352

11 2 Rates of Elimination

353

11 3 Extraction Ratio

353

TABLE OF CONTENTS

11 4 Clearances

355

11 4 1 Estimation of Clearance Using Theoretical Models

356

11 4 1 1 Well-Stirred Model

356

11 4 1 2 Parallel Model

358

11 4 1 3 Dispersion Model

358

11 4 2 Clearance Scale-Up in Mammalian Species

359

11 4 2 1 Extrapolation of Clearance from Animal to Human

359

11 4 2 2 Body-Weight Dependent Extrapolation of Clearance in Humans

361

11 4 3 Clearance Estimation in Linear PK/TK

362

11 4 4 Clearance Estimation in Nonlinear PK/TK

363

11 4 4 1 Nonlinear Clearance in Target-Mediated Drug Disposition References

364 364

12 Approaches in PK/PD and TK/TD Mathematical Modeling

368

12 1 Introduction

368

12 2 Physiologically Based PK/TK Models

368

12 2 1 Description

368

12 2 2 Model Development

371

12 2 2 1 Flow-Limited (Perfusion-Limited) Models

372

12 2 2 2 Permeability-Limited (Membrane-Limited) Models

374

12 2 2 3 Variability of Physiological/Biochemical Key Parameters

375

12 2 3 Predictive Capability and Sensitivity Analysis 12 3 Linear PK/TK Compartmental Analysis 12 3 1 Linear Dose-Independent Compartmental Analysis

376 378 379

12 3 1 1 Mathematical Descriptions of a Xenobiotic Administered via an Extravascular Route of Administration: Time Course of the Amount Change at the Site of Absorption in the Body and the Eliminated Amount from the Body

379

12 3 1 2 Mathematical Description of a Xenobiotic Administered Intravenously – Time Course of the Amount Change in the Body, Formation of Metabolite(s), and Elimination from the Body

381

12 3 1 3 Mathematical Relationships of the Central Compartment for an Intravenously Administered Xenobiotic that Follows Multicompartment Model: Use of Input-Disposition Function and General Partial Fraction Theorem

383

12 3 1 4 Mathematical Relationships of the Peripheral Compartment for an Intravenously Administered Xenobiotic that Follows Multicompartment Model: Use of Input-Disposition Function and General Partial Fraction Theorem

385

12 3 1 5 Mathematical Relationships When a Xenobiotic and Its Metabolite(s) Follow Multicompartmental Model – Intravenous Bolus Dose 12 3 2 Dose-Dependent Compartmental Analysis 12 3 2 1 Compartmental Models with Michaelis–Menten Kinetics

386 388 388 xv

TABLE OF CONTENTS

12 4 Non-Compartmental Analysis Based on Statistical Moment Theory

392

12 4 1 Overview

392

12 4 2 Mean Residence Time and Mean Input Time

393

12 4 3 Total Body Clearance and Apparent Volume of Distribution

394

12 5 PK-PD and TK-TD Modeling

395

12 5 1 Overview

395

12 5 2 Xenobiotic–Receptor Interaction and the Law of Mass Action

396

12 5 3 Pharmacodynamic Models of Plasma Concentration and Response

398

12 5 3 1 Linear Pharmacodynamic Model

398

12 5 3 2 Log-Linear Pharmacodynamic Model

399

12 5 3 3 Nonlinear Hyperbolic Emax Model

400

12 5 3 4 Non-Hyperbolic Sigmoidal Model

400

12 5 4 PK/PD and TK/TD Models

402

12 5 4 1 Linking the Nonlinear Hyperbolic Emax Concept to Compartmental Models

404

12 5 4 2 Linking Non-Hyperbolic Sigmoidal Model to PK/TK Models with Different Inputs

406

12 5 5 The Effect Compartment

407

12 5 5 1 PK/TK Models Connected to the Effect Compartment

408

12 6 Physiologically Based PK/TK Models with Effect Compartment

411

12 7 Hysteresis Loops in PK/PD or TK/TD Relationships

412

12 8 Target-Mediated Drug Disposition Models

413

12 8 1 One-Compartment TMDD Models

414

12 8 2 Two-Compartment TMDD Models

415

References

417

13 Practical Applications of PK/TK Models: Instantaneous Exposure to Xenobiotics Single Intravenous Bolus Injection

423

13 1 Introduction

423

13 2 Linear One-Compartment Open Model – Intravenous Bolus Injection

423

13 2 1 Half-Life of Elimination

425

13 2 2 Time Constant

425

13 2 3 Apparent Volume of Distribution

425

13 2 4 Total Body Clearance

426

13 2 5 Duration of Action

426

13 2 6 Estimation of Fraction of Dose in the Body at a Given Time

427

13 2 7 Estimation of Fraction of Dose Eliminated by All Routes of Elimination at a Given Time

427

13 2 8 Determination of the Area Under Plasma Concentration–Time Curve after Intravenous Bolus Injection

427

13 3 Linear Two-Compartment Open Model with Bolus Injection in the Central Compartment and Elimination from the Central Compartment xvi

428

TABLE OF CONTENTS

13 3 1 Equations of the Two-Compartment Model

429

13 3 2 Estimation of the Initial Plasma Concentration and Volumes of Distribution, Two-Compartment Model 13 3 3 Estimation of the Rate Constants of Distribution and Elimination 13 3 4 Half-Lives of the Two-Compartment Model

432 433 433

13 3 4 1 Biological Half-Life – Two-Compartment Model

434

13 3 4 2 Elimination Half-Life – Two-Compartment Model

434

13 3 4 3 Half-Life of α – Two-Compartment Model

434

13 3 4 4 Half-Life of k12

434

13 3 4 5 Half-Life of k 21

434

13 3 5 Determination of the Area Under the Plasma Concentration–Time Curve, Volumes of Distribution, and Clearances – Two-Compartment Model

435

13 3 6 Assessment of the Time Course of Xenobiotics in the Peripheral Compartment – Two-Compartment Model

436

13 4 Linear Two-Compartment Open Model with Bolus Injection in the Central Compartment and Elimination from the Peripheral Compartment

438

13 5 Linear Three-Compartment Open Model with Intravenous Bolus Injection and Elimination from the Central Compartment

440

13 6 Linear Three-Compartment Open Model with Intravenous Bolus Injection in the Central Compartment and Elimination from a Peripheral Compartment

442

13 7 Model Selection

444

13 8 Applications and Case Studies

444

References

444

14 Practical Applications of PK/TK Models: Continuous Zero-Order Exposure to Xenobiotics Intravenous Infusion

446

14 1 Introduction

446

14 2 Compartmental Analysis

447

14 2 1 Linear One-Compartment Model with Zero-Order Input and First-Order Elimination

447

14 2 1 1 Estimation of the Time Required to Achieve Steady-State Plasma Concentration Using a Single Long-Term Infusion

449

14 2 1 2 Administration of Loading Dose with Intravenous Infusion to Achieve the Steady-State Level Without a Long Delay

450

14 2 1 3 Estimation of Plasma Concentration after Termination of Infusion

452

14 2 1 4 Estimation of Duration of Action in Infusion Therapy

453

14 2 2 Linear Two-Compartment Model with Zero-Order Input and First-Order Disposition

454

14 2 2 1 PK/TK Equations of Zero-Order Input into the Central Compartment with First-Order Elimination from the Central Compartment

454

xvii

TABLE OF CONTENTS

14 2 3 Simultaneous Intravenous Bolus and Infusions Administration into the Central Compartment of a Two-Compartment Open Model with First-Order Elimination from the Central Compartment

456

14 2 4 Linear Two-Compartment Model with Two Consecutive Zero-Order Inputs, as Loading and Maintenance Doses, with First-Order Elimination from the Central Compartment

456

14 2 5 Three-Compartment Model with Zero-Order Input into the Central Compartment and First-Order Elimination from the Central Compartment

458

14 2 6 Three-Compartment Model with Zero-Order Input into the Central Compartment and First-Order Elimination from a Peripheral Compartment

459

14 3 Applications and Case Studies

459

References

459

15 Practical Applications of PK/TK Models: First-Order Absorption via Extravascular Routes Oral Administration

461

15 1 Introduction

461

15 2 Compartmental Analysis

461

15 2 1 Linear One-Compartment Model with First-Order Input and First-Order Elimination

461

15 2 1 1 Initial Estimates of the Overall Elimination Rate Constant, K and Absorption Rate Constant, k a

464

15 2 1 2 Estimation of Time to Peak Xenobiotic Concentration – Tmax

473

15 2 1 3 Estimation of Peak Concentration (Cp max)

474

15 2 1 4 Estimation of the Area Under Plasma Concentration–Time Curve

476

15 2 1 5 Estimation of Total Body Clearance and Apparent Volume of Distribution

476

15 2 1 6 Fraction of Dose Absorbed (F) – Absolute Bioavailability

478

15 2 1 7 Duration of Action

479

15 2 2 Linear Two-Compartment Model with First-Order Input in the Central Compartment and First-Order Elimination from the Central Compartment

479

15 2 2 1 Equations of the Model

479

15 2 2 2 Interpretation of ka , α , and β

481

15 2 2 3 Parameters and Constants of the Two-Compartment Model with First-Order Input

482

15 2 2 4 Estimation of First-Order Absorption Rate Constant of a TwoCompartment Model – Loo–Riegelman Method

485

15 2 3 Linear Two-Compartment Model with First-Order Input in the Peripheral Compartment and First-Order Elimination from the Peripheral Compartment

487

15 2 4 Linear Three-Compartment Model with First-Order Input in the Central Compartment and First-Order Elimination from the Central Compartment

xviii

490

15 3 Applications and Case Studies

491

References

492

TABLE OF CONTENTS

16 Practical Application of PK/TK Models: Multiple Dosing Kinetics

495

16 1 Introduction

495

16 2 Kinetics of Multiple Intravenous Bolus Injections – One-Compartment Model

495

16 2 1 Equations of Plasma Peak and Trough Levels

496

16 2 2 Estimation of Time Required to Achieve Steady-State Plasma Levels

496

16 2 3 Average Steady-State Plasma Concentration

498

16 2 4 Loading Dose vs Maintenance Dose

499

16 2 5 Extent of Accumulation of Xenobiotics Multiple Dosing in the Body

500

16 2 6 Estimation of Plasma Concentration After the Last Dose

500

16 2 7 Design of a Dosing Regimen

501

16 2 7 1 Dosing Regimen Based on a Target Concentration

501

16 2 7 2 Dosing Regimen Based on Steady-State Peak and Trough Levels

502

16 2 7 3 Dosing Regimen Based on Minimum Steady-State Plasma Concentration 16 3 Kinetics of Multiple Oral Dose Administration

502 502

16 3 1 Peak, Trough, and Average Plasma Concentrations Before and After Achieving Steady-State Levels 16 3 2 Extent of Accumulation in Multiple Oral Dosing

503 504

16 3 3 Oral Administration of Loading Dose, Maintenance Dose and Designing a Dosing Regimen

505

16 4 Effect of Changing Dose, Dosing Interval, and Half-Life on the Accumulation in the Body and Fluctuation of Plasma Concentration

505

16 5 Effect of Irregular Dosing Interval on Plasma Concentrations of Multiple Dosing Regimen 16 6 Multiple Dosing Kinetics – Two-Compartment Model

505 505

16 6 1 Peak, Trough, and Average Plasma Concentrations Before and After Achieving the Steady-State Levels for Two-Compartment Model Xenobiotics Given Intravenously

506

16 6 2 Estimation of the Time Required to Achieve Steady-State Plasma Levels of Two-Compartment Model Xenobiotics Given Intravenously

510

16 6 3 Estimation of Fraction of Steady State, Accumulation Index, and Relationship Between Loading Dose vs Maintenance Dose 16 6 4 Evaluation of Plasma Level after the Last Dose

510 511

16 6 5 The Concept of Half-Life in Multiple Dosing Kinetics of Multicompartmental Models

512

16 7 Multiple Intravenous Infusions

513

16 8 Applications and Case Studies

513

References

514

17 Biopharmaceutics Provisions, Classifcations and Mechanistic Models

545

17 1 Introduction

515

17 2 Infuence of Physicochemical Properties on Absorption of Xenobiotics

515 xix

TABLE OF CONTENTS

17 2 1 Polymorphism

515

17 2 2 Partition Coeffcient

516

17 2 2 1 Rule of Five

516

17 2 3 Infuence of Particle Size, Porosity, and Wettability on Dissolution Rate at the Site of Absorption

517

17 2 3 1 Absorption of Particles

517

17 2 3 2 Infuence of the Particle Size on the Solubility/Dissolution at the Site of Absorption 17 2 3 3 Infuence of Wettability and Porosity on the Dissolution Profle 17 3 Formulation Factors

518 518

17 3 1 Solutions and Syrups

518

17 3 2 Suspensions

518

17 3 3 Emulsions

519

17 3 4 Soft and Hard Gelatin Capsules

519

17 3 5 Compressed Tablets (Uncoated and Coated)

519

17 3 6 Dosage Form Tactics for Poorly Soluble Compounds

519

17 4 Disintegration and Dissolution 17 4 1 Mathematical Models of Dissolution

520 520

17 4 1 1 Noyes–Whitney Model

520

17 4 1 2 Hixson–Crowell “Cube Root” Model

521

17 4 1 3 First-Order Kinetics Model

521

17 4 1 4 Kitazawa Model

522

17 4 1 5 Higuchi “Square Root of Time Plot” Model

522

17 4 1 6 Weibull–Langenbucher Model

523

17 4 1 7 Korsmeyer–Peppas Model

523

17 4 1 8 Nernst–Brunner Model

523

17 4 1 9 Baker–Lonsdale Model

524

17 4 1 10 Hopfendberg Model

524

17 4 2 In Vitro–In Vivo Correlation (IVIVC) of Dissolution Data

524

17 4 2 1 Level A Correlation

524

17 4 2 2 Level B Correlation

525

17 4 2 3 Level C Correlation

525

17 4 2 4 Multiple-Level C Correlation

525

17 5 Biopharmaceutics Classifcation System

xx

517

525

17 5 1 Absorption Number

525

17 5 2 Dissolution Number

525

17 5 3 Dose Number

526

17 5 4 Classes of Biopharmaceutics Classifcation System

526

17 5 4 1 Class I: Compounds with High Permeability and High Solubility

526

17 5 4 2 Class II: Drugs with High Permeability and Low Solubility

526

17 5 4 3 Class III: Drugs with Low Permeability and High Solubility

526

TABLE OF CONTENTS

17 5 4 4 Class IV: Drugs with Low Permeability and Low Solubility

526

17 5 5 Biowaivers

526

17 5 6 Biopharmaceutics Drug Disposition Classifcation System

527

17 6 Other Factors Infuencing Absorption of Xenobiotics

527

17 6 1 Chirality and Enantiomers

527

17 6 2 Effects of Food and Drink on Absorption of Xenobiotics

528

17 6 3 Effects of Disease States

529

17 6 4 Infuence of Genetic Polymorphism

529

17 6 5 Effects of Release Mechanisms from the Solid Dosage Forms

529

17 6 6 Infuence of Drug Administration Scheduling

529

17 6 7 Presence of Other Substances

529

17 6 8 Other Factors

529

17 7 Mechanistic Absorption Models

529

17 7 1 Absorption Potential Models

530

17 7 2 Dispersion Models

530

17 7 3 Compartmental Absorption and Transit Model

531

17 7 4 Gastrointestinal Transit Absorption Model

532

17 7 5 Advanced Compartmental Absorption and Transit Model

535

17 7 6 Advanced Dissolution, Absorption, and Transit Model

535

17 7 7 Grass Model

536

References

537

18 Bioavailability, Bioequivalence, and Biosimilarity

545

18 1 Introduction

545

18 2 Defnitions

546

18 2 1 Bioavailability

546

18 2 2 Pharmaceutical Equivalents

546

18 2 3 Pharmceutical Alternatives

546

18 2 4 Bioequivalent Drug Products (Bioequivalence)

546

18 2 5 Therapeutic Equivalents

546

18 2 6 Generic Drug Products

547

18 2 7 Absolute and Relative Bioavailability

547

18 3 Peak Exposure, Total Exposure, and Early Exposure 18 3 1 Estimation of Absolute Bioavailability from Plasma Data – Single Dose

547 548

18 3 2 Estimation of Absolute Bioavailability from Amount Eliminated from the Body – Single Dose 18 3 3 Estimation of Relative Bioavailability from Plasma Data – Single Dose

548 549

18 3 4 Estimation of Relative Bioavailability from Total Amount Eliminated from the Body – Single Dose

549

18 4 Bioavailability and First-Pass Metabolism

549

18 5 Linearity Validation of Relative or Absolute Bioavailability During Multiple Dosing Regimen

550 xxi

TABLE OF CONTENTS

18 6 Bioequivalence Evaluation

552

18 6 2 Overview of Statistical Analysis of PK/TK Data for Bioequivalence Study

553

18 6 3 Required PD/TD Data

553

18 7 Biosimilar (Biosimilarity and Interchabgeability)

553

18 7 1 Introduction

553

18 7 2 Comparability of Biosimilar and Application of PK/PD Parameters

554

References

555

19 Quantitative Cross-Species Extrapolation and Low-Dose Extrapolation 19 1 Cross-Species Extrapolation 19 1 1 Introduction: Interspecies Scaling in Mammals 19 1 2 Allometric Approach

561 561 561 561

19 1 2 1 Allometric Approach and Chronological Time

564

19 1 2 2 Application of Allometric in Converting Animal Dose to Human Dose

565

19 1 3 Application of PBPK or PBTK in Cross Species Extrapolation 19 1 3 1 Toxicogenomics

566 566

19 2 Low-Dose Extrapolation

567

19 2 1 Introduction

567

19 2 2 Threshold and Non-Threshold Models

568

19 2 2 1 The Probit Model

568

19 2 2 2 The Logit Model

569

19 2 2 3 The One-Hit Model

569

19 2 2 4 The Gamma Multi-Hit Model

569

19 2 2 5 The Armitage-Doll Multi-Stage Model

570

19 2 2 6 Statistico-Pharmacokinetic Model

570

References

570

20 Practical Application of PK/TK Models: Population Pharmacokinetics/Toxicokinetics

xxii

551

18 6 1 Required PK/TK Parameters and Other Provisions in Bioequivalence Study

574

20 1 Introduction

574

20 2 Fixed Effect and Random Effect Parameters

575

20 2 1 Fixed Effect Parameters

575

20 2 2 Random Effect Parameters

575

20 2 3 Linear and Nonlinear Mixed-Effect Models

575

20 2 3 1 Linear Mixed-Effects Model

576

20 2 3 2 Nonlinear Mixed-Effects Model

576

20 2 3 3 Partially Linear Mixed-Effect Model

578

20 2 3 4 Naïve-Pooled Data Approach

578

20 2 3 5 Naïve Average Data Approach

579

20 2 3 6 Standard Two-Stage Approach

579

20 2 3 7 Global Two-Stage Approach

579

20 2 3 8 Iterative Two-Stage Approach

579

TABLE OF CONTENTS

20 2 3 9 Bayesian Approach

579

20 3 Computational Tools for popPK/TK

579

References

580

21 Practical Application of PK/TK Models: Preclinical PK/TK and Clinical Trials

586

21 1 Introduction

586

21 2 Preclinical PK/TK

586

21 2 1 Estimation of the First Dose in Humans 21 2 2 PK/TK Preclinical Requirements

586 587

21 2 2 1 Safety Pharmacology and Toxicity Testing

588

21 2 2 2 Metabolic Evaluations in Preclinical Phase

589

21 3 PK/TK and Clinical Trials

590

21 3 1 Phase I-a Clinical Trial

590

21 3 2 Phase I-b Clinical Trial

590

21 3 3 Phase II-a Clinical Trial

591

21 3 4 Phase II-b Clinical Trial

591

21 3 5 Phase III Clinical Trial

591

21 3 6 Phase IV Clinical Trial

591

References

591

22 Adjustment of Dosage Regimen in: Renal Impairment, Liver Disease and Pregnancy 22 1 Renal Impairment

595 595

22 1 1 Introduction

595

22 1 2 Dosage Adjustment for Patients with Renal Impairments

595

22 1 2 1 Estimation of the Overall Elimination Rate Constant or Half-Life of a Therapeutic Agent Based on the Estimated GFR

595

22 1 2 2 Adjustment of Multiple Dosing Regimen Using the Adjusted Elimination Rate Constant, K 22 1 2 3 Dosage Adjustment Based on the Steady-State Peak and Trough Levels 22 1 3 Applications and Case Studies 22 2 Liver Diseases 22 2 1 Introduction 22 3 Pregnancy

599 602 602 602 602 604

22 3 1 Introduction

604

22 3 2 Changes Impacting Oral Absorption during Pregnancy

604

22 3 3 Changes Infuencing Drug Distribution during Pregnancy

604

22 3 4 Changes in Drug Metabolism during Pregnancy

604

22 3 5 Changes in Renal Excretion during Pregnancy

604

22 3 5 1 Estimation of GFR during Pregnancy

604

22 3 6 Role of the Placenta

605

22 3 7 PK/TK Models

605

References

605

xxiii

TABLE OF CONTENTS

Addendum I – Part 1: Standard Terminologies for Routes of Administration

609

Addendum I – Part 2: Relevant Mathematical Concepts

615

Addendum I – Part 3: Abbreviation – Glossary – PK/TK Constants and Variables

626

Addendum II – Part 1

627

Addendum II – Part 1

629

Addendum II – Part 1

630

Addendum II – Part 2

631

Addendum II – Part 2

632

Addendum II – Part 2

635

Addendum II – Part 2

635

Addendum II – Part 2

636

Addendum II – Part 2

639

Addendum II – Part 2

640

Addendum II – Part 2

641

Addendum II – Part 2

643

Addendum II – Part 3

644

Addendum II – Part 3

647

Addendum II – Part 3

649

Addendum II – Part 3

650

Addendum II – Part 3

652

Addendum II – Part 3

654

Addendum II – Part 3

654

Addendum II – Part 3

655

Addendum II – Part 4

658

Addendum II – Part 4

659

Addendum II – Part 4

661

Addendum II – Part 4

662

Addendum II – Part 4

663

Addendum II – Part 4

665

Addendum II – Part 4

666

xxiv

TABLE OF CONTENTS

Addendum II – Part 4

668

Addendum II – Part 4

672

Addendum II – Part 5

674

Addendum II – Part 5

675

Addendum II – Part 5

676

Addendum II – Part 5

677

Addendum II – Part 5

678

Addendum II – Part 5

680

Addendum II – Part 5

682

Addendum II – Part 5

684

Addendum II – Part 5

685

Addendum II – Part 6

686

Addendum II – Part 6

691

Addendum II – Part 6

694

Addendum II – Part 6

697

Addendum II – Part 6

699

Addendum II – Part 6

701

Addendum II – Part 7

704

Addendum II – Part 7

705

Addendum II – Part 7

707

Addendum II – Part 7

708

Addendum II – Part 7

710

Addendum II – Part 7

714

Addendum II – Part 7

716

Addendum II – Part 7

717

Addendum II – Part 7

718

Addendum II – Part 8

720

Addendum II – Part 8

721

Addendum II – Part 8

722

Addendum II – Part 8

723 xxv

TABLE OF CONTENTS

Addendum II – Part 8

724

Addendum II – Part 8

725

Addendum II – Part 8

726

Addendum II – Part 8

727

Addendum II – Part 8

728

Addendum II – Part 9

729

Addendum II – Part 9

730

Addendum II – Part 9

731

Addendum II – Part 9

732

Addendum II – Part 9

733

Addendum II – Part 9

733

Index

735

xxvi

Preface The principles and methodologies of pharmacokinetics and toxicokinetics (PK/TK) have undergone steady growth and intricacy during the past fve decades. A cursory glance at the current publications reveals the increasing utility and recognition of PK/TK principles and methodologies in diverse scientifc research projects, regulatory decision-making processes and guidelines, and the multitude of applications in various medical felds. I undertook the writing of this handbook to provide a useful reference resource for research and a teaching text for graduate and undergraduate students. I have presented a substantial portion of the materials contained in this handbook in graduate and advanced undergraduate courses at the various universities and colleges at which I have had the pleasure of working with students of different disciplines and post-doctoral fellows. I have observed that the PK/TK subjects enthuse students, even those without a strong background in mathematics showing a keen interest in the subject, many of whom then undertook PK/TK studies at the research bench. This handbook is comprehensive enough to cover many areas of interest and permit integration of the methodologies in different research projects. The main emphases of the book are how the body deals with xenobiotics and how the xenobiotics interact with the elements of the body in vitro, in vivo, and in situ settings. I use the word ‘xenobiotics’ here and throughout the book as a general term for all natural and manufactured substances with therapeutic, toxic, and nontoxic properties. The frst chapter is an overview of PK/TK signifcance as it relates to other scientifc felds. Chapters 2–6 examine 19 essential routes of administration, highlighting their uniqueness and reviewing their PK/TK principles. Chapters 7–11 aid in understanding the kinetics of absorption, distribution, metabolism, and excretion (ADME) of xenobiotics and their related mathematical descriptions. Chapter 12 discusses various approaches and methodologies in PK/TK modeling and their important parameters and constants. Chapters 13–22 are about the practical applications of the models and their assumptions and concepts. The book includes two addendums: Addendum I includes the list of recognized routes of administration, selected mathematical topics relevant to the chapters of the book, and the abbreviations/glossary of acronyms, variables, and constants; Addendum II consists of case studies and the practical application of PK/TK principles. The book covers a wide range of basic concepts, and the focus is on the understanding of these concepts and the governing principles of their mathematics and kinetic interconnectivity. Therefore, I avoided topics like neural networks, advanced stochastic modeling, descriptions of statistical and numerical analysis of computer software, and computer-generated curves of published data. Finally, the editors and publisher deserve the author’s gratitude for their efforts toward the goal of producing a book of utility to researchers, educators, and students. Mehdi Boroujerdi

xxvii

1 Pharmacokinetics and Toxicokinetics 1.1 INTRODUCTION The word xenobiotic, a combination of two Greek words (xenos and biōtkós), literally means “foreigner or stranger to life” and refers mostly to an exogenous organic or inorganic chemical compound foreign to the human body or other living organisms. Xenobiotics, in small or large molecules, may act as therapeutic agents, food additives, toxins, carcinogens, pollutants, pesticides, etc., which upon direct or indirect interaction with various components of the body produce favorable or unfavorable biological response(s). The interaction occurs during the physiological and biochemical processes of absorption, distribution, metabolism (biotransformation), and excretion (ADME). The ADME processes facilitate the delivery and interaction of a xenobiotic with its receptor site or target tissues, the eventual outcome, and its removal from the body. The pharmacokinetics (PK) and toxicokinetics (TK) disciplines are the quantitative science of the processes of ADME for a given xenobiotic and the corresponding pharmacological and/or toxicological responses (Figure 1.1). The fundamental goals of both disciplines are to understand how the body handles the administered or exposed xenobiotic as a foreign substance and how to predict and measure the magnitude of its pharmacological or toxicological response or outcome. The result of pharmacokinetic/toxicokinetic (PK/TK) analysis is establishing appropriate criteria and approaches to avoid undesirable outcomes of the xenobiotics interaction with the body and to safeguard human life. The attainment of an acceptable outcome from interaction of a xenobiotic with the body can be as practical as: ◾ developing appropriate dosage forms or a dosage regimen for the purpose of achieving optimum therapeutic outcome or pharmacological response, or ◾ establishing regulatory guidelines/policies for consistency of the outcomes through bioequivalence and biosimilar evaluations, or ◾ as challenging as a low-dose extrapolation of toxicants, chemical carcinogens, and understanding of a safe dose in hazardous environmental pollutants or a new therapeutic entity in drug discovery and development. PK and TK use the same mathematical principles of kinetics, differential equations, statistical methodologies, and numerical analysis in data exploration, interpretation, and deduction or prediction. The difference between the two disciplines may be only their biological outcomes. 1.2 PHARMACOKINETICS AND PHARMACODYNAMICS The time course of xenobiotics in the body has been studied since 1927 (Widmark and Tandberg, 1924; Widmark, 1932; Widmark and Elbel, 1937 ; Teorell, 1937a; 1937b), but the term pharmacokinetics was frst introduced in 1953 (Dost, 1953). The history of the frst 50 years of pharmacokinetics is well chronicled in a paper published by John Wagner, another pioneer of pharmacokinetics in 1981 (Wagner, 1981). The following defnitions have gained wide acceptance in the scientifc feld: Pharmacokinetics is the study of the time course (i.e., kinetics) of ADME of drugs and their metabolites in the body (Figure 1.1) and the corresponding infuence(s) on the intensity and time course of the pharmacological response. Pharmacodynamics (PD) is the quantitative study of the pharmacological response and the therapeutic outcome of a drug and its mechanism of action in a biological system, with emphasis on the dose–response or drug concentration–effect relationships. In essence, pharmacokinetics and pharmacodynamics (PK/PD) are two integrated subjects that, in combination, elucidate how the body handles drug molecules and what therapeutic outcome results from their interaction with target tissue(s). PK/PD are studied using in vivo (animal or human whole body), ex vivo (using cells or tissues previously exposed to a drug), in vitro (on isolated cells and tissues), and/or in silico (computer modeling and simulation) methods, utilizing a wide range of analytical methodologies and computer programs. There are factors that can infuence the PK/PD of xenobiotics. The most notable ones are: ◾ genetic predisposition ◾ routes of administration DOI: 10.1201/9781003260660-1

1

1.2 PHARMACOKINETICS AND PHARMACODYNAMICS

Figure 1.1 Schematic illustration of the physiological processes of absorption, distribution, metabolism, and excretion (ADME); the arrows represent the direction of physiological or biochemical processes or fow, and the routes of administration include direct entry into the systemic circulation by intravenous injection (IV) or access after the absorption process that requires permeation of a compound through an absorption barrier; included examples in the fgure are gastrointestinal absorption (PO), intramuscular injection (IM), subcutaneous injection (SC), sublingual absorption (SL), rectal administration (PR), transdermal absorption (TD), and inhalation; biological samples of experimental animals are identifed by , and the noninvasive biological samples for human subjects include serum, plasma, whole blood, urine, saliva, sweat, milk, and hair; the biophase represents the target/receptor sites associated with organs, tissues, whole blood, or other milieus. ◾ physicochemical factors related to drug and administered dosage form (i.e., pharmaceutics and biopharmaceutics) ◾ drug–drug interaction, drug–occupational exposure interaction, and drug–herbal interaction ◾ age (including gerontology and pedology) ◾ gender (including hormonal infuence) ◾ body weight ◾ exercise ◾ nutrition and dietary factors and food–drug interaction ◾ alcohol intake ◾ tobacco or marijuana smoking ◾ pregnancy and lactation ◾ disease states that infuence renal, hepatic, gastrointestinal, cardiovascular, and immunological functions 2

PHARMACOKINETICS AND TOXICOKINETICS

◾ infection and infammation and diabetes ◾ stress, psychiatric status, behavior, and depression ◾ circadian and seasonal variation ◾ environmental factors, culture, and occupation ◾ trauma, surgery, etc. 1.2.1 Clinical Pharmacokinetics/Pharmacodynamics An applied feld of PK/PD is the clinical pharmacokinetics that focuses on the application of PK/ PD principles to the safe and effective therapeutic management of patients and individualized optimization of drug dosage and response. Clinical Pharmacokinetics/Pharmacodynamics (CPK/ PD) is achieved by using drug concentrations, pharmacokinetic principles, and pharmacodynamic criteria. CPK/PD is also called “therapeutic drug monitoring.” It is worth noting that not all CPK/ PD clinical or research projects involve patients, but their end goal is to optimize the therapeutic outcome and minimize the probability of toxicity. Ordinarily, the drugs that are monitored are those with a narrow therapeutic range that trigger a toxic response at plasma levels close to the maximum therapeutic level. Furthermore, all factors that can infuence the PK/PD of xenobiotics as noted in Section 1.2, especially the disease states, impact the ADME of many therapeutic agents, which necessitates the application of CPK/PD practice, and the safe and effective therapeutic management for the individual patient. Noteworthy disease states are those that signifcantly impair the organs and physiological processes involved in ADME. 1.2.2 PK/PD Modeling and Pharmacometrics The mathematical approaches of PK and PK/PD modeling with the related assumptions, logics and methodologies will be discussed in detail in forthcoming chapters of this book (Chapter 12). The PK/PD modeling links the ADME kinetics (PK) of a xenobiotic to its pharmacological response (PD). The kinetics and dynamics of optimum response are usually decided based on individual or collective observations of in vivo, in vitro, in situ and/or in silico studies. Over the last few decades, PK/PD modeling has continued to develop, offering higher levels of intricacy in order to: ◾ provide a quantitative ADME portrayal of a xenobiotic in the body ◾ summarize the observed data ◾ provide a useful framework that is founded on relevant observations and trustworthy for prediction ◾ develop mathematical relationship(s) that defne the connection between the magnitude of a xenobiotic exposure and its measurable response. The term “predication” applies to a whole host of benefts from the modeling, which include the following: ◾ extrapolation to a higher or lower input ◾ extrapolation from experimental animals to human ◾ predication of in vivo response from in vitro observations ◾ predication of response and ADME profle in disease states ◾ selection and optimization of a lead compound in drug discovery and development ◾ assessment of human health risk from exposure to environmental toxicants, or preclinical and clinical trials in drug discovery and development, and others. The PK/PD modeling has also grown into an advanced mathematical feld of pharmacometrics (PMX) that streamlines interpretation of complex physiological/pharmacological processes and conveys them in a quantitative manner. Pharmacometrics deals mostly with population PK/PD models and mechanistic models. It employs advanced graphical methods and computer programing, using statistical analysis and stochastic simulation to further clarify and quantify the complex PK/PD models and associated parameters and constants. The generated data from PMX analyses are most useful and practical in: 3

1.2 PHARMACOKINETICS AND PHARMACODYNAMICS

◾ drug discovery and development ◾ planning, conducting, and analyzing clinical trials and clinical data analysis (Williams and Ette, 2006; Koch et al., 2020; Akacha et al., 2021) ◾ quantitative expression of system pharmacology ◾ applied in silico models for drug development (Musuamba et al., 2020; Musuamba et al., 2021) ◾ regulatory review and decision-making (Garnett et al., 2011) ◾ drug approval and labeling decisions (Bhattaram et al., 2005), etc. 1.2.3 Population PK and PK/PD Modeling It has been proven that certain physiological, pathophysiological, biochemical, and demographical factors can alter the therapeutic outcome of a drug in specifc group(s) of patients. Consequently, the feld of population pharmacokinetics (popPK) has advanced and been applied in many areas, including drug discovery and development and pharmacogenetics (PGT) in recent years. PopPK helps to defne the PK variability among individuals in a target population sharing the same attributes, such as gender, age, race, disease, etc. and to study the impact of variability of drug behavior in the body (Aarons, 1991; FDA, 1999, 2022). Population PK and PK/PD analyses and modeling provide valuable quantitative assessment of the incorporated PK and PK/PD parameters and constants, including inter- and intraindividual and residual variability in drug concentration. The calculated data are then used to evaluate, validate, and optimize the dose, dosage regimen, or dosage adjustment of therapeutics agents (Kawaguchi et al., 2021; Stillemans et al., 2021; Krzyzanski et al., 2021) and biologics (Ogasawara et al., 2019; Wang et al., 2018). The popPK analyses are usually carried out in the following three phases: ◾ exploratory data analysis ◾ model development, and ◾ model validation. The interindividual variability in gene-encoding protein transporters, drug-metabolizing enzymes, some receptors, ion channels, immune molecules, etc. infuences the PK/PD of xenobiotics (Evans and Relling, 1999; Eichelbaum et al., 2006; Daly, 2010). Specifcally, genetic differences in drug metabolism (Evans and Relling, 1999), toxicity, and response have direct impact on metabolic clearance and formation of active or toxic metabolites. A pharmacogenetic polymorphism (Yang et al., 2013) occurs in a subgroup of a population when the mutated allele occurs at a level that causes different responses and/or different ADME outcomes. For example, when there is a drug metabolism polymorphism, the interindividual inconsistency manifests itself with greater variation compared to the rest of the population. Poor metabolizers will be at higher risk for toxicity when enzyme-mediated inactivation is failing, and fast metabolizers may reduce the drug concentration to a level below the therapeutic range, which will reduce effcacy of the therapeutic agent. It is essential to consider a large sample size for popPK studies when the goal is to identify slow and fast metabolizers. Various guidelines have been developed by working groups, regulatory agencies, and academic/ research institutions, conveying the signifcance of pharmacogenomics and pharmacogenetics in PK and PK/PD modeling and analyses. Notable among them are: ◾ US Food and Drug Administration (FDA) • https://www.fda.gov/regulatory-information/search-fda-guidance-documents/pharmacogenetic-tests-and-genetic-tests-heritable-markers) • https://www.fda.gov/drugs/science-and-research-drugs/table-pharmacogenomic -biomarkers-drug-labeling ◾ Clinical Pharmacogenetics Implementation Consortium (CPIC) • https://cpicpgx.org/guidelines/ ◾ Dutch Pharmacogenetics Working Group (DPWG) • https://www.pharmgkb.org/page/dpwg 4

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◾ European Medicine Agency (EMA) • https://www.ema.europa.eu/en/use-pharmacogenetic-methodologies-pharmacokinetic -evaluation-medicinal-products ◾ German Federal Institute for Drugs and Medical Devices (BfArM) • https://www.bfarm.de/EN/BfArM/Tasks/Research/Pharmacogenomics/_node.html ◾ and numerous related books and publications (Shekhani et al., 2020; Becquemont et al., 2011). Advances in PGX and PGT in combination with sophisticated PK/PD modeling and analysis will provide guidance for the practice of personalized medicine. 1.2.3.1 Infuences of Genetics and Genomics on PK/PD and TK/TD An important element infuencing the variability of PK/PD or toxicokinetics and toxicodynamics (TK/TD) analyses is the genetic makeup of the target population. The study of how humans respond to xenobiotics due to their genetic inheritance and the infuence of internal and external factors on their genetic makeup and PK/PD variability are explored in pharmacogenomics and pharmacogenetics. The infuence of genetics and genomics on PK and PD processes and the distinctions between PGX and PGT are presented in Figure 1.2. The following are very brief descriptions of the included titles in Figure 1.2, just to shed some light on their role in PK, PD, PGX, and PGT. For more comprehensive information on the titles, the relevant references and publications should be consulted. 1.2.3.1.1 Metabolomics and Metabonomics Metabolomics is the all-inclusive analysis of all metabolites in the body. The reason behind this undertaking is that certain metabolites are commonly associated with specifed pathogenic conditions and associated mechanisms (Gerszten and Wang, 2008). The scope of metabolomic analysis is way beyond the typical medical laboratory measurements and techniques. It includes very hydrophilic to very hydrophobic metabolites of endogenous and exogenous substances metabolism and enzymatic activities encoded by the human genome (Kuehnbaum and Brits-McKibbin, 2013) as well. This comprehensive metabolic enterprise provides opportunities for assessment of the complex nature of normal and abnormal chemical, physiological, and genetic factors and processes. The metabolomic knowledge provides valuable clues for discovery and development of new drug entities and forecasts the infuence of external factors on a system. The external factors also include the effect of xenobiotics on a system and their handling by the system after exposure. The term metabonomics is often used in conjunction with metabolomics. Metabonomics describes the time-dependent changes in metabolomics due to the involvement of pathophysiological conditions or genetic modifcations (Nicholson et al., 2007). 1.2.3.1.2 Proteomics Proteomics, in the same way as metabolomics, refers to the comprehensive study and identifcation of proteins and protein levels in a biological system/human body – those that correspond and correlate with the normal physiological state and those that can be used as biomarkers of disease states. The role of proteomics, in addition to acting as biomarkers, is to reveal and validate drug targets, identify toxicity markers, and clarify drug modes of action. The science of proteomics plays a signifcant role in drug discovery and development, environmental stressors like toxic chemicals, disease prevention and treatment, and many areas in biomedical science research, like apoptosis-related proteins and biomarkers in cancer research (Bai et al., 2011). 1.2.3.1.3 Transcriptomics As the title indicates, transcriptomics, is the omics of transcripts and the expression of gene regulation and provides principal understanding on gene structure, expression, and regulations (Lowe et al., 2017; Casamassimi et al., 2017). Transcriptomics offers information on RNA transcribed by the genome in a specifc organ or tissue or cell type under normal physiological conditions or pathophysiological states. The information reveals critical changes that initiate disease states and the molecular diagnosis that ultimately assists in providing a more effective therapeutic outcome (Byron et al., 2016).

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Figure 1.2 Interconnection between pharmacokinetics and pharmacodynamics with the collective infuence of genetic polymorphism, proteomics, metabolomics, epigenomics, and transcriptomics under the provisos of pharmacogenetics and pharmacogenomics. 1.2.3.1.4 Epigenomics The control and function of the genome are closely associated with the physical confguration of genomic DNA. This includes the way it is bundled into chromatin, encompassing histones, its binding sites, protein complexes, and noncoding RNAs (Kouzarides, 2007; Bernstein et al., 2007; Fraser and Bickmore, 2007). Chromatin is subject to alteration of its DNA and protein components that cause functional and structural changes. The distribution of the alteration and functional changes across the genome of a given system is called “epigenome,” and epigenomics is the study of characteristic alterations of gene expression, DNA methylation, and changes in chromatin that are associated with disease states (Jaenish and Bird, 2003; Jones and Baylin, 2007; Feinberg, 2007; Kabekkodu et al., 2017; Ushijima et al., 2021). 6

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1.2.3.2 Biomarkers The defnition of biomarkers has been evolving over the past two decades by various agencies and working group such as the Committee on Qualifcations of Biomarkers and Surrogate Endpoints in Chronic Disease from the Institute of Medicine (Micheel and Ball 2010) and the FDA-NIH Biomarker Working Group (FDA-NIH, 2016–2017) that offered the basic defnition for a biomarker as: “A defned characteristic that is measured as an indicator of normal biological processes, pathogenic processes to an exposure or intervention.” This defnition embodies the idea that biomarkers are molecular, physiologic, histologic, and radiologic traits of an intervention that can be assessed objectively, validated analytically, and appraised as an indicator of a normal biological process or a pathogenic manifestation. Furthermore, a pharmacological and/or toxicological response to a xenobiotic may generate endogenous or exogenous molecule(s) that can be considered a biomarker. In PK/PD and TK/TD modeling, the quantitative measurement of biomarkers is used to monitor the response and conceivably quantify the effect. When a biomarker is used to quantify the effect, instead of a clinical or medical endpoint, it is called a “surrogate endpoint.” A surrogate endpoint is expected and must be validated to predict beneft, harm, or no effect; and the validation must be based on scientifc, pathophysiologic, and/or population verifcation (Biomarkers Defnitions Working Group, 2001). Furthermore, a biomarker can only be identifed as a surrogate endpoint if it is correlated with the clinical outcome assessment, and any fuctuation in the measurements of the biomarker must correspond and refect variation in the measurements of the clinical endpoint. It is essential to differentiate between clinical outcome assessment and biomarkers. Clinical or medical outcome assessment, which is often associated with the clinical endpoint, is an important measurement for the therapeutic management and outcome. Whereas, the measurements of a biomarker, as the surrogate, is linked only to a projection of the clinical endpoint and exists between the infuence of therapeutic agent and therapeutic outcome. To rephrase it, a measurable biomarker may or may not refect the medical state of a patient as is determined by the medical endpoint. However, in the absence of an approved clinical outcome assessment, a validated biomarker can be used in drug discovery and development to accelerate the FDA approval process (FDA, 2018). Clinical endpoint measurements refect directly how a patient functions, endures, or survives the treatment. For example, for cancer patients, the endpoint is the disease-free period between the treatment or surgery and undetectable sign of the disease, or the improvement in quality of life, such as absence of pain and negative side effects of the therapeutic agent. A surrogate endpoint is laboratory or physical data such as cholesterol levels, bone density measurement, blood pressure, hemoglobin A1C measurement, etc. In ecotoxicological terms, surrogate endpoint biomarkers would be most valuable if they were linked to adverse effects in an individual as well as to population and ecosystem effects. It is common that a biomarker may represent the intermediate step between exposure and response. Biomarkers are used routinely in clinical practice and drug development to describe risk, select the appropriate dose, achieve maximum therapeutic effciency and minimum toxicity, and monitor the therapeutic outcome. Often, PK/TK parameters and constants associated with analysis of serum drug levels, e.g., area under the plasma concentration time curve, maximum plasma concentration, and time to achieve the maximum concentration are also considered biomarkers. The suggested (Califf, 2018; FDA-NIH, 2017) biomarker categories are: ◾ Diagnostic – reveals or confrms a disease or identifes patients with a disease. ◾ Monitoring – detects the effect of medical products/biologic agents for measuring pharmacodynamic response (e.g., measuring blood pressure). ◾ Pharmacodynamic/response – its measurements refect the response to administration of or exposure to a xenobiotic; useful in drug discovery and development. ◾ Predictive – indicates that its presence, absence, or change forecasts a favorable or unfavorable response in an individual or group of individuals; useful in the design and conduct of clinical trials in drug discovery and development. ◾ Prognostic – identifes the probability of disease progression or recurrence in patients with a given disease or medical condition. ◾ Safety – reveals the presence or extent of a toxic outcome or any harmful occurrence before or after exposure to a medical procedure or an environmental element/xenobiotic. ◾ Susceptibility/risk – indicates the possibility for an emerging disease or medical condition in an individual who does not have the condition/disease. 7

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◾ Digital – refers to sensors and personal electronic devices for detecting or collecting medical related data, such as exercise level and habit, cognitive abilities, psychological state, often called “digital phenotyping” (Insel, 2017). Biomarkers are used in pharmacology and toxicology for the evaluation of dose-response curves, for achieving the optimum therapeutic outcome, or toxicity dose-response curves, associated with safety and refning low-dose extrapolation. A few examples of biomarkers are: ◾ Metabolomic biomarkers for lung cancer diagnosis and prognosis: using AUC of fve most signifcant metabolites of lung cancer(palmitic acid, heptadecanoic acid, 4-oxoproline, tridecanoic acid, ornithine, etc.) shows potential for early prediction and diagnosis of lung cancer (Qi et al., 2021). The use of urinary 8-oxo-7,8-dihydro-2-deoxyguanosine, thymidine glycol, F2-isoprostanes, serum dehydroascorbic acid to ascorbic acid ratio and carotenoid concentrations can also be included in this category as biomarkers of the oxidative stress in the lung carcinoma (Lowe et al., 2013). ◾ Metabolomic biomarkers for the detection of obesity-driven endometrial cancer: glycerophospholipids (glycerophosphocholines and glycerophosphoethanolamine) have been found particularly important in differentiating endometrioid endometrial cancer from controls (Njoku et al., 2021). ◾ Proteomic biomarkers relating to Alzheimer’s disease: increase in Alpha-2-macroglobulin and fcolin-2 and decrease in fbrinogen gamma chain plasma levels (Shi et al., 2021). ◾ Proteomic biomarkers in conjunctiva: increased expression of human leukocyte antigen-Drelated (HLA-DR), a glycoprotein that is part of major histocompatibility complex class II cell surface receptor, is associated with ocular surface disease (Fernandez et al., 2015). ◾ Epigenomic biomarker of Alzheimer’s disease: microRNA (miRNA) expression can infuence the regulation of amyloid beta A4 precursor protein (APP), presenilin 1 (PSEN1), presenilin 2 (PSEN2), and beta-secretase 1 (BACE1) genes in the brain that are implicated in Alzheimer’s disease pathophysiology. A reduction in whole-blood miRNA expression is suggested to be signifcantly associated with an increased risk of Alzheimer’s disease (Yilmaz et al., 2016). ◾ Transcriptomics biomarkers of prostate cancer: differential expression of the genes HEATR5B, DDC, and GABPB1-AS1are considered potential biomarkers to predict prostate cancer. To predict progression of prostate cancer between stage II and subsequent stages of the disease, PTGFR, NREP, SCARNA22, DOCK9, FLVCR2, IK2F3, USP13, and CLASP1 genes are reported as the potential biomarkers (Alkhateeb et al., 2019). Many of the biomarkers of genomics, including metabolomic and proteomic, are expected to assist the implementation of personalized medicine or precision medicine. In developing the genomic biomarkers, in addition to the understanding of gene function at the DNA level, the transcriptional (RNA) and post-transcriptional (protein) data are also crucial. Gene expression at the RNA and protein levels is often nonlinearly correlated and somewhat inconsistent due to the translation, regulation, transport, and degradation. Genomic biomarkers will continue to signifcantly infuence the development and administration of therapeutic agents. However, their role in PK/TK modeling and analysis is yet to be accelerated. The use of biomarkers in environmental toxicology is essential as indicators of public health. For example, increased activity of enzymes like alanine aminotransferase and alkaline phosphatase is associated with liver damage; or reduction of cholinesterase activity is the biomarker for the organophosphate pesticides exposure, which causes inhibition of cholinesterase and breakdown in nervous system function; eggshell thickness as an indicator of organochlorine pesticide DDT, which interferes with calcium transport causing eggshell thinning and breaking (Eason and O’Halloran, 2002); C-reactive protein (CRP) as a marker of infammation (Pradham et al., 2001; Ridkar et al., 2002; Shacter and Weitzman, 2002); and also argininosuccinate synthase as a biomarker for infammatory conditions (Cao et al., 2013). Several chemical carcinogenicity biomarkers are reported, which in most cases are organ-specifc, such as ethylene oxide hemoglobin adducts or acetaldehyde-DNA and protein adducts for lung cancer (Hatsukami et al., 2006).

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1.3 TOXICOKINETICS AND TOXICODYNAMICS Toxicokinetics is the study of the time course of absorption, distribution, and elimination (metabolism and excretion) of a potentially toxic xenobiotic and/or its metabolites, leading to a toxic response in human or animals following exposure via a pertinent route of administration/exposure. The quantitative determination of the relevant TK parameters and constants is central to the toxicological risk assessment, bioaccumulation, and development of safety standards and regulations intended to protect humans. The TK quantitative determination is used in the safety assessment of pharmaceuticals, risk assessment of industrial chemicals (fertilizers, pesticides, biocides, carcinogens, nanomaterials, environmental chemicals, etc.), and safety of non-pharmaceutical ingredients added to food, cosmetics, and personal care products (ICH Guidance S3A, 1995; Baldrick, 2003; Ploemen et al., 2007; Bessems and Geraets, 2013; Wambaugh et al., 2015; FDA S3A Guidance, 2018; OECD Guidance, 2021). The application of toxicokinetics for the safety assessment of pharmaceuticals is an integrated part of the scholarship of drug discovery and development. The regulatory defnition of toxicokinetics according to the International Conference on Harmonization Guidance S3A (ICH, 1995) and FDA Guidance S3A (FDA, 2018) is “the generation of pharmacokinetic data, either as an integral component in the conduct of nonclinical toxicity studies or in specially designed supportive studies, to assess systemic exposure” Furthermore, the application of toxicokinetics in improvement of the chemical risk assessment has been recognized by organizations such as International Life Sciences Institute (ILSI) (Dybing et al., 2005); Organization for Economic Co-operation and Development (OECD, 2021); Health and Environmental Sciences Institute (HESI) (https://hesiglobal.org/development-of-methods-for-a -tiered-approach-to-assess-bioaccumulation-of-chemicals/), with specialized working groups on Bird Biotransformation and Toxicokinetics, Invertebrate Biotransformation and Toxicokinetics, Fish Biotransformation and Toxicokinetics; chemical industries research (Punt, 2018); and the US Environmental Protection Agency announcement of August 3, 2020 (https://www.epa.gov/newsreleases/epa-awards-4-million-develop-new-approaches-evaluating-chemical-toxicokinetics). Toxicodynamics (TD) can then be defned as the in vivo, in vitro, and in situ determination and quantifcation of the sequence of events at the cellular and molecular levels, leading to apoptosis, toxic risk, or lethal response after exposure to a chemical agent (Figure 1.3). 1.3.1 TK/TD Modeling, Population Toxicokinetics, and Toxicogenetics The development of TK/TD modeling for a xenobiotic elucidates the prediction of toxic response intensity as a function of the administered dose or chronic exposure. In other words, TK/TD models describe the kinetics of ADME of a toxic xenobiotic, either after a dynamic external dosing, as in clinical toxicology, or after long-term exposure, as in ecotoxicology, and connect the kinetic data to: ◾ the uptake by the body or an experimental/cellular system under observation ◾ the observed and measured toxicity, or ◾ to the overall apoptosis and system failure. The TK/TD models assist the extrapolation of the laboratory data developed under dynamic external dosing to long-term exposure useful in risk assessment. The models also have application in calculating the survival probabilities (Ashauer and Esher, 2010; Jager et al., 2011; Focks et al., 2018). A challenge in risk assessment of toxic xenobiotics is the quantifcation of interindividual variability for the compounds with complex ADME and multiple organ injuries. The toxicokinetic variability, whether due to genetic factors or other attributes can be characterized by the application of the Bayesian statistical inferences and physiological based toxicokinetic modeling (Bois et al., 1996; Chiu and Bois, 2006; Amzal et al., 2009; Thursby et al., 2018; Qing et al., 2021). The infuence of genotoxic xenobiotics on the DNA structure, a trait of TD evaluations, hinders the vital processes of DNA replication and gene transcription, which may cause gene mutations and chromosomal aberrations and contribute to the toxicogenetic damage in subjects exposed to genotoxic xenobiotics (Pereira et al., 2013; Matos et al., 2017). Furthermore, the interindividual variability in genes, infuencing the enzymes, transporters, and other proteins involved in ADME and toxic response, have direct impact on the clearance and other TK parameters and constants including the critical formation of active or inactive metabolites. Figure 1.4, like Figure 1.2, highlights the infuence of metabolomic/metabonomic, proteomics, epigenomics, and transcriptomics 9

1.4 BASIC CONCEPTS AND ASSUMPTIONS OF PK AND TK

Figure 1.3 A general schematic illustration of toxicokinetics and toxicodynamics of a toxic xenobiotic and the interrelationship between absorption, distribution, metabolism, and excretion (ADME) with the toxic response; the exposure in the fgure refers to routes of absorption and entry of a toxic xenobiotic into the body via various routes of administration. on toxicokinetics and toxicodynamics. Section 1.2.3.1 of this chapter provides further clarifcations on the parts of Figure 1.4. The TK/TD modeling application in population toxicokinetics together with assessment of genetic and genomic fundamentals in toxicogenetics and toxicogenomics have practical applications in the study of toxic effects of xenobiotics in living organisms and across species extrapolation (also see Chapter 19, Section 19.1.3.1), which includes enquiry of mechanisms, symptoms, and detection of toxic response in the areas of environmental, biochemical, regulatory, reproductive, forensic, analytical, clinical, food, pharmaceutical, nutritional, behavioral, dermato-, neuro-, occupational, immuno-, idiosyncratic organo-, nano-, and mechanistic toxicology. 1.4 BASIC CONCEPTS AND ASSUMPTIONS OF PK AND TK Based on the defnition of PK and TK, the response to the administration of a therapeutic xenobiotic or exposure to a toxic xenobiotic is the consequence of the: ◾ entry of the compound into the systemic circulation via absorption from a topical route of administration (i.e., transdermal, ophthalmic, otic, vaginal, rectal, and nasal route of administration), or oral route (i.e., gastrointestinal, sublingual, and buccal route of absorption), or pulmonary (i.e., nasopharyngeal, tracheobronchial, and pulmonary route of administration), or direct access via the injection of the compound into the systemic circulation (i.e., intravenous 10

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Figure 1.4 Schematic illustration of involvement of proteomics, metabolomics, epigenomics, and transcriptomics on interactions with biological targets and toxicodynamic parameters under the title of toxicogenomics and the infuence of genetic polymorphism on absorption, distribution, metabolism, and excretion of a toxicant and possible variability in toxicokinetic parameters and constants under the designation of toxicogenetics in broadened areas, like risk assessment and TK/TD analysis. or intra-arterial route of administration), or injection into the body other than systemic circulation (i.e., intramuscular, intraperitoneal, and other injections into the body for local or systemic effects) ◾ immediate distribution in the body via systemic circulation ◾ rapid equilibration of systemic circulation concentration with highly perfused organs and tissues (e.g., heart, liver, kidneys, lungs, pancreas, and other highly perfused tissues) and slow uptake by less perfused tissues (muscle, adipose tissue, and others) ◾ availability of the compound for biotransformation to its nontoxic water-soluble (i.e., inactivation process like glucuronidation, sulfation, glutathione, mercapturic acid, and cysteine conjugations), or reactive metabolites by the liver (e.g., CYP450 enzymes) and other sites of metabolism in the body 11

1.5 INTRODUCTION TO THE ROUTES OF ADMINISTRATION

◾ removal of unchanged parent compounds and water-soluble metabolites from the body via the kidneys and other routes, like sweat, perspiration, breast milk, etc. The absorption, biotransformation, and excretion progression of a xenobiotic occur concurrently with its distribution. The driving force behind the distribution of the xenobiotic is the orderly fow of the systemic circulation that transports the compound to different organs and tissues as a function of time. The response, therapeutic or toxic, is a function of the time course of its systemic circulation concentration and the uptake by the organs/tissue, which often house the receptor site. Thus, the fundamental assumption in both PK and TK is that the concentration in systemic circulation, blood/plasma, is proportional to the concentration of the xenobiotic at the receptor site; not equal but proportional, and the plasma concentration of the parent compound, or the main compound released from prodrugs, most often is the predictor of the response. However, the toxic response, in addition to the toxic dose of parent compounds, may also occur because of the primary metabolites, particularly the toxic reactive metabolites. There are analytical methodologies, such as positron-emission tomography (PET), liquid chromatography tandem mass spectrometry (LC-MS/MS), quadrupole/time-of-fight (TOF), Orbitrap, microdialysis, and magnetic resonance spectroscopy (MRS) (Longuespée et al., 2021), that allows one to measure the concentration of a xenobiotic at its target site, if the target site is known. The PK or TK analysis enables one to establish an understanding of the infuence of ADME processes on the time course of plasma concentration of a xenobiotic. The analysis reveals the extent of ADME infuence on the administered or exposed dose as manifested by the plasma levels of the xenobiotic and the ADME kinetic parameters and constants. It is worth noting again that the relationship between the dose and plasma concentration is also infuenced by the intrinsic factors, like genetic polymorphism (Yang et al., 2013), functional integrity of the liver and kidneys, and extrinsic factors, like the environment, xenobiotic–xenobiotic (drug–drug) interactions, and concurrent administration of the compound with beverages and food. The magnitude of an administered dose of a xenobiotic distinguishes its lethal dose from a safe/therapeutic one. Paracelsus (1493–1541), the Swiss physician and philosopher of the German renaissance, was frst to write: “Alle Ding sind Gift und nichts ohn’ Gift; allein die Dosis macht, das ein Ding kein Gift ist.” “All substances are poisons; there is none which is not a poison. The right dose differentiates a poison from a remedy.” During drug development, the main distinction between TK and PK analyses is the administered dose that is higher in TK studies and nominal in PK evaluation. Both evaluations are based on methodical approaches to yield information on drug plasma levels, ADME profle of the drug in the body, and the relevant PK/TK parameters and constants. For the environmental exposure, it is often diffcult to determine the exact dose, and the probability of adverse effects depends on the frequency, duration, and extent of exposure, which presents challenges for defning the effective dose in environmental toxicology. In addition to dose, the route of administration/entry into the body infuences the availability of a xenobiotic and thus the response. The oral route, because of the convenience of administration, remains the frst choice for most therapeutic agents. Advances in drug delivery systems, including nanoparticles and drug–device combinations, have led the administration of therapeutic agents to new levels of elegance in utilization of most available routes of administration. The unintended environmental exposure of xenobiotics and entry into the body are mainly through the skin (topical), lung (inhalation), eye (ophthalmic), and gastrointestinal (GI) tract (ingestion). The following chapters offer discussions on the uniqueness and intricacies of the major routes of administration/ exposure. 1.5 INTRODUCTION TO THE ROUTES OF ADMINISTRATION The potency and onset of action of a xenobiotic, its bioavailability and duration of action, and the nature and extent of its pharmacological and/or toxicological response depend in part on the route of its administration or exposure. The physiological and biochemical characteristics of a route of administration infuence the ADME of a compound: i.e., the choice of the route brings about the types of biological and physiological processes and barriers the body uses to deal with a xenobiotic. For instance, the parenteral route, in particular intravenous injection, avoid the 12

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absorption process and provide immediate onset of action; whereas, for compounds that permeate through a complex membrane to reach the systemic circulation and the receptor site, a delay to the onset of action may occur, and the rate of appearance in the systemic circulation and their availability depends on many factors. These factors include the complexity of the membrane, the physicochemical characteristics of the compound, and the type of administered dosage form. The presence of physiological barriers, such as the blood–brain barrier, the blood–placenta barrier, the blood–retina barrier, and other barriers in the body infuence the distribution of xenobiotics. The standard terminologies for routes of administration, harmonized by the US Food and Drug Administration (FDA) and of International Conference on Harmonization (ICH – E2B terms) are accessible in Addendum I – Part 1 and at the following sites: FDA: https://www.fda.gov/drugs/data-standards-manual-monographs/route-administration ICH: https://admin.ich.org/sites/default/fles/inline-fles/E2B_R2_Guideline.pdf The various descriptions of the routes of administration presented in this handbook are focused on the physiological, biochemical, and physicochemical characteristics of the route that affect the absorption, distribution, metabolism, and excretion of xenobiotics. Each description also highlights the relevant rate equations and PK/TK relationships. It should be noted that for comprehensive anatomical and physiological features of the sites of administration, beyond the objectives of this handbook, related textbooks and articles on anatomy and physiology should be consulted. REFERENCES Aarons, L. 1991. Population pharmacokinetics: Theory and practice. Br J Clin Pharmacol 32(6): 669–70. Akacha, M., Bartels, C., Bornkamp, B., Bretz, F., Coello, N., Dumortier, T., Looby, M., Sander, O., Schmidli, H., Steimer, J.-L. 2021. Estimand – What they are and why they are important for pharmacometricians. Clin Pharmacol Ther 10(4): 279–82. Alkhateeb, A., Rezaeian, I., Singireddy, S., Cavallo-Medved, D., Porter, L. A., Rueda, L. 2019. Transcriptomics signature from next-generation sequencing data reveals new transcriptomic biomarkers related to prostate cancer. Cancer Inform 18: 1–12. https://doi.org/10.1177 /1176935119835522. Amzal, B., Julin, B., Vahter, M., Wolk, A., Johanson, G., Åkesson, A. 2009. Population toxicokinetic modeling of cadmium for health risk assessment. Environ Health Perspect 117(8): 1293–301. Ashauer, R., Esher, B. I. 2010. Advantages of toxicokinetics and toxicodynamic modelling in aquatic ecotoxicology and risk assessment. J Environ Monit 12(11): 2056–61. Bai, Z., Ye, Y., Liang, B., Xu, F., Zhang, H., Zhang, Y., Peng, J., Shen, D., Cui, Z., Zhang, Z., Wang, S. 2011. Proteomics-based identifcation of a group of apoptosis-related proteomes and biomarkers in gastric cancer. Int J Oncol 38(2): 375–83. Baldrick, P. 2003. Toxicokinetics in preclinical evaluation. Drug Discov Today 8(3): 127–33. Becquemont, L., Alfrevic, A., Amstutz, U., Brauch, H., Jacqz-Aigrain, E., Laurent-Puig, P., Molina, M. A., Niemi, M., Schwab, M., Somogyi, A. A., Thervet, E., Maitland-van der zee, A-H., van Kuilenburg, A. Bp., van Schaik, R. Hn., Verstuyft, C., Wadlius, M., Daly, A. K. 2011. Practical recommendations for pharmacogenomics-based prescription: 2010 ESF-UB Conference on Pharmacogenetics and Pharmacogenomics. Pharmacogenomics 12(1): 113–24. Bernstein, B. E., Meissner, A., Lander, E. S. 2007. The mammalian epigenome. Cell 128(4): 669–81. 13

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2 PK/TK Considerations of Auricular (Otic) – Buccal/Sublingual, and Ocular/Ophthalmic Routes of Administration 2.1 AURICULAR OR OTIC ROUTE OF ADMINISTRATION 2.1.1 Overview The human ears, symmetrically positioned on either side of the head, are for detecting sound and maintaining the body position and balance. The ear is divided into the outer ear, the middle ear, and the inner ear (Figure 2.1) The outer ear consists of the following: i. Pinna, for collecting, amplifying, and directing sound to the auditory canal. ii. Auditory oval-shaped canal 25–31 mm long and 6–9 mm wide that ends at the eardrum. The canal is weakly acidic with pH of 5.0–5.7 to prevent bacterial growth (Martinez et al., 2003; Kim and Cho, 2009). iii. Eardrum, or tympanic membrane, which is 0.1 mm thick, has a total vibrating surface area of 55 mm2, and separates the middle ear from the outer ear. The eardrum is made up of three layers: a. the outer epidermis b. middle fbrous connective tissue, and c. an internal mucosal layer. The middle ear is an air-flled tympanic cavity consisting of: i. The ossicular chain of three bones: malleus, incus, and stapes. ii. The mucosal coating to keep the middle ear environment moist. iii. The connecting Eustachian tube (~30–40 mm long, named after 16th century anatomist Bartolomeo Eustachi, also called pharyngotympanic tube) between the middle ear and back of the throat (upper airway) that drains excess fuid and maintains air pressure. The inner ear with delicate membranous sensory is protected by the bony labyrinth and is comprised of: i. The vestibule, the central chamber of the bony labyrinth, which is about 4 mm long and houses two otolith organs (utricle and saccule) that are responsible for detecting motion, orientation, and body posture. ii. Three semicircular canals of horizontal, posterior vertical, and anterior vertical that orient orthogonally to one another. iii. The cochlea, a 32 mm long spiral-shaped bone, internally separated into three fuid-flled compartments, scala tympani (ST), scala vestibuli (SV), and scala media (SM). The upper and lower compartments (ST and SV) are both flled with perilymph, which is like cerebrospinal fuid (CSF), and the middle one (SM) is flled with endolymph, which is like the intracellular fuid. The overall function of the cochlea is in auditory transduction. At the basal end of the cochlea is the round window membrane (RWM) between the inner ear and middle ear that protects the inner ear from the middle ear (Spandow et al., 1988). The RWM is thinner in the middle and thicker at the periphery with slight curvature toward the scala tympani (Carpenter et al., 1989; Goycoolea, 2001) and is made up of the following three layers (Goycoolea and Lundman, 1997; Goycoolea, 2001): i. an epithelial layer facing the middle ear ii. a connective tissue layer that provides support and protection iii. a cellular layer facing the scala tympani. The exchange of xenobiotics between the middle ear and inner ear takes place across the RWM.

DOI: 10.1201/9781003260660-2

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Figure 2.1 Schematic diagram of different segments of the ear: the outer ear consisting of pinna, auditory canal, and ear drum; the middle ear comprising of ossicular bones of malleus, incus, and stapes; the moist environment covered with mucosal coating, and the Eustachian connecting pharyngotympanic tube; lastly the inner ear containing the vestibule, semicircular canals, cochlea, and the round window membrane (RWM). 2.1.2 Blood-Labyrinth-Barrier and Auricular Absorption, Distribution, Metabolism, and Excretion The environment of the inner ear is an isolated one. The perilymph and endolymph are static fuids and are not in contact with the systemic circulation. The blood-perilymph barrier and bloodendolymph barrier collectively form the blood-labyrinth-barrier (BLB), made of tight junctions between specialized epithelial cells in the vestibular and cochlear parts of the inner ear. The BLB, also called the blood-cochlea-barrier, functions the same as the blood-brain-barrier, protecting the dynamic equilibrium of inner ear fuids and thus limiting the delivery of xenobiotics to the inner ear (Hawkins, 1973; Juhn and Rybak, 1981; McCall et al., 2010; Liu et al., 2013; Nyberg et al., 2019). Thus, to achieve therapeutic levels in the inner ear, large doses of a drug must be administered systemically that may cause side effects and toxicity. To bypass the BLB and treat the inner ear directly, the specialized intratympanic or intracochlear administration is considered. It has also been suggested that in addition to blood-perilymph-barrier and blood-endolymph-barrier, other barriers such as cerebrospinal fuid-perilymph barrier, middle ear-labyrinth barrier, and endolymph-perilymph barrier also contribute to the function of the BLB barrier (Sun and Wang, 2015). 2.1.2.1 Syndromes and the Sites of Absorption The most common outer ear syndrome is acute external otitis, also known as swimmer’s ear, which is associated with infammation throughout the ear canal (Fedorova and Shardin, 2016). The cause is either bacterial infection or fungal infection. The treatment is local and includes the use of topical medications and cleaning the external auditory canal (Rosenfeld et al., 2014). The site of absorption is accessible beyond the infuence of the BLB. The common middle ear disorders (acute otitis media, otitis media with effusion, chronic suppurative otitis media, and adhesive otitis media), are caused by both viral and bacterial pathogens (Qureishi et al., 2014). The acute otitis media and otitis media with effusion are childhood diseases with infection and infammatory complications between the tympanic membrane and the inner ear. Aside from oral analgesics administered for the pain, antibiotic therapy is considered for the treatment of the infection. Locally administered delivery systems of antibiotics (e.g., polymeric, gels, pellets, or nanoparticle delivery systems) is a convenient administration of therapeutic 20

PK/TK CONSIDERATIONS OF AURICULAR, BUCCAL/SUBLINGUAL AND OCULAR ROUTES

agents. However, the trans-tympanic absorption i.e., the penetration of xenobiotics through the ear drum is the rate-limiting step for the uptake and low bioavailability in the middle ear. Furthermore, it has been shown that antimicrobial resistance, which is caused in part by bacterial bioflm (Won et al., 2021) decreases the effectiveness of the systemic administration of antibiotic therapy. The specialized direct injection of the drugs into the middle ear known as intratympanic injection is often considered for its effcient and middle-ear targeted administration. The downside of the injection is the requirement of perforating the ear drum. Inner ear infection causes the incursion of leukocytes and immunoglobulin into the endolymph and perilymph that often results in ionic composition instabilities and subsequently loss of hearing, loss of balance, and possibly ossifcation or conformity of auditory structures. In general, the inner ear diseases include: i. Hearing loss due to trauma, genetic disease, autoimmune disease, exposure to ototoxic xenobiotics. ii. Balance dysfunction disorders including Meniere’s disease, labyrinthitis, and positional vertigo. The delivery of therapeutic agents to the inner ear is not an easy task. Therapeutic agents given intravenously or orally encounter the presence of the BLB that limits the uptake by the inner ear signifcantly. The round window membrane, is the only site for the absorption of xenobiotics from middle ear into the inner ear. The permeability of the RWM has been the subject of numerous investigations in humans, rhesus monkeys, and chinchillas. It is suggested that this membrane is permeable to selected xenobiotics and adenoviral vector gene delivery systems (Goycoolea et al., 1988; Suzuki et al., 2003; Noushi et al., 2005; McCall et al., 2010). The diffusion of xenobiotics through the RWM depends on the thickness and pathological conditions of the membrane and on the molecular size and chemical characteristics of a xenobiotic e.g., its lipid solubility, diffusion coeffcient, etc. The current approaches to treating the inner ear diseases include systemic administration, intratympanic, or intracochlear injections. Because of the presence of the BLB, the maximum allowable therapeutic dose of most drugs administered systemically, via intravenous injection or oral administration, does not provide suffcient concentration within the inner ear to provide optimum response (Liu et al., 2013). Intratympanic injection is a preferred approach to deliver optimal concentration of a drug into the middle ear at the RWM site of absorption. This specialized injection is done in the central posterior quadrant over the RWM (Viglietta et al., 2020) with the objective of presenting enough drug at the site for diffusion through the membrane into the inner ear. The intratympanic injection, in addition to perforating the ear drum, suffers from inconsistencies of diffusion rate through the RWM and removal of the drug via the Eustachian tube. The highly specialized direct intracochlear microinjection provides the targeted delivery system but carries the danger of signifcant trauma to the inner ear (Yu et al., 2020). 2.1.2.2 Auricular Distribution, Metabolism, and Excretion The volumes of perilymph and endolymph remain mostly undisturbed, and their fow rates are extremely slow. The inner ear distribution refers to the movement of drugs through the fuid compartments of perilymph and endolymph and penetration from the fuid compartments to the various tissue compartments of the inner ear. The distribution of xenobiotics within the volumes depends merely on their size, electrical charge, and molecular diffusion. This allows a xenobiotic to spread along the length of the cochlea, distribute into tissues surrounding the scala, and ultimately eliminate from the inner ear by diffusion into the systemic circulation. When the drug is administered by intratympanic administration, the transfer through the RWM into the scala tympani is by passive diffusion. With the intracochlear method of administration, the distribution bypasses the RWM diffusion step, and the injected drug migrates to the scala tympani and vestibule. The binding of the molecules to proteins of perilymph also infuences the distribution and the therapeutic outcome. P-glycoprotein (Pgp) a member of the ATP-binding cassette (ABC) is also expressed in the capillary endothelial cells of the cochlea and vestibule, which is likely to protect the inner ear from xenobiotics distributing by systemic circulation (Saito et al., 1997). Several other transporters and ion channels are expressed in the inner ear that are mostly involved in K+ cycling and endolymphatic K+, Na+, Ca+, Cl− and pH homeostasis (Lang et al., 2007), e.g., K+ channels, Na+2Cl−-K+ cotransporter, Na+/K+-ATP, Cl− channels, connexins, and K+/Cl− cotransporters that are 21

2.1 AURICULAR OR OTIC ROUTE OF ADMINISTRATION

critical for the normal functioning and sensory transduction of the inner ear. It is worth noting that the endolymph has a high concentration of K+ that provides the electrochemical basis for sensory cell function. These transporters may not be directly involved in the distribution of xenobiotics, but their normal functioning contributes to the predictability of functioning in the inner ear. Based on a comparative study of the expression of drug metabolizing enzymes of cochlea vs liver in mice, it is reported that the expression of three Cytochromes P450 (CYP450), namely 1a1, 1b1, and 2c66 was found to be similar; but for 2c65, the cochlea expression was twice that of the liver. The expression of many other CYP450 isozymes was low in cochlea compared to the liver (Kennon-McGill et al., 2019). A comprehensive and methodical study of transporters and drug metabolizing enzymes of the inner ear in humans is yet to be fully established. Drug elimination from the middle ear is mainly through the Eustachian tube and subsequent losses to the pharynx. Drug molecules may also permeate from the middle ear into the vascular system. Elimination from the inner ear, however, is mainly through diffusion into the systemic circulation and cerebrospinal fuid. Drug molecules may also diffuse back from the inner ear into the middle ear and eliminate through the Eustachian tube. Clearly, this can only happen when the conditions for passive diffusion and concentration gradient from the inner ear into the middle ear are created. 2.1.3 Auricular Rate Equations and PK/TK Models If a xenobiotic is administered by intratympanic microinjection at the site of absorption, i.e., the RWM of the middle ear, and assuming the diffusion occurs simultaneously, the overall fux of the molecules through the RWM can be expressed by the following advection–diffusion equation (Watanabe et al., 2017): J total = J diffusion + J advection = -Dcoeff Ñc - vc

(2.1)

Where: J total = diffusion flux per unit area (mol/m 2 s) Dcoeff = diffusion coeefficient (m 2 /s) v = is the speed of the solution containing the mediccation (m/s) c = concentration mol/L Ñ represents gradient Under the assumptions of axisymmetric pore, uniform concentration gradient and the defnition of total fux, the following diffusion equation can be developed: Flux diffusion = - DArea ( c1 - c0 ) / h = -

D ´Area ( c1 - c0 ) h

(2.2)

The constants and variables of Equation 2.2 are: c c1 and c0 = concentration at intracochlear and donor solution, respectively. h = RWM thickness (m) Dcoeff = diffusion coefficient (m 2 /s)) Area = cross-sectional area of the pore Setting the permeability constant as P =

Dcoeff with unit of m / s , Equation 2.2 can be rewritten as: h

Flux diffusion = P ´ Area ´ (c0 - c1 )

(2.3)

Comparing Equation 2.3 with the fux equation from the injection, i.e., Flux diffusion = v ´ Area ´ c0

(2.4)

From Equation 2.4, one may conclude that the injection is independent of the inner concentration and the type of molecules injected. Using Equations 2.3 and 2.4, one can determine the permeability of the membrane by setting c1 = 0. 22

PK/TK CONSIDERATIONS OF AURICULAR, BUCCAL/SUBLINGUAL AND OCULAR ROUTES

A similar viewpoint for transfer of xenobiotics through the RWM is suggested based on the following equation (Zhang et al., 2016): Aabsorbed = Area ´ P ´ DC ´ t

(2.5)

Aabsorbed = Amount of a drug absorbed Area = designated surface area of the RWM DC = Concentration gradient between the middle ear and inneer ear on either side of RWM P = permeability coefficient of the xenobiiotic through the RWM Equation 2.5 assumes that the RWM is a semipermeable membrane and that the transfer follows frst-order kinetics. It is further suggested that a simple one-compartment model with frst-order input can be applied to the scenario of assuming the middle ear as the site of absorption and the inner ear as the receiver compartment (Figure 2.2). The Eustachian tube is the elimination route from the middle ear, and diffusion into systemic circulation is the elimination route from the inner ear, i.e., dAabsorbed dA perilymph dAeliminated = + dt dt dt

(2.6)

dAabsorbed = rate of absorption of xenobiotic dt dA perilymph = ratee of change of amount of xenobiotic in the perilymph dt dAeliminated = rate of removal from inner ear dt Defning Equation 2.6 in terms of perilymph concentration yields: dC dAabsorbed =V + k elVC dt dt

(2.7)

The integrated forms of Equation 2.7 between time 0 – t and 0 – ∞ are:

Figure 2.2 Schematic diagram of a linear one-compartment model assuming the middle ear as the donor compartment with frst-order transfer rate constant through the round window membrane (RWM) into the inner ear, the receiver compartment; the rate constants of absorption and elimination are ka and kel, respectively, V is the volume of perilymph, and C the concentration of drug in perilymph; the elimination of the administered dose in the middle ear is through the Eustachian connecting tube and in the inner ear is through the permeation into the systemic circulation. 23

2.2 BUCCAL AND SUBLINGUAL ROUTES OF ADMINISTRATION

( Aabsorbed )t = VCt + kelV ò

t

0

( Aabsorbed )¥ = kelV ò

¥

0

(2.8)

Ct dt

Ct dt = ( Clearance ) perilymph ´ AUC0¥

(2.9)

V = Volume of perilymph Ct = Concentration of the therapeutic agent in the perilymph at time t By employing Equations 2.8 and 2.9 and using the Wagner–Nelson method (consult the section on Calculation of Initial Estimate of Absorption Rate Constant in this handbook) the absorption rate constant through the RWM can be estimated as:

( Aabsorbed )t ( Aabsorbed )¥

=

Ct + k el k el

ò

ò

¥

0

t

0

Cdt

Cdt

t k a FA0 k a FA0 e -kelt - e -kat dt e -kelt - e -kat + k el V ( k a - k ell ) 0 V ( k a - k el ) = = 1 - e -kat ¥ k a FA0 -kelt -k at e dt k el -e 0 V ( k a - k el )

(

)

ò

(

ò

(

Fabsorbed = 1 - e -kelt or, ln (1 - Fabsorbed ) = -k at

)

)

(2.10)

The plot of Equation 2.10 (Wagner–Nelson equation for one-compartment model with frst-order input and frst-order output) i.e., the plot of the natural logarithm of fraction remaining to be absorbed versus time is a straight line with slope of the absorption rate constant through the membrane. Because the RWM is a non-uniform complex membrane, the relationship between the permeability coeffcient, the frst-order absorption rate constant, and the thickness of the membrane is defned as (Zhang et al., 2016): P = ka ´ h

(2.11)

2.2 BUCCAL AND SUBLINGUAL ROUTES OF ADMINISTRATION 2.2.1 Overview The mouth cavity with its sections of lips, tongue, hard palate, cheeks, soft palate, and foor of the mouth has been used for absorption of therapeutic agents and for intended or unintended absorption of toxicants. The lining of the cavity, known as oral mucosa, is comprised of the inner lining of the lips, the lining on the hard and soft palates, gingival, buccal, and sublingual areas. The oral mucosa is used to treat local conditions like fungal diseases (e.g., blastomycosis, candidiasis, coccidioidomycosis, etc.), ulcers (canker sores), and periodontal disease (gum disease). It is also used to achieve systemic effect mostly via buccal or sublingual sites of absorption. The oral mucosa is also ill-used by exposure to smoking, chewing tobacco, recreational drugs, and unintended exposure to environmental carcinogens/toxins, co-carcinogens, and cancer promotors. Among the different sites for the absorption of xenobiotics in the mouth cavity, the nonkeratinized buccal and sublingual sites are the most permeable of the oral mucosal membranes and are used for achieving systemic effect. The buccal site (BUC) with mucosal thickness of approximately 500–800 µm and surface area of 50.2 cm2 ± 2.9 cm2 is between the cheek and gums (gingival). The sublingual site (SL) with mucosal thickness of 100–200 µm and surface area of 26.5 cm2 ± 2.9 cm2 is under the tongue between the ventral surface of the tongue and foor of the mouth (Czerkinsky and Holmgren, 2012; Kraan et al., 2014). Both sites are highly vascular regions and allow quick entry into the systemic circulation. The absorption of xenobiotics through BUC and SL avoids pre-systemic elimination of GI tract, i.e., enzyme-mediated metabolism by CYP450 isozymes, e.g., CYP3A4, and/or gastric acid degradation, and circumvents the hepatic frst-pass effect in the GI tract. An essential factor in BUC and SL absorption for achieving an optimum systemic concentration and consistent bioavailability is the retention of the drug (i.e., the dosage form) at the site of absorption. This requires steady and prolonged contact of the compound/dosage form with the mucosa without unintended removal from the site of absorption by parallel loss processes like dilution in saliva and/or involuntary swallowing, which may exhibit inconsistent bioavailability. The fow of saliva (~0.5 ml/min) may wash away the compound from the 24

PK/TK CONSIDERATIONS OF AURICULAR, BUCCAL/SUBLINGUAL AND OCULAR ROUTES

site of absorption into the stomach and reduce the consistency of the rate and extent of absorption from both SL and BUC routes. The recent novel delivery systems, e.g., bioadhesive delivery systems, are promising ways to optimize the absorption and bioavailability of therapeutic agents administered through the BUC or SL routes. It is also worth noting that the absorption of a delivery system intended for the BUC or SL route may also dissolve in saliva and spread across the entire mouth cavity and thus the absorption would be from all sites of absorption in the cavity (Shojaie, 1998). The mucus layer of the sites of absorption in the mouth cavity is a protective gel layer comprised of water, antibodies, and mucins which are large glycoproteins and oligosaccharides. The SL mucosa can produce immunological tolerance, i.e., it is a tolerogenic environment, devoid of microfold cells, and its tissue has no specialized epithelium-associated follicles or organized lymphoid structures (Nagai et al., 2015). Because of its thinness, vaccines and protein antigens can cross the SL mucosa rather rapidly (~15 min) by paracellular or transcellular mechanisms to accumulate in submucosal regions to be taken up by immune cells or reaching the systemic circulation to induce both systemic and mucosal immunity (Paris et al., 2021). 2.2.2 Buccal and Sublingual ADME and Related Rate Equations The sublingual/buccal absorption of xenobiotics takes place in the presence of saliva in environments with pH of 6–7 through the mucosal barrier into the systemic circulation by passive paracellular and transcellular diffusions (Rathbone and Tucker 1993). The diffusion of a xenobiotic through the barrier is limited by its molecular weight (Walton, 1935), and its distribution is a function of the blood fow, reported as 22.78 mL/min/100g tissue (Heckmann et al., 2001; Xia et al., 2015). There has been no report on the presence of active absorption, facilitated transport, or pinocytosis at this site. The mucosal barrier of the mouth cavity is comprised of the following layers (Figure 2.3): i. mucus layer ii. variable keratinized layer (absent in BUC and SL)

Figure 2.3 Schematic depiction of mucosal barrier of the mouth cavity and portrayal of transcellular and paracellular permeation pathways; the variable keratinized layer is somewhat absent in buccal and sublingual sites. 25

2.2 BUCCAL AND SUBLINGUAL ROUTES OF ADMINISTRATION

iii. epithelial layer iv. basement membrane v. connective tissues vi. submucosal region. Thus, the BUC and SL absorption can be defned as a partitioning of a xenobiotic between saliva and top epithelial layer. The small-molecule xenobiotics can permeate through the epithelium by transcellular and paracellular permeation (see Chapter 7). If the compound remains free and unbound, its concentration at time t in the epithelial layers is proportional to the free concentration in the saliva, i.e.,

˜ Cepithelial °t ˛ ˜ Csaliva °t

(2.12)

Therefore, the ratio of epithelium concentration to saliva concentration at a given time is the fraction of xenobiotic in the epithelial layer, which is equal to the partition coeffcient of the compound between the epithelium and saliva: f epithelium ˜

° Cepithelium ˛t ˜ P ° Csaliva ˛t

coeff

(2.13)

The reciprocal of the partition coeffcient is thus the ratio of saliva concentration over the epithelium level. The mass balance of xenobiotic between epithelium and saliva at their interface is defned as in Equation 2.14 (Xia et al., 2015): ˙ d  Cepithelium 1 ° Vsaliva Dcoefff ˜ Vepithelium ˇˇ ˘  ˘ Area ˘  Cepithelium 1   Cepithelium 2  ˝˝   P dt h  epithelium  ˆ ˛ coeff 1

(2.14)

Vsaliva is the volume of saliva; Vepithelium is the volume of the epithelium at the interface; Pcoeff is the partition coeffcient; ˜ Cepithelium °1 is the concentration of the interface layer or the frst layer of the epithelium; Dcoeff is the diffusion coeffcient with unit of cm2/s; ˜ hepithelium °1 is the thickness of the frst epithelium layer; Area is the surface area with unit = cm2; ˜ Cepithelium °2 is the concentration of the subsequent layer. The sublingual site is more permeable than the buccal and palatal (roof of the mouth) region. The difference is attributed to the thickness of the barrier, vascularity of the region, the degree of keratinization of the barrier, and physicochemical properties of the drug (Kurosaki et al., 1991; Bartlett et al., 2012). However, in an evaluation of absorption of a fentanyl tablet administered buccally and sublingually, no signifcant differences were observed in human subjects (Darwish et al., 2008), which may well be related to the physicochemical characteristics of fentanyl. The important physicochemical properties of xenobiotics affecting the oral mucosal absorption include lipophilicity of the compound and its partition coeffcient, solubility in saliva, pH of the saliva, pKa of the compound, and binding to the oral mucosa, which reduces the absorption of the compound (Katz and Barr, 1955). The model presented in Figure 2.4 represents the ADME processes of a buccal/sublingual solid dosage form administered for achieving systemic effect of its active ingredient that follows a three-compartment model. The model takes into consideration the parallel removal from the site of the administered dose due to its dissolution in saliva and involuntary swallowing (see the caption of Figure 2.4). A modifed version of the compartmental model (Figure 2.4) in conjunction with noncompartmental analysis has been reported as a practical approach for estimation of absorption from sublingual or buccal mucosa (Wang et al., 2013).

˜ dAabs °i ˜ dAabs °i

dt

˛ ˜ ˆ °i ˝ ˜ Peff °i ˝ ˜V °i ˝ ˜ C1 (t) i ˙ C2 (t)i °

is the rate of absorption for compartments ABUC/SL and AGI (ith compartment); dt ° ˜ ˛i is identifed as the absorption scale factor; ˜ Peff ° i is the permeability of compound in the

Where,

26

(2.15)

PK/TK CONSIDERATIONS OF AURICULAR, BUCCAL/SUBLINGUAL AND OCULAR ROUTES

compartment with volume of (V ) with two concentration terms ( C1 (t) i - C2 (t)i ) representing the i

concentration at the site of administration and concentration in mucosa (Wang et al., 2013). The compartmental analysis of xenobiotics absorbed from the buccal or sublingual route is analogous to the models with frst-order input and output.

Figure 2.4 Schematic illustration of a three-compartment model with input from a sublingually or buccally administered dose; the model takes into consideration the partial transfer of the dose by involuntary swallowing into the GI tract and its possible absorption into the systemic circulation or removal from the body. The variables are ABUC/SL: amount at the site of absorption; AGI: amount removed from the site of absorption by dissolving in saliva and swallowing into GI tract; AR: amount removed from the body without being absorbed; A1: amount in the central compartment/systemic circulation; A2 and A3: amount in the peripheral compartments; Ael: amount eliminated from the body by renal excretion and metabolism. The constants are ks: frst-order rate constant for swallowing the medication; kR: rate constant of removal from the body without being absorbed; ka1 and ka2: frst-order absorption rate constants from the site of absorption and GI tract, respectively; k12, k21, k13, k31: distribution rate constants; and K: the overall elimination rate constant of absorbed drug. In addition to the risk of drug loss from the site of absorption due to involuntary swallowing or salivary washout, the oral mucosa contains proteases; and enzymes, such as aminopeptidase, carboxypeptidase, and esterase, have also been identifed in homogenates of human buccal epithelial cell culture and porcine buccal mucosa (Nielsen and Rassing, 2000). The use of tissue homogenates does not allow one to distinguish between the membrane and cytoplasmatic enzymes. However, the presence of these enzymes may reduce the bioavailability of the peptides and protein drugs administered buccally or sublingually. In general, the extent of absorption of peptides through buccal or sublingual route depends on their molecular weight, polarity, chemical and enzymatic stability, conformation, and dissociation of the molecule. 27

2.2 BUCCAL AND SUBLINGUAL ROUTES OF ADMINISTRATION

2.2.3 Saliva With a production rate of ~600 mL/day, saliva is secreted by salivary glands of sublingual, submandibular, and parotid glands to provide lubrication and protection for proteins (Proctor, 2016). It consists of water (94–99%), ions, and proteins. The oral cavity is continually bathed and covered with saliva, which plays an important role in dissolution of solid and semi-solid dosage forms to facilitate the absorption of xenobiotics for local or systemic effect. It also functions in the clearance of xenobiotics from the mouth cavity and clearance of food debris and sugar. It aggregates and eliminates microorganisms and is rich in biological markers like DNA, RNA, and proteins. It hosts viruses like SARS-CoV-2, cytomegalovirus, Zika virus, and COVID-19 and plays a signifcant role in the propagation of viruses and microorganisms during speaking, coughing, and sneezing, which also plays a minor role in the removal of therapeutic agents from the sites of administration. It is reported that each cough, or 5-minute conversation can spread ~3000 saliva droplets, and each sneeze disseminates ~40,000 droplets (Baghizadeh-Fini, 2020; Li et al., 2020). The saliva that is secreted by the serous acinar cells of mammalian parotid and submandibular glands contains salivary carbonic anhydrase isozyme that catalyzes the reversible reaction of carbon dioxide to bicarbonate in the presence of water, supplying buffer capacity to neutralize low acidity in the mouth cavity. Due to the presence of immunological and non-immunological agents, i.e., IgA, IgG, IgM proteins, mucins, and enzymes (like lactoferrin, lysozyme, and peroxidase), saliva manages its antibacterial action and controls oral microfora. The important enzyme of saliva involved in the digestion and breakdown of fat and starch is A-amylase (Humphrey and Williamson, 2001), which can alter xenobiotics with a chemically analogous structure to its substrates. It must be noted that the enzymes of saliva in periodontal infection include enzymes that are released from infammation, tissue damage, and/or bacterial cells. Among them are lactate dehydrogenase, creatine kinase, alkaline phosphatase, and aspartate and alanine aminotransferase (Ozmeric, 2004; Todorovic et al., 2006). These enzymes can metabolize and reduce the stability of xenobiotics in saliva, a signifcant consideration for the antimicrobial treatment of plaquerelated oral disease (Na et al., 2007). The salivary excretion is a route for elimination of compounds administered through other routes of administration (Wagner, 1971). Depending upon the molecular size, lipid solubility, and pKa, the unbound fraction of administered or exposed xenobiotic is transferred from the systemic circulation through the salivary glands into saliva. In the ductal system of salivary glands, secretion and reabsorption infuence the fow rate of saliva and the concentration of its solutes. The concentration of a xenobiotic in saliva is much lower than the urine or plasma. The degree of protein binding in saliva is negligible, and back-diffusion of the compound into the plasma depends primarily on the degree of ionization of a compound at the pH of saliva. Thus, the ratio of saliva/plasma concentrations is a function of pH of saliva: under equilibrium conditions between concentration of free drug in plasma and saliva using the Henderson–Hasselbalch principles (Henderson, 1908; Hasselbalch, 1917), the following equations can be used for weakly acidic and basic compounds: Acidic compounds :

pH -pK Saliva [ Asaliva ] ëé HAplasma ûù ( f u ) plasma 1 + 10( saliva a ) ( f u ) plasma = ´ = ´ pH p ( plasma -pK a ) ( f u ) Plasma éA ë plasma ûù [ HAsaliva ] ( f u )saliva 1 + 10 saliva

Saliva 1 + 10( a saliva ) ( f u ) plasma = ´ Plasma 1 + 10( pK a -pH plaasma ) ( f u )saliva pK -pH

Basic compounds:

(2.16)

(2.17)

Where [ Asaliva ] and éë Aplasma ùû are the total concentration of ionized and unionized molecules in saliva and plasma, respectively, i.e., [ A ] = é A - ù + [ HA ] and f u is the free fraction of the compound. ë û Saliva samples can be used as an alternative to blood samples in therapeutic or toxicity monitoring of xenobiotics. In general, saliva testing is used when the ease of collection, compared to urine or blood samples, outweighs the cost and the required sensitive methodology, like mass spectrometry (Kidwell et al., 1998). The detection of illicit compounds (e.g., amphetamines, PCP, cocaine, marijuana, etc.) in saliva has been established in forensic science and practice. Saliva screening for therapeutic agents is also used in pediatric patients and therapeutic drug monitoring of many agents, such as tolbutamide, propranolol, procainamide, etc.

28

PK/TK CONSIDERATIONS OF AURICULAR, BUCCAL/SUBLINGUAL AND OCULAR ROUTES

2.3 OCULAR/OPHTHALMIC ROUTES OF ADMINISTRATION 2.3.1 Overview The complex and delicate makeup of the eye, the presence of its physiological barriers, and the intricacy of its independence in the body offers signifcant challenges to the ocular pharmacokinetic/toxicokinetic analysis of xenobiotics. The physiological barriers, known as blood-ocular barriers includes blood-aqueous barrier (BAB) and blood-retinal barrier (BRB). Both barriers infuence the choice of route of administration, the type of medications, and the overall ADME profle of xenobiotics. There are also physiological phenomena, such as aqueous humor production and release, nasolacrimal drainage, blinking, and blood fow, that infuence the uptake and disposition of a xenobiotic. The most permeable site in the eye is the cornea, and the corneal absorption is by passive transcellular diffusion, paracellular transport, and carrier-mediated effux and infux (Rivers et al., 2015). Most eye diseases are related to aging and include elevated intraocular pressure and glaucoma, age-related macular degeneration (AMD), cataract, and ophthalmic sequelae of diabetes. The other common diseases are inherited retinal diseases, dry eye, and damage to conjunctiva and cornea caused by environmental elements, physical accidents; fungal, viral, and bacterial infections; and related allergic responses. 2.3.2 The Blood Aqueous Barrier The BAB is created by the tight junction of the vascular endothelium of the iris and tight junction of the nonpigmented ciliary epithelium. This barrier, also known as the iris-ciliary barrier, limits the transfer of large molecules, e.g., proteins, across the blood vessels and the backfow of aqueous humor. Some small molecules and water can go across this barrier. It has been suggested that leukocytes have access to aqueous humor through the ciliary epithelium. This supports the idea that BAB serves as an immune-skewing gate under steady state conditions (Shechter et al., 2013). The ATP-binding cassette family of drug transporters that are present in the iris-ciliary section include Pgp, breast cancer resistance protein, and several multidrug resistance proteins, like MRP1, MRP2, MRP3, MRP4, and MRP6 (Zhang et al., 2008; Chen et al., 2013; Dahlin et al., 2013; Lee and Pelis, 2016). Furthermore, The SLC family of drug transporters includes anion transporters, organic anion transporting polypeptides, organic cation transporters, and peptide transporters. The transporters reduce the permeation of xenobiotics from systemic circulation into aqueous humor – adding to the effectiveness of the tight junction barrier – and reduce the uptake and effcacy of therapeutic agents administered systemically and intended for the eye. An early kinetic analysis defnes the rate of penetration of xenobiotics into the eye at constant plasma concentration as (Davson and Matchet, 1952): dCa = kinCp - k outCa dt

(2.18)

dCa = Rate of diffusion dt nt Ca = Concentration in the aqueous compartmen kin = first-order transfer rate constant from plasma into aqueouss compartment Cp = constant plasma concentration k out = first-ordeer transfer rate constant from aqueous into plasma As t Þ ¥ : dCa =0 dt \

kin Ca = =r k out Cp

(2.19)

Where r is the steady-state ratio achieved at t = ∞ The integration of Equation 2.18 with constant plasma concentration yields: æ C ö k ln çç 1 - a ÷÷ = - in t rCp r è ø

(2.20) 29

2.3 OCULAR/OPHTHALMIC ROUTES OF ADMINISTRATION

Note: At t = 0, Ca = 0 æ C ö Thus, the plot of ln çç 1 - a ÷÷ against time is a straight line through the origin with the slope of rCp è ø k - in . The r value may be assessed by sustaining a constant plasma concentration of diffusing r compound for a long time and estimating the fnal steady-state ratio. Equation 2.20 is applicable, when the penetration is simple, and validates that the kinetics of the compound can be properly described in terms of the frst-order rate constant of kin . Using Fick’s law of diffusion, the penetration through the barrier into the aqueous humor can be defned as (Davson et al., 1949): dA = P ´ area ´ ( Cp - Ca ) dt

(2.21)

The defnition of constants and variables of Equation 2.21 are: dA = Rate of diffusing compound dt A = Amount of diffusing compound P = Peermeability constant area = Area of the barrier Cp = Plasma concentration a of the compound at time t Ca = Aqueous humor concentration of thee compound at time t Based on the assumption that the rate of penetration through the barrier is much slower than the rate of diffusion in plasma and aqueous humor, the integration of Equation 2.21 yields: Cp - ( Ca )0 1 P ´ area = ( Rate a ) permeability log = 2.303 ´Va t Cp - Ca

(2.22)

( Ca )0 = Initial concentration of the compound in the aqueous humor Va = Volume of the aqueous humor t of a compound into aqueous humor ( Ratea )permeability = Penetration rate The penetration rate in Equation 2.22 refers to the rate of passage through the BAB into the aqueous humor as a function of time. Its collective value, as is defned by the equation, circumvents the need for data on the area of the barrier and the volume of aqueous humor. 2.3.3 The Blood-Retinal Barrier The BRB is a tight and restrictive physiological barrier that controls and regulates the exchange of proteins, ions, and water in and out of the retina. It consists of an inner barrier and an outer barrier. The inner barrier (iBRB) is formed by the retinal vascular endothelium and, like the blood brain barrier it is in the retinal microvasculature with a microvascular endothelium lining. The outer layer (oBRB) consists of a retinal pigment epithelium cell layer, and its role is to regulate the exchange of solute and nutrients from the choroid to the subretinal space (Raviola, 1977; Campbell and Humpheries 2012). BRB separates the retina from the systemic circulation, regulates the distribution of molecules from plasma to retina, and maintains the retinal homeostasis. It prevents the transfer of macromolecules and xenobiotics into the retina, and its role is critical in maintaining healthy vision. Any damage to its integrity causes loss of vision. The diabetic retinopathy is the best example of diseases that affect the integrity of the barrier and trigger its breakdown (Navaratna et al., 2007; Klaassen et al., 2013). The mechanisms of transport across BRB include: i. Paracellular diffusion that is mainly controlled by the tight junction of microvascular endothelium lining. 30

PK/TK CONSIDERATIONS OF AURICULAR, BUCCAL/SUBLINGUAL AND OCULAR ROUTES

ii. Facilitated diffusion of preferred solutes by transporters, e.g., glucose transport via glucose transporter (GLUT1). iii. Energy requiring active transport against the concentration gradient with the help of transporters. iv. Transcytosis/pinocytosis that is absent in normal BRB (Chow and Gu, 2017). v. Solute modifcation, e.g., conversion of CO2 to HCO3−, for the purpose of permeation from the apical to the basal side. 2.3.3.1 BRB Effux Transporters 2.3.3.1.1 P-Glycoprotein (Pgp/ABCB1) Pgp is expressed largely at the iBRB and plays a signifcant role in the retinal distribution of xenobiotics that are considered its substrates, e.g., digoxin, quinidine, verapamil, etc. It has been shown that the blood-retinal barrier and blood-brain barrier exhibit dissimilar Pgp infuences on the distribution of its substrates (Chapy et al., 2016; Toda et al., 2011). 2.3.3.1.2 Breast Cancer Resistance Protein (BCRP/ABCG2) BCRP, like Pgp, is mainly expressed in iBRB and, to a lesser degree, in oBRB. It functions as the effux transporter of photosensitive xenobiotics in retinal tissue, including phototoxic compounds like pheophorbide and protoporphyrin IX (Asashima et al., 2006). 2.3.3.1.3 Multidrug Resistance-Associated Proteins (MRPs/ABCCs) MRPs are mainly expressed in oBRB and the effux of a wide variety of endogenous and exogenous compounds, including toxic and therapeutic xenobiotics. The elimination of substrates of MRPs is often in concert with the function of other infux and effux proteins. For example, benzylpenicillin is a substrate of MRP4 and OAT3 (Hosoya et al., 2009). 2.3.3.2 BRB Infux Transporters 2.3.3.2.1 Glucose Transporter 1 (GLUT1/SLC2A1) GLUT1 is located at both the iBRB and oBRB and is responsible for the transport of D-glucose and dehydroascorbic acid (Tomi and Hosaya, 2004; Fernandes et al., 2003; Minamizono et al., 2006). 2.3.3.2.2 Taurine Transporter (TAUT/SLC6A6) TAUT functions as the transporter of amino acid taurine, which is necessary for the preservation of retinal structure (Tomi et al., 2007). 2.3.3.2.3 Cationic Amino Acid Transporter 1 (CAT1/SLC7A1) CAT1 transports L-arginine (Tomi et al., 2009; Zakoji et al., 2015) and L-ornithin (Kubo et al., 2015) through iBRB. 2.3.3.2.4 L-type Amino Acid Transporter 1 (LAT1/SLC7A5) LAT1 transports large neutral amino acids including L-leucine (Tomi et al., 2005), L-histidine (Usui et al., 2013), and L-phenylalanine (Atluri et al., 2008). LAT 1 transporter is also involved in the retinal disposition of drugs like gabapentin, L-dopa, melphalan, and alpha methyldopa. 2.3.3.2.5 Creatine Transporter (CRT/SLC6As) CRT actively transports creatine from systemic circulation through iBRB to the retina to maintain ATP homeostasis from phosphocreatine (Tachikawa et al., 2007). 2.3.3.2.6 Monocarboxylate Transporters (MCTs/SLC16As) MCTs transport, L-lactic acid, pyruvate, ketone bodies, and other monocarboxylates across the BRB by MCT1, MCT2, MCT3, and MCT4 (Gerhart et al., 1999). 2.3.3.2.7 Nucleoside Transporters (ENT1/SLC29A1, ENT2/SLC29A2, CNT1/SLC28A1, CNT2/SLC28A2) Among the ENTs, ENT2 largely mediates the adenosine uptake and elimination of hypoxanthine from the retina. It also accepts nucleoside drugs, like gemcitabine, as substrates suggesting that ENT2 at the BRB would be involved in transferring nucleoside drugs from the systemic circulation to the retina (Yao et al., 2001; Ranganath et al., 2021). 31

2.3 OCULAR/OPHTHALMIC ROUTES OF ADMINISTRATION

2.3.3.2.8 Folate Transporters (RFC1/SLC19A1, PCFT/SCL46A1) Three types of folate transport proteins namely folate receptor-α (FRα), reduced folate carrier 1 (RFC1/SLC19A1), and proton-coupled folate transporter (PCFT/SLC46A1) transport water soluble folates from systemic circulation to the retinal site across the BRB (Bozard et al., 2010). 2.3.3.2.9 Organic Anionic-Transporting Polypeptides (OATP1A2, OATP2B1) The OATP1A2 mediates the cellular uptake of all-trans-retinol which is then converted to retinoic acid in vivo by retinal dehydrogenase. Xenobiotics, like chloroquine, hydroxychloroquine, and digoxin, inhibit the retinal function of OATP1A2 causing disfunction of the canonical visual cycle and toxic buildup of retinoids (Xu et al., 2016; Kinoshita et al., 2014). 2.3.3.2.10 Organic Cation Transporters (OCTN1, OCTN2) OCTNs are involved in the transfer of L-carnitine across iBRB for the generation of energy from long-chain fatty acids. Xenobiotics, such as quinidine, betaine, and tetraethylammonium, inhibit the function of OCTNs and signifcantly reduce the uptake of L-carnitine to the retina (Tachikawa et al., 2010). Cationic drugs, like verapamil, methadone, nicotine, tramadol, and oxycodone, are also transported by OCTNs (Kubo et al., 2013). 2.3.4 Kinetics of BRB Infux Permeability Clearance – Small Water-Soluble Compounds Given Systemically The BRB, which is composed of retinal capillary endothelial cells (iBRB) and retinal pigmented epithelial cells (oBRB), plays a central role in the infux and effux transport of required hydrophilic endogenous compounds and exogenous xenobiotics. The following relationships have been suggested for determination of infux permeability clearance of a small hydrophilic compound (Hosoya et al., 2004): The tissue uptake of a small water-soluble compound, like vitamin C, can be expressed as: dCtissue = ( K influx ) permeability ´ Cpt - ( K efflux ) permeability ´ ( Ctissue )t dt Where,

(2.23)

dCtissue is the rate of concentration change in the tissue, and ( K influx ) permeability and dt

( Kefflux )permeability are the infux and effux permeability clearance rate constant in tissue, respectively. Cpt and ( Ctissue ) are plasma and tissue concentration, respectively. t The integration of Equation 2.23 yields Equation 2.24 expressing the apparent concentration of the compound in the tissue:

( Ctissue )t = ( Kinflux )permeability ´ AUC0t + Vinterstitial ´ Cpt

(2.24)

The apparent tissue concentration at time t is defned as ( Ctissue ) with units of the amount of the t compound per gram of the tissue (i.e., amount/g tissue); the effux permeability clearance constant -1 is ( K influx ) permeability with unit of (time ); the area under plasma concentration-time curve from zero to time t is AUC0t with unit of (amount-time/volume); the volume of interstitial space is Vinterstitial with unit of (volume/g tissue); and Cpt is the plasma concentration of the compound at time t with unit of (amount/volume). Dividing both sides of Equation 2.24 by Cpt yields Equation 2.25: t ( Ctissue )t ( Kinflux )permeability ´ AUC0

Cpt

=

Cpt

Rt = ( K influx ) permeability ´

+

Vinterstitial ´ Cpt Cpt

AUC0t + Vinterstitial Cpt

(2.25)

Where Rt is the apparent tissue to plasma concentration ratio with unit of (volume/g tissue); The AUC0t AUC0t ratio of has unit of (time). A plot of Rt versus would yield a straight line with slope of Cpt Cpt

( Kinflux )permeability and Y-intercept of Vinterstitial . 32

PK/TK CONSIDERATIONS OF AURICULAR, BUCCAL/SUBLINGUAL AND OCULAR ROUTES

2.3.5 Recommended Ocular Routes for Drug Administration Many ocular diseases are treated with therapeutic agents that often have narrow therapeutic ranges. High concentrations of these agents are considered toxic systemically, and their levels below the therapeutic range at the site of action are deemed inadequate. Syndromes, like the posterior segment eye diseases, or vitreoretinal illnesses require drug treatments that are more targeted and effective, thus requiring a special route for administration. Giving the therapeutic agents systemically with the hope of achieving optimum response in the regions of the eye that are protected by BRB and BAB may not be an ideal approach to treating some ocular diseases. Bearing in mind that overcoming the impediments of the barriers is often diffcult and may require high concentrations in systemic circulation, which is not considered prudent. Recommended routes of administration for treatment of ocular diseases are presented and discussed in this section and are listed in Table 2.1. Each of these routes has unique ADME characteristics and PK/PD outcomes. 2.3.5.1 Conjunctival Route of Administration The conjunctiva (CONJUNC) is a semi-transparent mucous membrane that covers the ocular surface. It is formed by an outer epithelium, which exhibits tight junctions at the epical side, and inner connective tissue. It lines the inside of both lower and upper eyelids and covers the white surface of the eye known as sclera. It does not cover the cornea, and the border where it coalesces with the cornea and opaque sclera is known as the limbus. The conjunctiva has three recognized areas (Figure 2.5): i. The bulbar conjunctiva portion, which covers the outer surface of the eye and stops at the limbus. ii. Two palpebral portions that cover the inner surface of both the upper and lower eyelids. iii. The fornix, or forniceal conjunctiva that conjoins the palpebral and bulbar parts. Collectively, the conjunctiva is a nonuniform membrane that contains goblet cells and secrets mucin to provide the required lubrication and formation of the tear flm. The tear flm contains mucin, water, electrolytes (e.g., bicarbonate, sodium, potassium, chloride, magnesium, calcium, and traces of other electrolytes and amino acids), proteins (e.g., lysozyme, lipocalin, lactoferrin, and IgA), and lipid. In addition to the production of tear flm, the conjunctiva is involved in the clearance of selected xenobiotics into the systemic circulation and their transfer to other parts of the eye. It contains transport mechanisms that maintain the stability and constancy of water, solute, ions, and tear flm. Its surface area is larger than the cornea, thus the conjunctiva is considered a useful route of administration for therapeutic agents intended for the posterior sections of the eye (Hosoya et al., 2005). The transfer of xenobiotics through the conjunctiva, depending on their hydrophilic or hydrophobic characteristics, is by paracellular or transcellular diffusion. Furthermore, the conjunctiva is fully involved in the absorption of Na+ from its mucosa by the mechanisms of cotransporters like Na+-glucose, Na+-amino acids, and Na+-nucleoside; and active secretion of Cl− (Gukasyan et al., 2008). It is postulated that conjunctiva may express the isoforms of effux transporters, like P-glycoprotein (Yang et al., 2000). Drug absorption via conjunctiva leads to drug transfer to the sclera and the iris-ciliary body, bypassing the aqueous humor (Chien et al., 1990). Conjunctiva is a highly vascular tissue, but the vascularization varies in its different parts. It is more permeable than the cornea, but it is a major route for removal of xenobiotics from the

Table 2.1 Ocular/Ophthalmic Routes of Administration Type Conjunctival Intracameral Intracorneal Intraocular Intravitreal Ophthalmic Retrobulbar Subconjunctival Sub-tenon

Defnition

Acronym

Administration to the conjunctiva Administration into the anterior chamber of the eye Administration within the cornea Administration within the eye Administration within the vitreous body of the eye Administration to the external eye Administration behind the pons or behind the eyeball Administration beneath the conjunctiva Administration in between the sclera and Tenon’s Capsule

CONJUNC INTRACAM I-CORNE I-OCUL I-VITRE OPHTHALM RETRO S-CONJUNC S-TENON

33

2.3 OCULAR/OPHTHALMIC ROUTES OF ADMINISTRATION

Figure 2.5 Schematic of the eye to identify areas where a xenobiotic is administered either topically or by injection; the areas such as various locality of conjunctiva, intracorneal, intravitreal, the areas where the eye barrier exist, and tear layer or precorneal space. eye and transferring into systemic circulation after topical administration. Diffused xenobiotics into the palpebral conjunctiva end up in the systemic circulation; whereas, when diffused into the bulbar conjunctiva, they end up in different parts of the eye, including the posterior segment (Shikamura et al., 2016; Ramsay et al., 2017). 2.3.5.2 Subconjunctival Route of Administration Subconjunctival (S-CONJUNC) injection of a therapeutic agent is mainly to provide an initial high level in a targeted area to treat lesions in the cornea, vitreous, sclera, and anterior uvea (Figure 2.5). The injection is given under the bulbar conjunctiva, or underneath the palpebral conjunctiva. Although the initial concentration is high, the duration of action is rather short-term because of the rapid permeation of the drug from the site of injection. 2.3.5.3 Intracameral Route of Administration Intracameral (INTRACAM) administration refers to the injection of a therapeutic agent, typically an antibiotic, into the anterior chamber of the eye. The injection is considered a postoperative cautionary approach for prevention of endophthalmitis after cataract surgery. Endophthalmitis is infammation inside the eyeball caused by infection or trauma. It is a potentially visionthreatening complication that occurs with a statistic of 131 endophthalmitis in 100,876 cataract surgeries (Gower et al., 2013). 2.3.5.4 Intravitreal Route of Administration Intravitreal (I-VITRE) administration is the injection of a suitable dosage form into the posterior chamber of the eye known as the vitreous (Figure 2.5). This route is commonly used for the treatment of many diseases of the posterior segment of the eye like diabetic retinopathy and agerelated macular degeneration. Although retinal drug delivery can be accomplished using other routes of administration, intravitreal injection provides adequate drug concentrations in the retina and is currently considered the method of choice for treatment of retinal diseases and retinal drug delivery – in particular, the delivery of anti-vascular endothelial growth factor (anti-VEGF) proteins. To summarize, the benefts of the route are: 34

PK/TK CONSIDERATIONS OF AURICULAR, BUCCAL/SUBLINGUAL AND OCULAR ROUTES

i. Achieving the desired therapeutic concentration instantaneously in the vitreous and retina. ii. Limiting the removal of the drug from the site of action into the systemic circulation. The drawbacks, however, are: i. The procedure is invasive. ii. There is a risk of retinal detachment following multiple injections of therapeutic agents like anti-VEGF (Storey et al., 2019). The vitreous or vitreous humor is a colorless fuid between the lens and the retina. It is made up of: i. hyaluronan, an anionic hydrophilic polymer with MW of 2-4 million Daltons ii. structural proteins like fbrillin iii. collagen II, IX, V, and/or XI iv. cartilage oligomeric matrix protein (Bishop, 1996) v. non-structural proteins, like albumin, immunoglobulin, complement proteins, globulin, and transferrin (Chen and Chen, 1981; Laicine and Hadad, 1994; Ulrich et al., 2008). Furthermore, the proteomic analysis of human vitreous has identifed classes of proteins like protease, protease inhibitors, cytokines, complement and coagulation cascade hormones, apoptosis regulation, and proteins related to signaling activity and visual perception (Murthy et al., 2014). The average volume of the vitreous humor in human is about 4–4.5 ml, and the volume of injection is between 50–100 µL. Examples of the most common intravitreal medications with their recommended doses in permitted volumes include: amikacin 400 µg/100 µL; amphotericin B 5 µg/100 µL; voriconazole 50–100 µg/100 µL; dexamethasone 400 µg/100 µL; ceftazidime 2.25 mg/100 µL; vancomycin 1000 µg/100 µL; methotrexate 400 µg/100 µL; fomivirsen 330 µg/50 µL; foscarnet 1.2 mg/50 µL; clindamycin 1 mg/100 µL; ganciclovir 2 mg/50 µL; triamcinolone acetonide 2 mg/50 µL; brolucizumab 6 mg/50 µL; afibercept 2.0 mg/50 µL; ranibizumab 0.5 mg/50 µL; and bevacizumab (currently off-label) 1.25 mg/50 µL convection. The high concentration of the therapeutic agents given in small volume establishes the required concentration gradient in the vitreous for permeation and encountering the BRB. After the injection, the therapeutic agents diffuse and distribute in the vitreous and permeate into surrounding ocular tissues. In addition to diffusion, intravitreal convection may also play a role in the distribution of injected dose in the vitreous. The agents permeate frst in high concentration in the pericytes, which is the external layer of blood vessels in the normal retina, and then reach the endothelial inner layer after crossing the BRB (Giurdanella et al., 2015). 2.3.5.4.1 Intravitreal ADME and Rate Equations The overall elimination from vitreous is through the blood ocular barriers into the systemic circulation. It is worth nothing that binding of drugs to vitreous proteins and other components may infuence the clearance and elimination rate of a drug from vitreous into the systemic circulation, but it is considered marginal. The half-life of drugs administered intravitreally in humans is different from animal models, like rabbits. It is usually longer for compounds of large molecular weight (Urtti, 2006). Due to the presence of traces of metabolic enzymes in the vitreous humor, like peptidase and esterase, negligible metabolism may occur that is considered inconsequential in the elimination of a drug. In most intravitreal pharmacokinetic studies, the vitreous is considered a single compartment connected to the systemic model of the injected compound, which can be one, two, three, or more compartments (Makoid and Robinson, 1992). Although it may not be an easy task, the essential part of the in vivo PK study of an intravitreal injection is the simultaneous sampling of the vitreous and plasma and determination of the time course of the concentration changes. A proper systemic model, in addition to predicting the vitreous clearance of injected drug and its rate of elimination (i.e., the input rate into the systemic circulation), can provide an estimate of systemic clearance of the drug from the body, the uptake and degree of systemic exposure, and potential systemic toxicity. The rate of elimination of an intravitreally injected compound, assuming a linear onecompartment model for the vitreous as an isolated organ (Figure 2.6) is: 35

2.3 OCULAR/OPHTHALMIC ROUTES OF ADMINISTRATION

dAvitreous = -kvitreous Avitreous dt

(2.26)

The integrated form of the Equation 2.26 is:

( Cvitreous )t =

( Avitreous )t=0 Vvitreous

- k ´t ´ e ( vitreous )

(2.27)

Where ( Cvitreous ) is the drug concentration in vitreous at time t ; ( Avitreous ) is the injected amount t t=0 in vitreous at time zero (i.e., Dose); Vvitreous is the volume of distribution of drug in vitreous, which is different from the physiological volume of vitreous; and kvitreous is the frst-order rate constant of drug elimination from the vitreous. The drug’s half-life and clearance in vitreous may then be estimated as:

( T1/2 )vitreous =

ln 2 kvitreous

( Cl )vitreous = ( kvitreous ) ´ (Vvitreous )

(2.28) (2.29)

As was noted earlier, the elimination rate of an intravitreal injected drug is the rate of input into the systemic circulation; assuming all biological processes in the body follow frst-order kinetics, and the disposition of the drug follows the linear one-compartment model as depicted in Figure 2.6, the related rate equations are as follows (Xu et al., 2013; Zhang et al., 2014; Park et al., 2015; Vanhove et al., 2021): dAsys = ( kvitreous ´ Avitreous ) - KAsys dt Where ( Avitrreous )

(2.30)

= ( Dose )vitreous Equation 2.30 can be written as: t=0

dAsys = ( kvitreous ´ Avitreous ) - ClT Cp (2.31) dt dAsys The rate of change of amount in the body is , the elimination rate from vitreous (i.e., the rate dt of input from vitreous into the systemic circulation) is ( kvitreous ´ Avitreous ) , and the rate of elimination from the body is ClT Cp , or KAsys . It should be noted that for most, if not all, intravitreally injected drugs, the overall elimination rate constant of the drug from the body is greater than the rate constant of elimination from the vitreous, i.e., K >> kvitreous . This scenario is known as the “fip-fop model” where the frst-order rate

Figure 2.6 A typical one-compartment open model with the vitreous as the site of absorption of an intravitreal injection with elimination rate from the vitreous into the systemic circulation; kvitreous is the frst-order elimination rate constant from the vitreous and input rate constant into the systemic circulation; K is the overall elimination rate constant; Avitreous and Asys are the amount in the vitreous and systemic circulation as a function of time, respectively; and Aelim is the total amount eliminated at a given time. 36

PK/TK CONSIDERATIONS OF AURICULAR, BUCCAL/SUBLINGUAL AND OCULAR ROUTES

constant of input is less than the rate constant of the output, and the integrated equation based on the integration of Equation 2.30 is (see also Chapter 15, Figure 15.4): Cp =

F ´ Dvitreous ´ K -kvitreous -Kt e -e Vd ( K - kvitreous )

(

)

(2.32)

Where Cp is the plasma concentration at time t ; F is the fraction of the dose (injected in the vitreous) available in the body or the bioavailability of the administered dose; and Vd is the apparent volume of distribution of the compound, estimated from plasma concentration. If the tissue surrounding the vitreous acts as a separate compartment, depending on the disposition of the drug in the body, e.g., a two-compartment model, as depicted in Figure 2.7, the following linear differential equations can be established to predict the uptake and release of the compound by the tissue compartment of the vitreous: dAvit = -kvit ´ Avit dt

(2.33)

dAtissue = kvit ´ Avit - ktissue ´ Atissue dt

(2.34)

dAsys = ktissue ´ Atissue + k 21 ´ A2 - K ´ Asys - k12 ´ Asys dt

(2.35)

dA2 = k12 ´ Asys - k 21 ´ A2 dt

(2.36)

dAelim = K ´ Asys dt

(2.37)

Equation 2.33 represents the elimination rate from the vitreous; Equation 2.34 is the rate of change of drug level in ocular tissue compartment; Equation 2.35 is the rate of change of drug levels in the systemic circulation; Equation 2.36 is the rate of change of drug level in the peripheral compartment; and Equation 2.37 is the rate equation of elimination with respect to the total amount eliminated, that is why it has the positive sign. Using the Laplace transform (Addendum I, Part 2), the integrated forms of Equations 2.35–2.37 are essentially parallel to the model describing the frst-order input into the central compartment of a two-compartment model (see also Chapter 15, Section 15.2.2.1). 2.3.5.5 Intracorneal Route of Administration The major noninvasive delivery of drug into the eye is the permeation through the cornea (I-CORNE). The cornea is a complex, transparent, and protective barrier of the eye that provides two-thirds of the eye’s refractive power (Pepose and Ubels, 1992). In humans, it is comprised of the following six layers (Figure 2.8): i. Corneal epithelium, a multicellular thin tissue layer that acts as tight junctions and reduces the permeation of drug molecules. In addition, it contains multiple drug resistance proteins (MRP2, MRP5) and P-glycoprotein (Pgp) that collectively effux selected drug molecules out of the cornea (Hariharan et al., 2009; Karla et al., 2010). ii. Bowman’s layer, a subcellular fbrous meshwork made up mainly of collagen fbrils. iii. Stroma, a thick hydrated collagen-rich layer that forms 90% of the cornea’s thickness and allows only hydrophilic molecules to diffuse through this layer. iv. Dua’s layer, made up of 5–8 lamellae of compact collagen (Dua et al., 2013). v. Descemet’s membrane, a thin but strong layer formed mainly of collagen fbrils, supports the endothelial layer of the posterior cornea. vi. Corneal endothelium, which is structurally and functionally different from corneal epithelium. The thickness of the cornea is about 0.5 mm, its diameter is 12 mm, with a negatively charged surface area. It lacks blood vessels, and its oxygen and nutrients are provided by the aqueous humor and tear flm. Only small molecules with molecular weight of 500 Da can penetrate the tight junctions, and due to its negatively charged surface, the permeation of negatively charged molecules is less than neutral or positively charged molecules. 37

2.3 OCULAR/OPHTHALMIC ROUTES OF ADMINISTRATION

Figure 2.7 The diagram of a drug injected intravitreally, absorbed into the retinal tissue, and released into the systemic circulation where it distributes according to a two-compartment model; kvit is the rate constant of permeation into the tissue space of retina, ktissue is the rate constant of transfer from the tissue compartment into the systemic circulation, k12 and k21 are the distribution rate constants, and K is the overall elimination rate constant; Avit and Atissue are the amounts injected into the vitreous and the amounts penetrated into the tissue compartment, respectively; Asys and A2 are the amount in the systemic circulation (i.e., the central compartment) and peripheral compartment (i.e., the body tissue compartment), respectively; Aelim is the amount eliminated as a function of time. 2.3.5.5.1 Intracorneal Permeation and Related Rate Equations The therapeutic dose for intracorneal absorption is usually supplied as an eye drop of clear solution, a suspension of particles or nanoparticles, or an emulsion of oil in water. For selected therapeutic agents, the dose is also provided as a stationary polymeric delivery system/device for extended/controlled release. Harmful/toxic xenobiotics may enter the eye as liquids, particles/ nanoparticles, smoke, or gas which dissolve or mix in the tear volume upon entry. The normal resident tear is approximately 6–8 µL. The tear chamber can only hold around 10 µL, although a maximum of 20 µL is possible if one prevents spillover by avoiding blinking. The spillover effect is the result of drained away tears through the nasolacrimal duct or just running down the cheek. A standard eye drop of 40–50 µL activates the blinking refex, and excess volume that contains the drug are washed away within seconds. Thus, the calculation of the dose available at the site of corneal absorption must be estimated correctly. The two processes of dilution and spillover affect the bioavailability of an ophthalmic drug at the corneal site signifcantly. If the average eye drop is around 50 µL, and the resident tear volume is 7 µL, one would expect 10–20% reduction of instilled concentration just by the dilution effect. To achieve an acceptable bioavailability, an adequate dose needs to be incorporated in the eye drop, i.e., smaller drop size and higher concentration. The use of larger drops only increases the systemic exposure through the nasolacrimal duct. The following frst-order equation is suggested for the determination of the nasolacrimal rate (Le Merdy et al., 2019): Rnasolacrimal = K drainage ´ C pre-corneal ´ (Vt - Vphys ) + Ktear- flow ´ C pre -corneal

(2.38)

Where Rnasolacrimal is the nasolacrimal rate; K drainage is the drainage rate constant; C pre -corneal is the precorneal drug concentration; Vt is the volume at time t , or instantaneous precorneal volume; Vphys is the physiological or baseline volume of pre-cornea; Ktear - flow is the tear fow rate constant. The rate of change in volume of tear is estimated by the following relationship (Le Merdy et al., 2020):

(

dVtears Phys = -K drainage ´ Vtears (t) - Vtears dt 38

)

(2.39)

PK/TK CONSIDERATIONS OF AURICULAR, BUCCAL/SUBLINGUAL AND OCULAR ROUTES

Figure 2.8 Schematic diagram of six layers of cornea including the corneal epithelium containing the effux proteins; the collagen fbril of the Bowman layer; the hydrated collagen layer of stroma that is the site for intracorneal injection; the compact lamellae of the recently identifed Dua’s layer; the frm collagen layer of Decemet’s membrane; and fnally, the corneal endothelium with different structure and function than the epithelium. dVtears is the rate of change in volume of tear; K drainage is the drainage rate constant; Vtears (t) is the dt Phys changing volume of precorneal compartment; and Vtears is the same as Equation 2.38. The effect of dilution on the precorneal concentration of drug and its reduction as a function of time can be defned as (Mishima, 1981):

( Cprecorneal )t = ( Cprecorneal )t=0 e -K

drainge ´t

(2.40)

Where ( C precorneal )t is the precorneal drug concentration at time t ; ( C precorneal )t=0 is the initial precorneal concentration; and K drainage is the frst-order rate constant of loss due to the tears draining. The corneal transfer rate equation (i.e., transfer toward the posterior sites) can be defned as: dCcornea -K ´t = ( Ktear-cornea ) ´ ( C precorneal )t=0 e drainage - K coornea ´ Ccornea dt

(2.41)

dCcornea is the rate of drug transfer from cornea toward the posterior sites of the eye; dt Ktear-cornea is the transfer rate constant of drug from tear to cornea; ( C precorneal )t=0 is the initial precorneal concentration; K cornea is the rate constant of loss from the cornea; and Ccornea is the corneal concentration at time t . The integration of Equation 2.41 yields Equation 2.42 for the corneal concentration at time t (Mishima, 1981): Where

39

2.3 OCULAR/OPHTHALMIC ROUTES OF ADMINISTRATION

Ccornea =

( Ktear-cornea ) ´ ( Cprecorneal )t=0 K drainage - K cornea

(e

The time of the maximum corneal concentration, ( Tmax )

( Tmax )corneal =

-( Kcornea ) ´ t

corneal

- K ´t - e ( drainage )

)

(2.42)

can be estimated as:

ln ( K drainage K cornea )

(2.43)

K drainage - K cornea

Since K drainage ˜ K cornea , the corneal peak time, ( Tmax ) , occurs soon after the instillation in the eye. corneal Under the same condition, i.e., K drainage ˜ K cornea , the corneal concentration at time zero, which is the y-intercept of the extrapolated terminal portion of the biexponential curve of equation 2.42, is defned as:

( Ccornea )t=0 =

( Ktear-cornea ) ´ ( Cprecorneal )t=0 ( Ktear-cornea ) ´ ( Cprecorneal )t=0 K drainage - K cornnea

»

K drainage

(2.44)

Equation 2.44 and the calculated value of the y-intercept can be used to determine the tear-cornea transfer rate constant. Since the transfer through the cornea toward the posterior parts of the eye during the initial period after instillation is negligible, the rate of uptake by the cornea can also be defned as: dAcornea -K n ´t = Pepithelial ´ Areacornea ´ ( C precorneal )t=0 ´ e drainage dt

(2.45)

dAcornea Where the rate of drug uptake by the cornea (mass/time) is ; the permeability of the epithedt . lial barrier is Pepithelial ; and the surface area of the cornea is Areacornea The integration of Equation 2.45 yields the total amount of drug absorbed by the cornea, ( Acornea )t=0 , i.e.,

( Acornea )t=0 = Pepithelial ´ Areacornea ´

(Cprecorneal )t=0 K drainage a

(2.46)

Setting ( Acornea ) = ( Ccorneal ) ´ (V ) and using Equations 2.44 and 2.45, the permeability of the cornea t=0 t=0 epithelial barrier can be estimated as (Mishima, 1981): Pepithelial =

Vcornea ´ K drainage ( Ktear-cornea ) ´Vcornea = ( Ccornea )t =0 ´V Areacornea

Areacornea ´ ( C precorneal )t=0

(2.47)

Using Vcornea @ 0.08ml; Areacornea @ 120.3 ± 2.22mm2 ; K drainage @ 6h -1 ; and ( C precorneal )t=0 = concentration of the instilled drug, the assessment of the epithelial permeability can be estimated for most eye drops. The bioavailability of the instilled dose, estimated by the following relationship (Patton and Robinson, 1976), is: F=

K drainage ´Vaqh ´ AUCaqh Dinstill

(2.48)

Where F is the fraction of dose absorbed; K drainage is the rate constant of drug loss from precorneal layer; Vaqh is the estimated volume of aqueous humor (»0.3 ml); AUC is the area under the curve of aqueous humor estimated by intracameral injection (Tang-Liu et al., 1984); and Dinstill is the instilled dose. The clearance from aqueous humor is then estimated from the following relationship (Schoenwald, 1993): Claqh =

Dintracameral

( AUCaqh )0

(2.49)

¥

Where Dintracameral is the intracameral dose and ( AUCaqh )0 is the area under aqueous humor concentration-time curve from 0 ® ¥. The permeation through the cornea following the instillation of eye drops is often considered a less effcient route for required controlled and long-term treatment of certain ocular diseases, ¥

40

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particularly in the posterior of the eye. The advances in nanotechnology or biodegradable/nondegradable polymeric inserts have improved the bioavailability of eye medications for intracorneal absorption to some extent. Nonetheless, the controlled release of medication from polymeric inserts or facilitated permeation with the aid of nanoparticles would still face the junctures of absorption through the cornea, dilution in precorneal tear layer, and drainage through the nasolacrimal duct. It is worth noting that, due to its noninvasive procedure, the transcorneal permeation is a widespread method for many eye-related diagnostics and treatments. 2.3.5.5.2 Intracorneal Injection and Related Rate Equations The invasive intracorneal injection, also known as corneal intrastromal injection, is used mostly for the treatment of severe fungal and bacterial keratitis and corneal neovascularization (Hashemian et al., 2011; Liang and Lee, 2011; Niki et al., 2014; Hu et al., 2016). The technique is also advocated and used for gene therapy of selected corneal stromal abnormalities like transplant rejection, fbrosis, and corneal opacities (Hirsh et al., 2017; Vance et al., 2016). The corneal intrastromal injection bypasses the epithelial permeation, precorneal dilution, and nasolacrimal drainage. It increases drug concentration and bioavailability in the stromal layers very rapidly with less frequent application of precorneal dosing. As indicated earlier, the corneal stroma is a dense connective tissue covering about 90% of the cornea and consists of multiple collagenous lamellae dispersed in keratocytes. Keratocytes are responsible for the transparency of the stroma. The stromal environment favors permeation of hydrophilic compounds, and it is considered a barrier for diffusion of hydrophobic drug molecules. Thus, the permeation rate in stroma toward the endothelium and the anterior or posterior of the eye is different among the therapeutic agents injected in stroma. Due to the viscoelasticity of the stroma, the injected dose will form a depository in the stromal environment and gradually permeate toward the aqueous humor compartment. The stromal transfer rate into the aqueous humor compartment is faster than the return rate. The exit rates from the aqueous humor compartment include the elimination rate by aqueous humor fow and the other routes, and the transfer rate toward the posterior part of the eye. The rate equations of the model are: dAaqh = kSA Astroma - k AS Aaqh - k AE Aaqh - k AP Aaqh dt

(2.50)

dAstroma = kDSD + k AS Aaqh - kSA Astroma dt

(2.51)

dAstroma ; the rate of proliferation of dt the injected dose is kDSD; the injected dose is D; the rate of amount transferred from the stroma into the aqueous humor is kSA Astroma ; the amount in the stroma at time t is Astroma ; the return rate from aqueous humor compartment into the stroma is k AS Aaqh ; the amount in the aqueous humor compartment at time t is Aaqh ; the elimination rate from the aqueous humor within the anterior chamber is k AE Aaqh , which can involve the aqueous humor outfow distribution into the lens, or metabolism; and the rate of input into the posterior parts of the eye is k AP Aaqh . Where the rate of change of amount injected in the stroma is

2.3.5.6 Retrobulbar, Peribulbar, and Sub-Tenon Routes of Administration Most ophthalmic procedures are carried out under regional anesthetic nerve block. A traditional approach is the injection of anesthetic agent into the retrobulbar space located behind the eyeball (Figure 2.5). The injection provides sensory anesthesia of the cornea, uvea, conjunctiva, and loss of muscle control, thus prevents blinking and eye movement. The possible complications of retrobulbar injection include allergic reaction, which is related to the type of the anesthetic agent, hemorrhage, perforation of the eyeball, damage to the optic nerve, and systemic side effects, like drowsiness, vomiting, etc. The complications of retrobulbar injection have led clinicians to rely on the peribulbar route of administration. It is an alternative approach to retrobulbar administration and consists of injections into the extraconal space above and below the orbit. The injections avoid the risk of hemorrhage and damage to the optic nerve but may create a transient blindness. Overall, the peribulbar injection is considered by some experts to have fewer complications than the retrobulbar administration.

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2.3 OCULAR/OPHTHALMIC ROUTES OF ADMINISTRATION

Sub-tenon injection is another route for the administration for anesthetic agents. It involves the injection of the anesthetic into the sub-tenon space. It is considered a relatively low risk injection. It has gained more popularity with the refnement of the required cannula and the incisionless administration. The sub-tenon, retrobulbar, and peribulbar are specialized routes of administration used mostly for regional anesthetic nerve block and are not used routinely for administration of ocular therapeutic agents. REFERENCES Asashima, T., Hori, S., Ohtsuki, S., Tachikawa, M., Watanabe, M., Mukai, C., Kitagaki, S., Miyakoshi, N., Terasaki, T. 2006. ATP-binding cassette transporter G2 mediates the effux of phototoxins on the luminal membrane of retinal capillary endothelial cells. Pharm Sci 23(6): 1235–42. Atluri, H., Talluri, R. S., Mitra, A. K. 2008. Functional activity of a large neutral amino acid transporter (LAT) in rabbit retina: A study involving the in vivo retinal uptake and vitriol pharmacokinetics of L-phenyl alanine. Int J Pharm 347(1–2): 23–30. Baghizadeh-Fini, M. 2020. Oral saliva and COVID-19. Oral Oncol. https://doi.org/10.1016/j .oraloncology.2020.104821. Bartlett, J. A., Kees Voort Maarschalk, K. V. 2012. Understanding the oral mucosal absorption and resulting clinical pharmacokinetics of asenapine. AAPS Pharm Sci Tech 13(4): 1110–5. Bishop, P. 1996. The biochemical structure of mammalian vitreous. Eye 10(6): 664–70. Bozard, B. R., Ganapathy, P. S., Duplantier, J., Mysona, B., Ha, Y., Roon, P., Smith, R., Goldman, I. D., Prasad, P., Martin, P. M., Ganapathy, V., Smith, S. B. 2010. Molecular and biochemical characterization of folate transport proteins in retinal Müller cells. Invest Ophthalmol Vis Sci 51(6): 3226–35. Campbell, M., Humphries, P. 2012. The blood-retina barrier: Tight junctions and barrier modulation. Adv Exp Med Biol 763: 70–84. Carpenter, A., Muchow, D., Goycoolea, M. V. 1989. Ultrastructural studies of the human round window membrane. Arch Otolaryngol Head Neck Surg 15(5): 585–90. Chapy, H., Saubamea, B., Tournier, N., Bourasset, F., Behar-Cohen, F., Decleves, X., Scherrmann, J.M., Cisternino, S. 2016. Blood-brain and retinal barriers show dissimilar ABC transporter impacts and concealed effect of P-glycoprotein on a novel verapamil infux carrier. Br J Pharmacol 173(3): 497–510. Chen, C.-H., Chen, S. C. 1981. Studies on soluble proteins of vitreous in experimental animals. Exp Eye Res 32(4): 381–8. Chen, P., Chen, H., Zang, X., Chen, M., Jiang, H., Han, S., Wu, X. 2013. Expression of effux transporters in human ocular tissues. Drug Metab Dispos 41(11): 1934–48. Chien, D.-S., Homsy, J. J., Gluchowskil, C., Tang-liu, D. D. 1990. Corneal and conjunctival/scleral penetration of p-amino clonidine, AGN 190342, and clonidine in rabbit eyes. Curr Eye Res 9(11): 1051–9. Chow, B. W., Gu, C. 2017. Gradual suppression of transcytosis governs functional blood-retinal barrier formation. Neuron 93(6): 1325–33. Czerkinsky, C., Holmgren, J. 2012. Mucosal delivery routes for optimal immunization: Targeting immunity to the right tissues. Curr Top Microbiol Immunol 354: 1–18. https://doi.org/10.1007/82_2010 _112. 42

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3 PK-TK Considerations of Nasal, Pulmonary and Oral Routes of Administration 3.1 NASAL ROUTE OF ADMINISTRATION/EXPOSURE 3.1.1 Vestibule, Atrium, Valves, and Turbines The external nose, consisting of a group of muscle lying over the bony and cartilaginous structure of the organ, is the opening of the respiratory tract. The nasal cavity can be divided into three segments: the vestibule, the atrium, and the turbines. The initial part of the external nose is the vestibule, consisting of layers of thin fat cells known as stratifed squamous epithelium. At the upper part of the nasal cavity, the epithelium changes to a pseudostratifed column-like epithelium with long stiff hairs called vibrissae that flter particulate matter (Cauna, 1982). Serous mucus secretion is produced by the glands that are near the respiratory epithelium. The total volume of both nasal cavities is approximately 14–15 ml and the total surface area is about 140–160 cm2. The nasal cavity protects the body’s airway. Approximately 500–600 L/h of air transfers through the nasal cavity into the pulmonary tract. It is estimated that on average 25 million particles are processed by the nasal epithelium (Seaton et al., 1995). The narrow areas in both cavities that often cause diffculty in breathing are known as the nasal valves. There are two valves on each side, an internal and an external. The internal valve is inside the nose between the nasal septum and the upper cartilage, the external valve is the nostril that is supported by the cartilage in the tip of the nose. The nasal valves account for approximately 50–75% of total resistance of the airfow from the nostril to the pulmonary alveoli (Proctor and Adam, 1976; Yu et al., 2008). Thus, the nasal valves slow down the airfow rate and allow the inspired air to be in contact with warm nasal mucosa to reach 34°C by the time it reaches the nasopharynx (Lindemann et al., 2004). The nasal geometry facilitates the heat and humid transfer from nasal mucosa (Doorly et al., 2008). The valves also control the expired air to allow gas exchange (Hairfeld et al., 1987) in the lung. The airfow rate in nasal cavities is about 20–30L of airfow per minute. For larger volumes, oral breathing complements nasal breathing. The nasal cavity has different geometry among patients, and the dynamic of the airfow distribution is different (Yu et al., 2008). Another anatomical feature in the nasal passageway is the presence of turbines that flter, warm, and add moisture to the inspired air. There are three in each nostril know as inferior turbinate, middle turbinate, and superior turbinate (Figure 3.1). The inferior and middle ones are considered functionally the most signifcant. The turbines sit near the septum, the bone and cartilage that separates the nostrils, and, like the septum, add to the surface areas of the nasal cavity. The infammation of the turbines, which often is the result of seasonal allergens and/or environmental irritants or sinusitis, causes their enlargement that may trigger chronic swelling associated with nasal obstruction or stuffy nose (Swift and Proctor, 1977). 3.1.2 Mucosal Epithelium The frontal nasal cavity is covered by the transitional epithelium, the upper part by an olfactory epithelium, and the remaining portion by a mucosal epithelial layer that is anchored to the basement membrane by basal cells (Evans and Plopper, 1988). The mucosal epithelium is comprised of ciliated and non-ciliated columnar cells with microvilli, goblet cells, and sometimes eosinophils and lymphocytes (Jafek, 1983) (Figure 3.2). All ciliated and non-ciliated columnar cells are covered by microvilli, which are cytoplasmic extensions of the cells and expand the surface area of the mucosal epithelial layer. In the nasal cavity, the linear rate of inspiration is inversely proportional to the density of ciliated cells (Cole, 1982): the higher the compactness of ciliated cells, the lower the inspiration airfow rate. The mucosal epithelial cells form a tight junction with discontinuity only around the goblet cells (Carson et al., 1987). This discontinuity is the site of absorption of xenobiotics in aerosolized delivery systems. Goblet cells contribute to the volume of nasal secretion and respond to all types of irritation. There are about 4000–7000 goblet cells/mm2 that are also involved in the regulation of intrinsic immunity by modulating immunological responses to infection and allergens (Whitsett, 2018). It is worth noting that the genetic factors regulating goblet cell identity and its mucus production are different from ciliated and basal cells (Chen et al., 2014; Rajavelu et al., 2015). There are mechanisms for removal of particles and secretions from the nasal cavity. Leaving out the medical devices and pumps to clean the cavity, the physiological mechanisms include 50

DOI: 10.1201/9781003260660-3

PK-TK Considerations of Nasal, Pulmonary and Oral Routes

Figure 3.1  Schematic depiction of the nasal cavity identifying the locations of atrium, vestibule, and turbines (inferior, middle, and superior turbinate); the Eustachian canal’s opening (see Chapter 2) and nasopharynx pathway are near each other; the location of the olfactory epithelium at the roof of nasal cavity close to the superior turbinate and the brain is also shown. mucociliary transport, sneezing, and nose-blowing. Mucociliary transport is the main nasal clearance mechanism with two components of mucosal layer and ciliated epithelial cells. The layer, about 10 µm (range = 5–15 µm) thickness, covers the entire nasal cavity (Wilson and Allansmith, 1976) and is composed mainly of water (~95%), glycoproteins, immunoglobulins, and albumin together forming a double layer of water and gel, where the watery layer is in contact with ciliated cells and the gel layer is in contact with the air (Ali and Pearson, 2007) (Figure 3.2). The nasal mucous production rate is about 200 g/day or 2 L/day (Beule, 2010), and the mucociliary motion transports mucus with a flow rate of about 5 mm/min (also reported as 2–25 mm/min), and the mucus layer is renewed every 20 minutes (Ali and Pearson, 2007). The deposition of the particles in the gel layer occurs during inspiration and expiration (Wiesmiller et al., 2003). The nasal valves remove particles larger than 3 microns, and the mucociliary transport clears particles of 0.5–3 microns by mucosal flow into the nasopharynx. The deposition of small particles and gases happens to a much lesser extent, and they travel into the lower respiratory tract. The mucociliary transport is a vital defense mechanism for sustaining a healthy pulmonary system (Knowles and Boucher, 2002), and its impairment is a characteristic of diseases like chronic obstructive pulmonary disease (COPD), asthma, and cystic fibrosis wherein a viscous and thick mucus is problematic to transport, causing recurring infections and respiratory failure (Donaldson et al., 2007). Thus, one can conclude that the normal rate of the transport depends on the hydration and optimum viscosity of the mucus. Recent investigations also underline the significance of the interface between cilia and the watery layer of the mucosal layer (Boucher, 2019; Shaykhiev, 2019; 51

3.1 NASAL ROUTE OF ADMINISTRATION/EXPOSURE

Figure 3.2 Schematic illustration of mucosal epithelium highlighting the three layers of air and gel phase–water phase of the mucus in contact with ciliated, non-ciliated and goblet cells; the basal cells, basement membrane, and submucosal layer that form the foundation of the epithelium is also included; a single cell of the olfactory region represents the type of cells in the region that often extend beyond the mucosal layer. Voynow and Rubin, 2009). It sould also be noted that both temperature and humidity of inspired air are essential parts of the mucociliary transport (Kelly et al., 2021; Plotnikow et al., 2018; Kilgour et al., 2004). The transport of mucus also depends on the ciliary beat direction along the airway. All cilia have similar orientation (Sleigh, 1977), and their beat has two movements of cilia straightening (contacting the gel phase of the mucosal layer and pushing the mucus forward) followed by bending in the watery phase of the mucosal layer. This movement is the result of a circular order of the ciliary beating direction beneath the mucus (Khellouf et al., 2018). 3.1.3 Olfactory Epithelium At the apex of the nasal cavity, a short way down the septum and superior turbinate, is the olfactory epithelium that encompasses about 2–4 cm2, i.e., about 3–5% of the total surface area (Morrison and Constanzo, 1990), and contains approximately 10 × 106 receptor cells. Due to its small surface area, the olfactory epithelium does not contribute signifcantly to the absorption of xenobiotics, but it provides direct nose-to-brain access to the central nervous system. The mechanism of transport to the CNS is yet to be fully understood, but it may involve diffusion through the subarachnoid area. It is worth noting that the region’s neuroepithelium is the only exposed part of the CNS. This link between the nasal cavity olfactory neurons, and the CNS has been considered for direct transport of therapeutic agents to the CSF and brain, bypassing the blood brain barrier (Mignani et al., 2021). Olfactory dysfunction may cause anosmia (loss of sense of smell) which, along with ageusia (loss of taste), is confrmed as an early symptom in COVID-19 patients (Vaira et al., 2020). The olfactory epithelium holds bipolar neurons and is covered with a layer of mucus containing, secretory immunoglobulins (Mellert et al., 1992). The immune system protects the cribriform plate (the perforated bone through which the olfactory nerve fber enters into the nasal cavity) from pathogens. The nasal arterial blood supply is via the facial arteries (external carotid system) and via the ophthalmic artery (internal carotid system). The venous return is via facial and ophthalmic veins

52

PK-TK CONSIDERATIONS OF NASAL, PULMONARY AND ORAL ROUTES

into the jugular vein and directly into the right heart chamber, thus avoiding liver and other highly perfused organs. 3.1.4 Nasal ADME of Xenobiotics Nasal drug delivery has been used for decades and currently is considered a useful route for administration of selected therapeutic agents including peptides, vaccines, and nose-to-brain delivery systems (Cunha et al., 2021). The following are considered positive attributes of nasal route of adminbistration: i. patient safety and compliance, particularly in infants, children, and adolescence populations ii. convenience of administration iii. easy accessibility for direct permeation of small and large molecules iv. rapid onset of action v. no gastrointestinal enzymatic, chemical, or frst-pass degradation vi. bypassing the blood-brain barrier for direct nose-to-brain delivery vii. localized treatment of nasal conditions like rhinosinusitis (Okubo et al., 2020) viii. noninvasive route for achieving systemic effect (Vlerick et al., 2020). The absorption of xenobiotics in the nasal cavity takes place when the compound or particle is deposited on the surface of a mucosal layer. When the objective is to induce systemic effect, the deposited compound must pass through i) mucus layer, ii) epithelial layer, iii) basement membrane, and iv) capillary endothelium before reaching the systemic circulation. The pathway to achieve nose-to-brain transport, or gaining access to the CNS, has not been confrmed, but the transfer through olfactory epithelium (pseudostratifed columnar epithelium) is the required frst step followed by possible travel through the vessels into the CNS. The permeation of small molecules through the epithelium occurs by paracellular or transcellular passive diffusion. For compounds with molecular weights greater than 1 kDa, the endocytic transcellular absorption is proposed (Costantino et al., 2007). Peptides and proteins fall into the category of compounds with molecular weight greater than 1 kDa. In general, the nasal absorption of xenobiotics depends on the physicochemical characteristics of a compound and the limitations of the nasal cavity. The absorption of lipophilic compounds, in paticular those with molecular weight less than 1 kDa, is rapid with a bioavailability comparable to intravenous administration. The absorption of hydrophilic compounds is low, especially for those with high molecular weights. The pH of the nasal epithelium mucus is about 5.5–6.5 with little buffer capacity, and according to pH-partition theory, the absorption of compounds with a pKa that forms ionized molecules at this pH range would not be signifcant (see Chapter 2, Section 2.3.2, the Henderson–Hasselbalch equation for estimation of ionized vs unionized forms). In general, compounds with high pemeability and high solubility (Amidon et al., 1995) are the best candidates for nasal delivery with the expectation of systemic effect. Absorption via nasal cavity has limitations, such as i) restricted surface area and volume: e.g., administration of a single dose of 200 µL or higher of nasal solution will be subjected considerably to post-nasal removal, but administration of the same volume as multiple doses of 50 µL or less four times at intervals leads to more effcient deposition and absorption (Kundoor and Dalby, 2011); ii) the tight junction of the epithelium, olfactory, and respiratory protective mucus that reduces the permeability and absorption of xenobiotics; iii) the presence of effux transporter protein, Pgp, located in the apical region of ciliated epithelial cells and submucosal vessels of the olfactory region is confrmed, preventing the absorption of xenobiotics (Graff and Pollack, 2003, 2005). However, multidrug resistance proteins in the olfactory epithelium have not been confrmed in mammals (Kudo et al., 2010). It should be noted that in the nose there exist multimicrobial colonies, combinations of microfora and pathogens, that may contribute to the microbial degradation of administerd xenobiotics. Caution should be exrcised in the extrapolation of experimental absorption and direct-noseto-brain data from laboratory animals to human, as the differences in surface area and volume among the species is signifcant. The difference between the ratio of nasal cavity suface area (NCArea) to nasal cavity volume (NCVolume) is a meaningful indicator to highlight the magnitude of the differences between the species (Ruigrok and de Lang, 2015; Lochlead and Thorn, 2012; Gizurarson, 2012; Mygind and Dahl, 1998): 53

3.1 NASAL ROUTE OF ADMINISTRATION/EXPOSURE

Human =

160 cm 2 NCArea = = 6.4 cm -1 NCVolume 25cm 3

Rat =

(3.1)

13.4 cm 2 = 51.53 cm -1 0.26 cm 3

(3.2)

61cm 2 = 10.16 cm -1 6 cm 3

(3.3)

2.89cm 2 = 96.33 cm -1 0.03 3

(3.4)

Rabbit = Mice =

During absorption via the nasal epithelial barrier, xenobiotics may undergo signifcant metabolism with a broad range of enzymes including Phase II metabolic enzymes – such as gluthatione S-transferases, UDP-glucuronyltransferase – and Phase I metabolic enzymes, like epoxide hydrolases, carboxy esterases, aldehyde dehydrogenases, and cytochrome P-450 (CYP) enzymes (Dahl and Hadley, 1991; Dahl and Lewis, 1993). The expression of CYP1A1 and CYP1B1 are also reported to increase in the nasal epithelium of smokers (Zhang et al., 2010; Iskandar et al., 2013). In addition, proteolytic enzymes (like proteases) and aminopeptidases (such as aminopeptidiase A, aminopeptidase N, and dipeptidyldipeptidase) are present in the nasal cavity (Agu et al., 2009) and are considered major barriers to the absorption of peptides and proteins. 3.1.5 Nasal Rate Equations – PK/TK Models The nasal ADME of xenobiotics are infuenced by the anatomical features and physiological processes that are unique to this route. They include: i. Presence of surface mucus and the requirement for the molecules, particles, or droplets of dosage form or environmental pollutants to get deposited on the surface before the absorption process is initiated. ii. Unidirectional mucociliary movement of the deposited substance in the mucus toward the nasopharynx. iii. Elimination of the deposited compound from nasal cavity through nasopharynx into the GI tract for absorption, which then can be subjected to GI tract and liver metabolism. iv. Infuence of effux trasporters. v. Metabolism by nasal metabolic Phase I and Phase II enzymes. vi. Likely infuence of nasal multimicrobial degredation. vii. Transfer of small prticles or free molecules into the pulmonary tract for absorption. viii. Transfer to brain via nose-to-brain route of the olfactory epithelium. 3.1.5.1 A Nose-to-Systemic Circulation PK/TK Model If the nasal route is used to achieve systemic effect for a xenobiotic, under the following assumptions one could consider a model as depicted in Figure 3.3. The assumptions of the model are: i. There is no signifcant transfer to the pulmonary tract, and most of the molecules or particles of administered dose are deposited on the surface of nasal cavity. ii. There is no signifcant metabolism in the nasal cavity, and the compounds are either absorbed through mucosal epithelium into the capillaries or transferred by mucociliary movement into nasopharynx. iii. The mucociliary partial transfer of the administered dose into the nasopharynx is an assumed frst-order process. Although considering its physiological constant fow rate, it can also be regarded as a zero-order transfer rate process. iv. Assuming the disposition of the xenobiotic after reaching the systemic circulation follows a linear two-compartment model.

54

PK-TK CONSIDERATIONS OF NASAL, PULMONARY AND ORAL ROUTES

Figure 3.3 Schematic diagram of a PK/TK linear model taking into consideration the xenobiotic absorption through the mucosal epithelium and absorption from the GI tract, the fraction that is removed from the nasal cavity by the function of ciliary cells into the nasopharynx into the systemic circulation; all rate constants are considered frst-order and the administration of the dose is assumed instantaneous; ANC and A GI represent the amount of xenobiotic in the nasal cavity and GI tract at time t, respectively; A1 and A2 are the amounts in the central compartment (systemic circulation) and the peripheral compartment at time t, respectively; The rate constants include k MC, the constant of ciliated transfer into the nasopharynx; ka1 and ka2 are the absorption rate constants from epithelium and GI tract, respectively; k12 and k21 are the distribution rate constants between the central and peripheral compartments; and k10 is the overall elimination rate constant including renal excretion and systemic metabolism. The rate equations of the model are: dANC ˜ ° ˝ k MC ˛ k a1 ˙ ANC dt

(3.5)

dAGI ˜ k MC ANC ° k a2 AGI dt

(3.6)

dA1 ˜ k a1 ANC ° k a2 AGI ° k 21 A2 ˛ k12 A2 ˛ k10 A1 dt

(3.7)

dA2 ˜ k12 A1 ° k 21 A2 dt

(3.8)

The defnition of variables and rate constants of the equations are: dANC is the rate of change in the amount of deposited molecules or particles in the mucus of dt nasal cavity (NC); k MC and k a1 are the frst-order mucociliary transfer rate constant into the GI tract and the absorption rate constant through epithelium into the systemic circulation, respecdAGI tively; is the rate of change in the amount of the compound in the GI tract transferred via dt the nasopharynx; k a2 is the frst-order absorption rate constant from the GI tract into the systemic dA1 circulation; is the rate of change in the amount of the compound in the central compartment dt 55

3.1 NASAL ROUTE OF ADMINISTRATION/EXPOSURE

(systemic circulation) absorbed from NC and GI tract; k 21 and k12 are the frst-order distribution rate constant from central compartment into the peripheral compartment and from peripheral compartment into the central compartment, respectively; k10 is the overall elimination rate constant dA2 from the central compartment; is the rate of change of amount in the peripheral compartment. The integrated equations are: dt

˜ ANC °t ˛ ˜ ANC °t˛0 e ˝˜ k ˜ AGI °t ˛

MC ˙ k a1

°t

(3.9)

FD ˝ k MC ˙ka2 ˝t ˙ e ˙k MC ˝t e k MC ˙ k a2

˜

˜ A1 °t ˛ Ae˝ˇt ˝ Be˝˘t ˝ Ce˝(k

a1 ˙ka2

°

)ˆt

(3.10) (3.11)

Where ˜ ANC ° and ˜ AGI ° are the amount in the nasal cavity and GI tract at time t, respectively; t t ˜ A1 °t is the amount in the central compartment at time t; A, B, and C are the coeffcients that cor-

respond to the y-intercept of of the extrapolated line and residual lines, or exponential terms of a two-compartment model with frst-order input (see also Chapter 15, Section 15.2.3); α and β are the hybrid rate constants of the two-compartment model (see Chapter 15, Section 15.2.2.2). 3.1.5.2 An Inclusive Nose-to-Brain PK/TK Model The PK or TK analysis of direct nose-to-brain data and the true predictive values of calculated constants and variables depend largely on the actual measurement and time-course of a xenobiotic in various regions of nasal cavity. Often using limited data may generate calculated values that are of a hybrid nature. Inferring conclusions based on these values may present a false depiction of the actual facts. It is always a good practice to defne the assumptions of the PK or TK model clearly to offer a framework for the meaning of the calculated constants and parameters. For example, an important feature of any model for the nose-to-brain administration is the mass balance of the xenobiotic and eventual fate of the compound after reaching the brain and after delivering the pysiological response. If one uses the glymphatic principle (Jessen et al., 2015; Murtha et al., 2014; Iliff and Nedergaard, 2013; Groothuis et al., 2007; Koh et al., 2005) to defne the clearance of xenobiotics from the cerebrospinal fuid (CSF) (i.e., the compounds are cleared from the brain interstitium to the CSF where they eliminate into the systemic circulation via arachnoid granulations or peripheral lymphatics along cranial nerves) it would follow that the elimination from the brain compartment after nose-to-brain transfer would eventually enter into the systemic circulation. Thus, a model as depicted in Figure 3.4 and based on the glymphatic principle can be envisioned with following assumptions: i. There is no signifcant transfer to the pulmonary tract, and the administered dose is deposited on the surface of the nasal cavity. ii. The amount of xenobiotic (i.e., dose) in the nasal cavity will be transferred by three processes of 1) mucociliary clearance into nasopharyx, 2) absorption through the mucus into the systemic circulation, and 3) direct nose-to-brain transfer of the compound. iii. There is no signifcant metabolism in the nasal cavity and the compound is absorbed through epithelium into the capillaries, transferred by mucociliary movement into the nasopharynx, and transported to the brain. iv. The mucociliary clearance can be assumed frst- or zero-order transfer into the GI tract with subsequent absorption into the systemic circulation. v. The absorption through the epithelium is frst-order and the fraction of dose absorbed will ultimately reach the systemic circulation. vi. The amount transferred to the brain through the direct nose-to-brain passageway eventually, according to the glymphatic mechanism, reaches the systemic circulation with a subsequent encounter with the blood-brain barrier. vii. The disposition of the compound in the body follows the linear two-compartment model. The rate equations of the model are:

56

PK-TK CONSIDERATIONS OF NASAL, PULMONARY AND ORAL ROUTES

Figure 3.4 Schematic diagram of an inclusive PK/TK linear model representing the noseto-brain transfer, trans-epithelial absorption, and GI tract absorption of a fraction transferred through the nasopharynx into the GI tract; ANC represents the amount in the nasal cavity compartment with three exit rate constants of kNTB (frst-order rate constant of nose-to-brain transfer), ka1 (rate constant of trans-epithelial absorption), and k MC (rate constant of ciliated transfer into nasopharynx); AB is the amount transferred into the brain as a function of time; kB1 and k1B are the rate constants of exchange between the systemic circulation and the brain, kB1 is included in the model based on the glymphatic principle of eventual transfer of xenobiotic from the brain into the systemic circulation, and k1B is the rate constant facing the blood-brain-barrier; defnitions of other rate constants and compartments are the same as described in the caption of Figure 3.3. dANC = - ( k NTB + k a1 + k a2 ) ANC dt

(3.12)

dAbrain = k NTB ANC + k1B A1 - kB1 Abrain dt

(3.13)

dA1 = k a2 AGI + k a1 ANC + kB1 Abrain + k 21 A2 - k1B A1 - k12 A1 - k10 A1 dt

(3.14)

dAGI = k MC ANC - k a2 AGI dt

(3.15)

Where k NTB is the direct transfer rate constant of the compound from nose to the brain. dAbrain is the rate of change in amount of the compound in the brain. dt k1B is the rate constant of the transfer from systemic circulation to the brain that may encouter the blood-brain barrier. kB1 is the transfer rate constant of the compound from the brain into the systemic circulation based on the premise of glymphatic. Abrain is the amount of the compound in the brain as a function of time. kB1 Abrain is the rate of transfer from the brain into the systemic circulation according to the principle of glymphatic.

57

3.2 PULMONARY ROUTE OF ADMINISTRATION/EXPOSURE

k1B A1 is the rate of transfer from the systemic circulation into the brain after facing the blood-brain barrier. This rate is often a small value for most xenobiotics. k MC ANC is the rate of transfer from the nasal cavity through the nasopharynx into the GI tract. k NTB ANC is the transfer rate from the nasal cavity to the brain. k a1 ANC is the absorption rate of the xenobiotic through the mucus and epithelium into the systemic circulation. k10 A1 is the overall elimination rate that includes both the excretion rate and metabolism in the body. From the rate constants of the model, the following relationships can be developed to evaluate the amount of the xenobiotic that ultimately reaches each destination in the body: Setting K NC = k NTB + K a1 + k MC

(3.16)

Equation 3.16 represents the sum of all exit rate constants from the nasal cavity, which enables one to estimate the following fractions. f NTB =

k NTB K NC

(3.17)

Fraction of administered dose transferred to the brain by the nose-to-brain passageway. fA =

k a1 K NC

(3.18)

Fraction of administered dose absorbed through the mucus and epithelium into the systemic circulation. f MC =

k MC K NC

(3.19)

Fraction of administered dose transferred into the GI tract via mucociliary and nasopharynx route. Therefore, the total amount transferred from nasal cavity to any of the destination compartments can be estimated as: Total amount transferred to the brain:

( ANTB )¥ = f NTB ´ Dose

(3.20)

Total amount absorbed in systemic circulation: ( AA1 )¥ = f A ´ Dose

(3.21)

Total amount transfered to GI tract: ( AGI )¥ = f MC ´ Dose

(3.22)

For the integrated equation of the central compartment, use the methodology described in Addendum I, Part 2. 3.2 PULMONARY ROUTE OF ADMINISTRATION/EXPOSURE 3.2.1 Overview Pulmonary diseases like tuberculosis, lung cancer, chronic obstructive pulmonary disease (COPD), emphysema, cystic fbrosis, pneumonia, chronic bronchitis, asthma, and COVID-19 have the highest global mortality rates. As a result, treatments for these respiratory diseases are of the highest urgency, and the intrapulmonary route of administration is important in treating local and systemic targets. The permeability of cellular barriers with thin epithelial membranes, large surface areas, limited enzymatic activity, and highly vascularized sites for absorption have made the advancement of various pulmonary delivery systems possible. Examples include aerosol-based pulmonary delivery systems (Ari et al., 2021; Chow et al., 2021), nanoparticle delivery systems (Deng and Bae, 2020; Ndebele et al., 2021), and many more. Potential risks, safety, biocompatibility, and toxicity of these new dosage forms are yet to be addressed. Furthermore, the signifcance of the pulmonary route as a target in environmental toxicology and exposure to various airborne particles and heavy metal contamination, toxic gas, and 58

PK-TK CONSIDERATIONS OF NASAL, PULMONARY AND ORAL ROUTES

industrial solvents and carcinogens have been well researched and documented over the past decades and are ongoing (Nagar et al., 2014; Väisänen et al., 2019; Tao et al., 2021). The pulmonary route is a complex system with the nose and mouth comprising the upper part and the trachea and lungs comprising the lower part of the respiratory tract. The nasal route has been discussed in Section 3.1. This section (Section 3.2) focuses mainly on the lungs, right and left. The left is further subdivided into two lobes, and the right is subdivided into three lobes. Approximately 10,000 L of ambient air passes through the nasal airway daily, and about one liter of moisture is added to the air during this time. The alveolar surface is covered with a layer of fuid of about 0.2 µm in thickness, which contains a signifcant concentration of surfactants such as 1,2-dipalmitoyl phosphatidylcholine. The presence of surfactants, airfow, and moisture infuence the absorption of solution, inhaled gases, aerosols, and vapor of volatile solvents as well as their rate and extent of absorption. The interaction of surfactants with xenobiotics after deposition in the mucosa of tracheobronchial airways or alveolar region facilitates the solubility of lipophilic and cationic compounds (Wiedmann et al., 2000; Liao and Wiedmann, 2003). However, the interaction of surfactants with peptides, proteins, and other macromolecules often has the opposite effect, causing aggregation of the molecules and prevention of their absorption (Patton, 2007). 3.2.2 Morphological Differences of Airways Among Species The structure of the pulmonary tract, i.e., the lower respiratory tract, in humans is different from other species. The airways in human and a few primates have irregular dichotomous and trichotomous branching patterns, whereas in dogs and laboratory rodents the branching is mostly monopodial. The structural differences among the species infuence the deposition of particles, airfow distribution, and gas uptake. These differences must be taken into consideration when the data from laboratory animals is extrapolated to humans, e.g., in dosage form development that relies on the particles’ deposition or toxicity studies of organic solvents. Many scientifc publications have provided useful data on the structural differences of pulmonary tracts in experimental animals over the past decades. A few examples are guinea pig (Kliment et al., 1972; Schreider and Hutchens, 1980), mice (Mendez et al., 2010), rat (Yeh et al., 1979; Backer et al., 2009), human (Horsefeld and Cumming, 1968; Strum, 2010; Ahookhosh et al., 2020), and rabbit (Raabe et al., 1988). The breathing habit of laboratory animals is also different from humans, and the differences contribute to the variation in the deposition of particles. For example, rabbits and rodents inhale consistently from the nose – they are nose-breathing animals. Since humans are both nose- and mouth-breathing beings, particles larger than 5 µm are more likely to reach deep into the low respiratory tract via mouth-breathing, whereas they would be stopped and cleared by nose-breathing. 3.2.3 Pulmonary Microbiome Contrary to the tacit assumption that the human respiratory airways are germ-free and sterile, recent investigations have shown that the pulmonary tract conceals diverse communities of microbes. There are two terminologies used in microbiology that elucidate the microbial domain defnitions: 1) “microbiota” is used to defne the collection of microbes from a region, e.g., lung microbiota; 2) “microbiome” refers to the collection of all biotic residents including viruses, phages, fungi, and commensal and symbiotic microorganisms. The lung microbiota has generated signifcant and greater understanding of pathogenesis of infectious and noninfectious pulmonary diseases since it was recognized in 2010 (Hilty et al., 2010). In healthy subjects, the lung microbiome is maintained by the balance of microbial arrival and microbial elimination, with modest contributions from microbial reproduction (Venkataraman et al., 2015; Dickson et al., 2015). Analogous to the elimination of particles from lung, the elimination of microbiota is also mainly through mucociliary movement; coughing or sneezing also contribute to the removal process. The principal bacterial phyla in the healthy lungs, like the gut microbiota, include mainly Firmicutes, Bacteroidetes, Proteobacteria and Actinobacteria, and the most abundant types in humans are Prevotella, Streptococcus, Veilonella, Neisseria, Haemophilus, and Fusobacterium (Dickson et al., 2016; Sommariva et al., 2020, Georgiou et al., 2021). The new knowledge of identifying and understanding the lung’s microbiota and microbiome and establishing their role in pulmonary diseases are at the early stages of discovery and research. Most of the current knowledge has been made possible because of the advent of new technologies based on the high throughput sequencing of the 16S rRNA gene. Therefore, many questions related to the role of lung microbiota and/or microbiome in pulmonary diseases, chronic lung infammation, pulmonary immune 59

3.2 PULMONARY ROUTE OF ADMINISTRATION/EXPOSURE

response, infuence on metabolism of xenobiotics, their role in carcinogenesis, and many other important questions remain to be fully investigated. 3.2.4 ADME of Xenobiotics in the Pulmonary Tract 3.2.4.1 Pulmonary Absorption, Deposition, and Clearance The pulmonary absorption of xenobiotics is by active absorption, passive transcellular and/or paracellular diffusion, pore formation, and vesicular transport. The pulmonary route of administration provides a rapid onset of action locally and systemically for xenobiotics when their absorption from this route is thermodynamically favorable. Furthermore, the lower respiratory tract, like the nasal route, is considered a viable alternative to oral administration of compounds with poor bioavailability due to their low stability at acidic pH of stomach, and/or their biotransformation by the hepatic/intestinal frst-pass metabolism, and/or reduced absorption due to GI tract effux proteins. The shortcoming, however, is the optimization of the pulmonary administered dose, residence time, and absorption variability due to the random removal of the particles by exhalation, variability of airfow for absorption of gases and particles, and interindividual anatomical and physiological differences. The dissolution of the particles in the mucous layer of the pulmonary tract is also an important factor that contributes to the variability of the effective dose. Insoluble particles or particles with a slow dissolution rate will be removed by mucociliary clearance in the bronchial airway and microphage phagocytosis within a few days of initial deposition. The particle removal process in mice is assumed frst-order and is defned by a one-compartment model (Asgharian et al., 2014) as follows: d ( RM pulmonary ) Mice dt

= ( Rate )deposition - K pcl ( RM pulmonary ) Micee

( Rate )deposition = Cexposure ´Vtidal ´ ( fr )breath ´ f deposit K pcl = Where

Ln ( RM pulmonary ) Mice-1 - Ln ( RM pulmonary ) Mice-2 t2 - t1

(3.23) (3.24)

(3.25)

d ( RM pulmonary ) Mice

is the rate of change in the retained particles in pulmonary airway with dt respect to time; ( Rate )deposition is the deposition rate of particles according to Equation 3.24; K pcl is the particle clearance rate constant with unit of day-1; Cexposure is the exposure concentration of particles; Vtidal is the tidal volume that corresponds to the size change during breathing; ( fr )breath is the breathing frequency, f deposit is the deposition fraction in the pulmonary region; the slope relationship of Equation 3.25 is based on measuring the mass of deposited particle after a single dose administration in two time points in days apart. 3.2.4.2 Transport Proteins of Pulmonary Tract Transport proteins are present in many tissues and organs in the body and are involved in ADME of xenobiotics. In addition to their biological roles and functions, the transporters infuence the permeation, absorption, distribution, and elimination of xenobiotics that often cannot freely diffuse through cellular membrane. Throughout the pulmonary tract, membrane proteins particularly transporters have also a high relevance to the ADME of xenobiotics. The main pulmonary transporters are outlined in the following sections. 3.2.4.2.1 Organic Cation Transporters The pulmonary organic cation transporters are of the SLC22A1-A5 family that infuence the distribution and elimination of the xenobiotic in the respiratory tract (Salomon and Ehrhardt, 2012; Nickel et al., 2016). This family of transporters has two subgroups of OCTs or Na+ organic transporters and OCTNs or pH-dependent organic transporters (Horvath et al., 2007). The OCTs members include OCT1 or SLC22A1, OCT2 or SLC22A2, and OCT3 or SLC22A3. OCTs are mainly in the airway epithelium and involved in the transport of molecules like acetylcholine, dopamine, histamine, and serotonin. OCT1 and OCT2 are identifed in the apical membrane of

60

PK-TK CONSIDERATIONS OF NASAL, PULMONARY AND ORAL ROUTES

ciliated cells and OCT3 in the basolateral membrane (Lips et al., 2005; Bleasby et al., 2006; Barilli et al., 2020, 2021). The OCTNs members are OCTN1 or SLC22A4 and OCTN2 or SLC22A5. OCTNs transporters that have a broad spectrum in transporting organic cations, which includes both endogenous compounds and exogenous xenobiotics and zwitterions (Horvath et al., 2007). The roles OCTNs play in dealing with xenobiotics in the respiratory tract are not fully elucidated and require further research and understanding. 3.2.4.2.2 Organic Anion Transporters The pulmonary organic anion transporters are members of the organic anion transporting polypeptides (OATPs) superfamily (Hagenbuch and Meier, 2003) that are involved in the transport of a broad array of structurally unrelated compounds. Their substrates are large molecules of anions greater than 300 kDa; proteins and polypeptides fall into that category. The OATPs superfamily has subfamilies of OATP1, OATP2, OATP3 OATP4, OATP5, and OATP6. The members of the OATP1 family include OATP1A2, OATP1B1, OATP1B3, and OATP1C1. The members of the OATP2 family include OATP2A1 and OATP2B1. The members of the subfamilies involved in pulmonary tract include OATP1A2 (Schuster, 2002), OATP2B1, OATP3A1, and OATP4A1 (Obaidat et al., 2012). These transporters facilitate permeation of a wide range of endogenous and exogenous amphipathic compounds in the lungs. The examples of exogenous compounds include antihistamines, statins, anti-infammatory drugs, immunosuppressive, and anticancer agents (Bleasby et al., 2006; Hagenbuch and Gui, 2008; König, 2011). 3.2.4.2.3 ATP Binding Cassette Transporters Adenosine triphosphate (ATP) Binding Cassette (ABC) transporters are a superfamily of transmembrane proteins involved in transport of a large array of endogenous and exogenous compounds in an energy-dependent manner (Dean et al., 2001). The important members of this superfamily that are expressed at different levels in the lung tissue (Van der Deen et al., 2005; Bosquillon, 2010; Okamura et al., 2013; Nickel et al., 2016) are: ◾ p-glycoprotein (Pgp), or ABC transporter B1 (ABCB1/MDR1) ◾ multidrug resistance-associated protein 1 (ABCC1/MRP1) ◾ breast cancer resistance protein (ABCG2/BCRP). The above transporters are best known for effux of therapeutic agents and multidrug resistance that leads to the decrease in intracellular concentration of a therapeutic agent that is considered substrate for the members of this family of transporters. The positive aspect of their role is the protection of respiratory tract against toxic xenobiotics entering the lung. The roles these transporters play in lung cancer (Young et al., 2001), cystic fbrosis, asthma, and chronic obstructive pulmonary disease (Van der Deen et al., 2006) have been widely investigated (Park et al., 2020; Wang et al., 2021, Sharma et al., 2021). Depending upon the overexpression or inhibition of the effux proteins, their infuence on the disposition of xenobiotics in respiratory tract is signifcant and can alter the therapeutic or toxic outcome of xenobiotics. 3.2.4.2.4 Peptide Transporters Peptide transporters mediate the cellular permeation of dipeptides and tripeptides and many peptidomimetics. The ones that are present in the lung are PEPT1 and PEPT2 transporters that, in addition to their physiological function of handling endogenous substrates, both transport molecules that sterically resemble dipeptides and tripeptide molecules, e.g., β-lactam antibiotics and some prodrugs. They are present in bronchia, epithelial cells, alveoli, and endothelial cells (Groneberg et al., 2001, 2004). 3.2.4.3 Respiratory Tract Metabolic Enzymes – Lung Metabolism of Xenobiotics The metabolism of xenobiotics in the lung complements other respiratory tract clearances, such as mucociliary removal (Stahlhofen et al., 1990; Chivukula et al., 2020), phagocytosis by alveolar macrophages (Oberdorster, 1998; Roquilly et al., 2020), and other ancillary processes (Patton et al., 2004). It is often assumed that an advantage of using the pulmonary route to achieve systemic effect of a drug is the compound avoids the biotransformation of the GI tract, and thus the bioavailability of the compound can be improved. The assumption is valid if only the dug is not metabolized in the respiratory tract. The lungs carry out both Phase I and Phase II metabolism 61

3.2 PULMONARY ROUTE OF ADMINISTRATION/EXPOSURE

(Taylor, 1990; Spivack et al., 2003; Courcot et al., 2012; Oesch et al., 2019). The enzymes of Phase I metabolism of the lung are the cytochrome P450 isozymes (Shimada et al., 1994; Raunio et al., 1999; Ding and Kaminsky, 2003) including, but not limited to CYP1A1, CYP3A4, CYP3A5, CYP1B1, CYP2B6, CYP2C8, CYP2C18, CYP2D6, CYP2E1, CYP2F1, and CYP2J2 (Anttila et al., 1997; Kelly et al., 1997; Spivack et al., 2001; Hukkanen et al., 2002; Nebert et al., 2004; Zhang et al., 2006; Courcot et al., 2012; Peng et al., 2013; Oesch et al., 2019; Enlo-Scott et al., 2021). The expression levels of these isoforms in the lung are lower than the intestinal or hepatic enzymes and their activity is also generally lower than other major sites of metabolism. Within the lungs, most CYP enzymes are expressed in ciliated columnar epithelial cells, club cells, also known as clara cells, and bronchiolar epithelium (Oesch et al., 2019). The main substrates of the Phase I enzymes are polycyclic aromatic hydrocarbon benzo(a)pyrene from cigarette smoke, β2 agonist salmeterol, amiodarone (administered orally), and inhaled glucocorticoids like dexamethasone and beclomethasone dipropionate (Hukkanen et al., 2003). Other Phase I metabolism enzymes, such as favin-containing monooxygenase (e.g., FMO2); esterase (e.g., esterase 1); alcohol dehydrogenase (e.g., ADH1B, and ADH1C), have also been detected (yet to be confrmed), in human lung microsomes (Enlo-Scott et al., 2021). The Phase II metabolic conjugations also occur in the lungs with the help of enzymes, such as UDP glucuronosyltransferases (UGTs) (Zheng et al., 2002), glutathione S-transferases (GSTs) (Howie et al., 1990; Wang et al., 2003), and sulfotransferases (SULTs) (Somers et al., 2007). 3.2.4.4 Pulmonary Deposition and Disposition of Particles Particles of less than 100 nm may penetrate the epithelial membrane and absorb into systemic circulation by endocytosis (Kemp et al., 2008). The mechanism involves the transit of nanoparticles across epithelia of the respiratory tract and entering the circulation via lymphatic pathways (Medina et al., 2007). If the particles do not get deposited and remain afoat in the aerial milieu, they are remover by exhalation and outfow from the pulmonary tract. The deposition of particles, whether from the environment or from a dosage form like aerosol, depends on many particlerelated parameters including their size, composition, morphology, aqueous solubility, density, penetrability through the mucosal layer, charge and surface characteristics of the particles, and their ability to evade phagocytosis and mucociliary clearance. In addition to the particle-related parameters, there are respiration-related factors that infuence the deposition of particles including respiration volume, lung volume, breathing rate (whether mouth or nose breathing), and the vigor of the lung that depends on the well-being of the individual. The size of the particles is characterized by mass median aerodynamic diameter (MMAD) based on their distribution profle (Bosquillon et al., 2001). Particles with MMAD of approximately 0.5–5 µm are likely to be deposited in the lung, smaller particles in the alveolar space, and larger than 5 µm in the mouth cavity and throat (Sheth et al., 2015). The deposition of the particles occurs by mechanisms like impaction, sedimentation, interception, and/or diffusion (Smola et al., 2008). The epithelium of nasopharyngeal airway and tracheobronchial region (i.e., trachea, bronchi, and bronchioles) are the site of absorption for gases and deposited particles. However, the deposited particles in tracheobronchial region can be subjected to removal by the upward mucociliary movement of the region (Figure 3.5). If the mucous is swallowed, the absorption of particles would be partially through the GI tract. Thus, the clearance of particles from the site of absorption depends on the dissolution and absorption at the site, removal by the mucociliary transport and likely absorption at the GI tract, and pulmonary metabolism of the dissolved particles (Figures 3.5 and 3.6). These processes occur in parallel, and PK/TK analyses of the pulmonary absorption data without the correction for the effect of removal processes, such as mucociliary movement or metabolism or absorption from GI tract, may only generate a set of hybrid parameters and constants that may represent the collective effects of these parallel processes. This is particularly true when one intends to defne the true pulmonary absorption rate or rate constant and/or pulmonary elimination parameters and constants. For compounds with negligible GI absorption, the kinetic evaluation of pulmonary absorption can be accomplished with the standard approaches in PK/TK analysis. However, for drugs with signifcant GI absorption, the kinetics of pulmonary absorption can be defned by blocking gastrointestinal absorption with charcoal or by studying absorption during the frst 30 minutes post inhalation before appreciable oral absorption has occurred (Derendorf et al., 2001). As indicated earlier, the respiration-related factors, like the pattern of airfow, may infuence the pulmonary absorption of particles, which can limit their migration through the pulmonary system. The airfow depends on the physical activity of the individuals, which may also vary from 62

PK-TK CONSIDERATIONS OF NASAL, PULMONARY AND ORAL ROUTES

Figure 3.5 Depiction of the particles’ fate after meeting the epithelium of the tracheobronchial and nasopharyngeal airway; those that are subjected to dissolution followed by absorption, and those that are removed by the mucociliary upward transport, engaging ciliated cells. person to person, or species to species. The default value for a human is 20 m3/day, and when performing the occupational exposure assessment, it is 10 m3/8 h (USEPA, 1992, 2019). The maximum air capacity volume of the lung is about 5700 cm3, and the total air that moves in and out of the lung in active normal breathing is approximately 4500 cm3; therefore, the lung retains 1200 cm3 of air. The number of breaths per minute in humans is approximately 12–20. At rest, the volume of air is reduced to 500 cm3. The air velocity is very high in the nasopharyngeal region and becomes milder as it reaches the alveolar region. Large particles (5–30 µm) are retained in the nasopharyngeal region by inertial impaction. Small particles (1–5 µm) are retained in the trachea, bronchial, and bronchiolar region by sedimentation. Smaller particles (1 µm) penetrate deep into the pulmonary tree, reach the alveolar sacs, and dissolve in the available fuid followed by passive diffusion. There is always a greater tendency for smaller particles to be exhaled. Coughing or sneezing, with an air velocity of 75–100 miles per hour, can remove a signifcant number of particles from the site of absorption, particularly if they occur immediately after the administration of a compound. 3.2.4.5 Pulmonary Absorption of Gases and Vapors The absorption of gases is from the mucous membrane of nose, pharynx, trachea, bronchi, bronchioles, alveolar sacs, and alveoli. Therefore, a large surface area is available for the absorption of gases and vapors. The medical gases that often act as biological messenger molecules (Li et al., 2009) consist of hydrogen sulfde, carbon monoxide, and nitric oxide. The traditional ones include oxygen, hydrogen, carbon dioxide, and nitrous oxide. The signifcance of traditional gases like oxygen is well documented. It is required for most, if not all, metabolic rates. For example, although the brain represents 2% of the body’s weight, it receives 20% of the total body oxygen demands for its high metabolic rates. Hydrogen with its antioxidant capabilities is a free radical scavenger and cellular protective. The noble gases like xenon, argon, and helium are used as neuroprotectives following traumatic brain injury, acute ischemic stroke, and other neurologic 63

3.2 PULMONARY ROUTE OF ADMINISTRATION/EXPOSURE

Figure 3.6 Schematic of respiratory intake, deposition, absorption, distribution, metabolism, and excretion of xenobiotics; the sequential steps from the exposure dose to potential dose to internal dose and, ultimately, to the effective dose; k1 and k2 are the aerial rate constants of infow and outfow, respectively; kE1 is the local effect rate constant of amount absorbed from the upper respiratory tract, the nasal site; k3 and k4 are the rate constants of respiratory intake and respiratory outfow, respectively; k5 is the rate constant of mucociliary removal; k6 is the deposition and dissolution rate constant; k7 is the pulmonary metabolic rate constant; k8 is the discharged rate constant for partial removal of amount transferred by mucociliary process; k9 is the rate constant of ingestion of remaining amount transferred by mucociliary into the GI tract; ka is the absorption rate constant from GI tract into the systemic circulation; kE2 is the local effect rate constant of deposited amount of pulmonary site; kpa is the pulmonary absorption rate constant into the capillaries and systemic circulation; k12 and k21 are the distribution rate constant between the systemic circulation and peripheral compartments; k10 is the overall elimination rate constant (excretion and metabolism) from the body; kED is the rate constant of the effective dose that interact with the receptor site; kE3 is the systemic effect rate constant.

64

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dysfunctions (Coburn et al., 2009; Derwall et al., 2009). The PK/TK analysis of the above-mentioned gases, except xenon and argon, due to their rapid uptake and elimination has not been the focus of most investigations. Argon and xenon, however, are evaluated by a physiological model that divides the body into tissue compartments of the lung, brain, liver, muscle, poorly perfused tissue, and adipose tissue compartment. The model assumes that all exchanges between gas, blood, and tissues occur instantaneously, and the rate of tissue concentration changes with respect to time is defned by the following differential equation (Katz et al., 2015): dCtissue dt

æ Ctissue Qtissue ´ ç Carterial ç P coeff Ctissue - blood è = Vtissue

ö ÷÷ ø

(3.26)

Where Ctissue is the tissue concentration; Qtissue and Vtissue are the blood fow and the volume of the compartment, respectively; Pcoeff Ctissue -blood is the partition coeffcient for the blood and tissue compartment; and Carterial is the arterial concentration entering the compartment. Based on the assumption of the model, the venous concentration is estimated as: Cvenous =

Ctissue Pcoeff Ctissue -blood

(3.27)

The pharmacokinetics of inhalation anesthetics like enfurane, desfurane, isofurane, sevofurane, and halothane have also been evaluated using multicompartmental models without metabolic elimination, with the assumption that the uptake of the compounds are fow limited based on the rate equations of the inhaled anesthetics’ partial pressure (Bailey, 1997). The disposition of the anesthetics to determine how long the compounds remain in the body was also investigated using a physiologically based model with consideration for the metabolism and estimation of related Michaelis–Menten parameters (Lockwood, 2010); the reported rate of change of alveolar partial pressure is:

(

)

˜ ˜ ˜ dPA VAinsp ´ P1 - VAexp ´ PA + 0.9Q ´ l BG ( PA - PV ) = (3.28) dt FRC Where PA is the partial pressure of alveolar gas; V˜ Ainsp is alveolar inspired ventilation; P1 is partial ˜ 90% of cardiac output; l BG is the blood:gas partition coeffcient; P pressure in inspired gas; 0.9Qis V

is the partial pressure in mixed venous blood; and FRC is functional residual capacity assumed as 2 liters. The rate of concentration changes in each compartment based on the assumption of the perfusion-limited kinetics is suggested as (Lockwood, 2010):

dCcomp ˜ = Qcomp ´ l BG ´ ( Pa - Pcom ) (3.29) dt Where Ccomp is the compound concentration in a compartment; Q˜ comp is blood fow through the compartment; Pcom is partial pressure of the compartment; and Pa is the arterial partial pressure. In general, the solubility and reactivity of the gases determine their regional absorption. The upper pulmonary tract is the site for absorption of water-soluble and chemically reactive gases. The distal portion of the tract is the site for absorption of lipid soluble and non-reactive gases. The absorption of lipid soluble and volatile gases is very rapid, and the onset of systemic action is rather instantaneous. Because of their lipophilicity, high levels of these compounds can be accumulated in systemic circulation and tissue compartments, which can be measured by the blood:air or plasma:air partition coeffcient; they are assumed to be 100% bioavailable. The amount of organic soluble gas entrapped in the lung, a highly perfused organ, is mainly due to their extraction in the lung blood. There are signifcant intra-species differences, and caution should be exercised in data extrapolation from experimental animals to humans (Morris, 2012). The use of vapor as a drug delivery vehicle for pulmonary administration of therapeutic agents for local or systemic effects, or merely as vapor therapy are well established in treatments of emphysema, asthma, and COPD (e.g., bronchoscope vapor thermal ablation (BVTA)) (Perotin et al., 2021). However, the misuse of vapor as a vehicle of illicit drugs is also signifcant and fully recognized. The surge in the use of emerging tobacco-based products like heat-not-burn products (IQOS stands for I Quit Ordinary Smoking) and e-cigarettes are a few examples of the unsafe 65

3.2 PULMONARY ROUTE OF ADMINISTRATION/EXPOSURE

use of vapor that causes lung tissue injury. The expression of ‘vaping’ is created to signify the act of inhaling the vapor produced by an e-cigarette, which may include a solvent like propylene glycol, glycerin, possibly nicotine, favoring agents, and other unknown solvents in counterfeited products. The outbreak of severe lung diseases attributed to the e-cigarette is identifed as vaping product-associated lung illness (EVALI) and is also known as ‘vape lung’ (Carlos et al., 2019; Traboulsi et al., 2020). 3.2.4.6 Relevant Pulmonary Kinetic Parameters Because of the internal and external factors infuencing the pulmonary absorption and residence time of administered/exposed xenobiotic it is often diffcult to estimate the effective dose. An exposure equation recommended by (USEPA, 1997) for estimation of the average daily dose is Rateinhalation ´ tduration ( Daverage )daily = Cinhaled ´ BW ´t

(3.30)

ge averag

where ( Daverage )

daily

is the average daily dose (often denoted as ADD) in mg/kg/day, Cinhaled is the

chemical concentration in inhaled air in μg/m3, Rateinhalation is the inhalation rate in m3/day, tduration is the duration of exposure in days, BW is the body weight, and taverage is the average time with units of days. For non-carcinogenic effect taverage = tduration , and for carcinogenic or chronic effect, taverage is equal to 70 years = 25,550 days, that is, lifetime average daily dose (often denoted as LADD). The total potential dose may be estimated as:

( Dtotal )potential = Cinhaled ´ Rateinhalation ´ tduration

(3.31)

The approximated inhalation rate for children less than one year is 4.5 m3/day; for twelve-year-old children it is 8.7 m3/day; for adult females it is 11.3 m3/day; and for adult males it is 15.2 m3/day (USEPA, 1997). The calculations of the inhalation rate as a function of basal metabolic rate are proposed according to the following relationship (Layton, 1993): Rateinhalation = RateBM ´ A ´ H ´ VQ

(3.32)

where RateBM is the estimated basal metabolic rate in units of kcal/d (kilocalories/day) or MJ/d (mega joules/day) determined by the empirical equations developed through regression analysis (Layton, 1993). For example: Human(Male)18 £ Age £ 30 RateBM = ( 0.063 ´ bwt ) + 2.896

(3.33)

Human(Female)18 £ Age £ 30 RateBM = ( 0.062 ´ bwt ) + 2.036

(3.34)

Where bwt is the body weight in kg. The equations of other ages have also been reported (Layton, 1993). The parameter H has units of liter oxygen intake per units of energy (L/kJ or L/kcal). It represents the volume of oxygen at standard temperature and pressure dry air (STPD) consumed to produce one kJ of energy. It is also defned as the reciprocal of the energy yield of oxygen consumption equal to 0.0476, 0.0508, and 0.0529 L/kJ for carbohydrate, fat, and protein, respectively (McLean and Tobin, 1987; Layton, 1993). The parameter VQ is the ratio of volume minute (L/min) to the oxygen uptake rate (L/min), a unitless number that varies from individual to individual and represents the oxygen uptake capacity as a function of the lung physiology and metabolic processes. It is identifed as a ventilatory equivalent and varies from 25 to 30 with a geometric mean of ~27. Because RateBM is estimated from empirical equations based on the characteristics of population used to generate the equations, a multiplier or correction factor is needed to normalize the calculations according to the daily food-energy intake of a different population. This multiplier (denoted A) is called the basal metabolic rate multiplier and is estimated by dividing the average daily food-energy intake (denoted EFD) by the basal metabolic rate. EFD has the same units as the basal metabolic rate and thus, A is a fraction with no units, that is, A=

66

EFD RateBM

(3.35)

PK-TK CONSIDERATIONS OF NASAL, PULMONARY AND ORAL ROUTES

Equation 3.35 can also be corrected according to the intensity of physical activity during the active hours, that is, Aact = ëé( 24 - S ) F + Sùû 24

(3.36)

where Aact is the corrected value of A for activity and S is the number of hours the subject sleeps. F is the ratio of the rate of energy expenditure during active hours to the estimated basal metabolic rate and is calculated according to the following equation: F = éë( 24A - S ) ùû ( 24 - S )

(3.37)

3.2.4.7 Role of the Lungs in PK/TK of Xenobiotics: Pulmonary First-Pass Metabolism Being highly perfused organs, the lungs play an important role in the PK/TK profle of xenobiotics given by other routes of administration. Following the intravenous or oral administration of a compound, the concentration in the lung achieves equilibrium immediately with the systemic circulation. Thus, the lungs can metabolize and eliminate the compounds that are either retained or passing through. There are also compounds that are preferentially taken up by the lungs and are metabolized by the enzyme system of this organ; among them are amino compounds, for example, phenylethylamine (Gillis and Pitt, 1982; Junod, 1985) and basic drugs like imipramine, fentanyl, and chlorpromazine (Philpot, 1977), estrogen (Peng et al., 2013), hormones (Junod, 1975), prostaglandins (Bakhle and Ferreira, 2011), and peptides. The pulmonary frst-pass metabolism, although not as signifcant as the hepatic and intestinal frst-pass metabolism, nonetheless affects the bioavailability of a compound given orally or intravenously. The exposure dose (Figure 3.6), is estimated as: Exposure Dose =

C ´ IR ´ F ´ EF BW

(3.38)

where C is the infow concentration in mass/volume (e.g., mg/L, or parts per million), IR is the intake rate in units of volume/time (e.g., L/day), F is the bioavailability (unitless), BW is body weight in kg, and EF is called the exposure factor and represents the length and frequency of an exposure, that is, EF =

( Frequency of Exposure ) ´ ( Duration of Exposure ) Averaging Time

(3.39)

For short-term exposure, EF is approximately equal to 1. 3.2.4.8 Pulmonary Rate Equations Using Figure 3.6, the rate equations of the exposure dose, potential dose, and internal dose are: dAExposure = k1 AM + k 4 AP - AExposure ( k 2 + k 3 + kE1 ) dt

(3.40)

dAP = k 3 AExposure - AP ( k 4 + k 5 + k6 ) dt

(3.41)

dAR = k 5 AP - AR ( k8 + k9 ) dt

(3.42)

dAGI = k9 AR - k a AGI dt

(3.43)

dAdischarged = k8 AR dt

(3.44)

dAID = k6 AP - AID ( k7 + k pa + kE2 ) dt

(3.45)

dA1 = k pa AID + k a AGI + k 21 A2 - A1 ( k12 + k10 ) - kED A1 dt

(3.46)

dA2 = k12 A1 - k 21 A2 dt

(3.47) 67

3.2 PULMONARY ROUTE OF ADMINISTRATION/EXPOSURE

Where AExposure is the exposure dose; k1 and k 2 are the rate constant of infow and outfow, respectively; AP is the potential dose; k 3 is the input rate constant and k 4 , k 5 , and k6 are the output rate constants of the potential dose; kE1 is the local effect rate constant; AR is the amount removed by mucociliary transfer; k8 and k9 are the rate constants of removal due to discharge (spit/cough up) and swallowing, respectively; AGI is the swallowed amount in GI tract; k9 is the swallowing rate constant, and k a is the absorption rate constant from GI tract into systemic circulation; AID is the internal or absorbable dose in pulmonary tract; k6 is the hybrid rate constant of deposition and dissolution; k7 is the rate constant of pulmonary metabolism assuming linear process, or may change to the non-linear metabolism with Michaelis–Menten parameters of Vmax and K M ; k pa is the pulmonary absorption rate constant, and kE2 is the local effect rate constant; A1 is the amount in the systemic circulation at time t ; A2 is the amount in the peripheral tissues and organs at time t ; k12 and k 21 are the distribution rate constants between central (systemic circulation) and peripheral compartments; k10 is the overall rate constant of elimination representing combined excretion and metabolic rate constants. For volatile gases or lipophilic vapors, the potential dose is the internal dose and Figure 3.6 simplifes to Figure 3.7. Among different approaches to PK/TK analysis of xenobiotics in the body after the exposure by inhalation, the physiologically based modeling has been used more frequently. A typical model is presented in Figure 3.8, and more discussion on physiological modeling is provided in Chapter 12, Sections 12.2.2.1 and 12.2.2.2. The mass balance for the combined processes occurring in each organ/compartment of the model presented in Figure 3.8 can be described by a series of differential equations. The simplest set up of these equations defnes the rate of uptake by each organ as the difference between the rate of input (infow) and rate of output (outfow), as shown in Equation 3.48. For the organs with elimination (excretion/metabolism) process, the rate of uptake can be presented in the form of Equation 3.49. dAorgan dAin dAout = dt dt dt

(3.48)

dAorgan dAin dAout dAe = dt dt dt dt

(3.49)

dAorgan dAin dAout is the uptake rate of an organ with units of mass/time, is the input rate, dt dt dt dAe is the output rate, and is the elimination rate from an organ with the elimination process. dt The mass balance equations can be as simple as Equations 3.48 and 3.49 or as complex as Equation 3.50 (Willems et al., 2001):

Where

dAlung Alung-cap Vmax ´Vlung ´ Alung dAlung Qblood = ´ Qblood ´ Pcap ´ ´ Pcap ´ MPlung dt Vlung-cap K M ´Vlung + Alung Vlung PT :BCoeff

(3.50)

where Alung is the amount of xenobiotic in the lung (units of mass); Alung-cap is the amount in the lung capillary blood (units of mass); Vlung-cap is the volume of blood in the capillaries (units of volume); Qblood is the total blood fow (volume/time), Pcap is the capillary permeability constant (unitless); Vlung is the volume of lungs; PT :BCoeff is tissue:blood partition coeffcient (unitless); Vmax is the maximum rate of metabolism (units of [mass/ MPlung /time]); MPlung is the amount of microsomal protein (units of mass/volume of tissue); K M is the Michaelis–Menten constant (units of mass/ volume). Although Equation 3.50 seems to be signifcantly different from Equation 3.49, nonetheless it is A the same equation. The frst term ( lung-cap ´ Qblood ´ Pcap ) is the input rate with units of mass/time, Vlung-cap dA Q the second term ( lung ´ blood ´ Pcap ) is the output rate with units of mass/time, and the third Vlung PT :BCoeff Vmax ´Vlung ´ Alung term ( ´ MPlung ) is the rate of elimination due to metabolism of the compound K M ´Vlung + Alung (Willems et al., 2001). 68

PK-TK CONSIDERATIONS OF NASAL, PULMONARY AND ORAL ROUTES

Figure 3.7 Schematic of respiratory exposure, absorption and elimination of gases (volatile, nonvolatile, medical, or toxic) and the sequential steps from exposure dose to effective dose; k1 and k2 are the rate constants of gas infow and outfow, respectively; kE1 and kE2 are the rate constants of local effect related to the upper respiratory and lower respiratory exposure, respectively; k3 an k4 are the rate constants of exchange between the upper and lower respiratory tracts; the defnition of other rate constants are the same as described for Figure 3.6. A compartmental approach, proposed for simulation purpose of inhaled particles (Weber and Hochhaus, 2013), is presented in Figure 3.9. The model considers the lung as having two compartments: the central lung (LC) and peripheral lung (LP). Each of these compartments is divided further into two compartments of LC1, LC2 and LP1, LP2. The undissolved particles are in compartments LC1 and LP1 and dissolved particles, that is, in solution form, are in LC2 and LP2. The mucociliary clearance occurs from compartment LC1. The inhaled particles upon administration 69

3.2 PULMONARY ROUTE OF ADMINISTRATION/EXPOSURE

Figure 3.8 An example of a physiologically based pharmacokinetic model for absorption of xenobiotics from pulmonary tract and distribution to a selected group of organs and tissues; the arterial blood carries the xenobiotic absorbed from the pulmonary tract into the selected organs and/or tissues; the concentration differences between the input fow rate and output fow rate of an organ represents the uptake of the xenobiotic; the selection of the organs and tissues for constructing a physiological model depends mainly on the physicochemical characteristics of the compound, the knowledge of the target tissue/organ, and the affnity for reversible or irreversible binding to the macromolecules; the elimination of the compound is through metabolism in the liver and excretion by the kidneys. are fractionated into three fractions: deposited in the mouth, deposited in the lung, and exhaled from the mouth/lung (Figure 3.9). The exhaled fraction is considered negligible. The differential equations of the model are:

70

d ( LC1 ) = - ( kC1 C2 + k muc ) LC1 dt

(3.51)

d(LP1 ) = -k P1 P2 ´ LP1 dt

(3.52)

d(LC2 ) = ( kC1C2 ´ LC1 ) - ( kaLC ´ LC2 ) dt

(3.53)

d(LP2 ) = ( k P1P2 ´ LP1 ) - ( kaLP ´ LP2 ) dt

(3.54)

PK-TK CONSIDERATIONS OF NASAL, PULMONARY AND ORAL ROUTES

Figure 3.9 An example of compartmental analysis of the absorption of inhaled particles in the lungs; the model assumes each lung as having a central compartment (LC) and a peripheral compartment (LP); LC and LP are divided up into two chemical compartments of dissolved and undissolved compartments (LC2 and LP2 for dissolved and LC1 and LP1 for undissolved particles); the undissolved particles may partially reach the GI tract through the mucociliary action with the rate constant of kmuc; the dissolved particles following the absorption in the lungs and GI tract reach the central compartment or compartment 1 (i.e., the systemic circulation) with the rate constants of kaLP, kaLC, and kaGI; the dissolved xenobiotic distributes between the central and peripheral compartment with the distribution rate constants of k12 and k21; the elimination from the body is through the central compartment governed by the overall elimination rate constant of k10; Flung, FLP, FLC, and FGI are the fractions of the administered or exposed dose reaching the lung and GI compartments. dAGI ˜ ˝ °kaGI ˛ AGI ˙ ˆ ˝ k muc ˛ LC1 ˙ dt

(3.55)

dA1 ˜ °k10 A1 ° k12 A1 ˛ k 21 A2 ˛ kaLC LC2 ˛ kaLP LP2 ˛ kaGI AGI dt

(3.56)

dA2 ˜ k12 A1 ° k 21 A2 dt

(3.57)

Where kC1 C2 and k P1 P2 are the dissolution rate constant in the lung central and peripheral compartments, respectively; the tacit assumption of the model is that kC1 C2 is the same as k P1 P2 and represent the dissolution rate constant of the particles in the central and peripheral compartments (i.e., LC&LP ), as identifed in Figure 3.9 as k dissolution ; LC1 and LP1 are the amount in the respective compartments; k muc is the mucociliary rate constant of transfer from LC1 to the GI tract; LC2 and LP2 are the amounts in the lung peripheral compartments; kaLC and kaLP are the rate constants of absorption from the lung peripheral compartments into the systemic circulation or the body’s central compartment; AGI is the amount in the GI tract; kaGI is the absorption rate constant from GI tract into the systemic circulation/central compartment; A1 and A2 are the amounts in the central and peripheral compartments of the body; k12 and k 21 are the rate constants of distribution between the central and peripheral compartments of the d ˜ LC1 ° d ˜ LP1 ° , , body; and k10 is the overall elimination rate constant from the central compartment; dt dt d ˜ LC2 ° d ˜ LP2 ° dAGI dA1 dA2 , , , , and are the rate of change of amount of respective form of xenobidt dt dt dt dt otic in the related compartments as a function of time. The Laplace transform of Equations 3.51 through 3.57 are reported below (Weber and Hochhaus, 2013). Using the methodology in Addendum II, Part 2, one can generate the integrated equations representing the amount in each compartment of the lung and body. 71

3.2 PULMONARY ROUTE OF ADMINISTRATION/EXPOSURE

˜ 1 = ( LC1 )t=0 LC (s + k )

(3.58)

˜1 = ( LP1 )t=0 LP ( s + kdiss )

(3.59)

˜2 = k diss ´ ( LC1 )t=0 LC ( s + kaLC )( s + k )

(3.60)

˜2 = LP

k diss ´ ( LP1 )t=0

(3.61)

( s + kaLP )( s + kdiss )

æ k muc ´ ( LC1 )t=0 ( AGI )t=0 ö ˜ A + GI = FGI ´ ç ç ( s + kaGI ) ( s + k)) ( s + kaGI ) ÷÷ ø è ˜1 = A

(k

21

) (

) (

) (

˜2 + kaLC ´ LC °2 + kaLP ´ LP °2 + kaGI ´ A ° ´A GI

( s + k10 + k12 )

(3.62)

)

˜ ˜2 = k12 ´ A1 A + s k ( 12 )

(3.63) (3.64)

where FGI is the fraction of inhaled dose that reaches the GI tract; s is the Laplace operator; k diss is the dissolution rate constant; the initial conditions are: The lung central compartment:

( LC1 )t=0 = Dose ´ Flung ´ FC

(3.65)

( LP1 )t=0 = Dose ´ Flung ´ (1 - Fc )

(3.66)

( AGI )t=0 = (1 - Flung ) ´ Dose

(3.67)

The lung peripheral compartment: The GI absorption compartment:

Where Flung is the fraction of the administered dose by the inhaler that is deposited in the lung; and FC is the fraction of the deposited dose in the lung central compartment. Substituting Equation 3.58–3.62 and 3.64 into Equation 3.63 and setting the hybrid rate constants of the two-compartment model, a and b, as a ´ b = k10 ´ k 21 and a + b = k10 + k12 + k 21 yields the Laplace transform of the body’s central compartment as (Weber and Hacchhaus, 2013): ˜=A °1 + A °2 + A °3 + A °4 A

(3.68)

˜1 = kaLC ´ k diss ´ ( LC1 )t=0 ´ ( s + k 21 ) A ( s + a )( s + b )( s + kaLC )( s + k )

(3.69)

˜2 = kaLP ´ k diss ´ ( LP1 )t=0 ´ ( s + k 21 ) A ( s + a )( s + b )( s + kaLP )( s + kdiss )

(3.70)

˜3 = kaGI ´ FGI ´ ( AGI )t=0 ( s + k 21 ) A ( s + a )( s + b )( s + kaGI )

(3.71)

˜4 = kaGI ´ k muc ´ FGI ´ ( LC1 )t=0 ´ ( s + k 21 ) A ( s + a )( s + b )( s + kaGI )( s + k )

(3.72)

Exercise: Using the Laplace Transform Method of Integration (addendum I, Part 2), write the inverse Laplace transform of Equation 3.68, i.e., A = A1 + A2 + A3 + A4

72

(3.73)

PK-TK CONSIDERATIONS OF NASAL, PULMONARY AND ORAL ROUTES

3.3 GASTROINTESTINAL (ORAL) ROUTE OF ADMINISTRATION OR EXPOSURE 3.3.1 Overview The oral route is considered the most convenient route for administration of therapeutic agents. For economical, technical, and patient compliance reasons, the development of oral dosage forms is normally considered frst for a new small molecule drug, and it is commonly believed that it has relatively better medication adherence. Most often the sublingual and buccal routes of administrations are reported linked to the oral route of administration. However, because both sublingual and buccal routes have specifcations that are different from the oral route and have distinct advantages to avoid acid-catalyzed degradation, enzymatic breakdown, and hepatic frst-pass effect, they are discussed as separate routes in Chapter 2, Section 2.2. The details on anatomy and physiology of the GI tract are beyond the scope of this chapter, and readers are referred to anatomy and physiology textbooks and/or relevant references. The topics highlighted in this chapter are those that infuence the ADME of xenobiotics, directly or indirectly. To begin with the appraisal of the GI tract as a route of administration, a few of its biological roles are important to enumerate here. In general, the GI tract: i. Facilitates the absorption of required nutrients for furnishing needed energy for the body biochemical reactions and physiological tasks and, thus, contributes to preserving the overall functional integrity of the body. ii. Prevents absorption of uninvited entities like viruses, bacteria, foreign proteins, macromolecules, and harmful materials to protect the welfare of the body. iii. Self-regulates water absorption to maintain a stable internal environment for the body’s homeostasis. iv. Has the largest immune system (Mowat and Agace, 2014) and produces local immune systemlike IgA antibodies to counteract pathogens that are harmful to the body. v. Participates partially in the elimination of metabolites of hepatic biotransformation that enter the GI tract through the contraction of the gallbladder (and leave the body via fecal elimination), thus enhancing the overall health and well-being of the body. vi. Provides a passageway for the re-absorption of compounds or their metabolites, entering in the GI tract via biliary excretion, known as enterohepatic re-circulation. vii. Gives shelter to the multiplex system of the microbiota ecosystem in its environs for suppressing indigenous pathogens, digestion of some compounds and nutrients and collectively provides immune responses (Sender et al., 2016) to protect the body. viii. Has the intrinsic nervous system of the gut (ENS), that regulates its vital function like mucus secretion and communicating with the immune system (Yoo and Mazmanian, 2017; Chesné and Cardoso, 2019) that entails the release of mediators, like neurotransmitters, cytokines, chemokines, etc., to help maintain the physiological steady state within the GI tract (Verheijden and Boechxstaens, 2018). Moreover, the presence of diverse environments with different attributes within the GI tract combined with the existence of various physical and physiological processes make the absorption of xenobiotics from the GI route challenging. The physiological mechanisms and attributes of the route that infuence the absorption and thus the ADME of a xenobiotic in the body are gastric emptying rate, small intestinal transit time, actions of infux and effux proteins, regional pH of the GI tract and infuence on the ionization of xenobiotics, the surface area for absorption, hepatic frst-pass metabolism, pre-systemic GI metabolism, role of gastrointestinal microbiotas, effect of bile salts, etc. The factors linked to the physical and chemical characteristics of xenobiotics that infuence their permeation, absorption, and degradation are polymorphism and chemical nature of the compound, partition coeffcient and molecular properties, particle size, porosity, wettability, type of the dosage form (e.g., solid, solution, semisolid, nanoparticles, polymeric, extended release, etc.), disintegration and dissolution, chirality, and enantiomers. There are also ancillary factors, such as type of food and drink, disease states (Effnger et al., 2019), age, sex, genetic polymorphism, circadian infuences, and dosing regimens. The eventual outcome of using GI tract as route of administration for achieving systemic effect is the absorption and entry into the systemic circulation, which entails the permeation of drug from the lumen to gut wall and subsequent diffusion through capillary wall into the blood stream. 73

3.3 GASTROINTESTINAL (ORAL) ROUTE OF ADMINISTRATION OR EXPOSURE

Figure 3.10 Illustration of main factors infuencing the absorption of xenobiotics from the GI tract at four stages of esophagus, stomach, small intestine, and large intestine; the factors in stomach include the disintegration, dissolution, and solubilization of xenobiotics, the effect of acidic pH on ionization or degradation, gastric emptying rate or time, and partial absorption of unionized molecules; in the small intestine, the main site for absorption of administered or exposed xenobiotics, numerous factors like CYP450 isozymes and enzymes of Phase I and II metabolism, effux and infux proteins, interactions with other compounds and nutrients, hepatic frst-pass effect, and small intestinal transit time or rate; in the large intestine that facilitates the elimination of remaining compounds and their metabolites via fecal elimination, share the infuence of microbiotas with the distal portion of the ileum; the age, gender, and relevant disease states impact the absorption of xenobiotics through the GI tract. A schematic depiction of the factors infuencing the absorption of a xenobiotic from the gastrointestinal route of administration is presented in Figure 3.10. 3.3.2 Physiologic and Dynamic Attributes of the GI Tract Infuencing Xenobiotic Absorption 3.3.2.1 Regional pH of GI Tract and pH-Partition Theory Orally ingested compounds pass through two different and distinct environments: acidic in the stomach and alkaline in the small intestine. Aside from the anatomical differences of the stomach and intestine, gradual changes in the pH throughout the tract infuence the absorption of weakly acidic and weakly basic compounds. Gastric pH is highly acidic during the fasting state and is comparable to the pH of 0.15 M HCl containing pepsin. Age, disease states, and foods infuence the pH of the GI tract (Russell et al., 1993; Zimmermann et al., 1994; Charman et al., 1997; Martinez and Amidon, 2002). The presence of food alters the pH, content viscosity, and transit rates. It also infuences the disintegration and dissolution processes (Gai et al., 1997). Because of the daily intake of food and beverages, the gastric pH fuctuates during the day. The stomach pH increases signifcantly following the bariatric surgery and depending on the surgical procedure, it changes from 1.8 prior to surgery to 4.9 and 6.4 after sleeve gastrectomy and gastric bypass surgery, respectively (Porat et al., 2021). The alkaline environment of the small intestine, on the other hand, remains consistent during the day and throughout life. The regional pH of the small intestine starts from 5.5 to 6.0 in the duodenum and gradually increases to 7.0–9.0 in the jejunum and about 9–11 in the 74

PK-TK CONSIDERATIONS OF NASAL, PULMONARY AND ORAL ROUTES

ileum. The chemical stability, dissolution, and absorption of xenobiotics may also be affected by the pH along the GI tract (Charman et al., 1997; Ward, 2010). Most xenobiotics are either weakly acidic or weakly basic compounds, and according to the pH-partition theory, only the unionized form of molecules are able to partition through biological membrane; that is, the low pH of the stomach unionizes weakly acidic compounds, and the high pH of the intestines unionizes weakly basic compounds. The solubility of a compound also depends on its pKa and pH of the GI tract (Dressman et al., 2007). The classical Henderson– Hasselbalch equation links the pKa of a compound to the pH of the GI tract to estimate the degree of ionization and the ratio of unionized to ionized forms of the molecules at a given pH. æ [unionized] ö pKa - pH ) = 10( ç ÷ [ionized] ø acidic è

(3.74)

æ [unionized] ö pH -pKa ) = 10( ç ÷ [ionized] øbasic è

(3.75)

For example, consider compound A with a pKa of 3 and compound B with a pKa of 8 that are placed in pH 2 and 7, representing the pH of stomach and jejunum, respectively: æ [unionized] ö 3-2 = 10( ) = 10 ç ÷ è [ionized] ø drugA æ [unionized] ö 2-8 = 10( ) = 10 -6 ç ÷ è [ionized] ø drugB According to the calculation, at pH 2 for every 10 unionized molecules of A with a pKa of 3, only one molecule would be ionized. The ratio changes signifcantly for B with a pKa of 8, where for everyone unionized molecule 106 molecules would be ionized. At pH 7 the ratios for A and B are 10−4 and 10−1, respectively. Thus, at a higher pH, the weakly basic compounds are more unionized and acidic ones are more ionized, and the opposite is true at low pH. It should be noted, however, that although weakly acidic compounds are more unionized in the stomach, they are mainly absorbed from the intestine. The reasons are the limited surface area of the stomach, impermeability of the absorption barrier, and the short residence time in the stomach. Thus, the unionized form of the weakly acidic compounds at a high pH of intestine, although in low concentration compared to the ionized molecules, contributes more signifcantly to the overall absorption of weakly acidic xenobiotics. This is mainly due to the large surface area of the intestine and the long intestinal transit time. The longer residence time in the small intestine is considered a favorable condition for the absorption of weakly acidic compounds (Löbenberg and Amidon,2000 a, b). The absorption of neutral, acidic, basic, and amphoteric compounds has been studied in various models. The simultaneous chemical equilibria and mass transfer were studies in physical two- and multi-compartment unstirred diffusional models (Suzuki et al., 1970 a, b). The multidiffusional model consists of an aqueous compartment with a diffusion layer followed by n compartments and a perfect sink condition. The signifcance of the unstirred diffusional model, the role of pH and pKa, and the degree of ionization of a compound is discussed here based on the following ionic equilibria of an amphoteric compound in water: K1 K2 ¾¾ ¾¾ ® Rw0 + H + ¬ ¾¾ ¾¾ ® Rw- + 2H + Rw+ ¬ + w

w

(3.76)

0 w

where R , R , and R are the concentrations of the cationic, anionic, and unionized of the compound R, respectively. The subscript “w” represents the aqueous phase; K1 and K2 are the dissociation constants, and H+ is the hydrogen ion concentration. For a basic compound, K1 is the dissociation constant with K2 = zero. For an acidic compound, K2 is its dissociation constant with K1 = ∞. For a neutral compound, K1 = ∞ and K2 = zero (Suzuki et al., 1970 a, b). According to Fick’s First Law of Diffusion, the total fux of different species of the compound is æ d é R +w ù ö æ d é Rw- ù ö æ d é Rw0 ù ö + 0 J = - ( Dcoeff )w ç ë û ÷ - ( Dcoeff )w ç ë û ÷ - ( Dcoeff )w ç ë û ÷ ç dx ÷ ç dx ÷ ç dx ÷ è è è ø ø ø

(3.77)

Where J is the total fux, and ( Dcoeff ) is the diffusion coeffcient. 75

3.3 GASTROINTESTINAL (ORAL) ROUTE OF ADMINISTRATION OR EXPOSURE

The absorption of small-molecule xenobiotics are facilitated by having them in solution form at the site of absorption. The solubility of the compound at the site of absorption depends on the state of ionization of the molecules. The solubility of weak acids or bases is fully infuenced by the pH of the environment. Weak acids and bases are more soluble in their ionized (conjugate base or conjugate acid) forms than their unionized forms, so at a high pH, where the weakly acidic compounds are in ionized state, the solubility of the compounds increases, whereas the solubility of weakly basic drugs decreases. The reverse is also true for weakly basic compounds when their solubility enhances at low pH. When Equation 3.76 is at equilibrium, K2 represents the equilibrium constant equal to 2 é H + ù ´ é Rw- ù é H + ù ´ é Rw- ù K2 = ë 0 û ë + û = ë û 0 ë û é Rw ù é Rw ù é H ù ë û ë ûë û

(3.78)

K 2 éë Rw0 ùû é Rw- ù = ë û éH + ù ë û

(3.79)

Solving for é Rw- ù yields ë û

If the total solubility (S) consists of the concentrations of ionized and unionized forms of a compound, that is, S = éë Rw- ùû + éë Rw0 ùû

(3.80)

The substitution of Equation 3.79 in Equation 3.80 yields S=

K 2 ´ éë Rw0 ùû éH ù ë û +

æ K 2 ö÷ + éë Rw0 ùû = éë Rw0 ùû ç 1 + ç éH + ù ÷ ë ûø è

(3.81)

Setting é Rw0 ù as the pH independent solubility, that is, the solubility of unionized forms, Equation ë û 3.81 can be written as æ K 2 ÷ö S = S0 ç 1 + ç éH + ù ÷ ë ûø è

(3.82)

Where S is the pH-dependent total solubility of a compound and S0 is the pH-independent solubility of the compound. The solubility of ionized form is then equal to

( S - S0 ) = K 2

S0 éH + ù ë û

(3.83)

Equation 8.10 can be used to estimate pH below which the compound remains in unionized form by taking the logarithm of both sides: log(S - S0 ) = log K 2 + log S0 - log éë H + ùû

(3.84)

Setting K 2 = K a , where K a is the association constant, and solving for pH = -log é H + ù , yields ë û pH = pK a + log

S - S0 S0

(3.85)

Several conclusions can be made from Equation 3.85. For example, because food increases the pH of the stomach, giving weakly basic therapeutic agents (e.g., dipyridamole, ketoconazole, etc.) after the meal may not help their solubility, or co-administration of weakly basic drugs with proton pump inhibitors has the same outcome, or solubility of poorly soluble weakly acidic drugs (e.g., indomethacin, furosemide) occurs mainly in small intestine. Compounds in the free acid form are soluble at any pH of the GI tract. The application of the pH-partition theory for defning the passive absorption of unionized molecules in the GI tract has been debated and questioned in the literature. For example, the relationship between the solubility, partition coeffcient, and pH has been investigated with the suggestion that the product of intrinsic solubility and intrinsic octanol-water partition coeffcient is equal to 76

PK-TK CONSIDERATIONS OF NASAL, PULMONARY AND ORAL ROUTES

the product of total solubility of a partially ionized solute and its octanol-water distribution coeffcient at any pH.

( Solubility of a partially ionized solute ) ´ ( Octanol / water distribuution coefficient ) Thus, it may not be necessary to consider either pH of the GI tract or pKa of a compound necessary to predict or evaluate the passive absorption of orally administered compounds (Ni et al., 2002). A different view in permeation of ionized molecules is the effect of dynamic changes in the protonation state that may play an important role in permeation of ionized molecules (Yue et al., 2019). It should also be noted that not all xenobiotics are either weak acids or weak bases. The behavior of classes of compounds such as ampholytes and zwitterions are distinctively different. The ordinary ampholytes pK aacidic > pK abasic are both weak acids and weak bases. Thus, they remain unchanged at neutral pH, and their lipophilicity is similar to that of acids and bases. On the other hand, zwitterions at neutral pH have a negative and a positive charge and their lipophilicity is constant in a pH range defned by their two pKa (Pagliara et al., 1997; Bouchard et al., 2002). A more complex picture emerges when a compound is considered zwitterionic ampholytes. The discussion on the ionization equilibria of this group of compounds is beyond the scope of this chapter.

(

)

3.3.2.2 Absorptive Surface Area An orally administered compound travels frst through the esophagus, which is a tubular muscular structure 25–50 cm between the distal pharynx and gastroesophageal junction below the diaphragm. Because of its peristaltic contraction waves of the upper and lower sphincters and the related propulsive force with the help of gravity, the absorption from this part of GI tract is negligible. The next stop for the compound is the stomach with a normal empty volume of 45–50 mL, which lies on the left side of abdominal cavity between two smooth muscle sphincters: esophageal and pyloric. It has four distinct regions: the cardia (i.e., gastro-esophageal junction), the fundus (the upper curvature), the corpus (the central region) and the pylorus (the lower region). Its walls are composed of a mucous layer, sub-mucosa layer, and muscular layer consisting of longitudinal, circular, and oblique muscle fber, and serosa or connective tissues that fasten the stomach to peritoneum. The surface of the gastric mucosa is composed of a layer of columnar cells and secretory cells responsible for secretion of about two liters of gastric fuid per day containing hydrochloric acid, gastrin, pepsin, and mucus (Löbenberg and Amidon, 2000 a, b). Most weakly acidic compounds are expected to absorb in the stomach. Compounds like aspirin and alcohol absorb rather rapidly in the stomach. The empty stomach has several inner folds called rugae, which can expand signifcantly to accommodate the volume of food. In individuals with normal body weight, the surface area of acid-producing mucosa of the stomach is between 383 cm2 and 784 cm2, with an average of 583.48 cm2. After the stomach, the small intestine has the largest surface area for absorption of nutrients and xenobiotics; approximately 90% of all absorptive processes occur in the small intestine (Lennernäs, 1997; Fangerholm et al., 1996). Practically all mechanisms of absorption occur in the small intestine. Its walls are more permeable than the stomach, and because of the gradual variability of the pH, weakly basic and weakly acidic drugs permeate through the walls more readily than the stomach. The walls are folded into crinkles that increase the surface area by a factor of 3; its epithelial layer is folded into villi, which can increase the surface area by a factor of 10; and the microvilli folding of the villi enhances the surface area by a factor of 20. The epithelial cells that cover the surface of the villi are comprised of 90% enterocytes (Hillgren et al., 1995). The enterocytes have an apical side facing the lumen of the intestine, and a basolateral side or serosal side. The apical region of enterocytes has a brushed structure that forms microvilli. The enterocytes are connected by a tight junction. The main biological barrier for absorption of xenobiotics in the small intestine is the combination of membrane and tight junctions of enterocytes (Salana et al., 2006; Fangerholm et al., 2007). The entire surface of the intestine is covered by a thin layer of 0.1 mm thick glycoprotein known as the glycocalyx layer. The length of the small intestine for a 70 kg man is about 6–7 m with a diameter of 2.5–3 cm; approximately 5% of the total length is the duodenum (the initial segment), 50% is the jejunum (the mid-section of intestine), and 45% is the proximal and distal sections of the ileum (the terminal portion). Assuming the small intestine as a cylinder with a radius of 1.5 cm and length of 7 m, without its complex anatomical structure, its estimated surface area would be 77

3.3 GASTROINTESTINAL (ORAL) ROUTE OF ADMINISTRATION OR EXPOSURE

SA = 2pr ( r + h ) = 2 ´ 3.14 ´ 1.5 (1.5 + 700 ) = 6608 cm 2 = 0.66 m 2

(3.86)

Adding the surface of its complex anatomical structure (i.e., all the folding, villi, and microvilli), the theoretical surface area of the small intestine would be

( SA )SI = 0.66 m2 ´ 3 ´ 10 ´ 20 = 396 m2 = 4, 262.50 ft 2

(3.87)

where (SA)SI is the surface area of small intestine. According to these theoretical calculations, the total surface area of the small intestine approximately equals an NBA basketball court with dimensions of 50 × 94 ft. Because of this enormous surface area and because of being the hub of all enzymatic metabolism and transport proteins in the GI tract, the small intestine is the fnal stage and sole site for the absorption of nutrients and orally administered xenobiotics. The large intestine has a length of approximately 1.5 m (5 feet) and a diameter larger than the small intestine. It starts from the cecum (the pouch connected to the distal portion of the ileum) followed by the colon, which has four portions: ascending colon, traverse colon, descending colon, and the sigmoid colon, which is the fnal S-shape portion connected to the anal canal. The ileocecal valve located between the distal portion of ileum and cecum section serves as a barrier for guarding ileum against colonic content and microorganisms of the large intestine. The appendix, a closed tubular structure, is also in the cecum. The mucosal cells known as colonocytes have no microvilli and are covered with a thick layer (200–500 mm) of mucos and water, which prevents the absorption of xenobiotics. The major roles of the large intestine are absorption of residual water and electrolytes, providing an environment for microbiotas’ activities, and acting as a transport canal to get rid of waste products (Kararli, 1995; Wilson, 2000). In recent years, the colonic delivery systems of proteins and peptides have been developed for the localized and targeted treatment of disease (Patel et al., 2007). The rectum is the terminal part of the GI tract. It is a highly vascular region with limited surface area. It begins at the rectosigmoid junction and ends in anorectal junction. Its average length is about 13 cm (5 in), and because of its vascularity it is considered a signifcant route of absorption for therapeutic agents (see also Chapter 6). 3.3.2.3 Gastric Emptying and Gastric Accommodation The gastric emptying process, regulated by neural and hormonal controls, is a carefully synchronized contractile movement between stomach and small intestine. Following mastication (chewing) and deglutition (swallowing), the esophagus, through the coordinated contraction of its muscle, propels materials toward the stomach, and sphincters act in a coordinated manner with swallowing to prevent the materials from returning to the esophagus (Johnson, 2007). A sustained contraction is initiated and the content of the upper part of stomach is pressed toward its lower part. The proximal stomach is responsible for the transfer of gastric content known as chyme from the stomach to the upper part of the small intestine (duodenum). The lower part of the stomach triturates the content, prevents duodenal refux, and allows only liquids and small particles to pass into the duodenum. Peristaltic contractions of both the proximal and distal stomach are under neural control of vagus nerves and its autoregulation of acetylcholine release. The acetylcholine interacts with the appropriate receptor of the stomach muscle to stimulate the contraction and relaxes during the swallowing. Furthermore, several hormones participate in the stimulation or inhibition of the contraction. For example, cholecystokinin inhibits the contraction of the proximal stomach whereas it stimulates the contraction of the distal stomach; secretin and somatostatin both inhibit the contractions of the proximal and distal stomach. Gastric accommodation is referred to relaxation of the proximal stomach during meal consumption without a simultaneous increase in intragastric pressure. This relaxation has two parts of receptive and adaptive relaxations. Receptive relaxation is initiated by swallowing the food down the esophagus, the fundus-corpus widen, and the stomach responds immediately by expanding to accommodate the consumed food. Adaptive relaxation takes place following the receptive relaxation by dilation of fundus more fully to maintain the intragastric pressure at a minimum (Febo-Rodriguez et al., 2021). The gastric accommodation is mediated by the vagal refex and is important in food intake. This physiological refex function of the GI tract may not have a direct impact on the absorption of a xenobiotic, but it regulates partially the residence time of a molecule in the stomach. Loss of this refex function gives rise to rapid transit from proximal stomach into the antrum and leads to dyspeptic symptoms.

78

PK-TK CONSIDERATIONS OF NASAL, PULMONARY AND ORAL ROUTES

The gastric emptying rate is the rate at which the stomach empties its contents into the duodenum. The delay in the gastric emptying rate is responsible for a delay in the onset of action of certain therapeutic xenobiotics and dosage forms. For example, weakly basic drugs are in an ionized form in the stomach and therefore, a slow gastric emptying rate delays their onset of action. Furthermore, the gastric emptying rate often contributes to inconsistency in the absorption of an orally administered drug because of particle size, particle density, and viscosity of the content. The intensity of the gastric contraction and pyloric resistance changes during the fed state with a pattern of consistent low-medium amplitude contractions, which assist the small particles (2–7 mm) to be transferred to the small intestine. The infuence of particle size cutoff on gastric emptying is important, yet controversial. There are contradictory published data with results indicating that tablets and particles as large as 11 mm can leave the stomach during the fed state (Coupe et al., 1991 a, b). In fasting, the gastric emptying is a function of its cyclical contractions, which is characterized by the following phases: Phase I: Calm phase (quiescence) – duration: 30–60 minutes Phase II: Irregular phase with random medium-amplitude contractions – duration: 20–40 minutes Phase III: Intense contraction phase with high amplitude contractions – duration: 5–15 minutes Phase IV: Decelerate phase with gradual reduction of contractions leading to Phase I The cycle of fast-state contractions (Phase I–IV) are also known as “interdigestive migrating myoelectric complex” (IMMC). This cycle of contractions initiates from the part of the stomach adjoining the pylorus, known as pyloric antrum. Larger indigestible particles empty into the duodenum during Phase III of fast-state gastric emptying, when the gastric contractions are stronger with high amplitude. The IMMC continues during the fast state until the food is introduced into the stomach, which changes the IMMC to frequent contractions with medium-low amplitudes (Burks et al., 1985; Oberle et al., 1990). The kinetics of gastric emptying of non-nutrient liquids, such as polyethylene glycol (PEG 4000) or phenol red has been shown to follow a frst-order process after a lag time, attributed to a rapid emptying of the content at the beginning of fed-state contractions when the content volume is high. The literature data are not unanimous on the kinetics of the gastric emptying process. The following are important factors that infuence the gastric emptying rate: 1. Compounds that block gastric muscle receptors for acetylcholine delay the emptying rate (e.g., propanthelin). 2. The high acidity of gastric chyme also delays the emptying rate. 3. The chemical composition of chyme within the stomach determines the gastric emptying rate. In humans, liquids are emptied with a half-life of approximately 12 minutes and digestible solids at a half-life of about 2 hours, depending upon the chemical composition of the chyme. Carbohydrates are emptied faster than proteins, and proteins are faster than fats. 4. The gastric emptying rate is controlled by the stomach caloric content, such that the number of calories transferred to the small intestine remains constant for different nutrients (carbohydrates, proteins, or fats) over time, and the emptying process is slower when the meal is rich in calories. 5. The gastric emptying rate can be modulated by the meal size. For example, changing the solid meal size from 300 to 1692 g increases the half-life from 77 to 277 minutes; however, the size of the meal produces a pressure that stimulates the gastric emptying (Christian et al., 1980). 6. Stimulation of the small intestinal receptors (i.e., duodenojejunal receptors sensitive to osmotic pressure) by hypertonic or hypotonic solutions slows down the gastric emptying rate. 7. The temperature of the solid or fuid intake affect the gastric emptying rate. The temperature above or below the physiological 37°C proportionately reduces the gastric emptying rate. 8. Other factors, such as anger or agitation, may increase the rate, whereas depression, trauma, or injury has been suggested to reduce the gastric emptying rate. Position of the body such as standing or lying on the right side may facilitate the transfer of the content to the small intestine by increasing the pressure in the proximal part of the stomach. 79

3.3 GASTROINTESTINAL (ORAL) ROUTE OF ADMINISTRATION OR EXPOSURE

9. Disease states may cause disorders in gastric emptying, e.g., in patients with diabetes as the result of autonomous neuropathy infuencing the GI tract, in type 2 a moderate delay, and more extreme in type 1 patients when the disease is insuffciently controlled (Horowitz et al., 2002; Jones et al., 2001). Delayed gastric emptying, using gastro-retentive drug delivery systems, enhances the gastric absorption of therapeutic agents by increasing gastric retention time and reducing the gastric emptying rate (Lee et al., 1999; Streubel et al., 2003; Mundada et al., 2008). A few examples of these systems are: ◾ low-density foating systems ◾ gel-forming polymeric systems ◾ gas-generating systems ◾ expandable systems 3.3.2.4 Intestinal Motility: Small Intestinal Transit Time The major site of absorption in the GI tract is the small intestine. The stomach and colon have a small absorptive area and the colon has a lumen full of bacteria. The small intestine has a length of approximately 10–16 feet (about 300–500 cm) with an alkaline environment that starts from the pyloric sphincter and continues with the duodenum, jejunum, and proximal and distal portions of the ileum to the ileocecal valve of the large intestine. The duodenum is about 25 cm (10 inches) long, receives chyme from the stomach through the pyloric sphincter. Bile and pancreatic ducts deliver pancreatic juice (which contains a variety of enzymes, including trypsinogen, chymotrypsinogen, elastase, carboxypeptidase, pancreatic lipase, nucleases, and amylase), and bile (which contains bile acids and waste products, such as bilirubin) from the pancreas and liver, respectively. The jejunum is about 2.5 m (8 feet) long; it is the middle section of the small intestine. The ileum on average can reach 360 cm (12 feet) long and is the last section of the small intestine. It ends with the ileocecal valve (sphincter), which regulates the movement of chyme into the large intestine and prevents backward movement of material from the large intestine. As was noted earlier, the average pH of the duodenum is approximately 6 and increases gradually throughout the intestine. The average absorptive surface area of the small intestine is approximately 250 square meters, or about 2691 square feet, which is less than the theoretical value discussed earlier. The three features of its anatomy that contribute to its absorptive surface area are the 1) mucosal folds, 2) villi, and 3) microvilli. The microvillus border of intestinal epithelial cells is referred to as the “brush border.” The contents of the small intestine are highly rich in digestive enzymes such as lipase, protease, amylase, esterase, and nucleases. In addition, bile that is rich in micelles of bile salts is added to the content of the small intestine. The small intestinal motility is mostly segmental contractions that consist of mixing-contraction and propulsive-contraction. The intestinal motility has two different patterns of digestive and interdigestive activities. The interdigestive activity cycle starts within about 30–40 minutes of no activity in both the stomach and intestine, followed by increasing contractions that are circular and migrate along the small intestine (peristaltic). This pattern of contraction removes everything that is left in the small intestine. Finally, the contraction diminishes in occurrence and intensity, and the cycle starts again by the rest period followed by the contraction. One entire cycle may take between 90 and 120 minutes in healthy and normal subjects. However, it has been shown that intra-individual differences can be signifcant (Davis et al., 1986; Coupe et al., 1991 a, b). As was noted in the gastric emptying process, the ingestion of food suspends the interdigestive cycles and establishes the intestinal motility pattern for digestive cycle (van der Ohe and Camillari, 1992). The digestive cycle constitutes the contractions that are mainly mixing contractions with a few propulsive contractions. The small intestinal transit time (SITT) during the digestive cycle is rather diffcult to determine considering the variability among individuals and the types of food; it may not be as accurate as one would expect. The SITT of healthy individuals ranges from 3 to 5 hours. The esophago-duodenal-jejunalileal transit time in normal subjects receiving mashed potatoes with labeled sulfur has been reported as 378 ± 90 minutes. Various factors, including pharmaceutical excipients, such as mannitol, can reduce the SITT (Wilson, 2000). The transit time between the stomach and large intestine is one of the signifcant factors in bioavailability of xenobiotics.

80

PK-TK CONSIDERATIONS OF NASAL, PULMONARY AND ORAL ROUTES

Most of the factors that affect the gastric emptying time, such as the presence of fats or the volume of food, also infuences the intestinal motility. For example, a large meal rich in fats requires longer and stronger intestinal contractions. 3.3.2.5 Role of Bile Salts Hepatocytes in the liver secrete bile into canaliculi, which fows into the bile ducts, stored in the gallbladder, and ultimately secreted into the duodenum at meals. Its composition consists of mainly water, bile acids/salts, inorganic salts, electrolytes, cholesterol, phospholipids, bilirubin, xenobiotics, and their metabolites. Bile secretion into the duodenum is due to the chyme stimulation and secretion of secretin and cholecystokinin. Its total production in humans is about 400–800 mL per day. During fasting, the bile fow declines, and most of that is diverted into the gallbladder. The bile stored in the gallbladder, because of re-absorption of water and electrolytes, gets concentrated gradually and is used when needed. The secreted bile acids, by means of enterohepatic recycling, return to the liver. The process of recycling starts with an ileal sodium-dependent protein transporter known as apical Na+dependent bile acid transporter (ASBT) that transports the secreted bile acids into the intestinal absorptive columnar epithelial cells (enterocytes), where the ileal bile acid binding protein (IBABP) transports the bile acids from the apical side of enterocytes to its basolateral side, which is then released by organic solute protein transporter OSTα/β and delivered to the liver via portal vein. This complex and effcient recycling limits the excretion of the bile acids in feces to about 5% of the bile acids pool, an indicative of apical sodium-dependent bile acid transporter (ASBT) effectiveness in the absorption of bile acids (Deng et al., 2020). Using the bile acid transporter-mediated delivery is regarded as a feasible strategy to enhance the bioavailability of certain drugs with appropriate oral delivery systems. Bile is an important biological sample for xenobiotic in vivo metabolism studies, PK/TK analysis, and the mass balance studies of xenobiotics, and its components like bile salts/acids are critical for the digestion and absorption of lipid, lipophilic compounds including fat-soluble vitamins in intestinal milieu in the form of bile salt phospholipid mixed with micelles and vesicular colloids (Hofmann and Borgstrom, 1962; Nordskog et al., 2001). Bile salts are surface-active agents that aggregate in an aqueous solution forming micelles and mixed micelles with phosphotydylcholine, glyceraldehydes, and fatty acids in the small intestine (Hoffmann, 1962; Cabral and Small, 1989). It has been shown that biliary-derived endogenous fatty acids have a higher tendency to transport through the lymphatic network (Trevaskis et al., 2005). Bile salts are the sodium salt of bile acids and the polar derivatives of breakdown products of cholesterol in the liver by cytochrome P450-mediated oxidation. The important intermediates of the reaction are trihydroxycoprostanoate and cholyl CoA. The activated carboxyl carbon of cholyl CoA interacts with the amino group of glycine or taurine to form glycocholate or taurocholate, respectively. In humans, the most important bile acids are cholic acid, deoxycholic acid, and chenodeoxycholic. Bile salts are amphipathic compounds and have both hydrophilic (polar hydroxyl group) and hydrophobic (nonpolar methyl group) regions and are considered effective detergents in emulsifying lipids. Thus, bile acids/salts play several important roles in the body including elimination of cholesterol, removal of by-products of catabolism, emulsifying lipids and fat-soluble endogenous and exogenous compounds, control of microbiota, and removal of xenobiotics and their metabolites through fecal elimination. 3.3.2.6 Hepatic First-Pass Metabolism (Pre-systemic Hepatic Extraction) There are numerous factors that reduce the absorption of a compound and cause poor bioavailability of oral absorption from the GI tract; examples of these factors are poor solubility of the compound, incomplete intestinal absorption due to its poor permeability, intestinal metabolism, pre-systemic hepatic extraction, removal by effux proteins, degradation in the GI tract environs, digestion by proteolytic enzymes and microbiota enzymatic activity, etc. Among these factors, the pre-systemic metabolism is considered a major cause in low bioavailability of a compound. The intestinal metabolism and hepatic frst-pass metabolism are the main frst-pass extractions for an orally administered compound and are often diffcult to differentiate (Paine et al., 1996; Lee et al., 2001). The intestinal metabolism is discussed in the next section. The hepatic extraction occurs when xenobiotics and nutrients are absorbed into the submucosal capillaries, transfer into the intestinal vein, which joins with the veins from the spleen and pancreas, and fnally enter the hepatoportal vein moving toward the liver. In the liver, the nutrients, xenobiotics, small particles, 81

3.3 GASTROINTESTINAL (ORAL) ROUTE OF ADMINISTRATION OR EXPOSURE

and droplets from the hepatoportal vein are absorbed and hepatocytes metabolize free nutrients and xenobiotics before reaching the systemic circulation. The metabolic enzyme systems and pathways of the liver are discussed in Chapter 9. The quantity of xenobiotics eliminated by the hepatic frst-pass metabolism is usually expressed in terms of fraction or percentage removal, also known as extraction ratio, which is the ratio of the rate of metabolism to the rate of input into the liver: Rateof Input - Rateof Output = Rateof Extraction

(3.88)

Rateof Extraction = Extraction Ratio Rateof Input

(3.89)

(Qliver ´ Cliver )input - (Qliver ´ Cliver )output = Qliver ( ( Cliver )innput - ( Cliver )output )

(3.90)

(

Qliver ( Cliver )input - ( Cliver )output Qliver ( Cliver )input

) = (C

) - ( Cliver )output ( Cliver )input

liiver input

(3.91)

= ER

Where Q is the blood fow, C is concentration, and ER is the extraction ratio. The quantity that will be available for the systemic effect can therefore be estimated in terms of the fraction that escapes the liver frst-pass metabolism. That is, Favailable = 1 - ER

(3.92)

Under linear steady state condition, the rate of extraction by the liver is Rateof extraction = ( Hepatic blood clearance ) ´ ( Blood concentration n of thexenobiotic )

(3.93)

)

(

\Qliver ( Cliver )input - ( Cliver )output = ClH ´ Cblood ER =

(3.94)

ClH ´ Cblood Cl = H Qliver ´ Cblood Qliver

\Favailable = 1 -

(3.94)

ClH Qliver

(3.95)

Where ClH is hepatic blood clearance. From Equation 3.95, it can be concluded that the bioavailability of a compound after the hepatic frst-pass metabolism depends on the hepatic blood clearance. A large ClH is indicative of a smaller fraction that escapes frst-pass hepatic metabolism for distribution in the body and systemic effect. To expand on the defnition of hepatic blood clearance, one may apply the principles of mass balance to the amount of drug associated with different components of blood, namely, plasma and blood cells: Total Amount in Blood = Amount in Plasma + Amount Associated with Blood d Cells

(3.96)

\ (Vblood ´ Cblood ) = (Vplasma ´ Cp ) + (Vbc ´ Cbc )

(3.97)

where Vplasma and Vbc are the volume of plasma and blood cells, respectively, Cp and Cbc are the related concentrations. Dividing both sides of Equation 3.97 by Vblood and Cp yields Vp Cblood V ´ Cbc = + bc Vblood Vblood ´ C p Cp

(3.98)

The volume of blood cells divided by the total volume of blood is equal to the hematocrit (Hct) or the percentage of blood cells. Thus, the volume of plasma divided by the total volume of blood is equal to 1 – hematocrit (1 - Hct ) , that is, C Cblood = (1 - Hct ) + Hct ´ bc Cp Cp

82

(3.99)

PK-TK CONSIDERATIONS OF NASAL, PULMONARY AND ORAL ROUTES

Since the rate of elimination from the blood depends solely on the elimination from plasma, that is, (3.100)

Clblood ´ Cblood = Clp ´ Cp The total blood clearance would be equal to plasma clearance divided by Equation 3.99: Clblood = Clp ´

Clp Cp = Cblood æ Cblood çç è Cp

ö ÷÷ ø

=

Clp 1 Hct + C ) ( ) Hct ( Cbc Cp

(3.101)

Hepatic blood clearance is then equal to the fraction of the dose eliminated as metabolites ( f m ) multiplied by the blood clearance (Clblood ) ClH = f m ´ Clblood

(3.102)

Substituting Equation 3.101 into Equation 3.102 yields ClH =

f m ´ Clp Cp )

(3.103)

( Cblood

Replacing ClH in Equation 3.91 with Equation 3.103 gives the bioavailability equation for a compound after the hepatic frst-pass metabolism Favailable = 1 -

f m ´ Clp Qliver ´ ( Cblood C p )

(3.104)

The fraction of the dose eliminated as metabolite, f m , can also be estimated from the fraction of the dose excreted unchanged in urine, f e , that is, fm = 1 - fe

(3.105)

The above equations assume the elimination processes are linear and excretion and metabolism follow frst-order kinetics. Thus, for selected dose levels that follow linear PK/TK, in a specifed population, the fraction of dose metabolized and/or excreted unchanged is constant. However, when the metabolism and/or excretion are capacity-limited and nonlinear, the extraction ratio and other metabolic constants will be dose-dependent variables. In nonlinear PK/TK, the fraction available for systemic effect (i.e., bioavailability) increases signifcantly with increasing the dose. Under the nonlinearity conditions, hepatic frst-pass metabolism follows Michaelis–Menten equation kinetics. To evaluate the frst-pass metabolism, within the framework of non-compartmental analysis, the following relationships have been proposed (Ueda et al., 2002): d ( At ) portal dt

(

= (Qliver ) portal ( Ct ) portal - ( Ct )systemic

)

(3.106)

d ( At ) portal where is the rate of absorption from the intestinal vein into the portal vein, ( Ct ) portal dt and ( Ct )systemic are concentrations of xenobiotic in the portal and systemic vein, respectively, and (Qliver )portal is the portal vein blood fow rate. The local moment curve is then defned as æ d ( At ) portal ç ç dt è D

ö ÷dt ÷ æ ( AUC ) portatl - ( AUC )systemic ö 0 ø = Q (3.107) ÷ Fportal = ( liver )portal çç ÷ D è ø Where Fportal is the organ absorption ratio from the intestinal vein into the portal system. The mean absorption time from the GI tract into the portal system is

ò

¥

¥

tabsorption =

d ( At ) portal

ò t dt d(A ) ò dt 0

¥

t portal

æ ( MRT ) ö æ ( MRT ) portal systemic ç ÷-ç ç ( AU UC ) portal ÷ ç ( AUC )systemic ø è =è ( AUC )portal - ( AUC )systemic

ö ÷ ÷ ø

(3.108)

0

83

3.3 GASTROINTESTINAL (ORAL) ROUTE OF ADMINISTRATION OR EXPOSURE

Under the assumption that the hepatic pre-systemic extraction is linear, and the total body clearance is a constant, the hepatic recovery ratio, that is, the escaped fraction from the frst-pass metabolism, and the related mean absorption time MAT are æ AUCoral ö ç ÷ AUCintraarterial ø F FH = è = Fportal Fportal

(3.109)

MAT = MRToral - MRTintraarterial

(3.110)

Where MRToral andMRTintraarterial are the mean residence times of the compound given orally and intra-arterially, respectively. Under nonlinear conditions and a capacity-limited metabolism, the disposition rate is defned as dC V ´ Cout = Qliver ( Cin - Cout ) - max K M + Cout dt

Vd

(3.111)

The volume of distribution is Vd, the hepatic vein blood fow rate is Qliver , and the concentration of the compound in and out of the liver is Cin and Cout , respectively. The Michaelis–Menten parameters are the maximum rate of metabolism, Vmax , and Michaelis–Menten constant, K M . Setting Equation 3.111 equal to zero, and solving for the ratio of concentrations in and out of the liver, yields

FH =

Cout = Cin

æ Vmax ç Cin - K M Qliver è

2

ö æ Vmax ö ÷ + ç Cin - K M ÷ + 4Cin ´ K M Qliver ø è ø 2Cin

(3.112)

C where out is the hepatic recovery ratio, FH . Thus, the rate of amount reaching the systemic circulation is Cin d ( At )systemic dt

( Ct )out = (Qliver ) portal éê( Ct ) portal - ( Ct )systemic ùú û ( Ct ) ë

(3.113)

in

The integrated value is the total amount available in systemic circulation (Ueda et al., 2002), that is, A = FD =

ò

¥

0

(Qliver )portal éêë( Ct )portal - ( Ct )systemic ùúû

( Ct )out dt ( Ct )in

(3.114)

The mean absorption time is then estimated as ¥

MAT =

ò ò

0

t

¥

0

d ( At )systemic dt d ( At )systemic

(3.115)

dt

3.3.2.7 Gastrointestinal Metabolism – Role of CYP450 Isozymes The esophagus transfer of food to the stomach is where the main secretory and digestive processes begin. The metabolic enzymes of the stomach are few and include 1) alcohol dehydrogenase, which is involved in the frst stage of alcohol metabolism; 2) pepsinogens that is the zymogen, or inactive precursor of pepsin, the principal proteolytic enzyme of the stomach that can hydrolyze almost all peptide bonds and in particular peptide bonds involving leucine, tyrosine, and phenylalanine. Human gastric mucosa produces four types of pepsinogens known as pepsinogen I (PGA or PGI), pepsinogen II (PGC or PGII), cathepsin E, and cathepsin D. The therapeutic peptides and proteins encounter these enzymes when administered orally. Pepsin is stable with optimum activity only in low acidic pH 1.5–3.0 and denatures at neutral or alkaline pH; and 3) gastric lipase that is involved in the partial digestion of free fatty acids. The small intestine, in addition to being the major site of absorption of orally administered xenobiotics, has the capability of metabolizing many compounds administered orally (Paine et al., 2006). Many orally administered therapeutic agents are substrates for cytochrome P450 oxidative enzymes (CYPs). An important member of the CYP450 family in the small intestine is CYP3A4, a low specifcity isozyme that contributes to biotransformation of a signifcant number 84

PK-TK CONSIDERATIONS OF NASAL, PULMONARY AND ORAL ROUTES

of xenobiotics administered or exposed orally. Similar to the hepatic frst-pass metabolism, the intestinal pre-systemic metabolism infuences the PK/TK analysis (Mizuma, 2002). A large body of research both in vitro, in vivo, and in silico has established the intestinal metabolism of xenobiotics by CYP3A4 as a major determinant of the systemic bioavailability (Paine et al., 1996; Zhang et al., 1999; Kaminsky and Zhang, 2003; Melillo et al., 2019). The amount of CYP3A4 in the epithelial barrier of the small intestine is about 60–70% of all CYP450 isozymes in the small intestine, and inhibition or induction of this isozyme leads to variation in bioavailability, unintended drug–drug interaction, and other interactions, like herb–drug or food–drug interactions (Shang et al., 2018; Gupta et al., 2017). For example, the presence of inhibitors like ketoconazole, erythromycin, itoconazole, and grapefruit juice (Veronese et al., 2013) or inducers, like rifampicin or selexipag (Juif et al., 2017), can increase or decrease the bioavailability of orally administered CYP3A4 substrates. While CYP3A4 is the most abundant isozyme present in the duodenum for oxidative metabolism of xenobiotics, other members of CYP450 subfamilies are also present, but to a lesser extent (Shimada et al., 1994; Zhang et al., 1999). The presence of other members of the CYP3A subfamily, such as CYP3A5 (Lin et al., 2002) and CYP3A7, has been advocated and is yet to be confrmed in adults (Kivistö et al., 1996). Another subfamily of CYP450 present in the small intestine is CYP2C, which is reported as the second most abundant CYP450 subfamily (Klose et al., 1999; Obach et al., 2001; Lapple et al., 2003). Two members of this subfamily, CYP2C9 and CYP2C19, have been expressed in the small intestine of humans. Other CYP450s reportedly expressed in the human small intestine are: CYP2D6 (de Waziers et al., 1990; Madani et al., 1999), CYP1A1 (Prueksaritanont et al., 1996; Paine et al., 1999), CYP2S1, CYP4F12, and CYP2J2 (Matsumoto et al., 2002). Enzymes of conjugation, namely, glutathione S-transferase, UDP-glucuronosyl transferase and N-acetyltransferase, are also present in the small intestine (Gibbs et al., 1998; Fisher et al., 2000, 2001; Coles et al., 2002). There are polymorphisms in the coding and noncoding regions of the genes for expression of metabolic enzymes in humans, which explains the interindividual variability and inconsistencies for the presence or absence of some enzymes in the small intestine, for example, CYP3A5, CYP3A7, and CYP2D6 (Byeon et al., 2018). In addition to CYP450 isozymes, other enzyme systems such as pepsins, trypsin, chymotrypsin, elastase, carboxypeptidase A/B, endopeptidases, γ-glutamyl transpeptidase, and aminopeptidases are present in small intestine that in addition to their physiological roles of dealing with nutrients, metabolize structurally similar xenobiotics. An example is the metabolism of therapeutic peptides, polypeptides and proteins. In addition, Proteases are also present to the activity and availability of growth factors and cytokines and modulate the intestinal permeability (Spaendonk et al., 2017), which enhances the absorption of xenobiotics. When an orally administered xenobiotic undergoes hepatic and intestinal frst-pass metabolism, and its PK/TK is considered linear without enterohepatic recirculation, its systemic availability may be estimated by the following relationship: F˜ Where ˜Qliver °

arterial

°Qliver ˛arterial ˝ °Qliver ˛portal AUCoral ˜ °°Qliver ˛arterial ˙ °Clint ˛H ˛ °°Qliver ˛portal ˙ °Clint ˛GI ˛ AUCiv

(3.116)

and ˜Qliver ° portal are hepatic arterial and portal vein blood fows, respectively;

˜ Clint °H and ˜ Clint °GI are intrinsic clearances of hepatic and GI tract, respectively; AUCoral and

AUCiv are the area under the plasma concentration-time curve of the compound given orally and intravenously. Without the pre-systemic metabolism, Equation 3.116 is simplifed to Equation 3.95. In the presence of hepatic and intestinal pre-systemic metabolism, without taking into consideration the enterohepatic recirculation, the extent of absorption, that is, the area under the plasma concentration curve can be estimated by the following relationships: AUCoral ˜

Dose ° ˛Qliver ˝arterial ° ˛Qliver ˝ portal

˛

QliverHV ˆ ˛ Clint ˝H ˙ ˛ Clint ˝B ˇ˘

˝ ˛ ˛Q

˝

liver portal

˝

˙ ˛ Clint ˝GI ˙ ˛Qliver ˝ portal ° ˛ Clint ˝GI  

(3.117)

Where ˜ Clint ° is the intrinsic biliary clearance. B

° °

˛°

˛

Dose ˙ °Qliver ˛arterial ˝ ° Clint ˛H ˝ ° Clint ˛B °Qliver ˛ portala ˝ ° Clint ˛GI ˘ ˇˆ  AUCiv ˜ ˙ °Qliver ˛arterial ˇˆ ° Clint ˛H ˝ ° Clint ˛B °Qliiver ˛portal ˝ ° Clint ˛GI ˝ °Qliver ˛portal ° Clint ˛GI ˘

˛°

˛

(3.118) 85

3.3 GASTROINTESTINAL (ORAL) ROUTE OF ADMINISTRATION OR EXPOSURE

The AUC of oral and intravenous dose in the presence of combined hepatic metabolism, intestinal pre-systemic metabolism, and enterohepatic recirculation can be estimated as AUCoral =

AUCiv =

(

Dose ´ (Qliver )arterial ´ (Qliver ) portal

)

) ( ) ) + ( Cl ) ( Cl ) ûúù ) ( (Q ) + ( Cl ) )ùúû

(Qliver )arteriala éëê( Clint )H (Qliver )portal + ( Clint )GI + ( Clint )GI (Qliver )portatl + ( Clint )B ùûú

(

)(

Dose é (Qliver )arterial + ( Clint )H (Qliver ) portal + ( Clint GI ëê (Qliver )arterial éêë( Clint )H (Qliveer )portal + ( Clint )GI + ( Clint GI

(

)

int B

liver portal

int GI

(3.119)

(3.120)

int B

The following equation defnes the intestinal clearance when only the intestinal metabolism is present: Clintestinal =

QG ´ f u ´ ( Clint )GI

QG + f u ´ ( Clint )GI

(3.121)

where Clintestinal and QG are the intestinal clearance and blood fow to the intestine, respectively; f u is the free fraction of compound in plasma, and ( Clint ) is the intrinsic intestinal clearance. GI The fraction that escapes the intestinal metabolism is \F =

QG QG + f u ´ ( Clint )GI

(3.122)

3.3.2.8 GI Tract Infux and Effux Transport Proteins Intestinal enterocytes, in addition to forming a rigid epithelial monolayer preventing the paracellular diffusion of xenobiotics and expressing CYP450 subfamilies and other drug metabolizing enzymes, express membrane-associated effux/infux transport proteins. The effux proteins collectively establish a remarkable biochemical barrier to xenobiotic absorption through the GI tract wall (Bruyere et al., 2010). The ATP binding cassette (ABC) transporters, that is, effux proteins, such as permeability glycoprotein multidrug resistance proteins (Pgp/MDR1), multidrug resistance-associated proteins (MRPs), breast cancer resistance proteins (BCRP), in conjunction with the inhibition or induction of infux proteins can facilitate or limit the absorption (Dahan and Amidon, 2009a,b). The infux transport proteins located on the apical side of the membrane or brush border side of enterocytes are apical sodium dependent bile acid transporter (ASBT), concentrative nucleoside transporter (CNT1/2), monocarboxylate transporter (MCT1), peptide transporter (PEP1/2), organic anion transporting polypeptides (OATP1A2 and OATP2B1) and organic cation transporters (OCT3 and OCTN1/2). The effux proteins of the apical side are BCRP, MDR1, MRP2, and MRP4. These infux and effux proteins are in direct contact with the content of the small intestine that contains xenobiotics. On the basolateral side, the infux proteins are OATP3A1, OATP4A1, and OCT1/2, and effux proteins are MRP1, MRP3, MRP4, MRP5, MCT1, and equilibrative nucleoside transporter (ENT1/2), which are in contact with blood in capillaries. The apical and basolateral proteins can be induced, inhibited, or modulated in the presence of xenobiotics (Suzuki and Sugiytama, 2000; Chan et al., 2004; Kis et al., 2010). Among these transporters, permeability glycoprotein (Pgp/MDR1) has been studied in detail for its infuence on the bioavailability of single dose, multiple dose, and concomitantly administered xenobiotics and its signifcant role in chemoresistance and functional interaction with the intestinal CYP3A subfamily (Lown et al., 1997a,b; Ueda et al., 1999; Dietrich et al., 2003; Shugarts and Benet, 2009; Dahan et al., 2009, Otsuka et al., 2020). The high concentration of Pgp and CYP3A isozymes are present in the villus tip of enterocytes and share signifcant similarities in substrate specifcity. It has been well documented that extrusion of xenobiotics by intestinal Pgp allows CYP3A isozymes to have prolonged access to their substrates. Thus, Pgp reduces absorption of orally administered xenobiotics and can also modulate the infuence of inhibitors and inducers of CYP3A isozymes-mediated metabolism. Contrary to CYP3A4, which is present in a limited region of the small intestine, the expression of Pgp is present along the length of intestine. The function of Pgp and CYP3A4 are mutually exclusive, yet synergistic, and their function as intestinal barriers are of equal value and complementary

86

PK-TK CONSIDERATIONS OF NASAL, PULMONARY AND ORAL ROUTES

Figure 3.11 Diagram of activities of an enterocyte that include the function of major effux and infux transport proteins both at the apical and basolateral sides of the cell, and their involvement in exchange of xenobiotics between the small intestine and capillaries; also, the presence of Phase I and Phase II enzymes representing the internal metabolic activities of enterocytes. (Lown et al., 1997a, b; Ito et al., 1999; Perloff et al., 2000; Holmstock et al., 2012). Figure 3.11 depicts the localization of transport proteins and enzymes of enterocytes (see also Figure 3.10). In addition to functioning synergistically, Pgp and CYP3A4 are coregulated via an orphan nuclear receptor known as pregnane xenobiotic receptor (PXR) that regulates the metabolism of endobiotics and xenobiotics. The agonists of these orphan nuclear receptors often have potential to induce Pgp and CYP3A4 simultaneously (Elmeliegy et al., 2020). For example, compounds like rifampicin can induce both Pgp and CYP3A4. Because of this congruence between Pgp and CYP3A4, the US Food and Drug Administration recommends that data on CYP3A4 induction can be used to question whether a clinical investigation into Pgp induction is necessary (USFDA, 2020). In addition to xenobiotics, food and excipients of orally administered dosage forms may also interact with Pgp and infuence the bioavailability of orally administered drugs. The inhibitory effect of grapefruit and other citrus fruit juices on CYP3A4 and Pgp, which has been attributed to its favonoids and furanocoumarins, and the inhibitory effects of excipients, such as cremophor EL and cremophore RH 40, are a few examples (Lown et al., 1997 a, b; Wang et al., 2001; Xu et al., 2003; Dahan et al., 2009).

87

3.3 GASTROINTESTINAL (ORAL) ROUTE OF ADMINISTRATION OR EXPOSURE

MDR1 and other highly expressed effux transporters, for example, BCRP, MRP2, and MRP4, are present in the human jejunum and colon and extrude xenobiotics back into the lumen. The infux proteins facilitate the transport of xenobiotics into the enterocytes and systemic circulation. Both effux and infux transporters require ATP and function as active transport proteins. In the evaluation of the bioavailability of xenobiotics that are substrate for these proteins, the function of both groups of transporters should be taken into consideration. The kinetics of intestinal absorption of xenobiotics, taking into consideration the presence of effux and infux proteins and CYP3A4 are developed based on the model depicted in Figure 3.12 (Ito et al., 1999). According to the model: The concentration of xenobiotics in the lumen, Clum , and in epithelial cells, Cepi , are described by the following partial differential equations, which jointly represent the mass balance equations of the model (Ito et al., 1999): ˜Clum˝ x ,t ˙ ˜t

°˛

Vepi Qlum ˜Clum( x ,t ) Clinflux ˛ Clum˝ x ,t ˙ ˆ Clefflux ˇ ˇ Cepi˝ x ,0 ,t ˙ Arealum ˜x Vlum Vlum ˜Cepi˙ x , y ,t ˆ ˜t

where

° Dcoeff

˜ 2Cepi ˙ x , y ,t ˆ ˜y 2

˛ Clm ˝ Cepi ˙ x , y ,t ˆ

(3.123) (3.124)

˜Clum° x ,t ˛

is the partial change of the xenobiotic concentration in the gut lumen with respect ˜t to time, x represents the distance in the direction of luminal fow; Qlum is the luminal fow rate and Arealum is the sectional area of lumen; Clinflux is the apparent infux clearance from the lumen into the epithelial cells, Vlum is the volume of lumen and Clefflux is the apparent effux clearance from cells to the lumen; Vepi is the volume of epithelial cells, Cepi( x ,0,t )is the xenobiotic concentra∂Cepi( x ,y ,t ) tion in epithelial cells, and Dcoeff is the diffusion coeffcient in epithelial cells; is the ∂t partial change of the xenobiotic concentration in the epithelial cells with respect to time, distance ∂ 2Cepi( x ,y ,t ) of luminal fow, and the absorption direction; is the second partial derivative of the ∂y 2 concentration of the xenobiotic in the epithelial cells with respect to the distance in the direction of

Figure 3.12 Diagram of the model defning the kinetics of intestinal absorption of xenobiotics in the presence of effux and infux proteins and CYP450 isozymes from the small intestinal environment to enterocyte and absorption via the capillaries into the systemic circulation. 88

PK-TK CONSIDERATIONS OF NASAL, PULMONARY AND ORAL ROUTES

drug absorption, and Clm is the metabolic clearance in the epithelial cells; the xenobiotic concentration, Cepi( x ,y ,t ) , is considered with respect to the same three independent variables of the distance of luminal fow, x, the distance in the direction of drug absorption, y, and time, t. Integrating Equations 3.123 and 3.124 by the method of Laplace transform and setting the initial conditions at t = 0, i.e., Clum( x ,0) = 0 Cepi( x ,y ,0 ) = 0 the boundary conditions at the apical and basal membranes of epithelial cells are: Clinflux ´ Clum( x ,t ) = Cefflux ´Vepi ´ Cepi( x ,0,t ) - Dcoeff ´ Arealum Clabs ´Vepi ´ Cepi( x ,lepi ,t ) = -Dcoeff ´ Arealum ´

¶ epi( x ,0 ,t ) ¶C ¶y

¶Cepi( x ,lepi ,t )

(3.125) (3.126)

¶yy

Where Clabs is the absorption clearance from cells to the blood, and lepi is the length of each epithelial cell in the direction of drug absorption. The integrated equations under the assumption of steady-state drug concentration provide the fraction of a drug available in the lumen and the fraction absorbed as follows: ìï f Dcoeff ´ Clm (1 + ˜ ) ïü Flum = exp í ý ïî Clefflux ´ lepi ´ (1 - ˜ ) - Dcoeff ´ Clm (1 + ˜ ) ïþ

(3.127)

Where Flum is the fraction of drug in the lumen, f and ˜ are defned as: f=

Clinflux Qlum

(3.128)

æ Clm ö ÷ exp ç 2lepi ç Dcoeff lepi ´ Clabs - Dcoeff ´ Clm e ÷ è ø The fraction absorbed into the systemic circulation at a steady state, Favailable , is ˜=

lepi ´ Clabs + Dcoeff ´ Clm

(3.129)

æ ö Rate abs Favailable = (1 - Flum ) ´ ç ÷ è Rate abs + Rate metab ø Favailable =

(3.130)

2lepi ´ Clabs (1 - Flum ) æ Clm ïì í lepi ´ Clabs - Dcoeff ´ Clm expçç -lllum D coefrf ïî è

(

)

ö æ Clm ÷ + lepi ´ Clabs + Dcoeff ´ Clm expç lm ÷ ç D e coeff ø è

(

)

ö üï ÷ ÷ ýï øþ

(3.131) When (Dcoeff ˜ Clm ), and under the assumption that epithelial cells act as a well-stirred compartment, Equation 3.131 changes to æ Clabs Favailable = ç è Clabs + Clm

æ ö ìï Clabs + Clm ÷ ´ í1 - exp çç -f ´ Cl + Clabs + Clm efflux ø ïî è

ö üï ÷÷ ý øïþ

(3.132)

3.3.2.9 Role of Intestinal Microbiotas The intestinal microbiotas are prokaryotic (90%) and eukaryotic (10%) microorganisms of different kinds that become inhabitants of the intestinal environment soon after birth. They colonize not only in the large intestine but also in the distal portion of the ileum. All mammals, including humans, have a symbiotic life with microbial cells, which are implicated in immunology and nutrition of the host (MacFarlane and MacFarlan, 1997). Using ribosomal RNA gene sequencing methodology, eight prevailing kinds (phyla) have been identifed that may have a common evolutionary ancestor; they are Firmicutes, Bacteroidetes, Proteobacteria, Fusobacteria, Verrucomicrobia, Cyanobacteria, Spirochaeates, and Actinobacteria. The intestinal microbiotas are inhabited in the distal portion of ileum cecum, colon, rectum, and anal canal and interact 89

3.3 GASTROINTESTINAL (ORAL) ROUTE OF ADMINISTRATION OR EXPOSURE

extensively with the host through metabolic exchanges and the metabolism of mostly orally administered xenobiotics and their metabolites (Wang et al., 2005; Eckburg et al., 2005; Egert et al., 2006). The microbial content in the human GI tract includes lactobacilli, streptococci, and yeasts in the stomach and duodenum; lactobacilli, streptococci, enterobacteriacaea, bifdobacteria, bacteriodes, and fusobacteria in jejunum and ileum; and lactobacilli, streptococci, enterobacteriacaea, bifdobacteria, bacteriodes, viellonella, yeasts, proteus, staphylococci, pseudomonas, and fusobacteria in the colon (Holzapfel et al., 1998; Iannitti and Palmieri, 2010; Sasaki et al., 2021). The intestinal microbiotas are essential as a contributing factor to the well-being of humans, playing roles in metabolic, nutritional, physiological, and immunological processes in the body, and they are considered critical factors that regulate GI physiology and pathophysiology in a gender-independent manner (Ni et al., 2017; Chang and Lin, 2016; Vicentini et al., 2021). Over the past decade, the intestinal microbiotas have emerged as the focus of causes and treatments of several disease states. Among them are GI diseases related to blood fow, intestinal motility, mucosal immunity, and intestinal permeability (Clark et al., 2012). Also, a signifcant body of research portrays a bidirectional communication between the gut microbiota and the central nervous system identifed as the gut-brain axis (Dinan and Cryan, 2015, 2017; Rieder et al., 2017; Vicentini et al., 2021). It is proposed to affect cognitive functioning and mood through hormonal and metabolic pathways (Foster and Mcvey Neufeld, 2013). The premise of gut-brain axis is based on the production of certain neurotransmitters like serotonin, tryptophan, and GABA by microbiotas, which are involved in neuropeptide and GI hormone release (Dinan and Cryan 2015, 2017; Foster et al., 2017). Microbiotas, with their enormous metabolic capacity, can infuence the disposition and toxicity of xenobiotics (Sousaa et al., 2008). The variability in population of microbiotas, the inconsistency in their quantitative and qualitative metabolic activity, and their infuence on absorption and bioavailability of xenobiotics can be linked to many cases of inter-individual PK/TK differences. The microbiota carries out biochemical reactions that are mostly hydrolysis and, since the GI tract is practically an oxygen-free environment, no oxidation or reduction is expected. They are also capable of breaking down the conjugated metabolites, such as glucuronides, sulfates, glutathione, and other forms of conjugates. Other infuences are reducing the intestinal transit time, hydrolysis of macromolecules, and dietary polymers in the large intestine. In humans, the stomach, duodenum, jejunum, and proximal portion of ileum are somewhat free from microbiota. In experimental animals such as rats, mice, guinea pigs, and rabbits, however, because of their eating habits and types of food, their stomachs and small intestines are heavily colonized with different types of bacteria. Thus, caution should be exercised in quantitative extrapolation of microbiota experimental data from animal to human. For PK/TK studies, the plasma concentration–time profle of a compound is compared between a pseudo-germ-free animal model treated with antibiotics with that of nonantibiotic-exposed animals administered with the same compound. An approach in estimating the extent of bacterial metabolism in humans is by comparing the metabolic profle of a compound given orally with those estimated after the intravenous administration. A shortcoming of this approach is overlooking the biliary elimination of an intravenously administered compound and/ or its metabolites, which may also undergo secondary or tertiary metabolism by bacteria. Our current understanding of their enzyme systems, metabolism, and metabolomics; their effux and infux protein transporters and infuence on pharmacogenetics and pharmacogenomics; and their overall infuence on the ADME of xenobiotics and population variances need further investigation and elucidation. REFERENCES Agu, R. U., Ombimah, D. U., Lyzenga, W., Jorissen, M., Massoud, E., Verbeke, N. 2009. Specifc aminopeptidases of excised human nasal epithelium and primary culture: A comparison of functional characterictics and gene transcript expression. J Pharm Pharmacol 61(5): 599–606. Ahookhosh, K., Pourmehran, O., Aminfar, H., Mohammadpourfard, M., Sarafraz, M. M., Hamishehkar, H. 2020. Development of human respiratory airway models: A review. Eur J Pharm Sci 145: 105233. https://doi.org/10.1016/j.ejps.2020.105233. Ali, M. S., Pearson, J. P. 2007. Upper airway mucin gene expression: A review. Laryngoscope 117(5): 932–8. 90

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Whitesett, J. A. 2018. Airway epithelial differentiation and mucociliary clearance. Ann Am Thorac Soc 15(Suppl 3): S143–48. Wiedmann, T. S., Bhatia, R., Wattenberg, L. W. 2000. Drug solubilization in lung surfactant. J Control Release 65(1–2): 43–7. Wiesmiller, K., Keck, T., Leiacker, R., Sikora, T., Rettinger, G., Lindemann, J. 2003. The impact of expiration on particle deposition within the nasal cavity. Clin Otolaryngol Allied Sci 28(4): 304–7. Willems, B. A. T., Melnick, R. L., Kohn, M. C., Portier, C. J. 2001. A physiologically based pharmacokinetic model for inhalation and intravenous administration of naphthalene in rats and mice. Toxicol Appl Pharmacol 176(2): 81–91. Wilson, C. G. 2000. Gastrointestinal transit and drug absorption. In Oral Drug Absorption, Prediction and Assessment, eds. J. B. Dressman, H. Lennernäs, 1–10. New York: Dekker. Wilson, W. R., Allansmith, M. R. 1976. Rapid, atraumatic method for obtaining nasal mucus samples. Ann Otol Rhinol Laryngol 85: 391–3. Xu, J., Go, M. L., Lim, L.-Y. 2003. Modulation of digoxin transport across caco-2 cell monolayers by citrus fruit juices: Lime, lemon, grapefruit, and pummelo. Pharm Res 20(2): 169–76. Yeh, H. C., Schum, G. M., Duggan, M. T. 1979. Anatomic models of the tracheobronchial and pulmonary regions of the rat. Anat Rec 195(3): 483–92. https://doi.org/10.1002/ar.109195050308. Yoo, B. B., Mazmanian, S. K. 2017. The enteric network interactions between the immune and nervous systems of the gut. Immunity 46(6): 910–26. Young, L. C., Campling, B. G., Cole, S. P., Deeley, R. G., Gerlach, J. H. 2001. Multidrug resistance proteins MRP3, MRP1 and MRP2 in lung cancer: Correlation of protein levels with drug response and messenger RNA levels. Clin Cancer Res 7(6): 1798–804. Yu, S., Liu, Y., Sun, X., Li, S. 2008. Infuence of nasal structure on the distribution of airfow in nasal cavity. Rhinology 46: 137–43. Yue, Z., Li, C., Voth, G. A., Swanson, J. M. J. 2019. Dynamic protonation dramatically affect the membrane permeability of drug-like molecules. J Am Chem Soc 141(34): 13421–33. Zhang, Q. Y., Dunbar, D., Ostrowska, A., Zeisloft, S., Yang, J., Kaminsky, L. S. 1999. Characterization of human small intestinal cytochromes P-450. Drug Metab Dispos 27(7): 804–9. Zhang, J. Y., Wang, Y. F., Prakash, C. 2006. Xenobiotic metabolizing enzymes in human lung. Curr Drug Metab 7(8): 939–48. Zhang, X., Sebastiani, G., Liu, G., Schembri, F., Zhang, X., Dumas, Y. M., Langer, E. M., Alekseyev, Y., O’Conner, G. T., Brooks, D. R., Lenburg, M. E., Spira, A. 2010. Similarities and differences between smoking-related gene expression in nasal and bronchial epithelium. Physiol Genomics 41(1): 1–8. Zheng, Z., Fang, J. L., Lazarus, P. 2002. Glucuronidation: An important mechanism for detoxifcation of benzo(a)pyrene metabolites in aerodigestive tract tissue. Drug Metab Dispos 30(4): 397–403. Zimmermann, T., Yeates, R. A., Laufen, H., Pfaff, G., Wildfeuer, A. 1994. Infuence of concomitant food intake on the oral absorption of two triazole antifungal agents, itraconazole and fuconazole. Eur J Clin Pharmacol 46(2): 147–50.

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4 PK/TK Considerations of Intra-Arterial, Intramuscular, Intraperitoneal, Intravenous, and Subcutaneous Routes of Administration 4.1 INTRA-ARTERIAL ROUTE OF ADMINISTRATION 4.1.1 Overview Intra-arterial route of administration (I.A., or I-Arter) is used for specialized injection of selected therapeutic agents in certain disease states, like ischemic stroke and solid tumor reduction, and is not considered a common route for the administration of medications. Accidental or intentional injection of therapeutic agents is toxic, and the side effects start with instantaneous discomfort, intense pain, and irritation, followed by tissue necrosis, gangrene, and chronic pain. In certain cases, the extensive injury and toxicity is often associated with mortality. However, under the supervision of specialists, the intra-arterial route of administration has been used effectively for targeted treatment of certain illnesses, e.g., injection of neuroprotective agents in acute ischemic stroke to treat vessel occlusion (Maniskas et al., 2021), management of certain types of stroke by injection of therapeutic agents in nanocarriers to the brain (Grayston et al., 2021), advanced retinoblastoma (Abramson et al., 2016), and treatment of brain tumors (Rechberger et al., 2021). It is worth noting that in certain cases the conventional systemic chemotherapy achieved by the maximum allowable oral or intravenous dose of a chemotherapeutic agent may only provide sub-optimum concentration of the agent at the target site. For such cases, careful application of the intra-arterial route of administration can provide greater effcacy and lower systemic toxicity. 4.1.2 Intra-Arterial PK/TK Remarks The PK/TK considerations of the intra-arterial route of administration depend on the objectives of an investigation. The intra-arterial route may not be the best route for determining or verifying the systemic PK/TK parameters and constants of a xenobiotic. Its utilization to achieve a targeted exposure can only provide meaningful pharmacodynamic data. Also, the PK/TK parameters and constants estimated after an injection into the feeding artery of a target site would be meaningful only for optimizing the intra-arterial dose and its local effects. The comparison of these parameters and constants with those of an intravenously administered dose, considering the pattern of distribution, would be open to doubt. Dose-normalized concentration values would be meaningful if the uptake by the target organ is known. To restate, the main advantage of intra-arterial injection is to target a smaller dose than the systemic intravenous or oral dose for achieving optimum concentration at the target site/organ, thus avoiding systemic toxicity. Examples of using this advantage is hepatic arterial infusion of gemcitabine in unresectable pancreatic cancer patients (Shamseddine et al., 2005), or intra-arterial infusion of foxuridine with systemic gemcitabine and oxaliplatin in patients with unresectable intrahepatic cholangiocarcinoma (Cercek et al., 2020), or intra-arterial and intravenous injection of carboplatin for retinoblastoma (Daniels et al., 2021). After reaching the systemic circulation, the intra-arterially administered drugs follow the same processes of distribution and elimination as an intravenously administered dose. 4.2 INTRAMUSCULAR ROUTE OF ADMINISTRATION 4.2.1 Overview The intramuscular routes of drug administration (IM) are commonly used for injection of vaccines and medications like immunosuppressants, sedatives, hormonal therapy, and long-acting antipsychotics. The primary sites for IM injections are deltoid, triceps, gluteus maximus (dorsogluteal and ventrogluteal sites), rectus femoris, and vastus lateralis (thigh) muscles. All muscles have a higher blood perfusion, and the absorption from an intramuscular site of administration is faster than the subcutaneous region. Small molecules of xenobiotics absorb directly into the capillaries, whereas large molecules access the systemic circulation indirectly through the lymphatic capillaries. The deltoid muscle has a greater blood perfusion rate compared to the other muscle injection sites, but depending on the weight of the individual, it cannot painlessly hold, more than 1–2 mL (Covington and Tuttler, 1997) versus 4 ml in the dorsogluteal (Rodger and King, 2000), 2.5 ml in the ventrogluteal, 5 ml in the rectus femoris (Workman, 1999), and 1–5 ml in the vastus lateralis (Rodger and King, 2000). The recommended injection needles for human IM injection, depending on the type of solutions, are 21G (38 mm)–23G (32 mm) for oil-based (21G) and water-based (23G).

DOI: 10.1201/9781003260660-4

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Muscle blood fow at rest is about 3–4 mL per minute per 100 grams of muscle. It increases to a maximum of 80–90 mL/minute per 100 grams of muscle during activity. Thus, the muscular absorption of xenobiotics is faster in active individuals than those confned to bed. In general, xenobiotics are absorbed faster from the deltoid than gluteal muscle and the absorption from the gluteal muscle of individuals with lower fat/muscle ratio is faster than those with a higher ratio. The local uptake of lipophilic compounds, however, is signifcant in individuals with higher fat/ muscle ratio. The deltoid site is used more commonly for vaccination purposes, and the injected vaccine must be at least 5 mm deep into the muscle (Poland et al., 1997). Contrary to the gastrointestinal tract, where the absorption is a function of GI tract transit time, compounds injected into the muscle or subcutaneous region have no time limit for residence, and the injected dosage form (solution, suspension, or emulsion) forms a depot which gradually mixes with interstitial fuids and absorbs through capillaries until completely absorbed. The long-acting lipophilic intramuscular solutions, prepared by use of a suitable vegetable oil-containing vehicle and a lipophilic drug or prodrug, prolongs the duration of action and infuences the PK/TK performance of the compound in vivo (Murdan and Florence, 2000; Larsen and Larsen, 2009). It is reported that following the IM injection, a very rapid increase in blood fow occurs that lasts approximately six hours and then returns to normal (Ferré et al., 2005). The signifcant advantages of intramuscular route of administration are i) the absorption of high lipid-soluble molecules is rapid and complete; ii) the absorption of lipid-insoluble molecules is not as rapid, but is complete due to diffusion into interstitial fuids and through the pores of the capillary membrane; iii) the route can be used to bypass the low pH of the stomach and hepatic/ intestinal frst-pass metabolism; iv) the route is ideal for the sustained-release polymeric dosage form or crystalline suspension. However, there are some uncommon side effects and complications (Greenblatt and Allen, 1978), namely, sciatic nerve damage (Van Alstine and Dietrich, 1988; Mishara and Stringer, 2010), skin pigmentation, tissue necrosis, hemorrhage, abscesses (Rossi and Conen, 1995; Duque and Chagas, 2009; Hamann et al., 1990), cellulitis, quadriceps myofbrosis (Alvarez et al., 1980), and even gangrene. Furthermore, incomplete absorption may also occur in muscular environments because of precipitation or decomposition of drugs. Conditions such as hypotension or circulatory diseases may also reduce the rate and extent of absorption. 4.2.2 ADME of Intramuscular Route of Administration The ADME of intramuscular injection drugs is parallel to most routes of administration like oral, sublingual, etc., where a compound must pass through a biological barrier in a dose independent manner and enter the systemic circulation. A common characteristic of an intramuscularly injected drug, in contrast to most observations in oral absorption, is the slow absorption rate from the site of injection. As indicated earlier, most dosage forms for intramuscular injection, whether suspension, emulsion, or solution, establish a depot in the muscle for gradual release and absorption, which depends on the physicochemical nature of the injected drug. It is also important that the depot is formed in the vascular region of the muscle rather than in subcutaneous region, where the blood perfusion is less than the muscle and the absorption rate is lower than intramuscular injection. Also, to achieve an optimum therapeutic concentration for an intramuscularly injected drug, it is critical that the injection avoid the adipose tissue and reach the muscle, particularly in obese patients (Chan et al., 2006). Another aspect of intramuscular injection is the lack of environment to facilitate the drug–drug interaction, which is a common feature of the oral administration when multiple drugs are taken orally. Intramuscular injections rarely include more than one therapeutic agent. Also, in contrast to the GI tract route of administration, the muscle’s physiological pH remains constant and the injection bypasses the hepatic frst-pass metabolism. The xenobiotic metabolic enzyme systems of the skeletal muscle include CYP1B1 and CYP2E1. Other members of CYP450 subfamilies like CYP1A1, CYP1A2, CYP1B1, CYP2E1, and even CYP3A4 are present to a lesser extent and with signifcant interindividual variability (Molina-Ortiz et al., 2013). CYP2C9 is another member of CYP2 family that is involved in the regulation hyperemia and oxygen intake during exercise (Hillig et al., 2003). The presence of CYP450 isozymes in the muscle and their contribution to the overall metabolism of their substrate xenobiotics are negligible in comparison to the systemic and hepatic metabolism of the compounds. Thus, the main aspect of the ADME in the muscle, considering that there is no signifcant parallel elimination from the injection site, is the diffusion from the depot and absorption into the systemic circulation. When the compound reaches the capillary blood and systemic circulation, as with the other routes, its fate is determined by the systemic ADME process. The standard biochemical enzymes used as biomarkers and possibly pharmacodynamic parameters 108

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for diagnosis of muscle diseases or treatment/side effects of xenobiotics on the muscle, are creatine kinase (in three isoforms: MM for skeletal muscle, MB for cardiac muscle, and BB for the brain), lactate dehydrogenase, aspartate aminotransferase, aldolase, and alanine aminotransaminase. In linear PK/TK analysis of an orally administered dosage form, the rate constant of absorption is commonly greater than the overall elimination rate constant. For intramuscular injection due to the slow release from the depot and subsequent absorption, most often the overall elimination rate constant is greater than the absorption rate constant. Hence, the half-life of absorption would be greater than the half-life of elimination and create a scenario known as fip-fop kinetics (Yanez et al., 2011) (see also Chapter 15, Section 15.2.1). The analysis of PK/TK data following intramuscular administration is by using various methodologies in PK/TK modeling, which include compartmental analysis, non-compartmental analysis, physiologically based pharmacokinetic evaluations; in population pharmacokinetic analysis or applications in interspecies extrapolation, investigation is by mixed-effect pharmacokinetic modeling. These methodologies are discussed in other chapters of this handbook. In many respects, the PK/TK analysis of intramuscularly injected compounds mimics the models of oral administration without the complexities associated with the role of transport/effux proteins and intestinal/hepatic frst-pass metabolism in the GI tract. In linear PK/TK analysis it is assumed that the absorption of an IM injected xenobiotic takes place under the sink condition, and thus is governed by frst-order kinetics. 4.2.2.1 Rate Equations of Intramuscularly Injected Xenobiotics As discussed earlier, the comparative relationship between the absorption rate constant, k a , and the overall systemic elimination rate constant,K , of an intramuscularly administered drug is a critical consideration for defning the rate equations. When k a > K , assuming the absorption follows frst-order kinetics, the rate and integrated equations are the same as the linear models described in Chapter 15 with frst-order input and frst-order output. The slope of the terminal portion of the logarithm of plasma concentration vs the sampling-time curve, assuming one-compartment model, provides the estimated value of overall elimination rate constant, K , and applying the twocompartment model, the slope provides the estimated value of the disposition rate constant, b. The slope of log-linear residuals vs the sampling time in both models provide the estimated value of the absorption rate constant, k a (see Chapter 15). Every so often when a drug is administered via an extravascular route of administration like IM injection, it exhibits a prolonged absorption, which can be attributed to the physicochemical characteristics of the drug, the infuence of the factors affecting the absorption at the site of injection, or the formulation of the dosage form. The prolonged absorption causing the comparative relationship between k a and K changes to k a < K or k a @ K, which is referred to as a fip-fop system. Therefore, the estimation of K or b values from the terminal portion of the log-linear profle of plasma concentration vs time, or other numerical methodology based on the notion that k a > K or k a > b, would be incorrect. The frst question is how one can determine with certainty if there is a fip-fop display by the data. Since the half-life of elimination or disposition (i.e., 0.693 / K or 0.693 /b) for a given compound is constant and independent of the dose and route of administration, to validate the fip-fop system and authenticate the calculated values of K or b the simplest approach would be to compare the calculated constants with the same ones from an intravenously administered dose of the drug. The second question is what if one ignores or fails to detect the fip-fop kinetic behavior. To evaluate this question, consider a simple one-compartment model with frst-order absorption input and frst-order elimination output (Figure 4.1). The model assumes that the body behaves as a single homogeneous compartment, and the rate of change in plasma concentration following an intramuscularly injected compound is the difference between two linear differential equations: 1) the rate of absorption from the site of injection, the depot, into the systemic circulation, and 2) the rate of elimination from the body by metabolism and urinary excretion. dAdepot = -k a Adepot dt

(4.1)

dA = k a Adepot - KA dt

(4.2)

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4.2 INTRAMUSCULAR ROUTE OF ADMINISTRATION

Figure 4.1 The diagram of a simple one-compartment model for an intramuscular injection, displaying the formation of the drug depot in the muscle following the injection and gradual absorption into the systemic circulation as a frst-order process with the rate constant of k a and simultaneous frst-order elimination with the rate constant of K from the body; the relationship between the two rate constants characterizes the presence of a fip-fop model when k a ˜ K , the compartment related to the amount eliminated from the body represents metabolism and excretion of the injected drug. dAdepot dA is the rate of change in plasma level (both is the rate of absorption from muscle, dt dt rates have units of mass/time); the absorption rate constant is k a (time−1), Adepot is the absorbable amount at the site of injection at time t , k a Adepot is the rate of absorption (mass/time), the overall rate constant of elimination is K , and KA is the rate of elimination (mass/time). Under the condition of k a > K , the integration of Equation 4.2 yields the amount (A) or plasma concentration (Cp) in the body at time t . Where

A = Cp ´ Vd =

FD ´ k a -Kt -K at e -e ka ´ K

(

)

(4.3)

Where F represents the fraction of the dose absorbed; D is the injected dose; FDis the total amount absorbed; and Vd is the volume of distribution, Cp is plasma concentration at time t . The total body clearance (ml/min) is estimated as: ClT = K ´ Vd =

FD AUC

(4.4)

To answer the second question (i.e., what if one ignores or fails to detect the fip-fop kinetic behavior), initially consider the following invariants (Yáñez et al., 2011):

\ 110

I1 = k a + K

(4.5)

I 2 = ka ´ K

(4.6)

k I3 = a Vd

(4.7)

ClT I 2 = = KVd F I3

(4.8)

PK/TK CONSIDERATIONS OF I.A., I.M., I.P., I.V., AND S.C. ROUTES

Based on the provision of k a > K , setting k a = 0.2 h−1, K = 0.1 h−1, and Vd= 10 L, bring about the Cl invariant values I1 = 0.3, I 2 = 0.02, and I 3 = 0.02 that provide T = 1 L/h. F If K > k a , as is observed in fip-fop cases, setting the above values as K = 0.2 h−1 and k a = 0.1h−1, Cl the I 3 invariant value changes to 0.01 and T = 2 L/h, with the estimate of Vd= 20 L. This calculaF tion may signify the importance of identifying a fip-fop case after an intramuscular or any other extravascular route of administration. In some cases, the volume of distribution is concealed by the Fvalue (Yáñez et al., 2011). An issue of interest in a fip-fop model occurs when one looks at the mass balance equation and its instantaneous absorption rate (Wagner and Nelson, 1964), i.e., dCp dA = Vd + KVdCp dt dt

(4.9)

Writing Equation 4.9 in terms of a difference equation (Boxenbaum, 1998) yields DCp ö DA æ (4.10) = Vd ç KCp + Dt ÷ø Dt è Where Cpis the plasma concentration; DCpis the plasma concentration change during Dt ; and Cp DCp corresponds to the midpoint of . Dt DCp ö In situations where there is a fip-fop behavior, KCp ˜ çæ ÷ Equation 4.10 simplifes to è Dt ø Rateof Absorption =

Rate of Absorption = Vd ´ K ´ Cp = ClT ´ Cp = Rate of Elimination

(4.11)

Equation 4.11 denotes that under the stated conditions, rate of absorption approximates rate of elimination (Boxenbaum, 1998). In other words, fip-fop happens when rate of absorption is the rate-limiting step for ADME processes in the body. The above discussion may also be used for a two-compartment model with frst-order input and frst-order output with the following integrated equations (Figure 4.2): Cp =

ö k 21 - b k 21 - a k a FD æ k 21 - k a e -at + e -kat + e -bt ÷ ç ç ÷ V1 è ( a - k a )(b - k a ) ( ka - a )(b - a ) ( ka - b )( a - b ) ø

(4.12)

where V1 is the volume of the central compartment, k21 is the distribution rate constant from peripheral compartment into the central compartment, and α and β are the hybrid rate constants. When absorption is fast and complete, ka > α ≫ β, Equation 4.12 simplifes to Equation 4.13. Cp =

ö k a FD æ k 21 - b e -bt ÷ ç ÷ V1 èç ( k a - b )( a - b ) ø

(4.13)

However, under conditions where the absorption is the rate-limiting step for disposition and elimination, i.e., a > b ˜ k a Cp =

ö k a FD æ k 21 - k a e -kat ÷ ç ÷ V1 çè ( a - k a )(b - k a ) ø

(4.14)

The non-compartmental analysis of intramuscular absorption data is the same as oral administration (Chapter 15) and faces the same dilemma as compartmental analysis when it relates to fipfop pharmacokinetic behavior. The presence of fip-fop behavior in non-compartmental analysis is identifed by the comparison of mean residence time, mean absorption time and total body clearance of an intramuscularly injected dose with those of intravenously administered drug. The use of physiologically based pharmacokinetic models for intramuscularly injected compounds is like the models with absorption from an extravascular region. The selection of organs and tissues depends on the physiochemical characteristics of the compound, ADME profle of the compound in the body, and the uptake and infuences of the organs and tissues. A typical diagram of the model for the intramuscular injection is presented in Figure 4.3. The rate equations are discussed in Chapter 12.

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4.3 INTRAPERITONEAL ROUTE OF ADMINISTRATION

Figure 4.2 The schematic of a two-compartment model with intramuscular injection and the formation of the depot in the muscle followed by the frst-order absorption of the drug from the depot into the central compartment and simultaneous distribution to the peripheral compartment and elimination from the central compartment; the overall equation of the model is Cp = ae -at + be -bt + ce -kat , and the relationship between the absorption rate constant k a and the hybrid rate constants of a and b exhibits the presence of the fip-fop behavior where a ˜ b ˜ k a . 4.3 INTRAPERITONEAL ROUTE OF ADMINISTRATION 4.3.1 Overview The intraperitoneal (IP) route is used for administration of therapeutic agents and for removal of endogenous waste solutes and exogenous xenobiotics. Both the absorption and exsorption occur through the peritoneal membrane, also known as the peritoneum. Peritoneal membrane is a thin membrane that covers the abdominal cavity, GI tract, peritoneal muscles, and organs within the abdominal cavity. Anatomically, it is made up of one layer of squamous epithelial-like monolayer, known as mesothelium, and a layer of connective tissue attached to the mesothelium by a basement membrane. The mesothelium covers both the parietal and visceral surfaces of the perito9 2 neum and is comprised of a signifcant number of mesothelial cells ( » 1 ´ 10 cells/ » 1 - 2m ) (Rubin et al., 1988). The mesothelial cells are tightly attached to each other to form tight junctions with gaps that facilitate the passage of xenobiotics. The surface area of peritoneal cavity is about 1–2 m2. In males, the peritoneum is a closed system, whereas in females it is open with the fallopian tube and ovary connections. The physiological role of the mesothelium is to participate in maintaining the homeostasis of the peritoneal cavity (Nagy, 1996) by proving lubrication to facilitate the movement of organs within the abdominal cavity such as peristaltic movement of intestines, respiration, and often simultaneously opposing movement of the organs. It also produces lubricants like hyaluronic acid (Yung et al., 1994), proteoglycans like decorin and biglycan (Yung et al., 1995), and phosphatidylcholine (Dobbie et al., 1988). Mesothelium also acts as a barrier for the absorption of xenobiotics, although it is not considered as resistive as other physiological barriers. The mesothelium participates in the transfer of small and large molecules across the peritoneum by passive and active absorption (Cotran and Karnovsky, 1968; Gotloib and Shostak, 1989; 1995). The peritoneal membrane capillaries bring nutrients to and remove waste from the organs of the abdominal cavity. The vascular exchange of endogenous and exogenous molecules occurs by passing through capillaries with a diameter of 5–6 µm and venules with a diameter of 7–20 µm. In addition to the capillaries and venules, a web of lymphatic vessels is present in the peritoneal cavity that is responsible for the frst line of defense and the removal of fuid from the cavity to maintain a small volume of not more than 50 mL in the cavity (Mactier and Khanna, 2000). The excess fuid is then transferred to the systemic circulation. Compounds absorbed into the peritoneal 112

PK/TK CONSIDERATIONS OF I.A., I.M., I.P., I.V., AND S.C. ROUTES

Figure 4.3 Depiction of a physiological model with intramuscular injection and formation of a depot in the muscle with absorption into the capillary web of the muscle with ultimate absorption into the venous blood. capillary network reach the portal circulation and pass through the liver before the distribution in the body, thus the compound may undergo a hepatic frst-pass effect. 4.3.1.1 Applications of the IP Route of Administration The IP route is used for administration of therapeutic agents as well as for continuous ambulatory dialysis therapy for removal of endogenous and exogenous toxic molecules from the body. The long-term use of peritoneal dialysis is often associated with peritonitis and mesothelial fbrosis, that can be treated effectively (Wiggins et al., 2007). Furthermore, the peritoneum is susceptible to several disorders that impact its structure and function and include peritonitis (without dialysis application), peritoneal adhesion, different types of malignancies, and – the most concerning one – peritoneal carcinomatosis (Coccolini et al., 2013; Raptopoulos and Gourtsoyiannis, 2001). The chemotherapy of the peritoneal carcinomatosis requires high concentrations of chemotherapeutic agents and longer presence in the cavity, which is not achievable by giving large doses intravenously. Intraperitoneal injection of the drugs, known as IP chemotherapy, following the surgery has shown to provide the required high 113

4.3 INTRAPERITONEAL ROUTE OF ADMINISTRATION

concentration and longer residence time, which is mainly due to the mesothelial barrier and low clearance from the cavity (Hasovits and Clarke, 2012). New approaches for IP chemotherapy, like pressurized intraperitoneal aerosol chemotherapy (PIPAC) and hyperthermic PIPAC (Bachmann et al., 2021; Ploug et al., 2020; Alyami et al., 2019) are recommended to increase the effcacy of the treatment. Ovarian cancer is another aggressive type of cancer that spreads throughout the abdominal cavity and requires effective concentrations and presence of chemotherapeutic agents. A reported therapy approach is the IP injection of H19-DTA (BC-819), a DNA plasmid which is effective on oncofetal gene expressed in ovarian tumors, the paternally imprinted H19 (Lavie et al., 2017). Other approaches in treatment of ovarian cancer that are based on targeting effective doses of chemotherapeutic agents, e.g., cisplatin (70 mg/m2), at the site of action are IP chemotherapy and hyperthermic IP chemotherapy (van Driel et al., 2018; Wang et al., 2021). Although direct IP chemotherapy is more effective than intravenous injection in providing high concentration and longer presence in peritoneum, the clearance of small-molecule drugs in the peritoneum is considered rather fast, but the duration of action, though longer than the systemic administration, in certain cases may not be long enough. To extend the residence time of the drug in the cavity, various particulate formulations have been used to control the release within the cavity and thus extend the duration of action (Tsai et al., 2007; Bajaj and Yeo, 2010). An implantable, programmable micro-infusion reservoir for IP drug delivery is also under evaluation, in which the controlled release of medication by a pump may prolong the duration of action (Iacovacci et al., 2021, Xu et al., 2021). Regardless of the type of delivery system, whether simple injection of drug solution or suspension, particulate dosage forms, or implantable pumps, xenobiotics introduced into the abdominal cavity are absorbed into the systemic circulation by passive transcellular and paracellular diffusion through the peritoneum. They can also be removed from systemic circulation into a dialysate based on the concentration gradient between the capillaries of peritoneum and dialysate, which can then be removed from the body by drawing off the dialysate. The presence of active absorption or facilitated transfer may exist only for the transfer of nutrients included in the dialysate. 4.3.2 Kinetics of Intraperitoneal Transport of Xenobiotics The kinetics and dynamics of peritoneal-plasma transport are defned by a distributed model (Flessner et al., 1984) and its modifed version for diffusive transport in super-fused tissue applicable to cancer chemotherapy (Flessner et al., 2006; Lu et al., 2010). Both approaches are based on the assumptions that the peritoneum is a well perfused exchange site between the peritoneal cavity and the plasma, drug transfer through the barrier tissue is governed by the Fick’s second law of diffusion, and removal of a drug from the site of absorption is by permeation through the capillaries. The lymphatic uptake is considered signifcant in a distributed model. The schematic diagram of a distributed model is presented in Figure 4.4. According to the model (Flessner et al., 1984), the transfer of solute between the plasma and the distribution compartment is very rapid and collectively behave as a single distribution compartment. For IP injection, the transfer of solute occurs between the peritoneal cavity and peritoneal barrier, from the barrier into plasma and distribution compartment, and from the plasma to the body exchange compartment. The volumes of body exchange and distribution compartments are assumed constant with the rate of change equal to zero. dVI dVD = =0 dt dt

(4.15)

Diffusion through the tissue and permeation through the capillaries are assumed to be only a function of the molecular size of a compound. All compartments of the model are assumed wellmixed, and the peritoneal barrier is considered a distributive space. This theoretical model defnes the diffusion of a compound, such as anticancer drugs, from the peritoneal cavity. It also applies to the dynamics of peritoneal dialysis when a compound is removed from the distribution compartment and plasma, through the peritoneal barrier and into the dialysate infused in the peritoneal cavity. The differential equations of the absorption from the peritoneal cavity for well-mixed compartments (Flessner et al., 1984) are: d ( CPC ´VPC ) = -(RL + RPC ) dt

114

(4.16)

PK/TK CONSIDERATIONS OF I.A., I.M., I.P., I.V., AND S.C. ROUTES

Figure 4.4 Diagram of a distributed model based on the transport and exchange of watersoluble compounds between the peritoneal cavity and the plasma compartment, which requires diffusion and convection through peritoneal barrier; the transfer through the barriers is governed by Fick’s law of diffusion; lymphatic uptake from the peritoneal cavity into the systemic circulation bypasses the barrier; the amount of compound that reaches the plasma compartment distributes in the distribution compartment and body exchange compartment, and both compartments are assumed to have constant volume. \

d ( CPC ) -(RL + RPC ) = dt VPC

(4.17)

d ( CDVD ) = RPB + RL + RID - (RDI + RCp ) dt

(4.18)

d ( CD ) RPB + RL + RID - (RDI + RCp ) = dt VD

(4.19)

\

d ( CIVI ) = RDI - RID dt

(4.20)

d ( CI ) RDI - RID = dt VI

(4.21)

\

Where CPC , CD , and CI are the concentrations (mass/volume) in the peritoneal cavity, distribution compartment, and body exchange compartment, respectively; VPC is the volume in the peritoneal cavity, VD is the volume of distribution compartment, and VI is the volume of the body exchange compartment; the rates of mass transfer with units of mass/time are: RL for the lymphatic rate of mass transfer, RCp for the rate of mass transfer from the peritoneal cavity into the barrier, RPB for the rate of mass transfer from barrier to the distribution compartment via the plasma, RID and RDI for the rates of mass transfer between the body exchange and distribution compartments. The rate of volume change in the peritoneal compartment is a function of the lymphatic volume fow rate FL (volume/time) and peritoneal cavity to tissue volume fow rate FPC (volume/time), that is, dVPC = - ( FL + FPC ) dt \VPC = -

t

ò (F + F 0

L

PC

(4.22)

)

(4.23)

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4.3 INTRAPERITONEAL ROUTE OF ADMINISTRATION

Thus, the lymphatic mass transfer rate can be defned as RL = FL ´ CPC

(4.24)

To solve for the rate equation of transcapillary transport, that is the portion of the transport from peritoneal cavity through the tissue barrier and capillaries into the distribution compartment, Equation 4.17 can be expanded and modifed by substituting Equations 4.23 and 4.24 as follows: VPC

dCPC = CPC ( FL + FPC ) - ( RL + RPC ) = FPCCPC - RPC dt \

dCPC FPCCPC - RPC = dt VPC

(4.25) (4.26)

The initial conditions of the model are

¶CPB =0 ¶x Where CPB is the concentration of the barrier and x is the thickness of the barrier. Other mass transfer rate equations are defned as At t = 0 : CPC = 0, CD = 0, CI = 0, VPC = VPC (0); CPB ( x , 0 ) = g ( x ) = 0 and

RDI = kDI CDVD

(4.27)

RID = k IDCIVI

(4.28)

RCp = kCpCDVD

(4.29)

To make the model more practical and applicable to the clinical application of IP cancer chemotherapy, a simpler model without the body exchange compartment and distribution compartment has been suggested that considers the exchange between plasma, the peritoneal cavity, and barrier (Flessner et al., 2006). The diagram of the model is presented in Figure 4.5 with two sampling compartments of plasma and solution in peritoneal cavity. A new hybrid coeffcient that represents all the diffusive and solute transfer characteristics through the barrier is identifed in the model as the mass transfer area coeffcient (MTAC). The rate of mass transfer from the peritoneal cavity into the plasma is defned as

Figure 4.5 The diagram of a distributive model for transfer of a water-soluble xenobiotic from the peritoneal cavity into the plasma compartment; this model is the modifed version of the model presented in Figure 4.4 with the assumptions that the diffusion through the barrier is governed by Fick’s law of diffusion with two sampling compartments of plasma and fuid in the peritoneal cavity; MTCA is the product of mass transfer coeffcient (MTC) and the contact area in the peritoneal cavity. 116

PK/TK CONSIDERATIONS OF I.A., I.M., I.P., I.V., AND S.C. ROUTES

d ( CPC ´VPC ) = -MTAC ´ ( CPC - Cp ) dt

(4.30)

Where CPC and Cp are concentrations of peritoneal cavity and plasma, respectively, VPC is the volume of solution in peritoneal cavity, and MTAC is the product of mass transfer coeffcient and the fuid contact area in the peritoneal cavity, that is, MTAC = MTC ´ Area

(4.31)

MTC = Deff ´ Pa

(4.32)

When dealing with blood perfusion limitation (4.33)

MTC = Deff ´Q

where the blood fow is Q, Deff is the effective diffusion coeffcient of the solute, P is the capillary permeability, and a is the capillary surface area per unit of volume. The mass (M) of MTC is M = CPC ´VPC

(4.34)

Thus, the mass transfer rate in peritoneal cavity is Amount Removed dM = = -MTC ´ Area ( CPC -Cp ) dt Period of Experiment

(4.35)

The mass balance on the tissue side is defned based on the Fick’s second law (Flessner et al., 2006) ¶ ( qs ´ Cs ) ¶ é ¶Cs ù (4.36) + ( Cs ´ f ´ J v ´ qs ) ú + Rcap = Deff ¶x ¶x ëê ¶t û Where qs is the fraction of tissue accessible to the solute; Cs is the concentration of the solute in the interstitial space; x is the distance into the tissue from the cavity; f is the ratio of the solute rate of diffusion to the solvent rate of diffusion, identifed as the solute retardation factor; J v is the fuid fux through the tissue, which is dependent on the osmotic and hydrostatic pressure between the interstitial fuid and capillary network; and Rcap is the rate of solute exchange defned as Rcap = -Pa ( Cs - Cplasma )

(4.37)

In the case of blood perfusion limitation Rcap = -Q ( Cs - C plasma )

(4.38)

Under steady state condition, Equation 4.36 can be written as ¶ ( qs ´ Cs ) =0 ¶t = Deff

d 2C s é d ù + f ´ qs ê ( Cs ´ J v ) ú - Pa((Cs - Cp) dx ¶x 2 ë û

(4.39)

For isotonic solutions, J v @ 0 and Deff

d 2C s = Pa ( Cs - Cp ) dx 2

(4.40)

Deff

d 2C s = Q ( Cs - Cp ) dx 2

(4.41)

Or,

æ ç

ç Cs - Cp =e è CPC - Cp æ ç

ç Cs - Cp =e è CPC - Cp

Pa Deff

ö ÷x ÷ ø

Q ö÷ x Deff ÷ ø

(4.42)

(4.43)

117

4.3 INTRAPERITONEAL ROUTE OF ADMINISTRATION

As noted earlier, there is a renewed interest in the use of the intraperitoneal route of administration for targeting the cancerous tumor in the intraperitoneal cavity. The advantages include 1) the fexibility of using high concentration directly for targeting the cancerous cells in the peritoneal cavity and achieving a more effective therapeutic outcome, 2) averting the body’s normal tissues exposure to high plasma level for achieving the same outcome, 3) direct measurement of mass transfer at the site of absorption, 4) direct evaluation of pharmacological effect, 5) infuence of osmotic and hydrodynamic pressure on the absorption of therapeutic agent, 6) retrieval of the agent through peritoneal dialysis. These advantages offer research opportunities in PK/PD and TK/TD of xenobiotics. The diagram of a PK/PD model for analysis of intraperitoneal chemotherapy data of intraperitoneal carcinomatosis and advanced ovarian cancer is presented in Figure 4.6. The model includes an open two-compartment model with input from peritoneum to plasma. The central compartment of the model is connected to an effect compartment representing the therapeutic response of the drug and a dynamic model for the proliferation of the cellular marker of the surgery (Valenzuela et al., 2011; Pérez-Tuixo et al., 2015; Xie et al., 2020). The dose input into the peritoneal cavity can be fast or slow infusion with or without hyperthermia. The infusion rate of input, slow or fast, is k0 with unit of mass/time. The rate of change in the peritoneum compartment is dP = k0 - k a P dt

(4.44)

Where k a is frst-order absorption rate constant from peritoneal compartment into the central compartment, i.e., the systemic circulation, and P is the amount in the peritoneal compartment at time t.

Figure 4.6 The diagram of a general PK/PD model for the data analysis of intraperitoneal treatment of carcinomatosis or ovarian cancer using the intraperitoneal route of administration; the model is essentially a two-compartment model with zero-order intraperitoneal infusion of k0 into the peritoneal cavity and frst-order absorption with the rate constant of k a from peritoneal cavity into the central compartment A1 that have exchanged with the peripheral compartment with the distribution rate constants of k12 and k 21 ; other rate constants of the central compartment include k10 the overall elimination rate constant and the input rate constant into the effect compartment; the mechanistic portion of the model includes fve compartments of proliferation in series with the frstorder rate constant of ktr representing the infuence of the chemotherapeutic agent on the number of proliferative cells with feedback from the circulating cells compartment to the proliferative compartment with the rate constant of k prol, also refecting the impact of cytoreduction surgery (CRS); and MTT is the mean transit time. 118

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The rate of amount change in the central compartment is dA1 = k a P + k21 A2 - k12 A1 - k10 A1 dt

(4.45)

Where k12 , k 21 are the distribution rate constants between the central compartment and peripheral compartment; A1 and A2 are the amounts in the central and peripheral compartments, respectively; and k10 is the overall elimination rate constant for combined metabolism and excretion rate constants. The rate of amount change in the peripheral compartment is dA2 = k12 A1 - k 21 A2 dt

(4.46)

The mechanistic portion of the model is based on the model of chemotherapy-induced myelosuppression (Friberg et al., 2002) and consists, for this discussion, of fve compartments of proliferation, representing proliferative cells like stem cell or other originator cells, three transit compartments for maturing cells, and one compartment for circulating cells. The rate constants between the transit compartments are frst-order and account for the infuence of chemotherapy on the number of proliferative cells in each compartment at different stages of self-renewal or mitosis. The proliferation frst-order rate constant, k prol , determines the sell division rate with feedback from circulating cells. The related rate equations are g

dProlif æ CircB ö olif = k prolif ´ Prolif ´ ç ÷ ´ (1 - EDrug ) - ktr ´ Pro dt è Circ ø

(4.47)

dT1 = ( ktr ´ Prolif ) - ( ktr T1 ) dt

(4.48)

dT2 = ( ktr ´ T1 ) - ( ktr ´ T2 ) dt

(4.49)

dT3 = ( ktr ´ T2 ) - ( ktr ´ T3 ) dt

(4.50)

dC = ( ktr ´ T3 ) - ( kC ´ C ) dt

(4.51)

Where C is the circulating cells; T1 , T2, and T3 are the three transit compartments, and kC is the circulating cells rate constant; CircB is the baseline values of the circulating cells; g represents the infuence of the feedback function. The infuence of surgery, e.g., cytoreductive surgery, on the proliferation is identifed as CRS on Figure 4.6. The mean transit time (MTT) for proliferative cells to reach the circulation is estimated as MTT =

n+1 ktr

(4.52)

Where n represents the number of transit compartments. The model has been used in population PK/PD analysis of chemotherapy assessment of intraperitoneal carcinomatosis and advanced ovarian cancer treatments (Valenzuela et al., 2011; PérezTuixo et al., 2015; Xie et al., 2020). 4.4 INTRAVENOUS ROUTE OF ADMINISTRATION 4.4.1 Overview The intravenous route of administration is the reference route for pharmacokinetic and toxicokinetic investigations. Contrary to the oral administration, given xenobiotics intravenously provides immediate onset of action (~10–25 seconds after the injection), complete bioavailability (100%), and minimal hepatic frst-pass effect. A valuable aspect of intravenous administration and complete introduction of the dose into the systemic circulation is the titration of the administered dose according to its response, and determination of the limit of linearity of PK/TK behavior in the body more accurately. Because veins are not sensitive to irritation, it is the only route that highly irritant drugs, like anticancer agents, can be administered into the body without discomfort. Furthermore, the poorly soluble compounds can be dissolved in large volume and administered over a longer period of time. 119

4.4 INTRAVENOUS ROUTE OF ADMINISTRATION

4.4.1.1 Intravenous Injection Drawbacks The sites of intravenous injection for adults include the veins on the forearm, hand, and wrist, which connect to the antecubital fossa. For infants the sites include veins on the scalp, umbilical vessels, and the legs’ cursory veins. The major risks and complications associated with intravenous administration are toxicity due to overdose, infection, embolism, vascular injury, speed-shock, anaphylactic shock, phlebitis and thrombophlebitis, infltration, extravasation, and hematoma. Infection occurs when the epidermis bacteria, like staphylococcus, enter systemic circulation initiating systemic bacteremia. Phlebitis is a complication related to the infammation of the innermost layer of the vein with signs of swelling along with vein hardening, discomfort, and pain (Ung et al., 2002). Thrombophlebitis, the same as infection, is caused by introduction of the skin microorganism. Infltration and extravasation are caused by accidental administration of the drug into surrounding tissues. Speed shock occurs when the intravenous administration is fast and causes a systemic reaction and often circulatory overload with possible pulmonary edema (Dougherty, 2013). Anaphylactic shock is an immediate immunological response to the intravenous injection of a drug. Hematoma occurs when the blood is leaked out of the vein into the surrounding tissues. This happens when the needle or catheter is removed carelessly without applying adequate pressure. 4.4.1.2 Bolus Injection, Continuous Infusion, Intermittent Infusion The use of intravenous route of administration depends on the drug, desired therapeutic outcome, and patient’s condition. Drugs can be administered intravenously as bolus injection, continuous infusion, and intermittent infusion. The bolus injection is when the drug is given in small volume intravenously and rapidly without spending any time for injection i.e., kinetically the dose is in the syringe at time zero and the drug is in the systemic circulation at time zero, i.e., no rate for injection is presumed. The continuous infusion is the delivery of a medication in a large volume with zero-order constant rate over a predetermined period, e.g., a few hours to several days, if the treatment is required. The objective of the infusion is to precisely control and maintain the drug concentration at steady state for a long period of time, to attain consistent therapeutic outcome. The continuous infusion is also used without medication to replace fuids and electrolytes needed by the body. Possible drawbacks of the continuous infusion are fuid overload and incompatibility between the infusion solution containing one medication and other drugs administered through the same vascular access device. Intermittent infusion is the administration of a drug in a small volume of fuid (50–250 mL) over a short period of time (20–120 min) with regimen of once a day or multiple infusions during a 24-hour period. An intermittent infusion is used when the drug cannot be administered as an intravenous (IV) bolus and a peak plasma concentration is desired. The intention is not to achieve a steady-state level; rather, the intermittent infusion is used to achieve a peak concentration that can be maintained as steady state level with a continuous infusion. Both IV bolus injection and intermittent infusion are used to download a loading dose to attain a peak level to be maintained by continuous infusion maintenance dose (see also Chapter 14). The intermittent infusion may also cause irritation and phlebitis. Continuous and intermittent infusions are accomplished using pressure generated by gravity and electromechanical infusion pumps. When the dose is added to an IV bag, like normal saline or 5% dextrose in water, and the gravity is the driving force for the continuous dripping and infusing over several hours, it is called an “IV drip”; and when the dose is mixed in a separate small volume IV bag and connected to an existing port of the IV drip, it is called an “IV piggyback”. The common electromechanical pumps are peristaltic, cassette driven, and syringe driven. The newer type known as ‘smart’ infusion pumps include built-in drug error reduction software (DERS), which is based on a customized drug library that prevents errors in dose and rate calculation of a category of medications. 4.4.2 Intravenous PK/TK Analysis Most PK/TK profles of xenobiotics, whether using compartmental analysis (Chapters 13 and 14), physiological models (Chapter 12), non-compartmental analysis (Chapter 12), or dynamic models, are developed based on their intravenous administration. The process of assessing the risk associated with exposure to environmental chemical, extrapolation from animals to humans, and low dose extrapolation relies on the predictive power of PK/TK models that are developed following intravenous administration of xenobiotics. The drug administered intravenously will be subjected to all barriers, interaction with macromolecules and transporters, and permeation through the diverse membranes of various tissues and organs in the body that are linked with the systemic 120

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circulation (Chapters 12 and 13). Furthermore, the drug administered intravenously, as with the other routes of administration in which the drug reaches the systemic circulation, will undergo biotransformation by the liver and other organs of metabolism (Chapter 9). Ultimately, the drug will be eliminated as metabolites or unchanged by the kidneys and other routes of elimination (Chapters 9 and 10). 4.5 SUBCUTANEOUS ROUTE OF ADMINISTRATION 4.5.1 Overview The subcutaneous tissue is a layer of skin under the dermis that wraps the muscles and provides a passageway for blood and lymph vessels, the layer contains cutaneous network of nerves, and act as the connective tissue between the skin’s layers and muscles. The subcutaneous route of administration (SC) is used for injection of small volume vaccines or drugs, such as insulin (Mathieu et al., 2021), heparin (Li et al., 2020), local anesthetics, monoclonal antibodies (Rahimi et al., 2022), proteins, and peptides (Zou et al., 2021). SC route is also used to implant prolonged-release dosage forms such as polymeric rods, discs, or pellets (e.g., Oreton®, Percorten®, Norplant®, and Alzet® osmotic mini-pump). There is also newer technology under development using hyaluronidase (e.g., ENHANZE®) that facilitates the SC delivery of co-administered therapy by locally degrading hyaluronan, modifying the tissue permeability, increasing the dispersion and absorption of co-administered therapies, and allowing larger volumes to be injected subcutaneously (Loke et al., 2019; Soundararajan et al., 2020). The subcutaneous region is less vascular than the muscle, and thus has a lower rate of absorption than intramuscular administration (unless another compound is added to enhance absorption). The extent of absorption is somewhat similar for IM when given equally administered doses. As with muscle injection, the onset of action of subcutaneous administration depends on the physicochemical characteristics, diffusion, and permeation velocity of the compound in the region. In general, the rate and extent of subcutaneous absorption are infuenced by factors like: i. lipophilicity of the compound ii. size of the drug molecule iii. site and depth of administration, iv. injection volume of the dose v. dose concentration vi. effect of regional pH on pKa and the degree of ionization of the compound vii. type of delivery system viii. presence of surfactants ix. viscosity of the solution or emulsion containing the medication x. particle size of suspension, xi. etc. The volume of SC injection is 1.5–2 mL (Mathaes et al., 2016), as injecting larger volumes can cause pain due to edema, swelling, and blister formation. Furthermore, the intraindividual skin thickness and SC adipose layer vary in different sites of injection like abdomen, thigh (thinnest), buttocks (thickest), and arm of individual patients, and there are added interindividual differences related to gender, age, body mass index, and needle length (Hirsh et al., 2014). The mechanism of absorption in the subcutaneous region is by passive diffusion through the capillary wall into the systemic circulation. If the compound is lipophilic, it can diffuse directly through the membrane of capillaries by transcellular passive diffusion. Water-soluble drugs diffuse through cleft pores and vesicular channels of the membrane by paracellular passive diffusion. There has been no report to indicate that other mechanisms of absorption such as pinocytosis or active or facilitated transport may take place at the SC site of absorption. Proteins administered subcutaneously are absorbed into the systemic circulation either directly through the capillary wall or indirectly through the lymphatic capillaries within the interstitial space (Figure 4.7). Proteins larger than 16 kDa are absorbed mainly through the lymphatic capillaries (Supersaxo et al., 1990; Charman et al., 2000; McLennan et al., 2006; Kota et al., 2007). 121

4.5 SUBCUTANEOUS ROUTE OF ADMINISTRATION

4.5.2 Rate Equations of Subcutaneously Injected Xenobiotics The PK/TK data analyses for most compounds injected subcutaneously, other than insulin, are the same as the intramuscular approaches discussed in Section 4.2. The rate equations and PK/ TK models describing the plasma concentration–time profle of exogenous insulin are diverse and depend mainly on the type of insulin, concentration of injected dose, the expected pharmacological response, time to the onset of action, and the duration of action. For example, insulin lispro is a monomeric short-acting insulin analogue with a time to onset of action of 5–15 minutes and duration of action of 3–5 hours; regular insulin is also another short-acting insulin with a time to onset of action of 30–60 minutes and duration of action of 6–10 hours; whereas, insulin glargine is a long-acting insulin with a time to the onset of about 1–1.2 hours and duration of action of 24 hours. Most insulin models treat the diffusion in the subcutaneous region as an absorption compartment separate from insulin in plasma. Often the hexameric complex of the molecule must split up to a dimeric complex and then to a monomeric structure in the subcutaneous region, which adds to the complexity of the model in the absorption compartment. A few absorption models of insulin relevant to the objectives of this section are discussed below. 4.5.2.1 Subcutaneous Diffusion Rate-Limited Model The absorption site, the subcutaneous space holding the injected insulin, is assumed compartment one and the diffused molecules available for absorption (i.e., the diffusion compartment) as compartment two, analogous to the creation of an injection depot in intramuscular injection. The insulin available in compartment two is the absorbable amount reaching the systemic circulation.

Figure 4.7 An illustration of the important elements of the subcutaneous environment that facilitate the passive diffusion of small-molecule xenobiotics through the interstitial space into the capillaries and the absorption of protein/peptides of larger than 16 kDa via the lymphatic opening of the lymphatic vessels into the lymph. 122

PK/TK CONSIDERATIONS OF I.A., I.M., I.P., I.V., AND S.C. ROUTES

The decomposition of insulin is assumed to occur in both compartments (Kraegen and Chisholm, 1984; de Meijer et al., 1989). The equations of the model are dU 0 = k dU 0 - k diff U 0 dt

(4.53)

dUt = k diff U 0 - k aUt - k dUt dt

(4.54)

Ut =

U 0 (k a e

-(kd + kdiff )t

- k +k t - k12 e ( d a )

k a - k diff

(4.55)

Where Ut is the absorbable units of insulin in compartment two at time t, U 0 is the dose introduced in the subcutaneous region at time zero, k a is the absorption rate constant for transfer of insulin from the absorption compartment (compartment two) into the systemic circulation, k d is the decomposition rate constant, and k diff is the frst-order diffusion rate constant from compartment one to two (Figure 4.8). 4.5.2.2 Subcutaneous Dissolution Rate-Limited Model The assumptions of the model are ◾ The rate of delivery depends on the volume of injected dose and follows the cube root model of dissolution. ◾ The injected dose in the subcutaneous region distributes in a coin-shaped volume (de Meijer et al., 1989). The equation of the model is Ut = (U 0 - k12 ´ t ) + U rest 3

(4.56)

The parameter U rest was added to the equation to improve the ft (de Meijer et al., 1989). 4.5.2.3 Subcutaneous Capacity-Limited Model This model assumes that the movement of the molecules in the subcutaneous region is capacity limited, and the transfer rate constant at high concentrations is not governed by frst-order kinetics (de Meijer et al., 1989). dUt Vmax (Ut - U rest ) /V = dt K M + (Ut - U rest ) /V

(4.57)

dUt is the reduction rate of the injected dose, Vmax and K M are the maximum rate of transdt port and Michaelis–Menten constant, respectively, and V is the volume of injected dose.

Where

Figure 4.8 A diffusion rate-limited model for absorption of insulin from subcutaneous interstitial space into the diffusion compartment with the frst-order diffusion rate constant of k diff followed by the absorption from the diffusion compartment into the systemic circulation with the frst-order rate constant of k a ; the model takes into the consideration the decomposition of the injected insulin at the site of injection and at the diffusion compartment with a frst-order rate constant of k d . 123

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4.5.2.4 Subcutaneous Models Based on Diffusion Equations The compartmentalization of the process is the same as before, except the models take into consideration the hexameric and dimeric forms of insulin (Mosekilde et al., 1989; Trajanoski et al., 1993; Nucci and Cobelli, 2000; Tarín et al., 2005; Li and Kuang, 2009). These models represent mechanistic and conceptual relationships that are often diffcult to adapt to experimental data. The following are examples of these models (Tarín et al., 2005): ¶Cd ( t , p )

= DÑ 2Cd ( t , p )

(4.58)

= DÑ 2Ch ( t , p )

(4.59)

= db DÑ 2Cb ( t, p )

(4.60)

¶t ¶Ch ( t , p ) ¶t ¶Cb ( t , p ) ¶t

Where Cd and Ch are dimeric and hexameric concentration, respectively; Cb is a virtual insulin state concentration, identifed as bound concentration; and t is time, p is the position vector, and D is the diffusion constant, which is the same for both dimeric and hexameric forms. The diffusion is assumed to be homogeneous with rotational symmetry with respect to the site of subcutaneous injection with spherical coordinates, that is, ¶Cd ( t, r ) 1 ¶ æ 2 ¶Cd ( t, r ) ö = DÑ 2Cd ( t , r ) = D 2 çr ÷÷ ¶t ¶t r ¶r çè ø

(4.61)

¶Ch ( t , r ) 1 ¶ æ 2 ¶Ch ( t , r ) ö = DÑ 2Ch (t , r) = D 2 çr ÷÷ ¶t ¶r r ¶r çè ø

(4.62)

¶Cb ( t , r ) 1 ¶ æ 2 ¶Cb ( t, r ) ö = db DÑ 2Cb ( t, r ) = db D 2 çr ÷÷ ¶t ¶r r ¶r çè ø

(4.63)

Where r is the distance from the subcutaneous site of injection, and the concentration terms are defned locally as a function of time and r through a theoretical spherical model. The hexameric–dimeric dissociation at the site of absorption (Mosekilde et al., 1989; Tarín et al., 2005) is then identifed as k1 ¾¾ ¾ ® 3 Dimeric insulin Hexameric insulin ¬ ¾ k-1

The rate constants of dissociation and association are k1 and k−1, respectively. The partial differential equations representing the law of mass action are ¶Cd ( t , r ) 3 = k1Ch ( t , r ) - k -1Cd ( t , r ) ¶t

(4.64)

¶Ch ( t , r ) 3 = -k1Ch ( t , r ) + k -1Cd ( t , r ) ¶t

(4.65)

Because only the absorption of dimeric form is signifcant (Mosekilde et al., 1989), its absorption rate from subcutaneous region to the systemic circulation can be defned as ¶Cd ( t, r ) = -kadimericCd ( t , r ) ¶t

(4.66)

Where kadimeric is the absorption rate constant of dimeric insulin. The rate of change of insulin concentration in plasma can then be defned as (Lehmann et al., 2009) æ dCpinsulin ö Rate of insulin absorption (IU / time) ç ÷= dt Vinssulin ( volume ) è ø ´

124

1 me ´ kg ) Body Weight (kg) - KCpinsulin ( IU/volume ´ tim

(4.67)

PK/TK CONSIDERATIONS OF I.A., I.M., I.P., I.V., AND S.C. ROUTES

The concentration is expressed in international units (IU) per volume, and the rate as IU/time; Vinsulin is the relative volume of insulin distribution; K is the overall elimination rate constant; KCpinsulin is the rate of elimination. 4.5.2.5 Other PK Models for Subcutaneous Insulin There are other approaches to PK modeling of insulin administered subcutaneously; a few are grounded on nonlinear PK/PD presentation of controller synthesis, and some are based on linear step-response models (Parker et al., 1999). A few approaches also include a physiological model for insulin in conjunction with the physiological modeling of glucose for the purpose of defning a framework for an insulin sensor to be placed in subcutaneous adipose tissue (Bisker et al., 2015). Challenges in developing a PK model for insulin include 1) control of glucose uptake in the liver and interstitial compartment by local insulin; 2) production of the glucose in the liver which is controlled by insulin and glucagon concentrations in the liver; 3) the release of pancreatic insulin regulated by the blood sugar concentration. A suggested mass balance equation for insulin applicable to the physiological model and organ/ tissue designation follows (Sorensen, 1985; Bisker et al., 2015) For capillary blood: VC

dCCB = QC (C AB - CCB ) + PA ( CS - CCB ) - rRBC dt

(4.68)

dCI = PA ( CCB - CI ) - rT dt

(4.69)

For interstitial fuid: VI

dCCB is the accumuladt tion; QC is capillary blood rate; C AB and CCB are concentrations in arterial blood and capillary blood, respectively; QC (C AB - CCB ) is the convection; PA is permeability-area product and PA ( CS - CCB ) is the diffusion part of the equation; rRBC is the rate of red blood cell uptake, which is represented as dCI the metabolic sink; VI is interstitial fuid volume; CI is interstitial fuid concentration and VI is dt the interstitial fuid accumulation; PA ( CCB - CI ) is the convection; and rT is the rate of uptake by the tissue and is identifed as the metabolic sink. The permeability-area product, PA, can be expressed V as I , where T is the transcapillary diffusion time. A typical physiologic compartment relevant to T the above mass balance equations is presented in Figure 4.9.

Where VC is capillary blood volume; CCB is arterial blood concentration; VC

Figure 4.9 A unit compartment from physiological modeling of insulin representing the mass transfer between the capillary blood and interstitial fuid; VC and VI are capillary blood volume and interstitial fuid volume, respectively; C AB, CCB , and CI are concentrations of insulin in arterial blood, capillary blood, and interstitial fuid, respectively; QC is capillary blood fow rate and PA is the permeability-area parameter; and rRBC and rT are the rate of red blood cell uptake and tissue uptake of insulin, respectively. 125

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5 PK/TK Considerations of Transdermal, Intradermal, and Intraepidermal Routes of Administration 5.1 TRANSDERMAL ROUTE OF ADMINISTRATION 5.1.1 Overview The skin is considered the largest organ in the body, easily accessible for application of medications or absorption of toxic xenobiotics. The route is used for the local and systemic treatments of diseases; with the advantages of being safe, painless, simple to self-administer, and freed from hepatic frst-pass metabolism. However, because of its accessibility, the skin is also one of the available sites for the absorption of ◾ toxic alkylating agents, like sulfur mustard and similar chemicals that cause blisters, ulceration, and total loss of skin ◾ Pollutants, like aromatic hydrocarbons (e.g., benzo(a)pyrene), volatile organic solvents (e.g., benzene), or heavy metals, like arsenic with the skin, which cause cancer ◾ genotoxic elements like ultraviolet radiation and ozone which also react with skin and cause cancer. The skin, while accessible for absorption of all xenobiotics, provides a formidable barrier to their absorption and thus protects the body, meriting a tribute for preventing toxic xenobiotics from entering the body. This protection is also considered a major impediment for transdermal absorption of therapeutic agents. In addition to the absorption function, the skin barrier is a ◾ thermoregulator ◾ water loss controller ◾ protector against invading microorganism by its specialized immune cells ◾ guardian against diverse forms of trauma, like thermal, chemical, and ultraviolet radiation ◾ multidimensional host for its own benefcial microbiota ◾ capable synthesizer of vitamin D. The skin’s permeability is not homogeneous and uniform across the body. There are hairy areas that are more permeable, and non-hairy regions (e.g., palms of the hand, or soles of the feet, etc.) in which the skin is thicker and less permeable. Furthermore, the variation in sweat glands, hair follicles, and sebaceous ducts contribute to the permeability of the skin and undergo continuous cellular turnover to replace aging cells. The total surface area of the skin is a function of the body weight, and it changes about 1.3 m2 for a 90 lb. to ~1.8 m2 for a 143 lb. and ~2.2 m2 for a 198 lb. individual. Skin includes two layers, the epidermis and the dermis. Under these two layers lie the hypodermis and the subcutaneous region (see Chapter 4, Section 4.5). The epidermis layer is comprised of fve layers: stratum corneum, stratum lucidum, stratum granulosum, stratum spinosum, and stratum basale. The dermis layer is constituted of papillary and reticular layers. The papillary layer consists of collagen fbrils and elastic fbers and metabolically active fbroblasts cells. The reticular layer is denser than the papillary and anchors skin appendages. The hair follicles, sweat glands, and two types of glands, namely apocrine and eccrine are the appendages of the skin The skin’s barrier function exists in the epidermis and is mainly the role of the stratum corneum to provide the barricade. The passage of xenobiotic is mostly through the epidermis and the widely distributed hair follicle and eccrine glands on the surface of the body. 5.1.2 Stratum Corneum The stratum corneum, the outermost layer of the skin, is formed from 15–25 layers of biochemically inactive corneocytes originated from living keratinocytes (Odland, 1983); with estimated weight density of 1.3 to 1.4 gm/cm2 (Scheuplein, 1978; Scheuplein and Bronaugh, 1983); and normal water content of 25-30% (Caspers et al., 2001) which establishes the layer’s permeability and play crucial role in maintaining the health of skin, enzymatic activity and mechanical properties (Harding et al., 2000; Rawlings and Harding, 2004). The stratum corneum structurally has two distinct 132

DOI: 10.1201/9781003260660-5

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protein and lipoidal phases and plays an important part in making the skin a remarkable physiological barrier to the penetration of xenobiotics. This important physiological role is related to its anatomical structure, which is created by impermeable densely packed corneocytes, separated by lipid bilayers, thus minimizing the transdermal absorption of both hydrophobic and hydrophilic compounds (Figure 5.1). The lipid envelope that surrounds the corneocytes is a long-chain monolayer of ω-acylated-hydroxy-ceramides that is covalently bound mainly to the glutamate residues of involucrin, which is a protein involved in the formation of keratinocytes cells (Elias et al., 2014). Corneocytes are formed from keratinocytes by losing their cytoplasmic organelles like nucleus, endoplasmic reticulum, ribosomes, Golgi apparatus, etc., and are connected by desmosome to form a matrix (Figure 5.1). If the lipid bilayer of the skin is depleted bacteria can penetrate the skin and interact with the viable epidermis, trigger the infammation and onset of skin disease like atopic dermatitis (Lipsky et al., 2020). It is worthy of note that chemical penetration enhancers are often added to a formulation to enhance transdermal absorption of the active ingredient of the formulation (Osborne and Musakhanian, 2018). The addition of chemical enhancers like hydroalcoholic solvents or lipophilic solvents and similar solvents enhance the absorption of medication by replacing the skin water content and dissolving the lipid bilayer of the barrier. Thus, the absorption is achieved by degrading the structural integrity of the skin, and the outcome is a susceptible skin. The lipid layers of the stratum corneum are considered the main avenue for transdermal absorption. Various proposed models treat the stratum corneum as the brick-and-mortar models, organized as a fully aligned or fully staggered, or partially staggered two-dimensional structure (Michaels et al., 1975; Elias, 1983; Tojo, 1987; Lieckfeld and Lee, 1992; Heisig et al., 1996; Charalambopoulou et al., 2000; Frasch and Barbero, 2003), which all defne the permeation through the stratum corneum according to the assumptions of the model and hexagonal shape of the corneocytes. The applicability and validity of the models are based on how well they predict the observed data and the ease of parameter estimation.

Figure 5.1 Depiction of structural matrix of stratum corneum showing the impermeable biochemically inactive corneocytes connected to each other by adhesion protein desmosome and separated by lipid bilayers that are attached to residual proteins left from the keratinocytes the antecedent of corneocytes, making the skin impermeable to hydrophilic and hydrophobic xenobiotics; the dispersed water content maintains the well-being of the skin. 133

5.1 TRANSDERMAL ROUTE OF ADMINISTRATION

The diffusion of xenobiotics through the stratum corneum depends on diffusion through the lipid layer ( Dcoeff ) , diffusion through the protein ( Dcoeff ) , and the partition coeffcient of the lipid

protein

compound Pcoeff between the protein and lipid environments. Using these parameters, the diffusivity of the stratum corneum (s) is defned as æ ( Dcoeff ) ö (5.1) lipid ÷ s = ( Pcoeff ) ´ ç ç ( Dcoeff ) ÷ protein ø è 5.1.3 Epidermis After crossing the stratum corneum, a xenobiotic encounters the viable epidermis layer, also known as the Malpighian layer; a moist environment, with no capillaries, but with living cells that facilitate the transfer of xenobiotics to the lower layers (Scheuplein, 1976; Flynn, 1985). The viable epidermis poses a signifcant barrier to transdermal diffusion even in the absence of the stratum corneum (Andrews et al., 2013). It has a variable thickness of about 150–220 μm and holds two primary cell types: 1) keratinocytes that in ascending order include stratum basale, stratum spinosum, stratum granulosum, stratum lucidum, and then stratum corneum, and 2) non-keratinocytes cells such as Markel cells, melanocytes, and Langerhans cells (Figure 5.2). The basal cells of the stratum basale, also known as stratum germinativum, are the innermost layer of epidermis capable of generating new keratinocytes cells to be moved toward the outer membrane to replace the stratum corneum layer that are continually removed from the skin surface. The cells created by the stratum basale reach the stratum spinosum where they replace water with keratin flaments. In the stratum granulosum layer, the cells die and get compressed to form the stratum corneum. The live keratinocytes remaining in the basal layer is considered the epidermis stem cells. The keratinocytes make up approximately 75000 cells / mm2 (Hoath and Leahy, 2003; Bauer et al., 2001). The turnover rate in human and pig is about 30 days and in rodents it is much faster (Weinstein, 1966; Halprin, 1972; Bergstresser and Taylor, 1977).

Figure 5.2 Illustration of fve layers of epidermis where the live keratinocytes of stratum basale in their migration toward the stratum corneum replace their water content with keratin flament in stratum spinosum, die at stratum granulosum, change to corneocytes, are compressed at the stratum lucidum, and fnally form the stratum corneum matrix; other cells of epidermis include Markel cells and melanocytes that are mostly in stratum basale and Langerhans cells scattered in stratum spinosum. 134

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Carcinogenic agents/elements can cause irreparable damage to basal cells and trigger skin cancer, or squamous cell carcinomas. Melanocytes are dendritic cells present in the basal layer with a density of approximately 2000 cell / mm2 and their major role is to provide the skin with melanin. Cancerous melanocytes are identifed as melanoma. Markel cells are derivatives of the neural crest and axon and are in the basal region of the epidermis. They are mechanosensory cells that regulate touch receptors. Langerhans cells are derived from bone marrow and play an important role in the skin immune response. The epidermal living cells are joined together frmly and form another barrier to penetration of xenobiotics. Xenobiotics after crossing the stratum corneum must permeate through the lipoidal cell membranes of the epidermis. For ions and polar compounds this is not a thermodynamically favorable act, and the lipophilic compounds after permeating through the cell membranes face the aqueous cytoplasmic environment (Flynn, 1985). Thus, regardless of differences in physicochemical characteristics, the diffusion and permeation of xenobiotics through the live epidermis layer is not a straightforward process (Flynn, 1985; Masters et al., 1997). 5.1.4 Dermis Attached to epidermis by a thin basement membrane is the dermis, which is a connective tissue layer above the subcutaneous region. It has two distinct regions; one is attached to the epidermis called the papillary region, and the other is in the lower layer of the skin, a more vascular region known as the reticular region above the subcutaneous area (James et al., 2005). The dermis is formed by a network of elastic fbers, a matrix of collagen and elastic tissue, and a watery gel environment called ‘ground substance’ (Achterberg et al., 2014), comprised of glycosaminoglycans e.g., glycoproteins, proteoglycans, and hyaluronic acid, which maintain the skin’s homeostasis and protect the skin and subcutaneous region. The dermis assists in thermoregulation and helps the touch and heat sensations. The thermoregulation is mainly through a specialized structure called glomus bodies, which are complexes of glomus chemoreceptive cells, smooth muscle cells and vessels (Sethu and Sethu, 2016; Friske et al., 2016). 5.1.4.1 Dermis Cells The synthesis of the collagen in the dermis is by its fbroblast cells, which are also involved in wound healing. Other cells in the dermis include mast cells, which is a type of white blood cell that secretes vasoactive and proinfammatory mediators, also involved in wound healing (Wilgus and Wulff, 2014); histiocyte cells that help the immune system; and, fnally, adipocyte cells that originate from the adipose tissue of subcutaneous region and are involved in insulation, wound healing, and energy storage (Kruglikov and Scherer, 2016; Driskell et al., 2014). 5.1.4.2 Dermis Appendages The dermis holds many functional elements of skin such as capillary network, nerve ending and touch and heat receptors, pilosebaceous units that refer collectively to hair follicles and sebaceous gland, exocrine and apocrine sweat glands, and the lymphatic network (Figure 5.3), which collectively play a signifcant role in the absorption of xenobiotics. 5.1.4.2.1 Hair Follicle Hair is formed from dead keratinized cells, and its structure has two parts, the hair follicle and the hair shaft. There are two types of hair follicles, those that are androgen-free hair, like eye lashes and eyebrows (known as terminal follicle), and those that are hormone-dependent hair on the scalp, beard, chest, and pubic region (Verma et al., 2016). The active hair follicles exist between the surface of skin and dermis hence bypassing the stratum corneum, i.e., the frst skin barrier. In theory, the follicles should provide a direct passage to the dermis for all xenobiotics. However, the surface area of hair follicles opening on the skin is roughly about 0.1% of total surface area of skin, and for that reason it is considered a reliable but minor route for the absorption of xenobiotics. It should be noted that the hair follicles of different regions of the skin show variability in their opening size, diameter, and surface area. Thus, the percent of absorption based on the number of hair follicles and the surface area may well be more than 0.1% in different areas of the skin. In practice, the hair follicles contribute more signifcantly when the absorption is from a depot attached to the skin for an extended time. Furthermore, the hair follicles are highly vascular parts of the skin, and a xenobiotic molecule in its migration toward the dermis is expected to absorb into the follicle’s capillaries before reaching the end of its journey. 135

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Figure 5.3 Diagram of structural components of skin showing the combination of epidermis, dermis, appendages, and subcutaneous region; highlighting the presence of the hair follicle and shaft, sebaceous gland, eccrine and apocrine sweat glands, and their openings on the skin surface. Both parts of the hair, i.e., shaft and follicle, have functions such as protecting the skin from UV radiation, retaining heat, acting as body temperature regulators and indicators of change in temperature and safety. 5.1.4.2.2 Sebaceous Glands Sebaceous glands are attached to hair follicles and secret the oily sebum that lubricates the skin. The sebaceous glands, like sweat, saliva, and milk glands, are considered exocrine glands and are found in all areas of skin except areas, like the palms, lips, etc. 5.1.4.2.3 Sweat Glands (Eccrine and Apocrine Glands) There are two types of sweat glands, eccrine and apocrine. Eccrine glands are present in all areas of skin, except the lips and parts of external genitalia, and since their secretory gland is in the dermis with a connecting duct to the surface of the skin, there is no loss of cytoplasm in their secretion. The chemical composition of sweat consists of water, electrolytes, urea, and trace minerals, like iron, copper, zinc, etc., with acidic pH to protect the skin from microbial growth. Sweat plays a role in thermoregulation and cooling the body temperature. Furthermore, xenobiotics and/or their metabolites may be eliminated in the sweat by the eccrine glands; this elimination is part of the overall elimination of xenobiotics from the body. Apocrine glands are a group of sweat glands present only in certain areas of the body like the armpits, external genitalia, nipple, areola, etc.; and like the eccrine glands, their secretory part is in the dermis, and the duct opens on the surface of related skin areas. Apocrine glands have different sizes, their secretion is yellow, oily, viscous, and aromatic. These are hormonally controlled glands that are more active during puberty, and their secretion contains pheromones. 5.1.5 Transdermal Absorption, Metabolism, and Disposition 5.1.5.1 Transdermal Absorption The absorption of xenobiotics through the skin occurs by two major routes: trans-epidermal and trans-appendageal. Trans-epidermal is the diffusion through or between the epidermis cells and trans-appendageal is the passage via the appendages such as hair follicles or sebaceous glands, 136

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etc. For achieving systemic effect, xenobiotics must overcome the stratum corneum and associated layers underneath to reach the systemic circulation. Transdermal delivery of medications is safe, non-invasive, and painless; thus, it has improved patient compliance. Transdermal absorption has no major pre-systemic metabolism, manifest more predictable bioavailability, and in most cases exhibit longer duration of action; thus, it is a preferred route to the oral, subcutaneous, and other routes of administration. In addition to the administration of medications for therapeutic purposes, the skin is constantly exposed to chemicals like agricultural pesticides, air pollution, and hazardous chemicals in environment; cosmetics and healthcare products with absorbable xenobiotics, like sunscreen, hair dyes, etc., chemicals that can permeate and penetrate the skin and reach the capillaries and systemic circulation. Furthermore, exposure to chemicals can cause skin diseases, e.g., skin cancer, dermatitis, etc. The stratum corneum, which is a barrier for the absorption of drugs is considered a protector from exposure to harmful chemicals. The skin absorption of chemicals in various work environments is acknowledged as a major occupational health risk, which has generated signifcant research and understanding of the related risks. The absorption of both groups of compounds (i.e., therapeutic and non-therapeutic xenobiotics) is the same, and essentially both categories of compounds diffuse the skin barrier to reach the dermis, and most often their absorption is governed by frstorder kinetics. 5.1.5.1.1 Approaches to Enhance the Rate and Extent of Transdermal Absorption There are four different methodologies to enhance the rate and extent of absorption of therapeutic xenobiotics through the epidermis, including the stratum corneum and dermis, as follows: 1. The frst approach is the traditional application of a drug on the skin, either using a patch or semisolid dosage forms, relying on the physicochemical characteristics of medication, the condition and water content of the skin, and having the medication in a formulation that facilitates the diffusion through the stratum corneum or penetration through the appendages. The medication-related absorption factors include the diffusion coeffcient of drug through the skin, concentration of the compound in terms of thermodynamic activity on the surface of the skin, solubility and stability of the drug in the formulation, pKa of the compound, and pH of formulation; particle or molecular size; partition coeffcient of the compound in the formulation; and the viscosity of formulation. The formulation factors affect not only the physical state of compounds applied to the skin but also the physical characteristics of the skin, such as the thermodynamic activity of water in the skin and formulation. The environmental factors like temperature, dry vs humid weather conditions, and the fuctuation of temperature also infuence the rate and extent of absorption of xenobiotics. 2. The second approach is to add chemical penetration enhancers to the formulation to modify the composition of the stratum corneum and possibly other layers of the epidermis to facilitate the penetration of the compound through holes and cracks they create in the barrier to enable the medications to reach the dermis and its capillaries (Strati et al., 2021; Lopes et al., 2015). The presence of alcohol, polyols, or other organic solvents (Vasyuchenko et al., 2021); the presence of a surfactant (e.g., 5% sodium lauryl sulfate, or 0.50% polysorbate 80) to make the skin more permeable; the presence of penetration enhancers such as dimethyl sulfoxide (DMSO), dimethylformamide (DMF), dimethyl acetamide (DMA) and Azone; hydrogenated soybean phospholipid (Nishihata et al., 1988); alcohols with long carbon chains (C8-C14); cyclicmonoterpenes and n-octanol (Parsaee et al., 2002; Ho et al., 1994); nonionic surfactants (Iwasa et al., 1991); and propylene glycol and isopropyl myristate (Ho et al., 1994; Santoyo et al., 1995) are a few examples of chemical enhancers that, as indicated earlier, may change the structural integrity of the skin at the site of administration. The rate and extent of absorption of drugs in the presence of a chemical penetration enhancer is higher than the regular application of drug on the skin, i.e., the frst approach. 3. The third approach is to use the traditional formulation, or the formulation with a chemical penetration enhancer, in conjunction with physical enhancers like iontophoresis (Ita, 2015; Gratieri et al., 2013) or sonophoresis (Wang et al., 2021; Kaushik and Keck, 2021, Ita and Popova 2015). Physical enhancers are considered more effective than the chemical enhancers. Sonophoresis is based on the use of ultrasound with low, intermediate, or high frequency that increases skin permeability by mechanisms like cavitation and cellular effect, rectifed diffusion, etc. During electrophoresis, various mechanisms, such as electromigration and 137

5.1 TRANSDERMAL ROUTE OF ADMINISTRATION

electroosmosis, drive the ions across the skin and epidermis. The driving force is the repulsion of positively charged cations by anode and negatively charged anion by cathode. The rate and extent of absorption, depending on the physicochemical nature of the drug molecule, may increase signifcantly in the presence of the physical enhancers. 4. The fourth approach is to physically cross the stratum corneum using microneedle-based devices (Makvandi et al., 2021; Prausnitz and Langer, 2008; Barry, 2001). This emerging feld of transdermal delivery systems is considered less invasive and more effcient in absorption of drugs with diverse molecular weights that include macromolecules, like proteins and peptides. In this approach, the physicochemical characteristics of the medications or their formulation play a lesser role in crossing the skin barrier. The primary function of microneedles is to make a series of temporary holes in the stratum corneum to facilitate the absorption or migration of the applied medication. The microneedles are long enough to puncture only the stratum corneum and not to reach nerves and blood capillaries. The use of microneedles is mainly for achieving the systemic effect, and its mechanism of absorption is different from the simple diffusion through the barrier or trans-appendageal penetration. The absorption rate depends highly on the pore size generated by the microneedles and the concentration of applied medication. 5.1.5.2 Cutaneous Metabolism of Xenobiotics After surmounting the stratum corneum, the cutaneous metabolism, known as the skin’s frstpass metabolism, is the next stumbling block to consider in absorption of xenobiotics that may reduce the extent of absorption and/or generate metabolites that are more reactive than the parent compounds. Understanding of the skin metabolism is critical in drug discovery and is achieved through pharmacokinetics and toxicokinetics (PK/TK) analysis of all xenobiotics. The metabolizing enzymes detected in human skin that are responsible for the cutaneous metabolism of endogenous and exogenous compounds are also present in the liver, except at a much lower level than the liver. The following enzymes of Phase I and Phase II metabolism are detected in human skin (Oesch et al., 2007; Baron et al., 2008, Zhang et al., 2009; Kazem et al., 2019): Phase I metabolism includes the oxidation, reduction, and hydrolysis of xenobiotics using: 1. Multiple expression of CYP450 isozymes, including CYP1A1, CYP1A2, CYP1B1, CYP2A6, CYP2A7, CUP2B6, CYP2C8, CYP2C9, CYP2C18, CYP2C19, CYP2D6, CYP2E1, CYP2J2, CYP2R1, CYP2S1, CYP2U1, CYP2W1, CYP3A4, CYP3A5, CYP4B1, CYP 26, and CYP29. These enzymes are involved in aliphatic and aromatic hydroxylation; O-, N-, S-dealkylation; N-oxidation, and N-hydroxylation. 2. Proteases are referred to many diverse groups of hydrolytic enzymes involved in hydrolysis and breaking down proteins. There are about 30 proteases that are expressed by the keratinocytes and immune cells of the skin. 3. Alcohol dehydrogenases that are involved with oxidation of alcohol and interconversion between alcohols and aldehydes. 4. Aldehyde dehydrogenases are also oxidizing enzymes involved in the detoxifcation of endogenous and exogenous aldehydes by converting them into carboxylic acids, which ultimately leave the body. 5. Esterases and amidases are hydrolytic enzymes that function like proteases and cleave an ester or amide bond. 6. Epoxide hydrolase that hydrolyses the epoxide residue to hydroxyl residues. 7. NAD(P)H Quinone reductase is a favin adenine dinucleotide-containing enzyme that is involved in two-electron reductions of quinones and protects the body against free radicals. 8. Flavin-containing monooxygenase is involved in oxidation of xenobiotics with S- and N- groups. 9. Cyclooxygenase, also known as prostaglandin-endoperoxide synthase, is involved with prostaglandin synthesis and prostanoids, like prostacyclin and arachidonic acid. 10. Steroid 5-alpha-reductase is responsible for the metabolism of steroid and conversion of testosterone to dihydrotestosterone. 138

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Phase II metabolism is involved in conjugation with the xenobiotics and/or their metabolites to facilitate their elimination from the body: 1. UDP-Glucuronosyltransferase forms glucuronide conjugates through glucuronidation of lipophilic compounds, i.e., transfer of glucuronic acid to the hydrophobic molecules. 2. Sulfotransferases catalyzes the transfer of sulfate group from a donor molecule to hydroxyl or amine group of a compound. 3. Glutathione S-transferase catalyzes the conjugation of glutathione to several electrophiles. 4. N-acetyltransferase catalyzes the transfer of acetyl group to xenobiotics like aromatic amines, etc. 5.1.5.3 Skin Transport Proteins In addition to cutaneous metabolic activities that infuence the extent of absorption and elimination of xenobiotics, there are transport proteins that infuence the disposition (i.e., distribution and elimination) of the absorbed xenobiotics. These transporters are involved in translocation of both endogenous and exogenous compounds throughout the body including the skin (Kell and Oliver, 2014). The transport proteins that are expressed in skin are of the multidrug resistance proteins (MDR) family with its notable member MDR1, known as Pgp (ABCB1), and MDR3, MDR4, MDR5, and MDR6 (Randolph et al., 1998; Baron et al., 2001); multidrug resistance-associated transport proteins (MRPs) (ABCC2); organic anion transporting polypeptides (OATPs); organic cation transporters (OCTs); and oligopeptide transporters (PEPTs). Identifying the skin transporters with nomenclature of ATP-binding cassette (ABC) and solute carrier (SLC), the following are the most comprehensive ones detected to date (Takenaka et al., 2013; Osman-Ponchet et al., 2014; Fujiwara et al., 2014; Alriquet et al., 2015; Takechi et al., 2018; Al-Majdoub et al., 2020; Nielsen et al., 2021): 1. ABCA (2, 5, 6, 7, 8, 9, 10, 12), among them, ABCA2 is relatively common and ABCA7 is less common. 2. ABCB (1, 2, 3, 4, 6, 7, 8, 10, 11), ABCB1 is less common. 3. ABCC (1, 2, 3, 4, 5, 7, 9, 10, 11, 12), ABCC1 is relatively common, and ABCC3 is most abundant. 4. ABCD (1, 2, 3, 4). 5. ABCE (1). 6. ABCF (1, 2, 3). 7. ABCG (1, 2, 4), ABCG2 is less common. 8. SLCO1 (B1, B2). 9. SLCO2 (B1). 10. SLCO3 (A1). 11. SLCO4 (A1, C1). 12. SLC14 (1, 2). 13. SLC16 (A1, A4, A7). 14. SLC19 (A1). 15. SLC22 (A3), is most abundant. 16. SLC19 (A1). 5.1.6 Mathematical Interpretations of Transdermal Absorption of Xenobiotics While the skin with its complex structure is a site for transdermal delivery of therapeutic agents for local and systemic treatment or a site for exposure to environmental and/or occupational xenobiotics, it is essential to defne and predict the rate at which the administered or exposed compounds penetrate the skin. The mathematical modeling approaches that summarize the desired prediction include the physical models and pharmacokinetic models that include compartmental, physiological, and mechanistic modeling. The interpretation of the data is often complex and may not follow a unifed approach, it is noteworthy that the amount of a xenobiotic applied or 139

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deposited on the skin varies by the activity, lipophilicity, and other physicochemical characteristics of the compound, including the rate of deposition or application (Kissel et al., 1996). The sequence of xenobiotic permeation events through the skin after exposure can be summarized as: i. availability of the compound on the surface of the skin that may include the release from a device, or applied layer/patch, or permanency of exposure ii. partitioning on skin surface, i.e., migration of xenobiotic within the applied layer and between the applied layer and stratum corneum established by concentration gradient iii. diffusion and binding in stratum corneum, i.e., the frst phase of transferring enough compounds onto and gradually into the stratum corneum to build a concentration gradient with the lower layers of skin iv. permeation through the stratum corneum. v. partition in epidermis. vi. diffusion through the epidermis and dermis vii. cutaneous metabolism (dermis and viable epidermis) viii. permeation through the capillaries and uptake by the systemic circulation ix. Residual uptake, if any, by the subcutaneous layers including the adipose tissue. Inclusion of these events exclusively in a mathematical model, though doable, may reduce the applicability of the model due to the complexity of parameters estimation. Thus, most of the following models summarize these events to defne the overall process within acceptable limits to make it practical. 5.1.6.1 Diffusion Models The application of diffusion models to predict the skin absorption provides the basic understanding of permeation pathways and the infuence of formulations factors and environmental factors on the xenobiotic absorption. The assumptions of diffusion models are that the transdermal absorption is a passive process, the skin is a pseudo-homogeneous membrane, and xenobiotics are in contact with only one site of the membrane. The use of diffusion models often entails exact estimates of model geometry, affnity, and transport characteristics, and the models are mostly based on Fick’s frst or second law of diffusion. The second law is obtained from the frst law by introducing the idea of conservation of mass to do away with the fux J (Jost, 1952; Crank, 1975; Scheuplein, 1967; Scheuplein and Blank, 1973; Hada et al., 2005; Sugibayashi et al., 2010). The Fick’s frst law of diffusion can be defned as: A˜

Dcoeff ° Area ° time ° ˛C h

(5.2)

Where A is the amount of solute diffusing through the skin over time; Area and h are the subjected area of the skin to the solute and the thickness of membrane, respectively; ∆C is the concentration gradient within the membrane; and Dcoeff is the diffusion coeffcient. At steady state, the fux J ss across the membrane is defned as: J ss ˜

Dcoeff ° ˛C A ˜ h Area ° time

(5.3)

Since ∆C , the concentration gradient, is a conceptual item, one may use instead the product of concentration Capplied and partition coeffcient Pcoeff of the applied layer, i.e., J ss ˜

Dcoeff ° Pcoeff ° Capplied A ˜ Area ° time h

(5.4)

By dividing Equation 5.4 by Capplied , the permeability coeffcient K permeability is estimated as K permeability ˜

140

Dcoeff ° Pcoeff J ss ˜ h Capplied

(5.5)

PK/TK CONSIDERATIONS OF TRANSDERMAL, INTRADERMAL, INTRAEPIDERMAL ROUTES

Thus, to estimate the skin permeability, assuming the barrier is homogeneous, the diffusion coeffcient, partition coeffcient, and the thickness of the barrier must be known. The Fick’s second law of diffusion is defned as follows: ¶C Dcoeff ¶ 2C = ¶t ¶x 2

(5.6)

Where C is the concentration of permeating xenobiotic at time t and location x . Based on Fick’s second law of diffusion, the following general relationship summarizing the permeation and biotransformation of xenobiotic is proposed (Tojo, 1988).

{1 + B (C )} ¶¶Ct = 1r ¶¶r æçè D

coeff r

,r

¶C ö ¶ æ ¶C ¶C ö - F (C ) - M (C ) - ç Dcoeff x r ÷ - ux ÷ ¶r ø ¶x è ¶x ¶x ø

where B ( C ) is the binding term expressed as

p

(1 + qC )

2

(5.7)

in the stratum corneum, p and q are the

model parameters based on the Langmuir isotherm; F ( C ) is the facilitated transport through the stratum corneum in the cases where electrophoretic transport is applied (i.e., physical enhancer); and M ( C ) is the biotransformation defned by the Michaelis–Menten equation. The frst term after equal sign, 1 ¶ æç Dcoeffr , r ¶C ö÷ , stands for the radial diffusion; the second term, ¶ æç Dcoeff x r ¶C ö÷ , ¶r ø r ¶r è ¶x è ¶x ø ¶C stands for longitudinal diffusion; and the third term, ux stands for convective transport. ¶x As with the frst Fick’s law, providing known information and parameters would be necessary for the application of the model like the one presented in Equation 5.7. A common approach for predicting dermal exposure and absorption assumes that a fnite dose of xenobiotic is deposited on and absorbed by the skin. The application or exposure is uniform over the exposed skin surface area and the absorption is slow such that it can be differentiated from the application. The related relationship on the sequential processes of application and absorption is

(

)

(5.8)

Aabs = Area(cm 2 )´ Adeposited mg cm 2 ´ Fweight ´ Fabs

(

)

Where Aabs is the amount absorbed; Area cm 2 is the surface area available for contact; Adeposited is the amount applied on the skin; Fweight is the weight fraction of the compound in the formulation; and Fabs is the fraction of applied dose absorbed through the skin during the time of exposure. 5.1.6.2 Skin-Perm Model The permeation coeffcient through the stratum corneum for a lipophilic compound using the lipid fraction of the stratum corneum can be estimated by the following empirical equation (Fehrenbacher and ten Berge, 2000):

(

log PStratum = -1.326 + ( 0.6097 ´ log Pcoeff ) - 0.1786 ´ MW 0.5

)

(5.9)

Where log PStratum is the permeation coeffcient through the stratum corneum; Pcoeff is the partition coeffcient of the compound in an octanol-water system; and MW is its molecular weight. The empirical equations for estimating the permeation coeffcient of the protein fraction of the stratum corneum and the aqueous layer of epidermis are Pprotein ( cm h ) =

0.0001519 MW

Paqueous ( cm h ) =

2.5 MW

(5.10) (5.11)

The permeation coeffcient of skin from aqueous solution or from clear liquid is Pskin-water =

1 1 1 + PStratum + Pprotein Paqueous

(5.12)

141

5.1 TRANSDERMAL ROUTE OF ADMINISTRATION

The multiplication of the permeation coeffcient of the skin by the concentration of xenobiotic in the solution (mg/cm3) yields the permeation rate in mg/cm2/h (Lotens and Wammes, 1993; Wilschut et al., 1995). The absorption of gases or vapors is determined by multiplication of the aqueous-air partition coeffcient (Equation 5.13) with the skin-water permeation coeffcient (Equation 5.12): Pcoeff aq.air =

R ´ T ´ Wsb Vp ´ MW

(5.13)

Where R is the gas constant (8.314 Joule/mol/degree Kelvin, or Newton meter/mol/Kelvin) and T is the temperature in Kelvin. The product of Pcoeff aq. air with Pcoeff aq. air yields the permeation coeffcient for the interaction of gaseous compound and skin: ö æ ÷ ç 1 (5.14) ÷ Pskin-air = Pcoeff aq.air ç 1 1 ÷ ç ç PStratum + Pprotein + Paqueous ÷ ø è For gasses that absorb through the skin rapidly, the rate-limiting factor is the diffusion from the air to the skin. The stagnant air distance between the skin surface and the environment, d, is assumed to be 3 cm (Lotens and Wammes, 1993): æ

76 MW è

( Dcoeff )air = 360 çç Pair =

ö ÷÷ ø

( Dcoeff )air

Pskin-air - air =

(5.16)

d 1 1 Pskin-air

+

(5.15)

1 Pair

(5.17)

Where Dair is the air diffusion coeffcient in cm2/h; Pair is the permeation of the air layer in contact with the skin; and Pskin-air - air is the permeation coeffcient of the skin when the diffusion through the air is the rate-limiting factor for the absorption of gaseous compound. The multiplication of the permeation coeffcient of the skin with the concentration of gaseous compound in the air (mg/cm3) provides the rate of permeation. The skin-perm model provides an estimation of the skin permeation of vapors. The advantage of the model is to evaluate the skin permeation of gas or vapors from the permeation coeffcient of aqueous solutions. 5.1.6.3 One-Layered Diffusion Model The assumption of the model is the same as the diffusion models, i.e., the absorption of the xenobiotic in contact with skin follows the passive diffusion consistent with the compound’s activity gradient. According to the Fick’s frst law, the fux, J, of a substance through a one-layer barrier perpendicular to the direction of diffusion is directly proportional to the concentration gradient, that is, æ dC ö (5.18) J = -Dcoeff ç ÷ è dx ø Where J is the quantity of the compound diffusing per unit time through a unit area of the barrier, C is the concentration, and x is the distance the molecules travel. The quantity of the compound crossing the barrier at x in time dt is J dt whereas, the quantity leaving through the barrier at x + dx, ¶J ö at the same time is æç J + ´ dx ÷ dt . Thus, the net gain in the quantity of diffusing compound can be ¶x è ø expressed in terms of the difference between x and dx, that is, ¶J ö ¶J æ dCdx = J dt - ç J + ÷ dt = - ¶x dxdt ¶x è ø 142

(5.19)

PK/TK CONSIDERATIONS OF TRANSDERMAL, INTRADERMAL, INTRAEPIDERMAL ROUTES

The second law defnes the diffusion of the compound in the barrier as ¶ 2C ¶C = Dcoeff ¶x 2 ¶t

(5.20)

This can be integrated as the following fnite system: C = X ( x ) T ( t ) = [ Asin lx + B cos lx ] e

-l 2Dcoeff t

(5.21)

Under the assumption of homogenous one-layer membrane and sink condition at time t and distance x, the initial conditions at time t = 0 is as follows: f ( x ) = C 0 for 0 < x < h f (x) = 0

for h < x < L

The solution for this case is æ h 2 ¥ 1 - ( np L ) 2 Dcoeff t npx nph ö (5.22) e C = C0 ç + cos sin ÷ çL p ÷ L L n 1 è ø It may be necessary to evaluate Dcoeff as a function of time of the total solute diffused across the boundary. The absorbed amount through transdermal route of administration, ATD , is given by

å

ATD = -

t

òD

coeff

0

æ ¶C ö ç ¶x ÷ dt è øx = h

From Equation 5.22, æ ¶C ö is determined as ç ¶x ÷ øx=h è 2 æ nph ö ¥ -Dcoeff t ç ÷ ¶C nph 0 2 è L ø = -C e sin 2 ¶x L 1 L

å

(5.23)

(5.24)

Substitution of Equation 5.24 into Equation 5.23 yield the amount M as: M = C0

1 2p 2

¥

ån

1

sin 2

2

1

ìï Dcoeff n 2 p2t üïù nph é ê1 - exp íýú L2 L êë ïþúû ïî

(5.25)

The parameter ATD , the absorbed amount through the skin, can be evaluated for various ratios of h / L . For instance, when h / L = 1 / 2 , Equation 5.25 reduces to Equation 5.26: ATD =

C 0 L éê 8 14 ê p2 ë

¥

å m=1

ìï Dcoef ( 2m - 1)2 p2t üïù exp íýú 2 L2 ú ( 2m - 1) îï þïû 1

(5.26)

where m = 0, 1, 2, 3, … The average membrane concentration of xenobiotic is then estimated as Cave =

Pcoeff C ìï 8 í1 2 ï p2 î

¥

å m=1 1

æ Dcoeff ( 2m - 1)2 p2t ö ïü ÷ý ç exp 2 ÷ï ç L2 ( 2m - 1) øþ è 1

(5.27)

and the steady state as t Þ ¥ . 8 p2

¥

å m=1

æ Dcoeff ( 2m - 1)2 p2t ö ÷Þ0 exp ç ÷ ç L2 ( 2m - 1) ø è 1

(5.28)

2

Pcoeff C (5.29) 2 Equations 5.27 and 5.29 indicate that the average or steady-state concentration is directly proportional to the partition coeffcient of xenobiotics. \Css =

143

5.1 TRANSDERMAL ROUTE OF ADMINISTRATION

5.1.6.4 Two-Layered Diffusion Model This model resembles the anatomical features of the skin more closely. It takes into consideration the concentration of xenobiotic on the surface of the skin, in the stratum corneum, and the epidermis and dermis. The boundary conditions of the model between the stratum corneum and epidermis–dermis layers include the following (Sugibayashi et al., 2010): CED ˜ ° Pcoeff ˛Stratum ° Pcoeff ˛ED CStratum and dC Stratum ˛ DED ED ˜ Dcoeff °Stratum dCdx dx where ED stands for viable epidermis and dermis layer; Pcoeff is the partition coeffcient; Dcoeff and DED are the diffusion coeffcient of stratum corneum and epidermis–dermis; and C represents the concentration of the xenobiotic in different layers. The total permeability coeffcient of the whole skin includes the permeability coeffcient of the stratum corneum and that of viable epidermis– dermis layer (Ghanem et al., 1992): PT ˜

1 1 PStratum

°

(5.30)

1 PED

The reciprocal of the permeability coeffcient is referred to as the permeability resistance coeffcient. Thus, the total permeability of the skin PT is the reciprocal of permeability resistances of and viable epidermis–dermis layers ˜ PR ° , i.e., stratum corneum ˜ PR ° Stratum

ED

PT ˜

1

(5.31)

° PR ˛Stratum ˝ ° PR ˛ED

The permeability coeffcient of the stratum corneum is a function of its transcellular and paracellular permeations, that is, (5.32)

PStratum ˜ Ptrans ° Ppara The amount of xenobiotic absorbed per unit of area of stratum corneum is defned as ˆ

AStratum ˜

°P ˛



 ° Pcoeff ˛Stratum ˝ C ˝ LStratum ˘˘ 1 ˙ °RPStratum  R ˛T  ˇ

2 The amount of compound per unit area of viable epidermis–dermis layer is

(5.33)

°P ˛

° Pcoeff ˛ED ˝ C ˝ LED ˝ ° PR ˛ED R T

(5.34) 2 where LStratum and LED are the thickness of the stratum corneum and epidermis–dermis layer, respectively; ˜ PR ° is the total permeability resistance of the skin, that is, 1 PT , and C is the concenT tration of the xenobiotic on the surface of skin. Merging Equations 5.33 and 5.34 yields Equation 5.35, which defnes the total amount of xenobiotic per unit area of skin (Sugibayashi et al., 2010): AED ˜

AT ˜

ˆ ° PR ˛Stratum C  ˙ ° Pcoeff ˛Stratum ˝ LStratum ˘1 ˘ 2 ° PR ˛T ˇ 

 ° PR ˛ED   ˙ ° Pcooeff ˛ED ˝ LED ˝   ° PR ˛T  

(5.35)

Dividing Equation 5.35 by the total thickness of the skin yields the average concentration of the compound in the skin: Cave ˜

C 2LT

 ˆ ° PR ˛Stratum ˙ ° Pcoeff ˛Stratum ˝ LStratum ˘1 ˘ ° PR ˛T ˇ 

 ° PR ˛ED   ˙ ° Pcoeff ˛ED ˝ LED ˝   ° PR ˛T  

where LT is the total thickness of the skin, that is, LT ˜ LStratum ° LED . 144

(5.36)

PK/TK CONSIDERATIONS OF TRANSDERMAL, INTRADERMAL, INTRAEPIDERMAL ROUTES

The average concentration (Equation 5.36) in terms of the permeability is defned as Cave =

æ PT ö PT ïü C ïì ý í( Pcoeff )Stratum ´ LStratum ç 1 + ÷ + ( Pcoeff )ED ´ LED ´ P P 2LT îï ED ï Stratum ø è þ

(5.37)

5.1.6.5 Compartmental Analysis 5.1.6.5.1 Estimation of PK/TK Parameters and Constants of SkinAbsorbed Xenobiotic from Urinary Data – Infnite Dose The kinetics of systemic disposition of a transdermally absorbed xenobiotic becomes challenging when the magnitude of the absorbed dose is unknown. The compound can be rubbed off the skin or spread out on the skin, or the skin can be in contact with a large reservoir of the applied dose (Wurster and Kramer, 1961), etc. Regardless of type of application/exposure, the urinary data can be used to estimate the magnitude of absorbed dose. One approach is based on a single compartment model with an exit rate constant for excretion and metabolism (Cooper and Berner, 1985). The general differential equation of the model is the same as the one-compartment model with absorption from an extravascular route of administration (Figure 5.4) with the following adaptations: dA = ( Area skin ) J - KAt dt

(5.38)

Where A represents the amount of xenobiotic in the body and J is the fux across the skin at its inner layer; K is the overall rate constant of elimination from the compartment; and Area skin is the dA selected area of skin for application/exposed; and is the rate of change of amount in the comdt partment that represents the body with respect to time. The overall rate constant of elimination represents the excretion rate constant of the unchanged compound, k e , the rate constant of its metabolism, k m , and the non-renal, non-metabolic rate constant of elimination, which is a negligible quantity, is identifed as k nr . K = k e + k m + k nr

(5.39)

dAe dAm , and metabolic rate, ( i.e., the rate of dt dt change of the compartment identifed as Am in Figure 5.4), with respect to time, are:

The excretion rate of the unchanged compound,

dAe = k e At dt

(5.40)

dAm = k m A - k me Am dt

(5.41)

Figure 5.4 Diagram of a one-compartment model with absorption from an unknown dose applied on the skin surface and fux of J into the central compartment, At , with the exit rate constants of k e for excretion of unchanged xenobiotic, k m for frst-order metabolism, and k me for elimination of the metabolite(s); the absorbed dose is estimated from the applied dose by the known values of Ae and Ame with the related PK/TK parameters and constants of urinary data. 145

5.1 TRANSDERMAL ROUTE OF ADMINISTRATION

In the case of an incalculable dose, one approach is to rely on the urinary excretion and metabolism of the compound. The total rate of elimination from the body, based on the assumptions of the model (Figure 5.4) that all biological processes follow frst-order kinetics, and the effective dose for systemic effect is the amount absorbed in the systemic circulation is Total Rateof Elimination = KA = k e At + k me Am

(5.42)

The total amount of the applied dose eliminated unchanged and eliminated as metabolite(s) in the urine are the following integrated equations: Ae = k e

t

ò A dt

Ame = k me

(5.43)

t

0

t

òA

mt

0

(5.44)

dt

The asymptotic limit with an infnite dose is (Cooper, 1976; Cooper and Berner, 1985): æk AeArea = lim Ae = ( Area )skin ´ J ss ´ ç e Ae t ˜tlag èK

ö Ae ÷ ´ t - tlag ø

)

(5.45)

æk ö Area Ame Ame = lim = ( Area )skin ´ J ss ´ ç m ÷ ´ t - tlag me t ˜tlag è K ø

)

(5.46)

(

(

Where J ss is the steady-state fux, and the lag time of excretion and elimination of metabolite(s) are Ae tlag =

Ame tlag =

1 kDcoeff

+

1 K

1 1 1 + + kDcoeff k me K

(5.47) (5.48)

Therefore, the lag time of the system is the sum of lag time of excretion and lag time of elimination of metabolite(s). The missing fractional sum from the model is the amount eliminated through non-renal non-metabolic elimination that may include biliary elimination. The kDcoeff value in Equations 5.47 and 5.48 is kDcoeff =

6Dcoeff l2

(5.49)

Where Dcoeff and l 2 are the diffusion coeffcient and thickness of stratum corneum, respectively. Combining Equations 5.45 and 5.46 yields Equation 5.50, representing the total amount absorbed through the exposed area of skin into the systemic circulation that eliminates from the body by excretion and metabolism at steady-state fux: i.e.,

( At )Area = AeArea + AmeArea k e + k m ö é æ k e ö Ae æ k m ö Ame ù (5.50) ÷ tlag - ç ÷ tlag ú ÷ ´ êt - ç è K ø ë è ke + km ø è ke + km ø û Equation 5.50 assumes that the urinary excretion and metabolism reach a maximum, and if this maximum is suffciently high enough before declining with time, the steady-state fux is achieved. At steady state, the total amount absorbed through the skin is equal to the summation of the ‘total amount absorbed multiplied by the fraction excreted unchanged’ and ‘the total amount absorbed multiplied by the fraction eliminated as metabolite(s).’

( At )Area = ( Area )skin ´ J ss ´ æç

5.1.6.5.2 Estimation of PK/TK Parameters and Constants of SkinAbsorbed Xenobiotics from Urinary Data – Finite Dose The fnite dose implies that a known volume of dose is in contact with the skin surface, and the solvent or vehicle of the xenobiotic does not evaporate. The model is the same as discussed in the previous section, except the fux is determined by Equation 5.51: Vdose ´ 146

dCdose æ ¶C ö = Area ´ Dcoeff ´ ç skin ÷ dt è ¶x ø x=0

(5.51)

PK/TK CONSIDERATIONS OF TRANSDERMAL, INTRADERMAL, INTRAEPIDERMAL ROUTES

dCdose where Vdose is the volume of the dose, is the rate of change of dose concentration with respect dt to time, and æ ¶Cskin ö is the partial differential equation of the concentration migration in the ç ¶x ÷ ø x=0 è skin with the initial condition of x = 0. The partial differential equation of concentration change in the skin with respect to time is ¶ 2Cskin ¶Cskin = Dcoeff ¶x 2 ¶t

(5.52)

The initial conditions of Equation 5.52 are Cskin ( x, 0 ) = 0 Cskin ( l,t ) = 0 Cskin ( 0,t ) = Pcoeff Cdose ( t )

(5.53)

where Pcoeff is the partition coeffcient of xenobiotic between the dose and skin, Cskin Cdose . The assumptions of the model include 1) the rate of diffusion in the applied formulation of the dose on the surface of the skin is much faster than the diffusion in the skin, i.e., the diffusion through the formulation is not the rate-limiting step for diffusion through the skin, and 2) the systemic circulation in the viable epidermis and dermis presents a sink condition. The solution of Equations 5.51 and 5.52 based on the boundary conditions of Equation 5.53 is defned as follows (Carslaw and Jaeger, 1959): æ ¶C ö J = -Dcoeff ç skin ÷ = 2J ss è ¶x ø x =l

¥

åf e n

(5.54)

-g nt

n=1

Where f n and g n are defned as fn =

a 2n cos a n b + b2 + a n2

gn =

a 2nDcoeff a 2n = kDcoeff l2 6

(

(5.55)

)

(5.56)

where a n is the root of (a tan a = b ); and b in Equations 5.55 and 5.56 is defned as (Carslaw and Jaeger, 1959): b=

Pcoeff ´ lskin ldose

(5.57)

Other parameters of Equations 5.54 through 5.57 are (5.58)

ldose = Vdose Area

which is referred to as the thickness of the applied dose; J ss is the steady-state fux that is dependent on the concentration. Substitution of Equation 5.54 into Equation 5.38 followed by the integration yields in Equation 5.59, which defnes the amount in the body, at a given time with a fnite dose (Cooper and Berner, 1985): ¥

( At )Area = 2 ( Area ´ J ss ) å

fn

(K - gn ) n=1

(e

-g nt

- e -Kt

)

(5.59)

The total amount of the metabolite in the compartment Am (Figure 5.4) can then be estimated as Am = 2 ( Area ´ J ss )

¥

k m f n é e -g nt - e -kmet e -Kt - e -kmet ù ê ú ( kme - K ) ûú n ë ê ( k me - g n )

åK-g n=1

(5.60)

When the rate constant of elimination of metabolites is greater than the overall elimination rate constant and the parameter g n , Equation 5.60 reduces to Equation 5.61, and the total rate of elimination is defned as Equation 5.62. 147

5.1 TRANSDERMAL ROUTE OF ADMINISTRATION

lim ( Am )t =

kme ˜g n kme ˜K

km At k me

(5.61)

The total rate of elimination is ke + km ( Area ´ J ss ) = ke At + kme Am K

(5.62)

ke + km . K 5.1.6.5.3 Estimation of PK/TK Parameters and Constants of Absorption into the Stratum Corneum with Concurrent Loss from the Skin Surface The model is a simple one-compartment model with respect to the amount in systemic circulation and a one-compartment model with respect to the amount on the surface of skin, which has either zero- or frst-order loss from the surface of skin and parallel frst-order absorption into the systemic circulation compartment (Figure 5.5) (Guy and Hadgraft, 1982; 1984). The focus of the model is how much of the applied or exposed xenobiotic reaches the systemic circulation compartment. Hence, the elimination from the systemic circulation compartment or the distribution from the systemic circulation to the peripheral compartment(s) are excluded from the mathematical relationships (Guy and Hadgraft, 1984). where the total fraction of absorbed dose eliminated in the urine is

Part One: Zero-Order Loss from the Skin Surface with Parallel First-Order Absorption into the Stratum Corneum The differential equations of the model (Figure 5.5) are dCsurface = -k0 - k1Csurface dt dCStratum æ Vsurface =ç dt è VStratum

ö ÷ k1Csurface - k 2CStratum ø

dCblood æ VStratum ö =ç ÷ ( k 2CStratum ) dt è Vd ø

(5.63) (5.64) (5.65)

dCsurface is the rate of concentration change of the applied dose on the surface of skin; k0 is dt the zero-order rate or rate constant of loss from the skin surface; k1 is the frst-order transfer rate constant from the surface to the stratum corneum; Csurface is the concentration of the applied or dCStratum exposed dose on the surface; is the rate of concentration change in the stratum corneum; dt Vsurface , VStratum , and Vd are the volumes of the surface, stratum corneum and systemic circulation, Where

respectively; CStratum is the concentration of the applied or exposed xenobiotic in the stratum corneum; k 2 is the frst-order transfer rate constant from the stratum corneum compartment to the dCblood systemic circulation; and is the frst-order rate of entry into the systemic circulation. dt Integrating the differential equations (Equations 5.63–5.65), using the Laplace transform, yields the following relationship:

Figure 5.5 Description of concurrent absorption and loss of a xenobiotic applied on the surface of skin; the random removal or loss may occur with frst- or zero-order process, and the absorption is governed by frst-order kinetic (rate constant k1 ) into the stratum corneum; the absorbed compound in the stratum corneum ultimately absorbs into the systemic circulation (rate constant k 2 ); since the focus of the model is only on the amount that reaches the systemic circulation, the elimination and distribution from the systemic circulation are not included in the model. 148

PK/TK CONSIDERATIONS OF TRANSDERMAL, INTRADERMAL, INTRAEPIDERMAL ROUTES

˜ Asurface °N ˛ ˙ˇ 1 ˝ kk0 ˘ e k t  kk0

(5.66)

1

ˆ

˜ VStratum ˛˛ ° Vsurface ˜ Vd ˛˛ ° Vsurface

1



˝ k k1 e k1t  e k2t  0 ˆˆ ˇ AStratum ˘N  k 2  k1 k2 ˙

ˇ

˘

1

1  k2t k1t   1  k  k k1e  k 2 e 2 1  

ˇ

˘

(5.67)

 t  ˝ k k e k1t e k2t 1 k1e k2t  k 2 e k1t  k1k 2 k0   1 2 22  2  2 ˆˆ ˇ Abloodinput ˘N  1   k1 k 2 k1 ˇ k1  k 2 ˘ k 2 ˇ k1  k 2 ˘  k 2  k1  k1k 2 ˙

ˇ

˘

(5.68) Where ˜ Asurface ° , ˜ AStratum ° , and ° Ablood˜input ˛ are normalized amounts on the surface, stratum N N N corneum, and blood with respect to the initial amount on the skin at time zero, respectively. The values are also normalized with respect to the ratio of the volume of distribution of receiver/donor compartments. Equations 5.66 through 5.68 are valid if the xenobiotic is present on the surface of the skin. The time required for the total dose on the surface to be lost or absorbed into the stratum corneum (tduration ) is tduration ˜

ˆ 1 ˛ k1 ln ˙ ° 1 ˘ k1 ˝ k 0 ˇ

(5.69)

Part Two: First-Order Loss from the Skin Surface with Parallel First-Order Absorption into the Stratum Corneum The differential equations for stratum corneum and blood compartments (Figure 5.5) are the same as described in Part One. The differential equation of change in concentration of xenobiotic on the surface of skin due to a frst-order removal is described below dCsurface ˜ ° ˝ kloss ˛ k1 ˙ Csurface dt

(5.70)

Where kloss represents the frst-order rate constant for the surface loss. The integrated equations of the model using the Laplace transform are

˜ Asurface °N ˛ e ˝˜ k ˜ VStratum ˛˛ ° Vsurface ˜ Vd ˛˛ ° Vsurface

loss ˙k1

°t

˝ k1  t  k k e ˇ loss 1 ˘  e k2t ˆˆ ˇ AStratum ˘N  k k k   ˇ ˘ 2 1 loss ˙

ˇ

(5.71)

˘

 k o  k1 ˘ t  ˝ k 2 e ˇ loss  ˇ kloss  k1 ˘ e k2t  k1 k 2 1   ˆˆ ˇ Abloodinput ˘N  ˇ kloss  k1 ˘ k2  ˇ kloss  k1 ˘  k2 ˙ 

(5.72) (5.73)

Because Equations 5.71–5.73 are exponential, the time required for the total dose on the surface to be lost or absorbed into the stratum corneum, i.e., (tduration ) is estimated only as the time required for a certain fraction of the amount on the surface to disappear. For example, the time required for 90% of xenobiotic on the surface to disappear by frst-order loss and frst-order absorption into the stratum corneum is

˜ tduration °90% ˛

ln ˜10 ° kloss ˝ k1

(5.74)

5.1.6.5.4 PK/TK Parameters and Constants of Xenobiotics Disposition on Skin and in Plasma When Stratum Corneum Acts as a Reservoir The focus of the previous section was defning the effect of skin surface loss on absorption into the stratum corneum. Thus, the blood compartment was considered only as a receiver compartment as depicted in Figure 5.5 without exit rate(s). In this section, as depicted in Figure 5.6, the model includes the elimination from the systemic circulation and takes into consideration the preferential attachment of the xenobiotic to stratum corneum (Guy and Hadgraft, 1984), known as the “reservoir effect” (McKenzie and Stoughton, 1962; Vickers, 1963; Stoughton, 1965; Carr and Wieland, 1966; Winkelmann, 1969). The differential equations of the model are 149

5.1 TRANSDERMAL ROUTE OF ADMINISTRATION

Figure 5.6 Diagram of the compartmental model when absorption is from the skin surface with frst-order absorption rate constant of k1 into the stratum corneum that acts as a reservoir and has distributional exchange with the central compartment; the model is considered a two-compartment model with respect to the stratum corneum and blood compartment with distribution rate constants of k 2 and k 3 , and frst-order elimination from the blood compartment with a rate constant of k 4 ; each compartment is defned by its volume and concentration terms. dCsurface = -k1Csurface dt

(5.75)

Vd dCStratum Vsurface k1Csurface - k 2CStratum + k 3Cblood = VStratuum dt VStratum

(5.76)

dCblood VStratum k 2CStratum - ( k 3 + k 4 ) Cblood = dt Vd

(5.77)

Vd dCurine k 4Cblood = dt Vurine

(5.78)

The integrated form of Equation 5.75 is a simple monoexponential equation that represents the time course of concentration change on the surface of the skin. The integrated equations for the stratum corneum and blood compartment are presented below. The concentration terms as in previous section are normalized with respect to the initial concentration:

( CStratum )N =

Vsurface é ( k 3 + k 4 + k1 ) e -k1t ( k 3 + k 4 - a ) e -at ( k 3 + k 4 - b ) e -bt ù + + k1 ê ú VStratum ëê ( k1 - a ) ( k1 - b ) ( a - b ) ( a - k1 ) (b - k1 ) (b - a ) ûú

( Cblood )N =

é ù Vsurface e -k1t e -at e -bt + + k1 k 2 ê ú Vd êë ( k1 - a )( k1 - b ) ( a - b )( a - k1 ) (b - k1 )(b - a ) ûú

(5.79)

(5.80)

Where ( CStratum ) and ( Cblood ) are the normalized concentrations of the stratum corneum and N N blood, that is,

( CStratum )N = ( Cblood )N =

CStratum C0

(5.81)

Cblood C0

(5.82)

where the initial concentration on the surface of the skin at time zero is C0 ; the rate constant of permeation from the surface to the stratum corneum is k1 , estimated from the rate of disappearance from the surface; the rate constant of penetration into the skin is k 2 , which includes permeation through the viable dermis; the relative affnity rate constant of the compound for the stratum corneum is k 3 , in this model it is considered larger than k2; the rate constant of elimination from the blood compartment is k 4 ; the concentrations at each compartment is C ; and the volume of each compartment is represented by V . 150

PK/TK CONSIDERATIONS OF TRANSDERMAL, INTRADERMAL, INTRAEPIDERMAL ROUTES

The hybrid rate constants of a and b, the roots of the quadratic equation generated during the integration by the Laplace transform (Addendum I, Part 2), are equal to a ´ b = k2k4

(5.83)

a + b = k2 + k3 + k4

(5.84)

5.1.6.5.5 PK/TK Parameters and Constants of Percutaneous Absorption of Xenobiotics through Viable Epidermis with Parallel Penetration via Appendages In addition to permeation through the layers of epidermis and dermis, xenobiotics may penetrate the transappendageal pathways to reach the dermis (Wallace and Barnett, 1978). Figures 5.7 through 5.9 are the diagrams of models for the combined permeation and penetration into the dermis, capillaries, and ultimately the systemic circulation. The model presented in Figure 5.7, compartment A1 is the constant amount of compound in solution or suspension applied on the surface of the skin. Compartment A2 is considered the skin barrier as a single homogeneous layer. Compartment A3 is the receiver compartment with a negligible exit rate constant of k 32 . This makes compartment A3 compliable with the sink condition (Wallace and Barnett, 1978). The differential equations of the model based on the assumption of each compartment are

dA1 =0 dt

(5.85)

dA2 = k12 A1 - ( k 21 + k 23 ) A2 dt

(5.86)

dA3 = k 23 A2 dt

(5.87)

The integration of Equation 5.87 yields Equation 5.88, which is the amount in compartment A3 as a function of time. The model is suitable for in vitro assessment of percutaneous absorption of xenobiotics with compartment A3 as the sampling compartment.

( A3 )t =

- k +k t A1k12 k 23 æ 1 e ( 21 23 ) + çt k 21 + k 23 çè k 21 + k 23 k 21 + k 23

ö ÷ ÷ ø

(5.88)

- k +k t At steady state, when t ⇒ ∞ and e ( 21 23 ) Þ 0, Equation 5.88 reduces to the straight-line equation (Equation 5.89):

( A3 )ss =

ö A1k12 k 23 æ 1 çt ÷ k 21 + k 23 è k 21 + k 23 ø

(5.89)

Where the slope of the line is the diffusion rate (i.e., the fux), and the y-intercept is the lag time of absorption

Figure 5.7 Diagram of a three-compartment model suitable for in vitro evaluation of transdermal absorption of xenobiotics, consisting of skin surface compartment containing the compound of interest in solution or suspension ( A1 ), skin that includes epidermis layers, dermis, and appendages (A2 ); and a receiver compartment with negligible exit rate (A3 ); the model assumes the A2 compartment is homogeneous and that the A3 compartment conforms with the sink condition; the rate constants of xenobiotic exchange between the formulation and skin k12 and vice versa k 21 , the absorption rate constant k 23 , and the negligible exit rate constant from the sampling compartment k 32 follow frst-order kinetics. 151

5.1 TRANSDERMAL ROUTE OF ADMINISTRATION

Figure 5.8 Diagram of a compartmental model highlighting the transappendageal pathway for penetration of xenobiotics directly into the sampling compartment A3 with the migrating rate constant of k13 , combined with concurrent permeation from skin surface compartment A1 into the skin compartment A2 , and penetration into the receiver or sampling compartment. The assumptions of the model are the same as Figure 5.7.

Figure 5.9 Diagram of a typical compartmental model based on permeation through different layers of skin in combination with the transappendageal penetration into compartment 3, the dermis; the exchanges between the compartments include between the skin surface (compartment 1) with the frst tissue compartment that represents the epidermis (compartment 2), and between epidermis and dermis (compartments 2 and 3) with fnal absorption into the sampling compartment (compartment 4); compartment 3, the dermis, acts as a reservoir compartment for the sampling compartment 4. J=

A1k12 k 23 k 21 + k 23

tlag =

(5.90)

1 k 21 + k 23

(5.91)

The transfer rate from compartment A1 to compartment A2 is k12 A1 , and the total amount in compartment A2 at steady state is the ratio of the rate to the exit rate constants:

( A2 )ss =

A1k12 k 21 + k 23

(5.92)

The model presented in Figure 5.8 characterizes the absorption from compartment A1 through the skin layers, compartment A2 , and direct entry via the transappendageal pathway from compartment A1 into compartment A3 . The differential equation of compartment A3 , representing the rates of input from compartments A1 and A2 into compartment A3 is dA3 = k13 A1 + k 23 A2 dt

(5.93)

The integration of Equation 5.93 yields the amount of the compound in the sampling compartment as a function of time as presented in Equation 5.94 (Wallace and Barnett, 1978).

152

æAk k A3 = A1k13t + ç 1 12 23 è k 21 + k 23

öæ 1 e ( 21 23 ) + ÷ çç t ø è k 21 + k 23 k 21 + k 23 - k +k

t

ö ÷ ÷ ø

(5.94)

PK/TK CONSIDERATIONS OF TRANSDERMAL, INTRADERMAL, INTRAEPIDERMAL ROUTES

The diffusion rate of the model (fux) at steady state includes the bypass passageway to compartment 3. æk +k k ö J ss = A1 ç 13 12 23 ÷ è k 21 + k 23 ø

(5.95)

The lag time of absorption in this model, due to the bypass passageway, is shorter than the previous model and is equal to tlag =

1 k13 k 21 + k 23 ( k 21 + k 23 ) k13 + k12 k 23

(5.96)

The amount in compartment A2 is the same as described in previous model. The model presented in Figure 5.9 is a more complex model based on permeation through different tissue compartments of the skin in combination with the transappendageal absorption. The skin layer is represented by three compartments (Barnett and Locko, 1977; Wallace and Barnett, 1978): Compartment 1 represents the skin surface and compartments 2 and 3 are the tissue compartments representing the stratum corneum and viable epidermis and dermis layers. The transappendageal transfer is into compartment 3 and is considered in both directions, an occurrence that may exist only for a few compounds. The combined entry into compartment 3 makes it a reservoir compartment with input into the sampling compartment. The complexity of the model lessens its practicality. 5.1.6.6 Diffusion–Diffusion Model and Statistical Moments for Percutaneous Absorption The diffusion–diffusion model assumes that the applied dose on the surface of the skin and the skin itself are two simple diffusion layers with thicknesses of xsurface and xskin , respectively (Kubota and Ishizaki, 1986). The xsurface value is equal to the real thickness of the layer applied on the skin multiplied by the bioavailability of the applied xenobiotics. The volume of the applied dose on the surface and that of the skin are defned as ( Area ´ xsurface ) and (Area ´ xskin ), and the diffusion coeffcient of each volume is ( Dcoeff ) and ( Dcoeff ) . The normalized diffusion coeffcient, a rate surface skin constant that simulate the frst-order absorption rate constant, is estimated as k diffusion =

( Dcoeff )skin ( xskin )

(5.97)

2

The surface and skin concentrations are defned as Csurface and Cskin , respectively. Thus, the fuxes of the xenobiotic on the surface and through the skin are defned as J surface = - ( Dcoeff )surface ´ Area ´ J skin = - ( Dcoeff )skin ´ Area ´

¶Csurface ¶xsurface

¶Cskin ¶xskin

(5.98) (5.99)

The partial differential equations according to the Fick’s second law for the surface and the skin are ¶Csurface ¶ 2Csurface = ( Dcoeff )surface 2 ¶t ¶xsurface

(5.100)

¶Cskin ¶ 2Cskin = ( Dcoeff )skin 2 ¶t ¶xskin

(5.101)

¶Csurface ( -xsurface ,t ) = 0 ¶t

(5.102)

Csurface ( 0, t ) Pcoeff = C ( 0, t )

(5.103)

J surface ( 0, t ) = J skin ( 0,t )

(5.104)

The boundary conditions are

153

5.1 TRANSDERMAL ROUTE OF ADMINISTRATION

J skin ˜ ° xskin , t ˛ ˜ ClcapillaryC ° xskin , t ˛

(5.105)

Csurface ˜ xsurface , 0 ° ˛ C0

(5.106)

Cskin ˜ xskin , 0 ° ˛ 0

(5.107)

The initial conditions are

where Clcapillary is the clearance of the compound from the dermis into the capillaries, and the product of Clcapillary × Cskin is the rate of removal of the compound from the dermis into the capillary blood fow. According to the diffusion–diffusion model and statistical moments, the mean residence time for the surface and skin are (Kubota and Ishizaki, 1986, 1985):

˜ MRT °surface ˛ The term

2 xsurface 1 ˜ 3 ° Dcoeff ˛

xsurface ˝ 1 ˆ xdiffusion ˆ˙ k diffusion

2 ˇ 1 xsurf face    ˘ 3 ˜ Dcoeff °surface

(5.108)

in Equation 5.108 is related to the diffusion through the layers applied on

surface

the surface of the skin and is not related to the diffusion through the skin.

˜ MRT °skin ˛

1 2k diffusion

kclearance =

˝

1

(5.109)

kclearance

Clcapillary Vskin

(5.110)

When k diffusion and kclearance are equal to zero, the mean residence time (MRT) and variable residence time (VRT) equations are defned as

˜ MRT °surface ˛

2 xsurface 1 ˝ 3 ˜ Dcoeff °

(5.111)

surface

˜ MRT °skin ˛

1 2k diffusion

˝

1

(5.112)

kclearance

MRT and VRT are discussed in Chapter 12 (Section 12.4), Non-compartmental Analysis Based on Statistical Moment Theory. Where xdiffusion is the apparent diffusion length of the skin; k diffusion is the normalized diffusion coeffcient, and kclearance is the normalized skin–capillary boundary clearance; Vskin is the volume of the skin. The variance of mean residence times (Yamaoka et al., 1978; Kubota and Ishizaki, 1985) is

˜VRT °surface ˛

4 xsurface 2 ˝ 45 ˜ Dcoeff °2

surface

˙

2xsurface xdiffussion

ˆ 1 1 1 ˙ ˙ 2 ˘˘ 2 ˇ 3k diffusion k diffusion kclearance kclearance

 2  ˙ ˜ MRTskin ° 

(5.113)

When k diffusion and kclearance are equal to zero:

˜VRT °surface ˛ ˜VRT °skin ˛

4 xsurface 2 ˝ 2 45 Dcoeff

˜

°

(5.114)

surface

1 2 1 1 ˝ ˙ ˝ 2 2 6k diffusion 3 k diffusion kclearance kclearance c

(5.115)

The overall mean residence time of surface and skin is

˜ MRT °overall ˛ ˜ MRT °surface ˝ ˜ MRT °skin ˝ MRTblood

(5.116)

and the overall variance of surface and skin is

˜VRT °overall ˛ ˜VRT °surface ˝ ˜VRT °skin ˝ ˜VRT °blood 154

(5.117)

PK/TK CONSIDERATIONS OF TRANSDERMAL, INTRADERMAL, INTRAEPIDERMAL ROUTES

Figure 5.10 Diagram of a physiological model PK/TK model for absorption and disposition of xenobiotics in different localities of the skin, from its surface to deep muscle and systemic circulation; the focus of the model is on the skin-related features and their role in absorption and disposition of a xenobiotic rather than its quantitative uptake by various tissues and organs of the body; each physiological region of the model is characterized by its volume, concentration of xenobiotic, and respective blood fow input; the sequence of the compartments is indicative of vertical diffusion of xenobiotic with the assumption of pseudo equilibrium between the well-stirred physiological region and its blood fow. The MRTblood and (VRT )

blood

are estimated from the plasma concentration–time curve.

5.1.6.7 Physiological Modeling of Percutaneous Absorption of Xenobiotics A typical physiologically based PK/TK model of drug absorption and disposition in different skin layers, muscle, and systemic circulation is depicted in Figure 5.10. The parameters and constants of the model include the concentration of absorbed xenobiotic in different regions of skin; clearance from each physiological region; blood fow rates; and the physiological volumes (Singh and Roberts, 1993). The assumptions of the model are the presence of pseudo equilibrium between physiological compartments and their blood fow, negligible horizontal diffusion under the surface, and all compartments associated with the vertical diffusion behave as well-stirred compartments with no elimination. The general rate equations of the model are (Singh and Roberts, 1993): Vsurface

dCsurface - k t 0 = ( Cls« d ) Csurface e ( s® d ) dt

(5.118) 155

5.1 TRANSDERMAL ROUTE OF ADMINISTRATION

(Vu )dermis

d ( Cu )dermis dt

(

) (

- k t 0 = Cls«d ´ Csurface e ( s«d ) + Qdermis ´ ( f u )blood ´ Cblood

(

)

- Qdermis ´ ( f u )dermis ´ Cdermis + Cld«sc

(( f )

u sc

)

´ Csc - ( f u )dermis ´ Cdermis

(5.119)

)

0 Where Csurface is the initial concentration on the surface, k s« d is the transfer rate constant of surface is the apparent volume of distribution of unbound xenobiotic in derto dermis layer, (Vu ) dermis

mis, ( f u )dermis is the fraction of unbound compound in dermis, ( Cu )dermis is the concentration of the

unbound compound in dermis, Cls« d is the clearance from the surface to the dermis layer, Qdermis is the blood fow to dermis, ( f u )blood is the fraction of unbound compound in the blood, Cdermis is the

concentration in dermis, Cld « sc is the clearance from dermis to subcutaneous layer, ( f u )sc , and Csc are the fractions of the unbound compound and concentration in subcutaneous layer, respectively. ( fu )blood = ( Pcoeff )db , where ( Pcoeff )db is the partition coeffcient between When ( f u )dermis = ( f u )sc and ( fu )dermis

dermis and blood, Equation 5.119 reduces to Equation 5.120:

(Vu )dermis

{(

dCdermis t - k 0 = Cls« d ´ Csurface e ( s« d ) dt

)( f )

u dermis

(

)

+ Qdermis ´ Cblood ´ ( Pcoeff )db - (Qdermis ´ Cdermis )

(5.120)

+Cld«sc ( Csc - Cdermis )} ( f u )dermis

Integrating Equation 5.120 by the Laplace transform and rearranging the constants and parameters yields Equation 5.121 for the area under the zero-moment curve (Singh and Roberts, 1993):

( AUC )dermis =

(Cl

s«d

) (

) (

´ ( Pcoeff )db + Cld«sc ´ ( AUC )sc ´ ( AUC )surface ´ ( f u )dermis + Qdermis ´ ( AUC )blood l

)

Qdermis + Cld«sc (5.121)

The equation of the mean residence time of dermis is also defned as:

( MRTint )dermis + éëCls«d ´ ( fu )dermis ( AUCsurface ( MRT )dermis =

( AUCblood

AUCdermis ) MRTsurface ùû + Qdermis ´ ( Pcoeff )db ´ r

AUCderemis ) MRTblood + Cld « sc ( AUCsc AUC Cdermis ) MRTsc Qdermis + Cld « sc (5.122)

Where ( MRT ) is the intrinsic mean residence time of the compound in dermis and is estimated dermis as

( MRTint )dermis =

Vdermis ( fu )dermis (Qdermis + Cld«sc )

(5.123)

5.1.6.8 Six-Compartment Intradermal Disposition Kinetics of Xenobiotics with Contralateral Compartments The six-compartment model is proposed to refne further the physiological factors infuencing the intradermal permeation of xenobiotics (Higaki et al., 2002; Jepps et al., 2013). The diagram of the model is depicted in Figure 5.11. The model is considered dose-independent with the following linear differential equations (Higaki et al., 2002): 1. Surface (applied dose): Vsurface 2. Viable skin (vs):

156

dCsurface = -Clsurface ®vs ´ Csurface dt

(5.124)

PK/TK CONSIDERATIONS OF TRANSDERMAL, INTRADERMAL, INTRAEPIDERMAL ROUTES

Figure 5.11 Diagram of a six-compartmental model for intradermal permeation and disposition of xenobiotics with contralateral compartments; the model’s assumption includes dose-independent PK/TK with linear frst-order biological processes; the clearance terms represent the rate constant of each compartment multiplied by the volume of the compartment; the numerical subscripts of the clearances are further defned as: Cl12 = Clsurface ® skin Cl23 = Clviable skin ®muscle Cl34 = Clmuscle ®pplasma Cl56 = Clc - viable skin ® c -muscle Cl32 = Clmuscle®viable skinn Cl43 = Clplasma ® muscle Cl65 = Clc -muscle ® c - viable skin Cl24 = Clviiable skin ® plasma Cl45 = Clplasma ® c -viable skin Cl46 = Clplasma ®c-muscle Cl42 = Clplasma®viable skin Cl54 = Clc - viable skin ® plasma C 64 = Clc - muscle ® plasma Cl

Vvs

dCvs = ( Clsurface - vs ´ Csurface ) + ( Clmuscle -vs ´ Cmuscle ) + ( Clplasma p -vs ´ Cp ) - ( Clvs-muscle + Clvs- plasma ) Cvs dt

(5.125)

3. Muscle: Vmuscle

dCmuscle = ( Clvs-muscle ´ Cvs ) + ( Clplasma-muscle ´ Cp ) - ( Cllmuscle -vs + Clmuscle - plasma + Clmuscle ) ´ Cmuscle (5.126) dt

4. Contralateral viable skin (cvs): Vcvs

ö æ ö V V V dCcvs æ = ç Clmuscle -vs ´ c - muscle ´ Cc - muscle ÷ + ç Clplasma-vs ´ cvs Cp ÷ - ( Clvs-muscle + Clvs - plasma ) ´ cvs ´ Ccvs V V dt Vvs muscle vs è ø è ø (5.127)

5. Contralateral muscle (c−muscle): Vc - muscle

ö ö æ V V dCm æ = ç Clvs -muscle ´ cvs ´ Ccvs ÷ + ç Clplasma - musccle ´ c - muscle ´ Cp ÷ V V dt vs muscle è ø ø è

(5.128)

V - ( Clmuscle -vs + Clmuscle - plasma + Clmuscle ) ´ c - muscle ´ Cc - muscle Vmuscle

157

5.2 INTRADERMAL ROUTE OF ADMINISTRATION

6. Plasma Vd

dCp æ æ ö ö V V = Clvs- plasma ´ ç Cvs + cvs ´ Ccvs ÷ + Clmuscle - plasma ç Cmuscle + c - muscle ´ Cc - muscle ÷ V V dt vs muscle è è ø ø ö æ V Vcvs +Clplasma-muscle c - musclee + Cltotal ÷ ´ Cp - ç Clplasma-vs + Clplasma a -muscle + Clplasma-vs V V muscle vs è ø

(5.129)

The model has been applied successfully to ft the data from ten compounds with different physicochemical characteristics (Higaki et al., 2002). 5.2 INTRADERMAL ROUTE OF ADMINISTRATION 5.2.1 Overview The intradermal route of administration is used for the injection of therapeutic agents, like insulin or antibodies, and for allergy testing, which is the result of immune response toward an administration of rich concentration of an antigen. The route is also used for intradermal vaccination utilizing the immune system of the skin by injecting antigen directly to the dermal environment containing dendritic cells, also known as antigen-presenting cells. The current applications of intradermal vaccines include Bacille Calmette-Guérin (BCG) tuberculosis vaccine, infuenza, COVIT-19, rabies, hepatitis A, hepatitis B, measles, polio, yellow fever, and so on. Intradermal injection of medications circumvents the stratum corneum and considering the environment of dermis (Section 5.1.4), the absorption takes longer than the deeper injections. Compared to intramuscular or subcutaneous injections, the intradermal route is considered less invasive and provides faster onset of action in contrast to transdermal applications on the surface of skin. The traditional intradermal injection is achieved by using traditional needle and syringe, or using new devices designed for direct introduction of drug into the dermis. The traditional injection, that is commonly used for testing and immunization of tuberculosis (Dacso, 1990; Hickling et al., 2011) or for immunization against rabies (World Health Organization, 1997) employs the Mantoux technique, named after Charles Mantoux, who described the method at the beginning of 20th century (Mantoux, 1909). The traditional Mantoux method (i.e., inserting about 1 mm of needle at a specifed shallow angle of 5–15° into the dermis) requires training to execute the injection correctly, which may often be inconsistent. Various systems have been developed to overcome the diffculties of the traditional Mantoux injection, among them are microneedle microinjection systems like INSTANZA®, IDfu®, SoluviaTM, and microinfusion (Laurent et al., 2007; Holland et al., 2008; Leboux et al., 2021). There are other systems in use or in development, like tattoo guns (Potthof et al., 2009; Pokorna et al., 2009) and dissolving needles using several different polymers (Sullivan et al., 2008; Lee et al., 2008), etc. 5.2.2 PK/TK Parameters and Constants of Drug Absorption from Intradermal Space to Blood Few investigations have focused on the PK/TK analysis of intradermally administered xenobiotics. Most PK data and models are related to the disposition of insulin, in particular, fast-acting insulin from subcutaneous and intradermal routes of administration (Lv et al., 2015). Conceptually, the data treatment of xenobiotics administered intradermally shares the same approach as the PK/TK analysis of subcutaneously administered insulin or xenobiotics as discussed in Chapter 4 (Sections 4.5.2.1–4.5.2.5). A PK model that is applicable for insulin data analysis, given intradermally or subcutaneously, is presented in Figure 5.12 with the following rate equations dA1 = k d1 A0 - k d2 A1 - k a1 A1 dt

(5.130)

dA2 = k d2 A1 - k a2 A2 dt

(5.131)

dA3 = k a1 A1 + k a2 A2 - k e A3 dt

(5.132)

According to the model in (Figure 5.12), the intradermally administered insulin dose forms a reservoir in the dermis that gradually diffuses and permeates with a rate constant of k d1 into compartment A1 that represents the amount of medication stationed around the capillaries, which 158

PK/TK CONSIDERATIONS OF TRANSDERMAL, INTRADERMAL, INTRAEPIDERMAL ROUTES

Figure 5.12 Diagram of a PK model to describe the disposition of insulin following an intradermal injection; the injection forms an insulin reservoir in the dermis that permeates slowly in the dermis milieu toward the capillaries and either absorbs into the systemic circulation compartment A3 or remains in a transitional compartment before diffusion into the capillaries and systemic circulation compartment; thus, the insulin absorbs directly in compartment A3 with the rate constant of k a1 and indirectly from the transitional compartment with the rate constant of k a2 ; both absorption rate constants, the diffusion rate constants k d1 and k d2 , and the elimination rate constant from the body,k e , follow frst-order kinetics.

then is either absorbed with a rate constant of k a1 into the systemic circulation identifed as compartment A3 or further diffuses with a diffusion rate constant of k d2 into a transitional compartment A2 . The amount in the transitional compartment will ultimately absorb completely with the rate constant of k a2 into the systemic circulation, compartment A3 . The only elimination is from compartment A3 , with the elimination rate constant of k e . All biological processes of diffusion, permeation, absorption, and elimination follow frst-order kinetics. 5.3 INTRAEPIDERMAL ROUTE OF ADMINISTRATION 5.3.1 Overview As discussed in Sections 5.1.2 and 5.1.3 and depicted in Figure 5.3, the epidermis, the nonvascular region of the skin, is comprised of a horny layer of stratum corneum, a clear layer of stratum lucidum, a granular layer of stratum granulosum, a spinous layer of stratum spinosum, and a basal layer of stratum basale. The physiological role of these layers from stratum basale to stratum lucidum is to form the horny layer of the skin. The intraepidermal route of administration is not considered a routine route for administration of therapeutic agents. The site has numerous nerve fbers and nerve endings that are present in the layers of human epidermis, in particular the stratum granulosum. Some of the nerve endings continue to the surface of the skin that are considered sensory in nature. These sensory neurons are part of the peripheral nervous system that pass through the dermis, enter the epidermis, and form the free nerve ending that makes the epidermis the sensory layer of the skin that modulates pain (Misery et al., 1999; 2003) and touch (Reinisch and Tschachler, 2005). The loss of epidermal nerve fbers plays an important role in the development of different types of neuropathies, and the measurement of nerve density in the epidermal layer is used as a diagnostic tool for often-debilitating neuropathy (Smith et al., 2005). An example of using the intraepidermal route of administration for local effect is in regenerative medicine, where epidermal cells of the interfollicular epidermis are used to treat burns (Jackson et al., 2017), melanomas (Hoerter et al., 2012), vitiligo (Khodadadi et al., 2010), and other relevant skin diseases or disorders. The intraepidermal injection of therapeutic agents are mostly intended for local effect, and considering the limitation of space and closeness to the skin surface, the route is considered less frequently for single injection to achieve systemic effect. The use of a high-dose intraepidermal drug reservoir, polymeric, or nanotechnology systems may provide sustained and targeted local 159

5.3 INTRAEPIDERMAL ROUTE OF ADMINISTRATION

drug delivery that may generate systemic effect. However, as discussed in Section 5.1.5.1.1 (Part 4), the emerging feld of microneedle-based devices that can only bypass the stratum corneum is considered an effcient intraepidermal route of administration that facilitates the transdermal absorption of medications for the purpose of attaining systemic effects. The PK/TK treatment of the data with the use of microneedle devices is the same as discussed in the transdermal route of administration. REFERENCES Achterberg, V. F., Buscemi, L., Diekmann, H., Smith-Clerc, J., Schwengler, H., Meister, J. J., Wenk, H., Gallinat, S., Hinz, B. 2014. The nano-scale mechanical properties of the extracellular matrix regulate dermal fbroblast function. J Invest Dermatol 134(7): 1862–72. Al-Majdoub, Z. M., Achour, B., Couto, N., Howard, M., Elmorsi, Y., Scotcher, D., Alrubia, S., El-khateeb, E., Vasilogianni, A. M., Alohali, N., Neuhoff, S., Schmitt, L., Roatami-Hodjegan, A., Barber, J. 2020. Mass cpectrometry-based abundance atlas of ABC transporters in human liver, gut, kidney, brain, and skin. FEBS Lett 594(23): 4134–50. Alriquet, M., Sevin, K., Gaborit, A., Comby, P., Ruty, B., Osman-Ponchet, H. 2015. Characterization of SLC transporters in human skin. ADMET DMPK 3(1): 34–44. Andrews, S. N., Jeong, E., Prausnitz, M. R. 2013. Transdermal delivery of molecules is limited by full epidermis, not just stratum corneum. Pharm Res 30(4): 1099–109. Barnett, G., Locko, V. 1977. Transport across epithelia: A kinetic evaluation. Biochim Biophys Acta 464(2): 276–86. Baron, J. M., Höller, D., Schiffer, R., Frankenberg, S., Neis, M., Merk, H. F., Jugert, F. K. 2001. Expression of multiple cytochrome P450 enzymes and multidrug resistance-associated transport proteins in human skin keratinocytes. J Invest Dermatol 116(4): 541–8. Baron, J. M., Wiederholt, T., Heis, R., Merk, H. F., Bockers, D. R. 2008. Expression and function of cytochrome P450-dependent enzymes in human skin cells. Curr Med Chem 15(22): 2258–64. Barry, B. 2001. Novel mechanisms and devices to enable successful transdermal drug delivery. Eur J Pharm Sci 14(2): 101–14. Bauer, J., Bahmer, F. A., Wörl, J., Neuhuber, W., Schuler, G., Fartasch, M. 2001. A strikingly constant ratio exists between Langerhans cells and other epidermal cells in human skin: A stereologic study using optical dissector method and the confocal laser scanning microscope. J Invest Dermatol 116(2): 313–18. Bergstresser, P. R., Taylor, J. R. 1977. Epidermal turnover time a new examination. Br J Dermatol 96(5): 503–9. Carr, R. D., Wieland, R. G. 1966. Corticosteroid reservoir in the stratum corneum. Arch Dermatol 94(1): 81–4. Caspers, P. J., Lucassen, G. W., Carter, E. A., Bruinining, H. A., Puppels, G. J. 2001. In vivo confocal Raman microspectroscopy of the skin: Noninvasive determination of molecular concentration profles. J Invest Dermatol 116(3): 434–42. Carslaw, H. S., Jaeger, J. C. 1959. Conduction of Heat in Solids, 128. Oxford: Clarendon Press. Charalambopoulou, G. Ch., Karamertzanis, P., Kikkinides, E. S., Stubos, A. K., Kanellopoulos, N. K., Papaioannou, A. Th. 2000. A study on structural and diffusion properties of porcine stratum corneum based on very small angle neutron scattering data. Pharm Res 17(9): 1085–91. 160

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Cooper, E. R. 1976. Pharmacokinetics of skin penetration. J Pharm Sci 65(9): 1396–7. Cooper, E. R., Berner, B. 1985. Finite dose pharmacokinetics of skin penetration. J Pharm Sci 74(10): 1100–2. Crank, J. 1975. The Mathmatics of Diffusion. New York: Oxford University Press Inc. Dacso, C. C. 1990. Skin testing for tuberculosis. In Clinical Methods: The History, Physical, and Laboratory Examination, Thrid Edition, eds. H. K. Walker, W. D. Hall, J. W. Hurst. Chapter 47, 1–5, Boston: Butterworths. Driskell, R. R., Jahoda, C. A., Choung, C. M., Watt, V., Horsley, V. 2014. Defning dermal adipose tissue. Exp Dermatol 23(9): 629–31. Elias, P. M. 1983. Epidermal lipids, barrier function, and desquamation. J Invest Dermatol 80(s6): 44–9. Elias, P. M., Gruber, R., Crumrine, D., Menon, G., Williams, M. L., Wakefeld, J. S., Holleran, W. M., Uchida, Y. 2014. Formation and function of the corneocyte lipid envelope (CLE). Biochim Biophys Acta 1841(3): 314–18. Fehrenbacher, M. C., ten Berge, W. F. 2000. Dermal exposure modeling in Charles. In Mathematical Models for Estimating Occupational Exposure to Chemicals, ed. B. Keil, 65–74. Falls Church: AIHA (A Publication of the American Industrial Hygiene Association). Flynn, G. L. 1985. Mechanism of percutaneous absorption from physiochemical evidence. In Percutaneous Absorption, eds. R. L. Bronaugh, H. I. Maibach. 17–52, New York: Marcel Dekker, Inc. Frasch, H. F., Barbero, A. M. 2003. Steady-state fux and lag time in the stratum corneum lipid pathway: Results from fnite element models. J Pharm Sci 92(11): 2196–207. Friske, J. E., Sharma, V., Kolpin, S. A., Webber, N. P. 2016. Extradigital glomus tumore: A rare ethiology for wrist soft tissue mass. Radiol Case Rep 11(3): 195–200. Fujiwara, R., Takenaka, S., Hashimoto, M., Narawa, T., Itof, T. 2014. Expression of human solute carrier family transporters in skin: Possible contributor to drug-induced skin disorders. Sci Rep 4: 5251. https://doi.org/10.1038/srep05251. Ghanem, A. H., Mahmoud, H., Higuchi, W. I., Liu, P. 1992. The effects of ethanol on the transport of lipophilic and polar permeants across hairless mouse skin: Methods/validation of a novel approach. Int J Pharm 78: 137–56. Gratieri, T., Pujol-Bello, E., Gelfuso, G. M., de Souza, J. G., Lopes, K. F. V., Kalia, Y. N. 2013. Iontophoretic transport kinetics of ketorolac in vitro and in vivo. Demonstrating local enhanced topical drug delivery to muscle. Eur J Pharm Biopharm 86(2): 219–26. Guy, R. H., Hadgraft, J. 1982. A pharmacokinetic model for percutaneous absorption. Internat J Pharmaceut 11: 119–29. Guy, R. H., Hadgraft, J. 1984. Percutaneous absorption kinetics of topically applied agents liable to surface loss. J Soc Cosmet Chem 45: 103–13. Guy, R. H., Hadgraft, J. 1984. Prediction of drug disposition kinetics in skin and plasma following topical administration. J Pharm Sci 73(7): 883–7. Hada, N., Hasegawa, T., Takahashi, H., Ishibashi, T., Sugibayashi, K. 2005. Cultured skin loaded with tetracycline HCl and chloramphenicol as dermal delivery system: Mathematical evaluation of the cultured skin containing antibiotics. J Control Release 108(2–3): 341–50. 161

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Halprin, K. M. 1972. Epidermal turnover time a new examination. Br J Dermatol 96: 503–9. Harding, C. R., Watkinson, A., Rawlings, A. V., Scott, I. R. 2000. Dry skin, moisturization and corneodesmolysis. Int J Cosmt Sci 22: 21–52. Heisig, M., Lieckfeldt, R., Wittum, G., Mazurkevich, G., Lee, G. 1996. Non-steady-state descriptions of drug permeation through stratum corneum. I. The biphasic brick-and-mortar model. Pharm Res 13(3): 421–6. Hickling, J. K., Jones, K. R., Friede, M., Zehrung, D., Chen, D., Kristensen, D. 2011. Intradermal delivery of vaccines: Potential benefts and current challenges. Bull World Health Organ 89(3): 221–6. Higaki, K., Asai, M., Suyama, T., Nakayama, K., Ogawara, K.-I., Kimura, T. 2002. Estimation of intradermal disposition kinetics of drugs: II. Factors determining penetration of drugs from viable skin to muscular layer. J Int Pharm 239(1–2): 129–41. Ho, H. O., Huang, F. C., Sokolaski, T. D., Sheu, M. T. 1994. The infuence of cosolvents on the invitro percutaneous penetration of diclofenac sodium from a gel system. J Pharm Pharmacol 46(8): 636–42. Hoath, S. B., Leahy, D. G. 2003. The organization of human epidermis: Functional epidermal units and phi proportionality. J Invest Dermatol 121(6): 1440–6. Hoerter, J. D., Bradley, P., Casillias, A., Chambers, D., Weiswasser, B., Clements, L., Gilbert, S., Jiao, A. 2012. Does melanoma begin in a melanocyte stem cell? J Skin Cancer 2012: 571087. https://doi.org /10.1155/2012/571087. Holland, D., Booy, R., De Looze, F., Eizenberg, P., McDonald, J., Karrasch, J., McKeirnan, M., Salem, H., Mills, G., Reid, J., Weber, F., Saville, M. 2008. Intradermal infuenza vaccine administered using a new microinjection system produces superior immunogenicity in elderly adults: A randomized controlled trial. J Infect Dis 198(5): 650–8. Ita, K. 2015. Transdermal iontophoretic drug delivery: Advances and challenges. J Drug Target 24(5): 386–91. Ita, K., Popova, I. E. 2015. Infuence of sonophoresis and chemical penetration enhancers on percutaneous transport of penbutolol sulfate. Pharm Dev Technol 21(8): 1086373. https://doi.org/10.3109 /10837450.2015.1086373. Iwasa, A., Irimoto, K., Kasai, S., Okuyama, H., Nagai, H. 1991. Effect of nonionic surfactants on percutaneous absorption of diclofenac sodium. Yakuzaigaku 51: 16–21. Jackson, C. J., Tønseth, K. A., Utheim, T. P. 2017. Cultured epidermal stem cells in regenerative medicine. Stem Cell Res Ther 8(1): 155. https://doi.org/10.1186/s13287-017-0587-1. James, W. D., Berger, T., Elston, D. 2005. Andrews Diseases of the Skin: Clinical Dermatology, Tenth Edition, Vol. 1, 11–12. Philadelphia: Saunders Elsevier. Jepps, O. G., Danci, Y., Anissimov, Y. G., Roberts, M. S. 2013. Modeling the human skin barriertoward a better understanding of dermal absorption. Adv Drug Deliv Rev 65(2): 152–68. Jost, W. 1952. Diffusion in Solids, Liquids, Gases. New York: Academic Press Inc. Kazem, S., Linssen, E. C., Gibbs, S. 2019. Skin metabolism phase I and phase II enzymes in native and reconstructed human skin: A short review. Drug Discov Today 24(9): 1899–910. Kaushik, V., Keck, C. M. 2021. Infuence of mechanical skin treatment (massage, ultrasound, microdermabrasion, tape stripping and microneedling) on dermal penetration effcacy of chemical compounds. Eur J Pharm Biopharm 168: 29–36. 162

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6 PK/TK Considerations of Rectal, Vaginal, and Intraovarian Routes of Administration 6.1 RECTAL ROUTE OF ADMINISTRATION 6.1.1 Overview The rectum is the terminal portion of colon between the end of sigmoid colon and beginning of anal canal, and it lacks any sacs, follicles, villi, or appendices. In adults it has a length of 15–20 cm, surface area of 200–400 cm2, and a pH range of 7.4–8.0 (Evans et al., 1988; Purohit et al., 2018); the pH range in children is 7.2–12.2 (Jantzen et al., 1989). The anal canal, the terminal portion of the intestine is approximately 4 cm and at the midpoint of the canal there is a saw-toothed epithelial junction known as the dentate line, where the anal glands open into the canal. There are approximately 4–10 anal glands in the canal (Boutros and Gordon, 2017). The epithelium of the rectum consists of columnar cells and goblet cells. The anal canal lining above the dentate line is also columnar cells, but below the dentate line is the squamous epithelium, which at the anal verge lining changes to regular skin with hair follicles, apocrine glands, and other skin histologic features (see Chapter 5). The sensory nerve supply of the anal canal is the inferior rectal nerve with sensory nerve endings in the vicinity of the dentate line. Physiologically, the rectum acts as a reservoir for more dense portions of feces, its strong contractions complete the process of defecations and, in healthy subjects, can accommodate signifcant volumes of stool with little change in pressure (Palit et al., 2012). The rectal mucosa is pink and the color above the dentate line is deep purple because of the hemorrhoidal plexus. The rectal site of absorption is a vascular region with the superior rectal artery as the main artery, and its venous system includes the upper, middle, and lower rectal vein network. The upper rectal vein is linked with the hepato-portal vein system; the middle and lower veins enter the inferior vena cava. Therefore, drugs that are absorbed from the upper part of the rectum enter the upper rectal vein and may be subject to frst-pass metabolism in the liver before distribution. Drugs that are absorbed in the middle and lower rectal veins avoid the liver and distribute in the body following the absorption into the systemic circulation (Van Hoogdalem et al., 1991). The inferior rectal arteries, a branch of the inferior iliac artery, supply the anal canal and its muscles. Furthermore, the lymphatic system of the rectum is signifcant, starts from the lower part of the rectum, and drains via the superior rectal lymphatics to the inferior mesenteric nodes. The absorption of lipophilic compounds in the lymphatic system is signifcant (Jannin et al., 2014; Nunes et al., 2014; Purohit et al., 2018), particularly since the absorbed compounds avoid hepatic frst-pass metabolism (de Boer et al., 1982; van Hoogdalem et al., 1991). 6.1.2 Pharmacokinetic Considerations of the Rectal Route of Administration The rectal route of administration is considered a practical alternative for administration of nonirritant medications that absorb by passive diffusion and exhibit low bioavailability when they are given orally. Its pH range favors the absorption of xenobiotics with pKa near the physiological range of 7.4–7.6. The rectal route is currently used to treat local syndromes like infammation, hemorrhoids, and constipation, and it is used for systemic illnesses like fever, nausea and vomiting, pain, allergies, and sedation. The rectal route is often the only route of administration for certain medications in the pediatric population and has shown a proven effcacy in the treatment of variety of diseases. A good example is the use of the rectal route for administration of quinine to treat uncomplicated malaria with the outcome better than intramuscular or intravenous administration; even in severe cases of malaria, it is the only alternative where intramuscular or intravenous administration cannot be performed (Barennes et al., 1998; 1996; 1995). The rectal mechanisms of absorption are transcellular and paracellular passive diffusion. There is no active site for absorption, and there is no proof of carrier-mediated transport. The rectal absorption is rapid enough to be substituted for intramuscular, oral, or intravenous routes of administration of certain medications in adult and pediatric patients. There are some clear advantages in using the rectal route of administration. Among them are: ◾ The rectum has a constant temperature of 37°C, and it has limited buffer capacity; thus, its physiological pH may fuctuate depending on the pH of its content or administered dosage form, but in general, the rectal environment is invariable and a stable environment for the absorption of drugs. The pH, viscosity, and temperature are rather constant; the rectal motility DOI: 10.1201/9781003260660-6

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is very low, and the residence time of drugs is long and only limited by defecation. Overall, the condition is suitable for administration of both regular and controlled-release dosage forms. ◾ The lymphatic circulation of the region is signifcant; thus, the site can be used effectively for lymph-targeted drugs and, as indicated earlier, for absorption of lipophilic compounds. ◾ The rectum has no enzymes; and its residual enzymatic activity is very low to negligible. The metabolic activity of the colonic microbiota continues to be present, but metabolites of xenobiotics in the rectum, per se, may not infuence the systemic disposition of a compound signifcantly. ◾ It is an alternative route for administration of drugs that exhibit low bioavailability following oral administration, due to the hepatic frst-pass effect, decomposition in the acidic environment of the stomach, or other reasons as discussed in Chapter 3. ◾ It is a convenient route for patients who cannot take solid dosage forms orally, like infants, adults with dysphasia (diffculty swallowing), or even when the bitter taste of medication cannot be masked by formulation methods. There are, however, some limitations in using the rectal route of administration, such as inconsistent absorption and bioavailability of a dosage form and consequent inter- and intra-individual variability that can be attributed to certain factors such as: ◾ The infuence of the fecal content on the absorption of rectally administered drugs is a factor that may contribute to the variability of bioavailability data in humans, or the functional constipation and delayed bowel movement, which often occurs in geriatric patients. ◾ The absorption into the upper rectal vein and hepatic frst-pass metabolism. ◾ Formulation of the administered dosage form, for example, suppositories require time for liquefaction and dispersion of the suppository bases before the release and absorption of the active ingredient; or administration of solution, suspension, emulsion, and micro-enemas, that the medication can migrate to the region of upper rectal vein. ◾ Presence of surfactants in the dosage form can infuence the rectal absorption of the administered drug. ◾ The volume and retention of a medicated enema if the drug is in solution form. ◾ pH of the environment and pKa of the drug. ◾ The disease states that impact the rectal absorption of drug. ◾ Excessive metabolism by microbiota at the site of absorption. ◾ Untimely discharge of the dosage forms due to the irritant nature of active and/or inactive ingredients. The diffusion through the barrier and general pharmacokinetic approaches for rectal administration of therapeutic agents is similar to absorption of xenobiotics through buccal and sublingual routes, where the input into the systemic circulation is through extravascular sites. Based on thepH -partition theory, the barrier between the site of absorption in the rectum and the systemic circulation allows only the permeation of an unionized form of a xenobiotic, and, e.g., for a weak base medication, the following relationship provides the steady-state distribution of unionized molecules between the rectal environment and the systemic circulation (Kakemi et al., 1965): rectum ) 1 + 10( (6.1) ( pka - pH systemic ) 1 + 10 If the site of absorption has the same pH as the systemic circulation, the steady-state distribution ratio, ( Ratio ) , would be equal to one, and for the lower rectal, the ratio would be greater than ss unity. The assumption of Equation 6.1 is that only unionized molecules can cross the barrier, and ionized molecules do not. However, there are drug molecules that in their ionized form can cross the barriers, though the absorption is restricted. Under the assumption that both unionized and ionized moieties are absorbed with permeability of Punionized and Pionized , the ratio of the steady-state distribution, Equation 6.1, can be modifed to Equation 6.2.

( Ratio )ss =

168

pKa - pH

PK/TK CONSIDERATIONS OF RECTAL, VAGINAL, AND INTRAOVARIAN ROUTES

( Ratio )ss =

pKa - pH systemic pKa - pH rectum ) ) +P y 10( 1 + 10( unionized Pionized ´ ( pKa - pHrectum ) (pka - pH systemic ) 1 + 10 + Punionized Pioonized 10

(6.2)

To include the concept of permeability of absorbable moieties in the actual rate constant of absorption for the passive diffusion at the site of rectal absorption, one approach would be to assume a stagnant-aqueous diffusion layer at the absorption side of the barrier and a blood-sink condition on the systemic circulation side of the barrier, using the approach recommended for the gastrointestinal tract (Suzuki et al., (Part I)1970a; (Part II), 1970b; Do et al., 1972), the following defnition for the frst-order absorption rate constant can be formulated. ka =

Area ´ V

Pa æ Pa ö 1 + çç ÷÷ è f ´ Pb ø

(6.3)

Area is the geometrical surface area/volume of the absorption site; Pa and Pb are the permeV ability coeffcient for the aqueous drug at the site of absorption and the lipoidal permeability of the barrier, respectively; f represents the fraction of unionized drug at the site of absorption. A different approach in defning kinetics of rectal absorption is by using the following relationship (Riegelman and Crowell, 1958):

Where

log

At - Aend = -k at A0 - Aend

(6.4)

Where At is the amount in the body at time t; Aend is the amount remaining at the site of absorption after the end of observation (i.e., residuum in rectum); A0 is the total amount (i.e., dose) at time zero; and k a is the pseudo frst-order diffusion and absorption rate constant. Equation 6.4 does not take into consideration the permeation of molecules through the membrane and ignores the overall elimination of absorbed drug from the body. The kinetics of absorption through other mucous membranes, such as different segments of the GI tract, may also apply to rectal absorption. The various approaches in pharmacokinetic and pharmacodynamic analysis of drug absorption from an extravascular route - including noncompartmental analysis, compartmental analysis, and physiological modeling – are applicable for the analysis of rectally administered therapeutics. For bioavailability and bioequivalence estimations of rectally administered drugs, most investigators prefer non-compartmental analysis (see also the chapter on noncompartmental analysis). 6.2 VAGINAL ROUTE OF ADMINISTRATION 6.2.1 Overview The vaginal route of administration, analogous to sublingual, buccal, and rectal routes of drug administration, is used for achieving local and systemic effects. The site of absorption is the vagina that performs diverse and complex physiological functions, such as menstruation, pregnancy, menopause, immune defense, etc., and it is a signifcant part of the female reproductive system. A characteristic of this route of administration is the large variability in shape, axis, and dimension of the organ. The average length of the fbromuscular structure of the vaginal canal is 6–12 cm long (Srikrishna and Corozo, 2013) between the vulva (external genitalia) and the cervix; it is positioned between the urethra and bladder anteriorly and the rectum posteriorly to the urethra. The vaginal surface area is 34 cm2 to 164 cm2 (72 cm2 ± 21 cm2), and the mean ± SD of width at fve evenly spaced locations are (17 mm ± 5 mm), (24 mm ± 4 mm), (30 mm ± 7 mm), (41 mm ± 9 mm), and (45 mm ± 12 mm) (Luo et al., 2016). The structure of the vaginal site of absorption includes 1) the inner mucosal epithelial stratum, 2) a thin layer of connective tissue, known as lamina propria, containing thin-walled veins, 3) the intermediate muscular layer, known as muscularis stratum, and 4) the outer layer of fbrous connective tissue, known as the adventitial layer. The inner mucosal epithelial stratum adheres frmly to the muscular layer. The mucosal epithelium is estrogen-dependent and changes during the menstrual cycle. The epithelium in an adult is rich in glycogen, particularly during ovulation, which is fermented by Döderlein’s bacillus also known as Lactobacillus acidophilus, lowering the vaginal pHand making the environment acidic. Lactic acid produced from glycogen breakdown by the Lactobacillus acidophilus acts like a buffer to maintain the vaginal pH between 3.8 and 4.5, which is an important protection feature of the site against pathogens. Setting 4.5 as the demarcation pH, the normal healthy vagina haspH £ 4.5; pH > 4.5 169

6.2 VAGINAL ROUTE OF ADMINISTRATION

is indicative of the presence of vaginal infection (Mania-Pramanik et al., 2008), or of decreasing levels of estrogen. Therefore, in the absence of infection, the vaginal pHcan be used to diagnose menopause (Moradan et al., 2010; Panda, et al., 2014; Makwana et al., 2020). The connective tissue containing thin-walled veins (the lamina propria) contributes to the diffusion of the vaginal fuid across the epithelium. The fuid seeps out of the capillaries due to vasocongestion between vaginal epithelial cells and enter into the lumen (Dawson et al., 2015). The fuid also helps the dissolution of the semisolid dosage forms at the site of absorption. The lamina propria makes the vaginal site a highly vascular region for the absorption of drugs. The muscularis stratum is made up of autonomically smooth muscle fbers holding signifcant numbers of transmitters; the function of majority of these transmitters is not clear yet. 6.2.2 Vaginal Microbiota The human vaginal microbiota refers to all organisms that colonize the vaginal tract. They impact the health of individual and can infuence the uptake or metabolism of medications administered intravaginally for local or systemic effect. There is signifcant interindividual diversity of vaginal microbiota. Furthermore, the variation in microorganisms during the female life span are rather substantial; and intraindividual variations is often infuenced by hormonal fuctuations (Miller et al., 2017; Hickey et al., 2013). Factors that may modify the vaginal microbiota include aerobic or anaerobic pathogens that cause vaginitis, age, menopause, diabetes, pregnancy, and smoking (Brotman et al., 2014). Microorganisms originate from the mother and colonize in the daughter with low levels of lactobacilli and high levels of microbial diversity (Kalia et al., 2020; Gajer et al., 2012). Members of genus Lactobacillus dominate the vaginal microbiota, and in healthy women, the genus represents 95% of the bacterial community (Ravel et al., 2011). The Lactobacillus species involved in the production of vaginal lactic acid include Lactobacillus crispatus, Lactobacillus gasseri, Lactobacillus iners, and Lactobacillus jensenii (Ravel et al., 2011; Van De Wijgert et al., 2014). The antimicrobial and immunomodulatory properties of lactic acid help to maintain the well-being of the vaginal route of administration (Tachedjian et al., 2017). Lactic acid by maintaining the acidic vaginal pH of 3.8–4.5 has been shown to eradicate a variety of vaginal pathogens like Escherichia coli, Neisseria, gonorrhoeae, and Chlamydia trachomatis (Hummelen et al., 2017; Gong et al., 2014; Graver and Wade, 2011; Tomas et al., 2003). The stability of the vaginal environment is dependent on maintaining the acidic pH and the symbiotic actions of organisms like Lactobacillus species. Changes caused by the alteration of the Lactobacillus species, results in changes in pH, viscosity of the fuid, and change in the composition of discharge. There exist other bacterial species in the vaginal microbiota, such as Bifdobacterium, Veillonela, Bacteroids, Actinomycetes, Fusobacterium, Propionibacterium, Staphylococcus aureus, Staphylococcus epidermidis, Peptococcus, Peptostreptococcus, Streptococcus viridians, Enterococcus faecalis, Prevotella bivia, and Gardnerella vaginalis but at a very low proportion with respect to genus Lactobacillus (Chen et al., 2017; Van De Wijgert et al., 2014; Rave et al., 2011). 6.2.3 Pharmacokinetic Considerations of the Vaginal Route of Administration There are certain advantages in using the vaginal route for achieving local and systemic effects. In a stable and healthy vaginal route, they include 1) The vaginal route is a highly vascular route with ample blood circulation. 2) The reachable surface area is reasonably large, and the permeability of small-molecule drugs is high, and by knowing the pKa of the compound and the pH of the environment, or lipophilicity of the molecule, the permeability and absorption is often predictable. It is noteworthy that by generating lactic acid and sustaining an acidic vaginal canal, Lactobacillus species infuences the absorption of weakly acidic drug molecules in the vagina. 3) Although the vaginal route has its own Phase I and Phase II metabolic enzymes, which do not impact the administered dose signifcantly, a vaginally absorbed drug by-passes hepatic frst-pass metabolism, which is considered a favorable aspect of the route for delivering better bioavailability compared to the other routes like oral. Other advantages of using this route for achieving systemic effect include avoiding the low pH of the stomach and frst-uterine-pass effect. There are two aspects associated with the metabolism of therapeutic agents in the vaginal route of administration: metabolism by microbiota and metabolism by Phase I and Phase II enzymes of the site of absorption. The role of the microbiome in biotransformation of drugs has been sporadically studied, and very few examples are known. A few documented examples are the metabolism of antiretroviral tenofovir by vaginal microbiota (Carlson et al., 2017) and alteration of therapeutic levels of HIV prevention strategies that attributed to the metabolism by microbiota (Cheu et al., 2020). The vaginal Phase I and Phase II metabolic enzymes are also sparsely studied, and the few reported data indicate that the Phase I CYP450 isozymes like CYP1A1, CYP1B1 and Phase II 170

PK/TK CONSIDERATIONS OF RECTAL, VAGINAL, AND INTRAOVARIAN ROUTES

enzymes, like UGT1A1, are highly expressed in the human female genital tract (Zhou et al., 2013). The vaginal basal layer has shown the presence of protease enzymatic activity (Lee, 1988), which is a hindrance for absorption of peptides and proteins. Studies on the expression of transporters in the vaginal route of administration are indicative of the presence of Pgp, BCRP, MRP1, MRP2, MRP3, MRP4, MRP5, MRP7, OAT2, OCT1,2,3, and ENT1,2 with localization in epithelium and submucosal immune cells (Hu et al., 2015; Grammen et al., 2014; Zhou et al., 2013; Finstad et al., 1990; Nicol et al., 2013). Most vaginal dosage forms are intended for local effects. However, this site can also be used effectively for achieving the systemic effect for certain medications such as progesterone and progesterone-containing polymetric dosage forms. The vaginal absorption is predominantly by transcellular and paracellular passive diffusion. Menopausal changes decrease the regional vascularity and thin the mucosa, which leads to a signifcant increase in permeability of the barrier. The disadvantages of using this route of administration include vaginal changes during menstrual cycle, inter- and intra-individual absorption variability and mucosal irritation of drugs intended for systemic effect. However, the vagina is considered an effective site for the administration of drugs important to women’s health. It is used for administration of contraceptives, antimicrobials, and antifungals. As indicated earlier, the vaginal absorption of therapeutic agents is infuenced by their lipophilicity, degree of ionization, chemical structure and molecular weight, and interaction with the vaginal barrier and secretions. The dosage forms appropriate for this route include ointments, creams, emulsion, suspension, solution, foams, douches, sprays, gels, solid dosage forms, vaginal suppositories, and polymeric devices such as a vaginal ring. Depending on the objectives of a project, PK analysis of data from a vaginally administered drug for achieving systemic effect can be carried out by any of the PK modeling methodologies, i.e., physiologically based pharmacokinetics, compartmental analysis, or non-compartmental analysis. An example of the physiologically based pharmacokinetics is presented in Figure 6.1. The assumptions of the model are: 1) The compartments of the site of absorption and the compartment represented as the highly perfused organs (i.e., liver, kidney, and lungs) are homogenous, even the compartment that includes the remaining organs of interest. Depending on the objectives of the project, the pooled tissue compartments (i.e., remaining organ(s) of interest) can be deleted or expanded to identify one or more specifc organs. 2) The absorption of the drug into the systemic circulation is based on permeation and penetration of the drug from lamina propria of the site of absorption into the vaginal capillaries, identifed as k LC . 3) The elimination of the drug from the body is by metabolism and excretion via the liver and kidney, identifed by the rate constants of k m and k e , respectively. The fuid and layers of the site of absorption represent the sequential absorption of an administered solid, semisolid, and/or polymeric dosage forms/devices. For each layer of the sequential absorption, a reverse rate constant is considered (i.e., kEF , k LE ) that, depending on the physicochemical characteristics of the drug, can be considered negligible. The dissolution or the release rate constant of the dosage form in vaginal fuid is kDF . If the rate of release is equal to the rate of absorption (i.e., kDF FD @ k LC FD) it may imply, under linear condition, that the absorption is continuous, prompt, and indicting the compound is highly lipophilic; under this condition, the rate constants of k LE and kEF are negligible and can be deleted. 4) The rate constants of the sequential absorption are assumed frst-order. If the release of drug from the dosage form or device is nonlinear, the layers of absorption based on appropriate nonlinear relationship of the release would be applicable. 5) None of layers involved in the absorption process can act as a rate-limiting step. The related rate equations are: Dosage form release rate =

dA = -kDF A0 dt

(6.5)

A0 = absorbable dose of administered dosage form/device dAVF = kDF A0 - k FE AVF + kEF AES dt

(6.6) 171

6.2 VAGINAL ROUTE OF ADMINISTRATION

Figure 6.1 Schematic of a physiologically based pharmacokinetic model with release of a drug from a solid or semisolid vaginal dosage form dissolved into the vaginal fuid with the rate constant of kDF ; the permeation from vaginal fuid into the epithelium stratum is shown by the rate constant of k FE , and the return from the epithelium stratum into the vaginal fuid, if permitted by a favorable concentration gradient, is identifed by the rate constant of kEF ; the exchange between epithelium stratum and lamina propria are the next phase in the overall permeation shown by the rate constants kEL and k LE ; the absorption occurs from the lamina propria into the vaginal capillaries under the absorption rate constant of k LC ; following the absorption into the systemic circulation, the compound is subjected to the general distribution to the highly perfused organs and the poorly perfused tissues with elimination from the body by the liver metabolism and urinary excretion; it is worth noting that the metabolism by the vaginal Phase I and Phase II enzymes and the transformation by the vaginal microbiota are not included in the model. dAES = k FE AVF - AES ( kEL + kEF ) + k LE ALP dt

(6.7)

dALP = kEL AES - ALP ( k LC + kEL ) dt

(6.8)

dCvaginal = Qvaginal ( Cvenous - Carterial ) dt

(6.9)

dCorgan = Qorgan ( Carterial - Cvenous ) dt

(6.10)

dCLiver = QLiver ( Carterial - Cvenous )Liver - Clm ( Cvenous-Liver ) L dt

(6.11)

Vvaginal Vorgan VLiver 172

PK/TK CONSIDERATIONS OF RECTAL, VAGINAL, AND INTRAOVARIAN ROUTES

VKidney

dCKidney = QKidney ( Carterial - Cvenous )Kidney - Clrenal ( Cvenous-Kidney ) dt

(6.12)

Where AVF , AES , and ALP are the amounts of the administered drug in vaginal fuid, epithelial stratum, and lamina propria, respectively; Vvaginal is the volume of the vaginal capillaries; Equation 6.10 is the general equation for the organs without elimination; Vorgan is the volume of each organ, and Qorgan is the blood fow of each organ; Clrenal and Clm are renal and metabolic clearance, respectively. 6.3 INTRAOVARIAN ROUTE OF ADMINISTRATION 6.3.1 Overview The ovaries are two spheroidal organs confned to the peritoneal cavity, connected by ligament to the uterus and the pelvis. As discussed in Chapter 4, Section 4.3 - intraperitoneal route of administration - the ovarian syndromes are often treated by both intraperitoneal injection and intraovarian injection. The intraovarian injection is for the purpose of local effect and is considered for specialized cases of treatment. The ovaries are each comprised of an inner medulla and an outer cortex with follicles and stroma. The surface of the organ is covered by columnar cells, known as germinal epithelium, and the stroma with follicles is located underneath this epithelium. The dominant follicular cells produce estradiol, and after ovulation, the residual part of the follicle after oocyte, known as corpus luteum, produces progesterone for approximately two weeks. Both hormones prepare the uterus for implantation of a human embryo. The discussion on folliculogenesis and the chronology of the process is beyond the scope of this book, and related topics in anatomy and physiology books and references should be consulted. An important vascular feature of the ovaries is the cross-connection between uterine and ovarian arteries that infuence both organs. A similar connection is also formed by the vaginal and uterine arteries. The utero–ovarian connected arteries supply about 10% of the blood to the uterus in non-pregnant women. The main blood supply of the uterus is essentially from its right and left arteries. The connected arteries, also called communicating arteries, are considered a secondary route for blood supply to the uterus; in most women, they are smaller in diameter than the uterine or ovarian arteries (Kozik et al., 2002; Kozik, 2000). The blood in the communicating arteries fows from the uterine to the ovarian circulation and from ovary to the uterine circulation, and its direction depends on the resistance differences between the two circulations. During pregnancy, the blood fow increases signifcantly to the uterine artery, and the diameter of the communicating arteries also increases signifcantly to supply the uterus with the volume of blood it needs. An example of intraovarian administration is related to the treatment of premature ovarian insuffciency (Aiman and Amentek, 1985). The therapeutic agents for the injection are cellular based, e.g., platelet-rich plasma (Cakiroglu et al., 2020) or platelet-rich plasma with gonadotropin (Hsu et al., 2020). The intraovarian injection data is mostly for pharmacodynamic evaluation, and the pharmacokinetic considerations are very limited. Distinction should be made between the systemic administration of therapeutic agents for achieving ovarian effect vs intraovarian administration for achieving the local effect. The systemic administration can have full pharmacokinetics and pharmacodynamics evaluation (Rose et al., 2016). REFERENCES Aiman, J., Amentek, C. 1985. Premature ovarian failure. Obstet Gynecol 66(1): 9–14. Barennes, H., Kahiatani, D., Clavier, F., Meynard, D., Njifountawaouo, S., BarennesRasoanandrasana, F., Amadou, M., Soumana, M., Mahamansani, A., Granic, G. 1995. Rectal quinine, an alternative to parenteral injections for the treatment of childhood malaria. Clinical, parasitological, and pharmacological study. Med Trop (Mars) 55(4): 91–4. Barennes, H., Munjakazi, J., Verdier, F., Clavier, F., Pussard, E. 1998. An open randomized clinical study of intrarectal versus infused Quinimax for the treatment of childhood cerebral malaria in Niger. Trans R Soc Trop Med Hyg 92(4): 437–40. Barennes, H., Pussard, E., Mahaman, S. A., Clavier, F., Kahiatani, F., Granic, G., Hanzel, D., Ravinet, L., Verdier, F. 1996. Effcacy and pharmacokinetics of a new intrarectal quinine formulation in children with Plasmodium falciparum malaria. Br J Clin Pharmacol 41(5): 389–95. 173

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Graver, A., Wade, J. J. 2011. The role of acidifcation in the inhibition of Neisseria gonorrhoeae by vaginal lactobacilli during anaerobic growth. Ann Clin Microbiol Antimicrob 10: 8. https://doi.org/10 .1186/1476-0711-10-8. Hickey, R. J., Abdo, Z., Zhou, X., Nemeth, K., Hansmann, M., Osborn, T. W., III, Wang, F., Fomey, L. J. 2013. Effect of tampons and menses on the composition and diversity of vaginal mictobial communities over time. BJOG 120(6): 695–704. Hsu, C.-C., Hsu, L., Hsu, I., Chiu, Y.-J., Dorjee, S. 2020. Live birth in woman with premature ovarian insuffciency receiving ovarian administration of platelet-rich plasma (PRP) in combination with gonadotropin: A case report. Front Endocrinol 11: 50. https://doi.org/10.3389/fendo.2020.00050. Hu, M., Patel, S. K., Zhou, T., Rohan, L. C. 2015. Drug transporters in tissues and cells relevant to sexual transmission of HIV: Implications for drug delivery. J Control Release 219: 681–96. Hummelen, R., van der Westen, R., Reid, G., Petrova, M. I., Lievens, E., Younes, J. A. 2017. Women and their microbes: The unexpected friendship. Trend Bicrobiol 26: 16–32. Jannin, V., Lemagnen, G., Gueroult, P., Larrouture, D., Tuleu, C. 2014. Rectal route in the 21st century to treat children. Adv Drug Deliv Rev 73: 34–49. Jantzen, J. P., Tzanova, I., Witton, P. K., Klein, A. M. 1989. Rectal pH in children. Can J Anaesth 36(6): 665–7. Kakemi, K., Arita, T., Muranishi, S. 1965. Absorption and excretion of drugs. XXV. On the mechanism of rectal absorption of sulfonamides. Chem Pharm Bull (Tokyo) 13(7): 861–9. Klia, N., Singh, J., Kaur, M. 2020. Microbiota in vaginal health and pathogenesis of recurrent vulvovaginal infections: A critical review. Ann Clin Microbiol Antimicrob 19(1): 5. https://doi.org/10.1186/ s12941-020-0347-4. Kozik, W. 2000. Arterial vasculature of ovaries in women of various ages in light of anatomic, radiologic and microangiographic examinations. Ann Acad Med Stetin 46: 25–34. Kozik, W., Czerwinski, F., Pilarczyk, K., Partyka, C. 2002. Arteries of the hilum and parenchymal part of the ovary in reproductive age in microangiographic studies. Ginekol Pol 73(12): 1173–8. Lee, V. H. L. 1988. Enzymatic barriers to peptide and protein absorption. CRC Crit Rev Ther Drug Deliv Sys 5(2): 69–97. Luo, J., Betschart, C., Ashton-Miller, J. A., DeLancey, J. O. L. 2016. Quantitative analyses of variability in normal vaginal shape and dimension on MR images. Int Urogynecol J 27(7): 1087–95. Makwana, N., Shah, M., Chaudhary, M. 2020. Vaginal pH as a diagnostic tool for menopause: A preliminary analysis. J Mid Life Health 11(3): 133–6. Mania-Pramanik, J., Kerkar, S. C., Mehta, P. B., Potdar, S., Salvi, V. S. 2008. Use of vaginal pH in diagnosis of infections and its association with reproductive manifestations. J Clin Lab Anal 22(5): 375–9. Miller, E. A., Livermore, J. A., Alberts, S. C., Tung, J., Archie, E. A. 2017. Ovarian cycling and reproductive state shape the vaginal microbiota in wild baboons. Microbiome 5(1): 8. https://doi.org/10 .1186/s40168-017-0228-z. Moradan, S., Ghorbani, R., Nasiri, Z. 2010. Can vaginal pH predict menopause? Saudi Med J 31(3): 253–6.

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Nicol, M., Fedoriw, Y., Mathews, M.,Prince, H. M., Patterson, K. B., Geller, E., Mollan, K., Mathews, S., Kroetz, D. L., Kashuba, A. D. 2013. Expression of six drug transporter in vaginal, cervical and colorectal tissues: Implications for drug disposition in HIV prevention. J Clin Pharmacol 54(5): 574–83. Nunes, R., Sarmento, B., das Neves, J. 2014. Formulation and delivery of anti-HIV rectal microbicides: Advances and challenges. J Control Release 194: 278–94. Palit, S., Lunniss, P. J., Scott, S. M. 2012. The physiology of human defecation. Dig Dis Sci 57(6): 1445–64. Panda, S., Das, A., Singh, A. S., Pala, S. 2014. Vaginal pH: A marker for menopause. J Mid Life Health 5(1): 34–7. Purohit, T. J., Hanning, S. M., Wu, Z. 2018. Advances in rectal drug delivery systems. Pharm Adv Technol 23(10): 942–52. Ravel, J., Gajer, P., Abdo, Z., Schneider, G. M., Koenig, S. S., McCulle, S. L., Karlebach, S., Gorle, R., Russell, J., Tacket, C. O., Brotman, R. M., Davis, C. C., Ault, K., Peralta, L., Forney, L. J. 2011. Vaginal microbiome of reproductive-age women. Proc Natl Acad Sci U S A 108(Suppl 1): 4680–7. Rose, T. H., Röshammar, D., Erichsen, L., Grundemar, L., Ottesen, J. T. 2016. Population pharmacokinetic modelling of FE999049, a recombinant human follicle-stimulating hormone, in healthy women after single ascending doses. Drugs R D 16(2): 173–80. Riegelman, S., Crowell, W. J. 1958. The kinetics of rectal absorption III. The absorption of undissociated molecules. J Pharm Sci 47(2): 127–33. Srikrishna, S., Cardozo, L. 2013. The vagina as a route for drug delivery: A review. Int Urogynecol J 24(4): 537–43. Suzuki, A., Higuchi, W. I., Do, N. F. H. 1970a. Theoretical model studies of drug absorption and transport in the gastrointestinal tract (I). J Pharm Sci 59: 644–51. Suzuki, A., Higuchi, W. I., Do, N. F. H. 1970b. Theoretical model studies of drug absorption and transport in the gastrointestinal tract (II). J Pharm Sci 59: 651–9. Tachedjian, G., Aldunate, M., Bradshaw, C. S., Cone, R. A. 2017. The role of lactic acid production by probiotic Lactobacillus spcies in vaginal health. Res Microbiol 168(9–10): 782–92. Tomás, M. S. J., Ocaña, V. S., Wiese, B., Nader-Macías, M. E. 2003. Growth and lactic acid production by vaginal Lactobacillus acidophilus CRL 1259, and inhibition of uropathogenic Escherichia coli. J Med Microbiol 52(12): 1117–24. Van De Wijgert, J. H. H. M., Borgdroff, H., Verhelst, R., Crucitti, T., Francis, S., Verstraelen, H., Jespers, V. 2014. The vaginal microbiota: What we learned after a decade of molecular characterization? PLOS ONE 9(8): e105998. https://doi.org/10.1371/journal.pone.0105998. van Hoogdalem, E. J., De Boer, A. G., Breimer, D. D. 1991. Pharmacokinetics of rectal drug administration, part I. General considerations and clinical applications of centrally acting drugs. Clin Pharmacokinet 21(1): 11–26. Zhou, T., Hu, M., Cost, M., Poloyac, S., Rohan, L. 2013. Short communication: Expression of transporters and metabolizing enzymes in the female lower genital tract: Implication for microbicide research. AIDS Res Hum Retrovir 29(11): 1496–503.

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7 PK/TK Considerations of Absorption Mechanisms and Rate Equations 7.1 INTRODUCTION The fundamental considerations of pharmacological activity or organ toxicity include those aspects of biochemical and physiological processes which infuence the handling of xenobiotics in the body. These biological processes occur concurrently, impact the overall pharmacokinetic or toxicokinetic profle of xenobiotics, and most are predictable in normal populations except when the genetic differences are the variability factors among humans. Collectively, these processes are considered under absorption, distribution, metabolism, and excretion (ADME). The simplest mechanism of xenobiotic absorption is passive diffusion, which represents the partitioning of a compound through a barrier. The barrier can be as simple as the lipid bilayer of a cell membrane, as complex as the wall of gastrointestinal tract, or as shielding as the multilayer barrier of the skin. The partitioning is the result of the compound interacting with the barrier. The nature of such interaction can be as simple as transcellular or paracellular diffusion or as intricate as active or facilitated transport. The interaction was frst observed from the relationship between the rate of penetration and the lipid/water partition coeffcient (Overton, 1902). The concept was further investigated (Hober and Hober, 1937; Travel, 1940; Hogben et al., 1959; Hogben, 1960), refned, and presented as the concept of pH partition hypothesis. The hypothesis, however, has been shown to have certain limitations, which do not reduce the signifcance of the hypothesis, yet obscure the understanding of the absorption process. Examples are the complete absorption of weakly acidic drugs in the alkaline environment of the small intestine and the absorption of quaternary ammonium compounds at the different pH of the GI tract. Several explanations have been suggested to defne these deviations in support of the pH partition theory. For example, the absorption of weakly acidic drugs in an alkaline pH is because of the presence of an acidic microclimate adjacent to the apical surface of the intestinal epithelium (Tsuji et al., 1978). However, the observed transportation of charged species of xenobiotics via paracellular diffusion weakens the validity of the intestinal acidic microclimate theory (Palm et al., 1999; Neuhoff et al., 2003). Thus, the controversy regarding the exceptions to the pH partition theory remain to be clarifed. Other mechanisms of absorption are as intricate and involved as the pH partition theory. The absorption of xenobiotics from absorption sites such as buccal, sublingual, rectal, etc., is explained by passive diffusion. However, for routes such as the pulmonary or GI tracts, there are other mechanisms of absorption, which are as essential as the passive diffusion. The evaluations of these mechanisms within the context of the absorption sites are the focus of this chapter. The permeation/transport of a compound through physiological barriers is governed by one of the following mechanisms. The GI tract is the only site of absorption that utilizes all these mechanisms: ◾ passive diffusion ◾ transcellular and paracellular diffusion • carrier-mediated transcellular diffusion, facilitated diffusion, or passive-mediated transport ◾ transcellular diffusion subject to P-glycoprotein effux ◾ active transport ◾ pinocytosis and receptor-mediated endocytosis ◾ solvent drag, osmosis, and two-pore theory ◾ ion-pair absorption 7.2 PASSIVE DIFFUSION 7.2.1 Transcellular and Paracellular Diffusion The permeation of xenobiotics through biological barriers is frequently governed by passive diffusion. The driving force for the diffusion is the concentration gradient with no requirement for chemical energy according to Fick’s law of diffusion. The mechanism of transport is the passage DOI: 10.1201/9781003260660-7

177

7.2 PASSIVE DIFFUSION

of molecules from an aqueous or hydrophilic surroundings into the lipophilic environment of the barriers. The degree of lipophilicity of the molecules is quantifed by its partition coeffcient or distribution coeffcient. As indicated earlier, the pH-partition (Shore et al., 1957; Hogben et al., 1957, 1959; Bekett and Triggs, 1967; Becket et al, 1968; Becket and Moffat 1970) assumes lipophilic or unionized molecules of a xenobiotic can transfer across biological barriers. The theory still considered the encompassing feature of passive diffusion (Schaper et al., 2001). Transcellular and paracellular passive diffusions are mainly in the GI tract, nasal, buccal, pulmonary, rectal, and vaginal routes of administration (Figure 7.1). Small hydrophilic or hydrophobic molecules may absorb through vaginal, buccal, and rectal routes by paracellular diffusion. In the GI tract, the extent of paracellular absorption of the small molecules is as signifcant as transcellular diffusion (Figure 7.1). The nasal route, however, has been shown to have a nonlinear correlation between the lipophilicity of a compound and its permeation, which may be related to the complex structure of the nasal epithelium (see Chapter 3, Section 3.1). Diffusion of xenobiotics through the cornea and its component layers (epithelium, stroma, and endothelium, see Chapter 2, Section 2.3) is also by passive transcellular and paracellular permeability through its different layers and depends on the molecular size of the compound and its distribution coeffcient (Edwards and Prausnitz, 2001). The transcellular passive diffusion is the preferred absorption for xenobiotics. The absorption by the paracellular route is limited to the size of the paracellular opening and the presence of a tight junction barrier; despite this limitation, it is an important mechanism for the absorption of hydrophilic compounds with little membrane permeability. 7.2.1.1 Transcellular and Paracellular Transport Rate Equations To differentiate kinetically between transcellular diffusion and paracellular pathway, the following three-compartment closed model (Figure 7.2) is proposed for the accumulation and transport in Caco-2 cells (Bourdet et al., 2006). The combined differential equations representing the transcellular and paracellular pathways are dA1 ˜ ° ˝ k12 ˛ k13 ˙ A1 ˛ k 2 A2 dt

(7.1)

dA2 ˜ k12 A1 ° ˝ k 21 ˛ k 23 ˙ A2 ˛ k 32 A3 dt

(7.2)

dA3 ˜ k 23 A2 ° k13 A1 ˛ k 32 A3 dt

(7.3)

Figure 7.1 This illustration epitomizes the passive diffusion of small hydrophilic compounds by transcellular diffusion, running through the intracellular environment, and paracellular diffusion, permeating across the disrupted tight junctions of the barrier to reach the capillaries and ultimately the systemic circulation; which is the existing absorption process across membrane barriers of various routes of administration, with or without the microvilli, including the nasal route, GI tract, buccal, sublingual, rectal, etc., and even the Caco-2 cells in vitro systems. 178

PK/TK CONSIDERATIONS OF ABSORPTION MECHANISMS AND RATE EQUATIONS

Figure 7.2 Schematic of a kinetic model to differentiate between transcellular permeation and paracellular penetration of xenobiotics in Caco-2 cells. The focus of the model is on the transport of a xenobiotic between the apical (compartment A1 ), intracellular accumulation (compartment A2 ), and basolateral (compartment A3 ) of the system by paracellular passive diffusion; the related frstorder rate constants are k12 and k 32 (the input rate constants in A2 compartment), k 21 and k 23 (the exit rate constants from A2 ), and k13 (the input rate constant from apical to the basolateral compartment). where A1 is the amount in the apical compartment; A2 is the amount in the cells; A3 is the amount in the basolateral compartment; k12 is the frst-order input rate constant in the cellular compartment; k 21 is the frst-order input rate constant from the cellular compartment into the apical compartment; k 23 is the rate constant from the cellular compartment into the basolateral compartment; k13 is the input rate constant from apical into the basolateral compartment; k 32 is the rate constant dA1 from the basolateral compartment into the cellular compartment; is the rate of amount change dt dA2 in the apical compartment with respect to time; is the rate of amount change in the cellular dt dA3 compartment with respect to time; and is the rate of amount change in the basolateral comdt partment with respect to time. The following differential equations set apart the rate equations of paracellular from transcellular diffusion: Paracellular rate equations: dA1 ˜ °k13 A1 dt

(7.4)

dA3 = k13 A1 dt

(7.5)

dA1 ˜ °k12 A2 ˛ k 21 A2 dt

(7.6)

dA2 ˜ k12 A1 ° ˝ k 21 ˛ k 23 ˙ A2 ˛ k 32 A3 dt

(7.7)

dA3 ˜ k 23 A2 ° k 32 A3 dt

(7.8)

Transcellular rate equations:

The specifc input rate constants into the cellular compartment, k12 and k 32 , are estimated by the following equations, taking into consideration the saturable and non-saturable components of the cellular uptake (Bourdet et al., 2006): dA2 Vmax ° A1 K d A1 ˜ ˛ VAP dt K M ˛ A1

(7.9)

dA2 Vmax ° A3 K d A1 ˛ ˜ VBL dt K M ˛ A3

(7.10)

Or,

The rate of input from the apical compartment or basolateral compartment into the cellular compartment is Rate of input from A1 into A2 = k12 A1

(7.11)

Or, 179

7.2 PASSIVE DIFFUSION

Rate of input from A3 into A2 = k 32 A1

(7.12)

Setting Equation 7.9 equal to Equation 7.11 and solving for k12 yields the following relationship: k12 ˜

Vmax K ° d K M ° A1 VAP

(7.13)

Setting Equation 7.10 equal to Equation 7.12 yields the following relationship for k 32 : k 32 ˜

Vmax K ° d K M ° A3 VBL

(7.14)

where Vmax is the maximum uptake rate; K M is the Michaelis–Menten constant; K d is the nonsaturable parameter of the uptake with units of (volume/time)/mg of protein; and VAP or VBL is the volume of the donor compartment, A1 or A3 . The transcellular absorption is facilitated by the ability of xenobiotics to partition into the cell membrane. In other words, the molecules should have optimum lipophilicity to cross the membrane and not be too lipophilic to be retained in the lipid environment of the membrane. The lipophilicity of a compound in terms of its chemical structure is defned as Lipophilicity = Hydrophobicity – Polarity. Hydrophobicity makes up the hydrophobic and dispersion forces and polarity comprises hydrogen bonds, orientation, and induction forces (Liu et al., 2011). A measure of the lipophilicity of a xenobiotic molecule is the magnitude of its partition coeffcient and distribution coeffcient. In certain cases, the distribution coeffcient replaces the diffusion coeffcient. As will be discussed in this section, the diffusion coeffcient is inversely proportional to the molecular weight of the compound. Thus, small hydrophobic molecules have a better chance of getting absorbed by passive diffusion faster and more completely. 7.2.2 Partition Coeffcient The partition coeffcient implies the extent xenobiotics partition between a hydrophilic phase and a lipophilic phase. It represents the degree of lipophilicity or hydrophobicity of a compound. There are various in vitro methodologies to estimate a realistic value for the partition coeffcient of xenobiotics. These methodologies include 1) the classical n-octanol/water partitioning system; 2) liposome (Betageri and Rogers, 1988); immobilized artifcial membrane partitioning systems (Pauletti and Wunderli-Allenspach, 1994; Taillardat-Bertschinger et al., 2002); 3) high-performance liquid chromatography partitioning systems using stationary phases (Pidgeon et al., 1995; Ong et al., 1995); 4) quantum-mechanical prediction of partition coeffcients and acid dissociation constants for small drug-like molecules (Findik et al., 2021). The partition coeffcient determined by the octanol/water system or ODS (Octa Decyl Silane column) system corresponds only to the hydrophobic nature of a compound and not the interaction of its molecules with the structural characteristics of a barrier, for example, polar head groups of the lipid bilayer. The liposome partitioning systems provide the estimates that include both partition coeffcient and the membrane interaction. The discussion on comparison of different methodology to determine the partition coeffcient is beyond the scope of this chapter. The following discussion is focused on the data generated from the classical n-octanol/water system. Using the octanol/water system, the aqueous phase (water) is a buffer containing the xenobiotic with concentrations of Caqueous . It represents the hydrophilic environment at the absorption site. The organic or lipid phase is n-octanol, and it represents the lipophilic barrier with concentrations of Clipid . At the lipid-aqueous (octanol/water) interfaces, a discontinuity in solute concentration exists at equilibrium, which is the result of the molecular hindrance that exists for a solute molecule migrating from the aqueous phase into the lipid phase. The discontinuity between Caqueous and Clipid is represented by the lipid-aqueous partition coeffcient Pcoeff as the ratio of concentration in the hydrophobic core, Clipid , to its concentration in the hydrophilic phase, Caqueous (Equation 7.15). The ratio represents the relative affnity of a compound for the lipid environment of the biological barrier (Figure 7.3). Pcoeff =

180

Clipid Caqueous

(7.15)

PK/TK CONSIDERATIONS OF ABSORPTION MECHANISMS AND RATE EQUATIONS

Figure 7.3 Depiction of a partitioning occurrence of unionized and ionized molecules from the aqueous layer (e.g., water or buffer that represents the hydrophilic environment at the absorption site) into the organic layer (e.g., lipid phase or organic solvents like octanol that correspond to the lipophilic barrier), considering three scenarios of A) when ionized and unionized molecules are present in the aqueous phase, and the unionized form a partition into the organic phase; B) when the unionized molecules are present in both the aqueous and lipid phase, and ionized molecules are present only in the aqueous phase; and (C) when the ionized and unionized molecules are present in both aqueous and lipid phase. Partition coeffcient, Pcoeff , is a unitless fraction, and its value may vary according to the experimental conditions (pH, temperature, ionic strength, etc.) and selection of organic solvent (octanol, chloroform, dichloromethane, etc.). Thus, a validated procedure is essential in determining the partition coeffcient. Its logarithm, log Pcoeff , also signifes the lipophilicity of xenobiotics and provides numbers that are more convenient to use. Like pKa , log Pcoeff (Equation 7.16) is also an important descriptor in understanding the permeability behavior of xenobiotics. ° Clipid ˙ log Pcoeff ˜ log ˝˝ ˇˇ ˛ Caqeous ˆ

(7.16)

˜ Clipid ˝ The log Pcoeff is log10 scale, thus when it is equal to zero the ratio ˛ ˛ Caqeous ˆˆ is equal to 1, which indi° ˙ cates that the compound under evaluation is equally present in both the lipid and aqueous layer. When log Pcoeff is a positive number, for example, 2 or 5, it means that the compound has 100 or 100,000 times more affnity for the lipid phase; and when it is negative, for example, −1 or –2, the compound is 10 or 100 times more soluble in the aqueous phase. Several different approaches have been recommended for the theoretical calculation of log Pcoeff . These theoretical approaches are based mostly on predicting lipophilicity of a compound from the structural parameters of the molecule or solvent accessible surface area (SASA). These approaches 181

7.2 PASSIVE DIFFUSION

include CLOGP (Leo et al., 1975; Hansch and Leo, 1979; Leo, 1987, 1991, 1993), the acronym stands for calculated log P (described below); the Σ f-system (Nys and Rekker, 1973, 1974; Rekker, 1977; Rekker and de Kort, 1979) is used to study the aliphatic hydrocarbon/water partitioning and is f , where f is the fragmental constant; KLOGP (Klopman based on the relationship of log P ˜

°

and Iroff, 1981; Klopman et al., 1994; Zhu et al., 2005), or Klopman LOGP, is based on the atomic composition of the molecule; KOWWIN (Meylan and Howard, 1995) – the KOW part of the acronym stands for KOW (representing the partition coeffcient of the oil/water system), and the system is based on hydrophobic contribution of atomic structure and molecular fragments of a compound; ACD/LOGP (Petrauskas and Kolovanov, 2000), or Advanced Chemistry Development Inc LOGP, is also used for the quantitative measure of lipophilicity of a neutral molecule; AB/LOGP (Japertas et al., 2002), is based on an Algorithm Builder (AB) for all computational steps; XLOGP (Wang et al., 1997), developed by the Institute of Physical Chemistry of Peking University, is based on classifying atoms by their hybridization states and their neighboring atoms; X is used in the acronym as the unknown LOGP; CLIP (Gaillard et al., 1994), or Calculated Lipophilicity Potential, is based on the atomic lipophilic system and molecular lipophilicity potential; HINT (Kellogg et al., 1991, 1992; Abraham and Kellogg, 1994; Kellogg and Abraham, 1999, 2000), or Hydrophobic Interactions is based on hydrophobic fragment constants and could be used to assess interactions between small and large molecules; MLOGP (Moriguchi et al., 1992, 1994), or Moriguchi LOGP, uses the sum of hydrophilic and hydrophobic atoms (described below); VLOGP (Gombar and Enslein, 1996; Gombar, 1999) is a model developed based on the value of linear free energy relationship and uses molecular topology; TLOGP (Junghans and Pretsch, 1997), or Topological LOGP, is also based on topological and substructure coding of molecule description; AUTOLOGP (Devillers et al., 1998) is derived from heterogenous organic chemicals and used especially for highly lipophilic substances; BLOGP (Bodor and Huang, 1992; Klopman and Iroff, 1981) is based on parameters, such as geometric descriptors, including molecular surface area, ovality, and charge density and dipole moment; QLOGP (Bodor and Buchwald, 1997), or Quantum LOGP, is based on quantum mechanical semi-empirical calculations of molecular size; ClogPalk (Kenny et al., 2013), or Calculated logP of alkane/water partition, is based on the relationship between alkane/water partition coeffcient and molecular surface area. The two most often used methodologies are briefy described below. 7.2.2.1 CLOGPcoeff CLOGP stands for calculated partition coeffcient and is determined empirically by the CLOGP program developed by Pomona Med Chem. The manual of the program is available at http://www .daylight.com/dayhtml/doc/clogp. The calculated value is based on molecular structure and values of different fragments of the molecule. The program essentially uses the following relationship in conjunction with Hansch and Leo’s database (Hansch and Leo, 1979; Leo, 1993) in Equation 7.17: log Pcoeff ˜

˛ a f ° ˛b F n n

m m

(7.17)

where a is the number of occurrences of fragment f of type n, b is the number of occurrences of correction factor F of type m. 7.2.2.2 MLOGPcoeff MLOGP or the Moriguchi partition coeffcient is calculated by using the following relationship (Moriguchi et al., 1992; Moriguchi et al., 1994): MLOGPcoeff ˜ °1.014 ˛ 1.244 ˝ FCX ˙

0.6

° 1.017 ˝ NO ˛ N N ˙

0.9

0.8 ˛ 0.406 FPRX ° 0.145NUNS ˛ 0.511I HB ˛ 0.268 N POL

°2.215FAMP ˛ 0.912I ALK ° 0.392II RNG ° 3.684FQN ˛ 0.474N NO2 ˛ 1.582FNCS ˛ 0.773IˆL (7.18) There are 13 structural parameters in the above regression equation (Equation 7.18), which are identifed as independent Moriguchi based lipophilicity molecular descriptors. The defnition and values of the descriptors (i.e., regression coeffcients and variable of Moriguchi model) are as follows (Todeschini and Consonni, 2000). The y-intercept, is b0 = −1.014; the coeffcient of FCX is 1.244, and FCX is the total number of carbon and halogen atoms weighted by carbon = 1.0, fuorine = 0.5, chlorine = 1.0, bromine = 1.5, and 182

PK/TK CONSIDERATIONS OF ABSORPTION MECHANISMS AND RATE EQUATIONS

iodine = 2.0; the coeffcient of (NO + N N) is −1.017 and (NO + N N) is the total number of oxygen and nitrogen atoms; the coeffcient of FPRX is 0.406 and FPRX is the proximity infuence of oxygen and nitrogen for N-O, or O-N: FPRX = 2, for N-A-O where A = C, or S or P, FPRX = 1, and for carbon-amide bond FPRX = −1; the coeffcient of NUNS is −0.145, and NUNS represents the total number of unsaturated bonds, except those in NO2; the coeffcient of IHB is 0.511, and IHB stands for the presence of intramolecular H-bond; the coeffcient of NPOL is 0.268, and NPOL is the number of polar substituent; the coeffcient of FAMP is −2.215, and FAMP represents the amphoteric property: for α-amino FAMP = 1.0, for aminobenzoic acid or pyridine carboxylic acid FAMP = 0.5; the coeffcient of IALK is 0.912, and IALK is a variable for alkanes, alkenes, cycloalkanes, cycloalkenes (hydrocarbons with 0 or 1 double bond); the coeffcient of IRNG is −0.392, and IRNG is a variable for the presence of ring structures (not benzene and its condensed rings); the coeffcient of FQN is −3.684, and FQN represents quaternary nitrogen, FQN = 1 for quaternary nitrogen, FQN = 0.5 for nitrogen oxide; the coeffcient of N NO2 is 0.474, and N NO2 is the number of nitro groups in the molecule; the coeffcient of FNCS is 1.582, and FNCS = 1 for −N = C = S group, and FNCS = 0.5 for −S − CN group; the coeffcient of Iβ L is 0.773, and Iβ L stands for the presence of β-lactam in the molecule. MLOGcoeff and CLOGcoeff , as well as other approaches are used to estimate lipophilicity of chemicals and predict the solubility, permeability, and absorption of xenobiotics. One of these applications is Lipinski’s Rule of Five (see Chapter 17). 7.2.3 Distribution Coeffcient The estimation of the partition coeffcient for neutral molecules is straightforward and follows Equation 7.15 (Figure 7.3-A). However, for ionizable molecules, monoprotic, weakly acidic, or basic xenobiotics or polyprotic ampholytes (for example, peptides), the partition coeffcient depends on the pH and the degree of ionization of their functional group(s). Thus, their true partition coeffcient, known as the distribution coeffcient, is estimated based on the presence of both ionized and unionized forms in the lipid phase and aqueous phase (Stopher and McClean, 1990; Avdeef, 1993; Scott and Clymer, 2002; Davies and Flower, 2013). It is often assumed that the ionized form of a molecule may not partition into octanol or a biological barrier, but both octanol and biological barriers maintain a certain percent of water that makes it possible for ionized molecules to partition. Octanol is known to absorb water, and the solubility of water in octanol is about 26.4 mol% (or 4.73% w/w) (Margolis and Levenson, 2000), which is similar to the water content of biological barriers. When the unionized molecules are present in both the aqueous and lipid phase, and ionized molecules are present only in the aqueous phase, the following relationship applies, which is known as the distribution coeffcient (Figure 7.3-B):

( Dist )coeff

=

éëClipid ùû unionized éëCaqueous ùû éëCaqueous + o ù û ionized unionized

(7.19)

Comparing Equation 7.19 with Equation 7.15 indicates that ( Dist )coeff < Pcoeff Dividing the numerator and denominator of Equation 7.19 by éCaqueous ù and bearing in ë û unionized mind Equation 7.20, the dissociation constant is defned as Equation 7.21. H O

2 + ˜˜˜˜ ° éëCaqueous ùû ˛ ˜˜˜ ˜ éëCaqueous ùû ionized + H unionized

Kd =

H + + éëCaqueous ùû ionized

(7.20) (7.21)

éëCaqueous ùû unionized

The distribution coeffcient can then be estimated as

( Dist )coeff ( Dist )coeff

Pcoeff K 1 + d+ H Pcoeff = H+ 1+ Kd =

For weak acid: (7.22)

For weak base: (7.23)

When the ionized and unionized molecules are present in both aqueous and lipid phase, the distribution coeffcient is estimated as (Figure 7.3-C) 183

7.2 PASSIVE DIFFUSION

( Dist )coeff

éëClipid ùû + éëClipid ùû ionized unionized éëCaqueous ùû + éëCaqueous ùû ionized unionized

=

(7.24)

Therefore,

( Dist )coeff

=

( Dist )coeff

=

Pcoeff Pcoeff + + Kd 1+ + 1+ H H Kd

for weak acids: (7.25)

Pcoeff Pcoeff + H + 1 + Kd 1+ H+ Kd

and for weak bases: (7.26)

Where Pcoeff is the partition coeffcient of ionized molecules. For molecules with single ionizable group, the relationship between distribution coeffcient and partition coeffcient can be defned as Equations 7.27 and 7.28: for weak monoprotic acid:

log ( Dist )coeff = log Pcoeff - log é1 + 10( ë

pH -pKa )

ù û

(7.27)

and for weak monoprotic bases:

log ( Dist )coeff = log Pcoeff - log é1 + 10( ë

pKa - pH )

ù û

(7.28)

For molecules with more than one ionizable group, the relationships become more complex, such as amphoteric compounds with both acidic and basic functional groups. To avoid the pH-related variability for estimating the distribution coeffcient, buffers with a physiological pH of 7.4 or 7.2 are often selected in PK/TK inferences. 7.2.4 Diffusion Coeffcient Diffusion coeffcient, Dcoeff , is another constant required for defning the passive diffusion/absorption. This coeffcient is not expressed in terms of amount or concentration and may have multiple defnitions in different felds of physics or engineering. For biological systems, it is defned as a constant that refects diffusivity of a compound with a unit of distance2/time, or area/time. A better understanding of the diffusion coeffcient is based on the solution of Fick’s second law of diffusion ¶ 2C ¶C = Dcoeff ¶x 2 ¶t

(7.29)

Equation 7.29 is also known as a one-dimensional diffusion equation. The three-dimensional equation (Equation 7.30) is known as a diffusion equation ¶C ¶C ¶C ¶C + + = Dcoeff DC = ¶t ¶x 2 ¶y 2 ¶z 2 Where

D=

¶ ¶ ¶ + + ¶x 2 ¶y 2 ¶z 2

(7.30) (7.31)

For one-dimensional diffusion in the x direction, the assumption is that the barrier is uniform in the y and z direction and ¶ = ¶ = 0 ¶y 2 ¶z 2 A solution for Equation 7.29 is C=

1

( 4pDcoeff t )

12

e

-x 2 4Dcoeff t

(7.32)

The mean square displacement of a diffusing molecule is defned by the indefnite integral of Equation 7.32 (Rubinow, 1975; Michalet, 2010) x2 = Setting a = 184

1

( 4pDcoeff t )

12

ò

¥

é e -x 2 4Dcoeff t x 2 dx ù ê ûú ë -¥

1 and defnite integration between 0 ® ¥, Equation 7.33 yields 4Dcoeff t

(7.33)

PK/TK CONSIDERATIONS OF ABSORPTION MECHANISMS AND RATE EQUATIONS

x 2 = 2Dcoeff t

(7.34)

x2 2t

(7.35)

˜Dcoeff °

Thus, the diffusion coeffcient with units of distance2/time, or area/time is equal to the mean square displacement of a diffusing molecule during the time interval divided by 2. The diffusion coeffcient was also defned and interpreted by the Brownian movement theory (Einstein, 1905) and is known as the Stokes–Einstein relation. The Stokes–Einstein relation is a form of the fuctuation–dissipation theorem and relates the diffusion coeffcient of colloidal particles to the viscosity via the thermal energy. Dcoeff ˜

k BT 6°˛r

(7.36)

Equation 7.36 defnes the relationship between the diffusion coeffcient, Dcoeff ; the solvent viscosity, η ; the Boltzmann’s constant (1.38 × 10−23 J/K), kB ; the absolute temperature of the solution, T ; and the radius of the solute molecule, r . The radius is related to the molecular weight according to the following equation: °4 ˙ MW ˜ N˘V ˜ N˘ ˝ r 3 ˇ ˛3 ˆ

(7.37)

Solving for r: 1

˝ 3 ° MW ˛ ˇ 3 r˜ˆ  ˙ 4N ˘

(7.38)

where N is the Avogadro’s number, V is the molal volume of solute, r is the hydrodynamic radius – that is, solvent bound to solute – and ρ is the density. Equation 7.36 applies to the systems where the solute molecule is large compared to the solvent molecule and is assumed to have a spherical shape. Although it is developed with the diluted systems in mind, the relationship works well at higher densities (Bonn and Kegel, 2003). 7.2.5 Permeation and Permeability Constant Based on the defnition of diffusion coeffcient, partition coeffcient, and distribution coeffcient, one can describe the permeation of a compound, Pm , by passive diffusion through a biological barrier as a process that is directly proportional to the partition coeffcient and diffusion coeffcient and inversely proportional to the thickness of the barrier, or distance, it permeates through Pm ˜

Dcoeff ° Pcoeff x

(7.39)

Where Pm is the permeation with units of distance/time. Standardizing the permeation (Equation 7.39) with respect to the surface area of permeation yields the permeability constant P . P˜

Dcoeff ° Pcoeff ° Area Pm ° Area ˜ x x

(7.40)

Thus, permeability constant with units distance3/time, is the three-dimensional consideration of the barrier for permeation per a unit of time (Equation 7.40). According to the Fick’s frst law of diffusion, the driving force of passive diffusion is the concentration gradient, which is the movement of molecules, or mass transfer, from high concentration to the lower concentration region. The frst law defnes the mass fux as J ˜ °Dcoeff

dC dx

(7.41)

J ˜ °Dcoeff

˛C ˛x

(7.42)

or,

or, 185

7.2 PASSIVE DIFFUSION

DC Dx

J = -Dcoeff

(7.43)

Where Dcoeff has units of area/time; dC or DC represents change in concentration, ¶C is partial change in concentration, and all three terms have units of mass/volume; dx, or Dx, or ¶x is the distance that the molecules permeate with respect to their initial location. The negative sign states that the molecules permeate in the direction of decreasing concentration of the high concentration zone (i.e., gradient is negative). Rearranging Equation 7.41 and integrating with respect to the thickness of membrane (0 ® x ) and concentration (Coutside ® Cinside ) yield

ò

x

0

Jdx = -Dcoeff

ò

Cinside

Coutside

(7.44)

dC

Jx = -Dcoeff [Cinside - Coutside ] = Dcoeff [Coutside - Cinside ]

(7.45)

Dcoeff (7.46) [Coutside - Cinside ] x Dcoeff Equation 7.46 is a straight-line equation with a slope of , which approximates the permeabilx ity constant. \J =

The mass fux has units of mass/time and surface area

mass 1 mass x2 ´ 3 ´ = 2 time x x x ´ time

(7.47)

Thus, according to the Fick’s frst law of diffusion, the mass fux is related to the concentration gradient over the diffusion distance. The mass fux may also be defned as the “rate of diffusion/ area” or “rate of absorption/area.” Thus, a higher concentration gradient leads to a faster rate of absorption and mass fux (Figure 7.4). The rate of absorption of a compound by passive diffusion across a barrier with a thickness of x and known surface area according to the Fick’s law of diffusion (Figure 7.4) is æ Dcoeff ´ Pcoeff ´ Area ö dA = -ç ÷ DC = -PDC dt x è ø

(7.48)

dA is the amount absorbed per unit of time, that is, the rate of absorption, and ΔC is the dt concentration gradient of the compound with a partition coeffcient of Pcoeff across a barrier with thickness of x and a defned surface Area. The permeability constant P , as presented by Equation 7.40, is equal to where

P=

Dcoeff ´ Pcoeff ´ Area x

(7.49)

The unit analysis of the absorption rate (Equation 7.48) is

( ) ÷ö DC(Amount / x ) = Amount / time

æ Dcoeff ( x 2 / time) ´ Pcoeff ´ Area x 2 dA = -ç ç dt x è

3

(7.50)

÷ ø

Equation 7.49 is applied to various absorption processes through different biological barriers. It is applicable only where the rate-limiting step is diffusion across the barrier. Equation 7.48 is also used to estimate the diffusion coeffcient or diffusivity of a compound through a biological barrier (Equation 7.51):

Dcoeff

æ dA ö ç dt ÷ ( x ) ( Amount / time )( x ) = Area / time è ø = = 2 ( Pcoeff ) ( Area )( DC ) x Amount / x 3

( )(

)

(7.51)

It is important to recognize that not all biological barriers are the same. Using a gastrointestinal tract (GI) as an example, the driving force of diffusion is the concentration gradient between the GI tract and the systemic circulation. Considering the large volume of systemic circulation 186

PK/TK CONSIDERATIONS OF ABSORPTION MECHANISMS AND RATE EQUATIONS

Figure 7.4 Diagram of mass fux governed by the concentration gradient DC according to Fick’s frst law of diffusion when the mass fux, defned as the rate of diffusion divided by the surface area, occurs over the diffusion distance from high concentration space to low concentration space, membrane proteins are passive in the absorption process. and the small volume of GI fuid, the concentration of drug (mass/volume) at the site of absorption will always be greater than the concentration of free drug in the systemic circulation, that is, CGI ˜ C plasma and DC = CGI - C plasma = CGI . Thus, the rate of absorption (Equation 7.48) for passive

(

)

diffusion under the sink condition across the GI tract is dA = -PDC = -PCGI dt

(7.52)

The sink condition can exist for most, if not all, routes of administration with a biological barrier between the site of absorption and the systemic circulation, that is, the concentration of xenobiotics at the site of absorption is always higher than its concentration in the systemic circulation. Under dA ) is equal to the permeability constant multiplied by the this condition, the rate of absorption ( dt concentration at the absorption site (Equation 7.52). When the rate of absorption is then equal to a constant (permeability constant) multiplied by a variable (concentration), the absorption is governed by frst-order kinetics. 7.2.5.1 Estimation of Apparent Permeability Constant Using Caco-2 Cells Caco-2 cells, the human colon adenocarcinoma cell line (HTB-37), are used broadly for prediction of xenobiotic absorption in the intestine. The cells undergo spontaneous differentiation and polarization in culture to have an apical and basolateral surface with tight junctions and many normal functions of enterocytes. The culture is used in the evaluation of paracellular and transcellular absorption (Hidalgo et al., 1989; Artursson, 1991; Balimane et al., 2000; Le Ferrec et al., 2001; Balimane and Chong, 2005; Sambuy et al., 2005; Farrell et al., 2012; Ding et al., 2021) and other features including CYP3A4 and P-glycoprotein expressions and absorption of zwitterions (Pieksaritanont et al., 1998; Arias et al., 2014). Several theoretical relationships for permeation through in vitro biological barriers have been developed (Camenisch et al., 1998; Hubatsch et al., 2007; Farrell et al., 2012), which are based on Caco-2 cells and the assumption that permeation is mostly through paracellular and transcellular diffusion. The apparent permeability coeffcient according to these observations is described as Papp =

Vapical dA ´ basal dt Area ´ ( Aapical )t=0

(7.53)

187

7.3 CARRIER-MEDIATED TRANSCELLULAR DIFFUSION

Where Papp is the apparent permeability constant (distance/time) for an apical donor compartment with volume of Vapical (cm3), surface area of Area (cm2), and an initial amount of ( Aapical )t=0 in dA the donor compartment multiplied by ( basal ), which is the rate of change of amount in the basal dt receiver compartment. To estimate a realistic value for Papp, the amount in the basal compartment should exceed 10% of the apical compartment (Hubatsch et al., 2007), indicative of a constant concentration gradient. The relationship between the apparent permeability constant and partition coeffcient, or distribution coeffcient, is often defned by the following sigmoidal relationship: log Papp = alog Dcoeff - b log (1 + bDcoeff ) + c

(7.54)

where a, b , b, and c are the data-related curve-ftting coeffcients, and Dcoeff is the diffusion coeffcient, which is often replaced by the distribution coeffcient. 7.3 CARRIER-MEDIATED TRANSCELLULAR DIFFUSION The facilitated diffusion, also known as passive mediated transport or carrier-mediated transcellular diffusion, refers to the absorption mechanism that is facilitated by transport proteins also known as carrier proteins, or absorbed through ion channels of biological barriers. The absorption is still a passive diffusion, and the role of transport proteins is just to provide a passageway for the molecules to permeate across a barrier without requiring energy. This means that the diffusion is not due to the binding of xenobiotics to the membrane protein and subsequent energy requiring conformational change of the protein, but rather related to the kinetic energy of moving molecules through the passageway. Compounds that are quite insoluble in lipids and/or are larger than the barrier’s pores may use this mechanism of absorption (Figure 7.5). In contrast to passive diffusion, mass fux, J, for facilitated diffusion is not just directly proportional to the concentration gradient of the compound but rather to the quantity of the protein passageway present for the absorption, the diffusion coeffcient, and the rate of reversible interaction and splitting between the protein and the compound, if any. Any of these factors can act as a rate-limiting step for the absorption of compounds by carrier-mediated diffusion. This type of diffusion and its complex parameters mathematically require a boundary condition that deals with proteins that are stationary and do not move from the membrane with the compound. This scenario can only be defned by a nonlinear second-order differential equation that has no exact solution, and the numerical analysis of this type of equations mostly provides an approximation of actual diffusion. To elucidate further, as indicated earlier, even though the facilitated transport is a non–energy-requiring process, and the xenobiotic molecules do not cross the barrier against the concentration gradient, there is a marked contrast with passive diffusion in that the facilitated process depends on the amount of carrier present in the barrier and, thus, a saturation phenomenon may occur when the passage becomes the rate limiting step. Therefore, the kinetics of absorption may become nonlinear at high concentrations of a xenobiotic. The following relationship may clarify the distinction between the passive and carrier-mediated permeability (Usansky and Sinko, 2003). The relationship is based on the in vitro apical to basolateral transport in the brain micro-vessel endothelial cell (BMEC) monolayers. It is used to experimentally estimate the combined processes of permeability by passive diffusion and carriermediated transport (Figure 7.5). Pcombined =

J max + Ppassive K M + Capical

(7.55)

The parameter Pcombined is the combined permeability of passive diffusion and carrier-mediated transport; J max and K M are the maximum fux and the Michaelis–Menten constant, respectively, for the carrier-mediated transport, and Ppassive is the permeability for the passive diffusion. When K M ˜ Capical , the relationship is expressed as: Pcombined =

188

J max + Ppassive = ( Pint rinsic )carrier-mediated + Ppasssive KM

(7.56)

PK/TK CONSIDERATIONS OF ABSORPTION MECHANISMS AND RATE EQUATIONS

Figure 7.5 Depiction of facilitated diffusion, also known as passive mediated transport or carrier-mediated transcellular diffusion, an absorption mechanism that is facilitated by transport proteins that provide a passageway into the cell without requiring energy; certain compounds of various sizes, including those that are insoluble in lipids or have large sizes may penetrate the barrier using the passageway; the mass fux depends solely on the quantity of proteins that provide a passageway for a xenobiotic at the site of absorption, and high concentrations of a compound may overwhelm the passageways changing a linear absorption mechanism into a saturated nonlinear process. 7.4 TRANSCELLULAR DIFFUSION SUBJECTED TO P-GLYCOPROTEIN EFFLUX 7.4.1 Overview P-Glycoprotein (Pgp) is a member of the adinosine triphosphate (ATP)-binding cassette (ABC) transporter superfamily, capable of transporting a wide range of structurally unrelated xenobiotics from the intracellular environment into the extracellular space (Gatmaitan and Arias, 1993; Schinkel, 1997, Wang et al., 2021). The ABC superfamily represents the largest family of transmembrane proteins. Pgp was frst isolated from Pgp-overexpressing multidrug-resistant (MDR) tumor cells. However, detectable amounts of this protein have also been identifed in the normal tissues and organs of several mammalian species including rats and humans (Thiebaut et al., 1987; Cordon-Cardo et al., 1990; Storelli et al., 2020; Yamazaki et al., 2022;). Subsequent immunochemical studies on Pgp have confrmed the existence of highly expressed Pgp family members in the adrenal cortical cells, the brush border of the renal proximal tubule epithelium, the luminal surface of biliary hepatocytes, small and large intestine mucosal cells, and pancreatic ductules (Thiebaut et al., 1987, 1989; Cordon-Cardo et al., 1990). The capillary endothelial cells of the brain, testis, heart, placenta, lung, prostate, and stomach have also been shown to express Pgp, although to a lower extent (Matsuoka et al., 1999; Regina et al., 2002; Breedveld et al., 2005; Leslie et al., 2005; Kodaira et al., 2010; Storelli et al., 2020). In the GI tract, (the small intestine, in particular) Pgp is expressed on the brush-border membrane of enterocytes where it pumps xenobiotics out of the cytosol into the lumen environment of the intestine. Thus, the effux role of Pgp counteracts the absorption of xenobiotics in the GI tract and reduces the rate and extent of absorption. This section focuses only on the infuence of Pgp on the absorption and permeability models of xenobiotics and highlights the role it plays in PK/TK evaluation and bioavailability studies. 7.4.2 Pgp Structure and Function The following is a brief review of the structural characteristics and function of Pgp to elucidate the infuence of this protein on the rate and extent of absorption of xenobiotics. Pgp is a 170 kDa glycosylated transmembrane protein, composed of about 1200 amino acids (1280 in humans). Studies on human Pgp have shown that this protein is synthesized as a 150 kDa glycosylated intermediate 189

7.4 TRANSCELLULAR DIFFUSION SUBJECTED TO P-GLYCOPROTEIN EFFLUX

in the endoplasmic reticulum. Modifcation of the carbohydrates of this intermediate molecule in Golgi apparatus results in the mature protein, and glycosylation occurs at 91, 94, and 99 positions of the frst transmembrane loop (Loo and Clarke, 1998, 1999a, 2001; Loo et al., 2003). Reversible phosphorylation has been postulated to infuence the function of Pgp. A linker region between the frst nucleotide binding site and the second transmembrane domain of the protein has been identifed as a potential phosphorylation site. The linker region contains several serine residues that are phosphorylated by protein kinase C and/or protein kinase A in humans (Chambers, 1998; Dong et al., 1998; Cordon-Cardo et al., 1990). Primary structure studies using electron microscopy and image analysis techniques suggest four domains for the polypeptide. There are two hydrophobic transmembrane domains and two nucleotide-binding domains located at the cytoplasmic site of the membrane (Figures 7.6 and 7.7). Each of the hydrophobic domains consists of six membrane-spanning segments separated by hydrophilic loops, and each contains either the amino or the carboxylic terminal of the polypeptide chain (Rosenberg et al., 1997, 2003; Jones and George, 1998; Loo and Clarke, 1999b). The proposed structural studies identifed 14 anti-parallel transmembrane β-strands and 5 intracellular α-helices for each half of the hydrophobic domains as well as two β-strands and one α-helix for each nucleotide binding domain (Jones and George, 1998). According to this model, each half of the molecule can be assembled as a 16 β-sheet transmembrane barrel connected to a 6 α-helix bundle by short loops. Charged residues of the hydrophobic domains are often located immediately at the extracellular site of the membrane, which enables them to interact with the negative phospholipid head groups in the bilayer or with the cations capable of establishing an association with them. Using electron microscopy (Rosenberg et al., 1997, 2001, 2003) the overall shape of Pgp is reported to be cylindrical, about 10 nm in diameter, with a maximum height of about 8 nm. In comparison to the thickness of the lipid bilayer, which is about 4 nm, it is suggested that about one-half of the Pgp molecule is embedded in the membrane. Studies on Pgp structure using fuorescent resonance energy transfer (FRET) also suggested a cylindrical structure of 10 nm wide and 8 nm deep with two L-shaped nucleoside binding domains (NBD1, NBD2). This model also identifes two nonidentical putative drug binding sites (D1, D2) within the transmembrane region of Pgp, which interact with Pgp substrates and modulators (Sharom, 1995; Sharom et al., 1998, 1999; Wang et al., 2000; Marcoux et al., 2013). The two nucleotide binding domains (NBD1, NBD2) demonstrate ATPase activity, which is considered essential in the effux activity of Pgp. Studies on the Pgp structure using FRET have also shown that the two nucleotide binding domains (NBD1, NBD2) are partially embedded in the lipid bilayer in its cytosolic side (Figure 7.7). The two nonidentical putative drug binding sites (D1 and D2) within the transmembrane region are large enough to accommodate at least two structurally different drug substrates simultaneously. The interaction of xenobiotics with D1 and D2 is not fully clarifed yet. Using a cysteinefree human Pgp, it has been shown that there are two conformational changes: 1) when there is a

Figure 7.6 Portrayal of a Pgp structure within the transmembrane region with two hydrophobic nonidentical domains, D1 and D2, assumed as xenobiotic binding sites that interact with Pgp substrates and modulators; and two nucleotide-binding domains, NBD1 and NBD2, that are partially embedded in the lipid bilayer and essential in the effux activity of Pgp in generating and using ATP. 190

PK/TK CONSIDERATIONS OF ABSORPTION MECHANISMS AND RATE EQUATIONS

Figure 7.7 Schematic confguration of a cylinder-like Pgp molecule within the lipid bilayer of the biological membrane with a diameter of about 10 nm and height of about 8 nm; the two nucleotide binding domains (NBD1, NBD2) are partially embedded in the lipid bilayer and involved in ATPase activity generating energy to transfer the xenobiotic molecule from intracellular milieu to extracellular space; the two nonidentical putative drug-binding sites (D1 and D2) within the transmembrane region are large enough to accommodate at least two structurally different xenobiotics simultaneously. central cavity between the N- and C-terminal halves of the molecule, and 2) when the two halves have moved sideways closing the cavity (Lee et al., 2008). The consensus is that dimerization of NBDs driven by nucleotide binding is an essential step in the effux cycle, and the details of how this is linked to ATP hydrolysis and substrate transport remains unclear (Siarheyeva et al., 2010). Although the two NBDs appear to be located close to one another, they do not appear to be tightly associated, and, thus, this structure does not provide any additional information on their mode of interaction during catalysis (Aller et al., 2009). Two types of Pgp have been identifed in humans: Type I, termed multidrug resistance type 1 Pgp (MDR1Pgp, encoded by the MDR1 gene) and type II, MDR2Pgp (encoded by MDR2 gene). In rodents, the genes are identifed as mdr1a and mdr1b, which together are expressed roughly as the single MDR1 gene in humans and perform the same set of functions as MDR1Pgp. Pgp at the absorption site having binding sites for interaction with a variety of compounds, including its inhibitors/modulators, coadministration of multiple drugs, or exposure to various compounds, may enhance or reduce the absorption or bioavailability of the compound of interest. For example, it is well documented that the oral bioavailability of fexofenadine increases signifcantly when ketoconazole or erythromycin, known inhibitors of Pgp, are coadministered in humans. In addition to absorption, Pgp has signifcant infuence on the disposition of drugs in the body, for example, reduction of biliary elimination of vincristine when it is coadministered with verapamil (another known inhibitor of Pgp). However, several in vitro and in vivo studies have shown that coadministration of doxorubicin and Pgp modulators such as verapamil, cyclosporine A and amiodarone results in an increase in the cardiotoxicity of doxorubicin. There are numerous xenobiotics that can act as modulator/inhibitor of Pgp; among them are favonoids and other polyphenols, vinca alkaloids and taxenes. As it was indicated earlier, Pgp 191

7.4 TRANSCELLULAR DIFFUSION SUBJECTED TO P-GLYCOPROTEIN EFFLUX

is involved in secretory transportation of its substrates through its strategic location in the apical membrane of the proximal tubule of nephrons (Ernest and Bello-Reuss, 1998). Thus, the inhibitors of this transporter can signifcantly decrease renal excretion of its substrates and consequently have an impact on pharmacokinetics and pharmacodynamics of xenobiotics and are considered an important factor in drug–drug interaction. The synergistic role of metabolic enzyme (CYP 3A4) and Pgp in the small intestine is well documented and recognized as a major factor in limiting oral absorption and bioavailability of drugs. 7.4.3 Pgp Computational Equations The Pgp-mediated in vitro effux rate in cell culture is governed by the following equations, which represent the saturable nature of the effux (Kuh et al., 2000; Jang et al., 2001): J Pgp ˜ mediated °

J max ˛ C fcell K MPgp ˝ C fcell

(7.57)

Where J Pgp − mediated is the effux rate per cell; J max is the maximum effux rate per cell; K MPgp represents the dissociation constant of Pgp-mediated effux; and C fcell is the free concentration of xenobiotic in the cells. The mass balance equations for the amount inside the cells and in medium as a function of time are dV dC dAtotalcells ˜ Vcells ° totalcells ˛ Ctotalcells ° cells dt dt dt

(7.58)

dAtotalcells ; dt the total volume of cells is Vcells , and the total concentration of free and bound is Ctotalcells . Equation 7.58 assumes the volume of cells changes with time. Given that Rate = Clearance × Concentration, the equation can also be written as

where the rate of change in the total amount (i.e., bound and free) as a function of time is

J max ° C fcell ˘ dAtotalcells ˙ ˜ ˇCl f ° C fmedium ˛ Cl f ° C fcell ˛  ° N cells dt K MPgp ˝ C f cell  ˆˇ 

(7.59)

where the clearance of free xenobiotic by passive diffusion is Cl f , the free concentration of the compound in the medium and inside the cell is C fmedium and C fcell , respectively, and N cells is the number of cells. The rate of change in the total amount of compound in the medium is dC dAtotalmedium ˜ Vmedium ° totalmedium dt dt

(7.60)

where Ctotalmedium represents the total concentration (free and bound) in the medium and Vmedium is the volume of the medium. In comparison with Equations 7.58 and 7.59, the change in the volume of the medium is assumed negligible, and in terms of clearance, Equation 7.60 can be written as J max ˛ C f cell dAtotalmedium ˙ ˜ ˇ °Cl f ˛ C fmedium ˝ Cl f ˛ C fcell ˝ dt K MPgp ˝ C f cell ˇˆ

˘  ˛ N cells 

(7.61)

The rate equations are further expanded to defne the time-dependent changes in intracellular and extracellular drug concentration as a function of cell volume, binding affnity and capacity, and Pgp-mediated effux (Kuh et al., 2000; Jang et al., 2001) by identifying the total concentration (bound and free) in the cells, Ctotalcells , and in the medium, Ctotalmedium , as functions of free concentration in the cells, C fcells , and the medium, C fmedium , as follows Ctotalcells ˜ C fcells °

Bmaxcells ˛ C fcells ° BNS ˛ C fcells K Mcells ° C fcells

Ctotalmedium ˜ C fmedium °

Bmax medium ˛ C fmedium K Mmedium ° C fmedium

(7.62) (7.63)

Where Bmaxcells is the maximum binding to the cellular components; K Mcells is the corresponding Michaelis–Menten constant; BNS stands for binding to the cells’ non-saturable binding sites;

192

PK/TK CONSIDERATIONS OF ABSORPTION MECHANISMS AND RATE EQUATIONS

Bmax medium is the maximum binding to the medium; and K Mmedium is the medium Michaelis–Menten constant. The total volume of cells Vcells is estimated as (Equation 7.64): (7.64)

Vcells = Vcell ´ N cells

where Vcells is the volume of one cell and N cells is the number of cells at time t. Therefore, the time course of Vcells can be defned as Vcells = (Vcell ´ Nt=0 ) e (

kcell ´t )

(7.65)

Where Nt=0 is the initial number of cells; and kcell is the rate constant for changes in the cell number which, depending on the type of xenobiotic and its infuence on the cells, kcell can be positive when a xenobiotic mediates cell proliferation and negative when the compound reduces the number of cells. For compounds where there is a linear enhancement in the number of cells with respect to time, the relationship between Bmaxcells at time t and zero can be defned as (Kuh et al., 2000)

( Bmax )t = ( Bmax )0 ´ (1 + kB cells

cells

max cells

´ t)

(7.66)

where the cellular maximum binding sites at time t and 0 are ( Bmaxcells ) and ( Bmaxcells ) , respectively. t

0

The rate constant for increases in the cellular binding site is kBmaxcells , which is a concentrationdependent rate constant. Substitution of Equations 7.62–7.63 and 7.65–7.66 in Equations 7.59 and 7.61 yields the following relationships, which defne the time-course of intracellular and extracellular concentrations of a xenobiotic as a function of Pgp-mediated effux binding capacity and affnity and total cellular and medium volumes: J ´Y ù dCtotalcells é 1 = êCl f ´ X - Cl f ´ Y - max - ( kceell ´ Ctotalcells ) ú´ K MPgp + Y úû Vcell dt êë

(7.67)

J ´ Y ù Nt=0 e kceell ´t dCtotalmedium é = ê -Cl f ´ X + Cl f ´ Y + max ú´ K MPgp + Y ûú Vmedium dt êë

(7.68)

Where X and Y are defned as X = éê - ( K Mmedium + Bmax medium - Ctotalmedium ) + ë

(KM

medium

2 + Bmax medium - Ctotalmedium ) + 4 ´ K Mmedium ´ Ctotalmedium ùú / 2 û

(7.69) Y=

-Z + Z 2 + 4 (1 + BNS ) ´ K Mcells ´ Ctotalcells 2 (1 + BNS )

Z = (1 + BNS ) ´ K Mcells + ( Bmaxcells )0 ´ (1 + K Mcells ´ t ) - Ctotalcells

(7.70) (7.71)

The independent parameters of Pgp in the above relationships are Bmax medium , Bmaxcells , K Mmedium , K Mcells , BNS , and Cl f

(7.72)

Accordingly, the Pgp-mediated effux rate can be defned as RatePgp - efflux = Cl f ´ ( C freemedium - C freecells ) =

J max ´ C fcells K MPPgp + C fcells

(7.73)

and the effux clearance of xenobiotics from the intracellular environment through Pgp effux is ClPgp =

J max K MPgp + C fcells

(7.74)

The overall infuence of Pgp effux on in vivo PK/TK is discussed in in Chapter 17, Section 17.5.4 and 17.5.6.

193

7.5 ACTIVE TRANSPORT

7.5 ACTIVE TRANSPORT Active transport generally refers to the transport of molecules against the concentration gradient. It requires interaction with protein carriers and the expenditure of energy. Thus, the process is an endergonic one that must be joined with an exergonic process, namely, the hydrolysis of ATP (Figure 7.8). Both processes occur in the presence of enzymes and are inhibited with toxins, such as dinitrophenol and fuoride. The active transport occurs only at certain sites, such as the intestines, liver, lung, and kidneys. Due to the energy requirements of the process and limited concentration of protein carriers, the active transport is apt to produce a saturation occurrence. Most often, the kinetics of active absorption follow the Michaelis–Menten equation and exhibit nonlinear characteristics. Furthermore, the protein carriers are subject to competitive inhibition by compounds with similar structures. æ dC ö ç ÷ ´C dC è dt ømax = dt KM + C where

(7.75)

dC æ dC ö is the rate of absorption, ç is the maximum rate of absorption, K M is the ÷ dt è dt ømax

Michaelis–Menten constant, and C is the concentration at the site of absorption.

Figure 7.8 Illustration of the stepwise energy requiring active transport of molecules against the concentration gradient through the biological barriers; the active transport occurs only at certain routes of administration like intestine, liver, lungs, and kidneys, and is available only for specifc xenobiotics; the absorption is saturable, often exhibits nonlinear characteristics, and the carriers are subject to competitive inhibition by compounds with similar structure. 194

PK/TK CONSIDERATIONS OF ABSORPTION MECHANISMS AND RATE EQUATIONS

Depending upon the formation and use of ATP, the active absorption is identifed as either a primary or secondary active transport. The primary active transport refers to the movement of ions across biological barrier by protein transporters that combine the transport of the substrate with hydrolysis of ATP (Figure 7.8). Calcium pumps, sodium pumps, and proton pumps are examples of primary active transport. The secondary active transport occurs when the transport of ions by primary active transport stores and releases enough energy that forms ion gradient and transports other ions across the barrier. This is also called the co-transport process. The direction of the secondary transport can be the same or opposite that of the primary transport and is identifed as uniport, symport, or antiport (Figure 7.9). A well-known example of secondary transport is the co-transport of glucose and sodium in the small intestine. The GI tract is the main site of active absorption in the body. The pathways of active transport in the intestine are structure-specifc and exist for the purpose of transferring certain nutrients such as uracil, choline, folate derivatives, bile salts, monosaccharides, 1-amino acids, certain vitamins, etc. Xenobiotics that are structurally similar to these nutrients use the same pathways for absorption. For example, the anticancer drug 5FU (5-Fluorouracil) uses the uracil pathway, and methotrexate, another anticancer drug, uses the folate derivative pathway; other examples include iron salts, levodopa, propylthiouracil, and fuorouracil. Active transport is more common for the disposition of endogenous compounds such as glucose, amino acids, and certain ions. The intestinal peptide transport, known as PepT1, has a wide range of substrate specifcity and transports, not only di- and tri-peptide of food products (Mathews and Adibi, 1976; Furst et al., 1990; Gilbert et al., 2008; Yuri et al., 2020; Killer et al., 2021; ) but also a number of xenobiotics with structures similar to peptides, for example, angiotensin-converting enzyme (ACE) inhibitors (Kim et al., 1994; Covitz et al., 1996; Moore et al., 2000) or beta-lactam antibiotics (Snyder et al., 1997). Substrate mimicry, that is, development of a therapeutic agent with a 3-D structure similar to the substrate of PepT1, is an approach for the development of new therapeutic peptides. The apical sodium-dependent bile acid transporter (ASBT), is another transporter involved in the reabsorption of bile acid. It is noteworthy that from the total amount of bile acids in a human

Figure 7.9 Depiction of co-transport active absorption when the primary active transport stores and releases enough energy to transport other ions across the barrier; the direction of the secondary transport can be the same or opposite that of the primary transport identifed as uniport, symport, or antiport. 195

7.7 SOLVENT DRAG, OSMOSIS, AND TWO-PORE THEORY

(about 3–5 g), which circulates 6–10 times a day with a turnover of 20–30 g, only 0.2–0.5 g eliminates through fecal elimination (Zhang et al., 2002). The amount eliminated is replenished by the synthesis of new molecules. ASBT plays an important role in maintaining the balance of bile acids and governs the enterohepatic recirculation of bile acids (Hofmann, 1976, 1989; Erlinger, 1987). Compounds that are structurally similar to bile acids are also absorbed via this transporter. The molecular mechanism of the primary active transport at the protein level has not been fully elucidated. There are, however, certain postulations that have been accepted as plausible scenarios. For example, a protein transporter may interact with a xenobiotic molecule, undergo the energy-requiring conformational change, rotate 180°, and release the molecule at the opposite side (Figure 7.8). Another scenario is that when the transport protein is a large molecule with multiple binding sites and functional groups on its surface, can shuttle the molecule between the functional groups to the opposite side. 7.6 ENDOCYTOSIS AND PINOCYTOSIS In a nutshell, endocytosis is the process of transferring molecules or particles, here referred to as the “load,” forming a vesicle in the intracellular environment, and transferring it toward lysosomes and lytic vacuoles for cellular assimilation (Figure 7.10). The stages of transfer from extracellular milieu to intracellular environment include 1) the cell membrane forms a cavity or pouch to encase the load, 2) the membrane completely conceals the load in the intracellular environment, yet attaches to the membrane inside the cell, 3) the membrane-covered load, known as the endosome, separates itself from the cell membrane and joins the contents of the cell and other endosomes, and 4) the load, through the intracellular transcytosis, reaches the appropriate components of the cell including the lysosomes and vacuoles. The endocytosis of particles and molecules known as phagosome, is referred to as phagocytosis that can handle particles as large as 20 µm (Germain, 2004). The endocytosis of small particles in solution forming small vesicles is known as pinocytosis. Pinocytosis, or “cell-drinking,” is a nonselective physical absorption that forms small vesicles inside the cell from extracellular fuids before interacting with lysosomal acidic hydrolases and breaking down into smaller particles. Pinocytosis occurs continuously in nearly all cells. For small particles to be absorbed by pinocytosis, they must be associated with the extracellular fuid. The process, from forming the vesicles to breaking down by lysosomes, is an energy-requiring absorption. Compounds, such as proteins/ polypeptides, and particulate entities, like therapeutic or nontherapeutic nanoparticles, lipophilic carriers, such as small unilamellar liposomes, or droplets of micro-emulsions, etc., may cross the membrane by pinocytosis. The process of pinocytosis may include proteins that facilitate the endocytosis pathway. One of these proteins is clathrin (Pearse, 1976), which is involved in intracellular traffcking by coating the vesicle containing the load. There are two types of pinocytosis – those that are clathrin-dependent endocytosis and those that are clathrin-independent endocytosis (Conner and Schmid, 2003). Distinction and the types of endocytosis involved in the absorption of nanoparticles, proteins, and polypeptides depend to a large extent on their physicochemical characteristics including the particle size and geometry of the particles (Nel et al., 2009), surface chemistry, and shape and chemical composition. Distinction should be made between pinocytosis, which is a nonselective transport, and the receptor-mediated endocytosis, which is a selective process and depends on the specifc role of receptor on the membrane (Figure 7.10). Briefy, ligands bind to specifc receptors on the membrane form ligand-receptor complexes; then, like pinocytosis, it continues with the membrane coating and transferring inside the cells. Some examples of receptor-mediated endocytosis include nutrients, growth factors, hormones like insulin, viruses, and antigens. The intracellular coating is also by clathrin (Pearse and Robinson, 1990). 7.7 SOLVENT DRAG, OSMOSIS, AND TWO-PORE THEORY The transfer of appropriately sized xenobiotic molecules by solvent through the pores and the loosely connected biological barrier is referred to as solvent drag. The basic principles of solvent drag are somewhat intertwined with the principles of osmosis and the two-pore theory (Kedem and Katchalsky, 1958; Rippe and Haraldsson, 1994; Hammel and Schlegel, 2005). The following equation (Kedem and Katchalsky, 1958) shows the link between the solvent drag and osmosis: J S = Lp D p - sLp Dp 196

(7.76)

PK/TK CONSIDERATIONS OF ABSORPTION MECHANISMS AND RATE EQUATIONS

Figure 7.10 Illustration of dissimilarities between pinocytosis, a nonselective transport vesicle with and without clathrin, and the intracellular traffcking protein; phagocytosis that can handle particles as large as 20 µm, and the receptor-mediated endocytosis, which is a selective process and depends on the specifc role of receptor on the membrane; all vesicles and particles have membrane coating and go through the intracellular transcytosis to reach their destination in the cell, e.g., lysosome. where J S is the solvent fux; Lp is the fltration coeffcient of the solvent; D p and Dpare the hydrostatic and osmotic pressure differences, respectively; s is a coeffcient of refection used in the estimation of the solvent drag coeffcient as described below. The fux of solute, J solute , by the solvent drag is defned as J solute = J S (1 - s )( Cave )solute + Psolute Dp

(7.77)

The coeffcient of solvent drag is (1 - s ) ; the average concentration of solute is ( Cave )solute ; the solute permeability is Psolute , estimated by the ratio of the apparent transcapillary diffusion coeffcient to the diffusion distance; and Dp is the osmotic pressure differences across the barrier, which also represents the solute concentration gradient, that is, J solute = J S (1 - s )( Cave )solute + Psolute DC

(7.78)

DC = ( Coutside - Cinside )

(7.79)

Integration of Equation 7.78, according to the boundary conditions across the membrane, yields the following equation: J solute = J S (1 - s )

Coutside - Cinside e -Pe 1 - e -Pe 197

7.8 ION-PAIR ABSORPTION

(Pe ) is a modifed Peclet number equal to Pe =

JS (1 - s ) Psolute

(7.80)

Therefore, the solvent fux is approximately equal to JS =

Pe ´ Psolute 1- s

(7.81)

When Pe is larger than 3, the transport of the solute is mainly by solvent convection. Thus, one can conclude that the fux of a solute absorbed by passive paracellular diffusion and solvent drag is the sum of both fuxes. The microvascular endothelium permeability has been suggested to have small pores, represented by clefts in the intracellular junctions. In contrast, macromolecules use pores that are larger in size and are considered opening between cells or intracellular vesicular systems (Pappenheimer et al., 1951; Eleni et al., 2011). In fact, most of the macrovascular walls have two types of pores: 1) the small functional pores that are abundant, but restrictive toward large molecules, such as proteins; and 2) non-size-selective pathways that are scarce but permit the passage of large molecules (Rippe and Haraldsson, 1994). Large pores are predicted to make up 1 part per 10,000 to 30,000 of the total number of pores. The convection through the large pore size is considered an essential element of the permeation of proteins by solvent drag. In general, the role of solvent drag is considered more signifcant for the fux of large molecules than the small ones. The reason is the signifcant diffusive fux of small molecules compared to the solvent drag movement and the related velocity of solute in one direction. An interesting but unresolved aspect of the solvent drag is whether the back fux of an absorbed drug from blood into the intestinal lumen can be ignored while the back fux of water cannot be ignored. 7.8 ION-PAIR ABSORPTION The pH-partition theory postulates that only unionized lipophilic compounds can penetrate through the biological barriers whereas ionized forms of drugs cannot readily diffuse through the barrier. However, it can occur that a drug is completely ionized at the site of absorption in the GI tract, yet it is signifcantly bioavailable. This observation cannot be explained by the pH-partition theory. Thus, it has been postulated that such absorption can be a consequence of the ion-pair formation at the site of absorption (Jonkman and Hunt, 1983; Neubert et al., 1987; Neubert, 1989; Härtl et al., 1990). This hypothesis has been a highly debated subject in the literature. According to this hypothesis, oppositely charged molecules may interact and form a neutral complex. The neutral complex will then cross the biological barrier by passive diffusion. Since the formation of a neutral complex is a simple chemical equilibrium, an excess of one ion can increase the formation of the complex. Thus, the formation and absorption of the complex depends on the concentration of one or both ions, and this dependency contributes to an erratic absorption of the drug (Neubert et al., 1988). This means that the formation of the complex is the rate-limiting step in the absorption process. The breakdown of the complex is also based on the chemical equilibrium. Once the neutral complex is inside the membrane, a new equilibrium will be established, and all three species of positive, negative, and neutral will be present in the membrane at the same time (Irwin et al., 1969; Lippold and Schneider, 1974; Lippold and Lettenbauer, 1980; van der Giesen and Janssen, 1982; Boroujerdi, 1987; Neubert et al., 1988). As the charged molecules leave the barrier, more of the neutral complex breaks down into the charged molecule. The absorption of the neutral complex is by passive transcellular or paracellular diffusion. Thus, one can assume that the ion-pair formation increases the lipophilicity of hydrophilic ionized compounds and enhances their partition coeffcient. It has been shown that the partition coeffcient of hydrophilic compounds, like buformin, bretylium, and pholedrine, is markedly increased by lipophilic ions, such as hexyl salicylate, or the lipophilicity of doxorubicin is enhanced in the presence of the counter ions dioctyl sulfosuccinate and dodecyl sulfonate in the low pH range. There are several other counter ions, which have shown to increase the lipophilicity of ionized molecules. However, the argument against use of these compounds is that they are too harsh for in vivo applications. Among the counter ion compounds are: alkyl carboxylase, cholate, n-alkyl sulfates, n-alkyl carbonates, and trichloroacetate, which are considered absorption 198

PK/TK CONSIDERATIONS OF ABSORPTION MECHANISMS AND RATE EQUATIONS

enhancers of hydrophilic quaternary ammonium compounds. The application of this hypothesis in absorption of peptides from different sites (Quintanar-Guerrero et al., 1997; Soares et al., 2007), enhancement of prodrugs absorption (Samiei et al., 2013) and antiviral drugs (Miller et al., 2010) has renewed interest in ion-pair formation as a means of enhancing absorption of ionized molecules. The following are the relationships of lipophilic behavior of ion-pair complex species, according to pH-partition theory in a simple model of the aqueous-organic phase system. The dissociation constant of reversible interaction of an ionized drug with the ion pair agent to form an ion-pair complex in the aqueous phase (van der Giesen and Janssen, 1982), and organic phase are presented in Equations 7.82 and 7.83, respectively. é A - ù éB+ ù K[ AB] w = ë û w ë û w [ AB]w

(7.82)

The dissociation constant in the organic phase is é A - ù éB+ ù K[ AB] o = ë û o ë û o [ AB]o

(7.83)

where é A - ù and é A - ù are the concentrations of the ionized drug in aqueous phase (w) and ë ûw ë ûo organic (o), respectively. éB+ ù and éB+ ù are the positively charged ion-pair agents; [ AB]w and [ AB]o ë ûo ë ûw are the ion-pair complexes. The partition coeffcient of the ion-pair complex between the two layers can be defned as PAB =

[ AB]o [ AB]w

(7.84)

When the ion-pair agent is dissociated completely in the aqueous phase into charged ions, the apparent partition coeffcient, ( Pcoeff )app , is

( Pcoeff )app =

[ AB]o + éë A- ùû o

(7.85)

é A- ù ë ûw

To estimate ( Pcoeff )app in terms of measurable quantities éB+ ù and é A - ù , it would be convenient to ë ûw ë ûw ( Pcoeff )app use a quantity known as extraction constant, K ext , which is the ratio of K[ AB] w ( Pcoeff )app AB]o [ (7.86) K ext = = + K[ AB] w éB ù é A - ù ë ûw ë û w Combining Equations 7.83 and 7.86 gives rise to (7.87)

é A - ù éB+ ù = K ext K[ AB] o éB+ ù é A - ù ë ûo ë ûo ë ûo ë ûw

When the ion-pair complex is the only group of molecules that can pass through the barrier, then éB+ ù = é A - ù and Equation 7.87 can be written as ë ûo ë ûo

(

2

é A - ù = K ext K[ AB] o éB+ ù é A - ù ë ûo ë ûw ë ûw \ éë A - ùû = K ext K[ AB] o éëB+ ùû éë A - ùû o

)

(7.88) (7.89)

w

Combining Equations 7.85 ( Pcoeff )app , 7.86 K ext , and 7.89 é A - ù and solving for ( Pcoeff )app gives rise to ë ûo the following relationships:

(

-1 -1 æ Papp = éëB+ ùû ç K ext + K ext ´ K[ AB] o ´ éëB+ ùû ´ éë A - ùû w w è

)

1/2

ö ÷ ø

(7.90)

If K[ AB] o = 0 , then 199

7.8 ION-PAIR ABSORPTION

Figure 7.11 Plot of normalized apparent partition coeffcient with respect to positively charged ion-pair agent against reciprocal of square root of drug molecule and ion-pair agent for the purpose of estimating the ion-pair complex in aqueous and lipid layers from known values of the experiment. Papp = éëB+ ùû K ext w When K[ AB] o ¹ 0 (is not equal to zero), the equation can be written in terms of known quantities of the system

(

-1

-1

-1

(

A plot of Papp éB+ ù against éB+ ù ´ é A - ù ë ûw ë ûw ë ûw

(

K ext and slope of K ext ´ K[ AB] o

)

1/2

)

1/2

)

-1 1/2

Papp éëB+ ùû = K ext + K ext ´ K[ AB] o ´ éëB+ ùû ´ éë A - ùû w w w

(7.91)

should generate a straight line with the y-intercept of

(Figure 7.11).

The ion-pair absorption has been tested for permeation of compounds, like zanamivir heptyl ester and guanidino oseltamivir (Miller et al., 2010), for percutaneous absorption of loxoprofen (Hui et al., 2016), and dexmedetomidine (Wang et al., 2020). REFERENCES Abraham, D. J., Kellogg, G. E. 1994. The effect of physical organic properties on hydrophobic felds. J Comput Aid Mol Des 8(1): 41–9.

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8 PK – TK Considerations of Distribution Mechanisms and Rate Equations 8.1 INTRODUCTION After gaining access to the systemic circulation through one of the routes of administration, xenobiotics distribute in different tissues and organs of the body. Depending on their physicochemical characteristics, a series of physical and physiological processes occur simultaneously that shape the distinctive pattern of their distribution in the body. An example of physical processes is the simple dilution in the intracellular and extracellular fuids; and the examples of physiological processes are protein binding, tissue uptake, permeation of the compound through different biological barriers in the body, interaction with transport proteins in conjunction with the processes of metabolism and excretion. The distribution processes transfer the compound to various regions of the body, including the receptor sites, target organs, excretory organs, and other tissues. Thus, if the compound is not actively or passively targeted to a receptor site in the body, all organs and tissues will have to endure the presence of the xenobiotic. 8.2 FACTORS INFLUENCING THE DISTRIBUTION OF XENOBIOTICS IN THE BODY The distribution of a xenobiotic in the body is infuenced by factors such as I. Total body water (TBW) II. Blood fow and organ/tissue perfusion a. Transcapillary exchange of xenobiotics b. Perfusion-limited distribution c. Permeability-limited distribution III. Binding to plasma proteins a. Parameters of protein binding IV. Physicochemical characteristics of the compounds V. Extent of penetration through physiological barriers and parallel elimination processes Physiological barriers a. Blood–brain barrier b. Blood–lymph barrier c. Placental barrier d. Blood–testis barrier e. Blood–aqueous humor barrier VI. Body weight and composition VII. Disease states 8.2.1 Infuence of Total Body Water on Xenobiotic Distribution The total body water and its distribution between intracellular water and extracellular water is an important parameter in the distribution of xenobiotics in the body and has signifcant importance clinically in: abmormal hydration, malnourished patients, obesity, pregnancy, renal impairment, accurate assessment of fuid overload in patients on hemodialysis, monitoring body fuid balance in hospitalized patients who are susceptible to dehydration as well as overhydration that may inversely infuence the pharmacokinetics/pharmacodynamics (PK/PD) of medications, and other concerns. Various methodologies are used for measurement of total body water; among them are: 1) Bioelectrical impedance analysis (Segal et al., 1991; Kyle et al., 2004), which is based on the measurement of total body electrical resistance by skin electrodes placed on selected areas of the body like the hands and feet. This step is followed by conversion of the measured resistance into total body water volumes based on empirically derived algorithms. Some inaccuracies have been reported that are related to the use of the algorithm. 2) Use of the radioisotope dilution technique of deuterium. This methodology also has shortcomings that include the length and time 210

DOI: 10.1201/9781003260660-8

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consumption of the procedure, expense, may be invasive, and the short-term body volume changes are undetectable. 3) Use of biomarkers like natriuretic peptide and copeptin. The procedure is noninvasive, affordable, easy, and fast, but it suffers from poor reproducibility and high variability (Tan et al., 1989; Morgenthaler et al., 2006). In healthy adults, the total volume of water in the body responsible for the distribution of xenobiotics and nutrients is about 50–75% body weight. Approximately 40% of the TBW is in extracellular environment, and 60% is in the intracellular environment. The extracellular portion is divided between three sites in the body: 75% is in interstitial space, 20% in intravascular plasma, and 5% is transcellular fuid that includes cerebrospinal fuid and synovial fuid (within the joints), which have different solute compositions. A few of the empirical equations for estimation of TBW are: Watson Formula (Watson et al., 1980): For men: TBW(L) = 2.447 - ( 0.09156 ´ age ) + ( 0.1074 ´ Height(cm)) + ( 0.3362 ´ Weigh ht(kg ))

(8.1)

For women: TBW(L) = -2.097 + ( 0.1069 ´ Height(cm)) + ( 0.296785 ´ Weight(kg))

(8.2)

Hume Formula (Hume and Weyers, 1971): For men: TBW(L) = ( 0.194786 ´ Height(cm)) + ( 0.296785 ´ Weight(kg )) - 14.012934

(8.3)

For women:

(

)

TBW(L) = ( 0.34454 ´ Height ( cm ) ) + 0.183809 ´ Weight ( kg ) - 35.27

(8.4)

In newborns, TBW constitutes 75% of the body weight, and it reduces to the normal value within the frst 10–12 years of life. Using the above percentages, for a 70 kg male of standard height, if approximately 60% (a value between 50% and 75%) of the body weight is TBW, the calculated value would be 42 L. Therefore, the intracellular fuid is about 25.2 L, extracellular fuid ≈ 16.8 L, interstitial fuid is ≈ 12.6 L, intravascular plasma ≈ 3.36 L, transcellular fuid ≈ 0.84 L. The total blood volume is about 12.5% of total body water (≈ 5 L), that is, the blood volume for the 70 kg subject is approximately 71.4 mL/ kg body weight. The calculated values change in the adult population depending on sex, weight, and disease states. The fve liters of blood consists of approximately three liters of plasma and two liters of blood cells. An equation that is often used to determine blood volume is

( volume )blood =

( volume )plasma 1 - hematocrit

(8.5)

The calculated value is close to 79 mL/kg for mouse, 65 mL/kg for rat, and 85 mL/kg for dogs. The total blood volume is an essential physiological factor in determining the pharmacokinetics and toxicokinetics (PK/TK) of xenobiotics. When it is necessary to remove multiple large volumes of sample blood, the total reduction in the blood volume may infuence the distributional profle of the administered compound. Thus, the gradual replenishment of the volume may often be necessary. 8.2.2 Effect of Blood Flow and Organ/Tissue Perfusion on Xenobiotic Distribution Blood fow is the volume of blood that circulates in the cardiovascular system and reaches different organs and tissues in the body per unit of time. It is expressed in volume/time and its magnitude differs in the different regions of the body. The total blood fow is equal to 5000 mL/ min that correspond to the cardiac output at rest. The cardiac output is the volume of blood pumped by the heart per minute. In addition to the cardiac output, an important factor in blood fow is the variable volume of blood in the different networks of systemic circulation. The heart usually holds 7% of total blood; the pulmonary system maintains 9%; arterial capacity is about 13%; the arterioles and capillaries contain 7%; the remaining 64% is in the veins, venules, and venous system. The capillaries with their permeable walls allow the exchanges of xenobiotics, nutrients, etc., with the interstitial fuids of the organ/tissue and merge with the venule network that gradually converges into larger veins. During this transcapillary exchange, xenobiotics are transported across the capillary wall into the tissue. This transport may result from the 211

8.2 FACTORS INFLUENCING THE DISTRIBUTION OF XENOBIOTICS IN THE BODY

difference in pressures on the inside and outside of the capillary wall, such as osmotic pressure or hydrostatic pressure, or may result mainly from a concentration gradient across the capillary wall. The distribution of a compound in a region of the body depends on the blood fow to that region and how fast the drug is transferred from the site of administration to the region. Table 8.1 is a representation of the differences in volume and blood fow of different organs/tissues of the human body for a 30–39-year-old man with 70 kg body weight and 1.73 m2 body surface area. The blood fow essentially determines how fast a compound is delivered to an organ/tissue and refects the relative rate a compound in the organ/tissue is expected to achieve at equilibrium with blood. The amount of xenobiotic that can be stored or distributed in a tissue depends on the mass of tissue or the size of the organ as well as physicochemical characteristics of xenobiotic, such as the partition coeffcient or distribution coeffcient. Figure 8.1 exhibits the infuence of blood fow on the disposition time course of a compound injected intravenously and disseminated to various tissues. The accumulation in various regions is a function of blood fow, as well as other physiological factors and physicochemical characteristics of the compound. Physiologically, the blood fow is determined by the Ohm’s law, that is, DP (8.6) R Where Q is the fow rate in volume/time; DP is the pressure difference between the two ends of a vessel (cylinder) in mmHg; and R is the resistance in mmHg × time/volume. According to Equation 8.6, the fow rate is directly proportional to pressure differences and inversely proportional to the resistance. This principle can be applied to a single cylinder (vessel) or a network of cylinders (e.g., the vascular network of an organ). Resistance to blood fow infuences not only the cardiovascular function but also the distribution and transfer to organs/tissues. It has been suggested that in small microvessels, in addition to the rheological variability of blood, the interaction between the inner vessel surface and blood components also causes high fow resistance (Pries et al., 1994, 1997). In general, the resistance to blood fow depends on the radius and length of vessels and viscosity of blood according to the Hagen–Poiseuille equation (Hagen, 1839; Poiseuille, 1838; Sutera and Skalak, 1993): Q=

R=

8 ´ Length ´ h p ´ r4

(8.7)

where h is blood viscosity and r is the inner radius of the vessel. Thus, a small change in radius, raised to the fourth power, will increase the resistance signifcantly. Substitution of Equation 8.7 into Equation 8.6 yields the following equation applicable to blood fow:

Table 8.1 Blood Flow to Different Organs/Tissue for a 30–39-Year-Old Man at Rest with 70 kg Body Weight and 1.73 m2 Body Surface Area Organ/Tissue Adrenals Kidneys Thyroid Gray Matter Heart Liver + Portal System Red Marrow Muscle Fat Fatty Marrow Bone Cortex Total

212

Volume (L)

Blood Flow (mL/min)

Blood Flow (mL/ min)/100 mL Tissue

Volume of Blood in Equilibrium with Tissue (mL)

0.02 0.30 0.02 0.75 0.30 3.90

100 1240 80 600 240 1760

500 410 400 80 80 45

62 765 49 371 148 976

1.40 30.00 10.00 2.20 6.40

120 600 200 60 0 5000

9 2 2 2.70 0

74 370 123 37 0

PK/TK CONSIDERATIONS OF XENOBIOTICS SYSTEMIC DISTRIBUTION

Figure 8.1 General profle of the infuence of blood fow on the distribution of a xenobiotic following an intravenous bolus injection; the systemic circulation curve exhibits the exponential decline of the compound in the body due to the disposition and uptake by various tissues and organs; the highly perfused skewed bell-shape curve displays the rapid uptake of the injected compound delivered by the blood fow to the organs that reach to an equilibrium faster than the rest of the body; the time course of the uptake by the less perfused tissues and muscle represents the gradual uptake and a maximum concentration less than the highly perfused tissues; and depending on the lipophilicity of the injected compound a gradual uptake and possibly short-term storage is expected. Q=

DP ´ p ´ r 4 8 ´ length ´ h

(8.8)

Thus, the blood fow is directly proportional to the radii of vessels and pressure differences and inversely proportional to the viscosity of blood and length of the vessels. Depending upon the physicochemical characteristics of a xenobiotic, the transcapillary exchange and storage in tissue is either a blood fow-limited phenomenon known as perfusion-limited distribution, or a transfer-limited phenomenon known as permeability-limited distribution. 8.2.2.1 Perfusion-Limited Distribution and Permeability-Limited Distribution (Transcapillary Exchange of Xenobiotics) To differentiate the dynamics of perfusion-limited distribution from permeability-limited distribution, the blood capillary can be assumed as a cylinder of length Land radius of r in which blood fows with a velocity of v in the positive direction of x . The blood fow carries the xenobiotic with a concentration of Cblood , which transfers the compound to the tissue bed with a concentration of Ctissue . The compound diffuses across the capillary membrane by passive diffusion because of the concentration gradient of the compound between the blood and tissue. If we consider a segment of the capillary between x and x + Dx, the difference in mass fux of the compound between the

213

8.2 FACTORS INFLUENCING THE DISTRIBUTION OF XENOBIOTICS IN THE BODY

beginning and the end of Dx would be equal to the mass fux across the capillary wall into the surrounding tissue (Figure 8.2). If the cross-sectional area of the capillary is pr 2 , the surface area of diffusion is 2rpDx , and the permeability constant for the compound in the capillary membrane is P , the following rate equation would defne the movement of the amount in the capillary: pr 2v ( dCblood / dx ) = 2rpP [Cblood - Ctissue ]

(8.9)

As the limit of Dx → 0, Equation 8.9 changes to pr 2v ( dCblood / dx ) = -2rpP [Cblood - Ctissue ]

(8.10)

The cross-sectional area multiplied by the velocity pr 2v is equal to the blood fow Q . The surface area of the capillary under observation SArea is equal to 2rpL . Substitutions of Q and SArea in Equation 8.10 yields dC PS = - Area ( Cblood - Ctissue ) dx QL

(8.11)

Where ( Cblood - Ctissue ) is the concentration gradient between the capillary and surrounding tissue. Integration of Equation 8.11 yields the following relationship: Cblood ( x ) - Ctissue = ( Carterial - Ctissue ) e -PSx/QL

(8.12)

Figure 8.2 Illustration of a tissue bed with intracellular- and extracellular fuid and capillaries highlighting the distinction between perfusion-limited distribution versus permeability-limited distribution by assuming a segment of capillary as a cylinder between two points of x and Dx , radius of r , cross sectional area of pr 2 blood velocity of (v), transporting xenobiotic concentration to the tissue. 214

PK/TK CONSIDERATIONS OF XENOBIOTICS SYSTEMIC DISTRIBUTION

Setting x = L and Cblood ( L ) = Cvenous , Equation 8.12 transforms to Cvenous = Ctissue + ( Carterial – Ctissue ) e -PS/Q

(8.13)

The mass fux across the capillary wall into the tissue J tissue in terms of net mass-current leaving the capillary per length of L, is J tissue = 2rpP

L

ò [C 0

blood

( x) - Ctissue ]dx = QCarterial - QCvenous

(8.14)

Substituting Equation 8.13 into 8.14 yields

(

J tissue = (Carterial - Ctissue )Q 1 - e -PS/Q

)

(8.15)

Equation 8.15 indicates that the mass fux is directly proportional to the blood fow and concentration gradient of the compound across the capillary wall. Dividing the mass fux across the capillary wall by the concentration gradient yields the capillary clearance (Clcap ), which represents the volume of blood from which the compound is extracted and distributed into the tissue per a unit of time Clcap =

J tissue = Q 1 - e -PS/Q Carterial - Ctissue

(

)

(8.16)

Where the extraction ratio (ER) of the distribution is ER = 1 - e -PS Q

(8.17)

Thus, the capillary clearance is equal to the blood fow times the extraction ratio. The extraction ratio is also considered an equilibration fraction between the xenobiotic concentration in the tissue, arterial capillaries, and the venous side of the capillaries, i.e., ER =

Carterial - Cvenous Carterial - Ctissue

(8.18)

Equations 8.16 through 8.18 present the following two scenarios: SCENARIO 1: WHEN Q > PS The blood fow is greater than the permeability-surface area product. The permeability-surface area product, PS, characterizes the diffusion of the compound from the blood vessels into the tissue environment. Under this scenario, the extraction ratio is less than one, and the distribution of the compound is limited by how fast it permeates from the blood, through the capillary wall into the surrounding tissue during its transit by the blood fow (Figure 8.2). Under this condition, the distribution is permeability-limited (or diffusion-limited) distribution, and the driving force for the transfer of compound into tissue is the concentration gradient; the diffusion is slow and limited by the membrane permeability and physicochemical characteristics of the compound; and the mass fux into the tissue is defned as J tissue = PS ( Carterial – Ctissue )

(8.19)

SCENARIO 2: WHEN Q < PS In this situation, the extraction ratio is equal or close to one, the uptake of xenobiotic by the tissue is thermodynamically more favorable than its residence in the blood, and the limit of distribution is how fast the compound can be delivered to the tissue. When the blood fow brings the xenobiotic to the tissue bed, its uptake is immediate. This type of distribution most often applies to highly lipophilic compounds (Figure 8.2) and is known as perfusion-limited distribution or blood fow-limited distribution. The mass transfer into the tissue can be defned as J tissue = Q ( Carterial - Ctissue )

(8.20)

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8.2 FACTORS INFLUENCING THE DISTRIBUTION OF XENOBIOTICS IN THE BODY

Thus, the blood fow is the principal factor in distribution, and concentration gradient plays little or no role in the mass transfer of a xenobiotic. The distribution of a compound, whether it is perfusion-limited or permeability-limited, includes diffusion into the extracellular fuid, followed by diffusion through the cell membrane, which may include reversible interaction with cell membrane, followed by internalization and intracellular handling of the compound by the cells, which may include metabolism and exocytosis. The overall distribution profle of xenobiotics, after reaching the systemic circulation and before diffusion into the interstitial fuid, starts with interaction with blood components, namely, plasma proteins and blood cells. 8.2.3 Effect of Binding to Plasma Proteins on Xenobiotic Distribution Most xenobiotics, if not all, bind reversibly to plasma proteins, particularly albumin and α-acid glycoprotein. Binding to plasma proteins infuences not only the distribution of xenobiotics in the body but also their pharmacologic and/or toxic response as well as their elimination. The binding creates a xenobiotic–protein complex with a molecular size equal to the sum of the molecular weights of the compound and protein. In general, it is the unbound or free molecules of a xenobiotic that diffuse through the capillary walls to the extravascular environment where the interactions with cellular receptors and passage through the membrane takes place. The xenobiotic–protein complex, because of its large molecular weight, does not permeate easily through the barriers; thus, its distribution is different from the free compound and remains mainly in the systemic circulation. Even if the complex crosses the capillaries and enters interstitial fuid, its diffusion into the intracellular fuid is doubtful unless the complex dissociation constant favors the formation of free molecule at the site of cellular diffusion. A protein–drug complex is pharmacologically inactive; its formation affects the distribution of xenobiotics in the body and infuences the therapeutic/toxic response. For this reason, the concentration of an unbound drug in plasma is used mainly for PK/TK analysis. A critical question is whether the protein binding of a compound in systemic circulation remains constant during exposure to a compound or administration of small molecule pharmaceuticals during a multiple dosing regimen. Typically, the free molecules or free concentration of a therapeutic agent is defned by “free fraction” a single number that often is considered a constant and used as a characteristic of the compound. Whereas an alteration in protein binding due to intraindividual variations or interindividual differences lowers the signifcance of calculating a free fraction. The calculations are discussed in the upcoming paragraphs. The plasma protein binding of a compound administered solo is often different when it is administered concurrently with other compounds. The interaction of administered compounds involves the affnity of two or more compounds for binding to the same protein and/or the same binding site (Mattos and Ring, 1996). This type of interaction may cause displacement of one compound by the other at the binding site of protein, which increases the free concentration of displaced compound in systemic circulation. The outcomes in most cases would be changes in PK/ TK parameters, like an increase in the area under plasma concentration–time curve that would be associated with increase in bioavailability or increase in maximum plasma concentration, that would enhance the intensity of the outcome, or the increase in concentration at the receptor site that infuences the pharmacological or toxicological response. The displacement occurs when the interaction of one compound with the binding site of a protein is thermodynamically more favorable than the other. Similar displacement can also occur at the cellular levels with other proteins/ enzymes. Most displacement studies are done using in vitro systems. A safe practice would be to validate whether the observed displacement in vitro refects the corresponding occurrence in vivo (Aarons and Rowland, 1981). Human serum albumin (HSA) is the major protein of plasma and consists of 585 amino acids forming a single polypeptide of a known sequence with a molecular weight of 66 kDa with three homologous helical domains with each domain having at least two sub-domains (Ghuman et al., 2005) and a catabolic half-life of approximately 17–18 days. Albumin is produced exclusively by the hepatocytes of the liver and is the major protein of plasma and most tissues. A wide range of exogenous compounds, metals, particles (Herve et al., 1994; Kratochwil et al., 2002; Hao et al., 2021; Song et al., 2021), and endogenous compounds (Curry et al., 1998; Bhattacharya et al., 2000; Petitpas et al., 2001; Xie and Guo, 2021) bind to albumin reversibly. Its binding capacity is vast, and surmounting its capacity is a major challenge in drug discovery and development (Ghuman et al., 2005). In spite of its large molecular size, it can diffuse into the extravascular compartment. The 216

PK/TK CONSIDERATIONS OF XENOBIOTICS SYSTEMIC DISTRIBUTION

interaction of small molecule xenobiotics with HSA has been extensively investigated (Bowmer and Lindup, 1978; Ekman et al., 1980; Aarons et al., 1983; Nowak and Shaw, 1995). Albumin possesses both negatively and positively charged binding sites. Most acidic, neutral, and basic compounds bind to HSA. The strength and extent of the binding depends on the structure of the compound. The interaction of drugs with HSA occurs by hydrogen binding, hydrophobic binding, and van der Waals forces. The extent of protein binding signifcantly infuences not only the distribution of a compound but also its renal elimination as well. The kidneys do not flter albumin, and therefore, drugs that are bound to albumin also remain unfltered. Furthermore, since only free molecules of xenobiotics can be taken up by the hepatocyte of the liver and interact with metabolic enzyme systems, the extent of binding also infuences the rate and extent of metabolism. In general, plasma protein binding infuences the PK/TK profles of xenobiotics profoundly and has been widely studied for decades (Koch-Weser and Sellers, 1976; Wagner, 1976; Yacobi et al., 1976; Shary et al., 1978; McNamara et al., 1979; Rowland, 1980; Greenblatt et al., 1982; Buur et al., 2009; Smith et al., 2010). Other proteins of plasma are fbrinogen, globulins such as γ-globulins that are mainly antibodies (IgA, IgM, IgG, IgE, and IgD); α 1 globulin (α 1-antitrypsin, orosomucoid [α 1 acid glycoprotein]) (Edwards et al., 1982); α 2 globulin (ceruloplasmin, haptoglobulin); α 2 macroglobulin, β 1, and β 2 globulins (transferrin hemopexin plasminogen fbrinogen); and α and β lipoproteins. The normal plasma protein level is 63–83 g/L; albumin constitutes 31.5–41.5 g/L; α 1 globulin ≈ 3 g/L; α 2 globulin ≈ 1.6 g/L; α 2 macroglobulin ≈ 3 g/L; β 1 and β 2 globulins ≈ 8.2 g/L; and the remaining balance is γ-globulins and lipoproteins (Vallner and Chen, 1977). The physicochemical characteristic of a compound infuences its binding to a specifc protein in plasma. Neutral molecules that are highly lipophilic tend to bind mostly to lipoproteins. Acidic and weakly acidic compounds in ionized form with notably moderate lipophilicity bind mainly to albumin. Ionized basic compounds with moderate lipophilicity bind mostly to α 1 acid glycoprotein, and those with noticeable lipophilicity bind not only to α 1 acid glycoprotein but also to albumin and lipoproteins. Globulins usually interact negligibly with most drugs. The hydrophobic nature of compounds is the key factor for the interaction between compounds and proteins. Cationic drugs are of particular interest because they constitute 70–80% of all drugs, and they constitute most drugs showing a high frst-pass effect. It is worth noting that a wide variety of factors, such as pregnancy, surgery, age, sex, and interethnic and racial differences, affect the interaction of drugs with proteins. Also, metabolites of certain compounds, either the reactive metabolites of Phase I metabolism or conjugates of Phase II metabolism, interact signifcantly with plasma proteins. In certain cases, depending on the affnity of metabolites for interaction with a protein, the metabolites may interact competitively with the protein and displace the parent compound. 8.2.3.1 Estimation of Protein-Binding Parameters The binding of xenobiotics to plasma proteins is usually determined in vitro under physiological pH and temperature. The methodology includes equilibrium dialysis, dynamic dialysis, ultrafltration, gel fltration chromatography, ultracentrifugation, microdialysis, electrophoresis, and the methodologies for high throughput measurements. The objective of all these techniques is to estimate the concentration of the free xenobiotic that is in equilibrium with its protein complex. The free concentration is then used to determine the free fraction, or fraction unbound, f u . As discussed earlier, the assumption for calculated free fraction is that the binding to plasma proteins is constant, which may not fully refect the fuctuation of free concentration of a xenobiotic due to protein binding during treatment or observation fu =

freexenobiotic concentration ( C free )

total(i.e., bound + free)) xenobiotic concentration ( Ctotal ) \C free = f u ´ Ctotal

(8.21) (8.22)

According to Equations 8.21 and 8.22, changes in free fraction depend on (Ctotal ) and (C free ). Thus, if there is in vivo fuctuation in protein binding, for estimation of C free (Equation 8.22), f u cannot be considered a constant, and changes in its value should go together with the corresponding change in Ctotal . 217

8.2 FACTORS INFLUENCING THE DISTRIBUTION OF XENOBIOTICS IN THE BODY

As indicated earlier, most xenobiotic-protein complexes, in particular HSA, are held together by the weak chemical interactions such as electrostatic interactions, van der Waal’s forces, and hydrogen binding, which tend to split under non-physiological pH, temperature, and osmotic pressure. The methodology and conditions of the protein-binding study should preserve the stability of the complex and equilibrium such that the concentration of free molecules is not overestimated by breakdown of the complex during the measurement. To estimate the parameters of protein binding, a brief description of in vitro methodology is necessary. They all share the same objective of determining the bound and unbound fractions. Choosing the conventional and classical method of equilibrium dialysis, plasma, or HSA at a physiological pH (7.2–7.4), is spiked with different concentrations of xenobiotic. The mixture is then dialyzed at 37°C through a dialysis membrane with molecular cutoffs ≈ 12,000–14,000 Da against an equivalent volume of a phosphate buffer (≈ 67 mM, pH 7.2–7.4), made isotonic with NaCl. The dialysis membrane is used in the form of a bag that holds protein and drug, and the bag is placed in the buffer solution. A modifed version of the bag is a prefabricated two-compartment unit that is separated by a dialysis membrane. The equilibrium of free molecules across the membrane is usually achieved in about 2–3 hours. The concentration of free molecules is measured in the buffer side, that is, outside of the bag or compartment separated by the membrane, which is assumed to be equal to the concentration of free molecules inside the bag or compartment. The concentration of free molecules inside the bag or compartment is assumed to be in equilibrium with the bound molecules to protein. The binding parameters, such as free fraction or association constant, can then be determined by the following application of the Law of Mass Action: 1 ˜˜˜ ° xenobiotic + protein ˛ ˜˜ ˜ éë xenobiotic - protein ùû complexx k

(8.23)

k1 ˜˜˜ ° that is, C free + C protein ˛ ˜˜ ˜ Ccomplex k

(8.24)

k

2

2

The association constant of the interaction (Equation 8.24) is Ka =

Ccomplex k = 1 C free ´ C protein k 2

(8.25)

(C free ) is the concentration of free molecules in moles, (C protein ) is the concentration of protein with free binding sites, (Ccomplex ) is the concentration of the xenobiotic–protein complex, and ( k1 ) and ( k 2 ) are the rate constants of forward and reverse reactions, respectively. The reciprocal of the association constant is known as the dissociation constant. Kd =

C free ´ C protein k 2 1 = = k1 K a Ccomplex

(8.26)

The magnitude of the association constant, K a , represents the extent of protein binding. Compounds that bind extensively to plasma proteins usually have a very large association constant. From Equation 8.25, the concentration of complex is Ccomplex = K aC freeC protein

(8.27)

The concentration of total protein (CTprotein ), known at the start of the experiment, and the estimated value of the complex (Ccomplex ), are used to determine the concentration of protein at the equilibrium (C protein ). C protein = CTprotein - Ccomplex

(8.28)

Substitution of Equation 8.28 in 8.27yields Ccomplex = K aC free ( CTprotein - Ccomplex )

(8.29)

\C free (1 + K aC free ) = K a ´ C free ´ CTprotein

(8.30)

Therefore, the number of moles of drug bound per moles of protein at equilibrium (r) is r=

K a ´ C free Ccomplex = CTprotein 1 + K aC free

When there are n identical binding sites per mole of protein, the equation is multiplied by n 218

(8.31)

PK/TK CONSIDERATIONS OF XENOBIOTICS SYSTEMIC DISTRIBUTION

Figure 8.3 Diagram of Langmuir isotherm with dependent variable r , the number of moles of xenobiotic bound per moles of protein, versus C free , the free concentration of xenobiotic; the plateau y – intercept is the number of identical binding site, n.

r˜

n ˛ K a ° C free ˝ 1 ˙ K aC free

(8.32)

Equations 8.31 and 8.32 are known as Langmuir equations (Langmuir, 1916) and plot r versus C free yields the Langmuir hyperbolic isotherm (Figure 8.3). The straight-line modifcations of Equation 8.32 are: I. Double reciprocal plot (Figure 8.4) 1 1 1 ˜ ° r nK aC free n

(8.33)

r ˜ nK a ° rK a ˜ K a ˛ n ° r ˝ C free

(8.34)

II. Scatchard plot (Figure 8.5)

III.Plot of ˜˛ Ccomplex ˛ C free °

˝ versus Ccomplex (Figure 8.6) ˆˆ ˙

Linear version of Equation 8.31 is Ccomplex ˜ Ccomplex K aC free ° nCTprotein K aC free

(8.35)

Ccomplex ˜ nK aCTprotein ° K aCcomplex C free

(8.36)

219

8.2 FACTORS INFLUENCING THE DISTRIBUTION OF XENOBIOTICS IN THE BODY

Figure 8.4 Diagram of the linear relationship between the reciprocal of moles of xenobiotic bound per mole of protein (1/ r ) versus the reciprocal of free concentration (1/C free ) for estimating the number of identical binding sites per mole of protein (i.e., reciprocal of the y-intercept) and the association constant K a , estimated as slope/y-intercept, this plot is known as double reciprocal plot or Lineweaver Burk plot.

Figure 8.5 Diagram of linear relationship of normalized value of r with respect to the free concentration of xenobiotic ( r C free ) versus the number of moles of xenobiotic bound per mole of protein r , the slope of the line is -K a , y-intercept is nK a , and x -intercept is n; this plot is known as Scatchard plot, useful for estimation of number of binding sites per mole of protein, and the association constant of the xenobiotic with the protein that represents the extent of protein binding. 220

PK/TK CONSIDERATIONS OF XENOBIOTICS SYSTEMIC DISTRIBUTION

Figure 8.6 Diagram of linear relationship of xenobiotic–protein complex concentration to the complex concentration ratio (Ccomplex C free ) as dependent variable versus concentration of the complex (Ccomplex ) for estimation of the association constant from the slope of the line (-K a ), the number of binding sites, n , from y-intercept ( nK aCTotal protein ) or the x -intercept ( nCTotal protein ); worth noting that the total concentration of the protein, CTotal protein , is a known value. IV. When the protein has multiple binding sites (e.g., albumin has two positively charged main binding sites with high association constants of 104–106 M−1, which binds largely anionic drugs) and the interaction of xenobiotic is with different functional groups, the following Langmuir equation is preferred: r=

n1K a1 C free n2 K a2 C free nn K an C free + +……… + 1 + K a1 C free 1 + K a2 C free 1 + K an C free

(8.37)

where n1 and K a1 are the parameters of one type of identical binding sites, n2 and K a2 are the parameters of the second type of identical binding sites and so on. For example, the –COO – of aspartic acid or glutamic acid residue can be one type of binding site and –S – of cysteine or –NH2+ of histidine residue, the second type of binding site. When the drug molecule has affnity for two types of binding sites, the Scatchard plot of r C free versus r is not a straight line as shown in Figure 8.7, but the extrapolation of the initial and terminal linear segments of the curve yields two straight lines with the following equations r = n1K a1 + rK a1 C free

(8.38)

r = n2 K a2 + rK a2 C free

(8.39)

As the affnity and capacity of the two binding sites are different, the line with greater y-intercept and smaller x-intercept is identifed as a “high-affnity, low-capacity” site, whereas the line with smaller y-intercept and greater x-intercept represents the “low-affnity, high-capacity” binding sites.

221

8.2 FACTORS INFLUENCING THE DISTRIBUTION OF XENOBIOTICS IN THE BODY

Figure 8.7 Scatchard plot of binding to a protein with two different classes of binding sites; the line with y-intercept of n1K1 represents the “high-affnity, low-capacity” class of binding sites, and the line with y-intercept of n2 K 2 represents the “low-affnity, high-capacity” class of binding sites; the x intercepts of n1x and n2 are the number of low capacity and high capacity binding sites, respectively. 8.2.4 Infuence of Physicochemical Characteristics of Xenobiotics on Their Distribution Lipophilicity, among the physicochemical attributes of a compound, plays a principal role in its distribution in the body. The permeation of lipophilic compounds in the highly perfused organs of the liver and kidney (perfusion-limited distribution) and adipose tissue (permeability-limited distribution) is signifcant. When the plasma concentration declines, the amount stored in the adipose tissue is released into the systemic circulation for redistribution, elimination, and re-uptake by the adipose tissue, although it may not be signifcant. The redistribution often creates a lingering pharmacological response, such as long-term drowsiness in the case of certain barbiturates or anesthetics. As discussed in Chapter 7 the ability of a lipophilic compound to permeate through the capillary wall or partition into an organ/tissue is expressed in terms of its partition coeffcient. The degree of ionization also plays an important role in the distribution of xenobiotics. Most compounds are salts of weak acids or bases and, according to pH–partition theory, only the unionized form of drugs can penetrate through the capillary wall and cellular barrier. Therefore, the pKa of xenobiotics and pH of the environment are also important in their permeation and distribution in the tissue (see also Chapter 7). 8.2.5 Infuence of Extent of Penetration Through the Physiological Barriers, and Parallel Removal Processes on Xenobiotic Distribution The biological mechanisms discussed in Chapter 7 not only govern the absorption and permeation of xenobiotics through biological barriers but also infuence their distribution and elimination. For example, effux and transport protein in organs of elimination, mainly the liver and kidneys, not only infuence the elimination but also the distribution of xenobiotics in the body. To signify these shared characteristics among absorption, distribution, metabolism, and excretion processes, terms such as disposition are used to imply complementary and collective processes of distribution and elimination. Elimination is also the terminology used for the combined processes of metabolism 222

PK/TK CONSIDERATIONS OF XENOBIOTICS SYSTEMIC DISTRIBUTION

and excretion. Thus, disposition is a term used to refer to the distribution, metabolism, and excretion of xenobiotics. The expression and activity of effux proteins (P-glycoproteins) and their distribution throughout the body play a critical role in the disposition of xenobiotics (Ayrton and Morgan, 2001; Fromm, 2003; Fenneteau et al., 2009). The effux proteins in the kidneys maintain the total body homeostasis by selective elimination of endogenous and detoxifcation of exogenous compounds, and those in the liver coordinate the detoxifcation processes with the role of metabolizing enzymes. In addition to the important role effux proteins play in the disposition of xenobiotics (Silverman, 1999; Fromm, 2000; Litman et al., 2001; Schinkel and Jonker, 2003), their presence in all physiological barriers, which morphologically are different from each other, are also signifcant and infuence the overall disposition profle of a compound. 8.2.6 Physiological Barriers The function of physiological barriers in the body is essentially to protect different organs/ regions. There are libraries of literature on each of these barriers. The main physiological barriers, which infuence the distribution and PK/PD or TK/TD parameters, are briefy discussed here. 8.2.6.1 Blood–Brain Barrier The distribution of xenobiotic into the brain and the permeation from the blood into the extracellular space is highly regulated and is limited by the blood–brain barrier. The tight junctions of endothelial cells of cerebral capillaries create this barrier. Nutrients are selectively transported from the blood to the brain through specifc receptors or transporters. These transporters are located on the luminal and abluminal of the endothelium. They regulate the passage of nutrients like glucose or amino acids and prevent the permeation of xenobiotics. Examples are the glucose transporter (GLUT-1), large neutral amino acid transporter, organic anion transporters (OATP), and the monocarboxylic acid transporter (MCT1). The transfer of xenobiotics by pinocytosis through this tight barrier is very rare, and the transport of water-soluble drugs, like nutrients, requires a specifc carrier or receptor. Viruses and large molecules, like peptides and proteins, can be transferred by fuid phase endocytosis (Hawkins and Egleton, 2007; Gosselet et al., 2021), receptor-mediated transcytosis, and adsorptive mediated transcytosis (Abbott, 2004; Pulgar, 2019; Zhu et al., 2019). The permeation of lipophilic compounds occurs by passive transcellular diffusion (Abbott et al., 2006), and therefore their passage is dependent on their lipophilicity and molecular size (Alavijeh et al., 2005; Chancellor et al., 2012; Waring, 2009). The higher the partition coeffcient, the easier is the passage through the barrier. However, this permeation is not absolute even for highly lipophilic compounds if their molecular weight is above 500–600 Da (Pardridge, 1998). Because of the presence of tight junctions, the paracellular diffusion practically does not occur. However, under certain pathological conditions leakage in the junctions may occur, which may allow the permeation of proteins or xenobiotics (Cirrito et al., 2005; Abbott et al., 2006, 2010; Bennett et al., 2010; Pittet et al., 2011). Brain capillary endothelial cells express P-glycoprotein (Pgp) (Sun et al., 2003; Fromm, 2003; Löscher and Potschka, 2005; Bauer et al., 2012, Noak et al., 2018); thus, the low permeation may also be explained by the occurrence of this effux protein at the luminal face of the blood–brain barrier. It has been shown that the brain uptake of the xenobiotics can be enhanced when the body is subjected to the Pgp inhibitors (Sadeque et al., 2000; Bauer et al., 2005; Shan et al., 2022). MDR1Pgp and MDR2Pgp are the two types of Pgp in humans. MDR1Pgp is responsible for xenobiotics effux at the blood–brain barrier. In rodents both mdr1aPgp and mdr1bPgp are present in the brain, but in two different locations: mdr1a is localized in brain capillaries, whereas mdr1b is only in brain parenchyma. It is believed that Pgp is expressed by the gene encoding the protein (MDR1) at the apical membrane of brain capillary endothelial cells in humans and other mammals. There are other transport proteins in the brain, including the multidrug resistance-associated protein (MRP) family, which has several members, and they transport a wide range of anticancer drugs out of the cells, including the tumor cells. MRP2 has been identifed as an important MRP in the barrier. Contrary to Pgp, which interacts with diverse groups of xenobiotics, MRPs interact with and transport anionic and neutral compounds conjugated to acidic ligands. A recently identifed effux protein in the brain is the breast cancer resistance protein (BCRP). All the infux and effux proteins are expressed in the blood–brain barrier and the blood-CSFbarrier (BCSFB) (Figure 8.8). They become over-expressed in certain diseases or conditions, which adds to the challenges of overcoming the multiple drug resistance (Löscher and Potschka, 2005). It should also be mentioned that continuous production of cerebrospinal fuid (CSF) is an important 223

8.2 FACTORS INFLUENCING THE DISTRIBUTION OF XENOBIOTICS IN THE BODY

factor complementing the role of the barrier by transporting xenobiotics across arachnoid granulations into the systemic circulation. The function of effux and most infux proteins is an active and energy-demanding process. However, this active transport is not uniform for all xenobiotics, and it depends on the permeability-limited or fow-limited characteristic of the molecule. In summary, Pgp, MRPs, and BCRP in association with the tightly joined capillary endothelial cells severely restrict and limit the uptake and distribution of xenobiotics in the brain (Figure 8.8).

Figure 8.8 Depiction of collective restrictive functions of ABC effux transporters P-glycoprotein (Pgp), multidrug resistance protein (MRP) family, and breast cancer resistance protein (BCRP) in conjunction with the tightly joined capillary endothelial cells to prevent the uptake of xenobiotic by the brain. The two important parameters of PK/TK analysis of brain uptake are identifed as the extent of brain equilibrium with plasma concentration and the time to achieve this equilibrium (Liu et al., 2005; Padowski and Pollack, 2011; Spreafco and Jacobson, 2013). The extent of uptake depends largely on the partition coeffcient of the xenobiotic, which in turn depends on 1) the free concentration of the compound in plasma, 2) the fraction removed by effux proteins, 3) the extent of metabolism in the brain and 4) removal by the cerebrospinal fuid. It has been suggested that the ratio of brain to plasma concentration at steady state is a more relevant partition coeffcient to refect the brain uptake. Among the different PK/TK approaches and models to analysis and estimate parameters and constants of the brain uptake and the infuence of blood–brain barrier (Peng et al., 2001; Tunblad et al., 2003; Liu et al., 2005), the following is a functional model (Liu et al., 2005) with assumptions of: 1) the brain intracellular and extracellular compartments are considered two separate and interconnected compartments, where the intravascular space is in exchange with the central compartment of a two compartment model, 2) only free concentration in plasma may cross the blood–brain barrier and equilibrate in each compartment, 3) the uptake and effux clearance is equal to a parameter identifed and quantifed as a permeability-surface area product (PS) representing the distribution clearance across the barrier by passive diffusion, 4) CSF fow does not signifcantly infuence the compound’s disposition in the brain. The model can be perceived as a four-compartment model, where the central compartment (plasma and highly perfused organs/tissues) is in exchange with the peripheral compartment (less perfused tissues) and brain intravascular space (Figure 8.9). 224

PK/TK CONSIDERATIONS OF XENOBIOTICS SYSTEMIC DISTRIBUTION

Figure 8.9 Diagram of a four-compartment model for estimation of parameters and constants of the brain uptake and infuence of blood–brain barrier; the central compartment of the model include the plasma and highly perfused organs/tissues; the central compartment is in exchange with the peripheral compartment (less perfused tissues) and the intravascular space of the brain; the intravascular compartment is in exchange with the extravascular space of the brain; the input into the central compartment is governed by frst-order absorption rate constant from an extravascular route of administration; the rate constants between the central and peripheral compartment are expressed in terms of clearances, the exchange between the central compartment and the brain intravascular space is expressed in terms of blood fow containing the compound and the exchange between the intravascular space and extravascular space is defned by a permeability-surface area product, PS, that represents the distribution clearance across the blood brain barrier. The input is frst-order absorption from a site of administration, the fraction of dose absorbed is F and total amount absorbed is equal to FDose. The rate equation of the central compartment after administration of the dose and assuming complete absorption is

(

)

dCp dA1 = ( k a ´ F Dose ) - ( Cl ´ Cp ) - Q Cp - ( Cbrain )int - Cl12 ( Cp - C2 ) = V1 dt dt æ Cl ö

Cp =

Dose -çè V1 ÷øt e V1

Cl = k10 V1 Cb =

ö æ PS´( f u ) æ æ Cl ö n ÷t ö brain -ç ÷ ÷ ç Vb ç -çè V1 ÷øt ø è e -e ç ÷ Cl ö ç ÷ - ÷è ø V1 ÷ø

Dose ´ PS ´ ( f u ) plasma

æ PS ´ ( f u ) brain Vb ´V ´ 1ç ç Vb è

( fu )brain Vb

= k out

PS ´ ( f u )brain Vb

(8.40)

= k out
k10 )

(

) 225

8.2 FACTORS INFLUENCING THE DISTRIBUTION OF XENOBIOTICS IN THE BODY

dCp (mass/time); the dt −1 absorption rate constant is k a (time ); the plasma concentration is Cp (mass/volume); the systemic clearance is Cl (volume/time); the rate of elimination from the body is Cl ´ Cp (mass/time) and the cerebral blood fow is Q (volume/time); the concentration in the intravascular compartment of the brain is ( Cbrain ) ; the concentration of the peripheral compartment is C2 ; and the distribution clearwhere the change rate of the amount in the central compartment is V1

int

ance between the central and peripheral compartment is Cl12 . The rate equation of the peripheral compartment is dA2 æ dC = V2 ç 2 dt è dt

ö ÷ = k12 A1 - k 21 A2 = Cl12 ( Cp - C2 ) ø

(8.41)

dA2 ; the volume of the peripheral comdt partment is V2 ; the concentration of the peripheral compartment is C2 ; and the distribution rate where the rate of change in the peripheral compartment is

Cl12 Cl , and k 21 = 12 . V1 V2 The rate equation of the intravascular compartment of the brain:

constants are k12 and k 21 , also estimated as k12 =

æ C ö dCint dAint (8.42) = Vint = Q ( Cp - Cint ) - PS ( f u ) plasma çç Cint - ext ÷÷ Kp ø dt dt è where the amount in the brain intravascular compartment is Aint ; the volume of the intravascular dCint compartment is Vint ; the change rate of amount in the intravascular compartment is Vint the dt ; the xenobiotic concentration in the intravascular and extravasfree fraction in plasma is ( f u ) plasma

cular compartments are Cint and Cext , respectively; the equilibrium brain/plasma partition coef( fu )plasma ; and the uptake-effux clearance between intravascular and extravascular fcient is K p = ( fu )brain compartments is PS. The rate equation of extravascular compartment of the brain is

(

) æççè C

ö (8.43) ÷÷ ø where the volume and concentration of the extravascular compartment are Vext and Cext , respectively. The initial condition of the differential Equations 8.40 through 8.43 is defned as all concentration terms at time zero are equal to zero. Dose If the dose is injected intravenously then the plasma concentration at time zero is Cpt=0 = . V1 If the dose is administered intravenously, the rate equations are dCext dAext = Vext = PS ´ ( f u ) plsma dt dt

V1 Vb

int

-

Cext Kp

dCp = -Cl ´ Cp dt

)(

(

dCb = PS ´ ( f u ) plasma ´ Cp - PS ´ ( fu )brain ´ Cb dt

(8.44)

)

(8.45)

The volume of brain tissue is Vb , and the concentration of compound in the brain is Cb . The integration of Equations 8.44 and 8.45 yields equations that defne the time course of xenobiotic concentration in plasma and the brain. æ Cl ö

Cp =

Cb = 226

Dose -çè V1 ÷øt e V1 æ PS´( f u ) ö æ æ Cl ö brain ÷t ö -ç ç ÷ ÷ V ç -èç V1 ÷øt b ø -e è çe ÷ ö Cl ç ÷ - ÷è ø V1 ÷ø

Dose ´ PS ´ ( f u ) plasma

æ PS ´ ( f u ) brain Vb ´ V1 ç ç V b è

(8.46) (8.47)

PK/TK CONSIDERATIONS OF XENOBIOTICS SYSTEMIC DISTRIBUTION

By setting

Cl = k10 , the overall elimination rate constant from the central compartment, and V1

PS ´ ( f u )brain

= k out the rate constant of elimination from the brain, the ratio of brain concentration Vb to plasma concentration can be estimated by dividing Equation 8.47 by Equation 8.46: Cb PS ´ ( f u ) plasma = 1 - e -(kout -k10 )t Cp Vb ( k out - k10 )

(

)

(8.48)

Therefore, it can be concluded that when k out < k10 , the ratio increases over time, the compound resides longer in the brain with a half-life longer than plasma, and never reaches a plateau. When k out > k10 , the ratio increases initially, reaches a plateau with a half-life equal to the plasma concentration. 8.2.6.2 Blood–Lymph Barrier Physiologically, the balance of body water is maintained by the lymphatic system. This system drains the extra volume of interstitial fuids with all their waste products and proteins, known as lymph, into the blood circulation. The lymphatic system, like the blood system, is formed from lymphatic capillaries that join together and form lymphatic vessels. The vessels usually follow the same route as the blood vessels. They go through successive sets of lymph nodes to flter the lymph and fnally join blood vessels at the junction of the jugular and left subclavian veins. The exchange of fuids containing electrolytes, hormones, gasses, nutrients, and xenobiotics between blood, tissue, and blood–lymph barrier is coordinated effectively to maintain interstitial fuid and cellular homeostasis. For example, in the liver, water and solutes are transported into the sinusoidal space and interstitial area where they drain into lymphatic collecting vessels (Henriksen et al., 1984). Therefore, the liver blood–lymph barrier consists of a series of areas, barriers, and pathways connected sequentially, which are lining the walls of sinusoids, sinusoidal space, and interstitial fuid ending with terminal lymphatic capillaries. Because of its extensive network, the lymphatic system plays an important role in the distribution of lipophilic compounds. Small polar molecules are usually excluded from distribution by the lymphatic system, which is mainly due to the anatomical structure of the barrier and lymphatic capillaries. Proteins with an effective radius of about 120 Å with a large molecular weight can cross this barrier. In large experimental animals (e.g., dogs) solid spherical particles with a radius of up to 700 Å can cross the blood–lymph barrier. 8.2.6.3 Placental Barrier The placenta receives nutrients and oxygen from maternal arteries for distribution through fetal capillaries and disposes fetal waste products and carbon dioxide into maternal veins. The placenta barrier is essentially a dynamic fetal:maternal interface for regulating the exchange of essential nutrients, hormonal levels, gases, metabolic wastes, toxins, and drugs. The barrier is between fetal and maternal systemic circulations and is made up of various cell layers consisting of an endothelial cell layer, a thin layer of connective tissue, and a continuous syncytiotrophoblast (epithelial covering the highly vascular region) with some individual trophoblasts underneath, which are thicker at the early phase of pregnancy and gradually become thinner near term, such that the distance of over 50 µm at the late second month may get reduced to less than 5 µm by the 37th week of pregnancy. Nutrients such as glucose, amino acids, minerals, some vitamins, purines, and pyrimidines are actively transported across the placenta against the concentration gradient. The barrier, although considered remarkable for its function, often acts as a leaky flter for certain xenobiotics and teratogenic compounds. Xenobiotics may actively or passively cross the barrier and enter the fetal circulation. Small molecular weight xenobiotics, uncharged and lipophilic, can cross the placenta barrier readily. The amniotic fuid around the fetus is the site for the distribution of the absorbed xenobiotics. Several prominent effux transporters have been identifed in the placenta, suggesting a more active role for the barrier in regulating exposure of the fetus to xenobiotics circulating in the maternal circulation. The effux proteins and enzyme systems of the developing fetus may not be able to adequately extrude or metabolize the compound and may cause embryotoxicity or teratogenicity. The distribution of xenobiotics within the fetus follows essentially the same pattern as in the maternal body, with some differences related to the volume of drug distribution, 227

8.2 FACTORS INFLUENCING THE DISTRIBUTION OF XENOBIOTICS IN THE BODY

plasma–protein binding, blood circulation, and greater permeability of other membranous barriers. In addition to the presence of multilayer cells that form the physical fetal:maternal barrier, the placenta and the fetus are equipped with enzymes of phase I and phase II metabolism. The metabolism of xenobiotics complements the role of the placental barrier in protecting the fetus (Ganapathy et al., 2000; Syme et al., 2004). Cytochrome P450 (CYP) enzymes are characterized in the placenta at the level of mRNA, protein, and enzyme activity and CYP1A1, 2E1, 3A4, 3A5, 3A7, and 4B1 have been detected in the term placenta (Syme et al., 2004). Furthermore, the infux and effux proteins are abundant in the placenta (Allikmets et al., 1998; Myllynen et al., 2007; Vähäkangas and Myllynen, 2009) and include: ATP-binding cassette transporters (ABC transporters: ABCB1/MDR1/Pgp, ABCC1-3/MRP1-3, ABCG2/BCRP [breast cancer resistance protein]) (Hitzl et al., 2004; Ceckova-Novotna et al., 2006; Behravan and Piquette-Miller, 2007); organic anion transporters (OAT: SLC21A8/OATP-8, SLC21A12/OATPA-E, SLC22A11/OAT4) (Babu et al., 2002); organic cation transporters (OCT: SLC22A3/OCT3, SLC22A4/OCTN1, SLC22A5/ OCTN2); 5-HT (serotonin) transporter (SLC6A4/SERT), as well as a noradrenaline transporter (SLC6A2/NET) (Bottalico et al., 2004; Ganapathy and Prasad, 2005). The effux proteins are mostly expressed on the apical membrane of syncytiotrophoblast facing maternal blood, and thus their role is to protect the fetus against xenobiotics. There are signifcant interspecies differences in the expression of transport proteins, which should be taken into consideration in investigative teratogenicity of therapeutic agents or chemicals intended for use in the environment. During pregnancy in humans, the placenta is inaccessible for PK/TK sampling and analysis. Most qualitative information is obtained from the umbilical cord at birth, which represents only one sample and one observation. The interspecies variability in enzyme systems and transport proteins, and other genetically controlled endogenous compounds and processes, makes the extrapolation from experimental animal to human problematic and questionable (Bouazza et al., 2019; Schmidt et al., 2015). 8.2.6.4 Blood–Testis Barrier The capillary endothelium and the Sertoli cells mainly create this barrier against the uptake of xenobiotics, and it is often called Sertoli cell barrier (Russell, 1977). The Sertoli cells, which play an important role in spermatogenesis, are tightly joined together and form an additional layer after the endothelium of capillaries, making this barrier similar to the blood–brain barrier. The barrier has a complex pattern of transport protein expressions, where several proteins are highly expressed, whereas others are expressed at lower levels or are absent in Sertoli cells. A differential expression of protein transporters in Sertoli cells of adult Sprague Dawley rats has been observed (Augustine et al., 2005), which may play an important role in the distribution of xenobiotic in the male reproductive system. The over-expressed effux and infux protein in Sertoli cells include multidrug resistant proteins 1, 7, and 8 (Mrp1, Mrp7, Mrp8); testis specifc transporter 1 and 2 (Tst1, Tst2); organic cation/carnitine transporter (OCTN2); Wilson’s disease zinc transporter 1 (Znt1); and equilibrative nucleoside transporter 1 and 2 (Ent1 and Ent2). Testicular toxicity due to heavy metal (lead, cadmium, arsenic, etc.) poisoning occurs often in various species, but with variable sensitivity (Liu et al., 2001). The testis, due to the presence of metabolizing enzyme systems, such as isozymes of Cytochrome P450, can metabolize the molecules that succeed in crossing the barrier. The testis microsomal incubation of various experimental animals has shown interspecies differences in CYP1A1, CYP1A2, and CYP1B1 isoforms (Smitha et al., 2007). The barrier itself and the gonadal metabolism limit the distribution of xenobiotics into the testis. As with the blood–brain barrier, the transfer of drugs by pinocytosis through this tight barrier is very rare and the permeation of lipophilic small molecular weight compounds is by passive diffusion. The testis is susceptible to heavy metal poisoning, mutagens, and cancerous cells. 8.2.6.5 Blood–Aqueous Humor Barrier (BAB) – also Read Chapter 2, Section 2.3.2 The ocular barrier may not have a major impact on the overall distribution of xenobiotics in the body, but it limits the distribution to a vital region of the body. For delivery systems of therapeutic agents intended for the retina, it poses a major challenge particularly for retinal disorders and related visual impairment. Even the systemic administration of the compounds is strictly regulated by ocular barrier system. Upon access of a xenobiotic to the systemic circulation, a fraction of a dose may diffuse into the anterior chamber of the eye; however, some compounds do not appear there at all. This restriction, which is mostly for small hydrophilic compounds and 228

PK/TK CONSIDERATIONS OF XENOBIOTICS SYSTEMIC DISTRIBUTION

large molecular weight proteins and polymers, is analogous to the function of the blood–brain barrier. Similar to the blood–brain barrier, the presence of tight junctions of capillary endothelium and transport proteins impede the free exchange of drugs between blood and the barriers. The ATP-binding cassette (ABC) protein such as Pgp, and solute carrier (SLC) transporters, such as ABCB, ABCC, ABCG, SLC7, SLC16, SLC19, SLCO/SLC21A, SLC22A, and SLC29 transporters are all present and infuence the permeability of drugs across the blood–ocular barriers (Tomi and Hosoya, 2010). Blood Retinal Barrier (BRB): Read Chapter 2, Section 2.3.3 and 3.3.4 Blood Labyrinth Barrier (BLB): Read Chapter 2, Section 2.1.2 8.2.7 Effect of Body Weight and Composition on Xenobiotic Distribution The distribution of xenobiotics in the body is infuenced by body weight. A lower lean body mass and higher adipose tissue would increase the physiological volume of distribution and uptake of lipophilic compounds. Doses of therapeutic agents are usually adjusted according to the Ideal Body Weight (IBW), or Body Surface Area (BSA). Body Mass Index (BMI) and Lean Body Mass (LBM) are also used. The following are the equations used to determine these quantities: 8.2.7.1 Ideal Body Weight (IBW in kg) The dosing regimen of therapeutic agents that may have poor lipophilicity, but narrow therapeutic indexes are often based on the ideal body weight (IBW) for obese patients. There are several empirical equations to estimate the IBW, a few are (Devine, 1974): Men: IBW (kg) = 50 kg + 2.3 kg for each inch over 5 feet

(8.49)

Women: IBW ( kg ) = 45 + 2.3 kg for each inch over 5feet

(8.50)

Men: IBW ( kg ) = 52kg + 1.9kg for each inch over 5feet

(8.51)

Women: IBW ( kg ) = 49kg + 1.7 kg for each inch over 5feet

(8.52)

Men: IBW ( kg ) = 56.2kg + 1.41kg for each inch over 5feet

(8.53)

Women: IBW ( kg ) = 53.1kg + 1.36 kg for each inch over 5feet

(8.54)

(Robinson et al., 1983):

(Miller et al., 1983):

8.2.7.2 Body Surface Area (BSA in m 2) The body surface area is used essentially to normalize variables in PK/TK or physiological functions with respect to body size and conformation. The following are the examples of empirical equations: (Du Bois and DuBois, 1916): BSA = 0.007184 ´ Weight ( kg )

0.425

´ Height ( cm )

0.725

(8.55)

(Mosteller, 1987): æ Height ( cm ) ´ Weight ( kg ) ö BSA = ç ç ÷÷ 3600 è ø

1/2

(8.56)

It is also common to use the values reported in Table 8.2 for children of different age groups. 8.2.7.3 Body Mass Index (BMI in kg/m 2) The body mass index is used to determine and diagnose obesity. It is based on weight and the height, is easy to determine, and variability of the measurements is not signifcant.

(

)

BMI kg/m 2 =

Weight ( kg )

Height ( m )

2

(8.57)

229

8.2 FACTORS INFLUENCING THE DISTRIBUTION OF XENOBIOTICS IN THE BODY

(

)

BMI kg/m 2 =

Weight ( lb )

Height ( inch )

2

(8.58)

´ 703.07

8.2.7.4 Lean Body Mass (LBM in kg) Lean Body Mass, or Lean Body Weight is the estimate of body weight without adipose tissue. It is different from Ideal Body Weight (IBW) that takes into consideration the normal proportion of fat required for an ideal body weight. The commonly used empirical equation is (Hume, 1966):

(

(

)

Men: LBM ( kg ) = 1.10 ´ Weight ( kg ) - 128 ´ Weight 2 /( 100 ´ Height(m))

(

)

2

(

)

Women: LBM ( kg ) = 1.07 ´ Weight ( kg ) - 148 ´ Weight 2 /( 100 ´ Height(m))

2

(8.59)

)

(8.60)

An alternative equation is

(

)

(8.61)

(

)

(8.62)

LBM ( kg ) = 0.32810 ´ Weight ( kg ) + ( 0.33929 ´ Height ( cm ) ) - 29.5336 LBM ( kg ) = 0.29569 ´ Weight ( kg ) + ( 0.41813 ´ Height(cm)) - 43.2933

8.2.8 Impact of Disease States on Xenobiotic Distribution Certain disease states can modify the normal distribution of xenobiotics in the body, and the PK/ TK parameters can change to values different from the normal states. Briefy they are 8.2.8.1 Congestive Heart Failure (CHF) CHF is associated with hypoperfusion to various organs and tissues, particularly highly perfused organs such as the liver and kidneys. Thus, CHF causes change in the apparent volume of distribution, the overall elimination and clearance of xenobiotics, and also drug metabolizing activity. 8.2.8.2 Chronic Renal Failure (CRF) Chronic renal failure and end-stage renal disease alter the distribution and elimination of xenobiotics signifcantly. CRF impairs the plasma protein and tissue binding, reduces the systemic and renal clearance of parent compound and its water-soluble metabolites, and decreases nonrenal clearance signifcantly. Without the dosage adjustment, the linear PK profle of a compound may change to capacity-limited and nonlinearity of PK/TK parameters. 8.2.8.3 Hepatic Diseases Hepatic diseases, such as hepatitis, alcohol liver, hepatic cancers, liver cirrhosis, hepatic steatosis, and many other genetic or non-genetic liver diseases can modify the metabolic profle of xenobiotics signifcantly and reduce the rate of formation of metabolites, thus, increasing the free concentration in plasma, modifying the equilibrium of protein binding, and changing not only the profle of elimination but also their distribution in the body. 8.2.8.4 Cystic Fibrosis (CF) CF decreases the deposition of particulate compounds administered through inhalation, for example, aerosols and other pulmonary delivery systems. It also modifes the distribution and free concentration of xenobiotics in the body, in particular aminoglycosides and other antibiotics.

Table 8.2 Body Surface Area of Different Age Groups Children Age (year) Newborn 2 9 10 12–13

230

Adults ♂ (m )

BSA (m ) 2

0.25 0.5 1.07 1.14 1.33

2

1.6 Average 1.73 (m2)

♀ (m2) 1.9

PK/TK CONSIDERATIONS OF XENOBIOTICS SYSTEMIC DISTRIBUTION

The disposition profle of xenobiotics in CF patients is usually different than in individuals with healthy lungs. 8.2.8.5 Other Conditions Obesity, dehydration, edema, ascites, pregnancy, malnutrition, etc., can modify the disposition of xenobiotics in the body signifcantly. 8.3 APPLICATIONS AND CASE STUDIES The applications and case studies related to Chapter 8 are posted in Addendum II – part 1. REFERENCES Aarons, L., Grennan, D. M., Siddiqui, M. 1983. The binding of ibuprofen to plasma proteins. Eur J Clin Pharmacol 25(6): 815–19. Aarons, L. J., Rowland, M. 1981. Kinetics of drug displacement interactions. J Pharmacokinet Biopharm 9(2): 181–90. Abbott, N. J. 2004. Prediction of blood–brain barrier permeation in drug discovery from in vivo, in vitro and in silico models. Drug Discov Today Technol 1(4): 407–16. Abbott, N. J., Patabendige, A. A. K., Dolman, D. E. M., Yusof, S. R., Begley, D. J. 2010. Structure and function of the blood–brain barrier. Neurobiol Dis 37(1): 13–25. Abbott, N. J., Ronnback, L., Hansson, E. 2006. Astrocyte-endothelial interactions at the blood–brain barrier. Nat Rev Neurosci 7(1): 41–53. Alavijeh, M. S., Chishty, M., Qaiser, M. Z., Palmer, A. M. 2005. Drug metabolism and pharmacokinetics, the blood–brain barrier, and central nervous system drug discovery. Neurotherapeutics 2(4): 554–71. Allikmets, R., Schriml, L. M., Hutchinson, A., Romano-Spica, V., Dean, M. 1998. A human placentaspecifc ATP-binding cassette gene (ABCP) on chromosome 4q22 that is involved in multidrug resistance. Cancer Res 58(23): 5337–9. Augustine, L. M., Markelewicz, R. J., Jr., Boekelheide, K., Cherrington, N. J. 2005. Xenobiotic and endobiotic transporter mRNA expression in the blood–testis barrier. Drug Metab Dispos 33(1): 182–9. Ayrton, A., Morgan, P. 2001. Role of transport proteins in drug absorption, distribution and excretion. Xenobiotica 31(8–9): 469–97. Babu, E., Takeda, M., Narikawa, S., Kobayashi, Y., Enomoto, A., Tojo, A., Cha, S. H., Sekine, T., Sakthisekaran, D., Endou, H. 2002. Role of human organic anion transporter 4 in the transport of ochratoxin A. Biochim Biophys Acta 1590(1–3): 64–75. Bauer, B., Hartz, A. M. S., Fricker, G., Miller, D. S. 2005. Modulation of P-glycoprotein transport function at the blood–brain barrier. Exp Biol Med (Maywood) 230(2): 118–27. Bauer, M., Zeitlinger, M., Karch, R., Matzneller, P., Stanek, J., Jäger, W., Böhmdorfer, M., Wadsak, W., Mitterhauser, M., Bankstahl, J. P., Löscher, W., Koepp, M., Kuntner, C., Müller, M., Langer, O. 2012. Pgp-mediated interaction between (R)-[11C]verapamil and tariquidar at the human blood–brain barrier: A comparison with rat data. Clin Pharmacol Ther 91(2): 227–33. Behravan, J., Piquette-Miller, M. 2007. Drug transport across the placenta, role of the ABC drug effux transporters. Expert Opin Drug Metab Toxicol 3(6): 819–80.

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Bennett, J., Basivireddy, J., Kollar, A., Biron, K. E., Reickmann, P., Jefferies, W. A., McQuaid, S. 2010. Blood–brain barrier disruption and enhanced vascular permeability in the multiple sclerosis model EAE. J Neuroimmunol 229(1–2): 180–91. Bhattacharya, A. A., Grüne, T., Curry, S. 2000. Crystallographic analysis reveals common modes of binding of medium and long-chain fatty acids to human serum albumin. J Mol Biol 303(5): 721–32. Bottalico, B., Larsson, I., Brodszki, J., Hernandez-Andrade, E., Casslén, B., Marsál, K., Hansson, S. R. 2004. Norepinephrine transporter (NET), serotonin transporter (SERT), vesicular monoamine transporter (VMAT2) and organic cation transporters (OCT1, 2 and EMT) in human placenta from pre-eclamptic and normotensive pregnancies. Placenta 25(6): 518–29. Bouazza, N., Foissac, F., Hirt, D., Urien, S., Benaboud, S., Lui, G., Treluyer, J. M. 2019. Methodological approaches to evaluate fetal drug exposure. Curr Pharm Des 25(5): 496–504. Bowmer, C. J., Lindup, W. E. 1978. Binding of phenytoin, L-tryptophan and o-methyl red to albumin. Unexpected effect of albumin concentration on the binding of phenytoin and L-tryptophan. Biochem Pharmacol 27(6): 937–42. Buur, J. L., Baynes, R. E., Smith, G. W., Riviere, J. E. 2009. A physiologically based pharmacokinetic model linking plasma protein binding interactions with drug disposition. Res Vet Sci 86(2): 293–301. Ceckova-Novotna, M., Pavek, P., Staud, F. 2006. P-glycoprotein in the placenta: Expression, localization, regulation and function. Reprod Toxicol 22(3): 400–10. Chancellor, M. B., Staskin, D. R., Kay, G. G., Sandage, B. W., Oefelein, M. G., Tsao, J. W. 2012. Blood– brain barrier permeation and effux exclusion of anticholinergics used in the treatment of overactive bladder. Drugs Aging 29(4): 259–73. Cirrito, J. R., Deane, R., Fagan, A. M., Spinner, M. L., Parsadanian, M., Finn, M. B., Jiang, H., Prior, J. L., Sagare, A., Bales, K. R., Paul, S. M., Zlokovic, B. V., Piwnica-Worms, D., Holtzman, D. M. 2005. P-glycoprotein defciency at the blood–brain P-glycoprotein defciency at the blood–brain barrier increases amyloid-beta deposition in an Alzheimer disease mouse model. J Clin Invest 115(11): 3285–90. Curry, S., Mandelkow, H., Brick, P., Franks, N. 1998. Crystal structure of human serum albumin complexed with fatty acid reveals an asymmetric distribution of binding sites. Nat Struct Biol 5(9): 827–35. Devin, B. J. 1974. Gentamicin therapy. Drug Intell Clin Pharm 8: 650–5. Du Bois, D., Du Bois, E. F. 1916. A formula to estimate the approximate surface area if height and weight be known. Arch Intern Med 17(6): 863–71. Edwards, D. J., Lalka, D., Cerra, F., Slaughter, R. L. 1982. Alpha acid glycoprotein concentration and protein binding in trauma. Clin Pharmacol Ther 31(1): 62–7. Ekman, B., Sjodin, T., Sjoholm, I. 1980. Binding of drugs to human serum albumin. XV. Characterization and identifcation of the binding sites of indomethacin. Biochem Pharmacol 29(12): 1759–65. Fenneteau, F., Turgeon, J., Couture, L., Michaud, V., Li, J., Nekka, F. 2009. Assessing drug distribution in tissues expressing P-glycoprotein through physiologically based pharmacokinetic modeling: Model structure and parameters determination. Theor Biol Med Modell 6: 2. http://www.ncbi .nlm.nih.gov/pmc/articles/PMC2661039/. Fromm, M. F. 2000. P-glycoprotein: A defense mechanism limiting oral bioavailability and CNS accumulation of drugs. Int J Clin Pharmacol Ther 38(2): 69–74. 232

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Fromm, M. F. 2003. Importance of P-glycoprotein for drug disposition in humans. Eur J Clin Investig 33(Suppl 2): 6–9. Ganapathy, V., Prasad, P. D. 2005. Role of transporters in placental transfer of drugs. Toxicol Appl Pharmacol 207: 381–7. Ganapathy, V., Prasad, P. D., Ganapathy, M. E., Leibach, F. H. 2000. Placental transporters relevant to drug distribution across the maternal–fetal interface. J Pharmacol Exp Ther 294(2): 413–20. Ghuman, J., Zunszaina, P. A., Petitpasa, I., Bhattacharya, A. A., Otagirib, M., Curry, S. 2005. Structural basis of the drug-binding specifcity of human serum albumin. J Mol Biol 353(1): 38–52. Gosselet, F., Loiola, R. A., Roig, A., Rosell, A., Culot, M. 2021. Central nervous system delivery of molecules across blood-brain barrier. Neurochem Int 144: 104952. https://doi.org/10.1016/j.neuint .2020.104952. Greenblatt, D. J., Sellers, E. M., Koch-Weser, J. 1982. Importance of protein binding for the interpretation of serum or plasma drug concentrations. J Clin Pharmacol 22(5–6): 259–63. Hagen, C. H. L. 1839. Uber die Bewegung des Wassers in engen cylindrischen Rohren. Ann Physik Chem 42: 423–42. Hao, L., Zhou, Q., Piano, Y., Zhou, Z., Tang, J., Shen, Y. 2021. Albumin -binding prodrugs via reversible iminoboronate forming nanoparticles for cancer delivery. J Control Release 330: 362–71. Hawkins, B. T., Egleton, R. D. 2007. Pathophysiology of the blood–brain barrier: Animal models and methods. In Current Topics in Developmental Biology, ed. P. S. Gerald, 277–309. New York: Academic Press. Henriksen, J. H., Horn, T., Christoffersen, P. 1984. The blood–lymph barrier in the liver. A review based on morphological and functional concepts of normal and cirrhosis liver. Liver 4(4): 221–32. Herve, F., Urien, S., Albengres, E., Duche, J. C., Tillement, J. P. 1994. Drug binding in plasma. A summary of recent trends in the study of drug and hormone binding. Clin Pharmacokinet 26(1): 44–58. Hitzl, M., Schaeffeler, E., Hocher, B., Slowinski, T., Halle, H., Eichelbaum, M., Kaufmann, P., Fritz, P., Fromm, M. F., Schwab, M. 2004. Variable expression of P-glycoprotein in the human placenta and its association with mutations of the multidrug resistance 1 gene (MDR1, ABCB1). Pharmacogenetics 14(5): 309–18. Hume, R. 1966. Prediction of lean body mass from height and weight. J Clin Pathol 19(4): 389–91. Hume, R., Weyers, E. 1971. Relationship between total body water and surface area in normal and obese subjects. J Clin Pathol 24(3): 234–38. Koch-Weser, J., Sellers, E. M. 1976. Binding of drugs to serum albumin. N Engl J Med 294(10): 526–31. Kratochwil, N. A., Huber, W., Muller, F., Kansy, M., Gerber, P. R. 2002. Predicting plasma protein binding of drugs: A new approach. Biochem Pharmacol 64(9): 1355–74. Kyle, U. G., Bosaeus, I., De Lorenzo, A. D., Deurenberg, P., Elia, M., Gómez, J. M., Heitmann, B. L., Kent-Smith, L., Melchior, J.-C., Pirlich, M., Schafetter, H., Schols, A. M. W. J., Richard, C. 2004. Bioelectrical impedance analysis – Part I: Review of principles and methods. Clin Nutr 23(5): 1226–43.

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Langmuir, I. 1916. The constitution and fundamental properties of solids and liquids. Part I. solids. J Am Chem Soc 38(11): 2221–95. Litman, T., Druley, T. E., Stein, W. D., Bates, S. E. 2001. From MDR to MXR: New understanding of multidrug resistance systems, their properties and clinical signifcance. Cell Mol Life Sci 58(7): 931–59. Liu, J., Corton, C., Dix, D. J., Liu, Y., Waalkes, M. P., Klaassen, C. D. 2001. Genetic background but not metallothionein phenotype dictates sensitivity to cadmium-induced testicular injury in mice. Toxicol Appl Pharmacol 176(1): 1–9. Liu, X., Smith, W. J., Chen, C., Callegari, E., Becker, S. L., Chen, X., Cianfrogna, J., Doran, A. C., Doran, S. D., Gibbs, J. P., Hosea, N., Liu, J., Nelson, F. R., Szewc, M. A., Van Deusen, J. 2005. Use of a physiologically based pharmacokinetic model to study the time to reach brain equilibrium: An experimental analysis of the role of blood–brain barrier permeability, plasma protein binding, and brain tissue binding. J Pharmacol Exp Ther 313(3): 1254–62. Löscher, W., Potschka, H. 2005. Blood–brain barrier active effux transporters: ATP-binding cassette gene family. J Am Soc Exper Neurother 2(1): 86–98. Mattos, C., Ringe, D. 1996. Locating and characterizing binding sites on proteins. Nat Biotechnol 14(5): 595–9. McNamara, P. J., Levy, G., Gibaldi, M. 1979. Effect of plasma protein and tissue binding on the time course of drug concentration in plasma. J Pharmacokinet Biopharm 7(2): 195–206. Miller, D. R., Carlson, J. D., Loyd, B. J., Day, B. J. 1983. Determining ideal body weight (letter). Am J Hosp Pharm 40: 1622. Morgenthaler, N. G., Struck, J., Alonso, C., Bergmann, A. 2006. Assay for the measurement of copeptin, a stable peptide derived from the precursor of vasopressin. Clin Chem 52(1): 112–19. Mosteller, R. D. 1987. Simplifed calculation of body-surface area. N Engl J Med 317(17): 1098. Myllynen, P., Pasanen, M., Vähäkangas, K. 2007. The fate and effects of xenobiotics in human placenta. Expert Opin Drug Metab Toxicol 3(3): 331–46. Noak, A., Gericke, B., von Köckritz-Blickwede, M., Löscher, W. 2018. Mechanism of drug extrusion by brain endothelial cells via lysosomal drug trapping and disposal by neutrophils. Pro Natl Acad Sci (PNAS). https://doi.org/10.1073/pnas.1719642115. Nowak, I., Shaw, L. M. 1995. Mycophenolic acid binding to human serum albumin: Characterization and relation to pharmacodynamics. Clin Chem 41(7): 1011–17. Padowski, J. M., Pollack, G. M. 2011. Infuence of time to achieve substrate distribution equilibrium between brain tissue and blood on quantitation of the blood-brain barrier P-glycoprotein effect. Brain Res 1426: 1–17. Pardridge, W. M. 1998. CNS drug design based on principles of blood–brain barrier transport. J Neurochem 5(5): 1781–92. Peng, B., Andrews, J., Nestorov, I., Brennan, B., Nicklin, P., Rowland, M. 2001. Tissue distribution and physiologically based pharmacokinetics of antisensephosphorothioate oligonucleotide ISIS 1082 in rat. Antisense Nucleic Acid Drug Dev 11(1): 15–27. Petitpas, I., Grüne, T., Bhattacharya, A. A., Curry, S. 2001. Crystal structures of human serum albumin complexed with monounsaturated and polyunsaturated fatty acids. J Mol Biol 314(5): 955–60. 234

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Pittet, C. L., Newcombe, J., Prat, A., Arbour, N. 2011. Human brain endothelial cells endeavor to immunoregulate CD8 T cells via PD-1ligand expression in multiple sclerosis. J Neuroinfammation 8: 155. Poiseuille, J. L. M. 1838. Ecoulement des Liquides: Societe Philomatique de Paris. Extraits des ProcésVerbaux des Séances Pendant I’Année 1838, 1–3, 77–81. Paris: René et Cie. Pries, A. R., Secomb, T. W., Gessner, T., Sperandio, M. B., Gross, J. F., Gaehtgens, P. 1994. Resistance to blood fow in microvessels in vivo. Circ Res 75(5): 904–15. Pries, A. R., Secomb, T. W., Jacobs, H., Speradio, M., Osterloh, K., Gaehtgens, P. 1997. Microvascular blood fow resistance: Role of endothelial surface layer. Am J Physiol 273(5): H2272–9. Pulgar, V. M. 2019. Transcytosis to cross the blood brain barrier, new advancement and challenges. Front Neurosci. https://doi.org/10.3389/fnins.2018.01019. Robinson, J. D., Lupkiewicz, S. M., Palenik, L., Lopez, L. M., Ariet, M. 1983. Determination of ideal body weight for drug dosage calculations. Am J Hosp Pharm 40(6): 1016–19. Rowland, M. 1980. Plasma protein binding and therapeutic monitoring. Ther Drug Monit 2(1): 29–37. Russell, L. 1977. Movement of spermatocytes from the basal to the adluminal compartment of the rat testis. Am J Anat 148(3): 313–28. Sadeque, A. J. M., Wandel, C., He, H., Shah, S., Wood, A. J. J. 2000. Increased drug delivery to the brain by P-glycoprotein inhibition. Clin Pharmacol Ther 68(3): 231–7. Schinkel, A. H., Jonker, J. W. 2003. Mammalian drug effux transporters of the ATP binding cassette (ABC) family: An overview. Adv Drug Deliv Rev 55(1): 3–29. Schmidt, A., Morales-Prieto, D. M., Pastuschek, J., Frohlich, K., Markert, U. R. 2015. Only human have human placentas: Molecular differences between mice and humans. J Reprod Immunol 108: 65–71. Segal, K. R., Burastero, S., Chun, A., Coronel, P., Pierson Jr., R. N., Wang, J. 1991. Estimation of extracellular and total body water by multiple frequency bioelectrical-impedance measurement. Am J Clin Nutr 54(1): 26–9. Shan, Y., Cen, Y., Zhang, Y., Tan, R., Zhao, J., Nie, Z., Zhang, J., Yu, S. 2022. Effect of P-glycoprotein inhibition on the penetration of ceftriaxone across the blood-brain barrier. Neurochem Res 47(3): 634–43. Shary, W. L., Aarons, L. J., Rowland, M. 1978. Representation and interpretation of drug displacement interactions. Biochem Pharmacol 27(1): 139–44. Silverman, J. A. 1999. Multidrug-resistance transporters. Pharm Biotechnol 12: 353–86. Smith, D. A., Di, L., Kerns, E. H. 2010. The effect of plasma protein binding on in vivo effcacy: Misconceptions in drug discovery. Nat Rev Drug Discov 9(12): 929–39. Smitha, T. L., Merry, S. T., Harris, D. L., Ford, J. J., Ike, J., Archibong, A. E., Ramesh, A. 2007. Speciesspecifc testicular and hepatic microsomal metabolism of benzo(a)pyrene, an ubiquitous toxicant and endocrine disruptor. Toxicol Vitro 21(4): 753–8. Song, S., Li, Y., Liu, Q. S., Wang, H., Li, P., Shi, J., Hu, L., Zhang, H., Liu, Y., Li, K., Zhao, X., Cai, Z. C. 2021. Interaction of mercury ion (Hg2+) with blood cytotoxicity attenuation by serum albumin binding. J Hazard Mater 412: 125158. https://doi.org/10.1016/j.jhazmat.2021.125158.

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Spreafco, M., Jacobson, M. P. 2013. In silico prediction of brain exposure: Drug free fraction, unbound brain to plasma concentration ratio and equilibrium half-life. Curr Top Med Chem 13(7): 813–20. Sun, H., Dai, H., Shaik, N., Elmquist, W. F. 2003. Drug effux transporters in the CNS. Adv Drug Deliv Rev 55(1): 83–105. Sutera, S. P., Skalak, R. 1993. The history of Poiseuille’s law. Annu Rev Fluid Mech 25(1): 1–19. Syme, M. R., Paxton, J. W., Keelan, J. A. 2004. Drug transfer and metabolism by the human placenta. Clin Pharmacokinet 43(8): 487–514. Tan, A., Kloppenborg, P., Benraad, T. 1989. Infuence of age, posture, and intra-individual variation on plasma levels of atrial natriuretic peptide. Ann Clin Biochem 26(6): 481–6. Tomi, M., Hosoya, K. 2010. The role of blood–ocular barrier transporters in retinal drug disposition: An overview. Expert Opin Drug Metab Toxicol 6(9): 1111–24. Tunblad, K., Jonsson, E. N., Hammarlund-Udenaes, M. 2003. Morphine blood–brain transport is infuenced by probenecid co-administration. Pharm Res 20(4): 618–23. Vähäkangas, K., Myllynen, P. 2009. Drug Transporters in the human blood–placental barrier. Br J Pharmacol 158(3): 665–7. Vallner, J. J., Chen, L. 1977. β-Lipoproteins, possible plasma transport proteins for basic drugs. J Pharm Sci 66(3): 420–1. Wagner, J. G. 1976. Simple model to explain effects of plasma protein binding and tissue binding on calculated volumes of distribution, apparent elimination rate constants and clearances. Eur J Clin Pharmacol 10(6): 425–32. Waring, M. J. 2009. Defning optimum lipophilicity and molecular weight ranges for drug candidates–molecular weight dependent lower logD limits based on permeability. Bioorg Med Chem Lett 19(10): 2844–51. Watson, P. E., Watson, I. D., Batt, R. 1980. Total body water volumes for adult males and females estimated from simple anthropometric measurements. Am J Clin Nutr 33(1): 27–39. Xie, H., Guo, C. 2021. Albumin alters the conformational ensemble of amyloid-β bt promiscuous interactions: Implications for amyloid inhibition. Front Mol Biosci. https://doi.org/10.3389/fmolb .2020.629520. Yacobi, A., Stoll, R. G., DiSante, A. R., Levy, G. 1976. Intersubject variation of warfarin binding to protein in serum of normal subjects. Res Commun Chem Pathol Pharmacol 14(4): 743–7. Zhu, X., Jin, K., Huang, Y., Pang, Z. 2019. Brain drug delivery by adsorption-mediated transcytosis. Brain Target Drug Deliv Sys. https://doi.org/10.1016/B978-0-12-814001-7.00007-X.

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9 PK/TK Considerations of Xenobiotic Metabolism Mechanisms and Rate Equations 9.1 INTRODUCTION It can be said with some degree of confdence that metabolism and metabolizing enzymes and their genes have existed on our planet way before the frst primitive primate appeared. The timeline of the existence of metabolic enzymes may even go way back to the emergence of the frst functional and surviving single cells on earth. It is also safe to say that the metabolic enzymes of Homo sapiens probably had not been as diverse as it currently is in humans, and it may be presumed that the diversity of the current metabolic enzymes and biotransformation processes are the results of the ongoing evolutionary processes in humans over centuries in response to dietary changes and/or natural or man-made xenobiotics, either as environmental contaminants or therapeutic agents. The principal mechanism for biotransformation and elimination of xenobiotics, occurs at multiple sites in the body. Although the liver is the main site for metabolism, other organs, such as the kidneys, GI tract, skin, and nasal mucosa are also capable of contributing quantitatively to the overall metabolism of xenobiotics. 9.2 LIVER The liver weighs about one fftieth of the body weight (≈ between 1 and 1.5 kg), and has two lobes; the right lobe is approximately six times larger than the left one, and further into the liver are functional lobules around a central vein. Several factors infuence its weight in the normal population. Its weight is higher in the young male group than female or aged population. There are differences among races, for instance, Asians have greater visceral adiposity than Caucasians. Using the following empirical equations, one can estimate the weight of liver:

(

)

Standard Liver Weight ( g ) = bwt ( kg ) + 218 ´ 12.3 + ( GF ) ´ 51

(9.1)

Body weight is abbreviated as (bwt ), and GF stands for ‘Gender Factor’ expressed as ‘one’ for male and ‘zero’ for female, thus the Equation 9.1 changes to Equations 9.2 and 9.3 for male and female, respectively:

( Men ) Liver Weight(g) = ( bwt ( kg ) + 218 ) ´ 12.3 + 51

(9.2)

(Women ) Liver Weight ( g ) = ( bwt(kg) + 218 ) ) ´ 12.3

(9.3)

Another measurement of the liver volume (LV) estimated by the following empirical (Equation 9.4) and allometric (Equation 9.5) equations.

( )

LV ( mL ) = 706.2 ´ BSA m 2 + 2.4 LV ( mL ) = 2.223 ´ bwt ( kg )

0.426

´ height ( cm )

(9.4) 0.682

(9.5)

Physiologically, the liver is responsible for numerous functions that can be summarized in the three categories of metabolic, synthetic, and storage functions and include ◾ gluconeogenesis, storage, and breakdown of glycogen ◾ formation of urea from proteins ◾ formation of cholesterol and triglycerides from fats ◾ endocrine functions including the insulin-like growth factor (IGF) and hepatocyte growth factor (HGF) synthesis and catabolism of other hormones) ◾ exocrine function (including synthesis of bile acids and plasma cholinesterase) ◾ regulation of homeostasis (including coagulation and fbrinolysis) ◾ energetic and structural metabolism (including biosynthesis and metabolism of carbohydrates, lipids, and proteins) DOI: 10.1201/9781003260660-9

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◾ synthesis of albumin and proteins like α1-acid glycoprotein, protein S and anti-thrombin ◾ immunomodulation and regulation of infammatory response ◾ storage of glycogen, nutrients, vitamins A, D, K, B12 and folate and minerals like iron and copper ◾ biotransformation of endogenous and xenobiotics (including enzymatic reactions of Phase I and Phase II metabolism) ◾ function of transporters in the liver, often identifed as Phase III transfer (Nahmias et al., 2006; Arias et al., 2009; Mazzoleni and Steimberg, 2012; Döring and Petzinger, 2014); essentially the transporters transfer the metabolites for further metabolism, or transfer conjugates into the bile for elimination, and so on The vessels carrying blood to the liver are the hepatic portal vein and the hepatic artery. The former conveys venous blood from splenic, inferior, and superior mesenteric veins, which originate from the small and large intestines, spleen, pancreas, stomach, the double layer of peritoneum (omentum) that is attached to stomach, and all organs of the abdomen including the gallbladder. The liver receives approximately 30% of cardiac input. The portal vein supplies approximately 75% of blood volume to the liver, rich in nutrients but poor in oxygen, and about 50% of oxygen supply. The remaining oxygen requirement is supplied by the liver artery. The liver does not control the blood fow of the hepatic portal vein; the fow rate of the vein is the combination of blood fow out of all organs of the abdomen that forward the blood to the liver. Xenobiotics administered orally and absorbed via the gastrointestinal tract enter the capillaries and the portal blood, transfer to the liver, and depending on their physicochemical characteristics, may be metabolized by the enzymes of the liver. This process known as frst-pass metabolism reduces the bioavailability of the administered dose (Galetin and Houston, 2006). The frst-pass metabolism is often associated with the liver and hepatoportal vein system. However, the frstpass metabolism is also a feature of organs, like lungs and other metabolically active routes of administration (Uwe et al., 2009). The hepatic artery, which originates from the aorta by way of the celiac trunk, provides arterial blood rich in oxygen to the liver. Its fow rate depends on the fow rate of hepatic portal vein such that it maintains the balance of total blood fow through the liver and total blood volume of about 20–25%. Based upon the oxygen supply, the acinus is functionally divided into three zones. Zone 1, known as periportal, consists of parenchymal cells encircling the portal tracts where portal and arterial blood join and mix. Oxidative energy metabolism; glucose release; secretion of bile acids and bilirubin; fatty acid oxidation; amino acid utilization, degradation, and conversion to glucose; and urea formation predominantly take place in zone 1. Zone 3 is the periphery region around the hepatic central vein and zone 2 is the region between zone 1 and 3. All outlets of the liver are in zone 3. The major function in zone 3 includes glucose uptake, glycolysis, ketogenesis, synthesis of glycogen from glucose, synthesis of bile salts, lipogenesis, and biotransformation (Thurman and Kauffman, 1985; Jungermann, 1995). The vascular structure is such that the transfer through parenchymal cells is facilitated from zone 1 to zone 3, but not from zone 3 to zone 1. The basic structure of the liver lobule and its essential components are depicted in Figure 9.1. Approximately 78% of the cells are hepatocytes, 3% endothelial cells, 2% Kupffer cells, and 1% Ito cells (fat-storing cells); the remaining 16% is related to spaces, which includes the sinusoidal lumen, bile passageway, canaliculi, and the space between sinusoidal endothelium and hepatocytes, known as the space of Disse. Zone 3 has a higher quantity of CYP450 subfamilies and the highest xenobiotic-related toxicity; for example, necrosis caused by acetaminophen, pyrrolizidine alkaloids, mushroom poisoning, hydrocarbons, halothane, carbon tetrachloride (CCl4), etc. Toxicity in zone 2 is very rare. Zone 1 has also toxicity manifestations, like necrosis, caused by allyl alcohol and phosphorous. The function of hepatocytes in the metabolism depends on what zone the cells are located (Haussinger, 1988; Bhatia et al., 1996). In addition to hepatocytes that comprise approximately 80% of the liver, there are other cells, at least a dozen, which play important functional roles in the normal and diseased liver (Malarkey et al., 2005). The liver has the capacity to regenerate and recover from injuries. Loss of liver tissue is usually restored, and the hepatocytes have the unlimited capacity to regenerate (Michalopoulos and Bhushan, 2021). 238

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Figure 9.1 Schematic of hepatic acinus, the functional unit of the liver; the objective of the illustration is to highlight the main features of the unit’s roles in metabolism of xenobiotics; the liver has about 100,000 of these units and approximately 2/3 of the cells in this functional unit are hepatocytes that perform diverse metabolic, endocrine, and secretory functions; in addition to the hepatic central vein in each unit there are branches of hepatic artery, portal vein, bile duct, and lymphatic veins at each corner of this hexagonal structure; through the capillaries, the corner branches of the artery provide oxygenated blood, branches of portal vein supply nutrition and the absorbed xenobiotics from the GI tract, and branches of bile duct and lymphatic vein receives the secreted compounds from hepatocytes; depending on the proximity to the oxygen supply, the unit is functionally divided into three zones of 1, 2, and 3; zone 3 has higher quantity of CYP450 subfamilies and the highest xenobiotic-related toxicity; the zonal concept of the liver is used for the zonal liver model (section 9.4.5.4 of this chapter). 9.3 METABOLIC PATHWAYS Most, if not all, xenobiotics after entering the body encounter one or more metabolic pathways in the liver, and the outcome of encounter is the formation of chemically modifed molecules of xenobiotics known as metabolites. The end products of the metabolic pathways are hydrophilic derivatives of the parent compound that are more easily eliminated through urinary or biliary elimination. This biotransformation occurs with the help of several enzyme systems that are utilized for metabolism of dietary constituents. Genes encoding these enzyme systems have functions in every eukaryotic cell. At a larger scale, each person has their own unique alleles coding for metabolic enzyme systems. Interindividual variations in metabolism of xenobiotics, mostly heritable, have given rise to the feld of pharmacogenetics. Humans are continuously exposed to all sources of xenobiotics, from food constituents and additives to cosmetics to herbal medications to environmental pollutants and therapeutic agents. All undergo biotransformation by the same enzyme systems and may set off a series of xenobiotic–xenobiotic interactions based on the inhibition or induction of the metabolizing enzyme systems (Sugimura and Sato, 1983). The metabolites can be primary or secondary metabolites. There are compounds whose primary metabolites are more hydrophobic and reactive than the parent compound and require further metabolism for elimination. This is a paradoxical role of metabolism because elimination of xenobiotics from the body is useful for activation of inactive therapeutic xenobiotics known as prodrugs. The reactive or electrophilic metabolites may also interact with the cellular membrane or nucleophilic macromolecules, such as DNA, RNA, and protein, forming repairable or 239

9.3 METABOLIC PATHWAYS

non-repairable adducts (Dipple, 1983). The interaction of the reactive metabolites with DNA may result in the mutation of genes and possibly initiation of cancer in a region of the body. The metabolic pathways catalyzed by the enzyme system occurs in two distinct phases known as Phase I and Phase II metabolism. Xenobiotics are subjected to one or multiple biotransformation pathways offered by the Phase I and II enzyme systems in succession. Depending upon the chemical structure of a xenobiotic, enzymes of Phase I metabolism create functional groups such as NH2, −OH, −SH, -O-, or −COOH in the molecular structure, whereas the enzymes of Phase II metabolism add additional groups, such as the glucuronic acid, glutathione, sulfate, acetyl, or methyl group to inactivate and increase the water solubility of a xenobiotic and/or its metabolites. There are signifcant differences among species in Phase I and II metabolisms of xenobiotics, attributable to the dietary and environmental conditions during evolution. The differences in species metabolism can be qualitative or quantitative. The qualitative differences exist when the species use different pathways of metabolism to metabolize a compound. The quantitative differences are when the kinetic parameters, like the rate of metabolism, metabolic rate constant, or metabolic clearance of the same pathways are different. 9.3.1 Phase I Metabolism Phase I metabolic reactions include the modifcation of the molecular structure of xenobiotics by creating new functional group(s) and/or modifying existing functional group(s). The products of Phase I metabolism, also known as primary metabolites, can be reactive and toxic or nontoxic and hydrophilic. In general, any enzymatic reaction that forms nonreactive and nontoxic metabolites from a toxic parent compound, or toxic and reactive primary metabolites, is known as a detoxifcation pathway, which includes certain reactions of Phase I and most, if not all, pathways of Phase II metabolism. The Phase I metabolic enzyme systems are categorized as follows: 9.3.1.1 Flavin-Containing Monooxygenases The favin-containing monooxygenases (FMOs) catalyze the oxygenation of structurally diverse molecules containing nitrogen, sulphur, phosphorus, and selenium via the bound cofactor favin. FMO is considered a complementary enzyme system to the CYP450 family of enzyme systems (Cashman, 2002). The metabolites of FMO systems are polar molecules, which eliminate easily from the body. These characteristics are considered advantages in designing therapeutic agents that are metabolized by FMO and not by CYP450 isozymes. However, sometimes FMOs generate metabolites that are reactive materials and can cause toxicity (Cashman and Zhang, 2006). There are at least six FMO in the mammalian system. FMO1 mediates oxidative metabolism of lipophilic compounds, such as imipramine and orphenadrine (Koukouritaki et al., 2002). It is FMO form 3 (FMO3) that is the prominent form in the adult human liver and is likely to be associated with the bulk of FMO-mediated metabolism (Larsen-Su and Williams, 1996; Phillips and Shephard, 2020). FMO1 and FMO3 S-oxygenate a number of nucleophilic sulfur-containing substrates and are sensitive to steric features of the substrate and aliphatic amines with linkages between the nitrogen atom and a large aromatic group, such as a phenothiazine. Small amine molecules such as phenethylamines are effciently N-oxygenated by human FMO3. For this reason, the cimetidine S-oxygenation or ranitidine N-oxidation is used as a functional probe of human FMO3. Trimethylamine is another amine molecule metabolized by FMO3 in the body to non-odorous trimethylamine N-oxide (TMA N-oxide). Mutation of the FMO3 gene reduces or eliminates TMA N-oxide formation, which results in trimethylaminuria, a disorder manifested by a body odor for affected individuals (Motika et al., 2009). In addition to being the prominent form, FMO3 is species- and tissue-specifc, and contrary to human cytochrome P450, it is not easily inducible or inhibited, which minimizes drug–drug interaction and provides opportunities in design of molecules that metabolize by this enzyme system (Cashman and Zhang, 2006). There are signifcant interindividual variations in FMO3dependent metabolism of xenobiotics, an important issue in population PK/TK analysis. Although selective substrates of FMO enzyme systems are known, selective inhibitors of the systems are yet to be determined (Harper and Brassil, 2008). More recently, another isoform, FMO5, is implicated in the activation of the intermediate metabolite of 3-hydoxy nabumetone to the active metabolite of 6-methoxy-2 naphthylacetic acid in vitro (Matsumoto et al., 2021). 9.3.1.2 Flavin-Containing Amine Oxidoreductases The members of this enzyme system are polyamine oxidase (PAO), L-amino acid oxidases (LAO), and various favin-containing monoamine oxidases (MAO). Polyamine oxidase (PAO) with a molecular weight of 62 kDa is a FAD-dependent enzyme present in most tissues of vertebrates (Salvi and Tavladoraki, 2020). Molecules with two positively charged amino groups, with one or 240

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both with alkyl substituent, separated by a short carbon chain are substrates for PAO. Spermine and the monoacetyl derivatives like N1-acetylspermine and N1-acetylspermidine are natural substrates of PAO (Seiler, 1995). PAO plays a major role in the intracellular regulatory system and maintenance of polyamine homeostasis in non-proliferating cells such as brain cells. The dynamics of polyamine homeostasis, identifcation of PAO, and spermine oxidase sequences in conjunction with metabolic and cellular responses have provided new opportunities to evaluate clinically relevant polyamine analogues and inhibitors (Vujcic et al., 2003). L-amino acid oxidases (LAO) catalyze most amino acids of an L confguration. Methionine and other sulfur-containing molecules, for example, cystine, homocystine, and thialysine are the best substrates of this enzyme system (Cavallini et al., 1982). LAO have antibacterial, antifungal, antiprotozoal, antiviral, antiproliferative, and antitumor properties, with an unclear effect on platelet aggregation (Lukasheva et al., 2011; Kasai et al., 2021). MAO is an FAD-containing enzyme of the outer mitochondrial membrane. In humans, there are two isozymes, which are identifed as MAO-A and MAO-B. They differ in their substrate specifcity and are inhibited by different inhibitors (Shih, 1991; Gargalidis-Moudanos et al., 1997). Their important role is in regulating the intracellular levels of amines via their oxidation; these include various neurotransmitters, neurotoxins, and trace amines. MAO-A is found in the liver, GI tract, placenta, neurons, and astroglia; MAO-B is found in blood platelets, neurons, and astroglia. MAO-A metabolizes melatonin, adrenaline, noradrenalin, and serotonin, and MAO-B metabolizes phenethylamine and benzylamine. Both forms break down dopamine, tyramine, and tryptamine. 9.3.1.3 Epoxide Hydrolases Epoxide hydrolases, also known as epoxide hydratases (EHs), function in the detoxifcation of metabolites of xenobiotics and have a multifaceted role in human health and disease (Gautheron and Jéru, 2021). They catalyze the hydration of reactive endogenous and exogenous epoxides to dihydrodiols, which can then be conjugated and excreted from the body (Oesch and Arand, 1999; Arand and Oesch, 2002; Arand et al., 2003). Epoxide hydrolase 1, identifed as mEH, is localized in the endoplasmic reticulum and microsome, and epoxide hydrolase 2, a cytosolic enzyme known as soluble EH or sEH, plays a critical role in defending the body against highly reactive, toxic, and often carcinogenic epoxides. Epoxides are mostly produced as intermediate metabolites of lipophilic compounds that are considered carcinogenic. Many epoxides are strong electrophiles and react readily, with the DNA forming repairable or non-repairable DNA adducts which cause mutation and initiate cancer (Boroujerdi et al., 1981). EHs detoxify a broad range of structurally very different epoxides such as benzo (a) pyrene or styrene oxide. Members of EHs are used as therapeutic targets for regulation of blood pressure, infammation, cancer progression, and the onset of several other diseases (Morisseau et al., 2005). 9.3.1.4 Cytochrome P450 Cytochrome P450 (CYP450) represents a superfamily of heme-containing enzymes with the potential to catalyze dehydrogenation, hydroxylation, epoxidation, oxygenation, dealkylation, and ring-opening (Porter and Coon,, 1991; Guengerich, 1997). To represent the diversity of CYP450 family members, a nomenclature has been created to identify different families and subfamilies (Nebert et al., 1991). This nomenclature is based on the amino acid sequence of the protein and their assignment to different gene families. When 40% of amino acids of CYP450 are identical, they belong to a family, for example, CYP1, CYP2, CYP3, CYP4, etc. When more than 55% of amino acids are identical, they are in a subfamily labeled with ending letters, for example, CYP1A or B or C, or CYP2A or B or C, etc. The fnal identifcation number is a cataloging number to identify the individual enzyme as a member of the subfamily, for example, CYP1A1, CYP2B6, etc. The CYP450 family members are present practically in all mammalian organs, but predominantly in the liver, GI tract, skin, nasal epithelia, lung, kidney, and other organs (Nebert et al., 1991; Nelson et al., 1993, 1996; Agundez, 2004). The following sections are a brief review of the role, characteristics, and related metabolic reactions of each member of CYP450 family. 9.3.1.4.1 CYP1A Subfamily The members of CYP1A subfamily are predominantly involved in detoxifcation and bioactivation of environmental pollutant and endogenous metabolism (Goldstone and Stegeman, 2006, Lu et al., 2020). In humans, the subfamily has two members, CYP1A1 and CYP1A2. CYP1A1 is expressed at low levels in the liver and variably in extrahepatic tissues, for example, lung, placenta, lymphocytes, and intestines. CYP1A2, on the other hand, is highly expressed in the liver (Roins 241

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and Ingelman-Sunderg, 1999). Both enzymes are inducible, but CYP1A1 is more sensitive to the inducers (Whitlock, 1999). The induction of CYP1A enzymes may also lead to catalytic activation of xenobiotics to their reactive metabolites, which may result in toxicity (Digiovanna et al., 1979; Gelboin, 1980). CYP1A1 and CYP1A2 catalyze the metabolic activation of two important classes of environmental carcinogens, polycyclic aromatic hydrocarbons, and arylamines (Conney, 1982; Kamataki et al., 1983; Shimada, 2006, Jimma et al., 2019). The CYP1A subfamily members are also expressed in experimental animals (mice, rats, dogs, and monkeys), and are involved in stereoselective metabolism of certain compounds in Caco-2 cells (Ishida et al., 2009). 9.3.1.4.2 CYP1B Subfamily The members of CYP1B family are present in human and experimental animals like mouse, rat, dog, and monkey. In humans, CYP1B1 catalyzes estrogens to active 4-hydroxylated derivatives that may cause breast cancer. CYP1B1 is expressed in most organs including the liver, heart, lung, kidney, placenta, prostate, and brain (Sutter et al., 1994; Willey et al., 1997; Rieder et al., 1998; Cheung et al., 1999; McFadyen et al., 1999a, b; Muskhelishvili et al., 2001; Spivak et al., 2001). Most of the human tissues that are positive for CYP1B1 are those that are receptive to steroid hormones like prostate, testes, endometrium, and mammary tissue (Eltom et al., 1998; Larsen et al., 1998; Tang et al., 1999; Bofnger et al., 2001). Its expression is much higher in tumor cells (Murray et al., 1997; McFadyen et al., 1999a, b, 2001), and its induction in the body is considered a biomarker for determining a hormone-mediated risk of cancer and as a target for the development of anticancer agents (McFadyen et al., 1999a, b; Doostdar et al., 2000; Bofnger et al., 2001; Potter et al., 2002). 9.3.1.4.3 CYP2A Subfamily The members of CYP2A subfamily have diverse metabolic roles in different species. In humans, the two important functional members of this subfamily are CYP2A6 and CYP2A13 (Yamano et al., 1990; Su et al., 2000). CYP2A6 is expressed in the liver, whereas CYP2A13 is expressed in extrahepatic tissues, mainly in the respiratory tract (Koskela et al., 1999; Gu et al., 2000; Su et al., 2000). Two members of this subfamily in mice are CYP2a4 and CYP2a5. The former hydroxylates steroid, and the latter encodes coumarin 7-hydroxylase (Negishi et al., 1989). CYP2A6 of human functionally is like the CYP2a5 of mice (Yamano et al., 1990). CY2A6 and CYP2A13 participate in oxidation of nicotine, cyclophosphamide, and afatoxin B1, hydroxylation of coumarin, and metabolism of tobacco-specifc nitrosamines derivatives (Yamano et al., 1990; Tiano et al., 1993; Nakajima et al., 2006; von Weymarn and Murphy, 2003; Bao et al., 2005). Several xenobiotics such as phenobarbital, rifampicin, dexamethasone, and nicotine can induce both isoforms in humans (Honkakoski and Negishi, 1997). Other members of the same subfamily, such as CYP1A2, have overlapping substrate specifcity as CYP1A13 (Fukami et al., 2007). Other isozymes of this subfamily are CYP2A7 in human; CYP2A4, CYP2A5, CYP2A12 and CYP2A22 in mice; CYP2A1, CYP2A2, and CYP2A3 in rats; CYP2A13 and CYP2A25 in dogs; and CYP 2A23 and CYP2A24 in monkeys. 9.3.1.4.4 CYP2B Subfamily The isozymes of CYP2B subfamily are present in the liver and other organs such as kidney, brain, intestine, and lung (Gervot et al., 1999). In rodents, they are in the intestine, and two isozymes of CYP2B1 and CYP2B2 are expressed in the rat brain (Rosenbrock et al., 2001). In humans, CYP2B6 is the major isozyme of the subfamily (Ekins et al., 1997, 1998; Desta et al., 2021). It is responsible for the metabolism of several therapeutic agents including propofol (Court et al., 2001); promazine (Wójcikowski et al., 2003); tamoxifen (Sridar et al., 2012); cyclophosphamide (Chang et al., 1993; Melanson et al., 2010); efavirenz (Lindfelt et al., 2010); testosterone and androstenedione (Domanski et al., 1999); bupropion (Marok et al., 2021) and metabolism of carcinogens, such as dibenzanthracin and afatoxin B1 is also by human CYP2B isozymes. The metabolism of antidepressant/antismoking agent bupropion is exclusively by CYP2B6 and thus it is used as a marker for CYP2B6 (Faucette et al., 2000). The specifc inhibitors of CYP2B6 include clopidogrel (Richter et al., 2004) and thiotepa (Rae et al., 2002; Richter et al., 2004). Signifcant interindividual variability has been observed between men and women (♀ ≫ ♂) and between females of different ethnicities (Shimada et al., 1994; Code et al., 1997; Lang et al., 2001; Lamba et al., 2003; Elens et al., 2010; Jamshidi et al., 2010) for CYP2B6. Another isozyme of this subfamily in humans is CYP2B7, which is present in normal and neoplastic lung tissues (Czerwinski et al., 1994, Desta et al., 2021). Other isozymes of the subfamily in experimental animals are CYP2B9 and CYP2B10 in mice; CYP2B1, CYP2B2, and CYP2B3 in rats; CYP2B11 in dogs; and CYP2B17 in monkeys. 242

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9.3.1.4.5 CYP2C Subfamily The members of the CYP2C subfamily are involved in the metabolism of a signifcant number of xenobiotics (Goldstein and de Morais, 1994). The four members of the human CYP2C subfamily, namely CYP2C8, CYP2C9, CYP2C18, and CYP2C19, are expressed in different organs in the body: CYP2C18 is mostly in the skin (Zaphiropoulos, 1997) and is a biomarker for infammatory bowel disease (Du et al., 2021). CYP2C8, CYP2C9, and CYP2C19 are mainly in the liver. CYP2C8 is responsible for metabolism of paclitaxel, arachidonic acid, benzo[a]pyrene (Rahman et al., 1994; Klose et al., 1999), and hydroxychloroquine/chroquine (Takahasi et al., 2020). CYP2C9 is the major isozyme and most abundant one involved in the metabolism of antidiabetic, anticonvulsant, anti-infammatory, and antihypertensive therapeutic agents, such as tolbutamide, ibuprofen, phenytoin, warfarin, diclofenac, piroxicam, tenoxicam, mefenamic acid, losartan glipizide, and torasemide (Scott and Poffenbarger, 1979; Goldstein and de Morais, 1994; Kidd et al., 1999; McCrea et al., 1999; Miners et al., 2000; Theken et al., 2020). Because of the importance of this isozyme and its substrate specifcity, humanized mouse models are available for the investigation of metabolism and disposition of CYP2C9 substrates (Scheer et al., 2012). CYP2C19, present in the liver and duodenum, is involved in the metabolism of many xenobiotics including antidepressants, proton pump inhibitors, and anxiolytic drugs, such as omeprazole, imipramine, diazepam, and many other compounds (Kupfer and Branch, 1985; Ward et al., 1989, 1991; Andersson et al., 1992; Sindrup et al., 1993). Compounds like phenobarbital, rifampicin, and dexamethasone induce CYP2C8, CYP2C9, and CYP2C19, whereas antifungal drugs fuconazole, voriconazole inhibit their activity. The members of the CYP2C subfamily are more complex and numerous in rodents and expressed mostly in the liver and extrahepatic tissues. In mice, the members of the subfamily include CYP2C9, CYP2C37, CYP2C38, CYP2C39, CYP2C40, CYP2C44, CYP2C50, CYP2C54, and CYP2C55; in rats the isozymes are CYP2C6, CYP2C7, CYP2C11, CYP2C12, CYP2C13, CYP2C22, and CYP2C23; in dogs the isozymes are CYP2C21 and CYP2C41; and in monkeys they are CYP2C20 and CYP2C43. In dogs, CYP2C family members are neither well developed nor similar to humans. The CYP2C43 of monkeys is structurally similar to CYP2C9 of humans, but functionally is comparable to CYP2C19 (Matsunaga et al., 2002). 9.3.1.4.6 CYP2D Subfamily The members of the CYP2D subfamily are responsible for monooxygenation of several xenobiotics; examples are methadone, propranolol, tamoxifen, dextromethorphan, sparteine, bufuralol, and desipramine in humans and experimental animals (Bogaards et al., 2000; Hiroi et al., 2002; Teh et al., 2012). In humans, there are three isozymes of CYP2D6, -2D7 and -2D8, but only CYP2D6 is the most active and studied isozyme of the subfamily in humans (Nofziger et al., 2020). It is present not only in the liver but also in the kidney, lung, intestine, placenta, breast, and brain (Niznik et al., 1990; Huang et al., 1997). Genetic polymorphism of this isozyme is the cause of variability in tamoxifen metabolism, which attributes to the lack of therapeutic effcacy in certain patients and reoccurrence of breast cancer (Teh et al., 2012). The impact of its polymorphism is also reported on the clinical response to metoprolol (Meloche et al., 2020). This individual variation is the result of inherited mutant CYP2Ds alleles as an autosomal recessive trait in approximately 7–10% of the Caucasian population (Mahgoub et al., 1977). Fluoxetine (Prozac) and its active metabolite, norfuoxetine, inhibits CYP2D6 in the liver (Crewe et al., 1992; Stevens and Wrighton, 1993), and quinidine is its most potent inhibitor (Smith and Jones, 1992). There are more members of this subfamily in experimental animals. In mice the notable isozymes are: CYP2D9, CYP2D10, CYP2D11, CYP2D12, CYP2D13, CYP2D22, CYP2D26, CYP2D34, and CYP2D40. The mouse CYP2D22 function is somewhat similar to the human CYP2D6 (Blume et al., 2000). In rats there are six isoforms: CYP2D1, CYP2D2, CYP2D3, CYP2D4, CYP2D5, and CYP2D18. The isozyme CYP2D1 is the rat orthologue of CYP2D6 of the human. The major isozyme in dogs is CYP2D15, which enzymatically is like the human CYP2D6 (Zuber et al., 2002) and is inhibited by quinidine and the HIV-I protease inhibitor. The CYP2D isozymes in monkeys are strain-related. The CYP2D42 of the Rhesus monkey, CYP2D30 of the Marmoset monkey, and CYP2D17 of the Cynomolgus monkey are an orthologue of human CYP2D6. 9.3.1.4.7 CYP2E Subfamily A member of the CYP2E subfamily, CYP2E1, is present in the liver, lung, and the tissues of the nose and pharynx. CYP2E1 has dual physiological roles of providing nutritional support and protective detoxifcation (Lieber, 1997) and, under certain conditions, may generate superoxide radical with pathological consequences of liver injury and interaction with cellular macromolecules 243

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(Wu and Cederbaum, 2003). It plays two major protective roles in the body. It detoxifes certain xenobiotics, particularly organic solvents, such as ethanol (Wu and Cederbaum, 2003), and plays an important role in the nutritional balance of the body during starvation, metabolizing lipids, and converting ketones to glucose (Lieber, 2004). It has been demonstrated that mitochondrial disfunction leads to abnormal activation of CY2E1 and formation of free radical and lipid peroxidation (Guan et al., 2019; Wang et al., 2021). It is also involved in the formation of reactive metabolites of carcinogens, like acrylonitrile, styrene, nitrosamine, and benzene, which subsequently interacts with cellular DNA, forming carcinogen-DNA adducts (Siegers et al., 1983; Hetu et al., 1983; Dai et al., 1993; Wu and Cederbaum, 2003). To prevent liver injury by CYP2E1-mediated hepatotoxicity induced by compounds like isoniazid, coadministration of compounds like kaempferol is recommended (Shih et al., 2013). CYP2E1 metabolizes xenobiotics such as chlorzoxazone, caffeine, and acetaminophen. These compounds are also considered markers for CYP2E1 activity (Lofgren et al., 2004). Ethanol induces CYP2E1 in all species. Rodents are considered relevant animal models to study the CYP2E1 metabolism of xenobiotics in humans (Zuber et al., 2002). CYP2E1 of rats is one the few enzymes that functionally is similar to humans, and extrapolation seems to be warranted. However, the comparative data of CYP2E1-dependent metabolism of dogs and monkeys with those of humans have reported to be inconsistent (Bogaards et al., 2000). 9.3.1.4.8 CYP3A Subfamily The members of CYP3A subfamily are central to the metabolism of most xenobiotics. They are expressed in several organs and tissues including the liver, gastrointestinal tract, lung, and kidney. They metabolize a wide range of therapeutic agents such as benzodiazepines, lidocaine, quinidine, dextromethorphan, midazolam, triazolam, carbamazepine, nifedipine, erythromycin, terfenadine, tacrolimus, daridorexant, fnerenone, and many more (Nebert and Russell, 2002; Zuber et al., 2002, Gehin et al., 2022; Wendl et al., 2022). Furthermore, with their broad substrate specifcity, they are also involved in oxidation of endogenous compounds like steroids (Hirita and Matsubara, 1993), retinoic acid, and bile acids, and oxidation of potent carcinogens, notably afatoxin B1, afatoxin G1, and benzo (a) pyrene. The substrates of this subfamily or their metabolites are also substrates for several other subfamilies like CYP1A, CYP2A, CYP2E, and so on. The CYP3A subfamily members in humans are CYP3A4, CYP3A5, -3A7, and CYP3A43. Among the members of the subfamily and because of their presence in the liver, stomach, lungs, kidneys, and duodenum, jejunum, and ileum, CYP3A4 and CYP3A5 are the most prominent isozymes. CYP3A4 is abundantly present in the human liver (Kolars et al., 1994; Dresser et al., 2000) and the tip of the enterocytes, and plays a signifcant role in the intestinal and hepatic frst-pass metabolism, lessening the bioavailability of xenobiotics signifcantly (Paine et al., 1997, Eisenmann et al., 2022). Pgp is present in the same region of the small intestine and they often share the same substrate specifcity. The presence of both proteins in the same locality of the GI tract prevents the absorption of xenobiotics. The effux of xenobiotics by Pgp increases their concentration in the lumen and enhances their interaction with CYP3A4, suggesting a cooperative role between the two proteins for removal and metabolism of xenobiotics. CYP3A4 is present in the periphery of the endoplasmic reticulum of the enterocytes, and Pgp is present at the apical membrane of the enterocyte. The Pgp of the liver also effuxes the xenobiotics into the bile ducts, which enters the small intestine and will be available for interaction with CYP3A4 (Van der Valk et al., 1990; Guengerich, 1992; McKinnon et al., 1995; Hall et al., 1999; Ueda et al., 1999; Benet et al., 2003; Liu and Pang, 2005). The other members of the subfamily are: CYP3A5 – more prominent in stomach and colon and CYP3A7 – more expressed in fetal liver. CYP3A43 has limited activity in the liver (Domanski et al., 2001; Gellner et al., 2001) and shows some overlap in substrate specifcity with CYP3A4, which is attributed to amino-acid sequence similarity between the two enzymes (Domanski et al., 2001, Pouget et al., 2014). Different species have diverse and dissimilar CYP3A isozymes. In mice, there are six isozymes of CYP3A11, CYP3A13, CYP3A16, CYP3A25, CYP3A41, and CYP3A44 (Yanagimoto et al., 1992; Sakuma et al., 2000; Schellens et al., 2000). The afatoxin B1 metabolism by CYP3A of mice is similar to humans (Gallagher et al., 1996). In rats the isoforms include CYP3A1, CYP3A2, CYP3A9, CYP3A18, CYP3A32, and CYP3A62 (Kirita et al., 1993; Strotkamp et al., 1995; Wang et al., 1996; Matsubara et al., 2004). Although CYP 3A1 is expressed in the liver, its induction or inhibition does not resemble the human CYP3A4, and in general, rats are not considered appropriate animal models for human CYP3A studies. In dogs, there are two members of the subfamily, CYP3A12 and CYP3A26, which are expressed in the liver and show similar characteristics as the CYP3A4 and 244

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CY3A5 of humans (Fraser et al., 1997). The CYP3A isoform identifed in the Cynomolgus monkey is CYP3A8, which is to some extent similar to human CYP3A4 and CYP3A5 (Komori et al., 1992; Ohmori et al., 1993). The importance of the CYP3A subfamily in the metabolism of endogenous compounds and xenobiotics is well established and recognized. However, the species differences in the metabolism of xenobiotics are signifcant and the induction and/or inhibition of the isoforms are quite dissimilar. Thus, the extrapolation of metabolic data of the CYP3A subfamily among species is debatable. 9.3.1.4.9 CYP4A Subfamily CYP4A subfamily members are fatty acid hydroxylases, present in most mammalian organs, and responsible for ω-hydroxylation and ω-1-hydroxylation of endogenous saturated and unsaturated fatty acids, prostaglandin, and arachidonic acid (Sharma et al., 1989; Aoyama et al., 1990; Su et al., 1998; Okita and Okita, 2001; Cowart et al., 2002; Savas et al., 2003). The isozymes facilitate the elimination of excess fatty acids to produce energy. Their role in xenobiotic metabolism is not quite clear. CYP4A11 and CYP4A22 are the isoforms of the subfamily in humans. CYP4A11, present in the human liver and kidney, acts as fatty acid ω-hydroxylating enzyme to regulate cell and/or organ physiology by conversion of arachidonic acid to the vasoactive and natriuretic eicosanoid 20-hydroxyeicosatetraenoic acid. Four members of the subfamily in rats include CYP4A1, CYP4A2, CYP4A3, and CYP4A8; in rabbit they are CYP4A4, CYP4A5, and CYP4A7. There are functional and structural similarities between CYP4A11 of human, CYP4A1 and CYP4A3 of rats, and CYP4A6 of rabbits (Imaoka et al., 1993). CYP4A1 is the major fatty acid hydroxylase of the rat liver, regulated by physiological conditions, including diabetes and fasting, and induced by different compounds, such as fatty acids and peroxisome proliferators like hypolipidemic. These effects are mediated via the peroxisome proliferator-activated receptor alpha (PPAR) (Savas et al., 2003). PPARs are ligandactivated nuclear receptors and exist as three different subtypes in mammals (PPARα, PPARδ, and PPARγ [with isoforms γ1, γ2, and γ3]). The nuclear receptors are involved in lipid homeostasis regulation of body weight and food intake, control of infammation, wound healing, and induction of CYP4A isoforms. 9.3.1.4.10 CYP4B Subfamily The members of this subfamily belong to the CYP4 family of the mammalian enzyme system, which includes CYP4A, CYP4F, CYP4V, CYP4X, and CYP4B1 subfamilies. The subfamilies have the capacity of participating in ω-hydroxylation of medium-chain fatty acids. CYP4B1, a member of CYP4B subfamily, is present in humans, rats, mice, dogs, pigs, chimpanzees, goats, rabbits, and cow. Furthermore, CYP4B2 is identifed in goats and cattle, and CYP4B3 in wallabies. CYP4B1 also participates in the metabolism of some structurally unrelated xenobiotics that are protoxins, such as aromatic amines, valproic acid, 4-ipomeanol, 3-methylindole, and 3-methoxy-4-aminoazobenzene. The metabolites of these compounds cause tissue- or organ-specifc toxicity (Baer and Rettie, 2006). The pathway of hydroxylation arylamines not only is mediated by CYP4B1 but also by CYP1A1 and CYP1A2 (Windmill et al., 1997). Due to heterologous expression of CYP4B1, its metabolic capabilities are less clear in different human populations. 9.3.1.4.11 CYP4F Subfamily The members of this subfamily are present in different locations in the body. For example, CYP4F2, CYP4F3, CYP4F11, and CYP4F12 are expressed in the liver and kidney (Christmas et al., 1999, 2001, 2003; Cui et al., 2000; Hirani et al., 2008), whereas, CYP4F3A is found only in myeloid tissue, and CYP4F8 is expressed exclusively in seminal vesicles (Bylund et al., 2000). Like the enzymes of CYP4 family, CYP4F subfamily is involved in ω-hydroxylation of long and very longchain fatty acids and the metabolism of vitamin E (Sontag and Parker, 2002; Sanders et al., 2006). The isozymes have the unique ability to oxidize or ω-hydroxylate the terminal methyl group present in saturated and unsaturated fatty acids of different chains. The isoform CYP4F2 participates in the metabolism of arachidonic acid into 20-hydroxyeicosatetraenoic acid (20-HETE), a strong vasoconstrictor, inhibitor of ion transport, and cellular proliferation agent. 9.3.1.5 Alcohol Dehydrogenase Alcohol dehydrogenase (ADH), a member of the oxidoreductase family of enzymes, is present in high concentrations in the human liver and kidney. There are at least nine known isoforms of ADH in the liver and a form known as the sigma form in the GI tract (Farrés et al., 1994). In humans, the cytosolic fraction of the liver and other tissues contains several isozymes. The 245

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isozyme active in metabolism of ethanol is known as “class I ADH.” The primary role of this subfamily is the metabolism and detoxifcation of alcohols. The products of metabolism are often more toxic than the parent compounds, but quickly undergo detoxifcation through secondary metabolism; for example, ethanol and methanol metabolize to the more toxic acetaldehyde and formaldehyde, respectively (Hammes-Schiffer and Benkovic, 2006). For ethanol, the interaction is Ethanol + NAD ® Acetaldehyde + NADH

(9.6)

The human “class I ADH” is a zinc metalloenzyme, consisting of three subunits of α, β, and γ either as homodimer or heterodimer isozymes encoded by the ADH1, ADH2, and ADH3 genes. All isozymes have two different monomers of type E for ethanol active and type S for steroid active and the possible combinations of monomers are EE, SS, and ES. Furthermore, there are fve classes of ADH in humans, identifed as Classes I–V. The isoforms are inhibited by heavy metals and chelating agents. Class I is the primary enzyme system in the liver. Each monomer has two subunits, and each subunit has one binding site for NAD+ and two binding sites for Zn2+. The coenzyme NAD+ is necessary for the conversion of ethanol, which converts ethanol to acetaldehyde by proton transfer, while zinc takes two hydrogens away from the ethanol. In addition to ethanol and methanol, the isoforms of this enzyme metabolize other alcohols, retinol, steroids, and fatty acids (Hellgren et al., 2007; Duester, 2008). The catalysis is in the presence of NAD+ or NADP+ as the electron acceptor and form primary, secondary, or tertiary alcohols, aldehyde, and ketone. Their natural role is to protect the body against toxins. The metabolites of this enzyme system are substrates for other enzymatic reactions in the body. 9.3.1.6 Diamine Oxidase (Histaminase) Diamine oxidase is present in high concentrations in the intestinal mucosa and plasma, and its role is the deamination of diamines, for example, cadaverine and putrescine. This enzyme is specifc only for diamines and does not deaminate monoamines. The level of this enzyme in plasma correlates with its intestinal level. In disease states that cause mucosal damage, the level of the enzyme in plasma decreases parallel to enzyme level of intestinal mucosa. Thus, the plasma level is a marker to evaluate the integrity of intestinal mucosa (Luk et al., 1980). Patients with medullary thyroid carcinoma tissue, that has spread into the lung, exhibit an elevated level of blood histaminases (Baylin et al., 1975). In acute viral hepatitis, the plasma level of the enzyme declines signifcantly (Gäng et al., 1976). In addition to its role of deamination of diamines, it generally plays the role of biomarker for different disease states. 9.3.1.7 Aldehyde Dehydrogenases Different forms of the aldehyde dehydrogenase (ALDH) enzyme metabolize aldehydes to less reactive forms. Aldehydes, the by-products of metabolism of endogenous and exogenous compounds (e.g., metabolism of amino acids, certain xenobiotics, carbohydrate, retinoic acid, vitamins, steroids, and lipids form different aldehydes), are substrates for the following enzymes of the ALDH family: ◾ ALDH1A1: Involved in the formation of secondary metabolites of alcohols and helps transparency of cornea (Jester et al., 1999). ◾ ALDH1A2: Catalyzes the synthesis of retinoic acid, the derivative vitamin A from retinaldehyde (Ono et al., 1998). ◾ ALDH1A3, ALDH1B1, ALDH3B1, ALDH3B2, and ALDH7A1: Play major roles in detoxifcation of aldehyde metabolites of alcohols and lipid peroxidation (Hsu and Chang, 1996; Hsu et al., 1997; Yoshida, 1993; Yoshida et al., 1998; Wang et al., 2009; Brocker et al., 2011). ◾ ALDHL1 and ALDHL2: Catalyze the metabolism of 10-formyltetrahydrofolate in the presence of NADP and water to tetrahydrofolate, NADPH, and CO2 (Hong et al., 1999; Krupenko and Oleinik, 2002). ◾ ALDH2: Catalyzes the formation of acetic acid from acetaldehyde (Hempel et al., 1985; 1987; Agarwal and Goedde, 1987; Seitz and Meier, 2007). ◾ ALDH3A1: Metabolizes various aldehydes and participates in the metabolism of neurotransmitters, corticosteroids, and biogenic amines (Rekha et al., 1998; Yang et al., 2002). ◾ ALDH3A2: Involved in a neurocutaneous disease known as SLS or Sjögren–Larsson syndrome (Freedberg, 2003). 246

PK/TK CONSIDERATIONS OF XENOBIOTICS BIOTRANSFORMATION

◾ ALDH4A1: Catalyzes the formation of glutamate from purroline-5-carboxylate, which is the second phase of proline catalysis (Goodman et al., 1974; Geraghty et al., 1998). ◾ ALDH5A1: Oxidizes aldehydes to carboxylic acids and is present in the liver and many other tissues (Crabb et al., 2004). ◾ ALDH6A1: Involved in the catabolic pathways of pyrimidine and valine (Kuiper et al., 2005). ◾ ALDH8A1: Involved in biosynthesis of 9-cis-retinoic acid biosynthesis (Lin and Napoli, 2000). ◾ ALDH9A1: Catalyzes the dehydrogenation of gamma-aminobutyraldehyde to GABA, gammaamino butyric acid (McPherson et al., 1994; Kikonyogo and Pietruszko, 1996; Lin et al., 1996). ◾ ALDH16A1: Involved in the interaction with SPG21 protein ACP33/maspardin and spastic paraplegia (Hanna and Blackstone, 2009). ◾ ALDH18A1: Involved in biosynthesis of proline, ornithine, and arginine (Baumgartner et al., 2001, 2005). The isoforms of ALDH are all detoxifying enzyme systems of Phase I metabolism and are in three different classes in the mammalian system: Class 1 (cytosolic), Class 2 (mitochondrial), and Class 3 (expressed in tumors, stomach, and cornea). 9.3.1.8 Xanthine Oxidase Xanthine oxidase (XOD) has a large and very complex structure with a molecular weight of about 270 kDa. It has two molybdenum atoms, two favin nucleotides, and eight iron atoms, which are present as ferredoxin iron-sulfur groups and four redux centers that carry electrons to oxygen to yield superoxide ion (O2- ) (Enroth et al., 2000, Hadizadeh et al., 2009). As a cytosolic enzyme, this metalloprotein catalyzes the metabolism of purine bases and hypoxanthine to xanthine and fnally to uric acid. Because of its dual metabolic role, this enzyme system is often called xanthine oxidoreductase. In certain disease states, such as gout, hypouricemia, and chronic heart failure, the production of purine is increased, and for therapy, the major need is to reduce serum uric acid levels (Landmesser et al., 2002; Gibbings et al., 2011). Thus, this enzyme is the target for therapeutic agents like allopurinol to inhibit the enzyme in the related disease states (Pacher et al., 2006; Puntoni et al., 2013). The parent compound, allopurinol, not only is a substrate for xanthine oxidase, its metabolite oxypurinol (alloxanthine) is also an inhibitor. The complexity of interaction and the dual role of this enzyme in conjunction with the investigation of its inhibition in various disease states have encouraged continued research and inquiries by many researchers in different felds (Guthikonda et al., 2003; Becker et al., 2005; Shiina et al., 2022). 9.3.1.9 Carboxylesterases The members of the carboxylesterase (CES) enzyme system catalyze the hydrolysis of many structurally diverse endogenous and exogenous esters, amides, thioesters, and carbamates in humans (Satoh and Hosokawa, 1998, 2006). The diversity of the members among different species is signifcant. In humans, the members include CES1A1, CES1A2, CES1A3, CES2A1, CES3A1, CES4C1, and CES5C1. They are present in the endoplasmic reticulum of the liver, kidney, lung, testis, and small intestine (Hosokawa et al., 1990, 1995, 2001). Environmental pollutants and other xenobiotics induce the members of this enzyme system. The metabolism of esters by this enzyme system is of interest in the development of the ester prodrug. The ester prodrug is therapeutically inactive until it is hydrolyzed to its therapeutically active metabolite(s) by the enzyme system in the body. CES also catalyzes the hydrolysis of several therapeutic agents to its inactive metabolites – among them are opiate analgesics, anticancer drugs, angiotensin-converting enzyme inhibitors, and antiplatelet drugs; it also catalyzes the hydrolysis of the illicit recreational drugs cocaine and heroin (Redinbo and Potter, 2005). The transesterifcation of cocaine by this enzyme system in the presence of alcohol generates cocaethylene, a toxic metabolite. The impact of inhibition or induction of CES enzymes on metabolism and pharmacokinetics of xenobiotics are yet to be clarifed. 9.3.1.10 Peptidase (Protease/Proteinase) The enzymes of this family hydrolyze the peptide bonds of proteins and peptides. Based on the target link of amino acid in a protein or peptide, subfamilies of this enzyme system have been categorized as aspartate peptidases/proteases, cysteine peptidases/proteases, glutamic acid peptidases/proteases, metalloproteases/metallopeptidases, serine peptidases/proteases, and threonine 247

9.3 METABOLIC PATHWAYS

peptidases/proteases. Each subfamily has its own members, and each member has its own isoforms. For example, a member of serine peptidase is dipeptidyl peptidase-4 (DPP-IV) (Mentlein et al., 1993), DPP-7, DPP-8, DPP-9, etc. Dipeptidyl peptidase-4 (DPP4), also known as adenosine deaminase complexing protein 2 or CD26 (cluster of differentiation 26) is an antigenic enzyme expressed on the surface of most cell types and is associated with immune regulation, signal transduction, and apoptosis. The substrates of CD26/DPPIV are proline (or alanine)-containing peptides and include growth factors, chemokines, neuropeptides, vasoactive peptides, and gastric inhibitory peptide/glucose-dependent insulinotropic peptide (GIP) with terminal Tyr-Ala, as well as gluca gon-like peptide-1amide/insulinotropin [GLP-1(7–36) amide] and peptide histidine methionine (PHM) with terminal His-Ala hydrolyzed by dipeptidyl-peptidase IV. Proteases like thrombin and plasmin are present in blood; acid proteases, such as pepsin, are present in the stomach; serine proteases like trypsin and chymotrypsin are present in small intestine; and elastase and cathepsin G are proteases in white blood cells (Demuth, 1990; Hedstrom, 2002; Barrett et al., 2003). The peptidases/proteases hinder the delivery of therapeutic peptides or proteins. An example of this hindrance is the enzymatic barrier of nasal epithelium for nasal delivery of peptide drugs. 9.3.2 Phase II Metabolism: Conjugation All xenobiotics and their metabolites with appropriate functional groups can undergo metabolic conjugation in the body by interaction with charged ligand such as glucuronic acid, glycine, sulfate, or glutathione. The conjugated compounds are less reactive, more soluble than the parent compounds and eliminate easier from the body. The enzymes of Phase II metabolism, which catalyze the conjugation reactions, belong to a family of transferases with broad specifcity. The conjugation reactions are the inactivation processes or detoxifcation pathways. However, conjugation may not always result in detoxifcation or full inactivation of active substrates. The conjugation reactions, the related enzymes, and the appropriate functional groups of reactive compounds are 1. Glucuronidation: Catalyzed by uridine 5′-diphospho-glucuronosyltransferase (UDPglucuronosyltransferase) in endoplasmic reticulum. The required functional groups of substrates to interact with the glucuronic acid are −OH, −NH2, −COOH, −SH, =NH, and ≡CH. 2. Sulfation: Catalyzed by sulfotransferase in cytosol. The required functional groups are −NH2 and −OH. 3. Methylation: Catalyzed by methyltransferase in cytosol and endoplasmic reticulum. The required functional groups are −OH, −NH2, and −SH. 4. Acetylation: Catalyzed by acetyltransferase in cytosol. The required functional groups are − NH2 and −OH. 5. Glutathione conjugation: Catalyzed by glutathione S-transferase in cytosol and endoplasmic reticulum. The required functional groups are epoxide and organic halide. 6. Amino acid conjugation (glycine, glutamine, arginine, or taurine conjugate): Catalyzed by acyl-CoA (amino acid N-acyltransferase) in mitochondria. The required functional group is −COOH. 9.3.2.1 Glucuronidation Glucuronide conjugation is one of the most important and common detoxifcation pathways in the mammalian system (Kaivosaari et al., 2011). The body relies heavily on this pathway for the elimination of a signifcant number of exogenous and endogenous compounds and their metabolites (Kroemer and Klotz, 1992). Xenobiotics that eliminate by glucuronidation include nonsteroidal anti-infammatory compounds, sedatives, analgesics, and antidepressants (Baron and Sandler, 2000; Kuehl et al., 2005; Smith, 2009). Examples of endogenous substrates are thyroxin, steroids, bilirubin, catecholamines, vitamins, and bile acids. Interspecies difference in hepatic glucuronidation is rather signifcant. The glucuronide conjugates are formed in two rather simultaneously occurring steps of activation followed by transfer reaction. The activation step is the formation of glucuronic acid by binding α-D-glucose 1-phosphate to UTP in the presence of pyrophosphorylase to produce uridin-5′-diphopho-α-D-glucose (UDPG). Through the process of dehydrogenation in the presence of UDPG-dehydrogenase, UDPG is then changed to uridine-5′-diphospho-α-D-glucuronic acid (UDPGA), which is the donor of glucuronic acid for interaction with reactive metabolites or compounds. The transfer reaction, 248

PK/TK CONSIDERATIONS OF XENOBIOTICS BIOTRANSFORMATION

catalyzed by UDP-glucuronyl transferase, forms glucuronide conjugate and UDP is released, as summarized below: Activation: Pyrophodphorylase Glucose1 - Phosphate + UTP ¾¾¾¾¾¾¾¾¾¾ ® UDPG + PP

(9.7)

UDP - glucuronyltranferase UDPG + 2NAD + + H2O ¾¾¾¾¾¾¾¾¾¾¾¾¾® UDPGA + 2NADH 2 + 2H+

(9.8)

Transfer Reaction: UDP - glucuronyltransferase UDPGA + Reactive Xenobiotic ( RX ) ¾¾¾¾¾ ¾¾¾¾¾¾¾¾¾¾ ®RX - glucuronic acid + UDP (9.9) Depending on the functional group of a reactive xenobiotic or endobiotic, the types of glucuronide conjugation can be divided into the following categories: 9.3.2.1.1 O-Glucuronide Conjugation O-Glucuronides are formed when UDPGA interacts with primary, secondary, and tertiary alcohols, phenols, and carboxylic acid-containing molecules. Carboxylic acid-containing molecules form “ester,” whereas alcohols and phenols form “ether.” O-Glucuronides eliminate from the body through biliary excretion into the small intestine. Although it is generally accepted that glucuronide metabolites are pharmacologically inactive, for compounds like morphine, the formation of glucuronide metabolites, namely morphine-6-glucuronide and morphine-3-glucuronide, have signifcant pharmacological activities with interactive ability to bind to opioid receptors (Milne et al., 1996). O-Glucuronidation is an important elimination pathway for many xenobiotics, including morphine, naphthol, chloramphenicol, acetaminophen, codeine, hexobarbital, salicylic acid, bilirubin, diclofenac, naproxen, valproic acid, etc. The β-glucuronidase present in small intestine may hydrolyze the glucuronide conjugates transferred by the bile and release their primary xenobiotics or their metabolites. The released xenobiotic may then reabsorb and enter into the systemic circulation or form a chemically modifed compound in the alkaline environment of the intestine. There are marked species regioselectivity differences in O-glucuronidation. This selectivity is due to the genetic differences between humans and experimental animals, namely human UDPglucuronosyltransferase (UGT) genes and the related subfamilies. 9.3.2.1.2 N-Glucuronide Conjugation The substrates of N-glucuronidation that interact with UDPGA are alkylamines, arylamines, primary aromatic amines, tertiary aliphatic amines, hydroxylamines, amides, and various aromatic heterocyclic compounds. In general, the substrates can be classifed into compounds that form non-quaternary N-conjugates, such as sulfonamides, arylamines, and alicyclic, cyclic, and heterocyclic amines, and those that form the quaternary nitrogen conjugates, identifed as N+-glucuronides (Kaku et al., 2004). In humans, N+-glucuronidation plays a signifcant role in the metabolism of many aliphatic tertiary amines, such as tricyclic antipsychotics, antihistamines, and antidepressants. N-Glucuronides are generally safe, water-soluble metabolites but may contribute to the carcinogenicity of primary arylamines (Green and Tephly, 1998). The interspecies variability in N-glucuronidation is high, particularly for aliphatic tertiary amines and aromatic N-heterocyclic compounds. All common experimental animals (mouse, rat, guinea pig, rabbit, and dog) have N-glucuronidation pathways (Chiu and Huskey, 1998; Soars et al., 2001; Kaji and Kume, 2005; Xu et al., 2006; Shiratani et al., 2008). However, the variability among them stems from compound-dependency of their N-glucuronidation. For tertiary amines, most notably the tricyclic antidepressant and antihistamine drugs, N+-glucuronidation is commonly observed in chimpanzees, marmoset monkeys, humans, and rabbits, but the metabolic pathway for the compounds is generally absent in rats, mice, guinea pigs, and cynomolgus monkeys (Kojima et al., 2022). Another example is N-glucuronidation of molecules with the triazole ring, which occur at the same position of the ring in humans, rats, and dogs, but results in the generation of four different conjugates (Nakazawa et al., 2006). A better understanding of interspecies differences remains to be documented from the glucuronosyltransferase genes of animal species other than humans. 249

9.3 METABOLIC PATHWAYS

N-Glucuronidation represents a major elimination pathway for many xenobiotics, including aniline, N-hydroxyarylamines, benzidine, imipramine lamotrigine, olanzapine, carbamazepine, amitriptyline, retigabine, tioconazole, ketotifen, etc. (Zenser et al., 1998). Similar to O-glucuronides, N-glucuronides can breakdown by the intestinal β-glucuronidase and release their substrates in the intestinal environment. 9.3.2.1.3 S-Glucuronide Conjugation Thiol-containing xenobiotics, such as diethyldithiocarbamate, thiophenol, 2-mercaptothiazole, etc., also interact with UDPGA to form S-glucuronides (Buchheit et al., 2011). The formation of this conjugate is less frequent than O- and N-glucuronide conjugates and the documented information on species differences is not as comprehensive. Like the other glucuronides, the elimination of S-glucuronides is through both renal and biliary elimination and can be subjected to enterohepatic recirculation. 9.3.2.1.4 C-Glucuronide Conjugation This is a rare glucuronide conjugation pathway in humans. Xenobiotics with 1,3 dicarbonyl functional group, such as phenylbutazone and sulfnpyrazone, can interact with human UDPGA to form C-glucuronide conjugates (Kerdpin et al., 2006). Human liver and kidney microsomes form C-glucuronide of phenylbutazone. The in vivo study is also indicative of a signifcant elimination of phenylbutazone C-glucuronide in the urine. Because of the rarity of compounds, with dicarbonyl functional group at positions 1 and 3, only a few examples are known or documented in humans. In experimental animals, an acetylenic C-glucuronide of the sedative-hypnotic drug ethchlorvynol is identifed as a major metabolite in rabbit’s urine. 9.3.2.2 Sulfation Sulfate conjugation, like glucuronide conjugation, is a critical metabolic pathway for elimination of xenobiotics. Sulfation is a major conjugation pathway for phenols, alcohols, arylamines, N-hydroxy compounds, and, to a lesser extent, thiols; examples of endogenous substrates are steroids and carbohydrates. Sulfation occurs in vertebrates, invertebrates, fungi, and bacteria. Similar to glucuronide conjugations, the transfer of sulfate to a reactive substrate occurs in two steps of activation followed by transfer reaction. The activation involves the interaction of sulfate with ATP in the presence of ATP-sulfotransferase to form adenosine-5′-phosphosulfate (APS), which then interacts again with ATP in the presence of APS-phosphokinase to form 3′-phosphoadenosine-5′phosphosulfate (PAPS) (Klaassen and Boles, 1997). The transfer reaction involves the transfer of sulfate from PAPS to the molecule of substrate in the presence of sulfotransferase and the release of 3′-phosphoadenosine-5′-phosphate (PAP). The activation and transfer reactions are summarized below: Activation: ATP - sulfutransferase SO p ( APS ) + PPi 4 + ATP ¾¾¾¾¾¾¾¾¾¾¾®adenosine - 5¢ - phosphosulfate

(9.10)

APS - Phosphokinase APS + ATP ¾¾¾¾¾¾¾¾¾¾¾ ® 3¢ - phosphoadenosine - 5¢ - phosphosulfate ( PAPS ) + ADP (9.11) Transfer reaction: Sulfotransferase ù PAPS + Reactive Xenobiotic ( RX ) ¾¾¾¾¾¾¾¾® éRX - SO 3 úû + PAP êë

(9.12)

The enzyme “sulfotransferase” catalyzes the transfer of sulfonate (SO3- ) and not sulfate. The reaction is often called sulfonation rather than sulfation. In general, sulfation occurs less frequently than does glucuronidation mainly because of the lower concentration of PAPS at the cellular level (75 mM) as compared to UDPGA (350 mM), the availability of sulfate, and the presence of the two enzymes of activation. Exogenous or endogenous compounds with functional groups of −OH (phenols and alcohols), −NH2 (arylamines) or −NH − OH (N-hydroxy compounds) form sulfate conjugates. This N-sulfation or O-sulfation can occur for xenobiotics or endobiotics with an appropriate functional group. Sulfation of xenobiotics increases the hydrophilicity of molecules prior to biliary and urinary eliminations. Similar to the other Phase II metabolic pathways, sulfate conjugation is an inactivation or detoxifcation reaction of the parent compound and/or its metabolites. 250

PK/TK CONSIDERATIONS OF XENOBIOTICS BIOTRANSFORMATION

However, sulfation can also lead to the formation of biologically active compounds (Glatt et al., 1998; Funk et al., 2001), for example: ◾ Various environmental chemicals are metabolized to chemically reactive sulfuric acid esters, which may covalently bind to cellular macromolecules and induce mutations and tumors. ◾ Hepatic xenobiotic sulfation can lead to activation of prodrugs. ◾ Sulfate conjugates may inhibit salt effux pump (e.g., troglitazone), or form DNA adducts (e.g., tamoxifen sulfate), etc. It is worth noting that the formation of pharmacologically active conjugates, or any metabolites, with the same pharmacological or toxicological response as the parent compound should be taken into consideration for quantitation of pharmacodynamic/toxicodynamic parameters used in the estimation of PK/PD or TK/TD parameters. 9.3.2.3 Methylation In comparison to glucuronide and sulfate conjugations, methylation of xenobiotics is considered an important but minor pathway and most often generates compounds with less solubility and/ or more pharmacological activity (e.g., formation of epinephrine from norepinephrine), but less toxicity. The pathway is an important biochemical reaction for endogenous compounds. The endogenous substrates of the pathway include biogenic amines, histamine, serotonin, dopamine, and catecholamines. Methylation of the polar functional group of xenobiotics precludes the molecules from further conjugation by other pathways. The transfer of the methyl group to the appropriate functional group of acceptor molecules (e.g., phenols, catechols, amines, N-heterocyclic compounds, and sulfhydryl-containing compounds) and metals (such as mercury and arsenic that form mono- and di-methylated conjugates, and inorganic selenium that forms trimethylated conjugate) take place in two steps: activation (formation of coenzyme), followed by transfer reaction Activation (formation of coenzyme): L - methionineadenosyltransferase ¾¾¾¾¾¾¾¾¾¾® S - adenosylmethionine ( SAM ) + PPi ATP + L - Methionine ¾¾¾¾¾¾¾ Mg 2 + (9.13) Transfer reaction: methyl transferase Acceptor Molecule + SAM ¾¾¾¾¾¾¾¾¾® methylatedmo olecule + S - adenosylhomocysteine (9.14) There are different methyl transferases in the body, which facilitate the transfer of the methyl group from SAM to the acceptor molecules. Depending on the functional group(s) of the acceptor molecules, the related reactions are ◾ O-Methylation: • catechol O-methyltransferase (COMT) • phenol O-methyltransferase (POMT) ◾ N-Methylation: • phenylethanolamine N-methyltransferase (PNMT) • imidazole N-methyltransferase (histamine N-methyltransferase (HNMT) Nicotinamide N-methyltransferase • nonspecifc N-methyltransferase • guanidinoacetic acid N-methyltransferase • guanidinoacetic N- methyltransferase ◾ S-Methylation: • thiopurine S-methyltransferase (TPMT) 251

9.3 METABOLIC PATHWAYS

• thiol methyltransferase (TMT) Catechol O-methyltransferase (COMT) is present in the liver, kidney, skin, and nerve tissues; it is a cytosolic and microsomal enzyme, specifc for catechols (e.g., dopamine, norepinephrine, epinephrine, dopamine, L-DOPA, catechol estrogens, etc.), but not phenols. The transfer of the methyl group from SAM by this enzyme requires the presence of divalent ions (Mg2+, Co2+, Fe2+, Mn2+, etc.) (Männistöl et al., 1999). Phenol O-methyltransferase (POMT) is a microsomal enzyme that catalyzes the transfer of the methyl group from SAM to phenols. Phenylethanolamine N-methyltransferase (PNMT) is present in adrenal tissue and is responsible for the methylation of the neurotransmitter norepinephrine, forming epinephrine. It also methylates norephedrine and normetanephrine (Axelrod and Daly, 1968). Imidazole N-methyltransferase is located mainly in the liver, but it also presents in other tissues, including the brain. This enzyme catalyzes the methylation of histamine and is often called histamine N-methyltransferase (HNMT) (Weinshilboum et al., 1999). Nicotinamide N-methyltransferase (NNMT) methylates serotonin, tryptophan, and pyridine-containing compounds, such as nicotinamide and nicotine (Aksoy et al., 1994). Nonspecifc N-methyltransferase methylate the demethylated products of Phase I metabolism (Saavedra et al., 1973). Guanidinoacetic acid N-methyltransferase and phosphatidylethanolamine N-transferase are specifc for methylation of endogenous compounds, for example, methylation of guanidinoacetic acid and formation of creatine (Caldeira Araújo et al., 2005). Thiopurine S-methyltransferase (TPMT) catalyzes the S-methylation of aromatic and heterocyclic sulfhydryl compounds; thiol methyltransferase (TMT) is an important pathway in the metabolism of xenobiotics with the sulfhydryl group (e.g., captopril) (Wang et al., 2005). 9.3.2.4 Acetylation (Acylation) Xenobiotics containing an aromatic amine (R-NH2) or a hydrazine group (R-NH-NH2) converted to aromatic amides (R-NH-COCH3) and hydrazides (R-NH-NH-COCH3), and carboxylic acid and alcohol (e.g., choline), undergo acetylation to form amide before elimination from the body. The endogenous substrate example is serotonin. The general sequence of acetylation is also in two steps: formation of acetyl-coenzyme A (acetyl-CoA) followed by transfer of acetyl moiety to the appropriate functional group of xenobiotics in the presence of N-acetyl transferases. Formation of acetyl-CoA: R - COOH + ATP ¾¾ ®R - CO - AMP + PPi CoA - S - acetyl transferase R - CO - AMP + CoA - SH ¾¾¾¾¾¾¾¾¾¾¾¾¾ ®R - CO - S - CoA + AMP Transfer reaction: R - amine N - acetyl transferase RCO - S - CoA + R ’NH2 ¾¾¾¾¾¾¾¾¾¾¾¾¾¾®RCO ONHR ’+ CoA - SH

(9.15) (9.16)

(9.17)

N-Acetyl transferases are cytosolic enzymes present in the liver, mainly in Kupffer cells, and many other mammalian tissues such as spleen, lungs, and gut. In humans, similar to rats and hamsters, two distinct enzymes with different substrate specifcity are present: N-acetyl transferase 1 and N-acetyl transferase 2 (NAT-1 and NAT-2) (Levy et al., 1998). Mice express three types of NAT and dogs are unable to form acetyl conjugate. The most common NAT in humans is NAT-1, which is present in several tissues and organs, whereas NAT-2 is present in the liver and gut. Although the primary aliphatic amines are rarely substrates for N-acetylation, cysteine can form acetyl conjugates, which convert to mercapturic acids. Similar to methyl conjugation, N-acetylation forms less polar molecules with low water solubility than the parent compound. Acetyl conjugates do not undergo further conjugation by other Phase II pathways. Thus, the acetylated less-polar molecules remain longer in the body; it may reside in the kidney and cause renal toxicity (e.g., acetyl conjugate of sulfonamide) or increase the half-life of this conjugate in the body, thus potentiating the activity of the parent compound (e.g., acetylated procainamide). Furthermore, there are signifcant interindividual genetic, gender, and age differences (Meisel, 2002). The polymorphism of NAT-2 gene is the reason for the interindividual variability as slow acetylators versus fast acetylators (Blum et al., 1991; Levy et al., 1998). The slow acetylators are less effcient in metabolism and elimination of xenobiotics containing primary aromatic amine or hydrazine groups such as caffeine, isoniazid, sulfamethazine, procainamide, and hydralazine, as well as carcinogens such as benzidine and 2-aminofuorene (Ilett et al., 1987; Pawlik et al., 2002). 252

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9.3.2.5 Glutathione Conjugation Glutathione (GSH) is a tripeptide formed by the interaction of L-γ-glutamate (Glu) and cysteine (Cys) in the presence of enzyme gamma-glutamylcysteine synthetase, forming L-γ glutamylcysteine dipeptide, which then interacts with glycine (Gly) in the presence of glutathione synthetase to form glutathione or L-γ-glutamyl-L-cysteinylglycine. This tripeptide is present mainly in intracellular environments with its two characteristic features of having a sulfhydryl (SH) functional group and a γ-glutamyl linkage. The thiol group (SH) of cysteine serves as a proton donor and is responsible for the biological activity of glutathione. It exists in both reduced GS) and oxidized GSSG states and the intracellular ratio of GSH:GSSH should be in favor of GSH (greater than 90%) to maintain the cellular integrity and is often used as an indicator of cellular toxicity of oxidative stress. The endogenous substrates of the conjugation are the metabolites of arachidonic acid. The signifcant physiological role of glutathione in the body is the formation of glutathione conjugates of reactive metabolites and harmful electrophilic compounds, for example, free radicals and oxygen radicals such as peroxides (H2O2), superoxides O2− and hydroxyl radicals. The free radicals and reactive oxygen species have the propensity to cause cellular damage with consequent modifcation of cellular function. Glutathione conjugation protects the body and prevents the damage to important cellular macromolecules. It is the major endogenous antioxidant. In addition to detoxifying electrophilic carcinogens and antitumor agents, the conjugation inactivates endogenous epoxides, aldehydes, and quinines. The GSH transferases have other biological roles in the body such as DNA repair process, amino acid transport, and biosynthesis of leukotrienes, progesterone, testosterone, prostaglandins, and protein biosynthesis (Hayes et al., 2005). The formation of glutathione conjugates occurs in two steps: formation of co-substrate followed by transfer of glutathione. However, the glutathione S-conjugates undergo further metabolic reaction catalyzed by glutamyltranspeptidase (also known as γ-glutamyltransferase) to form Cys-Gly S-conjugates, which is a dipeptide, and free γ-glutamyl moiety. The newly formed dipeptide can undergo additional metabolism catalyzed by several dipeptidases and aminopeptidases to free glycine and form cysteine S-conjugates. N-acetylation of cysteine conjugates in the presence of N-acetyltransferase forms N-acetylcysteine, which is also known as mercapturic acid. The summary of the reactions is as follows: Formation of co-substrate: glutamylcysteine synthetase L ˜ ˙ glutamate ° cysteine ˛˛˛˛˛˛˛˛˛˛˛ ˛˛˛˛ ˝L ˜ ˙ glutamylcysteine glutathionesynthetase L ˜ ˝ glutamylcysteinedipeptide ° glycine ˛˛˛˛˛˛˛˛˛˛˛˙ L ˜ ˝ ˜ glutamyl ˜ L ˜ cysteinylglycine ˆ glutathione ˇ

(9.18)

(9.19)

Transfer reaction: glutathioneS ° transferases Glutathion ˜ electrophiliccompound ˛˛˛˛˛˛˛˛˛˛˛˛˛ ˝glutathioneconjugate ˝

(9.20)

Further metabolism: 1. Formation Cys-Glyc dipeptide: glutamyltranspeptidase GlutathioneS ˜ conjugates ° H2O ˛˛˛˛˛˛˛˛˛ ˛˛˛˛ ˝ Cys ˜ Gly S ˜ conjugates ° glutamate (9.21) 2. Formation of cysteine conjugates: dipepdiases or aminopeptidases Cys ˜ Gly S ˜ conjugates °°°°°°°°°°° ° °°°° ˛glycine ˝ Cys S ˜ conjugates

(9.22)

3. Formation of N-acetylcysteine (mercapturic acid) conjugates: N ˜ acetyltransferase CysS ˜ conjugates ° acetylecoenzyme A ˛˛˛˛˛˛˛ ˛ ˛˛˛˝ N ˜ acetylcysteine ˙ mercapturic acid ˆ (9.23) Approximately 90% of GSH is synthesized in the cytosol of all mammalian cells, 10% in the mitochondria and a negligible amount in the endoplasmic reticulum. The conjugates of GSH forms in the cells and then transports out of the cells. In the liver, which is the major site of glutathione 253

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conjugation, the conjugates exit the hepatocytes and enter the bile. The metabolism of GSH conjugates to cysteine and mercapturic acid conjugates begins mainly in the kidney, followed by biliary system, and small intestine. The GSH conjugates are mainly in the bile, and mercapturic acid conjugates are predominantly in the urine. Although the major role of glutathione conjugates is detoxifcation of harmful molecules, occasionally the conjugate itself plays an important role in the formation of cytotoxic, genotoxic, or mutagenic metabolites (Koob and Dekant, 1991). For example, the glutathione conjugate of p-aminophenol is nephrotoxic (Klos et al., 1992); the polyphenolic-glutathione (GSH) conjugates retain the electrophilic and redox properties of the parent polyphenols and contributes to their nephrotoxicity, nephrocarcinogenicity, and neurotoxicity (Monks et al., 1998); or GSH conjugate of dibromoethane is a mutagen (van Bladeren et al., 1980). There are more than 20 different human glutathione S-transferases, some of which are in the cytosolic fraction and some in the microsomal fraction. The GSH-transferases in the cytosolic fractions are more specifc toward xenobiotics and those in the microsomal fraction are more involved in the metabolism of endobiotics. There are marked species differences in the activities of GSH transferases for a given compound. In humans there are seven subfamilies of cytosolic GSH-transferases: ◾ alpha (GSTA1 and 2) ⇒ α GSTA1 and α GSTA1 ◾ mu (GSTM1 through 5) ⇒ μ GSTM1, μ GSTM2, μ GSTM3, μ GSTM4, μ GSTM5 ◾ omega (GSTO1) ⇒ ω GSTO1 ◾ pi (GSTP1) ⇒ π GSTP1 ◾ sigma (GSTS1) ⇒ σ GSTS1 ◾ theta (GSTT1 and 2) ⇒ θ GSTT1, θ GSTT2 ◾ zeta (GSTZ1) ⇒ ζ GSTZ1 Population studies have shown the subfamilies are polymorphic (Long et al., 2006). The Null activity in the GSTT1 gene is associated with adverse side effects and toxicity in cancer chemotherapy with cytostatic drugs, and so on. 9.3.2.6 Amino Acid Conjugation Amino acid conjugation is a relatively minor and unique pathway of Phase II metabolism. In humans, it involves the initial formation of a xenobiotic acyl-CoA thioester, which then conjugates with amino acids glycine, glutamine, glutamate, arginine, or taurine, glycine being the principle amino acid (Hutt and Caldwell, 1990; Shirley et al., 1994). In rats and rabbits, glutamine plays the principal role. Thus, the essential functional group of xenobiotics for reaction is the carboxylic acid moiety, and the conjugation reaction is a form of N-acylation. Briefy, the carboxylic acid moiety interacts with ATP generating an acyl adenylate and pyrophosphate, which acyl adenylate interacts with coenzyme A forming xenobiotic-CoA thioester followed by catalytic transfer of the activated acyl group to the amino group of the acceptor amino acid in the presence acylCoAamino acid-N-acyltransferase. The summary is as follows: Activation reaction: R - COOH + ATP ¾¾ ®R - CO - AMP + PPi ATP - dependent acid CoA synthetase R - CO - AMP + CoASH ¾¾¾¾¾¾¾¾¾¾¾¾¾¾ ¾¾¾¾¾ ®R - CO - S - CoA + AMP Transfer reaction:

(9.24) (9.25)

acylCoA - amino acid - Nacyltransferase R - CO - S - CoA + R¢ - NH2 ¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾ ¾ ®R - CO - NH - R¢ + CoASH (9.26) Amino acid conjugation, similar to the other pathways of Phase II metabolism, increases the solubility of xenobiotics (e.g., herbicides, benzoates, salicylic acid, valproic acid, etc.), and/or their metabolites, and facilitates their biliary and urinary elimination. The endobiotic such as bile acids or branched chain fatty acids also form conjugates with amino acids.

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9.3.3 In Vitro Systems for Xenobiotics Metabolism Study The in vitro qualitative and quantitative screening methods of xenobiotics metabolism have broadened in recent years. The qualitative evaluations are to identify the metabolic profle of a compound, its major metabolic reactions, important enzymes, cofactors involved, and possibly the complete picture of its biotransformation (Houston, 1994; Houston and Carlile, 1997; Houston et al., 2012). The in vitro quantitative evaluations are mainly for the purpose of in vivo extrapolation/ correlation in humans, which requires a proper experimental design using preferably human biotissues. In general, the in vitro metabolism data are essential in areas such as preclinical evaluation in drug discovery and development, environmental health sciences, cancer research, drug metabolism and toxicology studies, etc. A short list of applications of in vitro metabolism studies are: ◾ evaluation of metabolic and kinetic profles of xenobiotics ◾ selection of the appropriate animal models to study metabolism and toxicity of a compound ◾ determination of confdence limits for metabolic kinetic parameters ◾ development of high throughput assays ◾ determination of intrinsic clearance of xenobiotics ◾ verifcation of reaction phenotyping ◾ prediction of potential for drug–drug interactions ◾ investigation of the infuence of inducers or inhibitors on metabolism of xenobiotics ◾ identifcation of metabolic activation and deactivation ◾ formation of reactive and nonreactive metabolites ◾ investigation of interspecies differences of xenobiotic metabolism The in vitro systems used in drug metabolism studies are discussed in the following sections of 9.3.3.1–9.3.3.7. 9.3.3.1 Subcellular Fractions The subcellular fractions include ◾ tissue or organ homogenate ◾ homogenate without mitochondria also known as the S9 fraction ◾ cytosolic fraction, and microsomal fraction For routine in vitro metabolic studies, the subcellular fractions are from the liver. Subcellular fractions of other tissues/organs are also used to evaluate the in vitro extrahepatic metabolism and initial screening of xenobiotics (Dallner, 1978). Except for the homogenate, which is used infrequently due to the bulkiness of the fraction and presence of debris and coarse fragments, other fractions are commonly used for in vitro metabolism studies. 9.3.3.1.1 S9 Fraction The S9 fraction (Figure 9.2) contains the cytosolic and microsomal enzymes without the mitochondrial fraction and contains a wide range of metabolic enzymes such as CYP450 isozymes, glucuronosyltransferase, sulfatase, glutathione S transferase, acetyltransferases, methyl transferases, favin-containing monooxygenase, carboxylesterases, epoxide hydrolases, and other enzymes. Because of the presence of Phase I and Phase II enzymes, the liver S9 fraction is used for mutagenicity, toxicity, and metabolic screening and evaluation of xenobiotics. However, the presence of too many enzymes with low fractional presence and activity, may make some metabolites imperceptible particularly for the compounds that have extensive spectrum of metabolites e.g., benzo(a) pyrene. 9.3.3.1.2 Cytosolic Fraction The centrifugation of S9 fraction at 105000–110000gmax (Figure 9.2) yields the supernatant known as cytosolic fraction, which contains soluble enzymes of Phase II metabolism like sulfotransferases 255

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(SULT), glutathione S transferases (GST), N-acetyltransferases, carboxylesterase (NAT), epoxide hydrolase, alcohol dehydrogenase, xanthine oxidase, and diamine oxidase. The highest concentrations mostly belong to SULT, GST, and NAT, and all three enzymes require a cofactor for their activities, like PAPS for the SULT and GST, and acetyl coA for NAT. 9.3.3.1.3 Microsomal Fraction Further centrifugation of the cytosolic fraction’s pellet yields the microsomes (Figure 9.2), which are vesicles of hepatocytes endoplasmic reticulum and contain cytochrome P450 isozymes, epoxide hydrolase, glucuronosyltransferase, favin monooxygenase, and carboxy esterases. Microsomes are used more frequently than the other in vitro tools for general screening, metabolic profling, enzyme inhibition and induction studies, and interindividual variability of metabolic pathways. Microsomal fraction is also prepared from other organs and tissues with metabolic activities, e.g., lung, kidney, intestine, etc. Microsomal incubation of the same xenobiotic in different species helps in understanding interspecies differences in metabolism and calculations of their PK/TK parameters and constants. The microsomal incubation for CYP450 isozymes needs NADPH and for UGT evaluation, UDPGA, and alamethicin, which is a pore-forming peptide that is used to activate UGT. Several model substrates used to quantify the CYP isozymes activity are reported in Table 9.1 and some of their inhibitors and inducers in Table 9.2 (Birkett et al., 1993; Emary et al., 1998; Crommentuyn et al., 1998; Tucker et al., 2001): In general, the microsomal fraction under a precise experimental design can be used qualitatively and affordably to evaluate metabolic stability, reaction phenotyping, species differences/

Figure 9.2 Chart of sequential preparation procedure for the isolation of subcellular fractions of S9, cytosol and microsomes from liver homogenate by differential centrifugation; a common homogenizing medium consists of isotonic sucrose containing 0.05M Tris-HCl, pH 7.5, 0.005M MgCl, 0.025 M KCl, and the addition of 0.008 M CaCl2 is often optional; all operations should be carried out between 0–4°C; frequent freezing and thawing reduce the activity of the enzymes signifcantly. 256

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Table 9.1 Model Substrate to Quantify the CYP Isozymes Activity Isozyme CYP1A1 CYP1A2 CYP2A6 CYO2B1/2 CYP2B6 CYP2C8 CYP2C9 (Liver) CYP2C18/19 CYP2D6 (Liver) CYP2D6 (Intestine) CYP2D6 (Kidney) CYP2E1 (Liver) CYP3A4 (Liver) CYP3A4 (GI tract)

Substrate 7-ethoxyresorufn O-deethylation Phenacetin O-deethylation – or -Caffeine N3-demethylation Coumarin C7-hydroxylation Pentoxyresorufn O-dealkylation (S)-mephenytoin N-demethylation – or – Bupropion hydroxylation Paclitaxel C6-α-hydroxylation (S)-warfarin C6-, C7 hydroxylation (S)-mephenytoin C4’-hydroxylation Bufuralol C1’-hydroxylation Dextromethorphan O-demethylation Codeine O-demethylation Chlorzoxazone C6-hydroxylation Midazolam C1’-hydroxylation Testosterone C6-β-hydroxylation

similarities, interindividual variances, and determination of the in vitro intrinsic clearance and/ or extrahepatic metabolism of a compound for CYP and UGT enzymes. The metabolic data from microsomal incubation does not quantitatively represent the in vivo biotransformation in humans or experimental animals. The subcellular fractions can also be prepared from pretreated animals with specifc enzyme inducers. The induced fractions are often used as the positive control for metabolism of compounds given on a chronic basis to humans or experimental animals. 9.3.3.2 Cellular Fractions – Hepatocytes The parenchyma cells of the liver known as hepatocytes accounts for approximately 70–80% of the liver cytoplasmic mass. They are involved in several signifcant physiological processes, including biotransformation of xenobiotics and carbohydrates, protein synthesis and storage of phospholipids, cholesterol, and bile salts; and formation of bile. The multiplicity and complexity of their role in the body is unmatched (Arber et al., 1988). Hepatocytes, isolated from the liver, are prepared by infusing the liver with a medium that lacks calcium to separate the cells from each other followed by infusion of a proteolytic enzyme, such as collagenase, to digest the extracellular proteins (Berry et al., 1991). The procedure for the separation and culture of hepatocytes requires special care to prepare the cells without bacterial contamination and free from damaged cells. The isolation of hepatocytes either from the whole liver or segments of the liver is by using traditional collagenase perfusion for whole liver and its modifed procedure for the liver segment (Howard et al., 1967; Puviani et al., 1998). The hepatocytes of different species, including humans, can be isolated using fresh or as cryopreserved hepatocytes (Hengstler et al., 2000; Brown et al, 2007). Hepatocytes are effective tools for the Phase I and Phase II in vitro metabolic investigation of xenobiotics and represent the ideal in vitro system for studying xenobiotics (Worboys et al., 1996). They are whole cell models, expected to have the required cofactors, and under experimental methodology that

Table 9.2 Commonly Used CYP Isozyme Inhibitors and Inducers Isozyme

Inhibitor

CYP1A1 CYP1A2

α-Naphthofavone Furafylline

CYP2A6 CYP2C9 CYP2C18/19 CYP3A4

Sulfaphenazole Sulfaphenazole Ticlopidine Ketoconazole Grapefruit juice

Inducer Polycyclic hydrocarbon Smoking 3-Methyl cholanthrene Rifampicine Barbiturates Rifampicin Rifampicin Rifampicin Barbiturates

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maintains their viability and in vivo functionality, thereby generating metabolic data suitable for in vitro/in vivo correlation and extrapolation. The following are different approaches for in vitro metabolism study using hepatocytes: 9.3.3.2.1 Hepatocytes Suspension and 2D Cultured Hepatocytes Monolayer The isolated primary hepatocytes can be used as suspension of hepatocytes (Berry et al., 1992; Cross et al., 1995; Bayliss et al., 1999, Lee et al., 2020), which remain viable for a few hours, or used as a cultured hepatocytes monolayer (Chenery et al., 1987; Le Bigot et al., 1987) with maximum viability of one month. The cultured hepatocytes monolayer loses its viability for CYP specifc function over time. Various methodologies in culturing are used to prevent the gradual reduction of its specifc function. One approach is to create a double-layer collagen gel sandwich, which is used for metabolism and transporter-mediated investigations. In addition to freshly prepared hepatocytes, cryopreserved hepatocytes also retain the Phase I and Phase II metabolic enzymes and are used effectively for qualitative metabolic investigation. Like subcellular fractions, the interindividual variability is also associated with the use of hepatocytes for metabolism studies. 9.3.3.2.2 3D-Cultures of Hepatocytes To improve the longevity of hepatocytes and create an environment for the cells that mimics the in vivo situation, 3D platforms from biomaterials have been created to support larger number of hepatocytes. The 3D cultures include other hepatic cell lines like fbroblasts, Kupffer cells, vascular and biliary duct endothelial and other physiologically relevant healthy cells (Leite et al., 2011; Kostadinova et al., 2013). The presence of the other liver-related cells in the platform enables the hepatocytes to act more like functional tissues, with better morphological and functional behavior, by developing interaction with other cells and the matrix of the platform. The hepatocytes 3D models have become an important and powerful tool for in vitro drug metabolism and toxicity investigations. 9.3.3.3 Organ Fractions (Precision Cut Liver Slices) The use of liver slices in drug metabolism study is about a century old. Initially it was prepared manually with razor blades, which resulted in thickness inconsistency of the slices. The developments of precision-cut tissue slicers and associated new incubation methodology have revived the application of liver slices for in vitro drug metabolism (Ekins, 1996). The advantages of liver slices over other methodologies, for example, hepatocytes, are (1) maintaining the normal arrangement and structural design of hepatic cells, and (2) requiring no special treatment, such as digestive enzymes that are used in preparation of hepatocytes. The shortcomings of slices are 1) insuffcient diffusion of the medium during incubation/exposure period, 2) presence of splintered cells on the outer edges, and 3) a short period of viability (maximum 8 h). For several xenobiotics (e.g., tolbutamide, caffeine, 7-ethoxycoumarin, phenytoin, etc.) the estimated intrinsic clearance using precision-cut liver slices is consistent with the one calculated using hepatocytes. In addition, qualitative similarity in the production of metabolites of a few compounds has been shown between human liver slices and microsomes (Harris et al., 1994; Carlile et al., 1999). Cryopreservation of liver slices, due to the diversity of cell aggregates, is rather complicated and requires the optimization of concentration and composition of buffer and cryoprotectant in addition to the cooling and warming rates. It is important to note that the development of high-precision tissue slicers (e.g., Krumdiek tissue slicer, or Brendel-Vitron unit) allows equally sized slices of less than 250 m thickness to be generated. As a result, there has been a surge of interest in using liver slices in metabolism studies of Phase I and Phase II enzyme systems (Prins et al., 2021; Dewyse et al., 2021). 9.3.3.4 In-Situ and Ex-Vivo Liver Perfusion Techniques The isolated perfused liver is a methodology used frequently to estimate metabolic clearance and to determine metabolic profle and toxicity of xenobiotics. Its use for the physiological and biochemical evaluation of endogenous compounds dates to the beginning of the twentieth century. Other isolated organs are also employed for the evaluation of metabolic as well as physiological and toxicological studies. The term “perfused organ” is often referred to as an “ex vivo” method of study. The objective of using isolated perfused organs, such as the liver, is to mimic in vivo conditions in an ex vivo environment with all variables associated with the performance of the organ in vivo. There are a few requirements for the evaluation of an isolated organ: 258

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◾ The function should be specifc to the organ and not contingent on the performance of the other organs in the body. ◾ The study requires suitable circulatory fuid in an environment comparable to the in vivo condition. ◾ The viability of the organ during perfusion should be tested and maintained. The ex vivo technique of isolated liver perfusion to study the metabolism of xenobiotics is different from the in situ technique. The ex vivo technique requires complete vascular separation of the organ from the body. The organ is then maintained viable by an appropriate fuid mechanically circulated through its vascular network. It requires surgical removal of the liver and, in most cases, attachment to a special contraption to facilitate perfusion. The binding of the parent compound and/or its metabolites to the glassware, tubing, and perfusion apparatus should be prevented or corrected during the experiment. By means of the in situ technique, however, the organ is isolated but maintained within the in vivo environment. The change and infuence of the in vivo environment on the function and performance of the liver should be explored and tested during the investigation to evaluate whether the in situ system represents the function of the liver in isolation without the infuence of other organs. The main advantage of the in situ procedure over the ex vivo technique is minimizing the physical trauma to the organ during surgery and attachment to the apparatus. The ex vivo techniques have several advantages over other in vitro methods: ◾ The role of the liver in metabolism of a compound can be established more defnitely by the ex vivo technique. ◾ Unlike the other in vitro systems, the structural and functional integrity of the liver is sustained. ◾ Investigator controls the parameters of the perfusion, such as the concentration of xenobiotic or rate of perfusion, and thus infuence of various parameters can be evaluated. ◾ The biliary elimination of a compound can also be evaluated. ▪ The hepatic clearance estimated by this method represents the role of metabolic enzymes and transport proteins, including all enzymes of Phase I and Phase II metabolism, and multidrug resistance protein 1 and 2, breast cancer resistant protein, bile sites export protein, organic anion transporting polypeptides, organic cation transporters and monocarboxylic acid transporter 2, etc. The main limitations of isolated perfused liver in a drug metabolism study are maintaining the organ viability during study, which most often is short; setting up the system, which requires special apparatus; expertise in dealing with the rarity of human liver; and diffculty in duplicating in vivo conditions in an ex vivo setting. The kinetic analysis of the isolated perfused liver is based on the known rate of input into the liver that is fow rate of perfusate, Qperfusate , multiplied by the concentration of xenobiotic, Cin , and the measured rate of output that is the exit fow rate multiplied by the concentration:

( Rate )input = Qperfusate ´ Cin

(9.27)

( Rate )output = Qperfusate ´ Cout

(9.28)

( Rate )biliary = Qbile ´ Cbile

(9.29)

The biliary excretion rate is

The average biliary rate is the total amount excreted in the bile divided by the time of collection. The biliary excretion ratio or biliary extraction ratio is estimated as Ebiliary =

( Rate )biliary ( Rate )input

(9.30)

The estimated extraction ratio and the fraction that escapes metabolism are

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Extraction Ratio =

Rateof output Qperfusate ( Cin - Cout ) ( Cin - Cout ) = = Rateof input Qperfuste ( Cin ) ( Cin ) F = 1 - Extraction Ratio

(9.31) (9.32)

9.3.3.5 Antibodies Against CYP Proteins Among various approaches to determine the precise role of an isoform of CYP450 in the metabolism of a xenobiotic, monoclonal antibodies (MAbs) specifc for individual isoform are used more confdently (Gelboin, 1993; Gelboin et al., 1996; Gelboin and Krausz, 2006, Khayeka-Wandabwa et al., 2021). These antibodies are used to determine the protein content in subcellular fractions, the role of a specifc isoform by its inhibition of substrate-specifc metabolism, and the specifc enzyme involved in metabolism of a compound. Although signifcant numbers of monoclonal antibodies have been identifed toward human CYP450 isoforms and are commercially available, the list is far from completion. There are several issues in the use of MAbs: ◾ The inhibition of the isoform must be complete. Any partial inhibition would defeat the purpose of using MAbs. ◾ Considering the closely resembled structure of different members of a CYP450 subfamily, for example, CYP3A4 and CYP3A5, the inhibition must be specifc to an isoform only. ◾ The complex MAbs-CYP-substrate should not have any enzymatic activities. It should be noted that understanding the role of a CYP450 isoform in the metabolism of a xenobiotic by using MAbs is important in the understanding of genetic polymorphism, drug–drug interaction, toxicity, and carcinogenicity. 9.3.3.6 bDNA Probes The bDNA probes (branched DNA) are designed to identify mRNAs encoding CYP450 isoforms and several other xenobiotic-metabolizing enzyme systems (Hartley and Klaassen, 2000; Ogasawara et al., 2016). The probes are mostly used in conjunction with hepatocytes, which provide different expressions of mRNA in a single 60 mm dish. Each probe, usually consisting of short oligonucleotides, is designed to various chemically inducible CYP450 mRNA transcripts and can capture specifc mRNA. The commercially available probes give a chemiluminescent signal with intensity proportional to the level of mRNA, which are normalized with the levels of glyceraldehydes-3-phosphate dehydrogenase (GAPDH) mRNA. For example, the probes can measure the induction of CYP1A2 in omeprazole treated hepatocytes, which is normalized to GAPDH. The probes are often used as a valuable high-throughput assay for measuring enzyme induction of xenobiotics. 9.3.3.7 Pure and Recombinant Enzymes Relatively pure enzymes are used individually or in combination when specifc information on metabolism of xenobiotics is investigated. The incubation of a xenobiotic with an enzyme provides a more accurate estimation of its kinetic parameters by the specifc enzyme. Recombinant human enzymes, in particular the members of CYP450 families and subfamilies, are expressed in bacteria using E. coli and are commercially available as Bactosomes®. 9.3.3.8 Cell Lines A less common tool in xenobiotic metabolism studies is the use of the liver cell lines that are mostly isolated from liver parenchyma after chronic hepatitis, cirrhosis, or liver tumor. These cell lines are used mostly in cytotoxicity studies related to metabolism of xenobiotics and often have incomplete expressions of metabolic enzymes. Several currently available cells are: ◾ Hep G2 originated from hepatocellular carcinoma, its known active enzymes are CYP1A and CYP3A subfamily and UGT. ◾ Hep 3B also originated from hepatocellular carcinoma, its culture’s active enzyme is CYP1A1 subfamily.

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◾ HepaRGTM produce early hepatic progenitor cells as well as mature human hepatocytes. The differentiated HepaRGTM cells generate hepatocyte-like cells surrounded by a monolayer of biliary-like cells. ◾ HepatoPac® is a bioengineered in vitro co-culture of hepatocytes and 3T3 murine fibroblasts. The co-culture displays functional bile canaliculi and metabolizes xenobiotics using active Phase I and Phase II enzymes at higher levels than other in vitro models. ◾ HµRELTM provides a co-culture of primary hepatocytes and non-parenchymal stromal cells and is used in long-term cultures. ◾ BC2 originated from hepatoma with active enzymes of CYP1A1, CYP 1A2, CYP 2A6, CYP 2B6, CYP 2C9, CYP 2E1, CYP 3A4, DST, and UGT. ◾ C3A originated from hepatoblastoma with active CYP3A subfamily. ◾ PLC/PRF/5 created from hepatoma with active GST. ◾ SNU-182, SNU-398, SNU-449, and SNU-475, all originated from hepatocellular carcinoma, with various potency of Phase I and II metabolic enzymes. ◾ Transgenic cell lines represent the recombinant expression of the human enzymes in a cell line, and the expression provides a high level of an enzyme to be used in metabolism study. They are commercially available and often allow the study of a single enzyme, but they are expensive. 9.3.4  In Vivo Samples for Xenobiotic Metabolism Study The in vivo metabolism study requires use of biological specimens such as serum, plasma, urine, bile, and tissues. In human subjects the accessible and allowable samples are urine, blood, blood cells, plasma, serum, saliva, expired air, milk, biological secretions like sweat, and occasionally biopsy specimens. In experimental animals in addition to blood, plasma, serum, and urine, other biological samples like bile, tissue, and organ samples, including the whole organs, are used routinely in research. Adherence to all relevant ethical and safety regulations and guidelines for the collection and use of biological samples in humans and experimental animals are essential in all in vivo studies. According to the federal guidelines, all experimental protocols for collection of invasive or noninvasive samples for metabolism studies from humans require advance approval by the Institutional Review Board (IRB). Furthermore, protocols for collection of same biological samples in experimental animals, including cannulation of bile ducts, jugular vein, portal vein, or placing probes such as microdialysis probes in tissues/organs, require approval of the Institutional Animal Care and Use Committee (IACUC) in accordance with the federal guidelines. The in vivo samples appropriate for the metabolism study are: 9.3.4.1  Serum and Plasma Samples A serum sample is prepared by allowing the blood sample to clot naturally without adding any anticoagulants. Whereas a plasma sample is prepared by adding an anticoagulant followed by centrifugation to separate plasma from blood cells (red, white, and platelets), preferably at a low temperature (4°C). Serum samples require longer time to form, but plasma can be prepared after adding the anticoagulant followed by centrifugation. The centrifugation should be at an optimum speed that preserves the integrity of blood cells and prevents release of hemoglobin from red cells into plasma. Both samples should be prepared within 60 minutes after blood collection. The difference between serum and plasma is fibrinogen, which is removed by clotting in the serum and retained in plasma by an anticoagulant agent. There are, however, other minor differences between the two samples. For example, glucose concentrations are lower in plasma than in serum partly because of the addition of anticoagulants. Several anticoagulants are used for preparing plasma samples, among them are sodium citrate, potassium EDTA, or lithium heparin. For metabolism studies, lithium heparin is preferred because EDTA or citrate can interfere with metabolic profiling and analysis. It is highly recommended that after preparation of serum or plasma, aliquots of samples are frozen and stored at −80°C until analysis. This would stabilize, to some extent, the unstable metabolites and reduce the activity of enzymes present in the plasma or serum and repeated freezing and thawing of the same aliquots should also be avoided. Because of the dilution in systemic circulation, the concentration of the primary metabolites is usually low, and their detection would require sensitive detectors and analytical methodology. 261

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9.3.4.2 Urine Samples The convenience of urine collection, the absence of proteins that can interfere with the analytical methodologies, and a high concentration of eliminated water-soluble primary and secondary (Phase I and II) metabolites, make urine an important sample in drug metabolism and PK/TK studies. Urine can easily be sampled in a serial manner allowing the time course of the formation and excretion of the metabolites to be evaluated. The urinary data that is based on the measurements of the Phase I and Phase II metabolites and the unchanged parent compound provides a realistic and valuable estimation of the urinary excretion rate, cumulative amount excreted over the time of observation and the in vivo mass balance study. The PK/TK analysis of the data yields the estimates of excretion and metabolic rate constants and related clearances, with a better understanding of linearity or nonlinearity of metabolism in the body. The presence of conjugates in urine provides quantitative values of Phase II in vivo metabolism. The enzymatic breakdown of the conjugates in collected urine samples by incubation with enzymes like β-glucuronidase, or sulfatases can also provide valuable information on the primary metabolites that go through the conjugations as substrates of Phase II metabolism. There are, however, a few challenges associated with the use of urine for metabolic profling. These challenges include large intraindividual and interindividual variability in ionic strength, pH, and osmolarity that may infuence the results of analytical methodology. Microbial contamination may occur that introduces microbial metabolites in the sample with interfering in metabolic profle. Also, the presence of the other xenobiotics can infuence the formation or storage of the main metabolites. Being a noninvasive procedure, the urine collection is often used for infants and children of young ages or patients confned to bed using urine bags. The kinetic analysis of the urinary data is discussed in Chapter 10, Sections 10.6.1–10.6.4. 9.3.4.3 Bile Samples A major function of the liver is the production of bile with a rate of about 600–1200 mL/day. Bile is formed at canaliculi, which represent specialized extracellular spaces between hepatocytes. The canaliculi unite to form bile preductules that lead to bile ductules and fnally the hepatic duct. The hepatic duct is joined with the gallbladder duct to form the bile duct that connects through the sphincter of Oddi to the duodenum. Like urine, bile is a valuable biological sample for evaluation of metabolism and elimination of xenobiotics. Bile can be collected through bile duct cannulation, which is an invasive method and is used in humans only in certain disease states, patient condition permitting. Most bile cannulations for metabolism studies are done on experimental animals, different strains of rats or mice, under anesthesia. Briefy, the procedure involves making an inch-long incision on the surgically scrubbed abdomen of the animal. The bile duct located under the lobes of the liver is then isolated by removing the surrounding connective tissues. Two pieces of silk suture are then passed under the bile duct; the distal suture is tied to prevent the bile fow to the intestine, followed by making a midline cut on the bile duct and inserting a polyethylene catheter (PE 10) toward the liver. The proximal silk suture then secures the catheter (Boroujerdi et al., 1981). Most xenobiotics undergo Phase II metabolism before biliary elimination. Thus, for most compounds, the bile samples provide valuable data on conjugated metabolites. However, there are xenobiotics (e.g., anthracyclines) that the parent compound and metabolites of Phase I metabolism also excrete in the bile, enter in duodenum, and may reabsorb and enter in systemic circulation. This occurrence is known as enterohepatic recirculation. The enterohepatic recirculation often disguises the biliary elimination of xenobiotics and their metabolites. An experimental procedure to determine with certainty the presence of enterohepatic recirculation is the linked-animal model. In this procedure, the bile duct of one animal (donor) is cannulated and linked to the duodenum of a second bile-cannulated animal (receiver). The donor receives the dose of xenobiotic, and biological samples of both animals are then evaluated for the presence of the xenobiotic and its metabolites. The detection of the administered compound or its metabolite(s) in plasma, urine, or bile of the receiver, would indicate the presence of enterohepatic recirculation (Behnia and Boroujerdi, 1998). The rate of biliary excretion depends to a large degree on the polarity of the molecules and their molecular weight. The polar and ionizable molecules can be conjugates of Phase II metabolism of the parent compounds, and/or primary metabolites of Phase I metabolism. Thus, the analysis of bile samples may show the presence of the parent compound and its Phase I and II metabolites. 262

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Protein-bound molecules also undergo dissociation in the liver, followed by metabolism of unbound molecules and excretion in the bile. Several biological factors such as sex, age, fasting, and pregnancy infuence the biliary excretion of xenobiotics in humans. It is worth noting that although excreted parent compounds and/or their metabolites in the bile eventually leave the body through defecation, use of feces as a biological sample to estimate biliary elimination is neither accurate nor analytically practical. 9.3.4.4 Portal Vein Cannulation The portal vein is one of the blood vessels that bring blood to the liver. It carries venous blood from the gut, spleen, pancreas, and gallbladder to the liver. By comparing the drug concentration of this vein before entering the liver with blood levels of systemic circulation, one can obtain important information about the absorption of drug and the liver frst-pass metabolism. 9.4 KINETICS OF IN VITRO METABOLISM At the molecular level, the basic interaction of a xenobiotic with an enzyme system is characterized by a two-step chemical catalysis (Equation 9.33). The frst step is the reversible formation of xenobiotic-enzyme complex, which is basically a fast, relatively weak hydrogen bonding, ionic or hydrophobic associations, and induce conformational change in the enzyme. The second step is the slower step of irreversible formation of metabolite from the enzyme–xenobiotic complex. The basics of kinetics of this catalysis, its relevance to estimation of intrinsic clearance and hepatic clearance are discussed in this section (Section 9.4). 9.4.1 Michaelis–Menten Kinetics Equation 9.33 is the interaction of one compound with one enzyme and does not represent the interaction of two compounds or more with one enzyme. Under controlled and optimized conditions, the in vitro data from subcellular fractions follow Equation 9.33. ¾ ¾¾ ® Enzyme - Xenobiotic Complex [ED D] Enzyme [E] + Xenobiotic [D] ¬ ¾ k1

k2

3 ¾k¾ ® Metabolite [ M ] + Enzyme [E]

(9.33)

where k1 and k2 are the rate constants of reversible reaction and k3 is the rate constant of irreversible formation of metabolite, which is also known as the catalytic rate constant. The following differential equations defne the kinetics of Equation 9.33: d [ D] = -k1 [E][D] + k 2 [E.D] dt

(9.34)

d [ E] = -k1 [E][D] + ( k 2 + k 3 ) [E.D] dt

(9.35)

d [E.D] = k1 [E][D] - ( k 2 + k 3 ) [E.D] dt

(9.36)

d[M] = k 3 [E.D] = v dt

(9.37)

Depending on the type of metabolic system under consideration, equilibrium or steady state, the rate of metabolism or the turnover rate of enzyme-xenobiotic complex to metabolite, as defned by Equation 9.37, can be considered either under assumptions of equilibrium or a pseudo steady-state condition. Under the assumptions of equilibrium K eq =

[E.D] = k1 [E][D] k2

(9.38)

where K eq is the equilibrium constant, [E.D] is concentration of enzyme-xenobiotic complex, and

[E] and [D]are concentrations of enzyme and xenobiotic, respectively. \ [E.D] = K eq [E][D]

(9.39) 263

9.4 KINETICS OF IN VITRO METABOLISM

The enzyme is a catalyst and its total concentration at any time during the experiment remains constant and is equal to the concentration of free and bound enzyme, that is

[E0 ]

= [E] + [E.D]

(9.40)

Therefore, the free concentration of the enzyme is:

[E] = [E0 ] - [E.D]

(9.41)

Substituting Equation 9.41 in Equation 9.39 yields

[E.D] = K eq [D]([E0 ] - [E.D])

(9.42)

ö æ 1 \ [E.D] çç + [D] ÷÷ = [D][E0 ] K eq ø è

(9.43)

[E.D] =

[E0 ][D] 1 + [ D] K

(9.44)

eq

According to Equation 9.33, the rate of the formation of metabolite, that is, the rate of metabolism is v = k 3 [E.D]

(9.45)

Substituting Equation 9.44 in Equation 9.45 yields the rate of metabolism as v=

k 3 [E0 ][D] 1 + [ D] K eq

(9.46)

Equation 9.46 leads to the traditional Michaelis–Menten equation, which is named after Leonor Michaelis and Maud Leonora Menten. The equilibrium assumptions offer k 3 [E0 ] as the maximum rate of metabolism known as Vmax (i.e., Vmax = k 3 [E0 ]) and 1 is known as the Michaelis–Menten K eq constant, K M (i.e., K M = 1 ). The fnal form of the equation after substituting K M and Vmax and setK eq ting the concentration of xenobiotic as CD is v=

Vmax ´ CD K M + CD

(9.47)

Under the steady-state condition, the rate of change of an enzyme–xenobiotic complex with respect to time is defned as d [E.D] = k1 [E][D] - k 2 [E.D] - k 3 [E.D] dt

(9.48)

d [E.D] is the rate of change of the enzyme-xenobiotic complex with respect to time, dt k1 [E][D] is the rate of formation of complex, k 2 [E.D], is the rate of breakdown of the complex to the enzyme [E] and xenobiotic [D], and k 3 [E.D] is the rate of conversion of the complex to metabolite and [E] . Under the steady-state condition the rate of formation of the complex is equal to the rate of its breakdown, plus its rate of conversion to metabolite, that is, where

k1 [E][D] = k 2 [E.D] + k 3 [E.D]

(9.49)

Therefore, as in any steady-state condition, the rate of the change of complex with respect to time (Equation 9.48) is equal to zero

Solving for [E.D] yields 264

d [E.D] =0 dt

(9.50)

\ k1 [E][D] - k2 [E.D] - k3 [E.D] = 0

(9.51)

PK/TK CONSIDERATIONS OF XENOBIOTICS BIOTRANSFORMATION

[E.D] =

k1 [E][D] k2 + k3

Substituting Equation 9.41 in Equation 9.52 and setting

(9.52)

k2 + k3 = K M yields k1

éE ù [D] - [E.D][D] 0û KM

[E.D] = ë

(9.53)

\ [E.D] ´ K M = éëE0 ùû ´ [D] - [E.D][D]

(9.54)

[E.D]( K M + [D]) = éëE0 ùû [D]

(9.55)

éE ù [ D] 0û M + [ D]

[ED] = Kë

(9.56)

According to Equation 9.33, the rate of metabolism is k13 [E.D], that is k13 multiplied by Equation 9.56 k éE ù [D] k13 [ED] = 13 ë 0 û K M + [ D]

(9.57)

where k13 [ED] = v, the rate of metabolism, and k13 éëE0 ùû = Vmax , the maximum rate of metabolism. V [D] Vmax ´ CD \ v = max = K M + [ D] K M + CD

(9.58)

Thus, the equation developed under the steady-state condition is the same Michaelis–Menten equation that was developed under the equilibrium assumption. The two approaches differ in the condition of the enzymatic reaction and the mathematical defnition of Michaelis–Menten constant. Accordingly, K M is not a rate constant, nor a dissociation constant. It is merely a constant of convenience and represents the concentration of drug at the half-maximum rate of metabolism, and has the units of concentration, that is Vmax Vmax ´ CD = 2 K M + CD \2CD = K M + CD

(9.59)

\K M = CD However, the magnitude of the Michaelis–Menten constant refects the affnity of an enzyme for its substrate. The larger the value of K M, the lower is the affnity. In other words, a small K M corresponds to tight binding between an enzyme and its substrate, and a large K M represents weak binding. The maximum rate of metabolism, (Vmax), has units of mass/time and it is a constant for the interaction of an enzyme with its substrate. It represents the rate when the binding sites of all molecules of an enzyme, that is, binding sites of [E0] are occupied by substrate molecules and are present as the enzyme–substrate complex, [ED]. The Michaelis–Menten equation defnes the relationship between the rate of metabolism and the concentration of xenobiotic. The equation predicts a rectangular hyperbolic curve that represents the capacity-limited nature of an enzymatic metabolism (Figure 9.3). The rate of metabolism asymptotically approaches Vmax as the xenobiotic concentration increases. The parameters K M and Vmax are useful in providing 1) estimates of intrinsic metabolic clearance, 2) quantitative comparison of different compounds as substrate for the same enzyme system, 3) comparison of different enzymes or isozymes from different organs or different species, 4) a determining factor in identifying linearity or nonlinearity of PK/TK metabolism of a xenobiotic. For example, when the concentration of a compound, CD, is much smaller than K M , which is the

265

9.4 KINETICS OF IN VITRO METABOLISM

Figure 9.3 A typical rectangular hyperbolic curve of the Michaelis–Menten equation; the early part of the curve where CD > K M, the curve reaches the plateau level of Vmax that corresponds to the maximum rate of metabolism and zero-order kinetics; the concentration at half-maximum rate corresponds to the Michaelis–Menten constant (K M); the segment of the curve highlighted by the shaded area (i.e., between the frst-order and the zero-order) follows the Michaelis–Menten equation. plasma concentration at half maximum, the rate of metabolism follows the frst-order kinetics and linear metabolic kinetics, that is K M ˜ CD v= The quotient of

Vmax CD = Clint ´ CD KM

(9.60)

Vmax in Equation 9.60 is the intrinsic metabolic clearance with units of volume per KM

time and it is considered a constant for a metabolic system. Vmax ( mass time )

K M ( mass / volume )

= Clint ( volume time )

(9.61)

The dosing regimen of most therapeutic xenobiotics provides concentrations that are lower than the K M value, which is an assumption of linear pharmacokinetic modeling. When the concentration of xenobiotic is much greater than the K M value, the rate follows zeroorder kinetics, that is, K M ˜ CD

(9.62)

v = Vmax The values of K M and Vmax for a xenobiotic and an enzyme system are determined by measuring the rate of metabolism at different concentrations of a xenobiotic and measuring the change in the parent compound concentration, metabolite(s) concentration, and time-dependent viability of one of the in vitro systems discussed in this chapter. 266

PK/TK CONSIDERATIONS OF XENOBIOTICS BIOTRANSFORMATION

9.4.2 In Vitro Intrinsic Metabolic Clearance The intrinsic clearance (Equation 9.61) estimated by an in vitro system like liver microsome is the measure of catalytic activity of an enzyme toward a compound and metabolic stability of the compound. It is independent of all physiological factors that may infuence the estimation of clearance in vivo, such as blood fow, binding to blood cells and/or proteins, etc. The intrinsic clearance (Clint) is also a constant of proportionality between the rate of metabolism and concentration of the free xenobiotic and remains constant if the metabolism of a compound follows frst-order kinetics (Equation 9.60). The free concentration of xenobiotic, CD , is a variable, and thus the rate of metabolism, v, is also a variable dependent on the free concentration. The ratio of the two interrelated variables yields the intrinsic clearance, which is a constant of proportionality with units of volume/time. For example, if it is calculated from the incubation of subcellular fractions, the units can be reported as volume/time/mg of protein in the fraction (e.g., L/min/mg of protein), or in the case of hepatocyte it would be reported as volume/time/million cells (e.g., L/min/# of cells). The correlation of in vitro intrinsic clearance with the in vivo metabolic clearance has remained a topic of interest for research. For most enzyme systems hitherto, there is no established scaling method for prediction of in vivo metabolic clearance from in vitro intrinsic clearance using in vitro human data or in vitro animal data. There have been various kinetic analyses related to the in vivo intrinsic clearance correlation with the hepatic clearance that is based on the well-stirred and parallel tube liver models. For example, the following relationships are based on the conversion of hepatic clearance from in vivo intravenous data to in vivo intrinsic clearance (Ito and Houston, 2005).

( Clint )invivo =

ClH f free æ Cl ö ´ ç1 - H ÷ Qliver ø RB/P è

(9.63)

is in vivo intrinsic clearance; ClH is the hepatic clearance; f free is the free fraction where ( Clint ) invivo of the compound in plasma; RB/P is the blood to plasma concentration ratio; and Qliver is the hepatic blood fow, which is 20.7 ml/min/kg (Yang et al., 2007, Brown et al., 2007). The assimilation of intrinsic clearance in hepatic clearance is discussed in Section 9.4.5. 9.4.3 The Catalytic Effciency and Turnover Number The ratio of the metabolic rate constant k 3 to K M at low concentrations of a compound, k 3 / K M , represents the catalytic strength or effciency of an enzyme. The ratio has units of 1/(time × concentration) or (time−1M−1) which corresponds to the units of a rate constant of a pseudo-secondorder reaction. The upper limit of the effciency can be in the range of 108 to 109 s−1 M−1. The turnover number is the ratio of the maximum rate of metabolism to the total amount of enzyme Turnover Number =

Vmax [E0 ]

(9.64)

The turnover number represents the number of molecules of parent compound converted to metabolite per units of time. 9.4.4 Estimation of the Michaelis–Menten Parameters Similar to protein-binding, there are various approaches for estimation of K M , Vmax , Clint , etc., that are based on changing Michaelis–Menten equation to a linear or any other defnable relationship. 9.4.4.1 Lineweaver–Burk Plot or Double Reciprocal Plot Taking the reciprocal of the Michaelis–Menten equation yields the straight line equation (Equation 9.65) (Lineweaver and Burk, 1934), where the dependent variable is the reciprocal of the metabolic 1 1 rate and the independent variable is the reciprocal of xenobiotic concentration, , with a slope v CD K 1 (Figure 9.4). of M and y-intercept of Vmax Vmax 1 KM 1 1 (9.65) = × + v Vmax CD Vmax

267

9.4 KINETICS OF IN VITRO METABOLISM

Figure 9.4 Graphical presentation of Lineweaver–Burk plot, or double reciprocal plot, shaped by plotting reciprocal of the metabolism rate versus reciprocal of xenobiotic concentration; the straight line offers a simpler approach in estimating Michaelis–Menten parameters of Vmax and K M from the slope and intercepts of the line; although the plot provides the best straight line under the assumption of ideal second-order kinetics, just by taking reciprocal of the values the random errors associated with the measurements become exaggerated for minor errors and minimized for large errors, which infuence the estimation of the Michaelis–Menten parameters even when using regression analysis. Although this method provides the best straight line, it suffers from the distortion of the errors associated with the measurements. The reciprocal of small errors appears as large deviations and the reciprocal of large errors appear as small deviations. Thus, the estimation of K M and Vmax from slope and y-intercept of the unweighted data is likely to contain signifcant errors. This method is used often to compare inhibition mechanisms. 9.4.4.2 Hanes–Woolfe Plot C Multiplying Equation 9.65 by the xenobiotic concentration, CD , yields Equation 9.66 with D as the v dependent variable and CD as the independent variable (Figure 9.5). CD K M C = + D v Vmax Vmax

(9.66)

The plot is known as the Hanes plot (Hanes, 1932), or the Hanes–Woolfe plot (Haldane, 1957) with 1 K the slope = and y-intercept = M Vmax Vmax The Vmax estimated from the slope of the line is more precise than the Lineweaver–Burk plot. However, the errors in measurements of metabolic rate (v) contribute to the uncertainty about the K precision of y-intercept M . Vmax 9.4.4.3 Eadie–Hofstee Plot Multiplying Equation 9.65 by (Vmax ´ v) provides the linear Equation 9.67, known as the Eadie– Hofstee plot (Eadie, 1942; Hofstee, 1959): 268

PK/TK CONSIDERATIONS OF XENOBIOTICS BIOTRANSFORMATION

Figure 9.5 Graphical presentation of Hanes–Woolfe plot, which is shaped by multiplying the double reciprocal equation by the concentration of xenobiotic and plotting CD/v versus CD; the line provides a better estimate of the Vmax, but the presence of reciprocal values of the rate of metabolism in the y-axis creates the same uncertainty as the double reciprocal plot. v = Vmax - K M

v CD

(9.67)

The dependent variable of the line is the rate of metabolism, v, and the independent variable is the v normalized rate of metabolism with respect to the concentration of substrate, with a slope of C D -K M and y-intercept of Vmax (Figure 9.6). The estimation of K M and Vmax by this method is straightforward and are obtained directly from the graph or the regression equation. However, the measurement of metabolic rate, v, a quantity that is subject to more random errors than the xenobiotic concentration, can infuence the values of the slope and y-intercept. 9.4.4.4 Direct Linear Plot In this method, the metabolic rate (v) is plotted on the y-axis and the corresponding concentration of xenobiotic, (CD ), is multiplied by –1 and plotted on the x-axis. The two points (i.e., +v and -CD ) are connected and extrapolated into a positive quadrant, that is, (+v ) and ( +CD ) (Henderson, 1992). Thus, for each pair of data a line is generated and extrapolated. The point of intersection of the lines in the positive quadrant will have the coordinates of K M and Vmax (Figure 9.7). If the lines form more than one point of intersection, the median of coordinates of intersections is selected for the estimation of the parameters of the Michaelis–Menten equation (Figure 9.8). In a direct plot, no transformation of v or CD is needed, and the estimation of K M and Vmax do not require regression analysis. However, the disadvantage of this method is that if the relationship between the rate of metabolism and the xenobiotic concentration departs from the simple Michaelis–Menten hyperbolic relationship, it would not be apparent in the direct plot. In general, the direct plot followed by the Hanes plot are preferred for estimation of Vmax and K M. 9.4.4.5 Hill Plot Often the relationship between the rate of metabolism (v) and the concentration of xenobiotic (CD ) is sigmoidal and does not exhibit the classical rectangular hyperbolic curve of the Michaelis– Menten kinetics. This may indicate that the enzyme responsible for the metabolism has more 269

9.4 KINETICS OF IN VITRO METABOLISM

Figure 9.6 Graphical presentation of Eadie–Hofstee equation that is formed by multiplying the double reciprocal equation by (Vmax x v) and plotting the rate of metabolism versus the normalized rate of metabolism with respect to the concentration of xenobiotic; the uncertainty associated with the random errors of the measurements continues to exit even in this approach. than one binding site for the compound and the binding of one molecule may facilitate (positive cooperativity) or delay (negative cooperativity) the binding of other molecules. Under this condition, the rate of metabolism is governed by the following relationship known as the Hill equation (Hill, 1910): Vmax ´ [CD ]

n

v=

(9.68)

K Hill + [CD ]

n

where K Hill and n are known as the Hill coeffcients. Cross multiplication of Equation 9.68, solving for K, and substituting back into the equation, and taking the logarithm of both sides yields the linear relationship of Equation 9.68. Vmax [CD ] = vK Hill + v [CD ] n

K Hill =

n

n [CD ] (Vmax - v )

v

æ ö v log ç ÷ = nlog CD - log K Hill V v è max ø

(9.69) (9.70) (9.71)

æ ö v A plot of log ç ÷ versus log CD yields a straight line with slope = n and the y-intercept = è Vmax - v ø log K Hill . Determination of K Hill and n requires a known value for Vmax. Setting v = Vmax 2 would æ ö v v equal to one and log ç ÷ equal to zero. Therefore, the zero of the y-axes corVmax - v è Vmax - v ø responds to the logarithm of the xenobiotic concentration, log CD , on the x-axis, that yields a rate of metabolism equal to Vmax 2 (Figure 9.9)

make

270

PK/TK CONSIDERATIONS OF XENOBIOTICS BIOTRANSFORMATION

Figure 9.7 Graphical presentation of Direct plot with one point of intersection; the plot depends on the measurements of the rate of metabolism and concentration of xenobiotic; the random errors, large or small, are embedded in the measurements, but no reciprocals are taken from the measurements; the plot does not require regression analysis and is based on the actual reading of the Michaelis–Menten parameters in the positive quadrant of the plot; in an ideal situation, one point of intersection appears that provides the measurements of the K M and Vmax from x- and y-axis, respectively. log

v =0 Vmax - v

(9.72)

nlog CD50 = log K Hill

(9.73)

n \K Hill = CD50

(9.74)

9.4.5 Assimilation of Intrinsic Clearance in Hepatic Clearance Using Liver Models The following are some of the theoretical models to delve into the complex function of the liver. Models 1,2,3,6, and 7 are those that have integrated the intrinsic clearance in the calculation of the hepatic clearance (i.e., organ clearance) and are discussed in this section. 1. well-stirred model, also known as venous equilibration model (Pang and Rowland, 1977) 2. parallel-tube model, also known as undistributed sinusoidal model (Pang and Rowland, 1977) 3. dispersion model (Roberts and Rowland, 1986) 4. tank-in-series model (Gray and Tam, 1987; Robert et al., 1988) 5. interconnected tube model (Anissimov et al., 1997, 1999) 6. physiological PK/TK Organ Model for the Liver (Sun et al., 2006; Pang et al., 2007) 7. zonal liver model (Abu-Zahra et al., 2000; Pang et al., 2007 271

9.4 KINETICS OF IN VITRO METABOLISM

Figure 9.8 Graphical presentation of Direct plot with more than one point of intersection in the positive quadrants; to determine the Michaelis–Menten parameters all the joints from plotting the direct measurements are read directly on the y- and x-axis; the fnal values of the parameters are determined from the median of all readings on the x- and y- coordinates. The assumptions of the models are ◾ The distribution of xenobiotics in the liver is perfusion-limited distribution (Chapter 8, Section 8.2.2.1). ◾ The diffusion of the compounds into the hepatocytes is instantaneous with no diffusional impediment. ◾ The enzyme interacts only with the free and unbound compound. ◾ The interaction of the enzyme with the compound is also instantaneous with no threshold. 9.4.5.1 The Well-Stirred Model (Venous Equilibration Model) This model is also discussed in more detail in Chapter 10. According to the well-stirred model, the hepatic clearance can be summarized as Equation 9.75 (Gillette, 1971; Rowland et al., 1973; Wilkinson Shand, 1975; Pang and Rowland, 1977; Benet, 2010): ClH =

QH ´ Clint ´ f u QH + Clint ´ f u

(9.75)

where ClH is the hepatic clearance, QH is hepatic blood fow, Clint is the intrinsic metabolic clearance, and f u is the free fraction of xenobiotic in the blood. As the names of the model imply, the assumptions are: 1) the liver acts as a single homogeneous and well-stirred organ, and 2) the concentration of the unbound compound in venous blood leaving the organ is in equilibrium with unbound concentration in the hepatic tissue. Equation 9.75 integrates the intrinsic clearance in the estimation of organ clearance using physiological parameters of blood fow and protein binding. The intrinsic clearance can also be used to estimate the extraction ratio of a compound by the liver:

272

PK/TK CONSIDERATIONS OF XENOBIOTICS BIOTRANSFORMATION

Figure 9.9 Graphical presentation of the Hill equation for plotting metabolic data when the rate of metabolism versus concentration is sigmoidal and the Michaelis–Menten equation is replaced by the Hill equation, identifed by Vmax and two Hill parameters of K Hill and n; the linear plot of the Hill equation is achieved when the logarithm of (v/(Vmax-v)) is plotted against the logarithm of the concentration of xenobiotics, the zero value of the y-axis corresponds to log CD50n where its inverse logarithm corresponds to K Hill; the slope of the line provides the estimate of n. ER =

f u ´ Clint Q + f u ´ Clint

(9.76)

9.4.5.2 The Parallel-Tube Model (Undistributed Sinusoidal Model) This model considers the liver a series of parallel tubes or sinusoids, with a concentration gradient present along the theoretical tubes with the irreversible removal of compound by the metabolic process. (Bass et al., 1976; Bass and Bracken, 1977, 1978; Pang and Rowland, 1977; Pang and Gillette, 1978; Bass, 1979; Bracken and Bass, 1979; Lau et al., 2002; Ito and Houston, 2004). The metabolic process occurs in the hepatocytes that encircle each tube, through which blood containing xenobiotic fows at a constant rate. The concentration of the compound along the tubes declines exponentially. The equation of the model that incorporates the intrinsic clearance is é f ´Clint ù ö æ -ê u ú (9.77) ClH = QH ç 1 - e ë QH û ÷ ÷ ç ø è Comparison of Equation (9.77) with Equation 9.78 identifes the hepatic extraction ratio in terms of intrinsic clearance (Equation 9.79):

ClH =

Q(Cin - Cout ) = Q ( ER ) Cin

(9.78)

é f ´Clint ù ö æ -ê u ú (9.79) ER = ç 1 - e ë QH û ÷ ç ÷ è ø 9.4.5.3 The Dispersion Model The dispersion model is a more complex model for defning the liver and integrating the intrinsic clearance with the hepatic clearance (also read Chapter 10). Its basic principles are from the

273

9.4 KINETICS OF IN VITRO METABOLISM

chemical engineering nonideal fow longitudinal dispersion model. There are two important parameters in this model: the effciency number and dispersion number (Wehner and Wilhelm, 1956; Roberts and Rowland, 1986; Roberts and Assimov, 1999). The effciency number (Rn) refects the irreversible removal of xenobiotic under frst-order condition from the blood by the liver Rn =

f uClint QH

(9.80)

When the number of compounds present in the system is more than one, for example the parent compound plus two primary metabolites, the equation changes to Rn =

( fu )i ( Clint )i ri QH

(9.81)

where (i) represents the ith compound andri is the transmembrane clearance of the ith compound (see also Chapter 10), that is, r=

Pcoeff Pcoef + Clint

(9.82)

There is a fow-independent parameter, representing the stochastic measure of dispersion for a xenobiotic during the time it passes through the liver, known as the axial dispersion number, Dn . According to this model, the dispersion in the liver is due to the nonideal blood fow through a maze of interconnected sinusoids, which causes variations in transit time of the drug through the liver. A common value of 0.17 has been recommended for this parameter. The formula for hepatic clearance according to the dispersion model is é ù 4a ú ClH = QH ê1 ê (1 + a )2 e ëé( a-1)/2Dn ûù - (1 - a )2 e ëé-( a -1)/2Dn ûù ú ë û a = (1 + 4RnDn )

1

(9.83)

(9.84)

2

When the axial dispersion number, Dn , approaches zero, Equation 9.83 changes to Equation 9.85, which is the same as Equation 9.77:

(

ClH = QH 1 - e -Rn

)

(9.85)

Alternatively, when the dispersion number approaches infnity (i.e., Dn → ∞), Equation 9.83 reduces to Equation 9.86 after substituting Equation 9.81 for Rn. Equation 9.86 defnes the hepatic clearance as ClH =

Rn 1 + Rn

(9.86)

It should be noted that no mathematical model could comprehensively delineate the physiological complexity of the liver and its numerous roles in the body. The above theoretical models may seem abstract, but they can approximate the hepatic clearance based on the accessible data. 9.4.5.4 Physiological PK/TK Organ Model for the Liver Based on the principles of physiological PK/TK modeling, such as blood fow, binding to plasma proteins and blood cells, and other important elements of the liver, such as bile fow and functions of enzymes and effux proteins; physiological organ models are proposed to refne the interrelationships between organ clearance and intrinsic clearance (Pang and Rowland, 1977; Sun et al., 2006; Pang et al., 2007) (Figure 9.10). A similar approach is also proposed for the kidney model (de Lannoy et al., 1990). The model depicted in Figure 9.10 is an example of an organ-related, physiologically based model for the liver that demonstrate the relationship between the organ clearance and intrinsic clearance of the liver (Pang et al., 2007):

( Clm )hepatic = 274

Qliver ´ f u ´ Clinflux ´ Clint Qliver ( Clefflux + Clint + Clsec ) + f u ´ Clinflux ( Clin + Clsec )

(9.87)

PK/TK CONSIDERATIONS OF XENOBIOTICS BIOTRANSFORMATION

Figure 9.10 Schematic of a liver model conceptualizing the interrelationship between hepatic clearance and intrinsic clearance; the model assumes that the free xenobiotic, identifed as A in capillaries and sinusoid, is in equilibrium with the bound blood cell (BC) and proteins; the blood fow rate distributes the bound and unbound compound to acinus and back to the reservoir; the free form of the compound is taken up by the infux clearance from the sinusoid into the hepatocytes to metabolize and form metabolite (M) and the free form is returned to the sinusoid by the action of effux proteins and to the bile by the secretory clearance that includes the intrinsic clearance; the biliary fow rate removes both the metabolite(s) and free form of the xenobiotic from the liver.

( Cl )biliary =

Qliver ´ f u ´ Clinflux ´ Clsec Qliver ( Clefflux + Clint + Cllsec ) + f u ´ Clinflux ( Clin + Clsec ) Qliver ´ f u ´ Clinflux ( Clint + Clsec ) liver ( Clefflu ux + Clint + Clsec ) + f u ´ Clinflux ( Clint + Clsec )

( Clorgan )liver = Q

(9.88) (9.89)

where Qliver is the blood fow through the liver with the unbound fraction of xenobiotic f u ; Clinflux and Clint are infux and intrinsic metabolic clearances, respectively; Clefflux and Clsec are the effux and intrinsic secretion clearances, respectively. Equation 9.89 also demonstrates that the total hepatic and the hepatic metaclearance that is, ( Clorgan ) is the summation of biliary elimination ( Cl ) bolic clearance ( Clm )

biliary

liver

hepatic

with intrinsic metabolic clearance incorporated in all clearance terms.

In the case of hepatocytes incubation, the assimilation of intrinsic metabolic clearance in hepatic clearance can be achieved using Equation 9.90 Clmedium = Clint

Clpassive + Clactive Clpassive + Clefflux + Clint

(9.90)

Furthermore, substituting Clmedium (Equation 9.90) as the intrinsic clearance into Equation 9.75 yields the equation of hepatic clearance based on the well-stirred model. ö æ Clpassive + Clactive QH ´ f u ´ Clint çç ÷÷ è Clpassive + Clefflux + Clint ø = QH ´ fu ´ Clmedium ClH = ö QH + fu ´ Clmedium æ Clpassive + Clactive QH + f u ´ Clint ç ç Clpassive + Clefflux + Clint ÷÷ è ø

(9.91)

275

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9.4.5.5 Zonal Liver Model As noted earlier, the liver has the most complex histochemical heterogeneity and physiological functions in the body. Its complex roles combined with the diversity of its transport proteins, effux proteins, and receptors and enzyme systems provide challenges in developing relevant theoretical and applicable models for hepatic metabolism of xenobiotics. An approach for refnement of the liver models is based on inclusion of metabolic zonation of the liver in the model (Knapp et al., 1988; Homma et al., 1997; Oinonen and Lindros, 1998; Tirona and Pang, 1999; AbuZahra et al., 2000; Pang et al., 2007). As shown in Figure 9.1, the hexagonal shaped liver lobule has a central vein and portal triad corners with a branch from the portal vein. The acinus extends from the portal triad toward the central veins. Three zones of metabolism are identifed as zones 1, 2, and 3 (Figure 9.1). Zone 1 is the periportal zone, zone 2 is the intermediary or midzonal zone, and zone 3 is the perivenous, pericentral, or centrilobular zone. Depending upon the involvement of the zones in the metabolism of a compound, each zone is included as sub-compartments of the liver model with their specifc intrinsic hepatic clearances (Figure 9.11). Contrary to well-stirred, parallel, and dispersion models, the zonal liver model is compound-specifc and is developed based on the compound’s intrinsic metabolism by different hepatic zones. Furthermore, the model takes into consideration the intrinsic infux and effux of the parent compounds and their metabolites by the transport and effux proteins of hepatocytes and sinusoid. The specifc information related to the chemical structure and characteristics of a xenobiotic in juxtaposition with specifc function of the relevant enzyme systems acting on the compound, transport proteins involved in specifc functions acting on the compound in a specifc zone are considered in this model when it is developed for a given compound. The zonal liver model, as indicated earlier, is a xenobioticspecifc approach. 9.4.6 Inhibition of Xenobiotic Metabolism The in vivo metabolic inhibition and/or induction of xenobiotics results in alteration of the overall disposition profle of a compound, which can potentially cause a life-threatening outcome. For therapeutic xenobiotics, the inhibition and/or induction of enzyme systems causes metabolic drug–drug interaction, which is the main reason for adverse effects or loss of drug effectiveness. Accurate and early forecast of metabolic drug–drug interaction is critical in drug discovery and development process, and the in vitro experimentations prior to clinical assessment are the fundamental part of the process. There are also other reasons for drug–drug interactions that include the infuence of one compound on the protein binding of the other, and/or modulation/inhibition of infux or effux proteins, such as P-glycoprotein (Pgp) , organic anion transporters (OAT), organic cation transporters (OCT), etc., or the competitive nature of binding to one binding site of a receptor by more than one compound. The in vitro methodologies discussed in this section for metabolic enzymes are also applicable to the interactions of xenobiotics with nonenzymatic proteins. The chronic exposure to certain environmental/industrial chemicals can also cause long-term induction or inhibition of enzyme systems and, in certain cases, can cause liver diseases that undermine the metabolic homeostasis of the liver and its role as a site for synthesis of proteins. The inhibition or induction of enzyme systems by chronic or acute exposure may infuence the PK/PD or TK/TD of prescribed therapeutic agents or other chemical exposures. There are examples of the inhibitors and inducers of enzyme systems in the environment that cause chronic low-dose exposure in air, food, and drinking water. These often get stored in various organs and tissues and exert their infuence in daily life by inhibiting or inducing or preoccupying the enzyme systems. Chemicals like trichloroethylene, used in products like wood stains, adhesives, and lubricants; 7, 12-dimethylbenz(a)anthracene of cigarette smoke and car exhaust; and 2,3,7,8-tetrachlorodibenzo-p-dioxin (TCDD), which is a by-product of herbicides, insecticides, and disinfectants and has a very slow rate of metabolism and excretion; and bisphenol-A that is widely used in polycarbonate plastics, epoxy resins, and dental sealant. Humans receive chronic exposure through interior coatings of food containers, baby formula bottles, and optical lenses; and fnally, β-naphthofavone, which is a chemopreventive agent but an inducer of enzyme systems like CYP450 and UGT; and many other examples. 9.4.6.1 Classifcations of Metabolic Inhibition Metabolic inhibition is categorized in competitive inhibition, simple noncompetitive inhibition, uncompetitive inhibition, mixed noncompetitive, and suicide inhibitions. Analysis shows that the 276

PK/TK CONSIDERATIONS OF XENOBIOTICS BIOTRANSFORMATION

Figure 9.11 Schematic of the zonal liver model with inclusion of the three zonal regions of 1, 2, and 3 (See Figure 9.1); the model is the broadened form of the one presented in Figure 9.10, with the exception of including the activities of the zonal regions emphasizing the diverse metabolic activities of each zone and differentiating the intrinsic metabolic clearance from the secretion, effux, and infux clearances in different hepatic zones; the model tracks the free form of the xenobiotic Areservoir to (A2)zone1, (A2)zone2, and (A2)zone3, followed by infux into hepatocytes identifed as (A3)z (1-3) and fnally A4 in the bile with metabolites Mz(1-3). inhibition of effux transporters, such as Pgp, follows the same approach as the enzyme inhibition. The inhibition of effux proteins and/or metabolic enzymes, as indicated earlier, can infuence disposition, effcacy, and safety of a xenobiotic. The inhibition can become harmful when the concentration of free compound of one xenobiotic unexpectedly reaches toxic levels, because of the change in enzyme levels. In certain cases, the inhibition is advantageous for example, when the inhibition of enzymes at the site of absorption, or inhibition of effux proteins increases the extent of absorption, or when the inhibition reverses the multidrug resistance in the body. 9.4.6.1.1 Competitive Inhibition In this scenario, the inhibitor (i.e., the second compound, which closely resembles to the chemical structure and molecular geometry of the frst compound) has a strong affnity for the same binding site of the frst compound, and if it were present in a concentration comparable to the frst compound, it would compete for binding to the same site. It is a competitive process between the 277

9.4 KINETICS OF IN VITRO METABOLISM

frst and second compounds. The enzyme can interact with only one compound, but not both at the same time, and none of the compounds can occupy all binding sites completely. Thus, during the metabolism there will be enzyme-compound and enzyme-inhibitor complexes. The related reaction is k1 3 ¾¾ ¾¾ ® Enzyme - compound [E.CD ] ¾k¾ Enzyme [E] + Compound [CD ] ¬ ® Metabolite [ M ] + Enzyme [E] k2

+ ¾¾ ¾¾ ® Enzyme - inhibitor [E.I ] Inhibitor [ I ] ¬ Ki

(9.92) According to the above reaction, the inhibitor binds only to an enzyme, not to the [E.CD] complex. Therefore, the total free enzyme at a given time is

[E] = [E0 ] - ([E.CD ] + [E.I ])

(9.93)

Applying the stepwise approach in the development of the Michaelis–Menten relationship (Equations 9.34 through 9.47), the rate of metabolism in the presence of inhibitor is v=

Vmax ´ CD K M ( 1 + I K i ) + CD

(9.94)

The double reciprocal (Lineweaver–Burk plot) of Equation 9.94 is 1 KM = v Vmax

æ I ç 1+ K i è

ö 1 1 + ÷ C V D max ø

(9.95)

1 1 According to Equation 9.95, a plot of versus yields a straight line with the following paramv C D eters (Figure 9.12): y - intercept = x - intercept =

1 Vmax

-1 æ æ I öö KM ç 1 + ç ÷ ÷ è è Ki ø ø

æ K öæ I ö Slope = ç M ÷ç 1 + ÷ Ki ø è Vmax øè

(9.96) (9.97)

(9.98)

Where I is the concentration of the inhibitor and K i is the dissociation constant of the enzymeinhibitor complex and is estimated as Ki =

[E][ I ] [E.I ]

(9.99)

In competitive inhibition, Vmax and y-intercept are constant for an enzyme, but the x-intercept is a variable and changes with the concentration of inhibitor. 9.4.6.1.2 Noncompetitive Inhibition Noncompetitive inhibition occurs when the second compound, the inhibitor, has an affnity for interaction with a different binding site on the same enzyme. Thus, there is no competition between the two compounds for interaction with the same binding site, and the inhibitor can interact not only with the enzyme [E] but also with the enzyme–drug complex [E.CD], that is,

278

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Figure 9.12 Graphical presentation of double reciprocal of competitive metabolic inhibition of one compound by a second one that chemically and structurally resembles the frst one and is present at a concentration that competes for the same binding site of the metabolic enzyme; the enzyme has only one binding site and can only bind one compound at a time and thus, in the enzyme–xenobiotic complex, one compound cannot interact with the second compound; in comparison to the double reciprocal of the Michaelis–Menten equation, it is worth noting that the parameters of the enzymatic interaction and inhibition in addition to Vmax and K M, include Ki and I, the dissociation constant of enzyme–xenobiotic complex, and the concentration of the inhibitor, respectively. ˛˛ ˛˛ ˝ ˜E.CD ° ˛k˛˝ ˜ M ° +˜E° ˜ E ° + ˜ CD ° ˙ k k1

3

2

ˆ

ˆ

˜I °

˜I °

ˇ˘ K i

(9.100)

ˇ ˘ Ki

˛ ˛˛ ˝ ˜E.CD .I ° ˜E.I ° ˆ ˜CD ° ˙ ˛ Where ˜E.CD .I ° is the concentration of the enzyme-substrate-inhibitor complex. The interaction of the inhibitor with the enzyme results in energy requiring a conformational change of the protein, which may infuence the binding of the compound to its binding site. The assumption here is that the affnity of the inhibitor to interact with an enzyme and enzyme-substrate complex is the same and the extent of inhibition is independent of xenobiotic concentration. The rate of metabolism with a noncompetitive inhibition is ˆVmax ° ˝1 ˛ ˝ I / K i ˙ ˙ ˘  CD  v˜ ˇ K M ˛ CD

(9.101)

The linear double reciprocal of Equation 9.101 is

279

9.4 KINETICS OF IN VITRO METABOLISM

1 KM = v Vmax The plot of

æ 1 I ö 1 + ç1 + ÷ K i ø CD Vmax è

æ I ö ç1 + ÷ Ki ø è

1 1 versus yields a straight line with the following parameters (Figure 9.13): v CD 1 æ I ö y - intercept = ÷ ç1 + Vmax è Ki ø slope =

KM Vmax

æ I ö ç1 + ÷ Ki ø è

x - intercept = -

1 KM

(9.102)

(9.103) (9.104) (9.105)

In noncompetitive inhibition, the value of K M remains constant; the inhibitor infuences k3, the metabolism slows down, and Vmax changes to

(Vmax )noncompetitive =

Vmax /(1 + I / K i )

(9.106)

9.4.6.1.3 Uncompetitive Inhibition The uncompetitive inhibition occurs when the inhibitor does not bind to the free enzyme directly but interacts with the enzyme after the formation of an enzyme–xenobiotic complex. In other words, due to the interaction of the compound with the enzyme and resulting conformational change of the protein, the binding site(s) for the inhibitor will be exposed and available for the

Figure 9.13 Graphical presentation of double reciprocal of noncompetitive metabolic inhibition, when the second compound interacts with a different binding site of the enzyme thus there is no competition to bind to the same binding site; this would indicate that the enzyme–xenobiotic complex can also interact with a second compound; the assumption of noncompetitive interaction is that the binding site of the enzyme for the second compound is accessible when the enzyme forms the complex with the frst compound, even though the interaction of the frst compound is energy requiring reaction and may cause conformational changes in the enzyme. 280

PK/TK CONSIDERATIONS OF XENOBIOTICS BIOTRANSFORMATION

binding of the inhibitor. The binding of the inhibitor yields an inactive [E.CD.I] complex and prevents the formation of a metabolite and further binding of substrate to the enzyme. ˛˛ ˛˛ ˝ ˜E.CD ° ˛k˛˝ ˜ M ° +˜E° ˜ E ° + ˜ CD ° ˙ k k1

3

2

ˆ

˜I °

(9.107)

ˇ ˘ Ki

˜E.CD .I ° The rate of metabolism and its double-reciprocal linear relationship are ˆVmax ° ˝1 ˛ ˝ I / K i ˙ ˙ ˘  CD  v˜ ˇ K M ° ˝ 1 ˛ ˝ I / K i ˙ ˙ ˛ CD

(9.108)

1 KM 1 1 ˛ I ˆ ˜ ° ˙1 ° ˘ v Vmax CD Vmax ˝ Ki ˇ

(9.109)

The parameters of the linear plot (Figure 9.14) are Slope = y ˜ intercept °

KM Vmax

1 Vmax

x ˜ intercept °

(9.110)

˝ I ˇ  ˆ1 ˛ Ki ˘ ˙

(9.111)

˜1 ˛ ˝ I ˙ / K i KM

(9.112)

In uncompetitive inhibition, both K M and Vmax are variables as a function of inhibitor concentration (Figure 9.14):

˜ K M °uncompetitive ˛ K M ˜1 ˝ ˙ I ˆ / Ki ° ˜Vmax °uncompetitive ˛ Vmax / ˜1 ˝ ˙ I ˆ

Ki

°

(9.113) (9.114)

9.4.6.1.4 Mixed Noncompetitive Inhibition The mixed noncompetitive inhibition is similar to the noncompetitive inhibition. The main difference is that the binding of inhibitor to the enzyme reduces the binding of the compound and vice versa. Thus, as shown in the reaction (Equation 9.115), the binding of the compound decreases when the inhibitor is present. ˛˛ ˛˛ ˝ ˜E.CD ° ˛k˛˝ ˜ M ° +˜E° ˜ E ° + ˜ CD ° ˙ KD

3

ˆ

ˆ

˜I °

˜I °

ˇ˘ K i

(9.115)

ˇ ˘ Ki

˛˛ ˛˛ ˝ ˜E.CD .I ° ˜E.I ° ˆ ˜CD ° ˙ KD

The rate of metabolism for this type of inhibition is Vmax ° CD ˝ I ˇ I ˇ ˝ KD ˆ 1 ˛  ˛ CD ˆ 1 ˛  Ki ˘ Ki ˘ ˙ ˙ The linear conversion of Equation 9.116 is v˜

(9.116)

281

9.4 KINETICS OF IN VITRO METABOLISM

Figure 9.14 Graphical presentation of double reciprocal of uncompetitive metabolic inhibition when the second compound does not interact with the free enzyme until the interaction of the frst compound causes the conformational change of the enzyme molecule and reveals the second compound’s binding site; thus, the second compound interacts only with the enzyme–xenobiotic complex; the interaction of the second compound prevents the formation of the metabolite; in this type of inhibition K M and Vmax values are dependent on the concentration of the second compound, i.e., the inhibitor. æ I ö 1 æ I ö 1+ ç1 + ÷ K i ÷ø Ki ø 1 Vmax çè 1 KD è = + v Vmax Slope CD y - intercept

(9.117)

When K i ¹ K i , K M and Vmax are defned a æ K ´I ö æ I ö K M = ç KD + D ÷ ÷ ¸ 1+ K i ø çè Ki ø è Vmax =

Vmax I 1+ Ki

(9.118) (9.119)

The quandary here is how to estimate the parameters explicitly from the double reciprocal plot (Figure 9.15). To overcome this dilemma, most often the slope and y-intercept of the line is determined at different concentrations of the inhibitor, and the values of slope and intercept are then plotted against inhibitor concentrations. The x-intercept of plot of slope values versus the concentrations of the inhibitor is K i and the plot x-intercept of y-intercept values versus the concentration of the inhibitor is K i . 9.4.6.1.5 Suicide Inhibition When an inhibitor with a structure similar to the substrate interacts with a functional group of the binding site covalently, that is, irreversibly, it is called suicide inhibition. This type of interaction 282

PK/TK CONSIDERATIONS OF XENOBIOTICS BIOTRANSFORMATION

Figure 9.15 Graphical presentation of double reciprocal of mixed noncompetitive metabolic inhibition, which mimics the noncompetitive inhibition except that the binding of the second compound (i.e., the inhibitor) reduces the binding of the frst compound and vice versa, thus the binding of either compound to the enzyme decreases when the other compound interacts with the enzyme; the parameters of the mixed noncompetitive inhibition in addition to K M and Vmax include two dissociation rate constants, one for the interaction of the enzyme with the inhibitor and the second one for the interaction of the inhibitor with the enzyme–frst compound complex. Worth noting is that in this scenario either compound behaves as an inhibitor. disables the enzyme from binding to the compound. Thus, the metabolism is reduced merely because of the reduction of free enzyme. The rate and double reciprocal equations and plots resemble the noncompetitive inhibition. 3 ˝˝ ˝˝ ˙ ˜E.CD ° ˝k˝ ˙ ˜ M ° ˛ ˜ E° ˜ E ° ˛ ˜ CD ° ˆ k2 k1

˛

(9.120)

irreversible SI ˝˝˝˝ ˙ E.S SI

Where SIis the suicide inhibitor and E.SI is the inactive enzyme. 9.4.6.1.6 Bimolecular Enzymes There are enzymes that can interact with two different compounds and form two different metabolites. The interaction can occur in an orderly manner or in a random mode. The orderly interaction is when the enzyme interacts with both compounds according to the following formula: ˝˝ ˝ ˙ ˜E.CDA .CDB ° ˆ ˝˝ ˝ ˙ ˜E.M A .MB ° ˜E° ˛ ˜CDA ° ˛ ˜CDB ° ˆ ˝ ˝ k k k1

k3

2

4

k 5 ˇ˘ k6

(9.121)

E ˛ M A ˛ MB

283

9.4 KINETICS OF IN VITRO METABOLISM

where CDA and M A are compound A and its metabolite, and CDB and MB are compound B and its metabolite, respectively. All steps of the interactions are reversible and sequential. This interaction is also known as a single displacement mechanism, both compounds should interact with the enzyme before any metabolites are released. The interaction can also occur as a double displacement reaction known as the “ping-pong mechanism” in which the interaction of the frst compound modifes the enzyme structurally; the modifed enzyme will then interact with the second compound ˝ ˝˝ ˙ ˜E.CDA ° ˆ ˝˝ ˝ ˙ ˇE .M A  ˆ ˝˝ ˙ E ˛ ˜ M A ° ˜E° ˛ ˜CDA ° ˆ ˝ ˝ ˝ ˘  ˝ k k k k1

k3

2

4

k5 6

˝˝ ˙ ˜ E ° ˛ ˜ MB ° ˝˝ ˙ ˇE .CDB  ˆ ˝˝˝ ˙ ˜E.MB ° ˆ ˝ ˝˝ E ˛ ˜CDB ° ˆ ˝ ˘  ˝ k8 k10 k12 k7



k9

(9.122)

k11

When the interaction is uncompetitive, the rate equation is v˜

Vmax °CDA ˛°CDB ˛ K A °CDB ˛ ˝ K B °CDA ˛ ˝ °CDA ˛°CDB ˛

(9.123)

When the interaction is a mixed one, the rate is v˜

Vmax °CDA ˛°CDB ˛ °CDA ˛°CDB ˛ ˝ K A °CDB ˛ ˝ KB °CDA ˛ ˝ K A KB

(9.124)

The single displacement kinetics can be differentiated from the ping-pong model by measuring the rate of the reaction at a different concentration of one compound while keeping the concentration of the second compound constant. Then, repeating the same experiment by changing the concentration of the second compound while keeping the concentration of the frst compound constant. For the single displacement (i.e., sequential interaction), the linear plot of the data generates a series of straight lines that interconnect to the left of y-axis (Figure 9.16). The linear equation for the plot of 1/v versus 1/CDA (i.e., when [CDA] is kept constant and [CDB] is changed) is as follows: 1 1 ˜ v Vmax

B ˙ A KA ° KM ° ˇˇ K M ˛CDB ˝ ˆ

Slope ˜

1 Vmax

˘˙ 1 1 °  ˇ ˇ ˛CDA ˝ Vmax ˆ B ˙ A KA ° KM ˇˇ K M ° ˛CDB ˝ ˆ

B ˘˘ ˙ KM ˇˇ 1 °  ˛CDB ˝   ˆ

˘  

(9.125

(9.126)

The coordinates of the point of interconnection of the lines on the left quadrant (quadrant II) is x˜° y˜

1 KA

A ˆ 1 ˛ KM ˙1 ° ˘ KA ˇ Vmax ˝

(9.127) (9.128)

1 When the binding of ˜CDB ° is independent of ˜CDA °, and the double reciprocal lines of versus v 1 meet on the x-axis with the x-intercept of ˜CDA ° 1 x˜° A (9.129) KM In random interaction, that is, random sequential, there is no preference for binding of either compound to the enzyme, and the interaction follows the scheme presented in Figure 9.17, as compared to ordered sequential. The rate and double reciprocal equations for the random sequential are similar to the ordered sequential, and it is diffcult to differentiate between the random and ordered sequential by using the linear plot.

284

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Figure 9.16 Graphical presentation of double reciprocal of two compounds (A and B) interaction with the same enzyme, bimolecular enzymes, forming two different metabolites in an orderly or random manner (Figure 9.17); the interaction of both molecules occur at the same time forming enzyme–compound A–compound B complex in a reversible manner, followed by formation of enzyme–metabolite A–metabolite B complex in a reversible manner, and eventually forming two different metabolites and the free enzyme; a different type of interaction known as the “pingpong mechanism” occurs when the interaction of the frst compound with enzyme forms the frst metabolite but modifes the confguration of the enzyme such that it can interact with the second compound; the graph in this fgure represents the interaction of both compounds and the rate equation is developed by keeping the concentration of compound A constant and concentration of compound B variable. 9.4.6.1.7 Product Inhibition The product inhibition occurs when the metabolite has a chemical structure similar to the parent compound and interacts with the enzyme at the binding site of the parent compound. Thus, the metabolite acts as an inhibitor and competes with the parent compound for binding to the free enzyme. ¾¾ ¾ ® [E.CD ] ¾¾ ® [ M ] + [ E] [ E ] + [ CD ] ¬ ¾ + ¾¾ ® [E.M ] ¾ [ M]¬ ¾

(9.130)

At low concentrations of substrate, the interaction resembles the competitive inhibition. However, as the concentration of the parent compound increases, more metabolite is formed, and the rate of metabolism declines, and the double reciprocal plot loses its linearity. 9.4.7 Induction of Xenobiotic Metabolism The induction of enzyme systems by xenobiotics can occur through two general mechanisms: stabilization of the enzyme or mRNA, and/or augmented gene transcription. The induction increases the amount of enzyme, elevates its metabolic activity, amplifes its intrinsic metabolic clearance, 285

9.5 APPLICATIONS AND CASE STUDIES

Figure 9.17 Depiction of random-sequential and ordered-sequential for a bimolecular interaction where in random sequential the enzyme interacts with either compound randomly without any preference; most often it is diffcult to differentiate between a random and an orderly sequential; thus, the double reciprocal equations and plot presented in Figure 9.16 are considered applicable to the random sequential. and ultimately enhances the total body clearance of its substrates. The induction can have a profound effect on the PK/TK parameters and constants of xenobiotics both in vitro and in vivo. The in vitro parameters including the amount of enzyme available for interaction with a substrate, the metabolic parameters of Vmax and K M, and the intrinsic clearance may increase due to the induction. The in vivo PK/TK constants, such as the metabolic rate constant, the overall elimination rate constant, the total body clearance, and half-life may increase signifcantly. Whereas, in PK/ TK variables, such as the area under plasma concentration–time curve (AUC), maximum plasma concentration, and bioavailability decrease noticeably. Consequently, the induction of the enzyme responsible for metabolism of a compound lowers its optimum plasma-free concentration, reduces its duration of action, and thus, infuences negatively on the pharmacological response and its therapeutic outcome. An example of infuence of enzyme induction on PK/TK parameters is the coadministration of St. John’s Wort, a known inducer of CYP3A4, with midazolam, a substrate of the enzyme. The coadministration decreases the maximum plasma concentration by 65% and AUC by 79% (Mueller et al., 2006). This scenario is an example of having two compounds, one being the substrate and the other being the inducer. There are compounds, for example, carbamazepine, which induce the enzyme system responsible for their metabolism and exhibit autoinduction (Bertilsson et al., 1980; Simonsson et al., 2003). In drug discovery and development, the assessment of lead compound metabolism by the following members of the CYP450 subfamilies are recommended: all members of the CYP3A subfamily and CYP1A2, CYP2B6, CYP2C8, CYP2C9, CYP2C19, and CYP2D6 (Zhang et al., 2009). 9.5 APPLICATIONS AND CASE STUDIES The applications and case studies related to Chapter 9 are posted in Addendum II – part 2. REFERENCES Abu-Zahra, T. N., Wolkoff, A. W., Kim, R. B., Pang, K. S. 2000. Uptake of enalapril and expression of organic anion transporting polypeptide 1 in zonal, isolated rat hepatocytes. Drug Metab Dispos 28(7): 801–6. Agarwal, D. P., Goedde, H. W. 1987. Human aldehyde dehydrogenase isozymes and alcohol sensitivity. Isozymes-Curr T Biol 16: 21–48.

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10 PK – TK Considerations of Renal Function and Elimination of Xenobiotics - Estimation of Parameters and Constants 10.1 INTRODUCTION The kidneys are the major organs of excretion for water-soluble endogenous and exogenous compounds. In addition to the excretion, kidneys have the capacity to metabolize xenobiotics, play a central role in the clearance and PK/TK of xenobiotics and their metabolites, and act as the principal organs in management of metabolic waste of endobiotic (Hall et al., 1983; Masereeuw et al., 2000; Shitara et al., 2005). The renal function maintains the systemic homeostasis, maintain body pH, water, and electrolyte balance, and regulates the blood pressure. The physiological processes of renal elimination, which contribute to the extent of its clearance, consist of four different processes of glomerular fltration, active tubular secretion, active reabsorption, and passive reabsorption (Figure 10.1). In PK/TK analysis, urinary data are important in mass balance studies, simultaneous curve ftting in compartmental analysis, and provides information on the metabolic profle and clearances that are not readily attainable from other biological samples. The urinary data since 1977 are considered by the United States Food and Drug Administration as an alternative approach if the bioanalytical method lacked the appropriate sensitivity to characterize a pharmacokinetic profle of a compound from plasma concentration–time data. Furthermore, as it will be discussed in this chapter, the integrity of the renal function is essential for achieving an optimum therapeutic outcome from normal dosage regimen of therapeutic agents and removing xenobiotics and their metabolites from the systemic circulation. 10.2 GLOMERULAR FILTRATION The rate of fltration of catabolic products, xenobiotics, other impurities, excess, and unnecessary compounds from a volume of blood by the glomeruli of the kidney is referred to as the glomerular fltration rate (GFR). Thus, GFR is defned as the volume of blood that is fltered per units of time. The fltration occurs through permeability of the capillary wall of the glomerulus that is approximately 25 times more permeable than regular capillary walls. The large capillary pore sizes in association with the cardiac contraction force facilitate the transcapillary passage of water, nutrients, ions, and xenobiotics – ionized and unionized. The normal GFR is about 125 mL/min or 7.5 L/h or 180 L/day. GFR varies among healthy individuals with a range between 110 and 125 mL/min, and it represents the volume of 110 ml or 125 ml of blood that is fltered and cleared per minute. The normal rate of blood fow (RBF) through both kidneys is about 1100–1200 mL/min, and the normal rate of plasma fow (RPF) is approximately 650 mL/min, and thus about 10% of renal blood fow is fltered by the glomerular fltration (Leggett and Williams, 1991, 1995; Walton et al., 2004). This blood fow designates the kidney as a highly perfused organ. The normal GFR changes, and its variability refects the dynamic response of the kidney to factors like diet, body mass index, age, sex, and pregnancy (Cachat et al., 2015). The glomerular fltration of compounds that follow linear pharmacokinetics is equal to GFR multiplied by the free fraction of the compounds in plasma. The composition of the fltrate is similar to plasma but without the large molecular weight proteins. The extent of fltration of a compound depends on the molecular weight and protein binding. All compounds of low molecular weight ( ER > 0.3 and low extraction ratio when ER ≤ 0.3. A compound may have a high extraction ratio in the kidney, but low extraction ratio in the liver, or vice versa. Thus, ER can also be considered an organ-specifc number, which represents the relative ability of an organ to extract a compound from the systemic circulation. Dividing the elimination rate by the rate of input into the organ yields the ER of the compound by the organ (Figure 11.1), that is, Extraction Ratio = ER =

Rateof Elimination Rateof input

Q(C Arterial - Cvenous ) C Arterial - CVenous = QCArterial C Arteriala

(11.9) (11.10)

The physiological blood fow in and out of the organ is assumed constant. According to Equation 11.10, when Cvenous @ Carterial the extraction ratio is equal to zero, which is lack of extraction/elimination by the organ, and when Cvenous @ 0 the extraction ratio is equal to one, that is complete removal of the compound by the organ. The overall ER that represents the effciency of the body to eliminate a xenobiotic is the summation of all extraction ratios: ER =

( ER )hepatic + ( ER )renal + ( ER )pulmonary + .... n

(11.11)

Theoretically, the extraction ratio of a compound in a population should remain constant if the elimination processes follow frst-order kinetics (linear pharmacokinetics). The fraction of xenobiotic that escapes the extraction enters the general systemic circulation and will be available for distribution in the body (Equation 11.12). F = 1 - ER

(11.12)

When the removal of a xenobiotic occurs following the absorption and before distribution, such as oral administration, the fraction that escapes the elimination and enters the systemic circulation is the bioavailability of the orally administered compound, and it is defned as (Lee et al., 2001): Bioavailability = Fabs ´ (1 - ER )GI ´ (1 - ER )hepatic

354

(11.13)

ELIMINATION RATES AND CLEARANCES (EXCRETION + METABOLISM)

Where Fabs is the fraction of the administered dose absorbed into the GI tract wall, (1 - ER ) is GI the fraction that escapes gut metabolism and (1 - ER ) is the fraction that escapes hepatic hepatic metabolism. 11.4 CLEARANCES The effectiveness of an elimination process by an organ of elimination is expressed in terms of the organ’s clearance. The distinction should be made between clearance and the rate of elimination (Equations 11.4 – 11.8). The rate of elimination by an organ or the overall rate of elimination from the body expresses the elimination in terms of amount eliminated per units of time. Its magnitude depends on the administered dose or plasma/blood concentration. This dependency makes the rate of elimination a variable for linear and nonlinear PK/TK processes. In contrast to the rate of elimination, the clearance of an organ of elimination in linear pharmacokinetics is a constant representing the normalized rate of elimination with respect to plasma concentration. Therefore, it is independent of the amount or concentration, and it is defned as the volume of plasma or blood from which the compound is removed per unit of time by an organ of elimination. As a constant, clearance is a measure of an organ’s ability to eliminate a xenobiotic. In terms of body clearance, that is, the overall clearance of a compound, it is identifed as plasma clearance or blood clearance of free drug. The clearance is estimated as Clearance =

Rateof Elimination Concentration of Input

(11.14)

In terms of related variables, Equation 11.14 is defned as Clearance =

Q(Cinput - Coutput ) = Q(ER) Cinput

(11.15)

where the rate Q(Cinput - Coutput ) has units of mass/time (i.e., volume / time ´ mass / volume = mass / volume ) and the concentration Cinput has units of mass/volume. Thus, the clearance is expressed in units of volume/time, that is, mass time = volume time mass volume According to Equation 11.15, the clearance can also be defned as the product of blood fow (with units of volume/time) and extraction ratio (unitless). Thus, it represents irreversible extraction of a compound by an organ of elimination in a unidirectional input and output. The relationship between extraction ratio, blood fow, and clearance is depicted in Figure 11.2. Dividing a xenobiotic’s total body clearance by its volume of distribution yields the overall frstorder rate constant of elimination. overall rate constant of elimination =

plasma clearance volume of disstribution

11.16

Dividing the organ clearance like renal clearance, hepatic clearance, or pulmonary clearance, etc., by the volume of distribution of the compounds yields excretion rate constant, metabolic rate constant, pulmonary elimination rate constant, etc. Hence, the total body clearance often called plasma clearance, is the sum of all clearances and represents a measure of the body’s ability to eliminate a xenobiotic. Renal clearance, Clr , and metabolic clearance, Clm , are considered the most signifcant ones, but depending on the route of administration, physicochemical properties of the xenobiotic, and disposition profle of the compound, other clearances such as pulmonary clearance, Clp , sweat gland clearance, Clsg , mammary clearance, Clmilk , and metabolic clearances at sites other than liver, etc., are also parts of the total body clearance. Depending upon their physiological functions, these clearances are mutually independent, and collectively constitute the total body clearance, ClT . ClT = Clr + Clm + Clother

(11.17)

The reliable assessment of total body clearance of a xenobiotic is usually determined when the concentration of the compound is at steady-state level, where the rate of input is equal to the rate 355

11.4 CLEARANCES

Figure 11.2 Delineation of the relationship between the clearance, the blood fow, and the extraction ration (ER); by defnition, the clearance is the product of the blood fow and the ER; thus, if the blood fow remains constant, the clearance is a function of the organ’s extraction ratio, which is the rate of elimination of a xenobiotic divided by its rate of input into the organ. of output. The clearance is then determined by multiplying the rate of input by the steady-state plasma level. 11.4.1 Estimation of Clearance Using Theoretical Models As discussed in Chapter 9 (section 9.4.5), the topic related to the assimilation of intrinsic clearance in hepatic clearance, to defne clearance more clearly and consider the infuence of protein-binding on the elimination of xenobiotics, various theoretical models have been proposed that take into consideration the magnitude of protein-binding and its infuence on the clearance (Bass et al., 1976). These models are listed next. 11.4.1.1 Well-Stirred Model The well-stirred model (Gillette, 1971; Rowland et al., 1973; Wilkinson and Shand, 1975; Pang and Rowland, 1977; Benet and Hoener, 2002; Benet, 2010), also known as the venous equilibration model (Bass, 1979), assumes the organ of elimination, in particular the liver, is a single well-stirred compartment. This means that the compound distributes instantaneously and homogenously throughout the organ (e.g., liver), and the free concentration of fully mixed xenobiotic in the organ is in equilibrium with and identical to the free concentration in the plasma/blood. According to this model, the net clearance of an elimination organ is a function of blood fow Q to the organ, the free fraction of the compound in the plasma or blood, f u , and the intrinsic clearance of the organ, Clint (see also Chapter 9, Equation 9.72 for hepatic clearance):

( Clearance )organ =

Q ´ ( f u ´ Clint )

Q + ( f u ´ Clint )

(11.18)

The intrinsic clearance is the ability of the organ to remove free molecules with no limitations due to blood fow or binding to erythrocytes or other binding sites (see also Chapter 9). The relationship (Equation 11.18) essentially means the elimination of an organ that contributes to the overall clearance of a compound is a function of blood fow and governed by perfusion-limited distribution. 356

ELIMINATION RATES AND CLEARANCES (EXCRETION + METABOLISM)

When the elimination capacity of an organ f u ´ Clint is very high and larger than the blood fow, that is, f u ´ Clint ˜ Q, the Q term in the denominator of Equation 11.18 becomes negligible compared to f u ´ Clint , the two f u ´ Clint terms will be cancelled, and clearance of the organ will be nearly equal to the blood fow

( Clearance )organ =

Cancel ˜˛ °˛˝ Q ´ ( f u ´ Clint )

Q ˙

negligible

+ ( f u ´ Clint ) ˆ˛ˇ˛˘

@Q

(11.19)

Cancel

This means the perfusion of the organ limits the elimination of the compounds with high extraction ratio (ER ≥ 0.7). Conversely, if the elimination capacity is low in relation to the rate of input, that is, f u ´ Clint ˜ Q, the f u ´ Clint term in the denominator is negligible, Q terms will be cancelled, and the clearance will be equal to the elimination capacity of the organ

( Clearance )organ =

Q ´ ( f u ´ Clint )

Q + ( f u ´ Clint ) ˜˛°˛˝

@ f u ´ Clint

(11.20)

negligible

Most compounds with a low extraction ratio (ER ≤ 0.3) fall into this category. The initial model was based on the free fraction of xenobiotics in plasma with the tacit assumption that there is no binding to blood cells. However, for many xenobiotics, the magnitude of clearance exceeds the blood fow. One plausible explanation is that the uptake of xenobiotics by blood cells is signifcant, and the organ of elimination extracts the compound not only from the plasma but also from blood cells. Under this condition, it is more desirable to express the blood clearance rather than plasma clearance. The analytical determination of the concentration of a compound in whole blood is not an easy task. There are limitations that hinder an accurate assessment of the concentration in blood. One approach in overcoming this problem is to adjust the plasma levels by a correction factor, CF, that is based on the blood hematocrit, Hct, and drug concentration in plasma, Cp, and blood cells, CBC : CF = (1 - Hct ) +

CBC Cp

(11.21)

There have been several suggested modifcations to the well-stirred model that are essentially based on the defnition of free fraction and the role of blood cells in defning fu (Jansen, 1981; Masimirembwa et al., 2003; Yang et al., 2007). For example, if the free fraction in the blood is defned by Equation 11.22, the blood clearance is equal to Equation 11.23. (Masimirembwa et al., 2003) æ Cp ö f u blood = f u ç ÷ (1 - Hct ) è Cblood ø Clblood =

Q ´ ( f ublood Clint )

Q (1 - Hct ) + ( f ublood ´ Clint )

(11.22) (11.23)

The magnitude of binding to blood cells can be estimated from in vitro measurements, where total blood and isotonic phosphate buffer pH 7–7.2 are incubated with equal and known concentrations of xenobiotic simultaneously. After incubation, the concentrations of drug in the plasma portion of the blood and in the buffer are determined. Based on the assumption that the concentration of drug in the buffer is equal to the concentration of whole blood, the ratio of buffer concentration to plasma concentration is used as the correction factor to convert the plasma clearance to blood clearance as follows: æ Buffer Concentration ö Blood Clearance = Plasma Clearance ´ çç ÷ oncentration ÷ø è Plasma Co Clblood = Clplasma ´

[ buffer ] éë plasmaùû

(11.24) (11.25)

357

11.4 CLEARANCES

Hepatic microsomes, or hepatocytes, are used routinely to measure the in vitro intrinsic metabolic clearance (Clintinvitro ) of xenobiotics. The estimated values are then used to predict the in vivo metabolism profle of a compound and the elimination capacity of an organ (Houston, 1994; McGinnity and Riley, 2001). For the well-stirred model, the following equation is proposed to predict the in vivo hepatic clearance from in vitro data: æ Cl ´ SF ö QH ´ çç intinvitro ÷÷ ´ f u f u inc è ø ClH = æ Clint invitro ö QH + çç ÷÷ ´ f u è f uinc ø

(11.26)

where QH is the hepatic blood fow, SF is the number of hepatocytes or weight of microsomal protein, f u inc is the free fraction of the compound added to the incubation, and f u is the free fraction in plasma. Under linear conditions and with respect to microsomal protein concentration or hepatocytes density, the initial rate of metabolism is measured for different concentrations of xenobiotic at different time intervals. There are other theoretical models, like the well-stirred model, that are used to explore and defne clearance. Among them are the parallel-tube model and the dispersion model. 11.4.1.2 Parallel Model The parallel model, also known as sinusoidal perfusion model, has been fully investigated, theoretically and quantitatively (Pang and Rowland, 1977; Bass and Bracken, 1977; Pang and Gillette, 1978; Bass, 1979; Lau et al., 2002; Ahmad et al., 1983; Ito and Houston, 2004, 2005). The model assumes an organ of elimination, notably the liver, as a succession of a parallel tube, with evenly distributed metabolic enzymes around the tubes and declining concentration along the length of each tube (see also Chapter 9, section 9.4.5.2). The equation of the parallel-tube model, as defned in Chapter 9, Equation 9.74, is é æ f ´ Clint ö ù (11.27) Cl = Q ê1 - exp ç - u ÷ú Q êë è ø úû When the model is used for the hepatic metabolism, (Q) is the hepatic blood fow and Clint represents the liver intrinsic metabolic clearance (see Chapter 9). 11.4.1.3 Dispersion Model The dispersion model is based on the residence distribution of a compound in the organ according to the following differential equation, which here refects the dimensions of the liver viewed as a cylinder (Roberts and Rowland, 1986a, b): (see also Chapter 9, section 9.4.5.3) ¶ 2CN ¶CN ¶CN - RN CN = DN ¶dN 2 ¶dN ¶tN

(11.28)

DN is defned as the axial dispersion number representing the degree of dispersion as the compound migrates through the liver and is defned as DN =

Dcoeff Dcoeff A = vL QL

(11.29)

The term CN is the normalized concentration term with respect to the input concentration; dN is the normalized distance along the cylinder with respect to the length of the cylinder (L); tN is the time normalized to the mean residence time of substrate within the cylinder (i.e., liver); Dcoeff is the axial diffusion coeffcient, and A is the cross-sectional area of blood in the liver; v is the average velocity of blood in the liver Q A; and RN is calculated as RN = ( f u )blood ´ Clint ´ where ( f u )

blood

r Q

(11.30)

is the unbound fraction of the compound in the blood, Q is the hepatic blood fow

and r is estimated as r= 358

Pcoeff Pcoef + Clint

(11.31)

ELIMINATION RATES AND CLEARANCES (EXCRETION + METABOLISM)

Pcoef is the permeability coeffcient. The integration of Equation 11.28 at steady state (i.e., ∂ CN/∂ tN = 0), assuming frst-order process, yields the following equation (Wehner and Wilhelm, 1956): é ù 4a ú ClH = QH ê1 2 éë( a-1)/2DN ùû 2 éë -( a -1)/2 DN ùû ú ê (1 + a ) e - (1 - a ) e ë û

(11.32)

The term a is defned as: a = (1 + 4RN DN )

1

(11.33)

2

The concept of well-stirred, parallel, and dispersion models are the same as the chemical reaction engineering and reactors that are developed based on conservation laws and fow patterns (Benet et al., 2021; Jusko and Li, 2022). The comparative depiction of the models is presented in Figur  11.3. 11.4.2 Clearance Scale-Up in Mammalian Species 11.4.2.1 Extrapolation of Clearance from Animal to Human The conventional approach for scaling up the inter-mammalian-species extrapolation of PK/ TK parameters and constants is using the allometric methodology based on differences in body weight (also discussed in Chapter 19): Y = aW b

(11.34)

Where Y is the parameter or constant under evaluation, W is the body weight, a is the allometric coeffcient, and b is the allometric exponent. Both a and b are species-related coeffcients for (Y). The extrapolation of in vivo clearance from experimental animal to human is Cl = aW b

(11.35)

log Cl = log a + b logW

(11.36)

In logarithmic form For the extrapolation of metabolic clearance of experimental animal from in vitro data to human, the ratio of estimated in vitro clearances, æ Clintinvitro-human ö , is used as a correction factor in the followç ÷ ing relationship: è Clintinvitro-animal ø æ Cl ö æW ö Clmhuman = Clmanimal ´ ç intinvitro-human ÷ ´ ç human ÷ è Clintinvitro-animal ø è Wanimal ø

b

(11.37)

Depending on the species, the value of b is between 0.08 and 1.31. The allometric coeffcients, a and b, are estimated by linear least square analysis of log Cl versus logW based on Equation 11.36 (Figure 11.4). The b value for extrapolation from experimental animals to humans is defned as (Chiou and Hsu, 1988; Chiou et al., 1998): æ Cl ö log ç human ÷ Clrat ø è b= ö æW log ç human ÷ W rat è ø

(11.38)

There are other extrapolation approaches that involve using parameters such as maximum life span (MLP), or brain weight (BRW) for incorporation into Equation 11.35 (Boxenbaum, 1980, 1984, 1986; Davidson et al., 1986; Boxenbaum and Dilea, 1995; Feng et al., 1998, 2000; Mahmood and Bailian, 1996): Cl =

a.W b MLP

(11.39)

(

)

(11.40)

Cl = aW b BRW c

359

11.4 CLEARANCES

Figure 11.3 Representation of the three proposed models to defne the elimination of a compound by an organ of elimination, notably the liver; the models are based on chemical engineering descriptions of chemical reactions/reactors with logarithm of the concentration, as the y-axis, and distance or time as the independent variable; the concentrations of the input and output are the same for all three models, but the internal concentrations differ signifcantly; in the well-stirred model, the internal concentration remains constant in a homogeneous well-mixed environment, and the rate of elimination is a function of the unbound output concentration multiplied by the intrinsic clearance; in the parallel-tube model, the logarithm of internal concentration declines with time linearly, and the rate of elimination is a function of the log of input to output concentration the rate of elimination is the product of intrinsic clearance and the difference in input and output concentration divided by the logarithm of the ratio of input to output concentrations; the internal concentration of the dispersion model is more complex than the parallel model, and the rate of elimination is defned as the product of the intrinsic clearance with the complex internal concentration that is described as a value between the output concentration Coutput and the internal concentration of unbound xenobiotic as defned for the parallel model.

360

ELIMINATION RATES AND CLEARANCES (EXCRETION + METABOLISM)

Figure 11.4 The profle of the allometric relationship between the clearance and the body weight according to equation Clearance = aW b ; by taking the logarithm of both sides of the equation and plotting log clearance versus log of body weight, the coeffcients aand b are estimated from the y-intercept and the slope of the line, respectively. The parameter (MLP) has been shown to correlate well with BRW and W (Boxenbaum, 1986; Feng et al., 2000) MLP = 10.839 ´ BRW 0.636 ´W -0.225

(11.41)

The inclusion of MLP or BRW in Equation 4.35 is often based on the value of exponent b and the discretion of the investigator. For example, for b > 0.85, BRW is included, and for b ≤ 0.85, BRW is not included (Feng et al., 2000). 11.4.2.2 Body-Weight Dependent Extrapolation of Clearance in Humans For scaling the clearance value across a human life span, the b value is used as b = 2/3, or b = 3/4 (Kleiber, 1932; Brody et al., 1934; Stahl, 1967; Peters, 1983; Lavé et al., 1997). The values are estimated from the slope of log-linear plot of the basal metabolic rate against the logarithm of body weight. The allometric scaling of b = 3/4 has gained more acceptance for scaling PK/TK parameters and constants like clearance. The 3/4 allometric scaling is used not only in species extrapolation but also for extrapolation in the human life span and changes in body size. Equation 11.42 is a recommended equation for human body-weight scaling of clearance across the human life span is (Wang et al., 2012): b

æW ö (11.42) Cli = Clstd ´ ç i ÷ è 70 ø Where Cli is the estimated clearance of ith individual with body weight of Wi , and Clstd is the standard clearance with body weight of 70 kg. Using four different approaches to estimate b, it has been reported that, for example, for propofol clearance in populations of humans at different ages, the 3/4 allometric scaling was adequate for ages 1–81 years old with a body weight of 9–123 kg, but using the value underestimated the clearance in infants and overestimated in term and preterm neonates (Wang et al., 2012).

361

11.4 CLEARANCES

11.4.3 Clearance Estimation in Linear PK/TK If the elimination of a compound follows frst-order kinetics, the plasma clearance remains constant, and it is used as a proportionality constant to relate the rate of elimination of a drug to its unbound plasma concentration Rateof elimination = Clearance ´ Concentration

(11.43)

Depending on the type of compound, analytical methodology, and relevance of the calculated value to the objectives of the investigation, the term “Concentration” refers to plasma, blood, or systemic concentration of an administered xenobiotic at the time of measuring the rate, that is, dA dA = ClT ´ Cp or = ClT ´ Cblood dt dt

(11.44)

dA where is the rate of elimination; ClT is the total body clearance; and Cpand Cblood are plasma dt concentration at time t, respectively. and blood Based on Equations 11.43 and 11.44, the clearance is estimated as Clearance = Rateof Elimination ¸ Concentration ClT =

dA dt dA dt or ClT = Cp Cblood

Systemic concentration = Rateof elimination ¸ Clearance Cp =

dA dt ClT

(11.45) (11.46) (11.47) (11.48)

The following model-independent equation is often used to determine the total body clearance ClT when the elimination follows frst-order kinetics, and the clearance is a constant. ClT =

F ´ Dose AUC

(11.49)

where F is the fraction of dose absorbed, known as absolute bioavailability, with a value between zero and one; for intravenous bolus injection F= 1, and for the other routes, F ≤ 1. AUC is the area under the plasma concentration–time curve estimated by the trapezoidal or log trapezoidal rules (Addendum I). In linear compartmental analysis, depending on the number of compartments and the mode of administration, various equations are used that essentially estimate the same constant with model-dependent differences. One-compartment model – IV bolus: ClT = K ´ Vd

ClT =

ò

¥

dA dt

0

ò

¥

0

(11.50)

(11.51)

Cp

Where K is the overall elimination rate constant; Vd is the apparent volume of distribution;

ò

¥

0

dA dt is the total amount of the administered dose that ultimately is absorbed, which is equal to

the dose for intravenous administration and equal to F ´ Dose for other routes of administration; and

ò

¥

0

Cp is the area under the plasma concentration–time curve with units of mass time/volume.

One-compartment model – Zero-order input after achieving steady-state plasma concentration (e.g., intravenous infusion): ClT =

k0 Cpss

where k0 is the zero-order rate of input and Cpss is the steady-state plasma concentration. 362

(11.52)

ELIMINATION RATES AND CLEARANCES (EXCRETION + METABOLISM)

One-compartment model – First-order input (e.g., oral administration, rectal, sublingual, vaginal, etc.): FD

ClT =

ò

¥

0

(11.53)

Cpdt

Two-compartment open model – IV bolus: ClT = (Vd )b ´ b

(11.54)

ClT = V1 ´ k10

(11.55)

ClT =

Dose Dose = AUC A + B a b

(11.56)

where b is the frst-order hybrid rate constant of disposition; k10 is the frst-order overall exit rate constant from the central compartment; V1 and (Vd ) are the volumes of distribution of the central b

compartment and the overall volume of distribution, respectively, and are estimated as V1 =

Dose Dose = A + B Cp 0

(Vd )b =

Dose b ´ AUC

(11.57) (11.58)

where A and B are the y-intercepts of the terminal log-linear curve, i.e., the extrapolated line, and the residual line of the bi-exponential curve, which in combination represent the initial plasma concentration that is, A + B = Cp0; and A/α + B/β is equal to the area under the plasma concentration–time curve (also see Chapter 13, Section 13.3.5). Two-compartment open model – Zero-order input: The equation is the same as the onecompartment model, that is, equal to the rate of input divided by the steady-state plasma concentration. Two-compartment model – First-order input: The equation is like the previous equations of dose divided by the area under the plasma concentration–time curve ClT =

F ´ Dose

ò

¥

0

(11.59)

Cpdt

11.4.4 Clearance Estimation in Nonlinear PK/TK The kinetics of the compounds that are nonlinear is referred to as dose-dependent PK/TK, a scenario that may also occur in multiple dosing, intravenous infusions, or any chronic exposures. The dose-dependent PK/TK data deviate from linearity, which refects the involvement of one or more capacity-limited processes. This often occurs when the compound is extensively metabolized and overpower the enzymatic capacity to metabolize, or is subjected to renal tubular active reabsorption and/or secretion, or drug–drug interaction, or for xenobiotics that essentially have a highly complex metabolic profle, like benzo(a)pyrene, or presence of a disease state that negatively impacts the disposition of a compound. The nonlinear dose-dependent PK/TK is usually described by the Michaelis–Menten equation, where the rate of elimination and clearance are defned as follows: dA Vmax ´ Cp = K M + Cp dt ClT =

dA dt Vmax = Cp K M + Cp

(11.60) (11.61)

The clearance in dose-dependent PK/TK is a variable, directly proportional to the maximum rate of the nonlinear biological process, and inversely proportional to the plasma concentration and therefore to dose. Therefore, the calculation of clearance, based on the assumption that it is a constant, is no longer applicable (see Chapter 9, Section 9.3.2). 363

11.4 CLEARANCES

There exist cases where a combination of linear clearance and a nonlinear saturable clearance occur at the same time, and the rate of concentration or amount change in the body is defned as Vd

V ´ Cp dCp = -ClT ´ Cp - max dt K M + Cp

(11.62)

11.4.4.1 Nonlinear Clearance in Target-Mediated Drug Disposition The nonlinearity of the clearance occurs also in cases where a xenobiotic binds with high affnity to its receptor site to an extent that is large enough to impact the linearity of the disposition of the compound. This phenomenon is known as target-mediated drug disposition (TMDD), which occurs mostly with a saturable clearance mechanism for biologics like proteins, peptides, monoclonal antibodies, cytokines, and growth factors (Mager and Jusko, 2001; Mager and Krzyzanski, 2005; Dua et al., 2015; Stein and Peletier, 2018; An, 2020) and several small-molecule drugs (Levy, 1994; An, 2020). The products of biotechnology, in general, have a larger molecular weight; absorb at a much slower rate than the small-molecule xenobiotics; their distribution in the body is restricted; and their elimination is changeable. Their molecular structure is designed for interaction with a particular target in mind. Their targets are characteristically located on the surface of cell membrane and are available for immediate and high affnity binding. The targets on the cell membrane are usually very limited, thus they interact and get saturated very rapidly. This high affnity and saturable binding to the targets brings about the nonlinearity of the PK parameters, including the clearance (see also Chapter 12, section 12.8). REFERENCES Ahmad, A. B., Bennett, P. N., Rowland, M. 1983. Models of hepatic drug clearance: Discrimination between the well stirred and parallel-tube models. J Pharm Pharmacol 35(4): 219–24. An, G. 2020. Concept of pharmacologic target-mediated drug disposition (TMDD) in large molecule and small-molecule compounds. J Clin Pharmacol 60(2): 149–63. Bass, L. 1979. Current models of hepatic elimination. Gastroenterology 76(6): 1504–5. Bass, L., Bracken, A. J. 1977. Time-dependent elimination of substrates fowing through the liver or kidney. J Theor Biol 67(4): 637–52. Bass, L., Kieding, S., Winkler, K., Tygstrup, N. 1976. Enzymatic elimination of substrates fowing through the intact liver. J Theor Biol 61(2): 393–409. Benet, L. Z. 2010. Clearance (née Rowland) concept: A downdate and an update. J Pharmacokinet Pharmacodyn 37(6): 529–39. Benet, L. Z., Hoener, B. A. 2002. Changes in plasma protein binding have little clinical relevance. Clin Pharmacol Ther 71(3): 115–21. Benet, L. Z., Sodhi, J. K., Makrygiorgos, G., Mesbah, A. 2021. There is only one valid defnition of clearance: Critical examination of clearance concepts reveal the potential for errors in clinical drug dosing decisions. AAPS J 23(3): 67. https://doi:10.1208/s12248-021-00591-z. Boxenbaum, H. 1980. Interspecies variation I liver weight, hepatic blood fow, and Antipyrine intrinsic clearance: Extrapolation of data to benzodiazepines and phenytoin. J Pharmacokinet Biopharm 2(2): 165–76. Boxenbaum, H. 1984. Interspecies pharmacokinetic scaling and the evolutionary-comparative paradigm. Drug Metab Rev 15(5–6): 1071–121. Boxenbaum, H. 1986. Time concepts in physics, biology, and pharmacokinetics. J Pharm Sci 75(11): 1053–62.

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Boxenbaum, H., Dilea, C. 1995. First-time-in-human dose selection: Allometric thoughts and perspectives. J Clin Pharmacol 35(10): 957–66. Bracken, A. J., Bass, L. 1979. Statistical mechanics of hepatic elimination. Mathl Biol Sci 44(1–2): 97–120. Brody, S., Proctor, R. C., Ashworth, U. S. 1934. Basal metabolism, endogenous nitrogen, creatinine, and sulphur excretions as functions of body weight. Univ Mo Agric Exp Sta Res Bull 220: 1–40. Chiou, W. L., Hsu, F. H. 1988. Correlation of unbound clearance of ffteen extensively metabolized drugs between humans and rats. Pharm Res 5(10): 668–72. Chiou, W. L., Robie, G., Chug, S. M., Wu, T.-C., Ma, C. 1998. Correlation of plasma clearance of 54 extensively metabolized drugs between humans and rats: Mean allometric coeffcient of 0.66. Pharm Res 15(9): 1474–9. Davidson, I. W. F., Parker, J. C., Beliles, R. P. 1986. Biological basis for extrapolation across mammalian species. Regul Toxicol Pharmacol 6(3): 211–37. Dua, P., Hawkins, E., van der Graaf, P. H. 2015. A tutorial on target-mediated drug disposition (TMDD) models. Pharmacometr Syst Pharmacol 4(6): 324–37. Feng, M. R., Lou, X., Brown, R. R., Hutchaleelaha, A. 2000. Allometric pharmacokinetic scaling toward the prediction of human oral pharmacokinetics. Pharm Res 17(4): 410–18. Feng, M. R., Rossi, D., Strenkoski, C., Black, A., DeHart, P., Lovdahl, M., McNally, W. 1998. Disposition of cobalt mesoporphyrin in mice, rat, monkey, and dog. Xenobiotica 4: 413–26. Gillette, J. R. 1971. Factors affecting drug metabolism. Ann N Y Acad Sci 179: 43–66. Houston, J. B. 1994. Utility of in vitro drug metabolism data in predicting in vivo metabolic clearance. Biochem Pharmacol 47(9): 1469–79. Ito, K., Houston, J. B. 2004. Comparison of the use of liver models for predicting drug clearance using in vitro kinetic data from hepatic microsomes and isolated hepatocytes. Pharm Res 21(5): 785–92. Ito, K., Houston, J. B. 2005. Prediction of human drug clearance from in vitro and preclinical data using physiologically based and empirical approaches. Pharm Res 22(1): 103–12. Jansen, J. A. 1981. Infuence of plasma protein binding kinetics on hepatic clearance assessed from a Tube model and a well-stirred model. J Pharmacokinet Biopharm 9(1): 15–26. Jonsson, E. N., Karlsson, M. O., Wade, J. R. 2000. Nonlinearity detection: Advantages of nonlinear mixed-effects modeling. AAPS PharmSci 2(3): 114–23. Jusko, W. J., Li, X. 2022. Assessment of the Kochak-Benet equation for hepatic clearance for the parallel-tube model: Relevance of classic clearance concepts in PK and PBPK. AAPS J 24(1): 5. https://doi.org/10.1208/s12248-021-00656-z. Kleiber, M. 1932. Body size and metabolism. Hilgardia 6(11): 315–33. Lavé, T., Dupin, S., Schmitt, C., Chou, R. C., Jaeck, D., Coassolo, P. 1997. Integration of in vitro data into allometric scaling to predict hepatic metabolic clearance in man: Application to 10 extensively metabolized drugs. J Pharm Sci 86(5): 584–90.

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Lau, Y. Y., Krishna, G., Yumibe, N. P., Grotz, D. E., Sapidou, E., Norton, L., Chu, I., Chen, C., Soares, A. D., Lin, C. C. 2002. The use of in vitro metabolic stability for rapid selection of compounds in early discovery based on their expected hepatic extraction ratios. Pharm Res 19(11): 1606–10. Lee, Y.-H., Perry, B. A., Lee, H.-S., Kunta, J. R., Sutyak, J. P., Sinko, P. J. 2001. Differentiation of gut and hepatic frst-pass effect of drugs: 1. Studies of verapamil in ported dogs. Pharm Res 18(12): 1721–8. Levy, G. 1994. Pharmacologic target-mediated drug disposition. Clin Pharmacol Ther 56(3): 248–52. Mager, D. E., Jusko, W. J. 2001. General pharmacokinetic model for drugs exhibiting target-mediated drug disposition. J Phatmacokinet Pharmacodyn 28(6): 507–32. Mager, D. E., Krzyzanski, W. 2005. Quasi-equilibrium pharmacokinetic model for drugs exhibiting target-mediated drug disposition. Pharm Res 22(10): 1589–96. Mahmood, I., Bailian, J. D. 1996. Interspecies scaling: Predicting clearance of drugs in humans. Three different approaches. Xenobiotica 26(9): 887–95. Masimirembwa, C. M., Bredberg, U., Andersson, T. B. 2003. Metabolic stability for drug discovery and development: Pharmacokinetic and biochemical challenges. Clin Pharmacokinet 42(6): 515–28. McGinnity, D. F., Riley, R. J. 2001. Predicting drug pharmacokinetics in humans from in vitro metabolism studies. Biochem Soc Trans 29(2): 135–9. Pang, K. S., Gillette, J. R. 1978. A theoretical examination of the effects of gut wall metabolism, hepatic elimination, and enterohepatic recycling on estimates of bioavailability and hepatic blood fow. J Pharmacokinet Biopharm 6(5): 355–67. Pang, K. S., Rowland, M. 1977. Hepatic clearance of drugs I. Theoretical consideration of a well stirred model and a parallel tube model. Infuence of hepatic blood fow, plasma and blood cell binding, and the hepatocellular enzymatic activity on hepatic drug clearance. J Pharmacokinet Biopharm 5(6): 625–53. Peters, H. P. 1983. Physiological correlates of size. In The Ecological Implications of Body Size, eds. E. Beck, H. J. B. Birks, E. F. Conner, 48–53. Cambridge: Cambridge University Press. Roberts, M. S., Rowland, M. 1986a. A dispersion model of hepatic elimination; 1. Formulation of the model and bolus considerations. J Pharmacokinet Biopharm 14(3): 227–60. Roberts, M. S., Rowland, M. 1986b. A dispersion model of hepatic elimination; 1. Formulation of the model and bolus considerations. J Pharmacokinet Biopharm 14(3): 261–88. Rowland, M., Benet, L. Z., Graham, G. G. 1973. Clearance concepts in pharmacokinetics. J Pharmacokinet Biopharm 1(2): 123–35. Sedman, A. J., Wagner, J. G. 1974. Importance of the use of the appropriate pharmacokinetic model to analyze in vivo enzyme constants. J Pharmacokinet Biopharm 2(2): 161–73. Stahl, W. R. 1967. Scaling of respiratory variables in mammals. J Appl Physiol 22(3): 453–600. Stein, A. W., Peletier, L. A. 2018. Predicting the onset of nonlinear pharmacokinetics. Pharmacometr Syst Pharmacol 7: 670–7. Wang, C., Peeters, M. Y. M., Allegaert, K., van Oud-Alblas, H. J. B., Krekels, E. H. J., Tibboel, D., Meindert Danhof, M., Knibbe, C. A. J. 2012. A bodyweight-dependent allometric exponent for scaling clearance across the human life-span. Pharm Res 29(6): 1570–81.

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Wehner, J. F., Wilhelm, R. M. 1956. Boundary conditions of fow reactor. Chem Eng Sci 65: 89–93. Wilkinson, G. R., Shand, D. G. 1975. Commentary: A physiological approach to hepatic drug clearance. Clin Pharmacol Ther 18(4): 377–90. Yang, J., Jamie, M., Yeo, K. R., Rostami-Hodjegan, A., Tucker, G. T. 2007. Misuse of the well-stirred model of hepatic drug clearance. Drug Metab Dispos 35(3): 501–2.

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12 Approaches in PK/PD and TK/TD Mathematical Modeling 12.1 INTRODUCTION The goal in pharmacokinetic/toxicokinetic (PK/TK) modeling is to develop a scientifc, practical, and meaningful mathematical relationship that represents the function of physiological systems in handling xenobiotics and its time dependency. The achievement of this goal reveals how the components and compartments of the system interact and cooperate to yield the functional properties of the system and how the system ultimately handles the exposure to xenobiotics, whether they are therapeutic agents or environmental pollutants. The development of the model has immense practical implications for diagnosis and treatment of disease and quantifying human response and overall risk assessment of exposure to all kinds of xenobiotics. It is the precise and abstract nature of the mathematical terms that helps to explore the interconnection and infuence of physiological parameters and constants. Within the past fve decades, PT the PK/TK modeling has become progressively more complex, sophisticated, and predictive. The objectives of this chapter are to introduce the traditional and current approaches in PK/TK modeling, which can be categorized as: 1. physiologically based pharmacokinetic modeling 2. classical compartmental analysis a. linear compartmental models b. nonlinear compartmental models c. compartmental models with one- or two-tissue binding 3. non-compartmental analysis 4. population pharmacokinetics 5. pharmacokinetic/pharmacodynamic modeling 6. toxicokinetic/toxicodynamic modeling 7. stochastic and mechanistic modeling 12.2 PHYSIOLOGICALLY BASED PK/TK MODELS 12.2.1 Description The basic idea behind physiological modeling is to overlay xenobiotic data onto anatomical models encompassing relevant organs/tissues of the body and to incorporate quantitative parameters of each anatomical region in the mathematical description of the compound’s disposition (Bischoff et al., 1971; Bischoff, 1980, 1986). Thus, the physiologically based pharmacokinetics or toxicokinetics (PBPK or PBTK) and the related modeling incorporates parameters that are related to physicochemical properties of xenobiotics with those of the physiological processes based on numerical simulation of absorption, distribution, metabolism, and excretion (ADME). The physiological parameters include body weight, blood fow, cardiac output, organ/tissue volume or mass, tissue affnity, enzymatic and transporter activity, plasma protein-binding, membrane permeability, transport and elimination from an organ, and metabolism by the liver or other sites of metabolism. Xenobiotic-specifc parameters include solubility, partition coeffcient, permeability and permeation coeffcients, liver and kidney intrinsic clearances, metabolic rate constants, protein-binding constants and parameters, transport protein affnity and interaction, blood to plasma partitioning, and unbound fraction of the xenobiotic in the body. Therefore, based on xenobiotic-specifc measurements, and understanding of the xenobiotic interaction with the organs, tissues, and biological system, the absorption and disposition of the xenobiotic at different dose levels administered via a route of administration is estimated. Furthermore, since there are parallelisms in the anatomy of mammalian systems, such as having all organs between arterial blood and venous systems or most organs having comparable fractions of body weight, the physiological models of xenobiotics developed in experimental animals can be extrapolated to humans by changing the physiological parameters to those in a human (Dedrick et al., 1970; Harrison and Gibaldi, 1977; Nakashima et al., 1987; Chow, 1997; Rowland et al., 2004). The intended application of physiological models 368

DOI: 10.1201/9781003260660-12

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determines their complexity, and there is no limit to the number of relevant organs, tissues, or body fuids for inclusion in the model. Thus, the models can be highly complex and comprehensive and are intended to mimic the anatomical/regional geometry of the body with inclusion of all molecular and cellular events. A typical diagram of a semi-comprehensive model is presented in Figure 12.1. The models can be simplifed by “lumping” together various tissues. Lumping is the terminology used in constructing simplifed physiological models and refers to combining organs and tissues with similar kinetic behavior into one compartment (Figures 12.2–12.4). The simplifed models, known as “hybrid models” (Kawahara et al., 1999; Gallo et al., 2004) or “minimal models” (Henthorn et al., 1992; Cao and Jusko, 2012; Cao et al., 2013), are used to focus mainly on the disposition and uptake of specifc organs of interest (Figure 12.4). The selection of model, expanded or contracted, depends on the goals and objectives of an investigation; it is not an arbitrary decision and requires clear justifcations for expansion or contraction of the compartments. The important features and applications of physiological modeling are: ◾ to provide a detailed and quantitative portrayal of xenobiotic disposition in the body of mammalian species ◾ to improve the selection and optimization process of lead compound(s) in drug discovery and development

Figure 12.1 A typical diagram of a semi-comprehensive physiologically based pharmacokinetic (PBPK) model to highlight the input and output of each compartment and the routes of elimination namely the kidney for excretion of unchanged compound and elimination of water-soluble metabolites, the liver the major site for biotransformation of the compound, and the lung as a route of for the removal of the vaporous compound/metabolites from the body through the expired air; the rate of input to each organ/tissue compartment is the blood fow of the organ, Q, times the plasma/blood concentration of the administered xenobiotic. 369

12.2 PHYSIOLOGICALLY BASED PK/TK MODELS

Figure 12.2 Schematic of a simplifed fow-limited physiologically based PK/TK model for volatile organic compounds with the pooled compartments of rapidly and slowly perfused tissues, adipose tissue, skin, and the organs of eliminations; the objective of the model is to evaluate the general disposition of a pulmonary administered or exposed volatile compound in the body; most gaseous compounds are highly hydrophobic and are taken up by adipose tissue, skin and other organs; the elimination from the body in addition to the expired air is through the metabolism and elimination of metabolites through the urine and bile. ◾ to extrapolate more reliably the results from animals such as mice, rats, dogs, and primates upward to man. This extrapolation also known as “scaling up” may also be applicable in certain cases in the opposite direction, that is, scaling down to lower species ◾ to predict in vivo data based on in vitro observations ◾ to assess human health risk, including cancer risk following exposure to toxic environmental chemicals/carcinogens through different routes of administration and estimate the internal dose ◾ to extrapolate from one route to another route of administration and calculate the tissue/organ dose ◾ to predict the safe or low-dose exposure from high-dose administration, i.e., the low-dose extrapolation in chemical carcinogen/toxic compounds – the essence of health-risk assessment ◾ to provide a useful framework for predicting the infuence of PK/TK interactions from binary to more complex chemical mixtures ◾ to predict the time course of xenobiotic levels in different organs ◾ to facilitate simulation of drug disposition in disease states ◾ to predict fetal exposure from parental exposure to a xenobiotic 370

APPROACHES IN PK/PD AND TK/TD MATHEMATICAL MODELING

Figure 12.3 Schematic of an abbreviated physiologically based PK/TK model for a nonvolatile and lipophilic compound administered orally with elimination through hepatic and pulmonary biotransformation, which can be treated as linear or nonlinear metabolism governed by the Michaelis–Menten equation; the elimination through the fecal elimination is mostly through the unchanged metabolites eliminated through the biliary route into the small intestine; the measurement of the unchanged xenobiotic and its metabolites in the bile of experimental animals is done by using the bile cannulation procedure, which in most cases can equal the fecal route of elimination. 12.2.2 Model Development To develop a physiological model for a xenobiotic, a compilation of all relevant PK/TK data, related physiological constants and parameters, physicochemical values, and justifcation for new experimental data with appropriate animal model are necessary. Based on the collected data, an applicable diagram like Figure 12.1 is considered with relevant organs to describe the disposition of the compound. The number of organs and tissues included in the model is determined by the physicochemical characteristics of the compound, such as lipid solubility, plasma protein-binding, microsomal or hepatic intrinsic clearance, cell membrane permeability estimated from Caco-2 cells, degree of ionization of the compound and preferential binding by an organ, and physiological/biochemical parameters (Cowles et al., 1971; Rowland, 1985; Ritchel and Banerjee, 1986; Gallo et al., 1987; Brown et al., 1997, 1998; Corley et al., 2000; Blaauboer, 2003; Lipscomb and Poet, 2008). For example, if the compound is highly lipid soluble, the adipose tissue is included in the model, and if the compound is water-soluble, the adipose tissue is excluded. Therefore, the in vitro and in silico data are important in the selection of an appropriate model. It should be noted that the tacit assumption in physiological modeling is that all included organs and tissues are well-stirred compartments. Another important characteristic of physiological modeling is the presentation of each organ or region in terms of its sub-compartments. For example, the distribution and uptake of a compound 371

12.2 PHYSIOLOGICALLY BASED PK/TK MODELS

Figure 12.4 Diagram of a minimal physiologically based PK/TK model that all organs and tissue compartments are lumped into two compartments; the blood fow into the compartment is defned as the cardiac blood/plasma output, QCO , and fraction of the output for each compartment is defned as fQCO , the combined fractions of the compartments cannot exceed one. The total volume of the compartments, V1 + V2 is assumed as the total volume of the extravascular fuid; the uptake of the xenobiotic by either compartment is assumed to be dependent on the partition coeffcient of the compound in each compartment. in a region can be defned not only in terms of capillary concentration but also in terms of interstitial fuid and intracellular volume. 12.2.2.1 Flow-Limited (Perfusion-Limited) Models When a compound is taken up rapidly by the interstitial and intracellular sub-compartments of an organ, and they are essentially in equilibrium with the concentration in the capillaries, the limiting factor of the uptake is blood fow: that is, the magnitude of uptake by the organ depends on how fast the compound reaches the organ through blood fow. Under this condition, the uptake is considered a blood fow-limited process, and the organ is capable of not only taking up the compound rapidly but also of holding large quantities over longer periods. On the other hand, if the transfer of a drug between extracellular (i.e., blood and interstitial fuid) and intracellular fuids is limited by the permeation of the drug through the cell membrane, the process is considered a membrane-limited or permeability-limited process (see also Chapter 8, Section 8.2.2.1). For volatile organic compounds (VOCs), for example, a general seven-compartment, fow-limited model has been proposed that consists of blood, adipose tissues, skin, kidney, liver, rapidly and slowly perfused tissues, and a gas exchange compartment (Mumtaz et al., 2012) (Figure 12.2). The example of nonvolatile but lipophilic compounds is a six-compartment model for benzo(a)pyrene and benzo(def,p)chrysene (Anderson et al., 1987; Crowell et al., 2011), which includes blood, lung, adipose tissues, liver, and other highly perfused organs and slowly perfused tissues (Figure 12.3). Similarly, seven-compartment models have also been developed for xenobiotics intended for therapeutic use (Hudacheck and Gustafson, 2013). Once the relevant body regions have been identifed in a comprehensive physiological model, the mass balances for blood and the organs can be written by considering the rates of input and the rates of output. For example, the following is the overall mass balance equation for a fowlimited model: VB

ù Ckidneys C fat C C C dCBlood é + Q fat + ………ú = êQbrain brain + Qheart heart + Qliver liiver + Qkidney Rheart Rbrain dt Rliver Rkidneys R fat úû ëê

(12.1)

- éëQbrain + Qheart + Qliver + Qkineys + Q fat + ..........ùû Cblood + Dose(t) where C is the concentration, Q is the blood fow, V is the volume, and R is the distribution ratio of tissue/blood. The multiplication of Q (volume/time) by C (mass/volume) yields the units of rate in terms of mass/time; R has no units, and the multiplication of V (volume) by dC / dt (mass/ 372

APPROACHES IN PK/PD AND TK/TD MATHEMATICAL MODELING

volume time) yields the units of mass/time defning the rate of change in amount or concentration in the blood. The above rate equation requires that the dose be included as a function of time. The mass balance equations of organs with elimination are Liver - organ : Vliver Liver – Elimination: Vliver

é C ù dCliver C = Qliver êCblood - liver ú - Clm liver R dt Rliver liver û ë

V ´ Cliverr - venous dCliver = Qliver [Carterial - Cliver -venous ] - max dt K M + Cliver -venous

Kidney organ : Vliver Kidney Excretion : Vkidney

dCliver V ´ Cliver -venous = Qliver [Carterial - Cliver -venous ] - max dt K M + Cliver -venous dCkidney Ckidney -venous = Qkidney éëCarterial - Ckidney --venous ùû - Clr dt Rkidney

Lung : Vlung

é dClung Clung ù Clung = Qlung êCblood ú - Cll dt Rlung úû Rlung êë

(12.2) (12.3) (12.4) (12.5) (12.6)

where Clm, Clr, and Cll are metabolic, renal, and lung clearances, respectively, with units of volume/ time. The differential equations for other organs or regions also follow the general concept of: The rate of amount change in an organ = rate of input – rate of output That is, Vorgan

é dCorgan Corgan ù = Qorgan êCblood ú dt Rorgan úû êë

(12.7)

For highly perfused hpt and slowly perfused spt lumped compartments, the equations are Highly Perfused Tissues : Vhpt

dChpt = Qhpt ´ ( Carterial - Cvenous-hppt ) dt

(12.8)

Slowly Perfused Tissues : Vspt

dCspt = Qspt ´ ( Carterial - Cvenous-sppt ) dt

(12.9)

The most abbreviated PBPK/TK, known as minimal PBPK/TK, is presented in Figure 12.4. For the minimal PBPK model with only two lumped compartments (Figure 12.4), the differential equations of the model are (Cao and Jusko, 2012): dCp Dose C1 = + ( fQCO )1 ´ QCO ´ + ( fQCO )2 ´ QCO dt Vplasma P ( coeff 1 )tissue1 ´ Vplasma ´

C2

( Pcoeff 2 )tissue2 ´ Vplasma

( fQ )1 ´ QCO + ( fQ )2 ´ QCO + ClT - Cp ´ CO

(12.10)

CO

Vplasma

ö ( fQCO ) ´ QCO C1 dC1 æ 1 ÷´ = ç Cp dt ç V1 Pcoeff 1 )tissue1 ÷ ( ø è

(12.11)

ö ( fQCO ) ´ QCO 2 ÷´ ÷ V2 ø

(12.12)

C2 dC2 æ = ç Cp dt ç Pcoeff 2 )tissue2 ( è

where Vplasma is the volume of plasma, or blood; Cp is the concentration of compound in Vplasma ; QCO is cardiac blood fow, or plasma fow; ( fQCO ) and ( fQCO ) are fractions of cardiac output considered for compartments 1 and 2; ( Pcoeff 1 )

1

tissue1

and ( Pcoeff 2 )

2

tissue2

are partition coeffcients of tissue compart-

ment 1 and tissue compartment 2; ClT is the total body clearance; V1 and V2 are the volumes of tissue compartments 1 and 2. The assumptions of the model are:

( fQ )1 + ( fQ )2 £ 1 CO

CO

(12.13)

373

12.2 PHYSIOLOGICALLY BASED PK/TK MODELS

V1 + V2 + Vplasma = Total volumeof extracellular fluid

(12.14)

The model can be expanded to include specifc tissue compartments. 12.2.2.2 Permeability-Limited (Membrane-Limited) Models The permeability-limited physiologically based PK/TK model operates on a slower timescale and represents the disposition of many xenobiotics (Thompson and Beard, 2011; Huang and Rowland, 2012). For permeability-limited PBPK models, the compartments are subdivided into two subcompartments, and the differential equation of each organ is expressed in terms of two rate equations: 1) rate of change of xenobiotic concentration in extracellular fuid, and 2) rate of change of the concentration in the intracellular compartment. This means that each compartment of a model, like Figure 12.5, is subdivided into two compartments of extracellular and intracellular fuids as is shown in Figure 12.6. The related differential equations for tissue blood compartment and cellular matrix for a permeability-limited model are: Tissue Blood : Vtissue1

dCtissue1 (C - Ctissue2 ) = Qt éëCinput - Coutput ùû - [ PA ] tissue1 dt Pt.b.coeff

(12.15)

Equation 12.15 can also be presented as

Figure 12.5 Schematic of permeability-limited physiologically based PK/TK model for a xenobiotic that is given by oral administration; the permeability-limited nature of the disposition requires each compartment to be divided into two sub-compartments representing extracellular and intracellular environs; the shaded area represents the intracellular compartment and the major elimination of the xenobiotic from body is by metabolism governed by the Michaelis–Menten equation; the compound is mainly eliminated from the body through the biliary elimination. 374

APPROACHES IN PK/PD AND TK/TD MATHEMATICAL MODELING

Figure 12.6 Schematic of an isolated compartment of permeability-limited physiologically based PK/TK model, presented in Figure 12.5, with extracellular and intracellular sub-compartments; the input rate is the blood fow with concentration of the xenobiotic into the extracellular compartment; the exchange with the intracellular compartment is either governed by the driving force of concentration gradient and/or function of protein transporters. Ctissue2 PA æ dC1 Q = ( Cinput - Ctissue1 ) ç Ctissue1 Pt.b.coeff V1 çè dt V1 Cellular Matrix : Vtissue2

ö ÷÷ ø

dCtissue2 - Ctissue2 C = [ PA ] ´ tissue1 dt Pt.t b.coeff

(12.16) (12.17)

Ctissue2 ö dCtissue2 PA æ (12.18) = ç Ctissue1 ÷ Pt.b.coeff ÷ø dt V2 çè where Q is blood fow; Cinput is the concentration of input or infow; V1 is the vascular volume, PA is the membrane permeability-area coeffcient; Ctissue1 is the concentration in the vascular sub-compartment; Ctissue2 is the concentration in the extravascular sub-compartment; Ctissue1 - Ctissue2 is the concentration gradient across the membrane; and Pt.b.coeff is the partition coeffcient between Ctissue1 and Ctissue2 . Physiological models provide the versatility of identifying and evaluating potentially saturable processes like capacity-limited metabolism or active secretion. 12.2.2.3 Variability of Physiological/Biochemical Key Parameters Most of the estimated physiological parameters are based on allometric scaling functions. Various approaches have been used to establish the most realistic whole body PBPK models (Price et al., 2003; Kazama et al., 2003; Willman et al., 2003). For organ weight estimation, two of the recommended approaches are (Willmann et al., 2007; Upton et al., 2012):

( organ wt )individual

æ hgt individual = çç è hgt mean

ö ÷÷ ø

0.75

´ ( orrgan wt )mean

(12.19)

where “mean” parameters refer to the mean values of a virtual individual based on data reported by the International Commission on Radiological Protection (ICRP, 2002; Willmann et al., 2007). 1

æ ( wt ) ö (12.20) ( organ wt )individual =çç wt individual ÷÷ ´ ( orrgan wt )standard ( ) standard ø è The subscript “standard” refers to the values for a virtual standard man (Upton et al., 2012). Both “standard” and “mean” values were created based on the anthropometric reported data. The actual organ weights, based on autopsy measurements, have been reported to correlate better with the height of the body than body weight (de la Grandmaison et al., 2001). However, there are organs that are independent of height and weight, and organs that correlate well with the body weight, for example,

( brain wt )individual º ( brain wt )mean º ( brain wt )standard

(12.21)

375

12.2 PHYSIOLOGICALLY BASED PK/TK MODELS

The assumption here is the brain weight of individual is equivalent to the brain weight of virtual men. The skin weight, however, that is proportional to body surface area (BSA), correlates better with the body weight. 0.5 0

æ ( bwt ) ö individual ÷ ´ ( skin wt )Reference ( skin wt )individual =çç (12.22) ÷ è ( bwt )Reference ø The blood fow as a function of cardiac output (CO) is also estimated according to Equation 12.22 (Willmann et al., 2007). æ hgt individual ö ÷÷ è hgt mean ø

( CO )individual = çç

0.75

´ ( CO )mean

(12.23)

Equation 12.24 (Upton et al., 2012) is for the estimation of cardiac output for an individual within a population æ ( bwt )

( CO )individual = ( CO )standard çç

individual

è ( bwt )standard

ö ÷ ÷ ø

0.41

(12.24)

æ æ ( Age ) ( Age )individuaal individual - 0.06033ç ´ ç 1.0309 + 0.0295 ç ( Age ) ( Age )standard çç s tan dard è è

ö ÷ ÷ ø

2

ö ÷ ÷÷ ø

There are differences, though not signifcant as shown in Table 12.1, that may contribute to the cumulative random errors in estimates of physiological parameters, such as the volume or weight of an organ. The physiological models are often developed based on the disposition profle of a xenobiotic in experimental animals. To extrapolate the data from an experimental animal to a human or to another species, the data presented in Table 12.2 is taken from the literature (Gerlowski and Jain, 1983; Mordenti, 1986) are most helpful. 12.2.3 Predictive Capability and Sensitivity Analysis The analysis of experimental data according to physiological modeling is mostly based on simulation of the time course of a drug in different anatomical regions. The simulation is achieved by substituting in the equations of a selected model, the physiological parameters of the organs (blood fow, weight, etc.), the input function of the experimental conditions, and the physicochemical parameters of the compound. When used in PBPK/TK, the physiological parameters have usually been considered as exact values, ignoring uncertainties in the variability of the values and their impact on model prediction. The simulation provides the predicted concentrations of the drug. The observed data (i.e., the experimental values) are then judged based on their closeness to the predicted values. From the predicted and observed concentration–time profles, other pharmacokinetic parameters, such as clearances, area under plasma concentration–time curve, half-life, maximum plasma concentration, etc., can be estimated. The fundamental issues in PBPK/

Table 12.1 Comparison of Estimated Organ Blood Flow for “Mean” and “Standard” Men Organ

Q (L/min) Mean (Caucasian Virtual Man) 73 kg, 5 ft 8in Male, BMI = 23.6 kg/m2

Q (L/min) Standard (Virtual Man) 69 kg with 30% Fat and 40% Muscle

Lung Brain Kidneys Heart Liver Muscle Adipose

6.106 0.780 1.325 0.260 0.420 1.106 0.325

6.314 0.686 0.962 0.242 1.449 (Portal Drained Viscera + Artery) 1.377 1.035

Source: The “Mean” data are from Willmann, S. et al. 2007. J Pharmacokinet Pharmacodyn 34(3):401–431; and the “Standard” values are from Upton, R. N. et al. 2012. J Pharmacokinet Pharmacodyn 39:561–576.

376

APPROACHES IN PK/PD AND TK/TD MATHEMATICAL MODELING

Table 12.2 Species-specifc Physiological Parameters of Weight, Volume, and Flow Rate Parameter

Mouse

Body weight, g Body weight 22 Compartment volumes, mL Plasma 1.00 Muscle 10 Kidney 0.34 Liver 1.3 Gut 1.5 Gut lumen 1.5 Heart 0.095 Lungs 0.12 Spleen 0.1 Marrow 0.6 Plasma fow rate, mL/min Plasma 4.38 Muscle 0.5 Kidney 0.8 Liver 1.1 Gut 0.9 Heart 0.28 Lungs 4.38 Spleen 0.05 Marrow 0.17

Hamster

Rat

Rabbit

Monkey

Dog

Human

150

500

2330

5000

12000

70000

6.48 1.36 6.89 12.23 0.63 0.74 0.54 -

19.6 245 3.65 19.55 11.25 8.8 1.15 2.1 1.3 -

70 1350 15 100 120 6 17 1 47

220 2500 30 135 230 230 17 135

500 5530 60 480 480 120 120 36 120

3000 35000 280 1350 2100 2100 300 160 1400

40.34 5.27 6.5 5.3 0.14 28.4 0.25 -

84.60 22.4 1208 4.7 14.6 1.6 2.25 0.95 -

520 155 80 177 111 16 520 0.0 11

379 50 84 92 75 65 23

512 138 90 60 8105 60 512 13.5 20

3670 420 700 800 700 150 240 120

Data from Gerlowski and Jain, 1983; Mordenti, 1986

TK modeling, particularly when they are used in regulatory agencies, are the predictive capability of the model and its sensitivity analysis. The predictive capability of the model is usually assessed by estimating the percent error of prediction (PE%) (Sheiner and Beal, 1981; Wu, 1995): PE% =

( Value )predicted - ( Value )observed ( Value )observed

(12.25)

The precision of the prediction is measured by the median absolute prediction error (MAPE%): MAPE% = median ( PE%1 , PE%2 , PE%3 ,..., PE%n )

(12.26)

The bias of the prediction is measured by the median prediction error (MPE%): MPE% = median ( PE%1, PE%2, PE%3,……, PE%n )

(12.27)

An important attribute of PBPK/TK modeling is its predictive potential, particularly for toxic dose or toxic metabolites reaching the target organs. The PBPK/TK models incorporate signifcant numbers of biochemical, physiological, and physicochemical parameters related to linear and nonlinear biological processes. Each of these parameters carries some degree of error, and the interaction among them also contributes to the magnitude of the error. Contributing to the level of uncertainty of the model’s output includes dissimilarities in metabolism and nonlinear kinetics in the body and the impacts of the relationship between parameters of a model, such as body mass, organ mass, etc. To assess the infuences of these parameters, their interaction, and the presence of nonlinearity in the output, the sensitivity analyses are employed. Sensitivity analysis allows the evaluation of the model output uncertainty to be attributed to the model’s parameters and presents ways to establish consistency between the model structure and its output. There are various methodologies for sensitivity analysis that are grouped into “global” and “local” analysis 377

12.3 LINEAR PK/TK COMPARTMENTAL ANALYSIS

(McNally et al., 2011). The global analyses assess the contribution of a parameter over the set of all possible input parameters (McNally et al., 2011). The local sensitivity analyses, used more often, evaluate sensitivities that are associated with a specifc set of input parameters and are estimated by the normalized sensitivity coeffcient (NSC) (Bois et al., 1991; Hetrick et al., 1991; Clewell et al., 1994; Loccisano et al., 2012). The local sensitivity analysis involves the adjustment of one parameter at a time while keeping the other parameters constant. The NSC is the percent change in the output of the model produced by a fxed 1% percent change in the parameter (Plowchalk et al., 1997), that is, NSC =

( A + B) / B (C - D) / D

(12.28)

where A is the model prediction with a 1% increase in parameter value, B is the model prediction with original parameter value, C is the parameter value increased by 1%, and D is the original parameter value. Generally, NSC values of less than 0.15 (|NSC| < 0.15) are considered low sensitivity; the values between 0.15 and 0.5 (0.15 < |NSC| < 0.5) are of moderate sensitivity; and values greater than 0.5 (|NSC| > 0.5) refect high sensitivity. The positive values of NSC are indicative of direct relationship between output and parameter, and negative values represent an inverse relationship. The values greater than one (|NSC| > 1) may represent the augmentation of error (Allen et al., 1996). 12.3 LINEAR PK/TK COMPARTMENTAL ANALYSIS The classical compartmental analysis, the commonly employed methodology for interpretation of in vivo and in vitro data, is different from physiological modeling. In this approach, the body is assumed as a series of one, two, three, or more compartments, which may or may not have anatomical signifcance. The ADME characteristics of a compound are considered the outcome of collective contribution of various physiological processes in the body. It is assumed that after administration of a xenobiotic via any route of administration, it appears in the systemic circulation and distributes to different organs/regions, including the elimination organs and the receptor site. Along with the distribution of the compound, certain physiological processes infuence its concentration in blood or plasma and, thus, at the receptor site. These biological processes, including protein-binding, storage in adipose tissues, metabolism by the liver, excretion by the kidney, interaction with the receptor site, infuence by the effux proteins, removal by other organs of elimination, and, now and then, enterohepatic recirculation, are the most important factors (Chapter 1, Figure 1.1). The interaction of the compound with endogenous macromolecules, enzymes, and various tissues is mostly reversible and follows the law of mass action. Thus, the free amount in the blood or plasma can eventually reach to a quasi-equilibrium with the amount bound to macromolecules and tissues. If the equilibrium is achieved very rapidly, as is the case for the highly perfused organs like liver, heart, and kidney, these organs are considered indistinguishable from the systemic circulation and, with plasma/blood, are considered the central compartment. The PK/TK models of compartmental analysis are based on the mathematical description of the time course of xenobiotic concentration in biological samples, including blood, plasma, urine, bile, expired air, saliva, milk, and other biological samples. The change in time course of concentration/amount is the result of the physiological processes of ADME and related factors (see Chapters 7–10). Depending upon the chemical nature of the compound, its behavior in the body, and extent of its interaction with plasma proteins, its concentration is measured as a total (i.e., bound and free drug in blood or plasma), or just free concentration in plasma. The molecules of xenobiotic bound to plasma proteins, blood cells, or peripheral tissues are not free for interaction with the receptor site, and no pharmacological response is expected from the bound molecules or molecules that dwell in muscle or adipose tissues during the distribution. Hence, the changes in free concentration of a xenobiotic in plasma is equivalent to changes of the free concentration at the receptor site. According to the law of mass action and the equilibrium equation (Equation 12.29), the free concentration at the receptor site is proportional to the bound concentration to the receptor site and, consequently, the pharmacological or toxicological response.

( Cp ) free µ ( C free )Receptor site k1 ¾¾ ® C - Rcomplex Þ Response ¾ ( C free )Receptor site + R(Receptor)r ¬ ¾ k2

378

(12.29)

APPROACHES IN PK/PD AND TK/TD MATHEMATICAL MODELING

Based on the above deduction, the following are the assumptions of linear compartmental analysis: ◾ Only free xenobiotic interacts with the receptor site to provide the pharmacological or toxicological response. ◾ The concentration of free molecules of xenobiotics in the circulatory system is proportional to its concentration at the receptor site. ◾ In linear compartmental analysis, all physiological processes are governed by frst-order kinetics (Addendum I, Section A.6.1), thus the rate constants, half-lives, and clearances of biological processes remain constant during absorption and disposition (distribution and elimination (i.e., excretion and metabolism) of a compound). The above assumptions facilitate the development of mathematical formulas of linear compartmental models, taking into consideration the physicochemical characteristics of a xenobiotic and its disposition profle in the body. As the systemic circulation is connected to peripheral tissues and organs, it is assumed that changes in free plasma concentration refects changes in concentration of a drug in the peripheral compartments. Thus, the free plasma concentration–time curve can provide important information about the ways the body handles the drug. The basic approach in this quantifcation is to summarize the data of the free concentration–time curves and/or its metabolite(s) by mathematical equations or models. These mathematical relationships are based on conceiving the body as discrete compartments. This means that the body may be divided into one or more compartments without making any anatomical assumptions about the specifc content of these compartments. Contrary to the physiological models, the inputs and outputs of the compartments are not identifed as blood fow or diffusion fuxes but are defned in general forms of frst-order rate constants. The compartments are assumed as homogeneous well-mixed combinations of various organs or tissues that together handle a compound’s disposition (Rescigno and Serge, 1966; Jacquez, 1972; Wagner, 1975; Gibaldi and Perrier, 1982). As indicated earlier, the highly perfused tissues, because of their rapid blood fow rate and quick equilibration with blood, are pooled with systemic circulation and considered the central compartment. The central compartment is the major sampling compartment. To include other regions of the body that reach equilibrium with the concentration in the systemic circulation at a slower rate, or when the compound is taken up preferentially by a particular organ or region, more complex models with two, three, or four compartments are used, depending on the outcome of the curve-ftting analysis of the data. The optimum number of compartments for a xenobiotic is determined by ftting the plasma concentration–time curve and statistical considerations, such as Akaike information criteria (AIC) and/or Schwarz information criterion (also known as Bayesian information criteria, BIC) (see Addendum I, Part 2, Section A.7). Depending on the disposition profle of a compound, the simultaneous curve-ftting of the plasma concentration and cumulative urinary data, or plasma concentration and cumulative biliary data, or plasma concentration simultaneously ft with cumulative urinary and biliary data are considered more useful to achieve a more rational ft for data. All commercially available pharmacokinetic software packages have relevant statistical tests for goodness of ft to determine the optimum number of compartments in this type of modeling (Metzler, 1981). 12.3.1 Linear Dose-Independent Compartmental Analysis The mathematical description of a compartmental model is defned by a set of differential equations for concentration, or amount change in a compartment, and their related integrated equations. The descriptions of a few models are discussed here to review practical approaches for establishing the equations of the models. 12.3.1.1 Mathematical Descriptions of a Xenobiotic Administered via an Extravascular Route of Administration: Time Course of the Amount Change at the Site of Absorption in the Body and the Eliminated Amount from the Body The extravascular routes of administration are discussed in Chapter 2 (buccal and sublingual), Chapter 3 (nasal, pulmonary, and oral), Chapter 4 (intramuscular, intraperitoneal, and subcutaneous), Chapter 5 (transdermal, intradermal, and intraepidermal), Chapter 6 (rectal and vaginal). The models used for the oral route of administration are most often applicable to other extravascular routes of administration. 379

12.3 LINEAR PK/TK COMPARTMENTAL ANALYSIS

Figure 12.7 is a typical simple one-compartment model used for compounds administered by an extravascular route of administration. where compartment AD represents the amount at the site of absorption at time t after the administration of the dose, compartment At is the amount of free drug in the central compartment, which includes plasma and highly perfused tissues at time t and represents the product of plasma concentration of free drug Cpt and the apparent volume of distribution of the compartment (i.e., At = Cpt ´ Vd), and compartment Ael is the amount of the xenobiotic eliminated from the body at time t ). Although Figure 10.7 shows three compartments, the model is a one-compartment model with respect to the central compartment. The number of compartments represents the distributional regions in the body. The rate of change in the central compartment is the difference between the rate of input (i.e., the absorption rate) and the rate of output (i.e., elimination rate). The site of absorption has only an output rate, which is negative and represents the decline of the administered dose at the site of absorption with respect to time. The end compartment Ael has only one positive input rate that corresponds to the accumulation of the compound with respect to time outside of the systemic circulation. The input and output are defned by the rate constants k a (absorption rate constant) and K (overall elimination rate constant). Both biological processes of absorption and elimination are governed by frst-order kinetics, thus the rate constants have units of “time−1” (see Addendum I, Part 2, Section A.6.1), and the rate of amount change for each compartment is defned by the differential equation of the compartment and can be integrated by the Laplace transforms (Appendix I, Part 2, Section A.1). The integrated equations describe the measurable amount in a compartment as a function of time. Differential equation and Laplace transform of the site of absorption, AD : dAD = -k a AD dt ˜ D - AD0 = -k a A ˜ D Þ ( s + ka ) A ˜ D = AD0 Laplace transform : sA Differential equation :

0 ˜ D = AD \A s + ka

(12.30)

(12.31)

Differential equation and Laplace transform of the central compartment, At : Differential equation :

dAt = k a AD - KAt dt

˜ - 0 = k a AD - KA ˜ Laplace transform : sA ˜ D + (s + K ) A ˜ =0 \-k a A

(12.32) (12.33)

Differential equation and Laplace transform of the elimination compartment, Ael : Differential equation :

dAel = KA dt

(12.34)

Figure 12.7 Diagram of a one-compartment model with respect to the amount in the systemic circulation (At); absorbable amount at an extravascular site of absorption (AD); and the amount eliminated from the body (Ael ); the absorption rate constant ka and the overall elimination rate constant are assumed frst-order kinetic rate constants; the model is applicable for absorption of a xenobiotic that follows frst-order kinetics and absorbs from the extravascular routes of administration like buccal, sublingual, nasal, oral, intramuscular, intraperitoneal, subcutaneous, rectal, vaginal, etc. 380

APPROACHES IN PK/PD AND TK/TD MATHEMATICAL MODELING

˜ el - 0 = kA ˜ Laplace transform : sA ˜ + sA ˜e =0 \-KA

(12.35)

The inverse Laplace transform of AD (Equation 12.31) gives rise to the equation defning the amount of xenobiotic at the site of absorption as a function of time, t :

(

AD = AD0 e -kat

)

(12.36)

˜ ) and ( A ˜ el ) are the Laplace transform of At and Ael , respectively. The inverse Laplace transform (A ˜ of A , representing the amount in the body at time t, is determined from Equations 12.31 and 12.33, using a quotient of 2 × 2 determinants (Addendum I, Part 2, Section A.2) and the Laplace table (Addendum I, Table A.1). At time zero when the dose is introduced at the site of absorption, the ˜ is zero. systemic circulation concentration is zero, therefore the initial condition for A s + ka ˜ = -k a A s + ka -k a

AD0 0 AD0 ´ k a = 0 ( s + ka )( s + K ) s+K

(12.37)

Thus, the integrated equation describing the change in amount of the central compartment as a function of time is: \ At =

( A )(k ) 0 D

a

K - ka

(e

-Kt

- e -kat

)

(12.38)

To determine the amount eliminated from the body per unit of time, Ael , three Laplace transforms (Equations 12.31, 12.33, and 12.35) are solved simultaneously with the help of a quotient of 3 × 3 determinants (Addendum I, Part 2, Section A.2): s + ka -k a ˜ el = A

0 s + ka -k a 0

0 s+K -K 0 s+K -K

AD0 0 0 (AD0 )(K )(k a ) = 0 s(s + k a )(s + K) 0 s

(12.39)

Taking the inverse Laplace transform of Equation 12.39 (using Table A.1, Addendum I) provides the following equation representing the time course of amount eliminated from the body at time (t): æ 1 Ke -kat - k a e -Kt ö Ael = AD0 Kk a ç + ÷ k a K(k a - K) ø è kaK

(12.40)

æ Kee -kat k e -Kt ö or, Ael = AD0 ç 1 + - a ÷ (k a - K) (k a - K) ø è In the above integrated equations (Equations 12.36, 12.38, and 12.40), AD0 is the absorbable amount of administered dose at time zero, Ael is the total or cumulative amount eliminated up to the time t, AD represents the amount of the compound at the site of absorption at time t, At is the amount in the body at time t, and the rate constants of absorption and elimination are k a and K , respectively. The time course profle of AD , At , and Ael are presented as a linear plot and log linear plot in Figure 12.8. 12.3.1.2 Mathematical Description of a Xenobiotic Administered Intravenously – Time Course of the Amount Change in the Body, Formation of Metabolite(s), and Elimination from the Body A one-compartment model, like the one discussed in Section 12.3.1.1, is considered here for a xenobiotic that also metabolizes and eliminates partially as metabolite from the body, as depicted in Figure 12.9. The dose is introduced instantaneously by an intravenous bolus injection into the 381

12.3 LINEAR PK/TK COMPARTMENTAL ANALYSIS

Figure 12.8 Typical profles of the time course of the compartments in the model presented in Figure 12.7, which include the decline of administered amount (dose) at an extravascular site of absorption, the skewed bell-shape curve of the amount in the body or systemic circulation, and the cumulative amount excreted/eliminated from the body; the upper right corner inserted plots represent the profle of the changes when the amounts are plotted as logarithm of amount vs time (semilogarithmic plot); the linear portions of the plots exhibit the characteristics of frst-order kinetics, or dose-independent linear PK/TK.

Figure 12.9 Schematic of a one-compartment open model with respect to unchanged xenobiotic in the body/systemic circulation; the model includes the frst-order rate constant for the formation of metabolite, k m , the frst-order rate constant for the elimination of the metabolites from the body, k me , and the frst-order rate constant for the excretion of unchanged metabolites in the urine, k e . The intravenous bolus input is instantaneous and has no rate is associated with the administration of the dose; the compartments are: At the amount in the body/systemic circulation at time t ; Am , the amount of the metabolite in the body; Ame , the amount of metabolite eliminated out of the body; and Ae , the cumulative amount of the xenobiotic excreted unchanged in the urine. 382

APPROACHES IN PK/PD AND TK/TD MATHEMATICAL MODELING

central compartment At . There is no rate associated with the injection of the dose. Thus, the initial condition for the central compartment is equal to the administered dose. Compartment Ae is the cumulative amount excreted unchanged via the urine at time t. Compartment Am is the amount of metabolite formed in the body at time t. Compartment Ame is the cumulative amount of metabolite eliminated from the body at time t. The model is still one compartment with respect to the unchanged drug and one compartment with respect to the metabolite. If the compound has more than one metabolite, and the analytical methodology allows the measurement of the metabolites in plasma, for each metabolite, one additional compartment can be considered to the model if it is necessary and if it does not complicate the estimation of too many constants and parameters during the curve-ftting process. The Laplace transforms and integrated equations of the model are: Compartment At : ˜ = A

AD0 (s + K ) (12.41)

At = AD0 e -Kt Compartment Ae: 0 ˜ e = k e AD . A s(s + K)

Ae =

k e AD0 (1 - e -Kt ) K

(12.42)

Compartment Am: ˜m = A

k m AD0 , (s + k me )(s + K)

Am =

k m AD0 (e -kmet - e -Kt ) K - k me

(12.43)

Compartment Ame : ˜ me = A

k me k m AD0 s(s + k me )(s + K) k m AD0 K

é ù 1 (12.44) Ke -kmet - k me e -Ktt ú ê1 + û ë k me - K The frst-order rate constants include the excretion rate constant k e , the metabolic rate constant k m , elimination rate constant of the metabolite k me , and the overall elimination rate constant, K , is equal to k e plus k m . The input-disposition functions (Benet, 1972) can also be applied to determine the Laplace transforms of the compartments of Figure 12.9 as presented in Table 12.3 (also see Addendum I, Section A.3). Ame =

(

)

12.3.1.3 Mathematical Relationships of the Central Compartment for an Intravenously Administered Xenobiotic that Follows Multicompartment Model: Use of Input-Disposition Function and General Partial Fraction Theorem Application of the input-disposition function for obtaining the integrated mathematical relationship of a multicompartment model, as depicted in Figure 12.10, is described below. The model consists of a central compartment and three peripheral compartments interdependent with the central compartment. Because of the exit rate constants, all four compartments are considered drivingforce compartments with elimination into an outside environment, which may not be necessarily the same one. The overall exit rate constants are: Compartment A1 : E1 = k10 + k12 + k13 + k14

(12.45)

383

12.3 LINEAR PK/TK COMPARTMENTAL ANALYSIS

Table 12.3 Summary of Input-Disposition Functions of Model Depicted in Figure 12.9 Compartment A

Input Function

Disposition Function

Laplace Transform

A

1 s+K

Ae

˜ ke A

1 s

Am

˜ km A

1 s + k me

˜ ˜ m = km A A s + k me

˜m = A

Ame

˜m k me A

1 s

˜ ˜ me = k me Am A s

˜ me = A

0 D

0 D

˜= A A s+K ˜e = A

˜ ke A s(s + K )

0 ˜ e = k e AD A s ( s + ke )

k m AD0 ( s + kme )( s + K ) k me k m AD0 s ( s + k me )( s + K )

Figure 12.10 A representation of multicompartment mammillary model with a central compartment and three peripheral compartments, interdependent with the central compartment. The four compartments with exit rate to the outside environment, not necessarily the same environment, are considered the driving-force compartments and include A1 , A2 , A3 and A4 ; A1 is the central compartment representing the systemic circulation; the frst-order rate constants of distribution are k12 , k 21 , k13 , k 31 , k14 and k 41 ; the exit rate constants of the peripheral compartments are k 20 , k 30 , k 40 , and the overall elimination rate constant from the central compartment that includes the excretion of unchanged compound and elimination of the metabolites is k10 ; the input into the central compartment is bolus intravenous injection. Compartment A2 : E2 = k 20 + k 21

(12.46)

Compartment A3 : E3 = k 30 + k 31

(12.47)

Compartment A4 : E4 = k 40 + k 41

(12.48)

The disposition function of the compartment 1 according to Equation A.33 (Addendum I, Part 2) is as follows: (Disp.)s,1 =

(s + E2 )(s + E3 )(s + E4 ) (s + E1 )(s + E2 )(s + E3 )(s + E4 ) - k12 k 21 2 (s + E3 )( s + E4 ) - k13 k 31 ( s + E2 )( s + E4 ) - k14 k 41 ( s + E2 )( s + E3 ) (12.49)

384

APPROACHES IN PK/PD AND TK/TD MATHEMATICAL MODELING

Following the substitution of the miniature rate constants of the model for E1, E2 , E3 , and E4 (Equations 12.45–12.48) in the denominator of Equation 12.49, implementing mathematical manipulation and expressing a , b, g , and d as the hybrid rate constants of the new combination of miniature rate constants, the equation will change to the following disposition function: (Disp.)s1 =

(s + E2 )(s + E3 )(s + E4) (s + a)(s + b)(s + g)(s + d)

(12.50)

Multiplication of the disposition function (Equation 12.50) with any of the input functions yields the Laplace transform of the central compartment following that input. For example, the Laplace transform of compartment A1 with the intravenous bolus dose as the input is ˜ = Input ´(Disp.)s1 = Dose ( s + E2 )( s + E3 )( s + E4 ) L ( A1 ) = A ( s + a ) ( s + b) ( s + g ) ( s + d)

(12.51)

The inverse Laplace transform of the central compartment (Equation 12.51), using the partial fraction theorem (Benet and Turi, 1971) (see also Addendum I, Part 2, Section A.3) is A1 =

Dose(E2 - a)(E3 - a)(E4 - a) -at Dose(E2 - b)(E3 - b)(E4 - b) -bt e + e (b - a)( g - a)(d - a) ( a - b )( g - b )( d - b )

Dose(E2 - g)(E3 - g)(E4 - g) -gt Dose ( E2 -d )( E3 - d )( E4 - d ) -dt + e + e ( a - g )(b - g )( d - g ) ( a - d ) (b - d )( g - d )

(12.52)

where A1 is the amount in the central compartment at time t. The four exponential terms of the equation are created when s, in the Laplace transform of the compartment, is set equal to −α, −β, −γ, and −δ followed by multiplication with their exponential terms. 12.3.1.4 Mathematical Relationships of the Peripheral Compartment for an Intravenously Administered Xenobiotic that Follows Multicompartment Model: Use of Input-Disposition Function and General Partial Fraction Theorem Once the Laplace transform of the central compartment is established, the Laplace transform of the peripheral compartments can be obtained by using the method of substitution. For instance, for compartment A2 of Figure 12.10, the related equations are dA2 = k12 A1 - k 21 A2 - k 20 A2 dt

(12.53)

Setting the overall exit rate constant as E2 = k 21 + k 20

(12.54)

dA2 = k12 A1 - E2 A2 dt

(12.55)

Equation 12.53 changes to

é dA ù ˜ 2 - 0 = k12 A ˜ 1 - E2 A ˜2 L ê 2 ú = sA ë dt û ˜ ˜ 2 = k12 A1 A (s + E2 )

(12.56) (12.57)

˜ 1 , as defned in Equation 12.51, into Equation 12.57 yields the Laplace transform of Substitution of A the second compartment, A2 , as ˜ 2 = k12 [Dose(s + E2 )(s + E3 )(s + E4 )] A (s + E2 )(s + a)(s + b)(s + g)(s + d) =

k12 [Dose(s + E3 )(s + E4 )] (s + a)(s + b)(s + g)(s + d)

(12.58)

Therefore, the integrated equation is

385

12.3 LINEAR PK/TK COMPARTMENTAL ANALYSIS

A2 = +

E4 - b] -bt k12 [ Dose(E3 - a)(E4 - a)] -at k12 [ Dose(E3 - b)(E e + e (b - a)( g - a)(d - a) (a - b)( g - b)(d - b) k12 [ Dose(E3 - g)(E4 - g)] -gt k12 [ Dose(E3 - d)(E4 - d] -dt e + e (a - g )(b - g)(d (a - d)(b - d)( g - d) ( - g)

(12.59)

Thus, the input function into the second compartment is k12 multiplied by the Laplace transform of the central compartment. 12.3.1.5 Mathematical Relationships When a Xenobiotic and Its Metabolite(s) Follow Multicompartmental Model – Intravenous Bolus Dose Figure 12.11 is the depiction of the model when both parent compound and its metabolite(s) follow the two-compartment model. Compartment A1 is the central compartment representing the amount of the parent compound in the systemic circulation, which also represents the highly perfused tissues; compartment A2 is the amount of parent compound in the peripheral compartment; compartment A3 is the amount of metabolite(s) in the central compartment, and A4 is the amount of metabolite(s) associated with its peripheral compartment. The rate constant of formation of metabolite(s) is k13 , the elimination rate constants of the parent compound and its metabolite(s) are k10 and k 30 , respectively; the distribution rate constants of the parent compound between the central and peripheral compartment are k12 and k 21 . The distribution rate constants of the metabolite(s) between compartment A3 and A4 are k 34 and k 43 . The differential equations of the compartments are dA1 = k 21 A2 - k12 A1 - k10 A1 - k13 A1 dt

(12.60)

dA2 = k12 A1 - k 21 A2 dt

(12.61)

Figure 12.11 Diagram of a linear two-compartment model with respect to unchanged xenobiotic and two-compartment model with respect to its metabolite; a requirement for use of this model is a sensitive analytical methodology that can measure not only the plasma concentration of free xenobiotic, Cpt , but also the concentration of the metabolite, C3 , in plasma; A1 and A2 are the amount of unchanged xenobiotic as a function of time in the central and peripheral compartments, Ae is the cumulative amount of excreted xenobiotic in the urine; A3 and A4 are the amount of metabolite as a function of time in the central and peripheral compartments, and Ame is the cumulative amount of metabolite eliminated from the body; k12 and k 21 are the distribution rate constants of unchanged xenobiotic between central and peripheral compartments; k13 is the rate constant of metabolite formation; k 43 and k 34 are the distribution rate constants of the metabolite between the central and the peripheral compartment; and k 30 is the rate constant of the metabolite elimination; the input is the rapid intravenous bolus injection; V1 and V3 are the apparent volume of distribution of the central compartment for unchanged xenobiotic and its metabolite, respectively. 386

APPROACHES IN PK/PD AND TK/TD MATHEMATICAL MODELING

dA3 = k13 A1 + k 43 A4 - k 34 A3 - k 30 A3 dt

(12.62)

dA4 = k 34 A3 - k 43 A4 (12.63) dt The integrated equations for the unchanged xenobiotic and its metabolite in terms of their concentration in the central compartment (Cp = A1 / V1 and C3 = A3 /V3 ) are Cp =

Dose æ k 21 - a -at k 21 - b -bt ö e + e ÷ ç a -b V1 è b - a ø

(12.64)

where a and b are the hybrid rate constants representing:

C3 =

a + b = k13 + k12 + k 21

(12.65)

ab = k13 k 21

(12.66)

( k21 - b )( k43 - b ) e -bt k13 D é ( k 21 - a )( k 43 - a ) -at e + ê V3 êë (b - a )( d - a )( g - a ) ( a - b )( d - b )( g - b )

( k21 - d )( k43 - d ) e -dt + ( k21 - g )( k43 - g ) e -gt ù + ú ( a - d ) (b - d ) ( g - d ) ( a - g )(b - g )( d - g ) úû

(12.67)

The hybrid disposition rate constants of metabolite(s) correspond to the following relationships: d + g = k 30 + k 34 + k 43

(12.68)

dg = k 30 k 43

(12.69)

The complex models such as the one depicted in Figure 12.11, and defned by Equations 12.60– 12.63, are usually calibrated to suffciently defne the experimental data. When the compartmental model suffciently describes the data, it is important to know whether the parameters/constants of the model can be determined unambiguously from the amount and quality of experimental data. This requires further evaluation of the model predictions, which is known as identifability analysis. The methodology for identifability analysis is well established, and when the parameters of a model are determined explicitly, the model is considered uniquely identifable (Bellman and Åström, 1970; Cobelli and Mari, 1983; Chapman and Godfrey, 1987; Chappell et al., 1990; Dutta et al., 1996; van den Hof, 1998). 12.3.2 Dose-Dependent Compartmental Analysis The differentiation between linear and nonlinear PK/TK is critical in compartmental analysis. As in physiological modeling, the nonlinear pharmacokinetics exist when the absorption, metabolism, or excretion are by mechanisms that are potentially saturable. In contrast to the dose-independent PK/TK, where all biological processes follow frst-order kinetics, the saturable nonlinear biological process(es), also referred to as a dose-dependent PK/TK, are governed by the Michaelis–Menten equation (Chapter 9, Section 9.4.1). In linear dose-independent pharmacokinetics, all the rate constants, half-lives, and clearances, are constant, predictable, and independent of the administered dose, whereas in nonlinear PK/TK, parameters and constants are variable and their magnitude depends on the administered dose and plasma concentration. The nonlinearity of PK/TK data is often recognizable from graphical analysis of the data. Figure 12.12 depicts the logarithm of plasma concentration versus time when the compound, at high plasma levels, exhibits the nonlinear or dose-dependent kinetics and at low concentrations shows dose-independent behavior (Gibaldi and Perrier, 1982). The plot of plasma concentration versus time, not logarithmic, exhibits a profle that resembles a hockey stick (Figure 12.13) (Wagner, 1975). 12.3.2.1 Compartmental Models with Michaelis–Menten Kinetics The simplest model for consideration in this category is the one-compartment model, with intravenous bolus injection and the assumption that elimination of xenobiotic from the compartment is concentration-dependent, nonlinear, and governed by the Michaelis–Menten kinetics, and the 387

12.3 LINEAR PK/TK COMPARTMENTAL ANALYSIS

decline of the plasma concentration is dependent on the nonlinear rate of elimination from the compartment, that is, dCp V ´ Cp = - max K M + Cp dt

(12.70)

Taking the following steps to defne the rate -dCp ( K M + Cp ) = dt (Vmax ´ Cp ) -

dCpK M - dCp = Vmax dt Cp

(12.71) (12.72)

The integration yields: -KM ´ ln C = Vmax t + constant

(12.73)

Evaluation of the constant at t = 0 is constant = Cp 0 - K M ln Cp 0

(12.74)

Substitution of Equation 12.74 in Equation 12.73 and solving for the variables yield ln Cp =

Cp 0 - Cp V + ln Cp 0 - max t KM KM

(12.75)

Cp 0 = Vmax t Cp

(12.76)

or, Cp 0 - Cp + K M ln t=

Cp 0 Cp 0 - Cp KM ln + Vmax Cp Vmax

(12.77)

Figure 12.12 Plot of the logarithm of plasma concentration versus time for a xenobiotic that exhibits nonlinear dose-dependent kinetics at high concentrations and linear dose-independent at low concentrations of plasma; the darker shaded area highlights the nonlinear dose dependent kinetic behavior, and the lighter shaded area identifes the dose-independent linear part of the plot; the plot is helpful in identifying the presence of a capacity-limited process in the overall disposition of a compound. 388

APPROACHES IN PK/PD AND TK/TD MATHEMATICAL MODELING

Figure 12.13 Contour of plasma concentration–time curve of xenobiotics that display dosedependent nonlinear behavior; xenobiotic A exhibits faster decline of high plasma level than compound B, but the profle of both compounds resembles the hockey stick; plotting just the plasma concentration versus time may not clearly identify the nonlinearity of the data, however, combined with the log plasma concentration vs time (Figure 12.12), a clearer understanding can emerge. A modifed version of the Equation 12.77, appropriate for curve-ftting by setting the initial condition of both variables, is (Gibaldi and Perrier, 1982): t - t0 =

Cp 0 Cp 0 - Cp KM ln + Vmax Cp Vmax

(12.78)

Using the approaches described in Chapter 9, Section 9.4.4, the initial estimates for Vmax and K M are estimated from plasma concentration–time data. For example, Equation 12.70 can be modifed to Equation 12.79 as a difference equation; then by using the linear Lineweaver–Burk plot (Double Reciprocal), Hanes–Woolfe plot, or Eadie–Hofstee plot, the initial estimates of the parameters can be determined. In the following equations Cpmidpoint corresponds to the midpoint plasma concentraDCp tion that corresponds to the rate defned as Dt DCp V ´ Cp (12.79) = - max K M + Cp Dt The linear versions of the equation are KM 1 1 = + DCp Vmax Cpmidpoint Vmax Dt

(12.80)

Cpmidpoint K M Cpmidpoint = + Vmax DCp Dt Vmax

(12.81)

( DCp Dt ) K M DCp = Vmax Cpmidpoint Dt

(12.82)

When the elimination from the body is by multiple and parallel Michaelis–Menten processes, the rate of elimination is the pooled equation of all processes (Figure 12.14) (Wagner, 1975): 389

12.3 LINEAR PK/TK COMPARTMENTAL ANALYSIS

-

p V Cp dCp Vmax1 Cp V Cp V Cp = + max2 + max3 +…… + max n K Mn + Cp dt K M1 + Cp K M2 + Cp K M3 + Cp

(12.83)

Setting the boundary conditions: at t 0 , Cp = Cp 0 , and as t Þ ¥ , and Cp Þ 0, the dose-dependent parameters of Equation 12.83 can be estimated as (Wagner, 1975):

Vmaxn

é n æ Vmax ö ù é n æ Vmaxi i Cp 0 ê çç ÷ú ê ç 0 ÷ êë i=1 è K Mi + Cp ø úû êë i=11 è K Mi = n n æ Vmaxi ö æ Vmaxi ö ç ÷ ç ÷ K Mi ø i=1 çè K Mi + Cp 0 ÷ø i =1 è

å

å

å

é

n

æ

å ççè K êë i =1

n

å i =1

(12.84)

å

Cp 0 ê K Mn =

öù ÷ú ø úû

æ Vmax1 ö ÷ç è K Mi ø

Vmaxi 0 Mi + Cp

n

å i=1

öù ÷÷ ú ø úû

æ Vmaxxi çç 0 è K Mi + Cp

(12.85) ö ÷÷ ø

0

At Cp = Cp , when plasma has the highest concentration and is greater than K Mn , the parameters are n

Vmaxn =

å(V

maxi

)

(12.86)

i=1

n

å (V

maxi

K Mi =

i=1 n

)

(12.87)

æ Vmaxi ö ÷ Mi ø

å çè K i=1

If the body eliminates a xenobiotic by simultaneous dose-independent excretion and dose-dependent metabolism, the overall rate of elimination is the combination of both processes æ V Cp ö dC + k eCp ÷÷ = - çç max dt K Cp + è M ø \

dCp -k eCp ( K M + Cp ) - Vmax Cp = dt K M + Cp

(12.88) (12.89)

Figure 12.14 Diagram of a pharmacokinetic model with multiple and parallel nonlinear metabolism governed by the Michaelis–Menten equation; the model has no exit rate for excretion of unchanged xenobiotic and the biotransformation into three different metabolites by different enzyme systems are the only routes of elimination; the metabolites may leave the body in urine or bile; the dose is a single intravenous bolus injection and the plasma level vs time would be expected to exhibit the profle presented in Figures 12.12 and 12.13. 390

APPROACHES IN PK/PD AND TK/TD MATHEMATICAL MODELING

or,

dCp Cp ( -k e K M - Vmax - k eCp ) = dt K M + Cp

(12.90)

Similar approaches are used when the absorption of a compound follows frst-order kinetics, but the elimination is governed by dose-dependent metabolism. The rate of concentration change in the body is defned as ˜ dCp V Cp = k aCp 0 - max , and K M + Cp dt

(12.91)

˜ FD Cp 0 = Vd

(12.92)

where F is the fraction of dose absorbed and Vd is the apparent volume of distribution. When the xenobiotic follows a two-compartment model with parallel dose independent excretion and dose dependent metabolism, as depicted in Figure 12.15, the rate of concentration change in the central compartment can be defned as æ ö dCp V Cp = k 21C2 - çç k12Cp + max + k eCp ÷÷ dt K + Cp M è ø

(12.93)

The rate of concentration change in the peripheral compartment remains linear as discussed dC2 before, that is = k12Cp - k 21C2 . dt In the absence of the dose independent excretion of the compound, the rate of plasma concentration change in the central compartment will be a function of the distribution to the peripheral compartment and the concentration dependent metabolism, that is, dCp V Cp = k 21C2 - k12Cp - max dt K M + Cp

(12.94)

It is worth noting that depending on the type of PK/TK software, the experimental data are usually ftted directly to the appropriate rate equations (differential equations). The numerical analysis iterations of the parameters and constants are done in the same manner as the integrated equations. 12.4 NON-COMPARTMENTAL ANALYSIS BASED ON STATISTICAL MOMENT THEORY 12.4.1 Overview The principle of non-compartmental analysis assumes that the passage of a compound through the body is a random process (i.e., stochastic) and, thus, is governed by the principles of probability and can be defned by the probability function. In theory, a set of plasma concentration–time data is considered a statistical distribution. The equation of observations that defnes the distribution is called the probability function, which is characterized by the moments of the function. The moments of the function can be estimated about the origin or about the mean. If f ( x)is the function, the moments of the function are defned by integrating the function with a suitable power of the independent variable over an interval of [ a, b ]: mr =

b

ò x . f (x)dx

(12.95)

r

a

When r, the power of x, is equal to zero, the Equation 12.96 would defne the zero-moment function: zero moment: m 0 =

ò

b

a

f ( x)dx

When r = 1, Equation 12.96 defnes the frst-moment function first moment: m1 =

b

ò x f (x)dx a

When r = 2, it is known as the second-moment function 391

12.4 NON-COMPARTMENTAL ANALYSIS BASED ON STATISTICAL MOMENT THEORY

Figure 12.15 Diagram of a two-compartment model with parallel metabolism and excretion from the central compartment; the distribution between the central compartment and the peripheral compartment and the excretion from the central compartment are assumed linear and doseindependent; the formation of the metabolite is dose-dependent and nonlinear; the distribution rate constants, k12 and k 21 , and the excretion rate constant, k e , are all frst-order rate constants, the nonlinear biotransformation is governed by the Michaelis–Menten equation and the related K M and Vmax parameters; the apparent volume of distribution of the central compartment is V1 , and the amount of the xenobiotic in the central compartment at any time, At , is Cpt ´V1 ; The cumulative amount excreted unchanged and eliminated as metabolite are Ae and Am , respectively.

b

ò x f ( x ) dx

second moment: m 2 =

2

a

When r = n: n th moment: m n =

b

òx a

n

f ( x)dx

Parallel to the above statistical equations of moments, the following equations are developed for the statistical moments of the plasma concentration–time curve (Cutler, 1978; Yamaoka et al., 1978; Benet and Galeazzi, 1979; Riegelman and Collier, 1980; Cawello et al., 1999). The PK/TK based on plasma concentration–time data and the related moments of the origin within the observation interval of a = 0, b = ∞ are

ò

m0 =

m2 =

0

ò

m1 =

¥

0

ò

¥

0

¥

Cpdt =

tCpdt =

t 2Cpdt =

ò

¥

tCpdt +

ò

¥

t 2Cpdt +

ò

tn

0

ò

tn

0

ò

Cpdt +

ò tn

0

Cpdt = AUC0¥

(12.96)

tCpdt =AUMC0¥

(12.97)

t 2Cpdt =AUM2C0¥

(12.98)

tn

tn ¥

tn

where AUC0¥ is the area under the plasma concentration–time curve or the area under the zeromoment curve. This parameter is calculated by a model-independent numerical integration, for

392

APPROACHES IN PK/PD AND TK/TD MATHEMATICAL MODELING

example, linear trapezoidal, or log trapezoidal as described in Addendum I, Part 2, Section A.4, AUMC0¥ is the area under the frst-moment curve, and it refers to the area of observed plasma concentrations multiplied by the corresponding time points, as the dependent variable, plotted against the same time points as the independent variable. The method of estimation is the same as the AUC0¥ . The terminal area of AUMC is calculated as

( Terminal area )AUMC =

Cpntn Cp + 2 K K

The area under the second-moment curve, AUM2C0¥ , is constructed by multiplying each plasma concentration by the square of its time point and then plotted against the time point. The area under the second-moment curve provides a variety of residences for a xenobiotic in the body. The moments higher than AUM2C0¥ are used if the measures of skewness and kurtosis of the distribution need to be defned. The advantages of non-compartmental analysis are the simplicity of the approach, estimating parameters without model assumptions and limitations, and practical use in drug development, clinical trials, and FDA approval. However, the methodology does not elucidate fully the organ specifc function or the related physiological parameters and constants. The parameters are solely based on the observed data, and thus the analysis is sensitive to experimental protocol, such as frequency and duration of sampling. 12.4.2 Mean Residence Time and Mean Input Time Normalizing the frst and second moments (Equations 12.97 and 12.98) with respect to the zero moment (Equation 12.96), yields the following PK/TK parameters: m1 AUMC0¥ = AUC0¥ m0

(12.99)

m 2 AUM2C0¥ = AUC0¥ m0

(12.100)

MRT = VRT =

where MRT is the mean residence time (Benet, 1985; Cutler, 1987; Kasuya et al., 1987) and VRT is the variance of the residence time that may refect the magnitude of the random error. For xenobiotics that are given by bolus injection and follow the one-compartment model, Equation 12.99 can also be presented as Cp ˜ 2 1 (12.101) MRT = K = = 1.44 ´ T1 2 Cp ˜ K K Therefore, the mean residence time for xenobiotics that follow the one-compartment model is equal to 1.44T1 2 , which is also known as time constant, turnover time, transit time, or sojourn time (Shipley and Clark, 1972). The value of MRT is also estimated by dividing the area under the curve by the initial plasma concentration. MRT =

AUC0¥ Cp ˜

(12.102)

For compounds that follow the two-compartment model (Equation 12.103), the MRT equation can be estimated as Equation 12.104. Cp = ae -at + be -bt æ a b ö æ a bö MRT = ç 2 + 2 ÷ ç + ÷ a b è ø èa bø When α ≫ β, the mean residence time can be estimated as: MRT @

1 = 1.44 ´ ( T1/2 )b b

(12.103) (12.104)

(12.105)

393

12.4 NON-COMPARTMENTAL ANALYSIS BASED ON STATISTICAL MOMENT THEORY

If the system and data are considered linear and dose-independent, the mean residence time would represent the time required for 63.3% of the dose to be eliminated via all routes of elimination. Since in linear systems AUC is equal to Dose ClT , MRT can also be estimated as MRT =

AUMC ´ ClT Dose

(12.106)

When the dose is administered by zero-order infusion or frst-order absorption, the mean residence time is equal to the mean residence time of bolus injection plus the mean input time (MIPT) such as mean infusion time or mean absorption time, that is, T 2 where T represents the time of infusion and 0.5 T is the mean infusion time MIT . For oral administration MRTinfusion = MRTbolus +

MRToral = MRTbolus +

1 ka

(12.107)

(12.108)

where k a is the absorption rate constant and 1/k a represents the mean absorption timeMAT . The MAT is also estimated as: MAT = MRToral - MRTbolus =

1 = 1.44 ( T1 2 )absorption ka

(12.109)

When the absolute bioavailability is one or close to one, MAT represents the true mean absorption time (Cutler, 1978). If the bioavailability is low due to the pre-systemic metabolism or decomposition of the drug at the site of absorption, etc., the value of MAT is only an apparent value. In general, for the routes of administration such as oral route, where bioavailability is a concern, the MRT can be estimated as MRToral = MRTbolus ´

( AUC ) F ´ ( AUC ) ¥ 0

oral. ¥ 0 bolus

+ MIPT

(12.110)

12.4.3 Total Body Clearance and Apparent Volume of Distribution The total body clearance in non-compartmental analysis is estimated by the model-independent equation of Dose AUC0¥

(12.111)

F ´ Dose AUC0¥

(12.112)

( ClT )bolus = ( ClT )oral =

Equation 12.112 is also used for other routes of administration when bioavailability is a concern. Substituting AUC0¥ from Equation 12.99 into Equation 12.112 yields the following relationship for the total body clearance:

( ClT )bolus =

FD ´ MRT Dose ´ MRT or ( ClT )oral = AUMC AUMC

(12.113)

The apparent volume of distribution of the non-compartmental model (expressed as Vdss ), is estimated from total body clearance and the mean residence time

(Vd )ss = ClT ´ MRT

(12.114)

Substitution of Equations 12.113 and 12.99 in Equation 12.114 yields the following relationships for the apparent volume of distribution (Benet and Galeazzi, 1979):

(Vd )ss = çæ

Dose öæ AUMC ö ÷ç ÷ è AUC øè AUC ø

(12.115)

Dose ´ AUMC AUC 2

(12.116)

(Vd)ss =

394

APPROACHES IN PK/PD AND TK/TD MATHEMATICAL MODELING

Equations 12.114–12.116 are used for xenobiotics that are given by bolus injection regardless of the dispositional profle. Substitution of Equations 12.110 and 12.112 in Equation 12.114 provides the volume of distribution of a compound administered orally and other extravascular routes of administration Vdss =

(

(

)

é ù AUC0¥ F ´ Dose ê oral. ú MRT ´ + MIPT M bolus ¥ ¥ ê ú F ´ AUC0 AUC0 oral ë bolus û

(12.117)

Dose ´ MRTbolus F ´ Dose ´ MIPT + AUCbolus AUCoral

(12.118)

)

\Vdss =

(

)

The volume of distribution for compounds given by short-term infusion is defned as Vdss =

Ainfused ´ AUMC Ainfused ´ T AUC 2 2 ´ AUC

(12.119)

where Ainfused is the total amount administered by infusion, expressed as Ainfused = k0 ´ T , that is, the rate of infusion multiplied by the time of infusion. Therefore, Vdss is also presented as Vdss =

k0 ´ T ´ AUMC k ´T2 - 0 2 AUC 2 ´ AUC

(12.120)

12.5 PK-PD AND TK-TD MODELING 12.5.1 Overview As reviewed in Chapter 1, Sections 1.2 and 1.3, the quantitative evaluation of a xenobiotic’s interaction with its receptor site and its pharmacological response is known as pharmacodynamics (PD), and the evaluation of the interaction and resulting response at toxic level exposure is known as toxicodynamics (TD). As for any dynamic processes, the initial and fnal conditions of response are the important parameters in PD/TD analysis. In other words, at the initial phase of the evaluation when the dose is zero, no response is anticipated. When the dose is optimum, the desired outcome is expected, and when the dose is high and beyond the acceptable range, the toxic response is expected. The outcome, whether therapeutic or toxic, is a function of the amount or concentration of a xenobiotic and its affnity for interaction with its receptor. In most in vitro systems, for example in cell culture, the concentration of xenobiotic can be kept constant and the interaction with its receptor sites, which are mostly located on the cell membrane, is direct and rather rapid during optimized experimental conditions. Under in vivo conditions, however, the concentration of a xenobiotic at the receptor site, or biophase, is a function of time and the physiological processes of ADME. Therefore, the in vivo PD/TD observation is a function of the PK/TK profle of the compound. In PD/TD analysis, the response is measured directly or indirectly. The examples of direct response are the measurable physiological/biochemical changes such as lowering blood pressure, reducing intraocular pressure, change in the coagulation time of blood, binding to macromolecules, interaction with enzymes, etc. The examples of indirect response are the measuring-specifc biomarkers, such as serum levels of creatine kinase MB, lactate dehydrogenase, troponin T, and endothelin I in cardiomyopathy evaluation. In a dose–response relationship, when the dose is zero the response is zero, as the magnitude of the dose increases the magnitude of response increases until increasing the dose no longer increases the response. This maximum response is denoted as Emax and the dose that corresponds to half of Emax is known as D50 (Figure 12.16). The scenario is analogous to the Michaelis–Menten kinetics discussed in Chapter 9, Section 9.4.1. The plot of response versus logarithm of dose is sigmoidal as presented in Figure 12.17. The dose–response or dose–effect curves can also be presented as concentration–response or log concentration–response curve. 12.5.2 Xenobiotic–Receptor Interaction and the Law of Mass Action Xenobiotics interact with their receptor site according to the following relationship: ¾¾ ¾ ® [ RD] ¾¾ ® Effect [ R] + [ D] ¬ ¾

(12.121)

395

12.5 PK-PD AND TK-TD MODELING

where [ R] is the concentration of free receptor, [D] is the concentration of free compound in the environments of the receptor and [ RD] is the concentration of the xenobiotic–receptor complex. Equation 12.121 is a typical complexation reaction with the following assumptions: 1. [D] and [ R] interact rather rapidly to form [ RD] complex. 2. Only a single molecule of [D] interacts with a single receptor site. 3. [ R] , [D], and [ RD] are at equilibrium. 4. The interaction is reversible From the relationship presented in Equation 12.121, the association constant K A and dissociation constant K D are estimated as KA =

[RD] [R][D]

(12.122)

KD =

[R][D] [RD]

(12.123)

The density of free receptor [R] is the difference between total density [ RT ] (i.e., free and bound), and bound density [ RD]

[ R] = [ RT ] - [ RD]

(12.124)

Substitution of Equation 12.124 in Equation 12.123 yields:

Figure 12.16 Depiction of the hyperbolic dose-response plot implying that, in the absence of the dose, the response is zero; by increasing the dose the response intensifes until it reaches a plateau level representing the maximum response; the maximum response,Emax , may or may not be the optimum response for some therapeutic xenobiotics, but it is the maximum effect; the Emax represents the total occupancy of the receptors and increasing the dose would only increases the extracellular concentration at the receptor site without changing the response; the amount that corresponds to half-maximum effect (i.e., Emax 2) is referred to as D50 , and in terms of concentration, it refers to E50 , i.e., the free concentration of xenobiotic at which the effect is half-maximum and refects the potency of interaction between the xenobiotic and target site. 396

APPROACHES IN PK/PD AND TK/TD MATHEMATICAL MODELING

Figure 12.17 Depiction of the sigmoidal curve of response versus log of dose; similar sigmoidal curve is also obtained when the response is plotted against free concentration of xenobiotics and the x-intercept of the extrapolated linear part of the curve correspond to log Cp 0 ; the equation that n defnes the sigmoidal curve is known as Hill equation E = Emax ´ Cp n E50 + Cp n ; an important

(

)(

)

application of this plot is to determine how low is the safe dose for xenobiotics that are considered carcinogenic, or toxic agents; the other application of the plot is to determine the initial estimates of the parameters used in PK/PD or TK/TD models. KD =

([R ] - [RD]) ´ [D] = [R ][D] - [RD][D] T

T

[ RD] [ RD] [ RD](KD + [D] = [ RT ][D] [ R ][ D ] [ RD] = K T+ D [ ] D

(12.125) (12.126) (12.127)

Equation 12.127 indicates that [ RD] is directly proportional to [D] at the receptor site, and according to the relationship in terms of plasma concentration, it is also proportional to the plasma concentration, as in Equation 12.128.

[ RT ] éëCpùû éë RCpùû = K D + éëCpùû

(12.128)

An important assumption useful in defning Equations 12.127 and 12.128 is the “occupancy theory,” which states the intensity of the response is proportional to the number of receptors occupied by the compound, é RCpù (Ariens, 1964). Thus, the maximum response, Emax , occurs when all ë û available receptor sites [ Rt ] are occupied; in other words, the maximum response is proportional to complete occupancy of receptor sites, i.e., [ Rt ] µ Emax . Thus, Equation 12.128 can be presented as E=

Emax ´ [D] K D + [D]

(12.129)

Equation 12.129 can also be presented in the following linear version:

397

12.5 PK-PD AND TK-TD MODELING

æ ö E log ç ÷ = log[D] - log K D è Emax - E ø

(12.130)

æ ö E Thus, the plot of ç log ÷ versus [D] is a straight line with slope equal to one (slope = 1) and E E max è ø y-intercept is equal to log K D. E When the effect is half-maximum, that is, E = max , the constant KD will be equal to the concen2 tration that corresponds to 50% of maximum response (Figure 12.16), that is, Emax Emax ´ [D]50 = 2 K D + [D]50

(12.131)

Emax Emax ´ ëéCpùû 50 = 2 K D + éëCp ùû 50

(12.132)

K D = [D]50 = éëCpùû 50 = E50

(12.133)

The dose-response relationship is used in toxicodynamics to develop an understanding of the integrated biological processes underlying a response, hazard identifcation, and safety assessment. The determination of the lowest dose causing adverse effects, or determination of the threshold effect, or a No-Observed-Adverse-Effect-Level (NOAEL) in the dose-response evaluation of lead compounds in drug discovery and development, has been associated with analytical uncertainty and biological variability. PK/PD or TK/TD modeling in association with toxicogenomic and population PK/TK analysis has reduced this uncertainty signifcantly. 12.5.3 Pharmacodynamic Models of Plasma Concentration and Response When the response is measured as the outcome of a therapeutic dose or a known level of exposure, the magnitude of the outcome can be linked to plasma concentration through linear or nonlinear relationships. 12.5.3.1 Linear Pharmacodynamic Model The dependent variable in this model is the response/effect, which is assumed to be directly proportional to plasma concentration, that is, E = mCp + b

(12.134)

where b, the y-intercept, is the response at time zero, that is, b = E0 , and the model assumes a greater response is associated with higher plasma concentration. The maximum effect, Emax , cannot be estimated from Equation 12.134. The model is useful for the correlation of response with the low concentrations when the receptor sites are not fully occupied (Figure 12.18). 12.5.3.2 Log-Linear Pharmacodynamic Model This model assumes that the intensity of response is linearly dependent on the logarithm of plasma concentration (Levy, 1964, 1966) E = mlog Cp + b

(12.135)

Equation 12.135 defnes the linear segment of the sigmoidal curve presented in Figure 12.17. It follows that the response at time zero corresponds to the plasma concentration at time zero: E0 = mlog Cp 0 + b

(12.136)

0

Solving for log Cp in Equation 12.135, and log Cp in Equation 12.136 provides the following relationships (Equations 12.137 and 12.138):

398

log Cp =

E-b m

(12.137)

log Cp˜ =

E0 - b m

(12.138)

APPROACHES IN PK/PD AND TK/TD MATHEMATICAL MODELING

Substitution of Equations 12.137 and 12.138 in equation log Cp = log Cp 0 E - b E0 - b Kt = m m 2.303 \ E = E0 -

K ´t´m 2.303

Kt yields: 2.303 (12.139) (12.140)

Equation 12.140 represents the zero-order time course of response as a function of −(K/2.303), the slope of log plasma concentration versus time curve, and m the slope of the response E versus the log Cp curve. In other words, the plasma concentration–time curve of a xenobiotic declines exponentially according to frst-order kinetics, whereas the intensity of response decreases with time according to zero-order kinetics. 12.5.3.3 Nonlinear Hyperbolic Emax Model Based on Equations 12.129 and 12.131–12.133, the relationship between the effect and plasma concentration can be presented as Equation 12.141 (Wagner, 1968) E=

Emax ´ Cp E50 + Cp

(12.141)

where Emax is the maximum response and corresponds to total receptor-binding RT , and E50 is the half-maximum effect. Equation 12.141 indicates that the effect is a rectangular hyperbolic function of the free plasma level of the compound. When the effect is considered a rectangular hyperbolic function of receptor–xenobiotic concentration complex RCp , the relationship between the effect E and the complex RCp , using Equation 12.128, can be defned as E=

Emax ´ éë RCpùû K E + éë RCpùû

(12.142)

where K E corresponds to the concentration of RCp that brings forth the half-maximum effect.

Figure 12.18 Depiction of the linear pharmacodynamic model according to equation E = mCp + b that occurs at the low concentrations of free xenobiotic in plasma; the y-intercept, b, is the response at time zero, b = E0 ; the extrapolated line also represent the limit of linearity of the dose-response curve and the linear pharmacodynamic model. 399

12.5 PK-PD AND TK-TD MODELING

Combining Equation 12.128 and Equation 12.142 yields Equation 12.143, which shows that “when a hyperbolic agonist-receptor interaction feeds into a hyperbolic effector system, a hyperbolic curve result” (Black and Leff, 1983). E=

Emax [ RT ] Cp K D K E + ( RT + K E ) Cp

(12.143)

Analogous to other hyperbolic equations, for example, the Michaelis–Menten equation, at low concentrations when E50 ˜ Cp , the relationship between E and Cp is linear with a slope of Emax E50 , and the equation of the line is: E = (Emax/E50) Cp. At high concentrations of plasma, when Cp ˜ E50 , E = Emax , which represents the plateau line of the hyperbolic curve parallel to the x-axis. As presented in Figure 12.16, the rise in the magnitude of response versus concentration occurs at certain concentrations that are less than E50. The slope of this gradual increase in response is dE determined as a function of ln Cp (i.e., ). Since e ln Cp = Cp, Equation 12.141 can also be written d ln Cp E ´ e ln Cp as E = max ln Cp , the slope representing change in response as a function of ln Cp is E50 + e Emax ´ Cp ( E50 + Cp ) - Emax Cp 2 Emax ´ E50 ´ Cp dE (12.144) = = 2 2 d ln Cp ( E50 + Cp ) ( E50 + Cp ) By setting Cp = E50 , Equation 12.144 changes to Equation 12.145: dE E E2 E ´ E2 E = max 502 = max 2 50 = max d ln Cp ( 2E50 ) 4E50 4

(12.145)

Thus, the slope of the initial rise in response versus ln Cp is a function of Emax and equal to 1/4 of the total occupancy of the receptor sites for a given compound. Like Equation 12.130, the Emax model can also be presented in logarithmic form of Equation 12.146 (Figure 12.19): æ ö E log ç ÷ = log Cp - log E50 E E è max ø

(12.146)

12.5.3.4 Non-Hyperbolic Sigmoidal Model When more than one molecule of xenobiotic binds to a receptor, the interaction can be presented as ¾¾ ¾ ® [ R nD] ¾¾ ® Effect [E] [ R] + [ nD] ¬ ¾

(12.147)

The equilibrium constant, KD, of the interaction is

[ R][D] [DnR]

n

KD =

(12.148)

Mathematical manipulations like Equations 12.124–12.128, yield

[DnR] =

[ RT ] ´ [D] n K D + [D]

n

Emax ´ [D]

n

E=

K D + [ D]

n

=

Emax ´ Cp n n E50 + Cp n

(12.149)

(12.150)

Equation 12.150 represents the sigmoidal curve of response versus the concentration of the xenobiotic. It can also be used for the curve-ftting purpose of dose-response curves that are not visibly sigmoidal. In that case, the value of n may no longer represent the number of reacting molecules with the receptor site, but rather an element to improve the curve-ftting. The value of n infuences the slope of linear segment of the curve. Similar to Equation 12.145, the slope of rise in response as a function of ln Cp is

400

APPROACHES IN PK/PD AND TK/TD MATHEMATICAL MODELING

Figure 12.19 Plot of log ( E / ( Emax - E ) ) versus log dose according to the linear equation of log ( E / ( Emax - E ) ) = nlog D - log K D ; the slope of the line determines how many molecules of a xenobiotic binds to its receptor, and the y-intercept is the logarithm of the equilibrium constant of the interaction of molecule(s) of xenobiotic to its receptor site; when there is only one molecule of xenobiotic binding to one receptor site, the n value is equal to one. dE nE = max d ln Cp 4

(9.151)

Therefore, a larger n corresponds to a sharper slope and making the linear segment of the curve more perpendicular. Equation 12.150 can also be changed into a linear relationship through the following steps:

[ D] E = Emax K D + [D]n n

n [D] ( Emax - E ) = E ´ KD

KD =

n [D] ( Emax - E )

E

(9.152) (9.153) (9.154)

Taking a logarithm of both sides yields: log K D = nlog [D] + log

Emax - E E

(9.155)

Emax - E = -log K D + nlog D E

(9.156)

E = nlog [D] - log K D Emax - E

(9.157)

-log Therefore, log

E versus log [D] provides a linear relationship Emax - E with a slope of n and a y-intercept of log K D (Figure 12.19). According to Equation 12.157, a plot of log

When E =

Emax E Þ log =0 E 2 max - E) ( 401

12.5 PK-PD AND TK-TD MODELING

nlog [D] = log K D n log K D = nlog E50 or, K D = E50

The effect, E, in terms of xenobiotic concentration-receptor complex, RCp , for non-hyperbolic relationship is E=

Emax éë RCp ùû

n

K En + éë RCpùû

(12.158)

n

When n = 1, Equation 12.158 defnes the rectangular hyperbola of Equation 12.142. Substituting Equation 12.128 into Equation 12.158 yields the following non-hyperbolic relationship for the effect in terms of plasma concentration (Black and Leff, 1983): Emax [ RT ] éëCpùû n

E=

K

n E

(K

D

+ éëCpùû

n

) + [R ] n

n

T

(12.159) éëCpùû

n

12.5.4 PK/PD and TK/TD Models The delay in pharmacological or toxicological response following an administration of a therapeutic xenobiotic indicates that its response/effect is related to the ADME profle of the compound in the body. The relevance of linking the ADME kinetic profle of a compound to its pharmacodynamic or toxicodynamic relationships was initiated frst by introducing the concept of biophase, and later the hypothetical effect compartment linked to the PK or TK models (Segre, 1968; Wagner, 1968; Levy et al., 1969; Galeazzi et al., 1976; Dahlstrom et al., 1978; Kramer et al., 1979). If the change in the magnitude of a response is parallel with the rise and fall of plasma concentration, the linkage between their dynamic and kinetic models is direct and, often, as simple as the correlation between the measured plasma concentration and the magnitude of response. However, the accuracy of the direct correlation depends on the simultaneous measurements of plasma concentration and response. If the concentration of a xenobiotic at the effect compartment is proportional but not the same as plasma concentration, a separate compartment is needed to defne the amount/concentration in the effect compartment. Based on the assumption that the concentration at the effect compartment is the same as the plasma concentration, the following relationships for the compartmental models are developed. In the case of one-compartment model, the receptor sites or biophase is assumed to be associated with the systemic circulation (Figure 12.20). In the case of two-compartment model, the biophase can be associated with the central or peripheral compartment (Figure 12.21). This type of PK/PD modeling may be used to predict the response from the calculated or observed plasma concentration. 12.5.4.1 Linking the Nonlinear Hyperbolic Emax Concept to Compartmental Models 12.5.4.1.1 One-compartment Emax Model with IV Bolus The equation of model is Cpt = Cp 0 e -Kt

(12.160)

0

where Cp is the plasma concentration at time t ; Cp is the initial plasma concentration at t = 0; and K is the overall frst-order rate constant of elimination (i.e., metabolism and excretion). Substitution of Equation 12.160 in Equation 12.141 yields the linked model of one-compartment model with IV bolus as presented in Equation 12.161: E=

Emax ´ Cp0 e -Kt E50 + Cp0 e -Kt

(12.161)

Dividing the numerator and denominator of Equation 12.161 by Cp 0 generates Equation of 12.162: E=

402

Emax ´ e -Kt Emax ´ f b = E50 f b50 + f b + e -Kt 0 Cp

(12.162)

APPROACHES IN PK/PD AND TK/TD MATHEMATICAL MODELING

Figure 12.20 Illustration of direct linking of the receptor site, biophase, situated in a onecompartment model where the concentration of xenobiotic at the receptor site is the same as its concentration in plasma, with no transfer rate between the plasma and the receptor site; the reduction of plasma concentration of the xenobiotic, due to the excretion of unchanged molecules and metabolism, has direct infuence on the magnitude of response; thus a direct correlation can be established between the effect and the plasma concentration (not all xenobiotics follow this simple and direct relationship); the compartments of the model are identifed as At , the time -dependent amount of the xenobiotic in the body with plasma concentration of Cpt and the apparent volume of distribution of Vd, i.e., At = Cpt ´ Vd ; Ae and Am are the cumulative amounts of the xenobiotic excreted unchanged and eliminated as metabolite, governed by the frst-order rate constant of k e and k m , respectively.

Figure 12.21 Illustration of the direct linking of biophase association with the central compartment, peripheral compartment, or both compartments of a two-compartment model based on the assumption that the amount of xenobiotic at the receptor site is the same as the amount of the host compartment; if the receptor site is located in the central compartment, the concentration at the receptor site is the same as the plasma concentration; however, since the peripheral compartment is not considered a homogeneous compartment, we can only assume that the amount of a location within the peripheral compartment is the same as the receptor site located in the peripheral compartment; the compartments of the model are A1 and A2 as the central and the peripheral compartments; k10 is the overall elimination rate constant, i.e., k10 = k e + k m equal to the excretion and biotransformation rate constants; Ae + Am are the total amount eliminated from the body. 403

12.5 PK-PD AND TK-TD MODELING

where f b is the fraction of administered dose in the body at time t ; f b50 is the fraction of the administered dose in the body that provides an effect equal to 50% of Emax and represents the potency of the compound; the smaller f b50 represents a more potent compound. (Note: E50 in Equation of 12.141 was defned as “concentration of xenobiotic at half-maximum effect.”) 12.5.4.1.2 One-Compartment Emax Model with Zero-Order Input (IV Infusion) The equation of the model with zero-order input is k0 1 - e -Kt KVd

(

Cpt =

)

(12.163)

where Cpt is the plasma concentration at time t during the input (e.g., during the infusion); k0 is the zero-order input; and Vd is the apparent volume of distribution. Substitution of Equation 12.163 in Equation 12.141 yields the following linked model before achieving the steady state level of the zero-order input: E=

(

Emax ´ Cpss 1 - e -Kt E50 + Cpss (1 - e

-Kt

)=

)

(

)

Emax 1 - e -Kt Emax ( f ss ) = E50 f ss50 + f ss + 1 - e -Kt Cpss

(

(12.164)

)

where f ss is the fraction of the steady-state level, i.e., the plasma level during the infusion as a fraction of the steady-state level; plasma concentration as a fraction; f ss50 is the fraction of the steadystate plasma level that provides an effect equal to 50% of Emax. After attaining the steady-state level, the linked model is defned as: E=

Emax f ss50 + 1

(12.165)

12.5.4.1.3 One-Compartment Emax Model with First-Order Input (Oral Administration, Intramuscular Injection, Inhalation, etc.) The equation of the model with frst-order input is Cp = Setting

FDk a e -Kt - e -kat Vd(k a - K)

(

)

(12.166)

˜°° 0 FDka = Cp and substituting Equation 12.166 in Equation 12.141 yields: Vd ( k a - K ) ˜°° 0 Emax éCp e -Kt - e -kat ù - e -kat Emax e -Kt úû êë E= ˜°° 0 -Kt -k t = E 50 -Kt -k t E50 + Cp e - e a ˜ + (e - e a ) Cp 0

( (

) )

(

)

(12.167)

where k a is the frst-order absorption rate constant. Equation 12.167 determines the time course of the effect during the absorption process. When the absorption is complete, the term e -kat Þ 0 and the equation is modifed to E=

Emax e -Kt E50 -Kt ˜°° 0 + e Cp

(12.168)

FD -KTmax e The response corresponding to the maximum plasma concentration Cpmax = is defned Vd as

(

E=

Emax ´ ( FD Vd ) e -KTmax E50 + ( FD Vd ) e

-KTmax

=

Emax ´ FDe -KTmax ( E50 ´Vd ) + FDe -KTmax

)

(12.169)

2.303 k log a ka - K K Based on Equation 12.169, it can be stated that the only time the maximum response is associated with the maximum plasma concentration is when Cpmax ˜ E50 , and when Cpmax = E50 , the effect is half of the maximum response, E = Emax 2. where Tmax , the time point of Cpmax , is Tmax =

404

APPROACHES IN PK/PD AND TK/TD MATHEMATICAL MODELING

12.5.4.1.4 One-Compartment Emax Model Intravenous Bolus Multiple Dosing The peak and trough level equations after achieving the steady state are Cp 0 e -Kt

Cp 0

( Cpss )peak = 1 - e -Kt and ( Cpss )trough = 1 - e -Kt

(12.170)

The input rate is a fxed IV bolus given instantaneously at the beginning of a predetermined dosing interval and repeated long enough to achieve the steady state, or the steady-state fuctuation is achieved immediately by giving a loading dose. The fuctuation of response at steady state that corresponds to the peak and trough levels of plasma concentrations are

Epeak

Cp 0 Emax ´ Cp 0 Emax ´ Cp 0 1 - e -Kt = = = 0 Cp 0 E50 1 - e -Kt + Cp0 E50 ( f el )t + Cp E50 + -Kt 1- e Emax ´

(

)

(12.171)

where ( f el )t = 1 - e -Kt is the fraction of the dose eliminated in one dosing interval at steady state and Cp0 is the initial plasma concentration of a maintenance dose. The response corresponding to the trough level is Etrough

Cp 0 e -Kt Emax ´ Cp 0 1 - e -Kt = = 0 -Kt Cp e æ 1 - e -Kt ö E50 + E50 ç -Kt ÷ + Cp 0 -Kt 1- e e è ø Emax ´

=

(12.172)

Emax ´ Cp0 æ E50 ö 0 ç -Kt - 1 ÷ + Cp e è ø

The average response at steady state can be determined by linking the Emax model with an average Cp 0 plasma concentration, , at steady state: Kt Cp0 Emax ´ 0 Kt = Emax ´ Cp Eave = Cp0 E50 ( K t ) + Cp0 E50 + Kt (12.173) Emax ´ D E ´ AUC = max = E50 ( t ) + AUC E50 ( ClT ´ t ) + D For multiple-dosing kinetics with frst-order input, the Emax linked model is the same as Equations ˜°° 0 12.171–12.173 with the stipulation that Cp 0 is Cp , and the dose is equal to FD. 12.5.4.1.5 Two-Compartment Emax Model with a Single IV Bolus Injection The equation of the two-compartment model defning the plasma concentration of xenobiotic in the central compartment after an intravenous bolus dose is Cp = ae -at + be -bt

(12.174)

Substitution of Equation 12.174 in Equation 12.141 yields the linked two-compartment model with IV bolus input for the central compartment E=

( (

Emax ´ ae -at + be -bt E50 + ae -at + be -bt

) )

(12.175)

The response in the post-distributive phase can be estimated by E=

Emax ´ be -bt E50 + be -bt

(12.176)

405

12.5 PK-PD AND TK-TD MODELING

When the effect is associated with the amount of compound in the peripheral compartment, the relationship of the model can be presented as æk D Emax ´ ç 12 e -bt - e -a a -b è E= æ k12D -bt -a E50 + ç e -e è a -b

(

) ÷ö

(

)

ø ö ÷ ø

(12.177)

12.5.4.1.6 Two-Compartment Emax Model Intravenous Bolus Multiple Dosing The equations for prediction of response at steady state peak and trough levels of a drug that follows the two-compartment model are Epeak = where ( Cpmax ) = ss b

( Cpmax )ss = 1 - e -bt

Emax ´ ( Cpmax )ss

Emax ´ ( Cpmin )ss

Etrough =

E50 + ( Cpmax )ss

E50 + ( Cpmin )ss

(12.178)

a b ae -at be -bt and ( Cpmin ) = ; in post-distributive phase + + -at -bt -at ss 1- e 1- e 1- e 1 - e -bt and ( Cpmin ) = ss

be -bt . 1 - e -bt

12.5.4.2 Linking Non-Hyperbolic Sigmoidal Model to PK/TK Models with Different Inputs The linking is the same as was discussed for the Emax model. A few examples are 12.5.4.2.1 One-Compartment Linked Model with IV Bolus Equation of the linked model is Emax ´ f bn f bn50 + f bn

E=

(12.179)

12.5.4.2.2 One-Compartment Linked Model with Zero-Order Input Before achieving the steady-state level E=

Emax ´ f ssn n f ss50 + f ssn

(12.180)

Emax n f ss50 +1

(12.181)

At steady state level E=

12.5.4.2.3 One-Compartment Linked Model with First-Order Input During the absorption phase E=

(

Emax ´ e -Kt - e -kat æ E50 çç ˜ 0 è Cp

n

)

n

ö -Kt -k t ÷÷ + e - e a ø

(

(12.182)

)

n

In post absorptive phase E=

(

Emax ´ e -Kt n

)

n

æ E50 ö -Kt çç Cp˜ 0 ÷÷ + e è ø

(

(12.183)

)

n

At maximum plasma concentration E=

406

(

Emax ´ FDe -KTmax

( E50 ´Vd )

n

(

+ FDe

)

n

-KTmax

)

(12.184) n

APPROACHES IN PK/PD AND TK/TD MATHEMATICAL MODELING

12.5.4.2.4 One-Compartment Linked Model Multiple Dosing IV Bolus Injection At the peak and trough levels Epeak =

(

Emax ´ Cp 0

)

(

n

( E50 ( fel )t ) + ( Cp0 )

and Etrough =

n

n

Emax ´ Cp 0 n

)

n

æ E50 ö 0 ç e -Kt - 1 ÷ + Cp è ø

(

(12.185)

)

n

At the steady-state average plasma concentration Eave =

=

Emax ´ ( Cp ˜ )

( E50 ´ Kt )

n

n

+ ( Cp ˜ )

n

=

Emax ´ ( AUC )

( E50 ´ t )

n

n

+ ( AUC )

n

Emax ´ Dn

( E50 ´ ClT ´ t )

(12.186)

n

+ Dn

12.5.4.2.5 Two-Compartment Liked Model Single Intravenous Bolus Injection The general equation is E=

(

Emax ´ ae -at + be -bt

(

)

( E50 ) + ae -at + be -bt n

n

)

(12.187) n

During the post-distributive phase E=

(

Emax ´ be -bt

( E50 )

n

(

)

+ be -bt

n

)

(12.188) n

12.5.4.2.6 Two-Compartment Linked Model Multiple Dosing Intravenous Injection At the peak and trough plasma levels Emax ´ ( Cpmax )ss n

Epeak =

( E50 )

n

+ ( Cpmax )ss n

Emax ´ ( Cpmin )sss n

Etrough =

( E50 )

n

+ ( Cpmin )ss n

(12.189)

12.5.5 The Effect Compartment The tacit assumption of the models discussed in the preceding section (Section 12.5.4) was that the free concentration of xenobiotic at the receptor site is the same as its concentration in the central or the peripheral compartment. This means that the free molecules of the xenobiotic at the receptor environment which is in equilibrium with the xenobiotic–receptor complex is the same as the free plasma concentration. There are, however, receptor environments that do not reach rapid equilibrium with plasma, such as receptors in regions separated from plasma by a physiological barrier, or associated with the intracellular environment, or located in regions where blood perfusion is poor. For this type of receptors, the direct method of substitution may not be pertinent. One of the reasons is that the interaction of compounds with this type of receptor occurs after a lag time or a threshold of response, which cannot be predicted by the direct linking, as was discussed in Section 12.5.4. A different approach based on determining the actual amount or concentration of xenobiotic at the receptor site has been developed, which considers effect as a separate compartment to represent the receptor environments with the following assumptions (Sheiner et al., 1979; Colburn, 1981; Fuseau and Sheiner, 1984; Hochhaus and Derendorf, 1995; Vora and Boroujerdi, 1996; Vaidyanathan and Boroujerdi, 2000) in Figures 12.22 and 12.23: ◾ The effect compartment is in contact with the systemic circulation. ◾ The transfer of the xenobiotic from the systemic circulation to the effect compartment is governed by linear processes. ◾ The transfer rate constant from the systemic circulation to the effect compartment and the amount in the effect compartment are considered negligible, with respect to the overall disposition profle of the compound in the body.

407

12.5 PK-PD AND TK-TD MODELING

◾ The effect compartment has a frst-order exit rate constant that is also small and considered a negligible value. 12.5.5.1 PK/TK Models Connected to the Effect Compartment The approach is somewhat similar to the models described in the previous section. The effect compartment can be connected to the central compartment or the peripheral compartment, in the case of the multicompartmental models. A few examples are as follows: 12.5.5.1.1 One-Compartment Model with Intravenous Bolus Dose Connected to the Effect Compartment Following the administration of intravenous bolus dose, the rates of change in the amount of xenobiotic in the systemic circulation and the effect compartment, as depicted in Figure 12.22, are For centralcompartment ˜ systemic circulation ° : dA ˛ ˝KA dt

(12.190)

For theeffect compartment : dAE ˜ kCE A ° kEE AE dt

(12.191)

Solving the two differential equations by the Laplace transform (Addendum I, Part 2, Section A.1) yields the following integrated equations:

˜ A °t ˛ A0 e ˝Kt ˜ AE °t ˛

kCE A0 ˝Kt ˝ kEEt e ˝e kEE ˝ K

˜

(12.192)

°

(12.193)

where ˜ AE ° is the amount of the drug in the effect compartment at time t , K is the overall elimit nation rate constant (K ˜ k e ° k m ° kCE ; kCE is negligible compared to the other rate constants of elimination ), kCE and kEE are the input and exit rate constants of the effect compartment, respectively, and A0 is the administered dose. Calculations of concentration in the effect compartment requires the knowledge of its apparent volume of distribution. Since at steady state the concentration of xenobiotic in the effect

Figure 12.22 Illustration of the receptor environment as an effect compartment in contact with the systemic circulation; the transfer of xenobiotic from the systemic circulation to the effect compartment is considered linear with input rate constant, KCE, and exit rate constant, K EE ; the rate constant KCE, though negligible, is considered an addition to the overall elimination rate constant, i.e., K ˜ k e ° k m ° kCE ; one reason for establishing the effect as a separate compartment is to address the lag time or the threshold of response following the administration of a xenobiotic. 408

APPROACHES IN PK/PD AND TK/TD MATHEMATICAL MODELING

Figure 12.23 Illustration of a two-compartment model with the central compartment in exchange with the effect compartment; the input rate constant, k1E , from the central compartment,A1 , into the effect compartment and the exit rate constant from the effect compartment,kEE , are considered frst-order rate constants; the input rate constant of the effect compartment, k1E , is considered an exit rate constant from the central compartment with the other frst-order rate constants of k12 the distribution and k10 the elimination (i.e., excretion and metabolism) rate constants, the exit rate constant from the effect compartment, kEE , is assumed a negligible rate constant; k12 and k 21 are the distribution rate constants governing the exchange between the peripheral compartment, A2 , and the central compartment A1 ; k10 is the overall elimination rate constant equal to ke + km. compartment is assumed to be approximately equal to plasma concentration, one approach is to determine a relevant proportionality constant between the apparent volume of distribution of the compound in plasma and the apparent volume of the effect compartment. In other words, when a steady state is achieved between the systemic circulation and the effect compartment, the rate of input into the effect compartment is equal to the rate of output from the effect compartment, that is, kCE ´ ( Cp )t ´ Vd = kEE ´ ( CE )t ´ VE VE = Vd

kCE ´ ( Cp )t kEE ´ ( CE )t

(12.194) (12.194)

where VE is the apparent volume of the effect compartment. When ( Cp ) @ ( CE ) , the apparent volume of the effect compartment changes to t t VE = Vd

kCE , and kEE

(12.195)

the apparent concentration of the xenobiotic at the effect compartment can be estimated as

( C E )t =

( AE )t VE

(12.196)

12.5.5.1.2 One-Compartment Model with Zero-Order Input Connected to the Effect Compartment When the input in the central compartment is continuous and considered zero-order input, the rate equation of the effect compartment remains the same as Equation 12.191 and the rate equation of the central compartment is dA = k0 - k e A - k m A - kCE A = k0 - KA dt

(12.197)

The integrated equations of the model before achieving the steady-state condition are k

( Cp )t = K ´0Vd (1 - e -Kt )

(12.198)

409

12.6 PHYSIOLOGICALLY BASED PK/TK MODELS WITH EFFECT COMPARTMENT

( AE )t =

kCE k0 kCE k0 1 - e -Kt + 1 - e -kEEt kEE ( K - kEE ) K ( kEE - K )

(

)

(

)

(12.199)

After stopping the input, the integrated equations are k

( Cp )t = K ´0Vd (1 - e -Kt ) e -Kt’ ( AE )t =

(12.200)

kCE k0 kCE k0 1 - e - Kt e -Kt ’ + 1 - e - kEEt e -kEEt ’ kEE ( K - kEE ) K ( kEE - K )

(

)

(

)

(12.201)

After achieving the steady state Cpss =

( AE )t =

k0 KVd

(12.202)

kCE k0 kCE k0 + K ( kEE - K ) kEE ( K - kEE )

(12.203)

12.5.5.1.3 One-Compartment Model with First-Order Input Connected to the Effect Compartment For the frst-order absorption through the GI tract, intramuscular, sublingual, inhalation, etc., the rate equation of the central compartment can be defned as Equation 12.204 and the rate equation of the effect compartment as Equation 12.191. dA = k a FD - k e A - k m A - kCE A = kaFD - KA dt

(12.204)

The integrated equations are

( Cp )t = Vd ( AE )t =

kCE k a FD

( K - ka )( kEE - ka )

e -kat +

k a FD e -Kt - e -kat ( ka - K )

(

)

kCE k a FD

e -Kt +

( ka - K )( kEE - K )

(12.205) kCE k a FD

( ka - kEE )( K - kEE )

e - kEEt

(12.206)

12.5.5.1.4 One-Compartment Model with Multiple IV Bolus Dosing Connected to the Effect Compartment The integrated equations of the effect compartment following a multiple-dosing regimen at steady state are

( AE )peak = ( AE )trough =

1 ö 1 - e -kEEt ÷ø

(12.207)

kCED æ e -Kt e -kEEt ö ç ÷ -Kt 1 - e -kEEt ø ( kEE - K ) è 1 - e

(12.208)

kCED

æ

1

( kEE - K ) çè 1 - e -Kt

-

12.5.5.1.5 Two-Compartment Model with IV Bolus Connected to the Effect Compartment In dealing with multiple compartment models, the rate equations of the effect compartment remain the same, but the integrated equation will be different. For example, for a two-compartment model depicted in Figure 12.23, the integrated equation of the effect compartment is

( AE )t =

kCE

( kEE - a )

Ae -at +

kCE

( kEE - b )

Be -bt +

kCED ( k 21 - kEE ) - k e ( a - kEE ) (b - kEE )

EEt

(12.209)

12.6 PHYSIOLOGICALLY BASED PK/TK MODELS WITH EFFECT COMPARTMENT The connection of the effect compartment with a specifc organ or tissue in the body is the frst step for inclusion of the compartment in a physiologically based model. Depending upon the type of effect data, the rate constants or clearance exchange between the organ and the effect compartment is determined based on how the effect is perceived and measured. The effect compartment can be included as a separate single compartment attached to an organ, or two separate compartments representing the free receptors and bound receptors with the exchange between the two compartments, while both are associated with the organ (Figure 12.24).

410

APPROACHES IN PK/PD AND TK/TD MATHEMATICAL MODELING

When the association of the effect compartment with an organ is well established, a hybrid approach to PBPK/TK presented in Figure 12.24 is considered more practical in connecting the effect compartment to a PK/TK model (Johnson et al., 2011) (Figure 12.25). The rate equations of the model (Figure 12.25) are

dA3 æ Cl3 =ç dt è V1

Cl Q dA1 Q A2 - A1 - A1 = V1 V1 dt V2

(12.210)

Q dA2 Q A2 A1 = V2 dt V1

(12.211)

æ Cl3 ö ÷ A1 - ç V ø è 3

ö æ Cl4 ö æ Cl4 ö ÷ A3 - ç V ÷ ( f u ´ A3 ) + ç V ÷ ( f u ´ A4 ) è 4 ø ø è 3 ø

(12.212)

æ Cl5 ö æ Cl5 ö dA4 æ Cl4 ö æ Cl4 ö =ç ÷ ( f u5 ´ A5 ) ÷ ( f u ´ A3 ) - ç V ÷ ( f u5 ´ A4 ) - ç V ÷ ( f u5 ´ A4 ) + ç dt è 4 ø è V3 ø è 5 ø è V5 ø

(12.213)

æ Cl5 ö dA5 æ Cl5 ö =ç ÷ ( f u5 ´ A4 ) - ç V ÷ ( f u5 ´ A5 ) - ( k 56 ´ f u5 ´ A5 ( RT - RC5 ) ) + k65 A6 V dt è 4 ø è 5 ø

(12.214)

dAEffect = k 56 ´ f u5 ´ A5 ( RT - RC5 ) - k65 AEffect dt

(12.215)

Compartment A1 and A2 are the amount in the systemic circulation and peripheral tissues, respectively; A3 is the amount in the target organ’s vascular compartment; A4 is the amount of the target organ’s extravascular compartment; A5 is the amount in the receptor environments; AEffect is the amount bound to the receptor, or the effect compartment; V1 −V5 are the volumes of distribution of compartments 1–5; Q is the blood fow and Cl1- 5 are the compartmental clearance terms; f u and f u5 are xenobiotic free fractions in plasma and the target organ, respectively; RT is the receptor density; and RC5 is the concentration bound to the receptor. 12.7 HYSTERESIS LOOPS IN PK/PD OR TK/TD RELATIONSHIPS The PK/PD or TK/TD models attempt to relate the plasma concentration to the intensity of response. The direct plot of the magnitude of response against the plasma concentration according to their time order, yields a hysteresis loop which provides an understanding of the complexity of the effect and the disposition of a xenobiotic in the body. The plot can generate a straight line, a sigmoidal curve, a clockwise loop, or a counterclockwise loop. The straight line and sigmoidal curve are straightforward and easy to explain. However, if a loop is generated, whether counterclockwise or clockwise, it is indicative of a certain complexity in xenobiotic–receptor interaction. A counterclockwise phenomenon (Figure 12.26) is usually associated with the delay or presence of a threshold in response/effect. The delay in response can be related to the plasma concentration or the receptor environment. The plasma concentration-related reasons are 1) a slow equilibration of the plasma concentration with the receptor environments, which is caused by a holdup in distribution of the xenobiotic in the body and thus causing a delay in arrival at the receptor site or effect compartment; 2) time-dependent protein-binding that also may infuence the distribution of the compound in the systemic circulation; 3) the rate of appearance of the xenobiotic at the receptor site is slow due to the physiological considerations such as presence of a barrier; and 4) the presence of active agonist metabolite(s) in plasma/systemic circulation. The receptor-related reasons for counterclockwise loop can be related to the 1) upregulation of receptors and 2) slow interaction kinetics between the receptor and the xenobiotic. A clockwise direction occurs when the response to the initial concentration is rather high but declines rapidly even when the concentration increases, a scenario that is observed with the development of tolerance and/or downregulation of the receptor. Sometimes the presence of active antagonistic metabolite in plasma/systemic circulation infuences the interaction of a xenobiotic with its receptor. The relationship between the response and the plasma concentration is helpful in developing PK/PD or TK/TD models for the logical inclusion and location of the effect compartment in the model.

411

12.8 TARGET-MEDIATED DRUG DISPOSITION MODELS

Figure 12.24 Depiction of a physiologically based pharmacokinetic model with the effect compartment associated with the receptor density compartment, RT , of the brain; RCis the concentration of xenobiotic bound to the receptor site; the important part of the inclusion of an effect compartment in a physiologically based PK/TK model is to identify the organ or tissue that is in exchange with the receptor site; in this example, it is the brain, which is in exchange with the RT compartment of the receptor site; the effect compartment is the combination of bothRT and RC which are in equilibrium with each other. 12.8 TARGET-MEDIATED DRUG DISPOSITION MODELS In the interaction of xenobiotics with their receptor sites, as presented in Sections 12.5 and 12.6 of the present chapter, it is critical to remember that the amount of xenobiotic required for interaction with its receptor site to provide an optimum response must be at a level that does not impact the PK/TK profle of a xenobiotic. There are, however, circumstances in which the interaction of a xenobiotic with its target site occurs with high affnity and the target site is either present at a high level in the body or is produced continuously, such that the interaction of the xenobiotic with the target site modifes its disposition profle. The examples of this phenomenon are the interaction between monoclonal antibodies (mAb) and target antigens, biologics such as peptides and proteins, cytokines, and growth factors (Levy, 1994; Mager and Jusko, 2001; Mager and Krzyzanski, 2005; Dua et al., 2015; Stein and Peletier, 2018; An, 2020). Except a few small-molecule xenobiotics that manifest target-mediated drug disposition (TMDD) (Levy, 1994), this phenomenon occurs mostly with large-molecule biologics. The saturable clearance mechanism for biologics within the idea of TMDD is discussed in Chapter 11 (Section 11.4.4.1). The TMDD models have been applied to numerous cases and xenobiotics; a few are complex and target-specifc. The two most common models are the one-compartment TMDD model and the two-compartment TMDD model.

412

APPROACHES IN PK/PD AND TK/TD MATHEMATICAL MODELING

Figure 12.25 Schematic representation of an abbreviated physiological model focusing only on the organ that is in exchange with the effect compartment; compartments A1 and A2 are the central and the peripheral compartments of the model with clearance from the central compartment representing the elimination of the xenobiotic from the body; the exchange between the two compartments is governed by the blood fow and the concentration of the free compound in plasma; compartments A3 and A4 are the vascular and extravascular sub-compartments of the organ containing the receptor site; compartment A5 is the free amount of xenobiotic at the receptor site, RT , in equilibrium with bound ones, RC, in the effect compartment. 12.8.1 One-Compartment TMDD Models The TMDD models have been applied to several large and small molecules and have gone through modifcations in the past decades (Levy, 1994; Mager and Jusko, 2001, Aston et al., 2011, Byun and Jung, 2022). The assumptions of the model are: 1) the administered dose of xenobiotic, (bolus injection, or intravenous infusion, or frst-order absorption) provides optimum concentration of free xenobiotic in serum or plasma for interaction with the receptor site; 2) the compound has a high affnity for binding to the receptor site and interacts immediately to form xenobiotic–receptor complex; 3) the target site or receptor site may have a very high concentration of binding sites for interaction with the compound, or may have a zero-order production rate that provides ample receptor binding sites for the xenobiotic to form xenobiotic–receptor complex; 4) the free xenobiotic eliminates from the body by frst-order elimination; 5) the free receptors also degrade and become unavailable for binding with the xenobiotic, the degradation follows frst-order kinetics; 6) the formation of the xenobiotic–receptor complex may follow frst-order or second-order kinetics; the reverse reaction of the formation of complex is also governed by the frst-order kinetics; the xenobiotic–receptor complex under certain conditions or types of receptors, like enzymes, may degrade and eliminate. The schematic diagram of the model is presented in Figure 12.27. The ordinary differential equations of the model are: dCp = k off RCp - k elCp - k onCp.R dt

(12.216)

413

12.8 TARGET-MEDIATED DRUG DISPOSITION MODELS

Figure 12.26 Schematic plot of a counterclockwise hysteresis loop obtained when the magnitude of effect is plotted against corresponding plasma concentration; this phenomenon occurs when the response is delayed due to complexity of distribution of the plasma concentration of xenobiotic, like slow equilibration with the receptor environment, or certain events at the receptor site, such as upregulation of the receptors or slow interaction of the receptor with the xenobiotic. dR = k P - k deg R - k onCp.R + k off RCp dt d éë RCpùû dt

= k onCp.R - k off RCP - k out RCp

(12.217) (12.218)

dCp is the rate change of plasma concentration following the injection, k off RCp is the disdt sociation rate of the xenobiotic–receptor complex; k off is the frst-order rate constant of the complex dissociation; k el is the frst-order elimination rate constant; k onCp.R is the formation rate of the complex; k on is the rate constant for the formation of the complex, for certain xenobiotics and receptors dR it is considered a second-order rate constant; is the change in rate of receptor concentration dt with respect to time, k P is the zero order production/synthesis rate of the receptor, estimated as k P = k deg ´ R0 (Mager and Jusko, 2001), where R0 is the initial density of the receptor when it is at d é RCp ùû maximum; ë is the change in rate of the complex with respect to time; k out RCp is the elimidt nation rate of the complex. This rate may not be present for certain biologics. The dose of the model described here, Figure 12.27, is considered an intravenous bolus injection. If the dose is given by an intravenous zero-order infusion, a zero-order rate term would be added to the rate equation of the plasma concentration. If the dose is given by an extravascular route of administration, an extra compartment would be added to the model as the depot compartment to include the frst-order time course of the administered dose into the systemic circulation. where

414

APPROACHES IN PK/PD AND TK/TD MATHEMATICAL MODELING

Figure 12.27 Depiction of a one-compartment TMDD model with an instantaneous intravenous bolus injection; the free concentration of the xenobiotic in plasma, Cp, interacts with the receptors, R, to form the xenobiotic–receptor complex, RCp ; the affnity of the compound for the interaction with the receptors is considered very high such that the interaction infuences the PK/ TK profle of the injected compound; the high level of interaction is partly related to the density of the receptors, which in some cases is replenished by its zero-order production rate, k P , within the body; the frst-order elimination rate constant of the xenobiotic is k el , the frst-order rate constant of the free receptors degradation is, k deg , and the frst-order rate constant of degradation of the complex,RCp , is k out ; the association and formation of the RCp complex is governed by frst- or secondorder kinetics with the rate constant of k on , the frst-order dissociation of the complex is k off . 12.8.2 Two-Compartment TMDD Models The rate constants of the two-compartment model are the same as the ones defned for the onecompartment model presented in Section 12.8.1. Addition of the peripheral compartment adds two more rate constants, namely frst-order distribution rate constant for the exchange of xenobiotic between the central and peripheral compartments. For some xenobiotics, the addition of the peripheral compartment brings up the question of which of the two compartments is in exchange with the receptor site and whether both compartments contribute to the interaction with the target site. The common approach is to assume the target site is in exchange with the serum or plasma concentration, thus the central compartment model described is the place where the xenobiotic interacts with receptor site (Figure 12.28). The rate equations of the model are: dA2 ˜ k12 A1 ° k 21 A2 dt

(12.219)

A dC1 ˜ k 21 2 ° k off RC1 ˛ k elC1 ˛ k onC1 .R ˛ k12C1 V2 dt

(12.220)

dR ˜ k P ° k deg R ° k onC1 .R ˛ k eff RC1 dt

(12.221)

d ˜ RC1 ° ˛ k onC1 .R ˝ k off RC1 ˝ k out RC1 dt

(12.222)

where the k12 and k 21 are the distribution rate constants between the central and the peripheral dA2 , is defned in terms of the compartments. The rate equation of the peripheral compartment, dt mass/time, whereas equations 12.220–12.222 are defned in terms of concentration per time. It is worth noting that the amount of the central compartment is defned as A1 ˜ C1 °V1 , and for the peripheral compartment can also be defned as A2 ˜ C2 °V2 . However, since the peripheral compartment is a non-homogeneous compartment, the V2 is only a theoretical value, thus in use of TMDD models, the volume of the central compartment is also used as the volume of the peripheral compartment. The initial conditions of the compartments are: Central compartment = ˜ C1 °

t˛0

˛0

Peripheral compartment = ˜ A2 °

t˛0

˛0 415

12.8 TARGET-MEDIATED DRUG DISPOSITION MODELS

Figure 12.28 Depiction of a two-compartment TMDD model with an intravenous bolus injection into the central compartment; the amount in the central compartment is A1 = C1 ´V1 , where C1 is the free concentration and V1 is the volume of the central compartment; the distribution of the free xenobiotic between the central and the peripheral compartment is governed by frst-order kinetics and the rate constants of k12 and k 21 ; the free concentration also interacts with the receptors to form RC1 complex; the rate of change in free concentration of xenobiotic in the central compartment is infuenced by the rate of elimination of the compound k el ´ C1 ´V1 , by the rate of input from the peripheral compartment, k 21 A2, by the transfer rate to the second compartment, k12 A1 , by the dissociation rate of the complex, k off RC1 , and by the formation rate of the xenobiotic– receptor complex, k onC1 .R ; the rate of change in the density of the receptors in R compartment, the target site, is affected by the degradation rate of the receptors, k deg R , and the zero-order production rate of the receptors, k P ; the rate of change in the xenobiotic–receptor complex is a function of the degradation of the complex, k out RC1 , the rate of formation of the complex, kon C1.R, and the rate of dissociation of the complex, k off RC1 . Receptor site = Rt=0 =

kP kdeg

Complex compartment = ( RC1 )

t=0

=0

The description of other rate constants and the compartments are the same as described for the one-compartment model (Section 12.8.1) When the elimination from the central compartment is by frst-order elimination rate constant and a capacity-limited process, the equation of the model, Equation 12.220, is modifed to (Yan et al., 2010):

(Vmax V1 ) C1 A dC1 = k 21 2 - k elC1 - k12C1 V1 dt K M + C1

(12.223)

The input for the model here (Figure 12.28) is an instantaneous bolus intravenous injection; for other types of input, zero-order or frst-order, the equation of the model a function of xenobiotic input, In(t), will be added to the equation, that is,

(Vmax V1 ) C1 A dC1 = In(t) + k 21 2 - k elC1 - k12C1 V1 dt K M + C1

(12.224)

The total density of the receptor site is defned as Rtotal = R + RC1

(12.225)

The suggested initial estimates for Vmax and K M are (Dua et al., 2015): Vmax = Rtotal ´ k out K M = KD +

416

k out k on

(12.226) (12.227)

APPROACHES IN PK/PD AND TK/TD MATHEMATICAL MODELING

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13 Practical Applications of PK/TK Models: Instantaneous Exposure to Xenobiotics - Single Intravenous Bolus Injection 13.1 INTRODUCTION The instantaneous input such as intravenous bolus injection provides a clear evaluation of disposition of xenobiotics in the body without issues of absorption via an extravascular route of administration. Furthermore, there are other aspects of the bolus injection such as immediate pharmacological response or providing the reference data for bioavailability evaluations and similar considerations that are discussed in Chapter 4, Section 4.4. The term ‘disposition’ refers to the combined physiological processes of distribution and elimination, and the term ‘elimination’ refers to metabolism and excretion. The detailed description and assumptions of various models are described in Chapter 12. Compartmental modeling and analyses are the most applied, as well as traditional approaches in analysis of pharmacokinetic/toxicokinetic (PK/TK) data. A review of Section 12.3 of Chapter 12 would provide the required background information for compartmental analysis used in this chapter. Briefy, in compartmental analysis, the body is assumed as a collection of pools and/or compartments. Depending upon the type of PK/TK sampling data and analysis, sampling compartments are usually considered homogeneous. Each compartment is defned as a given amount of xenobiotic within a volume. The volume of the compartment may represent the size of the compartment, which may not be necessarily a physiological volume. When a model has compartments with exit to the environment external to the body, it is referred to as an open compartmental model. The model with no exit to the external environment is referred to as closed compartmental model. All compartmental models that are used to describe how the body handles the disposition of xenobiotics are considered open compartmental models. The closed compartmental models are typically used to defne the role and function of endogenous compounds. 13.2 LINEAR ONE-COMPARTMENT OPEN MODEL – INTRAVENOUS BOLUS INJECTION The one-compartment open model has the historical signifcance of being the model frst considered for disposition of xenobiotics in the body. It has also the signifcance of offering applicable methods of estimating PK/TK parameters and constants in research and particularly in clinical settings. The assumptions of the model are: 1) the input (intravenous dose) is mixed very rapidly with the systemic circulation and interacts with the elements of plasma and tissues, 2) the body behaves collectively as a single well-mixed and homogeneous compartment, 3) plasma, blood or serum, and urine are the reference samples, and any change in the concentration or amount of a compound in the reference samples refects the equivalent changes in the compartment (Gibaldi and Perrier, 1982), 4) the elimination (i.e., metabolism and excretion) is governed by frst-order kinetics. The model is depicted in Chapter 12, Figure 12.9. The exit rate of the compartment is KAt ; where At is the amount of the injected xenobiotic at time t , and the rate constant K is the frst-order overall elimination rate constant representing the renal excretion, hepatic metabolism, and other signifcant elimination rate constants, that is, K ˜ ke ° km ° kn

(13.1)

Where the frst-order excretion rate constant is k e ; the frst-order metabolic rate constant is k m ; and the combined frst-order rate constants of elimination via sweat glands, milk, exhaled air, etc. is k n. For most xenobiotics k n is considered negligible unless a compound specifcally eliminates through one of the minor routes of elimination. Thus, Equation 13.1 is always presented as K ˜ ke ° km

(13.2)

The amount of xenobiotic in the compartment at time t , is At , which is free concentration in plasma at time t , Cpt , multiplied by the apparent volume of distribution Vd: At ˜ Cpt ° Vd

(13.3)

The Cpt and At are variables, changing with time, and K and Vd are constant as long as the PK/ TK is dose-independent and linear for a given xenobiotic. The differential equation of the model, that is, the overall rate of elimination from the compartment is DOI: 10.1201/9781003260660-13

423

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dAt ˜ °KAt dt

(13.4)

According to Equation 13.4, the frst-order rate of elimination is a variable, but the rate constant of elimination K remains constant during the elimination process; the negative sign refects the reduction of the injected dose in the body because of the elimination process. The analogy can be made between the frst-order elimination and a hydrodynamic model of two tanks of water with a drain outlet at the bottom (Figure 13.1) (Boroujerdi, 2002). The diameter of the opening in tank 2 is twice that of tank 1. The fow of water from the tanks via the outlet per unit of time is analogous to the rate of elimination from the body. The fow rate of water from the tank depends on the volume of water and its hydrostatic pressure. As the volume of water declines, the pressure declines. This hydrostatic pressure of water is analogous to the amount of xenobiotic in the body. As the amount in the body declines, so declines the rate of elimination. The diameter of the outlets that remain unchanged during the removal of water from the tanks is analogous to K. A smaller diameter tends to have lower fow rate, which is analogous to a reduction of K in disease states, such as renal impairment and hepatic failure. The important assumption of the model, as was generalized in Chapter 12, is that only the free molecules of the injected xenobiotic can interact with receptors to provide pharmacological response. In other words, the xenobiotic concentration of free molecules in plasma or serum is proportional to its concentration at the receptor site, which is in equilibrium with the receptor-bound concentration. The integration of Equation 13.4 yields the following equations that defne the free amount or concentration of the xenobiotic in systemic circulation: ln A ˜ ln A0 ° Kt log A ˜ log A0 °

Kt 2.303

A ˜ A0 e °Kt

(13.5) (13.6) (13.7)

In terms of plasma or serum free concentration: ln Cp ˜ ln Cp 0 ° Kt log Cp ˜ log Cp 0 °

Kt 2.303

(13.8) (13.9)

Figure 13.1 Depiction of analogy between a hydrodynamic model and frst-order elimination from the body; the level of water in each tank bears a resemblance to the amount of a xenobiotic in the body; the diameter of the opening seems like the rate constant of elimination, K, that remains constant; the opening of tank 2 is twice the size of tank 1, equivalent to normal (tank 2) and reduced rate constant (tank 1), like the reduction in renal impairment and/or hepatic failure; the fow rate from tank 2 is much higher and comparison can be made between the fow rate and the rate of elimination; the higher hydrostatic pressure that changes with time corresponds to the higher amount of the xenobiotic in the body; thus, the rate of elimination is a variable as is the fow rate from the tank, the fow is proportional to the hydrostatic pressure that changes with time as the rate of elimination changes with the amount and time. 424

PRACTICAL APPLICATIONS

Cp ˜ Cp 0 e °Kt

(13.10)

where Cp and Cp are the plasma free concentrations at time t and initial concentration at t = 0. 0

13.2.1 Half-Life of Elimination A useful PK/TK constant of the model is the half-life of elimination. It is the time required for 50% of the compound at any initial level to be eliminated from the compartment, it does not have to be the initial concentration, that is, = At t T= 1 2 , Cp2

Cp1 A , or A2 = 2 2 2

(13.11)

Therefore, log

K ˛ T1 2 ˝ Cp1 ˜ log Cp1 ° 2 2.303

(13.12)

KT1 2 = 2.303 log 2

(13.13)

KT1 2 = 0.693

(13.14)

2.303 log 2 0.693 ln 2 , or T1 2 ° , or T1 2 ° K K K 2.303 log 2 0.693 ln 2 = and K = , or K , or K = T1 2 T1 2 T1 2 ˜T1 2 °

(13.15) (13.16)

Equations 13.15 and 13.16 indicate that the half-life and the rate constant of elimination for a xenobiotic that follows linear pharmacokinetics are constant, independent of the dose, amount, or concentration, and they are inversely proportional to each other. 13.2.2 Time Constant Occasionally it is more convenient to use the time constant rather than the half-life. The time constant is the reciprocal of the frst-order rate constant. The relationship between the time constant and the half-life is as follows: Time Constant =

1 K

1 1 = = 1.44T1 2 K 0.693 T1 2

(13.17) (13.18)

The time constant is also known as turnover time, and in non-compartmental analysis of intravenous bolus injection, it is referred to as mean residence time (MRT). The time constant or MRT, or turnover time, represents the time for 63% of an intravenous bolus dose to be eliminated from the body. The signifcance of the time constant or MRT is further discussed at the end of this chapter in association with the non-compartmental analysis. The relationship between MRT and T1/2 is T1 2 1 ˛ K 0.693

(13.19)

T1 2 ˜ 0.693 ° MRT ˛i.v.bolus

(13.20)

˜ MRT °i.v.bolus ˛

13.2.3 Apparent Volume of Distribution A useful and often misunderstood constant of the linear compartmental model is the volume of distribution, Vd. It is identifed as the apparent volume of distribution to differentiate the value from the physiological volumes of distribution. This constant has no relationship with the total body water (Chapter 8, Section 8.2.1), or other physiological volumes. It is only a constant of conversion for changing the concentration of a free xenobiotic to its total amount, or vice versa. The defnition is applicable not only for a one-compartment model, but also for multi-compartmental models (Chiou, 1981). A large volume of distribution usually indicates that the concentration of measurable free compound in plasma is low because of its vast distribution, localization, and association with regions that are not accessible or being a part of the sampling compartment. 425

13.2 LINEAR ONE-COMPARTMENT OPEN MODEL – INTRAVENOUS BOLUS INJECTION

The apparent volume of distribution is determined by dividing the amount by the plasma concentration: Vd =

At Cpt

(13.21)

At time zero, the amount of free xenobiotic in the body or compartment, is equal to dose. ˜Vd °

Dose Cp 0

(13.22)

The free concentration at time zero, or initial plasma concentration, Cp 0 , can be estimated by the extrapolation of the plasma concentrations to time zero graphically, or estimated from the calcuKt . lated equation of the line, e.g., the ftted line of log Cp ˜ log Cp 0 ° 2.303 13.2.4 Total Body Clearance The description and importance of clearance is discussed in detail in Chapters 10 and 11. For computational purposes, the total body clearance,ClT , can be estimated as the ratio of the rate of elimination of a xenobiotic from the body at a given time to its concentration at the same time. ClT ˜

° Rate ˛t

° Concentration ˛t

˜

dAt dt KAt ˜ ˜ K ˝ Vd Cp At Vd

(13.23)

Therefore, one approach in estimation of the total body clearance is the apparent volume of distribution multiplied by the overall elimination rate constant. It is also the sum of metabolic, renal, and residual clearances: ClT ˜ Vd ° K ˜ Vd(k e ˛ k m ˛ k n ) (13.24)

˜ Clrenal ˛ Clmetabolic ˛ Clresidual

13.2.5 Duration of Action The duration of action is the time that plasma concentration remains within the therapeutic range of a therapeutic xenobiotic, or at toxic level, which is above the maximum safe dose level. When the plasma concentration falls below the minimum effective level, no effective response should be expected from a therapeutic agent, and when the concentration of a toxic xenobiotic falls below the maximum safe concentration, the toxicity is at a controlled level. In general, the duration of action is a more relevant calculated value for therapeutic xenobiotics. For a single dose of intravenous bolus injection, the duration of action is estimated as = td

Cp 0 Cp 0 2.303 = log 3.3T1/2 log K CpMEC CpMEC

(13.25)

Where CpMEC is the minimum effective concentration of a therapeutic agent. As defned in Equation 13.25, the duration of action is inversely proportional to the overall elimination rate constant, K , and directly proportional to the half-life of elimination. This indicates that the duration of action of a compound in patients with renal impairment or hepatic failure is longer than normal. The duration of action is not directly proportional to the administered dose or the initial plasma concentration. In other words, if the dose is doubled, the duration of action will NOT be doubled. If the dose is doubled, Cp 0 will be doubled, but the duration of action will only increase by one half-life:

˜ td °1 ˛ 3.3T1/2 log ˜ td °2 ˛ 3.3T1/2 log ˙

˜ td °2 ˛ ˜ td °1 ˝ 3.3T1/2 ˇˇ log ˆ

Cp 01 CpMEC

˜

2 Cp 01

(13.26)

°

(13.27)

CpMEC

2Cp 0 Cp 0 ˛ log CpMEC CpMEC

˘  ˝ 1T1/2 

(13.28)

The amount of dose for Equation 13.27 is twice of Equation 13.26. Thus, if the dose is quadrupled with respect to the frst dose, the duration of action would only increase by two half-lives. 426

PRACTICAL APPLICATIONS

13.2.6 Estimation of Fraction of Dose in the Body at a Given Time The fraction of the dose in the body at any time after the administration of the dose is estimated by normalizing Equation 13.10 with respect to the initial plasma concentration, Cp 0

˜ fb ° t ˛

˝ Cp ˛ e ˝Kt ˛ e Cp ˜

0.693 t T1/ 2

t

˛ ˜ 0.5 ° T1/2

(13.29)

Based on Equation 13.29, the practical numerical values for the fraction of the administered dose in the body as a function of half-life can be estimated as:

˜ fb °t˛0 ˛ 1; ˜ fb °t ˛1T

12

˜ fb °t˛4.3 T

12

˜ fb °t˛2 T

˛ 0.5;

12

˜ fb °t˛6.6 T

˛ 0.05;

12

˛ 0.25;

˜ fb °t˛3.3 T

12

˛ 0.1;

˛ 0.01; ˜ f b °t ˛7 T ˝ 0 12

13.2.7 Estimation of Fraction of Dose Eliminated by All Routes of Elimination at a Given Time At time zero, t = 0, following an intravenous bolus dose, the fraction of the dose in the body is equal to one, as the dose eliminates from the body exponentially, the sum of fractions remaining in the body and eliminated from the body at time t is equal to 1, i.e., ˜ f el ° ˛ ˜ f b ° ˝ 1. Therefore, the t

t

fraction of the dose eliminated from the body by all routes of elimination at time t is Cp

˜ fel °t ˛ 1 ˝ ˜ fb °t ˛ 1 ˝ Cp0

˛ 1 ˝ e ˝Kt ˛ 1 ˝ e

˝

0.693 t T1/2

t

˛ 1 ˝ ˜ 0.5 ° T1/2

(13.30)

13.2.8 Determination of the Area Under Plasma Concentration– Time Curve after Intravenous Bolus Injection The area under plasma concentration–time curve, AUC, of an intravenously injected dose is a dose-dependent variable, useful in calculating other PK/TK variables and constants and particularly important in determining the absolute bioavailability of xenobiotics given by extravascular routes of administration. The AUC is estimated either by the model-dependent approach of integration of PK/TK model’s equation, or by the model-independent method of trapezoidal rule. The integration of Equation 13.10 provides the model-dependent general equation for estimation of AUC for the one-compartment model with intravenous bolus injection: AUC ˜

˝

˛

0

Cp 0 e °Kt ˜ Cp 0

e °Kt °K

˛

˜ 0

Cp 0 K

(13.31)

AUC is also estimated by the trapezoidal rule (Addendum I, Part 2, Section A.4) according to the following general equation ˜

AUC 0 ˛

n

˙ i˛0

Cpi ° Cpi°1 Cp (ti°1 ˝ ti ) ° last 2 K

(13.32)

Multiplying the numerator and denominator of Equation 13.31 by the apparent volume of distribution yields the following relationship:

°Cp ˛ ˝Vd ˜ Dose 0

AUC ˜

K ˝ Vd

ClT

(13.33)

Equations 13.31 and 13.33 imply that in linear pharmacokinetics (i.e., dose-independent pharmacokinetics) the AUC is directly proportional to the initial plasma concentration or initial amount, dose, and inversely proportional to the overall elimination rate constant and clearance. This means that if the dose is doubled, the AUC will be doubled, or if the clearance is declined by 50%, the AUC will be doubled (Figures 13.2 and 13.3). The plot of AUC versus Dose is used to determine the dose dependency of the PK/TK of a compound. A nonlinear relationship as presented in Figure 13.4 indicates that the rate of elimination of xenobiotic from the body is no longer frst-order kinetics, and the equations of linear pharmacokinetics as described in this chapter are not applicable. The nonlinear relationship between Dose and AUC occurs when saturable processes govern the excretion and/or metabolism. As demonstrated in Figure 13.4, in a nonlinear relationship if the dose is doubled, the AUC will increase disproportionately. 427

13.2 LINEAR ONE-COMPARTMENT OPEN MODEL – INTRAVENOUS BOLUS INJECTION

Figure 13.2 Schematic of the linear relationship in dose-independent PK/TK between the administered dose and the related area under the plasma concentration–time curve, AUC; the increase in AUC is directly proportional to the administered dose and the ratio of dose over AUC remains constant that corresponds to the total body clearance; plot of AUC versus Dose has no xand y-intercept, and the slope is equal to the reciprocal of the plasma clearance, ClT .

Figure 13.3 Schematic of the relationship between the initial plasma concentrations, Cp 0 , and the AUC in linear PK/TK; this linear relationship is the same as Figure 13.2, noting that AUC = Dose /ClT , therefore, dividing the numerator and denominator by the apparent volume of diction Vd, which is a constant, changes the relationship to AUC = Cp 0 / K ; the line passes through the origin with a slope of reciprocal of the overall elimination rate constant K , thus the ratio of Cp 0 / AUC remains constant at any dose level, however. 428

PRACTICAL APPLICATIONS

Figure 13.4 Diagram of nonlinear, dose-dependent plot of AUC versus dose; the initial part of the curve represents the high concentrations of a xenobiotics that is created by one or more capacity-limited elimination process(es); the overwhelmed system of elimination, whether hepatic metabolism or renal active secretion or reabsorption, continues at the maximum level of elimination until the system prevails and the concentration declines to a manageable level, which may continue at a linear condition; the main advantage of the plot is just to identify and confrm that the PK/TK behavior of the xenobiotic is dose-dependent and the relationship between AUC vs dose or vs initial plasma concentration is nonlinear; in a nonlinear relationship, the ratio of dose/AUC changes dramatically and increases with increasing the dose with no predictive interpretation at the initial high plasma levels. 13.3 LINEAR TWO-COMPARTMENT OPEN MODEL WITH BOLUS INJECTION IN THE CENTRAL COMPARTMENT AND ELIMINATION FROM THE CENTRAL COMPARTMENT Among the multi-compartmental models, the two-compartment model is a practical model for quantitative PK/TK analysis of most xenobiotics. The model also avoids the limitation of considering the body as a single compartment (Riegleman et al., 1968). The two-compartment model assumes the intravenously injected compound distributes in the central compartment and peripheral compartment in proportion to their blood perfusion and eliminates from the central compartment in proportion to its concentration. The “central,” or “frst,” compartment represents the systemic circulation, extracellular fuid, and highly perfused organs/tissues such as the liver, kidney, lungs, etc. (A1 ). The highly perfused tissues are included in the central compartment because they achieve a rapid equilibrium with the systemic circulation and are kinetically indistinguishable from the systemic circulation (Rescigno and Seger, 1966; Wagner, 1975; Gibaldi and Perrier, 1982). Thus, the central compartment is assumed kinetically as a homogeneous compartment. The “peripheral” or “second” compartment represents tissues and regions of the body that have reduced blood perfusion and are less accessible (A2 ). The examples of tissues that may be included in the peripheral compartment are adipose tissue, muscle, skin, etc. The peripheral compartment is not as homogeneous as the central one. In addition to the assumption of homogeneity of the central compartment, it is assumed that all biological processes of distribution, metabolism, and excretion follow frst-order kinetics. Following the bolus injection of a xenobiotic that follows the two-compartment model, the compound undergoes an initial very rapid mixing in the central compartment, with simultaneous distribution to the peripheral compartment, and removal from the central compartment by the major organs of elimination, which are highly perfused tissues and are part of the systemic circulation. The rate of elimination (metabolism and excretion) is proportional to the concentration of 429

13.3 LINEAR TWO-COMPARTMENT OPEN MODEL

the central compartment and the rate of distribution is proportional to the concentrations in each compartment. 13.3.1 Equations of the Two-Compartment Model The diagram of the model is depicted in Figure 13.5. The rate of change in the central compartment depends on the rates of distribution and the elimination: dA1 ˜ k 21 A2 ° k12 A1 ° k13 A1 1l dt

(13.34)

The input in the central compartment is instantaneous and has no rate. The rate of change in the peripheral compartment is a function of distribution rates in and out of the compartment with no elimination from the peripheral compartment: dA2 ˜ k12 A1 ° k 21 A2 dt

(13.35)

The variables A1 and A2 are the amounts of xenobiotics in the central and peripheral compartments at any time, respectively. The rate constants of distribution are k12 and k 21 , the frst subscript number represents the leaving compartment, and the second number represents the arrival compartment. By using the Laplace method of integration or method of input-disposition functions (Addendum I part 2, Section A1 and A3), the integrated equations of the model are defned as (Rescigno and Seger, 1966; Wagner, 1975, 1993; Gibaldi and Perrier, 1982) A1 ˜

D ˛ ˆ ° k 21 ˝ °ˆt D ˛ k 21 ° ˇ ˝ °ˇt e ˙ e ˆ °ˇ ˆ °ˇ

(13.36)

k12D °ˆt °˙t e °e ˙ °ˆ

(13.37)

A2 ˜

˛

˝

The frst-order hybrid rate constants, α and β, are the roots of a quadratic equation generated during the integration and development of the disposition function (Addendum I, part 2, Section A.2 Equation A.33), also shown by Equations 13.49–13.51. The hybrid rate constants of α and β are estimated by ftting the biexponential equation of the model to the plasma concentration versus time data. Their initial estimates can be obtained from

Figure 13.5 Diagram of a linear two-compartment model with intravenous bolus injection into the central compartment; the central compartment, representing the systemic circulation and highly perfused tissues, is considered a homogeneous compartment in contact with the peripheral compartment, less perfused and less accessible tissues and organs, by means of frst-order distribution; the injected xenobiotic eliminates from the central compartment by way of excretion and metabolism; all the rate constants are considered frst-order, and there is no rate associated with the injection of the dose; A1 and A2 are the amount of xenobiotic at time t in the central and the peripheral compartments, respectively; k12 and k 21 are the frst-order distribution rate constants between the two compartments, and k10 is the overall elimination rate constant (excretion + metabolism); A3 is the cumulative amount eliminated from the body by all routes of elimination. 430

PRACTICAL APPLICATIONS

the slope of the terminal portion of log plasma concentration–time curve (slope = ˜° 2.303) and the slope of the residual line (slope = ˜ ° 2.303) (see Addendum II for related case studies and applications). Dividing Equation 13.36 by the apparent distribution volume of the central compartment V1 yields the equation of the model in terms of plasma concentration Cp ˜

D ˛ ˆ ° k 21 ˝ °ˆt D ˛ k 21 ° ˇ ˝ °ˇt e ˙ e V1 ˛ ˆ ° ˇ ˝ V1 ˛ ˆ ° ˇ ˝

(13.38)

˙ D ˛  ° k 21 ˝ ˘ By setting the coeffcient of the frst exponential equal to ‘a’ i.e., ˇ a ˜  and coeffcient ˇ V1 ˛  °  ˝  ˆ ˙ D ˛ k 21 °  ˝ ˘ of second exponential equal to ‘b’ i.e., ˇ b ˜ , the biexponential equation of the model ˇ V1 ˛  °  ˝  (Equation 13.38) can be simplifed to ˆ Cp ˜ ae ˛˝t ° be ˛˙t

(13.39)

The semi-logarithm plot of plasma concentration versus time for a compound that follows the two-compartment model has two distinct phases (Figure 13.6). The early phase with steeper slope, referred to as the distributive phase, is formed by the rapid decline of plasma concentration in the central compartment. This rapid decline is the result of several physical and physiological processes that occur simultaneously and immediately after the injection. Among these processes are the rapid distribution and dilution of the xenobiotic into those fuids and tissues that make up the central compartment, transfer to the peripheral compartment, and eliminate from the central compartment. After a certain period, that varies from compound to compound, this initial phase comes to an end, and the xenobiotic concentration in various tissues and fuids declines steadily. This slower and gradual decline appears as a second phase of a semilogarithmic plot of plasma concentration versus time curve and is referred to as the post-distributive phase or disposition phase. During this phase, all physiological processes (excretion, metabolism, and distribution) contribute to the consistent decline of drug concentration in the central compartment, which is also a function of physicochemical characteristics of a drug and its interaction with biological molecules (Figure 13.6). Therefore, the rate constant β, calculated from the slope of the second phase of the curve (˜° 2.303) can only be identifed as the disposition (i.e., distribution and elimination (metabolism and excretion)) rate constant and the related half-life as physiological, biological, or disposition half-life. This is one of the differences between the one- and two-compartment models. Theoretically, extrapolation of the terminal portion of the curve to the y-axis gives the y-intercept, which is equal to the coeffcient ‘b’. This extrapolated line represents the biological processes that are involved in the disposition of a xenobiotic. Therefore, subtracting the data points of the extrapolated line that correspond to the early time points from the observed plasma concentrations of the distributive phase leads to several residual values that represent all other processes that contribute to the early sharp decline of plasma concentration. Plotting the residuals against the early time points forms a straight line with a slope of (˜ ° 2.303) and y-intercept of ‘a.’ Both α and β are frst-order hybrid rate constants; α represents the rate constant of all physical and physiological processes other than the distribution and elimination that occurs immediately and concurrently after the injection of a xenobiotic in the body during the early distributive phase. The rate constant β represents the hybrid rate constant of distribution and elimination (i.e., disposition). The initial estimates of a, b, α and β can be obtained by linear regression analysis of the extrapolated and residual lines; however, for the best estimate, specialized PK/TK software are recommended. The PK/TK software analyze the entire plasma level time data, using nonlinear regression analysis and the required statistical analysis/information criteria (see Addendum I – Part 2, Section A.7). 13.3.2 Estimation of the Initial Plasma Concentration and Volumes of Distribution, Two-Compartment Model The initial plasma concentration is estimated by setting t equal to zero in Equation 13.39: Cp 0 ˜ a ° b

(13.40)

Substituting the formulas for coeffcients a and b according to Equation 13.38 yields 431

13.3 LINEAR TWO-COMPARTMENT OPEN MODEL

Figure 13.6 Profles of plasma-concentration vs time and log plasma-concentration vs time of a xenobiotic that follows the two-compartment model for its disposition in the body; plot of Cpvs time shows a typical exponential curve without giving any obvious hint that whether the disposition is a single or multiexponential decline; plot of log Cp vs time exhibits the biphasic characteristics of the two-compartment model; the early phase of the biphasic plot is identifed as the distributive phase, and the second phase is the post-distributive or disposition phase; the slope of the disposition phase ( ˜° / 2.303) provides the value of β, one of the hybrid rate constants of the model; the y-intercept of extrapolated disposition phase determines the coeffcient of the small exponential β of the biexponential equation of the model; subtraction of the extrapolated values of the disposition phase from the distributive phase generate the residual line with slope of ˜° / 2.303 and the y-intercept of the coeffcient of the large exponential α of the biexponential equation of Cp ˜ ae ˛˝t ° be ˛˙t . Cp 0 ˜

D ˛ ˆ ° k 21 ˝ D ˛ k 21 ˙ ˇ ˝ D ° ˜ V1 ˛ ˆ ˙ ˇ ˝ V1 V1 ˛ ˆ ˙ ˇ ˝

(13.41)

D D ° Cp 0 a ˛ b

(13.42)

˜V1 °

Like the one-compartment model, the apparent volume of distribution of the central compartment is only a proportionality constant to determine the amount in the central compartment (Gibaldi et al., 1969):

˜ A1 °t ˛ Cpt ˝ V1

(13.43)

Equations 13.41 and 13.42 emphasize the importance of taking early plasma samples after the injection for the purpose of identifying the distributive phase more clearly. If the initial blood samples are taken late after the end of the distributive phase, which is often short, the profle of log Cp versus time would only show the disposition phase, which can be confused with the onecompartment model, thus assuming Cp 0 is equal to b.

432

PRACTICAL APPLICATIONS

13.3.3 Estimation of the Rate Constants of Distribution and Elimination The frst-order distribution rate constants are k12 and k 21 . The rate constant k 21 , representing the transfer rate constant from peripheral compartment to the central compartment, is determined from the a or b formula: b˜

˛ a ° b ˝ (k21 ˙ ˆ) ˛ˇ ˙ ˆ˝

˜k 21 °

a˝ ˛ b˙ a˛b

(13.44)

(13.45)

Because the denominator of Equation 13.45 is the initial plasma concentration, a and b of the numerator is normalized with respect to the initial plasma concentration as follows: aN ˜ a ˛ a ° b ˝

(13.46)

bN ˜ b ˛ a ° b ˝

(13.47)

˜k 21 ° aN˝ ˛ bN ˙

(13.48)

where aN and bN are the normalized values of a and b with respect to the initial plasma concentration. The frst-order hybrid rate constants α and β, are the roots of the quadratic equation generated during the integration and development of the disposition function (Addendum I, part 2, Section A.2 Equation A.33; Benet, 1972), as follows: (Disp)s ˜

s ° k 21 s ° E2 ˜ s2 ° ˛ k12 ° k 21 ° k13 ˝ ° k 21k13 ˛ s ° ˙ ˝˛ s ° ˆ ˝

(13.49)

The denominator of Equation 13.49, ° s ˜ ˝ ˛° s ˜ ˙ ˛ , following multiplication can be written as: s2 ˜ s ° ˝ ˜ ˙ ˛ ˜ ˝˙ which corresponds to s2 ˜ ° k12 ˜ k 21 ˜ k13 ˛ ˜ k 21k13 . The comparison of the two equations yields the following two relationships: ˜ ˛ ° ˝ k12 ˛ k 21 ˛ k13

(13.50)

˜° ˛ k 21k13

(13.51)

Therefore, the overall elimination rate constant, k13 , and the transfer rate constant from the central to the peripheral compartment, k12 , can be estimated by the equations 13.52 and 13.53. k13 ˜

°˛ k 21

k12 ˜ ˝ ° ˙ ˛ k13 ˛ k 21

(10.52) (13.53)

13.3.4 Half-Lives of the Two-Compartment Model Contrary to the one-compartment model with intravenous bolus injection that offers only one rate constant and one half-life of elimination, each rate constants of the two-compartment model provides an estimated half-life related to the function it represents. 13.3.4.1 Biological Half-Life – Two-Compartment Model The biological half-life, also known as half-life of β, or physiological half-life, or half-life of disposition represents the combined effect of distribution and elimination (i.e., disposition) and should not be mistaken with the elimination half-life. It is estimated as

˜ T1 2 °˝ ˛ 0.693 ˝

(13.54)

In the two-compartment model, the fractions of the dose remaining in the body or eliminated depends on the biological half-life and is estimated as:

433

13.3 LINEAR TWO-COMPARTMENT OPEN MODEL

˜ fel °t˛1˜T

°˙

˛ 0.50 ˝ ˜ f b °t˛1 T ˛ 0.50 ˜ 1 2 °˙

˜ fel °t˛2 ˜T

°˙

˛ 0.75 ˝ ˜ f b °t˛2 T ˛ 0.25 ˜ 1 2 °˙

12

12

˜ fel °t˛3.3 ˜T

°˙

˛ 0.90 ˝ ˜ f b °t˛3.3 T ˛ 0..10 ˜ 1 2 °˙

˜ fel °t˛4.3 ˜T

°˙

˛ 0.95 ˝ ˜ f b °t˛4.3 T ˛ 0.05 ˜ 1 2 °˙

˜ fel °t˛6.6 ˜T

°˙

˛ 0.99 ˝ ˜ f b °t˛6.6 T ˛ 0.01 ˜ 1 2 °˙

12

12

12

˜ fel °t˛7 ˜T

12

°˙

ˆ 1.00 ˝ ˜ f b °t˛77

˜T1 2 °˙

˛ 0.00

13.3.4.2 Elimination Half-Life – Two-Compartment Model Elimination half-life or half-life of k13 (often referred to as k10 ) represents the overall elimination rate constant, i.e., the combined half-lives of metabolism, excretion, and other residual elimination processes in the body:

˜ T1 2 °elim ˛ 0.k693

(13.55)

k13 ˜ k e ° k m ° k n

(13.56)

13

13.3.4.3 Half-Life of α – Two-Compartment Model The half-life of the hybrid rate constant α is estimated as

˜ T1 2 °˝ ˛ 0.693 ˝

(13.57)

Since α is estimated from the slope of the residuals, and residuals are calculated by subtraction of the extrapolated disposition line from the early distributive phase, the half-life of α can be defned as the half-life of physical and physiological processes other than distribution and elimination (disposition) that occur immediately after the injection of a xenobiotic during the distributive phase. This half-life of α is neither biological nor elimination half-life, it is just the half-life α used only to estimate α , if the half-life was known before the estimation of α . 13.3.4.4 Half-Life of k12 This half-life implies the half-life of xenobiotic transfer from the central to the peripheral compartment:

˜ T1 2 °k

12

˛

0.693 k12

(13.58)

13.3.4.5 Half-Life of k 21 This is the half-life of xenobiotic transfer from the peripheral to the central compartment:

˜ T1 2 °k

21

˛

0.693 k 21

(13.59)

13.3.5 Determination of the Area Under the Plasma Concentration–Time Curve, Volumes of Distribution, and Clearances – Two-Compartment Model The area under plasma concentration–time curve of a compound that follows the two-compartment model, and is given intravenously by bolus injection, can be estimated by the trapezoidal rule (Addendum II, part 2, Section A.4) or by the integration of Equation 13.39. The integration can be carried out from time zero to infnity or from time zero to any time t, or the last time point of sampling: AUC 0˜ ˛

434

˙

˜

0

ae °ˆt ˝ be °ˇt ˛

a b ˝ ˆ ˇ

(13.60)

PRACTICAL APPLICATIONS

AUC t0 ˜

t

˝ ae

˛˙t

0

° be ˛ˆt ˜

ae ˛˙t be ˛ˆt ° ˛˙ ˛ˆ

(13.61)

Substitution of a and b, as defned in Equation 13.41 in Equation 13.60, yields the following equation for AUC ∞0 ˜ AUC 0 °

°

D ˝ ˇ ˛ k 21 ˙ D ˝ k 21 ˛ ˘ ˙ ˆ V1 ˝ ˇ ˛ ˘ ˙ ˇ V1 ˝ ˇ ˛ ˘ ˙ ˘ ˘Cp 0 ˝ ˇ ˛ k21 ˙ ˆ ˇCp0 ˝ k211 ˛ ˘ ˙ ˝ ˇ ˛ ˘ ˙ k 21Cp 0 ° ˇ˘ ˝ ˇ ˛ ˘ ˙ ˝ ˇ ˛ ˘ ˙ ˇ˘ ˜ AUC ˛0 °

Cp 0 k13

(13.62)

Equation 13.62 can also be used to estimate k13 k13 ˜

Cp 0 ˛

AUC 0

˜

1 a°b ˜ a N bN a b ° ° ˝ ˙ ˝ ˙

(13.63)

The apparent volume of distribution, as it was discussed earlier in Section 13.2.3, is considered only a conversion factor that is used to change the concentration of a xenobiotic to the total amount in the body or vice versa. It has a very limited or no physiological signifcance, and its magnitude depends on the model and the assumptions of the related kinetic pool. For a two-compartment model, the volumes of distribution include the volume of the central compartment, the volume of the peripheral compartment, and the overall volume of distribution incorporating both compartments. The volume of distribution of the central compartment V1 was discussed earlier and can be estimated by Equation 13.42. The overall volume of distribution of a compound that follows the two-compartment model is known as ˜Vd ° or ˜Vd ° and is estimated from the model-independent relationship of area

˛

Clearance = Dose AUC :

˜Vd °area ˛

D D ˛  ˝ AUC0 ˆ a b ˘ ˙  ˇ 

(13.64)

The apparent volume of distribution of the peripheral compartment, often identifed as V2 , is diffcult to determine directly, mainly because the peripheral compartment of the model discussed here is not a sampling compartment, and the assumption of homogeneity, as it was assumed for the central compartment, is not applicable. The indirect estimation of this fctional volume term is by subtracting the volume of the central compartment from the overall volume of distribution. As a result, any “concentration” term for the peripheral compartment would also be a fctitious parameter. Therefore, it is preferred to use the “amount” to characterize the time course of a drug in the peripheral compartment rather than “concentration” in the peripheral compartment. The defnition of the total body clearance here is the same as described in Chapters 10 and 11. The useful equations of estimating the total body clearance of a xenobiotic that follows the twocompartment model are: ClT ˜

(13.65)

D AUC

° 0

ClT ˜ V1 ° k10

(13.66)

ClT ˜ °Vd ˛area ˝ ˙

(13.67)

Equations 13.66 and 13.67 indicate that the elimination rate from the body is the rate of elimination from the central compartment. By setting the total amount of xenobiotic in the body equal to the combined amount in the central compartment and peripheral compartment at any time t, that is, At ˜ A1 ° A2 , the fraction of the dose in the central compartment at any time would be 435

13.3 LINEAR TWO-COMPARTMENT OPEN MODEL

f central = A1 At

(13.68)

and the amount in the body and central compartment would be At = A1 f central

(13.69)

A1 ˜ f central ° At

(13.70)

The overall rate of elimination from the body can be defned as dA ˜ °˙A ˜ °k10 A1 ˜ °k10 ˛ f central ˝ A dt

(13.71)

and Cp ˙V1 A1 (13.72) ° k10 A A Because for the model discussed here (Figure 13.5) the elimination occurs only from the central compartment, the total body clearance can also be expressed in terms of ˜ ° k10 ˛ f central ˝ ° k10

˜A ° ˛Vd ˝area ˙ ˜ ° k10 ˙V1 Cp

(13.73)

By knowing the fraction of the dose eliminated from the body unchanged and fraction eliminated as metabolites, clearance terms, such as “renal” and “metabolic” are estimated as Clr ˜ f eClT ˜

D ° Am˛ Ae˛ ClT ˜ ClT D D

(13.74)

Clm ˜ f mClT ˜

Am˛ D ° Ae˛ ClT ˜ ClT D D

(13.75)

13.3.6 Assessment of the Time Course of Xenobiotics in the Peripheral Compartment – Two-Compartment Model In Equation 13.37, the amount of xenobiotic in the peripheral compartment at any time after the injection is based on the integration of differential Equation 13.35 and by using the method of input-disposition function, or the method of substitution, as described in Addendum I, Part 2. The uptake by the peripheral compartment for a given compound depends on several factors, such as lipid solubility of the compound, the extent of elimination from the central compartment, the degree of protein-binding, and the type of interaction and accumulation in the tissues (Figure 13.7). It should be noted that in linear PK/TK, the slopes of the terminal portion of log A2 versus time and log A1 versus time are the same and the lines should be essentially parallel (Figure 13.8). The area under the amount vs time curve of the peripheral compartment, ° AUC ˜0 ˛ peripheral , represents the extent of accumulation of xenobiotic in the peripheral compartment and is estimated as

° AUC ˛ ˜ 0

peripheral

˝

k12D ˆ 1 1  ˘ ˙   ˙ˇ   

(13.76)

The maximum amount in the peripheral compartment that corresponds to Tmax of the curve (Figure 13.7) can be estimated by the following equations

˜ Tmax °peripheral ˛ ˜ Amax °peripheral ˛

˙ 2.303 log ˆ ˙ ˝ˆ

k12D ˝ˆTmax e ˝ e ˝˙Tmax ˙ ˝ˆ

˜

(13.77)

°

(13.78)

At Tmax , the rate of input from the central compartment into the peripheral compartment, k12 A1 , is equal to the rate of output from the peripheral compartment into the central compartment, k 21 A2, which indicates a brief steady-state condition just at that moment. When the rate of input into the compartment is equal to the rate of output from the compartment, the rate of amount change in the compartment would be equal to zero: 436

PRACTICAL APPLICATIONS

Figure 13.7 Profle of the time course of amount of a xenobiotic in the peripheral compartment,A2 , in comparison to the amount in the central compartment, A1 , and cumulative amount eliminated from the body (Ae + Am ) following an intravenous bolus administration; the profle of the bell-shape curve of A2 represents the change in the rate of input, k12 A1 , and the rate of output, k 21 A2 during the disposition of the xenobiotic; before the maximum of the curve, the rate of input is greater than the rate of output, k12 A1 > k 21 A2, and at the maximum, the rates are equal at only a moment in time, and the later decline in the amount after the maximum is due to the rate of output being larger than the rate of input k 21 A2> k12 A1 , which is due to the elimination and decline of free amount in the central compartment.

dA2 ˜ 0 °k12 A1 ˜ k 21 A2 dt

(13.79)

Based on this concept, the volume of distribution at steady state is proposed and is derived according to the following assumptions and steps: At Tmax : A1 ˜ Cp ° ˛V1 ˝ss

(13.80)

A2 ˜ C2 ° ˛V2 ˝ss

(13.81)

˜Vd °ss ˛ ˜V1 °ss ˝ ˜V2 °ss

(13.82)

k12Cp ˜V1 °ss ˛ k 21C2 ˜V2 °ss

(13.83)

If Cp = C2

˝ k12 ˇ  V1 ˙ k 21 ˘

˜V2 °ss ˛ ˆ

˛k k ˆ ˛k ˆ ˜(Vd)ss ° ˙ 12 ˘ V1  V1 ° ˙ 12 21 ˘ V1 ˝ k 21 ˇ ˝ k 21 ˇ

(13.84) (13.85)

437

13.3 LINEAR TWO-COMPARTMENT OPEN MODEL

Figure 13.8 Depiction of a theoretical concept of parallel terminal phase of log amount of a xenobiotic versus time in the central and peripheral compartments; A1 is the amount in the central compartment; A2 is the amount in the peripheral compartment with two scenarios of higher and lower uptakes; the slope of the terminal phase of the central and peripheral compartments is a function of the disposition rate constant, which is a blend of frst-order distribution and elimination rate constants. It is worth noting that (Vd)ss , calculated by Equation 13.85, is different from (Vd)ss of the non-compartmental analysis. Equation 13.85 represents the volume of distribution at the point where the amount of xenobiotic in the peripheral compartment reaches to its maximum. 13.4 LINEAR TWO-COMPARTMENT OPEN MODEL WITH BOLUS INJECTION IN THE CENTRAL COMPARTMENT AND ELIMINATION FROM THE PERIPHERAL COMPARTMENT The diagram of the model is depicted in Figure 13.9. The differential equations of the model are dA1 ˜ k 21 A2 ° k12 A1 dt

(13.86)

dA2 ˜ k12 A1 ° k 21 A2 ° k 20 A2 dt

(13.87)

Integration by the Laplace transform, or input disposition function, yields the following integrated equation of the model (Rowland and Riegelman, 1968; Wagner, 1993): ˆ D ˝ k 21 ° k 20 ˛  ˙  ˛t ˆ D ˝ k 21 ° k 20 ˛  ˙  ˛t Cp ˜ ˘ e ° ˘ e ˘ˇ V1 ˝  ˛  ˙  ˘ˇ V1 ˝  ˛  ˙ 

(13.88)

Setting

438

a˜

D ˝ k 21 ° k 20 ˛ ˆ ˙ V1 ˝ ˇ ˛ ˆ ˙

(13.89)

b˜

D ˝ k 21 ° k 20 ˛ ˆ ˙ V1 ˝ ˆ ˛ ˇ ˙

(13.90)

PRACTICAL APPLICATIONS

Figure 13.9 Diagram of a linear two-compartment model with elimination from the peripheral compartment; the exchange between the central compartment, A1 , and peripheral compartment, A2 , and the description of each compartment are essentially the same as described for the model presented in Figure 13.5; the intravenous bolus injection is in central compartment, but the elimination from the body is via the peripheral compartment; this unique situation creates a dilemma for better understanding and assessment of the volume of distribution of the peripheral compartment, V2 , which depends on the description and assumptions of the model, physicochemical characteristics of the xenobiotic and the mechanism of elimination by the peripheral compartment; if all are defned clearly and logically, the clearance of the compound is estimated by multiplying the volume of the peripheral compartment with the rate constant of elimination. The abbreviated equation is the same as the model with elimination from the central compartment (Equation 13.39, Cp ˜ ae ˛˝t ° be ˛˙t ), but the miniature rate constants of elimination and distribution are estimated from the following relationships: ˜ ˛ ° ˝ k12 ˛ k 21 ˛ k 20

(13.91)

˜° ˛ k12 k 20

(13.92)

Equations 13.91 and 13.92 are different from Equations 13.50 and 13.51, and the distribution rate constant k12 is estimated as Equation 13.93, if the rate constant k 20 is known: k12 ˜

°˛ k 20

(13.93)

A more practical approach is using the following relationships: ˜˝

a° ˙ b˛ ˝ k 21 ˙ k 20 a˙b

(13.94)

k12 ˜ ˝ ° ˙ ˛ ˆ

(13.95)

k 21 ˜ ˛ ° k 20

(13.96)

Therefore,

k 20 ˜

°˛ k12

(13.97)

The volume of distribution of the central compartment V1 , the overall volume of distribution, and the total body clearance are estimated by the same equations discussed in Section 13.3 (Equations 13.42, 13.64, and 13.67). If the elimination from the peripheral compartment is by means of metabolism only, the metabolic clearance can be estimated as Clm ˜ V2 ° k 20

(13.98)

439

13.5 LINEAR THREE-COMPARTMENT OPEN MODEL

13.5 LINEAR THREE-COMPARTMENT OPEN MODEL WITH INTRAVENOUS BOLUS INJECTION AND ELIMINATION FROM THE CENTRAL COMPARTMENT The three-compartment model is often selected to defne the disposition of compounds that, in addition to the central and peripheral compartments, are taken up preferentially by an organ or tissue with a longer residence time that is kinetically a distinguishable pool. The preferential uptake may or may not be obvious and predictable biologically, but it can be detected through the analysis of experimental data and the curve-ftting process. Furthermore, the third compartment may represent a deep compartment associated with different tissues but with similar functioning (Kaplan et al., 1973; Wagner, 1993). A typical three-compartment model is depicted in Figure 13.10; the differential and integrated equations of the model are 1. Dispositional rate of the central compartment, A1 (i.e., rate of change in the amount of xenobiotic in the central compartment): dA1 ˜ k 21 A2 ° k 31 A3 ˛ k12 A1 ˛ k13 A1 ˛ k10 A1 dt

(13.99)

3. Rate of change in the amount of xenobiotic in the peripheral compartment A2 : dA2 ˜ k12 A1 ° k 21 A2 dt

(13.100)

5. Rate of change in the amount of xenobiotic in the peripheral compartment A3 :

Figure 13.10 Diagram of a linear three-compartment open model with elimination from the central compartment; the intravenous bolus dose is given rapidly into the central compartment, A1 , which is in exchange with the two peripheral compartments; the description of the central compartment is the same as fgure 13.5 and 13.9, i.e., the combination of the systemic circulation and highly perfused tissues, the second compartment, A2 , can be assumed as the combination of less perfused tissues and organs, and the third compartment, A3 , is either a mathematical necessity for the curve-ftting procedure, which may essentially points at an organ or region in the body that the xenobiotic has high affnity for and preferentially binds or gets taken up by the region and resides longer; the third compartment can be a highly perfused or a less perfused region; the magnitude of the hybrid rate constants of the data usually determines the sequence of the compartments, for example the second compartment can become the compartment in which the compound binds more and/or resides for a longer time; the frst-order distribution rate constants of the model are k12 , k 21 , k13 and k 31 ; and the overall elimination rate constant, k10 , represents the renal excretion and hepatic metabolism. 440

PRACTICAL APPLICATIONS

dA3 ˜ k13 A1 ° k 31 A3 dt

(13.101)

7. The integrated equation of the central compartment using the above differential equations and input-disposition function is Cp ˜

D ˛ k 21 ° ˆ ˝˛ k 31 ° ˆ ˝ °ˆt D ˛ k 21 ° ˇ ˝˛ k 31 ° ˇ ˝ °ˇt D ˛ k 21 ° ˘ ˝˛ k 31 ° ˘ ˝ °˘t e ˙ e ˙ e V1 ˛ ˆ ° ˇ ˝˛ ˘ ° ˇ ˝ V1 ˛ˇ ° ˆ ˝˛ ˘ ° ˆ ˝ V1 ˛ ˆ ° ˘ ˝˛ˇ ° ˘ ˝

(13.102)

By labeling the coeffcients of the exponential terms as a, b , and c ; and considering ˜ ˝ ° ˝ ˛ , Equation 13.102 can be simplifed to Equation 13.106: a˜

D ˛ k 21 ° ˙ ˝˛ k 31 ° ˙ ˝ V1 ˛ˆ ° ˙ ˝˛ ˇ ° ˙ ˝

(13.103)

b˜

D ˛ k 21 ° ˙ ˝˛ k 31 ° ˙ ˝ V1 ˛ ˆ ° ˙ ˝˛ ˇ ° ˙ ˝

(13.104)

c˜

D ˛ k 21 ° ˙ ˝˛ k 31 ° ˙ ˝ V1 ˛ ˆ ° ˙ ˝˛ˇ ° ˙ ˝

(13.105)

Cp ˜ ae ˛˝t ° be ˛˙t ° ce ˛ˆt

(13.106)

The initial plasma concentration of a xenobiotic that follows the three-compartment model and is administered by bolus injection can be estimated by setting t = 0 in Equation 13.106, i.e., Cp 0 ˜ a ° b ° c

(13.107)

Thus, the volume of distribution of the central compartment is V1 ˜

Dose a°b°c

(13.108)

Similar to the two-compartment model, the denominator of the disposition function determines the relationship between the hybrid rate constants of α , β, and γ with the miniature rate constants of distribution and elimination as follows: ˜ ˝ ° ˝ ˛ ˙ k10 ˝ k12 ˝ k13 ˝ k 21 ˝ k 31

(13.109)

˜° ˝ ˜˛ ˝ °˛ ˙ k10 k 21 ˝ k13 k 21 ˝ k10 k 21 ˝ k 21k 31 ˝ k12 k 31

(13.110)

˜°˛ ˝ k10 k 21k 31

(13.111)

Therefore, the frst-order rate constants of distribution and elimination can be estimated as ◾ The distribution rate constant from the central compartment to A2 peripheral compartment: k 21 ˜ ˆ °

b ˝ ˇ ˛ ˆ ˙˝ˆ ˛ ˘ ˙ ˝ˆ ˛ k13 ˙˝ a ° b ° c ˙

(13.112)

◾ The distribution rate constant from peripheral compartment A3 to A1 : k 31 ˜ ˆ °

c ˝ ˇ ˛ ˆ ˙˝˘ ˛ ˆ ˙ ˝ k31 ˛ ˆ ˙ ˝ a ° b ° c ˙

(13.113)

◾ The overall elimination rate constant: k10 ˜

°˛˝ k 21k 31

(13.114)

◾ The distribution rate constant from A1 to A2 :

441

13.6 LINEAR THREE-COMPARTMENT OPEN MODEL

2 ˆ k 21  k12 ˜ ˛ °  °  ˝ ˙ k 21 ˛  °  °  ˝ ˙ k10 k 31 ° ˘  k k ˙ 31 21 ˇ 

(13.115)

◾ The distribution rate constant from A1 to A3 : k13 ˜ ˆ ° ˇ ° ˘ ˛ ˝ k10 ˛ k12 ° k 21 ° k 31 ˙

(13.116)

The total body clearance is estimated according to the model independent Equation 13.65. 13.6 LINEAR THREE-COMPARTMENT OPEN MODEL WITH INTRAVENOUS BOLUS INJECTION IN THE CENTRAL COMPARTMENT AND ELIMINATION FROM A PERIPHERAL COMPARTMENT The differential equations of the model representing the rate of change in the amount of xenobiotic as a function of time for each compartment, as depicted in Figure 13.11, are dA1 ˜ k 21 A2 ° k 31 A3 ˛ k13 A1 ˛ k12 A1 dt

(13.117)

dA2 ˜ k12 A1 ° k 21 A2 ° k 20 A2 dt

(13.118)

dA3 ˜ k13 A1 ° k 31 A3 dt

(13.119)

The integrated equation of the central compartment in terms of plasma concentration using inputdisposition function and partial fraction theorem is Cp ˜

D ˝ k 20 ° k 21 ˛ ˆ ˙˝ k 31 ˛ ˆ ˙ ˛ˆt D ˝ k 20 ˛ k 21 ˛ ˇ ˙˝ k 31 ˛ ˇ ˙ ˛ˇt D ˝ k 20 ˛ k 21 ˛ ˘ ˙˝ k 31 ˛ ˘ ˙ ˛˘t e ° e ° e V1 ˝ ˆ ˛ ˇ ˙ ˝ ˘ ˛ ˇ ˙ V1 ˝ˇ ˛ ˘ ˙˝ ˆ ˛ ˘ ˙ V1 ˝ˇ ˛ ˆ ˙ ˝ ˘ ˛ ˆ ˙

(13.120)

Figure 13.11 Diagram of a linear three-compartment open model with intravenous bolus input into the central compartment and elimination from the peripheral compartment, A2 ; the general equation of this model, defning the decline of the plasma concentration in the body, i.e., Cp ˜ ae ˛˝t ° be ˛˙t ° ce ˛ˆt is like the model described in Figure 13.10, however the estimation of the miniature rate constants of distribution and elimination from the hybrid rate constants of α , β and γ is different; the descriptions of the frst-order rate constants of distribution and elimination are the same as described for the model presented in Figure 13.10; the elimination from the second compartment depends on the characteristics of the xenobiotic and mechanism of elimination from the compartment; a clear description of the apparent volume of distribution of the second compartment would be needed to defne the clearance of the xenobiotic. 442

PRACTICAL APPLICATIONS

The coeffcients of the exponential terms are different from the previous three-compartment model with elimination from the central compartment (Section 13.5), and are defned as a˜

D ˝ k 20 ° k 21 ˛ ˆ ˙˝ k 31 ˛ ˆ ˙ V1 ˝ˇ ˛ ˆ ˙˝ ˘ ˛ ˆ ˙

(13.121)

b˜

D ˛ k 21 ° k 21 ° ˙ ˝˛ k 31 ° ˙ ˝ V1 ˛ ˆ ° ˙ ˝˛ ˇ ° ˙ ˝

(13.122)

c˜

D ˛ k 20 ° k 21 ° ˙ ˝˛ k 31 ° ˙ ˝ V1 ˛ˆ ° ˙ ˝˛ ˇ ° ˙ ˝

(13.123)

Therefore, Cp ˜ ae ˛˝t ° be ˛˙t ° ce ˛ˆt

(13.124)

The general equation of the model, Equation 13.124, looks the same as the other three compartment models. However, the makeup of the coeffcients and the formulas for estimation of miniature rate constants of distribution and elimination are different (Nagashima et al., 1968; Wagner, 1993). The development of formulas for estimation of transfer and elimination rate constants is more complex than the previous model. The following approach is also based on the assumption of ˜ ˝ ° ˝ ˛ : ˜1 ° ˝ ˛ ˙ ˛ ˆ ° k12 ˛ k13 ˛ k 20 ˛ k 21 ˛ k 31

(13.125)

˜ 2 ° k12 ˝ k 20 ˛ k 31 ˙ ˛ k 20 ˝ k13 ˛ k 31 ˙ ˛ k 21 ˝ k13 ˛ k 31 ˙

(13.126)

˜ 3 ° k12 k 20 k 31

(13.127)

Normalizing the coeffcients of the exponential terms with respect to the initial plasma concentration yields X1 ˜

a a ˜ a ° b ° c Cp 0

(13.128)

X2 =

b Cp 0

(13.129)

X3 =

c Cp 0

(13.130)

Setting the following secondary substitutions as m1 ˜ k12 ° k13 ˜ ˝1 ˛ m2 ˛ k 31

(13.131)

m2 ˜ k 21 ° k 20 ˜ ˛ P ° S ˝ 2

(13.132)

m3 ˜ k 31 ˜ ˛ P ° S ˝ 2

(13.133)

P ˜ m2 ° m3 ˜ X1 ˝ ˆ ˛ ˇ ˙ ° X 2 ˝ ˆ ˛ ˘ ˙ ° ˇ ° ˘

(13.134)

2 S ˜ m2 m3 ˜ ˘ˆX1 ˛  °  ˝ ˙ X 2 ˛  °  ˝ ˙  ° ˇ ˙ 4X 2 ˛ °  ˝˛  °  ˝   

12

(13.135)

Finally, the individual frst-order rate constants of distribution and elimination are estimated as follows: ◾ Distribution rate constant from the central compartment to the peripheral compartmentA3 k13 ˜

˙ 3 ° m3 ˛ ˙ 2 ° m1m3 ° m2 m3 ˝ m3 ˛ m3 ° m2 ˝

(13.136)

Distribution rate constant from central compartment to the peripheral compartment A2 k12 ˜ m1 ° k13

(13.137) 443

13.8 APPLICATIONS AND CASE STUDIES

◾ Distribution rate constant from peripheral compartment A2 to the central compartment k 21 ˜

m1m2 ° m2 m3 ° m3 m1 ˛ ˝ 2 ˛ k13 m3 k12

(13.138)

◾ Overall elimination rate constant from the A2 compartment k 20 ˜ m2 ° k 21

(13.139)

◾ Distribution rate constant from A3 compartment to the central compartment k 31 ˜ m3 ˜ ˛ P ° S ˝ 2

(13.140)

13.7 MODEL SELECTION The selection of most relevant compartmental models for a compound is achieved by use of several statistical analyses, numerical analysis, and curve-ftting methodology of experimental data and the related goodness of the ft. The most implemented model selection approaches are the Akaike Information Criterion (AIC) (Akaike, 1974) and Schwarz criterion, also known as Bayesian Information Criterion (BIC) (Schwarz, 1978). These two methods enable one to simultaneously compare multiple models, assess the selection uncertainty, and allow for the realistic estimation of the model’s parameters and constants. In the absence of a perfect model, both AIC and BIC methods minimize the errors in estimation of parameters and constants by choosing the most realistic and appropriate PK/TK model. In comparing several models for a given set of data, the model with the lowest AIC or BIC is considered the most relevant model to describe a data set (see also Addendum I, Part 2, Section A.7). Commercially available PK/TK software have included the relevant statistical tests and criteria in their curve-ftting and numerical analysis of experimental data. 13.8 APPLICATIONS AND CASE STUDIES The applications and case studies of Chapter 13 are posted in Addendum II – Part 4. REFERENCES Akaike, H. 1974. A new look at the statistical model identifcation. IEEE T Automat Contr 19(6): 716–23. Benet, L. Z. 1972. General treatment of linear mammillary models with elimination from any compartment as used in pharmacokinetics. J Pharm Sci 61(4): 536–41. Boroujerdi, M. 2002. Pharmacokinetics: Principles and Applications, New York: McGraw Hill—Medical Publishing Division. Chiou, W. L. 1981. The physiological signifcance of the apparent volume of distribution, Vdarea or Vdβ , in pharmacokinetic studies. Res Commun Chem Pathol Pharmacol 33(3): 499–508. Gibaldi, M., Nagashima, R., Levy, G. 1969. Relationship between drug concentration in plasma or serum and amount of drug in the body. J Pharm Sci 58(2): 193–7. Gibaldi, M., Perrier, D. 1982. Pharmacokinetics, Second Edition, New York: Marcel Dekker, Inc. Kaplan, S. A., Jack, M. L., Alexander, K., Weinfeld, R. E. 1973. Pharmacokinetic profle of diazepam in man following single intravenous and oral and chronic oral administrations. J Pharm Sci 62(11): 1789–96. Nagashima, R. N., Levy, G., O’Reilly, R. A. 1968. Comparative pharmacokinetics of coumarin anticoagulants IV. Application of a three compartment model to the analysis of dose-dependent kinetics of bishydroxycoumarin elimination. J Pharm Sci 57(11): 1888–95. 444

PRACTICAL APPLICATIONS

Rescigno, A., Seger, G. 1966. Drug and Tracer Kinetics, Waltham: Blaisdell Publishing Co. Riegleman, S., Loo, J., Rowland, M. 1968. Shortcomings in pharmacokinetic analysis by conceiving the body to exhibit properties of a single compartment. J Pharm Sci 57(1): 117–23. Rowland, M., Riegelman, S. 1968. Pharmacokinetics of acetylsalicylic acid after intravenous administration in man. J Pharm Sci 57(8): 1313–19. Schwarz, G. E. 1978. Estimating the dimension of a model. Ann Statist 6(2): 461–4. Wagner, J. G. 1975. Fundamtals of Clinical Pharmacokinetics, First Edition, 57. Hamilton: Drug Intelligence Publication. Wagner, J. G. 1993. Pharmacokinetics for the Pharmaceutical Scientist, Basel: Technomic Publishing Company, Inc.

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:

14 Practical Applications of PK/TK Models: Continuous Zero-Order Exposure to Xenobiotics - Intravenous Infusion 14.1 INTRODUCTION The most relevant example of continuous zero-order exposure to xenobiotics is the intravenous administration of a solution containing a known concentration of a compound directly into the systemic circulation over time. In contrast to bolus injection, which is instantaneous and has no input rate, the intravenous infusion is a constant zero-order input. The specialized infusions through routes of administration such as epidural or intrathecal are not part of this discussion. Intravenous infusion is used commonly to treat illnesses that require long-term exposure at a constant concentration and long-term duration of action, or when the administration of bolus dose can be hazardous to the human subject or experimental animals, and thus it is infused over an interval. The long-term exposure to organic solvents that absorb through inhalation and enter the systemic circulation without impediment can also be considered zero-order input. An intravenous infusion requires that the compound remains stable and soluble in solution, and endurable by the subject. The vehicles are water of highest purity and sterility with dextrose, saline, or other specialized additives and/or medications. The intravenous (IV) infusion of therapeutic agents and fuids is an important part of treatment in hospitals, ambulatory infusion centers, home infusion therapies, home tele-infusion, and telecare industries. The IV infusion, also known as an IV drip, is usually administered by an infusion pump, which uses pressure to administer the solution at a constant rate, or a smart infusion pump with software capability to prevent medication and/or programming errors. The objectives of long-term and short-term intravenous infusion therapy are somewhat different. The objective of long-term infusions is to provide constant supply of the therapeutic agent to accumulate the compound in the body and achieve a steady-state level within the therapeutic range to attain a consistent pharmacological response and extended therapeutic outcome. It is the method of choice for the administration of compounds with a narrow therapeutic range and a short half-life. The objective of short-term infusions is often to substitute for bolus injection when rapid injection can cause side effects such as precipitation or crystallization at the site of administration, or speed shock. The objective of the short infusion is not to achieve the steady-state level, but to deliver the compound gently and safely. An advantage of using intravenous infusion is also to circumvent fuctuations of plasma concentration when the compound is given by multiple dosing administrations. Some applications of continuous infusion are ◾ total parenteral nutrition (TPN) or peripheral parenteral nutrition (PPN) therapy ◾ antineoplastic therapy ◾ pain management ◾ chelation therapy (e.g., removal of elevated levels of aluminum from plasma of patients on dialysis) ◾ chemotherapy with low therapeutic index drugs ◾ infectious disease therapy ◾ lifetime immunosuppressive therapy ◾ hydration therapy ◾ adjunctive therapies (such as simultaneous intravenous therapy with antiemetics, antidiarrheals, anticonvulsant, and anti-infammatories in AIDS patients to control the symptoms of disease and/or the side effects of medications) ◾ anesthesia ◾ programmable implantable infusion pumps

446

DOI: 10.1201/9781003260660-14

PRACTICAL APPLICATIONS OF PK/TK MODELS: ZERO-ORDER INFUSION

14.2 COMPARTMENTAL ANALYSIS 14.2.1 Linear One-Compartment Model with Zero-Order Input and First-Order Elimination The assumptions of the model are the same as discussed in Chapters 12 and 13. The difference is the administration of the dose, which is through a constant zero-order rate of input with units of mass/time, or volume/time for solutions with known concentrations of xenobiotic (Figure 14.1). The frst-order rate of elimination, KAt , depends on the amount or concentration of the infused compound in the body at a given time. Thus, at the beginning of an infusion when the amount, At , of the therapeutic agent in the body is low, the rate of elimination is less than the zero-order rate of input, that is, KA < k0 . As the infusion continues, the compound gradually accumulates in the body, and the rate of elimination increases accordingly and eventually equals the zero-order rate of input, KAt = k0 ; that is, the amount introduced into the body per unit of time is equal to the total amount eliminated from the body by all routes of elimination per unit of time. Therefore, the amount or concentration of a drug in the body remains constant. This plateau concentration is the steady-state plasma level and represents the maximum accumulation of a drug in the body that can be achieved with the selected zero-order rate of input (Figure 14.2). In long-term infusions, the objective is to achieve the steady-state level within the therapeutic range for achieving a consistent pharmacodynamic outcome. The steady-state level remains constant only if the same rate of infusion continues; after stopping the infusion, the amount and the rate of elimination in the body declines until total elimination of the infused compound is achieved. dAt , of a compound in the body, The following differential equation defnes the rate of change, dt which can also be considered the rate of accumulation of the compound in the body, before achieving the steady-state level:

æ dAt ö ç dt ÷ = k0 - KAt è øt

(14.1)

KAt = CptVdK = CptClT

(14.2)

The rate of elimination is defned as Therefore,

Figure 14.1 Depiction of a one-compartment model with zero-order input and frst-order output; the compartment represents the body and is assumed homogeneous representing the systemic circulation and highly perfused tissues/organ that are in equilibrium with each other and changes in plasma concentration refects the changes in the rest of the body; the dose of xenobiotic is dissolved in aqueous medium like water, normal saline, etc., thus, the infusion solution has constant concentration, and the rate infusion, k0 , has constant fow rate, with units of mass/time, or volume/time; after entry into the body, the compound will be subjected to elimination process governed by frst-order kinetics with the overall frst-order rate constant of K ; the elimination rate of the compound is a variable and its variability is a function of amount accumulated in the body, A1 ; the total amount eliminated, Ael , represents the combined cumulative amount excreted unchanged and eliminated as metabolite. 447

14.2 COMPARTMENTAL ANALYSIS

Figure 14.2 Schematic of plasma concentration–time profle of a zero-order infusion of a xenobiotic over an interval long enough to attain a steady-state plasma level; before reaching the steady-state level, the constant rate of infusion is greater than the changeable and rising rate of elimination, k0 > KA , at steady-state level the rate of elimination can no longer rise beyond the rate of infusion, and it equals the rate of infusion, k0 = KA ; thus, at the steady-state level, the rate of input is equal to the rate of output and steady-state level is directly proportional to the rate of infusion; abrupt modifcation of the rate of infusion at steady-state level causes fuctuation in the plasma level, but in due course a new steady-state level will be formed that is proportional to the revised rate of input. æ dA ö ç dt ÷ = k0 - ( ClT ´ Cpt ) øt è

(14.3)

where At and Cpt are the amount in the body and plasma concentration at time t , respectively; ClT is the total body clearance. At steady state, since the rate of input is equal to the rate of output, the rate of change of At is equal to zero. dAss =0 dt \ k0 = KAss = ClT ´ Cpss , and Ass =

k0 , or K

Cpss =

k0 ClT

(14.6)

Integration of Equation 14.1, by using the Laplace transform, yields the following relationships defning the amount and concentration at time t before achieving the steady-state level: At =

448

k0 1 - e -Kt K

(

)

(14.7)

PRACTICAL APPLICATIONS OF PK/TK MODELS: ZERO-ORDER INFUSION

Cpt =

k0 1 - e -Kt K ´ Vd

(

)

(14.8)

As infusion proceeds t Þ ¥, e -Kt Þ 0, Cpt Þ Cpss and Equation 14.8 changes to Equation 14.6. Substitution of Equation 14.6 into Equation 14.8 yields

(

)

Cpt = Cpss 1 - e -Kt Þ log

Cpss - Cpt Kt =CpSS 2.303

(14.9)

Since K = 0.693 T1/2 and e -0.693 = 0.5: t æ ö (14.10) Cpt = Cpss ç 1 - ( 1 2 ) T1 2 ÷ è ø The equation defning the plasma concentration at steady state, Equation 14.6, provides a more accurate estimate of the total body clearance (Chiou et al., 1978; Rowland and Tozer, 1994).

ClT =

k0 Cpss

(14.11)

Equation 14.11 is also used to determine the infusion rate, k0 , by multiplying the desired steadystate concentration with the total body clearance of the compound. k0 = Cpss ´ ClT

(14.12)

14.2.1.1 Estimation of the Time Required to Achieve Steady-State Plasma Concentration Using a Single Long-Term Infusion According to Equation 14.10, if the compound is infused for one half-life, tinfusion = T1 2 , the plasma concentration will be at 50% of the targeted steady-state level; if the time of infusion is twice the half-life, tinfusion = 2 ´ T1 2 , the plasma concentration will be at 75% of the steady-state level; at tinfusion = 3.3 ´ T1 2 , Cp = 90% Cpss ; at tinfusion = 4.3 ´ T1 2 , Cp = 95% Cpss ; and at tinfusion = 6.6 ´ T1 2 , Cp = 99% Cpss . The inverse of this concept is also useful, for example, if the ratio of Cpt Cpss is 0.75, that is the fraction of steady state, the length of infusion is twice the half-life. The fraction of steady-state concentration is estimated as: f ss =

Cpt = 1 - e -Kt Cpss

(

)

(14.13)

In Equation 14.13, the numeral 1 represents 100% of the steady-state level, e -Kt is the fraction left to reach to the steady-state level, and 1 - e - Kt (i.e., f ss ) is the fraction of the steady-state level (Figure 14.3). In practice, an infusion of a therapeutic agent for about four to fve half-lives is considered to have achieved the steady-state level, which is a practical assumption because 95–97% of a steadystate level is close enough to be considered 100%. Furthermore, for a relatively wide therapeutic range, any percent of the steady state would be considered effective if the level is within the therapeutic range. Mathematically, however, it takes about seven half-lives for Equation 14.13 to achieve 99–100% of a targeted steady-state level. If the duration of an infusion is less than seven half-lives of a compound, the plasma concentration at any time during the infusion is estimated by Equation 14.9 or 14.10. When the steady-state level is achieved, its magnitude depends on the rate of infusion; if the rate of infusion is doubled, the steady-state plasma concentration will be doubled and so on. However, the time required to achieve a steady-state level depends only on the half-life of the compound. In other words, by increasing or decreasing the rate of infusion, a different steady-state level is achieved, but it takes the same length of time to attain the steady-state levels (Figure 14.4). The following equation, derived from Equation 14.11, defnes the relationship between a steadystate level and half-life more clearly: Cpss =

1.44 ´ T1/2 ´ k0 k0 = K ´ Vd Vd

(14.14)

449

14.2 COMPARTMENTAL ANALYSIS

Figure 14.3 Illustration of fraction of the steady-state level achieved at different intervals of infusion as a function of the half-life of the compound, the plot is constructed based on the equation of Cp = Cpss 1 - e -Kt ; the ratio of plasma levels, Cp Cpss , provides equation 1 - e - Kt , which represents the fraction of steady-state; the numeral 1 is 100% of the steady-state level, and e -Kt is the fraction left to reach steady-state level; thus, at t = 1T1/2 , or 2T1/2 , or 3.3T1/2 , or 4.3T1/2 , or 6.6 T1/2 , e -Kt = 0.5, 0.25, 0.1, 0.05, 0.01, and 1 - e -KT = 0.5, 0.75, 0.90, 0.95, 0.99 ; infusion in seven half-life should achieve the target steady-state concentration.

(

)

Considering Equation 14.14 in patients with normal elimination functions, K , T1/2 and Vd are constant, thus the steady-state plasma level is only proportional to k0 . In disease states, like renal impairment or hepatic failure, etc., however, when the half-life of a compound is longer than normal or K is smaller, if the treatment of a patient is continued at a rate that is recommended for patients with normal renal and hepatic functions, a higher steady-state level would be achieved, which would require infusion rate adjustment. The time to reach to a predetermined steady-state level is also directly proportional to the half-life of the therapeutic agent in patients with renal or hepatic insuffciency, not the subjects with normal kidney/liver function. 14.2.1.2 Administration of Loading Dose with Intravenous Infusion to Achieve the Steady-State Level Without a Long Delay 14.2.1.2.1 Simultaneous Intravenous Bolus Injection and Infusion One approach in achieving the steady-state level without delay is to give a loading dose. The loading dose is given at the start to achieve a target concentration which then can be maintained by intravenous infusion (Wagner, 1974; Gibaldi and Perrier, 1982). To determine the bolus loading dose, the desired steady-state level is set equal to the initial plasma concentration of the bolus dose, that is, Cpss = Cp 0 . The loading dose would then be equal to: DL = Vd ´ Cpss

(14.15)

When Cp 0 is set equal to Cpss , the combination of intravenous bolus and infusion equations yields the steady-state level: Cptotal = Cpss e -Kt + Cpss (1 - e -Kt ) = Cpss

450

(14.16)

PRACTICAL APPLICATIONS OF PK/TK MODELS: ZERO-ORDER INFUSION

Figure 14.4 Illustration of the dependence of steady-state plasma concentration to the zeroorder rate of infusion; a higher rate of input results in a higher steady-state level; however, the time required for attaining the concentration is independent of the infusion rate, and it is a function of T1/2 and K ; thus the time to achieve a steady-state level remains the same for different input rates and different steady-state levels. Thus, for a compound that follows a one-compartment model, the simultaneous administration of intravenous and infusion provides an immediate steady-state level (Figure 14.5). 14.2.1.2.2 Combination of Fast and Slow Infusion There are circumstances that giving a bolus loading dose for reasons such as speed shock or precipitation, and/or crystallization of a drug at the site of injection, etc., may not be feasible. Under such contexts, the loading dose is administered by a fast infusion to achieve a desired accumulation of drug in the body and then followed by a slower infusion to maintain the plasma level. Because any change in the rate of infusion will have direct infuence on the plasma concentration of the xenobiotic, terminating one infusion at one rate and initiating a second infusion with a different rate creates a temporary fuctuation in plasma concentration (Figure 14.6) (Boroujerdi, 2002). To administer the loading dose as fast infusion, the following two parameters should be established initially: ◾ the time of fast infusion (tfast) ◾ target concentration (Cpt) at the end of fast infusion The apparent volume of distribution and the overall elimination rate constants are usually known for a compound. Thus, the fast infusion rate can be determined as Cpt =

( k0 ) fast =

( k0 ) fast ClT

(1 - e

-Kt fast

)

( Rate of Elimination )t fast Cp ´ ClT = -Kt fast 1- e ( fss )t fasst

(14.17) (14.18)

The plasma concentration following the fast and slow infusions without including the fuctuation of plasma is

451

14.2 COMPARTMENTAL ANALYSIS

Figure 14.5 Depiction of plasma concentration–time profles of simultaneous intravenous bolus injection and zero-order infusion to achieve immediate steady-state level (the dashed line); the main requirement is to set the initial plasma concentration of the intravenous bolus injection (the loading dose) equal to the steady-state target plasma concentration.

( Cptotal )t =

( k0 ) fast ClT

(1 - e

-Kt fast

) + ( Cl) (1 - e k0

slow

- Ktslow

T

)

(14.19)

Including the fuctuation of plasma in the equation, Equation 14.19 changes to

( Cptotal )t =

( k0 ) fast ClT

(1 - e

-Kt fast

)e

- Ktslow

+

( k0 )slow ClT

(1 - e

-Kt K slow

)

(14.20)

As tslow increases, e -Ktslow approaches zero and ( Cptotal ) will become the steady-state concentration of t the slow infusion, that is,

( Cptotal )ss =

k0slow ClT

(14.21)

14.2.1.3 Estimation of Plasma Concentration after Termination of Infusion After termination of infusion, whether the steady-state plasma level is achieved or not, the concentration in plasma, because of continuous frst-order elimination process, declines exponentially like the decline of plasma concentration after bolus injection (Loo and Riegelman, 1970). The appropriate equations for estimation of plasma concentration after the end of infusion are (Gibaldi and Perrier, 1982; DeVane and Jusko, 1986) Cpt ’ = éëCpss (1 - e -Ktinf )ùû e -Kt ’

(14.22)

Cpt ’ = Cpend e -Kt ’

(14.23)

Where t’ is any time after the termination of infusion and tinf is the period of infusion (Figure 14.7). Similarly, if the infusion is discontinued after the steady-state level is achieved, the plasma concentration at any time t′ after the termination of infusion is calculated as Cpt ’ = Cpss e -Kt ’

452

(14.24)

PRACTICAL APPLICATIONS OF PK/TK MODELS: ZERO-ORDER INFUSION

Figure 14.6 Depiction of the plasma concentration–time profles of a xenobiotic when the loading dose is administered as an infusion with faster rate and/or higher concentration rate followed by the slower rate that provides the intended steady-state level; the requirement here is to estimate the time and the rate for high concentration infusion such that the plasma level at the end of frst infusion equals the steady-state level expected with the second infusion; depending on the type xenobiotic, the switch from the frst infusion to the second one may create fuctuations in plasma that would be lower than the steady-state level of second infusion, but eventually the steady-state level will be achieved in a shorter waiting period; the plasma level fuctuations is less for compounds that follow a one-compartment model. 14.2.1.4 Estimation of Duration of Action in Infusion Therapy Duration of action of a compound during an infusion is the time that the plasma concentration is above the minimum therapeutic level within the therapeutic range. If an infusion is initiated without a loading dose, the initial period when the concentration is still rising to reach the minimum effective concentration, that is time to the onset of action, tonset , should be subtracted from the duration of infusion to provide an estimate of the beginning of the duration of action. In addition, after the termination of infusion, the plasma concentration, although declining, may remain above the minimum effective concentration. This post-infusion period is a part of the duration of action of infusion. The following stepwise calculations is to clear the concept of the duration of action of a therapeutic agent during infusion: 1. The minimum effective concentration in terms of fraction of steady-state plasma level is

( fss )MEC =

CpMEC = 1 - e - Ktonset Cpss

(14.25)

2. The time to onset of action is ln ( 1 - f ss )

(14.26) K 3. Subtraction of tonset from the time of infusion yields the interval in which the concentration is above the minimum effective level while the infusion is running tonset = -

( td )infusion = tinfusion - tonset

(14.27)

453

14.2 COMPARTMENTAL ANALYSIS

Figure 14.7 Depiction of the plasma concentration–time profle after ending the infusion, whether before or after achieving the steady-state level; the concentration at the end of infusion in one-compartment model declines mono-exponentially according to equation Cpt ’ = Cpend e -Kt ’ which enables one to estimate any concentration, Cpt ’ , after stopping the infusion at any time, t’. 4. The time that the plasma concentration remains above the minimum effective concentration after the discontinuation of infusion is tpost =

Cpss 2.303 log K CpMEC

or, tpost =

Cpend 2.303 log K CpMEC

(14.28) (14.29)

5. The actual duration of action is td = ( td )infusion + tpost

(14.30)

For infusions with an intravenous bolus-loading dose, the time to onset of action is zero (tonset = 0, therefore, td = tinfusion + tpost . 14.2.2 Linear Two-Compartment Model with Zero-Order Input and First-Order Disposition The description and assumptions of the two-compartment model are the same as discussed in Chapters 12 and 13. Briefy, a therapeutic xenobiotic after entry into the central compartment distributes between the central and peripheral compartments and simultaneously eliminates from the central compartment (Krüger-Thiemer, 1968; O’Reilly et al., 1971; Gibaldi and Perrier, 1982; Wagner, 1993). All biological processes are governed by frst-order kinetics; thus, the pharmacokinetics of the compound is considered linear. 14.2.2.1 PK/TK Equations of Zero-Order Input into the Central Compartment with First-Order Elimination from the Central Compartment The liver and kidneys, being highly perfused organs, are associated with the central compartment and metabolize and excrete the compound according to the principles of frst-order kinetics and the overall rate constant of elimination from the central compartment, i.e., k10 = k e + k m . The rate constants of distribution between the central and peripheral compartment and vice versa are also

454

PRACTICAL APPLICATIONS OF PK/TK MODELS: ZERO-ORDER INFUSION

governed by frst-order kinetics and the distribution is a function of concentration or amount in each compartment with different rate constants for the transfers (i.e., k12 ≠ k21). The rate of change in the amount infused compound in the central compartment with zeroorder input is dA1 = k0 - k 21 A2 - ( k10 A1 + k12 A1 ) dt

(14.31)

For the peripheral compartment, the rate is dA2 = k12 A1 - k 21 A2 dt

(14.32)

The disposition function of the central compartment is (Benet, 1972) d ( s )1 =

s + E2

s + E2

=

(14.33)

( s + E1 )( s + E2 ) - k12k21 ( s + a )( s + b )

Where E1 = k10 + k 21; E2 = k 21 ; a + b = k12 + k 21 + k10 and ab = k 21k10 The input function for constant rate of input (Addendum II, Part 2, Section A.3) is ins =

(

k0 1 - e -bs

)

(14.34)

s

where (b) is the exposure time or the infusion time. Thus, the Laplace transform and related integrated equation of the amount as a function of time t in the central compartment are a( s ) 1 =

( A1 )t =

(

(

)

k0 1 - e -bs ( s + E2 )

(14.35)

s ( s + a )( s + b )

)

k0 1 - e ab ( k 21 - a ) a (b - a )

(

k0 (b - k 21 ) 1 - ebb

e -at +

b (a - b)

)e

-btt

(14.36)

In the above equations, a and b represent the hybrid frst-order rate constants and b is the time of infusion. Dividing Equation 14.36 by the volume of distribution of the central compartment yields the equation of xenobiotic concentration in the central compartment as a function of the time:

( C1 )t =

(

)

k0 1 - e ab ( k 21 - a ) V1a (b - a )

e -at +

(

k0 (b - k 21 ) 1 - ebb V1b ( a - b )

)e

-bt

(14.37)

During the infusion, b = t, and the equation is modifed to

( C1 )t =

k0 ( k 21 - a ) -at k0 (b - k 21 ) -bt e -1 + e -1 V1a (b - a ) V1b ( a - b )

(

)

(

)

(14.38)

Given ab = k 21k10 , Equation 14.38 can also be presented as

( C1 )t =

b - k10 -at k10 - a -bt ö k0 æ e + e ÷ ç1 + V1k10 è b-a a -b ø

(14.39)

When the time of infusion approaches seven biological half-lives tinfusion ³ 7 ( T1/2 ) (four to fve halfb lives clinically), and since the exponential terms e -at and e -bt Þ 0, Equation 14.39 changes to the equation of steady-state plasma concentration: Cpss = Since (Vd )

area

=

k0 V1k10

Cp 0 Dose and AUC = , Equation 11.40 can also be presented as b ´ AUC k10 k0 Cpss = (Vd )area ´ b

(14.40)

(14.41)

455

14.2 COMPARTMENTAL ANALYSIS

Equation 11.37 is used to determine plasma concentrations after termination of infusion, that is,

( Cp )t ’ = a ’e -at ’ + b ’e -bt ’

(14.42)

Where a’ and b’are the coeffcients of the exponential terms a’ = b’ =

(

)

k0 1 - e ab ( k 21 - a ) V1a (b - a )

(

k0 (b - k 21 ) 1 - ebb V1b ( a - b )

)

(14.43)

(14.44)

The plasma concentration–time profle of an intravenously infused drug that follows the twocompartment model is the same as depicted as Figure 14.2. The rate of input is higher than the rate of elimination k0 > k10 A1 before achieving the steady state, and during the steady-state level, k0 = Cpss ´ (Vd )area ´ b . 14.2.3 Simultaneous Intravenous Bolus and Infusions Administration into the Central Compartment of a Two-Compartment Open Model with First-Order Elimination from the Central Compartment As discussed earlier, the immediate steady-state plasma level is achieved by simultaneous intravenous bolus and infusion administration. The combined equation of bolus and infusion is é Dbolus ( a - k 21 ) -at Dbolus ( k 21 - b ) -bt ù é k0 æ b - k10 -at k10 - a -bt ö ù e + e ú+ê e + e ÷ú Cptotal = ê ç1 + b-a a -b V1 ( a - b ) úû êë V1k10 è ø ûú ëê V1 ( a - b )

(14.45)

Expanding Equation 14.45, taking the common denominator, and simplifying the terms yields Cptotal =

k0 aDbolus - k0 -at (bDbolus - k0 )(( a - k10 ) -bt e + + e V1k10 V1k10 (b - a ) V1k10 ( a - b )

(14.46)

For Cptotal to be equal to Cpss , the last two terms of Equation 14.46 must equal zero. This condition can be achieved if e -at and e -bt are equal to zero (i.e., α t and β t are large values), or the numerator of each term is equal to zero. The conditions for the numerators to be equal to zero are not achievable because the only way they can be equal to zero is when aDbolus = bDbolus = k0 , which is contrary to the assumptions of a two-compartment model. Thus, the total plasma concentration can be equal to steady-state plasma concentration Cptotal = Cpss = k0 V1k10 if the exponential terms are equal to zero. Considering the plasma concentration–time profle of a compound that follows the two-compartment model, it should be expected that the combined administration may exhibit some fuctuation at the beginning of the coadministration, which is often due to the distributional phase of the loading dose (Figure 14.8). 14.2.4 Linear Two-Compartment Model with Two Consecutive Zero-Order Inputs, as Loading and Maintenance Doses, with First-Order Elimination from the Central Compartment When the loading dose to achieve a rapid steady-state level is an infusion with a faster rate than the maintenance infusion, the integrated equation of the model is different from combined bolus and infusion. The total amount or concentration in the central compartment during the second infusion is a function of the amount lingering in the body from the frst infusion and the amount introduced by the second infusion (Wagner, 1993). During the frst infusion, the plasma concentration at time t , defned by the following equation, is the same as Equation 14.39 with different arrangements of parameters and constants:

( k0 )1 é

k - b -at a - k10 -bt ù 1 - 10 e e ú a -b a -b V1k10 êë û The integrated equation for the total concentration in the central compartment is

( C1 )t =

456

(14.47)

PRACTICAL APPLICATIONS OF PK/TK MODELS: ZERO-ORDER INFUSION

Figure 14.8 Depiction of plasma concentration-time profles following simultaneous administration of a loading dose of intravenous bolus injection and simultaneous zero-order infusion of a xenobiotic that follows the two-compartment model; the requirement for the loading dose is, as described for Figure 14.5, the initial plasma concentration should be equal to the desired steadystate level; the fuctuation of the plasma concentration between the initial concentrations of the loading dose and infusion of the maintenance dose is more pronounced in the two-compartment model, which can be attributed to the initial rapid decline of concentration during the distributive phase.

Cptotal =

( k0 )2 êé ( k21 - a ) ( ( k0 )2 - a ( I1 )t ) - ak21 ( I 2 )t V1k10 ê a ( a - b ) V1 ê 1

ë

(

1

ù -a t -t e ( 2 1) ú ú úû

(14.48)

)

ù é ( k 21 - b ) ( k0 ) - b ( I1 ) - bk 21 ( I 2 ) 2 t1 t1 -b ( t2 - t1 ) ú e -ê ê ú b ( a - b ) V1 êë úû Where ( I1 ) and ( I 2 ) are t t 1

1

é 1 k - b -bt1 ù k - a -at1 + 21 - 21 e e ú a a b b k ( ) ( a - b ) ûú ëê 10

( I1 )t = ( k0 )1 ê 1

é

1 e -at2 e -bt2 ù + ú ëê k 21k10 a ( a - b ) b ( a - b ) ûú

( I 2 )t = ( k0 )1 k12 ê 1

(14.49)

(14.50)

The parameters and constants of Equations 14.48–14.50 are: ( k0 ) and ( k0 ) are the frst and second 1 2 zero-order infusion rates; t1 and t2 are the length of frst and second infusions; k10 is the overall frst-order rate constant of elimination; k12 and k 21 are frst-order distribution rate constants, a and b are the frst-order hybrid rate constants of the two-compartment model (a > b), and V1 is the volume of the central compartment. The profle of combined plasma concentration of consecutive fast and slow infusions is depicted in Figure 14.6.

457

14.2 COMPARTMENTAL ANALYSIS

14.2.5 Three-Compartment Model with Zero-Order Input into the Central Compartment and First-Order Elimination from the Central Compartment The assumptions and characteristics of the model (Figure 14.9) are the same as discussed in Chapters 12 and 13. The differential and integrated equations defning the time course of infused xenobiotic in the central compartment during the infusion are

( C1 )t =

(

dA1 = k0 + k12 A2 + k 31 A3 - A1 (k10 + k12 + k13 ) dt

(14.51)

dA2 = k12 A1 - k 21 A2 dt

(14.52)

dA3 = k13 A1 - k 31 A3 dt

(14.53)

)

(

)

(

)

+ab +bb +gb k0 éê ( k 21 - a )( k 31 - a ) e - 1 -at ( k 21 - b )( k 31 - b ) e - 1 -bt ( k 21 - g )( k 31 - g ) e - 1 -gt ùú e + e + e V1 ê a (b - a ) ( g - b ) b ( a - b )( g - b ) g ( a - g )(b - g ) ú ë û

(14.54) Where b is the length of infusion, and plasma concentration is a function of time, t , during the infusion. The descriptions of other constants are the same as described earlier.

Figure 14.9 Schematic of a three-compartment model with zero-order input in the central compartment, A1 , which as described before is a homogeneous compartment comprised of systemic circulation and highly perfused tissues/organs; of the two peripheral compartments, A2 and A3 , one is identifed as the pool of low-perfused organ/tissue, and the second is the organ or region toward which the compound exhibits preferential binding or uptake; the frst-order distribution rate constants are k12 , k 21 , k13 and k 31 ; the frst-order rate constant of overall elimination,k10 k 31 , represents the combination rate constants of excretion and metabolism, and Ael is the cumulative amount eliminated from the body. 458

PRACTICAL APPLICATIONS OF PK/TK MODELS: ZERO-ORDER INFUSION

14.2.6 Three-Compartment Model with Zero-Order Input into the Central Compartment and First-Order Elimination from a Peripheral Compartment The rate equations of the model are dA1 = k0 + k 21 A2 + k 31 A3 - A1 (k12 + k13 ) dt

(14.55)

dA2 = k12 A1 - A2 (k 21 + k 20 ) dt

(14.56)

dA3 = k13 A1 - k 31 A3 dt

(14.57)

The integrated equation defning the time course of the infused xenobiotic in the central compartment is

(

( C1 )t

)

(

)

é ( k 20 + k 21 - a )( k 31 - b ) 1 - e +ab ( k20 + k212 - b )( k31 - b ) 1 - e +bb -bt ùú ê e -at + e -b ( a - b )( g - b ) -a (b - a )( g - a ) ú k0 êê ú = V1 ê ú ( k20 + k21 - g )( k31 - g ) 1 + e +gb -gt ê ú + e ú ê -g a b g b ( ) ( ) û ë

(

)

(14.58)

The defnitions of parameters and constants are as defned before. 14.3 APPLICATIONS AND CASE STUDIES The applications and case studies of Chapter 14 are posted in Addendum II – Part 5. REFERENCES Benet, L. Z. 1972. General treatment of linear mammillary models with elimination from any compartment as used in pharmacokinetics. J Pharm Sci 61(4): 536–41. Boroujerdi, M. 2002. Pharmacokinetics: Principles and Applications. New York: McGraw Hill-Medical Publishing Division, Chapter 7. Chiou, W. L., Gadalla, M. A., Peng, G. W. 1978. Method for the rapid estimation of the total body drug clearance and adjustment of dosage regimen in patients during a constant-rate intravenous infusion. J Pharmacokinet Biopharm 6(2): 135–51. DeVane, C. L., Jusko, W. J. 1982. Dosage regimen design. Pharmacol Ther 17(2): 143–63. https://doi .org/10.1016/0163-7258(82)90009-2. Gibaldi, M., Perrier, D. 1982. Pharmacokinetics, Second Edition, 63–5. New York: Marcel Dekker, Inc. Krüger-Thiemer, E. 1968. Continuous intravenous infusion and multicompartment accumulation. Eur J Pharmacol 4(3): 317–24. Loo, J. C. L., Riegelman, S. 1970. Assessment of pharmacokinetic constants from postinfusion blood curves obtained after i.v. infusion. J Pharm Sci 59(1): 53. O’Reilly, R. A., Welling, P. G., Wagner, J. G. 1971. Pharmacokinetics of warfarin following intravenous administration to man. Thromb Diath Haemorrh 25(1): 178–86. Rowland, M., Tozer, T. N. 1994. Clinical Pharmacokinetics, Third Edition, 66. Media: Williams and Wilkins.

459

14.3 APPLICATIONS AND CASE STUDIES

Wagner, J. G. 1974. A safe method of rapidly achieving plasma concentration plateaus. Clin Pharmacol Ther 16(4): 691–700. Wagner, J. 1993. Pharmacokinetics for Pharmaceutical Scientist. Lancaster: Technomic Publishing Company, Inc.

460

15 Practical Applications of PK/TK Model: First-Order Absorption via Extravascular Route - Oral Administration 15.1 INTRODUCTION As discussed in Chapters 2, 3, 4, and 6, the absorption of xenobiotics via an extravascular route of administration involves permeation through a physiological barrier before reaching the systemic circulation. The barriers can be as complex as the gastrointestinal wall or as simple as a capillary wall in the intramuscular, rectal, or sublingual route of administration. The PK/TK profle of the compounds absorbed through extravascular routes involves all four biological processes of absorption, distribution, metabolism, and excretion (ADME). These processes occur simultaneously and not sequentially. The physicochemical characteristics of the xenobiotic, and behavior of the body in dealing with the compound, infuence the data, which in turn infuence the selection of the PK/TK model. The focus of the current chapter is to evaluate the application and practicality of the models used in PK/TK analysis of compounds absorbed from an extravascular route of administration. Although the focus of discussion is on the oral administration, the models and parameters are applicable to the other extravascular routes of administration. 15.2 COMPARTMENTAL ANALYSIS 15.2.1 Linear One-Compartment Model with First-Order Input and First-Order Elimination A diagram of the model is presented in Figure 15.1. The assumptions of the model are: 1) the xenobiotic is administered orally as a single dose at time zero; 2) depending on the type of dosage forms, the dose at the site of absorption absorbs gradually at a different rate into the systemic circulation, and its amount declines exponentially with time at the site of absorption; and 3) the driving force for absorption into the systemic circulation is the concentration gradient, and thus the absorption is linear, governed by passive diffusion, and follows frst-order kinetics. The rate of absorption or rate input into the systemic circulation is Rate of Absorption =

dAD = k a ´ ( AD )t dt

(15.1)

where k a is the absorption rate constant and AD is the absorbable amount of the dose at the site of absorption. As both processes of absorption and elimination follow frst-order kinetics, their rates depend on the amount of the compound available for each process (Garrett, 1993, 1994). The rate of absorption is higher at the early time points when the amount of xenobiotic is higher at the site of absorption. On the other hand, the rate of elimination (rate of output) at the early time points, because of the lesser amount in the body, is low but increases gradually as more of the administered dose is absorbed. Eventually the rate of elimination becomes equal to the rate of absorption and plasma concentration reaches to its maximum, Cpmax , at a time point known as Tmax . The rate of elimination at Tmax is the product of total body clearance and maximum plasma concentration, which is equal to the rate of absorption at Tmax . ClT Cpmax = k a ( AD )T

max

(15.2)

After Tmax , the amount at the site of absorption gradually declines, the rate of absorption lessens, the amount in the body steadily rises, and the rate of elimination becomes greater than the rate of absorption until no more absorbable xenobiotic is left at the site of absorption. The combination of the changing rates of input and output infuences the shape of the plasma concentration–time curve (Figure 15.2) to resemble a skewed bell-shaped curve. After the completion of absorption, the terminal portion of the plasma concentration–time curve is only a function of the elimination process. Thus, the rate of change in the amount of xenobiotic in the body can be defned as dAt = k a ( AD )t - K ( A )t dt

(15.3)

where k a ( AD )t is the rate of absorption; K ( A ) is the rate of elimination, which is also defned as t

ClT Cp or KCpVd ; ( AD )t is the absorbable amount at time t at the site of absorption; and At is the

amount in the body at time t.

DOI: 10.1201/9781003260660-15

461

15.2 COMPARTMENTAL ANALYSIS

Figure 15.1 Schematic of linear one-compartment model with frst-order input and frst-order elimination applicable to administration of xenobiotics via an extravascular route of administration; AD represents the absorbable amount of xenobiotic at the site of absorption, At is the amount of unchanged xenobiotic in the body at a given time, Ael is the amount eliminated from the body by all routes of elimination; the frst-order rate constants of the model include the absorption rate constant, k a and the overall elimination rate constant, K .

Figure 15.2 Depiction of the profle of plasma concentration of a xenobiotic administered via an extravascular route of administration, e.g., the oral route; the shape of the skewed bell-shaped curve is created by the changes in the rates of absorption from the site of administration and the rate of elimination from the compartment representing the systemic circulation and the highly perfused tissues; the rise in the concentration represents the larger rate of absorption due to the presence of high amount of the compound at the early time points at the site of absorption and lower rate of elimination from the body; at the maximum, the rates are equal and the subsequent decline of the plasma concentration represents the higher elimination rate when the amount at the site of absorption is dissipating due to the removal of the compound by the absorption process. At Tmax , ClT Cpmax = k a ( AD ) T

k a ( AD )T

max

= K ( A )T

max

max

that is, the rate of absorption is equal to the rate of elimination,

. Therefore, at Tmax the rate of change of amount in the body is equal zero that

dAt = 0. dt The time for complete absorption of a compound from the site of absorption into the systemic circulation is approximately 7 ´ ( T1 2 ) after this time, k a ( AD ) = 0 , and the rate of change of

is,

ka

462

t

PRACTICAL APPLICATIONS OF PK/TK MODELS: FIRST-ORDER ABSORPTION

dAt = -K ( A )t , which is the rate of elimination of the amount absorbed by all dt dAt routes of elimination. After 7 ´ ( T1 2 )K , the rate = 0. dt The integration of Equation 15.3 by the Laplace transform (Addendum I, Part 2) yields the following relationship:

amount in the body is

At =

(AD0 )(k a ) -Kt (e - e -kat ) K - ka

(15.4)

where At is the amount of xenobiotic in the body at time t; AD0 represents the absorbable amount of the administered dose at the site of absorption at t = 0, which is the same as (FD). where F corresponds to the fraction of the dose absorbed, a value that is equal or less than one (0 ≤ F ≤ 1), it is also known as absolute bioavailability and F × Dose, FD, represents the total amount absorbed or total absorbable amount of administered dose. In terms of plasma concentration of the administered xenobiotic, Equation 15.4 is defned as Cpt =

FDk a e -Kt - e -kat Vd ( k a - K )

(

)

(15.5)

Equations 15.4 and 15.5 represent the two biological processes of absorption and elimination that occur simultaneously during the absorption process: Cpt =

FDk a FDk a e -kat e -Kt Vd(k a - K) Vd(k a - K) Eliminatiion

(15.6)

Absorption

Plotting the logarithm of plasma concentration of the administered xenobiotic, log Cpt , versus time according to Equation 15.5, provides a skewed bell-shaped curve with a linear terminal portion, as shown in Figure 15.3. The slope of the linear terminal portion of Figure 15.3 is a function of the smallest rate constant. When k a > K , the slope is equal to -K / 2.303 ; the majority of compounds absorbed from the GI tract fall into this category and follow Equation 15.5. However, when K > k a , the slope of the linear segment of the curve equals -k a / 2.303, and Equation 15.5 is modifed to Equation 15.7: Cpt =

FDk a e -kat - e -Kt Vd ( k a - K )

(

)

(15.7)

Equation 15.7, representing the condition of K > ka, is known as “fip-fop” kinetics and occurs when the rate of absorption is slower that the rate of elimination and exhibits absorption ratelimited elimination kinetics (see Chapter 2, Section 2.3.5.4.1, Equation 2.32). The underlying mechanisms for such occurrences in the GI tract are most often related to the physicochemical characteristics of the compound, the formulation of therapeutic agent (for example, sustainedrelease dosage form), or species-related physiological/anatomical differences (Baggot, 1992). The comparative evaluation of fip-fop conditions K > k a with respect to the more common condition of k a > K is presented in Figure 15.4. Both Equations 15.5 and 15.7 generate the same bell-shaped curve. The fip-fop profle exhibits a smaller maximum plasma concentration and extended area under the plasma concentration– time curve (AUC). Identifying a fip-fop occurrence and managing the related data are important considerations in bioavailability assessment and estimation of the AUC. It has been recommended that an extended and frequent sampling may aid in a more accurate evaluation of the related parameters and constants (Byron and Notari, 1976; Bredberg and Karlsson, 1991; Boxenbaum, 1998; Neelakantan and Veng-Pedersen, 2005; Yáñez et al., 2011). It should be noted that estimation of K from an intravenously administered dose can help identify the fip-fop occurrence and differentiate between K and k a (Figure 15.5). When the two rate constants of absorption and elimination are equal, K = k a , Equations 15.5 and 15.7 do not apply, and it would be diffcult to demarcate a terminal linear slope from the logarithmic plot. Under this condition, the appropriate equation for plasma concentration is (Bialer, 1980): Cpt =

lFD -lt te Vd

(15.8) 463

15.2 COMPARTMENTAL ANALYSIS

Figure 15.3 A typical plot of logarithm of plasma concentration versus time according to the one-compartment model with administration via an extravascular route of administration under the assumption that ka > K, where the linear terminal portion of the curve provides two signifcant pieces of information about the administered xenobiotic, namely, the elimination rate constant, estimated from the slope of the line, and the estimate of y-intercept of the extrapolated line; the line represents the elimination of the compound from the body, and the magnitude of slope, or the elimination rate constant, indicates how fast the body eliminates the compound. where λ = K = ka and the logarithmic form of the equation is log Cpt = log

lFD lt + log t 2.303 Vd

(15.9)

When the absorption rate constant is signifcantly greater than the elimination rate constant, k a ˜ K , the absorption is fast and the plasma concentration–time profle resembles the intravenous bolus injection. Under this condition, e -ka t in Equation 15.5 approaches zero very rapidly as t increases; K would be negligible in comparison to k a , and the relationship is simplifed to Cpt =

FD k a

(

Vd k a - K

)(

)

e -kat - e -Kt =

FD -Kt e Vd

(15.10)

15.2.1.1 Initial Estimates of the Overall Elimination Rate Constant, K and Absorption Rate Constant, k a The commercially available PK/TK software packages have made it possible to determine the values of K , k a , and other parameters and constants of different models readily and without demur. The following are the principles and classical methodology of determination of the initial estimates of the rate constants for elimination and absorption under the assumption of k a > K . 15.2.1.1.1 Estimation of the Overall Elimination Rate Constant K For a set of ideal data with no random error, the plot of logarithm of plasma concentration versus time generates the bell-shaped curve with a linear terminal portion. The slope of the linear terminal portion can be determined by using two pair of data points in the slope relationship: Slope =

464

log Cpn-1 - log Cpn K =2.303 tn-1 - tn

(15.11)

PRACTICAL APPLICATIONS OF PK/TK MODELS: FIRST-ORDER ABSORPTION

Figure 15.4 Comparison of plasma concentration–time profle of the “fip-fop” condition, where the elimination rate constant is greater than the absorption rate constant K > ka, with the common condition where the absorption rate constant is greater than the elimination rate constant (ka > K); the fip-fop condition exists for various extravascular routes, where the permeation or penetration from the site of administration is slower than the rate of removal from the site or the body; detection of the fip-fop condition in oral administration is important for bioavailability evaluation and may require an estimate of K from an intravenous administration (see Figure 15.5). \ K = 2.303 ´ Slope

(15.12)

Extrapolation of the linear terminal portion of the curve with the slope calculated by Equation 15.11 to the y-axis provides the y-intercept of the extrapolated line. The extrapolated line represents the elimination component of Equation 15.5 (Figure 15.3). The y-intercept is equal to the coeffcient of Equation 15.5, and for an ideal set of data, can be estimated as ˜°°°° Cp FDk a (15.13) Cp 0 = -Ktnn = e Vd(k a - K) Therefore, the equation of the extrapolated line is ˜°°°° ˜°°° ˜°°°° ˜°°° Kt Cpt = Cp 0 e -Kt or log Cpt = log Cp 0 2.303

(15.14)

When the plasma data on the linear terminal portion are scattered, a better initial estimate of K is determined by using regression analysis Slope =

å éë( dt ) ( d log Cp )ùû = - K 2.303 å dt 2

(15.15)

By substituting the numerical value of slope and mean values of the ˜ °variables t and log Cpt in equation y = y + m ( x - x ) , the y-intercept of the extrapolated line, Cp 0 can be estimated, that is,

465

15.2 COMPARTMENTAL ANALYSIS

Figure 15.5 Depiction of a “fip-fop” occurrence and differentiation between K and ka by comparison with the overall elimination rate constant (K in the one-compartment model), or the disposition hybrid rate constant ( β in the two-compartment model) of a bolus injection of the same compound; if the slope of the residual line of the orally administered dose is the same as the slope of the terminal portion of the intravenous bolus injection, it would suggest the presence of a fip-fop condition. n é n æ öù ê t ÷ú ( dt ) ( d log Cp ) ç ê ç i=1 ÷ú log ( Cp )t = i=1 + ê i=1 n (15.16) ç t - n ÷ú n 2 ê ç ÷ú dt ) ( ê ç ÷ú i=1 è øû ë where n represents the number of data points located on the linear terminal portion of the curve (Curry, 1981). Substitution of the known values in Equation 15.16 gives rise to Equation 15.14. ˜ ° The parameter Cp 0 is a virtual value that cannot be measured experimentally. It is not the initial concentration of an orally administered dose. It can only be interpreted as the hypothetical y-intercept of an intravenous dose equal to FD. n

å

log ( Cp )t

å

å

å

15.2.1.1.2 Estimation of Absorption Rate Constant k a 1. The following four methods are used most commonly for estimation of the absorption rate of xenobiotics that follow one-compartment model with frst-order input and frst-order output. Method I: Using the slope of the residual line and estimation of lag time of absorption: This method is also known as the “feathering” or “peeling” method and involves the subtraction of Equation 15.14, the extrapolated line, from Equation 15.5. It yields Equation 15.17, which represents the absorption component of Equation 15.5, that is, ˜°°° ˜°° ˜°° FDk a (15.17) e -kat or Cp - Cp = Cp 0 e -kat (Cp - Cp)t = t Vd(k a - K)

(

466

)

PRACTICAL APPLICATIONS OF PK/TK MODELS: FIRST-ORDER ABSORPTION

˜° Plotting log Cp - Cp versus time gives the residual line with a slope of -k a 2.303 (Figure 15.6).

(

)

t

The method of residuals is applicable when there are enough early data points. The manual calculation formulas for the line of residuals are presented in Table 15.1. A y-intercept of the residual line greater than the y-intercept of the extrapolated line may indicate that the absorption from the site of administration starts after a delay, known as the lag time of absorption. There are various reasons for the delay, for example, the physicochemical characteristics of a compound, such as being a weak base, and formulation factors, such as coated tablets, slow disintegration and dissolution, and delayed-release dosage forms. A negative lag time of absorption may represent an inadequate sampling and experimental design. The lag time of absorption is estimated graphically (Figure 15.7), or calculated by setting Equations 15.14 equal Equation 15.17 and solving for the lag time (Boroujerdi, 2002): ˜°°°° ˜°°°° kt Kl Kt log Cp 0 - a l = log Cp 0 residual extrapolated 2.303 2.303

)

(

)

(

Figure 15.6 Depiction of the logarithm of plasma concentration–time curve following absorption from an extravascular route of administration, e.g., GI tract, showing the extrapolated and residual lines corresponding to the elimination and absorption processes, respectively; the extrapolated line is the linear terminal portion of the curve extended to the y-axis with a slope providing the initial estimate of the overall elimination rate constant; subtraction of the early observed data points from the corresponding values on the extrapolated line generates the residual values, plot of the positive residuals versus their time points creates the line of residuals with a slope providing the initial estimate of the frst-order absorption rate constant.

Table 15.1 Required Steps for Estimation of Residual Line Time t1 t2 t3 :

¬0 - Kt C¬ p = Cp e

C

¬ p1

C

¬ p2

C

¬ p3

:

Cp Cp1 Cp2

C¬ p - Cp ¬ p1

C - Cp1 ¬ p2

C - Cp2

Cp3

C - Cp3

:

:

¬ p3

( log ( C log ( C log ( C

) - Cp ) - Cp ) - Cp )

log C ¬ p - Cp ¬ p1 ¬ p2

¬ p3

:

1

2

3

467

15.2 COMPARTMENTAL ANALYSIS

Figure 15.7 Illustration of graphical detection and measurement of the lag time of absorption when the residual line has a greater y-intercept than the extrapolated line; reading the point of intersection on the x-axis provides the lag time of absorption; when the residual line has a y-intercept below the extrapolated line, or both lines meet at the same y-intercept, or the lag time is less than the gastric emptying time, the assumption is that there is no lag time of absorption. ˜°°°°

(log Cp ) 0

˜°°°° - log Cp 0

) ˜ ° Cp ) ( 2.303 t = log ˜°°°° k -K (Cp ) residual

(

extrapolated

æk -K ö = tl ç a ÷ è 2.303 ø

0

l

(15.18)

resaidual

0

a

extrapolated

Method II. Using Krüger–Thiemer method of combination of extravascular and intravenous data: The extravascular absorption rate constant of a compound can also be estimated from the y-inter˜ ° . The method requires that equal doses of a drug be given cept of the extrapolated line Cp 0

( )

extrapolated

both intravenously and orally in a crossover design. The estimated Cp 0 of the bolus dose is then substituted for D Vd in Equation 15.13 to develop Equation 15.20 for estimation of k a : ˜°°°° F ´ Cp 0 ´ k a Cp 0 = ( ka - K )

(15.19)

k a - K FCp 0 = ˜°°°° ka Cp 0 \ ka =

K FCp 0 1 - ˜°°°° Cp 0

(15.20)

This method is applicable when a compound can be administered both orally and intravenously. Method III. Wagner–Nelson method of using linear plot of percent remaining to be absorbed versus time: This method is based on the principle of mass balance for total amount absorbed at any time (Wagner and Nelson, 1964, 1970, 1975): 468

PRACTICAL APPLICATIONS OF PK/TK MODELS: FIRST-ORDER ABSORPTION

( AD )abs = At + Ael

(15.21)

where ( AD )abs is the cumulative amount absorbed at time t, At = total amount in the body at time t, and Ael is the cumulative amount eliminated from the body by excretion and metabolism. The differentiation of Equation 15.21 yields the rate equation (Equation 15.22), which defnes the rate of absorption in terms of adding up the rate of amount change in the body and the rate of elimination: dAD dAt dAel = + dt dt dt

(15.22)

Defning the rate Equation 15.22 in terms of concentration is achieved by substituting: At = Cpt ´ Vd dAel dt = Rate of Elimination = KAt = KVdCpt = ClT Cp in Equation 15.22 yields: dCp dAD = Vd + KVdCpt dt dt

(15.23)

Integration of Equation 15.23 yields:

( AD )t = VdCpt + KVd ò

t

0

Cpt dt = VdCpt + KVd ( AUC )0 t

(15.24)

Dividing both sides of Equation 15.24 by the apparent volume of distribution, Vd, yields Equation 12.25, known as the Wagner–Nelson equation:

( AD )t Vd

= Cpt + K

t

ò Cp dt

(15.25)

t

0

Stepwise calculations of the Wagner–Nelson method for initial estimation of the absorption rate constant are presented in Table 15.2. The total amount absorbed ( AD ) is obtained by integrating ¥ Equation 15.24 from zero to infnity:

( AD )¥ = (VdCp )0

¥

+ KVd

ò

¥

0

(15.26)

Cpt dt

For a compound given through the extravascular route of administration, the initial concentration, Cpt =0 , and infnity concentration, Cpt = ¥ , are both equal to zero, thus, the total amount of the dose that is ultimately absorbed is

( AD )¥ = KVd ò

¥

0

Cpt dt = KVd ( AUC )0

¥

(15.27)

Dividing Equation 15.24 by Equation 15.27 gives rise to the fraction of dose absorbed at time t

( AD )t ( AD )¥

VdCpt + KVd ( AUC )0 t

=

(15.28)

KVd ( AUC )0

¥

For an extravascularly administered compound, the product of total body clearance (KVd) and area under the plasma concentration–time curve from time zero to infnity is equal to the absorbable amount of the administered dose, that is, KVd ( AUC )0 = FDose ¥

(15.29)

Therefore, Equation 15.28 is the fraction of the absorbable amount of dose at time t, and by canceling the volume of distribution, it is expressed in terms of concentration

( AD )t ( AD )¥

Cpt + K ( AUC )0 t

=

(15.30)

K ( AUC )0

¥

Equations 15.28 or 15.30 can be expressed in terms of percent remaining to be absorbed, that is, the æ ( AD )t Percent of absorbable amount of dose remaining to be absorb bed = ç 1 ç ( AD )¥ è

ö ÷ ´ 100 ÷ ø

(15.31)

469

470

Cp

Cp1

Cp2

Cp3

Cp4

Cp5

Cp6

: : Cpn

t

t1

t2

t3

t4

t5

t6

: : tn

: : ((Cpn-1 + Cpn)/2) (tn – tn-1) = AUCn

((Cp5 + Cp6)/2) (t6 – t5) = AUC6

((Cp4 + Cp5)/2) (t5 – t4) = AUC5

((Cp3 + Cp4)/2) (t4 – t3) = AUC4

((Cp2 + Cp3)/2) (t3 – t2) = AUC3

((Cp1 + Cp2)/2) (t2 – t1) = AUC2

((0 + Cp1)/2) (t1 – 0) = AUC1

n AUCttn-1

AUC0n

: :

AUC06 = AUC05 + AUC6

AUC05 = AUC04 + AUC5

AUC04 = AUC03 + AUC4

AUC03 = AUC02+ AUC3

AUC02 = AUC01 + AUC2

AUC01 = AUC1

AUC0t

Vd

( ADn )abs

: :

Vd

( AD6 )abs

Vd

ü ï ï ï ý = Cpn + (K ´ AUC0n )ï ï ï þ

= Cp6 + (K ´ AUC06 )

= Cp5 + (K ´ AUC05 )

= Cp4 + (K ´ AUC04 )

= Cp3 + (K ´ AUC03 )

= Cp2 + (K ´ AUC02 )

= Cp1 + (K ´ AUC01 )

= Cp + (K ´ AUC0t )

( AD51 )abs

Vd

( AD4 )abs

Vd

( AD3 )abs

Vd

( AD2 )abs

Vd

( AD1 )abs

Vd

( AD )abs Vd

´

´

´

´

´

Plateau (p)

Vd

( AD6 )abs

Vd

( AD5 )abs

Vd

( AD4 )abs

Vd

( AD3 )abs

Vd

( AD2 )abs

Vd

´

( AD )abs

( AD1 )abs

%

100 p

100 p

100 p

100 p

100 p

100 p

Table 15.2 Stepwise Calculations of the Wagner–Nelson Method for Estimation of the Absorption Rate Constant

15.2 COMPARTMENTAL ANALYSIS

PRACTICAL APPLICATIONS OF PK/TK MODELS: FIRST-ORDER ABSORPTION

Equation 15.24 in terms of plasma concentration is defned as

( AD )abs

(

= Cpt + K ´ AUC0t

Vd

)

(15.32)

( AD )abs where the quantity of is the amount absorbed per unit of volume of distribution. Plot of Vd ( AD )abs versus time is a hyperbolic curve that represents the cumulative amount absorbed from Vd time zero to infnity. The curve reaches a plateau level that represents the total amount absorbed at t ⇒ ∞ (Figure 15.8). ( AD )abs values on the linear segment of the curve by the average values on the plateau Dividing Vd level results in fractions that after multiplying by 100 can also be plotted as percent amount ( AD )abs ( AD )abs . Plotting 100 - % , which corresponds to the percent absorbed per unit of volume, % Vd Vd amount remaining to be absorbed per unit of volume of distribution. The stepwise calculations are presented in Table 15.2. æ ( AD )abs ö versus time provides a straight line with a slope of -k / 2.303 A plot of log ç 100 - % a ÷ ç Vd ÷ø è (Figure 15.9). The Wagner–Nelson method of the ka estimation relies greatly on (K) and (AUC). An error in the estimation of K or the presence of truncation errors in AUC can signifcantly infuence the calculated value of the absorption rate constant (Wang and Nedelman, 2002). Method IV. Wagner–Nelson method of using the amount excreted unchanged in urine: The rate of urinary excretion of the administered dose of a compound is estimated as (Wagner and Nelson, 1964) dAe = k e At dt

(15.33)

where k e is the frst-order excretion rate constant and At is the amount in the body at time t, equal to VdCp, \

dAe = k eVdCp = Clr Cp dt

(15.34)

Solving for Cp yields: Cp =

dAe dt k eVd

(15.35)

Substitution of Equation 15.35 in Equation 15.23 provides the absorption rate equation in terms of urinary data æ dA dt ö dç e ÷ ( dAe dt ) k eVd ø è = Vd + KVd k eVd dt

d ( AD )abs dt

d ( AD )abs dt

=

d ( dAe dt ) k e dt

+

K ( dAe dt ) ke

(15.36) (15.37)

Integration of Equation 15.37 from zero to infnity generates the total amount absorbed at time t estimated in terms of the amount excreted in the urine

( AD )abs = Substitution of Equation 15.33,

1 æ dAe ö K + ( Ae )t k e çè dt ÷ø k e

(15.38)

dAe = k e At , in Equation 15.38 yields: dt

( AD )abs =

1 ke

( k A ) + keK ( A ) e

t

e t

= At +

K ( Ae )t ke

(15.39) 471

15.2 COMPARTMENTAL ANALYSIS

Figure 15.8 Depiction of a plot of a normalized cumulative amount absorbed with respect to the apparent volume of distribution versus time as a feature of Wagner–Nelson method of using a linear plot of percent remaining to be absorbed vs time; the fgure shows only the formation of ( AD )abs values by the average the plateau levels representing the total amount absorbed; dividing Vdof volume; multiplying of the plateau levels provides fractions of the amount absorbed per unit the fractions by 100, then subtracting from 100 gives the percent amount remaining to be absorbed (see Figure 15.9). At t = ¥ , At Þ 0 , and ( AD ) Therefore,

abs

Þ ( AD )¥

( AD )¥ =

K ( Ae )¥ ke

(15.40)

where ( AD ) is the total amount of drug absorbed or the absorbable amount, and ( Ae ) is the total ¥ ¥ amount excreted unchanged that corresponds to the plateau level of cumulative curve of urinary excretion data (see Chapter 10, Section 10.6). The proportion of Equations 15.38–15.41, gives the fraction of the absorbable amount of dose absorbed at time t , æ ( AD ) abs ç ç ( AD ) ¥ è

1 æ dAe ö K + ( Ae )t æç dAe ö÷ + K ( Ae )t ö k e èç dt ø÷ k e dtt ø =è ÷ = ÷ K K ( Ae )¥ øt ( Ae )¥ ke

æ ( AD ) abs At t = ¥,ç ç ( AD ) ¥ è

æ ( AD ) ö abs ÷ Þç ç ( AD ) ÷ ¥ è øt

(15.41)

ö ÷ Þ1 ÷ ø¥

(15.42)

Therefore, at time t : æ ( AD ) abs Fraction of dose remaining to be absorbed = 1 - ç ç ( AD ) ¥ è

472

ö ÷ ÷ øt

(15.43)

PRACTICAL APPLICATIONS OF PK/TK MODELS: FIRST-ORDER ABSORPTION

Figure 15.9 Depiction of the plot of log of percent remaining to be absorbed versus time, the fnal step in the calculation of the frst-order absorption rate constant according to the Wagner– Nelson method (see also Figure 15.8); the plot of percent remaining to be absorbed versus time is linear with a slope of -k a / 2.303. When the absorption process is linear and follows frst-order kinetics, a plot of the logarithm of fraction or percent of absorbable amount of the administered dose remaining to be absorbed versus time will be a straight line with a slope of - k a 2.303 (Wagner, 1975) (Figure 15.10). 15.2.1.2 Estimation of Time to Peak Xenobiotic Concentration – Tmax The three foremost parameters of the model used in the evaluation of bioavailability and bioequivalence are time of maximum plasma concentration, Tmax , maximum plasma concentration, Cpmax , and extent of absorption or AUC. The time to peak xenobiotic concentration is an indicator of how fast a compound absorbs, and it is the tacit refection of the rate of absorption. A short Tmax suggests a fast rate of absorption, and a long one signifes the slow absorption rate. The maximum plasma concentration Cpmax represents the highest concentration that can be achieved with a single dose. For therapeutic agents, it is preferable to achieve this concentration within the therapeutic range. The parameters Cpmax and Tmax profle the plasma concentration–time curve of an extravascularly administered dose by one point over the entire period of sampling and data collection. The AUC, on the other hand, is a parameter that encompasses the entire sampling time and beyond. It represents the totality of absorption. These and other parameters of oral absorption are presented graphically in Figure 15.11. The equation of Tmax is derived from Equation 15.5, using the calculus principle of determining the maximum or minimum of a function by setting its frst derivative equal to zero. As discussed earlier, at Tmax the rate of absorption equals the rate of elimination; i.e., the rate of change in plasma concentration, with respect to time at this time point, is equal to zero . Thus, taking the frst derivative of Equation 15.5, the integrated equation of the model, and setting it equal to zero should provide the Tmax relationship: ˜°°°° ˜°°°° dCp = -KCp 0 e -Kt + k a Cp 0 e -kat = 0 (15.44) dt \ Ke -KTmax = k a e - kaTmax

(15.45)

Dividing both sides of Equation 15.45 by K, taking the natural logarithm of both sides, and solving for Tmax yields: 473

15.2 COMPARTMENTAL ANALYSIS

Figure 15.10 Treating the amount excreted unchanged in urine by the Wagner–Nelson method yields the same outcome as exhibited in Figures 15.8 and 15.9 for estimation of the absorption rate constant; the fgure represents the fraction of the administered dose remaining to be absorbed, calculated from the amount excreted unchanged in urine, versus time, with the slope of -k a / 2.303. Tmax =

1 k ln a ka - K K

(15.46)

Tmax =

k 2.303 log a ka - K K

(15.47)

In terms of log base, 10 is

According to Equation 15.47, Tmax is a function of the rate constants of absorption and elimination. Therefore, disease states, physiological factors, diet, other xenobiotics, and environmental factors that infuence the absorption and elimination processes of a compound may change its numerical value. It should be noted that Tmax is a dose-independent parameter. The area under the plasma concentration–time curve from zero to Tmax known as the partial AUC (PAUC), is considered a sensitive parameter in comparative evaluations of two compounds or dosage forms (Chen, 1992; Bois et al., 1994; Rostami-Hojgan et al., 1994; Endrenyi et al., 1998a, b; Chen et al., 2011). 15.2.1.3 Estimation of Peak Concentration (Cp max) Contrary to Tmax , Cpmax is a dose-dependent parameter that corresponds to the intensity of the pharmacological response. It is a parameter used in bioavailability and bioequivalence studies and in the selection of the most appropriate extravascular route of administration for a lead compound in drug discovery and development. Like Tmax , Cpmax is an indirect measure of the absorption rate of a compound (Chen et al., 2001). Regulatory agencies favor the observed values of both parameters, and the non-compartmental analyses are often preferred in bioavailability assessment. This is mostly to avoid the biases associated with the assumptions of the models. However, the comparative evaluation of observed and calculated values of Tmax and Cpmax may add confdence in the selected model and the related calculated parameters and constants. The estimation of Cpmax , based on the model described in Section 15.2.1, is as follows (Gibaldi and Perrier, 1982). From Equation 15.45:

474

PRACTICAL APPLICATIONS OF PK/TK MODELS: FIRST-ORDER ABSORPTION

Figure 15.11 Illustration of the principal parameters of plasma concentration–time profle of a xenobiotic administered via an extravascular route of administration, e.g., oral administration; the minimum and maximum effective plasma concentrations are the attributes of a therapeutic agent; plasma concentrations less than the minimum effective level are inadequate, and concentrations above the maximum effective level are toxic and not recommended. The therapeutic effect of the agent starts at the onset of action when the concentration of absorbed dose in systemic circulation is above the minimum therapeutic level, and the safety of the administered dose is ensured by maintaining the maximum plasma level below the maximum effective level; the dose intensity is the difference between the maximum plasma level and minimum effective level; the duration of action is the time when the plasma level is within the therapeutic range; the time to peak concentration represents how fast the compound is absorbed; the area under the plasma concentration– time curve is a model-independent and dose-dependent parameter and a sensitive indicator of bioavailability. e -kat =

K -KTmax e ka

(15.48)

Substitution of Equation 15.48 in Equation 15.7 yields: Cpmax =

FDk a æ -KTmax K -KTmax ö - e ÷ çe ka Vd(k a - K) è ø

(15.49)

FD -KTmax e Vd

(15.50)

Cpmax =

(

)

Based on Equation 15.50, the maximum plasma concentration is a function of the total absorbable amount of dose, FD, and the fraction of dose in the body at Tmax , that is,

( fb )max = e -KT

max

(15.51)

Thus, the maximum amount of dose in the body at Tmax can be estimated as Amax = FD ´ f b max

(15.52)

475

15.2 COMPARTMENTAL ANALYSIS

15.2.1.4 Estimation of the Area Under Plasma Concentration–Time Curve The AUC is another important parameter of extravascularly administered compounds, in particular oral administration. It is considered a sensitive indicator of bioavailability and provides a collective understanding of the extent of exposure and magnitude of elimination. Theoretically, under similar conditions, equal doses of a compound should provide equal AUC. In addition to being a key parameter in bioavailability and bioequivalence assessment, it is also an indicator of pharmacological response (Krzyzanski and Jusko, 1998). The AUC of exogenous compounds has the initial condition of zero, that is, (AUC = 0 at t = 0), and it can be estimated from zero concentration until the elimination is complete, or it can be estimated between the boundary of any two points. For endogenous compounds, however, the initial condition of zero does not exist, and the approaches are different from xenobiotics (Scheff et al., 2011). The methods of AUC estimation, depending on area 0 → t or 0 → ∞, are 1. Using the model-independent equations: AUC0¥ =

FD Clt

(15.53)

2. Using the trapezoidal rule (Addendum 1, Part 2, Section A.4): n

AUC t0 =

å[(y + y i

) / 2](xi+1 - xi )

(15.54)

) / 2](xi+1 - xi ) + AUC ¥n

(15.55)

i+1

i=0

n

AUC ¥0 =

å[(y + y i

i+1

i=0

3. Using the integration method: AUC0¥ = AUC0t =

ò

t

0

ò

¥

0

˜°°°° Cp 0 e -Kt - e -kat

(

˜°°°° ˜°°°° æ 1 1 ö Cp 0 (e -Kt - e -kat ) = Cp 0 ç - ÷ è K ka ø ˜ ° ˜°°°° æ e -Kt e -kat öt Cp 0 0 Ke -kat - k a e -Kt = Cp ç + ÷ = k a ø0 K ´ k a è -K

)

(

(15.56)

)

(15.57)

The time t in AUC t0, methods 2 and 3, can be the duration of sampling or any time point, such as Tmax Like Cpmax , the AUC is a dose-dependent parameter. The dose-dependency of Cpmax and the AUC and dose-independency of Tmax are highlighted in Figure 15.12. The infuence of changing the rate of absorption on all three parameters of Tmax , Cpmax , and AUC is presented in Figure 15.13. A scenario that is relevant to the selection of the most appropriate extravascular route of administration, or the dosage form for a therapeutic agent. If the total amount absorbed remains the same but the rate of absorption changes with each dose, the slower the rate of absorption, the longer Tmax and lower Cpmax . The area under the curve depending on the extent of absorption, however, may or may not remain the same. Therefore, AUC, although considered a sensitive indicator of completeness of absorption, provides no information about the rate of absorption, Tmax , or Cpmax . 15.2.1.5 Estimation of Total Body Clearance and Apparent Volume of Distribution The defnitions of total body clearance and apparent volume of distribution are the same as discussed in previous chapters (see also Chapter 11). The following are practical methods of estimation of these constants within the context of the one-compartment model. Using the modelindependent Equation 15.53, the total body clearance is ClT =

F ´ Dose AUC0¥

(15.58)

Using a model-dependent equation, the clearance is ClT = K ´ Vd

476

(15.59)

PRACTICAL APPLICATIONS OF PK/TK MODELS: FIRST-ORDER ABSORPTION

Figure 15.12 Illustration of the dose-dependency of AUC and Cpmax and dose-independency of Tmax ; in linear PK/TK, there is a comparative relationship between the dose, AUC, and Cpmax ; that is, doubling the dose is refected by doubling the AUC and Tmax ; the time to peak concentration, however, remains unchanged; the main reason is that Tmax is a function of the rate constants of absorption and elimination.

Figure 15.13 Illustration of infuence of change in rate of absorption on Tmax and Cpmax ; the faster rate of absorption gives rise to higher maximum plasma level and a shorter time to the peak concentration; theAUCdepends on the extent of absorption thus lower absorption rate may or may not infuence the magnitude of the area under plasma concentration–time curve, if the extent of absorption remains the same. 477

15.2 COMPARTMENTAL ANALYSIS

The apparent volume of distribution, which can be estimated by using the following equations is: Vd =

F ´ Dose K ´ AUC0¥

FD ´ k a Vd = ˜°°°° Cp 0 ( k a - K )

˜ ° It should be noted that in most cases Cp 0 ¹ Cp 0

(

)

i.v.Bolus

(15.60) (15.61)

, unless F = 1 and k a ˜ K . If the F value is not

known, the volume of distribution can be reported as the following normalized value with respect to F: Vd D ´ ka = ˜°°°° 0 F Cp ( k a - K )

(15.62)

15.2.1.6 Fraction of Dose Absorbed (F) – Absolute Bioavailability The fraction of dose absorbed, also known as the absolute bioavailability, is the ratio of the area under the plasma concentration of an extravascularly administered dose to that of intravenous administration. It represents the absorbable fraction of the administered dose with the signifcance of formative data about the dosage form, route of administration, or interaction with other xenobiotics, therapeutics, environmental factors, herbs, or diet. It is also required for the calculation of the apparent volume of distribution or total body clearance (Equations 15.58 and 15.60). The following approaches are used to estimate the fraction of dose absorbed: 1. When equal doses are administered via intravenous and extravascular routes and total body clearances remain the same: F=

( AUC )extravascular ( AUC )intravenous

(15.63)

2. When the doses are different, the AUCs are normalized with respect to dose. F=

( AUC )extravascular ( Dose )extravascular ( AUC )extravascular ´ ( Dose )intravenous = ( AUC )intravenous ( Dosee )intravenous ( AUC )inntravenous ´ ( Dose )extravascular

3. From y-intercept of the extrapolated line: ˜ ° Cp 0 ´ Vd ´ (k a - K) F= D ´ ka

(15.64)

(15.65)

4. From Equation 15.58: ClT ´ AUC0¥ (15.66) D 5. The F value, as the dose descriptive in the biopharmaceutics classifcation system, is estimated as: F=

D - Adissolved - Aundissolved (15.67) D where D is the administered dose, Adissolved is the amount of dose dissolved at the site of absorption, and Aundissolved is the amount of the dose remaining to be dissolved (Charkoftaki et al., 2012). F=

6. Equation 15.67 is further simplifed to: 3

æ rp ö F = 1-ç ÷ -f è r0 ø

(15.68)

where r0 is the initial radius of drug particles; and rp and f correspond to the time equal to the mean intestinal transit time (Oh et al., 1993; Yu et al., 1996; Rinaki et al., 2004; Charkoftaki et al., 2012).

478

PRACTICAL APPLICATIONS OF PK/TK MODELS: FIRST-ORDER ABSORPTION

15.2.1.7 Duration of Action The equation of the extrapolated line can be used to estimate the duration of action of an orally administered dose if it is corrected for the time to the onset of action (Figure 15.11). The appropriate equation is: ˜ æ 2.303 Cp 0 ö (15.69) td = çç log ÷ - tonset CpMEC ÷ø è K where tonset is the time to the onset of action and approaches zero, tonset Þ 0 , when absorption is fast, and the rate constant of absorption is much greater than the rate constant of elimination. The duration of action of an extravascularly administered compound is the time required for the concentration of the extrapolated line that corresponds to the onset of action (point O in Figure 15.14) to decline to the minimum effective concentration (MEC) (point M in Figure 15.14). 15.2.2 Linear Two-Compartment Model with First-Order Input in the Central Compartment and First-Order Elimination from the Central Compartment The profle of the plasma concentration–time curve of compounds that are given via an extravascular route of administration and follow the two-compartment model refects the four biological processes of absorption, distribution, metabolism, and excretion (Teorel, 1937; Rescigno and Segre, 1966). The plasma concentration–time profle for a two-compartment model is similar to the profle of the one-compartment model. The distinction between the two bell-shaped profles is often diffcult to be made visually. However, depending on the values of the absorption rate constant in relation to the hybrid rate constants of the two-compartment model (a and b) and combined infuence of ADME processes, the two-compartment model may exhibit a distributional phase around the Cpmax , which is known as distributional nose. Frequently, when it is possible, the xenobiotic is also given intravenously to establish with certainty whether the drug follows the two- or more compartmental model. 15.2.2.1 Equations of the Model The diagram of the model is depicted in Figure 15.15. The model is comprised of the site of absorption, central compartment, and peripheral compartment. The central compartment, as discussed

Figure 15.14 Illustration of duration of action for a xenobiotic administered via an extravascular route of administration; depending on the rate of absorption, the time to the onset of action can be short or long, thus, as shown in the fgure, the duration of action should be corrected for the time to onset of action, when the concentration is below the therapeutic range; for compounds that absorbs fast, the correction may not be necessary. 479

15.2 COMPARTMENTAL ANALYSIS

Figure 15.15 Schematic of a two-compartment model with frst-order absorption, frst-order distribution, and frst-order elimination from the central compartment; the rate of absorption is a function of the dose administered at the site of absorption, and the removal from the site of absorption is governed by frst-order kinetics and related rate constant of absorption k a ; the distribution rate constants are k12 and k 21 , and the overall elimination rate constant is k10 , which includes both the excretion and metabolic rate constants; the central compartment, A1 , is the systemic circulation and highly perfused tissues, and the second compartment is the less perfused organs/ regions. for intravenous administration, represents the systemic circulation and highly perfused tissues. The peripheral compartment is the less accessible or slow equilibrating tissues and organs. The equations of the model and the related parameters and constants represent the overall outcome of the combined biological processes at the site of absorption, central compartment, and peripheral compartment. The rate equations of the model, representing the change in each compartment per units of time, are: Site of absorption: dAD = -k a AD dt

(15.70)

dA1 = k a AD + k 21 A2 - k12 A1 - k10 A1 dt

(15.71)

dA2 = k12 A1 - k 21 A2 dt

(15.72)

Central compartment:

Peripheral compartment:

The input function of the model is (Addendum I, Part 2, Section A.3.1): Input =

k a FD s + ka

(15.73)

The disposition function of the central compartment (Addendum I, Part 2, Section A.3.2) is:

( Disp )s1 =

480

s + E2

( s + E1 )( s + E2 ) - k12k21

=

s + k 21 s2 + s ( k 21 + k10 + k12 ) + k 211k10

=

s + E2 + s ( a )( s + b )

(15.74)

PRACTICAL APPLICATIONS OF PK/TK MODELS: FIRST-ORDER ABSORPTION

where E1 and E2 are the sum of the exit rate constants of the central and peripheral compartments, respectively. Multiplying the input with the disposition function yields the Laplace transform of the central compartment: L ( A1 ) =

k a FD ( s + k 21 ) ( s + ka )( s + a )( s + b )

(15.75)

The integrated form, defning the amount in the central compartment as a function of time, is: A1 =

k a FD ( k 21 - k a ) -k t k a FD ( k 21 - a ) -att k a FD ( k 21 - b ) -bt e + e + e ( a - k a ) (b - k a ) ( k a - a ) (b - a ) ( ka - b) ( a - b)

(15.76)

a

where F is the absolute bioavailability, D is the administered dose, k a is the frst-order absorption rate constant, k 21 is the frst-order distribution rate constant from peripheral compartment to the central compartment, and a and b are the frst-order hybrid rate constants estimated from the slope of the distributive and post-distributive phases. Dividing Equation 15.76 by the apparent volume of distribution of the central compartment yields Equation 15.77, which, by denoting the coeffcients of exponential terms as a*, b*, and c*, is further simplifed to Equation 15.78. Cp =

k a FD ( k 21 - k a ) -kat k a FD ( k 21 - a ) -at k a FD ( k 21 - b ) -bt e + e + e V1 ( a - k a )(b - k a ) V1 ( k a - a )(b - a ) V1 ( k a - b )( a - b )

(15.77)

Cp = a* e -at + b * e -bt + c * e -kat

(15.78)

That is, c* =

k a FD ( k 21 - k a ) V1 ( a - k a )(b - k a )

(15.79)

a* =

k a FD ( k 21 - a ) V1 ( k a - a )(b - a )

(15.80)

b* =

k a FD ( k 21 - b ) V1 ( k a - b )( a - b )

(15.81)

15.2.2.2 Interpretation of ka , a , and b As indicated earlier, depending on the frequency of plasma sampling and comparative magnitude of a and p , the plot of log plasma concentration versus time of an orally administered compound that follows the two-compartment model may exhibit a typical maximum peak or distributional nose, which is the combined infuence of distribution and absorption processes at the early time points (Figure 15.16). The only constant that can be identifed unambiguously from Equation 15.78 and Figure 15.16 is the smallest hybrid rate constant, b, or the disposition rate constant that corresponds to the slope of the terminal phase of the log Cp - time curve, that is, -b 2.303 . As in the two-compartment model bolus injection, the extrapolation of the linear segment provides the y-intercept b * . Theoretically, the other two rate constants, namely, a and k a , should be estimated from the residuals. The equation of the frst residual curve is determined by subtracting the extrapolated line equation from Equation 15.78: ˜°° Cp - Cp = a* e -at + b * e -bt + c * e -kat

(

)

residual-1

(

)

(

- b * e -bt ˜°

(Cp - Cp )

residual-1

)

(15.82) (15.83)

= a* e -at + c* e -kat

˜° Equation 15.83 represents the positive residual values, which when plotted as log Cp - Cp versus

(

)

time, exhibits the profle of a parabolic curve. The slope of the terminal linear portion of the curve is equal to -a 2.303 (if a > k a ) or equal to -k a 2.303 (if a > k a ). The y-intercept of the extrapolated line is either a* or c * depending on which rate constant (a or k a ) is smaller. The residual line of 481

15.2 COMPARTMENTAL ANALYSIS

Figure 15.16 Profle of log plasma concentration versus time of an orally administered xenobiotic that follows the two-compartment model and exhibits the distributional nose; this trend is often observed when the absorption of the compound is fast enough to cut in the expected distributive phase of the two-compartment model; most often when there are not enough samples at early time points, the distributional nose is overlooked. Equation 15.83 is calculated by subtracting the extrapolated line of the curve from Equation 15.83, that is, when a < k a Ü ˜ é ù * -at * k at * -at * -k at ê( Cp - Cp ) - Cp ú = a e + c e - a e = c e ë û

(

)

(15.84)

when a > k a Ü ˜ é ù * -at * k at * -k at (15.85) = a* e -at ê( Cp - Cp ) - Cp ú = a e + c e - c e ë û Depending on which rate constant is the largest, the slope of the second residual line (Equation 15.84 or 15.85) is either -a 2.303 or -k a 2.303, and the y-intercept of the line is the coeffcient associated with the largest rate constant (Figure 15.17). When a and k a are similar (i.e., a @ k a ), the log plasma concentration–time curve is similar to the profle of the one-compartment model with frst-order input (Ronfeld and Benet, 1977).

(

)

15.2.2.3 Parameters and Constants of the Two-Compartment Model with First-Order Input 15.2.2.3.1 Initial Plasma Concentration Setting t = 0 in Equation 15.78, the initial plasma concentration can be estimated as: Cp 0 = a* + b * + c*

(15.86)

15.2.2.3.2 Area under Plasma Concentration–Time Curve The integration of Equation 15.78 yields: AUC0¥ =

482

a* b * c* + + a b ka

(15.87)

PRACTICAL APPLICATIONS OF PK/TK MODELS: FIRST-ORDER ABSORPTION

Figure 15.17 Depiction of log plasma concentration vs time of an orally administered xenobiotic that follows the two-compartment model with graphical estimates of the hybrid rate constants of a and b, and the absorption rate constant, k a ; from the slope of the extrapolated line of the observed data, that is the linear terminal segment of the curve, the initial estimate of the disposition rate constant, b, is determined; there are two residual lines associated with the two-compartment model with frst-order input; the frst residual values are calculated by subtraction of the extrapolated line from the initial observed data points; the second residual values are estimated by subtraction of the initial values of the frst residuals from the extrapolated line of the frst residuals; without knowing the initial estimate of a from an intravenously administered dose, a dilemma will present itself as whether the slope of the frst residual line is a function of a or k a ; a similar dilemma also exists for the second residual line. The y-intercept of the lines determine the coeffcient of the three exponential terms of the equation of the model, i.e., Cp = a* e -at + b * e -bt + c * e -kat . Substitution of the equations for a*, b*, and c* (Equations 15.79–15.81) yields: AUC0¥ =

k a FD ( k 21 - b ) k a FD ( k 21 - k a ) k a FD(k 21 - a) + + aV1 ( k a - a ) (b - a ) bV1 ( k a - b ) ( a - b ) k aV1 ( a - k a ) (b - k a )

(15.88)

Mathematical manipulation of Equation 15.88 yields Equations 15.89 and 15.90. é ab ( k 21 - k a )( a - b ) + k ab ( k 21 - a )(b - k a ) ù ê ú -k aa ( k 21 - b )( a - k a ) ú k a FD êê ¥ ú ´ AUC0 = k aV1ab ê ú ( a - ka )(b - ka )( a - b ) ê ú ê ú ë û \ AUC0¥ =

(

2 2 2 2 2 2 é FD ê k 21 k a a - a k a - ab + a b + k ab - k a b V1ab ê k a2a - a 2 k a - ab2 + a 2b + k ab2 - k a2b êë

(

)

) ùú = FDk ú úû

(15.89)

21

V1ab

(15.90)

Substitution of ab = k 21k10 in Equation 15.90 provides the estimates of the area under the plasma concentration–time curve as: AUC0¥ =

FD ClT

(15.91)

483

15.2 COMPARTMENTAL ANALYSIS

AUC0¥ =

FD V1 ´ k10

(15.92)

AUC0¥ =

FD Vdb ´ b

(15.93)

The model-independent trapezoidal rule can always be used to determine the area under the curve from 0 → ∞ or from 0 → t. The estimated terminal area for AUC0¥ is achieved by dividing the last plasma concentration by b. 15.2.2.3.3 Apparent Volumes of Distribution The apparent volume of distribution of the central compartment and the overall volume of distribution, Vdb , are determined from Equations 15.92 and 15.93: V1 =

FD k10 ´ AUC0¥

(15.94)

FD b ´ AUC0¥

(15.95)

Vdb =

15.2.2.3.4 Total Amount Eliminated between 0 → t and 0 → ∞ The differential equation defning the total amount of drug eliminated from the body is: dAel = k10 A1 dt

(15.96)

The integration of Equation 15.96 between 0 → t gives

ò

t

t

ò Cpdt

(15.97)

\ Ael = k10V1 AUC0t = ClT AUC0t

(15.98)

Ael =

0

k10 A1dt = k10V1

0

when t approaches infnity Ael = FD, that is, the total amount eliminated between 0 → ∞ is equal to the total amount absorbed: FD = k10V1 AUC0¥ = ClT AUC0¥

(15.99)

Because Ael = Ae + Am , the total amount excreted unchanged between 0 → t and 0 → ∞ can be estimated as

(

)

Ae = f e k10V1 AUC0t = Clr ´ AUC0t

(

)

Ae¥ = f e k10V1 AUC0¥ = Clr ´ AUC0¥ = f e FD

(15.100) (15.101)

where f e is the fraction of absorbed dose that is excreted unchanged and Clr is the renal clearance. The fraction of absorbed dose excreted unchanged and eliminated as metabolites is: fe =

Ae¥ FD

(15.102)

FD - Ae¥ (15.103) FD The total amount of the absorbed dose eliminated as metabolites at time t and ∞ can be estimated as: fm =

( Am )t =

(

(

)

f m k10V1 AUC0t = Clm ´ AUC0t

)

Am¥ = f m k10V1 AUC0¥ = Clm ´ AUC0¥ = f m FD

(15.104) (15.105)

15.2.2.3.5 Estimation of Distribution and Elimination Rate Constants The rate constants of distribution and elimination are estimated according to methodology discussed in Chapter 13, Section 13.3.3.

484

PRACTICAL APPLICATIONS OF PK/TK MODELS: FIRST-ORDER ABSORPTION

15.2.2.4 Estimation of First-Order Absorption Rate Constant of a Two-Compartment Model – Loo–Riegelman Method This method is based on the mass balance equation in terms of amount or concentration between the site of absorption, the central compartment, the peripheral compartment, and the total amount eliminated from the body (Loo and Riegelman, 1968). The method requires a prior knowledge of the distribution rate constants, k12 and k 21 , and the overall elimination rate constant k10 . There are some similarities between this method and the Wagner–Nelson method (Chapter 15, Section 15.2.1.1.2). The Wagner–Nelson method is more applicable for the one-compartment model extravascular administration, although it can also be used for the two-compartment model (Wagner, 1974). The Loo–Riegelman method is preferred for the two-compartment model extravascular routes of administration with frst-order input. The Loo–Riegelman (Loo and Riegelman, 1968) method starts with the mass balance equation at time t:

( Atotal )t = ( A1 )t + ( A2 )t + ( Ae )t

(15.106)

where ( Atotal ) is the total amount absorbed at time t ; ( A1 ) and ( A2 ) are the amounts in the central t t t and peripheral compartments at time t, respectively; and ( Ae ) is the amount eliminated from the t body by all routes of elimination at time t. The total amount of a drug eliminated at time t and ¥, as described by Equations 15.97 and 15.99, is ( Ael )t = k10V1 AUC and Ael¥ = k10V1 AUC0¥ = FD . Therefore, the total percentage absorbed at time t is

( Atotal )t FD

é k ´ V1 ´ AUC0t ù ´ 100 = % ( A1 )t + % ( A2 )t + ê 10 ´ 100 ¥ ú ë k10 ´ V1 ´ AUC0 û

(15.107)

Equation 15.107 defnes the amount eliminated from the body in terms of the volume of the central compartment and the area under the plasma concentration–time curve from 0 ® tand 0 ® ¥ . Here, a prior knowledge of k10 is required. The second term of Equation 15.107 is the amount in the peripheral compartment, which for an orally administered dose is unknown, but by using the following equations and methodology, it can be expressed in terms of the amount in the central compartment, that is, the sampling compartment. The amount in the central compartment between two time points of tn and tn-1, where tn - tn-1 = Dt is: æ DA1 ö (15.108) +ç ÷ ( Dt ) è Dt ø The rate of the amount changed in the peripheral compartment is the same as described in Equation 15.72:

( A1 )t

n

= ( A1 )t

n-1

dA2 = k12 ( A1 )t - k 21 ( A2 )t dt Substituting Equation 15.108 for A1 in the above equation (Equation 15.72) yields:

( dA2 )t

n

dt

( dA2 )t

DA1 é = k12 ê( A1 )t + ( Dt )ùú - k21 ( A2 )tn n-1 Dt ë û

(15.109)

é DA ù (15.110) + k12 ê 1 ú Dt - k 21 ( A2 )t n ë Dt û Using the Laplace transforms (Addendum I, Part 2, Table A.1) yields the integrated form of Equation 15.110 as follows: dt

n

= k12 ( A1 )t

n-1

L ( 1) = 1 s L ( t ) = 1 s2 ˜1 L ( A1 ) = A

485

15.2 COMPARTMENTAL ANALYSIS

˜2 L ( A2 ) = A

( ) ( )

æ dA ö ˜2 - A ˜2 L ç 2÷=s A tn è dt ø Therefore, the Laplace transform of Equation 15.108 is:

( ) - ( A˜ )

˜2 s A

2

tn

t n-1

=

( )

˜1 k12 A s

2

2

tn

æ DA1 ö k12 ç ÷ Dt è Dt ø - k A ˜2 + 21 s2

( )

( ) yields: ˜ ˜ k (A A ( ) ) = +

˜2 Rearranging Equation 15.111 to solve for A

( A˜ )

tn-1

tn

12

tn-1

1

tn-1

s ( s + k 21 )

s + k 21

tn-1

(15.111) tn

æ DA1 ö D k12 ç Dt Dt ÷ø + 2è s ( s + k 21 )

(15.112)

The inverse Laplace transform of Equation 15.112 is:

( A2 )t

n

k k DA1 k é1 - e -k21Dt ù = ( A2 )t e -k21Dt + 12 ( A1 )t éë1 - e -k21Dt ùû + 12 DA1 - 12 ´ 2 ë û n-1 k 21 t k D k ˜˛ °˛˛ ˝ ˜˛˛˛ ˛n-1 21 21 ˛°˛˛˛˛ ˝ ˜˛˛˛˛˛ ˛°˛˛˛˛˛˛ ˝ 1st

2

nd

(15.113)

rd

e 3 term

To simplify Equation 15.113, the Taylor series expansion is applied to the exponential function of the third term, that is, 2 k 21 ( Dt ) 2 Substituting Equation 15.114 into Equation 15.113 yields:

2

e -k21Dt @ 1 - k 21Dt +

( A2 )t

(15.114)

k 21 ( Dt ) ù k k DA1 é k ê1 - k 21Dt + ú = ( A2 )t e -k21Dt + 12 ( A1 )t éë1 - e -k21Dt ûù + 12 DA1 - 12 ´ n-1 k 21 k 21 k 21 Dt ê 2 ˜˛ ˛n-1 °˛˛ ˝ ˜˛˛˛ úû ˛°˛˛˛˛ ˝ ˜˛˛˛˛˛˛˛ ë st ˛ °˛˛˛˛˛˛˛˛ ˝ 1 2

2

n

2

nd

(15.115)

3 rd term

Further simplifcation of the third term of Equation 15.115 yields the following relationship that represents ( A2 ) term: tn

( A2 )t

n

= ( A2 )t

n-1

e -k21Dt +

k k12 D ( A1 )tn-1 éë1 - e -k21Dt ùû + 12 DA1Dt k 21 2

(15.116)

Therefore, k k Atotal = ( A1 )t + ( A2 )t + 12 ( A1 )t éë1 - e -k 21Dt ùû + 122 DA1Dt + k˜ V1°˝˛ AUC0t 10˝ n-1 n-1 k 21 2 ˜°˛n ˜˝˝˝˝˝˝˝˝°˝˝˝˝˝˝˝˝˛ Ael A1

(15.117)

A2

Dividing Equation 15.117 by the volume of distribution yields Equation 15.118, which defnes the total concentration of the compound: Ctotal = ( Cp )t + ( C2 )t n

n-1

+

k k12 ( Cp )tn-1 ëé1 - e -k 21Dt ûù + 212 DCp1Dt + k10 AUC0t k 21

(15.118)

The use of the two-term Taylor expansion in simplifying the calculation of the absorption rate constant by the Loo–Riegelman method has been demonstrated to introduce errors in the calculation of the absorption rate constant (Boxenbaum and Kaplan, 1975; Zeng et al., 1983). Equation 15.118 is expressed in terms of plasma concentration, except for ( C2 ) , which is estimated by Equation tn-1 15.119. The initial condition of concentration in the second compartment is zero, ( C2 ) = 0 . tn-1

( C 2 )t

486

n

= ( C 2 )t

n-1

e

-k21Dt

k k + 12 ( Cp )t éë1 - e -k21Dt ûù + 12 DCpDt D n-1 k 21 2

(15.119)

PRACTICAL APPLICATIONS OF PK/TK MODELS: FIRST-ORDER ABSORPTION

Using Equation 15.117, the expression of “percent absorbed” (Equation 15.107) can be defned as k Atotal k é ù k V AUC0t ´ 100 = % ( A1 )t + % ê( A2 )t + 12 ( A1 )t ëé1 - e -k 21Dt ûù + 12 DA1Dt ú + 10 1 ´ 100 ¥ n n-1 n-1 k 21 FD 2 û k10V1 AUC0 ë

(15.120)

ù é A Plotting the logarithm of percentage remaining to be absorbed, that is, log ê100 - æç total ´ 100 ö÷ú è FD øû ë versus time, is linear with a slope of -k a 2.303. The numerical example of the Loo–Riegelman method is presented in Tables 15.3–15.6. The data presented in Table 15.3 are the plasma concentrations of a compound that follows the two-compartment model given orally and intravenously following administration of equal doses. The biexponential equation summarizing the intravenous data is Cp = 120e -4.145t + 91.41e -0.4145t . The calculated rate constants were estimated as k 21 = 2.0275h -1 , k10 = 0.8474 h -1 and k12 = 1.6846h -1 . The stepwise calculation of the absorption rate constant according to Equations 15.118 and 15.119, A as presented in Tables 15.4 through 15.6, generated the variable log æç 100 - % total ö÷ . Plotting this FD ø è -1 variable versus time provided the slope of (-k a 2.303 = -0.94 h ) and k a = 2.165h -1 15.2.3 Linear Two-Compartment Model with First-Order Input in the Peripheral Compartment and First-Order Elimination from the Peripheral Compartment The diagram of the model is depicted in Figure 15.18. The differential equations of the model are: Central compartment : Peripheral compartment :

dA1 = k 21 A2 - k12 A1 dt

dA2 = k a FD + k12 A1 - k 21 A2 - k 20 A2 dt

(15.121) (15.122)

The integrated equation of the time course of the compound in the central compartment, the sampling compartment, is: Cpt =

k + k 20 - b -btt FDk a é k 21 + k 20 - a -at k + k 20 - k a -kat ù e + 21 e + 21 e ú ê V1 ëê ( k a - a ) (b - a ) ( ka - b )( a - b ) ( a - ka ) (b - ka ) ûú

(15.123)

The abbreviated equation is: Cpt = a* e -at + b * e -bt + c * e -kat

(15.124)

The new coeffcients of the exponential terms are: a* =

FDk a ( k 21 + k 20 - a ) V1 ( k a - a ) (b - a )

(15.125)

b* =

FDk a ( k 21 + k 20 - b ) V1 ( k a - b )( a - b )

(15.126)

c* =

FDk a ( k 21 + k 20 - k a ) V1 ( a - k a )(b - k a )

(15.127)

Using the new coeffcients (Equations 15.125–15.127), the following equations provide the initial estimates of distribution and elimination rate constants: k 20 + k 21 =

a*bk a + b *ak a + c* ab a* ( k a - a ) + b ( k a - b )

k12 = a + b - ( k 20 + k 21 ) k 20 =

ab k12

(15.128) (15.129) (15.130)

487

15.2 COMPARTMENTAL ANALYSIS

Table 15.3 Data Related to Application Cp Oral (mg/L)

Time (h) 0.10 0.20 0.30 0.40 0.60 0.75 1.00 2.00 3.00 4.00 5.00 6.00 10.00 14.00

Cp IV (mg/L)

1.000 12.000 28.000 46.000 56.000 59.700 60.000 40.000 26.200 17.500 11.500 7.580 1.400 0.268

166.979 136.516 115.326 100.306 81.262 72.345 62.293 39.929 26.361 17.415 11.506 7.602 1.448 0.276

Table 15.4 Calculations of Absorption Rate Constant Using the Loo– Riegelman Method tn

tn−1

Δt

(Cp)tn

(Cp)tn−1

ΔCp

k12 DCpDt 2

k12 ( Cp ) tn - 1 éë1 - e -k21D k 21

0.10 0.20 0.30 0.40 0.60 0.75 1.00 2.00 3.00 4.00 5.00 6.00 10.00 14.00

0.0 0.10 0.10 0.10 0.10 0.20 0.15 0.25 1.00 1.00 1.00 1.00 1.00 4.00

0.10 0.10 0.10 0.10 0.20 0.15 0.25 1.00 1.00 1.00 1.00 1.00 4.00 4.00

1.000 12.000 28.000 46.000 56.000 59.700 60.000 40.000 26.200 17.500 11.500 7.580 1.400 0.268

0 1.000 12.000 28.000 46.000 56.000 59.700 60.000 40.000 26.200 17.500 11.500 7.580 1.400

1.000 11.000 16.000 18.000 10.000 3.700 0.300 −20.000 −13.800 −8.700 −6.000 −3.920 −6.180 −1.132

0.0842 0.9265 1.3477 1.5161 1.6846 0.4675 0.0632 −16.8460 −11.6237 −7.3280 −5.0538 −3.3018 −20.8217 −3.8139

0.0000 0.1525 1.8298 4.2694 12.7410 12.2014 19.7235 43.2887 28.8592 18.9027 12.6259 8.2970 6.2961 1.1629

k 21 = ( k 20 + k 21 ) - k 20

(15.131

The apparent volume of distribution of the central compartment can be estimated as

(

)

k aD / a* ( k 21 + k 20 - a ) V1 = F ( ka - a )(b - a ) V1 =

( k FD / a ) ( k a

*

21

+ k 20 - a )

( ka - a )(b - a )

(15.132)

(15.133)

The initial estimate of the overall volume of distribution is the same as the previous model. The plasma concentration–time curve of this model exhibits a more defned distributional nose.

488

PRACTICAL APPLICATIONS OF PK/TK MODELS: FIRST-ORDER ABSORPTION

Table 15.5 Calculations of Absorption Rate Constant Using the Loo– Riegelman Method continued

( C 2 )t

n-1

e -k21Dt

0 0.0687 0.9370 3.3594 7.4667 14.5943 20.1139 24.0351 6.6461 3.1443 1.9379 1.2521 0.8225 −4.1180

(C2tn)

n AUCttn-1

AUC0tn

k10 ´ AUC0tn

Ctotal

0.0842 1.1477 4.1145 9.1449 21.8923 27.2632 39.9006 50.4778 23.8816 14.7190 9.5100 6.2473 −13.7031 −6.7690

0.050 0.650 2.000 3.700 10.200 8.677 14.962 50.000 33.100 21.850 14.500 9.540 17.960 3.336

0.050 0.700 2.700 6.400 16.600 25.277 40.239 90.239 123.339 145.189 159.689 169.229 187.189 190.525

0.042 0.593 2.288 5.423 14.067 21.420 34.099 76.469 104.517 123.033 135.320 143.405 158.624 161.451

1.127 13.741 34.402 60.568 91.959 108.383 133.999 166.946 154.599 155.252 156.330 157.232 146.321 154.950

Table 15.6 Calculations of Absorption Rate Constant Using the Loo– Riegelman Method Atotal ´ 100 FD 0.722 8.808 22.053 38.826 58.948 69.476 85.897 107.017 99.102 99.521 100.212 100.790 93.795 99.327

100 - %

Atotal FD

99.278 91.192 77.947 61.174 41.052 30.524 14.103

A ö æ log ç 100 - % total ÷ FD ø è 1.99685 1.95996 1.89180 1.78657 1.61333 1.48464 1.14931

15.2.4 Linear Three-Compartment Model with First-Order Input in the Central Compartment and First-Order Elimination from the Central Compartment The diagram of the model is depicted in Figure 15.19. The assumptions of the model are, the same as discussed before, for intravenous administration. The central compartment represents the systemic circulation and highly perfused tissues/organs. It receives the absorbable dose after absorption from an extravascular route of administration. The number of peripheral compartments is determined either based on the physiological/biochemical circumstances confrmed by the curveftting data, or the number is determined solely through the curve-ftting process (Wagner, 1988; Wagner et al., 1990).

489

15.2 COMPARTMENTAL ANALYSIS

Figure 15.18 Schematic of a two-compartment model with frst-order input into the peripheral compartment and elimination from the peripheral compartment; the general equation of the model is the same as discussed for the model with input into the central compartment; however, the coeffcients of the exponential terms and the rate constant are different; the defnition of the compartments and the rate constants are the same as discussed before.

Figure 15.19 Schematic of a three-compartment model with frst-order input into the central compartment and frst-order elimination from the central compartment; the description of the site of absorption, the central compartment, and the rates and rate constants are the same as described for the two-compartment model; one of the peripheral compartments can be referred to as the heterogeneous combination of less accessible or less perfused organs/tissues; the extra peripheral compartment may be interpreted as the preferential uptake of xenobiotic by an organ or a defned region, or high binding affnity of the xenobiotic for a region/organ; the addition of an extra peripheral compartment that represents an additional exponential term in the equation of the model improves the curve-ftting of the observed data, but requires statistical justifcation for having the extra exponential term; due to the complexity of the model, most calculations of the parameters and constants and necessary statistical justifcations are determined by the PK/TK software. 490

PRACTICAL APPLICATIONS OF PK/TK MODELS: FIRST-ORDER ABSORPTION

The differential equations of the model are: dA1 = k a FD + k 21 A2 + k 31 A3 - A1 ( k12 + k13 + k10 ) dt

(15.134)

dA2 = k12 A1 - k 21 A2 dt

(15.135)

dA3 = k13 A1 - k 31 A3 dt

(15.136)

The integrated equation representing the time course of the absorbed dose in the systemic circulation is: é ( k 21 - a )( k 31 - a ) ( k21 - b )(( k31 - b ) e -bt ù e -at + ê ú k a b a g a a - b )( g - b ) ( k a - b ) ( ) ( )( ) ( a ú FDk a ê Cp = ê ú V1 ê k 21 - g )( k 31 - g ) k 21 - k a )( k 31 - k a ) ( ( -gt -k at ú e + e ê+ ( a - ka )(b - ka )( g - ka ) úû ë ( a - g )(b - g )( k a - g )

(15.137)

The abbreviated version of Equation 15.137 is: Cp = a* e -at + b * e -bt + c * e -gt + d * e -kat

(15.138)

The coeffcients of the exponential terms are: a* =

FDk a ( k 21 - a )( k 31 - a ) V1 (b - a )( g - a )( k a - a )

(15.139)

b* =

FDk a ( k 21 - b )( k 31 - b ) V1 ( a - b )( g - b )( k a - b )

(15.140)

c* =

FDk a ( k 21 - g )( k 31 - g ) V1 ( a - g )(b - g )( k a - g )

(15.141)

FDk a ( k 21 - k a )( k 31 - k a ) V1 ( a - k a )(b - k a )( g - k a )

(15.142)

d* =

The calculation of the absorption rate constant is by the Loo–Riegelman method (Loo and Riegelman, 1968; Wagner, 1983; Proost, 1985). 15.3 APPLICATIONS AND CASE STUDIES The applications and case studies of Chapter 15 are posted in Addendum II – Part 6. REFERENCES Baggot, J. D. 1992. Review bioavailability and bioequivalence of veterinary drug dosage forms, with particular reference to horses: An overview. J Vet Pharmacol Ther 15(2): 160–73. Bialer, M. 1980. A simple method for determining whether absorption and elimination rate constants are equal in the one-compartment open model with frst-order input. J Pharmacokinet Biopharm 8(1): 111–13. Bois, F. Y., Tozer, T. N., Hauck, W. W., Chen, M. L., Patnaik, R., Williams, R. L. 1994. Bioequivalence performance of several measures of rate of absorption. Pharm Res 11(7): 966–74.

491

15.3 APPLICATIONS AND CASE STUDIES

Boroujerdi, M. 2002. Pharmacokinetics: Principles and Applications, Chapter 8. New York: McGraw Hill-Medical Publishing Division. Boxenbaum, H. 1998. Pharmacokinetics tricks and traps: Flip-fop models. J Pharm Pharm Sci 1(3): 90–1. Boxenbaum, H. G., Kaplan, S. A. 1975. Potential source of error in absorption rate calculations. J Pharmacokinet Biopharm 3(4): 257–64. Bredberg, U., Karlsson, M. O. 1991. In vivo evaluation of the semi-simultaneous method for bioavailability estimation using controlled intravenous infusion as an “extravascular” route of administration. Biopharm Drug Dispos 12(8): 583–97. Byron, P. R., Notari, R. E. 1976. Critical analysis of “fip-fop” phenomenon in two-compartment pharmacokinetic model. J Pharm Sci 65(8): 1140–4. Charkoftaki, G., Dokoumetzidis, A., Valsami, G., Macheras, P. 2012. Elucidating the role of dose in the biopharmaceutics classifcation of drugs: The concepts of critical dose, effective in vivo solubility, and dose-dependent BCS. Pharm Res. https://doi.org/10.1007/s11095-012-0815-4. Chen, M. L. 1992. An alternative approach for assessment of rate of absorption in bioequivalence studies. Pharm Res 9(11): 1380–5. Chen, M.-L., Barbara Davit, B., Lionberger, R., Wahba, Z., Ahn, H.-Y., Lawrence, X., Yu, L. X. 2011. Using partial area for evaluation of bioavailability and bioequivalence. Pharm Res 28(8): 1939–47. Chen, M. L., Lesko, L. J., Williams, R. L. 2001. Measures of exposure versus measures of rate and extent of absorption. Clin Pharmacokinet 40(8): 565–72. Curry, S. H. 1981. Theoretical considerations in calculation of terminal phase half-times following oral doses, illustrated with model data. Biopharm Drug Dispos 2(2): 115–21. Endrenyi, L., Csizmadia, F., Tothfalusi, L., Balch, A. H., Chen, M. L. 1998a. The duration of measuring partial AUCs for the assessment of bioequivalence. Pharm Res 15(3): 399–404. Endrenyi, L., Csizmadia, F., Tothfalusi, L., Chen, M. L. 1998b. Metrics comparing simulated early concentration profles for the determination of bioequivalence. Pharm Res 15(8): 1292–9. Garrett, E. R. 1993. Simplifed methods for the evaluation of the parameters of the time course of plasma concentration in the one-compartment body model with frst-order invasion and frstorder drug elimination including methods for ascertaining when such rate constants are equal. J Pharmacokinet Biopharm 21(6): 689–734. Garrett, E. R. 1994. The Bateman function revisited: A critical reevaluation of the quantitative expressions to characterize concentrations in the one compartment body model as a function of time with frst-order invasion and frst-order elimination. J Pharmacokinet Biopharm 22(2): 103–28. Gibaldi, M., Perrier, D. 1982. Pharmacokinetics, Second Edition, 63–5. New York: Marcel Dekker, Inc. Krzyzanski, W., Jusko, W. J. 1998. Integrated functions for four basic models of indirect pharmacodynamic response. J Pharm Sci 87(1): 67–72. Loo, J. C. K., Riegelman, S. 1968. New method for calculating the intrinsic absorption rate of drugs. J Pharm Sci 57(6): 918–28.

492

PRACTICAL APPLICATIONS OF PK/TK MODELS: FIRST-ORDER ABSORPTION

Neelakantan, S., Veng-Pedersen, P. 2005. Determination of drug absorption rate in time-variant disposition by direct deconvolution using β clearance correction and end-constrained non-parametric regression. Biopharm Drug Dispos 26(8): 353–70. Oh, D. M., Curl, R. L., Amidon, G. L. 1993. Estimating the fraction dose absorbed from suspensions of poorly soluble compounds in humans: A mathematical model. Pharm Res 10(2): 264–70. Proost, J. H. 1985. Wagner’s exact Loo-Riegelman equation: The need for a criterion to choose between the linear and logarithmic trapezoidal rule. J Pharm Sci 74(7): 793–4. Rescigno, A., Segre, G. 1966. Drugs and Tracer Kinetics. Waltham: Blaisdell Publishing Company. Rinaki, E., Dokoumetzidis, A., Valsami, G., Panos Macheras, P. 2004. Identifcation of biowaivers among Class II drugs: Theoretical justifcation and practical examples. Pharm Res 21(9): 1567–72. Ronfeld, R. A., Benet, L. Z. 1977. Interpretation of plasma concentration-time curves after oral dosing. J Pharm Sci 66(2): 178–80. Rostami-Hodjegan, A., Jackson, P. R., Tucker, G. T. 1994. Sensitivity of indirect metrics for assessing “rate” in bioequivalence studies—Moving the “goalposts” or changing the “game”. J Pharm Sci 83(11): 1554–7. Scheff, J. D., Almon, R. R., DuBois, D. C., Jusko, W. J., Androulakis, I. P. 2011. Assessment of pharmacologic area under the curve when baselines are variable. Pharm Res 28(5): 1081–9. Teorell, T. 1937. Kinetics of distribution of substances administered to the body I. The extravascular mode of administration. Arch Int Pharmacodyn 57: 205–25. Wagner, J. G. 1970. “Absorption rate constant” calculated according to the one-compartment model with frst-order absorption: Implications in in vivo – In vitro correlation. J Pharm Sci 59(7): 1049. Wagner, J. G. 1974. Application of the Wagner-Nelson absorption method to the two-compartment model. J Pharmacokinet Biopharm 2(6): 469–86. Wagner, J. G. 1975. Fundamentals of Clinical Pharmacokinetics, 174–7. Hamilton: Drug Intelligence Publications, Inc. Wagner, J. G. 1983. Pharmacokinetic absorption plots from oral data alone or oral/intravenous data and an exact Loo-Riegelman equation. J Pharm Sci 72(7): 838–42. Wagner, J. G. 1988. Types of mean residence times. Biopharm Drug Dispos 9(1): 41–57. Wagner, J. G., Ganes, D. A., Midha, K. K., Gonzalez-Younes, I., Sackellares, J. C., Olsen, L. D., Affrime, M. B., Patrick, J. E. 1990. Stepwise determination of multicompartment disposition and absorption parameters from extravascular concentration-time data. Application to mesoridazine, furbiprofen, funarizine, labetanol and diazepam. J Pharmacokinet Biopharm 19(4): 413–55. Wagner, J. G., Nelson, E. 1964. Kinetic analysis of blood levels and urinary excretion in the absorptive phase after single doses of drug. J Pharm Sci 53: 1392–403. Wang, Y., Nedelman, J. 2002. Bias in the Wagner–Nelson estimate of the fraction of drug absorbed. Pharm Res 19(4): 470–6. Yáñez, J. A., Remsberg, C. M., Sayre, C. L., Forrest, M. L., Davies, N. M. 2011. Flip-fop pharmacokinetics—Delivering a reversal of disposition: Challenges and opportunities during drug development. Ther Deliv 2(5): 643–72.

493

15.3 APPLICATIONS AND CASE STUDIES

Yu, L. X., Crison, J. R., Amidon, G. L. 1996. Compartmental transit and dispersion model analysis of small intestinal transit fow in humans. Int J Pharm 140(1): 111–18. Zeng, Y. L., Akkermans, A. A. M. D., Breimer, D. D. 1983. Factors affecting the error in the LooRiegelman method for estimating the rate of drug absorption. Some suggestions for a practical sampling schedule. Arzneim Forsch 33(1): 757–60.

494

16 Practical Application of PK/TK Models: Multiple Dosing Kinetics 16.1 INTRODUCTION The outcome of prolonged exposure to xenobiotics, whether as a disciplined multiple dosing regimen of therapeutic agents or as random exposure to environmental pollutants, is the accumulation of xenobiotics in the body. The purpose of a multiple dosing regimen is to achieve a consistent pharmacological response for a period longer than the duration of action of a single dose (KrügerThiemer, 1966, 1969; Krüger-Thiemer and Bünger, 1965). The concept is the same as the zero-order input discussed in Chapter 14, which resulted in the accumulation of the infused compound in the body. When a xenobiotic is given on a multiple dosing basis of fxed dose and dosing interval, it accumulates in the body (van Rossum et al., 1968). Unlike the continuous infusion, the plasma concentration of administered compound in the multiple dosing regimen fuctuates after each dose reaches a maximum concentration, or peak level, and then declines to a minimum concentration or trough level (Wagner et al., 1965; Gibaldi and Perrier, 1982). In this type of regimented exposure, the dose is usually kept constant and dosing interval (t) is selected as a fxed interval approximately equal to the half-life of the compound. This means that during each dosing interval approximately 50% of amount in the body eliminates and 50% remains in the body. Thus, peak and trough levels of each dose will be higher than the previous one. Since dose and dosing intervals are kept constant, after a fnite time, the administered dose will become equal to the amount eliminated from the body. This is when the steady-state fuctuations (i.e., plateau peak and trough levels) are achieved, and the amount accumulated in the body is at its anticipated maximum level of the designed dosing regimen (Wagner, 1975; Gibaldi and Perrier, 1982; Buell et al., 1969). A welldesigned dosing regimen for therapeutic agents maintains the fuctuation at steady state within the therapeutic range (Gibaldi and Levy, 1976; Levy, 1974). The concept of attaining and maintaining the steady state is similar to intravenous infusion, that is, the rate of input is equal to the rate of output. In multiple dosing steady state, the administered dose is equal to the amount eliminated from the body in a dosing interval. The fxed dose that is given on a regular basis is also known as the maintenance dose. 16.2 KINETICS OF MULTIPLE INTRAVENOUS BOLUS INJECTIONS – ONE-COMPARTMENT MODEL In general, the kinetic descriptions of multiple dosing are based on geometric series (Addendum I, Part 2, Section A.5). The summary of plasma levels in terms of peak and trough levels, following a regimen of fxed intravenous dose and dosing interval, is presented in Table 16.1. In Table 16.1, A0 , Amax and Amin are the initial, maximum, and minimum amounts in the body, respectively. The last row of the table is the geometric series of the multiple dosing functions of maximum concentrations. As the number of doses increases, the number of exponential terms increases, the equations become longer, and using them for the purpose of calculating the concentration of the compound in plasma is cumbersome. To make the calculations more practical, the solution of geometric series for the nth dose is determined as follows: The difference between peak and trough levels of the nth dose is

( Amax )n - ( Amin )n = A0 ( Sn ) - A0 ( Sn ) e -Kt = A0 ( Sn ) (1 - e -Kt )

(16.1)

where Sn , is the geometric series of nth dose as identifed in Table 16.1. Also, the difference between ( Amin ) - ( Amax ) of the last two entries of Table 16.1 is: n

n

( Amin )n - ( Amax )n = A

0

(e

-K t

+ e -2K t + e -3K t + e -4K t

- K n-1 + e -5K t + ¼. + e ( ) t + e -nK t )

-A0 (1 + e -K t + e -2K t + e -3K t + e -4K t n-1 K +¼ + e ( ) t ) = e -nKt - 1

Therefore,

( Amax )n - ( Amin )n = 1 - e -nKt

(16.2)

Setting equation 16.1 equal to equation 16.2 and solving for Sn yields: DOI: 10.1201/9781003260660-16

495

16.2 Kinetics of Multiple intravenous Bolus injections – one-coMpartMent Model

Sn =

1 - e -nKt 1 - e -Kt

(16.3)

Equation 16.3 summarizes all the exponential terms of ( Amax ) in Table 16.1 and represents the n solution of the geometric series Sn of multiple dosing after nth dose. As n Þ ¥, the accumulation reaches the expected maximum level of steady state, e -nKt Þ 0, and Equation 16.3 changes to Equation 16.4, which represents the solution of the geometric series at steady state.

( Sn )ss =

1 1 - e -Kt

(16.4)

Equations 16.3 and 16.4 are often referred to as the multiple dosing functions to generate plasma levels before attaining and during the steady-state levels, respectively. 16.2.1 Equations of Plasma Peak and Trough Levels The peak and trough levels of the frst intravenous bolus dose are: Peak level of frst dose:

( Cpmax )1 = Cp0

(16.5)

Trough level of frst dose at the end of frst dosing interval:

( Cpmin )1 = ( Cpmax )1 e -tK = Cp0 e -Kt

(16.6)

Plasma level at any time t of the dosing interval between peak and trough levels of the frst dose (t < t):

( Cpt )1 = ( Cpmax )1 e -Kt = Cp0 e -Kt

(16.7)

Equation 16.7 represents the frst-order decline of plasma concentration as a function of time from peak to trough level of the frst dose. Multiplying Equations 16.5–16.7 by Equation 16.3, yields peak, trough, and any concentration at time t between the peak and trough level before achieving the steady-state level (Figure 16.1) (Gibaldi and Perrier, 1982): æ 1 - e -nKt ö ÷ è ø

(16.8)

æ 1 - e -nKt ö -Kt = ( Cpmax )n e -Kt ÷e è ø

(16.9)

( Cpmax )n = Cp0 ç 1 - e -Kt ( Cpmin )n = Cp0 ç 1 - e -Kt

æ 1 - e -nKt ö -Kt -Kt ÷ e = ( Cpmax )n e è ø

( Cpt )n = Cp0 ç 1 - e -Kt

(16.10)

Multiplying Equations 16.5–16.7 by Equation 16.4 generates equations of steady-state levels:

( Cpmax )ss = Cp0 çæ 1 - e - Kt ÷ö 1

è

ø

æ e -Kt ö

( Cpmin )ss = Cp0 ç 1 - e -Kt ÷ = ( Cpmax )ss e -Kt è

ø

æ e -Kt

ö

è

ø

( Cpt )ss = Cp ˜ ç 1 - e -Kt ÷ = ( Cpmax )ss e -Kt

(16.11) (16.12) (16.13)

In Equations 16.8–16.13, the initial plasma concentration, Cp 0 , is the initial plasma concentration of the maintenance dose, i.e., DM Vd, which also corresponds to the difference between the steadystate peak and trough levels (Figure 16.1). 16.2.2 Estimation of Time Required to Achieve Steady-State Plasma Levels In multiple-dosing kinetics, contrary to zero-order input (Chapter 14) where the amount and time of input are continuous, the regimen consists of several dosing intervals, and it is not expressed 496

practical application of pK/tK Models: Multiple dosing Kinetics

Table 16.1 Summary of Peak and Trough Levels in Multiple Intravenous Injections and the Related Geometric Series t

Dose

t1

D1

( Amax )1 = A0

( Amax )1 = A0 e -Kt

t2

D2

( Amax )2 = A0 (1 - e -Kt )

( Amin )2 = A0 ( e -Kt + e -Kt )

t3

D3

( Amax )3 = A0 (1 + e -Kt + e -2Kt )

( Amin )3 = A0 ( e -Kt + e -2Kt + e -3Kt )

t4

D4

( Amax )4 = A0 (1 + e -Kt + e -2Kt + e -3Kt )

( Amin )4 = A0 ( e -Kt + e -2Kt + e -3Kt + e -4Kt )

˜

˜

˜

˜

˜

˜

Amax

˜ tn

Dn

( Amax )4 = A

0

(1 + e

Amin

˜ -Kt

+e

-2Kt

+e

- n - 1 Kt +e -4 Kt + …… + e ( )

Geometric Series:

Sn

- n - 1 Kt = 1 + e - Kt + e -2Kt + e -3Kt + e -4Kt + …… + e ( )

( Amax )4 = A0 ( e -Kt + e -2Kt + e -3Kt + e -4Kt

-3Kt

)

- n -1 Kt +…… + e ( ) + e -nKt

)

Geometric Series:

Sn e -Kt

- n -1 K t = e -Kt + e -2Kt + e -3Kt + e -4Kt + …… + e ( ) + e -nKt

Figure 16.1 Illustration of fuctuation and gradual accumulation of plasma concentration resulting from multiple intravenous bolus injections of a xenobiotic; the orderly fuctuation is the result of giving a fxed maintenance dose every fxed dosing interval with the objective of maintaining predictable minimum and maximum concentrations within the known therapeutic range of therapeutic agents; the fgure shows that, for this dosing regimen, it takes fve dosing intervals or fve maintenance doses to have the peak and trough levels within the therapeutic range, but the steady-state levels take longer to attain; the therapeutic outcome is fully achieved when fuctuation is within the therapeutic range, the levels may or may not be at steady state. 497

16.2 Kinetics of Multiple intravenous Bolus injections – one-coMpartMent Model

as continuous t but rather as nτ. The equation of nτ, or the time needed to achieve any fraction of steady-state levels is determined by comparing the numerators of Equations 16.3 and 16.4. In both equations the numeral ‘1’ corresponds to 100% of steady-state level, ‘e ˜nK° ’ is the fraction left to reach the steady-state levels and ‘1 ˜ e ˜nK° ’ of equation 16.3 represents the fraction of steady-state levels: f ss ˜ 1 ° e °nK˛

(16.14)

Solving for nτ by taking the natural logarithm of Equation 16.14 yields: or

ln(1 ˛ f ss ) , K n˜ ° ˛3.3T1/2 log(1 ˛ f ss )

(16.15)

n˜ ° ˛

(16.16)

According to Equation 16.16 and analogous to zero-order input (Chapter 14), the time required for achieving any fraction of steady state is directly proportional to the half-life of the compound, or indirectly proportional to the overall elimination rate constant. For example, to achieve 50% of steady-state levels, that is, f ss = 0.50 n˜ ° ˛3.3T1/2 log(0.50) ° 1T1 2 f ss ° 0.75 n˜ ° ˛3.3T1/2 log(0..25) ° 2T1 2 f ss ° 0.90 n˜ ° ˛3.3T1/2 log(0.10) ° 3.3T1 2 f ss ° 0.95 n˜ ° ˛3..3T1/2 log(0.05) ° 4.3T1 2 f ss ° 0.99 n˜ ° ˛3.3T1/2 log(0.01) ° 6.6T1 2 16.2.3 Average Steady-State Plasma Concentration The average plasma concentration at steady state is a parameter developed based on the defnition of steady state. At steady state, the rate of input is equal to the rate of output. In the case of zeroorder input (Chapter 14) the rate of input is continuous, whereas in multiple dosing it is sequential, and the rate is defned as a maintenance dose given at every dosing interval. The rate of output in zero-order input at steady state, although frst-order, remains constant because of constant steady-state plasma level. In multiple dosing, the rate of output changes during a dosing interval and declines from a maximum rate to a minimum because of the change in plasma level. To defne a single rate of output and input during a dosing interval at steady state, like zero-order input, the average rates are calculated as follows: average rate of input ˜ maintenance dose ° dosing interval ˜ averagerateof output ˜ ClT ° ˛ Cpave ˝ss ˜ K ° ˛ Aave ˝ss

DM ˛

(16.17) (16.18)

where ˜ Cpave °ss and ˜ Aave ° are average plasma concentration and average amount in the body at ss steady state, respectively. Setting the rate of input equal to the rate of output and solving for ˜ Cpave °ss yields the following relationship, which can be presented in various forms: D

˜ Cpave °ss ˛ Cl M˝ ˙

(16.19)

t

D

˜ Cpave °ss ˛ K ˝ VdM ˝ ˙ .

˝T

˝ DM

˜ Cpave °ss ˛ 1 44 Vd1˝/2˙ Cp 0

˜ Cpave °ss ˛ K ˝ ˙ 498

(16.20) (16.21) (16.22)

practical application of pK/tK Models: Multiple dosing Kinetics

( Cpave )ss =

( AUC )t

(16.23)

t

The defnition of the average steady-state plasma concentration based on Equations 16.19–16.23 can be either “the ratio of average rate of input divided by clearance” or “the area under plasma concentration of a maintenance dose at steady state divided by the dosing interval” (Levy, 1974). Equations 16.19–16.23 are used in designing dosing regimens for therapeutic xenobiotics, in adjusting dosing regimens for patients with renal failure, and in therapeutic drug monitoring. The shortcoming of the Equations is that they provide no information about the fuctuation of peak and trough levels within the therapeutic range. The magnitude of the fuctuation, however, can be estimated from DM Vd. 16.2.4 Loading Dose vs Maintenance Dose To reduce the time required to achieve the steady-state fuctuations for certain medications, a loading dose (DL ) is often administered at the start of multiple dosing regimens. This one-time dose should be equal to the maximum amount in the body at steady state and provide an initial concentration equal to ( Cpmax )ss ; in other words, the loading dose is the product of maximum plasma concentration at steady state and the volume of distribution (Perrier and Gibaldi, 1973): DL = ( Cpmax )ss ´ Vd = ( Amax )ss

(16.24)

The maintenance dose (DM ) is then equal to the portion of the loading dose that leaves the body by all routes of elimination at steady state during the frst dosing interval after the administration of DL :

( fel )t

ss

= 1 - e -Kt

(

DM = DL 1 - e -Kt

(16.25)

)

(16.26)

The maintenance dose can also be defned as multiplication of steady-state fuctuation and the volume of distribution. The fuctuation is the difference between the peak and trough levels, i.e.

(

)

(16.27)

DM = ( Cpave )ss ´ ClT ´ t

(16.28)

DM = ( Cpmax )ss - ( Cpmin )ss ´ Vd It can also be estimated from Equation 16.19:

Equation 16.28 indicates that the maintenance dose is the “average rate of elimination at steady state (( Cpave )ss ´ ClT ) in one dosing interval multiplied by the dosing interval.” Thus, in a multiple-dosing regimen, if a loading dose is followed by a maintenance dose at every fxed dosing interval, the steady-state fuctuations can be achieved immediately and maintained if the administration of the maintenance dose continues a fxed-dose schedule (Figure 16.2). Using Equation 16.26, the relationship between the loading dose and maintenance dose is described as: DL =

DM DM = 1 - e -Kt ( f el )t

(16.29) ss

When the half-life is very short in comparison to the dosing interval ( t ˜ T1 2 ) the fraction eliminated from the body in one dosing interval approaches one, that is, ( f el ) Þ 1, representing tss approximately 100% elimination and DL = DM . An example would be a compound with wide effect range and very short half-life, say less than 30 minutes, given every four hours. When the dosing interval is equal to half-life ( t = T1/2 ), the fraction of loading dose eliminated in the frst dosing interval is half of the loading dose, that is DL = 2DM , or DM = DL / 2. When t ˜ T1/2 , ( f el ) = 1 - e » 0 @ 1 - 1 @ 0 , the denominator of Equation 16.29 approaches zero, and tss the relationship is undefned. An example is giving a compound with a half-life of 7 days once a day. A practical approach is to set the loading dose equal to the average amount at steady state and defne the relationship between DL and DM according to the equation: DL = (Aave )ss =

DM Kt

(16.30) 499

16.2 Kinetics of Multiple intravenous Bolus injections – one-coMpartMent Model

Figure 16.2 Profle of fuctuation of plasma concentration following administration of a loading dose and seven maintenance doses. Illustration of the signifcance of giving a loading dose to achieve immediate steady-state peak and trough levels, provides the required accumulation of the therapeutic agent in the body, and to get around the delay in achieving the steady-state levels by initiating the therapy with a maintenance dose; the amount of succeeding fxed maintenance doses is equal to the amount of the loading dose that eliminated from the body by all routes of elimination in one dosing interval; the initial plasma concentration of the loading dose is equal to the maximum plasma level at steady state, and the initial plasma concentration of the maintenance dose is the difference between the peak and trough levels at steady state. 16.2.5 Extent of Accumulation of Xenobiotics Multiple Dosing in the Body The degree of accumulation in the body at steady state is defned by a number known as the accumulation ratio, which is the ratio of the steady-state peak level to that of the frst dose, or the ratio of the steady-state trough level to the trough level of the frst dose (Equation 16.31). The frst dose here is referred to as the maintenance dose according to the profle of Figure 16.1. The ratio indicates the extent of accumulation in the body at steady-state peak or trough level with respect to the corresponding concentration of the frst maintenance dose of a dosing regimen. It also corresponds to the ratio of DL/DM, which is the reciprocal of the fraction of dose eliminated from the body by all routes of elimination during a dosing interval at steady state. For example, if the ratio is equal to 5, it is indicative of fve times more accumulation at steady state with respect to the frst dose, or ( Cpmax )ss = 5 ´ ( Cpmax )1 or ( Cpmin )ss = 5 ´ ( Cpmin )1 . R=

( Cpmax )ss ( Cpmax )ss ( Cpmin )ss = = ( Cpmax )1 ( Cp0 )D ( Cpmin )1 M

=

=

DL DM

1 1 = 1 - e -Kt ( f el )t

(16.31)

The extent of accumulation can also be determined by using the area under the plasma concentration–time curve (Colburn, 1983), where R = AUC0¥ ¸ AUC0t and R = AUC0t ¸ AUC0t .

(

) ( 1

)

1

(

) ( ss

)

1

16.2.6 Estimation of Plasma Concentration After the Last Dose After the last dose, whether the steady state is achieved or not, the plasma concentration declines exponentially until the total accumulated amount is completely removed from the body. The 500

practical application of pK/tK Models: Multiple dosing Kinetics

knowledge of plasma concentration after the last dose of the frst compound may help to determine when the second treatment with a different compound should be initiated. The following equations can be used to estimate the plasma concentration after the last dose of a multiple dosing regimen of a compound that follows the one-compartment model (Figure 16.3). Cpt’ = Cpmax e -Kt¢

or,

Cpt¢ = Cpmin e -K (t¢-t)

(16.32)

16.2.7 Design of a Dosing Regimen The purpose of designing a dosing regimen is to determine the loading dose, maintenance dose, and dosing interval for a given compound based on the PK/TK information. The following are a few of the practical approaches: 16.2.7.1 Dosing Regimen Based on a Target Concentration The target concentration is set equal to the average steady-state plasma level; a dosing interval is selected based on the half-life of the compound and the convenience of administration; the maintenance dose is estimated as follows: DM = ( Cpave )ss ´ Vd ´ t ´ K

(16.33)

If the dosing interval is set equal to half-life, DL = 2DM . If the dosing interval is different from the half-life, the loading dose is determined according to equation 16.29. For compounds with very a short half-life, the dosing interval is usually set equal to four hours for convenience of administration, and DL = DM.

Figure 16.3 Depiction of plasma concentrations decline after the last dose of a multiple dosing regimen before or after achieving the steady-state levels; the estimation of the plasma level at time t¢ after the last dose is based on the selected initial concentration and whether the time, t¢, is within the dosing interval or longer than the last dosing interval; if t¢ is within the dosing interval, the initial plasma level of the monoexponential decline, characteristic of the one-compartment model, is the last peak level; if t¢ is longer than the last dosing interval, the initial concentration of the decline can be the peak or the trough level; if the trough level is selected for the estimation of the Cpt¢ , the time of the exponential decline is t¢ - t. 501

16.3 Kinetics of Multiple oral dose adMinistration

It would be advisable to verify whether the peak and trough levels of dosing regimen are within the acceptable range. This is particularly important for therapeutic agents with a narrow therapeutic range. A quick check is: D

( Cpmax )ss = VdL

(16.34) D

( Cpmin )ss = ( Cpmax )ss - VdM

(16.35)

16.2.7.2 Dosing Regimen Based on Steady-State Peak and Trough Levels The fuctuation of plasma concentrations at steady state is set equal to the initial plasma concentration of the maintenance dose (Boroujerdi, 2002):

(

Fluctuation = ( Cpmax )ss - ( Cpmin )ss = Cp 0

)

DM

(16.36)

Therefore, DM = ( Cpmax - Cpmin )ss ´ Vd

(16.37)

A trough level equal to half of the peak level indicates that the dosing interval is equal to the halflife. The loading dose is then equal to twice of the maintenance dose. If the trough level is not half of the peak level, the dosing interval can be estimated as: t=

( Cpmax )ss 2.303 log K ( Cpmin )ss

(16.38)

16.2.7.3 Dosing Regimen Based on Minimum Steady-State Plasma Concentration When the minimum steady-state plasma concentration is known, the dosing interval is set preferentially equal to the half-life, and the maintenance dose is determined using Equation 16.12: DM =

(Cpmin )ss ´ Vd ´ (1 - e -Kt ) e -Kt

(16.39)

Depending upon the dosing interval, the loading dose is calculated as described earlier (Boroujerdi, 2002). 16.3 KINETICS OF MULTIPLE ORAL DOSE ADMINISTRATION The principle of kinetics of multiple oral administration is the same as described for multiple intravenous injections (section 16.2). The main difference is the infuence of absorption process on the plasma concentration of administered compound and the related geometric series. Like the elimination process, absorption of multiple dosing can also be defned in terms of a geometric series. For multiple dosing kinetics of oral administration there are two biological processes of absorption and elimination, thus, two geometric series and two solutions, one for each process. The solutions of the geometric series before achieving the steady state are: Elimination : ( Sn )K =

1 - e -nKt 1 - e -Kt

(16.40)

Absorption : ( Sn )k =

1 - e -nkat 1 - e -kat

(16.41)

a

At steady state, Equations 16.40 and 16.41 are modifed to:

( Elimination )ss : ( Sn )K =

1 1 - e -Kt

(16.42)

( Absorption )ss : ( Sn )k

1 1 - e -kat

(16.43)

a

=

The profle of plasma concentration versus dosing interval following multiple oral dosing is presented in Figure 16.4. The only noticeable difference is the bell-shaped characteristic of oral doses.

502

practical application of pK/tK Models: Multiple dosing Kinetics

Figure 16.4 Illustration of the concentration–time profle of the multiple oral dosing regimen after the administration of a loading dose to attain the steady-state levels immediately, followed by four maintenance doses, given in fxed dosing intervals, to maintain the steady-state peak and trough levels; this regimen is compared with the gradual accumulation of the compound in the body toward the steady-state fuctuation by giving fve maintenance doses; the signifcant differences of the oral multiple dosing profle with that of the intravenous are 1) the skewed bell-shape curve of the concentration–time profle due to the absorption process, 2) the absence of the sharp and immediate peak concentration, and 3) the probability of a lower accumulation in the body due to the bioavailability considerations of the oral dose, if the same dose is administered orally and intravenously; the half-life and the overall elimination rate constant remain the same for both routes of administration for a given compound in linear PK/TK. 16.3.1 Peak, Trough, and Average Plasma Concentrations Before and After Achieving Steady-State Levels Multiplying the equation of the frst dose by the solutions of the geometric series yields the trough levels of multiple oral administrations: Equation of trough level of the frst dose is: ˜ (16.44) (Cpmin )1 = Cp 0 e -Kt - e -kat

(

)

Equation of trough level after n dose before achieving steady state is (Gibaldi and Perrier, 1982): th

˜ éæ 1 - e -nKt ö -Kt æ 1 - e -nkat ö -kat ù e ú ÷e - ç -k at ÷ êè úû è 1- e ø ø ë

( Cpmin )n = Cp0 êç 1 - e -Kt

(16.45)

The plasma concentration at any time before achieving the steady state is: ˜ éæ 1 - e -nKt ö -Kt æ 1 - e -nkat ÷e - ç -k at êëè è 1- e ø

( Cpmin )n = Cp0 êç 1 - e -Kt

ö -kat ù ÷e ú úû ø

(16.46)

where n is the number of doses; t is the dosing interval and t is the time after the administration of nth dose. Equations 16.47 and 16.48 defne the trough level and plasma concentration at any time during the steady state: ˜ é e -Kt

e -kat ù

( Cpmin )ss = Cp0 ê 1 - e -Kt - 1 - e -k t ú ë

a

û

(16.47) 503

16.3 Kinetics of Multiple oral dose adMinistration

˜ é e -Kt

e -kat ù

( Cpmin )ss = Cp0 ê 1 - e -Kt - 1 - e -k t ú a

ë

û

(16.48)

When the absorption process is fast, and the dosing interval is equal or greater than 7 ( T1 2 )k , the a exponential term representing the absorption approaches zero (e -kat Þ 0), and the trough level will only be a function of elimination process (Equations 16.49 and 16.50). In other words, the maintenance dose is given when the absorption process is complete. In clinical practice, it can be assumed that the amount of dose left at the site of absorption is insignifcant after 4–5 half-lives of absorption. ˜ æ 1 - e -nKt ö -Kt ÷e è ø

( Cpmin )n = Cp0 ç 1 - e -Kt

(16.49)

˜ æ e -Kt ö

( Cpmin )ss = Cp0 ç 1 - e -Kt ÷ è

(16.50)

ø

Estimation of the peak levels depends on the value of Tmax . As the Tmax of steady-state levels remains constant and reproducible, it is considered a more consistent value than theTmax before achieving the steady state. By taking the frst derivative of Equation 16.48, the equation of Tmax for multiple oral dosing at steady state can be developed as follows: The frst derivative of equation 16.48 is d ( Cpt )ss dt

˜ é -Ke -Kt k e -kat ù + a -kat ú = Cp 0 ê -Kt 1- e û ë1 - e

(16.51)

Setting equation 16.51 equal to zero and solving for Tmax generates Equation 16.53 for calculation purposes:

( (

k a 1 - e -Kt e -KTmax = -k aTmax e K 1 - e -kat

( Tmax )ss =

) )

( (

(16.52)

æ k a 1 - e -Kt 2.303 log ç ç K 1 - e - k at ka - K è

) ö÷ ) ÷ø

(16.53)

Therefore, multiplying the equation of maximum plasma concentration after the frst dose by equation 16.42 and substituting ( Tmax )ss for t yields Equation 16.54 for the maximum plasma concentration at steady state (Gibaldi and Perrier, 1982): FD æ e -K (Tmax )ss ö -Kt ÷ è 1- e ø

( Cpmax )ss = Vd ç

(16.54)

The average plasma concentration at steady state for multiple oral dosing is similar to Equations 16.19–16.23, that is, FD

FD

( Cpave )ss = Cl ´ t = K ´ Vd ´ t = t

1.44 ´ T1/2 ´ FD ( AUC )t = Vd ´ t t

(16.55)

16.3.2 Extent of Accumulation in Multiple Oral Dosing The extent of accumulation at steady state is determined as described earlier by equation 16.31 for intravenous administration: When t < 7 ( T1 2 )k : a

R=

( Cpmin )ss 1 = -Kt Cp ( min )1 (1 - e )(1 - e -k t ) a

When t > 7 ( T1 2 )k : R = a

504

(

1 1 - e -Kt

)

(16.56) (16.57)

practical application of pK/tK Models: Multiple dosing Kinetics

16.3.3 Oral Administration of Loading Dose, Maintenance Dose and Designing a Dosing Regimen The procedure for designing a regimen for multiple oral dosing is the same as described for multiple intravenous injections, except for including the bioavailability factor, F, in the calculation of DM or DL ( Slattery et al., 1980). For example, Equation 16.33 changes to:

( Cpave )ss ´ Vd ´ t ´ K

(16.58) F The relationship between the loading dose and the maintenance dose, assuming F values are equal, can be described as: DM =

DL =

(

DM 1- e 1 - e -kat -Kt

)(

)

( when t < 7 (T ) ) 12

ka

(16.59)

Or, DL =

DM 1 - e -Kt

( when t > 7 (T ) ) 12

ka

(16.60)

The time required for reaching any fraction of steady state during multiple oral dosing is calculated by Equations 16.15–16.16. 16.4 EFFECT OF CHANGING DOSE, DOSING INTERVAL, AND HALF-LIFE ON THE ACCUMULATION IN THE BODY AND FLUCTUATION OF PLASMA CONCENTRATION The magnitude of the fuctuation of plasma concentration is the difference between the peak and trough levels; for intravenous multiple administration, it is equal to the initial plasma concentration of a single maintenance dose. The measure of accumulation in the body is the average amount at steady-state levels. For a given dosing regimen, if the dosing interval is kept constant and dose is increased, the following parameters increase: the peak and trough levels before and after achieving the steady-state levels, the average plasma concentration at steady state, the accumulation in the body, and the fuctuation of plasma concentration. Decreasing the dose would reduce the steady-state levels, plasma fuctuation, and drug accumulation in the body (Figure 16.5). If the dose is kept constant and dosing interval is altered, increasing the dosing interval would increase the fuctuation, lower the average steady-state level, and reduce the accumulation in the body. If the dosing interval is shortened, a reduced dosing interval would increase the accumulation, reduce the fuctuation, and increase the steady-state levels (Figure 16.6). In disease states, such as renal impairment or hepatic failure, the half-life is longer than the normal for a given compound. Recommending the dose and dosing interval that are designed for normal renal or hepatic function would increase the steady-state levels and accumulation in the body, and adjustment of dosage regimen is required (Figure 16.7). 16.5 EFFECT OF IRREGULAR DOSING INTERVAL ON PLASMA CONCENTRATIONS OF MULTIPLE DOSING REGIMEN The irregular change in dosing interval infuences the fuctuation, accumulation, and steadystate levels. The issue may not be crucial for therapeutic agents with a very wide therapeutic range. However, it should be considered important for compounds with low therapeutic index and narrow therapeutic range. Figure 16.8 depicts the plasma concentration fuctuations when a compound with half-life of six hours and narrow therapeutic range is administered on a regular dosing interval of 6-12-6-12 (q.i.d. around the clock or q6h) versus the plasma concentration of the same dose given on an irregular dosing interval 8-12-4-8 (so called 4 tablets a day). The latter schedule would create inconsistency in accumulation in the body and reduce the plasma levels below the minimum effective concentration. However, when the therapeutic range is wide, as shown in Figure 16.8, the inconsistency may not be of any concern, since the lowest concentration remains within the therapeutic range and both schedules are considered reasonable. 16.6 MULTIPLE DOSING KINETICS – TWO-COMPARTMENT MODEL The equations of multiple dosing kinetics of a compound that follows the two-compartment model are also based on solving the geometric series for each exponential term of the single dose 505

16.6 Multiple dosing Kinetics – two-coMpartMent Model

Figure 16.5 Illustration of the infuence of changing the dose on fuctuation and steady-state levels of a multiple intravenous dosing regimen when the dosing interval is kept the same; giving higher doses translates into higher steady-state levels, larger fuctuations, and a heightened probability of toxic response, especially in compounds with narrow therapeutic range; lower dose strength transforms the profle into lower steady-state levels, lower fuctuation, and the probability of forming the steady-state levels below the minimum effective concentration. equation. The multiple dosing relationships are then derived by multiplying each exponential term of the single dose equation by the multiple dosing functions, which are the solution of geometric series. The fast distributive phase and slow disposition phase, characteristics of the plasma concentration–time profle of the two-compartment model, repeats itself after giving each dose of the multiple-dosing regimen (Figure 16.9). The hybrid rate constants of a and bremain essentially the same as the frst dose; however, the coeffcients a and b change after each dose before achieving the steady-state levels. During the steady state, aand bremain relatively the same if the dosing interval is kept constant. 16.6.1 Peak, Trough, and Average Plasma Concentrations Before and After Achieving the Steady-State Levels for Two-Compartment Model Xenobiotics Given Intravenously Solutions of the geometric series of a and b exponentials for concentrations before and during steady-state levels are similar to the one-compartment model, that is, before achieving the steady state, the geometric solutions are:

(

Sn e -at

)

n

=

1 - e -nat 1 - e -at

( )

and Sn e -bt

n

=

1 - e -nbt 1 - e -bt

(16.61)

For steady state to be reached without giving a loading dose, several maintenance doses should be administered. This means that n in the numerator of Equation 16.61 increases, e -nat Þ 0 and e -nbt Þ 0, and thus the numerators of both exponential terms approach one and the steady-state solutions of the geometric series are:

(

Sn e -at

)

ss

=

1 1 - e -at

( )

and Sn e -bt

ss

=

1 1 - e -bt

(16.62)

The peak and trough levels of the frst dose are:

( Cpmax )1 = Cp0 = a + b 506

(16.63)

practical application of pK/tK Models: Multiple dosing Kinetics

Figure 16.6 Illustration of the infuence of changing dosing interval on fuctuation and steadystate levels of plasma concentrations when the dose is kept constant; the shorter the dosing interval, using half-life as a reference number, translates into higher steady-state levels, smaller fuctuation, and enhanced probability of toxicity in compounds with a narrow therapeutic range; and the longer dosing interval lowers the steady-state levels but increases the fuctuation of the plasma levels and the probability of having the fuctuation under the minimum effective concentration.

( Cpmin )1 = ae -at + be -bt

(16.64)

The equations of peak and trough levels before achieving steady state can be developed by multiplying Equations 16.63 and 16.6 by Equation 16.61: æ 1 - e -nat ö

æ 1 - e -nbt ö bt ÷ è 1- e ø

( Cpmax )n = a ç 1 - e -nat ÷ + b ç è

ø

æ 1 - e -nat at è 1- e

( Cpmin )n = a ç

æ 1 - e -nbt ö -at ÷e + bç -bt è 1- e ø

ö -bt ÷e ø

(16.65) (16.66)

It is important to recognize that when the dosing interval is equal or greater than seven half-life of a (i.e., t ³ 7 ( T1 2 )a ), e -at Þ 0 , and e -nat Þ 0 and Equations 16.65 and 16.66 will change to: æ 1 - e -nbt ö (Cpmax )n = a + b ç -bt ÷ è 1- e ø

(16.67)

æ 1 - e -nbt ö -bt e (Cpmin )n = b ç -bt ÷ è 1- e ø

(16.68)

To calculate plasma concentration at any time between the peak and trough levels of any dose before achieving steady state, the following relationship can be applied: æ 1 - e -nat at è 1- e

( Cpt )n = a ç

æ 1 - e -nbt ö -at ÷e + bç -bt è 1- e ø

ö -bt ÷e ø

(16.69)

When t ³ 7 ( T1 2 )a , Equation 16.69 is modifed to: 507

16.6 Multiple dosing Kinetics – two-coMpartMent Model

Figure 16.7 Illustration of the infuence of disease states, like renal and/or hepatic insuffciencies, that affect the overall elimination rate constant, or half-life of a compound; giving a dosing regimen without the required adjustment for the change in half-life or K, elevates the steady-state levels and increases the probability of having a toxic response, however the fuctuation remains somewhat similar to the normal fuctuation; the adjustment is with changing the dose, or dosing interval, or both; the ideal situation would be to select a regimen with lower fuctuation within the therapeutic range.

Figure 16.8 Illustration of the plasma-level fuctuations when the medication is taken on a regular fxed dosing interval around-the-clock versus irregular dosing regimen taken on an arbitrary schedule with periods of plasma falling below the minimum effective concentration. 508

practical application of pK/tK Models: Multiple dosing Kinetics

Figure 16.9 Depiction of plasma concentration–time profle of multiple intravenous dosing regimen of a compound that follows the two-compartment open model; the regimen is without giving a loading dose, and the profle refects the gradual accumulation of the therapeutic agent in the body based on the administration of fxed maintenance dose and fxed dosing interval; the time course of each injection refects the typical two-compartment model characteristics of having distributive and disposition phases; the coeffcient of the exponential terms, or the y-intercept of the extrapolated and residual lines, vary with each injection until the steady state is achieved; during steady state, the y-intercepts of the extrapolated and residual lines remain constant and different from the single-dose administration; the hybrid rate constants of the model, in particular the disposition rate constant, b, remain constant throughout the therapy. æ 1 - e -nbt ö -bt e (Cpt )n = b ç -bt ÷ è 1- e ø

(16.70)

A similar approach is used to develop equations of peak and trough levels after achieving the steady-state fuctuation, that is, a

b

ae -at

be -bt

( Cpmax )ss = 1 - e -at + 1 - ebt ( Cpmin )ss = 1 - e -at + 1 - e -bt

(16.71) (16.72)

When t ³ 7 ( T1 2 )a , Equations 16.71 and 16.72 convert to: b

( Cpmax )ss = a + 1 - ebt be -bt

( Cpmin )ss = 1 - e -bt

(16.73) (16.74)

Plasma concentrations between the peak and trough levels of any dose during steady state can be estimated by ae -at

be -bt

( Cpt )ss = 1 - e -at + 1 - e -bt

(16.75)

Whent ³ ( 7T1 2 )a , 509

16.6 Multiple dosing Kinetics – two-coMpartMent Model

(Cpt )ss =

be -bt 1 - e -bt

(16.76)

The average steady-state plasma concentration is also calculated similarly to that for the one-compartment model, with the same defnition (Levy, 1974; Gibaldi and Levy, 1976; Van Rossum and Tomey, 1968): D T ´t

(16.77)

AUC t

(16.78)

( Cpave )ss = Cl ( Cpave )ss =

D

( Cpave )ss = b ´ Vd

area

( Cpave )ss = k

10

´t

D ´ V1 ´ t

(16.80)

Cp ˜ 10 ´ t

(16.81)

( Cpave )ss = k ( Cpave )ss =

(16.79)

1.44 ´ ( T1/2 )biol ´ D Vdarea ´ t

(16.82)

16.6.2 Estimation of the Time Required to Achieve Steady-State Plasma Levels of Two-Compartment Model Xenobiotics Given Intravenously Analogous to Equations 16.14–16.16, the time required to attain steady-state fuctuations is estimated in terms of nt, that is, the fraction of steady-state levels is defned as f ss = 1 - e -nbt

(16.83)

nt = -3.3 ( T1 2 )b log ( 1 - f ss )

(16.84)

According to Equation 16.83, the time required to achieve any fraction of steady state is dependent solely on the biological half-life (Boroujerdi, 2002): To achieve percent of steady state 50%

Fraction

Required time in terms of half-life

f ss = 0.5

nt = 1( T1 2 )b

75%

f ss = 0.75

nt = 2 ( T1 2 )b

90%

f ss = 0.90

nt = 3.3 ( T1 2 )b

95%

f ss = 0.95

nt = 4.3 ( T1 2 )b

99%

f ss = 0.99

nt = 6.6 ( T1 2 )b

16.6.3 Estimation of Fraction of Steady State, Accumulation Index, and Relationship Between Loading Dose vs Maintenance Dose Equation 16.82 represents the fraction of steady state and can be used to estimate the trough levels before achieving the steady state, without relying on the values of a and a and solely based on the steady-state trough levels. The accumulation index and the relationship between the loading dose vs maintenance dose are similar to equations of the one-compartment model. The difference is replacing K with the disposition rate constant b. R=

510

( Cpmin )ss ( Cpmin )1

=

1 1 - e -bt

(16.85)

practical application of pK/tK Models: Multiple dosing Kinetics

DL =

DM = 1 - e -bt

DM 1 - (1 / 2)

(T1/2 )biol t

(16.86)

Combination of loading dose and maintenance dose provides immediate steady-state levels (Figure 16.10). Equation 16.86 indicates that for compounds with very short biological half-life no loading dose is necessary, that is, e -bt Þ 0 andDL = DM . By setting the dosing interval equal to biological half-life, the loading dose will be twice of the maintenance dose, t = ( T1/2 )b Þ DL = 2DM . For compounds with a very long biological half-life e -bt Þ e 0 Þ 1 the denominator of the the equation 16.86 approaches zero, making the relationship undefned. The loading dose under this condition is set equal to the average amount in the body at steady state, that is, DL =

DM bt

(16.87)

16.6.4 Evaluation of Plasma Level after the Last Dose After the last dose is given, regardless of whether the steady-state levels are achieved, the plasma concentration declines bi-exponentially according to the two-compartment model. When t¢ < t, that is when the selected time point after the last dose is less than the dosing interval, Equations 16.69–16.70 or 16.75–16.76 are used to estimate plasma concentration before or after achieving the steady state, respectively. However, when t¢ > t the trough level can be used as the initial concentration of a simple monoexponential decline with the slope of -b / 2.303: -b t¢-t Cpt¢ = Cpmin e ( )

log Cpt¢ = log Cpmin -

b ( t¢ - t ) 2.303

(16.88) (16.89)

Figure 16.10 Depiction of plasma concentration–time profle of multiple intravenous dosing administration of a therapeutic agent that follows the two-compartment model; the initial dose of the regimen is a loading dose that is followed by eight maintenance doses; the combination of the loading dose and the maintenance doses produces a steady-state fuctuation with constant peak and trough levels during the treatment; in contrast to Figure 16.9, the slopes and y-intercepts of the extrapolated and residual lines remain constant and predictable when the steady state is maintained. 511

16.6 Multiple dosing Kinetics – two-coMpartMent Model

Using Cpmax as the initial plasma concentration without taking into consideration the distributive phase of the decline, for certain drugs with distinct and rapid distributive phase, would add error to the calculated value of Cpt¢ . 16.6.5 The Concept of Half-Life in Multiple Dosing Kinetics of Multicompartmental Models In the design of a dosing regimen, setting the dosing interval equal to the half-life necessitates an accurate estimate of this constant. An accurate estimate of the constant becomes more important when the pharmacokinetics of the compound follows the multicompartment model. In the onecompartment model, the amount in the systemic circulation approximates the amount in the body, whereas in multicompartmental model, the half-life estimated from the central compartment concentration may not refect the true elimination half-life of the amount in the body. One approach in dealing with this dilemma is identifed as the ‘operational multiple dosing half-life’ (Sahin and Benet, 2008), which requires weighing the individual half-life by the fractional area under plasma concentration curve, that is, 1

( T1 2 )MD

=

n

f AUC , i

i=1

12

å (T

(16.90)

)i

where ( T1 2 ) MD is the multiple dosing half-life, f AUC , i is the fraction of the area under the curve corresponding to each half-life estimated as f AUC , i =

Li l i

(16.91)

n

åL

i

li

i=1

where l i andLi represent the exponential terms of the multicompartment model and the coeffcients of the exponential terms of the systemic concentration-time curve; and nis the number of exponential terms. Therefore, the multiple dosing half-life for the amount in the body, ( T1 2 ) Amt , can be characterized as: n

( T1 2 )Amt = å f AUC ,i ´ ( T1 2 )i

(16.92)

i=1

A different approach for setting a suitable dosing interval is the use of the sum of mean residence time of the central compartment and the mean residence time of the absorption site multiplied by a factor (Wagner, 1987). For the two-compartment model with IV bolus administration, the factor was determined to be 0.75 (Wagner, 1987). The value of dosing interval estimated by this approach will provide a ratio of peak/trough levels equal to 2, and the variation of steady-state levels remain within ±33%. For the intravenous administration, the MRT of the absorption site is equal to zero. The half-life of the amount in the body based on this approach is estimated as 0.75 ´ MRT . Whereas the half-life calculated from the mean residence time of the non-compartmental model is:

( T1 2 )Amt = 0.693 ´ MRT

(16.93)

ss ( T1 2 )Amt = 0.693 ´ Vd Cl

(16.94)

T

Another approach in estimating the half-life of the amount in the body for the design of a dosing regimen is the concept of “effective half-life” (EHL) (Kwan et al., 1984; Boxenbaum and Battle, 1995), which is based on the accumulation ratio, R , of the compound in the body estimated as R=

( AUCss )0®t

( AUCsingle dose )0®t ( AUCsingle dose )0®¥ R= ( AUCsingle dose )0®t

512

or

(16.95)

(16.96)

practical application of pK/tK Models: Multiple dosing Kinetics

where ( AUCss )0®t is the area under plasma concentration curve at steady state during a dosing interval; ( AUCsingle dose )

0®t

and ( AUCsingle dose )0®¥ are the area under a single dose during a time

equal to the dosing interval t and from time zero to infnity, respectively. The effective half-life is then estimated as a function of dosing interval.

( T1 2 )effective = ln é0.R693- 1´ tRù ë(

)

(16.97)

û

16.7 MULTIPLE INTRAVENOUS INFUSIONS Generating the multiple dosing equations of a compound that is administered by intravenous infusion is the same as described for the bolus or frst-order absorption. It involves the multiplications of the single dose equation by multiple dosing functions. The single dose equation as discussed in Chapter 14 (Equation 14.8) is k0 1 - e -Kt K ´ Vd

(

Cpt =

)

(16.98)

The multiplication yields the following relationship before achieving the steady state: æ 1 - e -nKt ö ÷ è ø

k

( Cp )n = K ´0Vd (1 - e -Kt ) ç 1 - e -Kt

(16.99)

where (t) is the time of infusion, n is the number of infusions and t is the dosing interval. The plasma level at time (t), i.e., the concentration at the end of nth infusion, is the peak level before the next dose and its trough level is the concentration at time ( t - t ) after the end of nth infusion, that is, æ 1 - e -nKt ö -K(t-t) ÷e è ø

k

( Cp )n = K ´0Vd (1 - e -Kt ) ç 1 - e -Kt

(16.100)

As the number of infusions increase, n - and e -nKt Þ 0, a steady-state fuctuation can be achieved with a peak level that corresponds to the concentration at the end of one infusion and a trough level that corresponds to the plasma level before the start of the next infusion. The related equations after achieving the steady state are: Single infusion steady-state equation (Chapter 14, Equation 14.6) is Cpss =

k0 ClT

(16.101)

The multiple dosing relationships for peak and trough levels at steady state are Peak : ( Cptinfusion ) = ss

Cpss k0 = 1 - e -Kt KVd 1 - e -Kt

(16.102)

Cpss e -Kt¢ 1 - e -Kt

(16.103)

(

Trough : ( Cpt¢ )ss =

)

The volume of distribution of the compound administered by multiple dosing infusion is estimated by the following equation (Sawchuk et al., 1977): Vd =

(

(

k0 1 - e -Kt

)

K Cpmax - Cppredose e -Kt

)

where (Cppredose ) is the concentration before the start of the succeeding infusion. 16.8 APPLICATIONS AND CASE STUDIES The applications and case studies of Chapter 16 are posted in Addendum II – Part 7.

513

16.8 applications and case studies

REFERENCES Boroujerdi, M. 2002. Pharmacokinetics: Principles and Applications. New York: McGraw Hill, Medical Publishing Division. Boxenbaum, H. G., Battle, M. 1995. Effective half-life in clinical pharmacology. J Clin Pharmacol 35(8): 763–6. Buell, J., Jelliffe, R., Kalaba, R., Sridhar, R. 1969. Modern control theory and optimal drug regimens, I: The plateau effect. Math Biosci 5(3–4): 285–96. Colburn, W. A. 1983. Estimating the accumulation of drugs. J Pharm Sci 72(7): 833–4. Gibaldi, M., Perrier, D. 1982. Pharmacokinetics, Second Edition. New York: Marcel Dekker, Inc. Krüger-Thiemer, E. 1966. Formal theory of drug dosage regimens, I. J Theo Biol 13: 212–35. Krüger-Thiemer, E. 1969. Formal theory of drug dosage regimens, II. The exact plateau effect. J Theo Biol 23(2): 169–90. Krüger-Thiemer, E., Bünger, P. 1965. The role of the therapeutic regimen, Part II. Chemotherapia 10: 129–44. Kwan, K. C., Bohidar, N. R., Hwang, S. S. 1984. Estimation of an effective half-life. In Pharmacokinetics: A Modern View, eds. L. Z. Benet, G. Levy, B. Ferraiolo, 147–62. New York: Plenum. Levy, G. 1974. Pharmacokinetic control and clinical interpretation of steady-state blood levels of drugs. Clin Pharmacol Ther 16(1part2): 130–4. Gibaldi, M., Levy, G. 1976. Pharmacokinetics in clinical practice: Applications. JAMA 235(18): 1987–92 Perrier, D., Gibaldi, M. 1973. Relationship between plasma or serum drug concentration and amount of drug in the body at steady state upon multiple dosing. J Pharmacokinet Biopharm 1(1): 17–9. Sawchuck, R. J., Zaske, D. E., Cipolle, R. J., Wargin, W. A. 1977. Kinetic model for gentamicin dosing with the use of individual patient parameters. Clin Pharmacol Ther 21: M362–5. Sahin, S., Benet, L. Z. 2008. The operational multiple dosing half-life: A key to defning drug accumulation in patients and to designing extended release dosage forms. Pharm Res 25(12): 2869–77. Slattery, J. T., Gibaldi, M., Koup, J. R. 1980. Prediction of maintenance dose required to attain a desired drug concentration at steady state from a single determination of concentration after an initial dose. Clin Pharmacokinet 5(4): 377–85. Van Rossum, J. M., Tomey, A. H. M. 1968. Rate of accumulation and plateau plasma concentration of drugs after chronic medication. J Pharm Pharmacol 20(5): 390–1. Wagner, J. G. 1975. Clinical Pharmacokinetics. Hamilton: Drug Intelligence Publications Inc. Wagner, J. G. 1987. Dosage intervals based on mean residence times. J Pharm Sci 76(1): 35–8. Wagner, J. G., Northam, J. I., Always, C. D., Carpenter, O. S. 1965. Blood levels of drug at equilibrium state after multiple dosing. Nature 207(5003): 1301–2.

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17 Biopharmaceutics Provisions, Classifcations and Mechanistic Models 17.1 INTRODUCTION The term “biopharmaceutics” is defned as “the study of the relationship between some of the physical and chemical properties of the drug and its dosage forms and the biological effects observed following administration of the drug in its various dosage forms” (Wagner, 1961). The absorption from extravascular routes of administration, in particular oral absorption, depends not only on the complex physiological factors of the routes but also on the physicochemical properties of xenobiotics and characteristics of dosage form used for therapeutic agents. These properties infuence and modify the pharmacokinetic/toxicokinetic (PK/TK) variables. The best example of PK/TK variables is the absolute bioavailability of a compound given via an extravascular route of administration. In the case of oral absorption, which is the complex extravascular route of administration (see Chapter 3, Section 3.3), in addition to the physiological factors, metabolic processes, infux and effux transport proteins, and variability of the pH, and other mechanisms of the GI tract that can impact the bioavailability of a compound, this PK/TK parameter can also be altered and engineered signifcantly by selecting compounds with apt physicochemical properties and/or well-chosen formulation ingredients for suitable dosage form. The topics of this chapter deal with the several fundamental physicochemical characteristics of xenobiotics that infuence the absorption, bioavailability, and the magnitude of physiological outcome. Most, if not all, of these factors and characteristics deal with the solubility in aqueous solution like the aqueous environment of a route of administration. For example, physiological intestinal fuid in the case of oral absorption. For all routes of administration, the aqueous solubility is essential and considered an indicator of the solubility at the site of absorption and availability of the molecules for permeation. The issue of aqueous drug solubility and permeability are particularly important in drug discovery and development, and the classifcations discussed in this chapter will be related to these topics. The two terminologies of solubility and dissolution, though connected, are somewhat different in their meanings. Solubility is an intrinsic property of a compound, and it is infuenced only by chemical modifcation of the molecules, examples are a different salt formation or a drug that has been chemically modifed to a more soluble prodrug. Solubility can be infuenced by the pH of the aqueous media (also see Chapter 3, Section 3.3.2.1), temperature, salt formation, polymorph type, solid-state presence (like crystalline vs amorphous), and composition of the aqueous media (Ni et al., 2002). Whereas dissolution is considered an extrinsic property of a compound that is infuenced by surface properties, complexation, or particle size modifcations. Disintegration and dissolution are discussed separately in this chapter. 17.2 INFLUENCE OF PHYSICOCHEMICAL PROPERTIES ON ABSORPTION OF XENOBIOTICS 17.2.1 Polymorphism The ability of a compound to crystallize to more than one crystalline form is known as polymorphism. The different crystalline forms, or polymorphs, occur due to the different space lattice arrangements of regularly packed atoms or molecules or ions with three-dimensional repeating patterns. The number of polymorphs for a compound poses the question of which of the forms would provide better solubility, dissolution, absorption, and thus better bioavailability. For example, sulfonamides have 30 and steroid hormones have 42 polymorphs. The study of polymorphic behavior of a compound is an important part of the preformulation of therapeutic dosage forms, and it includes both the active ingredients and excipients. A challenging part of the study is the phase transition, which is the process of transition from one polymorph into another during storage or manufacturing processes (Aguiar et al., 1967; 1969; Singhal et al., 2004). There are a few terminologies that are related to polymorphism and that deal with enhancing the solubility of polymorphs: ◾ If solvent molecules are incorporated in the lattice of a polymorph, it is known as solvates; the solvates also form their own polymorphs, which are known as pseudopolymorphs. ◾ If the solvent in the lattice is water, the polymorphs are known as hydrates. ◾ If the regularly packed atoms and molecules of crystals are changed into randomly arranged molecules, for example by lyophilizing, it changes to an amorphous state. DOI: 10.1201/9781003260660-17

515

17.2 INFLUENCE OF PHYSICOCHEMICAL PROPERTIES ON ABSORPTION OF XENOBIOTICS

In general, the amorphous compounds are more soluble than their corresponding crystalline form; the hydrates have a lower dissolution rate than the anhydrous crystalline form. The polymorphs or pseudopolymorphs exist in two forms: the ones with the lowest energy, lowest solubility, and highest melting point are considered stable polymorphs; and those with lower melting point, higher solubility, and higher energy are known as metastable. During dosage form development, the lowest energy polymorph, which is soluble and stable, is selected for the drug dosage form development. The selection of a stable polymorph is usually carefully investigated to rule out the metastable polymorphs before the fnal formulation is decided. The solubility and dissolution behavior of polymorphs is an important factor in the absorption and bioavailability of an orally administered drug. The absorption of drugs with low solubility rate is considered solubilitylimited absorption. It is common to choose a stable amorphous form of a compound to achieve the required solubility/dissolution and bioavailability. However, because of greater molecular mobility, the stability of amorphous is generally less than polymorphs. A theoretical methodology to determine the maximum absorbable amount based on the solubility of a compound is: Abmax = S ´ k a ´ SIVW ´ SITT

(17.1)

where Abmax is maximum absorbable amount if the small intestine could be saturated with the drug for 270 min (the average intestinal transit time), S is the solubility of the compound at pH 6.5 in mg/ml, k a is the frst-order absorption rate constant in min-1, SIVW is small intestinal water volume in ml (≈ 250 ml) and SITT is the small intestinal transit time in min. Equation 8.55 does not take into consideration the pre-systemic intestinal and hepatic metabolism. 17.2.2 Partition Coeffcient The octanol/water partition coeffcient, CLOG, MLOG, and distribution coeffcient (Moriguchi et al., 1992; 1994; Todeschini et al., 2000) are discussed in Chapter 7 (sections 7.2.2–7.2.5). They are quantitative measurements of lipophilicity or hydrophobicity of a compound. The MLOGPcoeff and CLOGPcoeff have been used in various theoretical approaches to predict the solubility and permeability of a compound. One of these approaches is known as Lipinski’s Rule of Five (Lipinski et al., 1997, 2016; Veber, 2002). 17.2.2.1 Rule of Five The Rule of Five (Ro5) is established based on the evaluation of known physicochemical characteristics of 2,245 drugs in United States Adopted Names (USAN) entries, and relies essentially on four parameters of the drugs, namely partition coeffcient, molecular weight, the number of H-bond donors, and the number of H-bond acceptors (Lipinski et al., 1997, 2001, 2016). It is called Rule of Five because the cutoff values for each of the four parameters are fve or a multiple of fve. According to this rule, a drug candidate has poor absorption or permeation when: ◾ The calculated logarithm of partition coeffcient exceeds 5, i.e., CLOGPcoeff > 5, (the cutoff for MLOGPcoeff is > 4.15). ◾ The molecular weight of a drug is over 500. ◾ There are more than fve NH bonds and OH bonds in the molecule. These bonds represent the hydrogen bond donating ability of the molecule. Therefore, if a drug molecule has less than fve H-bond donors of the type OH and NH collectively, the permeability of the drug would be enhanced, and if the H-bond donors are more than fve, the permeability would be reduced. ◾ The number of hydrogen bond acceptors of the type of Os (oxygen) and Ns (nitrogen) exceeds ten. The Rule of Five identifes several molecular properties that infuence the absorption of drugs and help to predict the bioavailability of a compound. However, the Rule has some limitations and cannot be used as blanket rule for all xenobiotics. For example, it is not applicable to the compounds that are considered substrates or modulators of protein transporters, or in certain cases, the molecular weight cutoff at 500 does not signifcantly separate compounds with poor oral bioavailability from those with acceptable bioavailability. It is a qualitative rule, which identifes the compounds as either having good or poor absorption (Andrews et al., 2000). In addition to the criteria defned in the Rule of Five, several other molecular properties have been identifed as important factors infuencing the absorption. The examples are: 516

BIOPHARMACEUTICS PROVISIONS, CLASSIFICATIONS AND MECHANISTIC MODELS

◾ Water complexation with amide bonds negatively infuences the bioavailability of a compound. ◾ Molecular fexibility helps the absorption and permeation of drugs through biological barriers. ◾ The polar surface area prevents the absorption of xenobiotics in the small intestine. ◾ Rotational bonds of 10 or fewer with polar surface area equal to or less than 140A° will have a high probability of acceptable bioavailability. ◾ Other than oral routes of administration, the application of Ro5 to select drug candidates for extravascular routes, like ophthalmic, transdermal, or inhalation, may require additional careful criteria and evaluation (Choy and Prausnitz, 2011; Karami et al., 2022). 17.2.3 Infuence of Particle Size, Porosity, and Wettability on Dissolution Rate at the Site of Absorption The particle size reduction, improved wettability, and higher degree of porosity of drugs enhance the solubility, accelerate the dissolution, and thus facilitate the absorption of a poorly water-soluble compound. Solid dispersion is often used to reduce the particle size and increase the surface area, resulting in an increased dissolution and improved bioavailability (Leuner and Dressman, 2000, Bikiaris et al., 2005). The infuence of particles size on absorption of xenobiotics may be considered from two different perspectives, 1) the absorption of particles and 2) the infuence of the particle size on the solubility/dissolution of particles at the site of absorption. 17.2.3.1 Absorption of Particles Particles of appropriate size (< 100 nm) can cross the intestinal barrier through two pathways: ◾ Absorption via microfold cells (M-cells) in the Peyer’s patches, which are lymphoid follicles in the mucus membrane of the small intestine, and isolated follicles of the gut-associated lymphoid tissue (GALT). ◾ Absorption through the intestinal enterocytes. Particles of larger size (> 1 µm) may get absorbed through GALT but will remain trapped in the Peyer’s patches. Physicochemical factors, such as surface charge and hydrophobicity, can also infuence absorption of the particles. The study of particulate absorption is of interest in areas such as traditional solid dosage forms, nanoparticles, polymeric delivery systems and biopharmaceuticals (Jani et al., 1990; Florence, 2005; Chen et al., 2011; Delon et al., 2022). The main dispute, however, is whether relying solely on absorption of particles would be suffcient to provide the intended optimum therapeutic outcomes. 17.2.3.2 Infuence of the Particle Size on the Solubility/Dissolution at the Site of Absorption The size of particles usually is interpreted as the surface area in contact with the environment that they are in. For a known amount of a solid compound, decreasing the particle size is synonymous with increasing their surface for contact with the dissolving fuid, thus reducing the time required for a compound to dissolve (i.e., a faster dissolution rate), and facilitate the absorption of the compound. In manufacture of the oral solid dosage form, milling or micronization for particle size reduction are commonly performed as approaches to improving the solubility based on the increase in surface area. The conventional methods of particle size reduction have long been employed to enhance the bioavailability of drugs. The size reduction limit of the conventional method is approximately 2–5 μm, which for some compounds may not be enough to signifcantly improve the drug solubility in the small intestine. Furthermore, solid powders with very small particle size have low fow properties and high adhesion characteristics and are often diffcult to handle. Small particles of hydrophobic compounds tend to establish static surface charges, which often causes agglomeration in aqueous environments. For this reason, solid dispersion methodology is used, which essentially is the molecular mixture of compounds with poor solubility in hydrophilic carriers and solidifying the mixture by cooling or evaporation (Hou et al., 2013; Patel et al., 2013; Ozeki et al., 2005; Higuchi, 1963). As a result of solid dispersion, the dissolution rates increase and the absorption in the GI tract is facilitated. The solid dispersion is also used to decrease the solubility and release of a compound to provide sustained release dosage forms. It should be noted that reduction of particle size is not always desirable, for example: 517

17.3 FORMULATION FACTORS

◾ Irritant compounds in fne particle size may cause more gastrointestinal irritation. ◾ Compounds with a low therapeutic index and fast dissolution of their fne particle size may raise their plasma concentration rapidly and bring on unexpected side effects. ◾ Reductions of particle size of compounds that are unstable in the GI tract accelerate the instability of the compounds in GI tract. To enhance the solubility of a compound at the site of absorption, in addition to the reduction of its particle size, the porosity of the particles with larger size can also increase the surface area and enhance the solubility. Particles in solid dispersions have been found to have a higher degree of porosity. Another factor that plays an important role in the solubility of hydrophobic particles at the site of absorption is the wettability of the particles. Additions of any agents that reduce the hydrophobic layer of the particles or carriers enhance their wettability and facilitate their solubility. Surfactants are often used to reduce the interfacial tension of particles and promote wetting of the particles. 17.2.3.3 Infuence of Wettability and Porosity on the Dissolution Profle The improvement of the drug wettability also enhances the drug solubility. Often to increase the wettability of a compound, carriers with surface active compounds like bile salts and cholic acid are used in solid dispersion, which enhances the dissolution by direct involvement or by cosolvent effects (Poulton, 2006, Kang et al., 2004). Though use of urea that has any surface activity as a carrier has increased the wettability (Levy, 1963). The increase in porosity also translates into higher solubility and better dissolution profle. Particles in solid dispersion have a higher degree of porosity. Solid dispersion with carriers, like linear polymers, results in increased porosity of drug particles. 17.3 FORMULATION FACTORS As an example of the extravascular routes, the oral route of administration is the most convenient route for administration of therapeutic agents, and it provides versatility for formulation of various dosage forms including ◾ solutions ◾ syrups ◾ suspensions ◾ emulsions ◾ soft and hard gelatin capsules ◾ compressed tablets (coated & uncoated) The formulation of dosage forms is beyond the scope of this book. However, their important characteristics, which infuence absorption of the compounds from the GI tract, are discussed briefy in this section. 17.3.1 Solutions and Syrups The active ingredient in these dosage forms is already in soluble form and its absorption is expected to be fast and complete. However, the chemical nature of some drugs and their chemical interaction with the acidic pH of the stomach may hinder their absorption. Examples are salts of acidic drugs soluble in neutral pH but insoluble as free acid in the stomach. Often the insoluble form aggregates and forms large particles, which further reduces the solubility of the compound. 17.3.2 Suspensions Unlike the solutions and syrups, the active ingredient of suspensions is not in soluble form and the dosage form is prepared by suspending the drug particles in a liquid medium. The absorption of the particles then requires the dissolution of particles in the GI tract. The advantage of this dosage form is circumventing the disintegration process, which is required for solid dosage forms like tablets, and having the suspended small particles ready to be dissolved at the site of absorption. 518

BIOPHARMACEUTICS PROVISIONS, CLASSIFICATIONS AND MECHANISTIC MODELS

However, if the dissolution were slow, the absorption would become a dissolution-rate limited process. Most suspensions have surfactants in their formulation, which helps the wettability of the particles and their dissolution. Another formulation factor that may infuence the absorption of the drug is the viscosity of the suspension due to the addition of hydrophilic polymers, such as Na-carboxy methyl cellulose, alginates, or gelatin, which may retard the passive diffusion of the drug and gastric emptying rate. 17.3.3 Emulsions Emulsions are prepared by dissolving hydrophobic compounds in oil and dispersing it in water to form an O/W emulsion or dissolving hydrophilic compounds in water and dispersing it in oil to form a W/O emulsion. There are also emulsions prepared by dispersing oil in water and then in oil O/W/O, or dispersing water in oil and then in water W/O/W emulsions. The oils used in emulsions are edible olive, corn, and peanut oil. Other compounds such as beeswax, long chain acids, and alcohol are also included. Surfactants and emulsifying agents, such as gelatin, acacia, or tragacanth are also added. The bioavailability of drugs given in emulsion is comparable or better than suspension. The only concern is the entrapment of the drug within the micelles, which may delay the absorption of the drug given orally and using this dosage form. In addition to the passive diffusion of compounds in emulsion form, the small droplets of emulsions or microscopic droplets of microemulsions can cross the barrier by pinocytosis. 17.3.4 Soft and Hard Gelatin Capsules The basic formulation of soft gelatin is encapsulation of drugs dissolved in an appropriate vehicle, e.g., polyethylene glycol or edible oils, in soft gelatin capsules. The soft-shell dissolves or disintegrates in the stomach and its content is released into the GI environment. If the content is formulated as a suspension, the particles must be dissolved and then absorbed. The hard gelatin capsules often contain powder, which after dissolution and/or breakdown of the hard gelatin shell in the GI tract undergo dissolution at the site of absorption. Because of the fast disintegration of the shell and quick availability of the loose powder for dissolution, the hard gelatin capsules usually have a better bioavailability than compressed tablets. Depending on its chemical nature, the compound released from the soft or hard gelatin capsules will be absorbed in stomach or small intestine by one of the absorption mechanisms discussed in Chapter 7. 17.3.5 Compressed Tablets (Uncoated and Coated) The uncoated compressed tablet consists of a drug powder and inert additives that are compressed together under pressure. Depending on the drug and formulation, the compressed tablet can be taken orally, which requires disintegration and dissolution in GI tract before the absorption, or if it designed to be dissolved in water before the administration, such as effervescent tablets. The dissolved molecules of the drug are then ready for absorption upon administration. In designing the uncoated tablets, the physicochemical properties of the drug, such as pKa, crystalline form, and amorphous form infuence the solubility and absorption of drugs, must be taken into consideration. The number of additives in a tablet (i.e., binders such as methyl hydroxyl ethyl cellulose, diluents like starch or Avicel, disintegrants such as Primogel, and lubricants like magnesium stearate) and the applied compression of the tablet machine must be carefully optimized to prevent ill-timed disintegration and dissolution of the tablet in the GI tract to avoid turning the disintegration and dissolution into rate-limiting steps for the absorption. Coated tablets are the compressed tablet coated with appropriate water-soluble polymers, such as cellulose acetate phthalate polymer or neutral methyl cellulose and hydroxyl propyl methyl cellulose or poly vinyl pyrrolidone, followed by an application of beeswax to polish the tablet. The water-soluble polymers dissolve in the GI tract and the compressed tablet will undergo disintegration and dissolution. The enteric-coated tablets are coated with polymers that are insoluble in acidic environment of stomach by preventing the disintegration and dissolution, but dissolve in alkaline environment of the intestinal tract. The lag time of absorption for enteric coated tablets are normally longer that the compressed tablet for the compounds that partially absorb from stomach. 17.3.6 Dosage Form Tactics for Poorly Soluble Compounds There are also creative approaches in formulation that enhance the solubility of poorly soluble compounds and/or improve their bioavailability. A few examples are: microemulsions, lyophilization, solid dispersion, liposomes, complexation with compounds like cyclodextrins, micellar and 519

17.4 DISINTEGRATION AND DISSOLUTION

surfactant systems, nanotechnology, cosolvent systems, chemically and physically targeted dosage forms, and other delivery systems. For more information, consult related publications. 17.4 DISINTEGRATION AND DISSOLUTION The conventional solid dosage forms (compressed tablets and capsules) are the most economical and convenient-to-administer dosage forms for most therapeutic agents. The feasibility of developing a solid dosage form for small molecules drugs is always considered at the outset for developing a new dosage form. The release of a drug from a solid dosage form, whether it is fast release or controlled release, is infuenced and characterized by parameters related to the physicochemical properties of the drug and the disintegration and dissolution characteristics of its dosage form (Hörter et al., 2001). The in vitro disintegration/dissolution testing does not guarantee therapeutic effcacy. However, dissolution testing has been accepted as a reliable in vitro methodology for qualitatively revealing the behavior of the solid dosage form in the GI tract and as an in vitro predictive marker for bioavailability and bioequivalence of the dosage form. The disintegration apparatus recommended by the United States Pharmacopeia (USP) is known as basket–rack assembly, which determines whether a solid dosage form disintegrates each time under experimental conditions and in a liquid medium and is described in the Pharmacopeia. Thus, the disintegration test neither implies nor tests for the complete solution of the drug or the dosage form. Likewise, there are four USP-recommended and standardized dissolution apparatuses (basket, paddle, reciprocating cylinder, and fow-through cell) with performance verifcation tests (PVTs), which are used to determine the concentration of active ingredient of dosage form in each medium at a specifed time. The most widely used one is the paddle apparatus. Dissolution testing is usually performed to measure the release from the solid dosage form as a test for quality assurance and compliance with the dissolution requirement in the individual monograph. In certain cases, in vitro–in vivo correlation can be established between the release of a therapeutic agent from the dosage form and absorption of the active ingredient (Dressman et al., 1998). The US Food and Drug Administration (FDA) requires that all drugs listed in the USP conform to the standards laid out in their respective monographs. Conforming to the standards ensures the reproducibility of the dosage forms and their bioavailability. 17.4.1 Mathematical Models of Dissolution Various mathematical relationships have been proposed to describe the dissolution rates of solid dosage forms and factors that can infuence the dissolution rate of a solid dosage. The following are the conventional models. 17.4.1.1 Noyes–Whitney Model This is the oldest and classic model for defning the dissolution rate of solid particles (Noyes and Whitney, 1897): dC = K Dis ´ SArea ( Csat - Ct ) dt

(17.2)

dC is the dissolution rate; K Dis is the dissolution constant as defned by Equation 17.3; S Area is dt the surface area of dissolving particles; Csat is the saturation concentration of drug in the diffusion layer, which is the stagnant layer around the dissolving particles; and Ct is the concentration in dissolution media at time t .

where

K Dis =

Dcoeff x

(17.3)

(Dcoeff ) is the diffusion coeffcient and x is the thickness of diffusion layer. When the concentration of drug in dissolution media is negligible compared to the saturation concentration, i.e., Ct ˜ Csat , it implies the sink condition, which is normally achieved by keeping the volume of media large. Under the sink condition, Equation 17.2 changes to: dC = K Dis ´ SArea ´ Csat dt

(17.4)

In calculating the rate of dissolution, the Noyes–Whitney relationship assumes Sarea a constant. 520

BIOPHARMACEUTICS PROVISIONS, CLASSIFICATIONS AND MECHANISTIC MODELS

17.4.1.2 Hixson–Crowell “Cube Root” Model This model is based on the concept of changing surface area with respect to time during dissolution without including it as a variable directly into the model, assuming the particle regular area is proportional to the cubic root of its volume (Hixson and Crowell, 1931): W01/3 - (W0 - Wt )

1/3

= Kt

(17.5)

where W0 is the initial weight or mass, Wt is the mass remained at time t , K is a complex constant proportional to diffusion coeffcient, solubility, the cube root of particles number, the particle size, and the thickness of the diffusion layer. This model is applied to the dissolution of monodispersed 1/3 powder consisting of particles with uniform size. The plot W01/3 - (W0 - Wt ) versus time will be linear with a slope of K . 1/3 The variable (W0 - Wt ) is the cube root of the amount dissolved at time t. Therefore, W01/3 - (W0 - Wt ) be written as:

1/3

= W01/3 - Wt1/3 is the cube root of mass remained at time t and Equation 17.5 can W01/3 - Wt1/3 = Kt

(17.6)

1/3 0

Dividing both sides of Equation 17.6 by W yields the following relationship, which describes the dissolution rate in terms of a fraction of mass at time zero: 1-

Wt1/3 Kt = W01/3 W01/3

\ ( 1 - ft )

1/3

= KNt

(17.7)

where ft is the fraction of dose dissolved, 1 - ft is the fraction of the initial mass remaining to be dissolved, and (K N ) is the normalized value of K with respect to the cube root of the initial mass. Thus, under the sink condition, a plot of the fraction of initial mass remaining to be dissolved versus time is linear with a slope of K N. 17.4.1.3 First-Order Kinetics Model There are two approaches in this model, and both are based on frst-order kinetics (Gibaldi and Feldman, 1967; Wagner, 1969). The frst approach is when, under the sink condition, the surface area changes with time and decreases exponentially, that is,

( SArea ) = ( SArea )0 e -k (t -t ) s

0

(17.8)

where ( SArea ) is the initial surface area at time zero, and k s is the frst-order rate constant for the 0 reduction of surface area. Under the sink condition, the rate of dissolution is equal to: dA = K ´ SArea ´ Cs dt

(17.9)

dA is the dissolution rate with dimension of mass/time, K is a constant with units of length/ dt time, SArea has units of length2, and Cs is the aqueous solubility of drug with units of mass/volume. Substituting Equation 17.8 in Equation 17.9 for t ³ t0 yields:

where

dA -k t -t = KCs ( SArea )0 e s ( 0 ) dt

(17.10)

Integration of Equation 17.10 fort ³ t0 yields: At = At0 + Setting M =

K -k t -t ´ Cs ´ ( SArea )0 é1 - e s ( 0 ) ù ë û ks

K ´ Cs ´ ( SArea )0 simplifes Equation 17.11 to 17.12: ks -k t-t At = At0 + M é1 - e s ( 0 ) ù û ë

(17.11)

(17.12) 521

17.4 DISINTEGRATION AND DISSOLUTION

where the amount dissolved at time t is At , at time zero is At0 , and M has units of mass. As t Þ ¥, i.e., when the dosage form is dissolved completely, (A¥ = At0 + M ), therefore: A¥ - At = Me

(

-ks ( t-t0 )

)

log A¥ - At = log M -

, or

ks ( t - t0 ) 2.303

(17.13) (17.14)

Equations 17.13 and 17.14 are linear frst-order equations in terms of amount remaining to be dissolved at time t . The second approach assumes that the surface area is a variable proportional to the amount remaining to be dissolved:

(

SArea = k p A¥ - At

)

(17.15)

where ( k p ) is a proportionality constant with units of area/mass. Substituting Equation 17.15 in Equation 17.9 yields: dA = KCs k p A¥ - At dt

(

)

(17.16)

Setting k = KCs k p changes Equation 17.16 to 17.17 dA = k A¥ - At dt

(

)

(17.17)

The constant (k) in Equation 17.17 has units of time -1 : length length 2 mass ´ ´ = time -1 time volume mass Integration of Equation 17.17 yields:

(

)

log A¥ - At = log A¥ -

kt 2.303

(17.18)

Equation 17.18 is another frst-order linear equation in terms of amount remaining to be dissolved. Both approaches assume frst-order kinetics under sink conditions. 17.4.1.4 Kitazawa Model This model is like the frst-order model (Kitazawa et al., 1975, 1977), and is defned as:

(

)

ln A¥ A¥ - At = K D ´ t

(17.19)

The variables have the same defnitions as described previously. In terms of the fraction dissolved, the equation changes to: ln 1 / ( 1 - ft ) = Kt

(17.20)

Equations 17.19 and 17.20 have a linear relationship under the sink condition, and a plot of ln A¥ A¥ - At or ln1 / ( 1 - ft ) versus t should yield a straight line with a slope equal to the dis-

(

)

solution rate constant. The model may exhibit a biphasic straight line, which, in the frst phase, is attributed to the disintegration and in the second phase, to the dissolution of the dosage form. 17.4.1.5 Higuchi “Square Root of Time Plot” Model This model is proposed for diffusion-controlled drug release from a matrix under a perfect sink condition (Higuchi, 1961, 1963). It assumes that the release of the drug is through diffusion from a planar surface of constant area. The equation for a homogeneous matrix is: As = Dcoeff ( 2Atotal - Cs ) Cst

(17.21)

where As is the amount of drug released per unit of surface area, Dcoeff is the diffusion coeffcient, Atotal is the total amount of drug in the homogeneous matrix, Cs is drug solubility in the matrix, and t is the time. 522

BIOPHARMACEUTICS PROVISIONS, CLASSIFICATIONS AND MECHANISTIC MODELS

Defning Equation 17.21 in terms of fraction of dose released from the matrix is: ft = K H t1 2

(17.22)

where K H is Higuchi’s release rate constant defned as: KH =

Dcoeff ( 2Atotal - Cs ) Cs Atotal

(17.23)

Under the assumptions of this model, a plot ft versus square root of time yields a straight line with a slope of K H. The Higuchi’s model for solid dosage forms with heterogeneous matrix is: Dcoeff E ( 2Atotal - ECs ) Cst t

As =

(17.24)

where E and t are the porosity and tortuosity of the matrix, respectively. For a homogeneous or heterogeneous matrix, a plot of fraction of drug released from the matrix versus square root of time is linear. 17.4.1.6 Weibull–Langenbucher Model This model is also known as Langenbucher’s model, and it is based on the following equation, which defnes the cumulative fraction of dose dissolved per units of time (Weibull, 1951; Langenbucher, 1972): ft = 1 - e

-

(t - Tlag )

b

a

(17.25)

¥

In Equation 17.25, ft is the ratio ofAt A , Tlag is the lag time of dissolution, “a” is a time scale constant, and “b” is a constant of the curvature; for exponential curves (b = 1), for S-shaped curves (b > 1), and for parabolic curve (b < 1). The linear version of the model is: log ëé - ln ( 1 - ft ) ùû = b log(t - Tlag ) - log a

(17.26)

For most solid dosage forms, Tlag is equal to zero. The time scale “a” is usually defned as a = ( Td ) , where Td is the dissolution time or the time point that corresponds to - ln ( -1 - ft )= 1, that is, ft = 0.693. Therefore, the dissolution time (Td ) is the time required for 63.2% of the solid dosage to dissolve in dissolution media. b

17.4.1.7 Korsmeyer–Peppas Model This model is also known as Power Law and relates the fraction of dose released from a polymeric dosage form to the time and mechanism of release (Korsmeyer et al., 1983): At (17.27) = at n A¥ where “a” is a combined constant of structural and geometrical characteristics of the dosage form, the parameter “n” is the release exponent and refers to the release mechanism of drug from the dosage forms. When the release is Fickian diffusion, “n” is equal to 0.5. When the release mechanism deviates from the Fick’s law of diffusion, it is identifed as non-Fickian release and “n” is between 0.5 and 1.0 (i.e., 0.5 < n < 1). Adjusting Equation 17.27 for the lag time of dissolution yields Equation 17.28 and log-linear Equation 17.29 ft =

n æ At - Tlag ö = a ( t - Tlag ) ç ¥ ÷ A è ø

(17.28)

æ At - Tlag ö = log a + n log ( t - Tlag ) log ç ¥ ÷ è A ø

(17.29)

17.4.1.8 Nernst–Brunner Model This model is a modifed version of the original Noyes–Whitney equation and defnes the dissolution as a diffusion-limited two-step process (Nernst, 1904; Brunner, 1904): 523

17.4 DISINTEGRATION AND DISSOLUTION

dC Dcoeff ´ Sarea = ( S - Ci ) dt V´x

(17.30)

dC where is the rate of dissolution, Dcoeff and Sarea are the diffusion coeffcient and surface area, dt V and x are the volume of dissolution media and thickness of the diffusion layer, S is respectively, the solubility, and Ci is the concentration of the dissolved drug. 17.4.1.9 Baker–Lonsdale Model This model assumes the release of solute is from a homogeneous spherical matrix and is defned by the following relationship (Baker and Lonsdale, 1974): é ( Areleased )t 3ê æ 1 - ç1 2 ê çè ( Areleased )¥ ë

ö ÷ ÷ ø

2/ 3

ù (A releasedd )t 3DmCms út = ú ( Areleased )¥ r02C0 û

(17.31)

where ( Areleased )t and ( Areleased )¥ are the amount of released solute at time t and infnity; Dm is the diffusion coeffcient from the matrix; Cms is the solubility of the compound in the matrix; r0 is the initial radius of the spherical matrix; and C0 is the initial concentration of the compound in the matrix. A similar relationship for the non-homogeneous matrix is proposed (Baker and Lonsdale, 1974), and both relationships can take the form of a straight line if the left-hand side of the equation is plotted against time, generating a slope equal to the complex coeffcient of time t . 17.4.1.10 Hopfendberg Model This model describes the drug release based on a surface eroding mechanism from the sphere, slabs, and sphere dosage forms exhibiting heterogeneous erosion (Hopfenberg, 1976; Katzhendler et al., 1997). The relationship is:

( Areleased )t ( Areleased )¥

é k t ù = 1 - ê1 - E ú ë C0 r0 û

n

(17.32)

where ( Areleased )t is the fraction of amount released and dissolved, and kE is the rate constant of ( Areleased )¥ erosion; other parameters are as defned in the Baker–Lonsdale model. 17.4.2 In Vitro–In Vivo Correlation (IVIVC) of Dissolution Data The dissolution data are the most appropriate in vitro property of a solid dosage form that can be correlated with relevant pharmacokinetic parameters of an orally administered solid dosage form. The examples of relevant PK/TK parameters are maximum plasma concentration and area under the plasma concentration–time curve. The defnition of IVIVC recommended by the United States Pharmacopeia (USP37-NF 32, 2014, General Chapter ) is “the establishment of a relationship between a biological property, or a parameter derived from a biological property, produced from a dosage form, and a physicochemical property of the same dosage form.” The method of establishing such a correlation is to use a dissolution test as a surrogate for bioavailability studies and to reduce the requirement for frequent bioequivalence studies in humans at the initial stages of the drug approval process (Dressman et al., 1998, 2007; Mizuma, 2002; Sirisuth et al., 2002; Emami, 2006). The FDA defnition of IVIVC is “a predictive mathematical model describing a relationship between an in vitro property of a dosage form and an in vivo response.” The FDA guidance classifes four levels of correlation (A, B, C, and multiple C) for IVIVC (FDA, 1995; 1997): 17.4.2.1 Level A Correlation Level A correlation is a point-to-point relationship between in vitro dissolution rate and the in vivo input rate. The relationship can be linear or nonlinear and is considered the most useful approach in defning a direct relationship between the in vitro dissolution rate and a pharmacokinetic parameter, such as the fraction of dose absorbed at different time intervals. The A level correlation indicates that the in vitro dissolution rate is indeed a surrogate for an in vivo response.

524

BIOPHARMACEUTICS PROVISIONS, CLASSIFICATIONS AND MECHANISTIC MODELS

17.4.2.2 Level B Correlation Level B correlation is based on the application of statistical moment theory. The correlation is with in vivo mean resibased on the comparison of mean in vitro dissolution time ( MDT ) dence time (MRT) or mean in vivo dissolution time ( MDT )

( MDT )invitro = ò

¥

0

MRT =

(A

¥

invitro

invivo .

)

- At dt A

¥

AUMC AUC

( MDT )invivo = ( MRT )solid - ( MRT )solution

(17.33) (17.34) (17.35)

where A¥ and At have the same defnitions as before; AUMC and AUC are area under the and moment curve and area under plasma concentration–time curve, respectively; ( MRT ) solid ( MRT )solution are the mean residence time of solid dosage form and solution of the same drug. Level B correlation relies on the comparison of calculated parameters, and the correlation is not as analytical as level A. 17.4.2.3 Level C Correlation Level C is based on the comparison of a dissolution time, for example, t50% , t63% , or t90% , which represents the time required for a certain fraction of solid dosage form to be dissolved in vitro, with the average of appropriate pharmacokinetic parameters, like AUC, Cpmax , or Tmax . This level of correlation establishes a single point relationship between a dissolution parameter and a PK parameter. 17.4.2.4 Multiple-Level C Correlation Multiple C relates the amount of drug dissolved at various time points to one or several appropriate pharmacokinetic parameters. This level of correlation establishes a relationship between several dissolution and pharmacokinetic parameters and is considered more realistic and dependable as level A. 17.5 BIOPHARMACEUTICS CLASSIFICATION SYSTEM BCS provides a scientifc basis for the correlation of in vitro dissolution of immediate release solid dosage form and in vivo bioavailability/bioequivalence studies. It helps to predict in vivo performance of drug products from in vitro measurements of permeability and solubility. The system recognizes fundamental parameters of dissolution along with solubility and permeability characteristics of a drug. According to this system, a drug has rapid dissolution when more than 85% of the dose dissolves within 30 minutes in a volume of less than 900 mL using one of the USP dissolution apparatuses. It is considered highly soluble when the highest dose strength is soluble in 250 mL of water or less over a pH range of 1–7.5 at 37°C. It is considered highly permeable when the extent of absorption in humans is greater than or equal to 90% of the administered dose or in comparison with an intravenous reference dose. The Biopharmaceutics Classifcation System (BCS) deals with the parameters of dissolution and absorption through a set of dimensionless numbers as follows: 17.5.1 Absorption Number Absorption number, An , is defned as the ratio of the residence time in the GI tract or GI transit time (TR ) to time required for complete absorption (Tabs ) (Löbenberg et al., 2000; Martinez et al., 2002): An =

Peff TR = ( TR ) Tabs R

(17.36)

where Peff is the effective permeability and R is the radius of the GI tract. 17.5.2 Dissolution Number The dissolution number, Dn is the ratio of residence time in the GI tract to dissolution time:

525

17.5 BIOPHARMACEUTICS CLASSIFICATION SYSTEM

TR æ 3Ddiff öæ S ö (17.37) =ç ÷ç ÷ ( TR ) Tdis è r 2 øè r ø where Ddiff is the diffusivity, r is the particle radius, S is the solubility (mg/mL), and r is the density (Löbenberg et al., 2000; Martinez et al., 2002). Dn =

17.5.3 Dose Number Dose number, N dose , is the ratio of drug in solution to its solubility, i.e., defning the dose as a function of its solubility: N dose =

( DH

Vwater ) S

(17.38)

(DH ) is the highest dose and Vwater is the volume of water taken with the dose (~250 mL). 17.5.4 Classes of Biopharmaceutics Classifcation System According to the BCS, drug substances are categorized into four classes (Amidon et al., 1995, Dahan et al., 2009): 17.5.4.1 Class I: Compounds with High Permeability and High Solubility The compounds in this category are expected to dissolve rapidly and absorb rather completely. It is assumed that the high permeability and high solubility allow the drugs in Class I to be present at high concentration at the absorption site. Thus, they can saturate effux, absorptive and metabolic proteins, and they reduce the clinical signifcance of interaction with these proteins. The drugs in Class I exhibit high absorption numbers and high dissolution numbers, and their IVIVC is expected when the dissolution rate is slower than the gastric emptying rate and is considered the rate-limiting step for absorption. When the rate-limiting step for absorption is the gastric emptying rate, no correlation is required. Examples of drugs in Class I are: propranolol, metoprolol, verapamil, midazolam, and diltiazem, most of which are considered Pgp substrates/modulators. 17.5.4.2 Class II: Drugs with High Permeability and Low Solubility The Class II compounds have a high absorption number but a low dissolution number. The ratelimiting step for absorption is drug dissolution. As a result, they have slower absorption than drugs in Class I; and because their low bioavailability is related to the low dissolution, IVIVC is expected for drugs in Class II, unless the dose is very high. The high permeability of drugs in this category indicates that they are lipophilic compounds and thus permeate rapidly into the enterocytes by passive diffusion. However, due to their low solubility, the amount present at the site of absorption is not at a level to saturate the intestinal effux and metabolic proteins, hence the proteins infuence their bioavailability (Löbenberg et al., 2000). Examples of drugs in Class II are nifedipine, ketoconazole, carbamazepine, phenytoin, ketoprofen, mefenamic acid, naproxen, danazol, and glibenclamide. Many of these compounds are substrates for Pgp and CYP3A4. 17.5.4.3 Class III: Drugs with Low Permeability and High Solubility The low permeability of compounds in this class is the rate-limiting step for their absorption. Thus, any variability in absorption of drugs would be due to the physiological factors governing the permeability of drugs in the GI tract and the role of intestinal apical transporters. Limited or no correlation is expected. Examples of drugs in Class III are acyclovir, cimetidine, ranitidine, neomycin B, atenolol, and captopril. 17.5.4.4 Class IV: Drugs with Low Permeability and Low Solubility Examples include furosemide, Taxol, and hydrochlorothiazide. The drugs in Class IV have poor bioavailability but may achieve suffcient solubility in the presence of biological surfactants and act like the compounds of Class III. Limited or no correlation is expected. 17.5.5 Biowaivers The term “biowaivers” refers to the exceptions to the requirement to perform bioequivalence studies. The exceptions, determined by regulatory authorities, are based on reasonable IVIVC and application of the BCS. Biowaivers reduce exhaustive study on humans, thus making the drug development process faster and more economical. Between the four levels of IVIVC, level A or 526

BIOPHARMACEUTICS PROVISIONS, CLASSIFICATIONS AND MECHANISTIC MODELS

multiple-level C correlation can serve as a surrogate for in vivo investigations. The guidance for the waiver of bioavailability and bioequivalence studies has been adopted by FDA under the title of the BCS. As it was mentioned earlier, the application of IVIVC is mostly to establish dissolution specifcations and criteria for biowaivers. The FDA guidance identifes fve categories of biowaivers (FDA, 2000; Yu et al., 2002): ◾ biowaivers without an IVIVC ◾ biowaivers using an IVIVC – non-narrow therapeutic index drugs ◾ biowaivers using an IVIVC – narrow therapeutic index drugs ◾ biowaivers when in vitro dissolution is independent of dissolution test conditions ◾ situations where an IVIVC is not recommended for biowaivers 17.5.6 Biopharmaceutics Drug Disposition Classifcation System As mentioned earlier, the Biopharmaceutics Drug Disposition Classifcation System (BDDCS) relies mainly on dissolution/solubility and permeability of drug products. Thus, the confdence in granting biowaivers is based on IVIVC and BCS prediction of the in vivo performance of the products. A modifed version of BCS classifcation has been suggested, which takes into consideration the overall disposition of drug in the body (Wu and Benet, 2005, Chen et al., 2011). The classifcation is known as Biopharmaceutics Drug Disposition Classifcation System (BDDCS). The disposition includes routes of drug elimination with emphasis on metabolism, infuence of infux and effux transport proteins, and food effect and transporter infuence on post-absorptive plasma levels following oral and intravenous dosing. Based on evaluated PK parameters, it is suggested (Wu and Benet, 2005) that: ◾ Class I and II are eliminated from the body predominantly through metabolism. ◾ Class III and IV are eliminated unchanged through bile and urine. ◾ The transporter effect is lowest for compounds in Class I. ◾ The effux transporter effect for Class II is predominant, with transporter-enzyme interplay and mainly CYP3A4 and Phase II metabolism. ◾ Absorptive transporter effect is predominant for Class III. ◾ Compounds in Class IV are substrates for absorptive and effux proteins. ◾ The effect of high-fat food on the extent of absorption for Class I compounds is not signifcant, but it increases the extent of absorption for Class II compounds, decreases for Class III, and is diffcult to predict for Class IV. It is further suggested that BDDCS expands the compounds in Class I and make them eligible for biowaivers (Wu and Benet, 2005; Chen et al., 2011). 17.6 OTHER FACTORS INFLUENCING ABSORPTION OF XENOBIOTICS 17.6.1 Chirality and Enantiomers A chemical characteristic of xenobiotics that can infuence PK/TK variables and constants signifcantly is the chirality of the molecular structure. A molecule or a system is considered chiral when it cannot be superimposed to its mirror image. Chirality is considered a property of many molecules in nature and biological systems and arises from a chiral atom; most often it is a carbon atom bonded to four different groups. The molecule and its mirror image are known as enantiomers. A mixture of equal amounts of enantiomers of a chiral xenobiotic is called a “racemic” mixture. Enantiomers with different 3-dimensional shapes behave differently in their interaction with proteins and biological systems. They have different rates and extents of absorption and have different pharmacological responses. Based on their optical rotation, the enantiomers are identifed as R (from the Latin word “rectus”), for the clockwise rotation, and S (from the Latin word “sinister”) for counterclockwise rotation. There is evidence that chiral molecules, such as nonsteroidal anti-inflammatory drugs (e.g., ibuprofen, fenoprofen, and 2 arylpropionic and benoxaprofen) may undergo enantiomer interconversion in the GI tract (Foster et al., 1980; Simmonds et al., 527

17.6 OTHER FACTORS INFLUENCING ABSORPTION OF XENOBIOTICS

1980; Hutt et al., 1984; Caldwell et al., 1988; Evans, 1992; Jamali, 1994; Geisslinger, 1995; Petrie and Camacho-Muñoz, 2020), and racemic mixtures of drugs, such as L-dopa, methotrexate, terbutaline, leucovorin, and cephalexin exhibit selectivity in the absorption of one enantiomer. Often a racemic mixture may exhibit complete absorption but a low bioavailability due to the extensive frst-pass metabolism of one of the enantiomers; examples are chiral molecules of beta-blockers and calcium channel blockers (Wade et al., 1973; Hendel et al., 1984; Echizen et al., 1985; Schilsky et al., 1989, Borgstrom et al., 1989; Seebauer et al., 2022). Thus, two enantiomers of a chiral xenobiotic may have 1) different rates and extents of absorption, 2) inconsistent interaction with metabolic enzymes, transporters, and receptors, and 3) different potency and effectiveness; and signifcantly variable bioavailability, PK/PD, and TK/TD. 17.6.2 Effects of Food and Drink on Absorption of Xenobiotics Food and drink can infuence the absorption of xenobiotics and complicate the PK/TK analysis (Lown et al., 1997; Gai et al., 1997; Custodio et al., 2008; Briguglio et al., 2018). The infuence is mostly on the rate and extent of absorption and therefore the bioavailability of an administered compound (Fleisher et al., 1999; FDA 2002). Depending on the type of xenobiotics, the rate and extent of absorption may delay, accelerate, or remain unchanged. There are certain physiological changes associated with consumption of food that have impact on the absorption of xenobiotics: ◾ Food may increase splanchnic blood fow and thus increase the absorption, bioavailability, and frst-pass effect (Liedholm and Melander, 1986). ◾ Food may interact with specifc transport proteins (Patil et al., 1998). ◾ Certain components of food may bind with the compound and reduce bioavailability (Leyden, 1985; Brown and Juhl, 1976). ◾ Food intake causes fuctuation of gastric pH; it rises initially and then declines. (Dressman et al., 1986; Kararli, 1995) ◾ Food stimulates bile secretion, which may enhance the absorption of lipophilic compounds (Kararli, 1995). ◾ Food prevents diffusion of poorly permeable compounds (Fleisher et al., 1999). ◾ Food reduces gastric emptying rate (Schmidt and Dalhoff, 2002; Sing, 1999). Food has no effect on BCS Class I compounds, may increase absorption of Class II compounds, decrease the absorption of Class III, and has mixed effect on Class IV (Fleisher et al., 1999; Radwan et al., 2012; Heimbach et al., 2013). The therapeutic agents with reduced absorption and bioavailability in the presence of food include the compounds that are unstable in gastric fuid and, because of the presence of food, stay longer in the stomach. Compounds that interact irreversibly with food components and are not absorbed or absorbed later in their passage through the small intestine, when they have already passed their site of absorption, also have reduced absorption and bioavailability. On the other hand, the absorption of drugs that are poorly soluble in GI fuids may increase in the presence of food because of the secretion of bile salts. The secretion of bile salts is the direct result of food intake, which releases the contents of the gallbladder into the small intestine. The classic examples of this type of compound are griseofulvin and cyclosporine, in which the extent of absorption increases in the presence of bile salts. Food with high fat content is expected to infuence the absorption and bioavailability of drugs that are substrates for Pgp and CYP3A4, for example Class II and III compounds of the BCS could be included in this category. However, the effect of a high-fat diet on the absorption of Class I compounds is expected to be insignifcant. Ordinarily, high-fat meals contain signifcant triglycerides, which are hydrolyzed in the GI tract to monoglycerides, known to inhibit Pgp. The type of food infuences the absorption of drugs, like theophylline, albuterol, and epinephrine. A high-fat meal may increase their bioavailability, whereas high-carbohydrate meals may reduce the extent of absorption. The main reason is the infuence of food constituents on the activity of proteins. The physical nature of food can also infuence the absorption of therapeutic agents in solid dosage forms. For example, meals that enhance the viscosity of GI tract fuid delay the disintegration and dissolution of solid dosage forms. Fruit juices (grapefruit, orange, and apple) are well known to infuence the bioavailability of drugs (Bailey, 2010). Grapefruit juice inhibits intestinal CYP3A4 (Veerman et al., 2020; Gjestad et al., 2019) and Pgp (Wang et al., 2001; Dahan and Amidon, 2009) and, thus, increases the bioavailability 528

BIOPHARMACEUTICS PROVISIONS, CLASSIFICATIONS AND MECHANISTIC MODELS

of substrates of these proteins. Herbal extracts containing compounds such as favonoids, antioxidants, curcumin (the principal curcuminoid of the popular Indian spice turmeric), etc. could also potentially infuence the absorption of drugs by inhibition of effux proteins. 17.6.3 Effects of Disease States The effect of pathophysiological factors of disease states and their infuences on the process of absorption is beyond the scope of this chapter. It is, however, noteworthy that certain disease states such as ulcerative colitis (Berends et al., 2018), thyrotoxicosis (a toxic condition resulting from excessive amounts of thyroid hormones, hyperthyroidism) (Wilkinson and Burr, 1984), achlorhydria (absence or reduction of hydrochloric acid in stomach), cirrhosis of liver, biliary obstruction, and Crohn disease (infammatory bowel disease) (Gesink-van der veer et al., 1997) can affect the absorption and bioavailability of drugs signifcantly. Furthermore, it has been reported that gastric emptying can be delayed for patients with diabetes mellitus (Narayanan et al., 2020). Therefore, depending upon the type of disease, different changes in the process of absorption of drug are expected. Infammation and infection also cause downregulation of CYP450 isozymes and transport proteins (Petrovic et al., 2007; Morgan, 2009), which directly infuence the PK/TK of xenobiotics. 17.6.4 Infuence of Genetic Polymorphism Genetic variation in humans and experimental animals has also been recognized as an important determinant of individual variability of absorption. The infuence of genetic polymorphism is through the control of metabolic enzymes and protein transporters. For example, ABCB1 gene encodes the effux protein, Pgp, which is highly polymorphic in different human populations. More than 20 variants of CYP3A4 are known across ethnic groups and have altered enzymatic activities, ranging from very low to high catalytic effciency. The mutation of breast cancer resistance protein (BCRP), which is an ABC transporter, ABCG2, is involved in the intestinal absorption and biliary excretion of xenobiotics and is widely present in ethnic groups (Grandhand and Kim, 2008; Zhou et al., 2008); there are many other similar examples of the infuence of genetic polymorphism on the absorption of xenobiotics. 17.6.5 Effects of Release Mechanisms from the Solid Dosage Forms Formulation of different dosage forms of a drug may result in different rates and extents of absorption. For example, a controlled-release oral dosage form has a different profle of absorption than the conventional solid dosage forms prepared by direct compression. 17.6.6 Infuence of Drug Administration Scheduling The timing of administration of drugs relative to food intake is also an important factor that may infuence the absorption of drugs. For example, taking penicillin G or erythromycin 1 hour before a meal or 2 hours after the meal improves the rate and extent of absorption of these drugs. In this case the reason is a reduction in the gastric emptying time and the length of exposure to the acidic environment of the stomach. The rate and extent of absorption for some drugs, such as halofantrine, are improved by taking the drug with fatty foods. 17.6.7 Presence of Other Substances The presence of other substances, such as environmental chemicals, other drugs, dietary constituents, tobacco smoking, and alcohol use are all known to infuence the absorption of other xenobiotics. Their infuences are mainly by induction or inhibition of drug-metabolizing enzymes, such as CYP3A4, and modulation of drug transporters, such as Pgp, which alters drug effcacy, induces drug–drug and drug–chemical interactions, and can cause drug side effects. 17.6.8 Other Factors Factors such as age, sex, pregnancy, exercise, starvation, and circadian rhythm can also contribute to individual variation of absorption and the PK/TK of administered drugs, in general. 17.7 MECHANISTIC ABSORPTION MODELS The mechanistic/predictive models are designed to establish strategies for the selection and formulation of lead candidates in drug discovery and development and for forecasting the rate and extent of oral absorption. They include: 529

17.7 MECHANISTIC ABSORPTION MODELS

◾ Absorption Potential Models ◾ Dispersion Models ◾ Compartmental Absorption and Transit (CAT) Model ◾ Gastrointestinal Transit Absorption (GITA) Model ◾ Advanced Compartmental Absorption and Transit (ACAT) Model ◾ Advanced Dissolution, Absorption and Transit (ADAM) Model ◾ Grass Model 17.7.1 Absorption Potential Models The absorption potential is a unitless number calculated by the following equation for predicting the fraction of administered dose absorbed by passive diffusion (Dressman et al., 1985):

( Abs )potential = log

Pcoeff ´ Sint ´ f u ´ Vlumen Dose

(17.39)

where Pcoeff is the partition coeffcient of the compound; Sint is the intrinsic solubility of the compound in the water, f u is the unionized fraction at pH 6.5; and Vlumen is the volume of lumen in a liter (» 250 mL = 0.25 L). The absorption potential estimated by Equation 17.39 correlated well with the fraction of dose absorbed in selected chemicals. Another suggested relationship for estimating absorption potential is (Balton et al., 1999):

( Abs )potential

6.8 6.8 éé ù ù ê êë( Dis )coeff ûú ´ ( S ) ´ Vlumen ú = log ê ú Dose ê ú ë û

(17.40)

6.8

6.8 where éê( Dis )coeff ùú is the distribution coeffcient (Chapter 7, Section 7.2.3) at pH 6.8, and ( S ) is the û ë solubility at pH 6.8. The next proposed relationship is based on the pH independency of the product of total solubility and the distribution coeffcient (Sanghvi et al., 2002). In other words, the product of distribution coeffcient and solubility at any pH is equal to the intrinsic values of partition coeffcient and intrinsic solubility. Thus, the relationship for absorption potential, identifed as modifed absorption potential, ( MAb ) potential , is:

é Pcoeff ´ Sint ´ Vlumen ù ú Dose ë û

( MAb )potential = log ê

(17.41)

For the lumen volume of 250 mL, the relationship can be simplifed to: é Pcoeff ´ Sint ù ú ë 4Dose û

( MAb )potential = log ê

(17.42)

17.7.2 Dispersion Models The main assumption of the dispersion model is considering the small intestine as a uniform tube with constant dispersion behavior, constant concentration, and axial velocity. The model postulates that the dynamic of drug absorption is based on the following convection–dispersion equation (Ho et al., 1983; Yu et al., 1996a; 1996b; Huang et al., 2009): ¶C ¶ 2C ¶C - gC =a 2 -v ¶z ¶z ¶t

(17.43)

where C is the concentration of drug in the GI tract; a is the hybrid dispersion coeffcient for mixing, molecular diffusion, and physiological factors; z is the axial distance from the stomach; v is the velocity in the axial direction; and g is the absorption rate constant. A different approach to the dispersion model is the inclusion of solubility and permeability effects on absorption and the assumption that the intestine is a tube with variable diameter and dimension, with species-dependent physiological parameters (Willman et al., 2003, 2004, 2007). The modifed model allows the estimation of the fraction of administered dose in a certain 530

BIOPHARMACEUTICS PROVISIONS, CLASSIFICATIONS AND MECHANISTIC MODELS

segment of the tube at time t . In the rat, the length of the duodenum is assumed as 10 cm, its initial pyloric diameter as 1.5 mm, and the proximal portion of jejunum as 2.0 mm. The lengths of jejunum and ileum are assumed 87 cm and 3 cm, respectively, with a diameter of 2.0 mm for jejunum and proximal portion of ileum, and 1 mm for the distal portion of ileum at the ileocecal connection. The amount of administered dose that is absorbed passively into the portal vein in the region of [ z , z + dz ] in a time interval of [t , t + dt ] is defned as: d 2 MPV ( z , t ) dAeff ( z) = PintestinalClumen ( z , t) dzdt dz

(17.44)

where d 2 M is the second derivative of administered dose per body weight; Pintestinal is the apparent intestinal permeability coeffcient of the drug; Aeff ( z) is the effective surface at intestinal position; and Clumen is the luminal concentration which can be treated as the compound’s intestinal solubility, that is, Clumen

ì Clumen ï =í ïS î int estinal

Clumen £ Sintestinal ü ï ý Clumen > Sintesstinal ïþ

if if

(17.45)

The dispersion models have been developed within the framework of physiology-based models for rats, monkeys, and humans. The related mathematical relationships have provided realistic predicted values for intestinal permeability, absorbable amount, luminal concentration, and other pertinent parameters when the pre-systemic metabolism is negligible (Willman et al., 2003; 2004; 2007; Huang et al., 2009). 17.7.3 Compartmental Absorption and Transit Model The Compartmental Absorption and Transit (CAT) model assumes the intestine as a physiological compartment in series with different volumes, different fow rates, and linear transfer kinetics. The small intestinal transit time (SITT) (the residence time) is assumed to be the same in all compartments (Yu et al., 1996a; Huang, 2009), that is, TSITT V1 V2 V V 1 = = ..... n = N = = Qn Q N K t N Q1 Q2

(17.46)

where V terms represent the volumes; Q terms are the fow rates; Kt is the transit rate constant; N is the number of compartments; and TSITT is the mean SITT. The rate equation of the model is (Yu et al., 1996b): dYn = KtYn -1 - KtYn - K aYn dt

,

n = 1, 2, 3..., 7

(17.47)

where n is the number of compartments; Yn is the percent of dose at the nth compartment; Kt and K a are the rate constants of transit and absorption, respectively. The model was frst developed based on the assumption that the drug is neither absorbed nor dY degraded in the GI tract, and the rate of percent dose entering the colon, c , that is the transit dt rate, is: dYc = KtYn dt

(17.48)

where Yc , the percent dose leaving the small intestine, is: N

Yc = 1 -

(K t) * t

n-1

å ( n - 1) ! e

-Kt*t

(17.49)

n -1

(N ) is the number of total compartment, Kt*t is the convolution of (Kt ) and (t ), and Kt is estimated as: Kt =

N TSITT

(17.50)

The small intestinal absorption rate is then defned as 531

17.7 MECHANISTIC ABSORPTION MODELS

N =7

å

dYa Yn = Ka dt n=1

(17.51)

The fraction of dose absorbed, Fa , can be calculated as: Fa = K a

7

Kt7

ò å Y dt = 1 - ( K + K ) %

0

n

t

n=1

(17.52) 7

a

The CAT model can easily be incorporated into the compartmental model, e.g., for a multicompartmental model with absorption rate into the central compartment and elimination from the central compartment, the differential equations are: dC1 D dYa = - ( k12 + k13 + ke ) C1 + k 21C2 + k 31C3 dt V dt

(17.53)

17.7.4 Gastrointestinal Transit Absorption Model The gastrointestinal transit absorption (GITA) model was frst developed for analysis and prediction of plasma concentration after oral administration of drugs given in solution to rats (Sawamoto et al., 1997). The model was then used for the solid dosage form (Fujioha et al., 2007) and applied to human data to predict the plasma concentration (Kimura et al., 2002). In the GITA model, the gastrointestinal tract is divided into eight sections, and the absorption parameters of each section are incorporated into a linear frst-order GITA kinetic model (Figure 17.1). The stomach is assumed as one section and the small intestine is divided into six segments of duodenum, upper jejunum, lower jejunum, upper ileum, lower ileum, and cecum; the large intestine is assumed as a separate section. The transit of drugs from the section that is not the absorption site of the drug to the next is assumed to follow frst-order kinetics. The absorption at the absorption site(s) is also assumed to follow frst-order kinetics (Figure 17.1). The gastric emptying rate and intestinal transit rate for each section is defned as:

(

)

æ dA ö = - k stomach + ( k a )stomach Astomach ç ÷ è dt ø stomach

(17.54)

where æ dA ö is the gastric emptying rate; ( Astomach ) = Dose ; k stomach is the emptying rate cont=0 ç dt ÷ è ø stomach stant; and ( k a ) is the rate constant of absorption from the stomach. stomach

The intestinal transit rate is: dAi+1 = Ai ki - ( ki+1 + k ai+1 ) Ai+1 dt

(17.55)

The subscript i stands for intestinal sections. The following three mathematical steps are used to predict and estimate plasma concentration: The time course of amount in each section can be estimated by using the convolution method. The Laplace transform of the amount of drug in segment i + 1 is: ˜ i+1 ( s ) = A

˜ i ( s) ki A s + ki+1 + k a i+1

(17.56)

The Laplace transform of absorbable fraction in each section is: f˜i+1 ( s ) =

ki f˜i (s) s + ki+1 + k ai+1

(17.57)

The Laplace transform of the fraction of the dose absorbed in each section is: f˜ai ( s ) = k ai f˜i ( s )

(17.58)

The prediction of plasma concentration following oral administration is accomplished by means of the convolution method and, assuming the total absorption rate-time data corresponds to 532

BIOPHARMACEUTICS PROVISIONS, CLASSIFICATIONS AND MECHANISTIC MODELS

Figure 17.1 Illustration of gastrointestinal transit absorption (GITA) model where the GI tract is divided into stomach and seven segments of the small and large intestines; the transit rate constant between the eight segments and the absorption rate constant from each segment into the systemic circulation are considered frst-order kinetics; the collective rate of absorption from all segments can be defned as FDk a , where FD is the total amount absorbed from different segments, though a few (like the stomach) may contribute less to the overall absorption of xenobiotics; k a is the rate constant of absorption comprised of all rate constants of absorption from different segments; A1 and A2 compartments are the central and peripheral compartments, with elimination from the central compartment that includes the systemic circulation and highly perfused organs/ tissues; the peripheral compartment represents the less accessible tissues; the distribution rate constants between the two compartments and the overall elimination rate constants are also governed by frst-order kinetics. input function and PK parameters after intravenous administration, corresponds to the transfer function. The Laplace transform of the plasma concentration is defned as: co ö öæ (17.59) ˜ iv (s) f ai (s)÷ Cp ÷ çç f a s (s) + ÷ øè i=d ø When the orally administered drug is in solid dosage forms, the dissolution and solubility of the dosage form can become the rate-limiting step in the absorption process. The modifed GITA model that takes into consideration the dissolution of the solid dosage form (Fujioka et al., 2007) is depicted in Figure 17.2. The Laplace transform of the amount of drug dissolved in the stomach and small intestine (Fujioka et al., 2007) is:

˜ oral ( s ) = æ Doseoral Cp ç è Doseiv

( A˜ ) sto

soln

( s) =

å

( ksto )dis ˜ sto ´ (A )solid ( s) s + ( ksto )soln

(17.60)

533

17.7 MECHANISTIC ABSORPTION MODELS

Figure 17.2 Illustration of the modifed GITA model that includes the dissolution and solubility of solid dosage forms in the overall absorption process from different segments of the GI tract; the signifcance of the model is its applicability to the cases when the dissolution or solubility can act as the rate-limiting step in the absorption of the active ingredient of the dosage forms; in this model, a solid dosage form at each state of its presence in the GI tract is subjected to both solubility and dissolution processes; the descriptions of the rest of the model are the same as described in Figure 17.1.

( A˜ ) i+1

( s) = soln

( ki )soln s + ( ki+1 )soln + k a

+

( ki+11 )dis

( )

˜i ´ A

i+1

s + ( ki+1 )soln + k ai+1

soln

( )

˜ i+1 ´ A

( s)

solid

(17.61)

( s)

( s ) and ( A˜ i+1 )solid ( s ) are the Laplace transform of solid dosage form in the stomach ( sto ) and sections of intestine( i + 1) , respectively; ( A˜ sto ) ( s ) and ( A˜ i+1 )soln ( s ) are the Laplace

( )

˜ sto where A

solid

soln

transform of the amount of solid dosage form dissolved in the stomach and sections of intestine, respectively; ( k sto )dis and ( ki+1 )dis are the dissolution rate constants; ( ksto )soln and ( ki+1 )soln are the

transit rate constants; and k ai+1 is the absorption rate constant in the intestine, absorption from the stomach is considered negligible. Based on Equation 17.61, the following relationship is developed to describe the Laplace transform of the fraction of dose in solution form available for absorption at the site of absorption (Fujioka et al., 2007):

f˜i+1 ( s ) =

( ki )soln s + ( ki+1 )soln + k a

i+1

534

( )

´ f˜i

( s) + soln

( ki+1 )dis s + ( ki+1 )soln + k a

i+1

( )

´ f˜i+1

solid

( s)

(17.62)

BIOPHARMACEUTICS PROVISIONS, CLASSIFICATIONS AND MECHANISTIC MODELS

( )

where f˜i+1 ( s ) is the fraction of dose available for absorption at the site of absorption; f˜i

( f˜ ) i +1

solid

soln

( s ) and

( s ) are the fraction of dose dissolved in section i and fraction remaining to be dissolved,

respectively. The Laplace transforms of the rate of absorption in section iand predicted plasma concentration are: f ai ( s ) = k ai ´ f˜i (s) Cp˜ oral ( s ) =

Doral ´ Div

ce

å f˜

ai

(17.63)

( s ) ´ Cp˜ iv ( s )

(17.64)

i=d

where Cp˜ oral ( s ) and Cp˜ iv ( s ) are the Laplace transforms of plasma concentration after oral and intravenous administration; Doral and Div are the oral and intravenous doses, respectively. A convolution program can then estimate the inverse Laplace transforms and (Cp - time ) profle following oral administration. 17.7.5 Advanced Compartmental Absorption and Transit Model The principles and assumptions of Advanced Compartmental Absorption Transit (ACAT) model are like those discussed for CAT and GITA models. The differences are the inclusion of frst-pass metabolism and colon absorption (Yu et al., 1996b; Agoram et al., 2001; Bolger et al., 2003; www.simulations-plus.com; Parrott et al., 2009). The model considers the GI tract as nine linked compartments. The frst and the last compartments are the stomach and the ascending colon, respectively. The seven other compartments, like GITA, represent the seven regions of small intestine (Figure 17.3). The transfer kinetics between the compartments is assumed linear; metabolism and transport are assumed nonlinear; and the model considers six states for a solid dosage form in the GI tract. The six states are: unreleased, undissolved, dissolved, degraded, metabolized, and absorbed (Huang et al., 2009). The physicochemical parameters of the drug (e.g., pKa, solubility, particle size, particle density, and permeability) and physiological parameters for each section/compartment (including pH, transit time, length, gastric emptying, luminal transport, and metabolism) are set according to the published data (Parrott et al., 2009; Huang et al., 2009). The linear and nonlinear features of the ACAT model with the relevant parameters are included in the commercially available simulation software GastroPlus ®. 17.7.6 Advanced Dissolution, Absorption, and Transit Model Advanced Dissolution, Absorption, and Transit Model is a similar model to GITA, CAT, and ACAT. It considers the GI tract as nine compartments that are anatomically identifable (Figure 17.4). The absorption process is described in terms of formulation, dissolution, precipitation, degradation, permeability, metabolism, transport, and transit through the sections of the GI tract. Like previous models, the transit is governed by frst-order kinetics. The commercially available software Simcyp® (http://www.simcyp.com) is the incorporation of ADAM in population-based pharmacokinetics. Two processes of absorption from stomach and metabolism at the colon are considered negligible, and the rates of other processes are defned by a set of ordinary differential equations (Jamei et al., 2009): dAF , n dAS , n dA = - dis , n - kt , n AS , n + kt , n -1 AS , n -1 + dt dt dt

(17.65)

dAD , n dAdis , n = - ( kdeg,n + k an + kt , n ) AD , n + kt , n -1 AD , n -1 + g nCLuint -T , n - f u gut Ce dt dt

(

dCent , n 1 Adis , n k an - Cent,nQent,n - f ugut Cent , n éëCluint-GG, n + Cluint -T , n ùû = dt Vent , n where AS , n is the amount of solid dose available for dissolution; rate;

)

(17.66) (17.67)

dAdis , n is the dissolution dt

dAF , n is the release rate of solid dosage form due to disintegration and de-aggregation; dt 535

17.7 MECHANISTIC ABSORPTION MODELS

Figure 17.3 Depiction of the advanced compartmental absorption and transit (ACAT) model with nine anatomical regions of GI tract as linked compartments with frst-order transfer kinetics including the stomach, ascending colon, and seven regional compartments for the small intestine. The model considers the four physical states of unreleased, undissolved, dissolved, degraded, and two biological processes of metabolized for the presence of and absorption of the active ingredient of a solid dosage form; the model includes the frst-pass metabolism through the hepatoportal vein and the overall elimination of metabolite(s) from the liver and excretion from the central compartment. The body is represented here by a three-compartment model, one central compartment, A1 , and two peripheral compartments, A2 and A3 ; the distribution rate constants between the three compartment are governed by frst-order kinetics and the elimination (excretion and metabolism) occurs only from the central compartment; the rate of input in the central compartment is frst-order with the rate constant of k a ; FD represents the total amount absorbed.

kdeg,n and k an are the degradation and absorption, respectively; g n is an adjustment factor for enterocyte uptake; CLuint -T ,n and Cluint-G , n are the intrinsic clearances due to the enterocyte’s effux and metabolism, respectively; f ugut is the unbound fraction in the enterocyte; Vent , n is the volume of enterocyte; Qent , n and Cent , n are the blood fow to the section and enterocyte concentration, respectively. 17.7.7 Grass Model The Grass model is another multiple-section model (stomach, duodenum, jejunum, ileum, and colon) for absorption of orally administered dosage forms. It takes into the consideration the three parameters of solubility, permeability, and tissue surface area. The software used in conjunction with the model is STELLA® (Structural Thinking Experimental Learning Laboratory with Animation) software (Grass, 1997).

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18 Bioavailability, Bioequivalence, and Biosimilarity 18.1 INTRODUCTION Bioavailability (BA), bioequivalence (BE) and biosimilar evaluations are the critical components in drug discovery and development and in the approval process for small-molecule pharmaceuticals and large-molecule biopharmaceuticals. The basic concepts of bioavailability are also applicable to nontherapeutic xenobiotics. Typically, the new drug applications (NDAs) contain bioavailability and, if necessary, the bioequivalence data. The abbreviated new drug applications (ANDAs), that most often are submitted for generic forms of marketed drug products, include the bioequivalence comparative testing. The biologic license applications (BLAs) that are submitted for biopharmaceuticals follow a different pathway for approval and usually contain the biosimilarity data that are discussed in Section 18.7 of this chapter. BA assessment is based on PK parameters and constants obtained from quantitative analysis of blood or plasma concentration of an active compound. The PK measurements of systemic exposure, such as AUC or Cpmax and other parameters and constants are used to defne the BA and BE (Chen et al., 2001). All the factors discussed in various chapters of this book that can infuence the absorption of a xenobiotic from a route of administration can also impact the bioavailability of xenobiotics. Physiological factors, such as gastric emptying rate, small intestinal transit time, blood fow rate, intestinal and hepatic frst-pass metabolism, inhibition or induction of infux and effux proteins, and physicochemical characteristics of the compound and formulation factors associated with the administered dosage forms including factors that infuence drug dissolution (Amidon et al., 1995); and inactive ingredients of a dosage form can all infuence the absorption and bioavailability of a compound. Furthermore, variation in splanchnic blood fow and biliary secretion may infuence the frst-pass effect; the magnitude of bile salts can affect the solubility of the lipophilic compounds (Fleisher et al., 1999; Karalis et al., 2008); species differences (Diamond et al., 2022); interindividual variability (McGilveray et al., 1990, Eker et al., 2020); specifc site-dependency of absorption (Vertzoni et al., 2019); the presence of food in the GI tract (Olanoff et al., 1986; Gupta et al., 1990; Bennett-Lenane et al., 2022) are the sources of variability in BA measurements. The regulatory guidelines related to bioavailability and bioequivalence evaluations are described in sections of Title 21 of Code of Federal Regulations; additional publications of the US Department of Health and Human Services Food and Drug Administration (FDA), Guidance for Industry – Bioavailability and Bioequivalence Studies Submitted in NDAs or INDs – General Considerations (FDA Guidance, 2014) prepared by the Offce of Communications Center for Drug Evaluation and Research (CDER); Center for Drug Evaluation and Research (CDER), and ICH-related guidelines provide valuable details of the BA and BE evaluation. Other FDA Core Guidance on Bioavailability and Bioequivalence are: ◾ US Food and Drug Administration, Title 21 Code of Federal Regulations (CFR) Part 320, Offce of Federal Register, National Archives and Records Administration (2001). Content current as of 03/22/2018. ◾ Food and Drug Administration Modernization Act, Public Law No. 105-115, 111 Stat. 2296, 1997 (http://www.fda.gov/opacom/7modact.html). ◾ US Food and Drug Administration, Center for Drug Evaluation and Research and Center for Biologics Evaluation and Research. Current as of 06/21/2022. ◾ Guidance for Industry – Providing Clinical Evidence of Effectiveness for Human Drugs and Biological Products. Offce of Training and Communications, Division of Communications Management, Drug Information Branch, HFD-210, Rockville, Maryland 20857. Current as of 08/24/2018. ◾ US Food and Drug Administration, Title 21 CFR 21 CFR 314.50(d) (1) – (6), Offce of Federal Register, National Archives and Records Administration. Current as of 03/22/2018. ◾ US Food and Drug Administration, Title 21 CFR 314.94, Offce of Federal Register, National Archives and Records Administration (up to date as of 09/09/2022). ◾ Waiver of in vivo bioavailability and bioequivalence studies for immediate-release solid oral dosage forms based on a biopharmaceutics classifcation system (12/26/2017).

DOI: 10.1201/9781003260660-18

545

18.2 DEFINITIONS

◾ Guidance for Industry: Bioavailability and bioequivalence studies for orally administered drug products – General considerations (FDA – Center for Drug Evaluation and Research (March 2003, 2006). ◾ Statistical approaches to establishing bioequivalence (published January 2001, current as of 04/29/2020). ◾ Bioanalytical methods validation – Guidance for industry (May 2018). ◾ Food-effect bioavailability and fed bioequivalence studies (current as 03/02/2018). ◾ Draft guidance for industry on topical dermatological drug product NDAs and ANDAs – In vivo bioavailability, bioequivalence, in vitro release, and associated studies (FDA 06/18/1998). ◾ Bioavailability and bioequivalence studies for nasal aerosols and nasal sprays for local actions (published 05/29/2020). 18.2 DEFINITIONS The key defnitions noted here are from Code of Federal Regulations Title 21 (21CFR) part 320.1, 314.3 to signify the applications of pharmacokinetic parameters and constants in drug approval and regulatory considerations: 18.2.1 Bioavailability The rate and extent of absorption of a xenobiotic into systemic circulation is referred to as bioavailability. The xenobiotic can be the active ingredient of a drug product, or environmental exposure to a chemical. The rate represents how fast a xenobiotic is absorbed from an extravascular route of administration and the extent represents how much of the compound reaches the systemic circulation and ultimately the site of action. 18.2.2 Pharmaceutical Equivalents “Drug products that contain the same active ingredient(s), are of the same dosage form, route of administration and are identical in strength or concentration.” In other words, two or more drug products with identical active ingredient(s) (i.e., the same salt or ester of the same therapeutic moiety that have the same quality, purity, identity, and strength according to USP or other standards) are considered pharmaceutically equivalent. This indicates that pharmaceutical equivalents may differ in their inactive ingredients (excipients) including preservatives, coloring, and favoring agents. They can also be different in release mechanism, expiration date, packaging, and labeling (with some limitation) and shape. 18.2.3 Pharmceutical Alternatives “Drug products that contain the same therapeutic moiety, or its precursor, but are different salts, esters, or complexes of that moiety, or are different dosage forms or strength.” Therefore, in contrast to pharmaceutical equivalents, pharmaceutical alternatives have different salts of the active ingredient(s) such as hydrochloride, sulfate, or phosphate, etc. They can also be of different strengths in different dosage forms. Thus controlled-release capsules and standardrelease tablets are pharmaceutical alternatives, or capsule of tetracycline phosphate is a pharmaceutical alternative to a tablet of tetracycline hydrochloride. 18.2.4 Bioequivalent Drug Products (Bioequivalence) The bioequivalent drug products are pharmaceutical equivalents or pharmaceutical alternatives that have the same bioavailability under the same experimental protocol of comparative studies. The above defnitions are intended for drug products that are absorbed into the systemic circulation. 18.2.5 Therapeutic Equivalents Therapeutic equivalents are pharmaceutical equivalents that are bioequivalents and can be expected to have the same clinical effect and safety profle when administered to patients under the conditions specifed in the labeling of the manufacturers

546

BIOAVAILABILITY, BIOEQUIVALENCE, AND BIOSIMILARITY

18.2.6 Generic Drug Products US Food and Drug Administration defnes a generic drug as “a copy of a brand-name drug that one company makes that was developed by another company.” The overall expectations from a generic drug product are to have: 1) the same active ingredient as the reference product, though its inactive ingredients can be different; 2) the same purity, quality, strength, and identity; 3) identical dosage form with the same strength and used via the same route of administration; 4) comparable bioavailability, i.e., bioequivalents; 5) the same use indications; 6) prepared under the US GMP practice regulations. 18.2.7 Absolute and Relative Bioavailability As noted in previous chapters, a low bioavailability can be the result of 1) hepatic frst-pass effect, 2) presystemic metabolism by intestinal CYP450, 3) infuence of P-glycoprotein, 4) simultaneous chemical degradation of drug at the site of absorption, 5) instability of drug in GI tract environment, and other infuences like the effect of food, disease states, and dosage form. The intravenously injected drugs (bolus or infusion), bypass the absorption related hindrances and enter in systemic circulation completely, thus their bioavailability is considered 100%, or a fraction of the dose reaching the systemic circulation is equal to one (F = 1). The comparison of bioavailability of an oral dosage form, or any other extravascular dosage form, with the bioavailability of intravenous injection is known as absolute bioavailability. It represents the fraction of a dose absorbed from an oral dosage form divided by fraction of dose available after the intravenous injection, which is equal to one that is Foral Fiv , orForal 1. The absolute bioavailability cannot exceed one and if Foral = 1, it implies that the absorption of the drug is complete and fully bioavailable. However, having a bioavailability equal to one does not mean that the absorption rate of the dosage form is fast; the rate may or may not be fast. It is worth noting that achieving an absolute bioavailability equal to one may not be attainable for many extravascular routes of administration, including the oral route. The relative bioavailability is often the comparison of a new formulation of a dosage form with its reference dosage form using the same or different extravascular route of administration (Colburn et al., 1986; Amini et al., 2020). The relative bioavailability also applies to the comparison of effect(s) of different physicochemical properties, like amorphous versus cocrystal form (Saharsrabudhe et al., 2022); conditions or events, such as fasting (Palaparthy et al., 2016); presence of food (Xin et al., 2018); types of food (Wang et al., 2022); exercise and/or drug interactions (Xin et al., 2018); disease states (Mukonzo et al., 2011); environmental factors (Liu et al., 2017); and age or the similar or different dosage forms intended for the same or different routes of administration (Colburn et al., 1985; Alrubia et al., 2022,). In contrast to absolute bioavailability, relative bioavailability can exceed one. A relative bioavailability greater than one indicates that more of the active ingredient is absorbed from the test formulation and the test has a higher bioavailability. And if the infuence of a condition, like disease state, or coadministration with another drug provides a relative bioavailability greater than one, it only indicates that the condition or other drug enhances the absorption of the therapeutic agent (Weidekamm et al., 1998; Eisenmann et al., 2022). In comparing two similar dosage forms, if relative bioavailability is less than one, it indicates that the absorption from the test formulation is not comparable to that of the reference, and it may also indicate the conditions of the investigation were not the same (Cassidy et al., 1999; Marathe et al., 1998). 18.3 PEAK EXPOSURE, TOTAL EXPOSURE, AND EARLY EXPOSURE As indicated earlier, the BA and BE assessments are based on the rate and extent of absorption. The extent of absorption is measured by the AUCof plasma concentration-time curve during the sampling time, that is, AUC0t , or to time infnity, AUC0¥ . The rate of absorption is commonly defned by Cpmax and Tmax , which are not the direct measures of the rate. They are estimated from the observed data or calculated by a relevant PK model equation, or directly from the observed data. Both Cpmax and AUC0¥ (or AUC0t ) are labeled by the FDA as the “peak exposure” and “total exposure,” respectively. The guidance also recommends the estimation of early exposure by determining the partial AUC between time zero and the population median Tmax , contingent upon having at least two plasma measurements before the Tmax for more accurate estimation of the partial area representing the early exposure. The early exposure has been shown to be more pertinent than Cpmax in expressing the absorption rate differences in relative bioavailability studies 547

18.3 PEAK EXPOSURE, TOTAL EXPOSURE, AND EARLY EXPOSURE

(Bois et al., 1994; Endrenyi et al., 1998; Chen et al., 1992; Endrenyi et al., 1998; Rostami-Hodjegan et al., 1994; Macheras et al., 1994). The rate constant of absorption, a model-dependent rate constant, is less commonly employed in BA and BE assessment (Chen et al., 2001). The preferred method for estimation of total or partial exposure is the trapezoidal rule (Addendum I, Part 2, Section A.4). The following sections are model-independent approaches for estimation of absolute and relative bioavailability. These approaches are applicable when all biological processes follow frstorder kinetics, and thus they are valid only for dose-independent pharmacokinetics. 18.3.1 Estimation of Absolute Bioavailability from Plasma Data – Single Dose Using the model-independent equations of intravenous injection and oral administration, the absolute bioavailability is estimated as follows:

( Dose )iv ( AUC )iv ( ClT )iv ´ ( AUC )iv = Fiv = 1 ( Dose )iv

(18.1)

( ClT )iv =

( ClT )oral =

(18.2)

Foral ´ Doseoral AUCoral

( ClT )oral ´ AUCoral

(18.3) (18.4)

= Foral

Doseoral

( ClT )oral ´ ( AUC )oral ´ ( Dose )i.v. Foral = Fabsolute = Fiv ( ClT )i.v. ´ ( AUC )i.v. ´ ( Dose )oral

(18.5)

Since in linear pharmacokinetics clearance is considered a constant and independent of the amount of xenobiotic in the body or route of administration, theoretically ( ClT ) is equal to oral

( ClT )iv , and if equal doses are administered orally and intravenously, Equation 18.5 is simplifed to ( AUC )oral (18.6) Fabsolute =

(AUC)i.v.

18.3.2 Estimation of Absolute Bioavailability from Amount Eliminated from the Body – Single Dose For assessment of bioavailability and evaluation of bioequivalence, primarily plasma concentration-time data is recommended (FDA, 1977). The use of urinary data is viewed as an alternative source of information. Contingent upon the feasibility of the urine collection, measurements of an unchanged parent compound and its metabolites, and sensitivity of analytical methodology to estimate the low concentration of the metabolites, the total amount excreted unchanged and total amount eliminated as metabolites can be used to estimate the bioavailability of a xenobiotic. The following relationship can be employed for the estimation of bioavailability of a compound from total amount eliminated in the urine:

( Dose )iv = ( Ae¥ )iv + ( Am¥ )iv

(18.7)

( F ´ Dose )oral = ( Ae¥ )oral + ( Am¥ )oral

(18.8)

Fabsolute =

(A (A

¥ e

+ Am¥

¥ e

¥ m

+A

) )

oral i .v .

´ ( Dose )i.v.

´ ( Dose )oral

(18.9)

Using equal doses Fabsolute = Equation 18.8 also yields 548

(A (A

¥ e

+ Am¥

¥ e

¥ m

+A

) )

oral i .v .

(18.10)

BIOAVAILABILITY, BIOEQUIVALENCE, AND BIOSIMILARITY

( F )oral =

(A ) ¥ e

oral

( )

+ Am¥

oral

Doseoral

(18.11)

It is also feasible to estimate the bioavailability from the unchanged compound in the urine using the following relationships: Intravenous administration: Ae¥ = f e ´ Dose

(18.12)

Ae¥ = f e ´ F ´ Dose

(18.13)

Oral administration:

Therefore, using Equations 18.12 and 18.13 yields the following relationship for the absolute bioavailability.

(A ) (A ) ¥ e

Fabsolute =

¥ e

oral i .v .

´ ( f e Dose )i.v.

´ ( f e Dose )oral

(18.14)

18.3.3 Estimation of Relative Bioavailability from Plasma Data – Single Dose The equations of relative bioavailability are the same as the ones for the absolute bioavailability estimation except for using parameters of reference product data instead of intravenous data. The test product data is compared to the reference data as follows: Ftest Freference

=

( ClT )test ´ ( AUC )test ´ ( Dose )reference ( ClT )refference ´ ( AUC )referencs ´ ( Dose )test

(18.15)

18.3.4 Estimation of Relative Bioavailability from Total Amount Eliminated from the Body – Single Dose As indicated earlier, the use of urinary data has been considered as an acceptable but secondary approach in bioavailability and bioequivalence assessments (FDA, 2003). It is important to point out that in using urinary data the presence of nonlinear renal clearance must be ruled out before the assumption and application of linear relationships (Thompson and Toothaker, 2004). The equations are like those used to estimate absolute bioavailability from total amount eliminated from the body: Test: Ftest ´ Dose = Ae¥ + Am¥

(18.16)

Freference ´ Dose = Ae¥ + Am¥

(18.17)

Reference:

Therefore, Ftest Freference

=

(A (A

¥ e

+ Am¥

¥ e

¥ m

+A

) )

test

´ ( Dose )reference

referrence

´ ( Dose )test

(18.18)

and Ftest Freference

=

(A ) (A ) ¥ e

¥ e

test

´ ( f e Dose )reference.

reference

´ ( f e Dose )test

(18.19)

18.4 BIOAVAILABILITY AND FIRST-PASS METABOLISM Several factors can infuence the bioavailability and reduce the amount of a xenobiotic reaching the systemic circulation. One possible contributing factor is the frst-pass intestinal and hepatic metabolism, which causes incomplete and often highly variable bioavailability, and intrasubject variability due to the metabolic enzyme inconsistent activity (Pang et al., 1978; Colburn, 1979; 549

18.5 LINEARITY VALIDATION OF RELATIVE OR ABSOLUTE BIOAVAILABILITY DURING MULTIPLE DOSING REGIMEN

Cassidy et al., 1980; Kwan, 1997; Jones et al., 2016). The general equation of infuence of intestinal and hepatic frst-pass metabolism on absolute bioavailability of an oral dosage form is: Fabsolute = Fabs ´ ( Fescaped )IFPE ´ ( Fescaped )HFPE where Fabsis the fraction at the site of absorption; ( Fescaped )

IFPE

(18.20)

and ( Fescaped )

HFPE

are the fractions that

escape the intestinal frst-pass effect (IFPE) and hepatic frst-pass effect (HFPE), respectively. The magnitude of fraction that escapes frst-pass metabolism depends on the rate and extent of frst-pass metabolism. A conceptual approach for estimating the fraction that escapes the hepatic frst-pass metabolism is:

where ( Aliver )

input

( Aliver )input = Q ´ AUCinput

(18.21)

( Aliver )output = Q ´ AUCoutput

(18.22)

Am = Clm ´ AUCinput

(18.23)

is the amount of xenobiotic entering the liver; ( Aliver )

output

is the amount escaping

the frst-pass metabolism; Q is the blood fow; and Am is the amount of frst-pass metabolism. Therefore, the fraction that is metabolized by the liver is the ratio of the metabolic clearance to the blood fow of the liver: Fm =

Clm ´ ( AUC )input

Qliver ´ ( AUC )input

=

Clm Qliver

(18.24)

The fraction that escapes the hepatic metabolism is

( Fescaped )HFPE = 1 - Fm

(18.25)

The following equations differentiate the gut and hepatic frst-pass effect of the orally administered compound (Lee et al., 2001): Fraction that escapes hepatic frst-pass effect: 1 - Fm = 1 - ( ER )liver =

( AUC )portal Div ´ ( AUC )iv Dportal

(18.26)

Fraction that escapes intestinal frst-pass effect: 1 - ( ER )GI =

( AUC )ID Dportal 1 ´ ´ Fabs ( AUC ) portal DID

(18.27)

where DID stands for intra-duodenal dose; and ER is the extraction ratio. The amount absorbed into portal vein can be estimated as Equations 18.28–18.29:

( Aabs )portal = DID ´ Fabs ´ (1 - ( ER )GI ) ( Aabs )portal = ò

t2

t1

Qportal ´ ( C portal - Cp )

(18.28) (18.29)

Therefore, the fraction that escapes the GI frst-pass effect can also be estimated by manipulating Equations 18.28 and 18.29 as Equation 18.30 (Lee et al., 2001; Tam-Zaman et al., 2004): 1 - (ER)GI =

1 1 ´ ´ DID Fabs

ò

t2

t1

Qportal ( C portal - Cp )

(18.30)

18.5 LINEARITY VALIDATION OF RELATIVE OR ABSOLUTE BIOAVAILABILITY DURING MULTIPLE DOSING REGIMEN To confrm that bioavailability of a compound remains the same during a multiple dosing regimen, the following relationships may be used to evaluate the dose independency of the compound during multiple dosing regimen at steady state. 550

BIOAVAILABILITY, BIOEQUIVALENCE, AND BIOSIMILARITY

%PTF =

( Cpmax )ss - ( Cpmin )ss ´ 100 ( Cpave )ss

%Change =

( Cpmax )ss - ( Cpmin )ss ´ 100 ( Cpmin )ss

(18.31)

(18.32)

where PTFis referred to as peak–trough fuctuation. For certain drugs, the single dose study may not be predictive of the steady-state peak and trough levels. The common reason is that, for certain drugs, the linear pharmacokinetic characteristics observed after a single-dose administration may change during multiple dosing to dosedependent or nonlinear pharmacokinetics. Thus, prediction of parameters of multiple dosing of such a drug based on the constants and parameters of the single dose administration is unsafe, and the data from a multiple dosing study are necessary to provide a complete bioavailability evaluation. The infuence of drug interactions may also modify the bioavailability of a compound in multiple dosing (Frassetto et al., 2013). One approach is to compare the area under the plasma concentration-time curve of the single dose, equal to the maintenance dose, from zero to infnity with the area under plasma concentration of a maintenance dose during one dosing interval at steady state. For drugs that follow linear pharmacokinetics, these two areas should be the same. It is worth reminding that the total amount eliminated during one dosing interval at steady state is equal to the maintenance dose. 18.6 BIOEQUIVALENCE EVALUATION Two or more drug products are often compared in a bioequivalence study. The objective of such a study is to determine whether a test drug product is equivalent to that of an existing reference product, in vivo. The reference product is an approved and marketed product. The test product is alleged to be interchangeable with reference product in clinical practice. This means that the test product is claimed to be therapeutically equivalent to the reference product. The reference product is often referred to as the “innovator product,” which indicates the product is authorized by the FDA to be marketed based on its documented safety, effcacy, and quality. If the reference is a brand-name product, the manufacturer is the patent holder of the product until the expiration of the patent. The equivalence testing is the absence of a signifcant difference between two product, which statistically is testing a null hypothesis, preferentially in a randomized, parallel or crossover design. The crossover design is usually preferred, and it requires the administration of the test and reference to the same group of subjects separated by a washout period. The washout period is determined based on the half-life or mean residence time of the active ingredient(s), and it is selected such that to ensure complete removal of the frst dose from the body (e.g., ≈ seven halflife) before the administration of the dose from the other product. When it is not feasible or possible to carry out a crossover trial, the parallel design is employed. The examples of bioequivalence trial with parallel design include drugs that are highly toxic, or compounds that have long mean residence time. An in-depth review of the relevant statistical analysis and experimental design can be found in statistical methodology books and references (Hauck et al., 1984; Selwyn et al., 1981; Steinijuans et al., 1990; Steinijans et al., 1992; Hauschke et al., 1990; Midha et al., 2005; Hauck et al., 2000; Anderson et al., 1990; Phillips, 1990; Dunnett et al., 1977; Farolf et al., 1999). In addition, other sources such as the following governmental resources should be consulted: ◾ Statistical Approaches Establishing Bioequivalence (USFDA, 2001, 2022) ◾ Bioavailability and Bioequivalence Studies Submitted in NDAs or INDs – General Considerations (USFDA, 2018–2022) ◾ ICH Guidance Documents – Food and Drug Administration (FDA, 2013, 2018) ◾ Biostatistical Methodology in Clinical Trials (EMA, 2022) ◾ Average, Population, and Individual Approaches to Establishing Bioequivalence, 1999a (FDA Center for Drug Evaluation and Research) ◾ BA and BE Studies for Orally Administered Drug Products – General Consideration, 1999b (FDA Center for Drug Evaluation and Research) ◾ etc 551

18.6 BIOEQUIVALENCE EVALUATION

The null hypothesis of the practical approaches in bioequivalence evaluation is briefy discussed here. Similar to null hypothesis for student t-test and ANOVA (i.e., H 0 : m Test = m Ref and H1 : m Test ¹ m Ref ), where m Test and m Ref are population means for test and reference products, for the bioequivalence trial the hypothesis is: H 0 : m Test - m Ref > D Þ bioinequivalence

(18.33)

H1 : m Test - m Ref £ D Þ bioequivalence

(18.34)

Often the distribution of pharmacokinetic parameters, such as AUC and Cpmax are positively skewed and exhibit heterogeneity of variances. Under this condition, a logarithmic transformation is usually considered to achieve a relatively homogeneous variance (Chow et al., 1991). Under the multiplicative models, the null hypothesis for the bioequivalence trial is: H0 :

m Test m D 2 Þ bioinequivalence m Ref m Ref H1 : D 1 £

m Test £ D 2 Þ bioequivalence m Ref

(18.35) (18.36)

The recommended values of D1 and D 2 by the FDA are 0.8 and 1.25, respectively. The two numbers refect the symmetrical normal distribution of ±0.223 around the central tendency of zero that corresponds to a 100% equivalent. In other words, ln 0.8 = -0.2331 and ln1.25 = +0.2231 falls on either side of the central tendency of ln1 = 0. The above hypothesis become additive after logarithmic transformation: quivalnece H 0 : lnm Test - ln m Ref < ln D1 or lnm Test - ln m Ref > ln D 2 Þ bioineq

(18.37)

H1 : ln D1 £ ln m Test - ln m Ref £ ln D 2 Þ bioequivalence

(18.38)

18.6.1 Required PK/TK Parameters and Other Provisions in Bioequivalence Study The required data for the testing of null hypothesis are the pharmacokinetic (PK) parameters and constants obtained from accessible biological samples representing a summary of the comparative data on absorption and disposition of both test and reference products. The measurements of the active compound and/or its metabolite(s) must be conducted under the Good Laboratory Practice (GLP) guidelines. The subjects of investigation should be 18 years old or older and from different age and racial groups with a balance of men and women, unless the drug is used in a specifc gender group or age group. If the drugs are intended for children, similar diversity should be followed in the recruitment of young patients, if it is feasible. At least 12–18 blood/plasma samples should be drawn at appropriate time points. The measurements should include control (pre-dose) measurement; early-time point samplings during the absorptive phase; and enough measurements during elimination phase. The sampling should span over at least three half-lives of elimination. Criteria for data deletion due to vomiting or infuence of food effect should be established and validated in advance. The parameters of single dose study include: ◾ the area under plasma concentration-time curve from time zero to the last measurement time point (time exposure): (AUC0t ) ◾ the area under plasma concentration-time curve from time zero to infnity (total exposure): AUC0¥ ◾ the area under the early exposure curve: AUC0Tmax ◾ maximum plasma concentration of single dose administration (peak exposure): Cpmax ◾ time to maximum plasma concentration: Tmax ◾ mean residence time: MRT ◾ mean absorption time: MAT ◾ the rate constant of elimination based on the terminal segment of the curve. 552

BIOAVAILABILITY, BIOEQUIVALENCE, AND BIOSIMILARITY

◾ volume of distribution: Vdss (estimated by non-compartmental analysis) ◾ total body clearance: ClT (estimated from model-independent equation ClT = The parameters of the multiple dosing study include:

Dose ) AUC

◾ the area under the plasma concentration-time curve of one dosing interval at steady state: AUCsst ◾ peak level at steady state: ( Cpmax ) ss ◾ trough level at steady state: ( Cpmin ) ss ◾ average steady state plasma concentration: ( Cpave ) ss

◾ time to maximum plasma concentration in one dosing interval at steady state: ( Tmax ) ◾ calculated percent of fuctuation at steady state: %Fluctuation =

( Cpmax )ss - ( Cpmin )ss

ss

( Cpave )ss

All the parameters and constants of single and multiple dosing are based on model-independent approaches, such as reading directly from the observed data or estimating by non-compartmental analysis. 18.6.2 Overview of Statistical Analysis of PK/TK Data for Bioequivalence Study For a conventional two-treatment comparison of test and reference using two-period, twosequence randomized, crossover design, the statistical analysis of the PK/TK data includes analysis of variance (ANOVA) with the source of variability among the groups, among the subjects in groups, and among periods of study and treatment. The “80/20 Power Rule” is used for sample size determination in the planning stage of a trial (Diletti et al., 1992). The rule requires that the sample size be large enough to provide an 80% probability of detecting 20% differences between the mean bioavailability characteristics of the two products. Except for theTmax , the concentration and concentration-dependent parameters and constants are based on logarithmic transformation. The parametric and/or non-parametric 90% confdence intervals of all parameters should fall within the window of 80% to 125%. For compounds with narrow therapeutic range, the window is closer, for example, 90% to 110%. 18.6.3 Required PD/TD Data The pharmacodynamic or toxicodynamic parameters are used instead of PK/TK data in a bioequivalence study when the measurement of plasma concentration or other biological samples are not feasible or relevant and/or when the measurements of the active compound and/or its metabolite(s) do not correspond to therapeutic response. The response must be relevant and determined under the Good Clinical Practice (GCP) guidelines. The response must be gauged by a validated dose-response relationship and the placebo effect should be taken under consideration. The experimental design is a crossover design unless it is necessary to use parallel design. The assessment of the outcome and related null hypothesis are the same as described earlier. 18.7 BIOSIMILAR (BIOSIMILARITY AND INTERCHABGEABILITY) 18.7.1 Introduction Biopharmaceutical medicines, also known as biological products, are typically complex protein molecules produced in living systems like microorganisms and other living cells. The examples are monoclonal antibodies, streptokinase, interleukin (IL)-2, plasminogen activator, granulocyte colony-stimulating factor, alpha and gamma interferon, erythropoietin, and many more emerging biologic moieties (De Lorenzo et al., 2009; Roovers et al., 2007; Alley et al., 2010; Bell et al., 2011; Horton et al., 2012). Contrary to the chemically synthesized small molecules, the “generic” form of a protein drug is not structurally identical to the reference product. The term “generic” does not even apply to the protein drugs with similar structure to the reference product. Thus, the similar products, not identical, are considered biologically similar or biosimilar products. Because minor modifcations in the structure of the protein drugs can modify their effcacy, potency, safety (e.g., immune response) 553

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and purity, the FDA has established guidance for Scientifc Considerations in Demonstrating Biosimilarity to a Reference Product: http://www.fda.gov/downloads/Drugs/GuidanceCom plianceRegulatoryInformation/Guidances/UCM291128.pdf and Scientifc Considerations in Demonstrating Biosimilarity to a Reference Product Guidance for Industry (fda.gov) The international agencies, such as World Health Organization, European Medicines Agency, and other agencies in different countries have similar policies related to biosimilars. Other helpful materials include the USP-NF chapters related to potency of biologic medicine (USP. USP-NF. http://www.usp.org/usp-nf. Sep 2013), USP has also agreed to work with National Institute for Biological Standards and Control and other governmental laboratories to assure that its biologic reference materials are in accord with national and international materials (Williams et al., 2014). The in vitro and in vivo PK analyses of therapeutic biologics are challenging and often inconsistent. Few investigations have shed light on the complex behavior of these compounds in the body (Faggioni, 1992; Vugmeyster et al., 2011; Vugmeyster et al., 2012; Fronton et al., 2014). For example, the PK analysis of monoclonal antibody has been investigated and various physiologically based pharmacokinetic (PBPK) or classical compartmental models or other mechanistic or dynamic models have been proposed, yet no clear consensus on disposition profle of the compound has emerged (Keizer et al., 2010; Vugmeyster et al., 2012; Jones et al., 2013; Xiao, 2012, Fronton et al., 2014). The challenges associated with PK analysis of therapeutic biologics in comparison to the small synthetic molecules include 1) physicochemical characteristics and properties of these large molecules, their size, shape, and complex molecular charge and different stability kinetics; 2) they are administered most often through intravenous, subcutaneous, or nasal route, and each route manifests inter- and intraindividual differences; 3) their distribution profle depends on their size, shape and charge, and binding to the target site; 4) their elimination is through proteolysis and processes such as nonspecifc endocytosis, Fc receptor-mediated clearance (Roopenian et al., 2007; Keizer et al., 2010; Kuo et al., 2011;), and target mediated clearance; 5) their target-binding and target-mediated clearance are capacity-limited, saturable, and kinetically nonlinear; 6) the subject variability, species differences in PK and PD profles, immunogenicity and off-target effects also contribute to the complexity of PK/PD analysis and human/species extrapolation; 7) the absence of validated in vitro systems for absorption or metabolism studies of biologics add to the complexity of PK/PD challenges. Concluding from the current language of the regulations, a biosimilar product can be defned as a therapeutic biologic agent that has similar in vitro and in vivo biological properties and characteristics, and similar clinical outcomes in effcacy, safety, and immunogenicity to the reference product. Interchangeability refers to the biosimilars that can be substituted for the reference products. The FDA defnitions of various terminologies like “highly similar,” “no clinically meaningful differences,” and “interchangeable products” can be retrieved from the following site: https://www.fda.gov/drugs/biosimilars/ biosimilar-and-Interchangeable Products | FDA 18.7.2 Comparability of Biosimilar and Application of PK/PD Parameters The following FDA Guidance for biosimilar comparability assessment provides comprehensive information on regulatory requirements: Scientifc Considerations in Demonstrating Biosimilarity to a Reference Product Guidance for Industry (FDA, 2017) The Guidance includes areas that are related to the structural analyses, such as amino acid sequence; functional assays including pharmacological activity, enzyme kinetics, binding assays, and functional effects on pharmacodynamic markers; quality characterizations and non-clinical studies; structure-activity relationship, animal PK/PD and toxicity data, animal immunogenicity results interpretation, safety and effcacy, comparative human pharmacokinetic, and pharmacodynamics studies; and, fnally, risk management plans (ICH, Q9, 2005). Depending on the type of the therapeutic proteins, peptides or polypeptides, the therapeutic proteins, the pharmacokinetic parameters discussed in bioequivalence studies are also applied to biosimilar assessments: area under the plasma concentration-time curve (0 ® t and 0 ® ¥), biological half-life, total body clearance, peak concentration, time to the maximum plasma concentration, volume of distribution, and dose-response curves. The comparison of PK/PD parameters is generally considered more sensitive than just the clinical endpoint. It is worth noting, however, that in biosimilar assessment, PK equivalence is not enough to determine the “no clinically meaningful differences” in PD, or effcacy, or safety. The reason is the large molecular size of most biologics that minor structural differences are hidden but can impact the PD/safety/effcacy. These minor structural differences 554

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may not have signifcant impact on PK evaluation, so a well-established PD biomarker is required to determine the clinical effcacy (Zhu et al., 2018). The study design protocol for the biosimilar compounds with a rapid response, a half-life of less than fve days, and a low immunogenicity is a crossover design. For half-life greater than fve days, a parallel design is considered suffcient. In general, biosimilars undergo the same laborious and thorough evaluation of effcacy and safety as the reference molecules. The only advantage for a biosimilar is the existing knowledge of the reference molecule. Thus, the evaluation revolves around the comparability of the biosimilar molecule to a reference molecule. Comparability of a biosimilar is the comparison of safety and effcacy risks. The comparability, also identifed as a critical quality attribute, is demonstrated by statistical analysis in three-tiered approach for comparison. They are (Burdick et al., 2017, Chow et al., 2016): Tier 1: Equivalence testing for biological activity and potency – the test is to assess analytical similarity. Tier 2: Range testing for in-process controls and less critical quality attributes. Tier 3: Raw data and graphical comparison. There are various statistical methodologies for each tier, such as, two one-sided t-test, K sigma, Weibull, Bayesian approach, etc. The description and discussion on the statistical approaches and their appropriateness of statistical analysis of biosimilarity are beyond the scope of this chapter, and published references and books should be consulted (Burdick et al., 2017, Chow et al., 2016; Tsong et al., 2017; Peck et al., 2022; Zhang et al., 2022). The global biosimilar assessment requirements and complexities are signifcantly the same as US FDA expectations (Ishii-Watabe and Kuwabara, 2019). REFERENCES Alley, S. C., Okeley, N. M., Senter, P. D. 2010. Antibody-drug conjugates: Targeted drug delivery for cancer. Curr Opin Chem Biol 14(4): 529–37. Alrubia, S., Mao, J., Chen, Y., Barber, J., Rostami-Hodjegan, A. 2022. Altered bioavailability and pharmacokinetics in Crohn’s disease: Capturing systems parameters for PBPK to assis with predicting the fate of orally administered drugs. Clin Pharmacokinet. https://doi.org/10.1007/s40262 -022-01169-4. Amidon, G. L., Lennernas, H., Shah, V. P., Crison, J. R. 1995. A theoretical basis for a biopharmaceutic drug classifcation: The correlation of in vitro drug product dissolution and in vivo bioavailability. Pharm Res 12(3): 413–20. Amini, M., Reis, M., Wide-Swensson, D. 2020. A relative bioavailability study of two misoprostol formulations following a single oral or sublingual administration. Front Pharmacol 11: 50. https:// doi.org/10.3389/jphar.2020.00050. Anderson, S., Hauck, W. W. 1990. Consideration of individual bioequivalence. J Pharmacokinet Biopharm 18(3): 259–73. Bell, D. A., Hooper, A. J., Burnett, J. R. 2011. Mipomersen, an antisense apolipoprotein B synthesis inhibitor. Expert Opin Investig Drugs 20(2): 265–72. Bennett-Lenane, H., Griffn, G. T., O’Shea, J. P. 2022. Machine learning methods for prediction of food effects on bioavailability: A comparison of support vector machines and artifcial neural networks. Eur J Pharm Sci 168: 106018. https://doi.org/10.1016/j.ejps.2021.106018. Bois, F. Y., Tozer, T. N., Hauck, W. W., Chen, M. L., Patnaik, R., Williams, R. L. 1994. Bioequivalence performance of several measures of rate of absorption. Pharm Res 11(7): 966–74. Burdick, R., Coffey, T., Gutka, H., Gratzl, G., Conlon, H. D., Huang, C.-T., Boyne, M., Kuehne, H. 2017. Statistical approaches to assess biosimilarity from analytical data. AAPS J 19(1). https://doi .org/10.1208/s12248-016-9968-0. 555

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19 Quantitative Cross-Species Extrapolation and Low-Dose Extrapolation 19.1 CROSS-SPECIES EXTRAPOLATION 19.1.1 Introduction: Interspecies Scaling in Mammals The reliance on experimental animals is a cornerstone of the preclinical phase of drug discovery and development and is considered the essential component of toxicological evaluation and risk assessment. The experimental animals are commonly considered a predictive model for humans, and the interspecies scaling is an important part of pharmacokinetic/toxicokinetic (PK/TK) analysis. All regulatory agencies and agricultural, food, chemical, environmental, and pharmaceutical/ biotechnology industries rely on the extrapolated data from experimental animals to make decisions about the safety of products, and the data are used for approval and marketing decisions. The application of interspecies scaling is based on the principles of similitude and dimensional analysis (Calder, 1984; Peters, 1983; McMahon and Bonner, 1983). There are many similarities in the anatomy and physiology of land mammalian species from the 3-gram shrew to the 6-ton African elephant. The blood fows in the same direction; the anatomical geometry is similar; the weight of each organ is a similar fraction of body weight, except for skin and skeleton; cellular structure is the same; and the general biochemistry and physiology for most parts are similar. This remarkable similarity has allowed interspecies scaling of physiological parameters and constants such as longevity, blood volume and fow, heart rate, and PK/TK parameters. There are two approaches to interspecies PK/TK scaling, 1) an allometric, or empirical, approach, and 2) a physiologic approach. The allometric methodologies are straightforward but frequently inaccurate and predict only the average values of parameters. In addition, they require signifcant data, and the prediction is only for the parent compounds. The physiologic extrapolations are grounded on physiologically based pharmacokinetic/ physiologically based toxicokinetic (PBPK/PBTK) models as discussed in Chapter12, Section 12.2. The PBPK/PBTK models are based on mechanistic descriptions of physiological processes and take into consideration the full PK/TK profle of a compound and associated variability and uncertainty and the formation of metabolites. 19.1.2 Allometric Approach The historical research by several scientists established the following empirical allometric relationship, which defnes the relationship between a physiological parameter and body weight (Adolph, 1949, Davidson et al., 1986, Voisin et al., 1990; Kleiber, 1932; Kleiber, 1947; Rubner, 1883). Y = aW b

(19.1)

Where (Y ) is the biological function, or PK/TK parameter of interest, W is body weight, a and b are the coeffcient and exponent of the relationship, respectively, which are considered speciesindependent constants. The geometric mean of b values is 0.82, in Table 19.1 (selected data are from Adolph, 1949, as presented in Mordenti, 1986). Taking logarithm of both sides of Equation 19.1 yields: log Y = log a + b logW

(19.2)

Therefore, a plot of the parameter versus the body weight on a log–log scale should generate a straight line with the slope of b and y-intercept of log a (Figure 19.1). The allometric exponent b is an indicator of the relationship between the parameter of interest, Y , and the body weight, W (Chappel and Mordenti, 1989). For instance, when the exponent is greater than zero (b > 0), the parameter Y decreases as the body weight increases; the analogy is the infuence of body weight on the heartbeat. When the exponent is equal to zero (b = 0), the parameterY is independent of the weight, a relevant example is the body temperature, which is independent of the weight. When b = 1, the increase in parameter Y is proportional to the body weight, for example, blood volume and body weight. When b > 1, the increase in parameter Y is faster than W (Mordenti et al., 1991). A beneft of allometric equations is that they can be used in algebraic operations, for example: The multiplication of two allometric equations yields : W x ´ W z = W x + z DOI: 10.1201/9781003260660-19

(19.3) 561

19.1 CROSS-SPECIES EXTRAPOLATION

Table 19.1 Allometric Equations of Physiological Parameters as Function of Body Weight Parameter (Y )

Units

Allometric Equation

Physiological Rates: Intake of Water

ml/hr

0.01W 0.88

Urine Output

ml/hr

0.0064W 0.82

Ventilation Rate Oxygen Consumption

ml/hr

120W 0.74

Basal

ml STP/hr

3.80W 0.734

Liver Slice Clearance

ml STP/hr

3.3W 0.77

Urea

ml/hr

1.59W 0.72

Inulin

ml/hr

1.74W 0.77

Creatinine

ml/hr

4.20W 0.69

Hippurate Physiological Period

ml/hr

5.4W 0.8

Heartbeat Duration

hr

1.19 ´ 10 -5 W 0.27

Breath Duration Organ Weight

hr

4.7 ´ 10 -5 W 0.28

Kidneys

g

0.0212W 0.85

Brain

g

0.081W 0.7

Heart

g

6.6 ´ 10 -3 W 0.98

Lung

g

0.0124W 0.99

Liver

g

0.082W 0.87

Thyroids

g

2.2 ´ 10 -4 W 0.80

Adrenal

g

1.1 ´ 10 -3 W 0.92

Pituitary

g

1.3 ´ 10 -4 W 0.76

Stomach and Intestine

g

0.112W 0.94

Blood

g

0.055W 0.99

The quotient of two allometric equations yields :

Wx = W x-z Wz

( )

The power of an allometric equation is : W x

(19.4) z

= W xz

(19.5)

Using the algebraic operation of the allometric relationships one parameter for example Y1 can be defned in terms of anther parameter Y2 (Adolph, 1949), that is

562

Y1 = a1W b1 Þ log Y1 = log a1 + b1 logW

(19.6)

Y2 = a2W b2 Þ log Y2 = log a2 + b2 logW

(19.7)

CROSS-SPECIES AND LOW DOSE QUANTITATIVE EXTRAPOLATIONS

Figure 19.1 Profle of the linear version of allometric equation log Y = log a + b logW , the y-axis is the logarithm of a biological function, or logarithm of a selected pharmacokinetic parameter, and x-axis is the logarithm of body mass with slope of “b” and y-intercept of “log a,” representing the exponent and log of coeffcient of the allometric equation of Y = aW b . Therefore, logW =

log Y1 - log a1 log Y2 - log a2 and logW = b2 b1 \

log Y1 - log a1 log Y2 - log a2 = b1 b2

(19.8) (19.9)

Solving for Y1 in terms of Y2 yields: log Y1 = log a1 +

b1 ( log Y2 - log a2 ) b2

(19.10)

b1

æ Y ö b2 Y1 = a1 ç 2 ÷ è a2 ø

(19.11)

The quotients of similar parameters can also provide useful information. For example, the ratio of renal blood fow to cardiac output yields: Renal Blood Flow (ml/min) 43.06W 0.77 = = 0.259 Cardiac Output (ml/min) 166W 0.79

(19.12)

This indicates that the renal blood fow is approximately 26% of cardiac output in mammalian species (Mordenti, 1986). It should be noted that the sample size, methods of measurement of the parameter of interest, and the variability of body weight contribute to the variation of empirical estimates of (a) and (b). The theoretical and statistical values of allometric approaches have been questioned and criticized for shortcomings, such as the sudden changes of slope in the allometry line; signifcant errors in extrapolation to extremes of biological parameters, such as body weight; deviations of cardiac output; and deviation of cardiac energetic, to name a few (Heusner, 1984; Smith, 1980; Yates, 1979; Loiselle et al., 1979; Günther et al., 1966; White et al., 1968). It has been suggested that body mass is not equivalent to body weight. Body mass has dimension of mass, while the body weight in 563

19.1 CROSS-SPECIES EXTRAPOLATION

accordance with Newton’s second law should be considered as a force. This distinction infuences the value of exponent b (Günther and Morgado, 2003). 19.1.2.1 Allometric Approach and Chronological Time The concept of time is a critical parameter in allometric interspecies extrapolation. The small animals have a rapid heartbeat, faster anabolism and catabolism, and a shorter life. The element of time associated with any of these processes is the chronological time. It is often assumed that one human year is equivalent to seven dog years. This assumption is based on the chronological life span of a dog and a human. In other words, a dog ages 7.14% of its life span per year, whereas a human ages 7.14% of its life in seven years (Ings, 1990). Thus, theoretically if time-dependent biological processes are measured according to each species’ biological clock, animals exhibit to have the same rate. The pioneering work on allometric scaling and the concept of invariant time was frst introduced when the disposition of methotrexate was evaluated in fve different mammalian species following intravenous administration of the compound (Dedrick et al., 1970). The transformation of chronological time to biological time, known as the Dedrick Time Equivalent Model, is achieved when the dependent variable on the y-axis is normalized by dividing plasma concentrations by dose (mg/kg) and body weight (kg), and 0.25 (see Equation 19.12) represents a constant for the conversion from chronological time to biological time as the independent variable on the x-axis. The following approach, known as the Elementary Dedrick Plot, is expressed as: y - axis =

Concentration Dose ( mg / kg ) /W ( kg ) x - axis =

Time W 1-b

(19.13) (19.14)

where (b) is the exponent of clearance. When considering the clearance and the volume of distribution, the plot is expressed as (Mahmood, 1999). y - axis =

Concentration Dose /W c

(19.15)

Time W c-b

(19.16)

x - axis =

where, b and c are exponents of clearance and volume of distribution, respectively. In the Elementary Dedrick Plot, the interspecies superimposability occurs only when y = 1 (Boxenbaum, 1984). When y ¹ 1, then both the intercept and slope will be species-specifc. The chronological time is an essential component of PK/TK analysis in interspecies scaling. The concept of time in physical relativity, psychological relativity, and biological relativity has been philosophically discussed, and various conceptual ideas have been presented (Boxenbaum, 1986). For example, the allometric mammalian heartbeat time (HBT ) is characterized by Equation 19.17 (Stahl, 1968; Gunther and DeLa Barra, 1966; Boxenbaum, 1982): HBT = 0.0428 W 0.28

(19.17)

For a 30-g mouse, the heartbeat time or “cardiochron” is 0.111 seconds and for a 70-kg human, it is 0.973 seconds; W is the body weight in grams. The breath time or the pulmonary cycle time (PCT ) identifed as “pneumatochron” is: PCT = 1.169W 0.28

(19.18)

The breath time for the mouse and the human are 0.438 and 3.841 seconds, respectively. Dividing PCT by HBT is approximately 4, that is,

564

0.438 æ PCT ö = = 3.946 ç HBT ÷ ø mouse 0.111 è

(19.19)

3.841 æ PCT ö = = 3.947 ç HBT ÷ ø human 0.973 è

(19.20)

CROSS-SPECIES AND LOW DOSE QUANTITATIVE EXTRAPOLATIONS

Thus, every species has four heartbeats per breath time. From comparison of allometric exponents for the duration of periodic phenomena, it can be concluded that its value remains relatively constant around 0.25, 0.28, etc. Based on this observation, the concept of biological time is then expressed as Equation 19.21 (Boxenbaum, 1986). (19.21)

tbiological = aW 0.25

where (a) is a constant. The allometric approach is used to extrapolate the PK/TK parameters and constants of small experimental animals to humans. The smaller animals with short life spans clear xenobiotics at a faster rate per unit of body weight than the larger animals with longer life spans. The life span of animals correlates well with their size and body weight. In general, the rate of absorption, distribution, metabolism, and excretion processes depends on the biological clock of the animals. In extrapolation of pharmacokinetic parameters and constants, each species has its own unique pharmacokinetic clock. Based on the concept of pharmacokinetic time, several units of time have been proposed (Boxenbaum and Ronfeld, 1983), which are discussed below. For example, “Kallynochron” is the unit of time that all species clear the same volume of plasma per body weight (kg). It is defned as: Kallynochron =

t W 1-b

(19.22)

where (b) is the exponent from the allometric equation for clearance. “Apolysichron” is another PK/TK time that refers to the combination of clearance and the apparent volume of distribution. In one apolysichron, species eliminate the same fraction of a compound from their bodies and clear the same volume of plasma per kg body weight. Apolysichron =

t W b¢- b

(19.23)

There are two additional PK/TK time units similar to apolysichrons called “dienetichrons” and “syndesichrons” except that these units take into consideration the maximum life potential (MLP) and brain weight (BW ) (see also Chapter 11, Section 11.4.2). The MLP can be estimated by the following empirical equation (Sacher, 1959): MLP(years) = 185.4 ´ ( BW )

0.636

´ W -0.225

(19.24)

where both brain weight (BW ) and body weight (W ) are in kilograms. The application of the allometric approach for extrapolation of PK/TK parameters, such as total body clearance, is discussed in Chapter 11, Section 11.4.2.1 – 11.4.2.2. 19.1.2.2 Application of Allometric in Converting Animal Dose to Human Dose An application of the allometric methodology is the scaling up of the animal dose to human dose. The approach is analogous to Equations 19.6–19.11, that is, MLP(years) = 185.4 ´ ( BW )

0.636

´ W -0.225

(19.25)

Dividing both sides of Equation 19.25 by Whuman and rearranging the equation to determine mg dose per kg of human body weight yields:

( Dose )human ( Dose )animal Whuman

=

Whuman

æW ö ´ ç human ÷ W è animal ø

b

(19.26)

Multiplying the numerator and denominator of the right side of Equation 19.26 by Wanimal yields:

( Dose )human ( Dose )animal (Whuman )b ´ Wanimal = ´ b Whuman Wanimal (Waniimal ) ´ Whuman

(19.27)

( Dose )human ( Dose )animal (Wanimal )1-b = ´ 1 1-b Whuman Wanimal (Whuman ) =

( Dose )animal Wanimal

æW ö ´ ç animal ÷ è Whuman ø

1-b

(19.28)

565

19.1 CROSS-SPECIES EXTRAPOLATION

19.1.3 Application of PBPK or PBTK in Cross Species Extrapolation As discussed in Chapter 12, Section 12.2, in physiologically based PK/TK, the body is described by physiological compartments that represent different compartment and physiological spaces. Every organ or region, if required by the objectives of the project, can be subdivided into interstitial space, plasma, and blood cells. Each organ and its subdivisions have physiological dimensions such as volume, weight, surface area, and other known quantities that are generally known for humans and experimental animals and are provided in PBPK/PBTK modeling software (Willmann et al., 2003; Sepp et al., 2019). The in vitro measurements for a given case, like kinetic parameters of an active process or lipophilicity or binding affnity, can also be included in the PBPK/PBTK models. For cross-species extrapolation, the physiological and biochemical values for the target species, which is often human, are integrated in the PBPK/TK models of a reference species, which is often an experimental animal. The essential part of the extrapolation is the prior knowledge of the physiological/biochemical parameters from data collection. A useful approach is to identify the required parameters for each species according to the following four domains (Thiel et al., 2015): ◾ Species-specifc basic physiological parameters mainly differ in measurement of the physiological functions like organ size and tissue composition (Lin, 1995) of the two species. ◾ Unbound fraction of the xenobiotic representing the species-specifc binding to plasma proteins is determined in the laboratory. Biotransformation kinetic parameters, mainly K M and Vmax (Ito et al., 2005; Wang et al., 2006; Bai et al., 2020). ◾ Tissue-specifc gene expression profles capturing differences in the abundance of enzymes and transporters (Meyer et al., 2012). The complexity of this approach depends mainly on the availability and the ease of retrieving the relevant reference information; it also depends on the sophistication of the PBPK/TK model and the number of parameters included in the model. The predictive capability of the model is often determined by the corresponding experimental data, statistical error analysis, and robustness analysis of the extrapolated results. 19.1.3.1 Toxicogenomics The prediction of human health hazards based on animal data is limited to the differences between human and experimental animal responses to xenobiotics. This includes the types of different responses the two species may have toward various dose levels. Thus, it is important to validate animal models for the regulatory processes, including environmental sciences and drug discovery and development. Toxicogenomics, a subdiscipline of genomics, contributes signifcantly to the selection of the appropriate animal model. Furthermore, the discipline has increased and improved: ◾ the knowledge of toxic mechanisms ◾ the understanding of in vitro and in vivo systems ◾ rapid screening of xenobiotic toxicity ◾ selection of the lead compounds in drug discovery ◾ the understanding between the genetic variability and response to xenobiotic exposure ◾ prediction of differences between experimental animals and human responses to chemicals These aspects collectively have increased the effciency of predictive toxicology and improved the accuracy and insightful knowledge of the safety profle of a xenobiotic, which in turn has improved the effciency of extrapolation of animal data. In addition, toxicogenomics plays an essential role in identifying the variability of response among humans to xenobiotics. Xenobiotics induce differential gene expression, which can be defensive, adaptive, or repairing, by up or down regulation. Humans show different basal and inducible patterns of gene expression when exposed to xenobiotics. Often the exposure to xenobiotics may lead to induction of genes which are not associated with their mechanism of toxicity; although they do not characterize a particular toxicity, however, the responses from these unrelated genes can indicate the stress response 566

CROSS-SPECIES AND LOW DOSE QUANTITATIVE EXTRAPOLATIONS

from exposure to both chemical and physical stress (Rieger and Chu, 2004). The inclusion of relevant genetic parameters in the allometric equations to improve their predictability is yet to be established. 19.2 LOW-DOSE EXTRAPOLATION 19.2.1 INTRODUCTION The risk assessment of xenobiotics with hazardous side effects is carried out through a process that includes 1) hazard identifcation, 2) exposure-response characterization, 3) exposure assessment, and 4) risk characterization. The focus of this section is mainly on low-dose or low exposure-response characterization of the hazardous xenobiotic and the involvement of PK/TK in the low-dose extrapolation. The process of risk assessment is beyond the scope of this book. The challenges of estimating toxic effects of xenobiotics at low-dose exposure have received considerable attention from diverse sciences, from drug discovery and development to cancer research and risk assessment, over the past decades. The low-dose extrapolation is one of the key issues in human health risk assessment, and it deals with the uncertainty of extrapolation of data beyond the observations and measurement of the response (Figure 19.2). Depending on the types of toxicity and toxicity responses, the approaches to low-dose extrapolation of noncarcinogenic compounds is identifed as threshold low-dose extrapolation and is different from carcinogenic compounds. The threshold models assume that there are safe low doses for certain toxic xenobiotics. The non-threshold low-dose extrapolation is the assumption for chemical carcinogenesis and cancer risk assessment. The assumption was frst developed on the idea that radiation-induced cancer had no threshold and doses of radiation can cause DNA damage, even if the damage is limited, and all doses were assumed to have mathematical chances of inducing cancer. This concept was extended to chemical carcinogenesis and, like radiation, it was assumed that chemical carcinogens induce cancer and cause genetic damage and mutations at any dose levels. Thought the linear concept of non-threshold assumption in chemical carcinogenesis is considered viable for safeguarding the public health, it overlooks the DNA repair process and hepatic biotransformation and metabolic detoxifcation. However, it is worth noting that the chronic exposure may overwhelm the defense mechanisms of the body and cause permanent genetic damage. Thus, as will

Figure 19.2 Depiction of the dose response curve, highlighting the uncertainty associated with the extrapolation beyond the limit of measurements, and the prediction of response to determine the safe low dose; the threshold profle is for the noncarcinogenic toxic xenobiotics predicting low doses that the body can neutralize with its defense mechanisms, whereas the non-threshold profle is for carcinogens with no plausible safe low dose. 567

19.2.1 INTRODUCTION

be discussed in this chapter, the low-dose extrapolation of carcinogens is different from noncarcinogenic hazardous compounds, and the models are mainly probability-based extrapolation, and the PK/TK parameters and constants are considered elements of probability. The National Research Council (NRC, 2009), in its 2009 report “Science and Decisions: Advancing Risk Assessment,” indicated that the exposure-response for cancer low-dose extrapolation is driven by stochastic events and does not incorporate information on human sensitivity, but the noncancer toxicity exposure is driven by the individual sensitivity and is not stochastic. A different scenario of dose extrapolation is the interspecies extrapolation of the data, for example, from experimental animals to human, the data must be generated from a dose level that is extrapolated from a high dose to a low dose, appropriate for the objectives of the experimental protocol. 19.2.2 Threshold and Non-Threshold Models There are different mathematical models for estimating the dose exposure. Most express the probability of response as a function of dose and start with the experimental evaluation of response at different dose levels in one or more species that are relevant to the objectives of the project and human exposure. The general probability equation is

( Pr )response =

f (Dose)

(19.29)

The models differ only in the choice of function f and refect two types of threshold and nonthreshold scenarios. The threshold models assume that below a threshold, which can be the magnitude of exposure, a dose, or concentration, there is no adverse health effect and there is a minimally effective dose. The non-threshold models assume there is no minimally acceptable dose and any nonzero dosage is considered unsafe, as in the Delaney clause for carcinogenic response. The Delaney clause was a 1958 amendment to the Food, Drugs, and Cosmetics Act of 1938, sponsored by Congressman Jim Delaney (New York), who verbalized the amendment as: “the Secretary of the FDA shall not approve for use in food any chemical additive found to induce cancer in man, or, after tests, found to induce cancer in animals.” The following are the models often used in low-dose extrapolation. The few models discussed in this chapter for low-dose extrapolation may also have other applications in biological sciences. Although they have no frm biological basis, their main purpose in low-dose extrapolation is to establish an acceptable balance between the risk and beneft of xenobiotics. ◾ the Probit model (Finney, 1952; Gad and Weil, 1986) ◾ the Logit model (Gad and Weil, 1986) ◾ the One-hit model (Hoel et al., 1975; Gad and Weil, 1986; Bailar et al., 1988) ◾ the Gamma Multi-Hit model (Cornfeld et al., 1979; Gad and Weil, 1986) ◾ the Armitage–Doll Multi-Stage model (Armitage and Doll, 1961; Gad and Weil, 1986) ◾ a simplifed Statistico-Pharmacokinetic model (Cornfeld, 1977; Gad and Weil, 1986) ◾ the Weibull model (Gad and Weil, 1986) ◾ the log-Probit model (Gad and Weil, 1986) ◾ the Steady-State model (Gad and Weil, 1986) ◾ the Gompertz function (Boxenbaum et al., 1988) ◾ the modifed Hill equation (Egorin et al., 1986, 1987) All these models, except one, are statistical in nature and do not rely on physiologic, biochemical, or PK/TK properties of a xenobiotic. The discussion here will be limited to a few of the above models. 19.2.2.1 The Probit Model The assumption of the model is that the logarithm of tolerances has a normal distribution with mean and standard deviation of m and s, respectively. The proportion of subjects responding to dose, P ( Dose ) , is:

568

CROSS-SPECIES AND LOW DOSE QUANTITATIVE EXTRAPOLATIONS

P ( Dose ) = F éë( log Dose - m ) s ùû (19.30)

= F ( a + blog Dose )

where F( x) is the standard normal integral from -¥ to x ; and a = -m s and b = 1 s. The doseresponse curve corresponding to the equation has P ( Dose ) near zero if Dose is close to zero, and as Dose increases, P ( Dose ) approaches one. A plot of probit dose-response is a sigmoidal with the quantity b as the slope of the probit line, where the equation of the line is: Y = F -1 éP ë ( Dose ) ùû = a + blog Dose

(19.31)

19.2.2.2 The Logit Model This model also generates a sigmoidal dose-response curve symmetric in about 50% of responses: P ( Dose ) =

1 - a + b log Dose ) 1+ e (

(19.32)

Like the Probit model, as Dose decreases P ( Dose ) also decreases and approaches zero more slowly than the Probit model, that is (19.33)

b lim é P ( Dose ) ( Dose ) ù = constant ë û

Dose®0

In general, the Logit model estimates a virtual safe dose lower than the Probit model. 19.2.2.3 The One-Hit Model This model is essentially a linear dose-response model with the assumption that only one hit, corresponding to one genetic change, is needed to transform a normal cell into a cancerous cell. The relationship of the model is: P ( Dose ) = 1 - e -lDose

(19.34)

where lis a rate constant of change of the dose response curve at Dose = 0 ; l Dose is the expected number of hit at dose level (Dose). Being a linear relationship, the dose-response curve of the model is linear. Thus, any dose of carcinogen presents a risk of cancer and only when Dose = 0 , P(Dose) = 0. Although the model tacitly refers to a biochemical interaction, it is considered a phenomenological model rather than a biochemical one. The probability of the phenomenon, that is a normal cell transformation into a cancerous cell, according to this model, follows the Poisson probability distribution. The model is viewed by some as simplistic and conservative. 19.2.2.4 The Gamma Multi-Hit Model This model is an extension of the one-hit model and assumes that if a number of hits (identifed as k hits) are required to induce cancer, then the probability of generating cancer as a function of exposure to a Dose is: P ( Dose ) = 1 -

k -1

å i=0

( lDose ) e -ld » ( lDose ) i

i!

k!

k

(19.35)

When k = 1, Equation 19.35 changes to Equation 19.34, which is indicative of the linear low-dose region of the plot. For k > 1 it is convex and for k < 1 it is concave. k For small l ( Dose ) , the model can be modifed to: P ( Dose ) = l ( Dose ) , or

(19.36)

log P ( Dose ) = log l + k log Dose

(19.37)

k

Therefore, k is the slope of a straight line generated from plotting log P ( Dose ) versus log Dose . Equation 19.35 describes a dose response curve that follows gamma distribution with k as the shape parameter. In addition to carcinogenesis, the multi-hit model is used in areas such as Alzheimer (Steele et al., 2022).

569

19.2.1 INTRODUCTION

19.2.2.5 The Armitage-Doll Multi-Stage Model The assumption of the model is that cancer originates from a malignant cell, formed through a multi-stage mutation. Thus, the effect of the agent is considered additive at those stages. This seven decades-old model is still used to defne carcinogenesis (Wilkins et al., 2023). The related probability function is: P ( Dose ) = 1 - e

é¥ iù -ê a i ( Dose ) ú ûú ëê i=0

å

for a i ³ 0

(19.38)

The model is also expressed as - b + g ( Dose ) )(bi+1 + g i+1 ( Dose ) )˜˜˜(bk + g k ( Dose ) ) P ( Dose ) = 1 - e ( i i

(19.39)

where bi ³ 0 , g i ³ 0 , and k is the number of mutational stages. In a simpler format, Equation 19.39 generates the Equation 19.40 that corresponds to the general Equation 19.38, that is, P ( Dose ) = 1 - e

2

-a 0 +a1 ( Dose ) +a 2 ( Dose ) +......+ g k ( Dose )

k

for a i ³ 0

(19.40)

The model also assumes that each stage of mutation follows a Poisson distribution. 19.2.2.6 Statistico-Pharmacokinetic Model This model takes into consideration the simultaneous and reversible activation and inactivation of the agent in the body. The probability of a response is assumed to be linearly proportional to the activated complex. If the total amount of the compound in the body is equal to S and if the deactivating agent is T , and the ratios of the back and forth rate constants of reactions governing the activation step are K , and deactivation step by K * , the model can then be expressed as: P ( Dose ) =

Dose - S éë P ( Dose ) ùû - y

Dose - S éë P ( Dose ) ùû - y + K

for Dose > T

(19.41)

The parameter y in Equation 19.41 is: y=

K éë P ( Dose ) ùû T

(19.42)

K ëé P ( D ) ûù + K ëé1 - P ( D ) ùû *

For Dose < T : P ( Dose ) @

Dose T ö æ S + K ç1 + * ÷ K è ø

(19.43)

At dose levels ofD < T , the dose-response curve closely resembles a linear plot. When K * = 0 , the dose response curve has a threshold at Dose = T , and for K * > 0 , the dose-response curve resembles a hockey stick. The constants S, T , K , and K * are considered species-dependent kinetic constants. REFERENCES Adolph, E. F. 1949. Quantitative relations in the physiological constitutions of mammals. Science 109(2841): 579–85. Armitage, P., Doll, R. 1961. Stochastic models for carcinogenesis. In Proceeding of the Fourth Berkeley Symosium on Mathematical Statistics and Probability, ed. J. Neyman. Vol 4, 19–38. Berkeley and Los Angeles: University of California Press. Bai, Y., Peng, W., Yang, C., Zou, W., Liu, M., Wu, H., Fan, L., Li, P., Zeng, X., Su, W. 2020. Pharmacokinetics and metabolism of naringin and active metabolite naringenin in rats, dogs, humans, and the difference between species. Front Pharmacol. https://doi.org/10.3389/fphar.2020 .00364. 570

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Bailar, J. C. 3rd, Crouch, E. A., Shaikh, R., Spiegelman, D. 1988. One-hit models of carcinogenesis: Conservative or not? Risk Anal 8(4): 485–97. Boxenbaum, H. 1982. Interspecies scaling, allometry, physiological time, and the ground plan of pharmacokinetics. J Pharmacokinet Biopharm 10(2): 201–27. Boxenbaum, H. 1984. Interspecies pharmacokinetics scaling and the evolutionary-comparative paradigm. Drug Metab Rev 15(5–6): 1071–121. Boxenbaum, H. 1986. Time concept in physics, biology, and pharmacokinetics. J Pharm Sci 75(11): 1053–62. Boxenbaum, H., Neafsey, P. J., Fournier, D. J. 1988. Hormesis, Gompertz functions and risk assessment. Drug Metab Rev 19(2): 195–229. Boxenbaum, H., Ronfeld, R. 1983. Interspecies pharmacokinetic scaling and the Dedrick plots. Am J Physiol 245(6): R768–74. Calder III, W. A. 1984. Size Function, and Life History. Cambridge: Harvard University Press. Chappel, W. R., Mordenti, J. 1989. The use of interspecies scaling in toxicokinetics. In Toxicokinetics and New Drug Development, eds. A. Yacobi, J. P. Skelly, V. K. Batra, 42–96. New York: Pergamon Press. Cornfeld, J., Carlborg, F. W., Van Ryzin, J. 1979. Setting tolerance on the basis of mathematical treatment of dose-response data extrapolated to low doses. In Proceedings of the First Internat Toxicol Congress on Toxicology. New York: Academic Press. Dedrick, R. L., Bischoff, K. B., Zaharko, D. S. 1970. Interspecies correlation of plasma concentration history of methotrsxate (NSC-740). Cancer Chemother Rep 54(2): 95–101. Davidson, I. W. F., Parker, J. C., Beliles, R. P. 1986. Biological basis for extrapolation across mammalian species. Regul Toxicol Pharmacol 6(3): 211–37. Egorin, M. J., Van Echo, D. A., Whitacare, M. Y., Forrest, A., Sigman, L. M., English, K. L., Aisner, J. 1986. Human pharmacokinetics, excretion and metabolism of the anthracycline antibiotic menogaril (7-OMEN, NSC 269148) and their correlation with clinical toxicities. Cancer Res 46(3): 1513–20. Egorin, M. J., Conley, B. A., Forrest, A., Zuhowski, E. G., Sinibaldi, V., Van Echo, D. A. 1987. Phase I study and pharmacokinetics of menogaril (NSC 269148) in patients with hepatic dysfunction. Cancer Res 47(22): 6104–10. Finney, D. J. 1952. Statistical Method in Biological Assay. New York: Hafner Publishing Co. Gad, S., Weil, C. S. 1986. Statistics and Experimental Design for Toxicologist. Caldwell: Telford Press. Günther, B., DeLa Barra, L. 1966. Physiometry of the mammalian circulatory system. Acta Physiol Lat Am 16(1): 32–42. Günther, B., Morgado, E. 2003. Dimensional analysis revisited. Biol Res 36(3–4). https://doi.org/10 .4067/S0716-97602003000300011. Heusner, A. A. 1984. Biological similitude: Statistical and functional relationships in comparative physiology. Am J Physiol 246(6 Pt 2): R839–45. Hoel, D. G., Galor, D. W., Kirschstein, R. L., Saffotti, V., Schneiderman, M. A. 1975. Estimation of risks of irreversible delayed toxicity. J Toxicol Environ Health 1(1): 133–51.

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Ings, R. M. J. 1990. Interspecies scaling and comparisons in drug development and toxicokinetics. Xenobiotica 20(11): 1201–31. Ito, Y., Yokota, H., Wang, R., Yamanoshita, O., Ichihara, G., Wang, H., Kurata, Y., Tagaki, K., Nakajima, T. 2005. Species differences in the metabolism of di(2-ethylhexyl)phthalate (DEHP) in several organs of mice, rats, and marmosets. Arch Toxicol 79(3): 147–54. Kleiber, M. 1932. Body size and metabolism. Hilgardia 6(11): 315–53. Kleiber, M. 1947. Body size and metabolic rate. Physiol Rev 27(4): 511–41. Lin, J. H. 1995. Species similarities and differences in pharmacokinetics. Drug Metab Dispos 23(10): 1008–21. Loiselle, D. S., Gibbs, C. L. 1979. Species differences in cardiac energetics. Am J Physiol 237(1): H90–8. Mahmood, I., Balian, J. D. 1999. The pharmacokinetic principles behind scaling from preclinical results to phase I protocols. Clin Pharmacokinet 36(1): 1–11. McMahon, T. A., Bonner, J. T. 1983. On Size and Life, 69–110. New York: Scientifc American Library. Meyer, M., Schneckener, S., Ludewig, B., Kuepfer, L., Lippert, J. 2012. Using expression data for quantifcation of active processes in physiologically based pharmacokinetic modeling. Drug Metab Dispos 40(5): 892–901. Mordenti, J. 1986. Man versus beast: Pharmacokinetic scaling in mammals. J Pharm Sci 75(11): 1028–40. Mordenti, J., Chen, S. A., Moore, J. A., Ferraiolo, B. L., Green, J. D. 1991. Interspecies scaling of clearance and volume of distribution data for fve therapeutic proteins. Pharm Res 8(11): 1351–9. National Research Council. 2009. Science and Decisions: Advancing Risk Assessment. Washington, DC: The National Press. Peters, R. H. 1983. The Ecological Implications of Body Size. Cambridge: Cambridge University Press. Rieger, K., Chu, G. 2004. Portrait of transcriptional responses to ultraviolet light and ionizing radiation in cells. Nucl Acids Res 32(16): : 4786–803. Rubner, M. 1883. Ueber den Einfuss der korpergrosse auf stoff und kraft Wechsel. Z Biol 1919: 535–62. Sacher, G. 1959. Relation of lifespan to brain weight and body weight in mammals. In Ciba Foundation Colloquia on Aging, eds. G. E. W. Wolstenholme, M. O’Conner, 115–33. London: Churchill. Sepp, A., Meno-Tetang, G., Weber, A., Sanderson, A., Schon, O., Berges, A. 2019. Computerassembled cross-species/cross-modalities two-pore physiologically based pharmacokinetic model for biologics in mice and rats. J Pharmacokinet Pharmacodyn 46(4): 339–59. Smith, R. J. 1980. Rethinking allometry. J Theor Biol 87(1): 97–111. Thiel, C., Schneckener, S., Krauss, M., Ghallab, A., Hofmann, U., Kanacher, T., Zellmer, S., Gebhardt, R., Hengstler, J. G., Kuepfer, L. 2015. A systematic evaluation of the use of physiologically based pharmacokinetic modeling for cross-species extrapolation. J Pharm Sci 104(1): 191–206. Stah, W. R. 1967. Scalling of respiratory variables in humans. J Appl Physiol 22(3): 453–60. 572

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Steele, O. G., Stuart, A. C., Minkley, L., Shaw, K., Bonnar, O., Anderle, S., Penn, A. C., Rusted, J., Serpell, L., Hall, C., King, S.,2022. A multi-hit hypothesis for an APOE4-dependent pathophysiological state. Eur J Neurosci 56(9): 5476–515. Voisin, E. M., Ruthsatz, M., Collins, J. M., Hoyle, P. C. 1990. Extrapolation of animal toxicity to humans: Interspecies comparisons in drug development. Regul Toxicol Pharmacol 12(2): 107–16. Wang, Q., Ye, C., Jia, R., Owen, A. J., Hidalgo, I. J., Li, J. 2006. Interspecies comparison of 7-hydroxycoumarin glucuronidation and sulfation in liver S9 fractions. In Vitro Cell Dev Biol Anim 42(1–2): 8–12. White, L., Haines, H., Adam, T. 1968. Cardiac output related to body weights in small mammals. Comp Biochem Physiol 27(2): 559–65. Wilkins, A., Corbett, R., Eeles, R. 2023. Age distribution and a multi-stage theory of carcinogenesis: 70 years on. Br J Cancer 128: 404–406, https://doi.org/10.1038/s41416-022-02009-9 Willmann, S. L. J., Sevestre, M., Solodenko, J., Fois, F., Schmitt, W. 2003. PK-Sim®: A physiologically based pharmacokinetic ‘whole body’ model. BIOSILICO 1(4): 121–4. Yates, F. E. 1979. Comparative physiology: Compared to what? Am J Physiol 237(1): R1–2.

573

20 Practical Application of PK/TK Models: Population Pharmacokinetics/Toxicokinetics 20.1 INTRODUCTION Most frequently, the profle of a xenobiotic concentration in biological samples like plasma, urine, and other specimens differ considerably among humans who are on similar dosing regimen, single or multiple dosing, and using similar route of administration. In general, the differences can be attributed and classifed as 1) the genetic polymorphism found in the general population 2) the degree of severity of the disease state for which the therapeutic agent is given, for instance hepatic failure or renal impairment, and interaction with other therapeutic agents; and 3) the lifestyle-related infuences like the type of food, drinks, regularity of exercise, being confned to bed, and so forth; and 4) dose-dependent, or dose-independent pharmacokinetic behavior of the compound in certain individuals with specifc dosing regimen (also see Chapter 1, Section 1.2.3). The quantifcation of the pharmacokinetic variabilities of these differences is referred to as population pharmacokinetics, popPK, and it is defned as “quantitative methodology” to explain the variability in drug concentrations among individuals who are the target population receiving clinically relevant doses of a drug of interest (Sheiner et al., 1977; Grasela and Sheiner, 1991; Aarons, 1991; Population Pharmacokinetics, FDA Guidance for Industry, 1999, 2022). The goals are mainly to obtain relevant pharmacokinetic parameters in a group of patients that represent the target population and measuring the variability between the subjects and linking it to various demographical, pathophysiological, genetic, and environmental factors. The aim is to optimize different aspects of therapy, like designing effective dosing regimens for a patient population that suffers from organ failure or a population of patients that requires a combination therapy and are vulnerable to drug–drug interaction, and perform comparative analysis between patients and healthy volunteers, between single and multiple dosing, and so forth. A typical example of a popPK study is a Phase III clinical trial or Phase IV post-marketing surveillance (Chapter 21, Section 21.3.5); where large sets of data are gathered from large representative groups of the target population at different medical centers with nonuniform and sparse data collection and with inter- and intra-subject variability (Sheiner, 1984, FDA Guidance for Industry, 1999). Many popPK study designs can infuence the study outcomes and related interpretations. For example, design factors such as sampling strategy (Ette et al., 1995), number of observations (Ette et al., 1995), sampling time recording (Sun et al., 1996), and study compliance (Girard et al., 1996) may infuence the outcome of a popPK investigation. Furthermore, the popPK/TK study outcomes are infuenced by factors that were generalized earlier in this section and are expanded as: ◾ Statistical characteristics of the population, such as age and body weight (Langenhorst et al., 2019; Crombag et al., 2019), ethnicity (Pétain et, 2019), gender (Ng et al., 2009), or surface area. ◾ Pathological factors, such as type of illness that infuence the PK/TK profle of xenobiotics in the body, like hepatic or renal impairment, atrial fbrillation (Ueshima et al., 2019), or having multiple diseases (comorbidity). ◾ Environmental factors, such as infuence of environmental waste products, job-related exposure to organic solvents, smoking, meals, exercise, diet, being bedbound, and other related habitual and environmental factors (Zou et al., 2022; Goti et al., 2018). ◾ Genetic polymorphism, for example, genetic factors infuencing enzyme systems like CYP 450 subfamily enzymes, for example, members CYP2D, CYP2C, or CYP3A, that are responsible for metabolism of xenobiotics in vivo and thus infuence the total body clearance of drugs or genetic phenotypes that affect the functions of infux, effux, and receptor proteins, and genetic interactions (Senek et al., 2020; Jeong et al., 2022). ◾ Physiological conditions like pregnancy, hyperthyroidism or hypothyroidism, and neonates (Pillai et al., 2015). ◾ Drug–drug, drug–herb, or drug–food interactions (Barnett et al., 2018). ◾ Circadian pattern (Kobuchi et al., 2018). 574

DOI: 10.1201/9781003260660-20

POPULATION PHARMACOKINETICS/TOXICOKINETICS-APPLIED PK-TK MODELS

◾ Formulation of medication (Thakkar et al., 2018). ◾ Other factors, like drug adherence or dosing regimen, that can impact and modify the PK profle of a compound. There are various approaches for analysis of population pharmacokinetic data. Some approaches, like the naïve pooled-data approach (discussed in this chapter), relies on isolated PK analysis of individual subject, overlooking the inter- and intraindividual variability; and other approaches, like the nonlinear mixed-effects model, where the PK analysis is an integrated component of a broader analysis, which takes into consideration some of the factors discussed above. Most popPK analyses use the latter approach, which considers the relationship between physiology (normal and disease states) and the PK/TK of a compound in a population of patients. It assesses the extent of interindividual variability in the population and its infuence on PK analysis. It identifes the demographic, pathophysiological, environmental, or drug-related origin factors that infuence the PK behavior of a drug. It is considered an enabling tool for pharmacokinetic analysis of sparsely sampled data, particularly when there are limitations in collecting the data, for example, AIDS patients (Pfster et al., 2003)), critical care patients (Georges et al., 2009), cancer patients (Sugiyama et al., 2010), and neonates (Urien et al., 2011; Shellhaas et al., 2013), to name a few. It is believed that the combination of heterogeneous data enhances the power of analysis in determining the linearity or nonlinearity of the data and achieves precision in analysis. It analyzes the residual interindividual variability and estimates the magnitude of unexplained variability. It allows for the combination of diverse sets of data from various sources, for example, limited plasma, serum, and/ or blood data from assorted sources analyzed with various but reliable analytical methodologies. Population analyses of PK and PD (popPK and popPD) data in support of new drug applications are often discussed and encouraged by the regulatory agencies. Various US Food and Drug Administration (FDA) and ICH guidelines have sections on popPK, with helpful guidance for the New Drug Application submission and approval process for new drug entities. Examples are ICH E5 guideline (ICH E5, 1998) related to the infuence of race factors on the PK assessment of a population; FDA guideline on the infuence of gender on popPK (FDA Guideline, 1993); ICH E7 guideline (ICH E7, 1993) related to the infuence of age and PK in geriatrics; ICH E11 guideline (ICH E11, 2000) pertaining to the evaluation of pediatric popPK; ICH E7 (ICH E7, 1993) and FDA guidance on pharmacokinetics in patients with impaired hepatic function (FDA Guideline, 2003); and FDA guidance on in vivo drug interaction studies (FDA Guidance, 2006), etc. 20.2 FIXED EFFECT AND RANDOM EFFECT PARAMETERS The emphasis in evaluation of PK/PD parameters by popPK analysis is the elucidation of the doseconcentration–response relationship of a drug based on inter- and intraindividual variability. In analysis of variability, two types of parameters are taken into consideration (Whiting et al., 1986; Karlsson et al., 1993; Sheiner et al., 1983): fxed effect and random effect parameters. 20.2.1 Fixed Effect Parameters Fixed effect parameters refer to the average values of PK parameters in a patient population, and/or average relationship between the PK parameters and the measurable biological factors, or biomarkers’, concentration. Thus, the defning feature of a fxed effect model is that all evaluations share a common effect size, and characteristics of the population are refected by the average values with no covariates. 20.2.2 Random Effect Parameters Random effect parameters quantify the random variability of the popPK data and represent the infuence of interindividual variability; inter-occasion variability, that is, the random differences in an individual between different occasions; and residual variability (the unexplained variability often referred to as intraindividual or within subject variability. The defning feature of the random effect model is that there is a distribution of true parameters, and the aim is to estimate the mean of this distribution. 20.2.3 Linear and Nonlinear Mixed-Effect Models The mixed-effects models incorporate two types of parameters, the fxed and the random parameters. These parameters are used when the population pharmacokinetics is investigated and the data from all subjects in the population are used simultaneously. The required elements for developing a population pharmacokinetics of a mixed-effect model are: 575

20.2 FIXED EFFECT AND RANDOM EFFECT PARAMETERS

◾ The data are from a population. ◾ The plasma concentration of drug in the population is described by an appropriate PK/TK model. ◾ The statistical model to defne random variability, such as between-individual, between-occasion, residuals, etc., and variability around the structural model are identifed. ◾ The covariate model includes important covariates that are related to the properties of the drug; for example, highly lipophilic or highly metabolized, CYP isozymes, measurable genetic factors, weight, or other covariates are identifed and incorporated. ◾ The modeling software capable of bringing the data from all elements of investigation together is selected. Like other statistical models, mixed-effects models describe the relationship between a response variable (like PK parameters) and covariates that have been measured in conjunction with the response. The challenge of popPK analysis in using these models is the identifcation of the most important covariate and its interaction with other covariates of the model (Ribbling et al., 2004). For example, the infuence of body weight on the clearance of a drug, which in turn affects the plasma concentration, is a relevant covariate. However, selecting the body weight as the only covariate defeats the purpose of popPK analysis and increases the bias in parameter estimation. The signifcance of using these models for population analysis is in the relationship between PK parameters and covariates and between the covariates, for example, body weight with gender, ethnicity, etc. An appropriate popPK model with relevant covariates can add to the knowledge of drug effcacy and optimum therapy. The required data for popPK analysis includes the following: ◾ plasma/serum/whole blood concentration data, optimized or sparse, determined by a validated analytical methodology ◾ known dose and dosing regimen ◾ same route of administration ◾ same formulation of the drug ◾ measurements of pharmacological response(s), PD/TD parameters ◾ data related to the selected covariates, for example, disease state(s), concurrent administration of other therapeutic agents, demographical data, environmental factors, and so forth ◾ accurate reporting of days and times of sampling and data collection ◾ report of any side effects and toxicological response. The mixed-effects models can be identifed as linear or nonlinear mixed-effects models. 20.2.3.1 Linear Mixed-Effects Model The model that describes response variable as a linear function of both the fxed and random effects, plus unit error term, is called the linear mixed-effects (LME) model (Laird et al., 1982). This model represents the relationship between a response and independent covariates with coeffcients that can change with respect to one or more covariates. The standard form of a linear mixed-effects model is: y = M F f + Mrb + e

(20.1)

Where (y) is the response, a vector of n observations; MF is the fxed-effects matrix; f is the fxed effects vector; Mr is the random effects matrix; b is the random effects vector; and e is the observation error vector. Thus, MF a is the fxed effects part of the equation; Mrb is the random component of the equation; and e is the error term. 20.2.3.2 Nonlinear Mixed-Effects Model When the response is expressed as a nonlinear function of both fxed and random effects (mixed effects), plus unit error term, it is identifed as nonlinear mixed-effects (NLME) model (Sheiner et al., 1980). In other words, the drug concentration or dependent variable is a nonlinear function of the model parameters and independent variable. Nonlinear mixed-effects model is a widely 576

POPULATION PHARMACOKINETICS/TOXICOKINETICS-APPLIED PK-TK MODELS

accepted popPK model (Phoenix®NLME, 2011). The detection and characterization of nonlinear processes, whether related to capacity-limited metabolism, absorption, or excretion, are important for the safe and effective use of a therapeutic agent (Ludden, 1991). In the NLME model, the PK model for estimation of PK parameters, is embedded in a statistical model (Prague and Lavielle, 2022). The assumptions of the statistical model should be determined in terms of covariates as they relate to the expected variability within and among individuals. For more extensive mathematical and statistical steps of the nonlinear mixed-effect model, relevant references should be consulted (Davidian and Giltinan, 1995, 2003; ; Bonate, 2005; Laffort et al., 2011; Nedelman, 2005). The initial basic setup is summarized below (Davidian, 2009) for a population of N individuals and several measurements for each patient: N = Number of individulas of i = 1,......, N ni = number of measurementsfor individual i PK/PD or TK/TD measurements of the outcome per individual is i = y i1 , y i2 , y i3 ,....y ini PK/TK measurements refer to plasma concentration, or other biological samples, and PD/TD measurements are the measurable responses or measurements of relevant biomarkers. The PK/PD or TK/TD measurements are taken at time pointsti1 , ti 2 , ti 3 ,....., tini . Thus, the measurement for ith individual at tij is y ij , where j = 1, 2, 3,...., ni The measurement–time relationship requires the input, which is also “within-individual covariates.” Setting it equal toQi , it can be defned as: For a single oral dose that is frst-order absorption administered via an extravascular route): Qi = Di

(20.2)

For zero-order input: Qi = Di infused over the time of infusion The “among-individual covariates,” which refer to demographical, environmental, and pathophysiological covariates, are identifed as X i , which theoretically do not change with time and remain constant during the observation and sampling. The collected data from N individuals of the population can be defned as: Yi , Xi , where i = 1,...., N Yi = ( y i1 ,....., y ini )

(20.3)

Xi = ( Qi , X i )

(20.4)

Therefore, the measurements of an individual subject are y ij = m ( tij , Qi , qi ) + eij , j = 1,...., ni

(20.5)

Where q corresponds to the key pharmacokinetic parameters, e.g., for a one-compartment model with frst-order input, there are three parameters (i.e., r = 3 ), and they correspond to the absorption rate constant, clearance, and the volume of distribution. qi = ( k a i , Vdi , Cli ) = ( qi1 , qi2 , qi3 )

(20.6)

Therefore, at the individual level, the one-compartment model with frst-order input with Qi = Di and r = 3 can be defned as: m ( ti , Qi , qi ) =

æ - çæ Cli ö÷t ö FDi k ai ç e è Vdi ø - e -kai t ÷ ÷ Vdi ( k ai - Cl Vdi ) ç è ø

eij = y ij - m ( tij , Qi , qi )

(20.7)

(20.8)

With the conditional expectation (E) of

(

)

E eij Qi , qi = 0

(20.9)

577

20.2 FIXED EFFECT AND RANDOM EFFECT PARAMETERS

At the population level, the individual parameters are defned as qi = d ( X i , b, bi ) i = 1,...., N ,

( r ´ 1)

(20.10) (20.11)

(d ) is the r -dimensional function of (qi ) and (X i ), in terms of b and bi , which corresponds to fxed effects ( p ´ 1) and random effects( q ´ 1) , respectively; Thus, the individual parameters at the population level are function of “among-individual covariates,” fxed and random effects. For example, if the “among-individual covariates” are age, weight, and creatinine clearance X i = ( wti , agei , Ccri ) ,

(20.12)

bi = ( bi1 , bi2 , bi3 ) for

(q = 3)

(20.13)

b = (b1 ,.....,b7 ) for

(p = 7)

(20.14)

Therefore k ai = qi1 = d1 ( X i , b, bi ) = eb1 +bi1

(20.15)

Vdi = qi2 = d2 ( X i , b, bi ) = eb2 + b4wti +bi 2

(20.16)

Cli = qi3 = d3 ( X i , b, bi ) = eb3 + b5wti +b 6 Ccri +b7 agei +bi 3

(20.17)

Taking logarithm of Equations 20.15–20.17 yields: log k ai = b1 + bi1

(20.18)

logVdi = b2 + b 4 wti + bi2

(20.19)

log Clib3 + b5 wti + b6 cri + b7 agei + bi3

(20.20)

The NLME models are complex with numerous assumptions about the structure of model and variability distributions. The NLME models rely heavily on statistical models, and having access to a well-designed popPK/PD or popTK/TD is the essential part of embarking on population pharmacokinetic analysis. 20.2.3.3 Partially Linear Mixed-Effect Model A partially linear mixed-effects (PLME) model, also known as semiparametric mixed-effects model, uses a set of spline basis functions that are treated as the random effect for the purpose of neutralizing the infuence of one variable, for instance, the time and measuring effect of another covariate on the response directly (Bonate, 2005; Rupert et al., 2003; Hardle et al., 2001). The advantages of PLME, in addition to the use of the spline basis function removing the need for a model that measures time, is that the model is based on a linear mixed-effects model and is faster to compute. However, there are some limitations associated with the approach, e.g., the model assumes the observations are independent and fails to consider the “within the subject” correlation. 20.2.3.4 Naïve-Pooled Data Approach The naïve-pooled and the two-stage approaches were commonplace prior to the introduction of population-based models (Sheiner et al., 1980). The naïve-pooled data analysis consisted of pooling all data points as though they belong to one individual and using least-square ftting to determine the parameters of the data set. Although the convenience of parameter estimation may be considered an advantage, nonetheless the methodology ignores the interindividual variability and may generate realistic estimates of parameters if the subject variability is very small; a scenario that can only be achieved in the lab with small experimental animals of the same strain. In general, the naïve-pooled approach is not considered an acceptable methodology for data analysis in humans. However, there have been reports indicating that the methodology can accurately estimate the mean parameters for several drugs used in anesthesia (Gustafsson et al., 1992; Shaefer et al., 1990; Dyke et al., 1993). 578

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20.2.3.5 Naïve Average Data Approach In this approach the mean value of the data for each time point of sample collection is calculated frst. Thus, all collected samples will change to one set of mean values. A model is then ftted to the mean values to determine the parameters of the population. This approach also ignores the subject and between-occasion variability and may lead to biased estimation of the population parameters. 20.2.3.6 Standard Two-Stage Approach The frst stage of the STS approach involves the calculation of model parameters for each subject individually. In the second stage, the parameters of the population are identifed as the arithmetic or geometric mean plus variance/covariance of all individually calculated parameters. The method does not take into consideration the variability and reliability of the individual estimates and has been shown to overestimate the parameters variance/covariance (Davidian and Giltinan, 1995). However, it is used commonly for PK parameter estimation in experimental animals. 20.2.3.7 Global Two-Stage Approach Like the Standard method, the parameters are estimated for each patient frst, and then iteratively optimized to estimate the population mean and variance/covariance of each parameter (Steimer et al., 1984). Qualitatively the Global Two-Stage (GTS) method is preferred to the Standard, but in dealing with large populations, it suffers from the same shortcomings as the Standard approach. 20.2.3.8 Iterative Two-Stage Approach The important feature of this approach is that the information from the sample mean and variance/covariance of the population are used as prior knowledge, in the context of Bayesian estimation, in quantifying the individual parameters from the data (Steimer et al., 1984). The method estimates the parameters more accurately, but the calculated data are not signifcantly different from the GTS method. In all two-stage approaches, accurate and precise estimates of model parameters are essential in the frst stage. This would require multiple and appropriately timed blood samples, which may not be feasible in some patient populations, and it is an inherent drawback of two-stage approaches. 20.2.3.9 Bayesian Approach This approach is computationally more demanding, and its solution is achieved through the application of Markov chain Monte Carlo methods (Racine-Poon, 1985; Wakefeld and Walker, 1997; Gelfand et al., 1990). Briefy, the calculated parameters are viewed as random variables with a distribution attached to each parameter to express the uncertainty with their values. The distribution is known as the prior distribution, which represent the prior knowledge about the parameter of interest. The prior distribution of the parameters across the population and the calculated data from an individual are used in estimation of an individual’s parameters (Dokoumetzidis et al., 2005). Prior distributions are incorporated in Bayesian analysis using Bayes’ Rule (Dempster et al., 1977). The estimation of mean and variance/covariance requires the estimates of the priors for the parameters. The method is used more often for therapeutic drug monitoring and to explore the covariate relationships (Dansirikul et al., 2005; Wendling et al., 2015). 20.3 COMPUTATIONAL TOOLS FOR POPPK/TK The commonly used software in popPK analysis includes NONMEM 7.5 (Bauer, 2011; Frame and Beal, 1998); S-ADAPT (Bulitta et al., 2011); MONOLIX (Chan et al., 2011); SAS (Galcki, 1998); and the new version of Phenix® NLME™ with QRPEM (Quasi-random Parametric Expectation Maximization), a method for likelihood expectation maximization (EM), which is considered an improvement in the population pharmacokinetic analysis of the nonlinear mixed-effect model (Leary et al., 2011); WinBUGS (BUGS stands for Bayesian inference Using Gibbs Sampling) is the software for Bayesian analysis using Markov Chain Monte Carlo (Lunn et al., 2009); and SimBiology for MATLAB. There have been certain critiques about these programs. For example, the programs such as NONMEM (Sheiner et al., 1982) and SAS macro NLINMIX use frst-order Taylor series expansion. The series expansion, also known as First-Order method, is considered the limitation of the program when dealing with nonlinear random effect parameters or nonlinear residual errors (Ette et al., 2007). The First-Order method of NLME has been reported to have potential in generating bias 579

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estimates (Beal, 1984; Sheiner et al., 1981; Sheiner, 1984; Grasela et al., 1986; Racine-Poon, 1985; Ette et al., 1994; 1995; Karlsson et al; 1993). For one- and two-compartment models, the bias has been shown to occur when the inter-subject variability is very high (White et al., 1991; Ette et al., 1998). The SAS macro NLINMIX uses the Alternative First-Order expansion for both the fxed effects and random effects, and the calculated parameters are like NONMEM. In general, the majority of these programs estimate PK parameters based on using maximum likelihood estimation. REFERENCES Aarons, L. 1991. Population pharmacokinetics: Theory and practice. Br J Clin Pharmacol 32(6): 669–70. Barnett, S., Ogungbenro, K., Ménochet, K., Shen, H., Lai, Y., Humphreys, W. G., Galetin, A. 2018. Gaining mechanistic insight into coproporphyrin I as endogenous biomarker for OATP1Bmediated drug–drug interaction using population pharmacokinetic modeling and simulation. Clin Pharmacol Ther 104(3): 564–74. Bauer, R. J. 2010. Introduction to NONMEM7. ftp://nonmem.iconplc.com/Public/nonmem712/int ro712.pdf. Bauer, R. J. 2011. ICON development solutions. NONMEM Users Guide: Introduction to NONMEM 7.2.0. Ellicott City. ftp://nonmem.iconplc.com/Public/nonmem720/guides/nm720.pdf. Beal, S. L. 1984. Population pharmacokinetic data and parameter estimation based on their frst two statistical moments. Drug Metab Rev 15(1–2): 173–93. Bonate, P. L. 2005. Covariate detection in population pharmacokinetics using partially linear mixed effects models. Pharm Res 22(4): 541–49. Bonate, P. L. 2005. Recommended reading in population pharmacokinetic pharmacodynamics. AAPS J 7(2): E363–73. Bulitta, J. B., Bingölbali, A., Shin, B. S., Landersdorfer, C. B. 2011. Development of a new pre- and post-processing tool (SADAPT-TRAN) for nonlinear mixed-effects modeling in S-ADAPT. AAPS J 13(2): 201–11. Chan, P. L., Jacqmin, P., Lavielle, M., McFadyen, L., Weatherley, B. 2011. The use of the SAEM algorithm in MONOLIX software for estimation of population pharmacokinetic-pharmacodynamic-viral dynamics parameters of maraviroc in asymptomatic HIV subjects. J Pharmacokinet Pharmacodyn 38(1): 41–61. Crombag, M. B. S., Dorlo, T. P. C., van der Pan, E., van Straten, A., Bergman, A. M., van Erp, N. P., Beijnen, J. H., Huitema, A. D. R. 2019. Exposure to Docetaxel in the elderly patient population: A population pharmacokinetic study. Pharm Res 36(12): 181. https://doi.org/10.1007/s11095-019-2706-4. Dansirikul, C., Morris, R. G., Tett, S. E., Duffull, S. B. 2005. A Bayesian approach for population pharmacokinetic modelling of sirolimus. Br J Clin Pharmacol 62(4): 420–34. Davidian, M. 2009. Non-linear mixed-effects models. In Longitudinal Data Analysis, eds. G. Fitzmaurice, M. Davidian, G. Verbeke, G. Molenberghs, 107–41. Boca Raton: Chapman & Hall/CRC Press. Davidian, M., Giltinan, D. M. 1995. Nonlinear Models for Repeated Measurement Data. Boca Raton: Chapman & Hall/CRC Press. Davidian, M., Giltinan, D. M. 2003. Nonlinear models for repeated measurement data: An overview and update. J Agr Biol Environ Stat 8(4): 387–419. 580

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Dempster, A. P., Laired, N. M., Rubin, D. B. 1977. Maximum likelihood from incomplete data via EM algorithm. J R Stat Soc B 39: 1–38. Dokoumetzidis, A., Aarons, L. 2005. Propagation of population pharmacokinetic information using a Bayesian approach: Comparison with Meta-Analysis. J Pharacokinet Phrmacodyn 32(3–4): 401–18. Dyck, J. B., Maze, M., Haack, C., Azarnoff, D. L., Vuorilehto, L., Shafer, S. L. 1993. Computercontrolled infusion of intravenous dexmedetomidine hydrochloride in adult human volunteers. Anesthesiology 78(5): 821–8. Ette, E. I., Howie, C. A., Kelman, A. W., Whiting, B. 1995a. Experimental design and effcient parameter estimation in preclinical pharmacokinetic studies. Pharm Res 12(5): 729–37. Ette, E. I., Kelman, A. W., Howie, C. A., Whiting, B. 1994. Infuence of ineranimal variability on the estimation of population pharmacokinetic parameters in preclinical studies. Clin Res Regul Aff 11(2): 121–39. Ette, E. I., Kelman, A. W., Howie, C. A., Whiting, B. 1995b. Analysis of animal pharmacokinetic data: Performance of the one point per animal design. J Pharmacokinet Biopharm 23(6): 551–66. Ette, E. I., Sun, H., Ludden, T. M. 1998. Balanced designs and longitudinal population pharmacokinetic studies. J Clin Pharmacol 38(5): 417–23. Ette, E. I., Williams, P. J., Ahmad, A. 2007. Population pharmacokinetic estimation methods. In Pharmacometrics: The Science of Quantitative Pharmacology, eds. E. I. Ette, P. J. Williams, Chapter 10, 263–85 Hoboken: John Wiley & Sons, Inc. FDA draft guidance on in vivo drug interaction studies. 2006. FDA fnal guidance for study and evaluation of gender differences. 1993. http://www.fda.gov/ downloads/RegulatoryInformation/Guidances/UCM126835.pdf. FDA fnal guidance on pharmacokinetics in patients with impaired hepatic function. 2003. http:// www.fda.gov/downloads/Drugs/GuidanceComplianceRegulatoryInformation/Guidances/ ucm072123.pdf. FDA fnal guidance on population pharmacokinetics. 1999. http://www.fda.gov/downloads/Drugs /GuidanceComplianceRegulatoryInformation/Guidances/ucm072137.pdf. FDA/ICH fnal guideline on clinical investigation of medicinal products in the pediatric population (E11). 2000. http://www.fda.gov/downloads/Drugs/GuidanceComplianceRegulatoryInformation/Guidances/ucm072114.pdf Frame, B., Beal, S.L. 1998. Non-steady state population kinetics of intravenous phenytoin. Ther drug Monitor 20(4): 408–416. Galecki, A. T. 1998. NLMEM: A new SAS/IML macro for hierarchical nonlinear models. Comput Methods Programs Biomed 5(3): 207–16. Gelfand, A. E., Smith, A. F. M. 1990. Sampling-based approaches to calculating marginal densities. J Am Stat Assoc 85(410): 398–409. Georges, B., Conil, J. M., Seguin, T., Ruiz, S., Minville, V., Cougot, P., Decun, J. F., Gonzalez, H., Houin, G., Fourcade, O., Saivin, S. 2009. Population pharmacokinetics of ceftazidime in intensive care unit patients: Infuence of glomerular fltration rate, mechanical ventilation, and reason for admission. Antimicrob Agents Chemother 53(10): 4483–9.

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Girard, P., Sheiner, L. B., Kastrissios, H., Blaschke, T. F. 1996. Do we need full compliance data for population pharmacokinetic analysis? J Pharmacokinet Biopharm 24(3): 265–82. Goti, V., Chaturvedula, A., Fossler, M. J., Mok, S., Jacob, J. 2018. Hospitalized patients with and without hemodialysis have marketly different vancomycin oharmacokinetics: A population pharmacokinetic model-based analysis. Ther Drug Monit 40(2): 212–21. Grasela Jr, T. H., Antal, E. J., Townsend, R. J., Smith, R. B. 1986. An evaluation of population pharmacokinetics in therapeutic trials. Part I. Comparison of methodologies. Clin Pharmacol Ther 39(6): 605–12. Grasela Jr, T. H., Sheiner, L. B. 1991. Pharmacostatistical modeling for observational data. J Pharmacokinet Biopharm 19(S3): S25–36. Guidance for industry (fda.gov) 2022. Gustafsson, L. L., Ebling, W. F., Osaki, E., Harapat, S., Stanski, D. R., Shafer, S. L. 1992. Plasma concentration clamping in the rat using a computer-controlled infusion pump. Pharm Res 9(6): 800–7. Hardle, W., Liang, H., Gao, J. 2001. Partially Linear Models. Rockville: Springer Verlag. http://www.ema.europa.eu/docs/en_GB/document_library/Scientifc_guideline/2009/09/ WC500002842.pdf. http://www.fda.gov/downloads/drugs/guidancecomplianceregulatoryinformation/guidances/ ucm189544.pdf. http://www.fda.gov/downloads/drugs/guidancecomplianceregulatoryinformation/guidances/ ucm292362.pdf. http://www.fda.gov/downloads/ScienceResearch/SpecialTopics/WomensHealthResearc UCM133184.pdf. http://www.fda.gov/OHRMS/DOCKETS/98fr/06d-0344-gdl0001.pdf. ICH fnal guideline on ethnic factors in the acceptability of foreign clinical data (E5). Released September 1998. ICH fnal guideline on special populations: Geriatrics (E7). 1993. http://www.ich.org/LOB/media/ MEDIA483.pdf Jeong, S.-H., Jang, J.-H., Lee, Y.-B. 2022. Population pharmacokinetic analysis of lornoxicam in healthy Korean males considering creatinine clearance and CYP2C9 genetic polymorphism. J Pharm Investig 52(1): 109–27. https://doi.org/10.1007/s40005-021-00550-y. Karlsson, M. O., Sheiner, L. B. 1993. The importance of modeling interoccasion variability in population pharmacokinetic analysis. J Pharmacokinet Biopharm 21(6): 735–50. Kobuchi, S., Yazaki, Y., Ito, Y., Sakaeda, T. 2018. Circadian variations in the pharmacokinetics of capecitabine and its metabolites in rats. Eur J Pharm Sci 112: 152–8. Laffont, C. M., Concordet, D. 2011. A new exact test for the evaluation of population pharmacokinetic and/or pharmacodynamic models using random projections. Pharm Res 28(8): 1948–62. Laired, N. M., Ware, J. H. 1982. Random-effect models for longitudinal data. Biometrics 38(4): 963–74.

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Langenhorst, J. B., Dorlo, T. P. C., van Maarseveen, E. M., Nierkens, S., Kuball, J., Boelens, J. J., van Kesteren, C., Huitema, A. D. R. 2019. Population pharmacokinetics of fludarabine in children and adult during conditioning prior to allogeneic hematopoietic cell transplantation. Clin Pharmacokinet 58(5): 627–37. https://doi​.org​/10​.1007​/s40262​- 018​- 0715​-9. Leary, R., Dunlavey, M., Chittenden, J., Brett Matzuka, B., Guzy, S. 2011. QRPEM A new standard of accuracy, precision, and efficiency in NLME population PK/PD methods. http://www​.pharsight​ .com​/library​/sign​_in​/whiteQRPEM​.pdf. Ludden, T. M. 1991. Nonlinear pharmacokinetics: Clinical implications. Clin Pharmacokinet 20(6): 429–46. Lunn, D., Spiegelhalter, D., Thomas, A., Best, N. 2009. The BUGS project: Evolution, critique and future directions. Stat Med 28(25): 3049–67. Nedelman, J. R. 2005. On some “disadvantages” of the population approach. AAPS J 7(2): E374–82. Ng, W., Uchida, H., Ismail, Z., Mamo, D. C., Rajji, T. K., Remington, G., Sproule, B., Pollock, B. G., Mulsant, B. H., Bies, R. R. 2009. Clozapine exposure and the impact of smoking and gender: A population pharmacokinetic study. Ther Drug Monit 31(3): 360–6. Pétain, A., Zhong, D., Chen, X., Li, Z., Zhimin, S., Zefei, J., Zorza, G., Ferré, P. 2019. Effect of ethnicity on vinorelbine pharmacokinetics: A population pharmacokinetics analysis. Cancer Chemmother Pharmacol 84(2): 373–82. https://doi​.org​/10​.1007​/s00280​- 019​- 03872​-9. Pfister, M., Labbé, L., Hammer, S. M., John Mellors, J., Bennett, K. K., Rosenkranz, S., Sheiner, L. B., the AIDS Clinical Trial Group Protocol 398 Investigators. 2003. Population pharmacokinetics and pharmacodynamics of efavirenz, nelfinavir, and indinavir. Antimicrob Agents Chemother 47(1): 130–7. Phoenix® NLME™ - Nonlinear mixed effects modeling, Phoenix NLME software review 2011. Phoenix; Pharsight. http://www​.certara​.com​/products​/pkpd​/phx​-nlme/. Pillai, V. C., Han, K., Beigi, R. H., Hankins, G. D., Clark, S., Hebert, M. F., Easterling, T. R., Zajicek, A., Ren, Z., Caritis, S. N., Venkataramanan, R. 2015. Population pharmacokinetics of oseltamivir in non-pregnant and pregnant women. Br J Clin Pharmacol 80(5): 1042–50. Prague, M., Lavielle, M. 2022. SAMBA: A novel method for fast automatic model building in nonlinear mixed-effects models. Pharmacometr Syst Pharmacol 11(2): 161–72. Racine-Poon, A. 1985. A Bayesian approach to nonlinear random effect models. Biometrics 41(4): 1015–23. Ribbing, J., Jonsson, E. N. 2004. Power, selection bias and predictive performance of the population pharmacokinetic covariate model. J Pharmacokinet Pharmacodyn 31(2): 109–34. Ruppert, D., Wand, M. P., Carroll, R. J. 2003. Semi Parametric Regression. Cambridge: Cambridge University Press. Senel, M., Nyholm, D., Nielsen, E. I. 2020. Population pharmacokinetics of levodopa gel infusion in Parkinson’s disease: Effect of entacapone infusion and genetic polymorphism. Sci Rep 10(1): 18057. https://doi​.org​/10​.1038​/s41598​- 020​-75052​-2. Shafer, S. L., Varvel, J. R., Aziz, N., Scott, J. C. 1990. Pharmacokinetics of fentanyl administered by computer-controlled infusion pump. Anesthesiology 73(6): 1091–102. Sheiner, L. B., Rosenberg, B., Marathe, V. V. 1977. Estimation of population characteristics of pharmacokinetic parameters from routine clinical data. J Pharmacokinet Biopharm 5(5): 445–79. 583

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Sheiner, L. B., Beal, S. L. 1980. The NONMEM system. Am Stat 34(2): 118–19. Sheiner, L. B., Beal, S. L. 1981. Evaluation of methods for estimating population pharmacokinetic parameters. I. Biexponential model and experimental pharmacokinetic data. J Pharmacokinet Biopharm 9(5): 635–51. Sheiner, L. B., Beal, S. L. 1982. Bayesian individualization of pharmacokinetics: Simple implementation and comparison with non-Bayesian methods. J Pharm Sci 71(12): 1344–8. Sheiner, L. B., Beal, S. L. 1983. Evaluation of methods for estimating population pharmacokinetic parameters III. Monoexponential model: Clinical pharmacokinetic data. J Pharmacokinet Biopharm 11(3): 303–19. Sheiner, L. B. 1984. The population approach to pharmacokinetic data analysis: Rationale and standard data analysis methods. Drug Metab Rev 15(1–2): 153–71. Shellhaas, R. A., Ng, C. M., Dillon, C. H., Barks, J. D., Bhatt-Mehta, V. 2013. Population pharmacokinetics of phenobarbital in infants with neonatal encephalopathy treated with therapeutic hypothermia. Pediatr Crit Care Med 14(2): 194–202. Steimer, J. L., Golmard, J. L., Boisvieux, J. F. 1984. Alternative approaches to estimation of population pharmacokinetic parameters: Comparison with the nonlinear mixed-effect model. Drug Metab Rev 15(1–2): 265–92. Sugiyama, E., Kaniwa, N., Kim, S. R., Hasegawa, R., Saito, Y., Ueno, H., Okusaka, T., Ikeda, M., Morizane, C., Kondo, S., Yamamoto, N., Tamura, T., Furuse, J., Ishii, H.,Yoshida, T., Saijo, N., Sawada, J. 2010. Population pharmacokinetics of gemcitabine and its metabolite in Japanese cancer patients: Impact of genetic polymorphisms. Clin Pharmacokinet 49(8): 549–58. Sun, H., Ette, E. I., Lundden, T. M. 1996. On the recording of sample times and parameter estimation from repeated measures pharmacokinetic data. J Pharmacokinet Biopharm 24(6): 637–50. Thakkar, N., Green, J. A., Koh, G. C. K. W., Duparc, S., Tenero, D., Goyal, N. 2018. Population pharmacokinetics of tafenoquine, a novel antimalarial. Antimicrob Agent Chemother 62(11). https://doi .org/10.1128/AAC. Ueshima, S., Hira, D., Tomitsuka, C., Nomura, M., Kimura, Y., Yamane, T., Tabuchi, Y., Ozawa, T., itoh, H., Horie, M., Terada, T., Katsura, T. 2019. Population pharmacokinetics and pharmacodynamics of apixaban linking its plasma concentration to intrinsic activated coagulation factor X activity in Japanese patients with atrial fbrillation. AAPS J 21(5): 80. https://doi.org/10.1208/s12248-019 -0353-7. Urien, S., Firtion, G., Anderson, S. T., Hirt, D., Solas, C., Peytavin, G., Faye, A., Thuret, I., Leprevost, M., Giraud, C., Lyall, H., Khoo, S., Blanche, S., Tréluyer, J. M. 2011. Lopinavir/ritonavir population pharmacokinetics in neonates and infants. Br J Clin Pharmacol 71(6): 956–60. Wakefeld, J. C., Walker, S. G. 1997. Bayesian nonparametric population models: Formulation and comparison with likelihood approaches. J Pharmacokinet Biopharm 25(2): 235–53. Wendling, T., Dumitra, S., Ogungbenro, K., Aarons, L. 2015. Application of a Bayesian approach to physiological modelling of mavoglurant population pharmacokinetics. J Pharmacokinet Pharmacodyn 42(6): 639–57. Whiting, B., Kelman, A. W., Grevel, J. 1986. Population pharmacokinetics: Theory and clinical application. Clin Pharmacokinet 11(5): 387–401.

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21 Practical Application of Pk/TK Models: Preclinical PK/TK and Clinical Trial 21.1 INTRODUCTION The PK/TK analysis is an essential part of assessing drug safety during drug discovery and development through preclinical investigations, clinical trials, and the post-marketing surveillance of a new drug entity. The preclinical toxicokinetic (TK) analysis is associated with the quantitative toxicity assessment of a new synthetic compound immediately after the initial phase of discovery. The analysis provides information and guidance in planning to optimize the dose and dosing regimen for human. The merged notion of PK/TK analysis is clearly defned and expressed in the ICH guidelines S3A as “… the generation of pharmacokinetic data, either as an integral component of the conduct of non-clinical toxicity studies or in specialty designed supportive studies, in order to assess systemic exposure.” Many of the failures of drug candidates in drug discovery and development are related to their undesirable PK/TK properties. For example, too long or too short T1/2; poor absorption; extensive hepatic frst-pass metabolism; Pgp-CYP3A4 coordinated intestinal metabolism; and nonlinear characteristics of lead compound following single or multiple dosing administration, and so on. The required preclinical PK/TK assessment includes 1) the in vitro metabolic studies and in vitro characterization of metabolite(s) when the measured metabolites are greater than 10% of the total drug-related exposure, which include hepatocytes, microsomal and/or cytosolic incubations; 2) repeated-dose toxicity, known as chronic toxicity studies; and 3) pharmacokinetics of the lead compound in experimental animals. The preclinical PK/TK in vivo and in vitro evaluations must be done before the clinical trials. 21.2 PRECLINICAL PK/TK 21.2.1 Estimation of the First Dose in Humans The determination of the frst dose in humans is an important step in the preclinical phase of drug discovery and development (DDD) and will remain a challenging aspect of the process (Lowe et al., 2007; Huang et al., 2008). The dose prediction for humans from preclinical PK/PD/TK data and biopharmaceutical properties is a multifaceted task. The challenge is to estimate an effcacious dose and dosing regimen for the frst time in a human (FTIH) based on the interspecies extrapolation of pharmacokinetic/pharmacodynamic (PK/PD) data (Miller et al., 2005) and preclinical data, such as physicochemical characteristics of the lead compound; its biopharmaceutical factors such as solubility, type of dosage form, dissolution of the solid dosage form; and all physiological factors of the GI tract including the intestinal and hepatic frst-pass metabolism of oral dosage form and other factors that infuence the permeability and absorption of the lead compound (Amidon et al., 1995; Lipinski, 2001; Li et al., 2005; Custodio et al., 2008). Various attempts have been made to predict the human PK/PD constants and parameters from preclinical data (Wajima et al., 2004; De Buck et al., 2007; Fura et al., 2008, Zhu et al., 2022). The projection of PK parameters and constants, such as the steady-state volume of distribution (Oie and Tozer, 1979; Obach et al., 1997, 2008; Mahmood, 2005; Hosea et al., 2009), bioavailability (Fura et al., 2008; Sinha et al., 2008; Hosea et al., 2009), and clearance (Oie and Tozer, 1979; Obach et al., 1997, 2008; Mahmood and Yuan, 1999; Stoner et al., 2004; Mahmood, 2005, 2006; Tang and Mayersohn, 2005; Hosea et al., 2009), etc., from preclinical data are mostly based on interspecies scaling using physiological modeling (Luttringer et al., 2003; Parrott et al., 2005; Jones et al., 2006; De Buck et al., 2007). The combined PK/PD/TK evaluation at the early phase of drug development is essential because of the development of a realistically safe dose for FTIH and toxicokinetic and risk identifcation (Ji et al., 2022). The old concept of LD50, the lethal dose required to kill 50% of experimental animals, is no longer considered a helpful measure of toxicity, and its use has been discontinued. The following are other measures which are recommended for the estimation of the frst dose: 1. NTEL dose (No Toxic Effect Level dose): The largest dose that produces no toxic response in the most sensitive experimental animals. 2. NOAEL dose (No Observed Adverse Effect Level dose): The largest dose with no undesirable side effects such as tissue toxicity, weight loss, or seizures. 3. MTD (Maximum Tolerated Dose): The maximum tolerated dose with no unwanted side effects in multiple dosing. 586

DOI: 10.1201/9781003260660-21

PRACTICAL APPLICATION OF PK/TK MODELS: PRECLINICAL PK/TK AND CLINICAL TRIALS

4. NOEL dose (No Observed Effect Level dose): The dose that produces the threshold of pharmacological or toxicological response. 5. Microdose trial. For exploratory clinical trials conducted at the initial part of the Phase I clinical trial, the ICH guideline (M3(R2), 2009, 2013) has recommended the use of microdose trials, which may involve human subjects, healthy volunteers, or patients from selected populations. The microdose regimen is supposed to produce limited exposure and is not intended for the measurement of any clinical outcomes. The related biological samples, however, can be used to investigate PK/TK or PD/ TD parameters and constants at a low dose, single administration, and multiple-dosing regimen. There are fve approaches for the starting dose: First approach: Single microdose trial, the subjects receive 100 μg/subject in one injection. This is the dose that can be used to target receptor interaction or disposition profle of the ligand in a PET ligands study. Second approach: Multiple microdose trial, the subjects receive fve administrations of the 100 μg/ subject for a total of 500 μg/subject. Third approach: Single sub-therapeutic administration of a dose yielding up to one-half of the NOAEL exposure. Fourth approach: A dosing regimen for a maximum of 14 days to attain the concentration within the therapeutic range. It is not intended to evaluate the therapeutic outcome of MTD. Fifth approach: A dosing regimen for a maximum of 14 days such that it achieves the concentration within the therapeutic range and it does not exceed the duration of dosing in non-rodent preclinical evaluation. The regimen is not intended to assess the pharmacological response of MTD. 21.2.2 PK/TK Preclinical Requirements The preclinical metabolic and pharmacokinetic data should include the data from two rodent species (usually the rat and mouse) and a non-rodent species (usually the dog). If nonlinear PK or dose dependency is observed in metabolic and pharmacokinetic or toxicity studies with one species, the same range of doses should be used in metabolic and pharmacokinetic studies with other species. The PK data are considered more effective if metabolism and pharmacokinetic studies are conducted in both sexes of young adult animals of the same species and strain. Commonly reported biological samples and data are from blood (RBCs, plasma, and serum), urine, bile, and feces. In addition, a few representative organ and tissue samples should be taken, such as liver, kidney, fat, and suspected target organs for uptake studies. Sampling times depend on the compound being tested and the selected route of administration. The number of blood samples should be taken frequently such that different phases of the concentration–time profle can be defned with certainty. Time spacing of samples will depend on the rates of uptake and elimination of the compounds. It is often recommended to take blood and tissue samples in a “power of 2” series, that is, 2, 4, 8, 16, and 30 minutes follows by 1, 2, 4, 8, and 16 hours; or seven time points using “power of 3,” that is, 3, 9, and 30 minutes and 1, 3, 9, and 24 hours. The sampling schedule for an oral dosing experiment can be established as 15 and 30 minutes and 1, 2, 4, 8, 24, 48, and 72 hours. When the absorption is rapid, more frequent initial sampling is recommended, and the measurement of concentration at the late time points depends on the sensitivity of the analytical methodology and clearance of the compound. The required supportive data for the in vivo PK/TK analysis include the in vitro metabolic analysis, related kinetic characteristics, and signifcant metabolic pathways and the intrinsic clearances. In addition, the in vitro measurements of protein-binding, transport across the cell membranes (e.g., data generated from Caco2 cells), and conjugate formation should be identifed. The integration of in vitro, in vivo, and in silico data generated during discovery and the preclinical phase of development has been facilitated by using PK modeling software, such as GastroPlus (Lukacova et al., 2009; Parrott et al., 2009); STELLA (Shono et al., 2009, 2010, 2011) and Simcyp (Shaffer et al., 2012; Vieira et al., 2012); (ADMET) PredictorTM (Agoram et al., 2001); and other similarly marketed tools.

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21.2.2.1 Safety Pharmacology and Toxicity Testing Other required data for the PK/TK analysis of a lead compound are the toxicity testing and evaluation. The focus of preclinical safety and toxicological evaluations are related to the structural and biochemical consequences of administration of a small new drug entity. The testing relies on the histopathological evaluation of organs and physiological and functional observations. An important part of the preclinical evaluation is the safety pharmacology data and observation. Safety pharmacology is the assessment of adverse effects of drugs on the central and peripheral nervous system, for example, seizures; adverse effects on the cardiovascular system for example, hypertension, hypotension, and arrhythmia; adverse effects on respiratory system, such as asthma and bronchoconstriction; adverse effects on the renal glomerular fltration rate; and adverse effects on the GI tract and its function (Pugsley et al., 2008). The ICH guideline S7A (Guidance for Industry S7A, 2001) defnes safety pharmacology as “those studies that investigate the potential undesirable pharmacodynamic effects of a substance on physiological functions in relationship to exposure in the therapeutic range and above.” The primary objectives of safety pharmacology are identifying potential pharmacodynamic risk to humans; investigating the mechanisms of risk posed by the lead compounds; correlating the PD response with peak drug levels; using the information to determine low-observed-effect level (LOEL) and maximum dose with no-observedeffect level (NOEL); and assuring human safety in advance of FTIH. 21.2.2.1.1 Acute Toxicity Studies An acute toxicity study deals with toxicity testing after single administration of a lead compound. The main purpose is to determine the degree of toxicity quantitatively and qualitatively (Robinson et al., 2008). Other purposes are to determine the level of exposure for chronic studies and to determine the onset and disappearance of toxic response, if it disappears. The acute toxicity study usually involves at least two mammalian animal species, normally rats (Sprague-Dawley, Fisher or Wistar) and mice or hamsters, using two different routes of administration: one is the route intended for human administration and the second preferred route is intravenous administration. The dose should be high enough to induce signifcant toxicity, its pH should be 7.4 with physiological osmolarity, and its volume cannot exceed 10 mL/kg bwt. Necropsy should be carried out on live and dead animals. Body weight, food and water consumption, urine and feces analysis, cardiovascular functions, hematological parameters, histological data, respiration, and other visible changes must be recorded. 21.2.2.1.2 Chronic Toxicity Studies The chronic toxicity studies are carried out to identify the target organs of toxicity and estimation of dose levels for clinical trials. The initial study is carried out on rodents between 2 weeks and 6 months and non-rodent between 2 weeks and 9 months in duration. The experimental protocol for systemic drugs usually involves one rodent (typically rat) and one non-rodent (e.g., dog). However, the selection of the animals is based on the kinetics and metabolic characteristics of the lead compound in the selected animals in comparison to humans. A lead compound may exhibit substrate specifcity for an enzyme of CYP450 family in the animal, which may not be present in humans. The strains of animals used in toxicity testing (acute and chronic) are: Swiss mice, NMR1 mice, Wistar rats, Sprague Dawley rats, beagle dogs, guinea pigs, Himalayan rabbits, New Zealand white rabbit, Cynomolgus monkeys, Rhesus monkeys, baboons, and mini pigs. The US Food and Drug Administration recommendations for the selection of the most appropriate species are 1) similarity in absorption, distribution, metabolism, and excretion (ADME) of the lead compound between species, 2) species should be predictive for a known pharmacological class, 3) the in vitro metabolic profle should be similar, 4) the species should have similar biochemistry and physiology to man, and 5) have similar pharmacology/receptor-binding properties to man. Other considerations in chronic toxicity study are standardization of time and daily administration of the dose; larger non-rodent species should be fasted before the administration of the compound; the behavioral and hematological, with complete clinical chemistry data, profle must be determined so that the symptoms of an illness are not confused with the symptoms of toxicity. At the end, all sacrifced animals go under macroscopic evaluation, including weight of each organ and histopathological evaluation. For anticancer drugs, the NCI recommendations are: single and daily lethality studies for fve days in mice; single and daily toxicity studies for 5 days in dog; and single and daily toxicity study for 5 days in rodents (optional).

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The assessment of systemic exposure in toxicity studies and objectives of undertaking toxicokinetic evaluation and the signifcance of the related parameters and constant are discussed in ICH - S3A guidelines (1994). The recommended toxicity screening assays include 1) adenosine triphosphate measurement using high-throughput methodology and viable cells in 96-well or 384-well plates with the lead compound at various concentrations to establish the dose–response curve; 2) release of the liverspecifc cytosolic enzymes alanine aminotransferase (ALT) and aspartate aminotransferase (AST) are commonly used as the endpoints for the evaluation of cytotoxicity; 3) macromolecular synthesis is an assay used to measure the dividing cells by measuring incorporation of labeled precursors (3H or 14C–labelled thymidine, uridine, or leucine) in DNA, RNA; 4) glutathione measurement which involves the measurement of GSA in exposed cells in comparison to nontreated cells; and 5) stem cell measurement. 21.2.2.2 Metabolic Evaluations in Preclinical Phase Metabolic processes infuence parameters that are relevant to the safety and effcacy of drugs, and the in vivo and in vitro metabolic data are important in the selection of viable drug candidates and evaluation of their bioavailability, systemic clearance, and toxicity. Identifcation of biotransformation pathways that might produce toxicity, and prediction of drug–drug interactions that might lead to alterations in the pharmacokinetic profles of coadministered drugs are a few challenges in the preclinical phase for PK/TK study. The design and conduct of drug metabolism studies and interpretation of results must take into consideration the advances in pharmacogenomics, pharmacogenetics, and transporters. In the discovery phase, metabolic data provide a basis for choosing chemical structures suited for lead compounds. In the preclinical phase, metabolic data aid in the development of clinical plans about human drug exposures and safety, and throughout the DDD process, metabolic data are needed for PK/PD analysis of the lead compounds. Drug metabolism methodologies employed during different stages of discovery and development can be vastly different, primarily because of the different needs and endpoints. In discovery, the primary purpose is to screen large numbers of compounds to select ideal candidates for development, hence, technologies with high throughput capabilities are required. The development studies require more in-depth analyses of a single compound, employing methods that have been thoroughly validated, from a good laboratory practice (GLP) perspective. The metabolic liabilities of lead compounds include formation of reactive metabolites which may exert adverse effects by inactivating the same drug-metabolizing enzyme that catalyzes its formation. Thus, it may raise the exposure/concentration of either the parent compound or other compounds, that rely on the same enzyme system for metabolism and clearance, and the possibility of binding to other proteins, such as DNA and RNA, causing genotoxicity and irreparable damage. The reactive metabolites may interact with glutathione (GSH) covalently, lowering GSH levels – a potential source of oxidative stress for the cell that may lead to various adverse outcomes. Furthermore, the reactive metabolites, by covalent binding to proteins in the cell, may cause apoptosis or necrosis. At the organ level, the formation of reactive metabolites in the liver and non-hepatic metabolism in the kidneys, adrenal glands, and skin may manifest as acute organ toxicity, changing the clearance of the compound by the organs of elimination. A different aspect of the interaction of metabolites of lead compound is their non-covalent, off-target binding, which may result in unexpected and often problematic pharmacological response. The importance of preclinical metabolism studies is evident in cases when a parent lead compound is precluded from the advancement into development phase, merely because it acts as inhibitor of CYP subfamilies, in particular CYP3A. In the drug discovery phase, depending on the physiological relevance, time, and cost, various in vitro, in silico, or in situ isolated organs, or in vivo methodologies, are employed (see Chapter 9). In the preclinical phase of drug development, metabolic studies require more in-depth analysis, and the employed in vitro and in vivo methodologies must be thoroughly validated under GLP guidelines. As discussed in Chapter 9, Section 9.3.3, the in vitro methodologies can be divided into two major groups: cellular fractions and organ fractions. The cellular fractions, derived from the endoplasmic reticulum of liver, include liver microsomes, cytosol, and S-9 fraction. They all require cofactors such as NAD, NADPH, FAD, and FAM during incubation with the lead compound. The application of cellular fractions in the preclinical phase includes the high throughput investigation of metabolism of lead compounds; investigation of the effect of inducers or inhibitors on metabolism of the compounds; drug–drug interaction; metabolic activation and deactivation; 589

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identifcation of metabolites; mutagenicity tests and comparative evaluation of a compound metabolism in different species. The major disadvantage of the cellular fractions is the lack cellular control; therefore, caution is exercised in extrapolation of the metabolic data from these systems to in vivo conditions. The organ fractions are the isolated intact perfused liver, liver slices, and intact hepatocytes (also see Chapter 9, Section 9.3.3). Among the organ fractions, the isolated intact perfused liver, due to the intact tissue and vascular systems, is a good model for metabolic investigation of drugs. However, the methodology is expensive, somewhat time-consuming, and the organ cannot be maintained viable over a long period of time. The liver slices are less desirable unless they are precision-cut (Chapter 9, Section 9.3.3.3), even with the precision cut slices, many cells in the system are damaged and there is limited availability of oxygen. Hepatocytes offer a novel tool in metabolic investigation of lead compound in the preclinical phase (also see Chapter 9, Section 9.3.3.2). They are multitasking parenchymal cells of the liver that are responsible for the metabolism of xenobiotics. Because the integrity of the cells is maintained, the metabolism profle mimics the in vivo profle more closely. The in vivo assessment of metabolism of a lead compound is based on the measurement of metabolites in urine, blood, bile, and portal vein blood (Chapter 9, Section 9.3.4). Generally, the concentration of metabolites in plasma for most drugs is below the sensitivity of the detection, and urine usually contains mostly Phase II metabolites. Bile duct cannulation and portal vein cannulation are useful techniques but are invasive for humans. Substituting feces for bile duct cannulation is neither accurate nor practical for the measurement of metabolites. Animal models are often used for determination of metabolic profle of a drug in the portal vein and bile. Comparison of drug concentration of the hepatoportal vein, before entering the liver with blood levels of systemic circulation, provides important information on the liver frst-pass effect and absorption of the lead compounds. 21.3 PK/TK AND CLINICAL TRIALS The general considerations and all technical requirements for clinical trials and the process of clinical development of pharmaceuticals for human use are established in ICH Guidelines (ICHE8, 1997). The ICH Guideline for Good Clinical Practice (GCP, E6 (R1)) defnes the Clinical Trial/ Study as Any investigation in human subjects intended to discover or verify the clinical, pharmacological and/or other pharmacodynamic effects of an investigational product(s), and/or to identify any adverse reactions to an investigational product(s), and/or to study absorption, distribution, metabolism, and excretion of an investigational product(s) with the object of ascertaining its safety and/or effcacy. The terms clinical trial and clinical study are synonymous. Clinical study of a lead compound starts with Phase I clinical trial, with the objective of evaluating the PK/TK safety of a drug candidate for the FTIH by giving single and multiple ascending doses. Phase I study has two phases, Phase I-a and I-b. 21.3.1 Phase I-a Clinical Trial During this clinical trial, the tolerability, PK/TK/PD, and safety of the lead compound is determined. It usually involves 40–60 healthy subjects in a randomized double-blind, placebo-controlled experimental design, with escalating dose, for about 6 months. The single ascending dose protocol involves giving the lowest dose to the frst group of subjects. If the safety and tolerability are acceptable, the dose will be increased for the second group, and if the conditions of safety and tolerability are met, the dose will be increased for the third group, and so on. 21.3.2 Phase I-b Clinical Trial This trial is also designed to evaluate safety, tolerability, adverse effects, and PK/TK/PD of the lead compound. The same number of healthy subject (≈40–60) is used in this study, and experimental design is the same. The difference is giving the escalating dose by multiple dosing regimens. The dose (D) escalation of I-a and I-b for a lead compound with low toxicity is based on exponential regimen of 1D, 2D, 4D, 8D, and 16D, and for highly toxic compounds, it is 1D, 2D, 4D, 6D, 8D, 10D, etc. The PK/TK/PD evaluations of Phase I are FTIH and include complete investigation of a lead compound’s ADME and its absolute bioavailability and bioequivalence study, if it is a new formulation. Further investigations of Phase I includes drug–drug interaction studies; geriatric PK 590

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and safety studies; ethnic population PK and pharmacogenomics; and hepatic and renal impairment PK and safety. The design of the escalating multiple dosing regimen of I-b is based on the PK analysis of the I-a phase of study. Phase II clinical trial is the crucial phase in drug discovery and development, with the main objective of “proof of concept” and effcacy of the lead compound. Phase II can also be divided into Phases II-a and II-b. 21.3.3 Phase II-a Clinical Trial This trial is an exploratory safety and effcacy study of the lead compound. The decision to continue with the lead compound as a viable therapeutic agent is made based on the results of this study. The human subjects are patients for whom the lead compound is intended. The study is based on placebo-controlled, randomized double-blind, or an open-label, and preferentially, crossover design using 50–200 patients. The intended dose and regimen are based on the data from the Phase I clinical trial. 21.3.4 Phase II-b Clinical Trial This trial is carried out on ≈200–500 patients. The objectives are confrmation and optimization of the therapeutic dose for patient population, followed by the safety, effcacy, and tolerability of the dosing regimen. The main purpose of the Phase II clinical trial (II-a and II-b) is proof of concept (PoC), that is, to determine whether the perceived hypothesis for the lead compound’s therapeutic outcome and receptor interaction is credible, logical, and consistent in the clinical reality of the disease state, and whether the differentiation points can be established for the lead compound with the marketed products. 21.3.5 Phase III Clinical Trial This study is to confrm the effcacy and safety of the lead compound in a larger and diverse population of patients. The study design is placebo-controlled, randomized double-blind, and crossover design (preferred) in 500–1000 patients in a multicenter and under the supervision of qualifed and approved medical staff. Patients are selected from different sexes, ages, and ethnicities. The dose for the study is the confrmed dosing regimen of Phase II clinical trial, and the clinical data are used to confrm the dose and dosing regimen for effcacy and safety of the fnalized dosing regimen in comparison to control group or existing therapy. 21.3.6 Phase IV Clinical Trial This phase is an obligatory post-marketing surveillance study in a large diverse population (>10,000) of patients. The objective of this study is to evaluate the adverse effects, drug–drug interaction, pharmacogenetic variances, etc., using the recommended therapeutic dose/dosing regimen in a parallel experimental design. All clinical studies and use of human subjects are expected to be carried out according to the ethical requirements of the declaration of Helsinki (World Medical Association, 1964, 2008), the ICH (E6) guidelines, and good clinical practice (GCP) guidelines. REFERENCES Agoram, B., Woltosz, W. S., Bolger, M. B. 2001. Predicting the impact of physiological and biochemical processes on oral drug bioavailability. Adv Drug Deliv Rev 50: S41–67. Amidon, G. L., Lennernaes, H., Shah, V. P., Crison, J. R. 1995. A theoretical basis for a biopharmaceutic drug classifcation: The correlation of in vitro drug product dissolution and in vivo bioavailability. Pharm Res 12(3): 413–20. Custodio, J. M., Wu, C.-Y., Benet, L. Z. 2008. Predicting drug disposition, absorption/elimination/ transporter interplay and the role of food on drug absorption. Adv Drug Deliv Rev 60(6): 717–33. De Buck, S. S., Sinha, V. K., Fenu, L. A., Nijsen, M. J., Mackie, C. E., Gilissen, R. A. H. J. 2007. Prediction of human pharmacokinetics using physiologically based modeling: A retrospective analysis of 26 clinically tested drugs. Drug Metab Dispos 35(10): 1766–80. 591

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Fura, A., Vyas, V., Humphreys, W., Chimalokonda, A., Rodrigues, D. 2008. Prediction of human oral pharmacokinetics using nonclinical data: Examples involving four proprietary compounds. Biopharm Drug Dispos 29(8): 455–68. Guidance for industry— M3(R2). Nonclinical safety studies for the conduct of human clinical trials and marketing authorization for pharmaceuticals–questions and answers (R2) ICH. 2009, 2013. http://fda.gov/downloads/Drugs/GuidanceComplianceRegulatoryInformation/Guidances/ UCM292340.pdf. Guidance for industry S7A safety pharmacology studies for human pharmaceuticals—ICH. 2001. http://www.fda.gov/downloads/Drugs/GuidanceComplianceRegulatoryInformation/Guidances/ ucm074959.pdf. Hosea, N., Collard, W. T., Cole, S., Maurer, T. S., Fang, R. X., Jones, H., Kakar, S. M., Nakai, Y., Smith, B. J., Webster, R., Beaumont, K. 2009. Prediction of human pharmacokinetics from preclinical information: Comparative accuracy of quantitative prediction approaches. J Clin Pharmacol 49(5): 513–33. Huang, C., Zheng, M., Yang, Z., Rodrigues, A. D., Marathe, P. 2008. Projection of exposure and effcacious dose prior to frst-in-human studies: How successful have we been? Pharm Res 25(4): 713–26. ICH guidelines E8–General considerations for clinical trials. 1997. http://www.ich.org/fleadmin/ Public_Web_Site/ICH_Products/Guidelines/Effcacy/E8/Step4/E8_Guideline.pdf. ICH guideline M3(R2) on non-clinical safety studies for the conduct of human clinical trials and marketing authorisation for pharmaceuticals EMA. 2009, 2013http://www.ema.europa.eu/docs/en _GB/document_library/Scientifc_guideline/2009/09/WC500002720.pdf. Ji, Y., Knee, D., Chen, X., Dang, A., Mataraza, J., Wolf, B., Sy, S. K. B. 2022. Model-informed drug development for immune-oncology agonistic and anti-GITR antibody GWN323: Dose selection based on MABEL and biologically active dose. Clin Transl Sci 15(9): 2218–29. https://doi.org/10.1111 /cts.13355. Jones, H. M., Parrott, N., Jorga, K., Lave, T. 2006. A novel strategy for physiologically based predictions of human pharmacokinetics. Clin Pharmacokinet 45(5): 511–42. Li, S., He, H., Parthiban, L. J., Yin, H., Serajuddin, A. T. M. 2005. IV–IVC considerations in the development of immediate-release oral dosage form. J Pharm Sci 94(7): 1396–417. Lipinski, C. A. 2001. Drug-like properties and the causes of poor solubility and poor permeability. J Pharmacol Toxicol Methods 44(1): 235–49. Lowe, P. J., Hijazi, Y., Luttringer, O., Yin, H., Sarangapani, R., Howard, D. 2007. On the anticipation of the human dose in frst-in-man trials from preclinical and prior clinical information in early drug development. Xenobiotica 37(10–11): 1331–54. Lukacova, V., Woltosz, W. S., Bolger, M. B. 2009. Prediction of modifed release pharmacokinetics and pharmacodynamics from in vitro, immediate release, and intravenous data. AAPS J 11(2): 323–34. Luttringer, O., Theil, F.-P., Poulin, P., Schmitt-Hoffmann, A. H., Guentert, T. W., Lave, T. 2003. Physiologically based pharmacokinetic (PBPK) modeling of disposition of epiroprim in humans. J Pharm Sci 92(10): 1990–2007. Mahmood, I. 2005. Interspecies scaling of drugs cleared by the kidneys and the bile. In Interspecies Pharmacokinetic Scaling: Principles and Application of Allometric Scaling, 105-143, ed. I. Mahmood. Rockville: Pine House Publishers.

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Mahmood, I. 2006. Prediction of human drug clearance from animal data: Application of the rule of exponents and ‘fu corrected intercept method’ (FCIM). J Pharm Sci 95(8): 1810–21. Mahmood, I., Yuan, R. 1999. A comparative study of allometric scaling with plasma concentrations predicted by species-invariant time methods. Biopharm Drug Dispos 20(3): 137–44. Miller, R., Ewy, W., Corrigan, B. W., Ouellet, D., Hermann, D., Kowalski, K. G., Lockwood, P., Koup, J. R., Donevan, S., El-Kattan, A., Li, C. S., Werth, J. L., Feltner, D. E., Lalonde, R. L. 2005. How modeling and simulation have enhanced decision making in new drug development. J Pharmacokinet Pharmacodyn 32(2): 185–97. Note for guidance on toxicokinetics: The assessment of systemic exposure in toxicity studies—ICH S3A. 1994. http://www.ich.org/fleadmin/Public_Web_Site/ICH_Products/Guidelines/Safety/S3A /Step4/S3A_Guideline.pdf. Obach, R. S., Baxter, J. G., Liston, T. E., Silber, B. M., Jones, B. C., Macintyre, F., Rance, D. J., Wastall, P. 1997. The prediction of human pharmacokinetic parameters from preclinical and in vitro metabolism data. J Pharmacol Exp Ther 283(1): 46–58. Obach, R. S., Lombardo, F., Waters, N. J. 2008. Trend analysis of a database of intravenous pharmacokinetic parameters in humans for 670 drug compounds. Drug Metab Dispos 36(7): 1385–405. Oie, S., Tozer, T. N. 1979. Effect of altered plasma protein binding on apparent volume of distribution. J Pharm Sci 68(9): 1203–5. Parrott, N., Jones, H., Paquereau, N., Lave, T. 2005. Application of full physiological models for pharmaceutical drug candidate selection and extrapolation of pharmacokinetics to man. Basic Clin Pharmacol Toxicol 96(3): 193–9. Parrott, N., Lukacova, V., Fraczkiewicz, G., Bolger, M. B. 2009. Predicting pharmacokinetics of drugs using physiologically based modeling—Application to food effects. AAPS J 11(1): 45–53. Pugsley, M. K., Authier, S., Curtis, M. J. 2008. Principles of safety pharmacology. Br J Pharmacol 154(7): 1382–99. Robinson, S., Delongeas, J. L., Donald, E., Dreher, D., Festag, M., Kervyn, S., Lampo, A., Nahas, K., Nogues, V., Ockert, D., Quinn, K., Old, S., Pickersgill, N., Somers, K., Stark, C., Stei, P., Waterson, L., Chapman, K. 2008. A European pharmaceutical company initiative challenging the regulatory requirement for acute toxicity studies in pharmaceutical drug development. Regul Toxicol Pharmacol 50(3): 345–52. Shaffer, C. L., Scialis, R. J., Rong, H. J., Obach, R. S. 2012. Using Simcyp to project human oral pharmacokinetic variability in early drug research to mitigate mechanism-based adverse events. Biopharm Drug Dispos 33(2): 72–84. Shono, Y., Jantratid, E., Dressman, J. B. 2011. Precipitation in the small intestine may play a more important role in the in vivo performance of poorly soluble weak bases in the fasted state: Case example nelfnavir. Eur J Pharm Biopharm 79(2): 349–56. Shono, Y., Jantratid, E., Janssen, N., Kesisoglou, F., Mao, Y., Vertzoni, M., Reppas, C., Dressman, J. B. 2009. Prediction of food effects on the absorption of celecoxib based on biorelevant dissolution testing coupled with physiologically based pharmacokinetic modeling. Eur J Pharm Biopharm 73(1): 107–14. Shono, Y., Jantratid, E., Kesisoglou, F., Reppas, C., Dressman, J. B. 2010. Forecasting in vivo oral absorption and food effect of micronized and nanosized aprepitant formulations in humans. Eur J Pharm Biopharm 76(1): 95–104.

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Sinha, V. K., De Buck, S. S., Fenu, L. A., Smit, J. W., Nijsen, M., Gilissen, A. H. J., VanPeer, A., Lavrijsen, K., Mackie, C. E. 2008. Predicting oral clearance in humans: How close can we get with allometry? Clin Pharmacokinet 47(1): 35–45. Stoner, C. L., Cleton, A., Johnson, K., Oh, D.-M., Hallak, H., Brodfuehrer, J., Surendran, N., Han, H. K. 2004. Integrated oral bioavailability projection using in vitro screening data as a selection tool in drug discovery. Int J Pharm 269(1): 241–9. Tang, H., Mayersohn, M. 2005. A novel model for prediction of human drug clearance by allometric scaling. Drug Metab Dispos 33(9): 1297–303. Vieira, M. L. T., Zhao, P., Berglund, E. G., Reynolds, K. S., Zhang, L., Lesko, L. J., Huang, S. M. 2012. Predicting drug interaction potential with a physiologically based pharmacokinetic model: A case study of telithromycin, a time dependent CYP3A inhibitor. Clin Pharmacol Ther 91(4): 700–8. Wajima, T., Yano, Y., Fukumura, K., Oguma, T. 2004. Prediction of human pharmacokinetic profle in animal scale up based on normalizing time course profles. J Pharm Sci 93(7): 1890–900. World Medical Association (WMA), Declaration of Helsinki. 1964, 2008. Adapted by the 18th WMA general assembly—Handbook of WMA policy. Declaration of Helsinki 2008 – WMA – The World Medical Association Zhu, M., Olson, K., Kirshner, J. R., Toroghi, M. K., Yan, H., Haber, L., Meager, C., Flink, D. M., Ambati, S. R., Davis, J. D., DiCioccio, A. T., Smith, E. J., Retter, M. W. 2022. Translational fndings for odronextamab: From preclinical research to a frst-in-human study in patients with CD20+Bcell malignancies. Clin Transl Sci 15(4): 954–66. https://doi.org/10.1111/cts.13212.

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22 Adjustment of Dosage Regimen in: Renal Impairment, Liver disease and Pregnancy 22.1 RENAL IMPAIRMENT 22.1.1 Introduction Pharmacokinetic analyses have been applied to predict and optimize dosage regimen of patients with diseases of different severity, and healthy individuals in special conditions. The data generated from population pharmacokinetic/toxicokinetic (PK/TK) analyses have also been helpful in identifying the covariates and refning the accepted in practice dosage regimen of certain therapeutic agents. The dosage adjustment in disease states becomes especially important when the elimination processes are infuenced, compromised, and reduced by the disease state. Under this condition, the removal of the drug from the body is hindered, the free fraction of the compound in the systemic circulation elevates, and the variation in total body clearance is signifcant. When the disease state is renal impairment, acute or chronic, and the drug is nil or partially excreted unchanged in the urine, and hepatic metabolism is intact, the dosage adjustment is a straightforward calculation. This is mainly due to the practical estimation of glomerular fltration rate (GFR) and other physiological functions of the kidney by the endogenous and exogenous biomarkers (see Chapter 10, Section 10.5). In contrast to renal elimination, the dosage adjustment in hepatic diseases or in general, defciencies in metabolism of xenobiotics in the body, bearing in mind that many organs can contribute to the metabolism, is not straightforward. This is mainly related to the fact that there is no overall clear index that refects the reduction in metabolic rate. In addition to disease states, there are conditions, like pregnancy, for which the dosage adjustment (particularly during the third trimester), is necessary but somewhat diffcult to predict. In this chapter a few of the approaches in disease states and conditions in individuals that may require dosage adjustment are examined. 22.1.2 Dosage Adjustment for Patients with Renal Impairments The following are the proposed approaches for the adjustment of dosage regimen based on the estimation of the overall elimination rate constant, or the half-life, in a patient population with renal insuffciency followed by adjusting the dosing regimen according to the calculated overall elimination rate constant (Chiou and Hsu, 1975; Slatlery et al., 1980; DeVane and Jusko, 1982; Ritschel, 1983; Burton et al., 1985; Van Dalen et al., 1986). 22.1.2.1 Estimation of the Overall Elimination Rate Constant or HalfLife of a Therapeutic Agent Based on the Estimated GFR Based on the assumption that the metabolic rate constant of patients with renal impairment remains unaffected, the overall elimination rate constant of drugs can be presented as One-compartment model: K ˜ ke ° km

(22.1)

k10 ˜ k e ° k m

(22.2)

Two-compartment model:

where k e is the reduced rate constant of urinary excretion. The assumption of unaltered metabolic rate constant is true only for therapeutic agents that maintain frst-order metabolism, and the increase in in plasma concentration of free drug due to the renal impairment has no impact on the linearity of Phase I and/or Phase II metabolism. In other words, the increase in the free concentration of the drug in the systemic circulation remains at a level which is less than Michaelis-Menten constant, i.e., (Cp ≪ K M). The adjustment of dosage regimen is expected to reduce the plasma concentration to the normal level. The frst step in adjusting the dose and dosing regimen is to estimate the overall elimination rate constant of a compound in patients with renal impairment according to their GFR, which can be estimated using creatinine clearance (Chapter 10, Section 10.5.2.1). The estimated GFR is then used to adjust the normal value of K, or half-life, to determine K , the overall elimination rate constant of a patient with renal impairment. The small-molecule therapeutic agents can be divided into three groups: DOI: 10.1201/9781003260660-22

595

22.1 RENAL IMPAIRMENT

22.1.2.1.1 Compounds Eliminated Entirely by the Renal Route For drugs that are eliminated entirely by the renal route of elimination and glomerular fltration and the overall rate constant of elimination is then equal to the rate constant of excretion, K = ke

(22.3)

dA dAe = = -KA = -k e A = ClT Cp = Clr Cp dt dt

(22.4)

and

Therefore, ClT = Clr K=

dAe dt dAe dt ClT = = , and A Cp ´ Vd Vd

(22.5)

dAe dt Cp

(22.6)

ClT =

22.1.2.1.1.1 Wagner’s Method When K = k e and Fraction of Dose Excreted Unchanged is f e = 1 The ratio of normal total body clearance to normal creatinine clearance is R=

ClT Ccr

\ClT = R ´ Ccr

(22.7) (22.8)

The excretion rate of the compound is then expressed as dAe = R ´ Ccr ´ Cp = ClT ´ Cp = K ´ Vd ´ Cp dt

(22.9)

Therefore, the normal overall elimination rate can be defned as K=

R Ccr Vd

(22.10)

Equation 22.10 indicates that for compounds that are eliminated through renal excretion, a plot of overall elimination rate constant, K, versus creatinine clearance Ccr yields a straight line through the origin with slope of R Vd (Figure 22.1). Thus, the product of slope and creatinine clearance of patient with renal impairment yields the overall elimination rate constant of subject with renal impairment, K (Wagner, 1975) æ R ö K =ç ÷ ´ Ccr è Vd ø

(22.11)

22.1.2.1.1.2 Dettli’s Method When K = k e and Fraction of Dose Excreted Unchanged is f e = 1 In this method k e is assumed to be proportional to glomerular fltration rate or Ccr, that is, k e = K = aCcr

(22.12)

As in Wagner’s method, a plot of K versus Ccryields a straight line through the origin (Figure 22.2) with slope of “a,” which is a proportionality constant estimated as (Dettli, 1974) a = K Ccr

(22.13)

a = K Ccr

(22.14)

Setting Equation 22.14 equal to 22.13 and solving for K yields K=K

Ccr Ccr

(22.15)

22.1.2.1.1.3 Giusti’s Method When K = k e and Fraction of Dose Excreted Unchanged is f e = 1) According to this method, the excretion rate constant of patients with renal failure can be defned in terms of normal excretion rate constant with the ratio of Ccr / Ccr as the correction factor:

596

DOSAGE ADJUSTMENT IN RENAL FAILURE, LIVER DISEASES AND PREGNANCY

Figure 22.1 Illustration of linear plot of the overall elimination rate constant versus creatinine clearance when the compound is excreted entirely by the renal route of elimination through the glomerular fltration, the tubular secretion and reabsorption are assumed negligible; when the creatinine clearance is normal and equals the ideal fgure of 120 ml/min, the y-intercept is equal to the overall elimination rate constant, which is equal to the excretion rate constant; the line goes through the origin, and when the GFR is equal to zero, and the administered dose remains in the body, the slope of the line is R (the ratio of total body clearance of the compound to creatinine clearance) divided by the volume of distribution. K= k= ke e

Ccr Ccr

(22.16)

Dividing both sides of Equation 22.16 by K yields (Giusti and Hayton, 1973) Ccr K = fe Ccr K

(22.17)

22.1.2.1.2 Compounds Eliminated Entirely by the Non-Renal Route The overall elimination rate constant is independent of renal function (i.e., k e = 0and K = k m ), and depends on hepatic and other organs of metabolism, and elimination of metabolites is through the biliary route of elimination. Therefore, unless there is a change in the linearity of elimination and pharmacokinetics of the compound, no adjustment would be necessary. In this scenario, plot of K versus Ccr is parallel to the x-axis. 22.1.2.1.3 Compounds Eliminated by Joint Action of Renal and Non-Renal Routes of Elimination Most therapeutic agents are eliminated by renal excretion and metabolism. Adding the frst-order metabolic rate constant to Equations 22.11, 22.15, and 22.16, or 22.17 yields the following relationships for estimation of the overall elimination rate constant in patient with renal impairment: 22.1.2.1.3.1 Wagner’s Method ˛ R ˆ K ˜ km ° ˙ ˘  Ccr ˝ Vd ˇ

(22.18)

597

22.1 RENAL IMPAIRMENT

Figure 22.2 This illustration is the same as Figure 22.1, except the slope of the line is labeled differently as ‘a’; the slope has the same defnition and magnitude as the line of Figure 22.1, which is the ratio of the overall elimination rate constant divided by creatinine clearance. 22.1.2.1.3.2 Dettli’s Method (Dettli, 1977) K = aCcr + k m \a =

K - km Ccr

(22.19) (22.20)

The slope of the line, a, is calculated by the equation ( y 2 - y1 ) ( x2 - x1 ) where ( y 2 - y1 ) = K - k m and ( x2 - x1 ) = Ccr = 120ml/min Substituting Equation 22.20 in 22.19 and solving for K yields K=

K - km Ccr + k m Ccr

(22.21)

Equations 22.18 and 22.21 represent a similar plot, as is shown in Figure 22.3, which indicates that when creatinine clearance is equal to zero (end-stage renal impairment), the y-intercept is equal to k m , and the metabolic rate constant is the only pathway for elimination in the body. When the clearance of creatinine is normal, that is, Ccr @ 120ml/min , the overall elimination rate constant represents the normal value (i.e., K = 0.693/T1/2). As the renal failure progresses and Ccr decreases, the overall elimination rate constant (K or k10 ) also decreases whereas the half-life, T1/2 , increases. 22.1.2.1.3.3 Giusti’s Method (Giusti and Hayton, 1973) Modifcation of Equation 22.16 to include the metabolic rate constant yields K = ke

Ccr + km Ccr

(22.22)

Dividing Equation 22.22 by K changes the equation into a relationship in terms of fraction of dose excreted unchanged Ccr K = fe + 1 - fe Ccr K 598

(22.23)

DOSAGE ADJUSTMENT IN RENAL FAILURE, LIVER DISEASES AND PREGNANCY

Figure 22.3 Illustration of a linear plot of the overall elimination rate constant versus creatinine clearance for the compounds that eliminate from the body by renal excretion and metabolism; the y-intercept of the line when creatinine clearance is equal to zero (that is, end-stage renal failure) is equal to the metabolic rate constant; the y-intercept of normal clearance is equal to the normal overall elimination rate constant; the slope of the line is based on the assumption that the metabolic rate constant remains the same and follows frst-order kinetics through different stages of renal failure, and is equal to the excretion rate constant that is K−km divided by the creatinine clearance. Since k e K = f e and k m K = f m = 1 - f e , Equation 22.23 can be written as Equation 22.24 and modifed to Equation 22.26 K Ccr = 1 - fe + fe K Ccr \

æ æ Ccr ö ö K = 1 - ç fe ç 1 ÷÷ ç Ccr ø ÷ø K è è

é æ æ Ccr ö ö ù K = K ê1 - ç f e ç 1 ÷ ÷ú ç Ccr ø ÷ø ú êë è è û

(22.24) (22.25)

(22.26)

The methods discussed here basically provide the same answer for the overall elimination rate constant of a patient with renal insuffciency. The cautionary notes about the methods of estimation of K based on creatinine clearance are: 1) the methods may provide a realistic estimate of K only when the serum creatinine concentration is at steady state; and 2) the pharmacokinetics of the compound under investigation remains linear and dose-independent during the adjusted therapy for the patients. 22.1.2.2 Adjustment of Multiple Dosing Regimen Using the Adjusted Elimination Rate Constant, K For patients with mild or moderate renal impairment, the rate of input is adjusted according to the reduced rate of input. The objective of the adjustment is to modify dose and dosing interval such that the accumulation of the compound in the body of patients with renal impairment at steady 599

22.1 RENAL IMPAIRMENT

state, Ass , equals that of the recommended level for patients with normal renal function, Ass , that is setting Ass = Ass

(

or, Aave

)

ss

= (Aave )ss

(22.27) (22.28)

Using the equation of average plasma concentration (Chapter 16, Section 16.3.1) for oral absorption, assuming bioavailability remains the same, Equation 22.28 can be expressed as FD FD = K´ t K´t

(22.29)

Therefore, assuming F values remain the same and cancel out, an adjusted dose can be estimated by using Equation 22.30, which is also usable for the adjustment of intravenous doses. D=

D´ K ´ t K´t

(22.30)

Using similar analogy for intravenous infusion, the infusion rate can be adjusted as follows: k0 k0 = K K

(22.31)

k0 k = 0 Cl ClT T

(22.32)

k0 = k0

Cl K = k0 T ClT K

(22.33)

The equations of dosage adjustment for multiple intravenous bolus doses assuming the one-compartment model are D D = K´ t K´t

(22.34)

D´ K ´ t K´t

(22.35)

D=

For intravenous bolus injection two-compartment model, the equations are D D = k10 t k10 t

(22.36)

Dk10 t k10 t

(22.37)

D=

Methods to achieve the dosage adjustment in multiple dosing kinetics are 1) by keeping the dosing interval the same and changing the dose, 2) by keeping the dose the same and changing the dosing interval, or 3) by changing both the dose and the dosing interval. If the dosing interval is kept the same as the recommended interval for patients with normal renal function ( t = t ) and the dose is changed, Equations 22.30 or 22.35 are simplifed to D=D

k 10 K or D = D k10 K

(22.38)

If the dose is kept the same as recommended for population with normal renal function, that is, D = D, and the dosing interval is changed, the equations are modifed to k10 (22.39) k 10 If both the dose and dosing interval are changed, Equations 22.30 or 22.35 will have two unknowns. One of the unknowns, either the dose or dosing interval, is then set to a convenient value, and the equation is used to solve for the second unknown, Equations 22.40 and 22.41. For example, if the standard dose is given every six hours around the clock (q.i.d.), the dosing interval t=t

600

K K

or t = t

DOSAGE ADJUSTMENT IN RENAL FAILURE, LIVER DISEASES AND PREGNANCY

can be set equal to eight hours, and the adjusted dose can be calculated based on three times around the clock (t.i.d.), with the hope that the fuctuations remain within the therapeutic range (also see Chapter 16). t=l

Þ

D=

D´ K ´l K´t

or D =

D ´ k 10 ´ l k10 ´ t

(22.40)

If the dose is set different from the standard regimen, the dosing interval is estimated as dose ´ K ´ t dose ´ k10 ´ t (22.41) or t = D´K D ´ k 10 The assumptions here are the fraction of dose absorbed and the volume of distribution of patient with renal insuffciency are the same as patients with normal renal function. If the volume of distribution changes in renal impairment, one option is to adjust the dosing interval according to the area under plasma concentration curve at steady state and then adjust the dose according to total body clearance, that is setting Cpave equations in terms of their area under plasma concentration of one dosing interval at steady state. D = dose

Þ

t=

AUC AUC = t t t=t D=

( AUC )

t

( AUC )t

t ´ D ´ ClT t ´ ClT

(22.42) (22.43) (22.44)

A different approach for adjusting dosing regimen is to modify the regimen based on the normal average amount accumulated in the body at steady state. The approach is also known as the “accumulation ratio” method and is accomplished by estimating the average accumulation at steady state of patient with normal renal function, that is

( Aave )ss =

F ´ D 1.44 ´ T1/2 ´ F ´ D = K´t t

(22.45)

followed by adjusting the dosing regimen of patient with renal impairment by selecting one of the options of changing the dose and keeping the dosing interval the same, or changing the dosing interval and keeping the dose the same, or changing both the dose and dosing interval. ◾ Solving for a new dose but keeping the dosing interval the same: D=

( Aave )ss ´ K ´ t (Aave )ss ´ t = 1.44 ´ T1/2 ´ F F

(22.46)

◾ Solving for a new dosing interval but keeping the dose the same: t=

F´D 1.44 ´ T1/2 ´ F ´ D = K ´ (Aave )ss (Aave )ss

(22.47)

◾ Changing both the dose and dosing interval: D=

(Aave )ss ´ K ´ t (Aave )ss ´ t = F 1.44 ´ T1/2 ´ F

(22.48)

Equation 22.48 has two unknowns; as was indicated earlier, one unknown is set equal to an acceptable value and then the equation is used to calculate the second unknown. A similar approach can also be used to adjust the infusion rate: Ass =

k0 = 1.44 ´ T1/2 ´ k0 K

k0 =

Ass 1.44 ´ T1/2

(22.49) (22.50)

601

22.2 LIVER DISEASES

22.1.2.3 Dosage Adjustment Based on the Steady-State Peak and Trough Levels The equations of peak or trough levels (Chapter 16, Section 16.2) can be used to adjust the standard dosing regimen by calculating the normal maximum amount of the compound in the body

( Amax )ss =

D D = -Kt 1- e ( fel )t

(22.51)

Followed by estimation of the dose by keeping the dosing interval the same:

(

D = ( Amax )ss 1 - e -Kt

)

(22.52)

or by changing the dosing interval giving the same normal dose: t=

( Amax )ss 2.303 log K ( Amax )ss - D

(22.53)

Equations 22.52 or 22.53 can be used to change both the dosing interval and the dose ( t and D), depending on which of the two unknowns is set equal to a known value. In the same manner as the peak level, the trough level is used to adjust the dosage regimen with respect to the minimum amount at steady state.

( Amin )ss =

D ´ e -Kt 1 - e -Kt

(22.54)

Estimation of the dose by keeping the dosing interval the same: D=

( Amin )ss (1 - e -Kt )

e -Kt Estimation of the dosing interval by giving the standard dose: t=

D - ( Amin )ss 2.303 log K ( Amin )ss

(22.55)

(22.56)

22.1.3 Applications and Case Studies The applications and case studies of Chapter 22 are posted in Addendum II – Part 9. 22.2 LIVER DISEASES 22.2.1 Introduction Acute and chronic liver diseases are widespread, causing a high rate of mortality and ill health. According to one report, the mortality rate is about two million per year globally and is expected to increase (Makdad et al., 2014; Beste et al., 2015). The major complications are related to cirrhosis, viral hepatitis, and hepatocellular carcinoma. In addition, alcohol-associated liver disease is another major contributing factor to liver disease, especially when it coexists with viral hepatitis. Cirrhosis rates are higher in populations with higher alcohol consumption (Naveau et al., 1997). Other contributing factors to liver disease are related to coexisting factors like obesity and diabetes, viral hepatitis, including hepatitis A, B, C, D, or E; and rare diseases, like autoimmune hepatitis 1 and 2 and Wilson’s disease; and fnally, drug-induced liver injury caused by xenobiotics of all kinds, such as antimicrobials, antibiotics, antilipemics, psychotropics, antineoplastics, immunomodulating agents, and so on (Sgro et al., 2002; Björnsson et al., 2023). Drug-induced liver injury is attributed to genetic susceptibility, which may require genetic risk validation before clinical application (Li et al., 2022). It is worth noting that drug–drug interaction may also increase the risk of hepatotoxicity, for example, the combination therapy of rifampicin, isoniazid, and pyrazinamide (Ortega-Alonso et al., 2016). The complexities of the abovementioned hepatic diseases and injuries, the myriad of liver functions in the body, and the numerous factors that can infuence the liver’s function have made it diffcult to identify an endogenous or even exogenous biomarker, like creatinine in renal failure, to determine the overall functionality of the liver. A combination of concentrations and ratios of several endogenous metabolites and the CYP3A5*3 genotype is suggested as a reliable predictive marker of hepatic CYP3A (Shin et al., 2013; Hibino et al, 2022). In general, as it is required by regulatory agencies too, when hepatic clearance exceeds 20% of the overall elimination of a drug in the body, 602

DOSAGE ADJUSTMENT IN RENAL FAILURE, LIVER DISEASES AND PREGNANCY

and the compound has a narrow therapeutic range, the PK/TK profle, including dosage adjustment approaches in renal and/or hepatic failure including the safety assessment should be evaluated. 22.2.2 Dosage Adjustment in Liver Cirrhosis Liver cirrhosis is a chronic liver disease that impacts not only the structural integrity of the liver but also the overall protein expressions, metabolic function, protein-binding, blood fow, and elimination of xenobiotics by the liver. It occurs in the late stages of different hepatic diseases like hepatitis, liver carcinoma, and alcoholic and non-alcoholic fatty liver. The data on protein expressions, enzymatic changes and metabolism, and magnitudes of protein transporters of the liver at different levels of cirrhosis severity are very limited. One approach for adjustment of dosage regimen in cirrhosis, which is also the FDA recommendation for classifcation of the degree of hepatic failure, is based on the Child-Turcotte-Pugh (CTP) score (Child and Turcotte, 1964; D’Amico et al., 2006; Pugh et al., 1973; Cholongitas et al., 2005; Kaplan et al., 2015). 22.2.2.1 Child-Turcotte-Pugh Score The CTP score is currently the standard assessment of cirrhosis severity. The score is based on fve scores. Three of the scores are considered objective scores based on medical laboratory data of the patient, and the other two are subjective scores based on the medical evaluation of the patient. The three objective laboratory scores are: 22.2.2.1.1 Serum Albumin Level If the measurement of serum albumin is greater than 35 g/L, the given score is (+1) point; if it is 28–35 g/L, the score is (+2) points; and if it is less than 28, the score is (+3). 22.2.2.1.2 Total Bilirubin Measurement If the measurement of total bilirubin is equivalent to 34 µmol/L, the score is (+1) point; if the measurement is 34–50 µmol/L, the score is (+2) points; and if the measurement is greater than 50 µmol/L, the score is (+3) points. 22.2.2.1.3 International Normalized Ratio The International Normalized Ratio (INR), also called standardized “prothrombin time” provides blood-clotting time information of the patient; if the ratio is less than 1.7, the score is (+1), for ratios between 1.7 and 2.3, the score is (+2) points, and for ratios greater than 2.3, the score is (+3) points. Other tests like fbrinogen level, platelet count, and other tests related to the risk of bleeding may be included in the INR lab request. The two subjective scores are related to the severity of the disease and include the following: 22.2.2.1.4 Hepatic Encephalopathy The stages related to hepatic encephalopathy involve the brain disorder and are divided into fve stages of 0) minimum changes in memory and concentration; 1) mild changes associated with mood changes and sleep problems; 2) moderate changes including trouble doing basic calculations, slurred speech, and inappropriate behavior; 3) severe changes including anxiety, severe sleepiness, and disorientation; 4) coma. The scores for hepatic encephalopathy are for stage 0 (+1 point); for stages 1 and 2 (+2 points); for stages 3 and 4 (+3 points). 22.2.2.1.5 Ascites Assessment when there is no ascites, the score is (+1) point; for mild to moderate ascites, the score is (+2) points; and for moderate to severe ascites, the score is (+3) points. The CTP score is designed as a mortality indicator, for example, if the total score is between 5 and 6, there is a 100% chance of survival in the frst year and 85% in the second year. If the total score is between 7 and 9 the frst-year survival is about 81%, and for the second year, it is 57%. If the total score is between 10 and 15 the probability of frst-year survival is 45%, and the second year is 35%. The CTP scores are used to predict the need for liver transplant and to establish an individual dosage adjustment (Safadi et al., 2022). However, the inclusion of CTP scores in PK/TK models, especially physiologically based pharmacokinetic models, is considered challenging and remains to be investigated (Han et al., 2021; El-Khateeb et al., 2021; Ravenstijn et al., 2022). A more practical approach would be to monitor free-drug concentration and adjust the regimen to maintain the free concentration of the administered dose within the therapeutic range. 603

22.3 PREGNANCY

22.3 PREGNANCY 22.3.1 Introduction The physiological changes during pregnancy are signifcant, and the absorption, distribution, metabolism, and excretion (ADME) profle of certain drugs may be different from the standard profles that have been developed mainly for the nonpregnant population. Appreciating these changes and their infuences on the PK/TK profle of therapeutic agents helps optimization of the treatment and adjustment of dosage regime. Some of the changes in the ADME of drugs during pregnancy are briefy described below. 22.3.2 Changes Impacting Oral Absorption during Pregnancy The reduction in gastric acid production and increase in mucus secretion elevate the gastric pH and infuence the ionization, absorption, and bioavailability of weakly acidic and weakly basic drugs (Vasicka et al., 1957; Waldum et al., 1980). In addition, the lower esophageal sphincter because of the action of progesterone lessens its tone, which causes heartburn, nausea, and vomiting (Richter, 2005; Law et al., 2010). Nausea and vomiting directly reduce the bioavailability of a drug. There is no unifed opinion on the changes in gastric emptying rate and small intestinal transit time, few observations have demonstrated delayed gastric emptying time, and others have reported no differences (Macfe et al., 1991; Levy, 1977; Sandhar et al., 1992). The hepatic artery blood fow remains the same as before pregnancy. 22.3.3 Changes Infuencing Drug Distribution during Pregnancy The changes in the cardiovascular system are signifcant during pregnancy. The parameters that exert infuence on the distribution of therapeutic agents include an increase in the cardiac output that reaches approximately to 7 L/min; an increase in stroke volume of about 20–30% with a heart rate of approximately 90 beats/min at rest (Pirani., 1978); and an increase in plasma volume of nearly 30–50%. Furthermore, the total body water increases by 5% in blood and 15% in extracellular fuid; and the serum albumin level decreases by about 20–40% (Pacheco et al., 2013; Tan and Tan, 2013; Jarvis and Nelson-Piercy et al., 2014). In addition to the decline in albumin concentration in plasma, the level of other proteins like alpha 1-acid glycoprotein decreases; hence in pregnancy, the protein-binding of xenobiotics is reduced, and their free fraction is increased (Hebert et al., 2013). 22.3.4 Changes in Drug Metabolism during Pregnancy The changes in the levels of metabolic enzymes of Phase I metabolism during pregnancy include induction of several members of the CYP450 family, like CYP2A6, CYP3A4, CYP2C9, and CYP2D6 (Dempsey et al., 2002; Tracy et al., 2005; Villani et al., 2006; Hodge and Tracy, 2007; Hebert et al., 2008; Anderson and Carr, 2009) and decreased activity of CYP1A2 (Hodge and Tracy, 2007). The changes in Phase II metabolism are mostly related to the induction of UGT1A1 and UGT1A4, which reduces the glucuronidation of substrates of this enzyme and increase their free fraction in plasma (Jeong et al., 2008; Pennell et al., 2004). 22.3.5 Changes in Renal Excretion during Pregnancy As other changes occur in the body during pregnancy, like increase in blood fow, plasma volume, and the free fraction of the systemic circulation, the renal blood fow also increases by about 80% and the GFR shows an increase of about 50% that remains high until the end of pregnancy (Dunlop, 1981; Davidson and Dunlop, 1980). Other functions of the kidneys, like secretion and reabsorption, sodium and water regulation, and the function of renal transporters, change during pregnancy. The outcome of the changes in the renal system is the signifcant variation in renal clearance, and when it is combined with the changes in metabolism, it can be concluded that the total body clearance of xenobiotics during pregnancy is signifcantly different from the nonpregnant population. 22.3.5.1 Estimation of GFR during Pregnancy The conventional approach for estimating GFR (namely, prediction of creatinine clearance from serum creatinine) during pregnancy is not as accurate as in the nonpregnant population. One reason can be related to the fuctuation of serum creatinine concentration during pregnancy. It has been reported (Harel et al., 2019) that the serum creatinine concentration declines rapidly in the 604

DOSAGE ADJUSTMENT IN RENAL FAILURE, LIVER DISEASES AND PREGNANCY

frst trimester, reaches a plateau level, and then increases during the third trimester. It is worth noting that estimation of creatinine clearance from serum creatinine is based on the assumptions of constant production rate and steady-state level of serum creatinine. Because of the hyper glomerular fltration during pregnancy, the GFR determination using creatinine clearance and empirical equations like Cockcroft-Gault and MDRD, that are developed mainly in nonpregnant patients, underestimate the real rate of fltration and have shown to differ signifcantly from experimentally determined values (Koetje et al., 2011). The estimates obtained from the MDRD formula are also signifcantly different from the estimates using cystatin C measurements (Larsson et al., 2010). (For all empirical equations of estimating creatinine clearance from serum creatinine, including Cockcroft-Gault and MDRD formulas, consult Chapter 10, Section 10.5.2.1.2; for cystatin C, see Chapter 10, Section 10.5.2.2). 22.3.6 Role of the Placenta The placenta, with its selective transport of xenobiotics from mother to fetus and vice versa add another aspect to the distribution and clearance of xenobiotics during pregnancy. The small lipophilic molecules can cross the placenta barrier and enter the amniotic fuid, which is considered another pool/compartment for distribution and accumulation of xenobiotics. In addition to the intracellular and transcellular passive diffusion, other mechanisms of absorption, like facilitated transport of certain drugs (such as cephalosporins and corticosteroids) and active absorption of molecules, like dopamine and norepinephrine, adds complexity to the overall ADME of xenobiotics (for the role of placenta in distribution of xenobiotics consult Chapter 8, Section 8.2.6.3). 22.3.7 PK/TK Models Mechanistic models and physiologically based pharmacokinetic models can play important roles in the determination of safe and effective dosage regimens during pregnancy. However, the exclusion of pregnant women from clinical studies due to ethical concerns of potential drug-related side effects, and/or the limited number of biological samples (e.g., blood samples) that can be drawn from a volunteer, have hindered the development of comprehensive models and data related to dosage adjustment during pregnancy. REFERENCES Anderson, G. D., Carr, D. B. 2009. Effect of pregnancy on the pharmacokinetics of antihypertensive drugs. Clin Pharmacokinet 48(3): 159–68. https://doi.org/10.2165/00003088-200948030-00002. Beste, L. A., Leipertz, S. L., Green, P. K., Dominitz, J. A., Ross, D., Ioannou, G. N. 2015. Trends in burden of cirrhosis and hepatocellular carcinoma by underlying liver diseases in US veterans, 2001–2013. Gastroenterology 149(6): 1471–82. Björnsson, E. S., Stephens, C., Atallah, E., Robles-Diaz, M., Alvarez-Alvarez, I., Gerbes, A., Weber, S., Stirnimann, G., Kullak-Ublick, G., Cortez-Pinto, H., Grove, J. I., Lucena, M. I., Andrade, P. J., Athal, G. 2023. A new framework for advancing in drug-induced liver injury research. The prospective European DILI registry. Liver Int43(1): 115–26. Burton, M. E., Vasko, M. R., Brater, D. C. 1985. Comparison of drug dosing methods. Clin Pharmacokinet 10(1): 1–37. Child, C. G., Turcotte, J. G. 1964. Surgery and portal hypertension. In The Liver and Portal Hypertension, ed. C. G. Child, 50–64. Philadelphia: Saunders. Chiou, W. L., Hsu, F. H. 1975. A new simple and rapid method to monitor the renal function based on pharmacokinetic consideration of endogenous creatinine. Res Commun Chem Pathol Pharmacol 10(2): 315–30. Cholongitas, E., Papatheodoridis, G. V., Vangeli, M., Terreni, N., Patch, D., Burroughs, A. K. 2005. Systemic review: The model for end-stage liver disease – Should it replace Child-Pugh’s classifcation for assessing prognosis in cirrhosis? Aliment Pharmacol Ther 22(11–12): 1079–89. 605

22.3 PREGNANCY

D’Amico, G., Carcia-Tsao, G., Pagliaro, L. 2006. Natural history and prognostic indicators of survival in cirrhosis: A systemic review of 118 studies. J Hepatol 44(1): 217–31. Davidson, J. M., Dunlop, W. 1980. Renal hemodynamics and tubular function normal human pregnancy. Kidney Int 18(2): 152–61. Dempsey, D., Jacob, P. III, Benowitz, N. L. 2002. Accelerated metabolism of nicotine and cotinine in pregnant smokers. J Pharmacol Exp Ther 301(2): 594–8. https://doi.org/10.1124/jpet.301.2.594. Detlli, L. 1974. Individualization of drug dosage in patients with renal disease. Med Clin North Am 58(5): 977–85. DeVane, C. L., Jusko, W. J. 1982. Dosage regimen design. Pharmc Ther 17(2): 143–63. Dunlop, W. 1981. Serial changes in renal hemodynamics during normal human pregnancy. Br J Obstet Gynaecol 88(1): 1–9. El-Khateeb, E., Achour, B., Al-Majdoub, Z. M., Barber, J., Rostami-Hodjegan, A. 2021. Nonuniformity of changes in drug-metabolizing enzymes and transporters in liver cirrhosis: Implication for drug dosage adjustment. Mol Pharm 8(9): 3563–77. https://doi.org/10.1021/1cs.molpharmaceut.1c00462. Giusti, D. L., Hayton, W. L. 1973. Dosage regimen adjustments in renal impairment. Drug Intell Clin Pharm 7(9): 382–7. Han, A. N., Han, B. R., Zhang, T., Heimbach, T. 2021. Hepatic impairment physiologically based pharmacokinetic model development: Current challenges. Curr Pharmacol Rep 7(6): 213–26. https:// doi.org/10.1007/s40495-021-00266-5. Harel, Z., McArtur, E., Hladunewich, M., Dirk, J. S., Wald, R., Garg, A. X., Ray, J. G. 2019. Serum creatinine levels before, during, and after pregnancy. JAMA 321(2). https://jamanetwork.com/. Hebert, M. F., Easterling, T. R., Kirby, B., Carr, D. B., Rutherford, T., Thummel, K. E., Fishbein, D. P., Unadkat, J. D. 2008. Effect of pregnancy on CYP3A and P-glycoprotein activities as measured by disposition of midazolam and digoxin: A University of Washington specialized center of research study. Clin Pharmacol Ther 84(2): 248–53. https://doi.org/10.1038/clpt.2008.1. Hebert, M. F., Zheng, S., Hays, K., Shen, D. D., Davis, C. L., Umans, J. G., Miodovnik, M., Thummel, K. E., Easterling, T. R. 2013. Interpreting tacrolimus concentrations during pregnancy and postpartum. Transplant J 95(7): 908–15. https://doi:10.1097/TP.0b013e318278d367. Hibino, H., Sakiyama, N., Makino, Y., Makihara-Ando, R., Horinouchi, H., Fujiwara, Y., Kanda, S., Goto, Y., Yoshida, T., Okuma, Y., Shinno, Y., Murakami, S., Hashimoto, H., Akiyoshi, T., Imaoka, A., Ohe, Y., Yamaguchi, M., Otani, O. 2022. Evaluation of hepatic CYP3A enzyme activity using endogenous markers in lung cancer patients treated with cisplatin, dexamethasone, and aprepitant. Eur J Pharmacol 78: 613–21. https://doi.org/10/1007/s00228-022-03275-5. Hodge, L. S., Tracy, T. S. 2007. Alterations in drug disposition during pregnancy: Implications for drug therapy. Expert Opin Drug Metab Toxicol 3(4): 557–71. https://doi.org/10.1517/17425225.3.4.557. Jarvis, S., Nelson-Piercy, C. 2014. Common symptoms and signs during pregnancy. Obstet Gynaecol Reorod Med Hiips 24(8): 245–49. https://doi.org/10.1016/j.ogrm.2014.05.006. Jeong, H., Choi, S., Song, J. W., Chen, H., Fischer, J. H. 2008. Regulation of UDPglucuronosyltransferase (UGT) 1A1 by progesterone and its impact on labetalol elimination. Xenobiotica 38(1): 62–75. https://doi.org/10.1080/00498250701744633.

606

DOSAGE ADJUSTMENT IN RENAL FAILURE, LIVER DISEASES AND PREGNANCY

Kaplan, D. E., Dai, F., Aytaman, A., Baytarian, M., Fox, R., Hunt, K., Knott, A., Pedrosa, M., Pocha, C., Mehta, R., Duggal, M., Skanderson, M., Valderrama, A., Taddoi, T. 2015. Development and performance of an algorithm to estimate the Child-Turcotte-Pugh score from a national electronic healthcare database. Clin Gastroenterol Hepatol 13(3): 2333–41. https://doi.org/10.1016/j.cgh.2015.07 .010. Koetje, P. M. J. L., Spaan, J. J., Kooman, J. P., Spaanderman, M. E. A., Peeters, L. 2011. Pregnancy reduces the accuracy of the estimated glomerular fltration rate based on Cockroft-Gault and MDRD formulas. Reprod Sci 18(5): 456–62. https://doi.org/10.1177/1933719110387831. Larsson, A., Palm, M., Hansson, L.-O., Axelsson, O. 2010. Cystatin C and modifcation of diet in renal disease (MDRD) estimated glomerular fltration rate differ during normal pregnancy. Acta Obstet Gynecol Scand 89(7): 939–44. Law, R., Maltepe, C., Bozzo, P., Einarson, A. 2010. Treatment of heartburn and acid refux associated with nausea and vomiting during pregnancy. Can Fam Phys 56(2): 143–4. Li, X., Tang, J., Mao, Y. 2022. Incidence and risk factors of drug induced liver injury. Liver Int 42(9): 1999–2014. https://doi.org/10.1111/liv.15262. Levy, G. 1977. Pharmacokinetics in renal disease. Am J Med 62(4): 461–5. Mcfe, A. G., Magids, A. D., Richmond, M. N., Reilly, C. S. 1991. Gastric emptying in pregnancy. Br J Anaesth 67(1): 54–7. https://doi.org/10.1093/bja/67.1.54. Mokdad, A. A., Lopez, A. D., Shahraz, S., Lozano, R., Mokdad, A. H., Stanaway, J., Murray, C. J. L., Naghavi, M. 2014. Liver cirrhosis mortality in 187 countries between 1980 and 2010: A systematic Analysis. BMC Med 12: 145. https://doi.org/10.1186/s12916-014-0145-y. Naveau, S., Giraud, V., Borotto, E., Aubert, A., Capron, F., Chaput, J. C. 1997. Excess weight risk factor for alcoholic liver disease. Hepatology 25(1): 108–11. Ortega-Alonso, A., Stephens, C., Lucena, M. I., Andrade, R. J. 2016. Case characterization and risk factor in drug induced liver injury. Int J Mol Sci 17(5): 714. https://doi:10.3390/ijms17050714. Pacheco, L. D., Constantin, M. M., Hankins, G. D. V. 2013. Physiological changes during pregnancy. In Clinical Pharmacology During Pregnancy, ed. D. R. Mattison, 5–16. New York: Academic Press. Pennell, P. B., Newport, D., Stowe, Z. N., Helmers, S. L., Montgomery, J. Q., Henry, T. R. 2004. The impact of pregnancy and childbirth on the metabolism of lamotrigine. Neurology 62(2): 292–5. Pirani, B. B. K. 1978. Smoking during pregnancy. Obstet Gynecol Surv 33(1): 1–13. Pugh, R. N. H., Murray-Lyon, I. M., Dawson, J. L., Pietroni, M. C., Williams, R. 1973. Transection of the oesophsgus for bleeding oesophageal varices. BT J Surg 60(8): 646–49. Ravenstijn, P., Chetty, M., Manchandani, P., Elmeliegy, M., Qosa, H., Younis, I. 2022. Design and conduct considerations for studies in patients with hepatic impairment. Clin Transl Sci. https://doi .org/10.1111/cts.13428. Richter, J. E. 2005. Review article: The management of heartburn in pregnancy. Aliment Pharmacol Ther 22(9): 749–57. Ritschel, W. A. 1983. A simple method for dosage regimen adjustment. Meth Find Exptl Clin Pharmacol 5(6): 407–12.

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Safadi, R., Rahimi, R. S., Thabut, D., Bajaj, J. S., Bhamidimarri, K. R., Pyrsopoulos, N., Potthoff, A., Bukofzer, S., Wang, L., Jamil, K., Devarakonda, K. R. 2022. Pharmacokinetics/pharmacodynamics of L-ornithine phenylacetate in overt hepatic encephalopathy and the effect of plasma ammonia concentration reduction on clinical outcomes. Clin Transl Sci 15(6): 1449–59. Sandhar, B. K., Elliott, R. H., Windram, I., Rowbotham, D. J. 1992. Peripartum changes in gastric emptying. Anaesthesia 47(3): 196–8. https://doi.org/10.1111/j.1365-2044.1992.tb02116. Sgro, C., Clinard, F., Quazir, K., Chanay, H., Allard, C., Guilleminet, C., Lenoir, C., Lemoine, A., Hilton, P. 2002. Incident of drug-induced hepatic injuries: A French population-based study. Hepatology 36(2): 451–55. Shin, K.-H., Choi, M. H., Lim, K. S., Yu, K.-S., Jang, I.-J., Cho, J.-Y. 2013. Evaluation of endogenous metabolic markers of hepatic CYP3A activity using metabolic profling and midazolam clearance. Clin Pharmacol Ther 94(5): 601–9. https://doi.org/10.1038/clpt.2013.128. Slatlery, J. T., Gibaldi, M., Koup, J. R. 1980. Prediction of maintenance dose required to attain a desired drug concentration at steady state from a single determination of concentration after an initial dose. Clin Pharmacokinet 5(4): 377–85. Tan, E. K., Tan, E. L. 2013. Alterations in physiology and anatomy during pregnancy. Best Pract Res Clin Obstet Gynaecol 27(6): 791–802. https://doi.org/10.1016/j.bpobgyn.2103.08.001. Tracy, T. S., Venkataramanan, R., Glover, D. D., Caritis, S. N. 2005. Teporal changes in drug metabolism (CYP1A2, CYP2D6 and CYP3A activity) during pregnancy. Am J Obstet Gynecol 192(2): 633–39. https://doi.org/10.1016/j.ajog.2004.08.030. Van Dalen, R., Vree, T. B., Baars, A. M., Termond, E. 1986. Dosage adjustment for ceftazidine in patients with impaired renal function. Eur J Clin Pharmacol 30(5): 597–605. Vasicka, A., Lin, T. J., Bright, R. H. 1957. Peptic ulcer and pregnancy, review of hormonal relationships and a report of one case of massive gastrointestinal hemorrhage. Obstet Gynecol Surv 12(1): 1–13. Villani, P., Floridia, M., Pirillo, M. F., Cusato, M., Tamburrini, E., Cavaliere, A. F., Guaraldi, G., Vanzini, C., Molinari, A., deg Antoni, A., Regazzi, M. 2006. Pharmacokinetics of nelfnavir in HIV1-infected pregnant and nonpregnant women. Br J Clin Pharmacol 63(3): 309–15. https://doi.org/10 .1111/j.1365.215.2006.02669.x. Wagner, J. G. 1975. Fundamentals of Clinical Pharmacokinetics. Hamilton: Drug Intelligence Publications, Inc. Waldum, H. L., Straume, B. K., Lundgren, R. 1980. Serum group I pepsinogens during pregnancy. Scand J Gastroenterol 15(1): 61–3.

608

Addendum I – Part 1 Standard Terminologies for Routes of Administration The following are the standard terminologies for routes of administration, harmonized by the US Food and Drug Administration (FDA) and the International Conference on Harmonization (ICH – E2B terms) available at: FDA: https://www.fda.gov/drugs/data-standards-manual-monographs/route-administration ICH: https://admin.ich.org/sites/default/fles/inline-fles/E2B_R2_Guideline.pdf

FDA Approved Routes of Administration NAME Auricular (Otic) Buccal

Conjunctival

Cutaneous Dental Electro-Osmosis

Endocervical

Endosinusial Endotracheal Enteral Epidural Extra–Amniotic Extracorporeal Hemodialysis Infltration

Interstitial

DEFINITION Administration to or by way of the ear. Administration directed toward the cheek, generally from within the mouth. Administration to the conjunctiva, the delicate membrane that lines the eyelids and covers the exposed surface of the eyeball. Administration to the skin. Administration to a tooth or teeth. Administration of through the diffusion of substance through a membrane in an electric feld. Administration within the canal of the cervix uteri. Synonymous with the term “intracervical.” Administration within the nasal sinuses of the head. Administration directly into the trachea. Administration directly into the intestines. Administration upon or over the dura mater. Administration to the outside of the membrane enveloping the fetus Administration outside of the body. Administration through hemodialysate fuid. Administration that results in substances passing into tissue spaces or into cells. Administration to or in the interstices of a tissue.

SHORT NAME

FDA CODE

NCI CONCEPT ID

OTIC

013

C38192

BUCCAL

030

C38193

CONJUNC

068

C38194

CUTAN

130

C38675

DENTAL

038

C38197

EL-OSMOS

357

C38633

E-CERVIC

131

C38205

E-SINUS

133

C38206

E-TRACHE

401

C38208

ENTER

313

C38209

EPIDUR

009

C38210

X-AMNI

402

C38211

X-CORPOR

057

C38212

HEMO

140

C38200

INFIL

361

C38215

INTERSTIT

088

C38219 (Continued)

609

ADDENDUM I – PART 1

NAME Intra-Abdominal Intra-Amniotic Intra-Arterial Intra-Articular Intrabiliary Intrabronchial Intrabursal Intracardiac Intracartilaginous Intracaudal Intracavernous

Intracavitary

Intracerebral Intracisternal Intracorneal

Intracoronal, Dental

Intracoronary Intracorporus Cavernosum Intradermal Intradiscal

DEFINITION Administration within the abdomen. Administration within the amnion. Administration within an artery or arteries. Administration within a joint. Administration within the bile, bile ducts, or gallbladder. Administration within a bronchus. Administration within a bursa. Administration within the heart. Administration within a cartilage; endochondral. Administration within the cauda equina. Administration within a pathologic cavity, such as occurs in the lung in tuberculosis. Administration within a nonpathologic cavity, such as that of the cervix, uterus, or penis, or such as that which is formed as the result of a wound. Administration within the cerebrum. Administration within the cisterna magna cerebellomedularis. Administration within the cornea (the transparent structure forming the anterior part of the fbrous tunic of the eye). Administration of a drug within a portion of a tooth which is covered by enamel and which is separated from the roots by a slightly constricted region known as the neck. Administration within the coronary arteries. Administration within the dilatable spaces of the corpora cavernosa of the penis. Administration within the dermis. Administration within a disc.

SHORT NAME

FDA CODE

NCI CONCEPT ID

I-ABDOM

056

C38220

I-AMNI

060

C38221

I-ARTER

037

C38222

I-ARTIC

007

C38223

I-BILI

362

C38224

I-BRONCHI

067

C38225

I-BURSAL

025

C38226

I-CARDI

027

C38227

I-CARTIL

363

C38228

I-CAUDAL

413

C38229

I-CAVERN

132

C38230

I-CAVIT

023

C38231

I-CERE

404

C38232

I-CISTERN

405

C38233

I-CORNE

406

C38234

I-CORONAL

117

C38217

I-CORONARY

119

C38218

I-CORPOR

403

C38235

I-DERMAL

008

C38238

I-DISCAL

121

C38239 (Continued)

610

ADDENDUM I – PART 1

NAME Intraductal Intraduodenal Intradural Intraepidermal Intraesophageal Intragastric Intragingival Intraileal

Intralesional Intraluminal Intralymphatic Intramedullary Intrameningeal

Intramuscular Intraocular Intraovarian Intrapericardial Intraperitoneal Intrapleural Intraprostatic Intrapulmonary Intrasinal Intraspinal Intrasynovial Intratendinous Intratesticular

DEFINITION Administration within the duct of a gland. Administration within the duodenum. Administration within or beneath the dura. Administration within the epidermis. Administration within the esophagus. Administration within the stomach. Administration within the gingivae. Administration within the distal portion of the small intestine, from the jejunum to the cecum. Administration within or introduced directly into a localized lesion. Administration within the lumen of a tube. Administration within the lymph. Administration within the marrow cavity of a bone. Administration within the meninges (the three membranes that envelope the brain and spinal cord). Administration within a muscle. Administration within the eye. Administration within the ovary. Administration within the pericardium. Administration within the peritoneal cavity. Administration within the pleura. Administration within the prostate gland. Administration within the lungs or its bronchi. Administration within the nasal or periorbital sinuses. Administration within the vertebral column. Administration within the synovial cavity of a joint. Administration within a tendon. Administration within the testicle.

SHORT NAME

FDA CODE

NCI CONCEPT ID

I-DUCTAL

123

C38240

I-DUOD

047

C38241

I-DURAL

052

C38242

I-EPIDERM

127

C38243

I-ESO

072

C38245

I-GASTRIC

046

C38246

I-GINGIV

307

C38247

I-ILE

365

C38249

I-LESION

042

C38250

I-LUMIN

310

C38251

I-LYMPHAT

352

C38252

I-MEDUL

408

C38253

I-MENIN

409

C38254

IM

005

C28161

I-OCUL

036

C38255

I-OVAR

354

C38256

I-PERICARD

314

C38257

I-PERITON

004

C38258

I-PLEURAL

043

C38259

I-PROSTAT

061

C38260

I-PULMON

414

C38261

I-SINAL

010

C38262

I-SPINAL

022

C38263

I-SYNOV

019

C38264

I-TENDIN

049

C38265

I-TESTIC

110

C38266 (Continued)

611

ADDENDUM I – PART 1

NAME Intrathecal

Intrathoracic

Intratubular Intratumor Intratympanic Intrauterine Intravascular Intravenous Intravenous Bolus Intravenous Drip

Intraventricular Intravesical Intravitreal Iontophoresis

Irrigation Laryngeal Nasal Nasogastric

Not Applicable Occlusive Dressing Technique

DEFINITION Administration within the cerebrospinal fuid at any level of the cerebrospinal axis, including injection into the cerebral ventricles. Administration within the thorax (internal to the ribs); synonymous with the term “endothoracic.” Administration within the tubules of an organ. Administration within a tumor. Administration within the aurus media. Administration within the uterus. Administration within a vessel or vessels. Administration within or into a vein or veins. Administration within or into a vein or veins all at once. Administration within or into a vein or veins over a sustained period of time. Administration within a ventricle. Administration within the bladder. Administration within the vitreous body of the eye. Administration by means of an electric current where ions of soluble salts migrate into the tissues of the body. Administration to bathe or fush open wounds or body cavities. Administration directly upon the larynx. Administration to the nose; administered by way of the nose. Administration through the nose and into the stomach, usually by means of a tube. Routes of administration are not applicable. Administration by the topical route which is then covered by a dressing which occludes the area.

SHORT NAME

FDA CODE

NCI CONCEPT ID

IT

103

C38267

I-THORAC

006

C38207

I-TUBUL

353

C38268

I-TUMOR

020

C38269

I-TYMPAN

366

C38270

I-UTER

028

C38272

I-VASC

021

C38273

IV

002

C38276

IV BOLUS

138

C38274

IV DRIP

137

C38279

I-VENTRIC

048

C38277

I-VESIC

128

C38278

I-VITRE

311

C38280

ION

055

C38203

IRRIG

032

C38281

LARYN

364

C38282

NASAL

014

C38284

NG

071

C38285

NA

312

C48623

OCCLUS

134

C38286

(Continued)

612

ADDENDUM I – PART 1

NAME Ophthalmic Oral Oropharyngeal Other Parenteral Percutaneous Periarticular Peridural Perineural Periodontal Rectal Respiratory (Inhalation)

Retrobulbar Soft Tissue Subarachnoid Subconjunctival Subcutaneous

Sublingual Submucosal Topical

Transdermal

DEFINITION Administration to the external eye. Administration to or by way of the mouth. Administration directly to the mouth and pharynx. Administration is different from others on this list. Administration by injection, infusion, or implantation. Administration through the skin. Administration around a joint. Administration to the outside of the dura mater of the spinal cord. Administration surrounding a nerve or nerves. Administration around a tooth. Administration to the rectum. Administration within the respiratory tract by inhaling orally or nasally for local or systemic effect. Administration behind the pons or behind the eyeball. Administration into any soft tissue. Administration beneath the arachnoid. Administration beneath the conjunctiva. Administration beneath the skin; hypodermic. Synonymous with the term “subdermal.” Administration beneath the tongue. Administration beneath the mucous membrane. Administration to a particular spot on the outer surface of the body. The E2B term “transmammary” is a subset of the term “topical.” Administration through the dermal layer of the skin to the systemic circulation by diffusion.

SHORT NAME

FDA CODE

NCI CONCEPT ID

OPHTHALM

012

C38287

ORAL

001

C38288

ORO

410

C38289

OTHER

135

C38290

PAREN

411

C38291

PERCUT

113

C38676

P-ARTIC

045

C38292

P-DURAL

050

C38677

P-NEURAL

412

C38293

P-ODONT

040

C38294

RECTAL

016

C38295

RESPIR

136

C38216

RETRO

034

C38296

SOFT TIS

109

C38198

S-ARACH

066

C38297

S-CONJUNC

096

C38298

SC

003

C38299

SL

024

C38300

S-MUCOS

053

C38301

TOPIC

011

C38304

T-DERMAL

358

C38305

(Continued)

613

ADDENDUM I – PART 1

NAME Transmucosal Transplacental Transtracheal Transtympanic Unassigned Unknown Ureteral Urethral Vaginal

614

DEFINITION Administration across the mucosa. Administration through or across the placenta. Administration through the wall of the trachea. Administration across or through the tympanic cavity. Route of administration has not yet been assigned. Route of administration is unknown. Administration into the ureter. Administration into the urethra. Administration into the vagina.

SHORT NAME

FDA CODE

NCI CONCEPT ID

T-MUCOS

122

C38283

T-PLACENT

415

C38307

T-TRACHE

355

C38308

T-TYMPAN

124

C38309

UNAS

400

C38310

UNKNOWN

139

C38311

URETER

112

C38312

URETH

017

C38271

VAGIN

015

C38313

Addendum I – Part 2 Relevant Mathematical Concepts A.1 THE LAPLACE TRANSFORM METHOD OF INTEGRATION The idea behind the Laplace transform is that every solution for any differential equation has a Laplace transform and the derivatives of the function can also be expressed in terms of a Laplace transform (Bauer and Nohel, 1973). This means that any complex differential equation can be converted into a linear algebraic equation and the resulting equation is then solved by simpler methods. The fnal solution of the algebraic conversion is subjected to an inverse transformation, which may often be facilitated by a table of Laplace transforms yielding the desired solution of the differential equation. To learn more about the basic properties and the uniqueness theorem of the Laplace transform, textbooks of ordinary differential equations should be consulted. The following discussion is for the beneft of understanding of application of the Laplace transform in PK/TK modeling. Briefy, assume our function isy = f (t). The Laplace transform of this function, represented by the notation L ˛ f ˜ t ° ˙ , can be obtained by multiplying the function, f (t), by e −st dt and integrating ˝ ˆ from time zero to ∞. Thus, the Laplace transform of the function f (t) is



L ˜° f (t)˛˝ ˙



0

e ˘st f ˆ t ˇ

(A.1)

where s is the Laplace operator and the function is converted from real variable t into a function of complex parameter s. As an example, consider the function of f (t) = 1. The Laplace transform of the function is: 



L ˛˝ f ˜ t ° ˙ˆ ˇ L ˜1° ˇ

0

e ˘st ˜1°dt ˇ



e ˘st ˘s

1 s

ˇ 0

(A.2)

1 1 Therefore, the Laplace transform of 1 is and the inverse Laplace transform of is 1; that is, s s (A.3). ˜1 ° 1 ˙ L ˝ ˇ˘1 ˛sˆ From the above transformation, it can also be concluded that the Laplace transform of any constant 5 5 A is A/s and the inverse Laplace transform of A/s is A. For example, L ˜ 5 ° ˛ andL ˜1 °˝ ˙ˇ ˘ 5 . s ˛sˆ The Laplace transforms of exponential functions such as f ˜ t ° ˛ e kt and f ˜ t ° ˛ e ˝ kt are:

˜ ° ˆ

L e kt ˛

˝ ˙ ˇ

L e ˜kt ˆ

°

0

˙

0

ˆ

e ˝st e kt dt ˛

e ˜st e ˜kt dt ˆ

˙

0

ˇ

°

0

e˜

k ˝ s°t

˛

e ˜ °t k˝s k ˝s

˙

˛ 0

˜ ˝ s ˛ k ˙t

e ˜ s˛k t e ˝ ˙ dt ˆ ˜˝s ˛ k ˙

1 s˝k °

ˆ 0

1 s˛k

(A.4)

(A.5)

Accordingly, the inverse Laplace transforms are: ° 1 ˙ kt L ˜1 ˝ ˇ˘e ˛s˜ k ˆ

(A.6)

˛ 1 ˆ (A.7) ˜kt L ˜1 ˙ ˘e ˝s° k ˇ Table A.1 contains a short list of Laplace transforms that are useful in the development of integrated equations of PK/TK. In Table A.1, A and B are constants; k, k1 , k 2 , and k 3 are rate constants in their general form (they are not related to any specifc rate constant and are included under the assumption that k ≠ k1 ≠ k 2 ≠ k 3 ); m is a power constant; and s is the Laplace parameter.

615

A.1 THE LAPLACE TRANSFORM METHOD OF INTEGRATION

Table A.1 Relevant Laplace Transforms Function

Laplace Transform

1

1 s

A

A s

t

1 s2

At

A s2

tm

m! sm+1

e kt

1 s-k

e -kt

1 s+k

t

A s-k

Ae -kt

A s+k A

Ate -kt

(s + k )

A 1 - e -kt k

(

A s(s + k )

)

A

At m e -kt

(s + k )

(

A 1 - e -kt æAö t çk÷ k2 è ø B 1 - e -kt k

(

)

A s2 ( s + k )

)

( B - Ak1 ) e -k t - ( B - Ak2e -k t ) 1

2

k 2 - k1 A e -k1t - e -k2t k 2 - k1

(

m +1

A k1 s + k 2

æ k2 ö

A - çè k1 ÷ø t e k1

Ae -kt -

2

)

( As - B ) s(s + k ) As + B

( s + k1 )( s - k2 ) A

( s + k1 )( s + k2 )

1 k e -k1t - k1e -k2t + 2 k 2 k1 k 1 k 2 ( k1 - k 2 )

1 s ( s + k1 )( s + k 2 )

A æ 1 - e -k2t 1 - e -k1t ö ÷ ç k1 - k 2 è k 2 k1 ø

A s ( s + k1 )( s + k 2 ) (Continued)

616

ADDENDUM I – PART 2

Table A.1

Continued

Function

Laplace Transform A s ( s + k1 )( s + k 2 )

æ 1 ö e -k1 e -k2t Aç + ç k1k 2 k1 ( k1 - k 2 ) k 2 ( k1 + k 2 ) ÷÷ è ø A t - A 1 - e -kt k

(

A s2 ( s + k )

)

( Ak1 - B ) e -k1t + ( Ak2 - B ) e -k2t B k1 k 2 k1 ( k1 - k 2 ) k 2 ( k1 - k 2 )

As + B s ( s + k1 )( s + k 2 )

( A - k1 ) e -k t + ( A - k2 ) e -k t + ( A - k3 ) e -k t ( k2 - k1 )( k3 - k1 ) ( k1 - k2 )( k3 - k2 ) ( k1 - k3 )( k2 - k3 ) 2

1

3

s+A

( s + k1 )( s + k2 )( s + k3 )

Among the theorems of Laplace transformation, the following are the most useful in PK/TK analysis: 1. If L ëé f1 ( t ) ùû = F1 ( s ) and L ëé f 2 ( t ) ùû = F2 ( s ) , then

(A.8)

L éë f1 ( t ) ùû + L éë f 2 ( t ) ûù = F1 ( s ) + F2 ( s )

(A.9)

L éë af ( t ) ûù = aL éë f ( t ) ùû = aF ( s )

(A.10)

af1 ( t ) + bf 2 ( t ) has Laplace transform aF1 ( s ) + bF2 ( s )

(A.11)

3. If a is a constant, then

5. If a and bare constants, then

dy is the Laplace transform of y minus the value of the func7. Laplace transform of derivative dx tion at t = 0 , i.e., é dy ù L ê ú = sy˜ - y 0 ë dx û

(A.12)

wherey˜ is the Laplace transform of y. A.2 DETERMINANTS AND CRAMER’S RULE In solving the PK/TK systems of linear differential equations with constant coeffcients using the Laplace transform, the coeffcients of the transformed functions and the constant terms of the initial conditions are important in solving the equations, as briefy described here. Any two- or three-variable linear equation can be written in the form of ax + by = c

(A.13)

ax + by + cz = Q

(A.14)

wherea, b, and c are the constant coeffcients and Q is also a constant. In solving the PK/TK system of linear differential equations by Laplace transform, one frequently encounters systems of simultaneous equations of the form ax + by = C dx + +ey = F

(A.15)

or,

617

A.2 DETERMINANTS AND CRAMER’S RULE

ax + by + cz = Q dx + ey + fz = R

(A.16)

gx + hy + iz = S and so on. The use of Cramer’s rule and determinants can make the process of solving the simultaneous set of differential equations simple. This is only one application of the rule, and a discussion of various applications and theorems of determinants is beyond the scope of this section. The frst step in solving the system of equations with three variables (Equation A.16) according to Cramer’s rule is to establish the determinant (D) of the system by using the coeffcients of the variables that are unknown: æa ç D =çd çg è

b e h

cö ÷ f÷ i ÷ø

(A.17)

The solution of this determinant is: D = aei + bfg + cdh - ceg - bdi - afh

(A.18)

The signs of the Equation A.18 are decided according to the following rule: Sign = ( -1)

No of Row + No of Column

That is: +11 -21 +31

-12 +22 -32

+13 -23 +33

where the frst number is the row number and the second is the column number. Therefore, position 32, for example, refers to the third row and second column. According to Cramer’s rule, if the solution of determinant D is not zero, then the equations have a unique solution, and each variable can be expressed as a fraction of two determinants. The numerator of the fraction is expressed by replacing the column of coeffcients of the variable under evaluation with the constants of the system. For example, for x in Equations A.16: æQ ç çR çS x=è æa ç çd çg è

b e h b e h

cö ÷ f÷ i ÷ø Qei + bfS + Rhc - Sec - Rbi b - Qfh = cö aei + bfg + chd - gec - dbi - afh ÷ f÷ i ÷ø

(A.19)

A similar arrangement is repeated for each of the unknowns. For instance, for y, the solution is as follows: æa ç çd çg y=è æa ç çd çg è

Q R S b e h

cö ÷ f÷ i ÷ø aRi + Qfg + dSc - cRg - Qddi - fSa = cö aei + bfg + chd - gec - dbi - afh ÷ f÷ i ÷ø

(A.20)

Similar calculations are used for the z variable. The unknowns of a set of simultaneous equations with two variables can also be determined in a similar way. For example, the solution for variables of Equations A.15 is:

618

ADDENDUM I – PART 2

æC ç F x=è æa ç èd

bö ÷ e ø Ce - bf = b ö ae - bd ÷ eø

(A.21)

æa ç d y=è æa ç èd

Cö ÷ F ø aF - dC = bö ae - bd ÷ eø

(A.22)

A.3 INPUT AND DISPOSITION FUNCTIONS In compartmental analysis, the Laplace transform of linear differential equations for any compartment can be developed by multiplication of its input function with its disposition function (Benet, 1972). This convolution method signifcantly facilitates the development of Laplace transforms of compartmental analysis. A.3.1 Input Functions The input functions via the major routes of administration into the body are: ◾ Intravenous bolus injection: input = Dose = AD0

(A.23)

f ( t ) = k0

(A.24)

◾ Intravenous infusion:

(k0 is the zero-order rate of infusion) L[ f (t)] =

ò

tinfusion

0

k0 e -st dt = -

k0 -stinfusion k0 e + s s

(A.25)

When the infusion starts at t = 0 and continues for a time long enough such that ,

tinfusion Þ ¥ input = L[k0 ] =

k0 s

(A.26)

When the infusion starts at t = 0 and continues for a time t < ¥, input = L [k0 ] =

(

k0 1 - e -stinfusion s

)

(A.27)

When the infusion starts at t ¹ 0 , input = L [ f (t)] =

ò

tinfusion

a

-st

k0 e dt =

(

k0 e -as - e - s´tinfusion

)

s

(A.28)

where, tinfusion is the time of infusion; a is a time different from zero. ◾ Oral absorption: f (t) = FDk a e -kat

(A.29)

(FD is the amount of drug absorbed, and ka is the frst-order absorption rate constant). L [ f (t)] = FDk a

ò

¥

0

e -kat e -st = FDk a

input =

FDk a s + ka

ò

¥

0

- ka+ s t e ( )

(A.30) (A.31)

619

A.4 TRAPEZOIDAL RULE

◾ The input function from any driving-force compartment (DFC) into other compartments is defnes as:

( exit rateconstant )DFC ´ ( Laplace transform )DFC

(A.32)

A similar approach can be taken for the development of other input functions. A.3.2 Disposition Functions In more complex multicompartmental linear models when all peripheral compartments are interdependent with the central compartment, the disposition function of the central compartment is determined by the following equation (Benet, 1972): n

Õ(s + E )

(A.33)

i

(Disp.)s =

i=2

n

n

Õ(s + E ) - å i

i=1

j=2

é ê ê k1j k j1 ê ë

ù ú (s + Em )ú ú m=2 m¹ j û n

Õ

where (Disp.)s is the disposition function of the central compartment as a function of Laplace operator, s; n is the number of driving force compartment; Ei is the overall exit rate constant from ith compartment; k1j is the frst-order rate constant from the central compartment to jth compartment; th are k j1 is the frst-order rate constant from j compartment to the central compartment; å and the symbols for summation and continued product, respectively.

Õ

A.3.3 Determination of Inverse Laplace Transform by the Method of Partial Fraction Theorem This method is used when the denominator of the Laplace transforms (i.e., multiplication of input and disposition functions) has no repeated terms and contains more “s” terms than the numerator. Under these conditions, the inverse Laplace transform can be estimated by substituting the hybrid rate constants for each “s” term in numerator and denominator followed by multiplication with the related exponential function, as shown below for a typical multicompartment mammillary model that consists of a central compartment and three peripheral compartments interdependent with the central compartment (Benet and Turi, 1971). All four compartments are considered drivingforce compartments and have elimination to the outside environment, although not necessarily the same environment (see Chapter 12, Section 12.3.1.3). ˜ = Input ´ (Disp.)s1 = L ( A1 ) = A A1 =

Dose ( s + E2 )( s + E3 )( s + E4 ) ( s + a )( s + b )( s + g )( s + d )

Dose(E2 - a)(E3 - a)(E4 - a) -at Dose(E2 - b)(E3 - b)(E4 - b) -bt e + e (b - a)( g - a)(d - a) ( a - b )( g - b )( d - b )

Dose(E2 - g)(E3 - g)(E4 - g) -gt Dose ( E2 - d )( E3 - d )( E4 - d ) -dt + e + e ( a - g )(b - g )( d - g ) ( a - d ) (b - d )( g - d )

(A.34)

(A.35)

A.4 TRAPEZOIDAL RULE The integration of linear differential equations is a straightforward concept used commonly in the development of PK/TK equations. For example, the integral of “rate” or “slope” relationship (i.e., a simple differential equation) is the equation of the line or curve, and the integral of equation of the line or curve is an area under the line or curve. Integration of these functions is referred to as antidifferentiation or reversing, and the following symbol is used to identify the operation:

ò f ( x ) dx = F ( x ) + C

(A.36)

When the interval of integration is not defned (Equation A.36), the operation is called the indefnite integration, and F ( x ) is the indefnite integral. The defnite integral, on the other hand, is the magnitude of an integral over a well-defned interval. Mathematically, if F ( x ) is the integral of f ( x )between an upper limit of x = b and lower limit of x = a (i.e., a £ x £ b), then the integration is defned as: 620

ADDENDUM I – PART 2

ò

b

a

f ( x ) dx = F ( b ) - F ( a )

(A.37)

An example of a defnite integral is the area under the plasma concentration-time curve of a compound between time of administration (a = 0) and six hours after the administration (b = 6h ). An example of an indefnite integral is when the area is between the time of administration (a = 0) and the time that the dose is eliminated from the body (b @ ¥). Both the upper and lower limits in defnite integral are real numbers. Therefore, the defnite integral is a real number, whereas the indefnite integral is a function. The trapezoid rule is a practical method for numerical approximation of defnite integrals. The method is based on dividing the interval of observation (i.e., a £ x £ b ) into subintervals that correspond to the time of sampling and measurements. The area between the curve and each subinterval on the x -axis is a quadrilateral with one pair of parallel lines and one pair of unparallel lines forming a trapezoid with an area equal to: æ y + y2 ö (A.38) Area = base ç 1 ÷ è 2 ø where y1and y 2 correspond to plasma concentration or logarithm of plasma concentration at two adjacent time points, and the base is the subinterval of x axis that corresponds to the time interval between the two y values (Figure A.1). The total area under the curve that corresponds to the indefnite integral of plasma concentration-time curve or log plasma concentration-time curve is then approximated by adding up all trapezoidal areas. The terminal area is estimated by dividing the last plasma data point by the overall elimination rate constant. The following equations are the summary relationships for area calculations of Cp - time or log Cp - time curve from time zero to time t or infnity:

Figure A.1 A graphical presentation of the trapezoidal rule for estimation of area under plasma concentration-time curve; AUC0t represents the area under the plasma concentration curve between time zero and the last data point; AUCt¥ is the terminal area between the last time point of sample collection and time infnity; AUC0¥ is the total area under the curve between time zero and infnity: it is equal to the summation of AUCt¥ +AUC0¥ . 621

A.6 KINETICS AND RATE EQUATIONS

n

AUC ¥0 =

å éë( y + y i

) / 2ùû ( xi+1 - xi ) + AUC ¥n

i+1

i=0

n

Cpi + Cpi+1 ö ÷ ( ti+1 - ti ) 2 ø

(A.40)

Cpi + Cpi+1 ö Cpn ÷ ( ti+1 - ti ) + K 2 ø

(A.41)

log Cpi + log Cpi+1 ö ÷ ( ti+1 - ti ) 2 ø

(A.42)

log Cpi + log Cpi+1 ö Cpn ÷ ( ti+1 - ti ) + K 2 ø

(A.43)

AUC t0 =

å æçè i=0

n

AUC ¥0 =

å èæç i=0 n

AUC 0 = t

å çæè i=0

n

¥ AUC 0 =

å æçè i=0

(A.39)

where n is the total number of observations, i identifes each measurement, i = 0 indicates that the interval starts from a = 0 and the initial values of x and y should be included, and AUCn¥ represents the terminal area between the last data point and infnity, estimated as the last yvalue divided by the rate constant estimated from the slop of the terminal phase of the curve. A.5 GEOMETRIC SERIES A geometric series is used in connection with multiple-dosing kinetics. A geometric series is a series for which the ratio of each two consecutive terms is a constant function of the summation; that is, Sn ( x) =

1 - ( x)n+1 1 - ( x)

(A.44)

Therefore, for a function like Cp = Cp 0 e -Kt , the geometric series is: Sn ( x) =

1 - e -n( K )t 1 - e -Kt

(A.45)

As n Þ ¥, e -n( K )t Þ 0 and the geometric series changes to: Sn ( x) =

1 1 - e -Kt

(A.46)

The application of geometric series is presented in Chapter 16, Section 16.2. A.6 KINETICS AND RATE EQUATIONS The activity of a chemical or biological system is defned either dynamically or kinetically. The dynamic principles often deal with the quantitative and qualitative evaluations of transformation between an initial condition and its intended end condition. The evaluations can only determine the dynamics of the system or process in terms of receiving an input and providing a response, like the principles of thermodynamics, pharmacodynamics, or toxicodynamics. Kinetics, on the other hand, is the study of time dependency of the transformation from the initial condition to the intended end condition in terms of associated rates or velocities. In PK/TK, the biological processes are explored in terms of their rate of mass fuxes and rapidity of distribution, biotransformation, and elimination. The mathematical expressions are the same as used in chemical kinetics. A.6.1 First-Order Kinetics When the rate of a process is proportional to the frst power of concentration, amount, or any measurable quantity under evaluation, the process is considered frst-order kinetics, which can be summarized as: Rate µ [ A ]

622

(A.47)

ADDENDUM I – PART 2

An equal sign and a proportionality constant, K, can replace the proportionality sign in Equation A6.1 and change the equation into a linear differential equation: dA = ±KA dt

(A.48)

Where Ais the amount and K is the frst-order rate constant. The plus sign symbolizes a positive outcome for the process such as “formation” or “appearance,” and the minus sign represents a negative outcome such as “decomposition” or “disappearance,” and dA dt is the differential equation. The integration of Equation A.48 yields the following relationships:

ò

¥

0

dA = ±K A

ò

¥

0

(A.49)

dt

ln A = ±Kt + constant

(A.50)

at t = 0, constant = ln At =0 = ln A0 ln A = ln A0 ± Kt

(

(A.51)

)

ln A A0 = ±Kt

(A.52)

ln A = 2.303 log A

(A.53)

log A = log A0 ±

(

)

log A A0 = ±

Kt 2.303

(A.54)

Kt 2.303

(A.55) 0

Raising Equation A.51 to the power of exponential ( e ln A = e ln A e ± Kt ) yields A = A0 e -Kt

or

A = A0 e Kt

(A.56)

Equation A.48 establishes that the rate of a frst-order process, dA dt, is a variable and changes with amount or concentration. Its unit is mass / time , and the unit of the rate constant, K , is time -1 . Furthermore, assuming the elimination process with negative slope, Equation A.51 implies that the plot of ln A versus time is linear with a slope of -K and y - intercept of ln A0 . Similarly, Equation A.54 has slope of - K 2.303 and y-intercept of log A0 . Equations A.52 and A.55 indicate that plots of ln A A0 or log A A0 versus time are also straight line passes through the origin with a slope -K or - K 2.303, respectively, and the time required for a frst-order process to advance to a certain level of completion (e.g., 50% complete or 90% percent complete, etc.) is independent of the amount. This property is used to estimate the half-life of the process under consideration. The half-life T1 2 is the time for the process to be half-completed, or it is the time required for one-half of the initial amounts or concentration of a drug to be used in a process; that is,

(

)

(

)

At t = T1 2 ; A = A0 2 ln

A0 KT1/2 = ln A0 ± 2 2.303 T1 2 =

ln 2 K

(A.57) (A.58)

or log A0 - log

A0 2 A0 = log 0 = log 2 = 0.301 2 A T1/2 =

0.693 K

(A.59)

and 623

A.7 AKAIKE AND SCHWARZ CRITERIA

K = 0.693 T1 2

(A.60)

For the processes that follow frst-order kinetics, the half-life and K are constant. A.6.2 Zero-Order Kinetics Contrary to frst-order kinetics, where the rate is a variable and directly proportional to the amount or concentration of the compound under consideration, the zero-order rate is constant; that is, the process occurs at a constant rate. Therefore, the rate is controlled by factors other than the compound’s concentration (e.g., saturation of enzyme when the rate of metabolism is at maximum, or infusion rate of intravenously administered therapeutic agents, which can be changed through the infusion pump, or by increasing the number of drops manually). The rate equation of a zero-order process is: dA = ±k0 dt

(A.61)

where k0 is the zero-order rate or rate constant, which has the units of mass/time. The integration of Equation A.61 yields A = A0 ± k0t

(A.62)

where the amount at time t is Aand the initial amount is A0. A plot of A versus t on a linear graph is a straight line with y - intercept = A0 and slope = k0 . The half-life of a zero-order process is not constant; it is directly proportional to 50% of the initial amount or concentration of the drug and is inversely proportional to the zero-order rate, that is, A0 -

A0 = k0T1/2 2

(A.63)

A0 2k0

(A.64)

T1/2 =

A.7 AKAIKE AND SCHWARZ CRITERIA In ftting models with different levels of complexity to a set of data, a better ft is usually achieved with a complex model with more parameters. However, a more complex model, because of the multitude of unsolvable parameters/constants, may not be the most relevant one, and a model with fewer parameters is usually preferred to the one with more. When comparing several conceivable models to select the most appropriate one, it is necessary to compensate for improvement of ft due to increased model complexity. The Akaike information criterion (AIC) (Akaike, 1974) and Schwarz criterion, also known as the Bayesian information criterion (BIC) (Schwarz, 1978), are useful for comparing compartmental models with any types of input. The AIC formula is a remarkably simple one: AIC = 2K - 2log ( L )

(A.65)

where ( K ) is the number of predictors (parameters to be estimated) and (L) is the maximum likelihood value. The ( 2K ) part of the formula controls the number of predictors, and the -2log ( L ) part of the equation is a large positive value. A model with the lowest AIC value is the preferred one. The BIC formula is also simple to use: BIC = -2ln L + K ln ( n )

(A.66)

where Lis the maximum likelihood value, K is the number of predictors, and nis the number of data points. When data are limited, BIC is usually preferred to AIC. Practically all PK/TK software used for ftting models use AIC and/or BIC criteria. REFERENCES Akaike, H. 1974. A new look at the statistical model identifcation. IEEE T Automat Contr 19(6): 716–23.

624

ADDENDUM I – PART 2

Benet, L. Z. 1972. General treatment of linear mammillary models with elimination from any compartment as used in pharmacokinetics J Pharm Sci 61(4): 536–41. Benet, L. Z., Turi, J. S. 1971. Use of a general partial fraction theorem for obtaining inverse Laplace transforms in pharmaceutical analysis. J Pharm Sci 60: 1593–4. Brauer, F., Nohel, J. A. 1973. Ordinary Differential Equations. Menlo Park: W. A. Benjamin. Schwarz, G. E. 1978. Estimating the dimension of a model. Ann Stat 6(2): 461–4.

625

A D D E N D U M I – PA R T 3

ABBREVIATION – GLOSSARY – PK/TK CONSTANTS AND VARIABLES

626

Addendum II – Part 1 CASE 1.1 - PROTEIN BINDING The total albumin concentration used in an experiment is 2 × 10−4 M. The concentrations of xenobiotic selected for the experiment ( CD ) is presented in Table 1.1.1. The concentration of drug–protein complex is estimated as presented in Table 1.1.2.

Table 1.1.1

9.9 × 10−4

CD ( M )

Table 1.1.2 CDP ( M )

Xenobiotic Concentration 2.67 × 10−3

6.00 × 10−3

1.6 × 10−2

Drug–Protein Complex Concentration 2.4 × 10−4

4.8 × 10−4

7.2 × 10−4

9.6 × 10−4

1.1.1 Calculate the association constant and the number of binding sites of the protein by using double reciprocal and Scatchard plots. Double reciprocal calculations are presented in Table 1.1.3.

Table 1.1.3 r 1/r 1/CD

Double Reciprocal Calculations 1.2 0.8334 1010.10

2.4 0.4167 374.53

3.6 0.2778 166.67

4.8 0.2083 62.54

Assuming ideal data and using slope relationship: 1 1 1 = + r nK aCD n Slope =

0.2083 - 0.8334 = 6.597 ´ 10 -4 M 62.54 - 1010.10

y - intercept =

1 = 0.8334 - 6.597 ´ 10 -4 ´ 1010.10 = 0.167 n

(

)

Therefore, K a = (1/ (6.597 × 10−4))(0.167) = 253.145 L/mole 627

ADDENDUM II – PART 1

n = 1/0.167 = 6 Using regression analysis yields the following equation: 1 1 1 = 0.1681 + \ n = 1 / 0.1681 = 5.9488 r 6.59 ´ 10 -4 CD K a = 0.1681 × (1/ (6.59 × 10−4)) = 255.08 L/mole 1.1.2 Scatchard plot calculations are presented in Table 1.1.4

Table 1.1.4 Scatchard Plot Calculations of Application 4.1 r/CD (M−1)

1212.12

898.88

600.00

300.19

Using slope relationship: Slope = (300.19–1212.12) / (4.8–1.2) =−253.31 = −K a K a = 253.31 M−1 y-intercept = nK a = 1212.12 + 1.2(253.31) = 1516.092 mole/L n = 1516.092/253.31 = 5.985

Using regression analysis: r/CD = 1511.00–252.90 r K a = 252.89 M−1 n = 1511.47 / 252.89 = 5.976

628

Addendum II – Part 1 CASE 1.2 – PROTEIN BINDING The binding of a xenobiotic to plasma proteins was evaluated using dynamic dialysis. The measured data are reported in Tables 1.2.1 and 1.2.2.

Table 1.2.1 Data Related to Application CD (mole/L) CDP (mole/L) r

4.590 13.928 0.612

9.792 27.234 1.102

26.316 46.206 2.065

77.724 70.533 3.060

0.0784

0.0394

Table 1.2.2 Data Related to Application 4.2 r / CD (L/mole)

0.1334

0.1125

1.2.1 Using a Scatchard plot, determine the association constant and the number of binding sites. Slope = (0.0394 − 0.1334) / (3.060 − 0.612) = −0.0384 L/mole = −K a K a = 0.0384 L/mole n =((r/CD) + r K a) / K a n = (0.1334 + (0.612 0.0384)) / 0.0384 = 4.08

1.2.2 Using equation

CDP = nK aCT P - K aCDP , calculate the association constant. CD

The related calculations are presented in Table 1.2.3.

Table 1.2.3 Data Related to Application 4.2 CDP/CD

3.034

2.781

1.755

0.907

Slope = (0.907−3.034) / (70.533−13.928) = −0.0375 L/mole = −K a K a = 0.0375 L/mole

629

Addendum II – Part 1 CASE 1.3 – PROTEIN BINDING The binding of a xenobiotic to albumin was investigated by the method of equilibrium dialysis. Albumin (1.2 × 10-3 M) mixed with the compound (0.5 × 10-3 M) was allowed to equilibrate for three hours. The measurement of the compound in the buffer compartment indicated a concentration of 0.2 × 10-3 M. 1.3.1 Determine moles of the xenobiotic bound per total protein. The buffer CD concentration that is in equilibrium with CD of the albumin compartment is equal to CD = 0.2 × 10-3 M. CD + CDP in the albumin compartment = (0.5 – 0.2) × 10-3 = 0.3 × 10-3 M Therefore, CDP = (0.3 × 10-3 M) – (0.2 × 10-3) = 0.1 × 10-3 M r = (0.1 × 10-3 M) / (1.2 × 10-3 M) = 0.0833 1.3.2 Calculate free and bound fractions. Free fraction = (0.2 × 10-3) / (0.3 × 10-3) = 0.67 Bound fraction = 0.1 × 10-3 M / (0.3 × 10-3) = 0.33

630

Addendum II – Part 2 CASE 2.1 – IN VITRO DRUG METABOLISM In a preliminary in vitro metabolic evaluation of a new drug using subcellular fractions, the initial rates of in vitro metabolism at different initial concentrations were determined as reported in Table 2.1.1.

Table 2.1.1 Initial In Vitro Rate of Metabolism at Different Initial Concentration of the Drug Initial Conc. of Drug (M)

Initial Rate of Metabolism (µmolL−1min−1)

2 × 10−2 2 × 10−3 2 × 10−4 1.5 × 10−4 1.25 × 10−5

150 150 120 112.5 30.0

2.1.1 Estimate the Michaelis-Menten constant of the observed data. Vmax ´ CD , indicates that when the K M + CD concentration of a xenobiotic is much less than K M, the metabolic rate behaves as a frst-order process; when the concentration is much higher than K M, it behaves like a zero-order process. According to the observed data (Table 2.1.1), Vmax = 150 µmolL−1min−1. Using the method of substitution, an estimate of the Michaelis-Menten constant can be specifed as: As discussed in Chapter 9, the Michaelis-Menten equation,v =

v CD = Vmax K M + CD (120/150) = (2 × 10 −4) / (K M + 2 × 10 −4) K M = 5 × 10 −5 M 2.1.2 What are the initial and maximum rates of metabolism at a concentration of 0.02 M? As a rule, when CD ³ 100 K M , the rate of metabolism follows zero-order kinetics. C 0.02 Since D = = 400 , it indicates that the rate of the metabolism is constant, follows the K M 5 ´ 10 -5 zero-order kinetics, and v = Vmax = 150 µmolL−1min−1. 2.1.3 What is the rate of metabolism at initial concentrations of (1) 5 × 10−5 M and (2) 2.5 × 10−5 M? The concentration of 5 × 10−5 M is equal to KM (i.e., CD = KM). Therefore, v = Vmax/2 = 75 µmolL−1min−1 When CD = 2.5 × 10−5 M v = (Vmax × CD) / (K M + CD) = (150 × 2.5 ×10 −5) / (5 × 10 −5 + 2.5 × 10 −5) v = 50 µmolL−1min−1 2.1.4 What would be the concentration of metabolite six minutes after addition of 0.08 and 0.1 M of the parent drug? Since CD /K M = 0.08 / (5 × 10 −5) = 1600 and 0.1 / (5 × 10 −5) = 2000, the rate of metabolism at these concentrations would follow zero-order kinetics (i.e., v = Vmax). Therefore, [Metabolite] = Vmax × time = 150 × 6 = 900 µmol/L = 900 × 10 −6 M = 9 × 10 −4 M 2.15 What would be the initial rate of metabolism at 2 × 10−4 M if the concentration of enzyme is doubled? The rate of metabolism is directly proportional to the concentration of the protein. If we double the enzyme concentration, then the initial rate will double. Therefore, at CD = 2 × 10 −4 with doubling the protein concentration: v = 120 × 2 = 240 µmolL−1min−1 631

Addendum II – Part 2 CASE 2.2 – DRUG METABOLISM The data presented in Table 2.2.1 are the results of an in vitro drug metabolism study.

Table 2.2.1 Initial In Vitro Rate of Metabolism at Different Concentrations of the Drug Drug Concentration (µmol/L)

Initial Rate of Metabolism (µmolL−1min−1)

10.00 12.00 15.00 20.00 24.00 30.00 39.60 60.00 120.00 240.00

16.56 19.20 22.92 28.56 32.04 36.96 43.44 53.40 68.64 80.04

2.2.1 Determine K M and Vmax using the Lineweaver-Burk plot. Lineweaver-Burk Method: The reciprocals of the data are presented in The regression analysis of 1/v versus 1/CD yields the following equation: 1/v = 0.0104 + 0.498 (1/ CD) Therefore, Slope = 0.498 min and y-intercept = 0.0104 µmol−1 L min Vmax = 1/0.0104 = 96.15 µmol L−1 min−1 K M = 0.498 × 96.15 = 47.88 µmol L 1 2.2.2 Determine K M and Vmax using the Hanes plot. The y and x values are presented in the following table.

The regression equation of CD /v versus CD is: CD /v = 0.497 + 0.0104 CD Therefore, Slope = 0.0104 µmol−1L min 632

ADDENDUM II – PART 2

Table 2.2.2 The Reciprocals of the Concentration and Related Rate According to the Linweaver-Burk Plot 1/v (µmol−1L min)

1/CD(L/µM)

0.060 0.052 0.044 0.035 0.031 0.027 0.023 0.0187 0.0145 0.0125

0.1 0.0833 0.0667 0.0500 0.0416 0.0333 0.0252 0.0166 0.0083 0.0041

y – intercept = 0.497 min Vmax = 1/0.0104 = 96.15 µmolL−1min−1 K M = 0.498 × 96.15 = 47.88 µmol L−1 2.2.3 Determine K M and Vmax using the Eadie-Hofstee plot. The data for this method are presented below.

Table 2.2.3 Estimation of the Normalized Concentration with Respect to the Rate as Dependent Variable To be Plotted against the Concentration According to the Hanes Plot CD/v (min) 0.600 0.624 0.660 0.700 0.744 0.810 0.911 1.123 1.747 3.00

CD (µmol/L) 10.00 12.00 15.00 20.00 24.00 30.00 39.60 60.00 120.00 240.00

633

ADDENDUM II – PART 2

The regression equation is: Therefore,

v = 96.00 – 47.8 (v/ CD) Slope = K M = 47.80 µmol L−1 y-intercept = Vmax = 96 µmol L−1min−1

Table 2.2.4 Estimation of the Normalized Rates with Respect to the Concentration to be Plotted as the Independent Variable of the EadieHofstee Plot v (µmol L−1 min−1) 16.56 19.20 22.92 28.56 32.04 36.96 43.44 53.40 68.64 80.04

634

v/CD (min 1) 1.656 1.600 1.528 1.428 1.335 1.232 1.096 0.890 0.572 0.333

Addendum II – Part 2 CASE 2.3 – IN VITRO DRUG METABOLISM The direct linear plot of a set of data provided more than one intersection of the estimates of K M and Vmax presented in the following table. 2.3.1 What pair would you select as the best estimate among the pair values? KM (µM/L) Vmax (µM/L min)

22.0 54.9

21.6 54.0

20.2 50.8

22.5 56.2

22.8 56.7

23.1 57.5

24.4 63.8

20.5 51.2

22.8 56.7

Answer: Arrange the data in ascending or descending order and choose the median pair as the best estimates of K M, Vmax. KM (µM/L) Vmax(µM/Lmin)

20.2 50.8

20.5 51.2

21.6 54.0

22.0 54.9

22.5 56.2

22.8 56.7

22.8 56.7

23.1 57.5

24.4 63.0

The best estimated values of K M and Vmax are 22.6 and 56.2, respectively.

635

Addendum II – Part 2 CASE 2.4 – IN VIVO DRUG METABOLISM The in vivo metabolism of a new drug was evaluated in a group of eight patients during an initial clinical trial. After an intravenous injection of 200 mg, plasma samples were collected at different times and assayed for the metabolite concentration. The preliminary investigations have shown the drug metabolizes to a metabolite and eliminates in bile without any further biotransformation. The mean values of metabolite concentration in the collected plasma samples are reported in Table 2.4.1; the standard deviations of the measurement are not included. The disposition profle of the compound is assumed as follows:

Table 2.4.1 Average Metabolite Concentration in Plasma Samples versus Time Time(h) Conc. (µg/ml)

1.0 0.292

1.5 0.482

2.0 0.601

3.0 0.706

4.0 0.714

6.0 0.625

8.0 0.511

m me At ¾k¾ ® Am ¾k¾ ® Ame

Where, k m and k me are frst-order rate constant and k m > k me

636

6.0 0.625

8.0 0.511

10.0 0.409

12.0 0.325

14.0 0.258

ADDENDUM II – PART 2

2.4.1 Estimate the rate constants of formation and elimination of the metabolite. Regression equation of the extrapolated line is:

( log Conc.)extrapolated = 0.09043 - 0.04825 t(h)

The regression equation of residual line is:

( log Conc.)residual = 0.3837 - 0.404 t

( h)

\ k me = 0.04825 ´ 2.303 = 0.1111h -1

( T1/2 )k

me

= 6.24 h

k m = 0.404 ´ 2.303 = 0.93 h -1

( T1/2 )k

m

= 0.644 h

The calculated data show that the drug metabolizes very rapidly, but the elimination of the metabolite from the body takes much longer, which can be attributed to several factors. One possible 637

ADDENDUM II – PART 2

explanation is that the chemical affnity of the metabolite to interact with various tissues and body elements is higher than the parent compound. The other explanation might be the slow biliary fow rate, as it is the major route of elimination of the metabolite. 2.4.2 How long would it take for the metabolite to be fully removed from the body?

( t )total-removal = 7 ´ 6.24 h = 43.90 h 2.4.4 How long would it take for the parent compound to be metabolized entirely?

( t )total - metabolism = 7 ´ 0.644 = 4.50 h

638

Addendum II – Part 2 CASE 2.5 – IN VITRO DRUG METABOLISM – TEST YOUR KNOWLEDGE The following are the ftted line equations of different sets of in vitro drug metabolism data according to the various linear versions of the Michaelis-Menten equation. Identify each version and determine the Michaelis–Menten constant, the maximum rate of metabolism, and intrinsic clearance (v is the rate of metabolism and CD is drug concentration). 1 v = 0.035mmol -1L min+

0.673 min CD

(

v = 54mmolL-1 min -1 - 54mmolL-1 1 v = 0.01mmol -1L min+

) Cv

D

0.2 min CD

(

)

CD v = 0.23 min+ 0.05mmol -1L min CD

(

v = 110mmolL-1 min -1 - 29mmolL-1

(

) Cv

D

-1

)

CD v = 0.096 min+ 0.075mmol L min CD

639

Addendum II – Part 2 CASE 2.6 – IN VITRO DRUG METABOLISM – TEST YOUR KNOWLEDGE An in vitro drug metabolism study generated the following experimental set of data for the purpose of estimating the Michaelis-Menten constant and the maximum rate of metabolism. The data set is real and carries random errors of measurement. Use the most appropriate approach to determine the parameters of the study.

640

Drug Concentration (CD) µM/L

Rate of Metabolism (v) µM/L/min

1.613 1.935 2.420 3.225 3.871 4.839 6.387 9.677 19.354 38.710

2.671 3.096 3.696 4.606 5.167 5.813 7.006 8.613 11.071 12.910

Addendum II – Part 2 CASE 2.7 – IN VITRO DRUG METABOLISM Discuss briefy the major factors infuencing the rate of in vitro xenobiotic metabolism. The rate of in vitro metabolism of a xenobiotic is often estimated by the rate of metabolite(s) formation, parent compound utilization, or both. Often incubation of a xenobiotic and the formation of the metabolite (s) may exhibit a deviation from the predicted line, as depicted in Figure 2.7.1.

The slope of the initial linear portion of the observed curve, which is the same as the slope of the predicted line, is known as the initial slope or initial rate of metabolism. The departure of the time course of observed curve from the theoretical line can be related to the following factors: 2.7.1 REDUCTION OF XENOBIOTIC CONCENTRATION If the departure from linearity is related to the diminution of the parent compound in the incubation, the addition of more parent compound should address the issue and prevent the fall-off of the curve. This scenario occurs when the concentration of xenobiotic is much smaller than the Michaelis-Menten constant, and the assay may not be highly sensitive for detection of negligible amounts of metabolite formation. 2.7.2 INHIBITION OF ENZYME ACTIVITY BY THE METABOLITE(S) Metabolites often reversibly inhibit the enzymatic reactions. Removal of metabolite from the incubation should prevent the departure. However, such a removal may not be feasible for all incubations. 2.7.3 EQUILIBRIUM The basic reaction is the reversible enzyme–xenobiotic interaction. When the rate of metabolite formation equals the rate of metabolite conversion to parent compound, that is the backward reaction; a departure from linearity can be observed that corresponds to the reduction of rate of metabolism. 641

ADDENDUM II – PART 2

2.7.4 INSTABILITY OF XENOBIOTIC AND/OR ENZYME UNDER INCUBATION CONDITION The practical approach for identifying this problem is to incubate the xenobiotic without protein or protein without xenobiotic under the same experimental condition and evaluate the stability of the xenobiotic or protein. It is important that the conditions of this pre-incubation be the same as the incubation of xenobiotic with protein. For instance, the pH, ionic strength of the buffer, temperature, and degree of illumination should be identical to the main incubation. The degree of illumination can create variability for light-sensitive compounds. 2.7.5 LIMITATION OF DETECTION If the detector(s) of the assay do not respond linearly to the high concentration of the metabolite, the departure from linearity would be created artifcially. Therefore, it would be necessary to establish the limit of detection with authentic standards of metabolite(s). 2.7.6 EFFECT OF XENOBIOTIC ON THE ENZYME The departure from linearity may also occur if the parent compound or metabolite infuences the activity and integrity of protein during the incubation. This would reduce the rate of metabolite formation during the incubation. The problem can be identifed and rectifed qualitatively by addition of more protein to the incubation. Obviously, for quantitative calculation of the rate of metabolism, the amount of added protein should be taken into consideration. 2.7.7 CHANGE OF INCUBATION CONDITIONS DURING THE INCUBATION There are several metabolic reactions in which the formation of metabolite is concurrent with the formation or utilization of hydrogen ions. This obviously would change the pH of the incubation during the incubation. The change in pH may gradually reduce the rate of formation of metabolite(s). It would be advisable, necessary, and a good practice to determine the pH of enzymatic incubation at the start and end of incubation. If the end pH is different from the start pH, then the progress curve will determine whether the change in pH had any negative infuence on the metabolic reaction. If the change in pH had no effect on the rate of metabolism, then the pH of incubation would be the start pH and not the average of the start and end pH. If the change in pH affected the rate of metabolism, then the condition of the incubation should be modifed. Using a buffer solution with suffcient capacity may prevent this problem.

642

Addendum II – Part 2 CASE 2.8 – IN VIVO DRUG METABOLISM – TEST YOUR KNOWLEDGE A xenobiotic is metabolized by CYP3A4, CYP3A5, and CYP3A7 in humans to hydroxyl metabolites 1′-OH and 4-OH. The compound eliminates unchanged and as glucuronide conjugate of 4-OH metabolite in the urine. The formation and elimination of the metabolites are governed by frstorder kinetics. The biliary excretion is considered negligible. The following data (Table 2.8.1) are collected from 12 volunteers following the administration of 500 mg.

Table 2.8.1 The Time Course of Amount of 1-OH Metabolite Eliminated in Urine, Its Concentration in Plasma and the Cumulative Amount of Xenobiotic Excreted Unchanged in Urine Time (h) 2.5 3.0 3.5 4.0 5.0 6.8 8.0 10.0 12.0 14.0 16.0 ∞

Am (mg)

Cm (mg/L)

Ae (mg)

65.34 97.78 115.29 122.22 124.44 117.15 98.00 80.00 65.24 52.80 43.02

2.90 4.35 5.12 5.43 5.53 5.21 4.35 3.55 2.90 2.34 1.91

2.075 6.195 13.120 20.409 35.734 49.920 96.569 114.062 127.911 139.573 196.800 231.112

Where Am is the mean values of the amount of 1′-OH metabolite in the body, Cm is the average plasma concentration of the metabolite (1′-OH), and Ae is the average amount of xenobiotic excreted unchanged in the urine. The standard deviations are not reported here. The average fractions of dose eliminated as 4-OH and its glucuronide conjugate in the urine are 0.15 and 0.078, respectively; the apparent volumes of distribution of unchanged drug and 1′-OH metabolite are 8.89L and 22.5L, respectively. 2.8.1 Determine the metabolic clearance of 1′-OH metabolite. 2.8.2 Calculate the total amount of the metabolites (1′-OH, 4-OH, and glucuronide conjugate) eliminated in the urine. Hint: ◾ Plot of log Cm versus time provides a biexponential curve of formation and elimination of the metabolite (skewed bell shape) with a linear terminal portion. The slope of the linear part is: (−k/2.303). ◾ Plot of the residual values of the curve is a straight line with scatter data points and a slope of (−km/2.303). ◾ The frst-order rate constant of metabolite elimination is greater than the rate constant of the metabolite formation (i.e., k > k m ).

643

Addendum II – Part 3 CASE 3.1 – ANALYSIS OF URINARY DATA After an intravenous bolus injection of 400 mg of a xenobiotic with an apparent volume of distribution of 20 L, the following urinary data were reported (Table 3.1.1). Determine the Rate and ARE equations of the line and estimate the total body clearance and the renal and metabolic clearances. Extrapolate from the urinary data to predict parameters of the plasma concentration, such as the area under the plasma concentration-time curve, plasma concentration 10 hours after the injection of the dose, fraction of the administered dose excreted unchanged, and fraction eliminated as metabolite. Calculations of the Rate plot is presented in Table 3.1.2:

Table 3.1.1 Cumulative Amount of Xenobiotic Excreted Unchanged in Urine at Different Intervals Time (h) 0 0.5 1.0 2.0 4.0 8.0 12.0 24.0 48.0 96.0

644

Cumulative Amount Excreted Unchanged in Urine (mg) 0 16.90 32.70 61.10 107.20 168.50 203.40 241.30 249.70 250.00

ADDENDUM II – PART 3

Table 3.1.2

Stepwise Calculation of the Rate Plot

DAe (mg)

Dt( h)

DAe Dt(mg / h)

tmidpoint ( h)

16.90 15.80 28.40 46.10 61.30 34.90 37.90 8.40 0.30

0.50 0.50 1.00 2.00 4.00 4.00 12.00 24.00 48.00

33.80 31.60 28.40 23.00 15.30 8.70 3.15 0.35 0.006

(0 + 0.50) / 2 = 0.25 (1 + 0.50) / 2 = 0.75 (2 + 1) / 2 = 1.50 (4 + 2) / 2 = 3.00 (8 + 4) / 2 = 6.00 (12 + 8) / 2 = 10.00 (24 + 12) / 2 = 18.00 (48 + 24) / 2 = 36.00 (96 + 48) / 2 = 72

DAe ö 3.1.1 The regression equation of Log çæ ÷ vs tmidpoint is: è Dt ø æ DAe ö Log ç ÷ = 1.503 - 0.0525tmidpoint è Dt ø -1 DAe = 31.623 e -0.12h ´ttmidpoint ( h ) Dt

3.1.2 Total body clearance: ClT = K ´ Vd = 0.12 h -1 ´ 20 L = 2.4 L / h = 40 ml / min 3.1.3 Renal and metabolic clearances: Clr = k e ´ Vd =

31.623 ´ 20 L = 1.58 L / h = 26.33 ml / min 400

Clm = ClT - Clr = 40 ml / min- 26.33 ml / min = 13.67 ml / min or , Clm = ( 0.12 - 0.0079 ) ´ 20 L = 0.82 L / h = 13.67 ml / min 3.1.4 Area under plasma concentration-time curve (AUC): AUC = Cp 0 / K = ( 400 20 ) ¸ 0.12 = 166.66 mgh / L 3.1.5 Plasma concentration 10 hours after the injection: Cp10 h = Cp 0 e -K(10 h) = 20e -1.2 = 6.024 mg / L 3.1.6 Fraction of the dose excreted unchanged: fe =

k e Clr 0.079 1.58 = = = = 0.66 i.e., 66% of the administerred dose 0.12 2.4 K ClT

3.1.7 Fraction of the dose eliminated as metabolite: fm =

k m Clm 0.041 13.6 = = = = 0.34 i.e., 34% of the administereed dose K ClT 0.12 40

645

ADDENDUM II – PART 3

Using the ARE plot (Sigma-minus), determine the equation of the line of log ARE vs time.

Table 3.1.3 Calculation of the Amount Remaining to be Excreted Unchanged in Urine Time (h)

ARE (mg)

0 0.50 1.00 2.00 4.00 8.00 12.00 24.00

250 – 0.00 = 250.00 250 – 16.90 = 223.10 250 – 32.70 = 217.30 250 – 61.10 = 188.90 250 – 107.20 = 142.80 250 – 168.50 = 81.50 250 – 203.40 = 46.60 250 – 241.30 = 8.70

3.1.9 The regression equation of log ARE versus time is: log ARE = 2.394 - 0.0605 t ARE = 248(mg)e -0.14 h

-1

´t

3.1.10 Fraction of the dose excreted unchanged and excretion rate constant: fe =

248 = 0.62 400

k e = 0.14 ´ 0.62 = 0.087 h -1 3.1.11 Fraction of dose eliminated as metabolites: fm =

400 - 248 = 0.38 400

Note: The differences in calculated parameters and constants are related to the truncation errors of the Rate plot calculations. The Rate plot is always more scattered than the ARE plot.

646

Addendum II – Part 3 CASE 3.2 – ANALYSIS OF URINARY DATA The cumulative amount of a new drug excreted unchanged in the urine of an 80-kg male subject is reported in Table 3.2.1. The dose was administered by intravenous bolus injection. Based on the analysis of the plasma samples, the initial plasma level and total body clearance were reported as 19.72 mg/L and 14L/h, respectively. Prepare a report on the calculated parameters and constants of urinary data. The cumulative urinary excretion data (Table 3.2.1) indicate that the collection of the urine samples was not long enough to provide an experimental value for the total amount excreted unchanged (i.e.,Ae¥ ). The plot of the data (Figure 3.2.1) also implies that the cumulative curve did not reach a plateau level. Thus, it was decided to use the Rate plot.

Table 3.2.1 Cumulative urinary excretion data Time (h) Amount (mg)

0.5 81

1.0 156

2.0 294

3.0 411

4.0 513

5.0 600

6.0 678

8.0 804

11.0 942

14.0 1029

647

ADDENDUM II – PART 3

3.2.2 The logarithmic and exponential equations of the Rate plot, generated by the regression analysis of the related data presented in Table 3.2.2, provide the estimated value of the overall elimination rate constant from the slope of the line and the total amount excreted unchanged from the y-intercept of the ftted line (Figure 3.2.2). æ DAe log ç è Dt or,

ö ÷ = 2.222 - 0.06 tmidpoint ø

DAe = 166.72 ( mg / h ) e -0.138´tmidpoint Dt k e Dose = 166.72mg / h

166.72 = 1208.12mg 0.138 3.2.3 The injected dose and the amount eliminated as metabolite(s) are: Ae¥ =

Dose = 19.72mg / L ´

14 L / h = 2000 mg 0.138 h -1

Am¥ = 2000 - 1208.12 = 792mg 3.2.4 From the calculated data, the percent of the dose excreted unchanged and eliminated as metabolites is: 166.72 = 0.083 h -1 ke = 2000 fe =

0.083 ged = 0.60 Þ 60% excretedunchang 0.138

f m = 1 - 0.60 = 0.40 Þ 40% eliminatedas metabolite(s) 1208.112 792 fm = @ 0.4 = 0.60 2000 2000 3.2.5 The renal and metabolic clearances are estimated as: 14 L / h Vd = = 101.45 L 0.138 or, f e =

Clr = 101.45 ´ 0.083 = 8.42 L / h Clm = 14 - 8.42 = 5.58 . L/h 3.2.6 Based on the calculated values, the equation of the amount remaining to be excreted (i.e., the equation of ARE plot) is also predicted as: ARE = 1208.12mg e -0.138h

-1

´t

or, log ARE = 3.08 - 0.3178 t

Table 3.2.2 Rate Plot Calculations DAe (mg) 81 75 138 117 102 87 78 126 138 87

648

Dt (h) 0.5 0.5 1.0 1.0 1.0 1.0 1.0 2.0 3.0 3.0

DAe Dt (mg/h) 162 150 138 117 102 87 78 63 46 29

tmidpoint (h)

log ( DAe Dt )

0.25 0.75 1.50 2.50 3.50 4.50 5.50 7.00 9.50 12.50

2.20952 2.17609 2.13988 2.06819 2.00860 1.93952 1.89209 1.79934 1.66276 1.46240

Addendum II – Part 3 CASE 3.3 – ANALYSIS OF URINARY DATA A beta-lactam antibiotic used in the treatment of bacterial infections caused by gram-positive organisms is administered intravenously to an infant. To avoid multiple blood sampling, the urine samples were collected by urine bag and replaced at multiple time intervals. The concentration of the compound in urine samples was measured in milligrams until there was no drug excreted unchanged in urine. The cumulative amounts excreted unchanged were calculated and the data were analyzed for the amount remaining to be excreted unchanged in urine. The following equation summarizes the time course of the drug excreted unchanged in the urine. log ARE = 2 - 0.301t(h) 3.3.1 If the administered dose was 500 mg, what would be the metabolic rate constant? 100 ö æ km = ç 1 ´ ( 0.301´ 2.303 ) = 0.554 h -1 500 ÷ø è 3.3.2 If the administered dose was 100 mg, what percent of the dose is excreted unchanged? æ antilog2 ö f e ´ 100 = ç ÷ ´ 100 = 100% è 100 ø 3.3.3 If the administered dose was 250 mg, what would be the excretion rate constant? ke =

100 ´ ( 2.303 ´ 0.301) = 0.277 h -1 250

3.3.4 If 45% of the dose is excreted as metabolite(s), what would be the dose? f e = 1 - 0.45 = 0.55 100 = 182mg 0.55 3.3.5 If 30% of the dose is eliminated as metabolite(s) and the volume of distribution is 10 L, what would be the renal clearance? Dose =

Clr = 0.7 ´ 10 L ´ 2.303 ´ 0.301 = 4.85L / h = 81ml / min 3.3.6 What were the rates of excretion six minutes and six hours after the administration of a 250 mg dose?

( Rate )6 min = ( 0.277 h -1 ´ 250 mg ) e -( 2.303´0.301)´0.1h = 64.61mg / h ( Raate )6 h = ( 0.277 h -1 ´ 250 mg ) e -( 2.303´0.301)´6 h = 1.081mg / h 3.3.7 How long does it take for the dose to be completely removed from the body? time @ 7T1/2 = 7 ´

0.693 @ 7 hours 0.693

649

Addendum II – Part 3 CASE 3.4 – ANALYSIS OF URINARY DATA A new ureidopenicillin, a broad-spectrum β-lactam antibiotic, is evaluated to determine its comparability to Pipracil (piperacillin injection), which was discontinued in 2002 by the manufacturer. The evaluation was focused mainly on PK analysis of the new agent and calculation of its parameters and constants in comparison to Pipracil. The evaluation protocol involved in intravenous injection of 4 g of the new antibiotic to a group of patients (N = 6) with normal renal function. Blood and urine samples were collected at different time intervals and analyzed for the unchanged drug in plasma and urine. Based on the preliminary evaluation of the urine data, the mean and standard deviation of the total amount excreted unchanged in 24 hours was reported as 2471 + 16 mg; from the plasma concentrations, the half-life of elimination was 0.9 + 0.06 h and the initial plasma concentration was 215 + 8 mg/L. Using only the mean values, calculate the following parameters and constants: 3.4.1 Metabolic rate constant: Since 24 h > 7 ´ 0.9 = 6.3 h Ae¥ = 2417 mg \ fe =

2417 = 0.6177 @ 0.62 4000

k m = (1 - 0.62 ) ´

0.693 = 0.29 h -1 0.9h

3.4.2 Excretion rate constant: K=

0.693 = 0.77 h -1 0.9h

k e = 0.77 - 0.29 = 0.48 h -1 or, k e = 0.62 ´ 0.77 = 0.48 h -1 3.4.3 Metabolic clearance: æ 4000 ö Vd = ç ÷ = 18.60 L è 215 ø Clm = 18.60 L ´ 0.29 h -1 = 5.40 L / h = 90 ml / min 3.4.4 Renal clearance: Clr = 18.60 ´ 0.48 = 8.93 L / h = 148.83 ml / min 3.4.5 Fraction of the dose remaining in the body three hours after the injection: f b = e -0.77 h

-1

´ 3h

= 0.1

or, 3 h @ 3.3T1/2 \ f b = 0.1 3.4.6 Percent of the dose eliminated from the body three hours after the injection: f el = 1 - 0.1 = 0.9 Þ 90% 3.4.7 Rate of elimination at t = 3 h:

( Rate )el = 0.1´ 4 g ´ 0.77 = 0.308 g / h = 308 mg / h 3.4.8 Area under plasma concentration-time curve: AUC0¥ = 215 ¸ 0.77 = 279.22 mgh / L

650

ADDENDUM II – PART 3

3.4.9 The above calculated data are compared with the published pipracillin data (Tjandramaga et al., 1978) in Table 3.4.1. The PK model assumed in Case 3.4 is a one-compartment model, and the published data are based on a two-compartment model. Except for the k el values, no signifcant statistical differences are observed between the PK parameters of the two ureidopenicillins. The term ClNR of the paper was assumed to be Clm.

Table 3.4.1 Comparison of the Calculated Constants and Parameters of the Case 3.4 Using a One-Compartment Model with Published Data Based on a Two-Compartment Model Parameter/Constant

Pipracil (Two-Compartment Model)

Case 3.4 One-Compartment

0.616

0.62

T1/2 ( h)

1

0.9 + 0.06

Vd(L)

18.6

18.60

AUC

250.3 + 12.5

279.22

k el (h−1) a

1.51 + 0.24

0.77

Clm (ml/min)

53.8 + 2.6

90.00

fe

a

The difference is due to the assumption of a two-compartment model.

REFERENCE Tjandramaga, T. B., Mullie, A., Verbesselt, R., De Schepper, P. J., Verbist, L. 1978. Piperacillin: Human pharmacokinetics after intravenous and intramuscular administration. Antimicrob Agents Chemother 14(6): 829–37.

651

Addendum II – Part 3 CASE 3.5 – ANALYSIS OF URINARY DATA The pharmacokinetics of a new cephalosporin derivative was investigated after an intravenous injection of 1000 mg. Urine was collected at various intervals and analyzed for unmetabolized drug. The following are the reported data:

Table 3.5.1 Urinary Excretion of the Derivative per Interval of Urine Collection Interval (h)

Amount of Unmetabolized Excreted/Interval (mg)

0–2 2–4 4–6 6–10 10–14

295 189 126 135 59

Ultimately, a total of 850 mg was excreted unmetabolized. Also, during the study, blood was drawn at 4 hours and the concentration was determined as 17.5 mcg/ml. Using the Rate plot, estimate the excretion rate constant, k e , metabolic rate constant, k m , the overall elimination rate constant K , renal clearance in L/h and ml/min, and the fraction of the administered dose excreted unchanged in the urine.

Table 3.5.2 Dt( h) 2 2 2 4 4

Stepwise Calculations of the Rate Plot

DAe (mg) 295 189 126 135 59

DAe Dt (mg / h)

tmidpoint ( h)

147.50 94.50 63.00 33.75 14.75

(0 + 2)/2 = 1 (2 + 4)/2 = 3 (4 + 6)/2 = 5 (6 + 10)/2 = 8 (10 + 14)/2 = 12

The regression equation of log (DA e / Dt)vs tmidpoint is: log (DA e / Dt) = 2.253 - 0.0905 tmidpoint

652

ADDENDUM II – PART 3

3.5.1 The estimate of the overall rate constant of elimination is: K = 2.303 ´ 0.0905 = 0.208 h -1 T1/2 =

0.693 = 3.3 h 2.303 ´ 0.0905

3.5.2 The rate constants of excretion and metabolism are estimated as: k e Dose = antilog 2.25 = 177.82 ke =

177.82 = 0.178 h -1 1000

k m = 0.208 - 0.178 = 0.03 h -1 3.5.3 Renal clearance in units of ml/min and L/h is: Cp 0 =

17.5 = 40.50 mg / L e -0.21´ 4

Vd = 1000 40.50 = 24.70 L 7 = 4.4 L / h = 73.30 ml / min Clr = 24.70 ´ 0.178 3.5.4 Fraction of dose excreted unchanged in urine is: f e = 0.178 0.21 = 0.847 i.e., 84.7% of thedoseis excreted unchanged.

653

Addendum II – Part 3 CASE 3.6 – ANALYSIS OF URINARY DATA A 75-kg hospitalized urology patient was given 4 g of an antibiotic intravenously by bolus injection. The half-life of the drug is 65 minutes, and the apparent volume of distribution is 0.3 L/ kg. The total amount of piperacillin excreted unchanged in the 24-hour urine specimen of this patient was 3.65 g. 3.6.1 Based on the above information, the Rate and ARE equations of the drug for this patient can be developed as: 24 h ˜ 7T1/2 \ Ae¥ = 3.65g f e = 3.65g / 4 g = 0.91 k e = 0.91 ´ ( 0.693 65min ) = 0.00097 min -1 = 0.58 h -1 K = 0.693 / 65min = 0.0107 min -1 = 0.64 h -1 -1 DAe = (0. 0 58 ´ 4 g )e -0.64 h ´tmidpoint = 0.32 ( g / h ) e -0.64´tmidpoint Dt

ARE = 3. 3 65 ( g ) e -0.64 h

-1

´t

3.6.2 The renal clearance of the patient can be estimated as: Clr = 0.58 h -1 ´ 0.3 L / kg ´ 75kg = 13 L / h = 216.7 ml / min

654

Addendum II – Part 3 CASE 3.7 – ANALYSIS OF URINARY DATA Following an intravenous bolus injection of 500 mg of a xenobiotic to a human volunteer, the following urinary excretion data of the unmetabolized compound are reported (see Table 3.7.1). Estimate the rate constants of elimination, excretion, and metabolism using the ARE plot.

Table 3.7.1 Cumulative Urinary Excretion of the Unchanged Xenobiotic in Intervals of Urine Collection Time (h)

Cumulative Amount of Unchanged Xenobiotic in Urine (mg)

0.5 1 2 3 4 5 6 8 11 14 20 40

27 52 98 137 171 200 226 268 314 343 375 400

3.7.1 Develop the ARE plot and generate the related Equation (see Table 3.7.2).

The regression equation of log ARE versus time is: log ARE = 2.60 - 0.0603 t

655

ADDENDUM II – PART 3

Table 3.7.2 Estimation of the Amount Remaining in the Body to be Excreted at the Intervals of Urine Collection Time (h)

Cumulative Amount (mg)

0.0 0.5 1.0 2.0 3.0 4.0 5.0 6.0 8.0 11.0 14.0 20.0

0 27 52 98 137 171 200 226 268 314 343 375

ARE (mg) 400 373 348 302 263 229 200 174 132 86 57 25

3.7.2 Estimated rate constants are: K = 2.303 ´ 0.0603 = 0.1388 h -1 k e = 0.8 ´ 0.1388 = 0.111h -1 k m = 0.1388 - 0.1 111 = 0.0278 3.7.3 Fractions of the dose excreted unchanged and eliminated as metabolites are: f e = 400 500 = 0.8 f m = 1 - 0.8 = 0.2 Why Ae¥ is assumed to be 400 mg: T1/2 = 0.693 0.188 = 5 h 40 h 5 h = 8 (i.e., the last urine sample was collected eight half-lives after the injection of the dose). 3.7.5 Develop the Rate plot and generate the related equation (see Table 3.7.3).

656

ADDENDUM II – PART 3

Table 3.7.3 Calculations of the Average Excretion Rate as a Function of the Midpoint of Urine Collection interval ΔAe (mg) 27 25 46 39 34 29 26 42 46 29 32

Δt (h)

ΔAe/Δt (mg/h)

tmidpt(h)

0.5 0.5 1.0 1.0 1.0 1.0 1.0 2.0 3.0 3.0 6.0

54 50 46 39 34 29 26 21 15.33 9.66 5.33

0.25 0.50 1.50 2.50 3.50 4.50 5.50 7.00 9.50 12.50 17.00

3.7.6. Determine the Rate equation and the estimated values of the related rate constants. log

DAe = 1.74 - 0.0598 t midpoint Dt

DAe = 54.954(mg / L) e -0.1377´tmiidpoint Dt K = 2.303 ´ 0.0598 = 0.1377 h -1 ke =

54.954 = 0.1099 = 0.11 h -1 500

k m = 0.1377 - 0.11 = 0.0277 h -1

657

Addendum II – Part 4 CASE 4.1 - TWO-COMPARTMENT MODEL – IV BOLUS After the injection of a single intravenous bolus dose of 6 mg/kg of theophylline to a 70-kg patient with normal renal function, the plasma samples were collected at different intervals and measured for theophylline concentration. The PK analysis of the measured concentrations in mg/L versus time (h) generated the following two-compartment model equation: Cp(mg / L) = 12(mg / L)e -6.0(h

-1

)´t

+ 18(mg / L)e -0.2(h

-1

)´t

Estimate the following constants and variable of the drug in this patient. 4.1.1 Total body clearance: AUC = ClT =

12 18 + = 92mgh / L 6 0.2

420 = 4.56 L / h 92

4.1.2 Biological half-life:

( T1/2 )b =

0.693 = 3.465 h 0.2

4.1.3 Elimination half-life: k 21 = k10 =

(12 ´ 0.2 ) + (18 ´ 6 ) = 3.68 h -1 30

1.2 = 0.326 h -1 3.68

0.693 = 2.12 h 0.326 4.1.4 Plasma concentration 60 minutes after the injection:

( T1/2 )k

10

=

Cp60 min = 12e -6.0 + 18e -0.2 = 14.76 mg / L 4.1.5 Total amount of the drug in the peripheral compartment 60 minutes after the injection: k12 = 6 + 0.20 - 3.68 - 0.326 = 2.194 h -1 A2 =

2.194 ´ 420 -0.2 -6 e - e = 129.70 mg 6.0 - 0.2

(

)

4.1.6 Total amount in the body 60 minutes after the injection: V1 =

420 = 14 L 12 + 18

( A1 )60 min = 14 ´ 14.76 = 206.64 ( Atotal )60 min = 129.770 + 206.64 = 336.34 mg

658

Addendum II – Part 4 CASE 4.2 – ONE-COMPARTMENT MODEL – IV BOLUS The pharmacokinetics of amdinocillin were compared in two subjects, A and C. Patient A has normal renal function, and Patient C endures severe renal impairment and requires hemodialysis. The drug was administered intravenously as a 10-mg/kg bolus injection to both patients.

Subject

Age (year)

Weight (kg)

Renal Function

Patient A Patient C

29 46

67 75

Normal Severe impairment

Blood samples were drawn, the drug concentrations were measured, and the data were analyzed according to the one-compartment model. The following equations are reported: 1. Patient A: log Cp = 1.26 - 0.258 t 2. Patient C: log Cp = 1.48 - 0.079 t ( befforedialysis ) 3. Patient C: Cp = 32e -0.533´t ( during dialysis ) Note: Plasma concentrations were in unit of mg/L and time points in hour. The dialysis was carried out four hours after the injection. 4.2.1 Calculate the volume of distribution for Patient C before and during dialysis. Before Dialysis : Vd =

750 750 = = 24.80 L antilog1.48 30.20

During Dialysis : Vd =

750 = 23.4 L 32

4.2.2 Calculate the fraction of the dose eliminated from the body by all routes of elimination at t = 4 h for Patient A and Patient C before dialysis. - 2.303´ 0.079´ 4 ) Patient C : f el = 1 - e -Kt = 1 - e ( = 0.52 - 2.303´ 0.258´ 4 ) Patient A : f el = 1 - e ( = 0.91

4.2.3 Calculate the peak concentration for Patient A if the dose had been infused over a 30-minute period. Compare the result with initial plasma concentration after bolus dose and briefy discuss the reason(s) for the difference, if any. k0 =

670 = 1340 ( mg / h ) 0.5 h

K = 0.258 ´ 2.303 = 0.59 h -1 Vd =

670 mg Dose = = 36.80 L Cp 0 118.19 mg / L

Cp30 min =

1340 1 - e -0.59´0.5 = 15.800 ( mg / L ) 0.59 ´ 36.80

(

)

659

ADDENDUM II – PART 4

4.2.4 If this drug is 30% bound to plasma protein, what would be the dose for Patient A? 670 = 957.14 mg 0.7 OR Total Concentration =

Free Concentration 18.19 = = 25.98 mg / L FreeFrraction 0.7

Dose = 36.80 ´ 25.98 = 956.06 mg

660

Addendum II – Part 4 CASE 4.3 – ONE-COMPARTMENT MODEL – IV BOLUS To compare the pharmacokinetics of azlocillin (Azlin) and piperacillin (Pipracil) in patients with normal and impaired renal functions, 4 g of each drug was administered intravenously (bolus) in a crossover design study. The following are the equations of the model ftted to the normalized plasma concentration-time data of the groups.

Group

Condition

Azlin

A

Normal renal function

B

Renal impairment

log Cp = 2.5 - 0.25t Cp = 224e

-0.16t

Pipracil log Cp = 2.35 - 0.34t Cp = 240e -0.34t

4.3.1 Estimate the elimination rate of Azlin at t = 10h for Group A only. Y = 2.5 - 0.25 t Rate = ClT ´ Cp10 h L/h ClT = 0.25 ´ 2.303 ´ ( 4 g anti log 2.5 ) @ 7.2L Cp10 h = Cp 0 e -Kt = 316.2e

(

- 0.57 h -1 ´10 h

) = 0.996 mg / L

Rate = 0.996 ´ 7.17 = 7.1 mg / h 4.3.2 What is the fraction of Pipracil dose in the body and related plasma concentration at t = T1/2 for both groups? At t = T1/2 ; f b = 0.5

( amount )T

1/2

= 2g

CpT1/2 = 0.5 ´ Cp 0 . = 111.9 mg / L ( Pip)A = 0.5 ´ 223.8

( Pip)B = 0.5 ´ 240 = 120 mg / L 4.3.3 Calculate the duration of action of Pipracil for Group A if the minimum effective concentration of the drug is 40 mcg/ml. td =

2.303 ( log 223.8 - log 40 )

= 2.2 h K 4.3.4 Calculate and compare the time constants of both antibiotics for Group B.

( Azl )B = 1/ K = 6.25 h (Pip)B = 2.94 h

661

Addendum II – Part 4 CASE 4.4 – ONE-COMPARTMENT MODEL – IV BOLUS Two hospitalized urology adult patients (A and B) were given 4 g of piperacillin intravenously in a single dose (IV bolus). Serial blood samples were obtained at different time intervals in an hour, free piperacillin plasma concentrations were measured and reported in unit of mcg/ml, and were then analyzed kinetically as a one-compartment model. The PK parameters and constants were estimated as:

Patient A B

ClT (ml/min)

Time Constant (h)

( fb )6h

220 105

1.50 3.30

0.02 0.16

4.4.1 Estimate the rate of elimination at t = 6 h for both patients.

( Rate )6 h

A

( Rate )6 h

B

æ 1 ö = 0.02 ´ 4 g ´ ç ÷ = 0.0533 g / h è 1.5 ø æ 1 ö = 0.16 ´ 4 g ´ ç = 0.194 g / h . ÷ø è 3.3

4.4.2 What fraction of the dose is eliminated by all routes of elimination at t = 6 h for patient A? f el = 1 - 0.02 = 0.98 4.4.3 Estimate plasma level for patient B three hours after the injection. æ 4000 mg öæ ö 1 Cp 0 = çç ÷ç ÷ç 198 min ÷÷ = 0.192mg / ml 105ml / min è øè ø Cp = 0.1192e -0.303 h

-1

´3 h

= 0.0775 mg / ml = 77.5mg / ml

4.4.4 Write the mono-exponential equation of the model for both patients. A : Cp = 0.200(mg / ml)e -0.66 h

-1

B : Cp = 0.192(mg / ml)e -0.303 h

662

´t

-1

´t

Addendum II – Part 4 CASE 4.5 – ONE-COMPARTMENT MODEL – IV BOLUS To compare the pharmacokinetics of Timentin (a combination of ticarcillin and clavulanic acid) in patients with normal and impaired renal functions, a dose of 5.2 g (5 g ticarcillin plus 0.2 g clavulanic acid) was administered intravenously (bolus). Plasma samples were collected at different time intervals in unit of hour, free concentrations were determined in unit of mg/L, and the data were analyzed according to the one-compartment model. The following are the reported equations of the model.

Patients

Ticarcillin

Clavulanic Acid

Normal renal function (NF)

Y = 2.65 - 0.004t

Cp = 9e -0.009t

Renal impairment (RI)

Y = 2.65 - 0.0005t

Cp = 9e -0.0013t

4.5.1 Calculate the fraction of the dose of clavulanic acid in the body one hour after the injection for both groups. f b = e -0.009 min fb = e

-1

-0.0013 min

( 60 min)

-1

= 0.58

( 60 min )

= 0.92

4.5.2 Calculate the peak concentration of ticarcillin for patients with normal renal function if the dose had been infused over 20 minutes. Compare the results with initial plasma concentration after the bolus dose, and briefy explain why the peak concentration of the infusion is less than the intravenous bolus dose. Cp 0 = anti log 2.65 = 447 mg / L Vd =

5000 = 11.185 L 447

Cp =

-1 250 1 - e -0.009 min ´20 min = 409mg / L -1 0.009 min ´ 11 1 .2L

)

(

( 447 - 409 ) ´ 11.185 = 425.03 mg total amount eliminated from the body during the infusion. 4.5.3 Calculate plasma concentration of ticarcillin at t = 3.3T1/2 after the bolus injection for patients with renal impairment. Cp3.3T1/2 = 447 mg / L ´ 0.1 = 4.47 mg / L (Note: The fraction of dose in the body 3.3T1/2 after the injection is f b = 0.1 . OR : log Cp = 2.65 - 0.0005 t K = 0.00115 min -1 = 0.069 h -1 1 min Cp = 446.7 e -0.00115

T1/2 =

-1

t

0.693 = 10 h 0.069 h -1

3.3T1/2 = 33 h 0 ´33 h Cp33 h = 446.7 e -0.069 = 4.47 mg / L

663

ADDENDUM II – PART 4

4.5.4 Estimate duration of action of ticarcillin for both groups if minimum effective concentration of the drug is 50 mg/L. Briefy explain the meaning and signifcance of the calculated values. NF : Cp 0 = antilog2.65 = 447 mg / L K = 0.0092 min -1 = 0.553 h -1 td =

2.303 447 log = 4h 0.5553 50

RI : Cp 0 = 447 mg / L K = 0.069 h -1 2.303 447 log = 31h 0.069 50 Duration of action of ticarcillin here is the interval when plasma concentration is between the minimum and maximum effective concentrations and provides the expected clinical response. td =

664

Addendum II – Part 4 CASE 4.6 – ONE-COMPARTMENT MODEL – IV BOLUS The therapeutic range of a drug is between 50 mg/L (minimum effective plasma concentration) and 300 mg/L (maximum effective plasma concentration). After an intravenous bolus injection of 2 g and regression analysis of plasma concentration of free drug (mg / ml ) versus time (h), the following equation was reported to summarize the data: log Cp = 2 - 0.1t 4.6.1 Fill in the blanks of the following Table.

Parameter/Constant

Answer

Units

Duration of action Total body clearance Total amount eliminated in six hours AUC Fraction of dose in the body at t = 6 h

Answers: Cp 0 = 100 mg / L K = 2.303 ´ 0.1 = 0.2303 h -1

(

)

td = Cp 0 Þ Cpmin = ( 100 Þ 50 mg / L ) \ td = T1/2 = 3 h æ 2000 mg ö ClT = çç . ml / min ÷÷ ( 0.2303 ) = 4.606 L / h = 76.76 è 100 mg / L ø

( Amount )6 h = 0.75 ´ 2 g = 1.5g AUC =

100 = 434.216 mgh / L 0.2303

( fb )2T

= 0.25

1/2

665

Addendum II – Part 4 CASE 4.7 – ONE-COMPARTMENT MODEL – IV BOLUS To compare pharmacokinetics of two drugs (A & B), equal dose is administered by bolus intravenous injection to a group of 12 patients in a crossover design. Drug A is administered frst, followed by a washout period before the administration of Drug B. The apparent volume of distribution is estimated as (VdA = 20 L) and (VdB = 40 L ). The reported overall elimination rate constant of Drug B is twice of Drug A (i.e., K B = 2 ´ K A ) , and the minimum effective plasma concentration is 5 mg/L for both drugs. The plasma concentration of Drug B is measured at 8.6 hours after the injection and reported as 1.25 mg/L. The fraction of dose of Drug B eliminated from the body during the 8.6-hour interval is estimated as 0.95. Based on the reported data: 1. The initial plasma concentration of Drug B is: a. 1.31 mg/L b. 2.50 mg/L c. 25.0 mg/L d. 250.0 µg/mL e. 1.187 µg/ml 2. The administered dose of Drug A is: a. 52.4 mg b. 100 mg c. 1 g d. 5 g e. 50 mg 3. The half-life of Drug B is: a. 2 hours b. 4 hours c. 1 hour d. 2.60 hours e. None of the above 4. The duration of action of Drug A is: a. 6.60 hours b. 4.00 hours c. 13.20 hours d. 3.3 hours e. None of the above 5. The washout period of the above clinical investigation is a. 12 hours b. 7 hours c. 14 hours

666

ADDENDUM II – PART 4

d. 28 hours e. None of the above 6. The total body clearance of Drug A is: a. 57.75 ml/min b. 6.93 L/h c. 115.5 ml/min d. 7.73 L/h e. None of the above 7. The total amount of Drug A eliminated from the body in duration of action, t = td is: a. 500 mg b. 250 mg c. 100 mg d. 1000 mg e. None of the above CASE 4.7 - ANSWERS

(

1. Cp 0

)

B

= 1.25 / 0.05 = 25 mg / L

2. ( D ) = ( 25mg / L ) ´ 40 = 1000 mg = 1g A 3. ( T1/2 ) = 8.6 ¸ 4.3 = 2 hours B

(

4. Cp 0

)

A

= 1000 mg / 20 = 50 mg / L

(

CpMEC = 5mg / L = 10% Cp0

)

A

\ td = 3.3 ( T1/2 ) A = 3.3 ´ 4 = 13.2 h 5. Washout = 7 ´ T1/2 = 7 ´ 4 = 28 h 6. K A = 0.693 / 4 = 0.1732 h -1 ClT = 20 L ´ 0.1732 = 3.465L / h = 57.75ml / min 7. Ael = 0.9 ´ 1000 mg = 900 mg

667

Addendum II – Part 4 CASE 4.8 – TWO-COMPARTMENT MODEL – IV BOLUS A clinically stable hospitalized urology adult patient receives 4 g of an antibiotic intravenously q4h. Serial blood samples are obtained after the frst dose, and serum concentrations are determined as presented in Table 4.8.1. The drug is known to follow the two-compartment model. Figure 4.8.1 and Figure 4.8.2 are a scatterplot of the data and the extrapolated line (last six data points), respectively.

Determine the initial estimates of the following: 4.8.1 Disposition rate constant: The regression equation of the last six data points, the post-distributive phase, or β phase, also known as the extrapolated line, is:

668

ADDENDUM II – PART 4

Table 4.8.1 Plasma Concentration of the Free Antibiotic in Serum Measured at Different Time Points Time (h)

Serum Concentration (g/ml)

0.1 0.3 0.4 0.5 0.7 1.5 2.5 4.0 5.0 6.0

520 310 250 190 120 72 36 18 7.2 6.0 ¬

log Cp = 2.232 - 0.254 t b = 2.303 ´ 0.254 = 0.585 h -1 ¬

Cp = 170.60(mg / L)e -0.585 h

-1

´t

4.8.2 Biological half-life is:

( T1/2 )b = 0.693 0.585 = 1.185 Table 4.8.2 Calculation of the Residual Line Using Initial Distributive Phase of the Plasma Levels t

˜° log Cp

˜° Cp (mg/L)

0.1 0.3 0.4 0.5

2.18016 2.13048 2.10564 2.08080

151.412 135.045 127.538 120.448

Cp (mg/L)

˜° Cp - Cp (mg/L)

˜° log Cp - Cp

368.588 174.955 122.462 69.552

2.56654 2.24293 2.08800 1.84231

520 310 250 190

(

)

4.8.3 The equation of the model is: The residual concentrations are presented in the following table:

669

ADDENDUM II – PART 4

The regression equation of the residual concentrations is: ˜°° log Cp - Cp = 2.76 - 1.77 t

(

)

˜°° -1 Cp - Cp = 575.44(mg / L)e -4.076 h ´t The biexponential equation of the model is: Cp = 575.44(mg / L)e -4.076 h

-1

´t

+ 170.60(mg / L)e -0.585 h

-1

´t

4.8.4 Half-life of elimination is: k 21 = k10 =

(170.60 ´ 4.076 ) + ( 575.44 ´ 0.585 ) = 1.40 h -1 575.44 + 170.44

4.076 ´ 0.585 = 1.70 h -1 1.4

( T1/2 )elimination = 0.693 1.70 = 0.40 h 4.8.5 Total body clearance is: V1 =

4000 mg

( 575.44 + 170.44 ) mg / L

= 5.36 L

1 83 = 152ml / min ClT = 5.36 ´ 1.70 = 9.11 L / h = 151. or AUC =

575.44 170.44 + = 432.53 mgh / L 4.076 0.585

Vdarea = a

4000 = 15.80 L 432.53 ´ 0.585

ClT = 15.80 ´ 0.585 = 9.24 L / h = 154 ml / min or ClT = 4000 mg 432.53 = 9.24 L / h 4.8.6 Total amount of drug in the central compartment one hour after the injection is:

(

)

A1 = 575.44e -4.076(1) + 170.44e -0.585(1) 5.36 = 561.30 mg 4.8.7 Plasma concentrations at t = 6 min and t = 7 h are: Cp6 min = 575.44e -4.076(0.1) + 170.44e -0.585(0.1) = 563.52mg / L

( T1/22 )a = 0.693 / 4.076 = 0.17h 7h ˜ 7 ( T1/2 )a \ Cp7 h = 170.44e -0.585(7) = 2..84mg / L 4.8.8 Distribution rate constants are: k 21 = 1.40 h -1 k12 = 4.076 + 0.585 - 1.40 - 1.70 = 1.56 h -1

670

ADDENDUM II – PART 4

4.8.9 Estimate the renal and metabolic clearances if the drug is cleared 75% unchanged via the kidneys and the balance by hepatic metabolism. f e = 0.75 Clr = 9.24 ´ 0.75 = 6.93 L / h Clm = 9.24 ´ 0.25 = 2.31 L / h

671

Addendum II – Part 4 CASE 4.9 – TWO-COMPARTMENT MODEL – IV BOLUS Cefotaxime is a semi-synthetic parenteral cephalosporin with activity against gram-positive cocci and gram-negative bacilli. The following two sets of data are obtained after intravenous injection of 1 g of cefotaxime to two patients, one with normal renal function and the other with mild renal impairment.

Time (h)

Serum Concentration (mg%) (Normal renal function)

Serum Concentration (mg%) (Renal failure)

1.050 0.920 0.820 0.720 0.620 0.500 0.420 0.295 0.145 0.071 0.035 0.017

1.100 1.050 0.960 0.900 0.820 0.740 0.530 0.455 0.330 0.240 0.175 0.125

0.05 0.10 0.20 0.30 0.40 0.50 1.00 2.00 4.00 6.00 8.00 10.00

The comparative estimated parameters and constants of both patients are:

Patient with Normal Renal Function

Patient with Mild Renal Failure

4.9.1 Disposition rate constant ˜° Cp = 0.6e -0.355 t

˜° Cp = 0.62e -0.16 t

B = 0.6 mg%

B = 0.62 mg%

b = 0.388 h -1

b = 0.16 h -1

4.9.2 Biological half-life

( T1/2 )biol =

0.693 = 1.95 h 0.355

4.9.3 Equation of the residual line ˜° Cp - Cp = 0.5e -3.5 t

(

)

(T ) 1/2

biol

=

0.693 = 4.40 h 0.16

˜°

(Cp - Cp ) = 0.48e

-1.7 t

4.9.4 Biexponential equation of the model Cp = 0.5e -3.5 t - 0.6e -0.355 t

Cp = 0.48e -1.7 t - 0.6e -0.16 t

4.9.5 Apparent volume of distribution in the central compartment Cp 0 = 1.1 mg%

Cp 0 = 1.1 mg%

V1 = 1000 mg / 1.1mg% = 90.90 L

V1 = 90.90 L

(Vd )area 672

ADDENDUM II – PART 4

AUC = 18 mgh / L

AUC = 42 mgh / L

(Vd )area = 156.5 L

(Vd )area = 148.80 L

4.9.7 Distribution and elimination rate constants k 21 =

( 0.5 ´ 0.355 ) + [0.6 ´ 3.5] 0.5 + 0.6

= 2.07 h -1 3.5 ´ 0.355 k10 = = 0. 0 6 h -1 2.07

k 21 = 1.0 h -1 k10 = 0.27 l -1 k12 = 0.59 h -1

k12 = 1.2 h -1 4.9.8 Half-life of elimination

( T1/2 )e lim = 0.693 / 0.6 = 1.2 h

( T1/2 )e lim = 0.693 / 0.27 = 2.6 h

4.9.9 Total body clearance

( Cl )T = 0.16 ´ 148.81 = 23.80 L / h

( Cl )T = 0.355 ´ 156.50 = 55.55 L / h or, ( Cl )T = 90.90 ´ 0.6 = 54.5 L / h

or, ( Cl )T = 90.90 ´ 0.27 = 24.50 L / h

673

Addendum II – Part 5 CASE 5.1 – ONE-COMPARTMENT MODEL – INTRAVENOUS INFUSION Several clinical trials have suggested that long-term infusions of Videsine (VDS) may be more effective than an IV bolus. Three patients with histologically documented neoplastic disease received a combination of VDS, cisplatin, and bleomycin. The brief protocols of the investigation are described in Table 5.1.1. The half-life of elimination of VDS was reported at 20 hours for patient A, 17.2 hours for patient B, and 28 hours for patient C. 5.1.1 If the steady-state plasma level for patient A is 10 mcg/L, what would be the total body clearance? ClT =

9mg /120 h 75mg / L k0 = = = 7.5 L / h Cpss 10mg / L 10mg / L

5.1.2 How long would it take for patient A to reach the concentration of 5 mcg/L of VDS? 5mg / L = ( 50% ) Cpss Thus, t = 1T1/2 = 20 h 5.1.3 How long would it take to reach 90% of the VDS steady-state level? 3.3T1/2 = 3.3 ´ 20 = 66 h 5.1.4 Calculate an intravenous bolus loading dose to achieve the steady-state level of 10 mcg/L in Patient A. DL = Cpss ´ Vd = Vd =

k0 K

7.5 L / h ClT = = 216.40 L K 0.0346 h -1

DL = 216.40 ´ 10 = 2164m 2 g = 2.164 mg OR, 0.693 = 0.0346 h -1 20

K=

75mg / h 6 = 2.167 mg = 2167.6mg 0.0346 5.1.5 Calculate plasma concentration 20 h after the start of long-term infusion without the loading dose for patient A. DL =

t = 20 h = 1T1/2 Þ 5mg / L 5.1.6 Calculate plasma concentration 20 hours after the end of infusion. (5 points)

(

)

- 120´0.0346 ) - 0.0346´20 ) Cp = 10 1 - e ( e ( @ 4.80 mg / L

Table 5.1.1 Brief Protocol of Cisplatin, Bleomycin and VDS Infusion for Patients of Case 5.1 Total Dose (mg)/Course of Treatment Patient A B C

674

Body wt. (kg) 80 80 70

Cisplatin 180 180 175

Bleomycin 50 55 50

VDS 9 9 9

Mode 5-day infusion 6-day infusion 5-day infusion

Addendum II – Part 5 CASE 5.2 – ONE-COMPARTMENT MODEL – INTRAVENOUS INFUSION A dose of 1.5 g piperacillin was injected intravenously to a 5-year-old pediatric patient. Plasma samples were collected at different time intervals in an hour and measured for the concentration of free drug in unit of mg/ml, and data were analyzed according to the one-compartment model. The following is the equation of the model. Cp = 0.25e -0.0173 t 5.2.1 The apparent volume of distribution of the drug for this patient is Vd =

1.5 g ´ 1000 mg/g = 6000 ml = 6 L 0.25mg / ml

5.2.2 The fraction of dose eliminated in 40 minutes is T1/2 =

0.693 = 40 min 0.0173 min -1

f el = 50% = 0.5 5.2.3 Estimate the renal clearance if the metabolic clearance is 40% of the total body clearance. Clm = 0.4 ClT

(

)

ClT = 0.0173 min -1 ´ ( 6 L ) = 0.104 L / min = 104 ml / min Clr = 0..104 - ( 0.104 ´ 0.4 ) = 0.0624 L / min = 62.4 ml / min 5.2.4 If the dose of 1.5 g piperacillin is infused over a 20-minute interval repeated four times in 24-h, what rate of infusion in mass/time would you use? k0 =

1.5 g = 0.075g / min = 75mg / min 20 min

5.2.5 Estimate the plasma concentration of piperacillin just before the start of the second infusion (i.e., six hours after the start of the frst infusion). Cp =

75mg / min

( 0.0173 min ) ´ ( 6000 ml ) -1

(1 - e

-0.0173´20

)e

-0.0173´340 4

= 5.89 ´ 10 -4 mg / ml = 59mg / ml 5.2.6 Calculate the steady-state plasma concentration with the rate calculated in Question 5.2.4. Cpss =

75 k0 = = 0.732 mg / ml ClT 0.0173 ´ 6000

5.2.7 Estimate the time required to reach 75% of the concentration calculated in Question 5.2.6. Time = 2 ´ 40 = 80 min 5.2.8 Calculate rate of elimination after achieving the steady-state concentration. At steady state :

dA = k0 = 0.075 g / min dt

675

Addendum II – Part 5 CASE 5.3 – ONE-COMPARTMENT MODEL – INTRAVENOUS INFUSION This case is the continuation of Case 4.5 If the dose of 5.2 g of Timentin was infused over 600 minutes in both groups of patients: 5.3.1 What would be the plasma concentrations of ticarcillin at the end of infusion for both groups of patients? Was the calculated rate of infusion suitable to provide a plasma concentration that exceeds 50 mg/L, the minimum effective concentration? Normal Renal Function (NRF) : T1/2 = 75min Þ 600 min > 7T1/2 Vd =

5000 mg = 11.2 L 447 4 mg / L

Cpss =

8.33 mg / min = 83 mg / L 0.009 ´ 11.2

Impaired RenalF Function (IRF) : T1/2 = 580 min 5g = 0.5g / h = 500 mg / h = 8.33 mg / miin 10 h

k0 = Cp =

(

)

-1 8.33 1 - e -0.0012 min ´578 @ 315mg / L 0.0012 ´ 11.2

5.3.2 Estimate the rate of elimination at the end of the infusion of ticarcillin for both groups of patients. NRF : Rateof elimination at steady state = Rateof input KA = k0 = 8.33 mg / min IRF : KA = KCpVd = 0.0012 min -1´ 315mg / L ´ 11.2 L = 4.23 mg / min 5.3.3 What is the plasma level 600 minutes after the end of the infusion of ticarcillin? NRF : 600 > 7T1/2 Þ Nothing left IRF : Cp = ( 315 mg / L ) e -0.0012´600 = 153.33 mg / L 5.3.4 Calculate the time required to reach 90% of the steady-state plasma concentration of ticarcillin for both groups of patients. NRF : ( Time )90%SS = 3.3 ´ 75min = 247.5min = 4.12 h IRF : ( Time )90%SS = 3.3 3 ´ 580 = 1914 min = 31.9 h

676

Addendum II – Part 5 CASE 5.4 – INTRAVENOUS INFUSION Continuous intravenous infusion of an analgesic is recommended for a patient with inoperable colon cancer and body weight of 110 pounds. The dose is prepared by dissolving 200 mg of the drug in 500 ml of 5% dextrose in water. The solution was infused over 24 hours. The drug has a half-life of two hours and an apparent volume of distribution of 3 L/kg. 5.4.1 What is the estimated plasma concentration 20 hours after the start of infusion? wt = 110 lb = 50 kg k0 = 200 / 24 = 8.34 mg / h K = 0.693 / 2 = 0.3465 h -1 Vd = 50 kg ´ 3 L / kg = 150 L ClT = 150 L ´ 0.3465 h -1 = 52L / h Cp20 h = Cpss = 8.34 / 52 = 0.16 m mg / L 5.4.2 What is the rate of infusion in ml/min? Concentration of drug solution = 200 mg / 500 ml = 0.4 mg / ml Rateof infussion in ml / min = 8.34 mg / h ¸ 0.4 mg / ml = 20.85ml / h 5.4.3 What is the plasma level six hours after the start of infusion?

(

)

Cp6 h = 0.16 1 - e -0.3465´6 = 0.14 mg / L 5.4.4 What is the estimated rate of elimination from the body at t = 21 h? At t = 21h, Cp = Cpss therefore, k0 = KA = 8.34 mg / h 5.4.5 Estimate the total amount of the analgesic in the body 15 hours after the start of infusion. Amount = 0.16 ´ 150 = 24 mg 5.4.6 Recommend a loading dose to achieve the steady-state concentration of 0.16 mg/L. DL = 0.16 ´ 150 = 24 mg

677

Addendum II – Part 5 CASE 5.5 – INTRAVENOUS INFUSION Three grams of ticarcillin and 100 mg of clavulanic acid were reconstituted in 100 ml of normal saline for injection. The solution was infused over 30 minutes to a male patient with ideal body weight of 70 kg. The pharmacokinetic data of the two drugs are presented in Table 5.5.1.

Table 5.5.1 Summary of the Selected PK/TK constants of Ticarcillin and Clavulanic Acid Constant

Units

Total body clearance Renal clearance Half-life

Ticarcillin

ml/min ml/min min

100 80 70

Clavulanic Acid 200 100 60

5.5.1 Determine the plasma concentration of ticarcillin and clavulanic acid at the end of a 30-minute infusion. æ ö æ 0.693 ö -ç ´ 30 ÷ ÷ 100 mg / min ç ç 1 - e è 70 ø ÷ = 0.26 m Cp = mg / ml 100 ml / min ç ÷ ç ÷ è ø = 260 mg / L = 260mg / ml Cp =

200 mg / L 1 - e -0.693´0.5 = 4.88 mg / L = 4.88mg / ml 12L / h

(

)

= 0.0049 mg / ml 5.5.2 Estimate the plasma level of ticarcillin 140 minutes after the termination of infusion and the plasma concentration of clavulanic acid one hour after the termination of infusion. Cp = 0.25 ´ 0.26 = 0.065 mg / ml Cp = 0.5 ´ 4.88 = 2.44 mg / L 5.5.3 Determine the volume of distribution and metabolic clearance of clavulanic acid. Vd =

(12 L / h ) = 17.31L 0.693

Clm = 200 ml / min- 100 ml / min = 100 ml / min 5.5.4 What is the total amount of ticarcillin eliminated during the 30-minute infusion? é æ 6 L / h öù Ael = 3000 mg – ê 260 mg / L ´ çç -1 ÷ ÷ ú = 374 mg êë è 0.594 h ø úû

678

ADDENDUM II – PART 5

5.5.5 Calculate the exact duration of action of the 30-minute ticarcillin infusion if its minimum inhibitory concentration is 60 mg/L.

(

CpMEC = Cpss 1 - e -Kt CpMEC = 1 - e -Kt Cpss

(

æ Cp ln çç 1 - MEC Cpss è

)

ö K ÷÷ = Kt ø

æ Cp tpre = ln çç 1 - MEC Cpss è tpost =

)

ö ÷÷ / K ø

Cpend 2.303 log K CpMEC

tduuring = tinfusion - tpre td = tduring + tpost CpMEC Cpss = 60 1000 = 0.06 tpre = tonset = ln (1 - 0.06 ) / 0.594 = 0.104 h = 6..24 min tduring = tduring theinfusion = 30 - 6.24 = 23.76 min = 0.396 h when Cp³Cp MEC

g ( 260 60 ) = 2.45 h tpost = tpost inf usion when CP³CPMEC = ( 2.303 0.594 ) log td = Duration of Action = 0.396 + 2.45 = 2.846 h

679

Addendum II – Part 5 CASE 5.6 – INTRAVENOUS INFUSION The maximum and minimum therapeutic concentration of a drug is 15 and 4 mg/L, respectively. It is decided to attain a target concentration of 10 mg/L immediately and maintain the level for 12 hours in a patient with 65 kg ideal body weight. The drug has a half-life of six hours with renal and metabolic clearances of 5 L/h and 6.55 L/h, respectively. 5.6.1 Determine the appropriate intravenous bolus loading dose and maintenance infusion rate to achieve and maintain the target concentration. K=

0.693 = 0.1155 h -1 6

ClT = 5 + 6.55 = 11.55 L / h Vd =

11.55 0.1155 = 100 L

DL = 100 ´ 10 = 1000 mg k0 = 10 mg / L ´ 11.55 L / h = 115.50 mg / h 5.6.2 What volume and concentration of the infusion solution is needed if the rate of input is 1 ml/min? 12 h ´ 60 ( ml / h ) = 720 ml k0 = 115.50 mg / h Amount = 115.50 ´ 12 = 1386 mg Conceentration = 1386 720 = 1.925 mg / ml 5.6.3 If the infusion was initiated without the loading dose, how long would it take to reach the plasma concentration of 5 mg/L (i.e., 50% of the target concentration)?

( tinfusion )50%Cp

ss

= oneT1/2 = 6 h

5.6.4 How long would it take to achieve 80% of the steady-state plasma concentration without the loading dose? f ss = 1 - e -Kt

(1 - fss ) = -e -Kt t = -ln ( 1 - f ss ) / K = - ln(0.2) / 0.1155 = 13.944 = 14 h 5.6.5 Estimate the plasma levels 1, 6, and 12 hours after stopping the infusion at plateau level of 10 mg/L. Cp1 = 100e -0.1155(1) = 8.91ng / L Cp6 = 10 / 2 = 5mg / L Cp12 = 0.25 ´ 10 = 2.50 m mg / L

680

ADDENDUM II – PART 5

5.6.6 If the patient develops acute renal failure and the renal clearance is reduced to 2 L/h, what bolus loading dose and maintenance infusion rate would you recommend to achieve the steady-state level of 10 mg/L? Assume metabolic clearance and the apparent volume of distribution remain the same. ClT = 6.55 + 2 = 8.55 L / h K = 8.55 / 100 = 0.0855 h -1 DL = 100 ´ 10 = 1000 mg k0 = 100 ´ 8.55 = 85.50 mg / h

681

Addendum II – Part 5 CASE 5.7 – INTRAVENOUS INFUSION A 20-year-old male patient with ideal body weight of 60 kg is receiving 950 mg of a cephalosporin infused intravenously over 15 minutes. The half-life of the drug is 45 minutes, and its apparent volume of distribution is 0.8 L/kg. 5.7.1 What is the rate of infusion? Rateof Input = k0 = dA / dt = 950 mg /15min = 63.30 mg / min = 3800 mg / h = 3.8 g / h 5.7.2 If the drug is supplied in vials of 100-ml solution with a concentration of 500 mg/ml, at what rate in milliliters per hour would you infuse to obtain the above infusion rate (Section 5.7.1)? Rate(volume / time) =

63.30 mg / min = 0.127 ml / min 500 mg / ml

5.7.3 Determine the plasma level at the end of infusion. Vd = 0.8 ´ 60 = 48 L K=

0.693 = 0.924 h -1 = 0.0154 min -1 0.75h

Cp15 min =

3800 mg m /l 1 - e -0.015´15 0.924 h -1 ´ 48 L

(

)

= 17.633 mg / L 5.7.4 What is the total amount of drug in the body at the end of infusion? A15 = (17.633 mg/L) (48 L) = 846.384 mg 5.7.5 Estimate the total amount of drug eliminated during the 15-minute infusion. Aeliminated = 950 - 846.384 = 103.616 = 104 mg 5.7.6 What would be the peak plasma concentration if the dose was administered as an intravenous bolus injection? Cp 0 =

950 = 19.8 mg / L 48

Note: The total amount eliminated during the infusion may also be calculated as:

(19.80 - 17.633 )Vd = 104 mg 5.7.7 If the minimum effective concentration of the drug is 10 mg/L, would the plasma concentration 5 minutes after the start of infusion be therapeutically effective? Cp5 min =

682

(

)

-1 3800 mg / h 1 - e -0.0154 min ´5 min = 6.35mg / L 0.924 h -1 ´ 48 L

ADDENDUM II – PART 5

Answer: No. 5.7.8 What steady-state level can be achieved with the rate calculated in Question 5.7.1? ClT = 0.924 ´ 48 = 44.35 L / h Cpss =

3800 mg / L = 85.70 mg / L 44.35 L / h

5.7.9 Estimate a single rate of infusion to achieve a plateau concentration of 150 mg/L. k0 = 150 mg / L ´ 44.35 = 6652.50 mg / h = 6.65 g / h 5.7.10 What is the rate of elimination after achieving the steady-state level calculated in Question 5.7.9? Rateof elimination at steady state = Cpss ´ ClT = KAss = k0 Cpss ´ ClT = 6.6 65 g / h = k0 5.7.11 How long does it takes to reach 90% of the steady-state plasma concentration of 85.7 mg/L or 150 mg/L?

( T95% )Cp

ss

= 3.3T1/2 = 3.3 ´ 0.75 h = 2.475 h

5.7.12 What bolus loading dose and zero-order maintenance dose are needed to attain and maintain plateau concentrations of 10 mg/L? DL = 10 mg / L ´ 48 L = 480 mg @ 0.5 g k0 = 0.924 h -1 ´ 48 L ´ 10 mg / L = 443.52 mg / h

683

Addendum II – Part 5 CASE 5.8 – INTRAVENOUS INFUSION A 40-year-old female patient with ideal body weight of 70 kg and complicated urinary tract infection is to be treated with 1 g of a third-generation cephalosporine infused over an hour, given every eight hours for ten consecutive days. The half-life of the drug is two hours, and the total body clearance is 110 ml/min. 5.8.1 If the frst infusion starts at 8:00 AM and stops at 9:00 AM, what would be the plasma concentration of the drug at 1:00 PM? k0 = 1000 mg / h ClT = 110 ml / min = 6.6 L / h Cp9 AM =

)

(

-1 1000 mg 1 - e -0.3465 h ´1h = 44.37 mg / L 6.6 L / h

Cp1PM = 44.37e -0.3465 h

-1

´4 h

= 11.10 mg / L

or, 4 h = 2T1/2 i.e. Cp1PM = 0.25 ´ 44.37 = 11.10 mg / L 5.8.2 What would be the plasma concentration just before the second infusion at 4:00 PM? Cp4 PM = 44.37e 0.3465 h

-1

´7 h

or, Cp4 PM = 11.10e -0.3465 h

= 3.92 mg / L -1

´ ´3 h

= 3.92 mg / L

5.8.3 What would be the total amount in the body at the end of the second infusion (i.e., at 5 PM)? Cp5 PM = 44.37 mg / L + 3.92e -0.3465 h Total amount = 47.14 . mg / L ´

684

-1

´1 h

= 47.14 mg / L

6.6 L / h = 897.90 mg 0.3465 h -1

Addendum II – Part 5 CASE 5.9 – INTRAVENOUS INFUSION A 65-year-old female patient weighing 70 kg was admitted to the hospital for hepatic encephalopathy and cirrhosis. On the fourth hospital day, she developed ventricular arrhythmias and lidocaine was ordered. The following PK data are known for lidocaine: Total Body Clearance

4.62mll / min/ kg

Apparent Volume of Distribution

0.6 L / kg

Fraction of the dose excreted unchanged

0.05

5.9.1 What loading and maintenance doses do you recommend for achieving a steady-state plasma level of 2 µg/ml? Vd = ( 0.6 L / kg )( 70 kg ) = 42L ClT = 4.62ml min -1/ kg ´ 70 kg = 323.4 ml / min DL = 2mg / L ´ 42L = 84 mg DM = k0 = ( 2mg / ml ) ´ 323 ml / min = 646.8 mg / min = 38.8 mg m /h 5.9.2 On the ffth hospital day, she developed premature ventricular contractions. Procainamide was then selected as the substitute medication, and the procainamide therapy started three hours after the end of the lidocaine infusion. Calculate the total amount of lidocaine in the body just before the start of the procainamide infusion. K = ( 19.4 L / h ) / 42L = 0.462 h -1 Cp3 h = ( 2mg / L ) e

(

)

- 0.462h -1 ( 3 h )

= 0.5mg g/L

A3 h = ( 0.5 mg / L )( 42L ) = 21mg

685

Addendum II – Part 6 CASE 6.1 – ORAL ADMINISTRATION An oral dose of 250 mg of a new anti-infammatory drug with an apparent volume of distribution of 70 L is given to an 80-kg patient. Plasma samples were collected at different intervals, measured in triplicate, and the mean values of the measurements are reported in Table 6.1.1 as observed data. The plasma concentration-time profle of the observed data is shown in Figure 6.1.1, and the scatterplot of the logarithm of plasma levels versus time is presented in Figure 6.1.2.

Table 6.1.1 Plasma Concentration of the Drug Measured at Different Time Points with Corresponding Log Values Observed Data Points Time ( h ) 0.2 0.4 0.6 0.8 1.0 1.6 2.0 2.5 3.0 3.5 4.0 6.0 8.0 10.0 12.0 14.0

686

Cp ( mg / L ) 0.72 1.25 1.63 1.89 2.06 2.28 2.27 2.18 2.06 1.92 1.78 1.29 0.93 0.67 0.48 0.35

logCp −0.142668 0.096910 0.212188 0.276462 0.313867 0.357935 0.356026 0.338456 0.313867 0.283301 0.250420 0.110590 −0.031517 −0.173925 −0.318759 −0.455932

ADDENDUM II – PART 6

Figure 6.1.2 reveals that the absorption process is concluded about four hours after the administration of the dose. Thus, the last six data points refect the decline of plasma concentration in the body due to the elimination process. The regression analysis of the last six data points (Table 6.1.2) is presented in Figure 6.1.3 with the equation that defnes the extrapolated line.

Table 6.1.2 The Terminal Loglinear Data Points of Plasma Concentration Forming the Extrapolated Line Last Six Data Points Time ( h ) 4 6 8 10 12 14

Cp ( mg / L ) 1.78 1.29 0.93 0.67 0.48 0.35

logCp 0.250420 0.110590 −0.031517 −0.173925 −0.318759 −0.455932

687

ADDENDUM II – PART 6

The calculation of the residual line is presented in Table 6.1.3 and Figure 6.1.4.

Table 6.1.3 Calculations of Residual Values and Loglinear Residual Line Time(h) 0.2 0.4 0.6 0.8

˜° log Cp

˜° Cp(mg/L)

0.520622 0.506444 0.492266 0.478088

3.31606 3.20955 3.10646 3.00669

Cp(mg / L) 0.72 1.25 1.63 1.89

˜° Cp - Cp

˜° log Cp - Cp

2.59606 1.95955 1.47646 1.11669

0.41431 0.29216 0.16922 0.04793

(

)

6.1.1 Assume the observed data are an ideal set of data (i.e., there are no random or systematic errors associated with the measurements, and there is no need for regression analysis and curve ftting). Estimate the overall elimination rate constant, absorption rate constant, time to maximum plasma concentration, and absolute bioavailability. • Estimation of the overall elimination rate constant: -

log 0.35 - log 0.48 K = -0.0686 h -1 = 2.303 14 - 12

K = 2.303 ´ 0.0686 = 0.158 8 @ 0.16 h -1 ◾ Estimation of the y-intercept of the extrapolated line: ˜°°°° 0.35 Cp 0 = -0.16´14 = 3.28 mg / L e ◾ Estimation of the absorption rate constant: ˜ ° Cp1 - Cp1 = 3.28e -0.16´0.2 - 0.72 = 2.456 mg / L ˜°°° Cp2 - Cp2 = 3.28e -0.16´0.4 - 1.25 = 1.826 mg / L -

log 1.826 - log 2.456 ka = = -0.643 h -1 2.303 0.4 - 0.22

k a = 2.303 ´ 0.643 = 1.48 h -1

688

ADDENDUM II – PART 6

◾ Estimation of the y-intercept of the residual line: y - intercept =

2.456 = 3.30 mg / L e -1.48´0.2

◾ Estimation of time to the maximum plasma concentration: Tmax =

2.303 1.48 log = 1.68 h 1.48 - 0.16 0.16

◾ Estimation of the absolute bioavailability if Vd = 70 L: ˜ ° Cp 0 ´ Vd ´ ( k a - K ) 3.28 ´ 70 ´ (1.48 - 0.16 ) F= = = 0.82 Dk a 250 ´ 1.48 6.1.2 Assume the data are real and, by using simple regression analysis, estimate the same constants and variables as in Section 6.1.1. • The regression equation of the extrapolated line and the related calculation of the elimination rate constant are: ˜° log Cp = 0.5348 - 0.07089 t(h) ˜°°° Cp 0 = antilog 0.5348 = 3.426 mg / L K = 2.303 ´ 0.07089 = 0.1632 h -1 T1/2 = 0.693 / 0.1632 = 4.25 h ◾ The regression equation of the residual line and the estimation of the absorption rate constant are: ˜° log Cp - Cp = 0.5364 - 0.611t(h)

(

)

Thus, the absorption rate constant can be estimated as: k a = 2.303 ´ 0.611 = 1.40 h -1 The half-life of absorption (i.e., the time required for 50% of drug to be absorbed) is:

( T1/2 )absorption = 0.693 / 1.40 = 0.495 h Comparison of the y-intercepts of extrapolated and residual lines indicates that there is no lag time of absorption. ˜°° Cp - Cp = antilog 0.5364 = 3.438 mg / L

(

)

t=0

˜°°° Cp 0 = 3.426 mg / L The time of the maximum plasma concentration, using data from the regression line, is: Tmax =

2.303 1.40 log = 1.74 h 1.40 - 0.1632 0.1632

Note: The calculated value of Tmax corresponds well to the observed time of maximum plasma concentration. To estimate the absolute bioavailability of the drug, the area under the plasma concentration-time curve (AUC) of the observed data should be calculated frst. The following are two approaches to estimate the AUC:

689

ADDENDUM II – PART 6

First approach: Using the classical trapezoidal rule (also see Addendum I, Part 2):

C2

C3

C4

Cp(t1 - tn )

Cp(t0 - tn-1 )

Dt

æ C2 + C3 ö ÷ C4 ç 2 ø è Area

0.72 1.25 1.63 1.89 2.06 2.28 2.27 2.18 2.06 1.92 1.78 1.29 0.93 0.67 0.48 0.35

0.00 0.72 1.25 1.63 1.89 2.06 2.28 2.27 2.18 2.06 1.92 1.78 1.29 0.93 0.67 0.48

0.2 0.2 0.2 0.2 0.2 0.6 0.4 0.5 0.5 0.5 0.5 2.0 2.0 2.0 2.0 2.0

0.0720 0.1970 0.2880 0.3520 0.3950 1.3020 0.9100 1.1125 1.0600 0.9950 0.9250 3.0700 2.2200 1.6000 1.1500 0.8300

C1 Time(h) 0.2 0.4 0.6 0.8 1.0 1.6 2.0 2.5 3.0 3.5 4.0 6.0 8.0 10.0 12.0 14.0

AUC0t

16.479 mgh / L

Terminal Area ( Cpn K = 0.35 0.1632 )

2.144 mgh / L

AUC0¥ = AUC0t + Terminal Area

18.623 mgh / L

˜ ° Second approach: Integrating the equation of Cp = Cp 0 e -Kt - e -kat : ˜°°°° æ 1 1 ö 1 ö æ 1 AUC = Cp 0 ç - ÷ = 3.426 ç = 18.54 mgh / L 0 1632 1 40 ÷ø . . K k è a ø è

(

)

6.1.4 To estimate the absolute bioavailability, one of the following relationships can be employed: Vd ´ K ´ AUC 70 ´ 0.1632 ´ 18.54 = = 0.850 D 250 ˜°°°° Cp 0 ´ Vd ´ ( k a - K ) 3.426 ´ 70 ´ (1.40 - 0.1632 ) II. F = = = 0.850 D ´ ka 250 ´ 1.40 I. F =

III. F =

690

( AUC )po 18.623 18.623 = = = 0.85 ( AUC )i.v. ( 250 / 70 ) / 0.1632 21.884

Addendum II – Part 6 CASE 6.2 – ORAL ADMINISTRATION To investigate the infuence of clavulanic acid on the pharmacokinetics of amoxicillin, a 30-yearold patient receives a single dose of 500 mg of amoxicillin orally on day one (Treatment A) followed by 500 mg of amoxicillin plus 125 mg clavulanic acid on the second day (Treatment B). The serum and urine samples are analyzed for amoxicillin and the data are reported as follows: Treatment A: Urine Data: Total amount excreted unchanged in the urine = 200 mg Serum Data: Concentrations of amoxicillin at different times were determined as reported in Table 6.2.1 and Figure 6.2.1.

Table 6.2.1 Plasma Concentration of Amoxicillin and the Related Log Values Time (min) 10 20 40 60 90 120 180 240 350

Plasma Concentration

log Cp

0.24 0.60 5.00 7.80 8.20 6.50 3.80 1.90 0.44

−0.619789 −0.221849 0.698970 0.892095 0.913814 0.812913 0.579784 0.278754 −0.356547

Treatment B: Urine Data: Total amount of amoxicillin excreted unchanged in the urine = 290 mg Serum Data: The equation of the model based on the serum concentrations of Treatment B is

(

Cp = 37 ( mg / L ) e -0.013 min

-1

(t)

- e -0.025 min

-1

(t)

) 691

ADDENDUM II – PART 6

6.2.1 Estimate the absorption and elimination rate constants of amoxicillin for both treatments. Treatment A: The regression equation of the extrapolated line is: ˜° log Cp = 1.59 - 0.0054 (t) Cp 0 = antilog 1.59 = 38.9 mg / L K = 0.0054 ´ 2.3303 = 0.0127 min -1 Calculations of the frst four points on the extrapolated line are: ˜° Cp10 min = 38.90 e -0.0127´10 = 34.24 mg / L ˜°° Cp 20 min = 38.90 e -0.0127´20 = 30.14 mg / L ˜°° Cp 40 min = 38.90 e -0.0127´40 = 23.35mg / L ˜°° Cp 60 min = 38.90 e -0.0127´60 = 18.09mg / L Calculations of the related residual concentrations are: ˜° Cp - Cp = 34.24 - 0.24 = 34.00 mg / L

( (

)

10 min

˜°° Cp - Cp

)

20 min

(Cp - Cp )

40 min

˜°° ˜°°

(Cp - CCp )

60 min

= 30.14 - 00.60 = 29.08 mg / L = 23.35 - 0.500 = 18.35mg / L = 18.35 - 8.20 = 9.89 mg / L

The regression equation of the line of residuals is: ˜° log(Cp - Cp) = 1.66 - 0.0108(t) Therefore, k a = 2.303 ´ 0.0108 = 0.0248 min -1 Summary: Treatment A K = 0.0127 min−1 ka = 0.025 min−1

Treatment B K = 0.013 min−1 ka = 0.025 min−1

6.2.2 Fraction of the dose absorbed if the volume of distribution of amoxicillin is 19 L: ˜ °

(Cp )Vd ( k - K ) = 38.90 ´ 19 (0.025 - 0.0127 ) = 0.727 = 0

FA

FB = 1.

692

a

Dose ´ k a

500 ´ 0.025

37 ´ 19 ( 0.025 - 0.013 ) = 0.675 500 ´ 0.025

Maximum plasma concentration of amoxicillin for both treatments:

( Tmax )A =

2.303 0.025 log = 55min 0.025 - 0.0127 0.0127

( Tmax )B =

2.303 0.025 log = 54.5 min 0.025 - 0.013 0.013

ADDENDUM II – PART 6

1.

( Cpmax )A =

0.727 ´ 500 -0.0127´ 55 e = 9.5 mg / L 19

( Cpmax )B =

0.685 ´ 500 -0.013´54.5 e = 8.87 mg / L 199

Renal clearance of amoxicillin: Clr = f eClT

( Clr )A = æç

200 ö ÷ ´ 19 ´ 0.0127 = 0.1327 L / min = 133 mll / min è 363.50 ø

( Clr )B = æç

290 ö ÷ ´ 19 ´ 0.013 = 0.2122 L / min = 212ml / min è 337.50 ø

Except for renal clearance, other PK parameters and constant remain the same.

693

Addendum II – Part 6 CASE 6.3 – ORAL ADMINISTRATION The infuence of high-fat and low-fat meals on the absorption of a new drug was evaluated in a crossover design on six volunteers, age 30 to 44 years old. No other drug or alcohol was allowed a week before and during the study. The experimental protocols were: Treatment A – Low-Fat Breakfast A single dose of 10-mg capsule of the drug was administered with 200 ml water immediately after a low-fat meal consisting of fruit juice, skim milk, cereal, and toast with jelly. Treatment B – High-Fat Breakfast A single dose of 10-mg capsule of the drug was administered with 200 ml water immediately after a high-fat meal consisting of whole milk, bacon, egg, sausage, and decaffeinated coffee. The averages of plasma concentrations of the drug are reported in Table 6.3.1.

Table 6.3.1 Average Plasma Concentration of the Drug Related to Treatments A & B Time (h) 0.10 0.20 0.30 0.40 0.6 0.75 1.00 2.00 3.00 4.00 5.00 6.00

CpTreatment- (ng/ml)

CpTreatment-(ng/ml)

1.0 12.0 28.0 46.0 56.0 59.7 60.0 40.0 26.2 17.5 11.5 7.60

0.70 4.10 11.1 32.5 42.1 42.9 39.6 31.0 21.5 14.2 9.70 6.60

The scatterplots of log Cp versus time for both treatments are shown in Table 6.3.2 and Figures 6.3.1 and 6.3.2.

694

ADDENDUM II – PART 6

Table 6.3.2 Logarithm of the Average Plasma Concentrations of Treatment A&B Time (h)

logCp-A

0.10 0.20 0.30 0.40 0.60 0.75 1.00 2.00 3.00 4.00 5.00 6.00

0.00000 1.07918 1.44716 1.66276 1.74819 1.77597 1.77815 1.60206 1.41830 1.24304 1.06070 0.88081

logCp-B −0.15490 0.61278 1.04532 1.51188 1.62428 1.63246 1.59770 1.49136 1.33244 1.15229 0.98677 0.81954

Estimate the following constants and variable for both treatments. The apparent volume of distribution of the drug is 120 L. 6.3.1 Absorption and elimination rate constants for both treatments 6.3.2 Time to maximum plasma concentration 6.3.3 Fraction of dose absorbed 6.3.4 Maximum plasma concentration 6.3.5 Area under plasma concentration-time curve 6.3.6 Total body clearance 6.3.7 Tabulate the calculated data and summarize your conclusions about the calculated data (Parts 1–6) in one paragraph. Treatment A ˜°° -0.4 ( h -1 ) t ExtrapolatedLine : Cp = 91.415 ( mg / L ) e ˜°° -3.455 ( h -1 ) t Residual Linee : Cp - Cp = 131.22 ( mg / L ) e 695

ADDENDUM II – PART 6

Tmax = F=

2.303 3.455 log = 0.70 h 3.455 - 0.40 0.40

. - 0.40)h -1 91.415 mg / L ´ 120 L ´ (3.455 = 0.97 -1 3.455 h ´ 10 mg ´ 1000mg / mg

Cpmax =

0.97 ´ 10 mg -0.40´0.70 e = 0.061mg / L = 61mg / L 1220 L

1 ö æ 1 AUC = 91.415 mg / L ç ÷ = 202mgh / L è 0.40 3.455 ø ClT =

0 mg / mg 0.97 ´ 10 mg ´ 1000 = 48 L / h 202mgh / L Treatment B

˜°° -0.389 ( h -1 ) t ExtrapolatedLine : Cp = 67 ( mg / L ) e ˜°° -2.945 ( h -1 ) t Residual Line : Cp p - Cp = 97.72 ( mg / L ) e Tmax = F=

2.303 2.945 log = 0.792 h 2.945 - 0.389 0.389

67.95 ´ 120 ´ (2.9455 - 0.389) = 0.707 2.945 ´ 10 4

Cpmax =

0.707 ´ 10 4 -0.389´0.792 = 43.30mg . e /L 120

1 ö æ 1 AUC = 67.95 ç ÷ = 151.60 mgh / L è 0.389 2.945 ø ClT =

0.707 ´ 10 4 = 46.63 L / h 151.6

Summary Treatment A B

Tmax (h)

F

Cpmax (mg/L)

AUC (mgh/L)

Clt (L/h)

0.706 0.792*

0.970 0.707*

61 43*

202.08 151.60*

48.02 46.63

* P 7 ( T1/2 )k thus, e -kat Þ 0 a

˜°°°° Cp = Cp 0 e -Kt = 34.20 e -0.0693´10 = 17 1 .10 mg / L Plasma concentration 2 hours after the administration of oral dose is:

(

)

Cp = 34.20 e -0.0693´2 - e -2.77´ 2 = 29.64 mg / L 6.5.6 Estimate the fraction of the oral dose in the body at t = 2 h. fb =

Cp2 h ´ Vd 29.64 ´ 24 = = 0.889 F ´ Dose 0.8 ´ 1000

699

ADDENDUM II – PART 6

6.5.7 Calculate the total body clearance. ClT =

F ´ Dose 800 = = 1.663 L / h = 27.72ml / min AUC 481

or, ClT = K ´ Vd = 0.0693 h -1 ´ 24 h = 1.663 L / h 6.5.8 Determine the rate of elimination at t = 2 h.

( Rate )2 h = ( dA / dt )2 h = ClT ´ Cp = 29.62mg / L ´ 1.663 L / h

700

Addendum II – Part 6 CASE 6.6 – ORAL ADMINISTRATION Drug BBW is an ester of drug ABW, absorbs from the GI tract, and provides serum levels comparable to those achieved after its equimolar intramuscular injection. A 9-month-old, 6.5-kg infant receives 10 mg/kg of BBW, equimolar to 7 mg/kg of drug ABW, every four hours. Blood samples were collected after giving the frst dose, and the concentration of drug ABW was determined in plasma, as reported in Table 6.6.1. The scatterplot of the logarithm of ABW plasma concentration versus time is shown in Figure 6.6.1

Table 6.6.1 Plasma Concentrations of the ABW Measured at Different Time Points Following Intramuscular Administration and the Related Log Values Time (h) 0.05 0.10 0.20 0.30 0.50 1.00 1.50 2.00 2.50 3.00 4.00 5.00 6.00

Cp of ABW (µg/ml)

Log Cp−ABW

0.20 0.50 2.00 7.00 7.50 6.50 4.30 2.85 1.95 1.25 0.55 0.25 0.10

−0.69897 −0.30103 0.30103 0.84510 0.87506 0.81291 0.63347 0.45484 0.29003 0.09691 −0.25964 −0.60206 −1.00000

701

ADDENDUM II – PART 6

6.6.1 Estimate the elimination half-life of ABW in this patient. The regression analysis of the last fve data points of log Cp-ABW yields the following equation: ˜° log Cp = 1.198 - 0.364 t ˜°°° i.e. log Cp 0 = 1.198 ˜°°° g/L Cp 0 = 15.776 mg Slope = -0.364 h -1 = -

K 2.303

K = 2.303 ´ 0.364 = 0.838 h -1 T1/2 = 0.8266 h Note: Using the slope relationship for a real set of data (i.e., scattered data points due to the presence of random errors) would provide different values for the slope depending on the selected pair of data points. Consider the following two examples: i.

Time points of 5 and 6 hours: Slope =

log 0.10 - log 0.25 = -0.398 h -1 6-5

K = 2.303 ´ 0.398 = 0.91 h -1 T1/22 = 0.76 h ii. Time pointsof 3 and 4 h : Slope =

log 0.55 - log 1.25 = -0.356 3 h -1 4-3

K = 2.303 ´ 0.356 = 0.821 h -1 T1/2 = 0.844 h 6.6.2 Estimate the absorption rate constant and half-life of absorption using the slope of the residual line. The estimated residuals of the frst fve data points: ˜°° Cp - Cp = 15.776e -0.838´0.05 - 0.20 = 14.92 mg / L

( ) ˜°° (Cp - Cp ) ˜°° (Cp - Cp ) ˜°° (Cp - Cp ) ˜°° (Cp - Cp )

1

702

2

3

4

5

(

)

(

)

(

)

(

)

(

)

= 15.776e -0.838´0.10 - 0.50 = 14.00 mg / L . .20 - 2.00 = 11.34 mg / L = 15.776e -0.838´0

= 15.776e -0.838´0.30 - 7.00 = 5.27 mg / L = 15.776e -0.838´0.50 - 7.50 = 2.87 m mg / L

ADDENDUM II – PART 6

The equation of the residual line is: ˜° log Cp - Cp = 1.302 - 1.70 t

(

)

k a = 2.303 ´ 1.70 = 3.915 h -1 0.693 6 = 0.177 h 3.915 The calculated and observed time to maximum plasma concentration, Tmax :

( T1/2 )k

Calculated value: Tmax =

a

=

2.303 4.0 log = 0.49 h 4.0 - 0.838 0.838

Observed value: 0.5 hours The area under the plasma concentration-time curve from time zero to infnity (AUC0¥ ) is: n

AUC0¥ =

å çèæ i=0

Cpi + Cpi+ 1 ö ¥ (Also see Addendum I, Part 2) ÷ ( ti+1 - ti ) + AUCn 2 ø

Calculation of Area under Plasma Concentration-Time Curve

ti+1 - ti

Cpi + Cpi+1 2

0.05 0.05 0.10 0.10 0.20 0.50 0.50 0.50 0.50 0.50 1.00 1.00 1.00 Total

0.100 0.350 1.250 4.500 7.250 7.000 5.400 3.575 2.400 1.600 0.900 0.400 0.175

AUCii+1 0.0050 0.0175 0.1250 0.4500 1.4500 3.5000 2.7000 1.7875 1.2000 0.8000 0.9000 0.4000 0.1750 13.510

AUC0t = 13.510 mgh / L AUC0¥ = 13.510 +

0.10 = 13.630 mgh / L 0.838

6.6.5 The apparent volume of distribution of ABW if the bioavailability is 0.8: Vd =

0.8 ´ 7 mg / kg ´ 6.5 kg FD = = 3.187 L K ´ AUC 0.838 h -1 ´ 13.63 mgh / L

703

Addendum II – Part 7 CASE 7.1 – MULTIPLE DOSING KINETICS A 60 kg hospitalized male patient with atrial fbrillation was given a cardiac glycoside by intravenous bolus injection every 24 hours for 10 consecutive days. Based on the analysis of plasma concentration, the following data are known about the patient: Vdarea = 7.38 L / kg ClT = 0.213 Lh -1 / kg

V1 = 1L / kg

( Cpmin ) = 1m

ClT = 0.2213 Lh -1 / kg

Estimate the following parameters and constants: 7.1.1 The disposition rate constant and biological half-life. ClT = 0.213 ´ 60 = 12.78 L / h b = 12.78 /(7.38 ´ 60) = 0.0289 h -1

( T1/2 )b = 0.6693 / 0.0289 = 24 h 7.1.2 The plasma concentration 72 hours after the last dose of steady-state levels. 72 h = 48 h after(Cpmin )ss \Cp72 = 1mg / L ´ 0.25 = 0.25mg / L 7.1.3 The peak level after the third dose? f ss = 1 - e -3 ´ 0.0289 ´ 24 = 0.875 (Cpmax )3 rd = 0.875 ´ 2 = 1.75mg / L 7.1.4 How long would it take to achieve 65% of the average steady-state plasma concentration? nt = -3.3 ´ 24 ´ log(1 - 0.65) = 36 h 7.1.5 How long would it take to achieve a maximum plasma concentration of 1 µg/L? 1mg / L = 50%(Cpmax )ss nt = one biological half-life

704

Addendum II – Part 7 CASE 7.2 – MULTIPLE DOSING KINETICS A 25-year-old woman receives 50 mg of an antibiotic intravenously q6h. The drug follows the twocompartment model, and the related data for the frst dose of this patient are as follows: a = 11.25mg / L b = 3.75 mg / L a = 1.8 h -1 b = 0.2 h -1 7.2.1 Estimate the average steady-state plasma concentration. k 21 =

(11.25 ´ 0.2 ) + ( 3.75 ´ 1.8 ) = 0.6 h -1 11.25 + 3.75

k10 =

0.2 ´ 1.8 = 0. 0 6 h -1 0.6

V1 =

50 = 3.34 L 15 50 mg

( Cpave )ss = 3.34 L ´ 0.6 h -1 ´ 6 h = 4.16 mg / L Or AUC =

11.25 3.75 + = 25 mgh / L 1.8 0.2

(Vd )area =

50 = 10 L 25 ´ 0.2 50 mg

( Cpave )ss = 10 L ´ 0.2 h -1 ´ 6 h = 4.16 mg / L 7.2.2 Determine the loading dose to attain the steady-state levels immediately. DL =

50 mg = 71.55 mg 1 - e -0.2´6

7.2.3 Estimate the trough level at steady-state.

( T1/2 )a = 0.385 \ t > 7 ( T1/2 )a æ e -0.2´6 ö

( Cpmin )ss = 3.75 ç 1 - e -0.2´6 ÷ = 1.62mg / L è

ø

7.2.4 Estimate plasma concentration 12 hours after the last dose of the steady-state levels. Cpt¢=12 h = 1.6 e

-0.2´(12-6 )

= 0.488 mg / L

7.2.5 Assess how long it takes to reach 80% of steady-state levels. nt = -3.3 ´ ( 0.693 / 0.2 ) ´ log (1 - 0.8 ) = 8 h 7.2.6 Calculate the accumulation index. R=

1 = 1.43 1 - e -0.2´6

705

ADDENDUM II – PART 7

7.2.7 Estimate the total amount excreted unchanged and eliminated as metabolite(s) in a dosing interval at steady-state if the fraction of dose eliminated as metabolite(s) is 0.46. f m = 0.46 f e = 1 - 0.46 = 0.54

( Am )ss

= 23 mg

( Ae )ss

= 27 mg

t

t

706

Addendum II – Part 7 CASE 7.3 – MULTIPLE DOSING KINETICS Recommend a reasonable multiple oral dosage regimen for a drug that has an elimination half-life of 4 hours and an absorption half-life of 0.5 hours. The dug is available as 150-mg and 300-mg capsules. The minimum and maximum effective therapeutic concentrations of the active ingredient are 15 mg/L and 55 mg/L. The apparent volume of distribution is 10 L, and its absolute bioavailability is about 0.9. Design the regimen based on a target concentration of 41 mg/L average plasma level. Furthermore, determine whether the fuctuation of your proposed regimen remains within the therapeutic range. K = 0.1732 h -1 k a = 1.386 h -1

( T1/2 )K = 4 h > 7 ( T1/2 )k

a

Set t = ( T1/2 )K = 4 h FDM = 41 mg / L ´ 10 L ´ 4 h ´ 0.1732 h -1 = 284.05mg DM = 284.005 / 0.9 = 315.6 mg @ 300 mg

(

)

DL = 315.60 1 - e -0.1732´4 = 631 @ 600 mg or, DL = 2 ´ DM ˜°°°° 0.9 ´ 300 ´ 1.386 Cp 0 = = 30.85 @ 31mg / L 10 (1.386 - 0.1732 )

( Tmax )sss = 1.146 h æ e -0.1732´1.146 ö

( Cpmax )ss = 24 ç 1 - e -0.1732´4 ÷ = 46.63 mg / L è

ø

e

-0.1732´4

( Cpmin )ss = 31 1 - e -0.1732´4

= 31mg / L

The fuctuation is within the therapeutic range.

707

Addendum II – Part 7 CASE 7.4 – MULTIPLE DOSING KINETICS Two grams of an antibiotic are given intravenously q12h to a 65-kg patient with severe staphylococcal infection. The patient has normal renal and liver functions. The following pharmacokinetic constants of the drug are known: Apparent volume of distribution = 26.2 L/100 kg Total body clearance = 116.4 ml/min per 70 kg (1.73 m2 BSA) Renal clearance = 82.2 ml/min per 70 kg Determine the trough level after the frst dose.

æ 65 ö Vd = 26.2 ç ÷ = 17.03 L è 100 ø æ 65 ö ClT = 116.4 ç ÷ = 108.08 ml / min = 66.4 L / h è 70 ø æ 65 ö Clr = 82.2 ç ÷ = 76.33 ml / min = 4.58 L / h è 70 ø K=

6.4 = 0.3376 h -1 17.03

ke =

4.58 = 0.269 h -1 17.03

T1/2 =

0.693 = 1.84 h 0.376

( Cpmin )1 =

2000 mg -0.376 ´ 12 e = 1.29 mg / L 17.03 L

(

)

7.4.2 Estimate the steady-steady peak level.

( Cpmax )1 = 117.44 mg / L ( Cpmax )ss = 117.44 æç 1 - e -0.376 ´ 12 ÷ö = 118.74 mg / L 1

è ø Note: Because the drug has a short half-life and the dosing interval is long compared to the half-life of elimination, the maximum steady-state plasma concentration of the drug is approximately the same as the maximum plasma concentration of the frst dose (i.e.,( Cpmax )ss @ ( Cpmax )1 ). In other words, most of the dose eliminates during one dosing interval and the accumulation in the body is not signifcant. 7.4.3 Determine the plasma levels 6 and 12 hours after the last dose, assuming the steady-state levels are achieved.

Cp6 h = 118.7e -0.376´6 = 12.43 mg / L 1 Cp12 h = ( Cpmin )ss = 118.7 e -0.376´12 = 1.3 mg / L

7.4.4 What is the average amount of the drug in the body at steady-state?

( Aave )ss =

708

2g D = = 0.44 g Kt 0.376 ´ 12

ADDENDUM II – PART 7

7.4.5 Determine the total amount eliminated per dosing interval at steady-state.

( Ael )t

ss

= DM = 2g

7.4.6 What is the total amount eliminated as metabolite(s) per dosing interval at plateau? æ 6.4 - 4.58 ö =ç ÷ ´ 2g = 0.575g 6.4 è ø 7.4.7 How long does it take to reach 60% of the average steady-state plasma concentration?

(A ) ¥ m

tss

nt = -3.3T1/2 log ( 1 - f ss ) = -3.3 ( 1.84 h ) log 0.4 = 2.42 h 7.4.8 How long does it take to reach 75% of the average plateau level? nt = 2 ´ 1.84 = 3.68 h

709

Addendum II – Part 7 CASE 7.5 – MULTIPLE DOSING KINETICS Three patients receive 500 mg intravenous bolus injection of an antibiotic every six hours. Their half-life and initial plasma concentrations of the frst dose are found to be different, as reported in Table 7.5.1.

Table 7.5.1 Initial Plasma Concentrations and Half-lives of the Drug Estimated for Patients A, B and C Patient A B C

(

)

T1/2 ( h )

Initial Plasma Level Cp 0 mg/Lmcg/ml

4 6 8

40 50 70

7.5.1 Determine the trough level after the third dose. Dose = 500 mg, t = 6 h K A = 0.173 h -1 , K B = 0.1155 h -1 , KC = 0.0866 h -1 Patient A : æ 1 - e -3´0.173´6 ö -0.173´ e 3 6 = 20.94 mg / L -0.173´6 ÷ è 1- e ø

( Cpmin )3rd = 40 ç Patient B :

æ 1 - e -3´0.1155´6 -0.1155´6 è 1- e

( Cpmin )3rd = 50 ç

ö -0.11155´6 = 43.75 mg / L ÷e ø

Patient C : æ 1 - e -3´0.0877´6 -0.0866´6 è 1- e

ö -0.00866´6 = 81mg / L ÷e ø 7.5.2 Calculate the steady-state peak and trough levels.

( Cpmin )3rd = 70 ç

Patient A : 40

( Cpmax )ss = 1 - e -0.173´6

= 61.92 mg / L

( Cpmin )ss = 61.92e -0- .173´6 = 21.57 mg / L Patient B : 50

( Cpmax )ss = 1 - e -0.1155´6 ( Cpmin )ss =

= 100 mg / L

100 = 50 mg m /L 2

Patient C : 70

( Cpmax )ss = 1 - e -0.0866´6

= 172.84 mg / L

( Cpmin )ss = 172.884 e -0.0866´6 = 102.67 mg / L 710

ADDENDUM II – PART 7

7.5.3 Estimate the plasma levels four and eight hours after the last dose, assuming the steady-state levels are achieved. Patient A : Cpt¢=4 h =

( Cpmax )ss 2

=

61.92 = 30.96 mg / L 2

Cpt¢ = 8 h = 25%(Cpmax )ss = 0.25 ´ 61.92 = 15.48 mg / L Patient B : Cpt¢= 4 h = 100 e -0.1155´ 4 = 63.00 mg / L Cpt¢=8 h = 100 e -0.1155´8 = 39.70 mg / L Patient C : Cpt¢=4 h = 172.84 e -0.0866´ 4 = 122.24 mg / L Cpt¢=8 h = 172.84 / 2 = 86.42 mg / L 7.5.4 Calculate the average steady-state plasma concentration. Patient A : ( Cpave )ss =

40 = 38.54 mg / L 0.173 ´ 6

Patient B : ( Cpave )ss =

500 = 72.15 mg / L 0.1155 ´ 6

70 = 134.72 mg / L 0.0866 ´ 6 7.5.5 Recommend a reasonable dosing regimen (DL, DM, and τ) to achieve the steady-state fuctuations within a therapeutic range of 50–150 mg/L. Patient C : ( Cpave )ss =

Patient A: First Method: Set (Cpave )ss = 100 mg / L , t = T1/2 = 4 h (Vd = 500 / 40 = 12.5L ) DM = 100 mg / L ´ 0.173 h -1 ´ 4 h ´ 12.5 L = 865mg DL = 2 ´ 865 = 1730 mg

( Cpmax )ss = 1730 / 12.5 = 138.4 mg / L ( Cpmin )ss = 138.4 / 2 = 69.2mg / L Second Method: Use the therapeutic range as the limit of fuctuation (i.e., fuctuation = 150 – 50 = 100 mg/L). This method carries the risk of exceeding the peak level. DM = 100 ´ 12.50 = 1250 mg t=

2.303 150 log = 6.35 @ 6 h 0.173 50

DL =

1250 = 1935.47 @ 1900 mg 1 - e -00.173´6 1900

( Cpmax )ss = 12.50 = 152mg / L Third Method: Use a preselected fuctuation within the therapeutic range (e.g., set fuctuation = 120 – 60 = 60 mg/L).

711

ADDENDUM II – PART 7

DM = 60 ´ 12.50 = 750 mg DL = 1500 mg t = T1/2

( Cpmax )ss = 120 mg / L ( Cpmin )ss = 60 mg / L Patient B: First Method: Set ( Cpave )ss = 100 mg / L , t = 6 h (Vd = 500 / 50 = 10 L ) DM = 100 ´ 6 ´ 0.1155 ´ 10 = 693 mg @ 700 mg DL = 1400 mg

( Cpmax )ss = 140 mg / L ( Cpmin )ss = 70 mg / L Note: The target plasma concentration may also be set equal to a value different from the average steady-state concentration. For example: Set ( Cpmax )ss = 80 mg / L and t = 4 h DM = 80 mg / L ´ 4 h ´ 0.1155 h -1 ´ 10 L = 369.600 mg @ 370 mg DL =

370 = 1000 mg 1 - e -0.1155´4

( Cpmax )ss = 100 mg / L

and ( Cpmiin )ss = 100 e -0.1155´ 4 = 63 mg

Second Method: Use the therapeutic range as the limit of fuctuation. Fluctuation = 150 - 50 = 100 mg / L DM = 10 L ´ 100 mg / L = 1000 mg t=

2.303 150 log = 9.51 @ 8 h 0.1155 50 5

DL =

1000 = 1658 @ 1500 mg 1 - e -0.1155´8

( Cpmax )sss = 1500 mg /10 = 150 mg / L ( Cpmin )ss = 150 e -0.1155´8 = 59.54 mg / L @ 60 mg / L Third Method: Use a fuctuation of 60 and 120 mg/L. DM = 10 L (120 - 60 ) = 600 mg DL = 1200 mg t = 6 h

( Cpmax )ss = 120 mg / L ( Cpmin )sss = 60 mg / L Patient C: First Method: Set ( Cpave ) = 100 mg / L t = T1/2 = 8 h Vd =

500 = 7.14 L 70

DM = 100 mg / L ´ 0.0866 h -1 ´ 8 h ´ 7.14 L = 494.66 @ 500 mg DL = 1000 mg m /L ( Cpmax )ss = 1000 / 7.14 = 140 mg

( Cpmin )ss = 140 / 2 = 70 mg / L 712

ADDENDUM II – PART 7

Note: A different regimen with less fuctuation of plasma concentration would be: Set ( Cpave )ss = 80 mg / L and t = 6 h . mg @ 300 mg DM = 6 h ´ 80 mg / L ´ 0.0866 h -1 ´ 7.14 L = 296.80 DL =

300 = 740.30 mg @ 750 mg 1 - e -0.0866´6

( Cpmax )ss = 750 / 7.14 = 105.04 mg / L ( Cpmin )ss = 105.04 e -0.0866´6 = 62.68 mg / L Second Method: Use the therapeutic range as the limit of the fuctuation. DM = 7.14 L (150 - 50 ) = 714 mg @ 700 mg t=

2.303 150 log = 12.68 = 12 h 0.0866 50

DL =

700 = 1083.15 @ 1000 mg 1 - e -0.0866´12

( Cpmax )ss = 1000 / 7.14 = 1400.05 mg / L ( Cpmin )ss = 140e -0.0866´12 = 50 mg / L Third Method: Use the fuctuation of 60 and 120 mg/L. DM = 7.14 ´ 60 = 428.40 mg @ 400 mg DL = 800 mg t = T1.2 = 8 h m /L ( Cpmax )ss = 112.04 mg

( Cpmin )ss = 60.02mg / L

713

Addendum II – Part 7 CASE 7.6 – MULTIPLE DOSING KINETICS The following equations summarize the decline of plasma concentration of an antibiotic in patients with normal renal function (Group A) and in patients with renal impairment (Group B), following intravenous injection of 0.5 g of the drug. The patients received the dose every four hours for three consecutive days. The ftted line equations are developed by the composite curve ftting of plasma concentrations (mg/L) of each group after the frst dose collected at different time points reported in unit of (hr). Group A: log Cp ˜ 0.6 ° 0.1˛ t ˝ Group B: Cp ˜ 5 e °0.0693˛ t 7.6.1 Determine whether the steady-state levels were achieved after giving the last dose of the third day, and estimate the plasma peak and trough levels of the last dose of the treatment.

˜ T1/2 °A ˛ 0.693 / 0.23 ˛ 3 h ˜ T1/2 °B ˛ 0.693 / 0.0693 ˛ 10 h 72 h ˝ 7 ˜ T1/2 ° A&BB Thus, the steady state levels were achieved. GroupA : 4

˜ Cpmax °ss ˛ 1 ˝ e ˝0.23˙ 4

˛ 6.67 mg / L

˜ Cpmin °ss ˛ 6.67e ˝0.23˙44 ˛ 2.67 mg / L GroupB : 5

˜ Cpmax °ss ˛ 1 ˝ e ˝0.0693˙ 4

˛ 20.65mg / L

˜ Cpmin °ss ˛ 20.65 e ˝0..0693˙4 ˛ 15.65 mg / L 7.6.2 Calculate the plasma peak and trough levels of the last dose of the frst day. GroupA :

˜ Cpmax °24 h ˛ ˜ Cpmax °ss ˛ 6.67 mg / L ˜ Cpmin °24 h ˛ ˜ Cpmin °ss ˛ 2..67 mg / L GroupB : ˙ 1 ˝ e ˝60.0693 4 ˘ ˛ 16.73 mg g/L ˝0.06934  ˆ 1˝ e 

˜ Cpmax °24 h ˛ 5 ˇ

˜ Cpmin °24 h ˛ 16.73 e ˝0.0693 4 ˛ 12.68 mg / L

714

ADDENDUM II – PART 7

7.6.3 Estimate the average steady-state plasma level. GroupA : 4

˜ Cpave °ss ˛ 0.23 ˝ 4 ˛ 4.347 mg / L GroupB : 5

˜ Cpave °ss ˛ 0.0693 ˝ 4 ˛ 18.03 mg / L 7.6.4 What is the accumulation index of the multiple dosing for both patients? GroupA : R˜

1 ˜ 1.44 1 ° e °0.23˛ 4

R˜

R ˜ 4.13 1 ° e °0.0693˛ 4

GroupB :

715

Addendum II – Part 7 CASE 7.7 – MULTIPLE DOSING KINETICS The PK/TK of a lead compound was investigated after oral administration of a 50-mg tablet to a group of volunteers (N = 5), age range of 68–75 years and weight range of 70–92 kg, with normal renal function. The total body clearance of the compound is 1.27 L/h, and the analysis of all plasma concentration-time data provided the following two equations: ˜°° -0.1155 ( h -1 ) t ExtrapolatedLine : Cp = 2.182 ( mg / ml ) e ˜° Residual Line : log Cp - Cp = 0.339 - 0.6 t

(

)

(Note: Units of plasma concentrations and time were “µg/ml” and “hour,” respectively.) 7.7.1 Predict the steady-state peak and trough levels if the compound is administered on a multiple dosing regimen every six hours. K = 0.1155 h -1 T1/2 = 0.693 / 0.1155 h -1 = 6 h k a = 2.303 ´ 0.6 = 1.382 h -1

( T1/22 )k

a

= 0.693 /1.382 = 0.5 h

Vd = 1.27 / 0.1155 = 11L F=

2.182mg / L ´ 11L ´ (1.382 - 0.1155) = 0.44 50 mg ´ 1.382

( Tmax )ss =

é 1.382 1 - e -0.1155´6 ù 2.303 log ê = 1.41 h ´ -1.382´6 ú 1.382 - 0.1155 ë 0.1155 1 - e û

( Cpmin )ss =

2.182 mg / L e -0.1155´6 = 2.18 mg / L 1 - e -0.1155´6 æ 0.44 ´ 50 mg m ö e -0.1155´1.41 = 3.4 mg / L -0.1155´6 ÷÷ 11L è ø1- e

( Cpmax )ss = çç

716

Addendum II – Part 7 CASE 7.8 – MULTIPLE DOSING KINETICS A 60-kg hospitalized male patient with atrial fbrillation was given a cardiac glycoside by intravenous bolus injection every 24 hours for 10 consecutive days. Based on the analysis of plasma and urine data, the following pharmacokinetic parameters and constants were reported: Vdarea : 7.38 L / kg V1 : 1 L / kg ClT : 0.213 Lh -1kg -1

( Cpmin )ss : 1mg / L ( Cpmaax )ss : 2mg / L 7.8.1 Calculate the maximum amount of drug in the central compartment after the third dose. V1 = 60 L b=

0.213 Lh -1kg -1 = 0.0289 h -1 7.38 Lkg -1

f ss = 1 - e -3´0.0289´244 = 0.875

( Amax )n=3 = ( 2mg / L ) ( 0.875 ) ( 60 L ) = 105 mg 7.8.2 Calculate plasma concentration 48 hours after the last dose.

( T1/2 )b =

0.693 = 24 h 0.0289

. /L ( Cp )48 h = 25% ( Cpmax )ss = 50% ( Cpmin )ss = 0.5mg

717

Addendum II – Part 7 CASE 7.9 – MULTIPLE DOSING KINETICS A new therapeutic agent is evaluated in a preclinical trial by giving 600 mg intravenously to two volunteers, A and B. The frst dose was injected at 6 AM of the frst day and continued every six hours for four days. The half-life and clearance of the drug are reported in the following table.

Patient

Half-Life (h)

Total Body Clearance (ml/min)

12 3

28.875 19.25

A B

7.9.1 Determine the steady-state fuctuation of plasma levels for both patients. Patient A: K=

0.693 = 0.058 h -1 12 h

ClT =

28.875 ml / min ´ 60 min/ h = 1.7325L L/h 1000 ml / L

Vd =

1.7325 L / h = 29.87 L @ 30 L 0.058 h -1

Cp 0 =

600 mg = 20 mg / L 30 L

Fluctuation is created by giving the maintenance dose. Thus, at steady-state the magnitude of the fuctuation is equal to the initial plasma concentration of the maintenance dose, that is:

(

Fluctuation = Cp 0

)

DM

= 20 mg / L

Now, by using the calculated values of peak and trough levels, determine the magnitude of fuctuation at steady-state: 20

( Cpmax )ss = 1 - e -0.058´6

= 68 mg / L

mg / L ( Cpmin )ss = 68 mg / L e -0.058´6 = 48 m

( Fluctuation )ss = 68 - 48 = 20 mg / L Patient B: K=

0.693 = 0.231h -1 = 0.00385min -1 3h

Vd =

19.25ml / min = 5000 ml = 5L 0.00385min -1

Cp 0 =

600 mg = 120 mg / L = Fluctuation 5L

( Cpmax )ss =

120 mg / L = 160 mg / L 1 - e -0.231´6

( Cpmin )ss = 160 e -0.231´6 = 40 mg / L

718

ADDENDUM II – PART 7

7.9.2 If the therapeutic range is 30–90 mg/L, for which patient would you modify the dosing regimen? What would be your recommendation? Based on the calculated peak and trough levels (Part 1), patient B will need a dosage adjustment to maintain the fuctuation within the therapeutic range of 30–90 mg/L. Setting ( Cpave ) = 60 mg / L and t = 6 h , the maintenance dose can be estimated as: ss

DM = 60 mg / L ´ 0.231h -1 ´ 5L ´ 3 h = 207.90 mg For practical purposesset DM = 200 mg Since t = T1/2 DL = 400 mg

( Cpmax )ss =

400 mg = 80 mg / L 5L

( Cpmin )sss =

80 mg / L = 40 mg / L 2

719

Addendum II – Part 8 CASE 8.1 – TEST YOUR KNOWLEDGE 1 Answer the following short questions without the use of a calculator. Assume a one-compartment open model with intravenous bolus input. 8.1.1 The initial rate of elimination of a drug (i.e., at t = 0) after an intravenous injection is 69.30 mg/h and its half-life is one hour. What is the dose? 8.1.2 The duration of action of a drug after intravenous injections of 500 mg and 4000 mg is 4 hours and 10 hours, respectively. What is the half-life of the drug? 8.1.3 The total amount of a drug eliminated in 43 hours after an intravenous dose of 500 mg is 475 mg. What is the half-life of the drug? Answers: 8.1.1: 100 mg 8.1.2: T1/2 = 3 h 8.1.3: T1/2 = 10 h

720

Addendum II – Part 8 CASE 8.2 – TEST YOUR KNOWLEDGE 2 Answer the following short questions. Assume one-compartment open model with intravenous bolus input. 8.2.1 A xenobiotic with a mean residence time of 1.443 hours is given intravenously. If 750 mg of the injected dose is eliminated in 2 hours, what would be the dose? 8.2.2 If the duration of action of a xenobiotic after intravenous injections of 200 mg and 1.6 g is 2 hours and 17 hours, respectively, what would be the half-life? 8.2.3 If the area under the plasma concentration-time curve of a xenobiotic is 72.151 mgh/L, the apparent volume of distribution is 10 L, and the half-life is 1 hour, what would be the total amount in the body 4.3 hours after the injection? 8.2.4 If 75% of an intravenous dose of a xenobiotic is eliminated in 12 hours, what would be the mean residence time? 8.2.5 If the rate of elimination 2 hours after the injection of a compound is 25 mg/h and the plasma concentration at that time is 10 mg/L, what would be the total body clearance in ml/ min? Answers : 8.2.1 : Dose = 1g 8.2.2 : T1/2 = 5 h 8.2.3 : ( Abody )4.3 h = 25mg 8.2. 2 4 : MRT = 8.64 h 8.2.5 : ClT = 41.67 ml / min

721

Addendum II – Part 8 CASE 8.3 – TEST YOUR KNOWLEDGE 3 The following is the PK/TK model equation of a new drug after oral administration of 250 mg to a group of six volunteers. The absolute bioavailability of the drug is reported at 0.85. -1.7325 ( h -1 ) t ö æ -0.0693 ( h-1 ) t Cp = 10mg / ml ç e -e ÷ è ø

8.3.1 What is the area under the plasma concentration-time curve? 8.3.2 What is the total body clearance of the drug in ml/min? 8.3.3 How long does it take for 75% of FD to be absorbed? 8.3.4 What is the apparent volume of distribution of the drug? 8.3.5 This new drug is pharmaceutically equivalent to an established, marketed drug product. Estimate the relative bioavailability of the new drug if the area under the curve of the marketed drug product is 300 mgh/L following oral administration of a 500-mg tablet. Assume the clearances are the same. Answers : 8.3.1 : 138.50 mgh / L 8.3.2 : 25.50 ml / min 8.3.3 : 48 min 8.3.4 : 22.14 L 8.3.5 : 1.083

722

Addendum II – Part 8 CASE 8.4 – TEST YOUR KNOWLEDGE 4 After an intravenous injection of 500 mg of a xenobiotic and analysis of plasma levels at different intervals in 12 subjects, the following two-compartment model equation was reported: Cp = 15 ( mg / ml ) e

( ) + 10 mg / ml e -0.231( h-1 )´t ( )

-3.465 h -1 ´t

8.4.1 What is the apparent volume of distribution of the central compartment? 8.4.2 What is the total amount of xenobiotic in the peripheral compartment at t = 0? 8.4.3 Estimate the total amount of the administered dose in the central compartment three hours after the injection. 8.4.4 Calculate the apparent volume of distribution of both compartments, Vdarea . Answers : 8.4.1 : V1 = 20L 8.4.2 : ( A2 )t=0 = 0 8.4.3 : ( T1/2 )b = 3 h; 3 h ˜ 7 ( T1/2 / ) a \( A ) 3 h = 100 mg 8.4.4 : Vdarea = 45.45L

723

Addendum II – Part 8 CASE 8.5 – TEST YOUR KNOWLEDGE 5 A 70 kg male patient received 500 mg of an antibiotic by bolus injection every six hours. The drug is known to follow the two-compartment model. The plasma concentration of the free drug after the frst injection was determined at different times, as reported in Table 8.5.1.

Table 8.5.1 Free Concentration of the Antibiotic Measured at Different Time Points Time (h) Cp (mg/L)

0.25 53.75

0.50 40.00

1.00 25.00

1.50 17.50

2.00 12.25

4.00 8.25

8.00 3.50

12.00 1.5

8.5.1 Using the slope relationship, determine the hybrid rate constants of α and β. 8.5.2 Estimate the biological and elimination half-lives; discuss the differences. 8.5.3 What is the apparent volume of distribution of the central compartment? 8.5.4 What is the estimate of total body clearance in ml/min? 8.5.5 Determine the rate constants of distribution between the central and peripheral compartments. Answers : 8.5.1 : a @ 1.7 h -1 , b @ 0.202 h -1 8.5.2 : ( T1/2 )biol @ 3.16 h, ( T1/2 / )elim @ 1.1 h 8.5.3 : V1 @ 6.7 L 8.5.4 : ClT @ 70.57 ml / min 8.5.5 : k12 @ 0.727 h -1 , k 21 @ 0.543 h -1

724

14.00 1.00

Addendum II – Part 8 CASE 8.6 – TEST YOUR KNOWLEDGE 6 The following equation represents the summary of the amount remaining to be excreted unchanged in the urine of six subjects after receiving a dose of xenobiotic intravenously. The equation is developed based on the amount of the xenobiotic measured in urine in mg at different time points in unit of (hr). log ARE = 2 - 0.1t Using the parameter and constants of the above equation, answer the following short questions. 8.6.1 If the dose was 500 mg, what would be the total amount of the xenobiotic eliminated as metabolite(s)? 8.6.2 If the dose was 100 mg, what would be the fraction of the dose excreted unchanged in the urine? 8.6.3 If the dose was 500 mg, what would be the excretion rate constant? 8.6.4 If 40% of the dose is excreted unchanged, what would be the dose? 8.6.5 If 30% of the dose is eliminated as metabolite(s) and the volume of distribution is 10 liters, what would be the renal clearance? Answers : 8.6.1 : Am¥ = 400 mg 8.6.2 : f e = 1 8.6.3 : k e = 0.046 h -1 8.6.4 : Dose s = 250 mg 8.6.5 : Clr = 1.612 L / h = 26.87ml / min

725

Addendum II – Part 8 CASE 8.7 – TEST YOUR KNOWLEDGE 7 A 60-kg male patient with chronic obstructive pulmonary disease is to be started on an intravenous multiple-dosing regimen of a systemic bronchodilator (a xanthine derivative). Assume the drug follows the one-compartment model, with the following information: Apparent volume of distribution = 1 L/kg Total body clearance = 57.75 ml h−1/kg Therapeutic range = 10–30 mg/L 8.7.1 Design a reasonable dosing regimen for this patient. (i.e., DL, DM, with a fuctuation within the therapeutic range) if the target plasma level is 20 mg/L and the dosing interval is equal to the half-life of the drug. Answers; 8.7.1 : DM @ 831.60 mg ( » 800 mg ) DL @ 1663.2 ( » 1.6 g ) t = 12 h

( Cpmaxx )ss = 27.72 mg / L ( Cpmin )ss = 13.86 mg / L

726

Addendum II – Part 8 CASE 8.8 – TEST YOUR KNOWLEDGE 8 A 50-kg hospitalized male geriatric patient receives 1 gram of an antibiotic intravenously every 8 hours for four consecutive days. The drug follows the one-compartment model, with an apparent volume of distribution of 0.5 L/kg, half-life of six hours, and a therapeutic range of 15–70 mg/L. 8.8.1 Determine peak and trough levels of the frst dose of the fourth day. 8.8.2 What would be the accumulation index of this regimen? 8.8.3 How many milligrams of the drug are eliminated during the last dosing interval? 8.8.4 What is the plasma concentration 12 hours after the last dose? 8.8.5 How long did it take to reach 60% of average steady-state concentration? Answers : 8.8.1 : ( Cpmax )ss @ 66.33 mg / L

( Cpmin )ss @ 26.33 mg / L 8.8.2 : R = 1.66 8.8.3 : ( Ael )t = 1000 mg 8.8.4 : Cp @ 16.58 mg / L 8.8.5 : nt = 7.92 h

727

Addendum II – Part 8 CASE 8.9 – TEST YOUR KNOWLEDGE 9 A hydrophilic antibiotic is given to an adult male patient (65 years old, 80 kg) by intravenous infusion. The elimination half-life is four hours, and the apparent volume of distribution is 2 L/kg. The drug is supplied in a 60-ml vial at a concentration of 10 mg/ml. One vial was added to 940 ml D/W 5% and infused over four hours. 8.9.1 What would be the plasma concentration of the drug at the end of infusion? 8.9.2 What rate in milliliters per hour would you infuse the solution to obtain the concentration calculated in Question 8.9.1? 8.9.3 What would be the plasma concentration of drug four hours after stopping the infusion? 8.9.4 What is the total amount eliminated from the body during the infusion? 8.9.5 Recommend a loading dose (bolus) and maintenance dose (infusion) for this patient if you decided to achieve and maintain a steady-state plasma level of 10 mcg/ml. 8.9.6 What would be the rate of elimination in Question 8.9.5? Answers : 8.9.1 : Cpend = 50%Cpss = 2.705 mg / L 8.9.2 : k0 = 250 ml / h 8.9.3 : Cp4 h = 1.35mg / L 8.9.5 : DL = 1600 mg k0 = 277.20 mg / h 8.9.6 : KA = k0 = 277..20 mg / h

728

Addendum II – Part 9 CASE 9.1 – DOSAGE ADJUSTMENT FOR PATIENTS WITH RENAL IMPAIRMENT A 42-year-old woman weighing 70 kg is to be treated with vancomycin for severe staphylococcus infection, which has failed to respond to penicillins and cephalosporines. The usual IV injection dose for patients with normal renal function is 500 mg every six hours. However, this patient has impaired renal function, as indicated by a serum creatinine level of 70 µg/ml. The normal and metabolic half-lives of vancomycin are six hours and 9.6 days, respectively. 9.1.1 Determine the overall elimination rate constant of vancomycin for this patient. K= km =

0.693 = 0.1155 h -1 6 0.693 = 0.003 h -1 9.6 d ´ 24 h / d

æ 0.003 h -1 ö f e = 1 - çç ÷ = 0.97 -1 ÷ è 0.1155 h ø æ (140 - 42 ) 70 ö Ccr = çç ÷÷ 0.85 = 11.57 ml / min 72 ´ 7 è ø æ æ æ 11.57 ml / min ö ö ö -1 K = 0.1155 ç 1 - ç 0.97 çç 1 ÷÷ ÷÷ ÷ = 0.0142 h ÷ ç ç 120 ml / min è ø øø è è 9.1.2 Using the Accumulation Ratio Method, recommend an appropriate dose for this patient if vancomycin is to be given once a day.

( Aave )ss = T1/2 = D=

1.44 ´ 6 h ´ 500 mg = 720 mg 6

0.693 @ 48 h 0.0142

720 ´ 24 = 250 mg 1..44 ´ 48

729

Addendum II – Part 9 CASE 9.2 – DOSAGE ADJUSTMENT FOR PATIENTS WITH RENAL IMPAIRMENT A 22-year-old, 60-kg woman with impaired renal function is being treated for tuberculosis with a combined therapy of streptomycin and isoniazid. The serum creatinine is 8 mg/dL. In patients with normal renal function, the intravenous dose of streptomycin is 2 g b.i.d., and its overall elimination rate constant is 0.27 h−1. The overall elimination rate constant for patients in the anuric state (end-stage renal function) is 0.01 h−1. 9.2.1 What is the overall elimination rate constant of streptomycin for this patient? Ccr =

(140 - 22 ) ´ 60 ´ 0.85 = 10.45 ml / min 72 ´ 8

æ 0.27 h -1 - 0.01h h -1 ö -1 K = 0.01h -1 + çç ÷÷ 10.45ml / min = 0.033 h è 120 ml / min ø 9.2.2 Calculate a new dosing regimen for streptomycin by keeping dosing interval the same and changing the dose. (3 points) D=

730

2g ´ 0.033 h -1 = 0.244 g 0.27 h -1

Addendum II – Part 9 CASE 9.3 – DOSAGE ADJUSTMENT FOR PATIENTS WITH RENAL IMPAIRMENT A 42-year-old man weighing 70 kg is to be treated with an antibiotic for a serious infection. The patient has a serum creatinine of 3.0 mg/100 ml. 9.3.1 If the antibiotic has an overall elimination rate constant of K = 0.3 h -1 in patients with normal renal function, and its metabolic rate constant is k m = 0.02 h -1 , determine its half-life of elimination in this patient. 9.3.1.1 Use the Cockcroft-Gault equation to estimate the creatinine clearance of this patient. Ccr =

(140 - 42)(70) = 31.76 ml / min 72 ´ 3

9.3.1.2Use the Dettli equation to calculate the overall elimination rate constant of the drug in this patient. K = aCcr + k m Slope = a =

( 0.3 - 0.02 ) = 0.0028 100

K = 0.02 + 0.0028 Ccr = 0.022 + 0.0028 (31.76 ml / min) = 0.109 h -1 T1/2 = 6.36 h

731

Addendum II – Part 9 CASE 9.4 – DOSAGE ADJUSTMENT FOR PATIENTS WITH RENAL IMPAIRMENT A 50-year-old woman weighing 70 kg (IBW) is to be treated with an antiemetic medication. The patient has impaired renal function, as indicated by a serum creatinine level of 4 mg/100 ml. 9.4.1 Estimate the creatinine clearance by using the Cockcroft and Gault equation. Ccr = 0.85 ´

(140 - 50 ) 70 = 18.6 ml / min 72 ´ 4 mg / dL

9.4.2 If the normal and anuric elimination rate constants for this drug are 1.4 h-1 and 0.06 h-1, respectively, calculate the overall elimination rate constant for the patient. K = km + aCcr ö æ æ 1.4 - 0.06 ö K = ç çç ´ 18.6 ÷ + 0.06 = 0.2667 h -1 ÷ ç 120 ml / min ÷÷ ø ø èè 0.693 = 2.6 h 0.267 9.4.3 If the normal dosing regimen is 1 g every six hours for patients with normal renal function, recommend a new dosage regimen for this patient by giving the same dose and changing the dosing interval. T1/2 =

æKö t = t = çç ÷÷ = 6 ´ (1.4 / 0.267 ) = 31.46 h èKø 9.4.4 Estimate the dose if the drug is to be given twice a day. D=

732

DK t 1g ´ 0.267 h -1 ´ 12 = = 0.381 g = 381mg Kt 1.4 h -1 ´ 6 h

Addendum II – Part 9 CASE 9.5 – DOSAGE ADJUSTMENT FOR PATIENTS WITH RENAL IMPAIRMENT For patients with normal renal function who suffer from urinary tract infections, an intravenous dose of 15–30 mg/kg body weight of a cephalosporine is usually given every 6–8 hours. For patients with chronic renal insuffciency, dosage depends not only on the type of infection but also on the degree of renal impairment. Generally, the elimination half-life increases with decreasing renal function, ranging from 48 min for patients with normal renal function to 22 h for patients with chronic end-stage renal function. Adjust the dosage regimen of 15 mg/kg q6h for the following patients: Patient A B C

Sex

Weight (kg)

Ccr(ml/min)

F M F

60 75 55

60 20 0

9.5.1 The following calculations are for patient A, using Giusti’s equation for estimation of the overall elimination rate constant. æ æ Ccr ö ö K = 1 - ç f e çç 1 ÷÷ ç Ccr ÷ø ÷ø K è è 0.693 = 0.866 h -1 0.8

K= km =

0.693 = 0.0315 h -1 2 h 22

fe =

0.866 - 0.0315 = 0.963 0.866

(

)

K = 1 - 0.963 (1 - ( 60 /100 ) ) = 0.615 K K = 0.532 h -1 D = ( 15mg / kg ) ´ ( 0.615 ) = 9.22mg / kg q6 h = 553 3 mg q6 h 9.5.2 The following calculations are for Patient B, using Dettli’s equation for K and Accumulation Ratio Method for D. æ 0.866 - 0.0315 ö -1 K = 0.0315 + ç ÷ ´ 20 = 0.198 h 100 è ø T1/2 = 0.693 / 0.198 = 3.5 h

( Aave )ss = ( 2.88 ) ´ ( 75 ) = 216 mg D = ( 216 mg ´ 6 ) / (1.44 ´ 3.5 ) = 257..14 mg q6 h 9.5.3 Calculations of K andD for Patient C. K = km = 0.315 h -1 T1/2 = 22 h D = 15mg / kg ( 0.0315 / 0.866 ) = ( 0.545 mg / kg ) ´ ( 55 5 kg ) = 30 mg q6 h 733

Index Note: Locators in italics represent fgures and bold indicate tables in the text. AB, see Algorithm Builder Abbreviated new drug applications (ANDAs), 545 ABC transporters, see ATP binding cassette transporters Absolute bioavailability; see also Bioavailability; Fraction of dose absorbed defnition, 547 estimation of from amount eliminated from the body, 548–549 from plasma data, 548 Absorption, distribution, metabolism, and excretion (ADME), 1, 2, 3, 10, 368, 461, 588 auricular ADME, 20–22 buccal and sublingual, 25–27 of intramuscular route of administration, 108–111 intravitreal ADME and rate equations, 35–37 nasal ADME of xenobiotics, 53–54 of xenobiotics in pulmonary tract ATP Binding Cassette (ABC) transporters, 61 organic anion transporters, 61 organic cation transporters, 60–61 peptide transporters, 61 pulmonary absorption, deposition, and clearance, 60 pulmonary absorption of gases and vapors, 63–66 pulmonary deposition and disposition of particles, 62–63 pulmonary frst-pass metabolism, 67 pulmonary rate equations, 67–72 relevant pulmonary kinetic parameters, 66–67 respiratory tract metabolic enzymes, 61–62 transport proteins of pulmonary tract, 60–61 Absorption mechanisms and rate equations, PK/TK considerations of, 177 active transport, 194–196 carrier-mediated transcellular diffusion, 188 endocytosis and pinocytosis, 196 ion-pair absorption, 198–200 passive diffusion diffusion coeffcient, 184–185 distribution coeffcient, 183–184 partition coeffcient, 180–183 permeation and permeability constant, 185–188 transcellular and paracellular diffusion, 177–180 P-glycoprotein (Pgp), 189 computational equations, 192–193 structure and function, 189–192 solvent drag, osmosis, and two-pore theory, 196–198 Absorption number, 525 Absorption rate constant, estimation of, 466–473 ACAT model, see Advanced Compartmental Absorption Transit model

Accumulation index, 510–511 Acetylation (acylation), 252–253 Acetyl-coenzyme A (acetyl-CoA), 252 Active transport, 194–196; see also Xenobiotics Acute toxicity studies, 588 ADAM, see Advanced dissolution, absorption, and transit model Adenosine-5′-phosphosulfate (APS), 250 Adenosine deaminase complexing protein 2, 248 Adenosine triphosphate (ATP) Binding Cassette (ABC) transporters, 61 ADH, see Alcohol dehydrogenase ADME, see Absorption, distribution, metabolism, and excretion Advanced Compartmental Absorption Transit (ACAT) model, 535, 536 Advanced dissolution, absorption, and transit model (ADAM), 535–536, 537 Afibercept, 35 AIC, see Akaike information criterion Airways, morphological differences of, 59 Akaike information criterion (AIC), 379, 444, 630 Alanine aminotransferase (ALT), 589 Albumin, 216–217 Alcohol dehydrogenase (ADH), 138, 245–246 Aldehyde dehydrogenases (ALDHs), 138, 246–247 ALDHs, see Aldehyde dehydrogenases Algorithm Builder (AB), 181 Allometric approach, 561 and chronological time, 564–565 in converting animal dose to human dose, 565 ALT, see Alanine aminotransferase Amidases, 138 Amikacin, 35 Amino acid conjugation, 254–255 Amphotericin, 35 ANDAs, see Abbreviated new drug applications Antibodies against CYP proteins, 260 Antiport, 195 Anti-vascular endothelial growth factor (anti-VEGF) proteins, 34 Apical sodium-dependent bile acid transporter (ASBT), 81, 195–196 Apocrine glands, 136 Apolysichron, 565 APS, see Adenosine-5′-phosphosulfate Aqueous-air partition coeffcient, 142 ARE Plot, 323–324, 324, 325 Armitage-doll multi-stage model, 570 ASBT, see Apical sodium-dependent bile acid transporter Ascites assessment, 603 Aspartate aminotransferase (AST), 589 Assumptions of PK/TK, 10–12 AST, see Aspartate aminotransferase ATP binding cassette (ABC) transporters, 86, 139 Atrium, 50 Auricular (otic) route of administration, 19

735

INDEX

auricular rate equations and PK/TK models, 22–24 blood-labyrinth-barrier (BLB) and auricular ADME, 20 auricular distribution, metabolism, and excretion, 21–22 syndromes and sites of absorption, 20–21 Axial dispersion number, 274 BA, see Bioavailability BAB, see Blood aqueous barrier Baker–Lonsdale model, 524 Bayesian approach, 579 Bayesian information criteria (BIC), 379, 444 Bayes’ Rule, 579 BCRP, see Breast cancer resistance protein BCS, see Biopharmaceutics Classifcation System BDDCS, see Biopharmaceutics Drug Disposition Classifcation System bDNA probes, see Branched DNA (bDNA) probes BE, see Bioequivalence Bevacizumab, 35 BIC, see Bayesian information criteria Bile salts role, 81 Bilirubin measurement, 603 Bioavailability (BA), 545 defnition, 546 and frst-pass metabolism, 549–550 Bioequivalence (BE), 545 defnition, 546 evaluation, 551 required PD/TD data, 553 required PK/TK parameters, 552–553 statistical analysis of PK/TK data, 553 Biological products, see Biopharmaceutical medicines Biologic license applications (BLAs), 545 Biomarkers, 7–8 Biopharmaceutics, 515, 553 absorption number, 525 absorption of particles, 517 biowaivers, 526–527 defnition, 515 dissolution, 520 in vitro–in vivo correlation (IVIVC), 524–525 mathematical models, 520–524 dissolution number, 525–526 dose number, 526 formulation factors, 518 compressed tablets, 519 dosage form tactics for poorly soluble compounds, 519–520 emulsions, 519 soft and hard gelatin capsules, 519 solutions and syrups, 518 suspensions, 518–519 mechanistic absorption models, 529 absorption potential models, 530 Advanced Compartmental Absorption Transit (ACAT) model, 535, 536 advanced dissolution, absorption, and transit model (ADAM), 535–536, 537 Compartmental Absorption and Transit (CAT) model, 531–532

736

dispersion models, 530–531 gastrointestinal transit absorption (GITA) model, 532–535, 533, 534 Grass model, 536 particle size infuence on solubility/dissolution, 517–518 partition coeffcient, 516 rule of fve, 516–517 polymorphism, 515–516 wettability and porosity infuence on dissolution profle, 518 xenobiotics, factors infuencing absorption of chirality and enantiomers, 527–528 disease states, effects of, 529 drug administration scheduling, infuence of, 529 effects of food and drink on absorption of xenobiotics, 528–529 effects of release mechanisms from solid dosage forms, 529 genetic polymorphism, infuence of, 529 Biopharmaceutics Classifcation System (BCS), 526 compounds with high permeability and high solubility, 526 drugs with high permeability and low solubility, 526 drugs with low permeability and high solubility, 526 drugs with low permeability and low solubility, 526 Biopharmaceutics Drug Disposition Classifcation System (BDDCS), 527 Biosimilarity, 553–555 Biotransformation, see Xenobiotics biotransformation, PK/TK considerations of Biowaivers, 526–527 BLAs, see Biologic license applications BLB, see Blood-labyrinth-barrier Blood aqueous barrier (BAB), 29–30, 229 Blood–brain barrier, 222–227 Blood-cochlea-barrier, see Blood-labyrinth-barrier (BLB) Blood fow, 212, 212, 213; see also Xenobiotics distribution Blood-labyrinth-barrier (BLB) and auricular ADME, 20 auricular distribution, metabolism, and excretion, 21–22 syndromes and sites of absorption, 20–21 Blood–lymph barrier, 227 Blood-placenta barrier, 227–228 Blood-retinal barrier (BRB), 30 effux transporters, 31 breast cancer resistance protein (BCRP), 31 multidrug resistance-associated proteins (MRPs), 31 P-glycoprotein (Pgp), 31 infux transporters, 31 cationic amino acid transporter 1 (CAT1), 31 creatine transporter (CRT), 31 folate transporters, 31 glucose transporter 1 (GLUT1), 31 L-type amino acid transporter 1 (LAT1), 31

INDEX

monocarboxylate transporters (MCTs), 31 nucleoside transporters, 31 organic anionic-transporting polypeptides, 31 organic cation transporters (OCTNs), 31 taurine transporter (TAUT), 31 kinetics of BRB infux permeability clearance, 32 Blood–testis barrier, 228–229 BMEC monolayers, see Brain micro-vessel endothelial cell monolayers Body mass index (BMI), 230 Body surface area (BSA), 229–230 Body-weight dependent extrapolation of clearance in humans, 361 Bolus injection, 120 Bowman’s layer, 37 Brain capillary endothelial cells, 223 Brain micro-vessel endothelial cell (BMEC) monolayers, 188 Branched DNA (bDNA) probes, 260 BRB, see Blood-retinal barrier Breast cancer resistance protein (BCRP), 31, 224, 529 Brolucizumab, 35 BSA, see Body surface area Buccal and sublingual routes of administration, 24 and related rate equations, 25–27 saliva, 27–28 Caco-2 cells, 178, 187–188 Calcium pumps, 195 Capillary endothelium, 228 Carboxylesterases (CES), 247 Carcinogenic agents/elements, 135 Carrier-mediated transcellular diffusion, see Facilitated diffusion CAT1, see Cationic amino acid transporter 1 Catalytic effciency and turnover number, 267 Catechol O-methyltransferase (COMT), 252 Cationic amino acid transporter 1 (CAT1), 31 Cationic drugs, 217 CAT model, see Compartmental Absorption and Transit model CD26, 248 Ceftazidime, 35 Cell-drinking, 196 Cell lines, 260–261 Cerebrospinal fuid (CSF), 56, 224 CES, see Carboxylesterases CF, see Cystic fbrosis C-glucuronide conjugation, 250 CHF, see Congestive heart failure Child-Turcotte-Pugh (CTP) score, 603 ascites assessment, 603 hepatic encephalopathy, 603 International Normalized Ratio (INR), 603 serum albumin level, 603 total bilirubin measurement, 603 Chromatin, 6 Chronic kidney disease (CKD), 340 Chronic renal failure (CRF), 230–231 Chronic toxicity studies, 588 Cimetidine, 314 CKD, see Chronic kidney disease Clearances, 355

estimation using theoretical models, 356 dispersion model, 358–359 parallel model, 358 well-stirred model, 356–358 linear PK/TK, clearance estimation in, 362–363 mammalian species, clearance scale-up in body-weight dependent extrapolation of clearance in humans, 361 extrapolation of clearance from animal to human, 359–361 nonlinear PK/TK, clearance estimation in, 363 target-mediated drug disposition (TMDD), 364 Clindamycin, 35 Clinical Pharmacokinetics/Pharmacodynamics (CPK/PD), 3 Clinical trials, 590 Phase I-a clinical trial, 590 Phase I-b clinical trial, 590–591 Phase II-a clinical trial, 591 Phase II-b clinical trial, 591 Phase III clinical trial, 591 Phase IV clinical trial, 591 Cochlea, 19 Collecting ducts, 311 Compartmental Absorption and Transit (CAT) model, 531–532 Compartmental analysis, see Extravascular routes oral administration, frst-order absorption via; Linear PK/TK compartmental analysis; Multiple dosing kinetics; Nonhyperbolic sigmoidal model; Nonlinear hyperbolic concept link to compartmental models; Xenobiotics intravenous infusion, continuous zero-order exposure to; Xenobiotics single intravenous bolus injection Competitive inhibition, 278 COMT, see Catechol O-methyltransferase Concepts of PK/TK, 10–12 Congestive heart failure (CHF), 230 Conjugation, 248 acetylation (acylation), 252–253 amino acid conjugation, 254–255 glucuronidation, 248 C-glucuronide conjugation, 250 N-glucuronide conjugation, 249–250 O-glucuronide conjugation, 249 S-glucuronide conjugation, 250 glutathione (GSH) conjugation, 253–254 methylation, 251–252 sulfation, 250–251 Conjunctival route of administration, 33–34 Continuous infusion, 120 Cornea, 37, 39 Corneal endothelium, 37 Corneal epithelium, 37 Corneocytes, 133 Co-transport process, 195 CPK/PD, see Clinical Pharmacokinetics/ Pharmacodynamics Cramer’s rule, 623–625 C-reactive protein (CRP), 8 Creatine transporter (CRT), 31

737

INDEX

Creatinine biosynthesis in mammalian systems, 314, 314 Creatinine clearance, 313 estimation by direct measurement of 24-h urine sample, 315 estimation from serum creatinine, 315–317 Agarwal and Nicar equation based on gender, age and weight, 316 Cockcroft and Gault equation based on gender, age and weight, 315 Hull et al equation based on gender, age and weight, 316 Jelliffe equation based on gender and age, 315 Mawer et al equation based on gender, age and weight, 315 MDRD and CKD-EPI equations based on age gender and race, 317 Salazar and Corcoran equation based on weight, age, gender, and height for obese patients, 316 Schwartz et al equation for pediatric patients based on height, age, body surface area, 316–317 Shull et al for pediatric patients based on age and body surface area, 317 Toto and Kirk equation based on gender, age and weight, 316 CRF, see Chronic renal failure Cross-species extrapolation, 561 allometric approach, 561 and chronological time, 564–565 in converting animal dose to human dose, 565 application of PBPK/PBTK in, 566 toxicogenomics, 566–567 interspecies scaling in mammals, 561 CRP, see C-reactive protein CRT, see Creatine transporter CSF, see Cerebrospinal fuid CTP score, see Child-Turcotte-Pugh score Cyclooxygenase, 138 CYP450, see Cytochrome P450 CysC, see Cystatin C Cystatin C (CysC), 317–318 Cystic fbrosis (CF), 231 Cytochrome P450 (CYP450), 84–85, 228, 241 CYP1A, 241–242 CYP1A1, 108 CYP1A2, 108 CYP1B, 242 CYP1B1, 108 CYP1 gene family, 338 CYP2, 108 CYP2A, 242 CYP2B, 242 CYP2C, 85, 243 CYP2C9, 85, 108 CYP2C19, 85 CYP2D, 243 CYP2E, 243–244 CYP2E1, 108 CYP2 gene family, 338 CYP3A, 244–245 CYP3A4, 84, 85, 87, 88, 108, 529

738

CYP3A7, 85 CYP3 gene family, 338 CYP4A, 245 CYP4A1, 245 CYP4A11, 245 CYP4A22, 245 CYP4B, 245 CYP4F, 245 CYP4 gene family, 338 CYP450, 22, 84–86, 108, 138, 240, 338, 588 CYP isozyme inhibitors and inducers, 257 CYP isozymes activity, 257 Cytosolic fraction, 256 DDD, see Drug discovery and development Dermis, 135 Dermis appendages, 135 hair follicle, 135–136 sebaceous glands, 136 sweat glands, 136 Dermis cells, 135 DERS, see Drug error reduction software Descemet’s membrane, 37 Detection limitation, 648 Determinants and Cramer’s rule, 623–625 Dexamethasone, 35 Dialysate composition, 341 Dialysis, 340; see also Glomerular fltration rate; Urinary data analysis dialysate, composition of, 341 effects on PK/TK parameters and constants, 343–346 hemodialysis, 341 peritoneal dialysis, 341 Dialysis clearance, 342–343 Diamine oxidase (histaminase), 246 Diffusion coeffcient, 184–185 Diffusion models, 140–141 Direct linear plot, 269, 271, 272 Dispersion model, 274, 358–359 Disposition functions, 626 Dissociation constant, 218 Dissolution, 520 mathematical models of, 520 Baker–Lonsdale model, 524 frst-order kinetics model, 521–522 Higuchi “square root of time plot” model, 522–523 Hixson–Crowell “cube root” model, 521 Hopfendberg model, 524 Kitazawa model, 522 Korsmeyer–Peppas model, 523 Nernst–Brunner model, 523–524 Noyes–Whitney model, 520 Weibull–Langenbucher model, 523 in vitro–in vivo correlation (IVIVC) of dissolution data, 524 level A correlation, 524 level B correlation, 525 level C correlation, 525 multiple-level C correlation, 525 Dissolution number, 525–526 Distal tubule, 311–312

INDEX

Distribution coeffcient, 183–184 Distribution mechanisms and rate equations, PK–TK considerations of, 210 applications and case studies, 231 xenobiotics distribution, 210 binding to plasma proteins, 216–220 blood–aqueous humor barrier (BAB), 229 blood–brain barrier, 222–227 blood fow and organ/tissue perfusion, 211–216 blood–lymph barrier, 227 blood–testis barrier, 228–229 body weight and composition, 229–230 disease states impact, 230–231 infuence of extent of penetration, 221–222 infuence of physicochemical characteristics of, 221 infuence of total body water, 210–211 physiological barriers, 222–229 placental barrier, 227–228 Döderlein’s bacillus, 169 Dose number, 526 Dosing regimen, 501 based on a target concentration, 501–502 based on minimum steady-state plasma concentration, 502 based on steady-state peak and trough levels, 502 Double reciprocal plot, see Lineweaver–Burk plot Drug discovery and development (DDD), 586 Drug error reduction software (DERS), 120 Drug metabolism (case), 638–640 Dua’s layer, 37 Eadie–Hofstee plot, 269, 270, 390, 639 Ear, 19, 20, 34 Eardrum, 19 Early exposure, 547 Eccrine glands, 136 E-cigarettes, 65–66 Effect compartment, 407, 408 one-compartment model with frst-order input connected to, 410 one-compartment model with intravenous bolus dose connected to, 408–410 one-compartment model with multiple IV bolus dosing connected to, 410–411 one-compartment model with zero-order input connected to, 410 two-compartment model with IV bolus connected to, 411 Effective half-life (EHL), 512 Effects of dialysis on PK/TK parameters and constants, 343–346 eGFR, see Estimated GFR (eGFR) EHL, see Effective half-life Elimination rate, 353 Endocytosis, 196 Enterocyte, 77 Enzyme, xenobiotic effect, 648 Enzyme activity inhibition by metabolites, 647 Epidermal living cells, 135 Epidermis, 134–135, 134 Epigenome, 6

Epigenomic biomarker, 8 Epigenomics, 6 Epoxide hydratases (EHs), see Epoxide hydrolases Epoxide hydrolases, 138, 241 Equilibrium, 647 ER, see Extraction ratio Esterases, 138 Estimated GFR (eGFR), 312 Estimation of Michaelis-Menten parameters, 267 Ethanol, 244 Eustachian tube, 19 Extraction ratio (ER), 82, 353–355 Extravascular routes oral administration, frst-order absorption via, 461 linear one-compartment model, 461 absorption rate constant, estimation of, 466–473 area under plasma concentration–time curve, 476 duration of action, 479 fraction of dose absorbed, 478 overall elimination rate constant, estimation of, 464–466 overall elimination rate constant and absorption rate constant, 464–473 peak concentration, 474–475 time to peak xenobiotic concentration, 473–474 total body clearance and apparent volume of distribution, 476–478 linear three-compartment model, 490–491 linear two-compartment model, 479 apparent volumes of distribution, 484 area under plasma concentration–time curve, 483–484 distribution and elimination rate constants, 485 equations of the model, 479–481 frst-order absorption rate constant, 485–487 initial plasma concentration, 482 in peripheral compartment, 487–490 total amount of drug eliminated from the body, 484 Ex-vivo liver perfusion techniques, 258–260 Facilitated diffusion, 188, 189 FDA, see Food and Drug Administration “Feathering” method, 466 Fick’s frst law of diffusion, 75, 140, 185, 186 Fick’s law of diffusion, 177, 186 Fick’s second law of diffusion, 114, 140, 141, 184 First-order kinetics model, 521–522 First-pass metabolism, 238 Fixed effect parameters, 575 Flavin-containing amine oxidoreductases, 240–241 Flavin-containing monooxygenases (FMOs), 138, 240 Flow-limited (perfusion-limited) models, 372–374 Fluorescent resonance energy transfer (FRET), 190 FMOs, see Flavin-containing monooxygenases Folate transporters, 31 Fomivirsen, 35 Food and Drug Administration (FDA), 615 Foscarnet, 35 Fraction of dose absorbed, 478

739

INDEX

FRET, see Fluorescent resonance energy transfer Fruit juices, 528 GALT, see Gut-associated lymphoid tissue Gamma multi-hit model, 569 Ganciclovir, 35 Gastric accommodation, 78–80 Gastric emptying process, 78–80 Gastrointestinal metabolism, 84–86 Gastrointestinal route of administration/ exposure, 73 xenobiotic absorption, GI tract infuencing, 74 absorptive surface area, 77–78 bile salts, role of, 81 gastric emptying and gastric accommodation, 78–80 gastrointestinal metabolism, 84–86 GI tract infux and effux transport proteins, 86–89 hepatic frst-pass metabolism, 81–84 intestinal microbiotas, role of, 89–90 intestinal motility, 80–81 regional pH of GI tract and pH-partition theory, 74–77 Gastrointestinal transit absorption (GITA) model, 532–535, 533, 534 GCP guidelines, see Good Clinical Practice guidelines Generic drug products, 547 Genetics and genomics on PK/PD and TK/TD, 5 epigenomics, 6 metabolomics and metabonomics, 5 proteomics, 5 transcriptomics, 5 Geometric series, 628 GFR, see Glomerular fltration rate GITA model, see Gastrointestinal transit absorption model Global Two-Stage (GTS) approach, 579 Globulins, 217 Glomerular fltration, 309 Glomerular fltration rate (GFR), 309, 312, 595 endogenous markers of creatinine clearance, 313–317 cystatin C (CysC), 317–318 estimation during pregnancy, 604–605 exogenous markers of, 312 inulin, 312–313 iohexol, 313 radioisotope-labeled compounds, 312 Glucose transporter 1 (GLUT1), 31 Glucuronidation, 248 C-glucuronide conjugation, 250 N-glucuronide conjugation, 249–250 O-glucuronide conjugation, 249 S-glucuronide conjugation, 250 GLUT1, see Glucose transporter 1 Glutathione (GSH) conjugation, 253–254 Glutathione S-transferase, 139 Good Clinical Practice (GCP) guidelines, 553 Grass model, 536 Ground substance, 135 GSH conjugation, see Glutathione conjugation

740

GTS approach, see Global Two-Stage approach Guanidinoacetic acid N-methyltransferase, 252 Gut-associated lymphoid tissue (GALT), 517 Hagen–Poiseuille equation, 212 Hair follicle, 135–136 Half-lives of two-compartment model, 433 biological half-life, 434 elimination half-life, 434 half-life of α, 434 half-life of k12, 434 half-life of k21, 434 Hanes–Woolfe plot, 268, 269, 390 Heartbeat time (HBT), 564 Heat-not-burn products, 65 Hemodialysis, 341 Henderson–Hasselbalch principles, 27 Hepatic acinus, 239 Hepatic artery, 238 Hepatic clearance, assimilation of intrinsic clearance in, 271 dispersion model, 274 parallel-tube model, 273 physiological PK/TK organ model for the liver, 274–276 well-stirred model, 272–273 zonal liver model, 276 Hepatic diseases, 231 Hepatic encephalopathy, 603 Hepatic frst-pass effect (HFPE), 550 Hepatic frst-pass metabolism, 81–84 Hepatocytes, 81, 257 hepatocytes suspension and 2D cultured hepatocytes monolayer, 258 3D-cultures of, 258 HFPE, see Hepatic frst-pass effect Higuchi “square root of time plot” model, 522–523 Hill plot, 270–271, 273 Histamine N-methyltransferase (HNMT), 252 Hixson–Crowell “cube root” model, 521 HNMT, see Histamine N-methyltransferase Hopfendberg model, 524 House-keeping gene, 317 HSA, see Human serum albumin Human serum albumin (HSA), 216 Hybrid models, 369 Hydrophobicity, 180 Hysteresis loops in PK/PD or TK/TD relationships, 412–413 IACUC, see Institutional Animal Care and Use Committee iBRB, see Inner blood-retinal barrier (iBRB) IBW, see Ideal body weight ICH, see International Conference on Harmonization Ideal body weight (IBW), 229 IFPE, see Intestinal frst-pass effect IMMC, see Interdigestive migrating myoelectric complex Incubation, change of incubation conditions during, 648 Inner blood-retinal barrier (iBRB), 30, 32 Inner ear, 19, 20

INDEX

Inner ear diseases, 21 Innovator product, 551 Input functions, 625–626 INR, see International Normalized Ratio In-situ liver perfusion techniques, 258–260 Institutional Animal Care and Use Committee (IACUC), 261 Insulin, 122 Interdigestive migrating myoelectric complex (IMMC), 79 Intermittent infusion, 120 International Conference on Harmonization (ICH), 615 International Normalized Ratio (INR), 603 Interspecies scaling in mammals, 561 Intestinal frst-pass effect (IFPE), 550 Intestinal microbiotas role, 89–90 Intestinal motility, 80–81 Intestinal peptide transport, 195 Intra-arterial route of administration, 107 Intracameral (INTRACAM) route of administration, 34 Intracorneal route of administration, 37 intracorneal injection and related rate equations, 41 intracorneal permeation and related rate equations, 38–41 Intradermal route of administration, 158–159 Intraepidermal route of administration, 159–160 Intramuscular route of administration, 107 ADME of, 108–111 rate equations of intramuscularly injected xenobiotics, 109–111 Intraovarian route of administration, 173 Intraperitoneal (IP) route of administration, 112, 118 applications of, 113–114 kinetics of IP transport of xenobiotics, 114–119 Intravenous infusion (case), 683–691 Intravenous route of administration, 119 bolus injection, continuous infusion, intermittent infusion, 120 intravenous injection drawbacks, 120 intravenous PK/TK analysis, 120–121 Intravitreal (I-VITRE) route of administration, 34 ADME and rate equations, 35–37 Inulin, 312–313 Inverse Laplace transform determination, 626 In vitro drug metabolism (case), 637, 641–649 In vitro intrinsic metabolic clearance, 267 In vitro–in vivo correlation (IVIVC), 524 level A correlation, 524 level B correlation, 525 level C correlation, 525 multiple-level C correlation, 525 Iohexol, 313 Ionized basic compounds, 217 Ion-pair absorption, 198–200 IP route of administration, see Intraperitoneal route of administration Iterative two-stage approach, 579 IV drip, 120 IVIVC, see In vitro–in vivo correlation IV piggyback, 120

Kinetics and rate equations, 628 frst-order kinetics, 628–630 zero-order kinetics, 630 Kitazawa model, 522 Korsmeyer–Peppas model, 523 Krüger–Thiemer method, 468 Kupffer cells, 252 Lactobacillus acidophilus, 169 Lactobacillus jensenii, 170 L-amino acid oxidases (LAO), 240, 241 Langmuir equations, 219 LAO, see L-amino acid oxidases Laplace transform, 71, 149, 326, 385, 448, 622–623, 626 Laplace transform method of integration, 621 LAT1, see L-type amino acid transporter 1 Law of Mass Action, 218 Lean body mass (LBM), 230 Linear mixed-effects (LME) model, 576 Linear one-compartment model, 447, 461 absorption rate constant, estimation of, 466–473 administration of loading dose with intravenous infusion, 450–452 area under plasma concentration–time curve, 476 duration of action, 479 estimation of duration of action in infusion therapy, 453–454 estimation of plasma concentration after termination of infusion, 452 estimation of time required to achieve steadystate plasma concentration, 449–450 fraction of dose absorbed, 478 overall elimination rate constant and absorption rate constant, 464–473 peak concentration, 474–475 time to peak xenobiotic concentration, 473–474 total body clearance and apparent volume of distribution, 476–478 Linear one-compartment open model, 423 apparent volume of distribution, 425–426 determination of area under plasma concentration–time curve, 427–428 duration of action, 426–427 estimation of fraction of dose eliminated by all routes of elimination, 427 estimation of fraction of dose in the body, 427 half-life of elimination, 425 time constant, 425 total body clearance, 426 Linear pharmacodynamic model, 398–399 Linear PK/TK, clearance estimation in, 362–363 Linear PK/TK compartmental analysis, 378 dose-dependent compartmental analysis, 388–391 linear dose-independent compartmental analysis, 379–387 Linear three-compartment model, 490–491 and frst-order elimination from central compartment, 458–459 from peripheral compartment, 459 Linear three-compartment open model in central compartment and elimination from peripheral compartment, 442–444

741

INDEX

and elimination from the central compartment, 440–442 Linear two-compartment model, 479 apparent volumes of distribution, 484 area under plasma concentration–time curve, 483–484 distribution and elimination rate constants, 485 equations of the model, 479–481 frst-order absorption rate constant, 485–487 initial plasma concentration, 482 in peripheral compartment, 487–490 simultaneous intravenous bolus and infusions administration into central compartment of, 456 total amount of drug eliminated from the body, 484 with two consecutive zero-order inputs, 456–457 with zero-order input and frst-order disposition, 454–456 Linear two-compartment open model assessment of time course of xenobiotics in peripheral compartment, 436–438 biological half-life, 434 in central compartment and elimination from peripheral compartment, 438–440 determination of area, 435–436 and elimination from the central compartment, 428–438 elimination half-life, 434 equations of, 429 estimation of initial plasma concentration and volumes of distribution, 432–433 estimation of rate constants of distribution and elimination, 433 half-life of α, 434 half-life of k12, 434 half-life of k21, 434 Lineweaver–Burk plot, 267–268, 268, 390, 639 Lipophilicity, 180, 221 Lipoproteins, 217 Liver, 237–238 Liver cirrhosis, dosage adjustment in, 602, 603 Child-Turcotte-Pugh (CTP) score, 603 ascites assessment, 603 hepatic encephalopathy, 603 International Normalized Ratio (INR), 603 serum albumin level, 603 total bilirubin measurement, 603 LME model, see Linear mixed-effects model Loading dose vs. maintenance dose, 510–511 Local moment curve, 83 LOEL, see Low-observed-effect level Logit model, 569 Log-linear pharmacodynamic model, 399–400 Loop of Henle, 311 Loo–Riegelman method, 485–487, 488, 489 Low-dose extrapolation, 567 threshold and non-threshold models, 568 Armitage-doll multi-stage model, 570 gamma multi-hit model, 569 Logit model, 569 one-hit model, 569 Probit model, 568–569

742

statistico-pharmacokinetic model, 570 Low-observed-effect level (LOEL), 588 L-type amino acid transporter 1 (LAT1), 31 Lung metabolism of xenobiotics, 61–62 Lungs’ role in PK/TK of xenobiotics, 67 Maintenance dose vs. loading dose, 510–511 Mammalian species, clearance scale-up in body-weight dependent extrapolation of clearance in humans, 361 extrapolation of clearance from animal to human, 359–361 Mantoux technique, 158 MAO, see Monoamine oxidases Markel cells, 135 Mass fux, 186 Mass median aerodynamic diameter (MMAD), 62 Mass transfer area coeffcient (MTAC), 116–117 Mass transfer coeffcient (MTC), 116 Mathematical concepts Akaike information criterion (AIC), 630 determinants and Cramer’s rule, 623–625 disposition functions, 626 geometric series, 628 input functions, 625–626 inverse Laplace transform, determination of, 626 kinetics and rate equations, 628 frst-order kinetics, 628–630 zero-order kinetics, 630 Laplace transform method of integration, 621 Laplace transforms, 622–623 Schwarz criterion, 630 trapezoidal rule, 626–628 Mathematical modeling, 368 hysteresis loops in PK/PD or TK/TD relationships, 412–413 linear PK/TK compartmental analysis, 378 dose-dependent compartmental analysis, 388–391 linear dose-independent compartmental analysis, 379–387 non-compartmental analysis based on statistical moment theory, 392 mean residence time and mean input time, 393–394 total body clearance and apparent volume of distribution, 394–395 physiologically based PK/TK models, 368 model development, 371–376 predictive capability and sensitivity analysis, 376–378 physiologically based PK/TK models with effect compartment, 411 PK-PD and TK-TD modeling, 395 effect compartment, 407–411 linking non-hyperbolic sigmoidal model to PK/TK models, 406–407 linking nonlinear hyperbolic concept to compartmental models, 404–406 pharmacodynamic models, 398–402 xenobiotic–receptor interaction, 396–398 target-mediated drug disposition (TMDD) models, 413

INDEX

one-compartment models, 414–415, 415 two-compartment models, 415–417, 416 Maximum life potential (MLP), 565 MCTs, see Monocarboxylate transporters MDR family, see Multidrug resistance proteins family Mean excretion time (MET), 337 Mean residence time (MRT), 425 Mean transit time (MTT), 119 Mechanistic absorption models, 529 absorption potential models, 530 Advanced Compartmental Absorption Transit (ACAT) model, 535, 536 advanced dissolution, absorption, and transit model (ADAM), 535–536, 537 Compartmental Absorption and Transit (CAT) model, 531–532 dispersion models, 530–531 gastrointestinal transit absorption (GITA) model, 532–535, 533, 534 Grass model, 536 Melanocytes, 135 MET, see Mean excretion time Metabolic inhibition, classifcations, 277 competitive inhibition, 278 mixed noncompetitive inhibition, 281–282 noncompetitive inhibition, 278–280 product inhibition, 285 suicide inhibition, 282–284 uncompetitive inhibition, 280–281 Metabolic pathways, 239 Metabolic rate constant, 327 Metabolism (in vitro), kinetics of, 263 catalytic effciency and turnover number, 267 hepatic clearance, assimilation of intrinsic clearance in, 271 dispersion model, 274 parallel-tube model, 273 physiological PK/TK organ model for the liver, 274–276 well-stirred model, 272–273 zonal liver model, 276 in vitro intrinsic metabolic clearance, 267 Michaelis–Menten kinetics, 263–267 Michaelis–Menten parameters, estimation of, 267 direct linear plot, 269, 271, 272 Eadie–Hofstee plot, 269, 270 Hanes–Woolfe plot, 268, 269 Hill plot, 270–271, 273 Lineweaver–Burk plot, 267–268 xenobiotic metabolism, induction of, 285–286 xenobiotic metabolism, inhibition of, 276 competitive inhibition, 278 mixed noncompetitive inhibition, 281–282 noncompetitive inhibition, 278–280 product inhibition, 285 suicide inhibition, 282–284 uncompetitive inhibition, 280–281 Metabolism Study, in vitro systems for xenobiotics metabolism study, 255 antibodies against CYP proteins, 260 bDNA probes, 260 cell lines, 260–261 cellular fractions, 257

hepatocyte 3d cultures, 258 hepatocytes 2D cultured monolayer 258 in-situ and ex-vivo perfusion, 258 liver perfusion, 258–259 in vivo samples for xenobiotic metabolism study, 261 bile samples, 262–263 portal vein cannulation, 263 serum and plasma samples, 261–262 urine samples, 262 organ fractions, 258 precision cut liver slice, 258 Phase I metabolism, 240 alcohol dehydrogenase (ADH), 245–246 aldehyde dehydrogenases (ALDHs), 246–247 carboxylesterases (CES), 247 cytochrome P450 (CYP450), 241–245 diamine oxidase (histaminase), 246 epoxide hydrolases, 241 favin-containing amine oxidoreductases, 240–241 favin-containing monooxygenases (FMOs), 240 peptidase (protease/proteinase), 247–248 xanthine oxidase (XOD), 247 Phase II metabolism: conjugation, 248 acetylation (acylation), 252–253 amino acid conjugation, 254–255 glucuronidation, 248–250 glutathione (GSH) conjugation, 253–254 methylation, 251–252 sulfation, 250–251 subcellular fractions, 255 cytosolic fraction, 255–256 microsomal fraction, 256 S9 fraction, 255 Metabolites, inhibition of enzyme activity by, 647 Metabolomic biomarkers, 8 Metabolomics, 5 Metabonomics, 5 Methotrexate, 35 Methylation, 251–252 Michaelis–Menten constant, 68, 84, 123, 188 Michaelis–Menten equation, 141, 194, 264, 265, 266, 267, 269, 363, 637, 645 Michaelis–Menten kinetics, 263–267, 388–391, 391, 390 Michaelis–Menten parameters, 65, 84, 267 direct linear plot, 269, 271, 272 Eadie–Hofstee plot, 269, 270 Hanes–Woolfe plot, 268, 269 Hill plot, 270–271, 273 Lineweaver–Burk plot, 267–268 Microbiome, 59 Microbiotas, 59, 90 vaginal, 170 Microsomal fraction, 256–257 Middle ear, 19, 20 Middle ear disorders, 20 Minimum steady-state plasma concentration, 502 Mixed noncompetitive inhibition, 281–282 MLP, see Maximum life potential MMAD, see Mass median aerodynamic diameter Monoamine oxidases (MAO), 240, 241

743

INDEX

Monocarboxylate transporters (MCTs), 31 MRPs, see Multidrug resistance-associated proteins MRT, see Mean residence time MTAC, see Mass transfer area coeffcient MTC, see Mass transfer coeffcient MTT, see Mean transit time Mucosal epithelium, 50–52, 52 Multidrug resistance-associated proteins (MRPs), 31, 139, 223 Multidrug resistance proteins (MDR) family, 139 Multiple dosing intravenous injection, 407 Multiple dosing kinetics, 495 case, 710–725 changing dose, 505 dosing interval, 505 half-life, 505 irregular dosing interval, effect of, 505 multiple intravenous infusions, 513 multiple oral dose administration, 502 extent of accumulation, 504 loading dose vs. maintenance dose, 505 peak, trough, and average plasma concentrations, 503–504 one-compartment model, 495 average steady-state plasma concentration, 498–499 dosing regimen, design of, 501–502 equations of plasma peak and trough levels, 496 estimation of plasma concentration after the last dose, 500–501 estimation of time required to achieve steadystate plasma levels, 496–498 extent of accumulation of xenobiotics multiple dosing in the body, 500 loading dose vs. maintenance dose, 499 two-compartment model, 505 accumulation index, 510–511 concept of half-life in multiple dosing kinetics, 512–513 estimation of time required to achieve steadystate plasma levels, 510 evaluation of plasma level after the last dose, 511–512 loading dose vs. maintenance dose, 510–511 peak, trough, and average plasma concentrations, 506–510 steady state, fraction of, 510 Multiple dosing regimen adjustment of, 599–601 relative/absolute bioavailability during, 550–551 N-acetyltransferase (NAT), 139, 252 NAD(P)H Quinone reductase, 138 Naïve average data approach, 579 Naïve-pooled data approach, 578 Nasal cavity, 51 Nasal rate equations, 54 inclusive nose-to-brain PK/TK model, 56–58 nose-to-systemic circulation PK/TK model, 54–56 Nasal route of administration/exposure, 50 mucosal epithelium, 50–52 olfactory epithelium, 52–53

744

vestibule, atrium, valves, and turbines, 50 xenobiotics, nasal ADME of, 53–54 NAT, see N-acetyltransferase National Kidney Disease Education Program (NKDEP), 312 NDAs, see New drug applications Nephrons, 310, 310 Nernst–Brunner model, 523–524 New drug applications (NDAs), 545 N-glucuronide conjugation, 249–250 Nicotinamide N-methyltransferase (NNMT), 252 NKDEP, see National Kidney Disease Education Program NLME model, see Nonlinear mixed-effects model NNMT, see Nicotinamide N-methyltransferase NOAEL, see No-Observed-Adverse-Effect-Level NOEL, see No-observed-effect level Noncompetitive inhibition, 278–280 Non-hyperbolic sigmoidal model, 400–402, 406 one-compartment linked model with frst-order input, 407 with IV bolus, 406 multiple dosing IV bolus injection, 407 with zero-order input, 406 two-compartment linked model multiple dosing intravenous injection, 407 single intravenous bolus injection, 407 Nonlinear hyperbolic concept link to compartmental models, 404 one-compartment model with frst-order input, 404–405 intravenous bolus multiple dosing, 405–406 with IV bolus, 404 with zero-order input, 404 two-compartment model with single IV bolus injection, 406 Nonlinear hyperbolic Emax model, 400 Nonlinear mixed-effects (NLME) model, 576–578 Nonlinear PK/TK, clearance estimation in, 363–364 No-Observed-Adverse-Effect-Level (NOAEL), 398 No-observed-effect level (NOEL), 588 Nose-to-brain PK/TK model, 56–58 Nose-to-systemic circulation PK/TK model, 54–56 Noyes–Whitney model, 520 Nucleoside transporters, 31 OATPs, see Organic anion transporting polypeptides oBRB, see Outer blood-retinal barrier (oBRB) Occupancy theory, 398 Octanol, 183 OCTs, see Organic cation transporters Ocular/ophthalmic routes of administration, 29 blood aqueous barrier (BAB), 29–30 blood-retinal barrier (BRB), 30 effux transporters, 31 infux transporters, 31 kinetics of BRB infux permeability clearance, 32 conjunctival route of administration, 33–34 intracameral route of administration, 34 intracorneal route of administration, 37 intracorneal injection and related rate equations, 41

INDEX

intracorneal permeation and related rate equations, 38–41 intravitreal route of administration, 34 intravitreal ADME and rate equations, 35–37 peribulbar routes of administration, 41–42 retrobulbar routes of administration, 41–42 subconjunctival route of administration, 34 sub-tenon routes of administration, 41–42 O-glucuronide conjugation, 249 Ohm’s law, 212 Olfactory epithelium, 52–53 Oligopeptide transporters, 139 One-compartment model IV bolus (case), 665–673 IV infusion (case), 680–682 TMDD models, 414–415, 415 One-hit model, 569 One-layered diffusion model, 142–143 Oral administration (case), 692–709 Oral route of administration/exposure, see Gastrointestinal route of administration/ exposure Organ fractions (precision cut liver slices), 258 Organic anionic-transporting polypeptides, 31 Organic anion transporters, 61 Organic anion transporting polypeptides (OATPs), 139 Organic cation transporters (OCTs), 31, 60–61, 139 Osmosis, 196–198 Outer blood-retinal barrier (oBRB), 30, 32 Outer ear, 19 Outer ear syndrome, 20 PAO, see Polyamine oxidase PAP, see 3′-phosphoadenosine-5′-phosphate PAPS, see 3′-phosphoadenosine-5′-phosphosulfate Parallel model, 358 Parallel-tube model, 273 Partial fraction theorem, 626 Partially linear mixed-effect (PLME) model, 578 Partition coeffcient, 180, 516 CLOGPcoeff, 181–182 MLOGPcoeff, 182–183 rule of fve, 516–517 Passive diffusion Caco-2 cells, 187–188 diffusion coeffcient, 184–185 distribution coeffcient, 183–184 partition coeffcient, 180–183 permeation and permeability constant, 185–188 transcellular and paracellular diffusion, 177–180 Passive mediated transport, see Facilitated diffusion PBPK model, see Physiologically based pharmacokinetic model PCT, see Pulmonary cycle time PD, see Pharmacodynamics Peak exposure, 547 “Peeling” method, 466 Peptidase, 247–248 Peptide transporters, 61 Percutaneous absorption diffusion–diffusion model and statistical moments for, 153–155

physiological modeling of, 155–156 Perfused organ, 259 Perfusion-limited distribution, 213–216 Peribulbar routes of administration, 41–42 Peritoneal dialysis, 341 Permeability-limited (membrane-limited) models, 374–375 Permeability-limited distribution, 213–216 Permeation and permeability constant, 185–188 Peroxisome proliferator-activated receptor alpha (PPAR), 245 P-glycoprotein (Pgp), 21, 31, 189, 223 computational equations, 192–193 structure and function, 189–192 PGT, see Pharmacogenetics Pharmaceutical equivalents, 546 Pharmacodynamics (PD); see also Absorption, distribution, metabolism, and excretion; Pharmacokinetics defnition, 1 linear model, 398–399 log-linear model, 399–400 non-hyperbolic sigmoidal model, 400–402 nonlinear hyperbolic Emax model, 400 PD/TD analysis, 395–396 plasma concentration and response, 398 Pharmacogenetics (PGT), 4 Pharmacokinetics (PK); see also Absorption, distribution, metabolism, and excretion; Intravenous infusion; Mathematical modeling; Multiple dosing kinetics; Oral administration; Pharmacodynamics; Population pharmacokinetics; Single intravenous bolus injection; Transdermal absorption; Urinary data analysis defnition, 1 modeling software, 587 PK/TK analysis, 120–121, 318–337 PK/TK mechanistic models, 338 PK/TK modeling, 8, 109, 274, 368 therapeutic biologics, analysis of, 554 Pharmacokinetics and pharmacodynamics (PK/ PD), 1 Clinical Pharmacokinetics/Pharmacodynamics (CPK/PD), 3 genetics and genomics, 5 epigenomics, 6 metabolomics and metabonomics, 5 proteomics, 5 transcriptomics, 5 of medications, 210 modeling and pharmacometrics (PMX), 3–4 population pharmacokinetics (popPK) and, 4–5 biomarkers, 7–8 genetics and genomics on PK/PD and TK/ TD, 5–6 Pharmacometrics (PMX), 3–4 Pharmceutical alternatives, 546 Phase I metabolism, 240 alcohol dehydrogenase (ADH), 245–246 aldehyde dehydrogenases (ALDHs), 246–247 carboxylesterases (CES), 247 cytochrome P450 (CYP450), 241

745

INDEX

CYP1A subfamily, 241–242 CYP1B subfamily, 242 CYP2A subfamily, 242 CYP2B subfamily, 242 CYP2C subfamily, 243 CYP2D subfamily, 243 CYP2E subfamily, 243–244 CYP3A subfamily, 244–245 CYP4A subfamily, 245 CYP4B subfamily, 245 CYP4F subfamily, 245 diamine oxidase (histaminase), 246 epoxide hydrolases, 241 favin-containing amine oxidoreductases, 240–241 favin-containing monooxygenases (FMOs), 240 peptidase (protease/proteinase), 247–248 xanthine oxidase (XOD), 247 Phase II metabolism, 248 acetylation (acylation), 252–253 amino acid conjugation, 254–255 glucuronidation, 248 C-glucuronide conjugation, 250 N-glucuronide conjugation, 249–250 O-glucuronide conjugation, 249 S-glucuronide conjugation, 250 glutathione (GSH) conjugation, 253–254 methylation, 251–252 sulfation, 250–251 Phenol O-methyltransferase (POMT), 252 Phenylethanolamine N-methyltransferase (PNMT), 252 Phosphatidylethanolamine N-transferase, 252 3′-Phosphoadenosine-5′-phosphate (PAP), 250 3′-Phosphoadenosine-5′-phosphosulfate (PAPS), 250 Physiologically based pharmacokinetic (PBPK) model, 368, 369 Physiologically based PK/TK models, 368 with effect compartment, 411 model development, 371 fow-limited (perfusion-limited) models, 372–374 permeability-limited (membrane-limited) models, 374–375 variability of physiological/biochemical key parameters, 375–376 predictive capability and sensitivity analysis, 376–378 Physiological PK/TK organ model for the liver, 274–276 Pinna, 19 Pinocytosis, 196 PK, see Pharmacokinetics PK/PD, see Pharmacokinetics and pharmacodynamics Placental barrier, 227–228 Placenta role, 605 Plasma clearance, 355 Plasma concentration and response, pharmacodynamic models of, 398 linear pharmacodynamic model, 398–399 log-linear pharmacodynamic model, 399–400 non-hyperbolic sigmoidal model, 400–402 nonlinear hyperbolic Emax model, 400

746

PLME model, see Partially linear mixed-effect model PMX, see Pharmacometrics PNMT, see Phenylethanolamine N-methyltransferase Polyamine oxidase (PAO), 240–241 Polymorphism, 515–516 POMT, see Phenol O-methyltransferase popPK, see Population pharmacokinetics (popPK) Population pharmacokinetics (popPK), 4 biomarkers, 7–8 genetics and genomics on PK/PD and TK/TD, 5 epigenomics, 6 metabolomics and metabonomics, 5 proteomics, 5 transcriptomics, 5 Population pharmacokinetics (popPK)/toxicokinetics, 574 Bayesian approach, 579 computational tools for, 579–580 fxed effect parameters, 575 Global Two-Stage (GTS) approach, 579 iterative two-stage approach, 579 linear mixed-effects (LME) model, 576 naïve average data approach, 579 naïve-pooled data approach, 578 nonlinear mixed-effects (NLME) model, 576–578 partially linear mixed-effect (PLME) model, 578 random effect parameters, 575 standard two-stage approach, 579 Population toxicokinetics, 9–10 PPAR, see Peroxisome proliferator-activated receptor alpha Preclinical PK/TK, 586, 588 estimation of frst dose in humans, 586–587 metabolic evaluations in preclinical phase, 589–590 safety pharmacology and toxicity testing, 588 acute toxicity studies, 588 chronic toxicity studies, 588–589 Pregnancy, adjustment of dosage regimen in, 604 drug distribution, changes infuencing, 604 drug metabolism, changes in, 604 GFR estimation during pregnancy, 604–605 oral absorption, changes impacting, 604 PK/TK models, 605 placenta, role of, 605 renal excretion, changes in, 604–605 Pregnane xenobiotic receptor (PXR), 87 Probit model, 568–569 Product inhibition, 285 Prostaglandin-endoperoxide synthase, see Cyclooxygenase Proteases, 85, 138, 247–248 Proteinase, 247–248 Protein binding (case), 633–636 Protein binding parameters, 217–222 Proteinuria, 311 Proteomic biomarkers, 8 Proteomics, 5 Proton pumps, 195 Pulmonary cycle time (PCT), 564 Pulmonary microbiome, 59–60 Pulmonary route of administration/exposure, 58

INDEX

morphological differences of airways among species, 59 pulmonary microbiome, 59–60 xenobiotics, ADME of lungs’ role in PK/TK of xenobiotics, 67 pulmonary absorption, deposition, and clearance, 60 pulmonary absorption of gases and vapors, 63–66 pulmonary deposition and disposition of particles, 62–63 pulmonary rate equations, 67–72 relevant pulmonary kinetic parameters, 66–67 respiratory tract metabolic enzymes, 61–62 transport proteins of pulmonary tract, 60–61 Pulmonary tract, 59 Pure and recombinant enzymes, 260 PXR, see Pregnane xenobiotic receptor Radioisotope-labeled compounds, 312 Random effect parameters, 575 Ranibizumab, 35 Rate plot, 320, 321–323, 322, 322, 324, 327, 328, 330, 331, 332, 334, 645, 646 stepwise calculations of (case), 658 Rectal route of administration, 167 pharmacokinetic considerations of, 167–169 Relative bioavailability defnition, 547 estimation of from plasma data, 549 from total amount eliminated from the body, 549 Renal function and elimination of xenobiotics, PK– TK considerations of, 309; see also Urinary excretion of unchanged xenobiotic, PK/TK analysis of collecting ducts, 311 dialysis, 340 dialysate, composition of, 341 dialysis clearance, 342–343 effects of dialysis on PK/TK parameters and constants, 343–346 hemodialysis, 341 peritoneal dialysis, 341 distal tubule, 311–312 glomerular fltration, 309 glomerular fltration rate (GFR), 309, 312 endogenous markers of, 313–318 exogenous markers of, 312–313 loop of Henle, 311 renal mechanistic models, 338–340 renal metabolism, 338 tubular reabsorption and secretion, 309–311 Renal impairment, dosage adjustment for patients with, 595 case, 729–733 compounds eliminated by joint action of renal and non-renal routes of elimination, 597 Dettli’s method, 598 Giusti’s method, 598–599 Wagner’s method, 597 compounds eliminated entirely by non-renal route, 597

compounds eliminated entirely by renal route, 596 Dettli’s method, 596 Giusti’s method, 596–597 Wagner’s method, 596 multiple dosing regimen, adjustment of, 599–601 steady-state peak and trough levels, dosage adjustment based, 602 Renal mechanistic models, 338–340 Renal metabolism, 338 Renal replacement therapy (RRP) dialysate, composition of, 341 dialysis clearance, 342–343 effects of dialysis on PK/TK parameters and constants, 343–346 hemodialysis, 341 peritoneal dialysis, 341 Respiratory tract metabolic enzymes, 61–62 Retrobulbar routes of administration, 41–42 Ro5, see Rule of fve Round window membrane (RWM), 19, 23 Routes of administration, 12–13 auricular (otic), 19–24 buccal and sublingual, 24–28 conjunctival, 33–34 FDA approved, 615–620 gastrointestinal, see Gastrointestinal route of administration/exposure intra-arterial, 107 intracameral (INTRACAM), 34 intracorneal, 37–41 intradermal, 158–159 intraepidermal, 159–160 intramuscular, 107–111 intraovarian, 173 intraperitoneal (IP), 112–119, 119, 118 intravenous, 119–121 intravitreal (I-VITRE), 34–37 nasal, see Nasal route of administration/exposure ocular/ophthalmic routes of administration, 29–42 peribulbar, 41–42 pulmonary, see Pulmonary route of administration/exposure rectal route of administration, 167–169 retrobulbar, 41–42 standard terminologies for, 615 subconjunctival (S-CONJUNC), 34 subcutaneous (SC), see Subcutaneous (SC) route of administration sub-tenon, 41–42 transdermal, see Transdermal route of administration vaginal, 169–173, 172 RRP, see Renal replacement therapy Rule of fve (Ro5), 516–517 RWM, see Round window membrane S9 fraction, 255 Safety pharmacology and toxicity testing, 588 Saliva, 27–28 SASA, see Solvent accessible surface area Schwarz criterion, 444, 630

747

INDEX

Schwarz information criterion, 379 SC route of administration, see Subcutaneous route of administration Sebaceous glands, 136 Sertoli cells, 228 Serum albumin level, 603 Serum creatinine, creatinine clearance estimation from, 315–317 S-glucuronide conjugation, 250 Sigma-Minus Plot, 320, 323–324 Single intravenous bolus injection, 407 Sinusoidal perfusion model, see Parallel model SITT, see Small intestinal transit time Skin; see also Transdermal route of administration structural components of, 136 transport proteins, 139 Skin-perm model, 141–142 SLC, see Solute carrier Small intestinal transit time (SITT), 80–81, 531 Small intestine, 84 Sodium pumps, 195 Solubility, 515 Solute carrier (SLC), 139 Solvent accessible surface area (SASA), 181 Solvent drag, 196–198 Space of Disse, 238 Standard temperature and pressure dry air (STPD), 66 Standard two-stage approach, 579 Statistical moment theory, non-compartmental analysis based, 392 mean residence time and mean input time, 393–394 total body clearance and apparent volume of distribution, 394–395 Statistico-pharmacokinetic model, 570 Steady-state peak and trough levels dosage adjustment based, 602 dosing regimen based, 502 Steroid 5-alpha-reductase, 138 Stokes–Einstein relation, 185 STPD, see Standard temperature and pressure dry air Stratifed squamous epithelium, 50 Stratum corneum, 132–134, 133, 137 Stroma, 37 Subcellular fractions, 255 cytosolic fraction, 256 microsomal fraction, 256–257 S9 fraction, 255 Subconjunctival (S-CONJUNC) route of administration, 34 Subcutaneous (SC) route of administration, 121, 122 rate equations of subcutaneously injected xenobiotics, 122 PK models for subcutaneous insulin, 125 subcutaneous capacity-limited model, 123 subcutaneous diffusion rate-limited model, 122–123 subcutaneous dissolution rate-limited model, 123 subcutaneous models based on diffusion equations, 124–125 Sublingual route of administration, 24–28

748

Sub-tenon routes of administration, 41–42 Suicide inhibition, 282–284 Sulfate conjugation, 250 Sulfation, 250–251 Sulfotransferases, 139 Surrogate endpoint, 7 Sweat glands, 136 Swimmer’s ear, 20 Symport, 195 Target concentration, dosing regimen based, 501–502 Target-mediated drug disposition (TMDD) models, 364, 413 nonlinear clearance in, 364 one-compartment TMDD models, 414–415, 415 two-compartment TMDD models, 415–417, 416 Taurine transporter (TAUT), 31 TAUT, see Taurine transporter TCDD, see 2,3,7,8-tetrachlorodibenzo-p-dioxin TD, see Toxicodynamics Testis, 228–229 2,3,7,8-Tetrachlorodibenzo-p-dioxin (TCDD), 276 Therapeutic drug monitoring, see Clinical Pharmacokinetics/Pharmacodynamics (CPK/PD) Therapeutic equivalents, 546 Thiol methyltransferase (TMT), 252 Thiopurine S-methyltransferase (TPMT), 252 3D-cultures of hepatocytes, 258 Threshold and non-threshold models, 568 Armitage-doll multi-stage model, 570 gamma multi-hit model, 569 Logit model, 569 one-hit model, 569 Probit model, 568–569 statistico-pharmacokinetic model, 570 TK, see Toxicokinetics TK/TD, see Toxicokinetics and toxicodynamics TMDD models, see Target-mediated drug disposition models TMT, see Thiol methyltransferase Tobacco-based products, 65 Total body water, 210–211 Total exposure, 547 Toxicodynamics (TD), 9 Toxicogenetics, 9–10 Toxicogenomics, 566–567 Toxicokinetics (TK), 9 Toxicokinetics and toxicodynamics (TK/TD) genetics and genomics, 5 epigenomics, 6 metabolomics and metabonomics, 5 proteomics, 5 transcriptomics, 5 modeling, 9–10 TPMT, see Thiopurine S-methyltransferase Trans-appendageal, 136 Transcellular absorption, 180 Transcellular and paracellular diffusion, 177–180 Transcriptomics, 5 Transcriptomics biomarkers, 8 Transdermal absorption, 136 approaches to enhance rate and extent of, 137–138

INDEX

Transdermal delivery of medications, 137 Transdermal route of administration, 132 dermis, 135 dermis appendages, 135 hair follicle, 135–136 sebaceous glands, 136 sweat glands, 136 dermis cells, 135 epidermis, 134–135, 134 skin transport proteins, 139 stratum corneum, 132–134, 133 transdermal absorption, 136 approaches to enhance rate and extent of, 137–138 xenobiotics, cutaneous metabolism of, 138–139 xenobiotics, transdermal absorption of, 139 compartmental analysis, 145–153 diffusion models, 140–141 one-layered diffusion model, 142–143 percutaneous absorption, 153–156 six-compartment intradermal disposition kinetics, 156–158 skin-perm model, 141–142 two-layered diffusion model, 143–144 Trans-epidermal, 136 Transgenic cell lines, 261 Transport proteins of pulmonary tract, 60 ATP Binding Cassette (ABC) transporters, 61 organic anion transporters, 61 organic cation transporters, 60–61 peptide transporters, 61 Trans-tympanic absorption, 21 Trapezoidal rule, 626–628 Triamcinolone acetonide, 35 Tubular reabsorption and secretion, 309–311 Turbines, 50 Two-compartment model IV bolus (case), 664, 674–679 with IV bolus connected to the effect compartment, 411 TMDD models, 415–417, 416 Two-layered diffusion model, 143–144 Two-pore theory, 196–198 Tympanic membrane, see Eardrum UDPG, see Uridin-5′-diphopho-α-D-glucose UDPGA, see Uridine-5′-diphospho-α-D-glucuronic acid UDP-Glucuronosyltransferase, 139 UDPGT, see Uridine diphosphate glucuronosyltransferases Uncompetitive inhibition, 280–281 Uniport, 195 Uridin-5′-diphopho-α-D-glucose (UDPG), 248–249 Uridine-5′-diphospho-α-D-glucuronic acid (UDPGA), 249 Uridine diphosphate glucuronosyltransferases (UDPGT), 338 Urinary data analysis (case), 650–663 Urinary elimination of xenobiotic metabolites, PK/ TK analysis of, 325 amount of metabolite remaining to be eliminated from the body, 326–327

metabolic rate constant, 327 Urinary excretion, PK/TK analysis of, 318 general equations of, 336–337 non-compartmental analysis, 337 of unchanged xenobiotic following frst-order absorption, 329–332 following two-compartment model, 333–336 following zero-order intravenous infusion, 327–329 intravenous bolus injection, 320–324 of xenobiotic metabolites, 325–327 Urinary excretion data cumulative amount of drug excreted unchanged in urine, calculations of, 319 general equations of PK/TK multicompartment analysis of, 336–337 PK/TK analysis of, 337 Urinary excretion of unchanged xenobiotic, PK/TK analysis of following frst-order absorption, 329 amount of xenobiotic remaining to be excreted in the urine, 331–332 into systemic circulation and estimation of absorption rate constant, 329–331 following two-compartment model, 333–336 following zero-order intravenous infusion, 327 after attaining the steady-state level, 327–329 cumulative amount of urinary excretion, 329 intravenous bolus injection, 320 ARE Plot aka Sigma-Minus Plot, 323–324, 324, 325 Rate Plot, 321–323, 328 Vaginal microbiota, 170 Vaginal route of administration, 169 pharmacokinetic considerations of, 170–173, 172 vaginal microbiota, 170 Vancomycin, 35 Variability of physiological/biochemical key parameters, 375–376 Vestibule, 19, 50 Volatile organic compounds (VOCs), 372 Voriconazole, 35 Wagner–Nelson method, 24, 469, 470, 471, 472, 485 Weibull–Langenbucher model, 523 Well-stirred model, 272–273, 356–358 Wilson’s disease, 602 Xanthine oxidase (XOD), 247 Xenobiotic absorption, GI tract infuencing, 74 absorptive surface area, 77–78 bile salts, role of, 81 gastric emptying and gastric accommodation, 78–80 gastrointestinal metabolism, 84–86 GI tract infux and effux transport proteins, 86–89 hepatic frst-pass metabolism, 81–84 intestinal microbiotas, role of, 89–90 intestinal motility, 80–81 regional pH of GI tract and pH-partition theory, 74–77

749

INDEX

Xenobiotic concentration reduction, 647 Xenobiotic effect on enzyme, 648 Xenobiotic metabolism induction of, 285–286 inhibition of, 276 competitive inhibition, 278 mixed noncompetitive inhibition, 281–282 noncompetitive inhibition, 278–280 product inhibition, 285 suicide inhibition, 282–284 uncompetitive inhibition, 280–281 Xenobiotic–receptor interaction and law of mass action, 396–398 Xenobiotics, 1, 228, 238, 252, 352, 354 cutaneous metabolism of, 138–139 factors infuencing absorption of chirality and enantiomers, 527–528 disease states, effects of, 529 drug administration scheduling, infuence of, 529 effects of food and drink on absorption of xenobiotics, 528–529 genetic polymorphism, infuence of, 529 solid dosage forms, effects of release mechanisms from, 529 instability of, 648 intramuscularly injected, 109–111 intraperitoneal transport of, 114–119 lung metabolism of, 61–62 lungs’ role in PK/TK of, 67 nasal ADME of, 53–54 subcutaneously injected, 122 PK models for subcutaneous insulin, 125 subcutaneous capacity-limited model, 123 subcutaneous diffusion rate-limited model, 122–123 subcutaneous dissolution rate-limited model, 123 subcutaneous models based on diffusion equations, 124–125 time course of, 1 transcapillary exchange of, 213–216 Xenobiotics, transdermal absorption, 139 compartmental analysis, 145–153 diffusion models, 140–141 estimation of PK/TK parameters and constants of absorption into stratum corneum, 148–149 of percutaneous absorption of xenobiotics, 151–153 of skin-absorbed xenobiotics from urinary data, 145–148 of xenobiotics disposition on skin and in plasma, 149–151 one-layered diffusion model, 142–143 percutaneous absorption diffusion–diffusion model and statistical moments for, 153–155 physiological modeling of, 155–156 six-compartment intradermal disposition kinetics of xenobiotics, 156–158 skin-perm model, 141–142 two-layered diffusion model, 143–144

750

Xenobiotics biotransformation, PK/TK considerations, 237 in vitro metabolism, kinetics of, 263 catalytic effciency and turnover number, 267 hepatic clearance, assimilation of intrinsic clearance in, 271–276 in vitro intrinsic metabolic clearance, 267 Michaelis–Menten kinetics, 263–267 Michaelis–Menten parameters, estimation of, 267–271 xenobiotic metabolism, induction of, 285–286 xenobiotic metabolism, inhibition of, 276–285 liver, 237–238 metabolic pathways, 239 in vitro systems for xenobiotics metabolism study, 255–261 in vivo samples for xenobiotic metabolism study, 261–263 Phase II metabolism: conjugation, 248–255 Phase I metabolism, 240–248 Xenobiotics distribution, 210 binding to plasma proteins, 216–220 blood fow and organ/tissue perfusion, 211–216 body weight and composition, 229 body mass index (BMI), 230 body surface area (BSA), 229–230 ideal body weight (IBW), 229 lean body mass (LBM), 230 disease states, 230 chronic renal failure (CRF), 230–231 congestive heart failure (CHF), 230 cystic fbrosis (CF), 231 hepatic diseases, 231 infuence of total body water, 210–211 parallel removal processes, 221–222 penetration through physiological barriers, 221–222 perfusion-limited distribution, 213–216 permeability-limited distribution, 213–216 physicochemical characteristics, 221 physiological barriers, 222 blood–aqueous humor barrier (BAB), 229 blood–brain barrier, 222–227 blood–lymph barrier, 227 blood–testis barrier, 228–229 placental barrier, 227–228 protein-binding parameters, estimation of, 217–220 Xenobiotics intravenous infusion, continuous zeroorder exposure, 446 linear one-compartment model, 447 administration of loading dose with intravenous infusion, 450–452 estimation of duration of action in infusion therapy, 453–454 estimation of plasma concentration after termination of infusion, 452 estimation of time required to achieve steadystate plasma concentration, 449–450 linear three-compartment model and frst-order elimination from central compartment, 458–459

INDEX

and frst-order elimination from peripheral compartment, 459 linear two-compartment model simultaneous intravenous bolus and infusions administration into central compartment of, 456 with two consecutive zero-order inputs, 456–457 with zero-order input and frst-order disposition, 454–456 Xenobiotics metabolism study in vitro systems for, 255 antibodies against CYP proteins, 260 bDNA probes, 260 cell lines, 260–261 cellular fractions–hepatocytes, 257–258 cytosolic fraction, 256 in-situ and ex-vivo liver perfusion techniques, 258–260 microsomal fraction, 256–257 organ fractions (precision cut liver slices), 258 pure and recombinant enzymes, 260 S9 fraction, 255 subcellular fractions, 255–257 in vivo samples for, 261 bile samples, 262–263 portal vein cannulation, 263 serum and plasma samples, 261–262 urine samples, 262 Xenobiotics single intravenous bolus injection, 423 linear one-compartment open model, 423 apparent volume of distribution, 425–426 determination of area under plasma concentration–time curve, 427–428

duration of action, 426–427 estimation of fraction of dose eliminated by all routes of elimination, 427 estimation of fraction of dose in the body, 427 half-life of elimination, 425 time constant, 425 total body clearance, 426 linear three-compartment open model in central compartment and elimination from peripheral compartment, 442–444 and elimination from the central compartment, 440–442 linear two-compartment open model assessment of time course of xenobiotics in peripheral compartment, 436–438 biological half-life, 434 in central compartment and elimination from peripheral compartment, 438–440 determination of area, 435–436 and elimination from the central compartment, 428–438 elimination half-life, 434 equations of, 429 estimation of initial plasma concentration and volumes of distribution, 432–433 estimation of rate constants of distribution and elimination, 433 half-life of α, 434 half-life of k12, 434 half-life of k21, 434 model selection, 444 XOD, see Xanthine oxidase Zonal liver model, 276

751