Pharmacokinetics: Basics to Applications 9811689490, 9789811689499

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Biswajit Mukherjee

Pharmacokinetics: Basics to Applications

Pharmacokinetics: Basics to Applications

Biswajit Mukherjee

Pharmacokinetics: Basics to Applications

Biswajit Mukherjee Department of Pharmaceutical Technology Jadavpur University Kolkata, West Bengal, India

ISBN 978-981-16-8949-9 ISBN 978-981-16-8950-5 https://doi.org/10.1007/978-981-16-8950-5

(eBook)

# The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

My beloved parents, whose love, affection, and blessings have always taught me to possess perseverance, dedication, devotion, and sincerity.

Preface

Pharmacokinetics: Basics to Applications is a textbook-cum-comprehensive guidebook for students and researchers of versatile fields of pharmaceutical and biomedical sciences that deal with pharmacokinetics as a subject and its application in research, industry, and clinical investigations. The book can be beneficial for undergraduate, graduate, and postgraduate students of pharmacy, pharmaceutical sciences, pharmaceutical technology, clinical pharmacy, pharmacy practice, medicine, faculties of the fields, researchers/scientists of pharmaceutical industries, and persons engaged in clinical investigations. I have initially introduced pharmacokinetics, followed by the idea of different pharmacokinetic parameters and the fundamental pharmacokinetic phenomena—absorption, distribution, metabolism, and elimination (ADME), including nonlinear kinetics. The explicit knowledge of the extent of drug absorption, bioavailability, clearance, bioequivalence, and protein binding, pharmacokinetic drug–drug interactions are also the parts of the book content. I have also explained the application of the subject in preclinical and clinical study designing and experiments. Right from the sample collection to its analysis, statistical data analysis guidelines, and the awareness of pharmacokinetic software have been included in the book. Various up-to-date pharmacokinetic experiments and the procedure of seeking clinical approval and clinical data collections have been included to provide real-time ideas and knowledge of the subject to make students well equipped to be true pharmacokinetic professionals. Finally, numerical problems and questions and answers have been incorporated to assist students in understanding and practicing the critical aspects of the chapters for examinations. My primary intention in writing the book is to present a complex subject in one easy way by sharing a vast more than 25 years of university-level teaching and research experience in the field. In its originality, the book can guide its readers through the sequential steps of learning pharmacokinetics quickly in a lucid language. I have always given the substantial effort and minute care to avoid unnecessarily making it complex to the readers. Adequate and requisite information has been included in the book. All the equations are vividly described without missing or jumping any steps. It would definitely help the beginners to the subject to understand and learn pharmacokinetics quite easily. This book has lots of illustrations, a few photographs, and necessary tables that further build the deep foundation and better clarification of the subject to a reader. All the figures have been self-drawn by the author alone so that the correct intended message through the illustrations can vii

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Preface

nourish the inquisitive minds of the readers. A few unpublished photographs have also been incorporated. It took little more than 8670 h to complete the book. I typed the entire book materials by myself to avoid mistakes or minimize the number of mistakes significantly. The information is being shared to clarify my dedication, devotion, and efforts instilled in the book to inculcate the knowledge of the subject with an easy presentation in pupils. I wrote the book based on the classical and pioneering work and some significant current reports to balance and finely tune the knowledge. Since it is a textbook, priority has been given to the established, wellaccepted knowledge. Necessary fundamental mathematics and statistics for easy and faster deduction and understanding of the equations have been incorporated in the book. The single book would guide the students in all necessary learning related to better assimilating the subject knowledge without additional support. I sincerely thank all the researchers who have enriched the field, particularly those whose publications have been considered in writing the book. Of course, without the support of my family members, it could have been impossible for me to write this book. Finally, I believe that this explicitly presented knowledge of pharmacokinetics will help the students acquire the foundation of pharmacokinetics and the growth of the subject knowledge and its applications in the field. Kolkata, India

Biswajit Mukherjee

Acknowledgment

To my beloved wife, and lovely son and daughter for their endless love, support, and encouragement, without which it would be hard for me to complete the book

ix

Contents

1

2

Fundamentals of Pharmacokinetics . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Fundamentals of Pharmacokinetics . . . . . . . . . . . . . . . . . . . . . 1.1.1 Pharmacokinetic Parameters and Blood Drug Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Multiple-Dose Regimen . . . . . . . . . . . . . . . . . . . . . 1.1.3 Apparent Volume of Distribution or Volume of Distribution (vd) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Steady-State Plasma Concentration of Drug . . . . . . . 1.1.5 Drug Accumulation Factor . . . . . . . . . . . . . . . . . . . 1.1.6 Krüger-Thiemer’s “Pharmacokinetic Factor” . . . . . . 1.1.7 Krüger-Thiemer Dose Ratio . . . . . . . . . . . . . . . . . . 1.1.8 Concept of a Loading Dose . . . . . . . . . . . . . . . . . . . 1.1.9 Relationship Between Elimination Rate Constant (KE) and Steady-State Drug Plasma Concentration (css) from Krüger-Thiemer Dose Ratio Concept . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Drug Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Drug Absorption and Determination of Drug Absorption Rate Constant “Ka” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Dominguez Equation and Its Importance . . . . . . . . . 2.1.2 Wagner–Nelson Equation and Method of Determination of Drug Absorption Rate Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Determination of Absorption Rate Constant (Ka) from Urinary Excretion Data . . . . . . . . . . . . . . 2.1.4 Nelson Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Wagner and Nelson Equation . . . . . . . . . . . . . . . . . 2.1.6 Loo–Riegelman Method for Determination of Drug Absorption Rate (Ka) . . . . . . . . . . . . . . . . .

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12 12 15 17 17 18

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18 19

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2.1.7

Method of Residual for Determination of Drug Absorption Rate . . . . . . . . . . . . . . . . . . . . . . 2.1.8 Flip-Flop Phenomenon . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4

5

Extent of Drug Absorption: Bioavailability, Clearance, Bioequivalence, and Protein Binding . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Extent of Drug Absorption: Bioavailability . . . . . . . . . . . . . . . . 3.1.1 Renal Clearance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Determination of Absolute Bioavailability . . . . . . . . . 3.1.3 Determination of Absolute Bioavailability by Urinary Excretion Data . . . . . . . . . . . . . . . . . . . . 3.1.4 Bioequivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Drug–Protein Binding . . . . . . . . . . . . . . . . . . . . . . . . 3.1.6 Reciprocal Plot or Klotz Reciprocal Plot . . . . . . . . . . 3.1.7 Scatchard Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.8 Sandberg Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pharmacokinetic Models and Drug Distribution . . . . . . . . . . . . . . 4.1 Various Pharmacokinetic Models and Drug Distribution . . . . . 4.1.1 Physiological Pharmacokinetic Model (Flow Model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Blood Flow–Limited Physiological Pharmacokinetic Model or Perfusion Model . . . . . . . . . . . . . . . . . . . 4.1.3 Physiological Pharmacokinetic Model with Drug–Protein Binding . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Membrane-Limited Model or Diffusion-Limited Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Statistical Moment Theory . . . . . . . . . . . . . . . . . . . 4.1.6 Compartment Models . . . . . . . . . . . . . . . . . . . . . . . 4.1.7 Some Mathematical Approaches for Easy Computation of Compartmental Equations and Their Applications . . . . . . . . . . . . . . . . . . . . . . 4.1.8 Details About Compartment Models . . . . . . . . . . . . 4.1.9 Drug Distribution Study Through Compartmental Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Drug Metabolism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Drug Metabolism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Hepatic Drug Metabolism . . . . . . . . . . . . . . . . . . . 5.1.2 Pharmacokinetic Compartmental Models and Equations for Assessing Hepatic “First-Pass” Effect of a Drug . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Hepatic First-Pass Effect Invariably Reduces Total Bioavailability of a Drug More When Administered Orally than By Its Intravenous Route of Administration . . . . . . . . . . . . . . . . . . . . . . . . .

31 32 35 37 37 38 39 41 42 45 46 47 48 50

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Contents

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5.1.4

Determination of Drug Metabolite Levels in Plasma Using Compartmental Model . . . . . . . . . . . . . . . . . . . 120 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Drug Elimination and Nonlinear Kinetics . . . . . . . . . . . . . . . . . . . 6.1 Drug Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Nonlinear Kinetics and Capacity-Limited Process . . . 6.1.2 Michaelis–Menten Equation . . . . . . . . . . . . . . . . . . 6.1.3 Capacity-Limited Process/Nonlinear Kinetics . . . . . . 6.1.4 Dose–Plasma Drug Concentration Relationship with Michaelis–Menten Constant Km for Drugs That Undergo Elimination Following Michaelis–Menten Nonlinear Kinetics . . . . . . . . . . . 6.1.5 Drug Elimination by More Than One Capacity-Limited Process . . . . . . . . . . . . . . . . . . . . 6.1.6 Sigma-Minus Method to Determine Elimination Rate Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.7 Bi-Exponential Absorption–Elimination Equation for Orally Administered Drugs Excreted Unchanged Through Urine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.8 Excretion Rate Method . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7

Pharmacokinetic Drug–Drug Interactions . . . . . . . . . . . . . . . . . . . 7.1 Pharmacokinetic Drug–Drug Interactions . . . . . . . . . . . . . . . . 7.1.1 Importance of Drug–Drug Interactions . . . . . . . . . . . 7.1.2 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Drug–Drug Interactions: Pharmacokinetic Type . . . . 7.1.4 Drug–Drug Interactions: Pharmacodynamic Type . . . 7.1.5 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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145 145 145 146 150 152 153 153

8

Pharmacokinetic Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Therapeutic Drug Monitoring and Dose Formula . . . . . . . . . . . . 8.1.1 Therapeutic Drug Monitoring . . . . . . . . . . . . . . . . . . 8.1.2 Physiological Effects on the Pharmacokinetic Drug Parameters and Available Dose Formula in Neonates, Infants, Children, Elderly Patients, Obese Patients, and Patients with Liver and Kidney Insufficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 Physiological Effects on the Pharmacokinetic Parameters of Drugs and Available Dose Formula in Elderly or Geriatric Patients . . . . . . . . . . . . . . . . . . 8.1.4 Physiological Effects on the Pharmacokinetic Parameters of Drugs in Obese Patients . . . . . . . . . . . .

157 157 157

6

125 125 125 125 129

. 131 . 132 . 135

. 137 . 142 . 143

159

161 163

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Contents

8.1.5

Physiological Effects on the Pharmacokinetic Parameters of Drugs and Available Dose Formula in Patients with Renal Insufficiency . . . . . . . . . . . . . 8.1.6 Chronological Pharmacokinetic Guidance for Preclinical and Clinical Studies and Research . . . . . . 8.1.7 Analysis of Pharmacokinetic Data . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

10

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Pharmacokinetic Sample Collection and Processing . . . . . . . . . . . 9.1 Pharmacokinetic Sample Collection and Processing for Preclinical and Clinical Experiments . . . . . . . . . . . . . . . . . 9.1.1 Pharmacokinetic Sampling . . . . . . . . . . . . . . . . . . . 9.1.2 Blood Sampling and Right Practices in Phlebotomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Urine Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.4 Other Tissue Samples . . . . . . . . . . . . . . . . . . . . . . . 9.1.5 Other Tissue Fluids . . . . . . . . . . . . . . . . . . . . . . . . 9.1.6 Processing of Fecal Samples . . . . . . . . . . . . . . . . . . 9.1.7 Extraction of Drug/Drug Metabolite from Biological Samples . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Important Bioanalytical Instrumental Techniques in Pharmacokinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Important Bioanalytical Instrumental Techniques . . . . . . . . . . 10.1.1 Liquid Chromatography with Tandem Mass Spectroscopy (LC-MS/MS) . . . . . . . . . . . . . . . . . . . 10.1.2 High-Performance Liquid Chromatography (HPLC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 163 . 164 . 169 . 173 . 175 . 175 . 175 . . . . .

175 179 179 181 181

. 182 . 184 . 185 . 185 . 185 . 192 . 197

Statistics in Pharmacokinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Statistics and Biometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Central Tendency . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Measure of Dispersion . . . . . . . . . . . . . . . . . . . . . . . 11.1.3 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.4 Some Important Definitions . . . . . . . . . . . . . . . . . . . . 11.1.5 The Null Hypothesis, Alternate Hypothesis, and Degree of Freedom . . . . . . . . . . . . . . . . . . . . . . . 11.1.6 Population and Sample . . . . . . . . . . . . . . . . . . . . . . . 11.1.7 Selection of Statistical Methods . . . . . . . . . . . . . . . . . 11.1.8 Different Statistical Methods Used in Pharmacokinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.9 Scattergram, Linear Regression Model, Correlation Coefficient, Nonlinear Regression Model . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

199 199 200 203 206 208 210 210 211 211 224 226

Contents

12

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Pharmacokinetic Software and Tools . . . . . . . . . . . . . . . . . . . . . . 12.1 Pharmacokinetic Software and Tools . . . . . . . . . . . . . . . . . . . 12.1.1 Software and Tools . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.2 Types of Software . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.3 Importance of Pharmacokinetic Software . . . . . . . . . 12.1.4 Open Source and Commercial Pharmacokinetic Software and Tools . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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227 227 227 227 228

Pharmacokinetic Laboratory-Based Experiments . . . . . . . . . . . . . 13.1 Pharmacokinetic Laboratory-Based Experiments for Undergraduate and Postgraduate Students . . . . . . . . . . . . . . . . 13.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.2 Helpful Information in Laboratory Work . . . . . . . . . 13.1.3 Drug-Plasma Protein (Albumin)-Binding Assessment with a Low Plasma Protein Binding Drug or a High Plasma Protein Binding Drug . . . . . . 13.1.4 Drug Metabolism . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.5 One-Compartment Model Following an Intravenous Bolus Dose Administration . . . . . . . . . . . . . . . . . . . 13.1.6 One-Compartment Model Following an Intravenous Infusion Administration . . . . . . . . . . . . . . . . . . . . . 13.1.7 A Two-Compartment Model Following the Administration of an Intravenous Bolus Dose . . . . . . 13.1.8 A Multi(Three)-Compartment Model Following the Administration of an Intravenous Infusion . . . . . . . . 13.1.9 Determination of Drug Absorption In Vitro . . . . . . . 13.1.10 Cellular Uptake of Drug . . . . . . . . . . . . . . . . . . . . . 13.1.11 In Vitro Drug Skin Permeation from Transdermal Drug Delivery System (Patch) . . . . . . . . . . . . . . . . . 13.1.12 Drug Assay from a Tablet Dosage Form Containing Paracetamol by HPLC Method . . . . . . . . . . . . . . . . 13.1.13 Estimation of Plasma and Urine Drug Concentrations by Reversed-Phase HPLC . . . . . . . . 13.1.14 Compare the Bioavailability of Ranitidine (300 mg) Tablets of Two Different Brands in Human Volunteers (Bioequivalence Study) . . . . . . . . . . . . . 13.1.15 Effect of Food Intake on Drug Bioavailability . . . . . 13.1.16 The Metabolism of Oxybutynin Hydrochloride to Its Metabolite N-Desethyl Oxybutynin Hydrochloride by Rat Liver Microsomes . . . . . . . . . 13.1.17 Simultaneous Determination of Drugs Spiked into Plasma from a Single Tablet Dosage, Using LC-MS System . . . . . . . . . . . . . . . . . . . . . . .

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. 269 . 270

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13.1.18

Determination of Metformin and Canagliflozin from a Single Tablet Dosage in Human Plasma by LC-MS/MS Method . . . . . . . . . . . . . . . . . . . . . . 13.1.19 Retrospective Data Collection . . . . . . . . . . . . . . . . . 13.1.20 Development of Clinical Trial Protocol . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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276 279 279 284

14

Pharmacokinetic Numerical Problems with Solutions . . . . . . . . . . . 287

15

Questions, and Questions and Answers for Practice . . . . . . . . . . . 15.1 The Practice of Questions and Answers on Topics Based on Pharmacokinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Multiple Choice Questions with Answers for Various Competitive Examinations . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Selected Questions from the University Examinations . . . . . . .

. 307 . 307 . 312 . 321

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

About the Author

Biswajit Mukherjee PhD, WBCS, FIC, FICS, is a Professor and former Head in the Department of Pharmaceutical Technology, Jadavpur University, Kolkata, India and a former faculty of the University Institute of Pharmaceutical Sciences, Panjab University, Chandigarh. He was a DAAD (German Academic Exchange Services) Fellow and Guest Scientist, German Cancer Research Center (DKFZ), Heidelberg. He was a visiting fellow of the School of Pharmacy, University of London, and an Indo-Hungarian Education Exchange Fellow, of the National Research Institute for Radiobiology and Radiohygiene, Budapest. He works on antisense technology, nanomedicine, and targeted drug delivery on pharmacokinetics and pharmacodynamics.

xvii

List of Figures

Fig. 1.1

Fig. 1.2

Fig. 1.3

The figure shows plasma drug concentration versus time curve that demonstrates how a drug behaves in the blood after its absorption upon a single-dose administration by any routes other than an intravenous route for systemic drug action. Various pharmacokinetic parameters of the drug in the blood are also seen. The Cmax is the maximum plasma drug concentration at Tmax (time taken to reach Cmax). The MEC is the minimum effective drug concentration to begin therapeutic drug action. The MTC is the minimum toxic drug concentration, and it is also known as the maximum safe drug concentration or the MSC. The plasma drug concentration between the MEC and the MTC is called the therapeutic window. After a drug administration by oral route or by any routes other than intravenous route, the time point at which the drug reaches minimum therapeutically effective drug concentration implies the onset of drug action . . . . . . . . . . . The figure shows plasma drug concentration versus time curve that shows how a drug behaves after its single-dose administration by intravenous route. The MEC is the minimum effective drug concentration to begin therapeutic drug action. The MTC is the minimum toxic drug concentration, and it is also known as the maximum safe drug concentration or the MSC . . . . . . . . . . . . . . . . . In a multiple-dose (repeated-dose) regimen for oral/any routes other than the intravenous route of drug administration for systemic drug action, plasma drug concentration versus time curve for the first dose and subsequent doses show peak and valley appearance between the minimum effective drug concentration (MEC) and the minimum toxic drug concentration (MTC) levels in the blood. The maximum plasma concentration (Cmax) and time to achieve Cmax (Tmax) are shown after the first dose in the drug absorption curve. Every individual peak represents the Cmax after each dose at the respective Tmax . . . . . . . . . . . . . . . . . . . . . . .

2

4

8

xix

xx

Fig. 1.4

Fig. 1.5

Fig. 1.6

Fig. 2.1

List of Figures

In a multiple-dose (repeated-dose) regimen of the intravenous route of drug administration, plasma drug concentration versus time curve for the first dose and subsequent doses shows peak and valley appearance between the minimum effective drug concentration (MEC) and the maximum safe drug concentration (MSC) levels in the blood . . . .. . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . . 8 The plasma drug concentration versus time curve upon drug administration by oral route/any route other than the intravenous route for systemic drug action shows that in each of the subsequent doses, the maximum and minimum plasma drug concentration values increase during drug absorption at the initial phase. After administering several doses, the maximum and minimum plasma drug concentration values become constant. The average plasma drug concentration at the zone is called steady-state plasma drug concentration (css). At the steady state, the maximum constant plasma drug concentration value is called steady-state maximum plasma drug concentration (css)max. Likewise, the minimum constant plasma drug concentration value is called the steady-state minimum plasma drug concentration (css)min. MEC indicates minimum effective drug concentration, MTC denotes minimum toxic drug concentration . . . . . . . . 13 The plasma drug concentration versus time curve upon drug administration by intravenous route shows that the maximum and minimum drug plasma concentration values increase upon drug administration at the initial phase in each of the subsequent doses. After administering many doses, the maximum and minimum plasma drug concentration values become constant. The average plasma drug concentration at the region is called steadystate plasma concentration (css). At the steady state, the maximum constant plasma drug concentration value is called steadystate maximum plasma drug concentration (css)max, and the minimum constant plasma drug concentration value is called the steady-state minimum plasma drug concentration (css)min. MEC indicates minimum effective drug concentration, MTC denotes minimum toxic drug concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Plasma drug concentration versus time curve after the drug absorption following a single dose by oral route/any routes other than the intravenous route of drug administration for systemic drug action. The area under the curve, shown by stars and RT depicted by 0 cp dt, shows bioavailability of the drug during the period of 0–T, after its administration. MEC indicates minimum effective drug concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

List of Figures

Fig. 2.2

Fig. 2.3

Fig. 2.4

Fig. 2.5

Fig. 2.6

Fig. 2.7

Fig. 2.8

xxi

Plasma drug concentration versus time curve after drug absorption following a single dose by oral route/any routes other than the intravenous route of drug administration for systemic drug action. The area under the curve, shown by stars and depicted by R1 c dt, shows the total bioavailability of the drug. MEC p 0 indicates minimum effective drug concentration . . . . . . . . . . . . . . . . . . Percentage of drug remaining to be absorbed when plotted against time on a semi-log paper, keeping “percentage of drug remaining to be absorbed” on a log scale and time on a linear K a scale, gives a straight line. The slope of the line is 2:303 where Ka provides the value of absorption rate constant . . . . . . . . . . . . . . . . . . . . . Percentage of drug remaining to be absorbed when plotted against time on a semi-log paper, keeping “percentage of drug remaining to be absorbed” on a log scale and time on a linear K a scale, gives a straight line. The slope of the line is 2:303 where Ka provides the value of absorption rate constant . . . . . . . . . . . . . . . . . . . . . Percentage of drug remaining to be absorbed when plotted against time on a semi-log paper, keeping “percentage of drug remaining to be absorbed” on a log scale and time on a linear K a scale, gives a straight line. The slope of the line is 2:303 where Ka provides the value of absorption rate constant . . . . . . . . . . . . . . . . . . . . . The figure shows the residual plot of the plasma drug absorption curve following the oral route of a single-dose drug administration. Plasma drug concentrations on the curve provide the actual drug absorption data (shown by “actual drug concentration”). The post absorption terminal linear portion down the curve when extrapolated to the plasma drug concentration axis (y-axis), the data on the line provide the extrapolated drug concentrations. The difference between the extrapolated drug concentration and the actual drug concentration gives the value of residual drug concentration (CR). When log CR is plotted against time, the K a slope of the line provides the value of 2:303 , where Ka is drug absorption rate. The “vs.” stands for “versus” . . . . . . . . . . . . . . . . . . . . . The figure shows that the extrapolated line and the feathered K a (residual) line with a slope 2:303 of the plot provide negative lag time. The meeting point of the extrapolated line and the residual line gives the value of lag time t0. In the graph, a negative lag time (which indicates erroneous findings) is shown. Here, logarithm values of residual drug concentration logcR are plotted against time t . .. . . .. . . . .. . . .. . . .. . . .. . . . .. . . .. . . .. . . .. . . . .. . . .. . . .. . . . . Residual plot of extravascular drug administration. Logarithm plasma concentration, logcp, data plotted against time that shows the slope of the terminal linear portion of the line provides the

24

25

28

30

31

33

xxii

Fig. 2.9

Fig. 3.1

Fig. 3.2

Fig. 3.3

Fig. 4.1

Fig. 4.2

List of Figures K a value of 2:303 where Ka is the rate of drug absorption. The slope of K E the feathered line gives the value of 2:303 where KE is the rate of elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Residual plot of intravascular drug administration. Logarithm plasma concentration, logcp, data plotted against time that shows the slope of the terminal linear portion of the line provides the K a value of 2:303 where Ka is the rate of drug absorption. The slope of K E where KE is the rate of the feathered line gives the value of 2:303 elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The figure shows Klotz reciprocal plot for the quantification for protein–drug binding. The data of 1r (where r is the ratio of protein-bound drug to total protein) are plotted against ½D1  (where [D] is the molar concentration of free drug in plasma), and the 1 . The extrapolated line gives slope of the line gives the value of νK 1 1 the value of the intercept ν on r -axis (y-axis). Here, “ν” is the number of binding sites of a protein molecule for drug molecules, and “K” is the equilibrium rate constant . . . . . . . . . . . . . . . The figure shows a Scatchard plot for the quantification for protein–drug binding. Data of ½Dr  (where r is the ratio of proteinbound drug to total protein and [D] is the molar plasma free drug concentration) are plotted against r, and the slope of the line gives the value of K. The intercept of the line provides the value with νK on ½Dr -axis (y-axis). Here, “ν” is the number of binding sites of a protein molecule for drug molecules, and “K” is the equilibrium rate constant . . .. . . .. . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. The figure shows the Sandberg plot for the quantification for b protein–drug binding. Data of D Df (where Db is the concentration of the protein-bound drug and Df is the free drug concentration) are plotted against Db, and the slope of the line gives the value of K. The intercept of the line provides the value with νK[Pt] “ ” b on D Df -axis (y-axis). Here, ν is the number of binding sites of a protein molecule for drug molecules. “K” is the equilibrium rate constant and [Pt] is the concentration of the total protein . . . . . . . . . The figure shows a physiological pharmacokinetic model of drug transportation to the extracellular fluid by blood capillaries connected to the arterioles, and drug molecules/drug metabolites that diffuse out of the cells to the extracellular fluid are transported back to the venules for removal . . . . . . . . . . . . . . . . . . . . . . . The figure shows a physiological pharmacokinetic model of drug elimination: (A) After the drug transportation to the extracellular fluid by blood capillaries connected to arterioles, the drug

34

34

47

48

49

52

List of Figures

Fig. 4.3

xxiii

diffuses into the cells, and drug molecules/drug metabolites that diffuse out of the cells to the extracellular fluid are transported back to the venules to venous blood for elimination. (B) After the drug transportation to the extracellular fluid by blood capillaries connected to arterioles, the drug diffuses into the cells, and drug molecules/drug metabolites diffuse out of the cells to the extracellular fluid from where the drug is eliminated through the organ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The figure shows that in a flow-limited physiological model, drug distribution from the arterial blood to different tissues for drug absorption, and then from the tissues, the drug molecules/ drug metabolites diffuse out to the vein. The rate of drug   dðcp vd Þ dx absorption dx dt is quantified by dt ¼ dt , where cp is the plasma drug concentration and vd is the volume of distribution. However, the rate of blood flow from the tissue to the venous

53

dQ

Fig. 4.4

Fig. 4.5 Fig. 4.6

Fig. 4.7 Fig. 4.8

Fig. 4.9 Fig. 4.10

Fig. 4.11

blood is tissue specific and quantified by dttissue . GIT denotes gastrointestinal tract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The figure depicts reversible drug–protein binding phenomena in the blood and the tissue and the exchange of drugs between those tissues . . .. .. . .. .. . .. .. . .. .. . .. .. . .. .. . .. .. . .. .. . .. .. . .. .. . .. .. . .. .. . .. .. One-compartment closed model that shows x0 amount of drug is administered to an entirely homogeneous body compartment . . . . One-compartment open model that shows x0 amount of drug is administered to an entirely homogeneous body compartment, and the drug is eliminated from it. k10 indicates the rate of drug elimination, and the arrowhead indicates the drug is eliminated outside the body compartment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The figure shows a catenary model where all the compartments are connected like the compartments of a train . . . . . . . . . . . . . . . . . . . . The figure shows a cyclic model. It is like a catenary model where two terminal compartments are connected to each other to form a closed structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The figure shows a mammillary model where all the compartments are connected to a central compartment . . . . . . . . . . . One-compartment open model that shows x0 amount of drug is administered intravenously at the time t0 to an entirely homogeneous body compartment, xB is the amount of drug present in the compartment at time t, and the drug is eliminated from it. KE indicates the rate of drug elimination, and the arrowhead indicates the drug is eliminated outside the body compartment . . Plot of logarithmic plasma drug concentration ( log cp) versus time on a semi-logarithmic paper. The slope of the line gives us KE  2:303 and the intercept on the y-axis of the extrapolated line gives us logc0 where c0 is the instantaneous plasma drug concentration upon the administration of x0 amount of drug . . . . . . . . .

57

58 61

62 62

63 63

86

88

xxiv

Fig. 4.12

Fig. 4.13

Fig. 4.14

Fig. 4.15

Fig. 4.16

List of Figures

The figure shows a one-compartment open model where a drug is administered by infusion at a rate k0. KE is the rate of drug elimination from the compartment and xB is the amount of drug present in the compartment at time t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of logarithmic plasma drug concentration versus time of a patient who received intravenous drug infusion and achieved the steady-state plasma drug level. Then, the infusion was stopped at time T. The time t0 elapsed after the stop of the drug infusion in the patient at time T. The time t is the total time the patient was supposed to receive the infusion and had steady-state plasma drug concentration. The dotted line shows the drop of plasma drug concentration after the stop of the infusion . . . . . . . . . . . . . . . . . . Plot of logarithmic plasma drug concentration versus time of a patient who receives intravenous drug infusion and should achieve the steady-state plasma drug level at time t, and any duration of time t0 before achieving the steady-state plasma drug concentration, the drug infusion is stopped for the patient at time T. The dotted line shows the drop of plasma drug concentration after the stop of the infusion . . .. . . . .. . . . . .. . . . .. . . . .. . . . . .. . . . .. . . . .. Two-compartment open models. Model I: x0 is the dose administered by the intravenous route to the central compartment and xc is the amount of drug present at the central compartment (compartment “1”) at time t and xp is the amount of drug present at the peripheral compartment (here designated as compartment “2”) at time t. Further, the drug is eliminated from the central compartment. Model II: xc is considered as the amount of drug present at the central compartment (compartment “1”) at time t and xp is the amount of drug present at the peripheral compartment (here designated as compartment “2”) at time t. x0 is the dose administered by intravenous injection into the blood (central compartment). Further, the drug is eliminated from both the central compartment and the peripheral compartment. Model III: x0 is the dose administered by the intravenous route to the central compartment and xc is the amount of drug present at the central compartment (compartment “1”) at time t and xp is the amount of drug present at the peripheral compartment (here designated as compartment “2”) at time t. Further, the drug is eliminated from the peripheral compartment. In all the models, K12, K21, K10, and K20 are the respective rate constants, and each arrowhead shows the direction of movement of the drug . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The figure shows the changes of logarithmic drug amount against time in the blood compartment (central compartment) and the tissue compartment (peripheral compartment) following a two-compartment model when the drug is administered by

89

90

92

93

List of Figures

Fig. 4.17

Fig. 4.18

xxv

intravenous route in a patient. Data show that the drug availability is more in the central compartment compared to the tissue compartment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 The figure shows the changes of logarithmic drug amount against time in the blood compartment (central compartment) and the tissue compartment (peripheral compartment) following a two-compartment model when the drug is administered by intravenous route in a patient. Data show that the drug availability is more in the peripheral (tissue) compartment compared to the central compartment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Plot of logarithmic tissue drug amount against the time of a two-compartment model. The slope of the line gives us the value  of β, and the intercept of the extrapolated line gives us log

Fig. 4.19

x0 K 12 αβ

, where α is the drug absorption rate, β is the drug

elimination rate, x0 is the dose, and K12 is the rate of drug transport from the central compartment to the peripheral compartment . . . .. . . . .. . . . . .. . . . .. . . . .. . . . .. . . . . .. . . . .. . . . .. . . . . .. . . . . 101 Three-compartment open models. Model I: Where x0 is the amount of drug administered by the intravenous route to the central compartment and xc is the amount of drug present at the central compartment (compartment “1”) at time t and xp and xq are the amounts of drug present at the two peripheral compartments (here designated as compartment “2” and “3,” respectively) at time t. Further, the drug is eliminated from the central compartment. Model II: Where x0 is the amount of drug administered by the intravenous route to the central compartment and xc is the amount of drug present at the central compartment (compartment “1”) at time t and xp and xq are the amounts of drug present at the two peripheral compartments (here designated as compartment “2” and “3,” respectively) at time t. Further, the drug is eliminated from the central compartment and the peripheral compartment number 3. Model III: Where x0 is the amount of drug administered by the intravenous route to the central compartment and xc is the amount of drug present at the central compartment (compartment “1”) at time t and xp and xq are the amounts of drug present at the two peripheral compartments (here designated as compartment “2” and “3,” respectively) at time t. Further, the drug is eliminated from the central compartment and peripheral compartments, numbers 2 and 3. Model IV: Where x0 is the amount of drug administered by the intravenous route to the central compartment and xc is the amount of drug present at the central compartment (compartment “1”) at time t and xp and xq are the amounts of drug present at the two peripheral

xxvi

List of Figures

compartments (here designated as compartment “2” and “3,” respectively) at time t. Further, the drug is eliminated from the central compartment and the peripheral compartment number 2. Model V: Where x0 is the amount of drug administered by the intravenous route to the central compartment and xc is the amount of drug present at the central compartment (compartment “1”) at time t and xp and xq are the amounts of drug present at the two peripheral compartments (here designated as compartment “2” and “3,” respectively) at time t. Further, the drug is eliminated from peripheral compartment number 2. Model VI: Where x0 is the amount of drug administered by the intravenous route to the central compartment and xc is the amount of drug present at the central compartment (compartment “1”) at time t and xp and xq are the amounts of drug present at the two peripheral compartments (here designated as compartment “2” and “3,” respectively) at time t. Further, the drug is eliminated from peripheral compartment number 3. In all the models, K12, K21, K13, K31, K10, K20, and K30 are the respective rate constants, and each arrowhead shows the direction of movement of the drug . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Fig. 5.1

Fig. 5.2

Fig. 5.3

Two-compartment model to investigate and compare the hepatic first-pass effect of drug administered by different routes. The self-explanatory model shows drug absorption and drug elimination from various compartments as explained under 5.1.2.1– 5.1.2.4 .. . . . . .. . . . . .. . . . . .. . . . . . .. . . . . .. . . . . .. . . . . .. . . . . . .. . . . . .. . . . . .. 114 The x0 amount of drug has been administered in the central compartment. xB is the amount of drug and MB is the amount of drug metabolite present in the central compartment (compartment “1”) at time t. Mu is the amount of drug metabolite eliminated through urine at time t. Kf, and Km are the respective rate constants of drug to metabolite conversion and the elimination of drug metabolite through the urine. Each arrowhead shows the direction of the movement of the drug metabolite . . . . . . . . . . . 121 Plot of logarithmic plasma drug metabolite concentration (logcm) versus time on a semi-logarithmic paper. The slope of KE the line gives us  2:303 and the intercept on the y-axis of the 0 extrapolated line gives us log vm ðKKmf xK where Kf and Km are EÞ the respective rate constants of drug to metabolite conversion and the elimination of drug metabolite through the urine. KE is the drug elimination rate, and vm is the volume of distribution of the drug metabolite upon the administration of x0 amount of drug . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

List of Figures

xxvii

Fig. 5.4

Plot of logarithmic plasma drug metabolite concentration (logcm) versus time on a semi-logarithmic paper. The slope of Km the line gives us  2:303 and the intercept on the y-axis of the x0 extrapolated line gives us log vm ðKKEf K where Kf is the rate of mÞ drug to metabolite conversion and Km is the elimination rate of drug metabolite through the urine. KE is the drug elimination rate, and vm is the volume of distribution of the drug metabolite upon the administration of x0 amount of drug . . . . . . . . . . . . . . . . . . . . 124

Fig. 6.1

A typical nonlinear kinetic plot is shown. The rate of reaction (v) is plotted against the concentration of substrate (c) (here drug) in an enzyme reaction. The first part of the curve shows a concentration-dependent first-order kinetic pattern that is followed by a zero-order kinetic pattern shown by a straight line parallel to the x-axis (concentration axis) when the enzymes are saturated with the substrates. The vmax is the maximum rate of reaction .. . .. .. . .. . .. .. . .. . .. .. . .. . .. .. . .. . .. .. . .. . .. .. . .. . .. .. . .. . .. .. The figure shows the rate of drug elimination is proportionally increased with the increase in drug concentration in the blood. The graph represents a linear plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonlinear kinetic “Michaelis–Menten” equation when rearranged and plotted as 1v against 1c gives a straight line. The slope of the line shows the value of vKm . It is called the max Lineweaver–Burk double reciprocal plot. Here, v is the rate of reaction, c is the concentration of substrate (here drug) in an enzyme reaction. The vmax is the maximum rate of reaction . . . . Nonlinear kinetic “Michaelis–Menten” equation when rearranged and plotted as cv against c gives a straight line. The 1 slope of the line shows the value of vmax . Here, v is the rate of reaction, and c is the concentration of substrate (here drug) in an enzyme reaction. The vmax is the maximum rate of reaction. The Km is the Michaelis–Menten constant . . . . . . . . . . . . . . . . . . . . . . . . Nonlinear kinetic “Michaelis–Menten” equation when rearranged and plotted as v against vc gives a straight line. The slope of the line shows the value of km. Here, v is the reaction rate, and c is the concentration of substrate (here drug) in an enzyme reaction. The Km is the Michaelis–Menten constant . . . Nonlinear kinetic “Michaelis–Menten” equation when rearranged and plotted as logc against t gives a straight line. vmax The slope of the line shows the value of  2:303 k m . Here, c is the substrate concentration (here drug) in an enzyme reaction. The vmax is the maximum rate of reaction. The Km is the Michaelis– Menten constant .. . . . . . .. . . . . .. . . . . .. . . . . .. . . . . . .. . . . . .. . . . . .. . . . . .. .

Fig. 6.2

Fig. 6.3

Fig. 6.4

Fig. 6.5

Fig. 6.6

126

126

128

128

129

131

xxviii

Fig. 6.7

Fig. 6.8

Fig. 6.9

List of Figures

  In the sigma-minus method, when log x1 u  xu is plotted  against time on a semi-log paper, keeping “ log x1 u  xu ” on a log scale and time on a linear scale, it gives a straight line. The kE where kE provides the value of slope of the line is  2:303 elimination rate constant. When the feathered line is drawn, the Ka slope of it gives the value of  2:303 where Ka is the absorption rate constant. The total unchanged drug to be eliminated through urine is x1 u and the unchanged drug eliminated through urine at any time t is xu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 The figure shows drug (xu) is excreted unchanged through urine with drug elimination rate (ke) following orally administered drug in a one-compartment model. Here, xA amount of drug is absorbed to the central compartment from the orally administered x0 amount of drug at an absorption rate ka at time t and xB is the amount of drug present at the central compartment at time t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Δx

Plot of logarithmic values of rate, logð Δtu Þ, of drug (xu) excreted unchanged through urine versus time on a semi-logarithmic e paper. The slope of the line gives us  2:303 and the intercept on

k

k k Fx

e 0 the y-axis of the extrapolated line gives us log ðKa K Þ where ka a

e

is the absorption rate of drug, ke is the elimination rate of drug excreted unchanged through urine, and F is the dose fraction drug absorbed upon the administration of x0 amount of drug . . .

143

Fig. 9.1 Fig. 9.2

Blood draw from antecubital fossa (elbow pit of the forearm) . . Urine collection funnel system for laboratory animals . . . . . . . . . .

177 180

Fig. 10.1 Fig. 10.2 Fig. 10.3

LC-MS/MS system along with its various components . . . . . . . . . A quadrupole assembly . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . . Passage of ion through a quadrupole assembly during ion filtering. Only the selected ions pass through to the mass analyzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . High-performance liquid chromatography (HPLC) system with its various components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An abridged classification of statistical data . . . . . . . . . . . . . . . . . . . . . . Statistical decision tree for continuous parametric data with normal distribution . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . Statistical decision tree for continuous parametric heterogeneous data . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . Statistical decision tree for nonparametric (not normal distributed) data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statistical decision tree for categorical or quantal response data . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .

186 186

Fig. 10.4 Fig. 11.1 Fig. 11.2 Fig. 11.3 Fig. 11.4 Fig. 11.5

189 193 209 212 213 214 215

List of Figures

Fig. 13.1

Fig. 13.2

Fig. 13.3 Fig. 13.4 Fig. 13.5 Fig. 13.6

Fig. 13.7

Fig. 13.8

Fig. 13.9

Fig. 13.10

xxix

Equilibrium dialysis method for studying the extent of drugprotein binding using dialysis tube. The upper figure shows that both the ends of the tube are clipped. The lower picture shows that both the ends of the tube are tied . . . . . . . . . . . . . . . . . . . . . Equilibrium dialysis chamber for studying the extent of drugprotein binding. The figure shows that a dialysis membrane separates the plasma compartment and the buffer compartment Laboratory scale ultrafiltration devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . An assembly for studying drug distribution and elimination in a one-compartment open model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A simple assembly for studying drug distribution and elimination in a one-compartment open model . . . . . . . . . . . . . . . . . . . . . . . . . . . An assembly for studying drug distribution and drug elimination of the peripheral compartment in a two-compartment open model system .. . . . . .. . . . . .. . . . . .. . . . . . .. . . . . .. . . . . .. . . . . .. . . . . . .. . . . . .. . . . . .. An assembly for studying drug distribution and elimination in a three-compartment open model system. The drug is eliminated from a peripheral compartment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . In vitro drug absorption using everted ileum. A portion of rat or fowl ileum is used. It is everted (keeping inside out) and clipped or tied at both the ends eventually after being filled with deionized water. Then it is dipped into a drug solution (reservoir) from where the drug is absorbed through the everted ileum into the water inside . .. . .. . .. .. . .. . .. . .. . .. . .. . .. .. . .. . .. . .. . .. . .. . .. .. . .. Cellular internalization of FITC-labeled drug product (green color) by c6 glioma cells cultured in DMEM (Dulbecco’s Modified Eagle Medium)—supplemented with 10% fetal calf serum and treated with 50 nM drug product. The cells were nuclear-stained with DAPI (blue color). The photograph of the cells incubated up to 4 h was taken with a confocal microscope A diffusion cell assembly for studying in vitro skin permeation of drugs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

238

239 241 249 251

254

256

259

262 264

List of Tables

Table 4.1 Table 4.2 Table 4.3 Table 7.1 Table 9.1 Table 9.2 Table 13.1

Table 13.2

Pharmacokinetic-related some selected Laplace transforms . . . . . . Logarithm table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Antilogarithm table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Drug–drug interactions and their systemic effects . . . . . . . . . . . . . . . Physiological zones of blood collection from small animals . . . Storage temperature of biological specimens . . . . . . . . . . . . . . . . . . . . Proforma for collection of retrospective patient data: Patients suffered from chronic kidney disorder, received dialysis and treatment in a hospital .. . . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . . Proforma for collection of retrospective patient data: Drug-drug interaction in intensive care unit (ICU) patients in a hospital . .

71 74 79 147 178 181

280 281

xxxi

1

Fundamentals of Pharmacokinetics

1.1

Fundamentals of Pharmacokinetics

The word “drug” was originated from the Greek word “pharmacon” or “pharmakon,” and the meaning of the word “kinetics” is “in motion” or “positional change concerning time.” Hence, the term “pharmacokinetics” implies that the drug is in motion in vivo. When the drug is administered to a patient, several phenomena occur in the body. When a patient takes a tablet as an oral dosage form of medicine, upon reaching the stomach, sometimes the intestine (e.g., an enteric-coated tablet intended to disintegrate in the intestine only), it disintegrates, and drug dissolution takes place in the stomach/intestinal fluid. Eventually, drug molecules are absorbed, and they reach the systemic circulation (in the blood). The drug molecules then provide therapeutically effective drug level (i.e., the concentration of drug in the blood on or above which drug can provide systemic therapeutic action/inhibitory action for antibiotics), which is also called minimum effective concentration (MEC) or minimum inhibitory concentration (MIC) of a drug in the blood (Fig. 1.1). Drug transportation simultaneously begins to deliver drugs with an effective concentration to the site of drug action. It produces its pharmacological effect(s) (also called “pharmacodynamic action,” that is, how the body disposes of an administered drug). While doing all such activities, the drug is also metabolized and eliminated as metabolite or drug as such or both from the body. The absorption, distribution, metabolism, and elimination (ADME) are the fundamental functional and usually common phenomena of the drug while in motion in a living system. All these processes involve the positional change of drugs in our bodies to time. Hence, a vivid description of these phenomena (ADME) of a drug in vivo with mathematical equations is called pharmacokinetics. However, memorizing the equations and using them for the calculation cannot help us understand the subject. Instead, only a proper and appropriate conceptualization can build up the foundation of knowledge to understand pharmacokinetics. ADME phenomena always vary for different drugs. ADME mainly depends on # The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B. Mukherjee, Pharmacokinetics: Basics to Applications, https://doi.org/10.1007/978-981-16-8950-5_1

1

2

1

Fundamentals of Pharmacokinetics

Fig. 1.1 The figure shows plasma drug concentration versus time curve that demonstrates how a drug behaves in the blood after its absorption upon a single-dose administration by any routes other than an intravenous route for systemic drug action. Various pharmacokinetic parameters of the drug in the blood are also seen. The Cmax is the maximum plasma drug concentration at Tmax (time taken to reach Cmax). The MEC is the minimum effective drug concentration to begin therapeutic drug action. The MTC is the minimum toxic drug concentration, and it is also known as the maximum safe drug concentration or the MSC. The plasma drug concentration between the MEC and the MTC is called the therapeutic window. After a drug administration by oral route or by any routes other than intravenous route, the time point at which the drug reaches minimum therapeutically effective drug concentration implies the onset of drug action

physicochemical characteristics of drugs such as drug dissolution, partition coefficient, surface activity, ionization of drug, isosterism, complexation, and biopharmaceutical characteristics such as drug–protein binding in the blood, drug transport, drug absorption across the gastrointestinal membrane, rate and extent of drug absorption, and degree of metabolism. Drugs directly administered into the blood (by intravenous [i.v.] route, e.g., i.v. injection, i.v. bolus injection [a single volume injection with a hefty dose of a drug for fast circulation in the blood], i.v. infusion) do not have any absorption phenomenon. Now we should understand why we require pharmacokinetic study. Several scientific data have been provided accurately with mathematics, such as the height of Mount Everest, the distance between the earth and the moon, and many physics and chemistry theories. It is done due to the correctness of calculation with the help of mathematics. It is because “mathematics depicts accuracy, provided the calculation is correct.” We determine the changes in drug concentration in the blood at different time intervals in patients by analyzing the drug concentration of the blood samples withdrawn from time to time. However, if we want to determine drug concentrations in the brain, liver, heart, or lung at various time points after drug administration in patients, the tissue samples to be taken from time to time from the patients for drug analysis are not practically feasible. Although using the radiolabeled drug, whole-body imaging at the different time points upon drug administration can be done, but the method has several drawbacks and lacuna. None of the

1.1 Fundamentals of Pharmacokinetics

3

methods is generally acceptable as a practical approach for such studies. Thus, without taking the tissue samples, by using pharmacokinetic equations and some pharmacokinetic parameters, we can also accurately predict or determine drug concentration in various tissues at different time points after administering a drug. Thus, the study of pharmacokinetics is essential to understand behavioral patterns of drugs in vivo without performing the actual experiments. Extremely logical and validated mathematical equations, practically applicable mathematical models, and appropriately selected set of conditions (assumptions) are the basis of pharmacokinetics.

1.1.1

Pharmacokinetic Parameters and Blood Drug Profile

(Concept of Cmax, Tmax, bioavailability, area under the curve (AUC), following intravenous bolus injection and infusion, therapeutic window, minimum effective concentration, maximum safe drug level/toxic concentration, lag time, onset of action, duration of action, first- and zero-order kinetics, biological half-life of drug) The blood level of a drug means the plasma concentration of the drug. While a conventional drug formulation for systemic drug effects is administered by any routes (such as intramuscular [i.m.], intraperitoneal [i.p.], oral, buccal, pulmonary, rectal, vaginal, nasal, etc.) other than the intravenous routes, drug upon administration is absorbed in the blood. Alteration of the drug amount in the blood varies with time and the route of administration. However, the trend of drug absorption generally remains the same. Initially, the amount of drug increases and eventually reaches the highest level (peak level), and then the drug level drops with the change in time (Fig. 1.1). When a drug is administered directly into the blood by intravenous (i.v.) injection or by intravenous infusion, the drug is distributed through the blood circulation and then from the blood to various tissues and organs, including the drug action site. Plasma concentration of drug alters differently from maximum to minimum levels with time (Fig. 1.2) since the drug absorption phenomenon does not occur here. Once a drug is absorbed and reaches the MEC, the therapeutic action of the drug is achieved. On and above the MEC, the drug provides its therapeutic action. However, there is an upper level of the drug, on and above which the drug concentration in the blood produces toxic action. It is called minimum toxic concentration or maximum safe drug concentration (MTC/MSC) of the drug. It is, therefore, essential to maintain the drug level between MEC and MTC to provide optimum drug action without any toxic manifestations. The plasma drug concentration between MEC and MTC of a drug is called its therapeutic window (Fig. 1.1). There are two essential terms, lag phase and the onset of drug action, in connection with a drug absorption phenomenon and blood level of the drug. Interestingly, if we want to define them, it seems similar. It is, therefore, important to understand the difference between the onset of drug action and the lag phase/lag time. When a drug is administered by any route other than the intravenous route, the length of time between the drug administration and achieving the MEC of the drug

4

1

Fundamentals of Pharmacokinetics

Fig. 1.2 The figure shows plasma drug concentration versus time curve that shows how a drug behaves after its single-dose administration by intravenous route. The MEC is the minimum effective drug concentration to begin therapeutic drug action. The MTC is the minimum toxic drug concentration, and it is also known as the maximum safe drug concentration or the MSC

(i.e., beginning of therapeutic action of the drug) is called the lag phase. On the other hand, onset is the time point at which the drug reaches MEC after the administration (interestingly, also measured as the length of time after drug administration at which drug reaches MEC). That is why, it states that the time point after drug administration to begin therapeutic drug action is called the onset of drug action. Therapeutically effective drug concentration starts when a drug reaches its MEC and continues up to when the drug from and above the MEC (after drug absorption) once again gets back the MEC (Fig. 1.1). It is also called the bioavailability of the drug. The bioavailability of a drug can be estimated in the most reliable way by determining the area under the plasma drug concentration–time curve (AUC) from its bioavailable region. The duration of drug action is the length of time when a drug provides therapeutic action after its administration in a sequence of reaching the drug level at the MEC, maintaining the drug level above the MEC, and then once again dropping it to the MEC. Within a therapeutic window, the time point at which plasma drug concentration reaches the maximum level after the administration of a drug is called peak time or Tmax, and the maximum plasma concentration of a drug is called Cmax. Further, for a single dose, a fraction of the dose absorbed (F) in the blood (bioavailable) at any time point is represented by multiplying dose by a ratio of drug absorbed in the blood at any time point to the total amount of drug (in a dose) administered. As stated earlier, drug administration directly into the blood by i.v. route does not possess any drug absorption phenomenon to the blood. For a single i.v. injection of a drug (Fig. 1.2), at the instantaneous moment of injection considering immediate diffusion, Cmax reaches, and that instantaneous time point is Tmax for a single i.v. injection of a drug. Again, the length of the period from the time of i.v. injection of a drug to reaching its MEC is the duration of action of the drug. However, as no absorption phenomenon is applicable here, the lag phase is equally not applicable for

1.1 Fundamentals of Pharmacokinetics

5

i.v. drug injection. The drug can provide therapeutic action immediately in the blood (e.g., antibiotics can start immediate action on bacteria present in the blood). But to reach the tissues to provide drug action, drug distribution from the blood to the tissues through the blood is necessary, and it needs additional time. The period also varies depending on the various physicochemical and biopharmaceutical properties of drugs. Pharmacokinetic ADME phenomena can be well described using various kinetic equations. The fundamental target of pharmacokinetics is to determine various rates, such as rate of drug absorption, rate of drug–protein bindings, rate of drug distribution, rate of drug metabolism, and rate of elimination of drug or its metabolites, or both, by using or developing rate equations. Besides, many mathematical equations and models are used to determine other drug-related phenomena such as steady-state plasma concentration, dose, drug plasma and drug tissue concentrations at various time points after the drug administration and many more. Different mathematical kinetic models follow drug behavior in our bodies. They include zero-order kinetics, first-order kinetics, mixed-order kinetics, and many more. The drug absorption or drug elimination sometimes undergoes essentially at a constant rate independent of the drug concentration. It is well represented as zero-order kinetics. In terms of the zero-order kinetic equation, if the amount of drug to be absorbed or eliminated is x at time t, then the rate of change of drug is represented by
dx dt ¼ k 0 , where k0 is the proportionality constant (here the rate of drug absorption/elimination, is always represented by the unit, mass/time). Minus (
) sign indicates that the amount of drug decreases from its initial amount in either process (absorption/elimination) with the increasing time. Now, if we integrate the equation from an initial time point of drug administration, i.e., say, “0” time to any time “t,” when at t ¼ 0, and the amount of drug present, say x0, and at a time “t,” the amount of drug in the blood, say, xt, then, we can write Z

t

Z dx ¼
k0

0

t

dt 0

or ½x

xt x0

¼
k0 ½t t0

or ½xt
x0  ¼
k0 ½t
0 or xt
x0 ¼
k0 t or xt ¼ x0
k0 t The half-life (t1/2) of a drug in zero-order kinetics is the amount of drug that becomes half of its initial amount, i.e., at half-life, xt becomes x20 , or

x0 ¼ x0
k0 t 1=2 2

6

1

or k 0 t 1=2 ¼ x0
or t 1=2 ¼

Fundamentals of Pharmacokinetics

x0 2

x0 2

k0

The unit of it is the unit of time, that is, hour or second. However, one of the kinetic models plays a significant role as it covers nearly 95% of drug behaviors in terms of rate equation in our body. It is the first-order kinetics. For most drugs, absorption, distribution, metabolism, and elimination follow the first-order kinetics as the drug concentration remains insufficient to saturate the kinetic mechanism of such phenomena (absorption, distribution, metabolism, and elimination) (Wagner 1970). It implies the amount-/concentration-dependent rate of change of drug in the body. According to the first-order kinetics, after the administration of drug directly into the blood (by i.v. route), the rate of change of the amount of drug (dx dt) at any time (t) is directly proportional to the amount of drug (x) present in the blood at that time. Thus,


dx /x dt

or
dx dt ¼ k:x, where k is the proportionality constant (here the rate of drug elimination from the blood) or


dx ¼ kdt x

Now, if we integrate the equation from an initial time point of injection, i.e., say, “0” time to any time “t,” when at t ¼ 0 (at the instantaneous moment of drug administration), the amount of drug present, say x0, and at a time “t,” the amount of drug in the blood, say, xt, then, we can write Z
0

t

dx dt ¼ k x

or
½ log e x

xt x0

Z

t

dt 0

¼ k½t t0

or
½ log e xt
log e x0  ¼ k ½t
0 or ½ log e xt
log e x0  ¼
kt or log e

xt ¼
kt x0

or xt ¼ x0 e
kt

ð1:1Þ

1.1 Fundamentals of Pharmacokinetics

7

During the elimination process, if the rate constant (k) is considered as KE and the amount of drug at time t is xt, the equation becomes xt ¼ x0 e
K E t In the case of an initial amount of drug (dose), (x0), is injected into the blood (by i. v. route) of a patient and after time “t,” the amount of drug in the blood is, say, xt, then xt ¼ x0 e
K E t where KE is the elimination rate constant and “e” is the base of the natural logarithm. Hence, xt ¼ x0 e
K E t may be considered as a significantly important and predominantly used equation of pharmacokinetics. This version of the equation is often used to describe different drug pharmacokinetic behaviors in our bodies. If the amount of drug in the blood at time t is xB, then xB ¼ x0 e
K E t

ð1:2Þ

From Eq. (1.1), we get ½ log e xt
log e x0  ¼
kt or kt ¼ log e x0
log e xt or kt ¼ log e

x0 xt

At the half-life (t1/2), the amount of drug becomes half of its initial amount, i.e., xt becomes x20 , or kt 1=2 ¼ log e

x0 x0 2

or kt 1=2 ¼ log e 2 or t 1=2 ¼

0:693 k

The unit of half-life (t1/2) is the unit of time, that is, second (s), minute (min), or hour (h).

1.1.2

Multiple-Dose Regimen

(Equations for determination of the minimum and maximum plasma drug concentrations in a multiple-dose regimen) In a multiple-dose regimen (when several doses of the same drug is given to a patient where each dose is given with a definite time gap, e.g., prolonged treatment of a patient with an antihypertensive drug/an antidiabetic drug, use of antibiotics,

8

1

Fundamentals of Pharmacokinetics

Fig. 1.3 In a multiple-dose (repeated-dose) regimen for oral/any routes other than the intravenous route of drug administration for systemic drug action, plasma drug concentration versus time curve for the first dose and subsequent doses show peak and valley appearance between the minimum effective drug concentration (MEC) and the minimum toxic drug concentration (MTC) levels in the blood. The maximum plasma concentration (Cmax) and time to achieve Cmax (Tmax) are shown after the first dose in the drug absorption curve. Every individual peak represents the Cmax after each dose at the respective Tmax

Fig. 1.4 In a multiple-dose (repeated-dose) regimen of the intravenous route of drug administration, plasma drug concentration versus time curve for the first dose and subsequent doses shows peak and valley appearance between the minimum effective drug concentration (MEC) and the maximum safe drug concentration (MSC) levels in the blood

etc.), the plasma concentration versus time curve commonly shows peak and valley appearance (Figs. 1.3 and 1.4). In a multiple-dose regimen, when a drug for systemic action is administered by any routes other than the i.v. route, while drug absorption takes place, the drug concentration in the blood eventually goes to a maximum level and then comes down

1.1 Fundamentals of Pharmacokinetics

9

to the MEC. The next administered dose should reach the MEC before the earlier dose touches at the MEC for maintaining a continuous plasma drug level for better therapeutic efficacy, and the same is true for subsequent doses and so on. These “up and down” curves from the doses provide a “peak and valley” appearance (Fig. 1.3). The length of time between administering the subsequent doses is called dose interval and denoted by “τ” (tau). When the drug is directly administered into the blood (i.v. route) in a multipledose regimen, the drug plasma concentration versus time curve also shows a “peak and valley” appearance (Fig. 1.4). In a multiple-dose (repeated-dose) regimen, the plasma concentration of a drug along with its maximum and minimum plasma levels against each dose can be determined. Let us consider that x0 is the amount of drug administered by i.v. route (i.v. dose) to a patient. We can then derive the equations for the plasma concentration of a drug along with its maximum and minimum plasma levels after each dose, as the maximum and minimum drug plasma levels vary in each case. The following equations are also applicable for the oral route of drug administration where rapid absorption is considered and x0 amount of drug available in plasma instantaneously after its oral administration. After the first dose, the maximum amount of drug (x1)max in the plasma is x0. If τ is the dosing interval, after the first dose, the minimum amount of drug in plasma is (x1)min and ðx1 Þmin ¼ x0 e
K E τ (where t = τ and τ is the time length at the end of which therapeutically effective minimum drug concentration is available) where KE is the elimination rate constant and “e” is the base of the natural logarithm. For the second dose, the maximum amount of drug, (x2)max, in plasma, is (x1)min + x0.   Therefore, ðx2 Þmax ¼ x0 þ x0 e
K E τ ¼ x0 1 þ e
K E τ Again, ðx2 Þmin ¼ ðx2 Þmax e
K E τ     ¼ x0 1 þ e
K E τ e
K E τ ¼ x0 e
K E τ þ e
2K E τ For the third dose, the maximum amount of drug, (x3)max, in plasma, is (x2)min + x0.   ðx3 Þmax ¼ ðx2 Þmin þ x0 ¼ x0 e
K E τ þ e
2K E τ þ x0   ¼ x0 1 þ e
K E τ þ e
2K E τ   ðx3 Þmin ¼ ðx3 Þmax :e
K E τ ¼ x0 1 þ e
K E τ þ e
2K E τ e
K E τ   ¼ x0 e
K E τ þ e
2K E τ þ e
3K E τ likewise, after the administration of n number of doses.

10

1

Fundamentals of Pharmacokinetics

For the nth dose, the maximum amount of drug, (xn)max, in plasma, is (xn
1)min + x0.   ðxn Þmax ¼ x0 e
K E τ þ e
2K E τ . . . . . . : þ e
ðn
2ÞK E τ þ e
ðn
1ÞK E τ þ x0   ¼ x0 1 þ e
K E τ þ e
2K E τ . . . . . . : þ e
ðn
2ÞK E τ þ e
ðn
1ÞK E τ For the nth dose, the minimum amount of drug, (xn)min, in plasma is ðxn Þmin ¼ ðxn Þmax e
K E τ   ¼ x0 1 þ e
K E τ þ e
2K E τ . . . . . . : þ e
ðn
2ÞK E τ þ e
ðn
1ÞK E τ e
K E τ us now consider that 1 þ e
K E τ þ e
2K E τ . . . . . . : þ e
ðn
2ÞK E τ þ e ¼s Then, (xn)max ¼ x0 s and ðxn Þmin ¼ x0 s:e
K E τ We know Let


ðn
1ÞK E τ

s ¼ 1 þ e
K E τ þ e
2K E τ . . . . . . : þ e
ðn
2ÞK E τ þ e
ðn
1ÞK E τ

ð1:3Þ

By multiplying both sides of the above Eq. (1.3) by e
K E τ , we get Eq. (1.4) (below):   e
K E τ s ¼ e
K E τ þ e
2K E τ þ e
3K E τ þ . . . . . . : þ e
ðn
2ÞK E τ þ e
ðn
1ÞK E τ þ e
nK E τ ð1:4Þ Now, subtracting Eq. (1.4) from Eq. (1.3), we get   s ¼ 1 þ e
K E τ þ e
2K E τ þ . . . . . . : þ e
ðn
2ÞK E τ þ e
ðn
1ÞK E τ   e
K E τ s ¼ e
K E τ þ e
2K E τ þ . . . . . . : þ e
ðn
2ÞK E τ þ e
ðn
1ÞK E τ þ e
nK E τ s
s:e
K E τ ¼ 1
e
nK E τ   s 1
e
K E τ ¼ 1
e
nK E τ s¼

1
e
nK E τ 1
e
K E τ

As we know, (xn) max ¼ x0 s, and ðxn Þ min ¼ x0 s:e
K E τ Therefore,

1.1 Fundamentals of Pharmacokinetics

11

  1
e
nK E τ ðxn Þmax ¼ x0 1
e
K E τ   1
e
nK E τ ðxn Þmin ¼ x0 :e
K E τ 1
e
K E τ Therefore, at any time t, the amount of drug present in the blood may be
K E τ determined using present outside the part h 
nK τ  i the equation for (xn)min, where τ of e 1
e E x0 1
e
K E τ is replaced by t, as the source of this t is from the variable time factor, 
nK τ  E portion is entirely a constant and obtained from the result of t. Further, 1
e 1
e
K E τ summation “s;” thus, replacement of any τ to t there leads to an error. Thus, the equation becomes   1
e
nK E τ ð xn Þ t ¼ x0 :e
K E t 1
e
K E τ (where (xn)t is the minimum amount of drug present in the plasma at any time t after the nth dose) Mostly as pharmacokinetic data, the plasma concentration of a drug is used instead of the amount of drug present in the blood/plasma (Gibaldi et al. 1969). If cp is the plasma concentration of drug and vd is the volume of distribution of the drug, then we can determine the minimum plasma concentration of the drug, maximum plasma concentration of the drug, and plasma concentration of the drug at any time t after the administration of “n” number of doses, by using the following equations:      x 1
e
nK E τ  cp max ¼ 0 : ,cp :vd ¼ x0
K τ E vd 1
e     x0 1
e
nK E τ cp min ¼ :e
K E τ vd 1
e
K E τ     x 1
e
nK E τ cp n ¼ 0 :e
K E t vd 1
e
K E τ     x0 1
e
nK E τ
K E τ cp min vd 1
e
KE τ :e   ¼ Therefore,   ¼ e
KE τ x0 1
e
nK E τ cp max
K τ vd

1
e

E

ð1:5Þ

ð1:6Þ

12

1.1.3

1

Fundamentals of Pharmacokinetics

Apparent Volume of Distribution or Volume of Distribution (vd)

The volume of drug distribution is more popularly known as the apparent volume of distribution of the drug in the body and is denoted by vd. The volume of drug distribution is a measure of the extent of drug distribution in the body. It is estimated by determining the resulting plasma concentration immediately when the drug is administered by rapid intravenous injection/quick bolus intravenous injection (Gibaldi and McNamara 1978; Levy and Yacobi 1974). The volume of drug distribution is represented by Vd ¼

amount of drug in the body plasma drug concentration

or Vd ¼

1.1.4

intravenous bolus dose immediately estimated plasma drug concentration

Steady-State Plasma Concentration of Drug

We have already seen above that the maximum and minimum plasma drug concentrations increase in the subsequent doses upon administering each dose in a multiple-dosage regimen. But if the doses are continued to administer at the correct dose intervals, the maximum and minimum plasma drug concentrations approach to constant levels at a particular time point. At this stage, the maximum plasma drug concentration becomes constant (a definite value), and the minimum plasma drug concentration also provides a constant value (another definite value) (Fig. 1.5). But those constant maximum and minimum plasma drug concentration values generally remain different for different drugs. At this stage, the average drug plasma level thus maintains a fairly consistent plateau level of the drug (Chiou 1979). This plasma drug concentration level is called steady-state plasma concentration/steady-state level of drug. It can be achieved by intravenous drug administration or drug administration by any route (e.g., oral route) (Figs. 1.5 and 1.6). Steady-state drug level is achieved as the overall intake of a drug remains fairly in a dynamic equilibrium with eliminating the drug. In a multiple-dosage regimen, generally, the time required to reach steady-state drug plasma concentration is equal to 4–5 biological half-lives (t1/2) of a drug. Steady-state plasma concentration/steady-state level is usually denoted by c or css. However, as stated earlier, it generally varies for different drugs. Thus, without any controversies, mathematically steady-state drug level must be achievable with infinite numbers of dosing, that is, “n” ¼ “ 1 . ” Therefore, at n ¼ 1, e
nK E τ approaches to zero (0); then,

1.1 Fundamentals of Pharmacokinetics

13

Fig. 1.5 The plasma drug concentration versus time curve upon drug administration by oral route/ any route other than the intravenous route for systemic drug action shows that in each of the subsequent doses, the maximum and minimum plasma drug concentration values increase during drug absorption at the initial phase. After administering several doses, the maximum and minimum plasma drug concentration values become constant. The average plasma drug concentration at the zone is called steady-state plasma drug concentration (css). At the steady state, the maximum constant plasma drug concentration value is called steady-state maximum plasma drug concentration (css)max. Likewise, the minimum constant plasma drug concentration value is called the steadystate minimum plasma drug concentration (css)min. MEC indicates minimum effective drug concentration, MTC denotes minimum toxic drug concentration

Fig. 1.6 The plasma drug concentration versus time curve upon drug administration by intravenous route shows that the maximum and minimum drug plasma concentration values increase upon drug administration at the initial phase in each of the subsequent doses. After administering many doses, the maximum and minimum plasma drug concentration values become constant. The average plasma drug concentration at the region is called steady-state plasma concentration (css). At the steady state, the maximum constant plasma drug concentration value is called steady-state maximum plasma drug concentration (css)max, and the minimum constant plasma drug concentration value is called the steady-state minimum plasma drug concentration (css)min. MEC indicates minimum effective drug concentration, MTC denotes minimum toxic drug concentration

14

1

ðcp Þ1 ¼

Fundamentals of Pharmacokinetics

  x0 1 :e
K E t ðPlease see the Eq:ð1:5Þ, where t is a variableÞ vd 1
e
K E τ ð1:7Þ

Similarly, we can obtain the maximum and minimum amounts of the drug in the blood and the maximum and minimum plasma concentrations of the drug at an infinite number of dosing (“n” ¼ “ 1 ”), while the drug is repeatedly administered in a regular dose interval at the steady state (Vaughan and Tucker 1976). Therefore, at “ ” n ¼ “ 1 , ” that is, at the steady state,  1
K τ 1
e E   1 The minimum amount of a drug in the blood is ðx1 Þmin ¼ x0 e
K E τ 1
e
K E τ   x 1 The maximum plasma concentration of a drug is ðc1 Þmax ¼ 0
K τ E vd 1
e The maximum amount of a drug in the blood is ðx1 Þmax ¼ x0



The minimum plasma concentration of a drug is ðc1 Þmin   x 1 ¼ 0 :e
K E τ
K τ vd 1
e E

ð1:8Þ

It is important to understand that (c1)max and (c1)min are the maximum and minimum plasma concentration values of a drug at the steady-state level (Fig. 1.5), and an average of these two concentration levels is the steady-state plasma concentration/steady-state level of a drug represented by c or css. In a multipledosage regimen, the average drug concentration in plasma at the steady state is very crucial as it provides the maximum efficacy of a drug without any toxicity. At the steady state, R in a τ dosing interval, c is represented by τ

c ¼

0

ðcp Þ1 dt τ

¼ ðAUC Þ,

at steady state during a dosing interval (AUC [area under Dose interval

the curve] here represents the drug-bioavailability during a dosing interval at the steady state)   At any time t, (cp)1 is again given by xv0d 1
e1
K E τ :e
K E t (where t is a variable; please see Eq. (1.7)). Z

τ 0

cp



Z

  x0 1
K E t dt
K E τ :e 0 vd 1
e    Z τ   x0 1 x0 1
K E t e dt , is entirely a constant value ¼ vd 1
e
K E τ 0 vd 1
e
K E τ

dt ¼ 1

τ

1.1 Fundamentals of Pharmacokinetics

15

 
K E t τ  
K E τ x0 1 e x0 1 e e
K E :0 ¼ or cp 1 dt ¼
vd 1
e
K E τ
K E 0 vd 1
e
K E τ
K E
K E 0  
K E τ   x0 1 e 1 x0 1 1 e
K E τ ¼ þ

¼ KE vd 1
e
K E τ KE vd 1
e
K E τ K E KE    x0 1 1
e
K E τ ¼ vd 1
e
K E τ KE Z τ   x or cp 1 dt ¼ 0 v KE d 0 R τ  cp 1 dt Therefore, c ¼ 0 τ x0 ¼ ðsteady‐state drug plasma concentrationÞ ð1:9Þ vd K E τ Z

τ



Then, c or css = vdxK0E τ [Please see Eq. (1.9).] In the case of intravenous infusion given at a fixed rate xτ0 ¼ k0 ðsayÞ Then, css ¼ or vd ¼

1.1.5

k0 vd K E

ð1:10Þ

k0 css K E

Drug Accumulation Factor

In a multiple-dosage regimen, accumulation of a drug in the body is usually expected. This excess drug accumulation in the body (Krüger-Thiemer 1968) often results in side effects and many unwanted harmful effects in a patient. The drug accumulation factor may be defined as a ratio of the steady-state plasma concentration of a drug to the average drug plasma concentration after its first dose in a multiple-dosage regimen. The drug accumulation factor can be calculated in advance and is a beneficial information to understand whether the drug would harm a patient due to its tissue accumulation upon its long-term use. The extent of drug accumulation in a patient can be estimated in different ways (Ronfeld and Benet 1977). One such deduction is described here. After the administration of n number of dosages of a drug in a patient with a regular dosing interval τ, the average plasma concentration of the drug is

16

1

Fundamentals of Pharmacokinetics

R τ  cp n dt cn ¼ 0 τ Again, plasma  drugconcentration at any time t after the nth dosing is   x0 1
e
nK E τ cp n ¼ vd 1
e
K E τ e
K E t [Please see Eq. (1.5).] Integrating the equation from 0 to τ, we get R τ  cp n dt cn ¼ 0 τ    Z τ Z τ x0 1
e
nK E τ x0 1
e
nK E τ
K E t ¼ dt ¼ e
K E t dt :e τvd 1
e
K E τ 1
e
K E τ 0 τvd 0     x 1
e
nK E τ , 0 is entirely a constant value τvd 1
e
K E τ   x0 1
e
nK E τ e
K E t τ ¼ τvd 1
e
K E τ
K E 0   
K τ x0 1
e
nK E τ e E e
K E :0 ¼
: τvd 1
e
K E τ
K E
K E  
K τ   x 1
e
nK E τ e E 1 x0 1
e
nK E τ 1 e
K E τ ¼ 0 þ

¼ KE τvd 1
e
K E τ KE τvd 1
e
K E τ K E KE    x 1
e
nK E τ 1
e
K E τ ¼ 0 τvd 1
e
K E τ KE  x0  1
e
nK E τ τvd K E  x0  Therefore, cn ¼ 1
e
nK E τ vd K E τ ¼

Again, the steady-state average plasma drug concentration css or c ¼ vdxK0E τ [Please see Eq. (1.9).]. Then, cn ¼ cð1
e
nK E τ Þ (by replacing the value of vdxK0E τ by c) or

 cn  ¼ 1
e
nK E τ c

When n is equal to 1, that is, after the first dose, cc1 ¼ ð1
e
K E τ Þ. Again, cc1 ¼ ð1
e1
K E τ Þ ¼ accumulation factor Therefore, accumulation factor } R} ¼ ð1
e1
K E τ Þ; since R is a ratio, it has no unit. Using this equation, merely using the drug elimination rate and the dosing interval in a multiple-(repeated) dose regimen, the extent of drug accumulation in

1.1 Fundamentals of Pharmacokinetics

17

a patient can be quantified. It is useful when a patient takes a particular medication regularly for an extended period, as seen for antidiabetic drugs, antihypertensive drugs, etc. Further, the factor may be used to calculate the loading dose of the drug in a repeated dosing regimen.

1.1.6

Krüger-Thiemer’s “Pharmacokinetic Factor”

Ekkehard Krüger-Thiemer established an equation of proportional relationship between the necessary effective drug concentration to the minimum inhibitory drug concentration at the equilibrium, that is, at the steady-state drug level. The proportionality constant is called Krüger-Thiemer’s “factor” or KrügerThiemer’s “pharmacokinetic factor.” It is denoted by “σ” (Krüger-Thiemer 1960; Krüger-Thiemer and Bunger 1965–1966). In a multiple-dose regimen, Krüger-Thiemer’s “pharmacokinetic factor” is a ratio of the necessary effective drug concentration to the minimum inhibitory drug concentration at the steady state: σ¼



or σ ¼ vd :

ðc1 Þ min

[Please see Eq. (1.8).]

x0 x0 vd

x0 vd

e
K E τ

1

ð1
e
K E τ Þ

or σ ¼

1
e
K E τ e
K E τ

The pharmacokinetic factor depends on the drug characters such as bactericidal, bacteriostatic, and degenerative and their antagonists. However, there is no rational method so far available to determine the factor. The value of the factor may be obtained by clinical observation and experience related to such drugs only.

1.1.7

Krüger-Thiemer Dose Ratio

In a multiple-dose regimen, the Krüger-Thiemer dose ratio (Krüger-Thiemer 1960; Krüger-Thiemer and Bunger 1965–1966) provides us with the concept of the loading dose. Krüger-Thiemer dose ratio is the ratio of a loading dose (x) to the maintenance dose (x0) of a drug. Thus, it is used to calculate the loading dose. Krüger1 Thiemer dose ratio is x x ¼ ð1
e
K E τ Þ ¼ R (drug accumulation factor). This equa0

tion enables us to determine the minimum fluctuation of drug level in plasma while the loading dose is administered in a multiple-dose regimen such as i.v. administration of antibiotics in severe infections.

18

1

1.1.8

Fundamentals of Pharmacokinetics

Concept of a Loading Dose

In the case of a severe infection or acute or severe health crisis, a high initial non-toxic dose of a drug is administered to control the progress of the disease quickly. It is then followed by the administration of maintenance dose (usual dose) of the drug to control the disease further. A loading dose is the sufficiently high first dose selected so that the minimum plasma concentration of which is equal to the minimum steady-state plasma concentration of the drug in its multiple-dose (repeated-dose) regimen. That is, (c1)min ¼ (c1)min Again, ðc1 Þmin ¼

 x0
K E τ

e ,ðc1 Þmin vd ¼ x0 e
K E τ , as for the first dose vd

and ðc1 Þmin ¼ xv0d ð1
e1
K E τ Þ e
K E τ ; when the first dose is a loading dose, x0 is replaced
K E τ by x* in the equation ðc1 Þmin ¼ xv0d e
K E τ . Then, for a loading dose, ðc1 Þmin ¼ x vd e
K E τ As per the definition of a loading dose, x ¼ xv0d ð1
e1
K E τ Þ e
K E τ vd :e or x ¼ x0 or

1 ¼ x0 R ð1
e
K E τ Þ

ð1:11Þ

x 1 ¼ ¼ R ðdrug accumulation factorÞ x0 ð1
e
K E τ Þ

The loading dose and the maintenance dose are determined using the following formula: v c Loading dose ¼ dF p, where F is the fraction of dose bioavailable, cp is the plasma concentration of drug, and vd is the volume of distribution Further, maintenance dose ¼ CL:cFss :τ (where CL is the clearance, css is the steadystate plasma drug concentration, τ is the dosing interval, and F is the fraction of dose bioavailable)

1.1.9

Relationship Between Elimination Rate Constant (KE) and Steady-State Drug Plasma Concentration (css) from Krüger-Thiemer Dose Ratio Concept

Since,

x 1 ¼ ¼R x0 ð1
e
K E τ Þ or x ¼ x0 R or

x x0 ¼ R vd v d

References

19



x x or css = cp R at any time t, is equal to css and 0 ¼ cp vd vd

 ð1:12Þ

Again, css ¼ τ vxd0K E [Please see Eq. (1.9).]  x ¼ css vd ¼ ¼

 x0 x v ¼ 0 τ vd K E d τ K E

  k0 x k0 ¼ 0 , that is, rate of infusion KE τ

ð1:13Þ

Again, the steady-state plasma concentration at any time t, 1 ½Please see Eq:ð1:12Þ: ð1
e
K E t Þ   or cp ¼ css 1
e
KE t

c or css ¼ cp :

ð1:14Þ

or cp ¼ css
css :e
K E t or e
K E t ¼

css
cp css

Taking natural logarithm, we get ln e
K E t ¼ ln

css
cp css

or
K E t ¼ ln

css
cp css

and converting the natural logarithm to the common logarithm (that is, the logarithm to the base 10), we get css
cp
K E t ¼ log 2:303 css or K E ¼


css
cp 2:303 log t css

ð1:15Þ

References Chiou WL (1979) Rapid compartment- and model-independent estimation of times required to attain various fractions of steady-state plasma level during multiple dosing of drugs obeying superposition principle and having various absorption or infusion kinetics. J Pharm Sci 68: 1546–1547

20

1

Fundamentals of Pharmacokinetics

Gibaldi M, McNamara PJ (1978) Apparent volumes of distribution and drug binding to plasma proteins and tissues. Eur J Clin Pharmacol 13:373–380 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 58:193–1997 Krüger-Thiemer E (1960) Dosage schedule and pharmacokinetics in chemotherapy. J Am Pharm Assoc 49:311–313 Krüger-Thiemer E (1968) Continuous intravenous infusion and multicompartment accumulation. Eur J Pharmacol 4:317–324 Krüger-Thiemer E, Bunger P (1965–1966) The role of the therapeutic regimen in dosage design. I. Chemotherapy 10:61–73 Levy G, Yacobi A (1974) Effect of plasma protein binding on elimination of warfarin. J Pharm Sci 63:805–806 Ronfeld RA, Benet LZ (1977) Interpretation of plasma concentration-time curves after oral dosing. J Pharm Sci 66:178–180 Vaughan DP, Tucker GT (1976) General derivation of the ideal intravenous drug input required to achieve and maintain a constant plasma drug concentration. Theoretical application to lignocaine therapy. Eur J Clin Pharmacol 10:433–440 Wagner JG (1970) “Absorption rate constants” calculated according to the one-compartment open model with first-order absorption: implications in in vivo-in vitro correlations. J Pharm Sci 59: 1049–1050

2

Drug Absorption

2.1

Drug Absorption and Determination of Drug Absorption Rate Constant “Ka”

When a drug is administered by any route other than the intravenous route for the systemic effect, the drug should be absorbed from the site of its administration to the blood. But for achieving the systemic effect, the drug should reach its minimum effective concentration (MEC) at least. Drug absorption is often a complicated process as it generally begins from the application site and continues till it comes to blood circulation. The process depends on the physicochemical and biopharmaceutical features of a drug. Drug absorption or rate of drug absorption is never determined directly from the amount of drug that reaches the blood. These data do not truly reflect drug absorption as other processes such as protein binding, drug distribution, drug elimination, drug metabolism, etc. take place simultaneously. Therefore, drug absorption or rate of drug absorption is determined from the drug amount remaining to be absorbed in various time points or by taking the residual drug concentrations. There are several methods available for the determination of drug absorption rate constant “Ka.” Some essential methods are discussed below. In a mass balance equation, xA ¼ xB þ xE: (where xA is the total amount of drug absorbed in the blood compartment, at any time t, xB is the amount of drug present in the blood compartment at time t, and xE is the amount of drug eliminated from the body, at time t). By differentiating both sides of the equation with respect to t, we get dxA dxB dxE ¼ þ dt dt dt

# The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B. Mukherjee, Pharmacokinetics: Basics to Applications, https://doi.org/10.1007/978-981-16-8950-5_2

ð2:1Þ

21

22

2

Drug Absorption

or, dxdtA ¼ dxdtB þ K E xB (, dxdtE ¼ K E xB , first-order elimination reaction, KE the elimi
 dðcp vd Þ nation rate constant) or, dxdtA ¼ dt þ K E cp vd (We know that mass ¼ concentration x volume; then, xB ¼ cp vd, where cp is the plasma drug concentration and vd is the volume of distribution of the drug. Again, KE and vd are constants for a drug.)

2.1.1

Dominguez Equation and Its Importance


 d ð cp Þ The equation dxdtA ¼ vd dt þ K E vd cp is called the original Dominguez equation (Dominguez 1934; Dominguez and Pomerene 1934, 1945). For the first time, Dominguez showed that the rate of absorption of a drug could be estimated as a function of time using the equation. He provided the concept of the volume of distribution of drugs (Dominguez 1934; Dominguez and Pomerene 1945).

2.1.2

Wagner–Nelson Equation and Method of Determination of Drug Absorption Rate Constant

Wagner and Nelson modified the Dominguez equation to develop the well-known Wagner–Nelson method (Wagner and Nelson 1963, 1964) to determine drug absorption rate, Ka, while the entire body is considered a single compartment, a homogeneous drug distribution system. By integrating the above equation for the duration “0” to “T,” the following equation is obtained:  ZT  ZT
 ZT
 d cp dxA dt þ K E vd cp dt ðsince vd and K E , are constantsÞ dt ¼ vd dt dt 0

0

or RT

RT 0

0

RT RT
 dxA ¼ vd dcp þ K E vd cp dt 0

(For the clarification of calculation,

0

dxA ¼ ðxA ÞT0 ¼ ðX A ÞT  ðX A Þ0 ¼ ðX A ÞT , as there was no drug absorption at

0

T ¼ 0. Thus, the integrated amount of drug absorbed in the blood at the duration “0” to “T” is (XA)T and similarly the concentration of drug in blood at “T,” say, (Cp)T),


or ðX A ÞT ¼ vd Cp

Z

 T

þ K E vd

T

cp dt 0

2.1 Drug Absorption and Determination of Drug Absorption Rate Constant. . .

23

Fig. 2.1 Plasma drug concentration versus time curve after the drug absorption following a single dose by oral route/any routes other than the intravenous route of drug administration for systemic RT drug action. The area under the curve, shown by stars and depicted by 0 cp dt, shows bioavailability of the drug during the period of 0–T, after its administration. MEC indicates minimum effective drug concentration

When T ¼ 1 (infinity), vd(Cp)T ¼ 0; since there would be no drug possibly available in the blood at an infinite time, the eliminated amount of drug would be maximum. Z Hence, ðX A Þ1 ¼ K E vd RT

1

cp dt 0

T (AUC, the area under the curve from “0” to “T,” 0 R1 indicates bioavailability of drug from “0” to “T”) (Fig. 2.1) and 0 cp dt ¼ Again,

0

cp dt ¼ ðAUCÞ

1 (AUC, the area under the curve from “0” to “1,” indicates the total 0 bioavailability of the drug) (Fig. 2.2). ðX Þ Therefore, part of the drug absorbed at time “T” is ðX AAÞ T ðAUCÞ

1






RT

vd Cp T þ K E vd 0 cp dt ðX A ÞT R1 ¼ ðX A Þ1 K E vd 0 cp dt
 RT C p T þ K E 0 cp dt ðX A ÞT R1 ð2:2Þ ¼ or ðX A Þ1 K E 0 cp dt RT   c dt ðCp ÞT þK ðX Þ R 1E 0 p The percentage of drug absorbed is ðX AAÞ T  100 ¼  100 1

KE

0

cp dt

24

2

Drug Absorption

Fig. 2.2 Plasma drug concentration versus time curve after drug absorption following a single dose by oral route/any routes other than the intravenous route of drug administration for systemic drug R1 action. The area under the curve, shown by stars and depicted by 0 cp dt, shows the total bioavailability of the drug. MEC indicates minimum effective drug concentration

h i ðX Þ Part of drug remaining to be absorbed ¼ 1  ðX AAÞ T 1

Percentage of drug remaining to be absorbed " #
 RT  C p T þ K E 0 cp dt ðX A ÞT R1 ¼ 1  100 ¼ 1   100 ðX A Þ1 K E 0 cp dt 

By plotting the data of the percentage of drug remaining to be absorbed (on a logarithmic scale) on the y-axis against time (on a linear scale) on the x-axis on a semi-logarithmic paper, a straight line is obtained. The slope of it gives the value of K a (Fig. 2.3). This method of obtaining the Ka value in a singleKa, where slope ¼ 2:303 compartment system (the whole body is considered a single compartment) is known as the Wagner–Nelson method. The Wagner–Nelson method involves the determination of Ka from the “percentage of drug remaining to be absorbed versus time” plot. The procedure does not require any drug absorption assumptions, such as zeroor first-order drug absorption kinetics. Thus, the Wagner–Nelson method allows us to determine drug absorption rate (Ka) without any pre-assumption of drug absorption kinetics. However, the Wagner–Nelson method requires the following assumptions: 1. The body behaves as a single homogeneous compartment. 2. Drug elimination obeys the first-order kinetics.

2.1 Drug Absorption and Determination of Drug Absorption Rate Constant. . .

25

Fig. 2.3 Percentage of drug remaining to be absorbed when plotted against time on a semi-log paper, keeping “percentage of drug remaining to be absorbed” on a log scale and time on a linear K a where Ka provides the value of absorption scale, gives a straight line. The slope of the line is 2:303 rate constant

2.1.3

Determination of Absorption Rate Constant (Ka) from Urinary Excretion Data

Upon administering a drug, the rate of absorption of drug “Ka” can be determined from urinary excretion data of unchanged drug excreted through urine. If xu is the amount of unchanged drug excreted through urine at time t, the fraction of unchanged drug and rate of elimination of unchanged drug excreted through urine can be deduced. Using those data, rate of drug absorption can be determined too (Wagner 1967). The rate of unchanged drug eliminated through urine is dX u dt ¼ K e xB (where Ke is the elimination rate constant of unchanged drug excreted through urine and the amount of drug present in the blood at time t is xB)
or dxdtu ¼ Ke x0 eKE t ,xB ¼ x0 eKE t ; please see Eq. (1.2)) Rt By integrating the equation from 0–t with respect to t, we have 0 dxdtu dt ¼ R t K t K e x0 0 e E dt Z or

t

Z

t

dxu ¼ K e x0

0

eK E t dt

0

or xu  x 0 at time “t” ¼ 0, x ⟶0 u

 Z t  0 if dxu ¼ ½xu t  ½xu 0 ¼ xu  x u 0 0 u

¼ K e x0

 K t  e E t K E 0

26

2

Drug Absorption

 K t   K t  e E eK E :0 e E 1 Therefore, xu ¼ K e x0  þ ¼ K e x0  KE K E K E KE   1 eK E t ¼ K e x0  KE KE    1  eK E t K x
¼ K e x0 ¼ e 0 1  eK E t KE KE ∴ xu ¼

K e x0 ð1  eK E t Þ KE

Again, at t ¼ 1, if the cumulative amount of drug excreted is x

x

1 u

¼

1 , then u

 K x K e x0
K x 1  eK E :1 ¼ e 0 ð1  0Þ ¼ e 0 KE KE KE ∴x

1 K e x0 ¼ KE u x

u x0 ¼ Ke KE

Then, x or

1

1 u K ¼ e x0 KE

ð2:3Þ

The above equation shows the total fraction ( f ) of unchanged drug excreted through urine. u Again, dX dt ¼ K e xB (where Ke is the elimination rate constant of unchanged drug excreted through urine and the amount of drug present in the blood at time t is xB) or


 dxu ¼ K e cp vd since xB ¼ cp vd dt or cp vd ¼

ð2:4Þ

1 dxu : ¼ xB K e dt

ð2:5Þ

Again, dxdtu ¼ K e vd cp ¼ KKEe vd cp K E [Please see Eq. (2.4) (by multiplying both the numerator and denominator of Eq. (2.4), by KE).] u or dX dt ¼ f K e vd cp (where f is the total fraction of unchanged drug excreted through 1 1 x

urine and f ¼

u x0

x

¼ KKEe ) [Please see Eq. (2.3) (since KxE0 ¼

u Ke

).]

2.1 Drug Absorption and Determination of Drug Absorption Rate Constant. . .

2.1.4

27

Nelson Equation

Again, dxdtA ¼ dxdthB þ dxdtE i[Please see Eq. (2.1).] or dxdtA ¼ dtd K1e : dxdtu þ K E : K1e : dxdtu ; [, dXdtE ¼ K E xB ¼ KKEe : dxdtu ] as xB ¼ K1e : dxdtu [Please see Eq. (2.5).] 2 or dxdtA ¼ K1e : ddtx2u þ KKEe : dxdtu . This is known as the Nelson equation (Nelson 1959; Nelson and Schaldemose 1959).

2.1.5

Wagner and Nelson Equation

The original Nelson equation was later modified by Wagner and Nelson (Wagner and Nelson 1964). " dX u ¼ f K e vd cp ; then cp ¼ We know dt

or

ðX A ÞT ðX A Þ1

¼

KE

see Eq. (2.6) is constantÞ.]

TþK E

dt f Ke V d

0 R 1 h dXdtu i 0

f Ke V d

(by canceling

dt

¼

f K e V d : dt

KE

dt 1 f KeV d

þK E

R 1 T

0

ð2:6Þ

f K e vd

RT c dt ðCp ÞT þK ðX Þ R 1E 0 p [Please see Eq. (2.2).] Again, ðX AAÞ T ¼ 1 KE cp dt  h dXu i 0 R h dXu i R T T dX u 1 dt f Ke V d

#

dX u dt

0

dX u 1 f K e V d : dt

dX u 1 f K e V d : dt



dt

[Please

dt

from the numerator and denominator as it


dx  u

¼


dx 
dx  R T
 RT u u þ K E 0 dxdtu dt þ K E ð xu Þ T dt T þ K E 0 dxu R R 1
dxu  ¼ dt T ¼ 1 K K dx K E 0 dt dt E ð xu Þ 1 E 0 u

dt T

Therefore, part of drug remaining to be absorbed ¼   ðdxu Þ þK E ðxu Þ 1  dt KT E ðxu Þ T

h i ðX Þ 1  ðX AAÞ T ¼ 1

1

Percentage of drug remaining to be absorbed " #
dx   u ðX A ÞT dt T þ K E ðxu ÞT ¼ 1  100 ¼ 1   100 ðX A Þ1 K E ð xu Þ 1 

Now by plotting the data of the percentage of drug remaining to be absorbed (on a logarithmic scale) on the y-axis against time (on a linear scale) on the x-axis on a semi-logarithmic paper, a straight line is obtained. The slope of the line gives the K a value of Ka, where slope ¼ 2:303 (Fig. 2.4). Here, Ka is determined based on the data of the unchanged drug excreted through urine in a single-compartment system

28

2

Drug Absorption

Fig. 2.4 Percentage of drug remaining to be absorbed when plotted against time on a semi-log paper, keeping “percentage of drug remaining to be absorbed” on a log scale and time on a linear K a where Ka provides the value of absorption scale, gives a straight line. The slope of the line is 2:303 rate constant

(whole body is considered a single compartment) using the Wagner–Nelson method (Wagner 1963).

2.1.6

Loo–Riegelman Method for Determination of Drug Absorption Rate (Ka)

The rate of drug absorption (Ka) can be determined by the Loo–Riegelman method (Loo and Riegelman 1968). The method is used for a two- or multi-compartment model. The whole body is virtually compartmentalized into several compartments, keeping the blood compartment as the central compartment. The compartment model has been discussed in detail in Chap. 4 in this book. However, plasma concentration–time data at various time points, both from oral and intravenous administrations of the same drug in the same subject, are required for the Loo– Riegelman method. In a mass balance equation, xA ¼ xB þ xP þ xE (where xA, xB, xP, and xE are the total amount of drug absorbed, amount of drug present at the central compartment, i.e., blood compartment, amount of drug present in the peripheral compartment, that is, in tissue/organ compartment, and amount of drug eliminated from the body, respectively, at any time point t). By differentiating both sides of the equation with respect to t, we get

2.1 Drug Absorption and Determination of Drug Absorption Rate Constant. . .

29

dxA dxB dxp dxE ¼ þ þ dt dt dt dt   dxp dx dx dx or A ¼ B þ þ K E xB , E ¼ K E xB dt dt dt dt

 d ð c vc Þ dxA d cp vd þ K E cp vd þ or ¼ dt dt dt (We know that mass ¼ concentration  volume; then, xB ¼ cp vd; xP ¼ c vc, where cp is the plasma concentration of the drug, vd is the volume of distribution of drug, c is the organ/tissue concentration of drug, and vc is the volume of distribution of drug at the tissue (peripheral) compartment. Again, KE, vd, and vc are constants.)

 d cp d ð cÞ dxA Therefore, þ vc þ K E vd cp ¼ vd dt dt dt ZT 

dxA dt

 dt ¼ vd

0

ZT
 ZT ZT
 d cp d ð cÞ dt þ vc dt þ K E vd cp dt dt dt 0

0

0

(as vd, vc, and KE are constants) ZT

ZT dxA ¼ vd

or 0

ZT dcp þ vc

0

ZT dc þ K E vd

0


 cp dt

0

If we integrate the equation as mentioned above from time “0” (initial time point considered as time zero) to “T,” we get


ðX A ÞT ¼ vd C p

Z

 T

þ vc ð C Þ T þ K E vd

T

cp dt 0

(where the integrated amount of drug absorbed in the blood at the duration “0” to “T” is (XA)T, the integrated amount of drug absorbed in the plasma compartment from “0” to “T” is (Cp)T, and the integrated amount of drug absorbed in the tissue compartment from “0” to “T” is (C)T). When T ¼ 1 (infinity), that is, at infinite time, vd(Cp)T ¼ 0, and vc(C)T ¼ 0, since there would be no drug possibly available in the blood and tissue compartments at the infinite time and eliminated amount of drug would be maximum. R1 ðX Þ Hence, ðX A Þ1 ¼ K E vd 0 cp dt; therefore, part of the drug absorbed is ðX AAÞ T 1

30

2

Drug Absorption

Fig. 2.5 Percentage of drug remaining to be absorbed when plotted against time on a semi-log paper, keeping “percentage of drug remaining to be absorbed” on a log scale and time on a linear K a where Ka provides the value of absorption scale, gives a straight line. The slope of the line is 2:303 rate constant

or

ðX A ÞT ðX A Þ1


 RT vd C p T þ vc ðC ÞT þ K E vd 0 cp dt ðX A ÞT R1 ¼ ðX A Þ1 K E vd 0 cp dt RT ðC p ÞT þvvc ðCÞT þK E cp dt 0 d R ¼ (by dividing the numerator and the denominator of 1 KE

0

cp dt

the equation by vd) ðX Þ

Percentage of drug absorbed is ðX AAÞ T  100 ¼ 1



ðC p ÞT þvvc ðCÞT þK E d

R1

RT 0

h K E 0 cp dt i ðX Þ Therefore, part of drug remaining to be absorbed ¼ 1  ðX AAÞ T

cp dt

  100

1

Percentage of drug remaining to be absorbed " #
 RT  C p T þ vvdc ðC ÞT þ K E 0 cp dt ðX A ÞT R1 ¼ 1  100  100 ¼ 1  ðX A Þ1 K E 0 cp dt 

Now by plotting the data of the percentage of drug remaining to be absorbed (on a logarithmic scale) on the y-axis against time (on a linear scale) on the x-axis on a semi-logarithmic plot, a straight line is obtained. The slope of it gives the value of Ka K a where slope ¼ 2:303 (Fig. 2.5).

2.1 Drug Absorption and Determination of Drug Absorption Rate Constant. . .

2.1.7

31

Method of Residual for Determination of Drug Absorption Rate

The method of residual is a commonly used pharmacokinetic method to determine the absorption rate (Ka) of a drug. Drug plasma concentration data upon extravascular drug administration that follows a one-compartment first-order kinetic model are considered for the residual method. The kinetic is usually represented by a biexponential equation as given below: cp ¼ AeK a t þ BeK E t After administering a drug by extravascular route, when plasma concentration data of drug absorption is plotted against time, a curve of total plasma bioavailability of the drug is obtained. The terminal linear portion of the curve is extrapolated to the plasma concentration axis (y-axis) (Fig. 2.6). The respective extrapolated values against the actual plasma concentrations of the drug are determined at all the drug

Fig. 2.6 The figure shows the residual plot of the plasma drug absorption curve following the oral route of a single-dose drug administration. Plasma drug concentrations on the curve provide the actual drug absorption data (shown by “actual drug concentration”). The post absorption terminal linear portion down the curve when extrapolated to the plasma drug concentration axis (y-axis), the data on the line provide the extrapolated drug concentrations. The difference between the extrapolated drug concentration and the actual drug concentration gives the value of residual drug concentration (CR). When log CR is plotted against time, the slope of the line provides the K a , where Ka is drug absorption rate. The “vs.” stands for “versus” value of 2:303

32

2

Drug Absorption

absorption time points studied. By subtracting the value of actual plasma concentra ! tion of the drug (c) from the extrapolated plasma concentration of the drug c at each absorption time point, the respective residual plasma drug concentration is determined. Therefore, the residual plasma drug concentration (cR) is given by !

cR ¼ c  c During the initial absorption phase, when the drug absorption was predominant, it has finite values, and the elimination was negligible, suggesting for BeK E t approaches to zero (0), and then, cR ¼ A eK a t or log cR ¼ log A 

Kat 2:303

If we plot log cR against time t, the equation provides
a straight  line (the feathered Ka line), and the slope of the line gives us the value of Ka  2:303 . The intercept on the plasma concentration axis (y-axis) gives us the value of logA. This residual method is also known as the feathering method (back projection), stripping method, or peeling method. The residual value is the difference between ! a value of extrapolated plasma drug concentration ( c ) and the corresponding actual plasma drug concentration (c) at each absorption time point. The determination of residual plasma drug concentration (cR) is like removing the peel from a fruit or separating one or more components of some material (stripping). That is why the method of residual is also named so. Lag time (t0) signifies that drug absorption does not occur immediately upon extravascular drug administration and is delayed. The intercept of the extrapolated terminal line and the feathered line obtained by plotting the residual concentration should ideally meet at the same point on the yaxis, and lag time is zero if drug absorption occurs immediately following its extravascular administration. A positive value of lag time (t0) signifies that drug absorption does not appear immediately upon drug administration by the extravascular route and is delayed. However, in many plots, these two lines meet across the yaxis and can thus produce a negative lag time (Fig. 2.7). The negative lag time that indicates drug absorption occurring before its administration is an error in a residual plot. Under such a situation, one should follow some other highly accurate mathematical methods such as the Wagner–Nelson method (single-compartment model), Loo–Riegelman method (considering single-compartment model), etc. to estimate the absorption rate constant.

2.1.8

Flip-Flop Phenomenon

The flip-flop phenomenon is usually associated with extravascular drug administration. Flip-flop occurs when absorption (Ka) is predominantly slower than the

2.1 Drug Absorption and Determination of Drug Absorption Rate Constant. . .

33

Fig. 2.7 The figure shows that the extrapolated line and the feathered (residual) line with a slope K a 2:303 of the plot provide negative lag time. The meeting point of the extrapolated line and the residual line gives the value of lag time t0. In the graph, a negative lag time (which indicates erroneous findings) is shown. Here, logarithm values of residual drug concentration logcR are plotted against time t

elimination rate (KE). The flip-flop phenomenon should not be ignored as it can lead to a misinterpretation of pharmacokinetic parameters. While suspecting a flip-flop disposition (the fate of the drug for undergoing ADME), a long-term sampling is desired to avoid overestimating the absorbed dose fraction (F). The important causes of the flip-flop phenomenon include sustained/controlled release formulations, physicochemical and biopharmaceutical characteristics of the drug, and those of the excipients. In pharmacokinetics, the flip-flop phenomenon often occurs between a sustained/controlled release formulation and an immediate-release drug formulation. For drugs showing flip-flop disposition, a switchover takes place in Ka for KE. Hence, the elimination phase of the drug profile exhibits the input Ka, instead of the output KE. Again, a much slower Ka becomes the rate-limiting step (KE  Ka), which enhances the biological half-life of the drug (Yáñez et al. 2011). The kinetic pattern of the two-compartment open model (described later in Chap. 4) with the first-order absorption is usually represented by a biexponential equation as given below: cp ¼ AeK a t þ BeK E t In the residual plot, when Ka  KE, the slope of the residual line determines KE instead of Ka, and the terminal slope determines Ka instead of KE (Fig. 2.8). More specifically, the slopes of the two lines interchange their meanings. It is called the flip-flop phenomenon. For extravascular (Fig. 2.8) and intravascular (Fig. 2.9) drug administration, such plots are given below.

34

2

Drug Absorption

Fig. 2.8 Residual plot of extravascular drug administration. Logarithm plasma concentration, logcp, data plotted against time that shows the slope of the terminal linear portion of the line K a where Ka is the rate of drug absorption. The slope of the feathered line provides the value of 2:303 K E gives the value of 2:303 where KE is the rate of elimination

Fig. 2.9 Residual plot of intravascular drug administration. Logarithm plasma concentration, logcp, data plotted against time that shows the slope of the terminal linear portion of the line K a where Ka is the rate of drug absorption. The slope of the feathered line provides the value of 2:303 K E gives the value of 2:303 where KE is the rate of elimination

References

35

References Dominguez R (1934) Studies of renal excretion of creatinine. III. Utilization constant. Proc Soc Exp Biol Med 31:1150–1154 Dominguez R, Pomerene E (1934) Studies of the renal excretion of creatinine. I. On the functional relation between the rate of output and the concentration in the plasma. J Biol Chem 104:449– 452 Dominguez R, Pomerene E (1945) Calculation of the rate of absorption of exogenous creatinine. Proc Soc Exp Biol Med 60:173–181 Loo JCK, Riegelman S (1968) New method for calculating the intrinsic absorption rate of drugs. J Pharm Sci 57:918–928 Nelson E (1959) Influence of dissolution rate and surface on tetracycline absorption. J Am Pharm Assoc 48:96–103 Nelson E, Schaldemose I (1959) Urinary excretion kinetics for evaluation of drug absorption I: solution rate limited and nonsolution rate limited absorption of aspirin and benzyl penicillin; absorption rate of sulfaethylthiadiazole. J Am Pharm Assoc 48:489–495 Wagner JG (1963) Some possible errors in the plotting and interpretation of semilogarithmic plots of blood level and urinary excretion data. J Pharm Sci 52:1097–1101 Wagner JG (1967) Method for estimating rate constants for absorption, metabolism, and elimination from urinary excretion data. J Pharm Sci 56:489–494 Wagner JG, Nelson E (1963) Percent absorbed time plots derived from blood level and/or urinary excretion data. J Pharm Sci 52:610–611 Wagner JG, Nelson E (1964) Kinetic analysis of blood levels and urinary excretion in the absorptive phase after single doses of drug. J Pharm Sci 53:1392–1403 Yáñez JA, Remsberg CM, Sayre CL, Forrest ML, Davies NM (2011) Flip-flop pharmacokinetics— delivering a reversal of disposition: challenges and opportunities during drug development. Ther Deliv 2:643–672

3

Extent of Drug Absorption: Bioavailability, Clearance, Bioequivalence, and Protein Binding

3.1

Extent of Drug Absorption: Bioavailability

The extent of drug absorption is determined to understand how much drug would be bioavailable from an administered dose. Bioavailability of drug means the amount of drug that has reached to provide drug concentration in the blood on and above the minimum effective concentration (MEC) upon its administration. The bioavailability is once again classified as absolute bioavailability and relative bioavailability. The ratio of the bioavailability of a drug administered by any route other than the intravenous route to the bioavailability of the drug administered by the intravenous route is known as absolute bioavailability. But in many circumstances, it is impossible to obtain the bioavailability data of a drug of the intravenous routes (as those drugs cannot be administered by i.v. route). In those cases, bioavailability of a drug given by any routes (obviously not i.v. route) is compared to that of the drug administered by any routes other than that route. When plasma or whole blood or urinary excretion data of a drug upon its administration by any routes other than the intravenous route are compared to those upon the drug administration by any routes other than the same route or the intravenous route, the ratio of the bioavailability of the drug is called relative bioavailability. R 1  0 cp dt oral ðoÞ  Absolute bioavailability ¼ R 1 0 cp dt intravenous ðIÞ   1 ðAUCÞ oral ðoÞ 0  ¼ : 1 ðAUCÞ intravenous ðIÞ 0

# The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B. Mukherjee, Pharmacokinetics: Basics to Applications, https://doi.org/10.1007/978-981-16-8950-5_3

37

38

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Extent of Drug Absorption: Bioavailability, Clearance, Bioequivalence, and. . .

  R 1  cp dt I and 0 cp dt O are the respective total bioavailability ðAUCÞ1 0 ] of a drug administered by intravenous route and oral or any other drug administration routes other than the intravenous route, respectively. R 1 0

ð

R1

0 cp dtÞoral ðoÞ Relative bioavailability ¼ R 1 ð 0 cp dtÞany route than the intravenous route

 ðAUCÞ ¼



1 0

 oral

1 any route than the intravenous route 0 R 1  R 1  0 cp dt o and 0 cp dt any other route than the intravenous route are the respective total bioavailability of a drug administered by oral route and any routes (e.g., intraperitoneal, intramuscular, etc.) of administration of the drug other than the intravenous route, respectively. Relative bioavailability may be determined by comparing the total bioavailability of any two routes other than the intravenous route. The area under the plasma drug concentration versus time curve (AUC) determines the bioavailability of a drug as seen earlier, and it is represented by AUC ¼ cτss. Other than the trapezoid rule, the area under the curve (AUC) can also be calculated by other methods, namely, weighing and platometry or cutting and weighing method, counting squares method, and integration of curve. Some important relationship of total bioavailability ((AUC)0  1) of a drug with its bioavailability from time 0 to time t, (AUC)0  t, is given by ðAUCÞ

ðAUCÞ01 ¼ ðAUCÞ0t þ

ct ke

ðAUCÞ01 ¼ ðAUCÞ0t þ 1:44 t 1=2 ct (where ct is the plasma drug concentration at time t, ke is the rate of urinary elimination of drug, and t 1=2 is the half-life of the drug). Renal clearance is well implicated with the bioavailability of drugs. Hence, the knowledge of renal clearance is necessary at this stage and is described below.

3.1.1

Renal Clearance

Clearance in pharmacokinetics is assessing the volume of plasma from which a substance (here, drug) is completely eliminated per unit time. About the renal clearance, the volume of blood that is cleaned from drug substance(s)/drug metabolite(s)/xenobiotics through the kidneys per unit time in a patient is known as renal clearance. Thus, more precisely, renal clearance/clearance (CL) is the volume of the blood containing the substance(s) such as drugs or metabolites or xenobiotics, which is/are

3.1 Extent of Drug Absorption: Bioavailability

39

filtered out of the blood per unit time by renal filtration in a patient (Rowland et al. 1973). Clearance is measured in L/h or mL/min. It quantitatively reflects the rate of drug elimination divided by plasma drug concentration. Although its unit suggests flow volume per unit time, it never refers to an actual volume; instead, it relates to the volume containing the substance that is eliminated/ cleared from the blood/plasma. Clearance ðCLÞ ¼ rate of excretion=plasma concentration   dxu dx ¼ dt = K e vd , u ¼ K e xB ¼ K e cp vd cp dt

x The equation of total renal clearance is K e vd ¼ R 1 0

1 u Cp dt

ð3:1Þ

1 (where x is the total amount of unchanged drug excreted through urine and u R1 0 cp dt is the total bioavailability of the drug in terms of concentration). Clearance (CL) is usually represented by vdke. There are other available equations to determine CL. Clearance ¼ UV P mL/min (where drug concentration in urine is U, the drug concentration in plasma is P, and volume of urine per minute is V ). ðDÞ F:D or AUC (for oral dose, F is the fraction of dose Similarly, Clearance ¼ Dose AUC bioavailable).

3.1.2

Determination of Absolute Bioavailability

We know dxdtu ¼ K e xB ¼ K e vd cp (,xB ¼ cpvd) By integrating the equation with respect to t, from 0 to 1, we have Z1 

dxu dt



Z1 dt ¼ K e vd

0

Z1

Z1 dxu ¼ K e vd

0

R1

 cp dt

0

or 1





 cp dt

0

1 cp dt (where x is the total amount of unchanged drug u u 0 excreted through urine) Here, or x

¼ K e vd

40

3

Extent of Drug Absorption: Bioavailability, Clearance, Bioequivalence, and. . .

1 x u vd ¼ R 1 K e 0 cp dt x but x

or

1 u K ¼ e x0 KE

1 u Ke

¼ Kx0E [Please see Eq. (2.3).] Then, vd ¼ Z

1

Therefore, 

¼



x R 10 0

cp dt

cp dt ¼

0



x0 vd K E

ð3:2Þ

x0 0 vd K E I is the total bioavailability of a drug by I

R 1  is the total bioavailadministration and 0 cp dt O ¼ F: vdxK0 E O

Thus, for a drug, intravenous drug

R 1

KE

cp dt

ability of the drug by oral drug administration. Drug absorption by the oral route is not considered 100%. Hence, F is the fractional multiple of the intravenous dose that is required by the drug given by the oral route to provide oral bioavailability. R1

x cp dtÞO F:ðvd K0 E ÞO ðDoseÞO ¼ ¼ Thus, absolute bioavailability ¼ R01 x0 ðvd K E ÞI ðDoseÞI ð 0 cp dtÞI

ð

Hence, F ¼ absolute bioavailability  or F ¼ absolute bioavailability 

ðK E ÞO ðK E ÞI

ðt ½ ÞI ðt ½ Þo

(x0)O and (x0)I are equal (that is, no F exists), and the drug is 100% bioavailable by both oral and intravenous routes, and the dose is the same by both the routes. When vd and KE values are also the same by both the routes, respectively, and are independent of rate and time of administration, the absolute bioavailability ¼ F. However, vd is constant for a drug and depends on the total amount of drug present in the body and the plasma drug concentration.

3.1 Extent of Drug Absorption: Bioavailability

3.1.3

41

Determination of Absolute Bioavailability by Urinary Excretion Data

For many drugs, a substantial amount of the unchanged drug excretes through urine. For those drugs, estimation of absolute bioavailability can be determined by comparing the data of the total drug amount excreted through urine upon the drug administration by any routes other than the intravenous route (say, oral route) to the intravenously administered drug amount excreted through urine.   1 x u O  Hence, absolute bioavailability ¼  1 x u I X

1

Further, we know K e vd ¼ R 1

u

[Please see Eq. (3.1).]

R1 1 or x ¼ K e vd 0 cp dt ¼ ðK e vd Þ V dxK0 E [Please see Eq. (3.2).] u 0

or

cp dt

    1 x0 ¼ ðK v Þ x vd K E I e d I u I t

x ⎸t21

u Renal clearance (Kevd) at any time interval t1 to t2 is given by ¼ R t2 t1

cp dt

(where

xu ⎸tt21 is the amount of Runchanged drug excreted through urine during the period t between t1 and t2 and t12 cp dt is the bioavailability of the drug during the period between t1 and t2)  or

x

1 u



 ¼ I

x0 V dKE

" I

# xu ⎸tt21 R t2 I t 1 cp dt

Again, for the oral route " of#drug administration,  

t 1 x ⎸2 0 Rut2 t1 o; drug absorption by the oral route is not considered x ¼ vF:x K d E O cp u O t1 100%. Hence, F is the fractional multiple of intravenous dose that is absorbed by the oral route to provide oral bioavailability.

42

3

Extent of Drug Absorption: Bioavailability, Clearance, Bioequivalence, and. . .

" #  

t xu ⎸t21 1 F:x0 R t2 o x vd K E O cp dt u O t1 " #  ¼ Then, absolute bioavailability ¼  x ⎸t2 1 u t x0 x R t2 1 I vd K E I u I cp dt t1 # " # "R t2     xu ⎸tt21 Fx0 vd K E t 1 cp dt R ¼ I o vd K E O x0 I t2 cp dt xu ⎸tt21 t1 F:ðK E ÞI ¼ ðK E ÞO

"

# # "R t 2 xu ⎸tt21 t 1 cp dt R t2 I o xu ⎸tt21 t 1 cp dt

Absolute bioavailability in terms of half  lives # ! " # "R t 2 F:ðt ½ ÞO xu ⎸tt21 0:693 t 1 cp dt R t2 ¼ I ,K E ¼ o t 1=2 ðt ½ ÞI xu ⎸tt21 t cp dt 1

3.1.4

Bioequivalence

The eventual growth of pharmaceutical industries has built up extensive competition among them to sell their formulation products containing the same active pharmaceutical ingredient (API)/therapeutically active substance/drug. Hence, the individual company has started using a different name (brand name) of a product with the same API for separate identification of its product among the other similar products of the different manufacturers. When the formulation is manufactured and sold in the non-proprietary name (generic name), i.e., the name of API itself, it is called a generic product. The United States Food and Drug Administration (USFDA) has expressed that generic medicine should be the same as its branded dosage form regarding safety, dose/power, delivery route, quality/standard, and efficacy. Thus, generic formulations should perform the same as branded medication. Generic medicine and brand medicine often have different manufacturers. For example, a generic bulk drug (API) is produced by a manufacturer, and other companies make the brand medicines using the API. Further, the same manufacturer can produce the generic and brand formulation products of the same API. Metoprolol is a generic antihypertensive medication, and Lopressor is its branded product. Thus, paracetamol/acetaminophen is the generic name of the brand names Crocin, Calpol, Metacin, etc. However, generic formulation products are generally cheaper than brand formulation products of the same API. It creates enormous confusion and doubts among the customers regarding the efficacies of the generic and brand formulations of the same API. It also generates a

3.1 Extent of Drug Absorption: Bioavailability

43

coveted apprehension in the minds of health professionals regarding the therapeutic equivalence of the products. Bioavailability and bioequivalence are literally woven tightly. Bioequivalence is as the comparison of the bioavailability of a drug from its different formulations of similar types administered by the same route. In the case of understanding of equivalence of various drug products (popularly known as formulations or dosage forms), the clarifications of the following terms are predominantly necessary. Hence, they are described below.

3.1.4.1 Chemical Equivalents When different drug formulations contain the same active ingredient (i.e., drug) in the same quantity, following the guidelines of the current official compendia, while the other ingredients (the excipients) in the formulations may differ, those formulations are called chemically equivalent drug products or chemical equivalents. 3.1.4.2 Pharmaceutical Equivalents According to the USFDA, the drug products with the same dosage form of the same active ingredient(s) with identical strength and administration routes are considered pharmaceutical equivalents. However, they may differ in shape, excipients, drugrelease mechanism, and even the label to an extent. 3.1.4.3 Bioequivalents When the rate and extent of drug absorption do not significantly differ between the test drug product (formulation) and the reference drug product (formulation) upon their administration with an identical strength of the same active ingredient in a single dose or multiple doses, under the same experimental conditions, the drug products are known as bioequivalent drug products or bioequivalents. However, in the case of bioequivalent drug products, the difference in the rate of drug absorption between the test drug product and the reference drug product is acceptable when the rate of drug absorption is modulated intentionally that clearly reflects on the proposed label of the test drug product. Further, the drug absorption essentially attains the minimum effective drug concentration for its chronic use, and the difference in the rate of drug absorption is treated as medically insignificant. 3.1.4.4 Therapeutic Equivalence Therapeutic equivalence indicates the same integrated outcome of bioequivalence and pharmaceutical equivalence of drug products and may be concisely represented as

44

3

Extent of Drug Absorption: Bioavailability, Clearance, Bioequivalence, and. . .

Therapeutic equivalence ¼ Bioequivalence þ Pharmaceutical equivalence When pharmaceutical equivalent drug products have the same efficacy and safety characteristics upon their administration to patients maintaining the advice and guidelines stipulated on the labels, the drug products are called therapeutic equivalents. According to the USFDA, the therapeutically equivalent products should be: (a) (b) (c) (d) (e)

Safe and effective Pharmaceutical equivalents Bioequivalents Adequately labeled Prepared according to the Current Good Manufacturing Practice regulations

The USFDA has the publication known as “orange book” that contains rating codes of the approved therapeutic equivalent generic medicines of various manufacturers as information for the public and health professionals.

3.1.4.5 Experimental Study Design in Bioequivalence The USFDA recommends comparing test formulation against reference or standard formulation, usually in a group of healthy volunteers of 18–55 years of age, alternatively in a crossover design with a wash-out period of 2 weeks between the two phases of treatments. An equal number of subjects receives treatment in all the phases with drugs of a long elimination half-life (approximately five half-lives). The number of treatments usually consists of the same number of periods (for example, three treatments, three periods). A parallel design is recommended for drug/metabolites having a long half-life. However, in a bioequivalence study, randomization is a primary criterion to receive unbiased outcomes. To avoid error and get a high precision level, experimental replication based on data variability and their differences is required. To compare two formulations, a replicated crossover design such as William’s and Balaam’s designs is suggested (Williams et al. 2000). Replicated crossover design is used for a product having a high variability, like modified drug-release products. The accepting parameters for bioequivalence studies are AUC ratio, Cmax ratio, difference in Tmax, etc. The confidence level (90%) for the AUC ratio is 0.8–1.25. A wider range of Cmax ratio is usually considered. Cmax is the concentration of drug at Tmax. Therefore, Cmax ¼

 ka :F:x0  K E :T max  eka :T max e vd ð k a  K E Þ

3.1 Extent of Drug Absorption: Bioavailability

45

Drug release and action related to adverse effects require evaluation of Tmax averages. 1 Tmax ¼ ka K ln KkaE , where ka is the absorption rate constant and KE is the E elimination rate constant Tmax ¼ k01k ln kk0 , where k0 is the initial rate constant and k is rate constant at any time t after drug absorption It can be presented as Tmax ¼ 2:303

c upper limit p lower limit

log cp

KE

or Tmax ¼ 3:32:t 1=2 : log

3.1.5

c

¼ 2:303

log cp

upper limit p lower limit 0:693 t

1=2

cp upper limit cp lower limit

Drug–Protein Binding

Drug–protein binding is a very significant phenomenon that is associated fundamentally with drug distribution. Besides, it also involves in drug metabolism and the elimination processes of many drugs. After the absorption of the drug in the blood, the drug binds reversibly with soluble blood proteins such as albumin, and the albumin-bound drug is then transported through the blood. It is ultimately distributed to various tissues, including the site of action of the drug, in the human body. Then, the drug is released to work. However, the drug–protein binding phenomenon and drug release from the protein-bound forms depend on the physicochemical characteristics of the drug along with some biopharmaceutical factors. Thus, drug– protein binding and drug-release phenomena from the protein-bound forms vary from drug to drug. It may decide the time length of its biological half-life. In the active transport (energy dependent), drugs bind to the carrier proteins for absorption. Upon protein binding, a protein-bound drug sometimes becomes the active form of the drug, or protein binding can activate the originally administered pro-drug molecules. Albumin and a plasma glycoprotein (α1-acid-glycoprotein) can bind with many drugs in plasma for transportation purposes. Albumin can bind with both acidic and neutral drug molecules, whereas α-acid-glycoprotein has a greater affinity toward basic and neutral drug molecules. In the drug–protein binding phenomenon, the amount of protein-bound drug, the amount of free drug, and the rate of drug–protein binding can be determined by various mathematical equations. Here, some selected fundamental equations have been described.

46

3.1.6

3

Extent of Drug Absorption: Bioavailability, Clearance, Bioequivalence, and. . .

Reciprocal Plot or Klotz Reciprocal Plot

Klotz reciprocal plot (Klotz 1946; KIotz 1982) has been developed from the following mass transfer reversible equation, where the free drug is D, free protein is P, and the protein-bound drug is PD. At equilibrium, P þ D
PD  Then, K ¼ ½½PPD ½D where [D], [P], and [PD] are the molar concentrations of the free drug, free protein, and protein-bound drug, respectively, and K is the equilibrium (association) constant.

Or K ½P½D ¼ ½PD If the total protein concentration is [Pt], then the total protein is equal to the amount of free protein and drug-bound protein. Or ½Pt  ¼ ½P þ ½PD Or ½P ¼ ½Pt   ½PD Now, by replacing the value of free protein concentration [P] in the equation K[P] [D] ¼ [PD], we get K f½Pt   ½PDg½D ¼ ½PD Or K ½Pt ½D  K ½PD½D ¼ ½PD Or K ½Pt ½D ¼ ½PD þ K ½PD½D Or ½PD þ K ½PD½D ¼ K ½Pt ½D ðby changing the sidesÞ Or ½PDð1 þ K ½DÞ ¼ K ½Pt ½D Or

½PD K ½D ¼ ½Pt  1 þ K ½D

K ½D ∴ r ¼ 1þK ½D , where r is the amount of protein-bound drug per unit amount of total protein (sum of free protein and drug-bound protein). Usually, its unit is expressed as mg of drug-bound protein per gram total protein or ng (nanogram) of drug-bound protein per mg total protein. In the above equation, when ν-independent protein-binding sites are available, the value is ν times the value of a single site. Therefore, the equation will be

3.1 Extent of Drug Absorption: Bioavailability

47

Fig. 3.1 The figure shows Klotz reciprocal plot for the quantification for protein–drug binding. The data of 1r (where r is the ratio of protein-bound drug to total protein) are plotted against ½D1  (where [D] is the molar concentration of free drug in plasma), and the slope of the line gives the 1 value of νK . The extrapolated line gives the value of the intercept 1ν on 1r-axis (y-axis). Here, “ν” is the number of binding sites of a protein molecule for drug molecules, and “K” is the equilibrium rate constant

r¼

νK ½D 1 þ K ½D 

Now, the equation is arranged in the following linear form that is suitable for graphical plotting. 1 1 1 r ¼ νK ½D þ ν ; this is known as Klotz reciprocal equation for a reciprocal plot where 1r is plotted on the y-axis and ½D1  is plotted on the x-axis to get a linear plot 1 and intercept on the y-axis of the (Fig. 3.1). The slope of the line gives the value of νK 1 extrapolated line provides the value of ν . Thus, from these values, the individual value of ν and K can be determined.

3.1.7

Scatchard Plot

Klotz reciprocal plot has some disadvantages. The fundamental disadvantage of the Klotz reciprocal plot is that the plot may misinterpret protein-binding behavior at a high concentration of the free drug. A high concentration of free drug leads to a very low or insignificant value of the factor νK1½D, resulting in real difficulty in plotting the data. The Klotz reciprocal equation can be modified as

48

3

Extent of Drug Absorption: Bioavailability, Clearance, Bioequivalence, and. . .

Fig. 3.2 The figure shows a Scatchard plot for the quantification for protein–drug binding. Data of r ½D (where r is the ratio of protein-bound drug to total protein and [D] is the molar plasma free drug concentration) are plotted against r, and the slope of the line gives the value of K. The intercept of the line provides the value with νK on ½Dr -axis (y-axis). Here, “ν” is the number of binding sites of a protein molecule for drug molecules, and “K” is the equilibrium rate constant

r þ rK ½D ¼ νK ½D to overcome this lacuna. Or ½Dr  ¼ νK  rK; the graphical plot of this equation is known as the Scatchard plot (Scatchard 1949). This equation, thus, can be used at a low concentration of the free drug. Here, ½Dr  is plotted on the y-axis and r is plotted on the x-axis to get a linear plot (Fig. 3.2). The slope of the line gives K and the intercept on the y-axis provides the value of νK. Thus, from these values, the individual value of ν and K can be determined.

3.1.8

Sandberg Plot

When the nature and amount of the protein are unknown, the Klotz reciprocal plot and the Scatchard plot cannot be used for drug–protein binding analysis (KIotz 1982). In such cases, the following plot, known as the Sandberg plot (Sandberg et al. 1966), is used where the molar concentration of a protein-bound drug is indicated by [Db] and that of free drug is denoted by [Df]. The equation can be developed from the equation of the Scatchard plot:

3.1 Extent of Drug Absorption: Bioavailability

49

Fig. 3.3 The figure shows the Sandberg plot for the quantification for protein–drug binding. Data b of D Df (where Db is the concentration of the protein-bound drug and Df is the free drug concentration) are plotted against Db, and the slope of the line gives the value of K. The intercept of the line D provides the value with νK[Pt] on Db -axis (y-axis). Here, “ν” is the number of binding sites of a f protein molecule for drug molecules. “K” is the equilibrium rate constant and [Pt] is the concentration of the total protein

r ¼ νK  rK, ½D  ½Db  or ½P½tD½bD f  ¼ νK  ½½DPbt  K (by replacing the value of r as r ¼ ½½PD Pt  ¼ ½Pt  )

or

½Db  νK ½Pt   ½Db K ¼ ½Pt ½Df  ½Pt 

or

½Db  ¼ νK ½Pt   K ½Db  ½Df 

or ½½DDbf  ¼ K ½Db  þ νK ½Pt . The plot obtained from this equation is known as the Sandberg plot. Here, ½½DDbf  is plotted on the y-axis and [Db] is plotted on the x-axis to get a linear plot (Fig. 3.3). The slope of the line gives the value of K, and the intercept on the yaxis provides the value of νK[Pt]. Drug–protein binding can be determined by various methods such as kinetic method, equilibrium dialysis, equilibrium gel filtration, ultracentrifugation, spectroscopic method, and many more. Some of these methods have been discussed in Chap. 13 of this book.

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Extent of Drug Absorption: Bioavailability, Clearance, Bioequivalence, and. . .

References KIotz IM (1982) Numbers of receptor sites from Scatchard graphs: facts and fantasies. Science 217: 1247–1249 Klotz IM (1946) The application of the law of mass action of binding to proteins. Interactions with calcium. Arch Biochem 9:109–117 Rowland M, Benet LZ, Graham GG (1973) Clearance concepts in pharmacokinetics. J Pharmacokinet Biapharm 1:123–136 Sandberg AA, Rosenthal H, Schneider SL, Slaunwhite WR (1966) Protein-steroid interactions and their role in the transport and metabolism of steroids. In: Nakao T, Pincus G, Tait JF (eds) Steroid dynamics. Academic, New York, pp 33–41 Scatchard G (1949) The attraction of proteins for small molecules and ions. Ann N Y Acad Sci 51(4):660–672 Williams RL, Adams WP, Chen M-L, Hare D, Hussain A, Lesko L, Patnaik R, Shah V, FDA Biopharmaceutics Coordinating Committee (2000) Where are we now and where do we go next in terms of scientific basis of regulation of bioavailability and bioequivalence? Eur J Drug Metab Pharmacokinet 25:7–12

4

Pharmacokinetic Models and Drug Distribution

4.1

Various Pharmacokinetic Models and Drug Distribution

The pharmacokinetic models are instrumental in describing the absorption, distribution, metabolism, and elimination (ADME) of drugs, determining the time course of drug action, understanding drug distribution patterns, improving drug therapy by enhancing drug efficacy, and minimizing adverse reactions by providing more accurate dose regimens. Other than the compartment models, predominant pharmacokinetic models are the physiological pharmacokinetic model, blood flow–limited or perfusion model, a physiological pharmacokinetic model with protein binding, diffusion-limited model, and statistical moment models.

4.1.1

Physiological Pharmacokinetic Model (Flow Model)

The physiological pharmacokinetic model consists of complex mathematical models of blood flow and tissue distribution of drugs. The human body has various organ systems. Each organ is composed of cells and extracellular aqueous fluid that surrounds the cells. In other words, the cells remain bathed or soaked in the extracellular fluid that acts as a medium for exchanging drugs and other endogenous and metabolic substances (Lutz et al. 1977). Inside the cells, intracellular fluid is present. The body is connected with the network of blood vessels (arteries and veins) and capillaries with access to the extracellular fluid. Drug and endogenous substances such as hormones, nutrients, and oxygen are transported to the extracellular fluid close to the cells by blood capillaries (Fig. 4.1). When the drug reaches the extracellular aqueous fluid, a quick equilibrium of drug concentration is established between its concentration in the capillary blood present in an organ and the extracellular fluid. The cellular uptake of drugs or other endogenous substances begins. Organ-specific drug accumulation depends on the blood flow to the organ and the rate of cellular internalization of the drug. Cells internalize drugs that permeate # The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B. Mukherjee, Pharmacokinetics: Basics to Applications, https://doi.org/10.1007/978-981-16-8950-5_4

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Pharmacokinetic Models and Drug Distribution

Fig. 4.1 The figure shows a physiological pharmacokinetic model of drug transportation to the extracellular fluid by blood capillaries connected to the arterioles, and drug molecules/drug metabolites that diffuse out of the cells to the extracellular fluid are transported back to the venules for removal

across the cell membrane by diffusion or binding with macromolecules (proteins) present on the cell membrane. Further, drugs may bind with the cytoplasmic proteins, which can enhance drug accumulation within the cells. An organ can function for both drug accumulation and drug elimination (Fig. 4.2). The elimination process generates space for drug accumulation (Himmelstein and Lutz 1979). However, the elimination can occur from other organs also. High blood flow is seen in the adrenal glands (highest blood flow), thyroid glands, and kidneys, and the minimum to no blood flow is seen in the skeleton. However, low blood flow is seen to teeth also. Drug accumulation in the tissue depends on the partitioning of drugs from the blood to the tissue. The partition coefficient of a drug from the blood to the tissue is represented as Ptissue or Ptissue/blood or Pt/b Ptissue=blood or Pt=b ¼

Drug concentration in tissue, C tissue C or Pt=b ¼ t Drug concentration in blood, Cblood Cb

The cells liberate their metabolic products such as carbon dioxide and other cellular wastes in the extracellular fluid. They reach through the blood capillaries and veins to the elimination organs such as the lungs, kidney, liver, and skin for elimination.

4.1 Various Pharmacokinetic Models and Drug Distribution

53

Fig. 4.2 The figure shows a physiological pharmacokinetic model of drug elimination: (A) After the drug transportation to the extracellular fluid by blood capillaries connected to arterioles, the drug diffuses into the cells, and drug molecules/drug metabolites that diffuse out of the cells to the extracellular fluid are transported back to the venules to venous blood for elimination. (B) After the drug transportation to the extracellular fluid by blood capillaries connected to arterioles, the drug diffuses into the cells, and drug molecules/drug metabolites diffuse out of the cells to the extracellular fluid from where the drug is eliminated through the organ

The size of the organ, blood flow rate to the organ, drug partitioning between the blood and extracellular fluid, cellular uptake, protein-binding capability and rate of protein binding of a drug in the blood and on the cellular surface, drug–protein binding in the intracellular fluid, fate of drug inside the cells, and capacity of the cells to internalize the amount of drug are some of the major factors, which play a significant role in the accumulation of the drug in an organ.

54

4.1.2

4

Pharmacokinetic Models and Drug Distribution

Blood Flow–Limited Physiological Pharmacokinetic Model or Perfusion Model

The blood flow–limited physiological pharmacokinetic or perfusion model is possibly the most simple physiological pharmacokinetic model based on blood flow to a tissue that becomes a rate-limiting process. In this model, the rate of blood flow to a tissue/an organ and the amount of drug accumulated to the tissue/organ are correlated to establish the equations for calculation (Dedrick and Forrester 1973). In this model, a large number of parameters can be used for calculation. The concentration of a drug in the tissue and drug–tissue partition coefficient can be determined in animal models. Hence, the blood flow–limited physiological pharmacokinetic model is reliable and realistic. However, the shortage of such data from human subjects has made this model insignificant for investigating pharmacokinetics in humans. In a mass balance
equation, at any time point t, the rate of change of the amount of t drug in a tissue dx ¼ xin  xout , where xin is the amount of drug entered into the dt tissue and xout is the amount of drug that is removed from the tissue at that time point. t Thus, at a particular time point t, dx dt ¼ [rate of flow of the blood (Q) in the tissue  drug concentration in the blood flowing in the tissue (Cin)  the rate of flow of the blood going out from the tissue  concentration of the drug in the blood flowing out (Cout)], at that time point. As the rate of flow of the blood is the volume of blood flows per unit time (vt ), therefore, rate of flow of the blood (Q)  concentration of the drug (C) ¼ vt  C ¼ v m m t  v ¼ t (, concentration (C) ¼ mass/volume, m/v). Therefore, at a particular time point, rate of flow of the blood (Q)  concentration of the drug (C) ¼ mt (mass transfer of drug per unit time). At the equilibrium state, the rate of flow of the blood in the tissue Qtissue in and the rate of flow of blood out of the tissue Qtissue out should be equal and may be considered as Qtissue. Further, Cin is the inflow of the blood to the tissue. Hence, it is considered arterial blood and renamed as Cartery and Cout is the outflow of the blood from the tissue; hence, it is considered venous blood and renamed as Cvein. Therefore, the mass balance equation at equilibrium state is
 dxt d ðV tissue C tissue Þ ¼ Qtissue ðCin  Cout Þ ¼ Qtissue C artery  C vein ¼ dt dt ðagain, amount of drug in the tissue at time t, xt ¼ V tissue Ctissue , as amount ðxÞ ¼ volume ðV Þ  concentration ðC ÞÞ Therefore,


 dðV tissue C tissue Þ ¼ Qtissue Cartery  C vein dt

ð4:1Þ

4.1 Various Pharmacokinetic Models and Drug Distribution

55

Here, an essential factor is partitioning a drug from the blood to the tissue and partitioning the drug from the tissue to the blood (Chen and Gross 1979). Again, the partition coefficient of a drug from the blood into the tissue Ptissue ¼

concentration of drug in the tissue, Ctissue concentration of drug in the blood, C blood

In this model, tissue drug concentration establishes equilibrium with the drug concentration in the blood outflow (venous blood) from the tissue. Hence, the above equation is arranged as Ptissue ¼

Ctissue C vein

∴Cvein ¼

C tissue Ptissue

By replacing the value of Cvein in Eq. (4.1), we get   d ðV tissue C tissue Þ C ¼ Qtissue C artery  tissue dt Ptissue The above equation is a general equation of drug distribution in a tissue or an organ that undergoes no elimination process in the blood flow–limited model. Thus, equations for a few such organs are given below. Indeed, these organs also undergo some elimination processes. The equation for the muscle is   dðV muscle Cmuscle Þ C muscle ¼ Qmuscle Cmuscle  dt Pmuscle The equation for the skin is   dðV skin C skin Þ C ¼ Qskin C skin  skin dt Pskin The equation for the heart is   d ðV heart Cheart Þ C heart ¼ Qheart C heart  dt Pheart However, the elimination parameters of an organ that undergoes the elimination process are considered to estimate drug removal in the mass balance equation. Further, the amount of drug inflow into the organ is always considered more than its elimination. Drug clearance from an organ (CLorgan) is the product of blood flow to the organ (Qorgan) and drug extraction ratio (Eorgan). Again, Eorgan is a ratio of the

56

4

Pharmacokinetic Models and Drug Distribution

concentration of drug accumulated in the blood to the concentration of drug in the blood inflow into an organ at a time point (Lane and Levy 1981). CLorgan ¼ Qorgan

C artery  Cvein ¼ Qorgan Eorgan Cartery

ð4:2Þ

As per the blood flow–limited model, the extraction ratio of an organ that undergoes elimination can be represented by Eorgan ¼

CLintrinsic Qorgan þ CLintrinsic

Where CLintrinsic is the intrinsic clearance of an organ that represents the maximum capability of the organ to clear the drug from the blood without any limitations of blood flow into the organ and drug–protein binding, with the substitution of Eorgan value in Eq. (4.2), we get,  CLorgan ¼ Qorgan

CLintrinsic Qorgan þ CLintrinsic



Hence, rate of change of the amount of drug in the organ at time t ¼ amount of drug accumulated in the organ at time t — clearance of the drug from the organ at time t.     dðV tissue Ctissue Þ C CLtissue ¼ Qtissue Cartery  tissue  C tissue dt Ptissue Ptissue Thus, the rate of change of drug in the kidney is
     d V kidney C kidney Ckidney CLkidney ¼ Qkidney Ckidney   C kidney dt Pkidney Pkidney However, the blood flows into the liver (Qliver) directly. The rate of blood flow to the kidney is Qkidney. Further, the flow of the blood from the gastrointestinal tract (GIT) (QGIT) and from the spleen (Qspleen) is also received by the liver (Fig. 4.3). Hence, the rate of change of drug in the liver is represented by  
 d ðV liver C liver Þ C GIT ¼ C liver Qliver  QGIT  Qspleen þ QGIT dt PGIT       Cspleen Cliver CLintrinsic þ Qspleen  Qliver  Cliver Pspleen Pliver Pliver Again, the blood flows into the lungs by the pulmonary vein and flows out by the pulmonary artery. Thus, the equation is represented by

4.1 Various Pharmacokinetic Models and Drug Distribution

57

Fig. 4.3 The figure shows that in a flow-limited physiological model, drug distribution from the arterial blood to different tissues for drug absorption, and then from the tissues,
the drug molecules/ drug metabolites diffuse out to the vein. The rate of drug absorption dx is quantified by dt dðcp vd Þ dt , where cp is the plasma drug concentration and vd is the volume of distribution.

dx dt

¼

by

dQtissue dt . GIT denotes gastrointestinal tract

However, the rate of blood flow from the tissue to the venous blood is tissue specific and quantified


   d V lung C lung C lung ¼ Qlung  C lung dt Plung Further, the rate of change of drug concentration in the blood is given by      Cmuscle Cheart Cskin þ Qheart þ Qskin Pmuscle Pheart Pskin       C kidney C lung Cliver þ Qliver þ Qkidney þ Qlung Pliver Pkidney Plung

dðV blood C blood Þ ¼ Qmuscle dt



 Qblood C blood The protein binding of a drug is a primary issue that causes alteration in drug distribution between the species. However, animal drug–protein binding data are used for the prediction and understanding of drug distribution in humans.

58

4.1.3

4

Pharmacokinetic Models and Drug Distribution

Physiological Pharmacokinetic Model with Drug–Protein Binding

Most drugs distribute exclusively in drug–plasma protein-bound forms. Proteins such as albumin (primarily) and globulin are responsible for the reversible drug– protein binding phenomenon in the plasma. The plasma protein-bound drug then reaches through blood capillaries at the cellular tissue microenvironment and dissociates the drug. Drug molecules then diffuse through the fluid present in the tissue microenvironment to reach the tissue cells. Drug molecules generally bind with the proteins of the tissue cells (Minturn et al. 1980) for their tissue-specific cellular internalization (Fig. 4.4). The phenomenon is more popularly called drug– tissue protein binding. Determination of drug–plasma protein binding is much easier for the availability of accessible experimental facilities. On the other hand, determining the drug–tissue protein binding is always more challenging (Harrison and Gibaldi 1977). Plasma drug concentration and tissue drug concentration establish equilibrium fast. At the equilibrium condition, tissue free drug concentration and free drug concentration of the blood (vein) coming out of the tissue are equal. If the fraction of tissue free drug concentration is ft and the fraction of free drug concentration of the blood (vein) is fb and the total drug concentration in the tissue is ct and the total drug concentration in the blood (vein) (cb) coming out of the tissue, at the equilibrium, we can write f t ½ct  ¼ f b ½cb  Again, the tissue partition coefficient Pt is the ratio of tissue drug concentration to the drug concentration in the blood (vein) (cb) coming out of the tissue.

Fig. 4.4 The figure depicts reversible drug–protein binding phenomena in the blood and the tissue and the exchange of drugs between those tissues

4.1 Various Pharmacokinetic Models and Drug Distribution

Hence, Pt ¼

59

f ct ¼ b ð, f t ½ct  ¼ f b ½cb Þ cb ft

Further, the rate of change of drug in the tissue is

dxtissue dt


 ¼ Qtissue C artery  Cvein


 d ðvtissue ctissue Þ ¼ Qtissue Cartery  C vein dt   d ðvtissue ctissue Þ c ¼ Qtissue Cartery  tissue Or dt Ptissue   d ðvtissue ctissue Þ fc Or ¼ Qtissue Cartery  t tissue dt fb Or

For hepatic drug metabolism, the blood flows into the liver (Qliver) directly. Further, the flow of the blood from the gastrointestinal tract (GIT) (QGIT) and from the spleen (Qspleen) is also received by the liver. Hence, the rate of change of drug in the liver as mentioned in the perfusion model is represented by cspleen d ðvliver cliver Þ c ¼ cliver ðQLiver  Qspleen  QGIT Þ  Qliver ð liver Þ þ Qspleen ð Þ dt Pliver Pspleen þ QGIT ð

4.1.4

cGIT CL Þ  C liver ð intrinsic Þ PGIT Pliver

Membrane-Limited Model or Diffusion-Limited Model

In the membrane-limited or diffusion-limited model, the primary consideration is the drug permeation through the cell membrane by diffusion at a slower rate than the blood flow. Thus, establishing the equilibrium between the blood drug concentration and tissue drug concentration requires time. Drug concentration in venous blood is more at the beginning than the tissue drug concentration, and eventually, the equilibrium between those concentrations is established. Drug permeation through the membrane becomes a rate-limiting step. The rate equations are complex in the process. The significant differences between the blood flow–limited model and the diffusion-limited or membrane-limited model are precisely stipulated below. A rapid drug diffusion, no membrane barrier, no drug–protein binding phenomenon, the same drug concentration in the tissue and the tissue outflow (venous blood), and blood flow as a rate-limiting step are the primary considerations for the blood flow–limited model. On the other hand, in the diffusion-limited or membrane-limited model, the prime concerns are cell membrane as a barrier, very fast blood flow, slow drug permeation, attainment of a drug concentration gradient between the tissue and the tissue outflow (venous blood), and drug permeation across the membrane as a rate-limiting step.

60

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Pharmacokinetic Models and Drug Distribution

Further, the blood flow–limited model uses simple mathematical equations due to its simple assumptions. On the contrary, the diffusion-limited or membrane-limited model needs very complex equations for its time-lag in establishing equilibrium between the tissue drug concentration and the blood drug concentration.

4.1.5

Statistical Moment Theory

Statistical moment theory relates alteration in a macroscopic event of its constitutive elements. When a drug (the constitutive element) is administered to a patient, the drug as tracer molecules will distribute in the body. Further, the drug as the tracer molecules should distribute well randomly without any interaction. Drug residence time in the body is a macroscopic event. The parameters showing characteristics of drug plasma concentration–time profile/drug concentration–time profile based on urinary excretion data following a single drug dose represent statistical moments (Yamaoka et al. 1978). The statistical moment theory is a non-compartment system for calculating absorption, distribution, and elimination parameters (statistical moments). Further, unlike the compartmental model, the method does not need any pre-assumptions. Alteration of plasma/blood/tissue drug concentrations with time can be determined by this method. 1 ðAUCÞ is the total area under the plasma concentration versus time curve from 0 zero time to infinite time that represents zero-moment drug concentration. However, product of the entire area under the plasma concentration versus time curve from zero time to infinite time and time provides the first-moment plasma concentration– 1 time profile, ðAUMCÞ . 0 Z 1 μm , mth moment t m f ðt Þdt 0

At the mth moment, at time t, the probability function is, say, f(t) At zero moment, m ¼ 0; hence, zero moment is Z μ0¼

1

f ðtÞdt

0

For a proper distribution probability function, the first moment gives the expected value. For the first moment, m ¼ 1; then, Z

1

μ1 ¼

Z t f ðt Þdt ¼ 1

0

For the second moment, μ2 ¼

R1 0

0

t 2 f ðt Þdt

1

t f ðt Þdt

4.1 Various Pharmacokinetic Models and Drug Distribution

61

μ2 is called distribution variance. Similarly, the moment μ3 and μ4 represent skewness and kurtosis, respectively. Thus, the moment equation can characterize moment curve families for distribution. The moment curve primarily calculates the mean residence time (MRT) of a drug and depicts drug distribution upon its administration.

4.1.6

Compartment Models

A mathematical model describes a mathematical concept–based system using computational language. With the increase in the number of compartments, the algebraic and numerical solutions of a model become more and more complicated. Different models are developed depending on the input/output features and the types of interlinking among the compartments. Some of them are discussed below. In a single-compartment model, the whole body is considered only one homogenous compartment in which drug distribution occurs almost instantaneously throughout the entire volume of distribution upon its administration, and input and output occur from this same volume of distribution. On the other hand, when the central compartment (blood compartment) and the peripheral compartments (tissue compartments) are distinct and the plasma drug concentration declines in multiple exponential phases, the model is called the multi-compartment model.

4.1.6.1 One-Compartment Closed Model One-compartment closed model describes the whole body as only one homogenous, highly perfused compartment in which drug distribution upon its administration takes place almost instantaneously throughout the entire volume of distribution, but there is no elimination of the drug from the body (Fig. 4.5). Fig. 4.5 One-compartment closed model that shows x0 amount of drug is administered to an entirely homogeneous body compartment

62

4

Pharmacokinetic Models and Drug Distribution

4.1.6.2 One-Compartment Open Model One-compartment open model describes the whole body as only one homogenous, highly perfused compartment in which drug distribution upon its administration takes place almost instantaneously throughout the entire volume of distribution. The elimination process of the administered drug takes place from the body (Fig. 4.6). 4.1.6.3 Catenary Model Catenary model represents a chain of interlinked compartments in which each compartment is connected to its neighboring compartment(s) (Fig. 4.7). It mimics the arrangement of the compartments in a train. 4.1.6.4 Cyclic Model The model has a similar arrangement of a chain of interlinked compartments like a catenary model except for the first and last compartments that are connected to each other (Fig. 4.8). The model needs at least three compartments. Fig. 4.6 One-compartment open model that shows x0 amount of drug is administered to an entirely homogeneous body compartment, and the drug is eliminated from it. k10 indicates the rate of drug elimination, and the arrowhead indicates the drug is eliminated outside the body compartment

Fig. 4.7 The figure shows a catenary model where all the compartments are connected like the compartments of a train

4.1 Various Pharmacokinetic Models and Drug Distribution

63

Fig. 4.8 The figure shows a cyclic model. It is like a catenary model where two terminal compartments are connected to each other to form a closed structure

Fig. 4.9 The figure shows a mammillary model where all the compartments are connected to a central compartment

4.1.6.5 Mammillary Model A mammillary model is a compartmental model in which a central compartment is connected to all other peripheral compartments, but the peripheral compartments are not interlinked to each other (Fig. 4.9). This model is widely accepted. This model needs at least two or more compartments.

4.1.7

Some Mathematical Approaches for Easy Computation of Compartmental Equations and Their Applications

(Matrix and determinant, Cramer’s rule, Laplace transforms, and use of log tables) Few mathematical methods are fundamental to working with compartmental models. Therefore, some basic concepts of matrix and determinant, Laplace transforms and anti-Laplace forms, and the idea of logarithms that would be useful

64

4

Pharmacokinetic Models and Drug Distribution

to ease the calculation related to compartmental model equations have been briefly discussed below.

4.1.7.1 Matrix and Determinant A matrix is defined as a regular ordered arrangement of numbers or functions called elements or entities of the matrix. In a matrix with m rows and n columns, the order of the matrix indicates m  n. For solving linear equations by a compact and simple method, the matrix is useful, and it is developed by obtaining the representative coefficients of linear equations that belong to a system. However, matrix operations are also helpful for other purposes, such as creating electronic spreadsheets, cryptography, and many physical processes such as magnification, rotation, and reflection. Two linear equations are represented below as a1 x þ b1 y ¼ c 1 a2 x þ b2 y ¼ c 2 where a1 and a2 are the coefficients of independent variable x and b1 and b2 are the coefficients of dependent variable y (as the value of y depends on the value of x), respectively. The values of c1 and c2 are the constants. Taking the coefficients of x and y, we can develop a matrix, say, matrix A.  a1 b1 Then, matrix A ¼ ; A is a matrix of 2  2. a2 b2 Now to identify whether a system of linear equations has a unique solution or not, the determinant of the matrix is obtained. The determinant of the above matrix is represented by Δ or D or ⎸A ⎸, and here, it is given below: Δ ¼ a1 b2  a2 b1 There is a difference between a determinant and a matrix. A determinant has a numerical value, but a matrix does not. The determinant of two rows and two columns is given by D, where D is given by

The value of this determinant is found by obtaining the difference between the diagonally down product and the diagonally up product, as shown by arrows in the above equation. Please see the example of solving a linear equation system by using determinants.

4.1 Various Pharmacokinetic Models and Drug Distribution

65

5x þ 2y ¼ 3 3x þ 2y ¼ 5

4.1.7.1.1 Cramer’s Rule Three determinants are to be created to solve this system. One is called the denominator determinant (say, D here); another is the x-numerator determinant (say, Dx) and the third is the y-numerator determinant, represented as Dy. The solution D of x is x ¼ DDx and the solution of y ¼ Dy : The solution mentioned above is done according to the famous Cramer’s rule, named after the mathematician who developed this method. So in the case of the system of linear equations, 5x þ 2y ¼ 3 3x þ 2y ¼ 5 The denominator determinant, D, is formed by taking the coefficients of x and y from the equations written in the standard form.  D¼

5 2



3 2

¼ ð5Þð2Þ  ð3Þð2Þ ¼ 10  6 ¼ 4

The x-numerator determinant is developed by taking the constant terms from the system, placing them in the x-coefficient positions, and putting the coefficients of y in the y-coefficient positions. Since we will solve for x here, we do not consider the coefficients of x here.  Dx ¼

3

2

5

2

 ¼ ð3Þð2Þ  ð5Þð2Þ ¼ 6  10 ¼ 4

The y-numerator determinant is developed by taking the constant terms from the system and placing them in the y-coefficient positions and keeping the x-coefficients in the x-coefficient positions. Since we will solve for y here, we do not consider the coefficients of y here.  ∴Dy ¼

 5 3 ¼ ð5Þð5Þ  ð3Þð3Þ ¼ 25  9 ¼ 16 3 5

y 16 Therefore, the value of x ¼ DDx ¼ 4 4 ¼ 1 and y ¼ D ¼ 4 ¼ 4 x ¼ 1 Therefore, the answer is y¼4 For solving three variables, x, y, and z, of the following equations,

D

66

4

Pharmacokinetic Models and Drug Distribution

ax þ by þ cz ¼ k

ð4:3Þ

dx þ ey þ fz ¼ l

ð4:4Þ

gx þ hy þ iz ¼ m

ð4:5Þ

where a, b, and c are the coefficients of x, y, and z, respectively, in Eq. (4.3); d, e, and f are the coefficients of x, y, and z, respectively, in Eq. (4.4); and g, h, and i are the coefficients of x, y, and z, respectively, in Eq. (4.5). Therefore, the denominator determinant matrix is formed by the coefficients of x, y, and z of the three equations as given below: 2

a

b

6 4d g

e h

c

3

7 f5 i

Therefore, the determinant Δ or D is  D¼a

e

f

h

i



 b

d

f

g

i



 þc

d

e

g

h

 ¼ aðei  hf Þ  bðdi  gf Þ þ cðdh  geÞ

The x-numerator determinant matrix is formed by taking the constant terms from the system, placing them in the x-coefficient positions, and keeping the y-coefficients and z-coefficients in their positions. Since we will solve for x here, we do not consider the coefficients of x here. 2

k

b

6 4l m

e h

c

3

7 f5 i

Then, the x-numerator determinant Dx is given by  Dx ¼ k

e

f

h

i



 b

l

f

m

i



 þc

l

e

m

h

 ¼ kðei  hf Þ  bðli  mf Þ þ cðlh  meÞ

The y-numerator determinant matrix is formed by taking the constant terms from the system, placing them in the y-coefficient positions, and keeping the x-coefficients and z-coefficients in their positions. Since we will solve for y here, we do not consider the coefficients of y here.

4.1 Various Pharmacokinetic Models and Drug Distribution

2

a 6 4d

k l

3 c 7 f5

g

m

i

67

Then, the y-numerator determinant Dy is given by 

l Dy ¼ a m

  f d k i g

  f d þc i g

 l ¼ aðli  mf Þ  k ðdi  gf Þ þ cðdm  glÞ m

The z-numerator determinant matrix is formed by taking the constant terms from the system, placing them in the z-coefficient positions, and keeping the x-coefficients and y-coefficients in their positions. Since we will solve for z here, we do not consider the coefficients of z here. 2

a 6 d 4

b e

g

h

3 k 7 l5 m

Then, the z-numerator determinant Dz is given by  Dz ¼ a



e

l

h

m

 b

d

l

g

m



 þk

d

e

g

h



¼ aðem  lhÞ  bðdm  glÞ þ kðdh  geÞ Solution of x is ¼ DDx ; solution of y ¼ ∴x ¼ y¼ z¼

Dy D

and solution of z ¼ DDz

kðei  hf Þ  bðli  mf Þ þ cðlh  meÞ aðei  hf Þ  bðdi  gf Þ þ cðdh  geÞ

aðli  mf Þ  kðdi  gf Þ þ cðdm  glÞ aðei  hf Þ  bðdi  gf Þ þ cðdh  geÞ

aðem  lhÞ  bðdm  glÞ þ k ðdh  geÞ aðei  hf Þ  bðdi  gf Þ þ cðdh  geÞ

Another example is given for solving the three variables x, y, and z for the following system of linear equations: 2x þ 3y þ 3z ¼ 5

ð4:6Þ

x  2y þ z ¼ 4

ð4:7Þ

3x  y  2z ¼ 3

ð4:8Þ

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4

Pharmacokinetic Models and Drug Distribution

The denominator determinant matrix is formed by the coefficients of x, y, and z of the three equations as given below: 2

2

3

6 41 3

2 1

3

3

7 1 5 2

Therefore, the determinant Δ or D is  D¼2

2

1

1

2



 3

1



1

 þ3

3 2

1

2

3

1



¼ 2fð2Þð2Þ  ð1Þð1Þg  3fð1Þð2Þ  ð1Þð3Þg þ 3fð1Þð1Þ  ð3Þð2Þg ¼ 2ð4 þ 1Þ  3ð2  3Þ þ 3ð1 þ 6Þ ¼ 2ð5Þ  3ð5Þ þ 3ð5Þ ¼ 10 þ 15 þ 15 ¼ 40 The x-numerator determinant matrix is formed by taking the constant terms from the system, placing them in the x-coefficient positions, and keeping the y-coefficients and z-coefficients in their positions. Since we will solve for x here, we do not consider the coefficients of x here. 2

5 6 4 4

3 2 1

3

3 3 7 1 5 2

Then, the x-numerator determinant Dx is given by  Dx ¼ 5

2

1

1

2



 3

4

1

3

2



 þ3

4

2

3

1



¼ 5fð2Þð2Þ  ð1Þð1Þg  3fð4Þð2Þ  ð1Þð3Þg þ 3fð4Þð1Þ  ð3Þð2Þg ¼ 5ð4 þ 1Þ  3ð8  3Þ þ 3ð4 þ 6Þ ¼ 5ð5Þ  3ð5Þ þ 3ð10Þ ¼ 25  15 þ 30 ¼ 40 The y-numerator determinant matrix is formed by taking the constant terms from the system, placing them in the y-coefficient positions, and keeping the x-coefficients and z-coefficients in their positions. Since we will solve for y here, we do not consider the coefficients of y here.

4.1 Various Pharmacokinetic Models and Drug Distribution

2

2 6 41 3

69

3 3 7 1 5 2

5 4 3

Then, the y-numerator determinant Dy is given by 

4 Dy ¼ 2 3

    1 1 1 1 5 þ3 2 3 2 3

4 3



¼ 2fð4Þð2Þ  ð1Þð3Þg  5fð1Þð2Þ  ð1Þð3Þg þ 3fð1Þð3Þ  ð3Þð4Þg ¼ 2ð8  3Þ  5ð2  3Þ þ 3ð3 þ 12Þ ¼ 2ð5Þ  5ð5Þ þ 3ð15Þ ¼ 10 þ 25 þ 45 ¼ 80 The z-numerator determinant matrix is formed by taking the constant terms from the system, placing them in the z-coefficient positions, and keeping the x-coefficients and y-coefficients in their positions. Since we will solve for z here, we do not consider the coefficients of z here. 2

2

6 41 3

3 2 1

5

3

7 4 5 3

Then, the z-numerator determinant Dz is given by  Dz ¼ 2

2

4

1

3



 3

1

4

3

3



 þ5

1

2

3

1



¼ 2fð2Þð3Þ  ð4Þð1Þg  3fð1Þð3Þ  ð4Þð3Þg þ 5fð1Þð1Þ  ð3Þð2Þg ¼ 2ð6  4Þ  3ð3 þ 12Þ þ 5ð1 þ 6Þ ¼ 2ð10Þ  3ð15Þ þ 5ð5Þ ¼ 20  45 þ 25 ¼ 40 Therefore, x ¼

Dx 40 ¼1 ¼ 40 D

y¼

Dy 80 ¼ 2 and ¼ 40 D

z¼

Dz 40 ¼ 1 ¼ 40 D

Therefore, x ¼ 1; y ¼ 2; and z ¼ 1

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4

Pharmacokinetic Models and Drug Distribution

4.1.7.2 Laplace Transform Laplace transform is a mathematical method or tool to integrate linear, ordinary, and differential equations by replacing independent variables (the time domain) with a complex domain (often frequency domain) “s” (known as Laplace operator). Finally, the complex domain (the transformed function) is replaced by the equivalent time domain with an ordinary algebraic equation (i.e., integrated equation). This method quickly solves linear, standard, and differential equations without creating much trauma in integral calculus. The process is similar to the logarithm. In logarithm, we initially replace a number with a logarithmic form for calculation, and after the calculation, we get back the result in real number by taking the antilogarithm. Laplace transform is an analogous operation to logarithm, conducted in the order of time-domain integral to a complex domain (transformed form) to time domain (algebraic integrated time function equation). A bar (-) is placed over the transformed dependent variable. For example, if x is the amount of drug present in the blood at time t, then x represents the Laplace transformed amount of x. The Laplace transform method uses a table of time functions, and their transform complex domains are used for solving integrals by the Laplace transform method. It is known as the Laplace table. The Laplace table contains two columns, one containing time-domain expression as f(t) and the other containing the corresponding complex-domain expression with Laplace operator “s” (Table 4.1). For example, if f(t) is a time domain represented by Aeat, where A is a constant and a is a rate constant, then the complex domain (Laplace transformed form) of it A is ðsþa Þ. Example In a patient, x0 amount of drug has been injected initially, and drug elimination follows first-order rate kinetics. At time t, the amount of drug present is, say, x, and KE is the rate of elimination, then dx ¼ K E :x dt Taking Laplace transform (denoted by L ), we get sLf ðxÞ  x0 ¼  K E Lf ðxÞ, for simplicity, L f ðxÞ ¼ x, where f ðxÞ is a function of x: s:x  x0 ¼ K E x Or x ðs þ K E Þ ¼ x0 or x ¼

x0 ðs þ K E Þ

ð1  eat Þ

A a

at

b

eðaÞt e f(t)

A a

A t eat

eat

Aeat

F(s  a)

ðs2 þa2 Þ2 n! ðsaÞnþ1 A ðsþaÞ 1 ðsaÞ A ðsþaÞ2 A sðsþaÞ A asþb

sðs2 a2 Þ

cos(at)  at sin (at)

tneat (n ¼ 1, 2, 3, ...)

2a3 ðs2 þa2 Þ2

ðs2 þa2 Þ2

sðs2 þ3a2 Þ

2as2 ðs2 þa2 Þ2

2s2 a s2 þa2 2as ðs2 þa2 Þ2 s s2 þa2 s2 a2 ðs2 þa2 Þ2

3

1 s A s 1 s2 m! smþ1 pffiffi π

(continued)

Laplace transforms F(s) or L[f(t)]a

sin(at)  at cos (at)

cos(at) + at sin (at)

sin(at) + at cos (at)

t cos (at)

cos (at)

t sin (at)

sin (at)

tm (m ¼ 1, 2, 3, ...) pffi t

t

A

Function f(t) or [L1{F(s)}]a 1

Table 4.1 Pharmacokinetic-related some selected Laplace transforms

4.1 Various Pharmacokinetic Models and Drug Distribution 71



e

þ

at

þ

e

bt

b2 AbþB bðbaÞ

AbB bðabÞ

ebt

a

(L1 represents anti-Laplace forms; L represents Laplace transforms). The relation between f(t) and F(s) is FðsÞ ¼ mentioned in the table

a2 AaþB aðbaÞ



B ab

at

e

AaB aðabÞ

t  aA2 ð 1  eat Þ

R1 0

f ðt Þ est dt, N.B. Convergence region not

s2 þAsþB sðsþaÞðsþbÞ

A s2 ðsþaÞ AsþB sðsþaÞðsþbÞ

A sðsþaÞðsþbÞ

As2 þBsþC ðsþaÞðsþbÞðsþcÞ

4

B ab

A a

1  PQR

e

{A + (B  Aa)t}


PðAa2  Ba þ C Þeat þ Q Ab2  Bb þ C ebt þ RðAc2  Bc þ C Þect (where P ¼ (b  c), Q ¼ (c  a), R ¼ (a  b); and a 6¼ b 6¼ c) h i 1 1 at 1 bt A ab þ aðab  bðab Þ e Þ e

AsþB ðsþaÞðsþbÞ

ðBAaÞeat ðBAbÞebt ðwhere b 6¼ aÞ ba
at  A bt e  e ð where b 6¼ aÞ ba at A ðsþaÞðsþbÞ AsþB ðsþaÞ2

Laplace transforms F(s) or L[f(t)]a a F(s) + b G(s) s F(s)  f(0) s2 F(s)  s f(0)  f 0(0)

Function f(t) or [L1{F(s)}]a a f(t) + b g(t) f 0(t) f 00(t)

Table 4.1 (continued)

72 Pharmacokinetic Models and Drug Distribution

4.1 Various Pharmacokinetic Models and Drug Distribution

73

The above figure shows that it is a complex domain (Laplace transformed form), Here, x0 ¼ A and KE ¼ a. The anti-Laplace form of the complex domain is Aeat, which for the above figure would be

A ðsþaÞ.

x0 eK E t

4.1.7.3 Use of Logarithm and Antilogarithm Tables Logarithms of a number may have one integral part called the “characteristic,” and a decimal part called “mantissa.” In a logarithm table (popularly called log table) (Table 4.2), each row containing some five (5) digits is placed against each number (from 10 to 99). Each of these five-digit numbers is called “mantissa.” The characteristic (also called index) has to be applied in each case. The “characteristic” when greater than unity is positive and less than unity is negative. In case the “characteristic” is positive, it is represented by one less than the number of figures to the left of the decimal point. Below are the examples: The “characteristic” of 65832 is 4. The “characteristic” of 6583.2 is 3. The “characteristic” of 658.32 is 2. The “characteristic” of 65.832 is 1. The “characteristic” of 6.5832 is 0. When the “characteristic” is less than unity (that is, negative), it is represented by one greater than the number of zeros that follow the decimal point and is represented by a bar (-) over the number (for example, 1). Below are the examples: The “characteristic” of 0.65832 is 1. The “characteristic” of 0.065832 is 2. The “characteristic” of 0.0065832 is 3. The “characteristic” of 0.00065832 is 4. There are mean differences against the numbers 1–9 for each row on the right side of the table, and they are used for the calculation of fourth and fifth decimal positions. How to find the logarithm value of a given number is shown below. Example 1 Determination of logarithm value of 6.4 (that is, log6.4): In log6.4, the characteristic is zero (0). Mantissa against 64 is 80618 (please see Table 4.2). So the value of log6.4 is 0.80618. Similarly, the value of log64 is 1.80618. The value of log640 is 2.80618. The value of log6400 is 3.80618.

0 00000

04139

07918

11394

14613

17609

20412

23045

25527

27875

30103 32222 34242 36173 38021 39794 41497 43136

10

11

12

13

14

15

16

17

18

19

20 21 22 23 24 25 26 27

30320 32428 34439 36361 38202 39967 41664 43297

28103

25768

23300

30535 32634 34635 36549 38382 40140 41830 43457

28330

26007

23553

20951

18184

15229

12057

08636

04922

2 00860

30750 32838 34830 36736 38561 40312 41996 43616

28556

26245

23805

21219

18469

15534

12385

08991

05308

3 01284

30963 33041 35025 36922 38739 40483 42160 43775

28780

26482

24055

21484

18752

15836

12710

09342

05690

4 01703

31175 33244 35218 37107 38917 40654 42325 42933

29003

26717

24304

21748

19033

16137

13033

09691

06070

5 02119

31387 33445 35411 37291 39094 40824 42488 44091

29226

26951

24551

22011

19312

16435

13354

10037

06446

6 02531

31597 33646 35603 37475 39270 40993 42651 44248

29447

27184

24797

22272

19590

16732

13672

10380

06819

7 02938

31806 33846 35793 37658 39445 41162 42813 44404

29667

27416

25042

22531

19866

17026

13988

10721

07188

8 03342

32015 34044 35984 37840 39620 41330 42975 44560

29885

27646

25285

22789

20140

17319

14301

11059

07555

9 03743

Mean difference 1 2 3 42 85 127 40 81 121 37 77 116 37 74 111 36 71 106 34 68 102 33 66 98 32 63 95 30 61 91 29 59 88 28 57 85 28 55 83 27 53 80 26 52 78 26 50 76 25 49 73 24 48 71 23 46 69 23 45 68 22 44 66 21 43 64 20 41 61 20 39 58 19 37 56 18 35 53 17 34 51 16 33 49 16 32 47 4 170 162 154 148 142 136 131 126 122 118 114 110 107 104 101 98 95 93 90 88 85 81 77 74 71 68 66 63

5 212 202 193 185 177 170 164 158 152 147 142 138 134 130 126 122 119 116 113 110 106 101 97 93 89 85 82 79

6 254 242 232 222 213 204 197 190 183 177 171 165 160 156 151 147 143 139 135 132 127 121 116 111 106 102 98 95

7 297 283 270 259 248 238 229 221 213 206 199 193 187 182 176 171 167 162 158 154 148 141 135 130 124 119 115 111

8 339 323 309 296 284 272 262 253 244 236 228 221 214 208 201 196 190 185 180 176 170 162 154 148 142 136 131 126

9 381 364 348 333 319 307 295 284 274 265 256 248 240 233 227 220 214 208 203 198 190 182 174 167 159 153 148 142

4

20683

17898

14922

11727

08279

04532

1 00432

Table 4.2 Logarithm table

74 Pharmacokinetic Models and Drug Distribution

44716 46240 47712 49136 50515 51851 53148 54407 55630 56820 57978 59106 60206 61278 62325 63347 64345 65321 66276 67210 68124 69020

0 69897 70757 71600 72428 73230

28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

50 51 52 53 54

1 69984 70842 71684 72509 73320

44871 46389 47857 49276 50650 51983 53275 54531 55751 56937 58092 59218 60314 61384 62428 63448 64444 65418 66370 67302 68215 69108

2 70070 70927 71767 72591 73400

45025 46538 48001 49415 50786 52114 53403 54654 55871 57054 58206 59529 60423 61490 62531 63548 64542 65514 66464 67394 68305 69197

45484 46982 48430 49831 51188 52504 53782 55023 56229 57403 58546 59660 60746 61805 62839 63849 64836 65801 66745 67669 68574 69461

4 70243 71096 71933 72754 73560

45332 46835 48287 49693 51054 52375 53656 54900 56110 57287 58433 59550 60638 61700 62737 63749 64738 65706 66652 67578 68485 69373

3 70157 71012 71850 72673 73480

45179 46687 48144 49554 50920 52244 53529 54777 55991 57171 58320 59439 60531 61595 62634 63649 64640 65610 66558 67486 68395 69285 5 70329 71181 72016 72835 73640

45637 47129 48572 49969 51322 52634 53908 55145 56348 57519 58659 59770 60853 61909 62941 63949 64933 65896 66839 67761 68664 69548 6 70415 71265 72099 72916 73719

45788 47276 48714 50106 51455 52763 54033 55267 56467 57634 58771 59879 60959 62014 63043 64048 65031 65992 66932 67852 68753 69636 7 70501 71349 72181 72997 73799

45939 47422 48855 50243 51587 52892 54158 55388 56585 57749 58883 59988 61066 62118 63144 64147 65128 66087 67025 67943 68842 69723 8 70586 71433 72263 73078 73878

46090 47567 48996 50379 51720 53020 54283 55509 56703 57864 58995 60097 61172 62221 63246 64246 65225 66181 67117 68034 68931 69810

30 29 29 28 27 26 25 24 24 23 23 22 21 21 20 20 20 19 19 18 18 18 9 70672 71517 72346 73159 73957

15 15 14 14 13 13 13 12 12 12 11 11 11 10 10 10 10 10 9 9 9 9

46 61 76 44 59 74 43 57 72 41 55 69 40 54 67 39 52 65 38 50 63 37 49 61 36 48 60 35 46 58 34 45 57 33 44 55 32 43 54 31 42 53 31 41 51 30 40 50 29 39 49 29 38 48 28 37 47 27 36 46 27 36 45 26 35 44 Mean difference 1 2 3 4 9 17 26 34 8 17 25 34 8 17 25 33 8 16 24 32 8 16 24 32 5 43 42 42 41 40

91 88 86 83 80 78 76 73 71 70 68 66 64 63 61 60 59 57 56 55 53 53 6 52 50 50 49 48

107 103 100 97 94 91 88 85 83 81 79 77 75 74 71 70 68 67 65 64 63 63 7 60 59 58 57 56

9 77 76 75 73 72

137 132 129 124 121 117 113 110 107 104 102 99 97 95 92 90 88 86 84 82 81 79

(continued)

8 69 67 66 65 64

122 118 114 110 107 104 101 98 95 93 90 88 86 84 82 80 78 76 74 73 72 70

4.1 Various Pharmacokinetic Models and Drug Distribution 75

55 55 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

0 74036 74819 75587 76343 77085 77815 78533 79239 79934 80618 81291 81954 82607 83251 83885 84510 85126 85733 86332 86923 87506 88081 88649 89209 89763 90309

1 74115 74896 75664 76418 77159 77887 78604 79309 80003 80686 81358 82020 82672 83315 83948 84572 85187 85794 86392 86982 87564 88138 88705 89265 89818 90363

Table 4.2 (continued)

2 74194 74974 75740 76492 77232 77960 78675 79379 80072 80754 81425 82086 82737 83378 84011 84634 85248 85854 86451 87040 87622 88195 88762 89321 89873 90417

3 74273 75051 75815 76567 77305 78032 78746 79449 80140 80821 81491 82151 82802 83442 84073 84696 85309 85914 86510 87099 87679 88252 88818 89376 89927 90472

4 74351 75128 75891 76641 77379 78104 78817 79518 80209 80889 81558 82217 82866 83506 84136 84757 85370 85974 86570 87157 87737 88309 88874 89432 89982 90526

5 74429 75205 75967 76716 77452 78176 78888 79588 80277 80956 81624 82282 82930 83569 84198 84819 85431 86034 86629 87216 87795 88366 88930 89487 90037 90580

6 74507 75282 76042 76790 77525 78247 78958 79657 80346 81023 81690 82347 82995 83632 84261 84880 85491 86094 86688 87274 87852 88423 88986 89542 90091 90634

7 74586 75358 76118 76864 77597 78319 79029 79727 80414 81090 81757 82413 83059 83696 84323 84942 85552 86153 86747 87332 87910 88480 89042 89597 90146 90687

8 74663 75435 76193 76938 77670 78390 79099 79796 80482 81158 81823 82478 83123 83759 84386 85003 85612 86213 86806 87390 87967 88536 89098 89653 90200 90741

9 74741 75511 76268 77012 77743 78462 79169 79865 80550 81224 81889 82543 83187 83822 84448 85065 85673 86273 86864 87448 88024 88593 89154 89708 90255 90795

Mean difference 1 2 3 8 16 23 8 15 23 8 15 23 7 15 22 7 15 22 7 14 22 7 14 21 7 14 21 7 14 20 7 13 20 7 13 20 7 13 20 6 13 19 6 13 19 6 12 19 6 12 19 6 12 18 6 12 18 6 12 18 6 12 17 6 12 17 6 11 17 6 11 17 6 11 17 6 11 17 5 11 16 4 31 31 30 30 29 29 28 28 27 27 26 26 26 25 25 25 24 24 24 23 23 23 22 22 22 22

5 39 39 38 37 37 36 36 35 34 34 33 33 32 32 31 31 31 30 30 29 29 29 28 28 28 27

6 47 46 45 44 44 43 43 41 41 40 40 39 38 38 37 37 37 36 35 35 35 34 34 33 33 32

7 55 54 53 52 51 50 50 48 48 47 46 46 45 44 43 43 43 42 41 41 41 40 39 39 39 38

8 63 62 60 59 58 58 57 55 54 54 53 52 51 50 50 50 49 48 47 46 46 46 45 44 44 43

9 70 69 68 67 66 65 64 62 61 60 59 59 58 57 56 56 55 54 53 52 52 51 50 50 50 49

76 4 Pharmacokinetic Models and Drug Distribution

81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99

90848 91381 91908 92428 92942 93450 93952 94448 94939 95424 95904 96379 96848 97313 97772 98227 98677 99123 99564

90902 91434 91960 92480 92993 93500 94002 94498 94988 95472 95952 96426 96895 97359 97818 98272 98722 99167 99607

90956 91487 92012 92531 93044 93551 94052 94547 95036 95521 95999 96473 96942 97405 97864 98318 98767 99211 99651

91009 91540 92064 92583 93095 93601 94101 94596 95085 95569 96047 96520 96988 97451 97909 98363 98811 99255 99695

91062 91593 92117 92634 93146 93651 94151 94645 95134 95617 96095 96567 97035 97497 97955 98408 98856 99300 99739

91116 91645 92169 92686 93197 93702 94201 94694 95182 95665 96142 96614 97081 97543 98000 98453 98900 99344 99782

91169 91698 92221 92737 93247 93752 94250 94743 95231 95713 96190 96661 97128 97589 88046 98498 98945 99388 99826

91222 91751 92273 92788 93298 93802 94300 94792 95279 95761 96237 96708 97174 97635 98091 98543 98989 98432 99870

91275 91803 92324 92840 93349 93852 94349 94841 95328 95809 96284 96755 97220 97681 98137 98588 99034 99476 99913

91328 91855 92376 92891 93399 93902 94399 94890 95376 95856 96332 96802 97267 97727 98182 98632 99078 99520 99957

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4

11 11 10 10 10 10 10 10 10 10 9 9 9 9 9 9 9 9 9

16 16 16 15 15 15 15 15 15 14 14 14 14 14 14 14 13 13 13

21 21 21 20 20 20 20 20 19 19 19 19 18 18 18 18 18 18 17

27 27 26 26 26 25 25 25 24 24 24 24 23 23 23 23 22 22 22

32 32 31 31 31 30 30 29 29 29 28 28 28 28 27 27 27 26 26

37 37 36 36 36 35 35 34 34 34 33 33 32 32 32 32 31 31 31

42 42 42 41 41 40 40 39 39 38 38 38 38 37 36 36 36 35 35

48 48 47 46 46 45 45 44 44 43 42 42 42 42 41 41 40 40 39

4.1 Various Pharmacokinetic Models and Drug Distribution 77

78

4

Pharmacokinetic Models and Drug Distribution

Example 2 Again, for the determination of the value of log645, the characteristic is 2; mantissa against 645 is 0.80956 (please see the log table and take the row value against 64 and under the column value 5). Thus, the value of log645 is 2.80956. For the determination of the value of log645.4, the characteristic is 2. Mantissa against 645 is: 0.80956 The mean difference against 4 of the number is: 27 ∴ Mantissa ¼ 0.80983 Therefore, the value of log645.4 ¼ 2.80983

Example 3 For the determination of the value of log64.543, the characteristic is 1. Mantissa against 645 is: 0.80956 The mean difference against 4 of the number is: 27 The mean difference against 3 of the number is: 2 0 ∴ Mantissa ¼ 0.80985 (0.80956 + 0.00027 + 0.000020) Therefore, the value of log64.543 ¼ 1.80985

Example 4 For the determination of the value of log6454.3, the characteristic is 3. Mantissa against 645 is: 0.80956 The mean difference against 4 of the number is: 27 The mean difference against 3 of the number is: 2 0 ∴ Mantissa ¼ 0.80985 (0.80956 + 0.00027 + 0.000020) Therefore, the value of log6454.3 ¼ 3.80985

Example 5 Calculate the value of log(6454.3)22.8 ¼22.8 log (6454.3) ¼ 22.8  3.80985 ¼ 86.86458 The number corresponding to a given logarithm number can be determined using the antilogarithm (antilog ) table (Table 4.3), as shown below. Example 1 To determine the natural number corresponding to the logarithm number, log1.85426

.00 .01 .02 .03 .04 .05 .06 .07 .08 .09 .10 .11 .12 .13 .14 .15 .16 .17 .18 .19 .20 .21

0 10000 10233 10471 10715 10965 11220 11482 11749 12023 12303 12589 12882 13183 13490 13804 14125 14454 14791 15136 15488 15849 16218

1 10023 10257 10495 10740 10990 11246 11508 11776 12050 12331 12618 12912 13213 13521 13836 14158 14488 14825 15171 15524 15885 16255

2 10046 10280 10520 10765 11015 11272 11535 11803 12078 12359 12647 12942 13243 13552 13868 14191 14521 14859 15205 15560 15922 16293

Table 4.3 Antilogarithm table

3 10069 10304 10544 10789 11041 11298 11561 11830 12106 12388 12677 12972 13274 13583 13900 14223 14555 14894 15241 15596 15959 16331

4 10093 10328 10568 10814 11066 11324 11588 11858 12134 12417 12706 13002 13305 13614 13932 14256 14588 14928 15276 15631 15996 16368

5 10116 10351 10593 10839 11092 11350 11614 11885 12162 12445 12735 13032 13335 13646 13964 14289 14622 14962 15311 15668 16032 16406

6 10139 10375 10617 10864 11117 11376 11641 11912 12190 12474 12764 13062 13366 13677 13996 14322 14655 14997 15346 15704 16069 16444

7 10162 10399 10641 10889 11143 11402 11668 11940 12218 12503 12794 13092 13397 13709 14028 14355 14689 15031 15382 15740 16106 16482

8 10186 10423 10666 10914 11169 11429 11695 11967 12246 12531 12823 13122 13428 13740 14060 14388 14723 15066 15417 15776 16144 16520

9 10209 10447 10691 10940 11194 11455 11722 11995 12274 12560 12853 13152 13459 13772 14093 14421 14757 15101 15453 15812 16181 16558

Mean difference 1 2 3 2 5 7 2 5 7 2 5 7 3 5 8 3 5 8 3 5 8 3 5 8 3 5 8 3 6 8 3 6 9 3 6 9 3 6 9 3 6 9 3 6 9 3 6 10 3 7 10 3 7 10 3 7 10 4 7 11 4 7 11 4 7 11 4 8 11 4 9 10 10 10 10 11 11 11 11 11 12 12 12 13 13 13 13 14 14 14 15 15

5 12 12 12 13 13 13 13 14 14 14 15 15 15 16 16 16 17 17 18 18 18 19

6 14 14 15 15 15 16 16 16 17 17 18 18 18 19 19 20 20 21 21 22 22 23

7 16 17 17 18 18 18 19 19 20 20 21 21 21 22 22 23 24 24 25 25 26 26

9 21 21 22 23 23 24 24 25 25 26 26 27 28 28 29 30 30 31 32 32 33 34 (continued)

8 19 19 20 20 20 21 21 22 22 23 24 24 25 25 26 26 27 28 28 29 30 30

4.1 Various Pharmacokinetic Models and Drug Distribution 79

.22 .23 .24 .25 .26 .27 .28 .29 .30 .31 .32 .33 .34 .35 .36 .37 .38 .39 .40 .41 .42 .43 .44 .45

0 16596 16982 17378 17783 18197 18621 19055 19498 19953 20417 20893 21380 21878 22387 22909 23442 23988 24547 25119 25704 26303 26915 27542 28184

1 16634 17022 17418 17824 18239 18664 19099 19543 19999 20464 20941 21429 21928 22439 22961 23496 24044 24604 25177 25763 26363 26977 27606 28249

Table 4.3 (continued)

2 16672 17061 17458 17865 18281 18707 19143 19588 20045 20512 20989 21478 21979 22491 23014 23550 24099 24660 25236 25823 26424 27040 27669 28314

3 16711 17100 17498 17906 18323 18750 19187 19634 20091 20559 21038 21528 22029 22542 23067 23605 24155 24717 25293 25882 26485 27102 27733 28379

4 16749 17140 17539 17947 18365 18793 19231 19679 20137 20606 21086 21577 22080 22594 23121 23659 24210 24774 25351 25942 26546 27164 27797 28445

5 16788 17179 17579 17989 18408 18836 19275 19724 20184 20654 21135 21627 22131 22646 23174 23714 24266 24831 25410 26002 26607 27227 27861 28510

6 16827 17219 17620 18030 18450 18880 19320 19770 20230 20701 21184 21677 22182 22699 23227 23768 24322 24889 25468 26062 26669 27290 27925 28576

7 16866 17258 17660 18072 18493 18923 19364 19815 20277 20749 21232 21727 22233 22751 23281 23823 24378 24946 25527 26122 26730 27353 27990 28642

8 16904 17298 17701 18113 18535 18967 19409 19861 20324 20797 21281 21777 22284 22803 23336 23878 24434 25003 25586 26182 26792 27416 28054 28708

9 16943 17338 17742 18155 18578 19011 19454 19907 20370 20845 21330 21827 22336 22856 23388 23933 24491 25061 25645 26242 26853 27479 28119 28774

Mean difference 1 2 3 4 8 12 4 8 12 4 8 12 4 8 12 4 8 13 4 9 13 4 9 13 5 9 14 5 9 14 5 10 14 5 10 15 5 10 15 5 10 15 5 10 16 5 11 16 5 11 16 6 11 17 6 11 17 6 12 18 6 12 18 6 12 18 6 13 19 6 13 19 7 13 20 4 15 16 16 17 17 17 18 18 19 19 19 20 20 21 21 22 22 23 23 24 24 25 26 26

5 19 20 20 21 21 22 22 23 23 24 24 25 25 26 27 27 28 29 29 30 31 31 32 33

6 23 24 24 25 25 26 26 27 28 29 29 30 31 31 32 33 34 34 35 36 37 38 39 39

7 27 28 28 29 30 30 31 32 32 33 34 35 36 37 37 38 39 40 41 42 43 44 45 46

8 31 32 32 33 34 35 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

9 35 36 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 53 54 55 56 58 59

80 4 Pharmacokinetic Models and Drug Distribution

28907 29580 30269 30974

1 31696 32434 33189 33963 34754 35563 36392 37239 38107 38994 39902 40832 41783 42756 43752 44771 45814 46881

28840 29512 30200 30903

0 31623 32359 33113 33884 34674 35481 36308 37154 38019 38905 39811 40738 41687 42658 43652 44668 45709 46774

.46 .47 .48 .49

.50 .51 .52 .53 .54 .55 .56 .57 .58 .59 .60 .61 .62 .63 .64 .65 .66 .67

2 31769 32509 33266 34041 34834 35645 36475 37325 38194 39084 39994 40926 41879 42855 43853 44875 45920 46989

28973 29648 30339 31046

3 31842 32584 33343 34119 34914 35727 36559 37411 38282 39174 40087 41020 41976 42954 43954 44978 46026 47098

29040 29717 30409 31117

4 31916 32659 33420 34198 34995 35810 36644 37497 38371 39264 40179 41115 42073 43053 44055 45082 46132 47206

29444 7 13 30130 7 14 30832 7 14 31550 7 14 Mean Difference 9 1 2 3 32285 7 15 22 33037 8 15 23 33806 8 15 23 34594 8 16 24 35400 8 16 24 36224 8 16 25 37068 8 17 25 37931 9 17 26 38815 9 18 27 39719 9 18 27 40644 9 19 28 41591 9 19 28 42560 10 19 29 43551 10 20 30 44566 10 20 30 45604 10 21 31 46666 11 21 32 47753 11 22 33

29376 30061 30761 31477

8 32211 32961 33729 34514 35318 36141 36983 37844 38726 39628 40551 41495 42462 43451 44463 45499 46559 47643

29309 29992 30690 31405

7 32137 32885 33651 34435 35237 36058 36898 37757 38637 39537 40458 41400 42364 43351 44361 45394 46452 47534

29242 29923 30620 31333

6 32063 32809 33574 34356 35156 35975 36813 37670 38548 39446 40365 41305 42267 43251 44259 45290 46345 47424

29174 29854 30549 31261

5 31989 32735 33497 34277 35075 35892 36728 37584 38459 39355 40272 41210 42170 43152 44157 45186 46238 47315

29107 29785 30479 31189 4 29 30 31 32 32 33 34 35 35 36 37 38 39 40 41 42 43 44

20 21 21 22 5 37 38 39 40 40 41 42 43 44 45 46 47 49 50 51 52 53 54

27 28 28 29 6 44 45 46 47 48 50 51 52 53 54 56 57 58 60 61 62 64 65

34 34 35 36 7 52 53 54 55 56 58 59 61 62 63 65 66 68 70 71 73 75 76

40 41 42 43

47 48 49 50 9 66 68 69 71 73 74 76 78 80 82 83 85 87 89 91 94 96 98

60 62 63 65

(continued)

8 59 60 62 63 65 66 68 69 71 72 74 76 78 80 81 83 85 87

54 55 56 58

4.1 Various Pharmacokinetic Models and Drug Distribution 81

.68 .69 .70 .71 .72 .73 .74 .75 .76 .77 .78 .79 .80 .81 .82 .83 .84 .85 .86 .87 .88 .89 .90 .91

0 47863 48978 50119 51286 52481 53703 54954 56234 57544 58884 60256 61659 63096 64565 66069 67608 69183 70795 72444 74131 75858 77625 79433 81283

1 47973 49091 50234 51404 52602 52827 55081 56364 57677 59020 60395 61802 63241 64714 66222 67764 69343 70958 72611 74302 76033 77804 79616 81470

Table 4.3 (continued)

2 48084 49204 50350 51523 52723 53951 55208 56494 57810 59156 60534 61944 63387 64863 66374 67920 69503 71121 72778 74473 76208 77983 79799 81658

3 48195 49317 50466 51642 52845 54075 55336 56624 57943 59293 60674 62087 63533 65013 66527 68077 69663 71285 72946 74645 76384 78163 79983 81846

4 48306 49431 50582 51761 52966 54200 55463 56754 58076 59429 60814 62230 63680 65163 66681 88234 69823 71450 73114 74817 76560 78343 80168 82035

5 48417 49545 50699 51880 53088 54325 55590 56885 58210 59566 60954 62373 63826 65313 66834 68391 69984 71614 73282 74989 76736 78524 80353 82224

6 48529 49659 50816 52000 53211 54450 55719 57016 58345 59704 61094 62517 63973 65464 66988 68549 70146 71779 73451 75162 76913 78705 80538 82414

7 48641 49774 50933 52119 53333 54576 55847 57148 58479 59841 61235 62661 64121 65615 67143 68707 70307 71945 73621 75336 77090 78886 80724 82604

8 48753 49888 51050 52240 53456 54702 55976 57280 58614 59979 61376 62806 64269 65766 67298 68865 70469 72111 73790 75509 77268 79068 80910 82794

9 48865 50003 51168 52360 53580 54828 56105 57412 58749 60117 61518 62951 64417 65917 67453 69024 70632 72277 73961 75683 77456 79250 81096 82985

Mean Difference 1 2 3 11 22 33 11 23 34 12 23 35 12 24 36 12 24 37 13 25 38 13 26 38 13 26 39 13 27 40 14 27 41 14 28 42 14 29 43 15 29 44 15 30 45 15 31 46 16 32 47 16 32 48 17 33 50 17 34 51 17 35 52 18 35 53 18 36 54 19 37 56 19 38 57 4 45 46 47 48 49 50 51 52 54 55 56 58 59 60 62 63 64 66 68 69 71 72 74 76

5 56 57 58 60 61 63 64 66 67 69 70 72 74 75 77 79 81 83 85 87 89 91 93 95

6 67 68 70 72 73 75 77 79 80 82 84 86 88 90 92 95 97 99 101 104 107 109 111 113

7 78 80 82 84 85 88 90 92 94 96 98 101 103 105 108 110 113 116 118 121 125 127 130 132

8 89 91 93 96 98 100 102 105 107 110 112 115 118 120 123 126 129 132 135 138 142 145 148 151

9 100 103 105 108 110 113 115 118 121 123 126 130 132 135 139 142 145 149 152 156 159 163 167 170

82 4 Pharmacokinetic Models and Drug Distribution

.92 .93 .94 .95 .96 .97 .98 .99

83176 85114 87096 89125 91201 93325 95499 97724

83368 85310 87297 89331 91411 93541 95719 97949

83560 85507 87498 89536 91622 93756 95940 98175

83753 85704 87700 89743 91833 93972 96161 98401

83946 85901 87902 89950 92045 94189 96383 98628

84140 86099 88105 90157 92257 94406 96605 98855

84333 86298 88308 90365 92470 94624 96828 99083

84528 86497 88512 90573 92683 94842 97051 99312

84723 86696 88716 90782 92897 95060 97275 99541

84918 86896 88920 90991 93111 95280 97499 99770

19 20 20 21 21 22 22 23

39 40 41 42 42 43 44 46

58 60 61 62 64 65 67 68

78 79 81 83 85 87 89 91

97 99 102 104 106 109 111 114

116 119 122 125 127 130 133 137

136 139 142 146 149 152 155 160

155 158 162 166 170 174 178 182

175 178 183 187 191 195 200 205

4.1 Various Pharmacokinetic Models and Drug Distribution 83

84

4

Pharmacokinetic Models and Drug Distribution

Antilog of 854 is: Antilog of 854 is: 0.71450 The mean difference against 2 of the number is: 33 The mean difference against 6 of the number is: 9 9 ∴ Mantissa ¼ 0.714929  0.71493 (0.71450 + 0.00033 + 0.000099) Therefore, the number corresponding to logarithm number, log1.85426 is 71.493.

4.1.7.4 Relationship Between Common and Natural Logarithms Natural logarithm implies logarithm with the base “e” and is denoted by ln. On the other hand, a common logarithm has the base 10, and it is usually denoted by log. Conversion: If y ¼ loga x, it means ay ¼ x For a common logarithm, if y ¼ log10x, it means 10y ¼ x By taking the natural logarithm on both sides of the equation 10y ¼ x, We get ln10y ¼ ln x Or y ln 10 ¼ ln x Or y ð2:303Þ ¼ ln x ðas the value of ln 10 ¼ 2:303Þ Or y ¼

ln x ¼ common logarithm 2:303

Therefore, common logarithm ¼

4.1.8

Natural Logarithm 2:303

Details About Compartment Models

(Definition, types, assumptions) The compartment model is a pharmacokinetic model in which the body is virtually compartmentalized into one/two/more (multi-) compartment(s). When the entire body is considered as a single compartment, it is called a one-compartment model. In one-compartment model, when the drug eliminates from it, it is called one (single)-compartment open model. When a body is compartmentalized into two/more (multi-) numbers of compartments, namely, one central compartment (blood compartment) and peripheral compartments (tissue compartments), they are called two-/multi-compartment models. In two-/multi-compartment models, while the drug eliminates from the body by any number of compartments, the compartment models are called two-/multi-compartment open models.

4.1 Various Pharmacokinetic Models and Drug Distribution

85

Some pre-considerations (assumptions) are required to provide accuracy of simulation of data in the two-/multi-compartmental model. They are called assumptions. For two-/multi-compartment open models, fundamental assumptions are mentioned below: • The body can be compartmentalized into two or more numbers of compartments. • The blood compartment is always considered the central compartment. • All the other compartments are considered peripheral compartments (tissue compartments) (for example, liver compartment, brain compartment, kidney compartment, lung compartment, etc.). • Drugs administered by any route into any compartment should reach into the central compartment (blood compartment). • The drug is distributed reversibly (represented by “
” ) between the central (blood) compartment and the tissue compartment/compartments and establishes equilibrium. • Drugs can eliminate from any one of the compartments or many or all of the compartments. • If a drug eliminates from our body, it will not come back into our system again. Hence, the elimination process is represented by a unidirectional arrowhead (shown by #), indicating drug/metabolite goes outside the body. • The central compartment is always numbered with one (1), while the numbering of the peripheral compartments (tissue compartments) starts with other positive consecutive numbers (such as 2, 3, 4,...). But unlike the central compartment, there is no fixed specific number choice for the tissue compartments. The numbers for tissue compartments are normally chosen by the person who draws the model. • The rate constant is recognized by two digits at its suffix. The first number at the suffix indicates the compartment where the drug is distributed, and the second number suggests the compartment where the drug would be distributed. For example, the drug is distributed from the central compartment (always numbered as 1) to a peripheral compartment (say, numbered as 2), and then, the rate constant is depicted as k12. • When a drug is eliminated from a compartment, the rate constant (k) is recognized by two digits also at its suffix. The first number at the suffix indicates the number of the compartment wherefrom the drug is eliminated, followed by the number zero (0) as the drug goes out of the system, and it is supposed that it would not come back into our system again. For example, the drug is eliminated from the central compartment (always numbered as 1), and then, the elimination rate is depicted as k10.

4.1.9

Drug Distribution Study Through Compartmental Models

Drug distribution can be well analyzed by compartmental models. The body is virtually compartmentalized into several segments for computing drug distribution

86

4

Pharmacokinetic Models and Drug Distribution

Fig. 4.10 One-compartment open model that shows x0 amount of drug is administered intravenously at the time t0 to an entirely homogeneous body compartment, xB is the amount of drug present in the compartment at time t, and the drug is eliminated from it. KE indicates the rate of drug elimination, and the arrowhead indicates the drug is eliminated outside the body compartment

after its administration. The description and analytical concept of one-, two-, and three-compartment models are given underneath.

4.1.9.1 One-Compartment Open Model In a one-compartment open model, the entire body is considered a single compartment that follows drug distribution and drug elimination controlled by first-order kinetics. 4.1.9.1.1 Intravenously Bolus Dose If x0 amount of drug administered intravenously as a bolus dose (Fig. 4.10), at time t, the amount of drug in the body is xB, the overall elimination rate constant is KE, and the rate of disappearance of drug from the body is dxdtB, then dxdtB ¼ K E X B (following first-order kinetics). ½ K E ðoverall elimination rateÞ ¼ ke ðurinary elimination rateÞ þk b ðbilliary elimination rateÞ þ k b ðmetabolic eliminationÞ þ . . . : Taking Laplace transform, we get sLf ðxB Þ  x0 ¼ K E Lf ðxB Þ, for simplicity, L f ðxB Þ ¼ xB , where f ðxB Þ is the function of xB : Or s xB  x0 ¼ K E xB Or xB ðs þ K E Þ ¼ x0

4.1 Various Pharmacokinetic Models and Drug Distribution

Or xB ¼

87

x0 ðs þ K E Þ

Taking anti-Laplace, we get xB ¼ x0 eK E t Or ln xB ¼ ln x0  K E t ðe  base logarithmic form, natural logarithmÞ Or log xB ¼ log x0 

KEt ð10  base logarithmic form, common logarithmÞ 2:303 Therefore, log xB ¼ log x0 

KEt 2:303

This equation has some limitations. We consider the total amount of drug in the body, and the body is a homogeneous, rapidly diffusible system. But in reality, neither the body is a homogeneous system nor do we determine the drug from the entire body. Instead, we choose data from blood, plasma, or tissue concentration. After the intravenous (i.v.) administration of a drug, the drug is distributed to the various organs in the body. If the drug concentrations in those organs are c1, c2, c3, . . .. . . . . cn and the volume of distribution of the drug in those organs and tissue are v1, v2, v3, . . .. . . . . vn, then xB ¼ c1v1 + c2v2+ .........+cnvn; and apparent volume of distribution (vd) is the summation of the tissue-specific volumes of distribution of the drug. That is, vd ¼ v1 þ v2 þ v3 þ . . . . . . :: þ vn In reality, the collection of such data is challenging, if not impossible. Further, log xB ¼ log x0  Or log

KEt 2:303

xB x K t ¼ log 0  E vd vd 2:303

(let us consider, xB ¼ cpvd and x0 ¼ c0vd where c0 drug plasma concentration at the instantaneous moment of administration of x0 amount of drug and cp is the plasma concentration of drug at any time t after the drug administration). Or log cp ¼ log c0 

KEt 2:303

It is a workable form of the equation when logcp is plotted against time on a semilog graph paper. The extrapolated straight line on y  axis or log cp  axis will give KE us the value of logc0 (Fig. 4.11). The slope of the line will give us the value of  2:303 , from where we can determine the value of KE. However, from the above relationship, we get

88

4

Pharmacokinetic Models and Drug Distribution

Fig. 4.11 Plot of logarithmic plasma drug concentration ( log cp) versus time on a semiKE and the intercept on the y-axis of the logarithmic paper. The slope of the line gives us  2:303 extrapolated line gives us logc0 where c0 is the instantaneous plasma drug concentration upon the administration of x0 amount of drug

x0 ¼ c0 vd Or vd ¼ xc00 ; this data also does not give a reliable volume of distribution of the drug.

4.1.9.1.2 Intravenous Infusion Intravenous infusion is the direct administration of the drug slowly at a constant rate into a patient’s vein for a longer duration. Some antibiotics, cardiovascular drugs, and drugs with a narrow therapeutic index are administered to the patients under direct monitoring and supervision of a specialized physician of the field. In the case of intravenous infusion, the drug is administered at a constant rate (here k0). Hence, we assume that the drug administration follows zero-order kinetics, in which drug administration rate is equal to k0. Further, at time t, the amount of drug in the compartment is xB, and KE is the overall elimination rate constant of the drug (Fig. 4.12). Therefore, the rate of change of the drug in the body at time t is dxB dt ¼ k 0  KExB (the rate at time t ¼ input of drug in the body  output of the drug from the body at that moment) Taking Laplace transform, we get s xB ¼

k0  K E xB s

4.1 Various Pharmacokinetic Models and Drug Distribution

89

Fig. 4.12 The figure shows a one-compartment open model where a drug is administered by infusion at a rate k0. KE is the rate of drug elimination from the compartment and xB is the amount of drug present in the compartment at time t

Or xB ðs þ K E Þ ¼ Or xB ¼

k0 s

k0 s ðs þ K E Þ

Taking anti-Laplace of the above equation, we get xB ¼

 k0
1  eK E t KE

Further, xB ¼ cp vd ¼


 k0
1  eK E t ,xB ¼ cp vd KE

Or cp ¼

 k0
1  eK E t K E vd

After 5–8 biological half-lives of a drug administered, the drug steady-state plasma concentration reaches. Mathematically, when t ¼ 1 (infinity), the drug plasma steady-state undoubtedly occurs. Hence, when the plasma drug concentration comes to the steady-state level, then eK E t ! 0, as t ! 1. Therefore, the steady-state plasma drug concentration c or css or cp ss ¼ KkE0vd .

90

4

Pharmacokinetic Models and Drug Distribution

4.1.9.1.3 Following One-Compartment Model, Determination of Overall Elimination Rate During the Time t0 Elapsed After the Stop of an Infusion at the Steady-State of a Drug to a Patient In a one-compartment model, the intravenous infusion was administered to a patient at a rate of k0 and the steady-state drug level was achieved. While the time t0 elapsed after the stop of the drug infusion to the patient at time T (Fig. 4.13), we want to determine the overall elimination rate after the stop of infusion. The time t is the total time during which the patient is supposed to receive the infusion. Therefore, t0 ¼ t  T. We know t0 ¼ t  T. Further, the rate of change of plasma drug concentration at any time t0 after the stop of infusion, after achieving steady-state drug level, is given by dcp ¼ K E cp dt 0 Taking Laplace transform, we get scp  cp ss ¼ K E cp (where cpss , steady-state drug plasma concentration at T, that is, at the initial time point of the declined phase)

Fig. 4.13 Plot of logarithmic plasma drug concentration versus time of a patient who received intravenous drug infusion and achieved the steady-state plasma drug level. Then, the infusion was stopped at time T. The time t0 elapsed after the stop of the drug infusion in the patient at time T. The time t is the total time the patient was supposed to receive the infusion and had steady-state plasma drug concentration. The dotted line shows the drop of plasma drug concentration after the stop of the infusion

4.1 Various Pharmacokinetic Models and Drug Distribution

91

Or cp ðs þ K E Þ ¼ cp ss k

0 cp ss Or cp ¼ ¼ K E vd ðs þ K E Þ ðs þ K E Þ

  k0 ,cp ss ¼ K E vd

Taking anti-Laplace, we get cp ¼

k0 K E t0 k e ¼ 0 eK E ðtT Þ K E vd K E vd 0

KEt Again, log cp ¼ log KkE0vd  2:303 . On a semi-log paper by plotting logcp against t0, we can determine the overall elimination rate, KE.

4.1.9.1.4 Following One-Compartment Model, Determination of the Overall Elimination Rate While an Infusion of a Drug to a Patient Is Stopped Before the Plasma Drug Concentration Reaches Steady-State Plasma Level In a one-compartment model, the intravenous infusion was administered to a patient at a rate of k0 and was stopped t0 time before it reaches the steady-state plasma level. In a one-compartment model, the intravenous drug infusion was administered to a patient at a rate of k0. Any duration of time t0 before the steady-state plasma drug concentration is achieved; the drug infusion is stopped for the patient at time T (Fig. 4.14). We want to determine the overall elimination rate after the stop of infusion. The time t is the total time period during which the patient is supposed to receive the infusion. Therefore, t0 ¼ t  T. Here, t 0 ¼ t  T dcp ¼ K E cp dt 0 Taking Laplace transform, we get scp  cp ðT Þ ¼ K E cp, where cp ðT Þ is the drug plasma concentration at time T, that is, at the initial time point of the declined phase scp 

 k0
1  eK E T ¼ K E cp K E vd

Or cp ðs þ K E Þ ¼ KkE0vd ð1  eK E T Þ (since, cpðtÞ ¼ KkE0vd ð1  eK E t Þ ; please see intravenous infusion, Sect. 4.1.9.1.2) k0

Or cp ¼ K E vd Taking anti-Laplace, we get

ð1  eK E T Þ ðs þ K E Þ

92

4

Pharmacokinetic Models and Drug Distribution

Fig. 4.14 Plot of logarithmic plasma drug concentration versus time of a patient who receives intravenous drug infusion and should achieve the steady-state plasma drug level at time t, and any duration of time t0 before achieving the steady-state plasma drug concentration, the drug infusion is stopped for the patient at time T. The dotted line shows the drop of plasma drug concentration after the stop of the infusion

cp ¼

dcp 0 k0 ð1  eK E T Þ eK E t ð, ¼ K E cp Þ K E vd dt 0 or cp ¼

 k0
1  eK E T eK E ðTtÞ K E vd 0

KEt Again, log cp ¼ log KkE0vd ð1  eK E T Þ  2:303 . On a semi-log paper by plotting 0 logcp against t , we can determine the overall elimination rate, KE.

4.1.9.2 Two-Compartment Open Model The two-compartment open model is a pharmacokinetic model in which our body is virtually compartmentalized into two compartments, the central compartment (blood compartment) and peripheral compartment (tissue compartment), and the drug can permeate reversibly from one compartment to another compartment and vice versa. It can eliminate from any one of the two compartments or both the compartments (Fig. 4.15). As per the model (Fig. 4.15, Model I), xc is considered as the amount of drug present at the central compartment (compartment “1”) at time t and xp is the amount of drug present at the peripheral compartment (here designated as compartment “2”) at time t. x0 is the dose administered by i.v. injection into the blood (central

4.1 Various Pharmacokinetic Models and Drug Distribution

93

Fig. 4.15 Two-compartment open models. Model I: x0 is the dose administered by the intravenous route to the central compartment and xc is the amount of drug present at the central compartment (compartment “1”) at time t and xp is the amount of drug present at the peripheral compartment (here

94

4

Pharmacokinetic Models and Drug Distribution

compartment). Further, the drug is eliminated from the central compartment (Fig. 4.15, Model I). Then, the rate of change of the amount of drug at the central compartment at time t is represented by dxc ¼ K 21 xp  K 12 xc  K 10 xc , ðinput  outputÞ dt By taking Laplace transform, we get s:xc  x0 ¼ K 21 xp  K 12 xc  K 10 xc (Please see Laplace transform discussed above.) or ðs þ K 12 þ K 10 Þ xc  K 21 xp ¼ x0

ð4:9Þ

Now, the rate of change of the drug amount at the peripheral compartment at time t dxp ¼ K 12 xc  K 21 xp , ðinput  outputÞ dt where xc is the amount of drug present at the central compartment (compartment “1”) at time t and xp is the amount of drug present at the peripheral compartment (here designated as compartment “2”) at time t. x0 is the dose administered to the central compartment, and drug elimination occurs from the central compartment only. By taking Laplace transform, we get s:xp  0 ¼ K 12 xc  K 21 xp (since no drug was available initially at the peripheral compartment) or ðs þ K 21 Þ xp  K 12 xc ¼ 0 or  K 12 xc þ ðs þ K 21 Þ xp ¼ 0

ð4:10Þ

To solve the values of xc and xp from Eqs. (4.9) and (4.10), different methods of solutions are given below:

ä Fig. 4.15 (continued) designated as compartment “2”) at time t. Further, the drug is eliminated from the central compartment. Model II: xc is considered as the amount of drug present at the central compartment (compartment “1”) at time t and xp is the amount of drug present at the peripheral compartment (here designated as compartment “2”) at time t. x0 is the dose administered by intravenous injection into the blood (central compartment). Further, the drug is eliminated from both the central compartment and the peripheral compartment. Model III: x0 is the dose administered by the intravenous route to the central compartment and xc is the amount of drug present at the central compartment (compartment “1”) at time t and xp is the amount of drug present at the peripheral compartment (here designated as compartment “2”) at time t. Further, the drug is eliminated from the peripheral compartment. In all the models, K12, K21, K10, and K20 are the respective rate constants, and each arrowhead shows the direction of movement of the drug

4.1 Various Pharmacokinetic Models and Drug Distribution

95

4.1.9.2.1 Method 1 (Simple Algebraic Method, Simultaneous Quadratic Equation) To solve for xc, multiplying the Eq. (4.9) by (s + K21) and the Eq. (4.10) by K21, we get the following Eqs. (4.11) and (4.12), respectively: ðs þ K 21 Þ ðs þ K 12 þ K 10 Þ xc  ðs þ K 21 Þ K 21 xp ¼ ðs þ K 21 Þ x0 K 21 K 12 xc þ K 21 ðs þ K 21 Þ xp ¼ 0

ð4:11Þ ð4:12Þ

Now by adding Eqs. (4.11) and (4.12), we get ðs þ K 21 Þ ðs þ K 12 þ K 10 Þ xc  K 21 K 12 xc ¼ ðs þ K 21 Þ x0 or, xc f ðs þ K 21 Þðs þ K 12 þ K 10 Þ  K 21 K 12 g ¼ ðs þ K 21 Þ x0 Therefore, xc ¼

ðs þ K 21 Þ x0 ðs þ K 21 Þ ðs þ K 12 þ K 10 Þ  K 21 K 12

Again, the denominator ðs þ K 21 Þðs þ K 12 þ K 10 Þ  K 21 K 12 ¼ s2 þ K 21 s þ K 12 s þ K 21 K 12 þ K 10 s þ K 21 K 10  K 21 K 12 2

¼ s þ ðK 10 þ K 12 þ K 21 Þ s þ K 21 K 10 xc ¼

ðs þ K 21 Þ x0 s2 þ ðK 10 þ K 12 þ K 21 Þ s þ K 21 K 10

4.1.9.2.2 ByMatrix and Determinant Method    x0 K 21 x0 K 21 0 ðs þ K 21 Þ  ¼ 0 ðsΔþ K 21 Þ xc ¼  ðs þ K 12 þ K 10 Þ K 21 K 12

ðs þ K 21 Þ  where Δ ¼

ðs þ K 12 þ K 10 Þ

K 21

K 12

ðs þ K 21 Þ



∴Δ ¼ ðs þ K 21 Þðs þ K 12 þ K 10 Þ  ðK 21 ÞðK 12 Þ 2

¼ s þ K 21 s þ K 12 s þ K 21 K 12 þ K 10 s þ K 21 K 10  K 21 K 12

96

4

Pharmacokinetic Models and Drug Distribution

2



x0 0

¼ s þ ðK 10 þ K 12 þ K 21 Þ s þ K 21 K 10  K 21 ¼ ðs þ K 21 Þ x0  0 ¼ ðs þ K 21 Þ x0 ðs þ K 21 Þ xc ¼

ðs þ K 21 Þ x0 s2 þ ðK 10 þ K 12 þ K 21 Þ s þ K 21 K 10

Now let us consider K10 + K12 + K21 ¼ α + β and K21K10 ¼ αβ; then, 2

2

2

s þ ðK 10 þ K 12 þ K 21 Þ s þ K 21 K 10 ¼ s þ ðα þ βÞs þ αβ ¼ s þ sα þ sβ þ αβ ¼ ðs þ αÞs þ ðs þ αÞβ ¼ ðs þ αÞðs þ βÞ By solving the quadratic equations, K10 + K12 + K21 ¼ α + β and K21K10 ¼ αβ, in the form of a quadratic equation α2  (K10 + K12 + K21)α + K21K10 ¼ 0 or β2  (K10 + K12 + K21)β + K21K10 ¼ 0 using the formula, x ¼

b 

where /¼ 12 ðk10 þ k12 þ k21 Þ þ

! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2  4ac 2 , of the equation ax þ bx þ c ¼ 0 2a

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðk10 þ k 12 þ k 21 Þ2  4ðk10 k 21 Þ

1 and β ¼ ðk10 þ k12 þ k21 Þ  2 ∴ xc ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðk10 þ k 12 þ k 21 Þ2  4ðk10 k 21 Þ

ðs þ K 21 Þ x0 As þ B in Laplace transform ; it is in the form of ðs þ aÞðs þ bÞ ðs þ αÞðs þ βÞ

Now taking anti-Laplace, we have xc ¼

ðK 21 x0  x0 αÞ eαt  ðK 21 x0  x0 βÞ eβt βα

or xc ¼

x0 ðK 21  αÞ eαt x0 ðK 21  βÞ eβt þ βα αβ

or xc ¼

x0 ðα  K 21 Þ eαt x0 ðK 21  βÞ eβt þ αβ αβ

If the apparent volume of distribution in the central compartment is vc, then the plasma concentration cp is

4.1 Various Pharmacokinetic Models and Drug Distribution

cp ¼

97

x0 ðα  K 21 Þ eαt x0 ðK 21  βÞ eβt þ vc ð α  β Þ vc ð α  β Þ

It is a bi-exponential series of equations in the form of cp ¼ Aeαt + Beβt where A ¼

x0 ðα  K 21 Þ x ðK  β Þ and B ¼ 0 21 vc ð α  β Þ vc ð α  β Þ

The average plasma drug concentration can be represented with the following equations, where cp is the average plasma drug concentration, F is the fraction of bioavailable dose, CL is clearance, and τ is the dosing interval: cp ¼ cp ¼ cp ¼

F:Dose CL:τ

F:Dose ð,CL ¼ ke vd Þ k e vd :τ

 F:Dose
when ke ¼ β, vβ is the drug distribution in the elimination compartment βvβ :τ

Mean residence time of a drug MRT ¼ AUMC AUC (AUMC is the area under the firstmoment curve; AUC is the bioavailability of the drug) 4.1.9.2.3 Drug Level in the Peripheral (Tissue) Compartment The rate of change of the amount of drug at the central compartment at time t is dxc ¼ K 21 xp  K 12 xc  K 10 xc ; dt Now, the rate of change of the amount of drug at the peripheral compartment at time t is dxp ¼ K 12 xc  K 21 xp dt where xc is the amount of drug present at the central compartment (compartment “1”) at time t and xp is the amount of drug present at the peripheral compartment (here designated as compartment “2”) at time t. x0 is the dose administered to the central compartment, and drug elimination occurs only from the central compartment. By taking Laplace transform of the rate equation of the central compartment, we get s:xc  x0 ¼ K 21 xp  K 12 xc  K 10 xc or ðs þ K 12 þ K 10 Þ xc  K 21 xp ¼ x0

ð4:13Þ

98

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Pharmacokinetic Models and Drug Distribution

By taking Laplace transform of the rate equation of the peripheral compartment, we get s:xp  0 ¼ K 12 xc  K 21 xp or ðs þ K 21 Þ xp  K 12 xc ¼ 0 or  K 12 xc þ ðs þ K 21 Þ xp ¼ 0

ð4:14Þ

By matrix and determinant method,  we get  ðs þ K 12 þ K 10 Þ x0 ðs þ K 12 þ K 10 Þ xp ¼ 

K 12 ðs þ K 12 þ K 10 Þ K 12  Now,

0  ¼ K 21 ðs þ K 21 Þ

ðs þ K 12 þ K 10 Þ

x0

K 12

0 

where Δ ¼

x0

K 12



0

Δ

 ¼ 0  ðx0 Þ ðK 12 Þ ¼ x0 K 12

ðs þ K 12 þ K 10 Þ K 21 K 12 ðs þ K 21 Þ



∴Δ ¼ ðs þ K 21 Þðs þ K 12 þ K 10 Þ  ðK 21 ÞðK 12 Þ 2

¼ s þ K 21 s þ K 12 s þ K 21 K 12 þ K 10 s þ K 21 K 10  K 21 K 12 2

¼ s þ ðK 10 þ K 12 þ K 21 Þ s þ K 21 K 10 ðby eliminating  K 21 K 12 and rearrangingÞ xp ¼

x0 K 12 s þ ðK 10 þ K 12 þ K 21 Þ s þ K 21 K 10 2

If K 10 þ K 12 þ K 21 ¼ α þ β and K 21 K 10 ¼ αβ x0 K 12 A then xp ¼ ðsþα ÞðsþβÞ ; it is in the form of ðsþaÞðsþbÞ in Laplace transform. Now taking anti-Laplace, we have

xp ¼ αt

βt

x0 K 12




eαt þ eβt αβ



12 e 12 e or xp ¼ x0 Kαβ þ x0 Kαβ ; it is an bi-exponential equation. After the completion of drug distribution, a parallel decline of drug concentrations occurs in the plasma (blood) compartment and the peripheral tissue compartment (Gibaldi et al. 1969) (Figs. 4.16 and 4.17). The bi-exponential equation is useful for correlating the pharmacological effect and the tissue drug concentration, although the relationship approximates the outcomes. The tissue level does not reflect accurate drug accumulation at the site of drug action.

4.1 Various Pharmacokinetic Models and Drug Distribution

99

Fig. 4.16 The figure shows the changes of logarithmic drug amount against time in the blood compartment (central compartment) and the tissue compartment (peripheral compartment) following a two-compartment model when the drug is administered by intravenous route in a patient. Data show that the drug availability is more in the central compartment compared to the tissue compartment

Drug elimination plays a significant role in various parameters of the two-compartment open model (Jusko and Gibaldi 1972). With the progress of time, drug absorption becomes insignificant, and the drug elimination becomes predominant. With the progressing of time eαt ⤑ 0; but eβt has a finite value, 

 x0 K 12 βt e αβ   x0 K 12 βt or xp ¼ e αβ

then xp 

Taking natural logarithms on both sides, we get  log e xp ¼ log e

x0 K 12 αβ

  βt

In common logarithm (to the base 10), the equation becomes  log xp ¼ log

x0 K 12 αβ

 

βt 2:303

100

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Pharmacokinetic Models and Drug Distribution

Fig. 4.17 The figure shows the changes of logarithmic drug amount against time in the blood compartment (central compartment) and the tissue compartment (peripheral compartment) following a two-compartment model when the drug is administered by intravenous route in a patient. Data show that the drug availability is more in the peripheral (tissue) compartment compared to the central compartment

If we plot logxp against time t, on a semi-logarithmic paper, the equation provides a straight-line, the slope of which gives us the value of β,  and the intercept on the yx0 K 12 axis of the extrapolated line will give the value of log αβ (Fig. 4.18).

4.1.9.3 Three-Compartment Open Model The three-compartment open model is a pharmacokinetic model in which our body is virtually compartmentalized into three compartments: the central compartment (blood compartment) and the two peripheral compartments (tissue compartments). The administered drug can permeate reversibly from the central compartment to the other tissue compartments. They can eliminate from anyone/two/all three of the compartments. Various models are given below (Fig. 4.19). Here, let us consider model I, where x0 is the dose administered by the intravenous route to the central compartment, and two peripheral compartments are numbered as 2 and 3. xc is the amount of drug present in the central compartment (compartment “1”) at time t, xp is the amount of drug present in the peripheral compartment (here designated as compartment “2”) at time t, and xq is the amount of drug present in the peripheral compartment (here designated as compartment “3”) at time t. Further, the drug is eliminated from the central compartment. The rate of change of the amount of drug in the central compartment at time t is

4.1 Various Pharmacokinetic Models and Drug Distribution

101

Fig. 4.18 Plot of logarithmic tissue drug amount against the time of a two-compartment model. The slope of the line gives us the value of β, and the intercept of the extrapolated line gives us log

x0 K 12 αβ

, where α is the drug absorption rate, β is the drug elimination rate, x0 is the dose, and K12

is the rate of drug transport from the central compartment to the peripheral compartment

dxc ¼ K 21 xp þ K 31 xq  K 13 xc  K 12 xc  K 10 xc ; dt By taking Laplace transform, we get s:xc  x0 ¼ K 21 xp þ K 31 xq  K 13 xc  K 12 xc  K 10 xc or ðs þ K 13 þ K 12 þ K 10 Þ xc  K 21 xp  K 31 xq ¼ x0

ð4:15Þ

Now, the rate of change of the amount of drug in the second (first peripheral) compartment at time t is dxp ¼ K 12 xc  K 21 xp dt By taking Laplace transform, we get s:xp  0 ¼ K 12 xc  K 21 xp or ðs þ K 21 Þ xp  K 12 xc ¼ 0

102

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Pharmacokinetic Models and Drug Distribution

Fig. 4.19 Three-compartment open models. Model I: Where x0 is the amount of drug administered by the intravenous route to the central compartment and xc is the amount of drug present at the central compartment (compartment “1”) at time t and xp and xq are the amounts of drug present at the two peripheral compartments (here designated as compartment “2” and “3,” respectively) at time t. Further, the drug is eliminated from the central compartment. Model II: Where x0 is the

4.1 Various Pharmacokinetic Models and Drug Distribution

103

Fig. 4.19 (continued) amount of drug administered by the intravenous route to the central compartment and xc is the amount of drug present at the central compartment (compartment “1”) at time t and xp and xq are the amounts of drug present at the two peripheral compartments (here

104

4

Pharmacokinetic Models and Drug Distribution

or  K 12 xc þ ðs þ K 21 Þ xp ¼ 0

ð4:16Þ

Now, the rate of change of the amount of drug at the third compartment (second peripheral compartment) at time t is dxq ¼ K 13 xc  K 31 xq dt By taking Laplace transform, we get s:xq  0 ¼ K 13 xc  K 31 xq or ðs þ K 31 Þ xq  K 13 xc ¼ 0 or  K 13 xc þ ðs þ K 31 Þ xq ¼ 0

ð4:17Þ

To solve the values of xc , xp , and xq , the matrix-determinant method (using Cramer’s rule) is used below for the equations: ðs þ K 13 þ K 12 þ K 10 Þ xc  K 21 xp  K 31 xq ¼ x0

ð4:15Þ

K 12 xc þ ðs þ K 21 Þ xp ¼ 0

ð4:16Þ

K 13 xc þ ðs þ K 31 Þ xq ¼ 0

ð4:17Þ

The matrix of the coefficients of xc , xp , and xq is

ä Fig. 4.19 (continued) designated as compartment “2” and “3,” respectively) at time t. Further, the drug is eliminated from the central compartment and the peripheral compartment number 3. Model III: Where x0 is the amount of drug administered by the intravenous route to the central compartment and xc is the amount of drug present at the central compartment (compartment “1”) at time t and xp and xq are the amounts of drug present at the two peripheral compartments (here designated as compartment “2” and “3,” respectively) at time t. Further, the drug is eliminated from the central compartment and peripheral compartments, numbers 2 and 3. Model IV: Where x0 is the amount of drug administered by the intravenous route to the central compartment and xc is the amount of drug present at the central compartment (compartment “1”) at time t and xp and xq are the amounts of drug present at the two peripheral compartments (here designated as compartment “2” and “3,” respectively) at time t. Further, the drug is eliminated from the central compartment and the peripheral compartment number 2. Model V: Where x0 is the amount of drug administered by the intravenous route to the central compartment and xc is the amount of drug present at the central compartment (compartment “1”) at time t and xp and xq are the amounts of drug present at the two peripheral compartments (here designated as compartment “2” and “3,” respectively) at time t. Further, the drug is eliminated from peripheral compartment number 2. Model VI: Where x0 is the amount of drug administered by the intravenous route to the central compartment and xc is the amount of drug present at the central compartment (compartment “1”) at time t and xp and xq are the amounts of drug present at the two peripheral compartments (here designated as compartment “2” and “3,” respectively) at time t. Further, the drug is eliminated from peripheral compartment number 3. In all the models, K12, K21, K13, K31, K10, K20, and K30 are the respective rate constants, and each arrowhead shows the direction of movement of the drug

4.1 Various Pharmacokinetic Models and Drug Distribution

2

ðs þ K 13 þ K 12 þ K 10 Þ 6 K 12 4 K 13

105

K 21 ðs þ K 21 Þ

K 31 0

0

ðs þ K 31 Þ

3 7 5

Therefore, the determinant of the matrix, Δ or D, is   ðs þ K 21 Þ 0 Δ ¼ ðs þ K 13 þ K 12 þ K 10 Þ  ðK 21 Þ 0 ðs þ K 31 Þ     K 12 ðs þ K 21 Þ 0 K 12  þ ðK 31 Þ K 13 ðs þ K 31 Þ K 13 0 Or Δ ¼ ðs þ K 13 þ K 12 þ K 10 Þðs þ K 21 Þðs þ K 31 Þ  K 21 K 12 ðs þ K 31 Þ  K 31 K 13 ðs þ K 21 Þ By taking (s + K31) as common from the first two terms, we get Or, Δ ¼ ðs þ K 31 Þfðs þ K 13 þ K 12 þ K 10 Þðs þ K 21 Þ  K 21 K 12 g  K 31 K 13 ðs þ K 21 Þ
∴Δ ¼ ðs þ K 31 Þ s2 þ K 13 s þ K 12 s þ K 10 s þ K 21 s þ K 21 K 13 þ K 21 K 12 þ K 21 K 10 K 21 K 12 Þ  K 31 K 13 ðs þ K 21 Þ
∴Δ ¼ s3 þ K 13 s2 þ K 12 s2 þ K 10 s2 þ K 21 s2 þ K 21 K 13 s þ K 21 K 10 s þ K 31 s2 þK 31 K 13 s þ K 31 K 12 s þ K 31 K 10 s þ K 31 K 21 s þ K 31 K 21 K 13 þ K 31 K 21 K 10 Þ K 31 K 13 ðs þ K 21 Þ By eliminating K21K12, and multiply the portion with (s + K31), we get, ∴Δ ¼ s3 þ K 13 s2 þ K 12 s2 þ K 10 s2 þ K 21 s2 þ K 21 K 13 s þ K 21 K 10 s þ K 31 s2 þ K 31 K 13 s þ K 31 K 12 s þ K 31 K 10 s þ K 31 K 21 s þ K 31 K 21 K 13 þ K 31 K 21 K 10  K 31 K 13 s  K 31 K 13 K 21 ∴Δ ¼ s3 þ K 13 s2 þ K 12 s2 þ K 10 s2 þ K 21 s2 þ K 31 s2 þ K 21 K 13 s þ K 21 K 10 s þ K 31 K 12 s þ K 31 K 10 s þ K 31 K 21 s þ K 31 K 21 K 10 ∴Δ ¼ s3 þ ðK 13 þ K 12 þ K 10 þ K 21 þ K 31 Þs2 þ ðK 21 K 13 þ K 21 K 10 þ K 31 K 12 þ K 31 K 10 þ K 31 K 21 Þs þ K 31 K 21 K 10 Now considering K13 + K12 + K10 + K21 + K31 to be “α + β + γ”; K21K13 + K21K10 + K31K12 + K31K10 + K31K21 to be “α β + β γ + γ α” and K31K21K10 to be “α β γ,” we have

106

4

Pharmacokinetic Models and Drug Distribution

∴Δ ¼ s3 þ ðα þ β þ γ Þ s2 þ ðα β þ β γ þ γ αÞ s þ α β γ ¼ s3 þ s2 α þ s2 β þ s2 γ þ α β s þ β γ s þ γ αs þ α β γ ¼ s3 þ s2 α þ s2 β þ α β s þ s2 γ þ γ αs þ β γ s þ α β γ ¼ s2 ðs þ αÞ þ sβ ðs þ αÞ þ sγðs þ αÞ þ β γ ðs þ αÞ
 ¼ ðs þ α Þ s2 þ s β þ s γ þ β γ ¼ ðs þ αÞfsðs þ βÞ þ γ ðs þ βÞg ¼ ðs þ αÞðs þ βÞðs þ γ Þ ∴Δ or D ¼ ðs þ αÞðs þ βÞðs þ γ Þ Now the matrix of xc is 2

x0

6 40 0

K 21

K 31

3

7 ðs þ K 21 Þ 0 5 0 ðs þ K 31 Þ

The determinant of the matrix of xc or Dx is given below:  Dx ¼ x0  

ðs þ K 21 Þ

0

ðs þ K 31 Þ  ðs þ K 21 Þ 0 0

0 0



  ðK 21 Þ



0

0

0

ðs þ K 31 Þ

Dx ¼ x0 ðs þ K 21 Þðs þ K 31 Þ þ K 21 :0  K 31 :0 ∴Dx = x0 ðs þ K 21 Þðs þ K 31 Þ Now the matrix of xp is 2 6 4

ðs þ K 13 þ K 12 þ K 10 Þ x0

K 31

K 12

0

0

K 13

0

ðs þ K 31 Þ

The determinant of the matrix of xp or Dy is given below:

3 7 5

þ ðK 31 Þ

4.1 Various Pharmacokinetic Models and Drug Distribution



0 Dy ¼ ðs þ K 13 þ K 12 þ K 10 Þ 0   K 12 0 þ ðK 31 Þ K 13 0

  0 K 12  ð x0 Þ K 13 ðs þ K 31 Þ

107

0 ðs þ K 31 Þ



Dy ¼ ðs þ K 13 þ K 12 þ K 10 Þ:0  x0 ðK 12 Þðs þ K 31 Þ  K 31 :0 ∴Dy ¼ x0 K 12 ðs þ K 31 Þ Now the matrix of xq is 2

3 ðs þ K 13 þ K 12 þ K 10 Þ K 21 x0 6 7 K 12 ðs þ K 21 Þ 0 5 4 K 13 0 0 The determinant of the matrix of xq or Dz is given below: 

  K 12 ðs þ K 21 Þ 0 Dz ¼ ðs þ K 13 þ K 12 þ K 10 Þ  ðK 21 Þ 0 0 K 13   K 12 ðs þ K 21 Þ  K 13 0

 0 þ ð x0 Þ 0

Dz ¼ ðs þ K 13 þ K 12 þ K 10 Þ:0 þ K 21 :0 þ x0 f0  ðK 13 Þðs þ K 21 Þg or Dy ¼ 0 þ 0 þ x0 ð K 13 Þðs þ K 21 Þ ∴Dz ¼ x0 K 13 ðs þ K 21 Þ D

Dz Dx y As per Cramer’s rule, xc ¼ Δ or D ; xp ¼ Δ or D; and xq ¼ Δ or D x0 K 12 ðsþK 31 Þ x0 K 13 ðsþK 21 Þ 0 ðsþK 21 ÞðsþK 31 Þ Therefore, xc ¼ xðsþαÞðsþβÞðsþγÞ ; xp ¼ ðsþαÞðsþβÞðsþγÞ and xq ¼ ðsþαÞðsþβÞðsþγÞ As þBsþc So xc in the Laplace transform form of ðsþaÞðsþbÞðsþcÞ AsþB xp in the Laplace transform form of ðsþaÞðsþbÞðsþcÞ AsþB xq in the Laplace transform form of ðsþaÞðsþbÞðsþcÞ Now taking anti-Laplace value, we have 2

xc ¼

x0 ðK 21  αÞðK 31  αÞeαt x0 ðK 21  βÞðK 31  βÞeβt þ ðβ  αÞðγ  αÞ ðγ  β Þðα  β Þ þ

x0 ðK 21  γ ÞðK 31  γ Þeγt ðα  γ Þðβ  γ Þ

Without the anti-Laplace table, calculation of anti-Laplace (also called inverse ðsþK 21 ÞðsþK 31 Þ Laplace) value of xc = xð0sþα ÞðsþβÞðsþγÞ is given below:

108

L1

4

n

x0 ðsþK 21 ÞðsþK 31 Þ ðsþαÞðsþβÞðsþγÞ

Pharmacokinetic Models and Drug Distribution

o indicates anti-Laplace of the given function to be

calculated. x0(s + K21)(s + K31)¼ A (s + β)(s + γ) + B(s + α)(s + γ) + C(s + α)(s + β) (where A, B, and C are constants) Comparing coefficient powers of s on both sides of the equation, we understand that three (3) becomes equivalent to three equations of the system with A, B, and C as the three unknown values. But there is a shortcut to determine the three unknowns. If we set s ¼  α, we obtain x0 ðs þ K 21 Þðs þ K 31 Þ ¼ Aðs þ βÞðs þ γ Þ or x0 ðα þ K 21 Þðα þ K 31 Þ ¼ Aðα þ βÞðα þ γ Þ or x0 ðK 21  αÞðK 31  αÞ ¼ Aðβ  αÞðγ  αÞ or A ¼

x0 ðK 21  αÞðK 31  αÞ ðβ  αÞðγ  αÞ

Similarly, we can obtain B and C by setting s ¼  β and s ¼  γ, at the respective equations and B¼

x0 ðK 21  βÞðK 31  βÞ x ðK  γ ÞðK 31  γ Þ and C ¼ 0 21 ðα  βÞðγ  βÞ ðα  γ Þðβ  γ Þ

x0 ðs þ K 21 Þðs þ K 31 Þ A B C þ þ ¼ ðs þ αÞðs þ βÞðs þ γ Þ ðs þ αÞ ðs þ βÞ ðs þ γ Þ





 x0 ðs þ K 21 Þðs þ K 31 Þ A B or L1 þ L1 ¼ L1 ðs þ αÞ ðs þ βÞ ðs þ αÞðs þ βÞðs þ γ Þ

 C þ L1 ðs þ γ Þ





 1 1 1 = A:L1 þ B:L1 þ C:L1 ðs þ αÞ ðs þ β Þ ðs þ γ Þ Therefore,

= A:eαt þ B:eβt þ C:eγt Now by replacing the values of A, B, and C, we obtain L21

x0 ðs þ K 21 Þðs þ K 31 Þ ðs þ αÞðs þ βÞðs þ γÞ



4.1 Various Pharmacokinetic Models and Drug Distribution

¼

109

x0 ðK 21  αÞðK 31  αÞeαt x0 ðK 21  βÞðK 31  βÞeβt þ ðβ  αÞðγ  αÞ ðγ  β Þðα  β Þ þ

x0 ðK 21  γ ÞðK 31  γ Þeγt ðα  γ Þðβ  γ Þ

Therefore, αÞðK 31 αÞeαt βÞðK 31 βÞeβt γÞðK 31 γÞeγt xc ¼ x0 ðK 21ðβαÞðγαÞ þ x0 ðK 21ðγβÞðαβÞ þ x0 ðK 21ðαγÞðβγÞ xp ¼

x0 K 12 ðK 31  αÞeαt x0 K 12 ðK 31  βÞeβt x0 K 12 ðK 31  γ Þeγt þ þ ðβ  αÞðγ  αÞ ðγ  β Þðα  β Þ ðα  γ Þðβ  γ Þ

xq ¼

x0 K 13 ðK 21  αÞeαt x0 K 13 ðK 21  βÞeβt x0 K 13 ðK 21  γ Þeγt þ þ ðβ  αÞðγ  αÞ ðγ  β Þðα  β Þ ðα  γ Þðβ  γ Þ

Then, the drug concentrations in various compartments are given below, when the volumes of drug distribution in the central (1) and the peripheral compartments (2, 3) are v1, v2, and v3, respectively. Cc ¼

x0 ðK 21  αÞðK 31  αÞeαt x0 ðK 21  βÞðK 31  βÞeβt þ v1 ðβ  αÞðγ  αÞ v1 ð γ  β Þ ð α  β Þ þ

x0 ðK 21  γ ÞðK 31  γ Þeγt v1 ðα  γ Þðβ  γ Þ

and Cp ¼

x0 K 12 ðK 31  αÞeαt x0 K 12 ðK 31  βÞeβt x0 K 12 ðK 31  γ Þeγt þ þ v2 ðβ  αÞðγ  αÞ v2 ð γ  β Þ ð α  β Þ v2 ð α  γ Þ ð β  γ Þ

Cq ¼

x0 K 13 ðK 21  αÞeαt x0 K 13 ðK 21  βÞeβt x0 K 13 ðK 21  γ Þeγt þ þ v3 ðβ  αÞðγ  αÞ v3 ð γ  β Þ ð α  β Þ v3 ð α  γ Þ ð β  γ Þ

and

Therefore, drug concentration in any compartment of a three-compartmental model as shown here is represented by a tri-exponential equation C ¼ Aeαt + Beβt + Meγt, where A, B, and M are the coefficients of eαt, eβt, and eγt, respectively, and they are constants. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Further, /¼ ðk10 þ k12 þ k21 Þ þ ðk10 þ k12 þ k 21 Þ2  4ðk10 k21 Þ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 and β ¼ ðk10 þ k12 þ k21 Þ  ðk10 þ k 12 þ k 21 Þ2  4ðk10 k 21 Þ 2 γ ¼ k a ðabsorption rate constantÞ

110

4

Pharmacokinetic Models and Drug Distribution

References Chen H-SG, Gross JF (1979) Estimation of tissue-to-plasma partition coefficients used in physiological pharmacokinetic models. J Pharmacokinet Biopharm 7:117–125 Dedrick RL, Forrester DD (1973) Blood flow limitations in interpreting Michaelis constant for ethanol oxidation in vivo. Biochem Pharmacol 22:1133–1380 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 58:193–1997 Harrison LI, Gibaldi M (1977) Physiologically based pharmacokinetic model for digoxin distribution and elimination in the rat. J Pharm Sci 66:1138–1142 Himmelstein KJ, Lutz RL (1979) A review of the applications of physiologically based pharmacokinetic modeling. J Pharmacokinet Biopharm 7:127–145 Jusko WJ, Gibaldi M (1972) Effects of change in elimination on various parameters of the two-compartment open model. J Pharm Sci 61:1270–1273 Lane EA, Levy R (1981) Metabolite to parent drug concentration ratio as a function of parent drug extraction ratio: cases of nonportal route of administration. J Pharmacokinet Biopharm 9:489– 496 Lutz RJ, Galbraith WM, Dedrick RL, Shrager R, Mellett LB (1977) A model for the kinetics of distribution of actinomycin-D in the beagle dog. J Pharmacol Exp Ther 200:469–478 Minturn M, Himmelstein KJ, Schroder RL, Gibaldi M, Shen DD (1980) Tissue distribution kinetics of tetraethylammonium ion in the rat. J Pharmacokinet Biopharm 8:373–409 Yamaoka K, Nakagawa T, Uno T (1978) Statistical moments in pharmacokinetics. J Pharmacokinet Biopharm 6:547–558

5

Drug Metabolism

5.1

Drug Metabolism

Drug metabolism suggests the biotransformation of drugs in the body for their easy elimination. Most drugs are unknown chemicals to the body. Our body eliminates them, considering them a threat to the system. Cells contain various enzymes. The primary purpose of drug metabolism is the chemical manipulation of drugs by some of those enzymes for their fast elimination from the body (Gillette 1971). Most metabolites possess polar groups that make them water soluble. The water-soluble metabolites are then eliminated by the kidney relatively easier than the metabolites formed with non-polar groups. A non-polar group such as the acetyl group makes the metabolites comparatively insoluble. Such types of metabolites may precipitate in urine and can cause crystalluria. Metabolites can sometimes be active drugs also. The drug codeine is metabolized to the active drug morphine. Amitriptyline, upon its metabolism, forms active nortriptyline. Primidone forms active phenobarbital upon its metabolism.

5.1.1

Hepatic Drug Metabolism

The drug is usually metabolized in the cells present in various organs and tissues. However, the principal organ that plays a vital role in drug metabolism is the liver. In the case of drug metabolism of orally administered drugs, drug absorption takes place through the hepatoportal route. As stated earlier, drug molecules are mostly foreign bodies to our systems. Hence, the body tries to eliminate them by forming metabolites that could be removed easily from our system. In the liver, the drug is metabolized by various enzymes (Gillette 1971), commonly called hepatic drugmetabolizing enzymes. Since those enzymes were discovered in relation to drug metabolism in the liver, they were named so. In reality, those enzymes involve the general metabolic function of the liver. After the oral administration of a drug, its # The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B. Mukherjee, Pharmacokinetics: Basics to Applications, https://doi.org/10.1007/978-981-16-8950-5_5

111

112

5

Drug Metabolism

absorption is first routed through the hepatoportal route into the liver (Perrier and Gibaldi 1974), and its metabolism follows in the organ (Colburn and Gibaldi 1978). That is why it is called hepatic first-pass metabolism, in which many a drug undergoes even 40–50% drug metabolism. The presence of some drugs can increase or decrease drug metabolism. Phenytoin and warfarin enhance the metabolism of theophylline, causing faster elimination of the drug. In contrast, warfarin reduces tolbutamide metabolism by competitive enzyme inhibition, and thus, it decreases tolbutamide elimination. The hepatic drug metabolism process involves some distinctive enzymatic chemical reactions. They include oxidation, reduction, hydrolysis, and conjugation. The reaction patterns are further categorized into (a) phase I type and (b) phase II type hepatic drug-metabolizing reactions. The enzymes that follow phase I type reactions are cytochrome P-450 monooxygenases, aryl hydrocarbon hydroxylase, esterase, monoamine oxidase, alcohol dehydrogenase, aldehyde dehydrogenase, etc., and uridine diphosphate (UDP)-glucuronyl transferase, glutathione S-transferase, N-acetyl transferase, methyltransferase, sulfotransferase, etc. belong to phase II category. Drugs that undergo phase I and phase II metabolism following the different chemical reactions are given below.

5.1.1.1 Phase I Oxidation: Oxidation refers to oxygen gain or hydrogen loss or loss of electron (s) from a molecule/atom/charged atom (ion). In the liver, oxidative drug metabolism mainly occurs in the endoplasmic reticulum of the hepatocytes. Phenytoin, phenobarbital (also known as phenobarbitone), phenacetin, 5-hydroxytryptamine, and chlorpromazine are some examples of drugs that are metabolized by oxidation. Alkyl group (–CH2 – CH3) of phenobarbital is oxidized to the corresponding alcohol (–CHOH – CH3) during its metabolism. Phenytoin is oxidized to the related phenolic group (PhOH) in its metabolic process. Chlorpromazine forms sulfoxide upon oxidation at its sulfur (S) moiety. The drug 5-hydroxytryptamine undergoes deamination and forms aldehyde (– CHO) during metabolism.
CH2 CH2 NH2 !
CH2 CHO: Oxidative dealkylation occurs in phenacetin during its hepatic metabolism. Reduction: Reduction implies hydrogen gain or oxygen loss or gain of electron (s) by a molecule/atom/charged atom (ion). The reduction of the ketone group inactivates warfarin to a hydroxyl group by reductase enzyme. Hydrolysis: When water molecules participate in a chemical reaction to split the bond(s) of a chemical to produce new compounds/moieties, it is called hydrolysis. Aspirin is hydrolyzed to salicylic acid and acetic acid, and it is a prevalent example of drug metabolism by hydrolysis. Another example is procainamide. This drug is hydrolyzed to corresponding amine and acetic acid during its metabolism.

5.1 Drug Metabolism

113

5.1.1.2 Phase II Reaction Conjugation is the most common type of phase II drug metabolism reaction. Conjugation is principally incorporating a part or whole of a systemic indigenous compound to drug molecule to form metabolite(s) that can be eliminated by the system quickly. Glucuronidation, acetylation, glycine conjugation, and salt (such as sulfate) formation are conjugation processes to form metabolites for drug elimination. Hydroxylated morphine undergoes conjugation with glucuronide during its metabolism. Sulfonamides are metabolized by acetylation. Glycine (NH2CH2COOH) addition to nicotinic acid and sulfate formation for both paracetamol and morphine are the conjugation processes to metabolize those drugs. The hepatic first-pass effect of drugs is primarily responsible for less bioavailability of drugs given by oral route (Perrier and Gibaldi 1972; McLean et al. 1978; Gibaldi et al. 1971) than by many other routes.

5.1.2

Pharmacokinetic Compartmental Models and Equations for Assessing Hepatic “First-Pass” Effect of a Drug

In a two-compartment open model, various models (Chiou 1975; Pang and Gillette 1978; Colburn and Gibaldi 1978; Shepard and Reuning 1997) used for computing the hepatic “first-pass” effect of a drug and its distribution are available in the literature. Here, some models have been presented in the figures (Fig. 5.1). However, model I and model II have been described and correlated below.

5.1.2.1 Model I x0 is the amount of drug administered by the intravenous route to the central compartment and xc is the amount of drug present at the central compartment (compartment “1”) at time t and xp is the amount of drug present at the peripheral (liver) compartment (here designated as compartment “2”) at time t. Further, the drug is eliminated from the peripheral (liver) compartment. 5.1.2.2 Model II x0 is the amount of drug administered by the oral route. xc is considered as the amount of drug present at the central compartment (compartment “1”) at time t and xp is the amount of drug present at the peripheral (liver) compartment (here designated as compartment “2”) at time t. Further, the drug is eliminated from the peripheral compartment. 5.1.2.3 Model III x0 is the amount of drug administered by the intravenous route to the central compartment. xc is the amount of drug present at the central compartment (compartment “1”) at time t and xp is the amount of drug present at the peripheral compartment (here designated as compartment “2”) at time t. Further, the drug is eliminated from both the central and peripheral compartments.

114

5

Drug Metabolism

Fig. 5.1 Two-compartment model to investigate and compare the hepatic first-pass effect of drug administered by different routes. The self-explanatory model shows drug absorption and drug elimination from various compartments as explained under 5.1.2.1–5.1.2.4

5.1 Drug Metabolism

115

Fig. 5.1 (continued)

5.1.2.4 Model IV x0 is the amount of drug administered by the oral route. xc is considered as the amount of drug present at the central compartment (compartment “1”) at time t and xp is the amount of drug present at the peripheral (liver) compartment (here designated as compartment “2”) at time t. Further, the drug is eliminated from both the central and peripheral compartments. In all the models, K12, K21, K10, and K20 are the respective rate constants, and each arrowhead shows the direction of movement of the drug.

5.1.3

Hepatic First-Pass Effect Invariably Reduces Total Bioavailability of a Drug More When Administered Orally than By Its Intravenous Route of Administration

In the model I, the rate of change of the drug amount in the central compartment at time t is dxc ¼ K 21 xp
K 12 xc ; dt where xc is the amount of drug present in the central compartment (compartment “1”) at time t and xp is the amount of drug present in the peripheral (liver) compartment (here designated as compartment “2”) at time t. x0 is the intravenously administered bolus dose to the central compartment. The drug here is eliminated from the peripheral compartment only and K20 is the elimination rate from the peripheral (liver) compartment (Fig. 5.1). By taking Laplace transform of the above equation, we get s:xc
x0 ¼ K 21 xp
K 12 xc

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Drug Metabolism

or, ðs þ K 12 Þ xc
K 21 xp ¼ x0

ð5:1Þ

Now, the rate of change of the amount of drug in the peripheral compartment at time t is dxp ¼ K 12 xc
K 21 xp
K 20 xp dt By taking Laplace transform, we get s:xp
0 ¼ K 12 xc
K 21 xp
K 20 xp (as no drug was available initially in the peripheral compartment) or
K 12 xc þ ðs þ K 20 þ K 21 Þ xp ¼ 0

ð5:2Þ

To solve for the value of xc from the Eqs. (5.1) and (5.2) using matrix and determinant, we have
xc ¼



K 21

x0






x0


K 21



0 ðs þ K 20 þ K 21 Þ 0 ðs þ K 20 þ K 21 Þ  ¼ Δ ðs þ K 12 Þ
K 21
K 12 ðs þ K 20 þ K 21 Þ
 ðs þ K 12 Þ
K 21 where Δ ¼
K 12 ðs þ K 20 þ K 21 Þ ∴Δ ¼ ðs þ K 12 Þðs þ K 20 þ K 21 Þ
ð
K 21 Þð
K 12 Þ

¼ s2 þ K 12 s þ K 20 s þ K 12 K 20 þ K 21 s þ K 21 K 12
K 21 K 12


x0 0

¼ s2 þ ðK 12 þ K 20 þ K 21 Þ s þ K 12 K 20 
K 21 ¼ x0 ðs þ K 20 þ K 21 Þ
0 ¼ x0 ðs þ K 20 þ K 21 Þ ðs þ K 20 þ K 21 Þ xc ¼

s2

x0 ðs þ K 20 þ K 21 Þ þ ðK 12 þ K 20 þ K 21 Þ s þ K 12 K 20

Now let us consider K12 + K20 + K21 ¼ α + β and K12K20 ¼ αβ; then, s2 þ ðK 12 þ K 20 þ K 21 Þ s þ K 12 K 20 ¼ s2 þ ðα þ βÞs þ αβ ¼ s2 þ sα þ sβ þ αβ ¼ ðs þ αÞs þ ðs þ αÞβ ¼ ðs þ αÞðs þ βÞ sþK 20 þK 21 Þ 0 K 20 þx0 K 21 ∴ xc ¼ x0 ððsþα ¼ x0 sþx ; it is in the form of ðsþαÞðsþβÞ ÞðsþβÞ transform. Now taking anti-Laplace, we have

AsþB ðsþaÞðsþbÞ

in Laplace

5.1 Drug Metabolism

xc ¼

117

ðK 21 x0 þ K 20 x0
x0 αÞ e
αt
ðK 21 x0 þ K 20 x0
x0 βÞ e
βt β
α

or xc ¼

x0 ðK 21 þ K 20
αÞ e
αt x0 ðK 21 þ K 20
βÞ e
βt þ β
α α
β

If the apparent volume of distribution in the central compartment is vc, then the plasma concentration cp is cp ¼

x0 ðK 21 þ K 20
αÞ e
αt x0 ðK 21 þ K 20
βÞ e
βt þ vc ðβ
αÞ vc ð α
β Þ

or cp ¼ Now

R 1 0

Cp dt

total 

x0 ðα
K 21
K 20 Þ e
αt x0 ðK 21 þ K 20
βÞ e
βt þ vc ð α
β Þ vc ðα
βÞ

bioavailability (by intravenous route) ¼ (AUC)

1 0

or

intravenous

Z

Z 1 x0 ðα
K 21
K 20 Þ e
αt x0 ðK 21 þ K 20
βÞ e
βt dt þ dt vc ð α
β Þ vc ð α
β Þ 0 0
Z 1  Z 1 x0
αt
βt ðα
K 21
K 20 Þ e dt þ ðK 21 þ K 20
βÞ e dt ¼ vc ð α
β Þ 0 0
 Z 1 Z 1 x0 ¼ ðα
K 21
K 20 Þ e
αt dt þ ðK 21 þ K 20
βÞ e
βt dt vc ð α
β Þ 0 0 ¼

1

[since x0, vc, α, K21, and K20 are constants] h
a:1 i 
ax  1 R1
a:0 (where a ¼ constant) ¼ e
a
e
a ¼ But we know that 0 e
ax dt ¼ e
a 0 h i h i   1 R 1
αt R1 0 e0 0 e0 1 dt ¼ α1 and 0 e
βt dt ¼ β1 ]
a

a ¼
a

a ¼ 0

a ¼ a; Thus, 0 e
 ðα
K 21
K 20 Þ ðK 21 þ K 20
βÞ x0 þ ¼ α β vc ð α
β Þ
 ðα
K 21
K 20 Þβ þ ðK 21 þ K 20
βÞα x0 ¼ αβ v c ðα
β Þ
 αβ
βK 21
βK 20 þ αK 21 þ αK 20
αβ x0 ¼ αβ vc ð α
β Þ
 αK 21 þ αK 20
βK 21
βK 20 x0 ¼ ðby eliminating and arrangingÞ αβ vc ð α
β Þ

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5

Drug Metabolism


 αðK 21 þ K 20 Þ
βðK 21 þ K 20 Þ x0 ¼ αβ vc ð α
β Þ
 ðα
βÞðK 21 þ K 20 Þ x0 ¼ αβ vc ð α
β Þ ¼

x0 ðK 21 þ K 20 Þ x0 ðK 21 þ K 20 Þ ðby replacing the values of αβÞ ¼ vc αβ vc K 12 K 20

The plasma level of a drug following the oral route of administering the same drug is calculated as given below. According to Fig. 5.1, model II, xc is the amount of drug present in the central compartment (compartment “1”) at time t and xp is the amount of drug present in the peripheral (liver) compartment (here designated as compartment “2”) at time t. x0 is the administered dose to any extravascular route (here, say, oral route). The drug here is eliminated from the peripheral compartment only and K20 is the elimination rate from the peripheral (liver) compartment. The rate of change of the amount of drug in the central compartment at time t is dxc ¼ K 21 xp
K 12 xc dt By taking Laplace transform, we get s:xc
0 ¼ K 21 xp
K 12 xc or ðs þ K 12 Þ xc
K 21 xp ¼ 0

ð5:3Þ

Now, the rate of change of the amount of drug in the peripheral compartment at time t is dxp ¼ K 12 xc
K 21 xp
K 20 xp dt By taking Laplace transform, we get s:xp
x0 ¼ K 12 xc
K 21 xp
K 20 xp or
K 12 xc þ ðs þ K 20 þ K 21 Þ xp ¼ x0

ð5:4Þ

To solve the value of xc from Eqs. (5.3) and (5.4) using matrix and determinant, we have
xc ¼


0


K 21






0

x0 ðs þ K 20 þ K 21 Þ x  ¼ 0 ðs þ K 12 Þ
K 21
K 12 ðs þ K 20 þ K 21 Þ


K 21 ðs þ K 20 þ K 21 Þ Δ



5.1 Drug Metabolism

119




ðs þ K 12 Þ
K 21 where Δ ¼
K 12 ðs þ K 20 þ K 21 Þ



∴Δ ¼ ðs þ K 12 Þðs þ K 20 þ K 21 Þ
ð
K 21 Þð
K 12 Þ ¼ s2 þ K 12 s þ K 20 s þ K 12 K 20 þ K 21 s þ K 21 K 12
K 21 K 12


0 x0

¼ s2 þ ðK 12 þ K 20 þ K 21 Þ s þ K 12 K 20 
K 21 ¼ 0
ð
x0 K 21 Þ ¼ x0 K 21 ðs þ K 20 þ K 21 Þ xc ¼

s2

x0 K 21 þ ðK 12 þ K 20 þ K 21 Þ s þ K 12 K 20

Now let us consider K12 + K20 + K21 ¼ α + β and K12K20 ¼ αβ; then, s2 þ ðK 12 þ K 20 þ K 21 Þ s þ K 12 K 20 ¼ s2 þ ðα þ βÞs þ αβ ¼ s2 þ sα þ sβ þ αβ ¼ ðs þ αÞs þ ðs þ αÞβ ¼ ðs þ αÞðs þ βÞ x0 K 21 A ∴ xc ¼ ðsþα ÞðsþβÞ ; it is in the form of ðsþaÞðsþbÞ in Laplace transform. Now taking anti-Laplace, we have x0 e
αt x0 e
βt þ K 21α
β or xc ¼ K 21β
α If the apparent volume of distribution of the drug in the central compartment is vc, then the plasma concentration cp is

cp ¼ or cp ¼

K 21 x0 e
αt K 21 x0 e
βt þ vc ðβ
αÞ vc ð α
β Þ


K 21 x0 e
αt K 21 x0 e
βt K 21 x0 e
βt K 21 x0 e
αt þ ¼
vc ð α
β Þ vc ðα
βÞ vc ð α
β Þ vc ð α
β Þ ¼

 K 21 x0 
βt e
e
αt vc ð α
β Þ

Now total bioavailability (by oral route) ¼ (AUC) K 21 x0 ¼ vc ð α
β Þ


Z

[since x0, vc, α, β, K21 are constants]

1

e 0


βt

Z

R 1  1 or 0 C p dt oral 0

1

dt


e 0


αt

 dt

120

5

Drug Metabolism

h
a:1 i 
ax  1 R1
a:0 (where a ¼ constant) ¼ e
a
e
a [But we know that 0 e
ax dt ¼ e
a 0 h i h i   1 R1 R1 0 e0 0 e0 1 ¼
a

a ¼
a

a ¼ 0

a ¼ a; thus, 0 e
αt dt ¼ α1 and 0 e
βt dt ¼ 1 β

.]
 K 21 x0 1 1
¼ vc ð α
β Þ β α
 K 21 x0 α
β ¼ vc ðα
βÞ αβ ¼

K 21 x0 vc αβ

K 21 x0 ðsince αβ ¼ K 12 K 20 Þ vc K 12 K 20 Z 1 K 21 x0 ∴ cp dt ¼ v K 12 K 20 c 0 oral ðOÞ

¼

F 2

R ¼ R

1

0 1

0



"

cp dt



oral

cp dt

¼

K 21 x0 vc K 12 K 20 x0 ðK 21 þK 20 Þ vc K 12 K 20

# (

F 2

is the comparative systemic drug

intravenous

availability by the two routes of drug administration) ¼

K 21 ðK 21 þ K 20 Þ

The numerical expression shows that the denominator value is more than the numerator value in the fraction. K21 and K20 are positive whole numbers. Hence, the value of the fraction is less than one. The value suggests that the total bioavailability of a drug administered by the oral route is always invariably less than the total bioavailability of the drug administered by the intravenous route. One substantial reason for this phenomenon is the hepatic first-pass effect.

5.1.4

Determination of Drug Metabolite Levels in Plasma Using Compartmental Model

According to Fig. 5.2, in a compartmental model, dose (x0) of a drug is administered to the central compartment by the intravenous route, and the drug is quickly converted to its metabolite in the blood. After time t, xB is the amount of drug present at the central compartment and MB is the amount of metabolite also present at the central compartment. Further, Mu is the amount of the metabolite excreted through urine at time t, and Kf and Km are the first-order rate constants of formation

5.1 Drug Metabolism

121

Fig. 5.2 The x0 amount of drug has been administered in the central compartment. xB is the amount of drug and MB is the amount of drug metabolite present in the central compartment (compartment “1”) at time t. Mu is the amount of drug metabolite eliminated through urine at time t. Kf, and Km are the respective rate constants of drug to metabolite conversion and the elimination of drug metabolite through the urine. Each arrowhead shows the direction of the movement of the drug metabolite

of metabolite in the blood and excretion of the metabolite through urine, respectively (Fig. 5.2). KE is the overall elimination rate constant of the drug. Then, the rate of change of metabolite in the blood at time t following first-order kinetics is given by dM B ¼ Rate of formation of metabolite at time t ðinputÞ dt
rate of elimination of metabolite at time t ðoutputÞ or,

dM B ¼ Kf XB
KmMB dt

By taking Laplace transform, we get s:M B
0 ¼ K f xB
K m M B (since the amount of metabolite was not available at the beginning) or ðs þ K m Þ M B ¼ K f xB or M B ¼

K f xB ðs þ K m Þ

ð5:5Þ

However, since two unknown variables exist at this stage, it is not possible to take anti-Laplace. Thus, we need first to eliminate one. As the rate of change of drug at the central (blood) compartment following firstorder kinetics is dX B dt ¼
K E xB , by taking Laplace transform, we get

122

5

Drug Metabolism

s:xB
x0 ¼
K E xB , or, ðs þ K E Þ xB ¼ x0 or xB ¼ Therefore, M B ¼

x0 ðs þ K E Þ K f x0 ðs þ K m Þðs þ K E Þ

ð5:6Þ

(by replacing the value of xB in the Eq. (5.5)) f x0 Now by taking anti-Laplace of the Eq. (5.6), we get M B ¼ ðKKE
K  mÞ
K m t
K E t ½e
e , where MB is the amount of metabolite in the plasma at time t. If vm is the volume of distribution of drug metabolite, then the concentration of drug metabolite in plasma (cm) is depicted below: cm ¼


K t  K f x0 e m
e
K E t vm ð K E
K m Þ

ð5:7Þ

In this bi-exponential equation, there is the possibility of two situations: case one Km  KE and the other case KE  Km. The drug metabolites such as sulfate, glycine conjugates are predominantly polar, and they eliminate faster than the parent drug (Km  KE). On the other hand, less-polar acetylated or many oxidized forms of metabolites usually are excreted slowly compared to the parent drugs (KE  Km). In case Km  KE, over time, a situation arises when the entire drug metabolite is excreted out, but some unchanged amount of parent drug remains in the blood/body. In such a case, cm 

K f x0 e
K E t vm ð K m
K E Þ

or log cm ¼ log

K f x0 K t
E vm ðK m
K E Þ 2:303

If we plot log cm against time t, on a semi-logarithmic paper, the equation provides a straight-line  K (Fig. 5.3) slope of which gives the value of elimination E . The intercept on the plasma concentration axis (y-axis) of rate constant KE
2:303 0 the extrapolated line gives us the value of log vm ðKK fmx
K . EÞ In case KE  Km, over time, a situation arises when there is no unchanged drug remains in the blood/body, but there is a huge amount of drug metabolite that remains in the blood/body. In such a case, cm 

K f x0 e
K m t vm ð K E
K m Þ

5.1 Drug Metabolism

123

Fig. 5.3 Plot of logarithmic plasma drug metabolite concentration (logcm) versus time on a semiKE and the intercept on the y-axis of the logarithmic paper. The slope of the line gives us
2:303 K f x0 extrapolated line gives us log vm ðK m
K E Þ where Kf and Km are the respective rate constants of drug to metabolite conversion and the elimination of drug metabolite through the urine. KE is the drug elimination rate, and vm is the volume of distribution of the drug metabolite upon the administration of x0 amount of drug

or log cm ¼ log

K f x0 K t
m vm ðK E
K m Þ 2:303

If we plot log cm against time t, on a semi-logarithmic paper, the equation provides a straight-line (Fig. 5.4), and the slope of the terminal phase of the line  Km . The intercept on the gives the value of the elimination rate constant Km
2:303 plasma concentration axis (y-axis) of the extrapolated line gives us the value of 0 log vm ðKK fEx
K . mÞ

124

5

Drug Metabolism

Fig. 5.4 Plot of logarithmic plasma drug metabolite concentration (logcm) versus time on a semiKm and the intercept on the y-axis of the logarithmic paper. The slope of the line gives us
2:303 K f x0 extrapolated line gives us log vm ðK E
K m Þ where Kf is the rate of drug to metabolite conversion and Km is the elimination rate of drug metabolite through the urine. KE is the drug elimination rate, and vm is the volume of distribution of the drug metabolite upon the administration of x0 amount of drug

References Chiou WL (1975) Quantitation of hepatic and pulmonary first-pass effect and its implications in pharmacokinetic study. I. Pharmacokinetics of chloroform in man. J Pharmacokinet Biopharm 3:193–201 Colburn WA, Gibaldi M (1978) Pharmacokinetic model of presystemic metabolism. Drug Metab Dispos 6:193–196 Gibaldi M, Boyes RN, Feldman S (1971) Influence of first pass effect on availability of drug on oral administration. J Pharm Sci 60:1338–1340 Gillette R (1971) Factors affecting drug metabolism. Ann N Y Acad Sci 179:43–66 McLean AJ, McNamara PJ, du Souich P, Gibaldi M, Lalka D (1978) Food, splanchnic blood flow, and bioavailability of drugs subject to first-pass metabolism. Clin Pharmacol Ther 24:5–10 Pang KS, Gillette JR (1978) A theoretical examination of the effects of gut wall metabolism, hepatic elimination, and enterohepatic recycling on estimates of bioavailability and of hepatic blood flow. J Pharmacokinet Biopharm 6:355–367 Perrier D, Gibaldi M (1972) Influence of first-pass effect on the systemic availability of propoxyphene. J Clin Pharmacol New Drugs 12:449–452 Perrier D, Gibaldi M (1974) Clearance and biologic half-life as indices of intrinsic hepatic metabolism. J Pharmacol Exp Ther 191:17–24 Shepard TA, Reuning RH (1997) An equation for the systemic availability of drugs undergoing simultaneous enterohepatic cycling, first-pass metabolism, and intestinal elimination. Pharm Res 4:195–199

6

Drug Elimination and Nonlinear Kinetics

6.1

Drug Elimination

Drug elimination (also called excretion) is a necessary process of several procedures to remove a drug or its metabolite(s) from the body. Glomerular filtration, tubular secretion, and passive reabsorption are the three predominant processes involved in renal excretion (Koeppen and Stanton 2000). After passing through the kidneys, drugs or their metabolites reach the urinary bladder and are eliminated through urine. Drugs are primarily biotransformed in the body, and soluble metabolites are formed. The urinary excretion process then removes them. Drugs and their metabolites are also excreted by the liver in the bile and ultimately eliminated in feces. Drugs can be eliminated by different other routes also.

6.1.1

Nonlinear Kinetics and Capacity-Limited Process

When the drug concentration is not proportional to dose (amount of drug administered) and/or the drug elimination is not proportional to the drug concentration in the blood, the drug exhibits nonlinear kinetics (Fig. 6.1). On the other hand, when a change of drug concentration in the blood is proportional to dose (amount of drug administered) and/or elimination of the drug is proportional to the drug concentration in the blood, the drug exhibits linear kinetics (Fig. 6.2).

6.1.2

Michaelis–Menten Equation

Many elimination processes such as active secretion, active reabsorption, and drug metabolism involve enzymes to act. In a biological system, the number or amount of enzymes available in a process is always limited. Thus, at a high concentration of substrate (here drug), the enzyme activity gets saturated. It means that the reaction # The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B. Mukherjee, Pharmacokinetics: Basics to Applications, https://doi.org/10.1007/978-981-16-8950-5_6

125

126

6

Drug Elimination and Nonlinear Kinetics

Fig. 6.1 A typical nonlinear kinetic plot is shown. The rate of reaction (v) is plotted against the concentration of substrate (c) (here drug) in an enzyme reaction. The first part of the curve shows a concentration-dependent first-order kinetic pattern that is followed by a zero-order kinetic pattern shown by a straight line parallel to the x-axis (concentration axis) when the enzymes are saturated with the substrates. The vmax is the maximum rate of reaction

Fig. 6.2 The figure shows the rate of drug elimination is proportionally increased with the increase in drug concentration in the blood. The graph represents a linear plot

6.1 Drug Elimination

127

rate is concentration dependent (following first-order kinetics) up to a level of particular drug concentration. Once the enzyme activity is saturated, the reaction rate becomes constant while more drug is available in the blood. Thus, the process is called capacity-limited process (Fig. 6.1). Michaelis–Menten equation is a perfect example of a nonlinear kinetic equation that runs with a capacity-limited process and is represented by C v ¼ Kvmax (Michaelis and Menten 1913) where v is the velocity/rate of enzyme– m þC substrate reaction, c is the substrate concentration (here in pharmacokinetics, drug concentration), vmax is the maximum reaction velocity, and Km is Michaelis–Menten constant. The Lineweaver–Burk plot (Lineweaver and Burk 1934) is a double reciprocal plot by the linear representation of the graph for the determination Km and Vmax, arranging the Michaelis–Menten equation (in Eq. (6.1) mentioned below): 1 K þc Km c Km 1 ¼ m ¼ þ ¼ þ v vmax c vmax c vmax c vmax c vmax 1 Km 1 ∴ ¼ þ v vmax c vmax Or

c v

ð6:1Þ

¼ vKmaxm cc þ vmaxc (by multiplying both sides of the equation by c) c K c ∴ ¼ m þ v vmax vmax Or

ð6:2Þ

c K þc ¼ m v vmax

Or vmax ¼

ðK m þ cÞv c

Or vmax ¼

K m v vc þ c c

Or vmax ¼

Kmv þv c

Or v ¼ 

K mv þ vmax c

ð6:3Þ

Now for the Eq. (6.1), by plotting 1v against 1c, we can get a straight line, the slope 1 of which is vKmaxm and the intercept on the y-axis is vmax (Fig. 6.3). From those values, Km and vmax can be determined. From the Eq. (6.2), by plotting cv against c, we can get a straight line, the slope of 1 which is vmax and the intercept on the y-axis is vKmaxm (Fig. 6.4). From those values, Km and vmax can be determined.

128

6

Drug Elimination and Nonlinear Kinetics

Fig. 6.3 Nonlinear kinetic “Michaelis–Menten” equation when rearranged and plotted as 1v against 1 Km c gives a straight line. The slope of the line shows the value of vmax. It is called the Lineweaver–Burk double reciprocal plot. Here, v is the rate of reaction, c is the concentration of substrate (here drug) in an enzyme reaction. The vmax is the maximum rate of reaction

Fig. 6.4 Nonlinear kinetic “Michaelis–Menten” equation when rearranged and plotted as cv against 1 c gives a straight line. The slope of the line shows the value of vmax . Here, v is the rate of reaction, and c is the concentration of substrate (here drug) in an enzyme reaction. The vmax is the maximum rate of reaction. The Km is the Michaelis–Menten constant

6.1 Drug Elimination

129

Fig. 6.5 Nonlinear kinetic “Michaelis–Menten” equation when rearranged and plotted as v against v c gives a straight line. The slope of the line shows the value of km. Here, v is the reaction rate, and c is the concentration of substrate (here drug) in an enzyme reaction. The Km is the Michaelis– Menten constant

From the Eq. (6.3), by plotting v against vc, similarly, we can get a straight line, the slope of which is Km and the intercept on the y-axis is vmax (Fig. 6.5). Thus, Km and vmax can be determined.

6.1.3

Capacity-Limited Process/Nonlinear Kinetics

The rate of drug elimination that follows a capacity-limited process can be determined by the famous Michaelis–Menten equation, arranged in the following form: vmax c dc  dc dt ¼ K m þc, where  dt is the rate of elimination, c is the plasma drug concentration at time t, vmax is the maximum reaction velocity, and Km is Michaelis–Menten constant. Now,  or  or 

dc v c ¼ max dt K m þ c

ðK m þ cÞ dc ¼ vmax dt c

K m dc c dc  ¼ vmax dt: c c

By integrating both sides of the above equation, we have

130

6

Z K m

dc  c

Drug Elimination and Nonlinear Kinetics

Z

Z dc ¼ vmax

dt

or  K m ln c  c ¼ vmax t þ i

ð6:4Þ

(where “ln” is the natural logarithm and “i” is the integration constant) If c ¼ c0 (say) at t ¼ 0, then the Eq. (6.4) becomes K m ln c0  c0 ¼ vmax :0 þ i or i ¼ K m ln c0  c0 Putting the value of i in the Eq. (6.4), it becomes K m ln c  c ¼ vmax t  K m ln c0  c0 or K m ln c0 þ c0  K m ln c  c ¼ vmax t

ð6:5Þ

or K m ln c0  K m ln c þ c0  c ¼ vmax t or t ¼

K m ln

c 

þ ð c0  cÞ ðThis is a time  concentration relationship:Þ vmax

0

c

or  K m ln c0  c0 þ K m ln c þ c ¼ vmax t [obtained by multiplying both sides of the Eq. (6.5) by minus one (1)] K m ln c  K m ln c0  c0 þ c ¼ vmax t c  c0 þ c ¼ vmax t c0   c or K m ln ¼ ðc0  cÞ  vmax t c0   ðc  cÞ vmax t c  or ln ¼ 0 Km c0 Km or K m ln

or log

ð6:6Þ

  ð c  cÞ c ¼ 0 2:303K m c0 v t  max ðconverting natural to common logarithm to the base 10Þ 2:303K m or log c  log c0 ¼

ðc0  cÞ v t  max 2:303K m 2:303K m

6.1 Drug Elimination

131

Fig. 6.6 Nonlinear kinetic “Michaelis–Menten” equation when rearranged and plotted as logc vmax against t gives a straight line. The slope of the line shows the value of  2:303 k m . Here, c is the substrate concentration (here drug) in an enzyme reaction. The vmax is the maximum rate of reaction. The Km is the Michaelis–Menten constant

or log c ¼ log c0 þ

ðc0  cÞ v t  max 2:303K m 2:303K m

The above equation is a time–concentration relationship. By plotting the change of plasma drug concentration during an elimination process on a logarithmic scale against time on a linear scale on a semi-logarithmic paper, we can determine the rate of elimination for those drugs eliminated following Michaelis–Menten kinetics. The plot provides us witha straightline (Fig. 6.6). The slope is determined from the line vmax and the slope value  2:303K gives the rate of elimination. m

6.1.4

Dose–Plasma Drug Concentration Relationship with Michaelis–Menten Constant Km for Drugs That Undergo Elimination Following Michaelis–Menten Nonlinear Kinetics

c K m ln ð c0 Þþðc0 2 cÞ Further, t = (This is a time–concentration relationship.) vmax This relationship can be represented as a dose of a drug. dose ðDÞ amount of drug, We know concentration ðcÞ ¼ volume of distribution, vd ¼ vd

Therefore, t ¼

K m ln

D 0  D

þ ðD0  DÞ vmax

ð6:7Þ

132

6

Drug Elimination and Nonlinear Kinetics

(where the initial amount of drug, dose at the time of administration, is D0 and D is the amount of drug [dose] present in the blood at time t). If two dosing rates (say, for intravenous infusion) are D1 and D2 and the respective plasma drug concentrations are c1 and c2, then D1 ¼

vmax c1 vmax c2 and D2 ¼ ð K m þ c1 Þ ð K m þ c2 Þ

Then, vmax c1 ¼ D1 ðK m þ c1 Þ and

ð6:8Þ

vmax c2 ¼ D2 ðK m þ c2 Þ

ð6:9Þ

Now multiply Eq. (6.8) by c2 and Eq. (6.9) by c1, we have vmax c1 c2 ¼ D1 c2 ðK m þ c1 Þ

ð6:10Þ

vmax c1 c2 ¼ D2 c1 ðK m þ c2 Þ

ð6:11Þ

Since the Eqs. (6.10) and (6.11) are equal to vmaxc1c2, we can write D1 c2 ðK m þ c1 Þ ¼ D2 c1 ðK m þ c2 Þ Or, D1 c2 K m þ D1 c2 c1 ¼ D2 c1 K m þ D2 c1 c2 Or, D1 c2 K m  D2 c1 K m ¼ D2 c1 c2  D1 c2 c1 Or, K m ðD1 c2  D2 c1 Þ ¼ ðD2  D1 Þc2 c1 Then, K m ¼

ðD2  D1 Þc2 c1 ðD2  D1 Þ ¼ ðD1 c2  D2 c1 Þ Dc 1cc2  Dc 2cc1 2 1

D  D1  Therefore, K m ¼  2 D1 D2  c1 c2

6.1.5

2 1

ð6:12Þ

Drug Elimination by More Than One Capacity-Limited Process

A drug elimination process may involve different enzymes that could obey different kinetics during their reaction processes. In the case of eliminating a drug, when more than one enzyme saturation kinetic (capacity-limited process) is involved, they follow multiple capacity-limited procedures (Fagerholm 2007). In an example, a drug may follow first-order kinetics and Michaelis–Menten kinetics simultaneously during its elimination process. At time t, if the concentration of the drug in the blood is c, then the rate of change of drug concentration in plasma is

6.1 Drug Elimination

133

 0

K ¼

E

0

dc v c ¼ K c þ max dt K mþc E 0

cðK m þ cÞ þ vmax c ¼

Km þ c 0

c K ¼

E 0

c K ¼ 0

Let us consider K

E

K

E

E

0

Kmc þ K

c2 þ vmax c E Km þ c !

0

c þ vmax E Km þ c

Km þ K

0

K m þ vmax þ K

!

c E

Km þ c

K m þ vmax ¼ a



dc ¼ dt

 c aþK



0

E Km þ c

c

To solve the above equation, we should take the reciprocals of both sides of the equation, and the following step is critical to bring the equation in an integrating form. ∴

or 

dt K þc ¼  m 0  dc c aþK c E

dt Km ¼  dc c aþK

0

E

1 þ c aþK



0

E

c

K dc dc or  dt ¼  m 0  þ  0  c aþK c aþK c E E K dc dc or dt ¼   m 0    0  c aþK c aþK c E E

134

6

Z

Z 

dt ¼ 

or

K m dc

Z 



0

c aþK

E

c

dc



aþK

 ðintegrating both sides of the above equationÞ

0

E

c 

Therefore, t ¼ 

Drug Elimination and Nonlinear Kinetics

aþK

Km ln a



0

E

c

c



 ln a þ K

1

0

K

0

E

 c þi

ð6:13Þ

E

(where i is the integration constant)



If we consider when t ¼ 0, c ¼ c0; then, i ¼ Kam ln



0

aþK

E

c0

þ

c0

1 0 K





0

ln a þ K

E

c0 E By replacing the value of “i” in the Eq. (6.13), we have the following equation: 

t¼

þ

Km ln a

0

or t ¼

c

1



0

K 

0

E

 ln a þ K

 K c þ m ln a E 0

 aþK



0

E

c0

c0

E

c0

E

Km ln a



E

c

 ln a þ K

1 K

aþK



0

1

0

K

E

 aþK

E

c0 

c0  ln a þ K





0



0

E

c

Km ln a

aþK c



0

E

c þ

1

0

K

E

 ln a þ K



0

E

c0

6.1 Drug Elimination

135



or t ¼

   0 a þ K c0 c a þ K c0 E E 1   þ ln  ln 0 0 0  aþK c a þ K c c0 K E E E ! 0

Km a

0

0

or t ¼

Km

0

K

K ln

K m þ vmax

E

K þ

1 0

K

ln

E

K

K m þ vmax þ K

K m þ vmax þ K

0

E

E

!

0

0

E

E

c0 c !

0

0

K

K m þ vmax þ K

E

E

c

c0

c0 ! ðby replacing the value of aÞ

0

K m þ vmax þ K

E

E

c

The above equation is a time–concentration relationship of a drug elimination process that follows more than one capacity-limited process. This relationship can be represented in dose (amount of drug) as Concentration ðcÞ ¼ 0 K

Therefore, t=

0 K

Km K m þvmax

ln

E

amount of drug, dose ðDÞ volume of distribution ðvd Þ ! 0

K m þvmax þK

0 K

K m þvmax þK

D0

E ! 0 D

D

K

þ D0

1 0 K

ln

E

K m þvmax þK

0 K

!

0

0

K þv

D0

E ! 0

þK

D

m max E E E E E (where the initial amount of drug, i.e., the initial dose, is D0 and D is the amount of drug (dose) present in the blood at time t).

6.1.6

E

Sigma-Minus Method to Determine Elimination Rate Constant

The sigma-minus method is another method of estimation of elimination rate of drugs excreted unchanged through urine. A graphical plot of the logarithmic amount of drug remaining to be excreted unchanged against time provides the drug elimination rate. The rate of elimination mostly fluctuates, and thus, the data are invariably scattered. It, therefore, often causes difficulties in the correct estimation of the elimination rate. Sigma-minus method has an advantage over it (Ahmed 2015).

136

6

Drug Elimination and Nonlinear Kinetics

The drug remaining to be excreted unchanged correlates to the cumulative amount (sum or sigma) of the drug already excreted through urine. The limit of the method is the difference (drug remaining to be excreted unchanged) (minus) till it becomes zero (0), that is, till the entire dose in the blood is eliminated completely. Thus, this method is called the sigma-minus method. The rate of unchanged drug (xu) eliminated through urine is given by dX u dt ¼ K e xB (where Ke is the elimination rate constant of unchanged drug excreted through urine and amount of drug present in the blood at time t is xB) or dxdtu ¼ K e x0 eK E t (, xB ¼ x0 eK E t ; please see Eq. (1.2)). By integrating equation with respect to t from a time interval 0–t, we have R t dx R t the K E t u dt ¼ K x e dt e 0 0 0 dt Z or

t

Z dxu ¼ K e x0

0

Z

t

Again

t

eK E t dt

0

dxu ¼ ½xu t  ½xu 0 ¼ xu  x

0

or xu  x

0 u

 K t 0 e E t ¼ K e x0 ; K E 0 u

0 at time “t” ¼ 0, x ⟶0 u  K t  K t e E eK E :0 e E 1  þ ¼ K e x0  KE K E K E KE  K E t 1 e ¼ K e x0  KE KE   1  eK E t K x  ¼ K e x0 ¼ e 0 1  eK E t KE KE

Therefore, xu ¼ K e x0

∴xu ¼

K e x0 ð1  eK E t Þ KE

Again, at t ¼ 1, if the cumulative amount of drug excreted is x

x

1 u

¼

 K x K e x0  K x 1  eK E :1 ¼ e 0 ð1  0Þ ¼ e 0 KE KE KE

ð6:14Þ 1 , then u

6.1 Drug Elimination

137

∴x

1 K e x0 ¼ KE u x

Then,

ð6:15Þ

1

u x0 ¼ Ke KE

Again, xu ¼ KKe Ex0 ð1  eK E t Þ [Please see the Eq. (6.14).] 1 1 K e x0 ¼ K E ; please see the Eq. (6.15)] ð1  eK E t Þ [, x or xu ¼ x u u or xu ¼ x

1 u

x

1 u

:eK E t

By changing sides and arranging the equation, we get 1 1 2 xu = x :e 2 KE t u u   1  xu indicates the amount of drug remaining to be In the equation, x u excreted. Now taking logarithm on both the sides, we get x

  1 1 K t  E  xu ¼ log x log x 2:303 u u In the sigma-minus method, during the elimination process of a drug that is excreted unchanged, the “amount of drug remaining to be excreted” is plotted on a logarithmic scale against “time” on a linear scale on a semi-logarithmic paper. The KE Þ, the plot provides us a straight line (Fig. 6.7), and from the slope of the line ( 2:303 rate of elimination (KE) can be determined. From the feathered line (residual line) Ka drawn using the method of residual, Ka can be obtained from the slope “ 2:303 ” of the feathered line.

6.1.7

Bi-Exponential Absorption–Elimination Equation for Orally Administered Drugs Excreted Unchanged Through Urine

Drugs usually undergo hepatic first-pass effect when administered by the oral route and are also subjected to other physiological degradation processes (Gibaldi et al. 1971; Gillette 1971). Hence, they are absorbed less than their administered doses. Thus, the bioavailability is also less. For oral drug administration, the rate of drug elimination can be deduced and determined as described below.

138

6

Drug Elimination and Nonlinear Kinetics

  Fig. 6.7 In the sigma-minus method, when log x1 u  xu is plotted against time on a semi-log  1  paper, keeping “ log xu  xu ” on a log scale and time on a linear scale, it gives a straight line. The kE where kE provides the value of elimination rate constant. When the slope of the line is  2:303 Ka feathered line is drawn, the slope of it gives the value of  2:303 where Ka is the absorption rate constant. The total unchanged drug to be eliminated through urine is x1 u and the unchanged drug eliminated through urine at any time t is xu

Fig. 6.8 The figure shows drug (xu) is excreted unchanged through urine with drug elimination rate (ke) following orally administered drug in a one-compartment model. Here, xA amount of drug is absorbed to the central compartment from the orally administered x0 amount of drug at an absorption rate ka at time t and xB is the amount of drug present at the central compartment at time t

x0 is the amount of drug administered orally. The total amount of drug bioavailable is F. x0, where “F” is the fraction of dose bioavailable. At time t, if the amount of drug absorbed is xA, the amount of drug eliminated unchanged is xu, and the amount of drug present in the blood is xB (Fig. 6.8), the rate of change of drug in the blood can be represented as

6.1 Drug Elimination

139

dxB dxA dxu ¼  dt dt dt

dxB ¼ k a xA  k e xB dt

ð6:16Þ

(considering the absorption and elimination processes follow first-order kinetics, and ka and ke are the respective absorption and elimination rate constants) According to the first-order kinetics, the rate of change of the amount of absorbed drug (dxdtA) at any time (t) is directly proportional to the amount of drug absorbed (xA) in the blood at that time. Thus,  dxdtA / xA (negative sign indicates with the ongoing absorption process, the initial drug amount is reducing) or 

dxA ¼ ka :xA dt

where ka is the proportionality constant (here the rate of drug absorption) or 

dxA ¼ ka dt xA

Now, if we integrate the equation from an initial time point of drug absorption, i.e., say, “0,” to any time “t,” when at t ¼ 0, the total amount of drug available for absorption is F. x0 and atRa time “t,” the R t amount of drug in the blood is xA. t Then, we can write  0 dxxAA dt ¼ ka 0 dt or  ½ log e xA 

xA ¼ ka ½t t0 F:x0

or  ½ log e xA  log e F:x0  ¼ k a ½t  0 or, ½ log e xA  log e F:x0  ¼ k a t or log e or

xA ¼ ka t F:x0

xA ¼ eka t F:x0

or xA ¼ F:x0 eka t

ð6:17Þ

Again, dxdtB ¼ ka xA  ke xB [mentioned above as Eq. (6.16)] Or dxdtB ¼ ka F:x0 eka t  ke xB [substituting the value of xA from Eq. (6.17)]

140

6

Or

Drug Elimination and Nonlinear Kinetics

dxB þ ke xB ¼ ka F:x0 eka t dt

Multiplying both sides of the above equation by eke t , we have dxB ke t e þ k e xB eke t ¼ ka F:x0 eka t eke t dt Or

dxB ke t e þ ke xB eke t ¼ ka F:x0 eðke tka tÞ dt

Or dxB eke t þ k e xB eke t dt ¼ ka F:x0 eðka ke Þt dt By integrating from time interval “0” to “t,” we have Z

t

 dxB eke t þ ke xB eke t dt ¼

Z

0

t

k a F:x0 eðka ke Þt dt

0

Z

t

or,

dxB e

ke t

Z



t

þ ke xB e dt ¼ ka F:x0 ke t

0

eðka ke Þt dt

0

Let us solve the left-hand side of the equation first. Z That is,

t

 dxB eke t þ k e xB eke t dt ¼

0 t

 dxB eke t þ

Z

0

¼

t

 dxB eke t þ

0

Z 

Z

t

ke xB eke t dt

Z

t

ke xB eke t dt



0



0

  Z t Z t  ke t  Z t t  kt  d e e dxB  dxB dt þ ke xB e dt dx 0 0 0 0

Z ek e t

ðusing integration by partsÞ " ) # "  k t # Z t( t t ee t ke t ke t ¼ e :½xB   ke e :½xB  dt þ k e xB ke 0 0 0 0  ¼

Z

t

e :xB  ke t

0



ke e :xB dt ke t





 kt ee eke ,0  þ k e xB ðsince at t ¼ 0, ke ke

there was no drug absorption in the blood, and at a time“ t, ” the amount of drug in the blood is xB Þ    kt Z t  k t e e 1 ¼ eke t :xB  ke :xB e e dt þ ke xB ke 0  kt  kt e e 1 e e 1 ke t ¼ e :xB  ke :xB þ k e xB ke ke

6.1 Drug Elimination

141

¼ eke t :xB The right-hand side of the above-equation is Z k a F:x0

t

eðka ke Þt dt

0

 ¼ ka F:x0

eðka ke Þt  ðk a  k e Þ

t 0



eðka ke Þt eðka ke Þ:0 ¼ ka F:x0  ðk a  k e Þ ðka  k e Þ  ðk k Þt e a e 1 ¼ k a F:x0 þ ðk a  ke Þ ðka  ke Þ  1 eðka ke Þt ¼ k a F:x0  ðk a  k e Þ ðka  ke Þ  1  eðka ke Þt ¼ ka F:x0 ðk a  k e Þ Therefore, we can establish the following relationship equating both the sides of the equation. Or xB eke t ¼

ka F:x0 ½ 1  eðka ke Þt  ðka  ke Þ

Dividing both sides of the equation by eke t , we have  ka F:x0 1 eðka ke Þt xB ¼  ð k a  k e Þ ek e t ek e t or xB ¼

k a F:x0 ke t  eke t ðeka t :eke t Þ  ½e ðk a  k e Þ

or xB ¼

ka F:x0 ½ eke t  eke tþke t eka t  ðk a  k e Þ

or xB ¼

ka F:x0 ½ eke t  e0 :eka t  ðka  ke Þ

Or xB ¼

k a F:x0 ðeke t  eka t Þ ðka  ke Þ

ð6:18Þ

142

6.1.8

6

Drug Elimination and Nonlinear Kinetics

Excretion Rate Method

The excretion rate method is an alternative method for determining the elimination rate of drugs excreted unchanged through urine. However, this method needs the data of the average rate of the amount of unchanged drugs excreted in the urine. It is only valid when urine samples are collected in short time intervals, and the unchanged drug eliminates following the first-order rate kinetics. When, the rate of urinary excretion of unchanged drug is dxdtu (where the amount of unchanged drug, xu), the amount of drug in the blood at time t is xB, and the firstorder elimination rate constant of the unchanged drug is ke, then their relationship is dxu ¼ k e xB dt k e t 0 Again, we know, xB ¼ ðkkaaF:x  eka t Þ [Please see the Eq. (6.18) above.] ke Þ ðe By replacing the value of xB in the expression dxdtu ¼ k e xB , we have

dxu k e k a F:x0 ke t  eka t Þ ¼ ðe dt ðka  ke Þ Since the rate of urinary excretion of unchanged drug ðdxdtu Þ cannot be determined practically, we use the average rate of the amount of unchanged drug excreted in u urine (Δx Δt ). But as mentioned above, the urine samples should be collected more frequently in the short intervals for computing the average rate of urinary excretion of unchanged drug. Otherwise, it may result in inaccurate findings. However, the expression becomes Δxu ke ka F:x0 ke t  eka t Þ ¼ ðe Δt ðk a  k e Þ For the most orally administered drugs, the rate of absorption (ka) is faster than the rate of elimination (ke). With the duration, while drug absorption completes, the drug elimination process carries out. For drugs, Ka  ke, the eka t approaches to zero (0) with the increasing time, while drug elimination continues. In such cases, the equation is represented as Δxu ke k a F:x0 ke t ¼ e Δt ðk a  k e Þ Taking logarithm on both the sides of the above equation, we have log

  Δxu k k F:x kt ¼ log e a 0  e Δt ðk a  ke Þ 2:303

By plotting the logarithmic values of the average rate of excretion of the unchanged drug against time on a semi-log paper, the slope of the obtained straight

References

143

u logðΔx Δt Þ , of drug (xu) excreted unchanged through ke urine versus time on a semi-logarithmic paper. The slope of the line gives us  2:303 and the ka k e Fx0 intercept on the y-axis of the extrapolated line gives us log ðK K Þ where ka is the absorption rate a e

Fig. 6.9 Plot of logarithmic values of rate,

of drug, ke is the elimination rate of drug excreted unchanged through urine, and F is the dose fraction drug absorbed upon the administration of x0 amount of drug

line (Fig. 6.9) provides us the elimination rate (ke). The intercept of the extrapolated F:x0 line on the y-axis obtained by extrapolating the line gives us the value of log kðke kaak : eÞ

References Ahmed TA (2015) Pharmacokinetics of drugs following IV bolus, IV infusion, and oral administration. InTech Open, Rijeka. https://www.intechopen.com/chapters/49459 Fagerholm UJ (2007) Prediction of human pharmacokinetics—renal metabolic and excretion clearance. J Pharm Pharmacol 59:1463–1471 Gibaldi M, Boyes RN, Feldman S (1971) Influence of first pass effect on availability of drug on oral administration. J Pharm Sci 60:1338–1340 Gillette R (1971) Factors affecting drug metabolism. Ann N Y Acad Sci 179:43–66 Koeppen BM, Stanton BA (2000) Renal physiology, 3rd edn. Elsevier, Amsterdam Lineweaver H, Burk D (1934) The determination of enzyme dissociation constants. J Am Chem Soc 56:658–666 Michaelis L, Menten ML (1913) Die Kinetik der Invertinwirkung. Biochem Z 49:333–369

7

Pharmacokinetic Drug–Drug Interactions

7.1

Pharmacokinetic Drug–Drug Interactions

Pharmacokinetic drug–drug interactions can be explained as the interaction between two or more different drug molecules after their coadministration or during their simultaneous systemic availability in a subject, at the absorption, distribution, metabolism, and elimination processes. Pharmacokinetic drug–drug interactions alter drug disposition, vary drug plasma level, and modulate the pharmacodynamic effect of drugs. The pharmacodynamic effect could be agonistic, synergistic, additive, or antagonistic (no or reduced drug effect is produced) and can produce adverse drug reactions. Sometimes, it could be toxic to the subjects/patients also. Level 1 drug–drug interaction is a severe and lethal interaction that requires a clinician to discontinue the medication order. It suggests for “hard stop” or “no further continuation” alert. Drug–drug interactions are often observed as the outcome of polytherapy, mostly in elderly patients. The medication error or erroneous prescription order is a primary cause of it. Other than the drug–drug interactions, there are also drug–food interactions, drug–beverage (alcohol) interactions, drug–disease interactions, etc. However, we will limit our discussion in pharmacokinetic drug–drug interactions.

7.1.1

Importance of Drug–Drug Interactions

Drug–drug interactions have broad implications in pharmacokinetics. Precisely, drug–drug interactions in patients can cause (a) Alteration of plasma drug level (b) Drug-related side effects or toxic effects (c) Deteriorating the existing medical condition # The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B. Mukherjee, Pharmacokinetics: Basics to Applications, https://doi.org/10.1007/978-981-16-8950-5_7

145

146

7

Pharmacokinetic Drug–Drug Interactions

7.1.1.1 Alteration of Plasma Drug Level Pharmacokinetic drug–drug interactions often modulate the plasma drug levels of many drugs. They either cause faster drug absorption or retard drug absorption. Increased drug absorption often produces drug-related side effects or toxicity. However, poor drug absorption also reduces drug efficacy. Drugs such as erythromycin, roxithromycin, and clarithromycin are shown to increase the blood level of digoxin, theophylline, or doxorubicin by p-glycoprotein-mediated enhanced drug absorption (Table 7.1). It may induce drug-related side effects. Omeprazole increases gastric pH and makes the stomach environment poorly acidic, reducing ketoconazole, itraconazole, or fluconazole absorption and the bioavailability of those drugs. 7.1.1.2 Drug-Related Side Effects or Toxic Effects Drug-related side effects or toxic effects are other expected outcomes of drug–drug interactions in patients. Coadministration of two or more drugs can produce cumulative effects that result in side effects or toxicity in patients. Coadministration of hydrocodone and diphenhydramine increases drowsiness due to the additive effect of the drugs. 7.1.1.3 Deteriorative Existing Medical Conditions Another common finding related to drug–drug interaction in vivo is the deterioration of the existing medical condition of patients. Poly-therapy in patients often results in interactions between the drugs that worsen the disease condition of the sufferers. Fenofibric acid (reduces triglycerides and cholesterol), when coadministered with warfarin (blood thinner, prevents the formation of arteriolar blood clots), enhances the warfarin (blood thinner) effect that causes more hemorrhage and frequent internal bleeding in patients. The beta-blockers such as atenolol, bisoprolol, metoprolol, and propranolol are antihypertensive drugs. Omeprazole, lansoprazole, and pantoprazole, when coadministered with the beta-blockers, reduce the bioavailability of the beta-blockers. It results in persistent enhanced blood pressure in patients, leading to fatal health consequences.

7.1.2

Categories

Pharmacokinetic drug–drug interactions are categorized into two predominant classes. (a) Pharmacokinetic Type When the drug–drug interactions involve in drug disposition (that is, absorption, distribution, metabolism, and excretion), modulate drug plasma levels, and modify drug actions or cause toxicity or ineffective treatment, the type of drug–drug interaction is pharmacokinetic type. (b) Pharmacodynamic Type When the drug–drug interactions potentiate or inhibit drug action and cause toxicity or treatment failure, the class of interaction is called the

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147

Table 7.1 Drug–drug interactions and their systemic effects Drugs Aluminum/magnesium hydroxide antacids, omeprazole, lansoprazole, pantoprazole, ranitidine, famotidine, atropine, benztropine mesylate, clidinium, cyclopentolate, darifenacin, dicyclomine, fesoterodine Heavy metal antacids such as aluminum/ magnesium hydroxide antacids, iron formulations (iron salts), calcium formulations (calcium salts), etc. Bile salts

Cholestyramine, colestipol

Cathartics such as magnesium citrate, magnesium sulfate, sodium sulfate, and magnesium hydroxide, sorbitol Metoclopramide, cisapride, bethanechol, loperamide, neostigmine Metoclopramide, cisapride, bethanechol, loperamide, neostigmine Erythromycin, roxithromycin, clarithromycin, omeprazole, lansoprazole, pantoprazole, esomeprazole, dronedarone, amiodarone, diltiazem, verapamil, cyclosporine

Affected coadministered drugs Atenolol, bisoprolol, metoprolol, propranolol, glibenclamide, tolbutamide, ketoconazole, itraconazole, fluconazole, posaconazole, ampicillin, atazanavir, clopidogrel, diazepam, methotrexate, vitamin B12, paroxetine, raltegravir Tetracycline, ciprofloxacin, penicillamine

Cause Reduced bioavailability by inducing gastric pH

Remarks Ogawa and Echizen (2010); Krishna et al. (2009)

Predominant reduction of drug absorption due to complex formations.

Ogawa and Echizen (2011); Bokor-Bratić and Brkanić (2000)

Cholestyramine, colestipol

Reduction of drug absorption due to binding with bile salts

Warfarin, acetylsalicylic acid, sulfadiazine, sulfamethizole, phenytoin, furosemide Neostigmine, metoclopramide, cisapride, bethanechol, loperamide

Reduction of drug absorption

Phillips et al. (1976); Scaldaferri et al. (2011) Phillips et al. (1976); Scaldaferri et al. (2011) Lee et al. (2000)

Digoxin, theophylline

Acetylsalicylic acid, acetaminophen, tetracycline, levodopa, Digoxin, theophylline, doxorubicin

Poor gastrointestinal drug absorption

Poor gastrointestinal drug absorption due to fast gastric emptying. Increased drug plasma level that may induce drug-related toxicity Increased blood level by p-glycoproteinmediated enhanced drug absorption that may induce drugrelated side effects

Lee et al. (2000); Johnson et al. (1984) Lee et al. (2000); Johnson et al. (1984) Rushing et al. (1994)

(continued)

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Table 7.1 (continued) Drugs Diclofenac

Affected coadministered drugs Warfarin

Propafenone

Metoprolol

Propafenone

Venlafaxine

Naloxone

Morphine

Omeprazole

Amitriptyline

Saquinavir, ritonavir, indinavir

Sildenafil

Tacrolimus

Posaconazole

Phenobarbital, lamotrigine, ibuprofen, mefenamic acid, etoricoxib, indomethacin, naproxen, diclofenac, celecoxib, mefenamic acid, amoxicillin

Methotrexate

Cause Increased blood level of free-warfarin for less warfarin–albumin binding due to competitively faster diclofenac–albumin binding. This may result in systemic hemorrhage Reduction of clearance of metoprolol that may induce drug-related toxicity Increase venlafaxine plasma concentration that causes visual hallucination and psychomotor agitation Naloxone blocks the function morphine by competitive receptor inhibition Development of delirium (confused thinking and reduction of environmental awareness) Drastic enhancement of sildenafil plasma concentration that causes drug-related toxicity Enhancement of posaconazole plasma concentration that may induce drugrelated side effects Methotrexate clearance is inhibited and this results in methotrexate-related toxicity such as leukopenia and thrombocytopenia

Remarks Sudlow et al. (1975); Cropp and Bussey (1997)

Wagner et al. (1987)

Gareri et al. (2008)

Newman et al. (2000)

Palleria et al. (2013)

Muirhead et al. (2000); Merry et al. (1999) Lewis et al. (2013)

Kristensen (1976); Ronchera et al. (1993)

(continued)

7.1 Pharmacokinetic Drug–Drug Interactions

149

Table 7.1 (continued) Drugs Probenecid

Affected coadministered drugs Penicillin, cephalosporin, ampicillin

Rifampicin

Quinidine, midazolam, cyclosporine A

Hydrochlorothiazide, indapamide, metolazone, chlorothiazide, chlorthalidone Hydrocodone

Lithium salts (e.g., lithium carbonate)

Calcium carbonate

Acetaminophen, diphenhydramine Dolutegravir

Fenofibric acid

Warfarin

Diltiazem

Simvastatin

Ibuprofen, mefenamic acid, etoricoxib, indomethacin, naproxen, diclofenac, celecoxib, mefenamic acid Iron salts

Lithium salts (e.g., lithium carbonate)

Levodopa, methyldopa

Omeprazole, etravirine

Clopidogrel (prodrug)

Cause Plasma concentration of penicillin/ cephalosporin increases that may produce drug-related side effects Increased biotransformation of quinidine, midazolam, and cyclosporine A and reduction of drug activities Increase lithium reabsorption and lithium toxicity

Liver injury Reduction of dolutegravir absorption from the gastrointestinal tract that causes poor bioavailability of the drug Enhanced warfarin effect that cause more bleeding Increased simvastatin blood level that may cause the drug-related side effects Increased lithium blood level that may cause lithium toxicity in patients Absorption of levodopa and methyldopa is inhibited Inhibits biotranformation of the prodrug that causes reduction of drug activity

Remarks Wu et al. (2010)

Koselj et al. (1994); Venkatesan (1992); Backman et al. (1996) Amdisen (1982); Handler (2009) Hendrickson et al. (2010) Song et al. (2015)

Guo et al. (2018) Kanathur et al. (2001)

Phelan et al. (2003); Danion et al. (1987) Greene et al. (1990)

Kakuda (2011); Angiolillo et al. (2011) (continued)

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Table 7.1 (continued) Drugs Levodopa

Affected coadministered drugs Dopamine

Quetiapine

Dopamine, serotonin

Donepezil

Acetylcholine

Fluoxetine

Serotonin

Omeprazole, lansoprazole, pantoprazole, esomeprazole

Metformin

Cause Increases dopamine synthesis and provides the agonistic function in Parkinson’s disease and schizophrenia Quetiapine acts as antagonist for dopamine and serotonin, in bipolar disorder and schizophrenia Donepezil provides agonist function to acetylcholine preventing the release of acetylcholine from synapse Fluoxetine provides antagonistic function to serotonin preventing serotonin release from synapse in depression and obsessive-compulsive disorder Modulate metformin transportation within our body

Remarks Bogetofte et al. (2020)

Prieto et al. (2010)

Cacabelos (2007)

Ni and Miledi (1997)

Nies et al. (2011)

pharmacodynamic type. It can be further categorized into three subclasses based on the way of functioning: (1) precisely by involving receptor activities, (2) by modulating biological or physiological regulatory function, and (3) by pharmacological agonistic/antagonistic action.

7.1.3

Drug–Drug Interactions: Pharmacokinetic Type

Drug–drug interactions of this type modify the drug absorption, distribution, metabolism, or excretion by the other coadministered drug(s).

7.1.3.1 Absorption Many drugs alter gastrointestinal pH that modulates drug absorptions of some other drugs, which need optimum pH values for their absorption. Aluminum/magnesium

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151

hydroxide antacids, omeprazole, lansoprazole, pantoprazole, ranitidine, famotidine, atropine, benztropine mesylate, clidinium, cyclopentolate, darifenacin, dicyclomine, and fesoterodine increase gastric pH that reduces the gastrointestinal absorption of atenolol, bisoprolol, metoprolol, propranolol, glibenclamide, tolbutamide, ketoconazole, itraconazole, fluconazole, ampicillin, atazanavir, clopidogrel, diazepam, methotrexate, vitamin B12, paroxetine, raltegravir, etc. More drug–drug interaction– related examples are mentioned in Table 7.1. Similarly, many drugs form complex during their interaction with the other drugs. The developed complexes often are not absorbable. Heavy metal antacids such as aluminum/magnesium hydroxide antacids, iron formulations (iron salts), calcium formulations (calcium salts), etc. predominantly reduce drug absorption of many drugs such as tetracycline, ciprofloxacin, and penicillamine due to non-bioavailable complex formations. Intestinal p-glycoprotein (p-gp or p-gp 120 [molecular weight]) often plays a significant role in transmembrane drug diffusion. Erythromycin, roxithromycin, clarithromycin, omeprazole, lansoprazole, etc. enhance blood levels of digoxin, theophylline, and doxorubicin by p-glycoprotein-mediated enhanced drug absorption that may induce drug-related side effects of digoxin, theophylline, and doxorubicin.

7.1.3.2 Distribution Once drugs are absorbed in the blood, except water-soluble drugs that remain soluble in the plasma and get transported through the blood, drug molecules need to bind with some blood proteins for their distribution. Acidic drugs primarily bind to albumin, and basic drugs bind to either /1acid glycoprotein or lipoproteins or both. Modulation of drug–plasma protein binding of coadministered drugs often produces drug-related side effects or toxicity. Diclofenac increases free-warfarin blood level by quicker competitive drug–albumin binding. Free-warfarin in the blood induces toxicity that reflects in systemic hemorrhage in patients. 7.1.3.3 Metabolism In poly-therapy, it is often seen that one drug interferes with the metabolism of the other coadministered drug and thereby causes modulation of the drug activity of the second drug. Omeprazole inhibits biotransformation of clopidogrel (prodrug) by inhibiting CYP2C19 that causes reduction of conversion of clopidogrel to its active form and decreases its anti-platelet activity. Rifampicin-mediated induction of many cytochrome enzymes such as CYP3A, CYP2A6, and CYP2B6 causes enhanced biotransformation of quinidine, midazolam, and cyclosporine A and reduction of their activities. 7.1.3.4 Elimination Elimination of drugs or their metabolites from the body protects us from drug-related toxicity. It is an essential phenomenon for the safety of drug users (patients). However, a drug often interferes with the bodily removal process (elimination) of another coadministered drug during poly-therapy. It often causes serious

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drug-induced toxicity in patients. The thiazides and thiazide-like drugs such as chlorothiazide, hydrochlorothiazide, indapamide, metolazone, etc. increase lithium reabsorption from lithium salts such as lithium carbonate and induce lithium toxicity. In another example, propafenone reduces the clearance of metoprolol that may induce drug-related toxicity. Ibuprofen, mefenamic acid, etoricoxib, indomethacin, naproxen, diclofenac, celecoxib, and mefenamic acid inhibit methotrexate clearance and induce methotrexate-related toxicity such as leukopenia, thrombocytopenia, etc.

7.1.4

Drug–Drug Interactions: Pharmacodynamic Type

Modifying drug action such as induction, reduction, failure, and inhibition of drug function by coadministration of drugs belongs to the pharmacodynamic-type drug– drug interactions. Alteration of drug pharmacodynamic activity due to drug–drug interactions largely occurs by variable use of receptor functions, changing biological or physiological function, or pharmacological additive or inhibitory function.

7.1.4.1 Involving Receptor Activities Coadministered drugs, after absorption, sometimes use the same receptor or mimicking receptor. When a drug occupies the receptor faster competitively, it causes poor cellular internalization of the other drug that uses the same receptor. Thus, the free drug cannot reach inside the cells and causes inefficient drug action and induction of the drug-related toxicity. Morphine and naloxone use the same receptor. Upon coadministration of morphine and naloxone, the drug naloxone blocks the function of morphine by competitive receptor inhibition. 7.1.4.2 Modulating Biological or Physiological Regulatory Function In poly-therapy, alteration of the pharmacodynamic effect of a drug by a coadministered drug is widespread. For example, propafenone decreases the clearance of metoprolol that may induce drug-related toxicity. 7.1.4.3 Pharmacological Agonistic/Antagonistic Action Agonists work by receptor activation and give the pharmacodynamic response. Conventional agonists enhance the degree of receptor activation. Levodopa, an agonist for dopamine, increases dopamine synthesis using dopamine receptors and provides the agonistic function in Parkinson’s disease and schizophrenia. Inverse agonists bind with the receptors. The receptors are then unavailable and become inactive for the other drugs to bind. Donepezil acts as an agonist for acetylcholine in Alzheimer’s disease. It blocks the receptor that operates in the process of acetylcholine removal from the synapse. Antagonists inhibit receptor activation. Antagonists (reversible or irreversible) enhance their function in the cells but antagonize the cellular function of the drug(s) that they antagonize. Reversible antagonists dissociate from the receptors, but the irreversible antagonists bind permanently with the receptor. Quetiapine blocks dopamine and serotonin receptors and acts as an antagonist for dopamine and serotonin in bipolar disorder and schizophrenia.

References

7.1.5

153

Data Collection

The collection of data on pharmacokinetic drug–drug interactions depends on various data sources. They include direct observations of hospitals or clinics, consultation records and documents, survey through questionnaires, discussion with the health care personnel, history of the patients, interviewing health care professionals and the patients, published scientific reports, findings of researchers, electronic media, and the various data sources such as PubMed, Embase, Medline, Toxline, Drug Interaction Checker, Cochrane library, Patient Manager CSC, etc.

References Amdisen A (1982) Lithium and drug interactions. Drugs 24:133–139 Angiolillo DJ, Gibson CM, Cheng S, Ollier C, Nicolas O, Bergougnan L, Perrin L, LaCreta FP, Hurbin F, Dubar M (2011) Differential effects of omeprazole and pantoprazole on the pharmacodynamics and pharmacokinetics of clopidogrel in healthy subjects: randomized, placebocontrolled, crossover comparison studies. Clin Pharmacol Ther 89:65–74 Backman JT, Olkkola KT, Ojala M, Laaksovirta H, Neuvonen PJ (1996) Concentrations and effects of oral midazolam are greatly reduced in patients treated with carbamazepine or phenytoin. Epilepsia 37:253–257 Bogetofte H, Alamyar A, Blaabjerg M, Meyer M (2020) Levodopa therapy for Parkinson’s disease: history, current status and perspectives. CNS Neurol Disord Drug Targets 19:572–583 Bokor-Bratić M, Brkanić T (2000) Clinical use of tetracyclines in the treatment of periodontal diseases. Med Pregl 53:266–271 Cacabelos R (2007) Donepezil in Alzheimer’s disease: from conventional trials to pharmacogenetics. Neuropsychiatr Dis Treat 3:303–333 Cropp JS, Bussey HI (1997) A review of enzyme induction of warfarin metabolism with recommendations for patient management. Pharmacotherapy 17:917–928 Danion JM, Schmidt M, Welsch M, Imbs JL, Singer L (1987) Interaction between non-steroidal anti-inflammatory agents and lithium salts. Encephale 13:255–260 Gareri P, De Fazio P, Gallelli L, De Fazio S, Davoli A, Seminara G, Cotroneo A, De Sarro G (2008) Venlafaxine-propafenone interaction resulting in hallucinations and psychomotor agitation. Ann Pharmacother 42:434–438 Greene RJ, Hall AD, Hider RC (1990) The interaction of orally administered iron with levodopa and methyldopa therapy. J Pharm Pharmacol 42:502–504 Guo C, Xue S, Zheng X, Lu Y, Zhao D, Chen X, Li N (2018) The effect of fenofibric acid on the pharmacokinetics and pharmacodynamics of warfarin in rats. Xenobiotica 48:400–406 Handler J (2009) Lithium and antihypertensive medication: a potentially dangerous interaction. J Clin Hypertens (Greenwich) 11:738–742 Hendrickson RG, McKeown NJ, West PL, Burke CR (2010) Bactrian (“double hump”) acetaminophen pharmacokinetics: a case series and review of the literature. J Med Toxicol 6:337–344 Johnson BF, Bustrack JA, Urbach DR, Hull JH, Marwaha R (1984) Effect of metoclopramide on digoxin absorption from tablets and capsules. Clin Pharmacol Ther 36:724–730 Kakuda TN, Schöller-Gyüre M, Hoetelmans RMW (2011) Pharmacokinetic interactions between etravirine and non-antiretroviral drugs. Clin Pharmacokinet 50:25–39 Kanathur N, Mathai MG, Byrd RP Jr, Fields CL, Roy TM (2001) Simvastatin-diltiazem drug interaction resulting in rhabdomyolysis and hepatitis. Tenn Med 94:339–341 Koselj M, Bren A, Kandus A, Kovac D (1994) Drug interactions between cyclosporine and rifampicin, erythromycin, and azoles in kidney recipients with opportunistic infections. Transplant Proc 26:2823–2824

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Krishna G, Moton A, Ma L, Medlock MM, McLeod J (2009) Pharmacokinetics and absorption of posaconazole oral suspension under various gastric conditions in healthy volunteers. Antimicrob Agents Chemother 53:958–966 Kristensen MB (1976) Drug interactions and clinical pharmacokinetics. Clin Pharmacokinet 1:351– 372 Lee HT, Lee YJ, Chung SJ, Shim CK (2000) Effect of prokinetic agents, cisapride and metoclopramide, on the bioavailability in humans and intestinal permeability in rats of ranitidine, and intestinal charcoal transit in rats. Res Commun Mol Pathol Pharmacol 108:311–323 Lewis RE, Ben-Ami R, Best L, Albert N, Walsh TJ, Kontoyiannis DP (2013) Tacrolimus enhances the potency of posaconazole against Rhizopus oryzae in vitro and in an experimental model of mucormycosis. J Infect Dis 207:834–841 Merry C, Barry MG, Ryan M, Tjia JF, Hennessy M, Eagling VA, Mulcahy F, Back DJ (1999) Interaction of sildenafil and indinavir when co-administered to HIV-positive patients. AIDS 13: F101–F107. https://doi.org/10.1097/00002030-199910220-00001 Muirhead GJ, Wulff MB, Fielding A, Kleinermans D, Buss N (2000) Pharmacokinetic interactions between sildenafil and saquinavir/ritonavir. Br J Clin Pharmacol 50:99–107 Newman LC, Wallace DR, Stevens CW (2000) Selective opioid agonist and antagonist competition for [3H]-naloxone binding in amphibian spinal cord. Brain Res 884:184–191 Ni YG, Miledi R (1997) Blockage of 5HT2C serotonin receptors by fluoxetine (Prozac). PNAS 94: 2036–2040 Nies AT, Hofmann U, Resch C, Schaeffeler E, Rius M, Schwab M (2011) Proton pump inhibitors inhibit metformin uptake by organic cation transporters (OCTs). PLoS One 6:e22163. https:// doi.org/10.1371/journal.pone.0022163 Ogawa R, Echizen H (2010) Drug-drug interaction profiles of proton pump inhibitors. Clin Pharmacokinet 49:509–533 Ogawa R, Echizen H (2011) Clinically significant drug interactions with antacids: an update. Drugs 71:1839–1864 Palleria C, Di Paolo A, Giofrè C, Caglioti C, Leuzzi G, Siniscalchi A, De Sarro G, Gallelli L (2013) Pharmacokinetic drug-drug interaction and their implication in clinical management. J Res Med Sci 18:601–610 Phelan KM, Mosholder AD, Lu S (2003) Lithium interaction with the cyclooxygenase 2 inhibitors rofecoxib and celecoxib and other nonsteroidal anti-inflammatory drugs. J Clin Psychiatry 64: 1328–1334 Phillips WA, Ratchford JM, Schultz JR (1976) Effects of colestipol hydrochloride on drug absorption in the rat II. J Pharm Sci 65:1285–1291 Prieto E, Micó JA, Meana JJ, Majadas S (2010) Neurobiological bases of quetiapine antidepressant effect in the bipolar disorder. Actas Esp Psiquiatr 38:22–32 Ronchera CL, Hernández T, Peris JE, Torres F, Granero L, Jiménez NV, Plá JM (1993) Pharmacokinetic interaction between high-dose methotrexate and amoxycillin. Ther Drug Monit 15:375– 379 Rushing DA, Raber SR, Rodvold KA, Piscitelli SC, Plank GS, Tewksbury DA (1994) The effects of cyclosporine on the pharmacokinetics of doxorubicin in patients with small cell lung cancer. Cancer 74:834–841 Scaldaferri F, Pizzoferrato M, Ponziani FR, Gasbarrini G, Gasbarrini A (2011) Use and indications of cholestyramine and bile acid sequestrants. Intern Emerg Med 8:205–210 Song I, Borland J, Arya N, Wynne B, Piscitelli S (2015) Pharmacokinetics of dolutegravir when administered with mineral supplements in healthy adult subjects. J Clin Pharmacol 55:490–496 Sudlow G, Birkett DJ, Wade DN (1975) The characterization of two specific drug binding sites on human serum albumin. Mol Pharmacol 11:824–832 Venkatesan K (1992) Pharmacokinetic drug interactions with rifampicin. Clin Pharmacokinet 22: 47–65 Wagner F, Kalusche D, Trenk D, Jähnchen E, Roskamm H (1987) Drug interaction between propafenone and metoprolol. Br J Clin Pharmacol 24:213–220

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8

Pharmacokinetic Applications

8.1

Therapeutic Drug Monitoring and Dose Formula

In clinical pharmacy practice, one significant area is the application of pharmacokinetics in therapeutic drug monitoring. Another predominant area is evaluating the physiological effect on the pharmacokinetics of drugs in neonates, infants, children, elderly patients, obese patients, and patients with liver and kidney insufficiencies.

8.1.1

Therapeutic Drug Monitoring

Therapeutic drug monitoring involves quantifying drug concentration in plasma, serum, blood, and other body fluids (saliva, milk) and interpreting the findings. Here, we must have a clear idea about plasma and serum of the blood.

8.1.1.1 Plasma Plasma is the single most significant component (55% of human blood) that possesses soluble salts, some proteins including antibodies, some enzymes, and water. Once the blood cells, thrombocytes, and other cellular components are separated from the blood, the straw yellow-colored transparent left-out liquid portion is known as plasma.

8.1.1.2 Serum When clotting factors (blood-clotting proteins such as fibrinogens), blood cells, and thrombocytes are separated from the blood, the left-out liquid that contains soluble salts, some proteins including hormones, antigens, antibodies, exogenous substances (e.g., drugs or microbes), and non-blood-clotting proteins is called serum. # The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B. Mukherjee, Pharmacokinetics: Basics to Applications, https://doi.org/10.1007/978-981-16-8950-5_8

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8.1.1.3 Required Sectors of Therapeutic Drug Monitoring Therapeutic drug monitoring becomes significantly crucial during the following cases: 1. For a drug with a narrow therapeutic window (it implies the MEC [minimum effective plasma drug concentration] and the MSC [maximum safe plasma drug concentration] are close to each other) 2. For those drugs that show a considerable variation in pharmacokinetic profiles in patients 3. While a direct correlation exists between a drug concentration in plasma/body fluids (saliva/milk) and the physiological effects of the drug, for example, the effects of absorption, distribution, metabolism, and elimination (ADME) of drugs in neonates and elderly patients 4. When the therapeutic effect of drugs cannot be assessed readily, for example, as seen in the case of antibiotics and antiviral drugs 5. When clinical symptoms such as seizures are avoided 6. When desired or expected therapeutic outcomes are not seen due to drug absorption or compliance problems 7. When a drug has a high hepatic first-pass effect 8. When accurate and inexpensive analytical methods for estimation of a drug from blood/biological fluids are available 9. When patients have hepatic or renal insufficiencies 10. When there is a possibility or doubt about drug–drug interaction in patients receiving multiple medications A physician consulting a pharmacist should recommend medicine(s) and dose (s) for a patient in ideal health care practice. For this, they require patient-related information such as age, sex, height, body weight, smoking and drinking habits, etc. Besides, clinical information such as blood pressure, pulse rate, history of diseases (if any) suffered from, existing liver or kidney problems (if any), whether currently suffering from a disease, blood profile, cardiac output, if under medications, the name of the medicine(s) being consumed by the patient, etc. are also required. Once the medication is decided, the dose is determined. For a dose, the loading dose and maintenance dose are often required to calculate. For complicated health conditions such as patients suffering from liver or kidney insufficiencies, hepatic or renal cancers, severe cardiac problems, patients on a life-support system, patients receiving dialysis, and patients in coma, accuracy in time of sample collections and their analysis provide a more accurate evaluation of pharmacokinetic profiles of drugs for fine-tuning and critical adjustment of the dose to be recommended for those patients.

8.1 Therapeutic Drug Monitoring and Dose Formula

8.1.2

159

Physiological Effects on the Pharmacokinetic Drug Parameters and Available Dose Formula in Neonates, Infants, Children, Elderly Patients, Obese Patients, and Patients with Liver and Kidney Insufficiencies

Radical changes in physiological processes usually occur in children and elderly patients. Such physiological changes interfere markedly in the drug absorption, distribution, metabolism, and elimination strategies in those patients. Thus, calculating the right amount of drug (dose) is essential for treating those patients with safety and avoiding any dose-related untoward effects.

8.1.2.1 Pediatric Patients Changes in physiological processes rapidly and markedly take place in neonates, infants, and children. In the very first year of human life, these changes vary radically and widely. There is always a sizeable inter-patient variation due to dramatic changes in physiological processes that alter drug disposition widely from patient to patient. Thus, proper and accurate dose adjustment is badly required for neonates (from birth up to 1 month of age), infants (from 1 month up to 1 year of age), and children (1 year to 12 years of age) by experienced/specially trained health professionals (physicians or pharmacists). While the physiological process undergoes maturation, the changes and variations in drug disposition in neonates, infants, and children continue throughout childhood. It causes changes in pharmacokinetic profiles of drug absorption, distribution, metabolism, and elimination in pediatric patients (Batchelor and Marriott 2015). Some of such changes are depicted below. 8.1.2.2 Absorption Until reaching 3 years of age, the newborn babies have less hydrochloric acid secretion (relative achlorhydria), increasing the enhanced bioavailability of acidsensitive drugs such as penicillin. The average gastric acid output in neonates is 0.15 mEq/10 kg body weight/h, while the value is 2 mEq/10 kg body weight/h for the average adults. Other features such as slow gastric emptying and irregular peristaltic movement often delay drug absorption in neonates and infants. The average gastric emptying time in neonates and infants is 87 min compared to 65 min in adults. 8.1.2.3 Distribution The total body water content (expressed in % of body weight) in neonates is 78% compared to 60% in adults. During the first year of life, the total body water content decreases. Average neonatal extracellular and intracellular water content values (expressed in % of body weight) are 44% and 34% compared to 19% and 41% in adults, respectively. The values suggest that extracellular body fluid is higher in the first year of life, and during this phase of life, the drug distribution volume is always significantly more. Thus, many drugs (e.g., tetracycline, ampicillin) show an

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increased drug distribution volume in neonates and infants. Further, blood albumin content (3.7 g/dL) in neonates is always found less compared to the average healthy adults (4.5 g/dL). It causes less plasma protein–drug bindings, resulting in faster removal of many drugs in their active forms through urine.

8.1.2.4 Metabolism Wide inter-patient variation, premature metabolic processes, and shortage of data concerning drug metabolism in neonates and infants complicate understanding pharmacokinetics and drug disposition issues in neonates and infants. Many exciting observations make the process further complex. Glucuronidation is predominantly inefficient, whereas sulfate conjugation (e.g., paracetamol) is quite functional at birth and in newborns. Thus, the drug (e.g., chloramphenicol) that needs glucuronidation for elimination eliminates slowly in newborns. During the first month after delivery, no caffeine metabolism is observed in neonates, but during 3–7 months, caffeine metabolism in neonates becomes similar to that of adults. 8.1.2.5 Elimination The rate of glomerular filtration greatly varies at neonatal ages. In the first 4 days of life, the average rate of glomerular filtration is 1 mL/min/m2 of body surface area, which reaches to 22 mL/min/m2 of body surface area in a 2-week duration, and in 1-year time, the value becomes at par to that of the adult value (70 mL/min/m2). The data suggest that during the first year of life, the elimination process of drugs radically differs and increases elimination time for drugs through the kidneys in neonates and infants. The dose is usually adjusted during the first few years of life based on the body surface area, body weight, and age. For children, the height parameter is also taken into consideration. Some of such popular formulas are given below (Elias et al. 2005). 8.1.2.6 Mosteller’s Equation


2

Body surface area of the infant or child m Child dose ¼



pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi height ðin mÞ  weight ðin kgÞ ¼ 60

body surface area ðsquare meterÞ  ðadult doseÞ 1:73

Since 1.73 m2 is considered as the body surface of an average adult weighing 70 kg. Body surface can be determined using the following formula also:
 Body surface area m2 ¼ ðbodyweight of the neonate in kgÞ0:7

8.1 Therapeutic Drug Monitoring and Dose Formula

Child dose ¼

161

 0:7 weight of child in kg  ðadult doseÞ 70

8.1.2.7 Clark’s Formula Child dose ¼

bodyweight of the child in pounds  ðadult doseÞ 150

8.1.2.8 Fried’s Rule for Infants Dose ¼

Age in months  ðadult doseÞ 150

For children with the age group of 2 years or more, Young’s formula is often used. Child dose ¼

Age of the child in years  ðadult doseÞ Age of the child in years þ 12

8.1.2.9 Cowling’s Formula Child dose ¼

8.1.3

Age of the child on next birthday in years  ðadult doseÞ 24

Physiological Effects on the Pharmacokinetic Parameters of Drugs and Available Dose Formula in Elderly or Geriatric Patients

Day by day, the elderly population is increasing in many parts of the globe. Simultaneously, geriatric patients are also increasing predominantly throughout the world. Although they consume medicines that cost nearly one-third of the total cost of drugs globally, nearly a sevenfold enhancement of drug toxicity–related incidences is observed in elderly patients compared to young patients. It may be because of the dearth of availability of drug toxicity–related data in geriatric patients and the availability of volunteers of higher age groups (70–85 years), which is always rare during clinical trials. Further, elderly patients often receive multiple medications that cause drug toxicity due to drug–drug interaction in them. Thus, various pharmacokinetic factors that affect drug disposition and drug response in elderly patients are often a challenge for medication on geriatric patients (Klotz 2009; McLean and Le Couteur 2004).

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8.1.3.1 Absorption Physiological changes such as decreased gastric acid secretion, low gastrointestinal blood flow, less pancreatic trypsin content, and decreased gastrointestinal motility generally occur in the elderly population with their increasing age. It reflects changes in drug dissolution and reduced and/or delayed drug absorption for many drugs (e.g., thiamine, tolbutamide, calcium, iron). 8.1.3.2 Distribution In the elderly population, decreased total body water content, lean body weight, and high body fat (20–30% increase, again female > male) decrease distribution volume of polar drugs in the blood and increase the distribution volume of non-polar lipidsoluble drugs. Plasma protein binding of acidic drugs is found to be more than the basic drugs in elderly patients. Decreased serum albumin, in some cases, increased alpha-1-glycoprotein, increased gamma-globulin, and often reduced RBC binding may be responsible for slower drug distribution and increased biological half-life for many drugs in elderly patients. Further, drug distribution is significantly reduced to the liver and kidneys in elderly patients due to less cardiac output. 8.1.3.3 Metabolism In the geriatric population, drug-metabolizing enzyme levels and activities are predominantly reduced along with the reduction of hepatic mass and hepatic blood flow. Such physiological phenomena potentially alter drug metabolism in elderly patients. Even some factors such as nutritional problems, smoking, and drinking alcohol strongly influence drug metabolism in elderly patients. Interestingly, phase I metabolism (oxidation, reduction) significantly affects elderly patients, whereas phase II metabolism (acetylation, glucuronidation) does not change distinctively. Thus, metabolism of phenytoin, theophylline, and lidocaine is reduced, whereas isoniazid and temazepam do not appear to undergo metabolic change greatly. 8.1.3.4 Elimination Decreased renal clearance of the drug in geriatric patients is an inherent problem that enhances the half-lives of many medicines. It often leads to drug-related toxic manifestations in elderly patients. In the kidneys, glomerular filtration rate, renal plasma flow, and active secretion are commonly decreased. Serum creatinine level is also reduced due to the reduction of muscle mass. There is a 30–35% reduction of nephrons in elderly patients. Thus, dose adjustment should be critically evaluated and decided before recommendation to elderly patients. Drugs such as chlordiazepoxide (half-life of the drug in young is 7 h, whereas that in elderly patients is 40 h), digoxin (half-life of the drug in young is 37 h, whereas that in elderly patients is 70 h), nitrazepam (half-life of the drug in young is 29 h, whereas that in elderly patients is 40 h) that have less clearance in elderly patients need a critical dose adjustment. Few formulas have been suggested for the recommendation of dose in elderly patients.

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8.1.3.5 Dose Calculation of Elderly Patients Þ0:7 ð140age in years of the patientÞ Dose ¼ ðbodyweight in kg of the elderly patient  ðadult doseÞ 1660 (Fesce and Fumagalli 2015)

8.1.4

Physiological Effects on the Pharmacokinetic Parameters of Drugs in Obese Patients

A person whose body weight is more than 25% of the ideal body weight is called obese. However, the ideal body weight varies from male to female. The ideal body weight for a male with a height of 150 cm is 50 kg. A 1-kg increase in body weight is considered for an increase of every 2.54 cm (1 inch) of his height. For females, the ideal body weight with a height of 150 cm is 45 kg. But a 1-kg increase in body weight is also considered for an increase of every 2.54 cm (1 inch) of her height. The apparent volume of distribution is usually proportional to the body weight of a patient. Obese patients have a higher proportion of adipose tissue and lower water content, causing a lower volume of distribution of the relatively polar drugs. For example, a 25% reduction in the apparent volume of distribution is observed for antipyrine in obese patients compared to the patients with normal body weight. Thus, due to wide pharmacokinetic and pharmacodynamic variations, dose adjustment in obese patients is essential.

8.1.5

Physiological Effects on the Pharmacokinetic Parameters of Drugs and Available Dose Formula in Patients with Renal Insufficiency

The number of patients with renal insufficiency is increasing day by day. For many patients who have diabetes, pancreatic cancer, renal cancer, and various other chronic and acute kidney diseases, the renal insufficiency is common. In those patients, drug elimination through the kidneys greatly suffers due to inefficient kidney function. Dose adjustment in those patients is often made based on the total clearance of the drug in the patients. A formula is given below to calculate the dose of a drug in a patient with renal insufficiency: x00 x CL 0 ¼ 0 τ τ0 τCLτ where x0, τ, and CLτ are the maintenance dose, dosing interval, and total clearance of a drug in an average person and x00 , τ0, and CLτ0 are the maintenance dose, dosing interval, and total clearance of the drug in a patient suffering from renal insufficiency. Thus, by obtaining the value of CLτ0 , dose in those patients is determined.

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Chronological Pharmacokinetic Guidance for Preclinical and Clinical Studies and Research

Safety of the investigations and accuracy of the outcomes of pharmacokinetic studies both in preclinical (in animals) and clinical (in human subjects) levels primarily depend on well-designed and carefully conducted studies with adequate data analysis.

8.1.6.1 Study Design The pharmacokinetic study is an essential and integral part of the development of new drugs and new formulations. It has two stages, namely, preclinical investigation and clinical investigation. Preclinical investigations are done on animal models (mice, rats, rabbits, dogs, monkeys, etc.). The studies are done to determine the safety and efficacy of a new drug/a new formulation by assessing various pharmacokinetic parameters. These investigations also provide us with the guidance of dose and dose escalation aspects, which may be translated into the clinical examination (that is, in human subjects). For example, a dose-escalation study guides us through the dose tolerance investigations in phase I clinical trials. Preclinical experiments often help us interpret toxicological data. As a whole, preclinical data are the primary guidance for clinical trials regarding safety, efficacy, and toxicological concerns about new drugs/new formulations. Successful clinical trials are essential for human use of a new drug/new formulation, and it requires receiving permission from the drug regulatory authority. In phase I clinical trial, dose-determination and dose-escalation studies are conducted. A dose-escalation study is done to determine the maximum dose tolerance to understand the safety aspect of a dose in human subjects. Randomized dose-based clinical trials in many patients are conducted in phase II and phase III clinical trials to understand the safety and efficacy of a new drug/a new formulation. Individual variation of pharmacokinetic parameters, patient-specific factors, populationspecific pharmacokinetic data, and individualization of dose are determined in these clinical trials. Once the data are statistically analyzed, they are then considered as the results of these studies. Based on the successful data of preclinical (in animals) investigations, clinical (in human) trials are permitted. Further, based on the successful data of clinical trials, the new drug/new formulation is allowed for human use. Thus, the study design for preclinical studies and clinical trials should be adequate to make the investigations successful. A vivid description is provided in the following paragraphs of adequate designing a preclinical study or a clinical investigation. Various factors that play a significant role in designing a preclinical/clinical study are given below:

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1. Group design: Formation of the various test group(s) and control group(s) so that the data can be analyzed scientifically to reach a reliable and accurate outcome of the investigation. 2. Sample type: For preclinical experiments, small or large animals, and for clinical trials, healthy volunteers and patients who are selected based on age, sex, disease, etc. 3. Sample size: The number of animals (for preclinical study)/subjects (for clinical study) is selected in a way so that it can at least fulfill the minimum sample size requirement for statistical analysis, perform the experiment adequately, and take care of financial constraint. 4. Route of drug administration. 5. Mode of drug administration. 6. Schedule of drug administration. 7. Time points of drug sampling. 8. Endpoints. 9. Parameter(s) to be analyzed. 10. Method of analysis. 11. Method of statistical analysis of data.

8.1.6.1.1 Group Design The selection of groups for a preclinical experiment or a clinical trial is significant. Groups should be selected in such a way that proper control group(s) should exist against the test group(s). It is required for obtaining reliable and accurate results, which can be statistically analyzed also. For Preclinical Investigation

In preclinical (in animals) investigation, the most critical issue in designing a study is the proper consideration of control group(s) that remains unexposed to the experimental variable. Data of the control group is considered as reference or standard. The comparison of the experimental (test) group data is made against the control group data to understand the changes in the study parameters. For example, a study is conducted to understand the anticancer efficacy of a new drug in a carcinogeninduced cancerous animal model, and the dose has already been determined by the pharmacological method. In this study, we need to run at least four groups. One group consists of normal animals (normal control group), the animals of the second group should receive carcinogen treatment only (carcinogen control group), the animals of the third group should receive carcinogen treatment along with the treatment of new drug (test group), and the fourth group of animals are normal animals that should receive the treatment of the new drug only (new drug control group). Therefore, out of the four groups, there are three control groups. Thus, selecting proper control group(s) is an essential criterion for an experimental study design to obtain reliable and accurate data that could be analyzed statistically to understand their significances.

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For Clinical Investigation Controlled Trial

In this clinical trial, a proper control group of the subjects is selected. The subjects, who belong to the control group, receive a placebo (no therapeutic value) or an existing standard treatment. It is called a parallel design when a test and control groups are run parallel in a controlled trial. However, when the subjects of the control group and test group are interchanged at least in two sets of the same trial, that is, the subjects of the test group will become the subjects of the control group and vice versa, in the second set, this is called a crossover design. Randomization

The distribution of the subjects in the test and control groups is done randomly by a preselected consideration or by a computerized program. It is, therefore, called the randomization design. Any biases of the investigator or the subjects for the inclusion of a specific group may be avoided. Blinding (Masking) Process

Blinding process design is done by concealing the information regarding the test and control samples by using codes. In this trial, the subjects of the test group and the control group are kept unaware of the treatment types (test and control samples; single-blind process), or both the subjects and the investigator are kept ignorant (double-blind process) of the test and control samples. A third party (may be an analyst) is involved in keeping the code. However, bias can come from analysts also. Thus, when all three parties are kept uninformed about the treatment types and codes are kept to a fourth party, this is called the triple-blind process. Cohort Study

A group of individuals with standard features such as age, concerns, and experiences is called a cohort group. Subjects of a group take a particular drug, and the group may be considered as a cohort group. The beneficial or adverse effect of a drug in the subjects of a group may be determined by a prospective study (which provides expected outcome for the future) or retrospective (from record) study. The cohort study is helpful to decide on the adverse effect of a drug being used. 8.1.6.1.2 Sample Type Sample types of a preclinical animal pharmacokinetic experiment are small animals such as mice, rats, and rabbits or large animals such as dogs and monkeys. The type of animals are generally selected depending on the endpoint(s) (e.g., drug concentration in the blood or tissue), based on the volume of the blood, the frequency of blood withdrawals, and the total number of time points. For example, a pharmacokinetic experiment that needs the withdrawal of greater blood volume with more time points should use small animals with greater body weight (means blood volume), such as rabbits. Withdrawal of blood volume (200–500 μL) with fewer frequencies and fewer time points may be made using rats. Mice can also be used to collect a greater blood volume by pooling of blood collected from 2 to 3 mice of the same group at each time point. However, receiving the Animal Ethics Committee approval can be an issue as more animals would be required.

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For clinical trials, phase I clinical trial is done on healthy human volunteers and, in some cases, on patients who have been suffering from some particular diseases such as cancer, AIDS, etc. For phase II and phase III clinical trials, only patients (related to new drug/formulation) are generally selected as subjects. 8.1.6.1.3 Sample Size From the perspective of ethical and financial issues, the least number of animals that can satisfy for statistically approved data comparison analysis is considered. However, for the animal experiments of long duration, some fewer additional animals than the minimum required number (bare minimum statistical requirement of animals) in each group are advisable to include, mainly when the chance of animal death is probable or predictable. For phase-wise (phase I, II, and III) clinical trials, the number of human subjects for each type of trial has been recommended by the drug regulatory authorities of each country and guided by some international organizations. Phase I clinical trial requires carrying out on 20–80 healthy human volunteers or, in some cases, patients who are suffering from diseases such as cancer and AIDS. Phase II clinical trial needs to conduct on 100–500 patients, whereas phase III clinical trial is conducted on 501–3000 patients. However, successful data of the larger number of subjects always give the additional credential during the evaluation of the project for drug approval. 8.1.6.1.4 Route of Administration For the pharmacokinetic study, the route of administration of a drug is vital for preclinical or clinical investigations. The pharmacokinetic profile of a drug greatly varies with an alteration of the route of drug administration. A drug administered orally shows lag time due to some physicochemical and biopharmaceutical processes such as dissolution, hepatic first pass, and absorption. On the other hand, by intravenous route of administration, the drug becomes immediately bioavailable and can avoid lag time. Again, the buccal route of administration of a drug provides a faster absorption than its oral administration. Thus, the drug administration route significantly alters the drug pharmacokinetic profile in animals or humans. The nature of a drug can also play an important role in selecting an appropriate route of drug administration. Many drugs (such as insulin, other proteins) degrade when given by the oral route, and hence, the oral route is avoided for the types of drug. Thus, the selection of a relevant delivery route for a drug product is enormously important for adequate bioavailability of the drug. 8.1.6.1.5 Mode of Drug Administration The mode of drug administration is equally crucial for the determination of the pharmacokinetic profile of a drug. For example, a drug selected for intravenous route of administration can be given by various modes such as intravenous bolus injection, continuous intravenous infusion, etc. For the oral route of drug administration, a drug can be administered by multiple modes such as tablets, capsules, oral solution, or suspension. In all the cases, variable pharmacokinetic profiles are observed. Even

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tablet formulations of a drug may be given orally as a non-coated tablet, coated tablet, sustained-release tablet, etc. Thus, the mode of drug administration also differs the pharmacokinetic profile of a drug. 8.1.6.1.6 Scheduling of Drug Administration In a pharmacokinetic study design, the preparation of a schedule of drug administration depends on the type of drug/formulation and its pharmacokinetic profile. In preclinical studies (animal experiments), various pharmacokinetic parameters are determined by administering the dose (usually predetermined by the pharmacological methods) of a drug or drug product. Such pharmacokinetic parameters include Tmax, Cmax, area under the curve (AUC), etc. Based on the preclinical pharmacokinetic data, a dose of the drug in humans is projected. The estimated dose is used in phase I clinical trial for dose-determination and dose-escalation investigation to determine the maximum tolerated dose in human subjects. Further, various pharmacokinetic parameters, as mentioned above, are determined and established in human subjects. Based on the data of phase I clinical trials, scheduling of drug administration is made for phase II and phase III clinical trials. However, several incidences of failure of phase I clinical trials due to drug absorption and bioavailability cause loss of time, effort, and money. Hence, the United States Food and Drug Administration (USFDA) and the European Medicines Agency (EMA) have recommended a phase “0” trial. In this trial, the drug is administered 1/100 of the estimated dose for humans or a maximum of 100 μg in a few healthy volunteers. The drug is estimated from the blood by a sophisticated analytical device such as LC-MS/MS. If the drug is detectable in the blood, the phase I clinical trial may be conducted. Thereby, the loss of time, effort, and money can be minimized. Thus, the phase “0” (micro-dosing) trial data guides phase I clinical trial. However, the study is not related to the determination of safe dose and frequency of drug administration required. The successful phase I clinical trial data are used to develop drug administration schedules such as dose, dosing interval, frequency of drug administration needed for a day, etc. 8.1.6.1.7 Time Points of Sampling Based on the dose, dosing interval, duration of drug action, and objective (endpoints) of the investigation, sampling time points are determined both in animals and humans. For example, for the determination of AUC after the oral administration of a drug, the time points are selected in such a way so that the data obtained could help plot the curve correctly for accuracy. A long gap between data sets cannot accurately determine many pharmacokinetic parameters such as Cmax, Tmax, etc. 8.1.6.1.8 Endpoints In a pharmacokinetic study, endpoints such as drug concentrations in urine and plasma drug concentrations at various time points after a drug administration are required to determine. Various other pharmacokinetic parameters, such as Cmax,

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Tmax, AUC, etc., are determined using such fundamental data. Therefore, endpoint (s) should be prefixed before the experiment/trial begins. 8.1.6.1.9 Parameters to Be Analyzed The pharmacokinetic parameters such as Cmax, Tmax, AUC, the area under the firstmoment curve (AUMC), Ka, Ke, biological half-life, etc. are determined. 8.1.6.1.10 Method of Analysis The method of drug analysis from biological samples such as tissues, serum, plasma, whole blood sample, urine, and other body fluids, namely, milk and saliva, is primarily selected based on the quantity of drug present in the samples. For the meager amount of drug, sophisticated analytical devices such as LC-MS/MS are used. However, a greater quantity of drugs can be analyzed by HPLC, spectrophotometer, etc. Thus, after the drug extraction from the blood or tissue samples, the drug should be analyzed using an adequate analytical device using a validated method that can produce reliable, reproducible, and accurate data. 8.1.6.1.11 Statistical Analysis An appropriate statistical analysis method is required for the selection for comparison of data to understand whether the data are statistically significant in terms of accuracy and reliability. The statistical methods that address pharmacokinetic data analysis are discussed in Chap. 11 later in the book.

8.1.7

Analysis of Pharmacokinetic Data

Once the pharmacokinetic data are obtained, the subsequent essential actions related to the data are data handling, data analysis, and preparing the pharmacokinetic report. Hence, they have been discussed below.

8.1.7.1 Data Handling Data handling is fundamentally a process of assembling, recording, documenting, and furnishing information to assist others or to a system. After receiving the pharmacokinetic data, data are now computerized in a spreadsheet usually obtained using Microsoft Excel, Lotus version 1-2-3, etc. It is a simple process and useful for individual data evalution, preclinical and clinical comprehensive, and routine studies. Primary data should be kept separated from model software that can cause unwanted data modification. 8.1.7.2 Data Editing Data editing is a process of reviewing and adjustment of collected assembled data using a set of methods. It reduces potential bias and assures a consistent estimate that adequately analyzes the collected data by correcting inconsistent data. It detects the unusable data (e.g., missing blood sampling time or dosing frequency to determine

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blood concentration of a drug) and corrects the errors. The useless data should be predefined in a pharmacokinetic study.

8.1.7.3 Outliers An outlier may be defined as a datum that is in no way close to the other data during random sampling in an observation. However, in a sense, data analysts should decide to consider the data to be abnormal. Thus, the values that are not fitted in a model are called outliers. Outliers are rejected from data analysis as those data are inappropriate. The reasons to consider the data as outliers should be disclosed and explained in a pharmacokinetic report. 8.1.7.4 Analysis of Pharmacokinetic Data The analysis of pharmacokinetic data should be segmented in the following parts: (a) (b) (c) (d) (e)

Noncompartmental data analysis Compartmental data analysis Data analysis of physiological/physiology-based pharmacokinetic model Data analysis of pharmacokinetic/pharmacodynamic model Population-based pharmacokinetic data analysis

8.1.7.4.1 Noncompartmental Data Analysis Unlike compartmental models, noncompartmental models do not need some definite assumptions. However, in noncompartmental models, a considerably large volume of data is indispensable for the reliable determination of pharmacokinetic parameters. For example, for terminal slope determination of terminal elimination rate constants, 3–4 observations (by logarithmic-linear regression to bring highly skewed data to a more evenly arranged data set) are required. Similarly, the area under the curve (AUC) and the area under the first-moment curve (AUMC) are calculated using “linear trapezoidal rule” (for equal or increasing concentration, as seen in the drug absorption phase in the drug plasma concentration versus time curve) or by logarithmic-linear trapezoidal rule (for decreasing concentration, as seen in the drug elimination phase in the drug plasma concentration versus time curve). For the determination of AUC or AUMC from zero to infinity time, the area from the last estimation to time infinity is extrapolated as an aggregate of individual trapezoidal areas. However, the extrapolated area should never outstrip 25% of the entire area. Once the primary pharmacokinetic parameters such as AUC, AUMC, and terminal elimination rate constant are estimated, they may be used to determine the other pharmacokinetic parameters such as steady-state plasma concentration and clearance using the available standard pharmacokinetic equations. Noncompartmental analysis parameters are calculated with industry-accepted techniques and generate plots of the whole data, the estimated parameter distributions, and correlations of variables with covariates. Graphical software is used to set parameters and rules easily, scrutinize the computation and exhibit the outcomes. A widely used programming language, “R” is often used for powerful scripting.

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Compartmental Data Analysis

A paradigm of pharmacokinetic models is a compartmental model. Through the compartmental data analysis, the pharmacokinetic parameters that represent the compartment dynamics are determined for every individual. The interactive plots are intuitive, and it is a fast process for such data interpretation. The entire setting should be reserved for reproducible outcomes. The merged validation software secures correct installation and computation. Model Selection

Compartmental models start from a simple one-compartment model but then complicate into the classic multi-compartmental models. As discussed earlier, drug distribution and drug elimination processes in the body can be described most easily by a one-compartment open model. However, multi-compartmental models provide a simulation of pharmacokinetic processes of drug absorption, distribution, metabolism, and elimination processes by developing kinetic equations and providing their solutions by data input such as dose, dosing time, duration, etc. Based on mass balance equations, simple linear equations are developed as ordinary differential equations. Sufficient data of drug concentrations in the blood or tissue are required from a subject to solve the equations. Initial Estimates

Differential equations are solved based on starting values of kinetic parameters. The exponential equation of the residual plot is a good example. Scattered data and fewer time points make it always challenging to draw a graphical plot. The initial values may converge to undesired points of implausible parameter estimates with local and absolute minima. Selection of Minimizing Algorithm

Minimization of difference between the observed values and predicted values of drug concentrations is the ultimate objective as it serves for a reliable and more accurate plot. Ordinary least-squares (OLS), weighted least-squares (WLS), and generalized least-squares (GLS) methods are commonly used for homoscedastic, heteroscedastic (but uncorrelated), and correlated and heteroscedastic data, respectively. Downhill sample or Nelder–Mead algorithm deals with the geometry of space. Powell method is based on one-dimensional local minimization. Channels’ matching (CM) algorithm used for testing the maximum mutual information (MMI) and the maximum likelihood of tests and Levenberg–Marquardt algorithm (LM) used for solving nonlinear problems of least squares are based on first derivatives. Newton-type methods use a second derivative. A genetic algorithm or simulated annealing by perturbation avoids local minima. Choice of Weight

When a few or a massive data span shows heterogeneity in the variance, the scheme of weight is employed. Weights are applied for observations for obtaining parameter estimates. With the parameter estimates, a model that can be fitted with low to high

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values is chosen. In many cases, weight is employed for each observation of reciprocal of the predicted data of the expected squares of the data. However, weighted residuals for analysis should be comparable to all data. Assessing Goodness of Fit

The goodness of fit is the arrangement between a model and its data and checked by the diagnostic plot. Residual plots are used as diagnostic tools. Residual data are required to distribute randomly around the predicted curve. Distinction Between the Models

Akaike information criterion, Schwarz criterion, and F-test are often used to determine differentiation between the models. F-test determines the variability between the mean values of the data groups of the models to provide judgments about the means. F-distribution identifies the model that best fits a data set under the null hypothesis (Goldstein 1964). F-distribution points out the probability of influence of additional parameters on the sum of the least squares. The least-squares method is a standard solution of regression analysis. The process minimizes the sum of the squares of the residual values of every single equation of over-determined systems. The Akaike information criterion is a method for selecting a model. A lower score as per the model suggests a better-fit model. The Akaike information criterion assesses the comparative loss of information by a given model (Akaike 1974). The most negligible loss of data presents the best model. The Schwarz criterion is also used for a selection of a model from a set of models. It is also called the Bayesian information criterion. The model selection is based on the score of the Bayesian information criterion. The lower is the score, the better is the model. 8.1.7.4.2 Physiological Pharmacokinetic Model This model represents physiology-based, and anatomy-based biochemical processes in an organism, within the tissue/organ compartments. Allometric scaling of dose, drug uptake, and drug clearance are used as assumptions. In this model, body compartments are connected to the flow network. Mass balance equations of dose, drug amount, or concentration as a function of time are built up. It is a more complex model than the compartment model and deals with a large number of parameters. 8.1.7.4.3 Pharmacokinetic/Pharmacodynamic Model The time-dependent behavior of drugs and their metabolites in pharmacokinetics and pharmacodynamics shows the pharmacological effects based on the concentration of the drug. Observations of pharmacokinetics and pharmacodynamics are correlated using the pharmacokinetic/pharmacodynamic model. The pharmacokinetic/pharmacodynamic model describes pharmacological and toxic effects of a drug after its administration during the time course of its action in the body. Repeated individual pharmacodynamic measurements may obtain reliable data related to pharmacodynamics in a preclinical investigation or a clinical trial.

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8.1.7.4.4 Population-Based Pharmacokinetic Data Analysis It is a standard two-stage approach. First, fitting of the available pharmacokinetic data is done in a pharmacokinetic model. Second, collected individual pharmacokinetic data related to pharmacokinetic parameters are subjected to statistical analysis. There may be an overestimation of individual variance, and it is not suitable to apply to fit in the individual model when data are too scattered or scanty. 8.1.7.4.5 Nonlinear Mixed Effect Modeling Approach Pharmacokinetic parameters and individual variance parameters are estimated to obtain mixed effects. Individual residual parameters are obtained as population parameters. Age, gender, body weight, creatinine clearance, and accompanying diseases are included in a population pharmacokinetic data analysis as relevant covariates. The quantitative relationship between covariates and pharmacokinetic parameters may be used to predict individual pharmacokinetics before obtaining the actual individual data. The covariate relationship model can identify a subgroup population who may require specific dosage recommendations.

8.1.7.5 Pharmacokinetic Report Pharmacokinetic data presented in a summarized form is called a pharmacokinetic report. They should include plots of observed data and predicted data (residual data) with the standard. However, mean concentration plots should not be ignored. An integrated report of pharmacokinetic and pharmacodynamic (clinical reports) findings should be presented. The integrated report may include an additionally produced supplement. Clinical reports and pharmacokinetic reports should interlink with each other.

References Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control 19: 716–723 Batchelor HK, Marriott JF (2015) Paediatric pharmacokinetics: key considerations. Br J Clin Pharmacol 79:395–404 Elias GP, Antoniali C, Mariano RC (2005) Comparative study of rules employed for calculation of pediatric drug dosage. J Appl Oral Sci 13:114–119 Fesce R, Fumagalli G (2015) Control of drug plasma concentration. In: Clementi F, Fumagalli G (eds) General and molecular pharmacology: principles of drug action. Wiley, New York, pp 74–89 Goldstein A (1964) Biostatistics—an introductory text. Macmillan, New York Klotz U (2009) Pharmacokinetics and drug metabolism in the elderly. Drug Metab Rev 41:67–76 McLean AJ, Le Couteur DG (2004) Aging biology and geriatric clinical pharmacology. Pharmacol Rev 56:163–184

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Pharmacokinetic Sample Collection and Processing

9.1

Pharmacokinetic Sample Collection and Processing for Preclinical and Clinical Experiments

9.1.1

Pharmacokinetic Sampling

Pharmacokinetic sampling is the process of selection and collection of specimens for pharmacokinetic analysis. The process begins at the point of collection of specimens and continues till the transfer of it to an analytical laboratory. The collection of a specimen is called primary sampling, and its transfer to the analytical laboratory till before processing it for analysis (including storage, if any) is called secondary sampling. Proper precaution and preservation methods should be adopted during primary and secondary sampling processes to obtain reliable and accurate data by minimizing changes between the collection of samples and their analysis. The changes of physical properties such as volatility, diffusion, adsorption, and chemical properties that include oxidation, thermal degradation, photodegradation, and microbial decomposition are commonly experienced with human specimens for analysis. Thus, sample-specific proper containers, freezing of samples, incorporation of stabilizers such as antioxidant, anticoagulant, and preservatives such as antimicrobial agents are used in practice. The sampling of different important pharmacokinetic specimens is mentioned below.

9.1.2

Blood Sampling and Right Practices in Phlebotomy

Blood is the maximally drawn specimen that is frequently used for various diagnosis and analysis purposes. At large, for quick sampling for a small quantity of blood and also for fast analysis with commercially available kits (e.g., blood glucose determination kit) and instruments (e.g., glucometer), capillary blood is probably the most frequently collected biological sample by puncturing fingertips, for analysis. # The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B. Mukherjee, Pharmacokinetics: Basics to Applications, https://doi.org/10.1007/978-981-16-8950-5_9

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However, occasionally, in little children, a small blood sample is also collected from heals, large toes, and even from ear lobules. Usually, the tip of the fourth finger on the left hand is punctured with a sterile needle to collect the blood. But if any reason, it is not possible, blood may be collected from any other fingers. Fingertip is first cleaned by ethanol and then by ether. The finger should be firmly but gently held and squeezed slightly upwards to the intended site of puncture. Generally, the left side of the fingertip-median line and a little below the nail is considered to be the site of puncture. A blood sample may be collected using a micropipette tip (sterile) that is often pre-treated with anticoagulant (chemicals that prevent blood clotting, e.g., heparin, sodium citrate). After the blood collection, cotton dipped in antiseptic liquid (e.g., tincture iodine) is placed on the wound area of the fingertip to stop bleeding and prevent any microbial growth at the site of the wound. Phlebotomy is the method of collecting blood specimens, usually from the vein by puncture/incision for diagnosis and analysis. Safety and proper health care for the patients, the health workers, and laboratory staff are very important in phlebotomy. Patient cooperation is also essentially required, and the patients/volunteers should be clearly informed about the phlebotomy procedure. An adequate supply of protective substances/equipment such as materials for maintaining hand hygiene (hand sanitizer, or alcoholic hand rub, soap, and water), gloves, masks, disposable (single-use) syringes, needles, or lancing devices with sterile needle and syringe, personal protective equipment (PPE), and the appropriate use of them can help health workers to avert many fatal infections such as COVID-19, SERS, MERS, HIV, hepatitis B, hepatitis C, and many more. In many cases, preimmunization of the health workers is necessary. Care is particularly more important for pediatric or neonatal patients. A good size, straight, and clearly visible vein is first located at the antecubital fossa (elbow pit of the forearm). The area is then cleaned with 70% alcohol for 20–30 s and allowed to dry completely. A tourniquet is applied. A tourniquet is a medical device to apply pressure to a limb to limit blood flow without stopping it. A correct size needle is inserted at the vein (venepuncture) (Fig. 9.1), avoiding any diverting zones of a vein, as that can lead to hematoma formation. Hematoma is the accumulation of blood under the skin from an injury in blood vessels in the surrounding tissues. In hospitalized patients, prior to a cannula attachment of intravenous fluids, blood may be withdrawn from the vein by insertion of an indwelling venous device. But once the intravenous fluid is administered, blood should not be collected from any peripheral vein access site, as that can give erroneous findings due to the presence of intravenous fluid. For the collection of blood from the limb (hand), a tourniquet is applied on the arm above the elbow. It limits the blood flow but does not stop it. A thumb is placed under the venepuncture site, and the needle of the injection syringe is introduced into the vein rapidly at an angle 30
or less, and blood is withdrawn. The patient is asked to form a fist during the time for the prominent visualization of the vein. Once sufficient blood is collected, the tourniquet is released prior to the withdrawal of the needle. Some guidelines recommend removing a tourniquet as soon as blood flow becomes normal (it should be done within 2 min). After the completion of the

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Fig. 9.1 Blood draw from antecubital fossa (elbow pit of the forearm)

withdrawal of blood, the needle is gently withdrawn, and gentle pressure is applied at the site with a dry cotton ball. The patient is suggested to hold the cotton ball in place, with the arm extended and raised, as bending the arm may result in a hematoma.

9.1.2.1 Processing of Blood Samples Blood samples thus collected are taken in the collection tubes and each tube is then closed with a stopper. After collecting the blood sample and depending on the purpose of analysis, pre-treatment is often required. For the analysis of whole blood, including the cells and blood components, and blood serum analysis, a blood sample immediately after collection from a vein is taken in the collection tube and mixed with a suitable anticoagulant (Adcock et al. 1997; Durham et al. 1995; Erwa et al. 1998; Goosens et al. 1991; Marlar et al. 2006) by gentle inversion (10–12 times) to stop blood clotting. Normally, 0.102–0.129 mol/L or 3.2–3.8% w/v sodium citrate solution or 0.13 M sodium oxalate solution is used. Sodium citrate or sodium oxalate binds with calcium ions of blood and prevents blood clotting. Heparin (2 mg/10 mL of blood) is also used to prevent blood clotting. Heparin interacts with blood proteins involved in the blood coagulation process and prevents blood clotting. For the collection of blood serum, blood is collected in a clean and dry collection tube (preferably glass tube) and refrigerated overnight for clotting. The straw-yellow colored blood serum remains above the clotted blood in the tube. The blood serum is carefully collected by micropipette for further analysis. The rupture of red blood cells releases hemoglobin from the red blood cells into the blood plasma. This is called hemolysis. The use of incorrect needle size, inefficient mixing in the collection

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tube, excessive suction, etc. are some of the predominant causes of hemolysis in the collected blood specimen. Upon hemolysis, certain analytes such as aspartate aminotransferase (AST) and lactate dehydrogenase (LDH) present more in the erythrocytes release in the plasma. The plasma that contains less AST, LDH than in erythrocytes, would now contain them in increased concentrations, and such plasma sample interferes with analysis and gives incorrect results. Hemolysis often shows false enhanced concentrations of other analytes such as glucose, calcium, and total bilirubin. It also interferes with the spectrophotometric method of analysis. Proper methodical collection of the blood specimen, the use of anticoagulants, and proper mixing of the blood sample immediately after collection by gentle inversion could be the best way to prevent such hemolysis. Proper labeling of the sample after the collection of blood is an absolute need. The container of the specimen should be labeled in the presence of the patient/volunteer. Then they are stored appropriately and transported to the place of analysis. Any blood or body fluid spillage should be cleaned as per the medical guidance with the necessary protection. Collected specimens should be delivered promptly to the laboratory for analysis.

9.1.2.2 Blood Collections from Small Animals The body parts from where blood is usually collected in small animals are given in Table 9.1. Animals are pre-anesthetized locally or centrally using isoflurane, xylazine, ketamine, and halothane using various routes such as inhalation, intramuscular injection, and intraperitoneal injection (Parasuraman et al. 2010).

Table 9.1 Physiological zones of blood collection from small animals Body parts/methods Tail vein Tail snip Orbital sinus Jugular vein Temporary cannula Blood vessel cannulation Tarsal vein Marginal ear vein/artery Saphenous vein Dorsal pedal vein Cardiac puncture (terminal experiment) Retro-orbital plexus (terminal experiment, nowadays animal ethics regulation does not permit it in most of the countries) Posterior vena cava

Animals Rats, mice Mice Rats, mice Rats, mice Rats, mice Rats, guinea pigs, ferrets Guinea pigs Rabbits Rats, mice, guinea pigs Rats, mice Rats, mice, guinea pigs, rabbits, ferrets Rats, mice Rats, mice

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Urine Specimen

Urine is probably one of the most frequently analyzed clinical samples. Urine is collected in a glass or polyethylene flask, usually by portion, and stored in a refrigerator until used. Commonly, little interference of the results of urine analysis is observed by bacterial contaminants. However, to prevent microbial growth in urine, the addition of preservatives (Nicar et al. 1987) such as toluene, boric acid, oxalic acid, acetic acid, hydrochloric acid, tartaric acid, thymol, and formaldehyde is not unknown. For the analysis of steroids or steroidal metabolites, urine is pre-treated with 1% (v/v) concentrated hydrochloric acid. Fresh urine should be always used for the determination of enzyme(s) from urine specimens. Collecting a urine sample from the experimental (laboratory) small animals such as mice, rats, rabbits, and guinea pigs, the animal is placed on a glass separatory funnel (Fig. 9.2) for 12/24 h. The animal’s urine passes through a sintered glass funnel. At its orifice, a glass nail is fitted to prevent fecal contamination of urine. Ultimately, urine passes through the funnel to a small container that collects urine.

9.1.4

Other Tissue Samples

Organ/tissue samples (biopsy samples, i.e., tissue taken from a living body) from a patient or volunteer are collected during the surgical intervention by scalpel, by needle biopsy, or by bioptome (a specialized cutting instrument) during endoscopy (endoscopy is a medical procedure to investigate the interior of a hollow organ or
body cavity by inserting a long and narrow tube device) and stored at 20 C until used. The tissue samples are thawed (bring the frozen substance to a nonfrozen soft/ melted state by keeping it usually at room temperature) and processed as per biochemical/clinical investigation requirements. Tissues may then be processed for histological or immunohistochemical investigations by following or developing appropriate methods. Biochemical reactions often involve and locate in cellular organelles such as cell membrane, mitochondria, endoplasmic reticulum. Hence, for such analysis, tissues are subjected to disintegration using various techniques such as tissue homogenization using a tissue homogenizer, by ultrasonication (for cells), etc. In a homogenizer, small pieces of the pre-weighed tissue sample are
triturated with a cold (at 4 C) solution of sucrose (0.25 M) by rotating pestle with speed controlled by an electrical motor. The collected tissue homogenate undergoes cold centrifugation (the supernatant gives cytosolic fraction) and ultracentrifugation of a cytosolic fraction (the precipitate gives microsomal fraction) to separate different cellular organelles. The required portion(s) of the homogenate thus obtained is processed for biochemical observation (e.g., enzyme activity, protein separation) using specific biochemical methods (Sarkar et al. 1994, 1995). For histological and histochemical analysis, the collected tissue after dissection is immediately chopped into small pieces. They may be snap-frozen in liquid nitrogen (195.8
C) or a mixture of liquid nitrogen and isopentane (160
C), and the tissue samples may be stored at 80
C deep freezer until further use. An effective and efficient tissue

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Fig. 9.2 Urine collection funnel system for laboratory animals

preservation process makes various components such as antigens, mRNA, and many low-quantity proteins well-preserved. Tissue samples may be fixed in 4% paraformaldehyde (10% formalin) in 1 phosphate buffer or 1 phosphate buffer saline. However, there are other fixative solutions, such as Bouin’s solution, Carnoy’s solution, Methacarn, Gendre’s solution, etc., that are also available. Paraformaldehyde and osmium tetroxide are used during vapor-fix freeze-drying of tissues. It is required during histological/biological electron microscopy. Fixation is done to protect the tissue from autolysis or putrefaction. Fixation also prevents

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morphological, chemical, and physical alterations of the tissues and eases the diagnosis. For preclinical experiments, animal tissues are also preserved similarly. Tissues (cryopreserved and/or undergone fixation) are then sectioned by cryo-cut microtome or microtome, and necessary test-specific processing is conducted for analysis.

9.1.5

Other Tissue Fluids

For enzyme assay from biological fluids, no pre-treatment is done. However, the presence of enzyme(s) if interferes the results of analysis by catalyzing the compound(s) of analysis, the biological fluid sample immediately after collection is pre-treated with enzyme inhibitor or the enzyme, and proteins are precipitated with a protein precipitator such as trichloroacetic acid, perchloric acid, barium hydroxide, and zinc sulfate. A sample is centrifuged at 3000–5000 rpm for 10–15 min in a cold
centrifuge (at 4 C) to precipitate the large protein molecules. The carefully collected supernatant is processed for further biochemical analysis using the available process. The microsomal precipitate may be collected from the cytosolic fraction by centri
fugation at 100,000  g for 2 h at 4 C, using an ultracentrifuge. For preclinical experiments, a similar procedure is followed. Various storage temperatures of biological specimens are given in Table 9.2.

9.1.6

Processing of Fecal Samples

Fresh human fecal samples are collected at room temperature, and within 2 h, the samples should be stored at 80
C until further use for analysis. After thawing the sample, a portion of the sample is taken, water is added to it, and the mixture is then homogenized (Xu et al. 2015). The required numbers of aliquots are taken at this stage before vacuum drying for extraction of drugs or metabolites using different solvents. The aliquots are taken for vacuum-drying by SpeedVac. The dry sample is then used for solvent extraction. Various solvents such as water, methanol, and Table 9.2 Storage temperature of biological specimens Storing method Deep-freezing Frozen Refrigerated Cold room Cool Room temperature Ambient temperature Controlled room temperature Warm

Temperature 80
C 20
C 2–8
C Not exceeding 8
C 8–15
C 15–25
C 20–25
C 20–25
C (in special cases may vary between 15 and 30
C) 30–40
C

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acetonitrile are used. The solvent extraction (using 100 μL solvent in each case) is done by following the sequence of ultrasonication (10 min), vortex-mixing (2 min), and centrifugation at 18,000 g for 15 min at 4
C (Xu et al. 2015). The supernatants are collected and combined. The extract blend is dried using a SpeedVac, and the dry sample is stored at 20
C until further use. The sample is further processed and analyzed by LC-MS/MS method using standard procedure.

9.1.7

Extraction of Drug/Drug Metabolite from Biological Samples

Quantitative measurement of drug/drug metabolite is required for various investigations related to pharmacokinetics, bioequivalence studies, forensic studies, illicit drug consumption issues, drug-/drug-metabolite-associated toxicological issues, doping of sportspersons, and many drug-related environment issues. For quantitative estimation of drug/drug metabolite from biological samples, it is essential to extract drug/drug metabolite from biological matrices such as blood, serum, plasma, urine, feces, other biological fluids (e.g., saliva), and various tissues. Sophisticated analytical instrumentation and methodologies are capable of estimating drug/drug metabolite at a very low concentration level with high accuracy. However, complete extraction of drug/drug metabolite is essential to get reliable and accurate data of bioanalysis. The major purposes of extracting drug/metabolite from biological samples are isolation and purification of drug/metabolite (the analyte) from interfering materials, dissolution of the analyte in a suitable solvent/solvent mixture, and enrichment of its concentration to a detectable limit for its appropriate measurement. Often and on, sample extraction and preparation consume a significant portion of the total time required for an entire analytical estimation method. Extraction may be defined as a process of removal of soluble materials from an insoluble residue using a suitable solvent/solvent mixture, and an extract is a concentrated preparation of soluble material(s) with solvent obtained upon extraction. Evaporation of all or nearly all of the solvent is often carried out to obtain the residual mass in a solid or powder form. Extraction of drug/drug metabolite from biological samples is generally based on two types of extraction principles—solid-liquid mass transfer process and liquidliquid mass transfer process. The solid-liquid mass transfer process mainly involves the dissolution of the solid soluble principle with a liquid solvent system. When a solid material comes into contact with the solvent, a thin film of a saturated layer from the solid is formed on the surface of the solid. This is known as the boundary layer of the solute. Transport of molecules from a solid into the bulk of the liquid (solvent) takes place in two steps—molecules from the solid move into and through the boundary layer by molecular diffusion phenomenon. Once molecules cross the boundary layer, they diffuse into the bulk of the solvent by natural convention. Natural convection may be varied by changing the temperature, agitation, and density of the solid material. The second type of diffusion (from the boundary layer to the bulk of the liquid) is known

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as Eddy’s diffusion. However, the passage of molecules through the boundary layer is a rate-limiting step that controls the rate of dissolution. Mass transfer depends on the concentration gradient of the solid-liquid system. The famous Noyes-Whitney equation can well explain the phenomenon that describes the dissolution of a solid in a suitable solvent system. The equation is given below. dc KDS ¼ ðcs  cÞ dt Vl where dc dt is the rate of dissolution, c is the concentration of solid (drug/drug metabolite) in the bulk of the solution, cs is the concentration of solid in the saturated boundary layer, l is the thickness of the diffusion layer, S is the surface area of un-dissolved solid, V is the volume of solution, D is the diffusion coefficient, and K is the proportionality constant. In general, if the volume of the solution is kept constant, the values of K, l and D also remain unchanged in the case of a particular solute. In that case, by 0 considering the factor KD Vl is a constant, K , the equation may be written as mentioned below. 0 0 dc dt ¼ ðcs  cÞK S ¼ K SΔc, where Δc is the concentration gradient. Hence, the drug dissolution rate depends on the surface area and concentration gradient of the solute (here drug). In the liquid-liquid extraction process, the mass transfer takes place between the two immiscible liquids. In this phenomenon, mass transfer of ingredients takes place from one liquid to the other (say, second liquid) due to the higher to lower concentration gradient and higher affinity of the solute to the second liquid. It is described by the equation of partition coefficient (P), which is equal to the ratio of the concentrations (c1 and c2) of solute in the two immiscible liquid solvents. log P ¼ log

c1 c2

Nernst law of distribution states that the ratio of distribution concentration of any neutral species between two immiscible solvents remains constant. The equation is represented as KD ¼

c1 , c2

where KD is distribution constant, c1 and c2 represent concentrations of solute in the two immiscible liquid solvents.

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References Adcock D, Kressin D, Martar RA (1997) Effects of 3.2% vs. 3.8% citrate concentration on routine coagulation testing. Am J Clin Pathol 107:105–110 Durham BH, Robinson J, Fraser WD (1995) Differences in the stability of intact osteocalcin in serum, lithium heparin plasma and EDTA plasma. Ann Clin Biochem 32:422–423 Erwa W, Bauer FR, Etschwaiger R, Steiner V, Scott CS, Sedlmayr P (1998) Analysis of aged samples with the Abott CD 400 hematology analyzer. Eur J Lab Med 6:4–15 Goosens W, van Duppen V, Verwilghen RL (1991) K2- or K3-EDTA: the anticoagulant of choice in routine haematology? Clin Lab Haematol 13:291–295 Marlar RA, Potts RM, Marlar AA (2006) Effect on routine and special coagulation testing values of citrate anticoagulant adjustment in patients with high hematocrit values. Am J Clin Pathol 126: 400–405 Nicar MJ, Hsu MC, Johnson T, Pak CY (1987) The preservation of urine samples for determination of renal stone risk factors. Lab Med 18:382–384 Parasuraman S, Raveendran R, Kesavan R (2010) Blood sample collection in small laboratory animals. J Pharmacol Pharmacother 1:87–93 Sarkar A, Mukherjee B, Chatterjee M (1994) Inhibitory effect of β-carotene on chronic 2-acetylaminofluorene induced hepatocarcinogenesis in rat: reflection in hepatic drug metabolism. Carcinogenesis 15:1055–1060 Sarkar A, Mukherjee B, Chatterjee M (1995) Inhibition of 30 -methyl-4-dimethylaminoazobenzeneinduced hepatocarcinogenesis in rat by dietary β-carotene: changes in hepatic anti-oxidant defense enzyme levels. Int J Cancer 61:799–805 Xu W, Chen D, Wang N, Zhang T, Zhou R, Huan T, Lu Y, Su X, Xie Q, Li L, Li L (2015) Development of high-performance chemical isotope labeling LC–MS for profiling the human fecal metabolome. Anal Chem 87:829–836

Important Bioanalytical Instrumental Techniques in Pharmacokinetics

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Important Bioanalytical Instrumental Techniques

Bioanalytical instrumental techniques are employed to quantitatively estimate drugs and metabolites from various biological samples, including blood, serum, plasma, saliva, urine, different tissues, fecal samples, etc. The data obtained are applied for pharmacokinetic computation, graphical presentation, and analysis. Few gold standard techniques used in bioanalysis of drugs and metabolites from the biological samples are chromatography amalgamated with mass spectrometry, liquid chromatography coupled with tandem mass spectroscopy, and high-performance liquid chromatography.

10.1.1 Liquid Chromatography with Tandem Mass Spectroscopy (LC-MS/MS) Liquid chromatography with tandem mass spectroscopy (LC-MS/MS) is a very sensitive and significant bioanalytical technique used to detect and quantify trace amounts of a drug in blood and other body fluids. LC-MS/MS also detects residual chemicals as confirmatory identification, which can be used to identify and detect contaminants and adulterants in food and pharmaceuticals. Detection sensitivity level varies from parts per million to many parts per billion. For the last one and a half decades, it has been used as a very sophisticated method to accurately determine a very small quantity of drugs in biological samples. In simple language, LC-MS/MS is a combination of liquid chromatography (LC) and mass spectrometry (MS) (Fig. 10.1). More precisely, one HPLC (high-pressure liquid chromatography) system is combined with two mass spectrometry detectors.

# The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B. Mukherjee, Pharmacokinetics: Basics to Applications, https://doi.org/10.1007/978-981-16-8950-5_10

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Fig. 10.1 LC-MS/MS system along with its various components

Fig. 10.2 A quadrupole assembly

10.1.1.1 Working Principle LC-MS/MS separates a mixture of compounds based on their physicochemical properties. The method identifies the compounds with the LC peaks and detects them by their mass spectra. Electrospray ionization (ESI) provides the interface between liquid chromatography and mass spectrometry. Parent ions and the fragments (ions) produced by ionization of the analyte molecules only can pass through the quadrupoles (Fig. 10.2) based on their “mass to charge ratio” (m/z) in the MS/MS compartment to the detector. Different components are first separated and concentrated by chromatography, and then they reach a mass spectrometer. There

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are four metallic rods in a quadrupole assembly. Each metallic rod is an electric charge-related or gravitational mass-related configuration that can provide a quadrupole moment. The ionized parent molecules and their fragments are then selected, isolated, and sent for measurement by a mass spectral detector. The repeated separations, obtainment of concentrated analytes, ionization, selective fragmentation, selective isolation, and highly accurate detection have made the LC-MS/MS a highly sensitive analytical instrument. The word tandem means here two pieces of equipment (here MS) that work together to achieve a result. The first mass spectrometer separates the ions by their m/z. The second mass spectrometer separates the fragments by their m/z and sends them to a detector to finally detect them.

10.1.1.2 Components and Their Applications Various components of LC-MS/MS and their functions are mentioned below. 10.1.1.2.1 Liquid Chromatography Part Liquid chromatography is a physical separation technique that employs the distribution of the components of a mixture on a polar stationary phase from a nonpolar mobile phase to isolate them by mass transfer. Liquid chromatography separates components from a mixture based on mass transfer also on a nonpolar stationary phase from a polar mobile phase. The types of liquid chromatography are partition chromatography, adsorption chromatography, affinity chromatography, ion-exchange chromatography, and molecular sieve chromatography. In addition, the widely applied reverse-phase-partition chromatography technique operates with a nonpolar (hydrophobic) stationary phase and a polar (aqueous) mobile phase. Mobile phase: Commonly utilized mobile phases are polar solvents or solvent mixtures, including water, acetonitrile, methanol, isopropanol, and their aqueous mixture or mixture with various buffers. Stationary phase and separating column: Solid granulated polymers or silica particles (average diameter 5 μm) produce the stationary phases by modifying the surface of regular- or irregular-shaped silica flakes with long-chain alkyl moieties such as n-octadecyl as seen in C18 (molecule has 18 carbon atoms) column. After the injection of the sample, the mobile phase (solvent or solvent mixture) transports the sample to the column packed with the stationary phase. Typically, a small volume of sample (20 μL) is injected into the mobile phase flow. As stated earlier, LC-MS/MS is basically HPLC with two mass spectrometry detectors. A pump is operated to generate high pressure (as in HPLC, instead of the gravitational pull for the solvent flow as seen in traditional LC) to carry the mobile phase to the column. Thus, it avoids any pressure drop in the column, expedites the separation process, and reduces the duration of separation. The components of the samples adsorb differentially on a stationary phase based on their relative affinity to the stationary phase. The adsorbed compounds are separated one by one based on their respective relative affinity to the stationary phase by passing a solvent (mobile phase) through the column. Compound with the most affinity needs the maximum time to separate.

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10.1.1.2.2 Sample Injector The sample injector injects a tiny (20 μL) sample in the mobile phase solvent stream passing with high pressure. Hence, the injector should withstand such high pressure. Most systems use solvent injection valves. The sample can be injected at the loop of the valve by autosampler or manually by microinjection. 10.1.1.2.3 Pump Device A pump is operated to generate a high pressure to carry the sample with the mobile phase stream to the separation column. Some commonly used pumps for the purposes are the following: Direct Gas-Pressure System

It is cheap and dependable. But the change of solvent is tough. Syringe Pump

It provides flow with a constant high pressure without any pulsing. Syringe pumps are precise and accurate, and have a large capacity. Pneumatic Intensifier

It provides flow with constant high pressure but often causes pulsing due to block that causes the pressure drop. Reciprocating Pump

It provides flow with constant high pressure but often causes pulsing. It is cheap. 10.1.1.2.4 Interface The significant contrast of the interface is that LC requires nearly 1000 times gas flow with high pressure (1 L/min) for separation of sample components in the liquid phase at an ambient temperature without a limitation of mass range. On the other hand, mass spectroscopy needs a vacuum with limited access of gas flow, 1 mL/min, at a high temperature, with a limitation of mass range. Besides, inorganic buffers are used for LC, and volatile buffers are the choice for MS. Thus, the LC-MS interface is very crucial and significant in the LC-MS and LC-MS/MS systems. The atmospheric pressure ionization procedures are used for the LC-MS interface. The atmosphericpressure ionization systems based on electrospray ionization, chemical ionization, and photoionization are commonly applied. 10.1.1.2.5 Atmospheric Pressure Ionization Technique In a modern LC-MS/MS system, one can employ the atmospheric pressure ionization technique or electrospray ionization approach (Cole 1997) and even use unmodified high-pressure without the flow-rate limitation. Atmospheric pressure ionization technique operates soft ionization processes and subsequent evaporation. Nebulization, ionization, desolvation (solvent removal), and ion evaporation are four distinctive processes involved in the technique. One can analyze small to large polar to nonpolar target compounds applying the atmospheric pressure ionization process.

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The atmospheric pressure ionization technique can also identify several volatile and nonvolatile compounds using precise molecular weight and fragmentation particulars. 10.1.1.2.6 Electrospray Ionization Collision-Induced Dissociation MS/MS system breaks ions that collide with other molecules into fragments by collision-induced dissociation process. The electrospray ionization can indeed produce collision-induced dissociation spectra. One can alter the voltage to achieve various fragments of the parent ions. The higher is the voltage, the more is the fragmentations of the parent ions. The application of voltage on lenses escorts the ions into the vacuum. One can select the ions to transmit into the vacuum by altering the voltage on the lenses. 10.1.1.2.7 MS/MS System Quadrupole Assembly

A quadrupole assembly is a set of four parallel metallic rods equally spaced from the central axis. In a quadrupole assembly, the rods are linked to radiofrequency (RF) and direct current (DC) power generators. A fluctuating electrical field is applied with RF and DC voltage. The same voltage is applied on the opposite rods. The fluctuating electrical field with RF and DC voltage builds up the trajectory for ions. The path of ions passing through the field is optimized to m/z by adjusting RF and DC potentials. It allows only the ions of interest with the desired m/z to form a stable trajectory and transmit to the mass spectrometer. The ions with the larger or smaller m/z strike the rods and get neutralized and removed (Fig. 10.3). The quadrupole setting is either static or scan type. The static setting allows ions of interest to transmit to the mass spectrometer, and the method has a high level of sensitivity. Static mode is commonly used for LCMS-MS analysis. The static setting is also known as the selected ion recording (SIR)

Fig. 10.3 Passage of ion through a quadrupole assembly during ion filtering. Only the selected ions pass through to the mass analyzer

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or the selected ion monitoring (SIM), as the specific ion type passes to the detector of the mass spectrometer for detection. In the mass scan mode, the RF amplitude and the DC voltage are scanned, with a constant ratio, so that ions with increasing m/z can transmit to the mass spectrometer detector. Demerits of the method are the comparatively much shorter duration of analysis, and only a small analyte reaches the detector following each scan cycle. In tandem equipment, two quadrupoles are used. Mass Spectra Detectors

There are many different mass spectra detectors, such as Faraday cup detector, array detector, Daly or post acceleration detector, microchannel plate detector, etc. Each has specific advantages and disadvantages. The desired properties of a detector are a fast response to the signal, low signal-to-noise ratio, high amplification, adequate signal collection efficiency, a wide dynamic range, equi-response to all masses, durable, and low cost. However, they generally work with the same basic principle. First, the mass of the target molecules is converted into gas-phase ions. Then, the resultant flux of ionic charge undergoes conversion into a proportional electrical current detected by the detector and read by the machine data system.

10.1.1.3 LC-MS/MS Method Optimization The steps of optimization of LC-MS/MS methods are given below. 10.1.1.3.1 Standard Solution A standard solution of a 100% pure compound (free from any impurities) is used for optimization. First, we prepare a stock solution of the standard compound and dilute it at a very low concentration range (50 ppb–5 ppm) depending on the instrumental sensitivity using a suitable instrument-friendly solvent or solvent mixture. Further, the solvent or solvent mixture used should not harm the device. 10.1.1.3.2 Optimization of MS/MS Operation and Parameters The standard (solution of a pure target compound of known concentration) or the test sample (compound solution of unknown concentrations) is passed through the LC compartment in the instrument. However, method optimization of a compound begins with obtaining the spectrum signal of the target compound from the MS/MS operation and optimizing its MS/MS parameters (Hoffmann and Stroobant 2007). For receiving the correct signal, the standard solution of the pure target compound should be passed through the LC compartment. The compound is then broken into fragments by ionization and collision in the MS/MS compartment. The MS/MS detector then detects the fragments of the compound. Hence, the determination of energy required for compound fragmentation for detection is a significant optimization step in LC-MS/MS. 10.1.1.3.3 Energy Optimization for Parent Ion Ionization After the arrival of the compound in the first mass compartment, electrospray ionization or atmospheric pressure chemical ionization turns the compound into an

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ion (also known as parent ion). Knowledge of the mass of the ion is essential for the optimization of parent ion parameters as the orifice voltage is optimized by scanning a range of voltages based on the mass of the parent ion. The optimized voltage provides to the parent ion to give the maximum response. Sometimes, ion that forms an adduct with component additives of the solvent provides an inadequate response. In those cases, the orifice voltage is optimized based on the mass of the adduct formed.

10.1.1.3.4 Collision Energy Optimization for the Fragmented Ions The parent ions form different charged fragments (daughter ions) with different spectra with collision energy ranges. By overlaying the spectra, the most abundant daughter ions/fragments are determined. The characteristic distribution of fragments (with the maximum response) from parent ions can be monitored by selecting a multiple reaction monitor by optimizing collision energy. The multiple reaction monitor selects at least two fragment pairs. However, more pairs ensure accuracy. One pair of the two fragment pairs of a compound chosen based on a multiple reaction monitor is then quantified, and the other pair is used for confirmation. The ratio of the two fragment ion pairs is for the confirmation of the standard compound. The test sample must contain the same fragment pairs. The ratio of the fragment pairs of the test sample selected by multiple reaction monitors should be the same as the standard compound for confirmation and optimization.

10.1.1.3.5 Liquid Chromatographic Optimization In the liquid chromatographic (LC) optimization, the selection of a suitable column that acts as a stationary phase and the appropriate solvent(s) such as acetonitrile, methanol, water, or their mixture, commonly called the mobile phase, is the primary criterion to optimize LC separation. The various LC conditions such as the flow rate, column temperature, particle size of the stationary phase, and the mobile phase gradient are also optimized to obtain a clear resolved peak that ensures accurate identification and correct quantification of the compound. For improving resolution, various additives such as ammonium salts or acids are also used. A broadened peak, distorted peak, or tailing peak should be avoided. The merging of several peaks due to a very rapid flow rate can produce broadened peaks.

10.1.1.4 Method Confirmation There are various available recommendations for method optimizations in LC-MS/ MS systems. One easy method is mentioned here. The method optimization for a target compound can be ensured by verifying the standard samples of various concentrations against a calibration curve. If the correct response with a clear resolved peak in each case is obtained, the method is suitable for estimating the target compound from the test samples.

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10.1.2 High-Performance Liquid Chromatography (HPLC) High-performance liquid chromatography (HPLC), also popular as high-pressure liquid chromatography, is an analytical technique applied for separating, identifying, and quantifying components from a mixture. It is a maximally used chromatography technique globally.

10.1.2.1 Working Principle HPLC is an upgraded and technologically advanced version of column chromatography. The mobile phase stream travels with high pressure and carries the injected sample through the separation column containing granulated solid adsorbent called the stationary phase. While passing through the column, the sample compounds separate by physicochemical interaction with the stationary phase based on their different affinities to the phase. The different affinities that depend on the chemical structures of the different compounds determine their duration of stay in the column (Lough and Wainer 1995). A mobile phase (also the eluent here) elutes the sample compounds at different intervals, and thus, the separation of the sample takes place. The separated sample components leave the column and reach the detector that recognizes and distinguishes the signals on absorbed-light/refractive index/conductivity changes/size distribution of the separated components. Finally, computer software converts the signal to a chromatogram. 10.1.2.2 Types of HPLC There are four most common HPLC varieties: normal-phase HPLC, reversed-phase HPLC, size-exclusion HPLC, and ion-exchange HPLC. 10.1.2.2.1 Normal-Phase HPLC Small silica particles in the column as polar stationary phase and a nonpolar solvent (e.g., chloroform) as mobile phase are used in the normal-phase HPLC. Nonpolar compounds in the mixture travel rapidly with high pressure through the column, whereas polar compounds stay longer to the polar silica. 10.1.2.2.2 Reversed-Phase HPLC Reversed-phase HPLC is the most common and widely applicable HPLC method. Polar silica particles are provided nonpolar character by modifying the surface with a long (8–18 C) alkyl chain. A polar solvent (e.g., water-methanol mixture) that acts as a mobile phase is used. Hence, polar compounds travel rapidly than nonpolar compounds for their high affinity toward polar solvents. Nonpolar compounds retain longer with the nonpolar stationary phase forming weak bonds such as van der Waals force of attraction, dipole moment, weak hydrogen bonds, etc. 10.1.2.2.3 Size Exclusion Chromatography Size exclusion chromatography (SEC), also called molecular sieve chromatography, separates molecules by their size and sometimes molecular weight by gel filtration.

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When the aqueous solution passes through the stationary phase column (e.g., dextrin, starch), they form a gel. The diffusion of the molecules depends on the pore size of the gel. While passing through the column, the molecules smaller than the pore size quickly pass through the column, and those bigger than the pores retain and are eluted as void volume. 10.1.2.2.4 Ion-Exchange HPLC Ion-exchange HPLC separates ions/polar molecules. Ionic functional groups present on the stationary phase constituents interact with oppositely charged analytes by Coulomb’s law. One can use cation-exchange chromatography when the target molecule is positively charged.

10.1.2.3 Different Components of an HPLC System The HPLC system consists of the following primary parts. They include a column of the stationary phase, mobile phase solvent or solvent mixture, pump, sample injector, detector, waste collector, and data processor and recorder (Fig. 10.4). 10.1.2.3.1 Column The fundamental and the most significant part of an HPLC system is its column. The column contains the stationary phase that efficiently separates the sample components based on their affinities to adsorb on the stationary phase by physicochemical interactions. The packing materials (the stationary phase) within the

Fig. 10.4 High-performance liquid chromatography (HPLC) system with its various components

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column are usually small granulated polymers or silica or surface-modified silica particles. The internal diameter and length of a column and the type and size of the stationary phase particles play a significant role in the sample-component separation process. Further, a constant temperature at the column secures a constant flow rate of the mobile phase through the stationary phase, ensuring correct retention times of the sample components in the column. 10.1.2.3.2 Mobile Phase The mobile phase is a particle-free degassed pure solvent, solvent mixture, or buffer mixture that is inert to the sample components. It travels at a constant rate with a high pressure generated by an HPLC pump and carries the injected sample components through the stationary phase column for their separation. 10.1.2.3.3 Pumps A pump is an essential component in an HPLC system to maintain a constant flow rate of the mobile phase with a constant pressure, while it carries the sample components through the stationary phase column. Variable flows can alter the elution duration of the sample compounds and lead to erroneous findings. 10.1.2.3.4 Injectors Injectors transport a small but definite volume of the sample into the mobile phase stream with accuracy. 10.1.2.3.5 Detector After leaving the column, the eluted components of the sample enter the detector device. Detectors detect the time of appearance and the quantity of each component, based on the alteration in the eluent composition, and convert the gathered information into electrical signals. The signals are then transmitted to a data processor and recorder. Some commonly used detectors are ultraviolet-visible light, refractometric, electrochemical, and fluorescence detectors. 10.1.2.3.6 Waste Collector After the detection is complete, the solvent mixture leaving the detector deposits in a waste collector container. 10.1.2.3.7 Data Processor and Recorder The electrical signals collected by a detector are processed by computer-based software, converting the data into chromatograms as output and recording the information. The software calculates the response signals and controls operating parameters such as injection sequence and volume, wash cycles, detection wavelength, etc. A chromatogram is a total number and the quantity of all the peaks obtained from the sample by chromatographic separation. HPLC shares qualitative information such as peak shape, the signal intensity to form the peak, time of appearance of the peak, and the quantification of the peaks, by estimating the area of each peak, which is proportional to its concentration in the sample. The

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computer-based software manages for such calculation, quantification, and recording of the information.

10.1.2.4 Tailing Factor The symmetric peaks of the compounds in a chromatogram ensure the correctness of the data. But the asymmetric peaks are often observed. The peak should be symmetric as a Gaussian curve. If we draw a centerline as height, the front half and back half of the peak (divided by height) should be the same. If the front half and back half widths
of the peaks are m and n, respectively, their ratio called the tailing factor T ¼ mn should be equal to 1 for symmetric peak. However, for the peak where T > 1 is called tailing. For those where T < 1 are called fronting. Pre-filtration of the solvent before introduction into the HPLC system and proper packing of the column with the pure stationary phase of the correct size particles can prevent tailing problems significantly. Injecting a large volume broadens the peak and, thus, often causes fronting. Therefore, sample injection volume should be less. 10.1.2.5 HPLC Method Optimization Optimization of HPLC method indicates optimization of chromatographic conditions (Swadesh 2000). They include compositions of stationary phase (or selection of column) and mobile phase, primary control variables such as flow rate, sample amounts, injection volume, temperature, diluents, etc. HPLC method optimization has some predominant steps, which include the following: Physicochemical properties of drugs Choosing chromatographic conditions Sample preparation Development of an analytical method Method validation

10.1.2.5.1 Physicochemical Properties of Drugs Physicochemical properties such as solubility, pH value in solution, pka (negative log of the acid dissociation constant), polarity, Log P (partition coefficient), etc., of drugs, are predominantly associated with the HPLC method development and optimization. The selection of the mobile phase also depends on drug polarity. Similarly, the correct pH selection of charged analytes provides a sharp peak without tailing or fronting.

10.1.2.5.2 Choosing Chromatographic Conditions The chromatographic conditions such as method type, column, mobile phase, and a detector are selected first based on the chemical to be separated. This selection is very important for detecting and measuring the analytes accurately.

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10.1.2.5.3 Selection of HPLC Method Type Method selections mainly depend on the physicochemical characteristics of drugs. Small organic drug molecules are vastly separated using reversed-phase chromatography. The reversed-phase HPLC also separates ionizable drugs (acids/bases) with a buffer as mobile phase. 10.1.2.5.4 Column The proper selection of columns provides us the accurate separation and quantification of a drug in HPLC. A column contains stationary phases such as polymers, zirconium, alumina, silica, surface modified silica, etc. These materials have their respective advantages and disadvantages. Silica particles have many benefits that include low cost, easy way to convert them spherical uniform size particles, ability to withstand high pressure, and application in many HPLC separations. Still, a significant disadvantage of silica is that it dissolves in alkaline pH. However, nowadays, some modified silica columns are available which can withstand alkaline pH. In most reversed-phase separations, the C18 column is used. The selection of a column depends on stationary phase chemistry, the particle size of the stationary phase, column dimensions, retention time, and temperature. The system column that runs with a column-temperature varying between 30 and 40  C shows good and reproducible outcomes. 10.1.2.5.5 Mobile Phase Optimization The selection of the mobile phase depends on the physicochemical characteristics of drugs and the stationary phase. Depending on the nature of the drug, the mobile phase is selected for adequate separation and appropriate retention of drug molecules on the stationary phase. Phosphate buffer, acetate buffer, formate buffer, etc. are commonly used. The buffer concentration range usually varies from 10 to 50 mM. The buffer pH closer to pKa value enhances buffer capacity. The buffer capacity increases with the increased molar concentration of the buffer. For ionizable drugs, pH of the mobile phase based on pKa value of a drug or alteration of pH of mobile phase is chosen. The organic modifiers such as acetonitrile, methanol, and tetrahydrofuran are also commonly selected in reversed-phase HPLC. Maximum 50% organic liquids (as mentioned above) may be mixed with buffer to prepare a mobile phase. Gradient elution is a choice for the sample with many drugs and metabolites. 10.1.2.5.6 Sample Preparation A crucial step in an HPLC method development is sample preparation. The sample should be free from contaminants. It is done by extraction, filtration, or centrifugation. However, care should be taken to avoid loss of the sample by adsorption, leaching, etc. The prepared sample in aliquot should be free from interferences, compatible with the HPLC system (including column) and the method, and should not interfere with the detection process. 10.1.2.5.7 Detectors A detector is chosen based on the purpose of analysis and the sample(s) to be detected. Besides, in the selection of a detector, interference, detection limit, and

References

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availability of the detector play an important role. Thus, various detectors, namely, ultraviolet, fluorescence, electrochemical, and refractive index detectors, are selected based on the requirement. The ultraviolet-visible light detectors are highly sensitive dual-wavelength detectors. Photodiode detector performs advanced optical detection. The fluorescence detectors are highly sensitive and selective to detect fluorescent compounds. The refractive index detectors are highly sensitive, reliable, and reproducible for detecting chemicals with little or no ultraviolet absorption. 10.1.2.5.8 Development of an Analytical Method In the method development stage following the selection of stationary phase, mobile phase, the other chromatographic parameters are standardized based on trial runs. For example, in the case of reversed-phase HPLC method, mobile phase flow rate, mobile phase pH, a number of theoretical plates (>2000), the retention time of a compound (>5 min), tailing factor (5% of the relative standard deviation of relative peak area), and detection wavelength are experimentally established. 10.1.2.5.9 Method Validation Validation of an analytical method indicates documentation that proves the method to generate acceptable data for the intended analytical application. It also ensures the reproducibility and reliability of the method. Distinctive validation characteristics, based on which a method is validated, include specificity, detection limit, quantification limit, range, linearity, accuracy, precision, robustness, system suitability, forced degradation, and stability. All these parameters should be standardized to validate an HPLC method.

References Cole RB (ed) (1997) Electrospray ionization mass spectrometry: fundamentals, instrumentation, and applications, 1st edn. Wiley-Interscience, New York Hoffmann E-d, Stroobant V (eds) (2007) Mass spectrometry principles and applications, 3rd edn. Wiley, The Atrium, Chichester, West Sussex, England Lough WI, Wainer IW (eds) (1995) High performance liquid chromatography-fundamental principles and practice, vol 10. Blaclue Academic and Professional, London Swadesh JK (ed) (2000) HPLC—practical and industrial applications, 2nd edn. CRC Press, Boca Raton

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Statistics and Biometry

Statistics belongs to the discipline of applied mathematics that deals with data, more precisely their collection methods, organization, presentation, analysis, interpretation, and conclusion. Statistical application in biology for data analysis to comprehend its significance and interpret the data for a decision is called biostatistics or biometry. We can also use it for appropriate study designing in biological experiments (Goldstein 1964). Statistics thus provides us with a superior understanding and accuracy of appropriate quantitative data and efficient method design to collect relevant and accurate data. The statistical analysis has a fundamental goal of collecting quality data, organizing them for proper display, and simplifying the data to comprehend their significance by data analysis. It helps to draw an appropriate conclusion. The statistical analysis of pharmacokinetic data describes the significance of the relationship of blood/plasma drug concentration-time course, determination of rate constants, drug-plasma protein binding, drug distribution, metabolism, and elimination, correlate pharmacokinetic and pharmacodynamic data, in vitro and in vivo data, data analysis through compartmental models, and evaluate the validity of assumptions of pharmacokinetic methods, and data transformation (Triola and Triola 2018; Hopkins and Goeree 2015). There are two fundamental areas of statistics. They are descriptive statistics and inferential statistics. Descriptive statistics deal with sample-based or populationbased data properties, and inferential statistics utilize those properties to evaluate the hypotheses and figure out conclusions. Descriptive statistics summarize sample/ population data exercising the mean or standard deviation. Components of descriptive statistics are mean, median, mode, variance, standard deviation, standard error, etc. Inferential statistics is a complex process and consists of various methods such as Student’s t-test, Regression analysis, Analysis of Variance (ANOVA), Statistical Equation Modelling, etc. # The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B. Mukherjee, Pharmacokinetics: Basics to Applications, https://doi.org/10.1007/978-981-16-8950-5_11

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The statistical distribution properties include mean, median, mode, standard deviation, range, etc. Distribution is a possible value of defined events, and variable values of events are called random variables. Two fundamental types of statistical distribution are discrete random variable of distribution and continuous random variable of distribution. A precise numerical value that belongs to a random variable is called a discrete random variable of distribution. When random variables can have infinitely multiple values, they are called continuous random variables of distribution. In pharmacokinetic data analysis, understanding the statistical terms is very important. Hence, they have been precisely discussed below. We often need to comprehend statistically significant quantity in an available data set that requires calculating mean, median, mode, data deviation from the average value, range, etc. We use this initial information to investigate primary cause trouble and apprehend data accuracy of a study design and acceptable data parameters. Representation of statistical data in a diagram is generally attractive, but it does not provide detailed information of a given data set. Bar charts, pie charts, histograms, frequency polygons, and cumulative frequency curves (ogive) are popular graphics for representing statistical data. Further, statistical data representation in tabular form is also not much effective. Hence features of data based on the observation are essential.

11.1.1 Central Tendency A single number that represents the entire observed data is known as the central value. The experimental values generally cluster around the middle of the distribution of the central value that gives a measure of central tendency. Central tendency is thus a central position of a frequency distribution clustered around by the other observed values. A large set of data in statistics is thus often represented with a single value (central tendency) that describes the “mean/average” value or median or “middle” value of the whole data set. Mean, median, mode, range, etc. are used to describe the central tendency of a data set.

11.1.1.1 Mean Mean here is the arithmetical average of all the terms in a data set of discrete random variables of a statistical distribution. Mean is calculated by dividing the sum of a data set by the number of values added. Values of all the terms are added, and then the sum is divided by the number of terms. The mean of continuous random variables of a statistical distribution is acquired by summing the product of variables and their frequencies, and then dividing the sum by the sum of the frequencies. It is also called the expected value, represented by “μ.” The arithmetic mean represents all the requisite properties of the accurate measure of central tendency. Simple arithmetic mean and weighted arithmetic mean are the two kinds of arithmetic mean.

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11.1.1.1.1 Simple Arithmetic Mean Two methods to calculate simple arithmetic mean are a direct method and a short-cut method. Direct Method

If x1, x2, x3,........ xn are the n number of values of the variable x, the arithmetic mean x is x þ x2 þ x3 þ :: . . . . . . þ xn x¼ 1 ¼ n

Pn

i¼1 xi

n

ði ¼ 1, 2, 3, . . . . . . :, nÞ

Short-Cut Method

If x1, x2, x3,........ xn are the n number of values of the variable x, and A is the assumed mean of the quantities that exist between the maximum value and the minimum value, and the deviation of each value i, di ¼ xi – A (i ¼ 1, 2, 3, . . .. . . ., n) Then mean x is Pn x¼Aþ

i¼1 d i

N



N¼

X  i

11.1.1.1.2 Weighted Arithmetic Mean Weighted arithmetic mean is the mean of a simple frequency. There are three methods to calculate weighted arithmetic mean. They are (a) direct method, (b) short-cut method, and (c) step derivation method. Direct Method

If x1, x2, x3,........ xn are the n number of values of the variable x, and f1, f2, f3,........ fn are their respective frequency, the weighted arithmetic mean x is Pn f xi f 1 x1 þ f 2 x2 þ f 3 x3 þ :: . . . . . . þ f n xn x¼ ¼ i¼1 i ði ¼ 1, 2, 3, . . . . . . :, nÞ f 1 þ f 2 þ f 3 þ :: . . . . . . þ f n N X Xn Here N ¼ f ¼ f i¼1 i Short-Cut Method

In large values of variable or frequency, the determination of mean by the direct method is time-consuming and difficult. If x1, x2, x3,........ xn are the n number of values of the variable x, and f1, f2, f3,........ fn are their respective frequency, and A is the assumed mean of the quantities that exist between the maximum and the minimum values, and the deviation of each value i, di ¼ xi – A (i ¼ 1, 2, 3, . . .. . . ., n), the weighted arithmetic mean x by this method is

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Pn Pn f i di i¼1 f i d i x ¼ A þ Pi¼1 ¼ A þ n N f i¼1 i

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N¼

n X

! fi

i¼1

Step Derivation Method

In the case of substantial values of variable or frequency, the determination of mean by the short-cut method becomes complicated. Thus, for simplifying calculation, the step deviation method is used. In this method, di is changed to ui which is determined by dividing di by HCF (highest common factor) (here considered as h) of all di values. ui ¼

di d x A ¼ i¼ i ð,d i ¼ xi  AÞ ði ¼ 1, 2, 3, . . . . . . :, nÞ h HCF of d i values h ! P P n X ð ni¼1 f i ui Þh h ni¼1 f i ui x ¼ A þ Pn ¼Aþ fi N¼ N i¼1 f i i¼1

11.1.1.1.3 The Arithmetic Mean of the Composite Group Three distribution groups of n1, n2 and n3 values have the respective means x1, x2 and x3 . The composite mean will be x¼

n1 x 1 þ n2 x 2 þ n3 x 3 n1 þ n2 þ n3

11.1.1.2 Standard Error of the Mean/Measurement (SEM) The SEM represents a function of the standard deviations (discussed later in the chapter) of the observed test scores. It measures how closely the observed scores are dispersed around the correct score. SEM ¼ pSDffiffiNffi, where SD is the standard deviations and N (also represented by n) is the number of observations. rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi error mean square or SEM ¼ N qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi witin a group It can also be represented as SEM ¼ error mean square . N

11.1.1.3 Median The median is the middlemost number of a discrete random variable in a distribution. It is the second measure of the central tendency. The number of terms in the distribution could be even or odd. In the case of an odd number, there is a single

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middle value. In comparison, there are two median values in the even number of terms in a distribution. When N is an odd number, the median is Nþ1 2 th value. The median of a group data set, when the median is the N2 th term and the value is ðN CÞh

¼ L1 þ 2 f L1 is the boundary limit of the lower class. C is the cumulative frequency of the class just preceding the median class. f is the frequency of the median class. h is the width of the median class. N is the total frequency. When the total frequency is even, the median value will be the arithmetic average  of the n2 th term and the n2 þ 1 th term. Therefore, the median in case of even number is Median ¼

value of n2 th term þ the value of 2

n 2

 þ 1 th term

11.1.1.4 Mode It is the third and last measure of the central tendency. The most frequently appearing value in a distribution containing a discrete random variable (i.e., in a data set) is called mode. There could be one or more than one mode (e.g., bimodal, trimodal distributions for two or three modes, respectively, of distributions/data sets) in a distribution of a discrete random variable or no mode at all. The mode of a continuous random variable provides the maximum value of the function. In a group distribution, f m1 Þ Mode¼ L1 þ h 2 f ðf m  f m1  f mþ1 m L1 is the boundary limit of lower class. fm is the frequency of the modal class. fm  1 is the class frequency just preceding the modal class. fm + 1 is the class frequency just following the modal class. h is the width of the modal class. Empirical relationship among mean, median, and mode is given below: Mean  Mode ¼ 3ðMean  MedianÞ

11.1.2 Measure of Dispersion The variability or spread of a distribution is called statistical dispersion. The measure of data dispersion is a measure of the scatterings of a distribution. The measure of dispersion is called the absolute measure of dispersion when it is expressed with the

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same unit of the variable value. On the contrary, the relative measures are calculated using the statistical percentage average (called dispersion coefficient). Variation coefficient indicates a relative measure of dispersion over the absolute measure of dispersion and can be represented as Variation coefficient ð%Þ ¼

Standard deviation  100 Mean

11.1.2.1 The Absolute Measure of Dispersion Fundamental four types of the absolute measure of dispersion are range, quartile deviation, mean deviation, and standard deviation (SD). 11.1.2.1.1 Range The range is the most straightforward absolute measure of dispersion and is defined as the difference between the maximum variable value and the minimum variable value of a distribution. As for an example, if x1, x2, x3,........ xn are the n number of values of the variable x in ascending order, the range of x is xn  x1, where xn is the maximum value and x1is the minimum value. 11.1.2.1.2 Quartile Deviation (Q) Quartile deviation is the half of the difference between the first quartile (Q1) and the third quartile (Q3) of a distribution. That is, Q ¼

1 ðQ3  Q1 Þ 2

11.1.2.1.3 Mean Deviation In a distribution of the values of a variable, the deviation indicates the difference between a value and the mean value. if x1, x2, x3,........ xn, are the n number of values of the variable x, and their mean is x, deviation of every value from the mean value is shown below. ðx1  xÞ, ðx2  xÞ, ðx3  xÞ, . . . . . . . . . ::ðxn  xÞ, and positive values of them are, jðx1  xÞj, jðx2  xÞj, jðx3  xÞj, : . . . . . . . . . jðxn  xÞj, Therefore, the mean deviation is jðx1  xÞj þ jðx2  xÞj þ jðx3  xÞj þ . . . . . . . . . þ jðxn  xÞj n Pn 1 Thus, the mean deviation is ¼ n i¼1 jðxi  xÞj ði ¼ 1, 2, 3, . . . . . . :, nÞ In the case of any distribution, median (M) may be considered instead of the mean value for calculation of the mean deviation also.

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Then the mean deviation is ¼ 1n

205

Pn

i¼1 jðxi

 M Þj ði ¼ 1, 2, 3, . . . . . . :, nÞ

11.1.2.1.4 Standard Deviation (SD) The most commonly used absolute measure of dispersion is the standard deviation (SD). It is expressed as positive square root of arithmetic mean of the square deviations of the “values from their arithmetic means” and is usually denoted by σ (sigma in small letter). It is also denoted by S or s or sd or SD. if x1, x2, x3,........ xn are the n number of values of the variable x, and their mean is x, deviation of each value from the mean is shown below. ðx1  xÞ, ðx2  xÞ, ðx3  xÞ, . . . . . . . . . ::ðxn  xÞ, Then, the squares of those deviations are ðx1  xÞ2 , ðx2  xÞ2 , ðx3  xÞ2 , . . . . . . . . . ::ðxn  xÞ2 Then arithmetic mean of the square of the deviation of the “values from their arithmetic mean” ð x1  xÞ 2 þ ð x2  xÞ 2 þ ð x3  xÞ 2 þ . . . . . . . . . þ ð xn  xÞ 2 n The mean of the square deviation values is also known as variance and is denoted by σ 2. Therefore, σ2 ¼

n 1 X ðx  xÞ2 ði ¼ 1, 2, 3, . . . . . . :, nÞ n i¼1 i

pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Then, the standard deviation or SD or σ ¼ σ 2 ¼ variance ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 1 i¼1 ðxi  xÞ , n If x1, x2, x3,........ xn are the n number of values of the variable x, and f1, f2, f3,........ fn are their respective frequency, n P

x¼

f i xi f 1 x1 þ f 2 x2 þ f 3 x3 þ :: . . . . . . þ f n xn i¼1 ¼ ði ¼ 1, 2, 3, . . . . . . :, nÞ f 1 þ f 2 þ f 3 þ :: . . . . . . þ f n N X Xn Here N ¼ f ¼ f i¼1 i

The mean square deviation is

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f 1 ðx1  xÞ2 þ f 2 ðx2  xÞ2 þ f 3 ðx3  xÞ2 þ . . . . . . . . . þ f n ðxn  xÞ2 N X Xn f ði ¼ 1, 2, 3, . . . . . . :, nÞ Here N ¼ f ¼ i¼1 i 1 Xn f ðx  xÞ2 i¼1 i i N pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Standard deviation, σ ¼ σ 2 ¼ variance rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Xn f ðx  xÞ2 ði ¼ 1, 2, 3, . . . . . . :, nÞ ¼ i¼1 i i N σ 2 ¼ variance ¼

11.1.3 Probability Probability numerically describes how likely the outcome (result) of a random (unpredictable) experiment occurs. The classical definition of probability represents the ratio of the number of favorable event points of an event to the number of possible event points when it is impossible to predict the outcome of any single event point. The probability (P) of an event consisting of n out of m possible equally likely cases is given by P ¼ n=m A random experiment of tossing an unbiased coin shows the outcome of the head or tail. It is impossible to predict the outcome of any single throw. Therefore, the chance of occurrence or the probability of the occurrence of a particular event (head or tail) is 50%. Similarly, in a single throw of a die, the chance of occurrence of any specific face (from 1 to 6) is 1/6 (16.66%). Thus, even with the known different outcomes, the experiment cannot predict which results will occur at any experiment performance. It is called a random experiment. A random experiment has a well-defined procedure that can be repeated, and the outcomes are known and determined, but the outcomes cannot be predicted. The results or outcomes of a random experiment are called the events of the investigation. The capital letters (A, B, C, D, E, etc.) are generally used to represent events. The events are primarily divided into two categories: elementary events and composite events. An elementary event or a simple event of a random experiment has a single outcome that cannot be decomposed. For example, when a single coin is tossed, it has a single outcome on top (head/tail). However, the composite event has more than one elementary event or the decomposed outcome. For example, the event has more outcomes when five coins are tossed together (with head/tail).

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11.1.3.1 Mutually Exclusive Events In a random experiment, two events that cannot occur simultaneously are called mutually exclusive events. One event (either red ball or blue ball) can happen in a single ball pick up from a bag containing blue balls and red balls. It is an example of a mutually exclusive event. 11.1.3.2 Impossible Events In a random experiment, an event that cannot occur at any performance is called an impossible event. If we roll a die with six faces having the numbers 1–6 and look for an event 8, it is an impossible event. 11.1.3.3 Equally Likely Event Equally likely events are the events with the same probability of happenings. If we throw a die, on its top, any number from 1 to 6 may appear. Therefore, all the outcomes from 1 to 6 are equally likely. 11.1.3.4 Sure or a Certain Event In a random experiment, the chance of an event is that which indeed occurs in any performance. In a random experiment, a certain event is that which indeed occurs in any performance. All the points of the sample space occur. If we throw a die, on its top, any number 1–6 comes. If we roll a die and look for an event 6 or less up to 1, it will be a sure or certain event. 11.1.3.5 Complementary Events If an event A happens to occur, then the complementary event of A (Ac) is that which does not happen to occur. For example, if picking up a heart card from a 52-card deck is event A, then the complementary event Ac is not picking up a heart card. Event A and the event Ac are mutually exclusive and exhaustive events. 11.1.3.6 Exhaustive Events A set of events in a random experiment is exhaustive when at least one event is sure to occur. Once a die is rolled, there is a possibility of the outcome of any number from 1 to 6. Events 1–6 is collectively called exhaustive as any performance at least one event from 1 to 6 is the outcome. 11.1.3.7 Sample Space or Event Space Sample space is a set of all probable results of a random experiment. It is represented by S. If a coin is tossed, there is a possibility to obtain either head (H) or tail (T). So its sample space is S ¼ {H,T}. Similarly, if a die is rolled, the possibility of outcomes on its top is any of the numbers from 1 to 6. Hence its sample space is S ¼ {1,2,3,4,5,6}. 11.1.3.8 Sample Points A sample point is one of the possible results of a random experiment. Therefore, any outcome of a sample space is known as a sample point. If a coin is tossed, there is a

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possibility to obtain either head (H) or tail (T). So its sample space is S ¼ {H,T}. Then H and T are its two sample points.

11.1.4 Some Important Definitions Figure 11.1 shows a brief classification of statistical data types. Some important definitions related to statistical data, the distribution of the statistical data, and some significant information are given below.

11.1.4.1 Continuous Data Continuous data are numerical data sets that can be quantified between two realistic values on an infinite scale. It refers to any possible measurements between the two numbers, for example, measurement of time, temperature, etc. 11.1.4.2 Discontinuous Data Discontinuous data, also called categorical or discrete data, mainly records sample behavior that occurs during each interval or some intervals or at the momentary time of observation(s) by dividing an observation into equal intervals. Discrete data only can take definite values. Human blood group and gender are the examples of the discontinuous data variable. 11.1.4.3 Parametric Data Parametric data are the sample data of a population appropriately modeled by a probability distribution with a fixed set of parameters. 11.1.4.4 Nonparametric Data Nonparametric data are those data that cannot be fitted in a recognized statistical probability distribution model. Nonparametric data do not require any assumptions on sample parameters or sample characteristics, and the data types (qualitative/ quantitative). For example, data of a survey on like or dislike of a cold cream by the consumers. 11.1.4.5 Normal Distribution Normal distribution, also known as Laplace–Gauss or simply Gaussian distribution, is the data distribution about the mean that gives a bell-curve pattern when represented graphically. The data are more frequent at the mean, and the data are symmetrically distributed on both sides of the mean. 11.1.4.6 Non-normal Distribution Non-normal distribution is a skewed distribution in which data distribution clumps up on one side and then tapers on the other side on a graph. It happens due to the outliers (observation with abnormal distance from the other data).

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Fig. 11.1 An abridged classification of statistical data

11.1.4.7 Homogeneous Population The types population, when identical, are known as homogenous population. A data set of mathematics test of boys of 12 years old in a school class is the example of a homogeneous sample. 11.1.4.8 Heterogeneous Population When the types of population are different, they are known as heterogeneous population. A data set of the length of long jumps of boys of height varied from 5 to 6 feet in a school class is the example of a heterogeneous sample. 11.1.4.9 Skewness Skewness indicates a lack of symmetry. More precisely, the left and right portions of the distribution curve are not equal concerning the central line. 11.1.4.10 Kurtosis Normal data distribute on both left and right sides of the distribution curve equally. From the centerline, data taper on either side. This decreasing data on either side of

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the centerline may be lightly tapered or heavily tapered. Kurtosis provides the measure of such light or heavily tapering data distribution for normal data distribution.

11.1.4.11 Decision Tree in Statistics A decision tree in statistics is a schematic representation to assist us in selecting the appropriate statistical methods of analyzing a set of data to reach a reliable statistical goal, decision, or outcome.

11.1.5 The Null Hypothesis, Alternate Hypothesis, and Degree of Freedom There are two crucial hypotheses. If there are k numbers of population and their mean variances are μ1, μ2, μ3, . . .. . .. . .. . ...,μk, according to the null hypothesis (H0), μ1 = μ2 = μ3. . .. . ... = μk. It suggests that there is no statistical difference among the types of the population. The null hypothesis indicates that there is no statistically significant difference between the groups. In the case of the less t-value than the critical value, the null hypothesis fails to reject. On the other, the alternate hypothesis (Ha) states that there must be at least one of the means different from the others in the sets of population. It implies a statistical rejection of the null hypothesis. Degree of freedom (df) means logical maximum independent values with freedom to vary in a data set/sample. Sequential steps for significance tests are: initially, we define H0 and Ha and determine critical value based on α ( p) value, and df from the distribution table. Then by drawing a distribution curve, rejection regions are decided. Once the null hypothesis is rejected, we calculate test statistics, and a conclusion is drawn based on the preselected critical value.

11.1.6 Population and Sample If there are 3 schools with the same number of students, 3000 each (the number of students in the 3 schools may differ), we have 3 population types here. From these 3 sets of population, we can take 20 students from each school for analysis. They are samples in each case. The sample groups are considered for analysis to conduct the process fast and economically. If there are three populations, and from there, three sample groups are considered, then the individual mean of the three groups and the overall mean (grand mean) of all three groups are calculated. We calculate the mean of each sample group and the overall mean of the three groups.

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11.1.7 Selection of Statistical Methods Appropriate selection of statistical methods is essential for the correct statistical analysis in a data distribution. When the mean represents the center of the distribution of a set of data more accurately and the sample size is large, the parametric test is conducted. However, when the median represents the center of the distribution of a data set more accurately, a nonparametric test is conducted even for a large data set. In the parametric tests of two groups, the data are processed for a scattergram plot that displays two data sets. However, in the parametric test of more groups, Bartlett’s homogeneity of variance is used (Fig. 11.2). Bartlett’s test of homogeneity of variances identifies whether there are equal variances of a continuous or discrete dependent variable in two or more groups of a categorical independent variable (Gad and Weil 1989). Homogeneity of variance is hypothesized as no difference in variances (i.e., homogeneous) among the groups. The significant correlation between the groups is considered heterogeneous data, and they are undergone for Scattergram plot. The nonsignificant (no difference in variances, i.e., homogeneous, among the group sets upon Bartlett’s test of homogeneity of variance, i.e., unrelated variables) data are then analyzed by analysis of variance (ANOVA) (Balakrishnan 2014). Duccan’s multiple range test further analyzes significant data sets by intercomparison of the groups. But the comparison is not made against a control group. The group sizes are more or less exact. The intercomparison between the groups within a range is done. However, Dunnett’s t-test is conducted to compare the groups versus a control group with the group sizes more-or-less exact. Data sets upon scattergram analysis if found to be normal are subjected to F-test analysis (Fig. 11.3). F-test significant data sets (heterogeneous) with equal N value (N1 ¼ N2) are subject to the analysis of Student’s t-test with a degree of freedom (N  1), and for unequal N values (N1 6¼ N2), Cochran test is performed. Non-normal data of a scattergram analysis are indicative of nonparametric data. Nonparametric data with two independent groups are compared by Wilcoxon ranksum test (Fig. 11.4). Three or more groups with variation within the groups for nonparametric data are compared by Kruskal Wallis test. For a significant test of continuous data distributions, multiple comparisons may be made. For quantal data (response/no response or alive/dead types of binary data from a set of data) for more than two groups, Fisher’s exact test or RXC Chi-square test is done (Fig. 11.5). For significant and continuous RXC Chi-square tested data sets, Fisher’s exact test may further analyze the data sets for more accuracy. The right method(s) of statistical analysis is selected using statistical decision trees based on data types or distribution patterns. Such schemes (Figs. 11.2, 11.3, 11.4, and 11.5) are provided for the appropriate selection of statistical methods to draw reliable statistical decisions and conclusions.

11.1.8 Different Statistical Methods Used in Pharmacokinetics The statistical analysis commonly used in pharmacokinetics is given below.

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Fig. 11.2 Statistical decision tree for continuous parametric data with normal distribution

11.1.8.1 Bartlett’s Test for Homogeneity Bartlett’s test for homogeneity provides a comparison of variance of parametric test among more than two groups of continuous data sets (e.g., data obtained from biochemical measurements, bodyweights, etc.). The corrected RXC Chi-square values are used in Bartlett’s test. P2 P 2 N χ ð χÞ N The variance (S2) is calculated as S2 ¼ where χ is the individual N1 datum of a group and N is the number of data in each group.

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Fig. 11.3 Statistical decision tree for continuous parametric heterogeneous data

2:303 χ2test ¼

P

P

 

P  df ðS2 Þg fP df log 10  df log 10 S2 df   , 1 1 1 P 1 þ 3ðK1Þ df  df

where K is the number of groups:

11.1.8.2 Cochran Test For two groups of continuous data sets of heterogeneous variance where many data within the groups are unequal (N1 6¼ N2), Cochran test is carried out. It is a nonparametric test. It is used to understand whether several treatments provide the same effects. In this test, two t-values, tobserved and texpected, are calculated. The texpected is usually denoted by t0. x1  x2 S2 ffi , where w ¼ SEM2 ¼ t observed ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N w1 þ w2 t 0 w þw t 0

t 0 ¼ 1 w11 þw22 2, where t 01 and t 02 are the values obtained from t-distribution table with a degree of freedom N  1 and p-value 0.05 (generally considered).

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Fig. 11.4 Statistical decision tree for nonparametric (not normal distributed) data

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Fig. 11.5 Statistical decision tree for categorical or quantal response data

11.1.8.3 F-Test F-test is conducted to identify homogeneity of variance of two groups of separate parametric continuous data sets. It suggests whether the data of a group and the subsequent parametric group is valid. It is calculated by dividing larger sample variance by smaller sample variance of the groups.

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P

x2 

ð

P

Þ

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N The variance (S ) is calculated as S ¼ , where χ is the individual N1 datum of a group and N is the number of data in the group.

2

F¼

2

larger sample variance variance between the groups or, F ¼ smaller sample variance variance within the group

11.1.8.4 Types of T-Tests (for Comparison of Means) There are three fundamental types of t-tests: (a) independent sample (when means of two groups are compared), (b) paired sample (when means of same groups are compared at different points), and (c) one sample type t-tests (When mean of a lone group is compared with a known mean value). 11.1.8.5 Student’s t-Test William Sealy Gosset invented this t-test method for statistical comparison of the barley production of two fields. He was not interested in publishing the technique by his name, and he published it by the name of student. Hence the t-test became popular as Student’s t-test. The t-test for independent samples is carried out assuming a normal distribution of the data in the groups. A data distribution curve is bellshaped with equal mean and median values in a normal distribution. We often come across the terms one-tailed or two-tailed test. What do they mean? In the two-tailed test, the mean values are unequivalent, and the direction of t-values toward the critical value approaches from either direction of the critical value. But one-tailed test indicates if the test (observed) mean is larger or smaller in value compared to the hypothesized mean value. For one-tailed test, direction of t-values toward critical value is the same for both groups, and the direction is known. The tvalues of data for the observed group are estimated and compared to the hypothesized mean value. More extremely observed t-values than the critical value (depends on the significance level) reject the null hypothesis. After the F-test, if the data variables are approximately equal, it requires a parametric test. However, when F-test shows homogeneous variables with equal numbers of data within two groups, Student’s t-test is conducted. t¼

xμ ðwhere x is sample mean, μ is population mean, n is sample size, pSffiffi n

S is standard deviationÞ: If F-test shows significance (heterogeneous data) with equal numbers of data within two groups, modified Student’s t-test is appropriate. The test is applicable for the comparison of two sets of small-size quantitative data. There should be a similar variance of each sample and the same number of data points in either sample. Further, a paired test is conducted to compare the means of the same group and with a specific mean value of a separate group. In the paired test, the variance between the groups is assumed to be unequal. On the other hand, in an

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unpaired test, two independent group means in which the variance between the groups assumed to be equal are compared. The t-value is calculated using the formula (independent t-test) x1ffiffiffiffiffiffiffiffi x2 q , where x1 and x2 are the means of the two groups, s1 t  value ¼ signal noise ¼ 2 2 s s 1 2 n1 þ n2

and s2, are the standard deviation values of the two groups, 1 and 2, and n1 and n2 are the respective numbers of observations in groups 1 and 2. The null hypothesis is tested based on the critical value. If the t-value is greater than the “critical value” that is obtained from the t-table based on the selected probability value ( p ¼ 0.05 or 0.01) and the degree of freedom, the rejection of the null hypothesis is made. Degree of freedom means logical maximum independent values with freedom to vary in a data set/sample and is expressed as df ¼ n1 + n2  2. When n1 ¼ n2, and F-value is significant, the degree of freedom represents df ¼ n  1.

11.1.8.6 Analysis of Variance (ANOVA) Analysis of variance (ANOVA) analyzes the statistically significant differences among and between the means of a minimum of three independent sets of population or three independent sample groups of the population. It usually compares distributed continuous data with homogeneous variance. Ronald Fisher, a renowned statistician, has developed the method for statistical analysis. The process compares the means of variance of three or more population groups to identify whether any statistical difference exists within or between the sample groups of comparison. However, it does not state which group differs statistically significantly. For this, further tests are required to be conducted. The critical value is chosen from the F-distribution table based on the degrees of freedom of the mean square of treatments and the degrees of freedom of the mean square errors. The degrees of freedom within and between the groups are also calculated. The degree of freedom between the groups (dfbetween) is k  1 (k is the number of population tested), whereas the degree of freedom within the group (dfwithin) is N  k (where N is the total number of population together in all the population or all the samples belonging to the population). One-way ANOVA is conducted by estimating variance between the means of the samples and variance within every sample mean. The ratio of the variance between the means of the samples and the variance within the sample means is presented by mean square of treatment F, where F ¼ The The . mean square error Therefore, the first variance between the mean squares of treatments and then the variance of the mean square error are calculated. Then F value is calculated, and the critical value is determined to interpret the results. The sum of squares regression (SSR) is calculated. We also calculate the sum of squares of errors (SSE). The sum of squares total (SST) is then determined where SST = SSR + SSE. Therefore, the sequence of the calculation of samples from k numbers of different population is given below.

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The first step, the means of each sample of the population (i.e., x1 , x2 , x3 , . . . , xk , for k number of samples of population) and the grand mean (x ) of the k samples of population are calculated, and the critical F-value is determined from the F-distribution table. The degrees of freedom within and between the groups are calculated using the formula mentioned above in the subsequent step. The degrees of freedom for the total, the sum of the degree of freedom within and between the groups, are determined as mentioned below. In the next step, the total sum of squares is determined using the by formula, 2 P SStotal = x2x Then, the sum of squares (SS)within the groups is calculated by  P P P P 2 2 2 2 xi 2 x1 Þ þ xi 2 x2 Þ þ xi 2 x3 Þ þ . . . þ xi 2 xk Þ , SSwithin = for k numbers of population, where xi is the data of each population, where i = 1,2,3, . . ..,n. Further, SS between the groups is calculated using the formula SSbetween = SStotal 2 SSwithin In the subsequent step, the mean of squares between the groups and the mean of squares within the groups are determined using the mentioned formula. MSbetween =

SSbetween df between

MSwithin =

SSwithin df within

between Finally, F-value is calculated using the relationship, F 2 value = MS MSwithin The calculated F-value less than the critical F value suggests that the statistical comparison of the k numbers of population or sample groups fails to reject the null hypothesis.

11.1.8.7 Post Hoc Tests Various post hoc tests (to identify specific differences upon the significant outcome of the omnibus F test) are available for the analysis of data following the ANOVA test. The tests such as Duncan’s multiple range test, Scheffe’s test, Dunnett’s t-test, Tukey’s test, etc. are popularly used post hoc in pharmacokinetic and toxicological data analysis. 11.1.8.7.1 Scheffe’s Test Scheffe’s test is used for the statistical comparison of three or more groups (especially with unequal numbers, N1 6¼ N2 6¼ N3, say, for three groups), having continuous and randomly distributed data, as a robust significant test, but it is less sensitive to the violation of normal and homogeneous data of variance of assumptions. We here compare the means of two sample groups at a time. It is not suitable for pairwise

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comparison but is mostly used for complicated comparisons where the unplanned tests cannot be applicable. ANOVA suggests a statistically significant difference exists. But it does not inform us between which groups the statistically significant difference exists. Scheffe’s test is used for knowing the specific groups which differ statistically significantly. Based on the degrees of freedom between the groups (dfbetween, k  1) or within the groups (dfwithin, N  k), with the selected p-value (F-critical value), we compare the test F-values. In a sequential event, we first determine the mean values of each sample group, determination of mean squares within the groups or variance within the groups (S2w ), and finally, test F-values are compared with the critical F-value. When the test F value is greater than the critical value, we reject the null hypothesis. It means the data is statistically significantly different. The F-critical value ¼ (k  1) (ANOVA critical value) 2 ðxi x j Þ  , where xi and x j are the means of the two sample The test F  value ¼  S2w

1 1 n1 þ n2

groups having i and j numbers of data when i ¼ 1, 2, 3, . . . . . . , n1 , and j ¼ 1, 2, 3, :: . . . n2 : S2w is variance within the groups. F test ¼

ð C 1 x1 þ C 2 x2 þ . . . þ C k xK Þ 2  2 , C2 C C2 ðK  1ÞMSwithin n11 þ n22 þ . . . þ nkk

Ck is comparison number such that C 1 þ C 2 þ . . . þ C k ¼ 0

11.1.8.7.2 Dunnett’s t-Test Dunnett’s t-test is a multiple comparison test when a mean (of a single control group) is compared with the means of several experimental groups. If the one-way ANOVA test shows significant findings, it never indicates the pairs of means that are statistically significantly different. Dunnett’s t-test, a post hoc test following the ANOVA, identifies the pairs with significant differences. The test is performed for the comparison of several sample groups against a fixed control group. When there is no control group, the Tukey’s test is performed. In Dunnett’s test, the Dunnett difference, DDunnett, is determined by DDunnett ¼ qffiffiffiffiffiffiffiffiffiffiffi 2MS

S=A t Dunnett n 1 x2 , MSwithin can be written also as MSS=A : x1 and x2 are the or, t Dunnet ¼ pxffiffiffiffiffiffiffiffiffiffiffi 2MS within n

resprctive mean test and control observations: Like many other post hoc test, in this test, we first determine DDunnett critical value or D critical value from Dunnett critical value table, which requires a number of sample groups including the control group, degree of freedom within the groups, and α value that is generally set at α ¼ 0.05. The time of residence tDunnett is calculated from Dunnett’s table, using the number of sample groups, degree of freedom within the groups determined from ANOVA source table at the selected α value as taken

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here, α ¼ 0.05. The value of mean squares within the groups MSS/A is determined from the ANOVA source table. The n depicts the number of individuals per group. Thus, we determine D critical value. When the difference between the mean of the control group and the mean of any test group is more than the D critical value, the data of the sample group differ statistically significantly from the control group. 11.1.8.7.3 Tukey's Test or Tukey Test Tukey test, also known as the honestly significant difference (HSD) test, is a post hoc test done after ANOVA. It determines if two groups of data are statistically significantly different in multiple comparison tests. In the Tukey test, the sample size remains the same. We use the following formula to calculate Tukey’s honestly significant difference. qffiffiffiffiffiffiffi HSD ¼ q

MSw nk ,

where q is a constant, MSw is mean squares value within the

groups, and nk is number of participants in each group. The q is determined from the studentized range q table using the degree of freedom within the groups and the numbers of treatments (groups). MSw is determined from the ANOVA summary table. After calculating the value of HSD, we compare each mean difference to HSD. When the mean difference is greater than the HSD value, the sample group differs statistically significantly at the selected p-value level. 11.1.8.7.4 Williams’ Test Williams’ test is a reliable statistical tool for normal data distribution in a balanced setup to provide critical values with upper 1 and 5% points for the treatment groups and variable degrees of freedom. The test determines the minimum effective dose with a monotonically increasing or decreasing dose series in an investigation. It measures the response at higher levels of monotonic order. For example, if x1 , x2 , x3 , . . . , xk , are the means for k numbers of sample in population, then, x1  x2  x3 ......... xk. But it does not occur frequently. Any mortality at a high dose seriously hampers the test also. Numbers of treatment groups with the same sample size n when receiving different Ki doses (i ¼ 1,2,3, ...., k) of treatments of a drug/substance of interest with control (0 treatment), then it is assumed that effect is the substance mediated enhancement of the mean response that is proportional to the dose size. First, the null hypothesis/alternate hypothesis is tested. Upon the rejection of the null hypothesis, T value is determined using the following formula. μi x0 ffiffiffiffiffiffiffiffi Ti ¼ p , where s is an unbiased estimate of standard deviation within the 1 1 s

n1 þ n0

group, statistically independent of xi, and μi, the maximum likelihood estimate of μi. It is expressed as9 8P v > > x jn j= < , where the most effective dose is the ith dose when j, α, and T > t μi ¼ j¼u v P > nj > ; : j¼u

value for j  i, where α, j, and t are in the upper percentage α critical value in the Tj distribution.

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11.1.8.7.5 Duncan’s Multiple Range Test Duncan’s multiple range test is a post hoc test to compare specific mean differences between any two treatment means for a set of pairs. This test is conducted following ANOVA. The test is protective in the case of false-negative errors but not against false-positive errors. In Duncan’s multiple range test, we first determine the means of each group, that is, x1 , x2 , x3 , . . . , xk , are the means for k number of samples (groups), and then Rp value is determined using the following qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi formula. Rp ¼ qαðp,νÞ

MSwithin r

¼ qαðp,νÞ

MSE r ,

where Rp is range, α is the level of signifi-

cance, p is Duncan’s factor, ν is the degree of freedom, r is the number of observations in each group, and MSE is mean square error. From Duncan’s multiple range table, we determine Rp values for each group, R2, R3. . . . . , Rk. In Duncan’s multiple range table, the degrees of freedom are noted in rows, and the p (Duncan’s parameter, starts from 2 and continues such as 2,3,4, and so on) values are read against the columns. First, we calculate the critical value q, at a fixed α value with the respective p-value and degree of freedom, using the table. Then weqdetermine the ffiffiffiffiffiffiffiffi value of MSE from the ANOVA table, and we calculate the value of

MSE r .

Finally,

we determine the respective Rp values (such as, R2 . R3,...). Then, the group means x1 , x2 , x3 , . . . , xk , are arranged in the order of their increasing values. The greatest mean value is considered as the extreme highest value, and the smallest mean value group is the lowest extreme value. The difference between the means of the two groups is calculated and compared with the respective Rp value to determine statistical significance between the groups. If the mean difference is greater than the corresponding Rp value, it is considered that the mean values between the groups differ statistically significantly.

11.1.8.8 Fisher’s Exact Test Fisher’s exact test is a parametric test applicable for discontinuous data and qualitative data. The small sets of data can be tested directly from the contingency table. However, large sets of data require computation. Frequency data such as expression of genes or proteins in histological findings, mortality incidences, etc. are used, and the p-value is expressed in ratios. The positive or negative responses are noted. As an example, data in two samples (groups) are demonstrated below with the numbers of positive, negative, and total responses. Two samples Group I Group II Total

Positive response r1 r3 r1 + r3

Negative response r2 r4 r2 + r4

Total responses r1 + r2 r3 + r4 r1 + r2 + r3 + r4 ¼ rtotal ¼ M

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ðr1 þ r2 Þðr3 þ r4 Þðr1 þ r3 Þðr2 þ r4 Þ! M!r1 !r2 !r3 !r4 !

The p value suggests whether one- or two-tailed exact test groups differ statistically significantly or not, at the selected level of p or α.

11.1.8.9 RXC Chi-Square Test RXC Chi-square test is the row  column Chi-square (χ 2) test or simply χ 2 test that is applied for qualitative data. The qualitative data implies non-numerical, nominal (values are the names), ordinal (values have intrinsic order), and binary (values belong to two categories) types. The χ 2 test is used to compare the observed frequencies for two or more groups with the expected frequencies that happen by chance. Fisher’s exact test is also conducted on qualitative data, but it is used to compare two groups. However, the χ 2 test is more accurate compared to Fisher’s exact test on large numbers of groups. On the other hand, the Fisher’s exact test has more accuracy than the χ 2 test on small samples (groups). Manual calculation related to the Fisher’s exact test is always very hard compared to the χ 2 test that is much easy to compute manually. The formula for determination of χ 2 value is given below. P ðOEÞ2 , where O is the observed frequencies and E is the expected χ2 ¼ E frequencies. The expected frequency is determined by the direct count formula as mentioned below. Þðcolumn totalÞ , the formula values (row total, The expected frequency ¼ ðrow total Grand total column total, and grand total) are obtained from the contingency table. However, the expected frequency can be obtained by the probability approach also. From the χ 2 table, the critical χ 2 value is determined based on the selected p-value and the degree of freedom. If the test χ 2 value is greater than the critical χ 2 value, the sample differs statistically significantly. 11.1.8.10 Mann–Whitney U Test/Wilcoxon Rank-Sum Test/Mann– Whitney–Wilcoxon/Wilcoxon–Mann–Whitney Test Mann–Whitney U test is a nonparametric rank statistics to compare two independent groups. It is an independent sample t-test performed on ordinal (ranked) continuous data. That is, data are not normal. The test is also known by some other names, such as Wilcoxon rank-sum test, Mann–Whitney–Wilcoxon, and Wilcoxon–Mann– Whitney test. It is used to identify if there is any statistically significant difference between the two treatments at α (statistical level of significance considered). When the null hypothesis and the alternate hypothesis test show difference exists between the groups of data, the test is conducted at a preselected α value (say, p ¼ 0.05). We then determine the z-critical value from the Z distribution table. The decision rules follow the Z distribution table, which needs a sample size of 20 at least, and it is a two-tailed test. Next in the U-statistics, the U points of the two treatment groups are determined based on the data ranks. The smallest to the highest rank orders are provided to the

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smallest to the highest data values. In the case of the same data values, we give them the average ranking. Then we determine U values for two groups (say, for A and B, U values are UA and UB, respectively). The formula of U-statistics is given below to determine UA and UB values. Þ U Statistic group ¼ Rank sum  nðnþ1 2 , where n is the number of observations. The smaller U value is selected as U value. The U-statistics suggests the degree of overlap in ranks between the groups. The smaller the degree of overlap, the smaller is the U value. At no degree of overlap, the U value is zero. Then the z test value is calculated using the following formula n n U A2 B ffi , where nA, nB are the number of observations for A and B z ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n n ðn þn þ1Þ A B A 12

B

treatment groups, respectively. When the z-value of a test sample is greater than the z-critical value, in that case, we reject the null hypothesis, and the data become statistically significantly different at the selected α value.

11.1.8.11 Analysis of Covariance (ANCOVA) Analysis of covariance (ANCOVA) is a linear regression analysis using a blend of ANOVA and regression to remove antecedent variable effects. ANCOVA is an extension of ANOVA. Here, dependable variable scores estimate the principal effects/interactions by adjusting the differences with a single covariate or some covariates. Covariates are measured before the measurement of dependant variance for the correlation with it. The assumptions used for ANOVA, such as variance homogeneity, normality, and random independent samples, are equally applicable for ANCOVA. Further, ANCOVA in comparison to ANOVA is a powerful test to compare mean differences between the groups in the presence of priming (introduction of one stimulus that stimulates the other stimulus), as it increases the power of the effects within the subject groups in multiple baseline situations. In ANCOVA, response variables are compared by a continuous independent variable and a factor. The critical purpose of ANCOVA is first to increase the sensitivity of the test to the prime effects and interaction by reducing the error terms (the errors of covariates that are considered as noise) and thereby, to make F-test used during computation more powerful. Secondly, it is used to adjust the means at the level of the dependable variables. Unlike ANOVA, the sum of squared errors is subdivided into two terms: squared errors and squared covariates. It reduces the sum of squares of errors, making the F test more powerful as the denominator becomes smaller. The F-value is represented as, F ¼ MSC MSE , where MSC and MSE are mean squared covariates and mean squared errors, respectively. 11.1.8.12 Kruskal–Wallis Test The Kruskal–Wallis test is a nonparametric rank-based one-way ANOVA test to compare three or more groups of equal or different sample sizes on an independent

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variable measured on a continuous or ordinal dependent variable. However, Kruskal-Wallis compares the mean ranks, and ANOVA compares means of values. In this test, we first define the null and alternate hypotheses, and the α ( p) value is generally set at 0.05. We then determine the degrees of freedom (df ¼ k  1), where k is the number of groups. In the following step, we state the decision rule by determining critical χ 2 value, from χ 2 table using α, and df. Then we determine the test χ 2 value. For this, the data ranks of the treatment groups are determined. The smallest to the highest rank orders are provided to the smallest to the highest data values. In the case of the same data values, we give them the average ranking. Then we determine H (test χ2 value) from the following equation. P r2i 12 2 H ¼ N ðNþ1Þ ni  3ðN þ 1Þ, where r i is the sum of squares of the rank of group i (where i ¼ 1, 2, 3, . . .. . ., n), n is the sample size of individual group i, and N is the total sample size of all the groups. Then we compare H with the critical χ 2 value. If H > critical χ 2 value, it rejects the null hypothesis, and we can conclude that a statistically significant difference exists between the distribution of the samples (groups).

11.1.9 Scattergram, Linear Regression Model, Correlation Coefficient, Nonlinear Regression Model A scatter plot is a Cartesian coordinate system plot representing data for two variables for a set of data. The numeric values of two variables are represented by dots on two perpendicularly intersecting axes in a Cartesian coordinate system to understand the relationship between the variables. More precisely, we want to determine if the two sets of data of two variables are related to one another or random. In the Cartesian coordinate system, we plot the data values as dots of the dependent variable against the data values of the independent variable. Then we try to understand dots of the plotted data produce any linear relationship or not. Here regression analysis can be performed to investigate and model the relationship between a response variable and the predictors. When a scattergram produces a tighter fit of observations around the regression line and establishes a rectilinear relationship between the two variables, the relationship is then called linear regression relationship or linear regression model. On the other hand, based on a scatter plot or even so a residual plot (non-evident or non-pattern), when linear regression model shows not to be the best choice, and the model better fits to nonlinear curvature, then the curvilinear relationship between the original two variables is called nonlinear regression relationship or nonlinear regression model. Non-evident or non-pattern shows a nonlinear regression relationship.

11.1.9.1 Linear Regression Model There are four possibilities: (a) the plotted data produce a linear relationship with a positive slope; (b) the plotted data produce a linear relationship with a negative slope; (c) data produce a line parallel to x- or y-axis (slope ¼ 0); and (d) the plotted

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data produce no evident relationship (random). When the plotted data produce a linear relationship, it suggests a correlation between the two variables. The correlation can be quantified by computing the correlation coefficient.

11.1.9.2 Correlation Coefficient (r) Correlation coefficient (r) is a measure to quantify the correlation between the two variables. To determine the correlation coefficient of a pair of variables, say, x and y, having n pairs of data, we first calculate the following values related to the variables x and y and arrange a tabular arrangement for easy calculation. The values to be determined are x2, y2, xy, ∑ x, ∑ y, ∑ x2, ∑ y2, ∑ xy. The linear equation of the relationship line is considered as y ¼ b axþb by. Then we determine the values of b a b and b, using the following formula, P P P P P P P ð xÞð yÞn xy ð xÞð xyÞð yÞð x2 Þ b P 2 P P 2 P b , and b ¼ . Thus, by putting the a¼ ð xÞ n x2 ð xÞ n x2 values of b a and b b , in the equation y ¼ b axþb by , we can establish the equation. Further, we determine the standard deviations of the variables xrand y. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P P 2 1 P 2ffi ðxxÞ2 x n ð xÞ The standard deviation of x, say, Sx and Sx ¼ ¼ , and n1 n1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 1 P 2ffi y n ð yÞ . the standard deviation of y, say, Sy, where Sy ¼ n1 Finally, we calculate the correlation coefficient (r) using either of the two equations. P 1 P P   xyn ð xÞ ð yÞ (a) r ¼ SSxy b . When r value is closed to 1, it a or (b) r ¼ ðn1ÞSx Sy suggests that the variables are strongly correlated. Values much less than 1 indicate a week correlation between the variables.

11.1.9.3 Nonlinear Regression Model The scatter plot or the residual plot (non-evident or non-pattern) fails to show the linear regression model as the best choice, and the model better fits its regression data to nonlinear curvature. The curvilinear relationship or nonlinear relationship between the original two variables is known as the nonlinear regression relationship. Nonlinear polynomial regression adds an extra independent variable at the power of the original variable as, x2, x3. . ., etc. The quadratic equation model is by ¼ b0 þ b1 x1 þ b2 x21 . Higher R2 value indicates more variance in the quadratic model. Further, low standard errors suggest that the observed values are closely fit around the quadratic regression line. The nonlinear quadratic model achieves more explained variance (William 2019), tighter fit around the regression line, reduced standard errors in the model, better confidence, and better prediction.

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References Balakrishnan N (ed) (2014) Methods and applications of statistics in clinical trials, vol 2: planning, analysis, and inferential methods. Wiley, Hoboken Gad SC, Weil CS (1989) Statistics for toxicology. In: Wallace Hayes A (ed) Principles and methods of toxicology, 2nd edn. Raven Press Ltd, New York, pp 435–483 Goldstein A (1964) Biostatistics—an introductory text. Macmillan, New York Hopkins RB, Goeree R (2015) Health technology assessment using biostatistics to break the barriers of adopting new medicines. CRC Press, Boca Raton Triola M, Triola M (2018) Biostatistics for the biological and health sciences, Global edn. Pearson, New York William WSW (2019) Multivariate time series analysis and applications. Wiley, Hoboken

Pharmacokinetic Software and Tools

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Pharmacokinetic Software and Tools

The advancements of pharmacokinetics and pharmacodynamics are strongly associated with the outburst of computer knowledge (Ekins 2006). Nowadays, pharmacokinetic software and tools are widely used in practice (Sensen and Hallgrimsson 2009; Saverno et al. 2010). The highly faster calculation, rapid estimation of various pharmacokinetic parameters, and often incredibly accurate prediction of pharmacokinetic data have made pharmacokinetic software and tools very popular among the students, researchers, and clinical scientists (Saverno et al. 2010), including many others.

12.1.1 Software and Tools Software or a programming tool is a program that instructs a computer or an electronic device (e.g., mobile device, calculator) how to operate the system to perform a task. Software development tools, or merely called tools, are also computer programmer software to develop other software. Examples of some software are Windows, Excel, Adobe Photoshop, CorelDraw, Android, etc.

12.1.2 Types of Software There are four primary types of software.

12.1.2.1 Application Software It is a common type of end user (people) software for performing an end task. As an example, Adobe. # The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B. Mukherjee, Pharmacokinetics: Basics to Applications, https://doi.org/10.1007/978-981-16-8950-5_12

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12.1.2.2 System Software System software runs application software as well as the entire computer system. One example is Microsoft Windows. 12.1.2.3 Programming Software Programming software assists computer programmers in writing code—for example, C++, Python, etc. 12.1.2.4 Driver Software Driver software runs devices that are linked to a computer to execute their desired functions. For example, a printer is plugged into a computer.

12.1.3 Importance of Pharmacokinetic Software Special type computer programming or software, more popularly known as pharmacokinetic and pharmacodynamic software, simulates various pharmacokinetic and pharmacodynamic parameters well in different pharmacokinetic and pharmacodynamic models. The various types of such software fulfill the need for fast computation of data to obtain different pharmacokinetic and pharmacodynamic parameters and provide the desired outcomes from the complex pharmacokinetic and pharmacodynamic models with ease and accuracy. The use of various types of pharmacokinetic and pharmacodynamic software helps us achieve correct study design, dose selection, determination of various pharmacokinetic parameters, developing linear and nonlinear kinetic graphics, development of well-simulated pharmacokinetic models, compartmental and noncompartmental data analysis, and many more (Lotsch et al. 2004). The software tools have a robust data storage capacity through a computer with a fast calculation of tedious pharmacokinetic complex equations. Hence, they are explicitly a choice for use in clinical trials, particularly in the later phases where it is done on substantial patient populations.

12.1.4 Open Source and Commercial Pharmacokinetic Software and Tools 12.1.4.1 PKQuest It is Java-based, fully integrated, freely available user-friendly pharmacokinetic software used to manage human clinical pharmacokinetic data for the non-compartment pharmacokinetic model, analysis of nonlinear kinetics, and absorption through the intestine. It is also applied for the in vivo drug absorption kinetics in individual subjects. The software can analyze and graphically represent the data. Further details of the software may be obtained from: http://www.pkquest.com/ home

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12.1.4.2 LAPKB The LAPKB or LAPK software (where LAPKB is an abbreviated form of Laboratory of Applied Pharmacokinetics and Bioinformatics) is freely available software for optimizing patient dose. It uses a nonparametric model based on Bayesian designs. It is an integrated tool. BestDose is used for dose optimization and intended for physicians and pharmacists, and Pmetrics is applicable for the scientists as a research tool. Further details of the software may be obtained from: http://www.lapk.org/ software.php 12.1.4.3 Lixoft The software is intended for population analysis in preclinical experiments and clinical trials based on nonlinear kinetic models. Lixoft has different software products. Monolix is advanced and mighty software to determine pharmacokinetic nonlinear mixed effect model-related parameters by estimation and graphical presentation. Simulux is used for the optimization of samples for phase III clinical trials. Pkanalix is applied for the pharmacokinetic compartment and non-compartment models. It is an interactive graphical tool. Further details of the software may be obtained from: https://lixoft.com/ 12.1.4.4 SwissADME It is a user-friendly and free web tool developed by the molecular modeling group, Swiss Institute of Bioinformatics. The software is used to determine physicochemical and biopharmaceutical parameters of small molecules or drug-like substances based on their absorption, distribution, metabolism, and elimination (ADME) parameters, thus assisting in drug discovery. It can predict gastrointestinal absorption of small molecules and the ability of small molecules to permeate across the blood-brain barrier to reach the brain. Further details of the software may be obtained from: http://www.swissadme.ch/ 12.1.4.5 Edsim++ Edsim++ is a visual and straightforward pharmacokinetic and pharmacodynamic design software. It may be used for teaching pharmacokinetics because it is straightforward and noncomplicated to operate. Edsim++ is designed with several advanced pharmacokinetic and pharmacodynamic programming tools, and the software can be used by scientists for advanced research purposes also. The software provides object-oriented modeling without the need for any standard operation-related programming. Further details of the software may be obtained from: http://www.medimatics.net/ projects/objsim 12.1.4.6 GraphPad Prism Prism is probably the simplest program for data analysis with the graphical solution. By fitting the data for the curve upon the selection of data parameters, desired

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graphics can be generated. Prism is equipped with a vast library of statistical tools for appropriately selecting tests for statistical analysis. Further details of the software may be obtained from: https://www.graphpad.com/ scientific-software/prism/

12.1.4.7 JGuiB A Java SE scripted graphical user interface tool runs the well-known pharmacokinetic and pharmacodynamic modeling program Boomer. In due course of its use, one can run Boomer with Microsoft Disk Operating System or Macintosh Operating System without JGuiB. JGuiB converts interactive mode to the graphical user interface-mediated application mode for Boomer, with the inclusion of simulation and Bayesian optimization, for pharmacokinetic and pharmacodynamic modeling. Further details of the software may be obtained from: http://pkpd.kmu.edu.tw/ jguib/ 12.1.4.8 PKSolver 2.0 PKSolver is a freely downloadable program tool for analyzing compartmental and non-compartmental pharmacokinetic and pharmacodynamic data. A user chooses inputs from the option list (at the menu) associated with the computer. It is written in Microsoft menu-driven “Visual Basic for Applications” programming language Visual Basic 6. Twenty encrypted, routinely applied common pharmacokinetic equations are accessible for calculation. Further details of the software may be obtained from: https://www.pharmpk.com/ soft.html or https://www.boomer.org/boomer/software/pksolver.zip 12.1.4.9 JPKD JPKD (also known as JavaPK) is software for therapeutic drug monitoring. The desktop programming tool for clinical pharmacokinetics can define the Bayesian pharmacokinetic estimation model with the population-based available pharmacokinetic parameters for dose adjustment. Further details of the software may be obtained from: http://pkpd.kmu.edu.tw/ jpkd/ 12.1.4.10 PCModfit PCModfit is popular software that has been used primarily on Phase I and II clinical investigations. PCModfit V7.0 is its recently updated version modified extensively to determine various non-compartmental pharmacokinetic parameters, including clearance, drug distribution volume, mean residence time, half-life, and bioavailability at various time points and maximum and minimum plasma drug concentration values in a repeat dosing regimen. The graphics are produced in Microsoft Excel and can be stored with the .png extension that creates the elegant appearance of the graphs. It is free software. Further details of the software may be obtained from: https://www.pcmodfit.co. uk/

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12.1.4.11 SimBiology SimBiology software has a wide range of applications in pharmacokinetic and pharmacodynamic modelings. It simulates dynamic systems for quantitatively analyzing physiologically based pharmacokinetic models. SimBiology develops such models with its block diagram editor or by MATLAB programming platform. The software is capable of analyzing ordinary differential equation-based complex and robust models. The programming tool can predict the efficacy and safety aspects of drugs and optimize their doses. The principal pathways and significant parameters can be picked up with available data to estimate bio-variation. Using the tool, we can also accomplish non-compartmental analysis. Further details of the software may be obtained from: https://www.mathworks. com/products/simbiology.html 12.1.4.12 acslXtreme The software is widely used to model and simulate various systems, from missile and aircraft to medical, toxicological, and biomedical systems, and solve complex nonlinear differential equations. Further details of the software may be obtained from: https://acslxtreme.software. informer.com or https://aegistg.com/ 12.1.4.13 ADAPT 5 ADAPT 5 software is used for pharmacokinetic and pharmacodynamic modeling of parametric populations by estimating maximum likelihood using the expectationmaximization algorithm with sampling. The analysis is based on the estimation module, simulation module, and sample schedule design module. The weighted least squares and maximum likelihood are the basis for estimation in the estimation module. On the other hand, a posterior Bayesian analysis and single and multisubject simulations are utilized for computing in the simulation module and sample schedule design module, respectively. Further details of the software may be obtained from: https://bmsr.usc.edu/ software/adapt/ 12.1.4.14 Kinetica Kinetica Streaming Data Warehouse is a high-performance database management system. The system uses graphic processing units. The software provides data availability for healthcare decisions from the level of drug discovery to patient data to optimize patient experiences to offer precision medicine. Further details of the software may be obtained from: http://www.thermo.com/ kinetica/ 12.1.4.15 Maxsim2 Maxsim2 is equipped with pharmacokinetic and pharmacodynamic familiar models. Checkboxes help users select several answers from the choices. Several pharmacokinetic and pharmacodynamic parameters that include the volume of distribution, clearance, and drug partitioning during absorption can be estimated by altering dosage regimens and drug plasma concentration-time profiles.

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Further details of the software may be obtained from: http://www.maxsim2.com/

12.1.4.16 Design–Expert Design–Expert is statistics-based software for designing experiments which include comparative tests, characterization, screening, optimization, mixture design, robust parameter design, and combined design as claimed by the US(Minnesota)-based company, the Stat-Ease Inc. Using a power calculator, the software determines the required number of tests to be conducted and the factors-associated primary effects and the interactions, if any, between the factors, by altering the factor values. Thus, ultimately, the operating parameters are optimized. ANOVA establishes statistical significance. The software furnishes many graphical and text-based outputs for the residual analysis. Further details of the software may be obtained from: https://www.statease.com

References Ekins S (ed) (2006) Computer applications in pharmaceutical research and development. Wiley, Hoboken Lotsch J, Kobal G, Geisslinger G (2004) Programming of a flexible computer simulation to visualize pharmacokinetic-pharmacodynamic models. Int J Clin Pharmacol Ther 42:15–22 Saverno KR, Hines LE, Warholak TL, Grizzle AJ, Babits L, Clark C, Taylor AM, Malone DC (2010) Ability of pharmacy clinical decision-support software to alert users about clinically important drug-drug interactions. J Am Med Inform Assoc 18:32–37 Sensen CW, Hallgrimsson B (eds) (2009) Advanced imaging in biology and medicine: technology, software environments, applications, 1st edn. Springer, Berlin, Heidelberg

Pharmacokinetic Laboratory-Based Experiments

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Pharmacokinetic Laboratory-Based Experiments for Undergraduate and Postgraduate Students

13.1.1 Introduction Laboratory-based practical work or research-related work, or clinical investigation, is mostly teamwork. To obtain reproducible, reliable, and accurate data, it needs good understanding, truthfulness, sharing of things, cooperation, sincere discussion on the topic, carefulness during an investigation, patience, skilled hands, devotion and dedication to the work, keen observation, calibrated instrument facilities, standard reagent facilities, other infrastructural facilities, accuracy in study design, and many more. A famous poem for encouragement for good teamwork is shared below. An anonymous poet of the Great Britten wrote this. There are four people named Everybody, Somebody, Anybody, and Nobody. There was an important job to be done, and Everybody was asked to do it. Everybody was sure that Somebody would do it. Anybody could have done it, but Nobody did it. Somebody was angry about that, because it was Everybody’s job. Everybody thought that Anybody could do it, but Nobody realized that Everybody would not do it. It ended up that Everybody blamed Somebody when Nobody did what Anybody could have done.

Therefore, along with excellent teamwork, sincerity with honest and ethical practice can produce correct findings. In the laboratory, discipline, good laboratory practice, knowledge of the laboratory hazards, and the necessary precautions are also equally very important.

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13.1.2 Helpful Information in Laboratory Work Some vital information is shared below. They are helpful for practical-based work.

13.1.2.1 Practical Concept of Molar Concentration (Conversion of Concentration to Molar Concentration) The number of moles of solute dissolved in 1 L of the solution is called its molar concentration. In 1 L solution, when 1 gram-mole of solute is dissolved, it is called one molar solution or 1 M solution. In simple words, molar mass (molecular weight), when represented in the concentration mg/mL, the concentration of the solution is 1 M. For example, a compound has molecular weight of 380. Then, if its solution contains 380 mg/mL, the concentration of the solution is 1 M. If 1 M solution is diluted 1000 times (i.e., 1 mL in 1000 mL), the concentration becomes 1 mM. For the above case, 380 μg/mL is 1 mM. If 1 mM solution is diluted 1000 times (i.e., 1 mL in 1000 mL), the concentration becomes 1 μM. For the above case, 380 ng/mL is 1 μ(M). If 1 μM solution is diluted 1000 times (i.e., 1 mL in 1000 mL), the concentration becomes 1 nM. For the above case, 380 pg/mL is 1 nM. 13.1.2.2 Buffers A buffer is an aqueous weak acid and its associated base or a weak base and its associated acid solution that prevents pH alteration upon incorporating acidic or basic constituents. Every buffer has its pH value. The pH is a scale to identify the acidity or basicity of an aqueous solution. It indicates the negative logarithm (base ten) of the hydrogen ion concentration and is represented by
log10[H+]. Composition of few commonly used buffers: Phosphate buffer, pH 7.4 Composition 0.075 mM Na2HPO4. 7 H2O 0.024 mM NaH2PO4. H2O

Quantity for 1000 mL/1 L 20.214 g 3.394 g

(Adjust pH with NaOH or HCl as required)

Phosphate buffer saline (PBS), pH 7.4 Composition 140 mM NaCl 2.7 mM KCl 10 mM Na2HPO4. 7 H2O 1.8 mM KH2PO4. H2O

Quantity for 1000 mL/1 L 8.18 g 0.20 g 1.4196 g 0.2449 g

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20 3 Saline sodium citrate (SSC) buffer, pH 7.0 Composition 3 M NaCl 300 mM sodium citrate

Quantity for 1000 mL/1 L 175.2 g 88.20 g

Tris-EDTA (TE) buffer, pH 8 (at 20  C) Composition 10 mM Tris.HCl 1 mM EDTA

Quantity for 1000 mL/1 L 1.576 g 0.372 g

Cell lysis buffer, pH 7.6 Composition 50 mM HEPES 10 mM tetrasodium pyrophosphate (Na4P2O7, 10H2O) 150 mM NaCl 1.5 mM EDTA 10% glycerine 1.5% triton-X 100

Quantity for 1000 mL/1 L 11.92 g 4.46 g 1.5 mL (1 M NaCl) 1.0 mL (0.5 M EDTA) 1.6 mL 15.0 mL

13.1.2.3 Preparation of Tissue Homogenate A pre-weighed tissue sample is homogenized in ice-cold 0.25 M aqueous sucrose solution (four times of tissue volume) using a tissue homogenizer, namely, a tefloncoated tissue homogenizer (Sarkar et al. 1994). 13.1.2.4 Preparation of Liver Microsomes The liver homogenate is centrifuged at 8000 g for 15 min at 4  C in a cold centrifuge to pellet the nuclei and other tissue debris. Take a supernatant aliquot (cytosolic preparation) and centrifuge at 105,000 g for 90 min to pellet the microsomes in an ultracentrifuge (Sarkar et al. 1994). 13.1.2.5 Revolution per Minute (RMP) and g Relationship g ¼ (1.118  10
5)r s2 where s is the centrifuge speed in revolution per minute and r is the radius in centimeters of a rotor being used. 13.1.2.6 Anticoagulants and Their Concentration for the Clinical Blood Sampling Procedure Anticoagulants are used to prevent clotting of blood. Various anticoagulants and their concentrations used to prevent blood clotting are given below (Adcock et al. 1997; Durham et al. 1995; Erwa et al. 1998; Goosens et al. 1991; Marlar et al. 2006).

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Concentration required for blood 1.2
2 mg/mL of blood 1:9 (v/v) for normal clinical samples and 1:4 (v/v) for Erythrocyte sedimentation rate test (Volume ratio indicates citrate: blood) 46
60 IU/mL of blood 10 mg/L of blood

13.1.2.7 Protein Precipitation The final concentration in tissue homogenate for perchloric acid is 1 M, and for trichloroacetic acid (TCA), 5% (w/v) is used for protein precipitation. However, the concentrations can vary based on the nature of the tissues. 13.1.2.8 MTT Assay The complete name of MTT is 3-(4, 5-dimethylthiazolyl-2)-2, 5-diphenyltetrazolium bromide. MTT assay is conducted for the determination of cellular viability and toxicity. Prepare the stock solution of MTT by dissolving it into isopropyl alcohol containing hydrochloric acid (4 mM) and 0.1% NP 40 reagent (commercially available reagent). Remove medium from the cultured cells. Wash with serum-free medium. Add 50 μL with serum-free medium and 50 μL MTT solution (5 mg/mL) (in case of 96-well plate). Incubate it at 37  C for 3 h in a CO2 incubator. Add 150 μL MTT solvent (DMSO) for dilution, and shake it for 15 min on an orbital shaker. Then read the absorbance at 590 nm. 13.1.2.9 Ethical Clearance Any experiments on human subjects/volunteers or patients or the biological samples taken from them cannot be conducted without receiving the institutional ethical clearance. Further, the volunteers/subjects should be informed about the objectives and the expected hazards related to the study. For animal experiments, pre-approval of the project work by the Institutional Animal Ethics Committee is required. 13.1.2.10 Relationship Between Common Logarithm and Natural Logarithm Common logarithm ¼

Natural logarithm 2:303

(For detailed calculation, please see Sect. 4.1.7.4 of Chap. 4 in this book)

13.1.2.11 Important Equations ðAUCÞ0
1 ¼ ðAUCÞ0
t þ or

ct ke

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ðAUCÞ0
1 ¼ ðAUCÞ0
t þ 1:44 t 1=2 ct The urinary excretion rate of an unchanged drug ¼ drug excreted through urine Renal clearance, CLð0
tÞ ¼

Δxu Δt ,

where xu is the unchanged

xuð0
tÞ AUCð0
tÞ

Dose Total operative body clearance ¼ CL F ¼ AUC, where F is oral bioavailability AUMC Mean residence time ¼ AUC , the AUMC is the area under the first moment curve; the AUC is the area under the plasma drug concentration versus time curve. The trapezoidal rule can calculate the AUMC.

13.1.2.12 Important Units ðAUCÞ0
1 , or ðAUCÞ0
t , ng:h=mL cmax , ng=mL T max , h CL, L=h

13.1.2.13 Volume of One Drop of Water Volume of 1 drop of water ¼ 0:05 mL

13.1.3 Drug-Plasma Protein (Albumin)-Binding Assessment with a Low Plasma Protein Binding Drug or a High Plasma Protein Binding Drug 13.1.3.1 Background A major portion of the systemic drug distribution occurs by the transportation of the plasma protein-bound drugs. A highly protein-bound drug has a long plasma residential time and low volume of distribution that limits the systemic availability of free drug and reduces its clearance. Many experimental methods such as equilibrium dialysis, ultrafiltration, and ultracentrifugation are commonly used for a drugplasma protein binding assessment. 13.1.3.2 Equilibrium Dialysis The equilibrium dialysis method is the gold standard as it avoids nonspecific drug binding, which is common to the other available methods. In equilibrium dialysis, a semipermeable dialysis membrane either forms a sac by clipping/tying at both the ends of a tubing dialysis membrane (Fig. 13.1) or separates the two chambers of a

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Fig. 13.1 Equilibrium dialysis method for studying the extent of drug-protein binding using dialysis tube. The upper figure shows that both the ends of the tube are clipped. The lower picture shows that both the ends of the tube are tied

dialysis system (Fig. 13.2). The membrane-based pore size prohibits the plasma proteins and the high molecular weight compounds from passing across it from the sac/the chamber where plasma is placed with a definite volume. However, small solute molecules (here, the drug molecules) can pass through the pore easily. They can reach the buffer solution outside the sac/the other chamber where an equivalent buffer volume (aqueous) is introduced. The drug solution is added to the plasma chamber and incubated at 37  C for an appropriate duration. The sac/plasma chamber of the dialysis system contains protein-bound drugs and free drugs, and the buffer solution outside the sac/the buffer chamber has a free drug (protein

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Fig. 13.2 Equilibrium dialysis chamber for studying the extent of drug-protein binding. The figure shows that a dialysis membrane separates the plasma compartment and the buffer compartment

unbound drug) that passes across the membrane. With an adequate dialysis time, a dynamic equilibrium is established. One can measure the free-drug concentration by analyzing an aliquot collected from the buffer solution outside the sac/buffer chamber. By analyzing the aliquot from the sac and the plasma chamber, one can obtain the total drug concentration that remains in the protein-bound form and free form. The test compound (drug) is quantified from each compartment by various analytical techniques such as HPLC, LC-MS/MS, spectrophotometer, etc. The fraction unbound drug ( fu) provides the extent of protein binding and can be calculated as mentioned
below. : In the equation, PC is the drug concentration in the plasma f u ¼ 1
PC
PF PC compartment, and PF is the drug concentration in the protein-free compartment (buffer compartment). Further, despite the whole plasma, 10 or 50% (v/v) plasma and buffer mixtures are also employed for drug-plasma protein binding analysis. The highly proteinbound drug is difficult to detect in the plasma protein-free buffer compartment. Hence, instead of whole plasma, 10% or 50% plasma in the buffer is used in the plasma compartment for the highly protein-bound drugs. When 10 or 50% plasma is

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used, the fraction unbound drug ( fu) concentration is required to convert at 100% plasma situation using the following conversion formula. f u100% ¼

f u10% f u50% ðfor 10%plasmaÞ and for 50%plasma, f u100% ¼ 10
9 f u10% 2
f u50%

For comparatively less highly protein-bound drugs, 50% plasma is often used. However, the whole plasma is applicable for all experimental stages of preclinical ADME studies. If the percentage of plasma protein-binding of a drug exceeds 90% or otherwise the percentage of the free or unbound drug is less than 10%, the drug-plasma protein binding is believed to be extensive and very high. Further, drug solubility is also an important factor. In a drug-plasma protein binding assay, a standard drug concentration 5 μM is used. The insoluble drug particles precipitate if a drug has a solubility of less than 5 μM in water or aqueous buffer at 37  C (the experimental conditions). It causes errors in the findings. Hence, those drugs are not suitable for this assay. Theoretically, the recovery should be 100%. However, we often get less value due to drug solubility problems or nonspecific binding of the drug to the equipment. The following formula can be used for recovery calculation.  Recovery% ¼

cBD þ cPD cBI þ cPI

  100

where cBD is the drug concentration after the dialysis (when the equilibrium is established) in the buffer compartment, cPD is the drug concentration after the dialysis (when the equilibrium is established) in the plasma compartment, cBI is the initial drug concentration in the buffer compartment, and cPI is the initial drug concentration in the plasma compartment. 13.1.3.2.1 The Donnan Effect In an equilibrium dialysis process, the ionic species often distributes unequally in the dialysis chambers and causes an alteration of the actual protein-binding data. The effect is called the Donnan effect. The Donnan effect can be more or less prevented by employing a high buffer concentration with an ionic strength of more than 0.10 M.

13.1.3.3 Ultrafiltration Unlike the equilibrium dialysis method, ultrafiltration has a fast equilibration time, and thus, it can determine the extent of drug-plasma protein binding quicker. The technique separates the unbound drug from an equilibrated solution of drug and plasma proteins through a semipermeable membrane by filtration (Fig. 13.3) under a vacuum or with a positive pressure. For a big system or in a large scale, the positive pressure by compressed nitrogen or argon is used. The protein cannot pass through the membrane. A significant disadvantage of the method is a recurrent and notable

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Fig. 13.3 Laboratory scale ultrafiltration devices

nonspecific binding to the membrane. Further, protein accumulation on the membrane reduces the porosity and permeability of the free drug across the membrane.

13.1.3.4 Ultracentrifugation The free-drug solution as supernatant is separated with a high centrifugal force 105 g or greater by precipitating the protein component from an equilibrated mixture of drug and plasma using an ultracentrifuge. The equipment is expensive. Here no semipermeable membrane is required. This method can avoid the membrane-related effect. However, the process often produces an error in the free-drug concentration due to back diffusion, sedimentation, viscosity, etc., as the procedure takes a prolonged duration. It is a very high-speed centrifugation technique by which protein-bound drug is precipitated, and the unbound drug remains in the proteinfree supernatant. The drug is estimated from the supernatant and the precipitate using standard analytical procedures and techniques such as spectrophotometry, HPLC, or LC-MS/MS. The unbound drug is estimated using the following formula: F f u ¼ DFDþDP where fu denotes the unbound drug fraction, DF is the amount of freedrug, and DP is the protein-bound drug.

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F Therefore, the percentage of unbound drug ¼ DFDþDP  100, and the percentage of DP bound drug ¼ DF þDP  100.

13.1.3.5 Experiment No. 1 13.1.3.5.1 Equilibrium Dialysis Method Using Tubing Dialysis Membrane and Albumin to Determine Diclofenac-Albumin Binding Materials and Equipment Diclofenac sodium 1 phosphate buffer, pH 7.4 Dialysis tubing cellulose membrane, molecular weight cutoff 14,000 Da Bovine serum albumin (BSA), molecular weight approximately 66,500 Da Deionized water UV-visible spectrophotometer pH meter Electronic balance Temperature control shaker-incubator or metabolic shaking incubator Hot plate 13.1.3.5.2 Membrane Pre-treatment Cut dialysis tubing membrane (about 2 inches in length) and dip in the water in a beaker and heat at 65
70  C for 8 h. Every 1 h interval, water should be replaced by fresh water for sulfur removal. 13.1.3.5.3 Preparation of Protein (BSA) BSA is water-soluble. Add a measured quantity of BSA to produce 20 μM BSA in phosphate buffer, pH 7.4, with a gentle stirring to avoid foam formation. Store in a closed container at 4  C until it is used. 13.1.3.5.4 Stock Diclofenac Sodium Solution Prepare 1 mM diclofenac sodium stock solution by dissolving the weighed amount of drug in the deionized water. 13.1.3.5.5 Preparation of Calibration Curve of the Drug Prepare various dilutions of stock solution with the solvent buffer to produce a range of concentration 1
10 μM of diclofenac sodium in BSA-phosphate buffer solution at pH 7.4. Read the absorbance of the solutions at 276 nm, using a UV spectrophotometer, operating as a reference sample, BSA-phosphate buffer solution (pH 7.4). The calibration curve is obtained by plotting the absorbance against the concentrations. Calculate drug concentration from the calibration curve developed. Plot the data in the Scatchard plot and determine the rate of drug-protein binding.

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13.1.3.5.6 Data Analysis In the Scatchard plot, ½Dr  is plotted on the y-axis, and r is plotted on the x-axis to get a linear plot. The slope of the line gives
K (the equilibrium rate constant), and the intercept on the y-axis provides the value of νK. Thus, from these values, the  individual value of ν and K can be determined. r ¼ ½½PD Pt  , where r is the amount of protein-bound drug per unit total protein (sum of free protein and drug-bound protein) (for details, please see Chap. 3 of this book).

13.1.3.6 Experiment No. 2 13.1.3.6.1 Rapid Equilibrium Dialysis (RED) Method Using Plasma Materials 1. Dialysis chamber system (various such systems are commercially available) 2. Test compound(s) Atenolol—low plasma protein binding Propranolol—high plasma protein binding 3. Solvent acetonitrile, methanol, DMSO 4. Isotonic 1 PBS pH 7.4 5. Mouse/rat plasma 6. 37  C incubator shaker (~300 rpm) 7. Tabletop centrifuge with a plate holder-capable rotor 8. 96-deep well plates, 2 mL (polystyrene) 9. Multichannel pipette and reagent reservoirs 10. 96-well autosampler for LC/MS or HPLC 13.1.3.6.2 Methods Rapid equilibrium dialysis (RED) is well-described by Waters and the co-workers (Waters et al. 2008; Brouwer et al. 2000; Clausen and Bickel 1993). However, for commercial systems [e.g., Thermo Scientific (Pierce)], the manufacturer’s protocol is required to follow. Assay Protocol

Many systems have base plates. If the base plate is there, prepare the base plate first. 1. Prepare the Teflon base plate wells with 20% ethanol (which acts as a selfpreservative) for 10 min and air-dry to prevent microbial contamination. 2. Rinse the RED chambers or inserts (commercial systems) twice with deionized water for 10 min each, and remove the water keeping open ends down (inverted condition) on a blotting paper. 3. Use it immediately before the membrane gets dried. 4. Place the required numbers of the RED inserts keeping the open ends at the top, into the wells of the Teflon base plate.

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Sample Preparation

Required pre-treatment of the stored plasma Thaw the previously collected and stored plasma and centrifuge at 2000 g for 10 min at 4  C to remove particulate matter. Decant the supernatant. Check and adjust pH if necessary to pH 7.4 using lactic acid. 1. Preparation of drug stock solution: Prepare 10 mM drug solution (propranolol in DMSO/atenolol in water). 2. Preparation of 5 μM drug in plasma: Add 15 μL of drug solution from the stock, and add to 285 μL acetonitrile (it gives 500 μM drug solution). Add 5 μL of the resultant solution to 495 μL plasma to produce 500 μL 5 μM drug plasma preparation where the final concentration of acetonitrile is 0.95% and DMSO is 0.05%. 3. Add 200 μL of sample into the sample chamber. 4. Add 350 μL of dialysis buffer (1 PBS pH 7.4) to the buffer chamber. 5. Seal the plate with self-adhesive lids, and incubate at 37  C on an orbital shaker at 300 rpm for 2
6 h (average 4.0 h), depending on the capability of the drug to establish the equilibrium. 6. Remove 50 μL each from both the buffer chamber and the plasma chamber, and dispense in a separate 96-deep well plate. Discard the insert. 7. Add 100 μL of plasma to the collected buffer samples and 100 μL of 1 PBS pH 7.4 to the collected plasma samples to produce identical sample matrices. 8. Add 300 μL acetonitrile mixed with 1 μM internal standard (say, midazolam or as per the applicable analytical method) to each sample to release the drug from the protein-bound stage by precipitating protein. 9. Vortex briefly and centrifuge for 10 min at 13,000 g. 10. Transfer 120 μL of the supernatant from each to the 96-well plate. 11. Add 120 μL of deionized water, and shake briefly on a plate shaker for the LC-MS/MS analysis. 12. Drug quantification can be done by spectrophotometer or HPLC also, using a validated standard protocol. Calculation

%Free drug ¼

C BC  100 C PC

CBC is the concentration of drug in the buffer compartment, and CPC is the concentration of drug in the plasma compartment Further, %bound drug ¼ 100%
%Free drug

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13.1.3.7 Experiment No. 3 13.1.3.7.1 Assessment of Drug-Protein Binding by Ultrafiltration Methods

Prepare a standard drug solution of diclofenac sodium as mentioned in experiment no. 1. 1. Prewash filter at 20  C for 20 min at 2000 g. 2. Weigh the top and the bottom compartments separately of the ultrafilter (column) device before the addition of the drug. 3. Add drug to 400 μL of mouse/rat/human plasma to obtain an ultimate drug concentration of 10 μM. 4. Collect two aliquots of 50 μL each from the same drug-plasma sample, and add them separately into the two prewashed (column) devices. 5. Process one of them immediately to determine initial drug concentration, and incubate the other device with the sample at 37  C, with 5% CO2 for 20 min. 6. Spin the microcentrifuge tube containing the column at 1000 g for 5 min at room temperature. 7. Weigh the top and the bottom compartments separately of the ultrafilter device after spinning to know the approximate volumes of the top plasma and the bottom ultrafiltrate, considering their density is the same. 8. Collect 50 μL of an aliquot from the top plasma and the bottom ultrafiltrate compartments separately for analysis. 9. To determine nonspecific binding, use phosphate buffer (pH 7.4) instead of the plasma, and follow the same method as mentioned above. 10. Use 50 μL of spike phosphate buffer (pH 7.4) for the determination of initial drug concentration, and use 50 μL of ultrafiltrate for the detection of unbound drug in the ultrafiltrate. 11. Add an equal volume (50 μL) of plasma to provide analytically identical matrices. 12. Analyze by LC-MS/MS or any suitable analytical techniques. For a syringe filter, two separate aliquots of the drug-serum/drug-protein (ultimate drug concentration 10 μM) are passed through syringe filters. The first one is passed immediately after mixing, and the second sample is passed after the incubation at 37  C for 6
8 h till the equilibrium is established. The filtrate contains a free drug, and the membrane contains the protein-bound drug. The free drug is then analyzed using a standard method.

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13.1.3.8 Experiment No. 4 13.1.3.8.1 Assessment of Paracetamol-Albumin Binding by Ultracentrifugation Materials and Equipment

Drug, paracetamol Bovine serum albumin (BSA)/human serum albumin (HSA) Trichloroacetic acid Deionized water 1 phosphate buffer (pH 7.4) Ordinary cold centrifuge Ultracentrifuge Methods

The drug and BSA/HSA mixture 5  10
5 M (final drug concentration), drug solution in buffer alone, and the BSA/HSA alone with a total volume of 6 mL in each case are incubated at 37  C for 12 h. 1. Take 5 mL volume, weigh for equal weight by adjusting with the corresponding solvent in each tube before centrifugation. 2. Centrifuge at 106,000 g for 2 h at 4  C. 3. Separate the supernatant in a test tube (it contains protein-unbound free drug). 4. Resuspend the precipitate in buffer (it contains protein-bound drug). 5. Separate drug from the protein-bound drug suspension by adding the same volume of 20% trichloroacetic acid (dilution factor should be considered at the time of calculation). 6. Vortex briefly, and put on ice for 10 min. 7. Vortex briefly, and centrifuge at 10,000 g for 10 min at 4  C to separate protein from the protein-bound drug. 8. Collect the supernatant that contains the drug-free from the protein-bound condition. 9. Collect 1.5 mL aliquot from each case in separate test tubes (samples mentioned in point no. 3 and point no. 8). 10. Determine absorbance taking phosphate buffer, pH 7.4, as blank. 11. Read absorbance using a spectrophotometer at 243 nm. Determine the Protein-Bound Drug Using the Following Formula

Amount of the protein bound drug ¼

AbD
AbPD þ AbP Molar absorbtivity of drug

where absorbance of free-drug is AbD, the absorbance of protein-bound-drug is AbPD, and the absorbance of drug-unbound protein (free protein) is AbP.

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On the other hand, prepare the calibration curve, and determine the drug concentration.

13.1.4 Drug Metabolism 13.1.4.1 Background The fundamental objective of drug metabolism is to form water-soluble metabolites to eliminate the drugs quickly from our bodies. The drug metabolism primarily occurs in various organs. The liver is the principal and significant organ of drug metabolism. In the liver, the drug is metabolized by various enzymes, commonly called hepatic drug-metabolizing enzymes. They are called hepatic drugmetabolizing enzymes as they have been discovered concerning drug metabolism in the liver. Once the drug is taken orally, it reaches the liver through the hepatoportal route. Drug metabolism occurs mostly by enzymatic oxidation, reduction, hydrolysis, and conjugation reactions by specific hepatic enzymes—the Phase I and Phase II types. Phase I enzymes such as cytochrome monooxygenases, aryl hydrocarbon hydroxylases, alcohol dehydrogenase, and esterase mainly involve oxidation, reduction, and hydrolysis types of reaction. On the other hand, Phase II drug-metabolizing enzymes such as uridine diphosphate (UDP)-glucuronyl transferase, glutathione S-transferase, and N-acetyl transferase mostly engage in conjugation reactions. However, drug metabolism takes place in other organs, too. Some prodrugs become active after drug metabolism. However, the body primarily uses drug-metabolism machinery for the removal of drugs from it. The experimental model of drug metabolism 1. In Vitro (a) Cell culture (primary cell culture of hepatocyte, Chang liver cells, WRL-68 cells) (b) Recombinant cells (cells artificially manipulated with human cytochrome gene, e.g., CYP2C18 in CHL-CYP2C18 cell line) (c) The enzyme (isolated enzyme) (d) Cellular fraction [preparation of microsome/S9 fraction (as defined by US National Library of Medicine and is used to estimate metabolism of drugs and xenobiotics)] (e) Organ parts 2. In Vivo (a) In humans (b) Humanized mouse (transgenic) (c) Rats, mice, and dogs (d) Primates (e.g., monkeys) 3. In Situ Model Perfused organs 4. In silico Using various software

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Cell culture model Add experimental substrate in the media, and allow it to get metabolized by the cells/enzymes. Analyze media and cells to quantify metabolites.

13.1.4.2 Experiment No. 5 13.1.4.2.1 Objective Drug biotransformation study by liver microsome 13.1.4.2.2 Methods Add calcitriol (1,25-dihydroxycholecalciferol) (which is involved in many gene expressions by activating nuclear receptors), 1 phosphate buffer (pH 7.4), NADPH, liver microsome, and the substrate (drug), and incubate the mixture at 37  C for 30 min in a CO2 incubator. 1. 2. 3. 4. 5. 6. 7.

Add acetonitrile to stop the reaction. Add internal standard. Vortex briefly. Carefully decant the organic layer. Dry in a vacuum drier. Reconstitute the dry extracted materials with methanol. Identify the metabolites against an authentic standard using LC-MS/MS.

13.1.5 One-Compartment Model Following an Intravenous Bolus Dose Administration 13.1.5.1 Background Potassium permanganate is here used as a model drug, and the dose is considered an intravenous bolus dose for the experiment. A constant-head water reservoir that maintains a constant flow of water to the plasma beaker and from the plasma beaker to the simulated urine collecting measuring cylinder (Fig. 13.4) is required to obtain first-order kinetics of changes of drug diffusion in the plasma beaker and the urine collecting measuring cylinder, for the model. The apparatus is designed with a beaker and a measuring cylinder connected by a bending hollow glass tube. The plasma beaker is connected to a constant-head water reservoir, and a flow rate of water is maintained at a fixed rate within 15
25 mL/min. This maintains a steady flow of water through the plasma beaker to the measuring cylinder. The bolus dose of the drug (5 mL solution containing 250 mg of drug) is added to the plasma beaker and stirred steadily using a magnetic bead and a magnetic stirrer. A dry measuring cylinder is immediately placed as a collecting cylinder for the simulated urine by replacing the cylinder that has been existed. The samples are periodically withdrawn from the plasma beaker and from the simulated urine collecting measuring cylinder. The samples may then be estimated

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Fig. 13.4 An assembly for studying drug distribution and elimination in a one-compartment open model

by spectrophotometric analysis at 530 nm or any other analytical methods such as HPLC to obtain drug concentration in the samples. 13.1.5.1.1 Experiment No. 6 Objectives

(a) Determination of the first-order kinetic equation of reduction of drug concentration in simulated plasma, (b) determination of the first-order kinetic equation of change of drug concentration in urine, (c) determination of rate of elimination of the drug, and (d) determination of elimination half-life of the drug Materials and equipment

Potassium permanganate, volumetric flasks (100 mL), beakers (500 mL), measuring cylinder, hot plate with a magnetic stirrer, bent hollow glass tubes, a constant-head water reservoir, and spectrophotometer Methods

(a) Prepare the calibration curve of the drug.

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(b) (c)

(d)

(e)

(f) (g) (h)

(i)

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The calibration curve should be prepared in the following way. Dissolve an accurately weighed amount (say, 100 mg) of the drug in water in a volumetric flask (say 100 mL) to prepare a stock solution (1 mg/mL). Using the stock solution, prepare numbers of standard drug solutions with different concentrations (in at least three sets) by diluting the sample of different drug concentrations taken from the stock solution. For example, each 1
10 mL stock solution is diluted to 100 mL in separate volumetric flasks to obtain standard drug concentrations, 10
100 μg/mL. If the working solution is not water, you should use the solvent of the standard solution as the working solution only. Then, read spectrophotometrically the absorbance of each sample at the drug absorption maxima (also known as λmax, or lambda maxima). When the λmax value is unknown, a minute quantity of drug should be dissolved to form a dilute drug solution. It is then scanned for absorbance by spectrophotometer to determine the value of the absorbance maxima of the drug. Using a constant-head water reservoir, maintain a fixed water flow rate (between 15 and 25 mL/min) to the plasma beaker. Turn on the magnetic stirrer. Add the bolus dose (here 5 mL that contains 250 mg of drug in the plasma chamber). Replace the existing measuring cylinder quickly with a new one to collect the drug-containing simulated urine. Withdraw 5 mL (may be any fixed volume between 2 and 5 mL) of the sample in each case from the plasma beaker at the different time intervals, such as from 5 min up to 65 min with an interval of 5 min, using Pasteur pipette or pipette with sample withdrawal system (such as Steripette). Collect them in the pre-labeled test tubes. Withdraw 5 mL (may be any fixed volume between 2 and 5 mL) of the sample in each case from the measuring cylinder at the different time intervals, such as from 7.5 min up to 67.5 min with an interval of 5 min, using a pipette with sample withdrawal system (such as Steripette) or Pasteur pipette. Collect them in the pre-labeled test tubes. Read the absorbance of each sample at the absorbance maxima (530 nm) by spectrophotometer. Calculate drug concentration in the samples from the calibration curve drawn previously. Plot simulated plasma drug concentration (cp) versus time (t) data on a semi-log graph paper. Determine the rate constant (K ) and the intercept on y-axis (cp-axis) on a linear scale. Develop the kinetic equation of the line using the formula, y ¼ mx + c. Note the rate of reduction of drug concentration (K ) in the plasma beaker. Report elimination rate constant, and find the half-life of the drug. Plot drug concentration in urine versus time (t) data on a semi-log graph paper. Determine the rate constant (K ) and intercept on the y-axis on a linear scale. Develop the kinetic equation of the line using the formula, y ¼ mx + c. Report the rate constant.

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13.1.6 One-Compartment Model Following an Intravenous Infusion Administration 13.1.6.1 Background Potassium permanganate is considered as a model drug here, and the dose is considered an intravenous infusion for the experiment only. A constant-head water reservoir maintains a constant flow of water to the plasma beaker and from the plasma beaker to the simulated urine-collecting measuring cylinder (Figs. 13.4 and 13.5) is required to obtain first-order kinetics of changes of drug diffusion in the plasma beaker and the urine-collecting measuring cylinder of the model. The apparatus is designed with a beaker and a measuring cylinder connected by a bending hollow glass tube. The beaker is used as a plasma beaker, and the simulated urine is collected in the measuring cylinder. An infusion bottle containing water/ normal saline is connected to act as a water reservoir, and a flow rate of water is maintained at a fixed rate within 15
25 mL/min. It maintains a steady flow of water through the plasma beaker to the measuring cylinder. The infusion dose (0.5 mL/ min) of the drug solution (30 mL solution containing 150 mg of drug) over 1 h is added to the plasma beaker and stirred steadily using a magnetic bead and a magnetic

Fig. 13.5 A simple assembly for studying drug distribution and elimination in a one-compartment open model

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stirrer. The measuring cylinder is immediately replaced with a dry measuring cylinder for collecting the simulated urine. To obtain sample drug concentrations, the periodically collected samples from the plasma beaker and the simulated urine samples in the collecting measuring cylinder may be estimated by spectrophotometric analysis at 530 nm. 13.1.6.1.1 Experiment No. 7 Objectives

(a) Determination of the first-order kinetic equation of reduction of drug concentration in simulated plasma, (b) determination of the first-order kinetic equation of change of drug concentration in urine, (c) determination of rate of elimination of the drug, and (d) determination of elimination half-life of the drug Materials and Equipment

Potassium permanganate, volumetric flasks (100 mL), beakers (500 mL), measuring cylinder, hot plate with a magnetic stirrer, bent hollow glass tubes, a constant-head water reservoir, and spectrophotometer Methods

(a) Prepare a calibration curve of the drug. (b) The calibration curve should be prepared in the following way. Dissolve an accurately weighed amount (say, 100 mg) of the drug in water in a volumetric flask (say 100 mL) to prepare a stock solution (1 mg/mL). Using the stock solution, prepare numbers of standard solutions of the drug with different concentrations (in at least three sets) by diluting the sample of different drug concentrations taken from the stock solution. For example, each 1
10 mL stock solution is diluted to 100 mL in separate volumetric flasks to obtain standard drug concentrations, 10
100 μg/mL. If the working solution is not water, you should only use the solvent of the standard solutions as the working solution. Then, read spectrophotometrically the absorbance of each sample at the absorption maxima (also known as λmax, or lambda maxima) of the drug. When the λmax value is not known, a minute quantity of drug should be dissolved to form a dilute drug solution, and it is then scanned for absorbance by spectrophotometer to determine the value of the absorption maxima of the drug. (c) Using the constant-head water reservoir, maintain a fixed water flow rate (between 15 and 25 mL/min) to the plasma beaker. (d) Turn on the magnetic stirrer. Add the infusion dose (0.5 mL/min) of the drug solution (30 mL solution containing 150 mg of drug) to the plasma chamber over 1 h. Replace the existing measuring cylinder quickly with a new dry one to collect the drug-containing simulated urine.

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(e) Withdraw 5 mL (may be any fixed volume between 2 and 5 mL) of the sample in each case from the plasma beaker at different time intervals, such as from 5 min up to 65 min with an interval of 5 min, using Pasteur pipette or pipette with sample withdrawal system (such as Steripette). Collect them in the pre-labeled test tubes. (f) Withdraw 5 mL (may be any fixed volume between 2 and 5 mL) of the sample in each case from the measuring cylinder at the different time intervals, such as from 7.5 min up to 67.5 min with an interval of 5 min, using a pipette with sample withdrawal system (such as Steripette) or Pasteur pipette. Collect them in the pre-labeled test tubes. (g) Read the absorbance of each sample at the absorbance maxima (530 nm) by spectrophotometer. (h) Calculate drug concentration in the samples from the calibration curve drawn previously. (i) Plot simulated plasma drug concentration (cp) versus time (t) data on a semi-log graph paper. Determine the rate constant (K ) and intercept on y-axis on a linear scale. Develop the kinetic equation of the line. Note the rate of reduction of drug concentration in the plasma beaker. Determine elimination rate constant and find the half-live of the drug. (j) Plot drug concentration in urine versus time data on a semi-log graph paper. Determine the rate constant and intercept on y-axis on a linear scale. Develop the kinetic equation of the line. Note the rate constant.

13.1.7 A Two-Compartment Model Following the Administration of an Intravenous Bolus Dose 13.1.7.1 Background In a multi-compartment system, the central compartment is connected to the peripheral compartments. The model is designed with two compartments where the central compartment is connected to a peripheral compartment, and the drug is eliminated from the peripheral compartment (Fig. 13.6). The peripheral compartment is connected to a measuring cylinder that measures the volume of the eliminated (mimicking urine clearance) liquid. A constant-head water reservoir is connected to the central compartment for water supply with a constant flow rate between 15 and 25 mL/min (approximately 300
500 drops/min). Each compartment has a magnetic bead, and the compartments are placed on magnetic stirrers with the heating facility to stir the liquid of the compartments and heat the liquid at 37  C. The bolus dose of potassium permanganate as a model drug (5 mL solution containing 250 mg of drug) is added to the plasma beaker and stirred steadily using a magnetic bead and a magnetic stirrer. A dry measuring cylinder is immediately placed as a collecting cylinder for the simulated urine by replacing the cylinder that has been existed. The samples are periodically withdrawn from the peripheral chamber and from the simulated urine collecting measuring cylinder. The samples

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Fig. 13.6 An assembly for studying drug distribution and drug elimination of the peripheral compartment in a two-compartment open model system

may then be estimated by spectrophotometric analysis at 530 nm. The concentrations should be read from the calibration curve. 13.1.7.1.1 Experiment No. 8 Objectives

(a) Determination of the first-order kinetic equation of drug concentration for the peripheral compartment, (b) determination of the first-order kinetic equation of change of drug concentration in urine, (c) determination of the rate of elimination of the drug, and (d) determination of elimination half-life of the drug Materials and Equipment

Potassium permanganate, volumetric flasks (100 mL), beakers (500 mL), measuring cylinder, hot plate with a magnetic stirrer, a constant-head water reservoir, and spectrophotometer Methods

Prepare the calibration curve of the drug. The calibration curve should be prepared in the following way. Dissolve an accurately weighed amount (say, 100 mg) of the drug in water in a volumetric flask (say 100 mL) to prepare a stock solution (1 mg/mL). Using the stock solution, prepare numbers of standard solutions of the drug with different concentrations (in at least three sets) by diluting the sample of different drug concentrations taken from the stock solution. For example, each 1
10 mL stock

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solution is diluted to 100 mL in separate volumetric flasks to obtain standard drug concentrations, 10
100 μg/mL. If the working solution is not water, you should use the solvent of the standard solution as the working solution only. Then, read spectrophotometrically the absorbance of each sample of different concentrations at the absorption maxima (also known as lambda maxima) of the drug. When the λmax value is unknown, a minute quantity of drug should be dissolved to form a dilute drug solution. It is then scanned for absorbance by spectrophotometer to determine the value of the absorbance maxima of the drug. (a) Maintain a fixed water flow rate (between 15 and 25 mL/min) to the plasma beaker using the computer-controlled pump. (b) Turn on the magnetic stirrer. Add the bolus dose (here 5 mL that contains 250 mg of drug in the plasma chamber). Replace the existing measuring cylinder quickly with a new dry one to collect the drug-containing simulated urine. (c) Withdraw 5 mL (may be any fixed volume between 2 and 5 mL) of the sample in each case from the peripheral chamber at the different time intervals, such as from 5 min up to 65 min with an interval of 5 min, using Pasteur pipette or pipette with sample withdrawal system (such as Steripette). Collect them in the pre-labeled test tubes. (d) Withdraw 5 mL (may be any fixed volume between 2 and 5 mL) of the sample in each case from the measuring cylinder at the different time intervals, such as from 7.5 min up to 67.5 min with an interval of 5 min, using a pipette with sample withdrawal system (such as Steripette) or Pasteur pipette. Collect them in the pre-labeled test tubes. (e) Read the absorbance of each sample at the absorbance maxima (530 nm) by spectrophotometer. (f) Calculate drug concentration in the samples from the calibration curve drawn previously. (g) Plot simulated drug concentration at the peripheral compartment versus time data on a semi-log graph paper. Determine the rate constant and intercept on yaxis on a linear scale. Develop the kinetic equation of the line. Note the rate of reduction of drug concentration in the plasma beaker. Determine elimination rate constant, and find the half-life of the drug. (h) Plot drug concentration in urine versus time data on a semi-log graph paper. Determine the rate constant and intercept on y-axis on a linear scale. Develop the kinetic equation of the line. Note the rate constant.

13.1.8 A Multi(Three)-Compartment Model Following the Administration of an Intravenous Infusion 13.1.8.1 Background In a multi-compartment system, the central compartment is connected to the peripheral compartments. The model is here designed as three compartments where the central compartment is connected to two peripheral compartments, and the drug is eliminated from one of the peripheral compartments (Fig. 13.7). One of the

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Fig. 13.7 An assembly for studying drug distribution and elimination in a three-compartment open model system. The drug is eliminated from a peripheral compartment

peripheral compartments has the outlet (for the elimination process) connected to a measuring cylinder that measures the volume of the eliminated (mimicking urine clearance) liquid. A fixed-rate water supply is connected to the central compartment for water supply with a constant flow rate between 15 and 25 mL/min (approximately 300
500 drops/min). Each compartment has a magnetic bead, and the compartments are placed on magnetic stirrers with the heating facility to stir the liquid of the compartments and heat the liquid at 37  C. The infusion dose (0.5 mL/min) of potassium permanganate solution (30 mL solution containing 150 mg of drug) over 1 h is added to the plasma beaker and stirred steadily using a magnetic bead and a magnetic stirrer. The measuring cylinder is immediately replaced with a dry measuring cylinder for collecting the simulated urine. The samples are periodically withdrawn from the peripheral chamber without the elimination outlet and the simulated urine-collecting measuring cylinder. The samples may then be estimated by spectrophotometric analysis at 530 nm. The concentrations should be read from the calibration curve. 13.1.8.1.1 Experiment No. 9 Objectives

(a) Determination of the first-order kinetic equation of change of drug concentration in the peripheral compartment not having the elimination outlet, (b) determination of

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the first-order kinetic equation of change of drug concentration in urine, (c) determination of rate of elimination of the drug, and (d) determination of elimination halflife of the drug Materials

Potassium permanganate, volumetric flasks (100 mL), beakers (500 mL), measuring cylinder, hot plate with a magnetic stirrer, a constant-head water reservoir, and spectrophotometer Methods

(a) Prepare the calibration curve of the drug. The calibration curve should be prepared in the following way. Dissolve an accurately weighed amount (say, 100 mg) of the drug in water in a volumetric flask (say 100 mL) to prepare a stock solution (1 mg/mL). Using the stock solution, prepare numbers of standard solutions of the drug with different concentrations (in at least three sets) by diluting the sample of different drug concentrations taken from the stock solution. For example, each 1
10 mL stock solution is diluted to 100 mL in separate volumetric flasks to obtain standard drug concentrations, 10
100 μg/mL. If the working solution is not water, you should use the solvent of the standard solution as the working solution only. Then, read spectrophotometrically the absorbance of each sample of different concentrations at the absorption maxima of the drug. When the λmax value is not known, a minute quantity of drug should be dissolved to form a dilute drug solution, and it is then scanned for absorbance by spectrophotometer to determine the value of the absorbance maxima of the drug. Then plot absorbance against the concentration to develop the calibration curve. (b) Using a fixed rate water supply, maintain a fixed water flow rate (between 15 and 25 mL/min) to the plasma beaker. (c) Turn on the magnetic stirrer. Add the infusion dose (0.5 mL/min) of the drug solution (30 mL solution containing 150 mg of drug) to the plasma chamber over 1 h. Replace the existing measuring cylinder quickly with the new dry one to collect the drug-containing simulated urine. (d) Withdraw 5 mL (may be any fixed volume between 2 and 5 mL) of the sample in each case from the peripheral chamber without the elimination outlet at the different time intervals, such as from 5 min up to 65 min with an interval of 5 min, using Pasteur pipette or pipette with sample withdrawal system (such as Steripette). Collect them in the pre-labeled test tubes. (e) Withdraw 5 mL (may be any fixed volume between 2 and 5 mL) of the sample in each case from the measuring cylinder at the different time intervals, such as from 7.5 min up to 67.5 min with an interval of 5 min, using a pipette with sample withdrawal system (such as Steripette) or Pasteur pipette. Collect them in the pre-labeled test tubes. (f) Read the absorbance of each sample at the absorbance maxima (530 nm) by spectrophotometer.

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(g) Calculate drug concentration in the samples from the calibration curve drawn previously. (h) Plot simulated drug concentration at the peripheral compartment versus time data on a semi-log graph paper. Determine the rate constant and intercept on yaxis on a linear scale. Develop the kinetic equation of the line. Note the rate of reduction of drug concentration in the plasma beaker. Determine elimination rate constant and find the half-life of the drug. (i) Plot drug concentration in urine versus time data on a semi-log graph paper. Determine the rate constant and intercept on y-axis on a linear scale. Develop the kinetic equation of the line using the formula, y ¼ mx + c. Note the rate constant.

13.1.9 Determination of Drug Absorption In Vitro 13.1.9.1 Background Drug absorption is essential for systemic drug action, while the drug is administered by any route other than the intravenous route. After the absorption, the drug molecules must reach the blood and achieve the minimum effective concentration level to provide drug action. Drug absorption is a complex process that starts from the absorption site to the blood. The process is associated with the physicochemical and biopharmaceutical features of the drug. As stated earlier, drug absorption or rate of drug absorption cannot be determined from the blood/plasma drug concentration values, as the data do not truly reflect drug absorption. During the drug absorption, different other processes, such as drug protein binding, drug distribution, elimination, drug metabolism, etc., take place simultaneously. Therefore, drug absorption or rate of drug absorption is determined from the drug amount remaining to be absorbed in various time points or by taking the residual drug concentrations. The ileum happens to be the largest part of the small intestine where the most absorption of nutrients, vitamins, bile salts, and drug molecules takes place before traveling into the large intestine. It has a larger surface with a more permeable membrane. A portion of ileum is used. It is everted (keeping inside out) and clipped or tied at both the ends eventually after being filled with deionized water (Fig. 13.8). Then it is dipped into a drug solution (reservoir) from where the drug is absorbed through the everted ileum to the water inside. With the increasing time, more drugs are absorbed, and the drug concentration gradually decreases in the drug reservoir. Drug concentration in the reservoir, thus, shows the drug concentration remaining to be absorbed. At the different time intervals, the samples taken from the drug solution reservoir are analyzed, and % drug remaining to be absorbed is plotted against time following the Wagner-Nelson method to determine the absorption rate (ka).

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Fig. 13.8 In vitro drug absorption using everted ileum. A portion of rat or fowl ileum is used. It is everted (keeping inside out) and clipped or tied at both the ends eventually after being filled with deionized water. Then it is dipped into a drug solution (reservoir) from where the drug is absorbed through the everted ileum into the water inside

13.1.9.1.1 Experiment No. 10 Objective

To quantify experimental data, draw a graph, and determine the rate of drug absorption by the Wagner-Nelson method using the ileum of experimental animals or fowl Materials

4
5 cm long ileum, distilled water, and Ringer’s solution Methods

1. Collect 4
5 cm long ileum from experimental animals (e.g., Sprague-Dawley rats, albino rats) or fowl (e.g., hens, cocks). 2. Clean it, and wash it with distilled water.

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3. 4. 5. 6. 7.

8. 9. 10. 11.

12. 13.

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Rinse with Ringer’s solution. Dip in Ringer’s solution. Gently and carefully evert the ileum. Clip or tie one end. Fill it with water or phosphate buffer (pH 7.4) or phosphate-buffered saline (pH 7.4) [if the drug is dissolved in phosphate buffer (pH 7.4) or phosphatebuffered saline (pH 7.4)]. Clip or tie the other end. Take the oxygenated drug solution in a beaker (acts as drug reservoir) with the known drug concentration. Dip the filled ileum sac in the reservoir. Heat the drug solution at 37  C and stir at 10
15 rpm. Withdraw the definite volume of drug-reservoir samples at the preselected time points, and add the equal volume of water/buffer (without drug) to the solution in the beaker. Analyze them by suitable analytical technique (e.g., spectrophotometry, HPLC, etc.). Plot percentage of drug remaining to be absorbed versus time, and determine rate of drug absorption.

13.1.10 Cellular Uptake of Drug 13.1.10.1 Background For functioning a chemical moiety inside a cell, its cellular/tissue internalization is essential. It can be investigated by both in vitro and in vivo methods. There are various popular methods available for quantifying the cellular uptake of drugs. LC-MS/MS, HPLC, flow cytometer (FACS), spectrofluorimetry, confocal microscopy, and fluorescent microscopy followed by quantification of fluorescence by software tool are some popular methods for quantifying cellular internalization of drug. Drug solution/suspension with a specific amount (dose) is incorporated into the medium of the culture Petri dish where the cells have been seeded. It is done according to the treatment plan (number of treatments/doses, time points at which samples are analyzed, etc.) along with their replica (at least three sets for each treatment at each time point). Different drug concentrations are used to determine concentration-dependent cellular uptake, whereas to assess time-dependent cellular uptake, the same dose is used for the treatment with variable duration (different time points). Both the concentration and time can be varied to know time-dependent/ concentration-dependent cellular uptake in a single experiment. Fluorescent drug or fluorescent agent (FITC/Biotin)-conjugated drug internalization can be quantified by flow cytometer, spectrofluorimetry, confocal microscopy, and fluorescent microscopy. Cellular internalization of a radiolabeled drug can be quantified by scintigraphy. After the completion of drug treatment, the cells are washed a number of times (three to five times) with ice-cold buffer, trypsinized, and separated. Then they are collected in a centrifuge tube. The cells are then lysed with the lysis buffer. Then

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perchloric acid/trichloroacetic acid is added to precipitate proteins. The sample is then centrifuged, the supernatant is carefully decanted, the drug content is analyzed by HPLC or LC-MS/MS using a standard method, and the concentration is determined against the standard drug sample. In the case of assessing drug-tissue internalization in vivo, fluorescent drug or fluorescent agent-conjugated drug or radiolabeled drug can be assessed by whole-body imaging to quantify the fluorescence/radiation by imaging software tool/scintigraphy. Otherwise, for non-fluorescent or unlabeled drugs, after dissecting the animals, the weighed amount of tissue sample is homogenized, the cells are lysed with lysis buffer, protein is precipitated by adding perchloric acid/trichloroacetic acid followed by centrifugation, and the drug content is analyzed from the supernatant by HPLC or LC-MS/MS method. 13.1.10.1.1 Experiment No. 11 Objective

Cellular drug uptake analysis by confocal microscope Methods

1. Seed the experimental cells on a coverslip, placed in a 35 mm tissue culture dish supplemented with the cell culture medium. For example, seed 1  105 C6 rat glioma cells on a coverslip, placed in a 35 mm tissue culture dish with supplemented DMEM medium (with 10% fetal calf serum). 2. Wash the cells with ice-cold phosphate buffer-saline (pH 7.4). Treat the cells with FITC-labeled drug/drug product (formulation, here we used FITC-labeled nanoparticle) with a dose [here, 50 nM [a dose response should be determined early by 3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide (MTT) assay] for a preseletced duration (chosen here, 1
6 h) in a serum-free medium (here DMEM). If the drug itself is a fluorescent compound (e.g., doxorubicin), it can be used directly without its conjugation with a fluorescent agent. 3. Incubate in a CO2 incubator for the mentioned duration. 4. Remove the treated medium once the incubation period is over. 5. Wash the cells with ice-cold phosphate-buffered saline (pH 7.4). 6. Fix the cells with 70% ethanol. 7. Process for counterstaining with 300 nM DAPI, and mount the coverslip on a slide for observation under a confocal laser microscope. 8. Capture image of cells (Fig. 13.9) using channels of FITC (excitation/emission 495 nm/519 nm) and DAPI (excitation/emission 359 nm/461 nm). 9. Quantify fluorescence intensity using the software too. 10. The cells without any treatment should be run parallelly as control. 13.1.10.1.2 Experiment No. 12 Objective

Cellular drug uptake analysis by flow cytometric analysis

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Fig. 13.9 Cellular internalization of FITC-labeled drug product (green color) by c6 glioma cells cultured in DMEM (Dulbecco’s Modified Eagle Medium)—supplemented with 10% fetal calf serum and treated with 50 nM drug product. The cells were nuclear-stained with DAPI (blue color). The photograph of the cells incubated up to 4 h was taken with a confocal microscope

Methods

Seed the experimental cells in the cell growth medium in 60 mm tissue culture dishes. For example, seed 1  106 HCT116 cells in Dulbecco’s Modified Eagle Medium (DMEM) in 60 mm tissue culture dishes, and incubate overnight in CO2 incubator. 1. Treat the cells with a required dose of FITC-labeled drug (say, 150 μg/mL medium, dose response is determined by MTT assay), and incubate for the necessary duration, as considered here 1
6 h, in a CO2 incubator. 2. Run a control group of cells treated with the media alone in parallel. 3. Remove the treated medium after the incubation period. 4. Wash the cells three to five times with ice-cold phosphate-buffered saline (pH 7.4). 5. Detach the cells by trypsinization, and redisperse them in a fresh medium. 6. Analyze the cell suspension in a FACS instrument, and process the data using the machine software for quantification of FITC-labeled drug.

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Note: For Tissue Drug Uptake

In the cases of in vivo model, the whole animals may be imaged by the fluorescent imager or radiolabel drugs by gamma scintigraphic imaging to know drug distributions in different tissues or organs in terms of percentage of the total quantity of drug (dose) administered. Otherwise, after the completion of the treatment duration, the animals may be dissected. The organs may be removed, a portion of the organs is weighed and homogenized in ice-cold 0.25 M aqueous sucrose solution. The cells are then lysed with the lysis buffer. Then perchloric acid/ trichloroacetic acid is added to precipitate proteins. The drug is extracted using a suitable organic solvent. Then the solvent is evaporated, and the drug is dissolved in the mobile phase and quantified by HPLC or LC-MS/MS.

13.1.11 In Vitro Drug Skin Permeation from Transdermal Drug Delivery System (Patch) 13.1.11.1 Background Transdermal drug delivery systems are intended to deliver drugs through the intact skin to maintain sustained plasma drug levels for systemic drug action. Transdermal drug delivery systems include transdermal patches, transdermal hydrogel, transdermal gels, etc. Once the formulation is applied to the skin, sweat and sebum accumulate locally to neutralize the heat effect produced by the occlusive layer of the formulation. The sweat and sebum diffuse into the formulation matrix by capillary action, and the drug molecules diffuse out onto the skin surface through the biphasic liquid mixture. Drug permeates across the uppermost skin barrier layer, the stratum corneum, into the dermis and then to the systemic blood circulation. There are three distinct drug permeation routes through the skin. Drug mostly permeates through the transcellular route, followed by the intercellular and transappendageal routes in a much less quantity. Before testing the transdermal formulations on animals and then on the human volunteers, in vitro skin permeation data essentially guide us on the suitability of the formulation toward success. A skin permeation study is conducted on cadaver skin, animal skins that include rat, mouse, pork, etc., and cellophane. In vitro, skin permeation study liquids include various receptor fluids, for example, phosphate buffer, pH 7.4, phosphate buffer saline, pH 7.4, and phosphate buffer, pH 7.4, or phosphate buffer saline, pH 7.4 with polyethylene glycol 400 [10
30%, v/v, to provide biphasic character, hydrophilic (sweat) and hydrophobic (sebum), to the liquid]. Various skin-permeation diffusion cells such as Franz diffusion cell, Keshary-chein diffusion cell, and different modified diffusion cells are used. The study is conducted at 37  C that provides the skin temperature at 34  C in vitro. The receptor medium is stirred by a magnetic bead using a magnetic stirrer. A diffusion cell consists of two different compartments—a donor compartment and a receptor compartment. A formulated patch is placed in the donor compartment on the upper (stratum corneum) side of the skin in such a manner that the drug-release side of the patch remains in touch with the skin and the inner side (dermis side) of the skin

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Fig. 13.10 A diffusion cell assembly for studying in vitro skin permeation of drugs

remains in contact with the drug release medium in the receptor compartment (Arora and Mukherjee 2002). The temperature of the receptor compartment is maintained by circulating water from a water bath at 37  0.5  C to its external jacket (Fig. 13.10). Samples are withdrawn at different intervals, and the same volume is replaced with a fresh medium on each occasion to maintain the sink condition.

13.1.11.1.1 Experiment No. 13 Objective

To study in vitro skin permeation of a drug and to test the drug release kinetics Materials

Diffusion cell, permeation membrane (skin, cellophane), magnetic stirrer, magnetic bead, temperature-controlled water bath, small pump, rubber tubes, solvents, and transdermal patches

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Methods

Prepare transdermal patches according to the methods available (Arora and Mukherjee 2002; Mukherjee et al. 2005), select the receptor study fluid, and method of drug analysis accordingly. Prepare the Calibration Curve

Prepare a stock solution by dissolving an exact amount of drug in the solvent mentioned in the method. Prepare different standard solution concentrations in 25 mL volumetric flasks by diluting the different volumes of the stock solution with the solvent. Read the absorbance against the concentrations spectrophotometrically at the absorption maxima, keeping the solvent without the drug as blank. Plot absorbance against concentrations to develop the calibration curve. Skin Permeation

1. Attach cadaver/rat or mouse skin/cellophane at the donor compartment. For skin, keep the stratum corneum side upward. 2. Fill the receptor compartment with skin permeation study liquid (receptor medium). 3. Place the donor compartment on the receptor compartment, and clip them or attach them with glue tape. The dermis side of the skin/cellophane should remain in touch with the receptor fluid. 4. Circulate warm water (37  C) through the external jacket. 5. Place the patch in the receptor compartment on the skin, keeping the drug release side toward the stratum corneum. 6. Withdraw 1 mL test sample from the receptor compartment, and replace the same volume with the fresh receptor medium in each case through the sampling port at the different time points, namely, initially after every half an hour up to 4 h and then at an interval of 1 h till 8 h, and then at 10, 12, 16, 24, 48, and 72 h, and store them in a refrigerator. 7. Analyze the samples spectrophotometrically at the absorption maxima (λmax), keeping the receptor medium without the drug as blank. Find out the concentrations. 8. Plot the cumulative drug concentrations permeated through the skin against time, and test in various kinetic models such as zero-order, first-order, and Higuchi kinetic model to determine respective regression coefficient (R2) value of three such sets.

13.1.12 Drug Assay from a Tablet Dosage Form Containing Paracetamol by HPLC Method 13.1.12.1 Background Acetaminophen (popularly known as paracetamol) is N-(4-hydroxyphenyl)acetamide or N-(4-hydroxyphenyl)ethanamide. The drug is used as an antipyretic (to reduce body temperature) and analgesic (to reduce pain). It is available in different dosage forms such as tablets, suspension, syrup, etc. The tablet form is

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the most common solid dosage form containing excipients and the active ingredient (s). The molecular weight of paracetamol is 151.16. Paracetamol is soluble in n-alcohols, but the solubility decreases with the increasing carbon chain lengths. Paracetamol has the approximate solubility 1.3 g/100 mL in water. As per various pharmacopoeias, if the drug content in paracetamol-tablets varies within 5%, of its dose, it is acceptable for consumption. It means that if the dose is 500 mg in a tablet, the permissible limit to pass a formulation for consumption by a patient is from 475 to 525 mg. To assay the drug from the tablet, the method to be employed is HPLC. Highperformance liquid chromatography (HPLC), also called high-pressure liquid chromatography, is an analytical technique to identify and quantify components from a mixture. HPLC is a technologically upgraded, advanced version of column chromatography (for details, see Chap. 10). The injected sample is carried in a flow of the mobile phase with high pressure through a separation column that contains a granulated solid adsorbent known as the stationary phase. While traveling through the column, the sample compounds separate on the stationary phase. The mobile phase elutes the compounds at different intervals based on their affinity to the stationary phase. The separated sample components then reach the detector that recognizes and distinguishes the signals on absorbed-light/refractive index/conductivity changes/size distribution of those compounds. Finally, computer software converts the signal to a chromatogram. One HPLC unit works with the following primary components: a column (the stationary phase), mobile phase solvent or solvent mixture, a pump, a sample injector, a detector, a waste collector, and data processor and recorder. 13.1.12.1.1 Experiment No. 14 Objective

To estimate paracetamol from a tablet dosage form by HPLC method (Youssef et al. 2019) and report the drug content Materials

Pure paracetamol drug, paracetamol tablets, solvents (HPLC grade water, HPLC grade methanol, HPLC grade acetic acid), HPLC unit with a detector and recorder Methods

1. Prepare the mobile phase by mixing HPLC grade water, methanol, and acetic acid at a ratio (v/v) 65: 30:5. Adjust the pH at 3.4. Keep the mixture in a tightly closed glass container. 2. Preparation of the standard solution: Weigh accurately 50 mg of pure paracetamol, and dissolve in the mobile phase in a 50 mL volumetric flask. Adjust the volume accurately. Mix it thoroughly, and filter by Whatman filter paper. Take 1 mL filtrate in a 25 mL volumetric flask, and adjust the volume up to 25 mL using the mobile phase and mix. Filter the solution using a membrane filter, and

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the sample is now ready for injection. Inject the sample into the HPLC unit, and take the reading. 3. Preparation of the test solution: Take five paracetamol tablets (containing 500 mg paracetamol each). Crush them, and blend them in a mortar with a pestle. Take 10 mg accurately weighed powder. Dissolve in 10 mL volumetric flask with the mobile phase. Accurately adjust the volume up to 10 mL. Filter by Whatman filter paper. Take 1 mL filtrate in a 25 mL volumetric flask, and adjust the volume up to 25 mL using the mobile phase and mix. Filter the solution using a membrane filter, and the sample is now ready for injection. Inject the sample into the HPLC unit, and take the reading at 257 nm. 4. Record the area under the curve (AUC) of the test sample and the AUC of the standard sample. 5. Calculate % purity using the formula: %purity ¼

AUC of the test  100 AUC of the standard

Take the average of three separate readings and use the average AUC in each case.

13.1.13 Estimation of Plasma and Urine Drug Concentrations by Reversed-Phase HPLC 13.1.13.1 Background In various disease states, for example, hepatic, renal, cardiac diseases, diabetes, etc., fluid accumulation takes place in our body. It often leads to edema. Frusemide is used for removal of such extra fluid accumulation. It is a diuretic drug and also administered to control blood pressure. 13.1.13.1.1 Experiment No. 15 Objective

To determine plasma and urine frusemide concentrations in volunteers treated with 40 mg frusemide single tablet by reversed-phase HPLC and calculate various pharmacokinetic parameters Methods Volunteers, Treatment, and Sample Collection Time Points

Quantification of the drugs in the plasma and urine samples of human volunteers can be conducted only after receiving the institutional ethical clearance. The volunteers should be informed about the objectives and the expected hazards related to the study. A single tablet (40 mg) should be taken orally with 250 mL of water after the overnight fasting of 10 h.

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After the drug (single dose) administration, the volunteers should take breakfast, lunch, and dinner at 2 h, 7 h, and 12 h after the drug single dose. Volunteers should take food and beverages without caffeine, alcohol, and they will not do any strenuous exercise. Blood samples (6 mL each sample) should be collected from the volunteers at 30 min, 1 h, 1 h 30 min, 2 h, 2 h 30 min, 3 h, 3 h 30 min, 4 h, 4 h 30 min, 5 h, 5 h 30 min, 6 h, 8 h, 12 h, and 16 h, after the drug administration. Urine samples should be collected before the drug administration and during first 1
10 h after the administration of the drug. The volume should be noted and stored at
20  C straightway. Drug Estimation The plasma and urine frusemide determination method by reversed-phase HPLC (Abou-Auda 1998) is given below. 1. Centrifuge the collected blood samples quickly at 4000 rpm for 10 min adding EDTA (anticoagulant). 2. Separate the plasma from each sample, and store at
20  C in a deep freezer (if necessary) for few days till further use. After Thawing the Samples 1) Add 100 μL acetonitrile containing the internal standard (clonazepam, 9 μg/mL) to 100 μL plasma (acetonitrile deproteinizes the plasma). 2) Centrifuge at 4000 rpm at 4  C for 10 min. 3) Inject 25 μL aliquot from the supernatant into HPLC. 4) Detect with variable wavelength (190
660 nm) detector. 5) Monitor the column effluent at 275 nm at the ambient temperature. HPLC Condition

Column M-bondpack C18 reversed-phase column, 10 μm average silica particle size Mobile phase 30% acetonitrile in 0.01 (M) in sodium acetate buffer, pH 3.4 Limit of detection 5 ng/mL Calibration curve concentration range: 0.05
2.5 μg/mL Urine: 1. 2. 3. 4. 5. 6.

Spike the IS (1.8 μg/mL) with 0.5 mL aliquot of urine. Add 10 mL diethyl ether and mix briefly in a vortex-mixer. Centrifuge at 1000 g for 10 min. Remove ether to evaporate to dryness using nitrogen gas flow. Reconstitute with 200 μL mobile phase. Inject 25 μL of it to HPLC. Limit of quantification: 20 ng/mL

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Calibration curve: Concentration range: 0.5
50 5 μg/mL Determine: ðAUCÞ0
1 ¼ ðAUCÞ0
t þ

ct ke

or ðAUCÞ0
1 ¼ ðAUCÞ0
t þ 1:44 t 1=2 ct Dose Total operative body clearance ¼ CL F ¼ AUC , where F is oral bioavailability and cmax, Tmax, and KE.

13.1.14 Compare the Bioavailability of Ranitidine (300 mg) Tablets of Two Different Brands in Human Volunteers (Bioequivalence Study) 13.1.14.1 Background Ranitidine is a histamine-2 receptor blocker. It prevents secretion of gastric acid. It is used for the treatment of peptic ulcer, duodenal ulcer, gastric acid reflux disease, Zollinger-Ellison syndrome, etc. Ranitidine (300 mg) tablets are available in the different brand names (e.g., Histac, Aciloc, Rantac, etc.). By comparing the bioavailability of the drug from two of its branded products and their pharmacokinetic parameters, the rate and extent of drug absorption in the two formulations can be compared. If they do not significantly differ upon their administration with an identical strength of the same active ingredient in a single dose (here) under the same experimental conditions, we can consider the drug products as bioequivalent. 13.1.14.1.1 Experiment No. 16 Objectives

To compare the bioavailability of ranitidine (300 mg) tablets of two different brands in human volunteers, determine plasma drug concentrations by HPLC, and compare the pharmacokinetic parameters Methods Volunteers

A single tablet should be taken orally with 250 mL of water after the overnight fasting of 10 h. After the drug (single dose) administration, the volunteers should take breakfast, lunch, and dinner at 2 h, 7 h, and 12 h. Volunteers should take food and beverages without caffeine, alcohol, and they will not do any strenuous exercise.

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Blood samples (6 mL each sample) should be collected from the volunteers at 30 min, 1 h, 1 h 30 min, 2 h, 2 h 30 min, 3 h, 3 h 30 min, 4 h, 4 h 30 min, 5 h, 5 h 30 min, 6 h, 8 h, 12 h, and 16 h after the drug administration. 1. Centrifuge the collected blood samples rapidly at 4000 rpm for 10 min adding EDTA (anticoagulant). 2. Separate the plasma from each sample, and store at
20  C in a freezer (if necessary) for a few days till further use. HPLC detection method with UV detector at 330 nm (Bawazir et al. 1998). 1. Add the internal standard (procainamide) into plasma, and mix with vortexer briefly. 2. Extract the drug and the internal standard (procainamide) from the alkaline plasma into dichloromethane. HPLC Conditions

Column: M-bondpack C18 column Injection volume: 25 μL Study temperature: Ambient temperature Mobile phase: Acetonitrile—0.1 (M) potassium hydrogen phosphate (8: 92 v/v) pH 5.1 (to be adjusted with phosphoric acid) Flow rate: 2.0 mL/min Standard solution concentration range: 20–2000 ng/mL Limit of quantification: 20 ng/mL Draw plasma concentration versus time curves, and determine the following pharmacokinetic parameters: ðAUCÞ0
1 ¼ ðAUCÞ0
t þ

ct ke

or ðAUCÞ0
1 ¼ ðAUCÞ0
t þ 1:44 t 1=2 ct Dose Total operative body clearance ¼ CL F ¼ AUC, where F is oral bioavailability cmax, and Tmax.

13.1.15 Effect of Food Intake on Drug Bioavailability 13.1.15.1 Background Omeprazole is a drug of choice for managing frequent acid reflux from the stomach to the mouth through the esophagus, gastric ulcer, Zollinger–Ellison syndrome, and duodenal bleeding. Food-drug interaction and stomach emptying play a significant

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role in the drug absorption process. Thus, the drug pharmacodynamic effect often hampers seriously. The study is intended to investigate the effect of food in the absorption and bioavailability of omeprazole from its orally administered dosage form. 13.1.15.1.1 Experiment No. 17 Objective

To compare the bioavailability of omeprazole with or without the food intake by the HPLC method Materials

Omeprazole (highly pure grade), omeprazole capsule (20 mg), carbamazepine, HPLC grade cyclohexane, dichloromethane, acetonitrile, and methanol Methods Ethical Clearance

Any experiments on human subjects/volunteers or patients or the biological samples taken from them cannot be conducted without receiving the institutional ethical clearance. Further, the volunteers/patients should be informed about the objectives and the expected hazards related to the study. At least three volunteers for each group (two groups, with or without food) are required for statistical analysis. Volunteers Each volunteer of a group should receive a single omeprazole capsule (20 mg) with 250 mL of water orally after the overnight fasting of 10 h (then breakfast after 1 h), and the other group should receive the same treatment 15 min after the standard breakfast (four slices of butter toast, two eggs, 200 mL orange juice, and 100 g curd, a standard size banana) at the same time point (say, at 8 am). Crossover design may be conducted after a 7-day washout period. The volunteers should take lunch and dinner at 5 and 12 h after the drug (single dose) administration. Volunteers should take food and beverages without caffeine, alcohol, and avoid any strenuous exercise. Blood samples (5 mL each sample) should be collected from the volunteers, namely, initially after every half an hour up to 4 h and then at an interval of 1 h till 8 h, and then at 10, 12, 16, and 24 h after the drug administration. 1. Centrifuge (at 4000 rpm for 10 min) the collected blood samples adding EDTA (anticoagulant). 2. Separate plasma from each sample and store at
20  C in a freezer (if necessary) till further use. HPLC Analysis (Pillai et al. 1998; Amantea and Narang 1988)

Column Hypersil ODS column (particle size 5 μm) Internal standard carbamazepine

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Injection volume 100 μL Study temperature: Ambient temperature Mobile phase: 45% potassium phosphate buffer 0.025 M, 30% acetonitrile, and 25% ethanol (pH adjusted to 7.3) Flow rate: 1.1 mL/min Standard solution concentration range: 40
600 ng/mL with IS (75 ng/mL) Limit of quantification: 10 ng/mL Detection at 302 nm 1. Separate omeprazole along with the IS from plasma from the tests and calibration samples (prepare with the incorporation of the defined amount of drug and the IS in human plasma) with a 60:40 v/v mixture of cyclohexane and dichloromethane. 2. Evaporate the solvent mixture to dryness. 3. Reconstitute the residue by the mobile phase. 4. Inject the sample into the HPLC system for analysis. 5. Determine plasma concentrations of the test samples with the help of a calibration curve. 6. Plot plasma concentration versus time curves for both the test samples. 7. Determine (AUC)0
t, (AUC)0
1, Tmax, cmax. 8. Compare the bioavailability of the drug taken by the volunteers after the breakfast and the volunteers on an empty stomach.

13.1.16 The Metabolism of Oxybutynin Hydrochloride to Its Metabolite N-Desethyl Oxybutynin Hydrochloride by Rat Liver Microsomes 13.1.16.1 Background Oxybutynin acts against the over-reactive urinary bladder to stop frequent urination in adults and children. Oxybutynin hydrochloride forms its metabolite n-desethyl oxybutynin hydrochloride by cytochrome P-450 (CYP) monooxygenases. CYP3A4 and CYP3A5 predominantly metabolize oxybutynin hydrochloride to n-desethyl oxybutynin hydrochloride by n-deethylation (Lukkari et al. 1998). 13.1.16.1.1 Experiment No. 18 Objectives

To investigate and quantify metabolism of oxybutynin hydrochloride and its metabolite n-desethyl oxybutynin hydrochloride by rat liver microsomal CYP3A4 and CYP3A5 using LC-MS/MS technique Materials

Highly pure grade oxybutynin hydrochloride, and the metabolite, n-desethyl oxybutynin hydrochloride (reference standards), and diphenhydramine (internal

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standard, IS), HPLC grade methanol, acetonitrile, n-hexane, formic acid, and ammonium acetate Methods Preparation of Microsome

1) Homogenize the weighed amount of a portion of normal liver in four volumes of ice-cold 0.25 M sucrose solution (Sarkar et al. 1994, 1995) (please see Sect. 13.1.2.3). 2) Separate microsome by differential centrifugation (Sarkar et al. 1994, 1995) (please see Sect. 13.1.2.4). 3) Suspend the final microsome pellet in 0.1 M phosphate buffer with an approximate protein concentration 20 mg/mL. Incubation of Oxybutynin with Microsome and Cytochrome Isoform-Specific Inhibitors

1. Incubate oxybutynin 100 μM in alcohol with microsomal suspension 10 μM (approximate protein concentration 20 mg/mL) in the following incubation solution with or without containing different cytochrome isoform-specific inhibitors. [1 mL incubation solution (Lukkari et al. 1998) contains: 2 M KCl (50 mM), 0.1 M magnesium chloride (2.5 mM), 0.03 M glucose-6-phosphatase (1.5 mM), 2.5 mM NADP (62.5 μM), 40 units glucose-6-phosphatase dehydrogenase (1 unit), phosphate buffer, pH 7.4, (10 mM), 0.1 M tris buffer, pH 7.8, (59.9 mM), and albumin 12 mg/mL (1.2 mg)]. 2. Preincubate at 37  C for 2 min. 3. Add inhibitors to the solution (for inhibitor control samples). [inhibitor concentration and preparation: ketoconazole 100 μM in dimethyl sulfoxide (DMSO) against CYP3A4 and CYP3A5] 4. Incubate the experimental samples containing microsomes with and without the inhibitors at 37  C for 45 min. 5. Stop the reaction by adding 2.5 mL of methanol. 6. Centrifuge briefly in closed cap container. 7. Store the supernatant solution at
20  C, for a few days until the analysis is done. Method of Estimation of Oxybutynin and N-Desethyl Oxybutynin Liquid Chromatography Conditions

Column ACQUITY UPLC BEH C18 column [100 mm  2.1 mm  1.7 μm (internal diameter)] Column temperature: 40  C Mobile phase: methanol—water (90:10, v/v) (water contains 2 mM/L ammonium acetate and 0.1% formic acid) Injection volume: 10 μL Mass Spectrometry Conditions The triple quadrupole tandem mass spectrometer

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Electrospray ionization interface: positive mode Quantification

SRM mode (selected reaction monitoring mode), m/z 358.0!71.6, 358.0!123.9, 358.0!141.9 for oxybutynin hydrochloride, m/z 330.0!95.7 for n-desethyl oxybutynin hydrochloride, and m/z 256.0 ! 166.9 for IS, respectively (Tian et al. 2019). Scan time: 0.10 s per transition Capillary voltage 2.5 kV Cone voltage: 30 V for oxybutynin hydrochloride and n-desethyl oxybutynin hydrochloride Cone voltage: 20 V for IS Desolvation temperature: 400  C (nitrogen for cone gas, 30 L/h and desolvation, 450 L/h) Source temperature: 110  C Collision gas: Argon at 0.278 Pa Collision energy: 35 V, 25 V, and 20 V for m/z 71.6, 141.9, and 123.9, respectively, for oxybutynin hydrochloride, and 15 V for n-desethyl oxybutynin hydrochloride and IS (Tian et al. 2019) Preparation of Standard Samples

Prepare the stock solutions for oxybutynin hydrochloride, n-desethyl oxybutynin hydrochloride, and IS to prepare the standard solutions in methanol at the concentrations of 94.4, 60.16, and100 μg/mL, respectively. Dilute them with a methanol-water mixture (90:10, v/v) to provide the standard working solutions at desired concentrations. IS concentration is 10.0 ng/mL. Store at 4  C until further use. Prepare calibration standards by evaporating 50 μL of working standard solutions to dryness under a gentle stream of nitrogen (same is done with the test samples also). Reconstitute in 100 μL of methanol-water (90:10, v/v) and vortex-mix. Inject an aliquot (10 μL) into the LC-MS/MS system. Quantify the drug and its metabolite from the samples incubated with or without the inhibitor. Report the outcomes.

13.1.17 Simultaneous Determination of Drugs Spiked into Plasma from a Single Tablet Dosage, Using LC-MS System 13.1.17.1 Background Atorvastatin and ezetimibe reduce blood cholesterol levels. Here, the method (Elawady et al. 2021) shows how to simultaneously determine the drugs from human/animal plasma upon administering a single tablet dosage that contains the drugs, using the LC-MS technique.

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13.1.17.1.1 Experiment No. 19 Objective

Simultaneous determination of atorvastatin and ezetimibe plasma levels from a single tablet dosage spiked into plasma, using LC-MS system Materials

Highly pure grade drugs (atorvastatin and ezetimibe). Atoreza®10/10 or Attor Ez® 10 mg/10 mg tablets, acetonitrile, formic acid, millipore water, drug-free human/rat plasma Instrumentation

Quadrupole MS (Agilent 6120) integrated with HPLC (Agilent 1200 series) using ChemStation software for data recording, C18 solid-phase extraction (SPE) column, Autosampler vials Standard Solutions Preparation

1. Prepare a standard stock solution of each drug (atorvastatin and ezetimibe) and internal standard (IS) solution separately in methanol to produce 1 mg/mL for each drug and IS. 2. Prepare different working standard solutions by diluting the stock solutions with the mobile phase to get the final concentrations of 5–25 ng/mL for ezetimibe and 1.5–25 ng/mL for atorvastatin and 1 μg/mL for IS. For Tablet Solution Preparations

1. Take 10 Atoreza®10/10 or Attor Ez® 10 mg/10 mg tablets, crush them, and take the equivalent weight of powder that contains 10 mg of each drug. 2. Dissolve the drugs from the powder in 10 mL methanol by sonication for 10 min in a screw-capped tube and extract the drugs. Filter using 0.45 μm nylon filter to get final drug concentration 1 mg/mL. 3. Dilute the solution in a 10 mL-volumetric flask with the mobile phase as mentioned below at point no. 11, after adding 1.5 mL IS to get the ultimate concentrations of 10, 20, and 25 ng/mL of ezetimibe and atorvastatin. Application to Spiked Plasma

1. Add the standard drug solutions to each 1 mL of human/rat drug-free plasma to prepare a final concentration range of 0.5–7.5 ng/mL for both the drugs by diluting with ultrapure water. 2. Pass methanol (3 mL) first, then aqueous 3% formic acid solution (2 mL) through the C18 SPE column for primary conditioning. 3. Load the column with 200 μL of spiked (by adding known quantities of the drugs) plasma and then aqueous 3% formic acid solution (50 μL) followed by the internal standard (75 ng/mL, 100 μL), and apply vacuum that allows the sample to exit from the stationary phase. 4. Close the tap for 2 min.

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5. Wash the sample with 1 mL of aqueous 3% formic acid solution. 6. Elute the retained compounds first with 1 mL of acetonitrile and then with 1 mL of ethyl acetate. 7. Evaporate the solution at 40  C in dry nitrogen gas to dryness on a heat block. 8. Reconstitute the dry sample in 50 μL of methanol. 9. Inject 20 μL of the reconstituted sample for analysis. 10. Carry out an experiment without the drug at the same time. Analysis Conditions

11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

Mobile phase: acetonitrile-aqueous formic acid (0.1%) in the ratio of 65:35 v/v Flow pattern and rate—isocratic at 0.5 mL/min. Injection volume—20 μL Standard drug solutions—10 μg/mL Internal standard (IS) (diclofenac sodium) solution—10 μg/mL Capillary voltage: 4000 V Nebulizer pressure: 35 psig Drying gas flow: 12 L/min Desolvation temperature: 350  C Dwelling time: 71 ms Fragmentor voltage: 135 V for atorvastatin and ezetimibe Internal standard: 80 V

MS Parameters

The m/z values of atorvastatin and ezetimibe are at 559.3 and 392.1, respectively. The m/z value of IS is 296. Find the drug concentrations in plasma samples spiked with drugs from the tablets.

13.1.18 Determination of Metformin and Canagliflozin from a Single Tablet Dosage in Human Plasma by LC-MS/MS Method 13.1.18.1 Background Metformin (MET) and canagliflozin (CFZ) in combination (500 mg/50 mg) are used against type-2 diabetes to control hyperglycemia in patients. Brand names of such formulations are Vokanamet (CFZ and MET, 50 mg/500 mg) Tablet, Invokamet Tablet, etc. CFZ acts as a sodium-glucose cotransporter inhibitor, and MET lowers liver glucose production and enhances glucose utilization in the gastrointestinal tract. Various analytical methods are available to determine the plasma drug concentration in patients. Here the LC-MS/MS method (Dalia et al. 2019) is used.

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13.1.18.1.1 Experiment No. 20 Objective

Determine plasma drug concentrations in volunteers received a single tablet dosage containing metformin and canagliflozin, by LC-MS/MS method and determine pharmacokinetic parameters of the drugs, such as Cmax, Tmax, elimination t 12 , (AUC)(0
t), (AUC)(0
1) Materials

Metformin hydrochloride, canagliflozin, and propranolol, tadalafil, Vokanamet (canagliflozin and metformin 50 mg/500 mg) tablets, or Invokamet (canagliflozin and metformin 50 mg/500 mg) tablet, human blank plasma, HPLC grade methanol, and acetonitrile, formic acid and ethyl acetate Equipment

Agilent 1260 HPLC system with mass spectrometry using a triple quadrupole API 4000 with Analyst 1.6.3 software Standard Stock Solutions

Dissolve MET (50 mg)/CFZ (10 mg) separately in 100 mL of methanol to achieve the concentration 500 μg/mL for MET and 100 μg/mL for CFZ. Dilute them (at a ratio 1:10, v/v) further with methanol to prepare two working standard solutions (WS 1 and WS 2) with MET concentrations, 50 μg/mL and 10 μg/mL and CFZ concentrations, 10 μg/mL and 2 μg/mL. Prepare different MET (0.5–50 μg/mL) and CFZ (0.1–10 μg/mL) solutions by diluting WS 1 and WS 2 with methanol as various reference standards. Store the solutions in a refrigerator for few days until use, if necessary. Internal Standard (IS)

In methanol, prepare propranolol (100 μg/mL) and tadalafil (700 μg/mL) separately as IS solutions for MET and CFZ, respectively. Calibration Standards

Spike 50 μL in each from MET/CFZ solutions from lower to higher concentrations with their respective IS (50 μL) into 400 μL human drug-free plasma to prepare the standard solutions. Use plasma without the drugs and internal standard as a blank sample. Test Sample Collection and Processing

Quantification of the drugs in the plasma samples of human volunteers or patients can only be conducted after receiving the institutional ethical clearance. The volunteers/patients should be informed about the objectives and the expected hazards related to the study.

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1. After administering one tablet (canagliflozin and metformin 50 mg/500 mg) each at zero time orally to each volunteer, collect 5 mL each blood sample initially after every half an hour up to 2 h and then at an interval of 1 h till 8 h, and then at 10, 12, 16, 24, 48, and 72 h. 2. Centrifuge the collected blood samples straight away at 4000 rpm for 10 min adding EDTA (anticoagulant). 3. Separate the plasma samples. and store them at
20  C, or at
80  C in a deep freezer (if necessary) till further use. Extraction protocol 1. Spike the plasma with the standard drugs and the internal standards. 2. Precipitate protein by adding 1 mL acetonitrile and vortex-mixing of the standard/ the test samples. 3. Extract the drugs in ethyl acetate (3 mL), mix briefly by vortex mix, and centrifuge at 4000 rpm for 5
6 min. 4. Concentrate the supernatant carefully at 60  C. 5. Reconstitute with HPLC grade methanol. 6. Inject aliquot (2 μL) of the reconstituted solution for analysis. 7. Draw the plasma concentration-time curves from the obtained plasma drug concentration, and calculate pharmacokinetic parameters, Cmax, Tmax, AUCð0
tÞ , AUCð0
1Þ , and t1 . 2

Analysis Conditions

Column: Zorbax C18 Temperature: At ambient temperature Mobile phase: Acetonitrile-formic acid (0.1%) (mix at 2:3, v/v) Flow rate: 500 μL/min Run time for LC: 5 min Operate positive-mode electrospray ionization. The m/z 130.2 ! 60.1 for metformin, m/z 462.3 ! 191.0 for canagliflozin, m/z 260.2 ! 183.0 for propranolol, and m/z 390.2 ! 268.2 for tadalafil (Dalia et al. 2019). The parameters of mass spectrometer: 1. 2. 3. 4. 5. 6.

Nebulizer gas: Air (zero grade) Auxiliary gas: Nitrogen Curtain gas pressure: 10 psi Collision gas pressure: 10 psi Temperature and voltage of ion spray: 400  C, 2000 V The ion source gas pressure: For one, 25 psi, and for the second, 45 psi

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13.1.19 Retrospective Data Collection 13.1.19.1 Background Data collection is a process of systematically accumulating and storing data for analysis and utilizing the outcomes of the analysis for current and future use. In clinical pharmacy and pharmacovigilance, data collection is primarily carried out as a prospective or a retrospective study. A prospective study, also called prospective cohort (group) data collection or a longitudinal cohort study, collects and analyzes the data of the enrolled participants before the investigating outcomes appear in them. In a prospective study, the participants are observed over time to collect the data. In a retrospective study, previous data are collected from the sampled individuals, and at the time of data assembling, the events or the outcomes have already happened. Retrospective data are generally collected from various available or recorded sources. Hence, the study is less expensive compared to the prospective cohort study. Moreover, suspected risk, medical errors that cause patient’s harm, and protection factors are some areas that can be well-investigated through the retrospective study. 13.1.19.1.1 Experiment No. 21 Objective

Collection of retrospective data of sampled patients or sampled individuals in a tabular format for further scientific and statistical analysis Method of Collection of Patient Data for Retrospective Study

The tabular sample formats (Tables 13.1 and 13.2) that may be used for the collection of retrospective patient data are included. After the collection of a large number of data, they are analyzed for the outcomes. Similar need-based data collection tables may be drawn as per the objective of a particular retrospective study. Finally, data are scientifically and statistically analyzed for conclusive outcomes.

13.1.20 Development of Clinical Trial Protocol 13.1.20.1 Background The clinical protocol should clearly define the objectives and the endpoints and the sufficient detailed, well-explicated, and easily understandable methodologies that are essentially indispensable for the regulatory bodies for its approval decision. Clinical trials are essential for the approval of a drug for its application in humans. It starts with the Phase I clinical trial study generally in healthy human volunteers to generate the safety and pharmacokinetic data of a new chemical. Then, in the subsequent trial phases (II and III), the safety and efficacy of the patients are determined. The minimum number of subjects/patients is fixed for the phases.

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Table 13.1 Proforma for collection of retrospective patient data: Patients suffered from chronic kidney disorder, received dialysis and treatment in a hospital Patient numbers ! Patient identification no. for first patient

Patient identification no. for second patient

Patient identification no. for third patient

Patients Parameters Sex Age Kidney disorder type Dialysis type Dialysis frequency per week Dialysis duration in min/h Diabetes mellitus Cardiovascular disease Hypertension Chronic liver disease Gender identity disorder Neurological disorder Rheumatoid Others Drug(s) used Route(s) Intravenous iron Oral iron Combination therapy Immunosuppresant Blood transfusion Hemoglobin 1 Hemoglobin 2 Hemoglobin 3 Payer status

13.1.20.1.1 Experiment No. 22 Objective

The objective of developing a clinical trial protocol is to receive the approval for conducting a clinical investigation to collect quality data within a time frame by protecting the participating volunteers/patients in the program.

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Table 13.2 Proforma for collection of retrospective patient data: Drug-drug interaction in intensive care unit (ICU) patients in a hospital Prescription audit format (ICU) Prescription audit proforma (inpatient medicine card) Patient registration no.! Details Specialty Consultant name mentioned Patient details Age Height Weight Blood group Drug hypersensitivity Diet sticker attached Previous medications mentioned As required medication properly mentioned Once daily and premedication drugs mentioned Regular prescription (clinical parameters) All drugs written in capital letters and legible All drug orders are signed with reg no. Dose mentioned correctly (normal/based on hepatic carcinoma/based on hepatic impairment) Route of administration mentioned properly with the frequency of dosing Number of antibiotics prescribed Number of restricted antibiotics prescribed Number of drugs written in generic name Number of high-risk medication prescribed (Avg use/prescription) Presence of therapeutic duplication Fixed dose combination mentioned (no of fixed dose) Error-prone abbreviations/illegible handwriting (if any) Drug interaction Suspected Food Drug Interaction documented (if any)/ information sticker attached Drug-drug interaction (DDI) documented in DDI form by pharmacologist. No of serious suspected DDI informed Name and signature of Auditor:____ Date and Time:____

Patient registration no. Remarks

Patient registration no. Remarks

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Methods Developing a Clinical Trial Protocol as per ICH Guidelines

(ICH-GCP E6: Guidance for Industry, E6 Good Clinical Practice: Consolidated Guidance, Revision 1 (R1) June 1996) The actual contents of a clinical trial protocol are mentioned below. Additional information may be included in separate pages. (a) General Information The protocol-related general information such as title, identity number, date, amendment if any, of the protocol, and the names, communication addresses with telephone number, email, etc., of the sponsor, monitor, authorized person, medical expert of the sponsor, details of the investigators, the physician for all trial-related decisions, clinical laboratory(ies), institute(s) engaged in trials, or any technical sections involved in the study. (b) Background Information Background information provides the investigational product name along with its description. Nonclinical and clinical investigations that show potential support for the clinical trial are also included. Potential risks and benefits related to human subjects should be disclosed. Further, dosage, drug administration route, dose regimen, duration of treatment, population, and the justification of those selections must be depicted. A declaration should be provided mentioning the compliance of the approved clinical protocol and the ICH GCP guidelines (ICH-GCP E6: Guidance for Industry, E6 Good Clinical Practice: Consolidated Guidance, Revision 1 (R1) June 1996). The published work and supportive data which have trial relevance may also be included. (c) Trial Objectives and Purpose This section defines the clear objectives and the intended purpose of the clinical trial. (d) Trial Design The trial design is significantly essential for the correct findings, data accuracy, study integrity, and a reliable conclusion. The study should describe the primary endpoints and any other endpoints with a detailed design with blinding and randomization to avoid bias such as placebo-controlled, double-blind, triple-blind, parallel design, etc., preferably with a schematic stage-wise procedural design. Further, we should include detailed information related to the investigational product (dosage form, placebo), packaging and label of the product, and trial-related information such as duration of subject participation, randomization code, the procedure of break code, trial discontinuation criteria, electronic record of data, etc. (e) Selection and Withdrawal of Subjects Inclusion and exclusion criteria of volunteers, the data collection time, procedure of subject replacement, and withdrawal of treatment should be included. (f) Treatment of Subjects The detailed treatment protocol, including the product name, dose, mode of treatment, route, frequency of administration, dosage regimen, duration of

13.1

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(p)

(q)

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therapy in subjects, and method to assess subject compliance, should be included. Besides, the permitted and non-permitted medications during the trials should be included, too. Assessment of Efficacy Parameters with time points of assessment, efficacy determination process, and data recording methods are included. Assessment of Safety Safety parameters with the assessment time points, safety determination procedure, and data recording methods should be described. Further, recording and reporting methods of adverse happenings and the detailed follow-up program should be mentioned. Statistics Detailed statistical methods include subject selections, numbers of subjects in a uni- or multicenter (sample size of each center) trial, significance level, assessment of spurious data, the procedure of reporting of deviation from the selected statistical method, and the justification in all relevant areas should be described. Direct Access to Source Data/Documents The protocol or a written agreement must be included stating that authority can monitor, review, audit, and inspect trials and trial-related documents and data. Quality Control and Quality Assurance The process and the assessment parameters related to the trial quality control and quality assurance should be described. Ethics Ethics is the primary and fundamental issue in a clinical trial that indicates the trial’s reliability and authenticity. A descriptive trial-related ethical consideration should be provided. Data Handling and Record-Keeping The detailed procedure of data handling, recording, and protecting them with privacy and methods of data disposition should be included. Financing and Insurance Financing and insurance policy are the two critical areas that should be mandatorily disclosed in the protocol or as a separate agreement. Publication Policy There is a need for disclosure of the publication policy either in the protocol or arranged separately. Supplements The other relevant information as per the ICH Clinical Study Reports Guideline available may be attached as additional sheets. Investigator-Initiated Multisite Research Studies In a multicentered trial, the investigator must describe how to ensure the same protocol version to be used in each center.

Assignment

Prepare a protocol with any new compound (say X), and get it checked and corrected by your professor.

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References Abou-Auda HS (1998) Comparative phramacokinetics and pharmacodynamics of furosemide in Middle eastern and in Asian subjects. Int J Clin Pharmacol Ther 36:275–281 Adcock D, Kressin D, Martar RA (1997) Effects of 3.2 % vs. 3.8 % citrate concentration on routine coagulation testing. Am J Clin Pathol 107:105–110 Amantea MA, Narang PK (1988) Improved procedure for quantitation of omeprazole and metabolites using reversed-phase high-performance liquid chromatography. J Chromatogr 426:216–222 Arora P, Mukherjee B (2002) Design, development, physicochemical, and in vitro and in vivo evaluation of transdermal patches containing diclofenac diethylammonium salt. J Pharm Sci 91: 2076–2089 Bawazir SA, Gouda MW, El-Sayed YM, Al-Khamis KI, Al-Yamani MJ, Niazy EM, Al-Rashood KA (1998) Comparative bioavailability of two tablet formulations of ranitidine hydrochloride in healthy volunteers. Int J Clin Pharmacol Ther 36:270–274 Brouwer EJ, Verweij J, De Bruijn P, Loos WJ, Pillay M, Buijs D, Sparreboom A (2000) Measurement of fraction unbound paclitaxel in human plasma. Drug Metab Disposition 28:1141–1145 Clausen J, Bickel M (1993) Prediction of drug distribution in distribution dialysis and in vivo from binding to tissue and blood. J Pharm Sci 82:345–349 Dalia M, Elshahed MS, Tamer N, Nageh A, Zakaria O (2019) Novel LC–MS/MS method for analysis of metformin and canagliflozin in human plasma: application to a pharmacokinetic study. BMC Chem 13(1):82. https://doi.org/10.1186/s13065-019-0597-4 Durham BH, Robinson J, Fraser WD (1995) Differences in the stability of intact osteocalcin in serum, lithium heparin plasma and EDTA plasma. Ann Clin Biochem 32:422–423 Erwa W, Bauer FR, Etschwaiger R, Steiner V, Scott CS, Sedlmayr P (1998) Analysis of aged samples with the Abott CD 400 hematology analyzer. Eur J Lab Med 6:4–15 Goosens W, van Duppen V, Verwilghen RL (1991) K2- or K3-EDTA: the anticoagulant of choice in routine haematology? Clin Lab Haemat 13:291–295 Lukkari E, Taavitsainen P, Juhakoski A, Pelkonen O (1998) Cytochrome P-450 specificity of metabolism and interactions of oxybutynin in human liver microsomes. Pharmacol Toxicol 82: 161–166 Marlar RA, Potts RM, Marlar AA (2006) Effect on routine and special coagulation testing values of citrate anticoagulant adjustment in patients with high hematocrit values. Am J Clin Pathol 126: 400–405 Mukherjee B, Mahapatra S, Gupta R, Patra B, Tiwari A, Arora P (2005) A comparison between povidone-ethylcellulose and povidone-eudragit transdermal dexamethasone matrix patches based on in vitro skin permeation. Eur J Pharm Biopharm 59:475–483 Pillai GK, Hassan MM, Salem MS, Najib NM (1998) Effect of food on the bioavailability of omeprazole. Int J Pharm Med 12:99–203 Sarkar A, Mukherjee B, Chatterjee M (1994) Inhibitory effect of β-carotene on chronic 2-acetylaminofluorene induced hepatocarcinogenesis in rat: reflection in hepatic drug metabolism. Carcinogenesis 15:1055–1060 Sarkar A, Mukherjee B, Chatterjee M (1995) Inhibition of 30 -methyl-4-dimethylaminoazobenzeneinduced hepatocarcinogenesis in rat by dietary β-carotene: changes in hepatic anti-oxidant defense enzyme levels. Int J Cancer 61:799–805 Tian Y, Wen Y, Sun J, Zhao L, Xiong Z, Qin F (2019) Simultaneous quantification of oxybutynin and its active metabolite N-desethyl oxybutynin in rat plasma by ultra-high-performance liquid chromatography-tandem mass spectrometry and its application in a pharmacokinetic study of oxybutynin transdermal patch. Biomed Chromatogr 33(4):e4456. https://doi.org/10.1002/bmc. 4456 Waters NJ, Jones R, Williams G, Sohal B (2008) Validation of a rapid equilibrium dialysis approach for the measurement of plasma protein binding. J Pharm Sci 97:4586–4595

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Youssef SH, Mohamed D, Hegazy MAM, Badawey A (2019) Analytical methods for the determination of paracetamol, pseudoephedrine and brompheniramine in Comtrex tablets. BMC Chem 13(1):78. https://doi.org/10.1186/s13065-019-0595-6

Pharmacokinetic Numerical Problems with Solutions

14

Sums 1. A patient with body weight of 75 kg received an intravenous bolus dose. For optimal treatment, plasma drug concentration should be maintained between 20 mg/mL and 5 mg/mL. The apparent volume of distribution of the dose is 0.35 L/kg. Rate of elimination is 0.15 h
1, and clearance is 0.12 L/kg.h. Find out dosing interval. Solution The volume of distribution (vd) of the drug in the patient ¼ 75  0.35 ¼ 26.25  26 L
 cp min
 ¼ e
K E τ cp max ∴e
K E τ ¼


 5 ¼ 0:25 or e
ð0:15Þτ ¼ 0:25 K E ¼ 0:15 h
1 20

Taking natural logarithm (ln), we get, ln (0.25) ¼
(0.15) τ or
1:3862 ¼
ð0:15Þ τ Therefore, τ ¼ 9.24 h 2. A loading dose and then a maintenance dose of a drug are administered to a patient by intravenous injection. The drug optimally functions between the drug plasma concentration level 30 and 12 mg/L. The drug has an elimination rate of 0.18 h
1, and the volume of distribution 35 L. Find loading dose, maintenance dose, and dosing interval. Solution

# The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B. Mukherjee, Pharmacokinetics: Basics to Applications, https://doi.org/10.1007/978-981-16-8950-5_14

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 Here cp max ¼ 30 mg=mL, cp min ¼ 12 mg=L, vd ¼ 35 L, K E ¼ 0:18 h
1 x ¼ ?; x0 ¼ ?; τ ¼ ?;
 cp min 12
 ¼ e
K E τ or, e
K E τ ¼ ¼ 0:4 30 cp max or e
ð0:18Þτ ¼ 0:4 Taking natural logarithm (ln), we get, ln (0.4) ¼
(0.18) τ or
0:9162 ¼
ð0:18Þ τ ∴τ ¼ 5 h   x 1 ðc1 Þ max ¼ 0 vd 1
e
K E τ   x 1 30 ¼ 0 35 1
0:4 ∴x0 ¼ 30  35  0:6 ¼ 630 mg     1 1 x 630 ¼ 0 ¼ ¼ 1050 mg; or lading dose ðxÞ ¼ x0 ¼ x 0 1
0:4 0:6 0:6 1
e
K E τ 3. The volume of distribution of a drug is 28 L, and rate of elimination of the drug in a patient is 0.2 h
1. The plasma concentration of the drug is required to maintain between 40 and 12 mg/L. Find loading dose, maintenance dose, drug accumulation factor, and dosing interval. Solution

 Here cp max ¼ 40 mg=mL, and cp min ¼ 12 mg=L, vd ¼ 28 L, K E ¼ 0:2 h
1 x ¼ ?; x0 ¼ ?; τ ¼ ?; R ¼ ?
 cp min 12
 ¼ e
K E τ or e
K E τ ¼ ¼ 0:3 40 cp max or e
ð0:2Þτ ¼ 0:3 Taking natural logarithm (ln), we get, ln (0.3) ¼
(0.2) τ or
1:20 ¼
ð0:2Þ τ

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Pharmacokinetic Numerical Problems with Solutions

289

∴τ ¼ 6 h   x 1 ðc1 Þ max ¼ 0
K τ E vd 1
e   x 1 40 ¼ 0 28 1
0:3 ∴x0 ¼ 40  28  0:7 ¼ 784 mg     1 1 x 784 or lading dose ðxÞ ¼ x0 ¼ 0 ¼ ¼ 1120 mg; ¼ x 0
K τ 1
0:3 0:7 0:7 1
e E R¼

1 1 1 ¼ ¼ 1:42 ¼
K τ E 1
0:3 0:7 1
e

4. Rate of intravenous infusion of a drug is 20 mg/h, and it provided plasma concentration of the drug at 8 h and 30 h to be 9 mg/L and 14 mg/L, respectively, in a patient. The volume of distribution of a drug is 28 L, rate of elimination halflife of the drug in the patient varied between 4 and 6 h. Find the volume of distribution and the elimination half-life of the drug. Solution Rate of intravenous infusion (k0) ¼ 20 mg/h At t ¼ 8 h, c ¼ 9 mg/L and css ¼ 14 mg/L We know,
K E ¼

2:303 c
c 2:303 14
9 log ss log ¼ 0:2878: log 0:3572 ¼ t css 8 14

¼ ð0:2878Þð
0:4471Þ ¼
0:1286 K E ¼ 0:1286 h
1   0:693 0:693 ¼ 5:38 h ,t 12 ¼ Therefore t 12 ¼ 0:1286 KE Again, css ¼ Therefore, vd ¼

k0 vd K E

k0 20 ¼ ¼ 11:1 L css K E 14 ð0:1286Þ

5. Find out the intravenous infusion rate of a drug administered to a patient who achieved the steady-state plasma concentration of the drug 18 mg/L. The apparent volume of distribution of the dose is 15 L, and the elimination rate is 0.15 h
1. Find out how long the drug requires achieving 85 and 98% of the steady-state

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Pharmacokinetic Numerical Problems with Solutions

plasma concentration in the patient? When the rate of elimination is 0.12 h
1, what should be the rate of infusion then? Solution css ¼ vkd K0 E , where k0 is the rate of infusion or, k 0 ¼ css vd K E ¼ 18  15  0:15 ¼ 40:5 mg=h
 c ¼ css 1
e
K E t When plasma concentration of the drug becomes 85% of the steady-state 85 plasma concentration of the drug (css), the equation is 100 css ¼
K E t 85% css ð1
e Þ
 or 0:85css ¼ css 1
e
K E t85%
 or 0:85css ¼ css 1
e
K E t85%
 or 0:85 ¼ 1
e
K E t85% or e
K E t85% ¼ 1
0:85 ¼ 0:15 or ln e
K E t85% ¼ ln 0:15 or
K E t 85% ¼ ln 0:15 t 85% ¼

ln 0:15
0:1897  2:303 4:369 ¼ ¼ 29:12 h ¼
K E
0:15 0:15

When plasma concentration of the drug becomes 98% of the steady-state 98 plasma concentration of the drug (css), the equation is 100 css ¼
K E t 98% css ð1
e Þ
 or 0:98css ¼ css 1
e
K E t98%
 or 0:98css ¼ css 1
e
K E t98%
 or 0:98 ¼ 1
e
K E t98% or e
K E t98% ¼ 1
0:98 ¼ 0:02 or ln e
K E t98% ¼ ln 0:02 or
K E t 98% ¼ ln 0:02

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Pharmacokinetic Numerical Problems with Solutions

t 98% ¼

291

ln 0:02
3:912  2:303 9:009 ¼ ¼ 60 h ¼
K E
0:15 0:15

When rate of elimination is 0.12 h
1, k0, the rate of infusion, is or k 0 ¼ css vd K E ¼ 18  15  0:12 ¼ 32:4 mg=h 6. A drug with an elimination half-life of 5 h in a patient has been administered by an intravenous dose of 150 mg for a prolonged period, and the dosing interval of the formulation is 8 h. The apparent volume of distribution of the dose is 25 L. Find out the maximum and minimum plasma concentration of the drug in the patient at the steady-state. Calculate the loading dose of the drug for the patient. Solution Here, x0 ¼ 150 mg; vd ¼ 25 L t 1=2 ¼ 5, then K E ¼

0:693 0:693 ¼ ¼ 0:1386 h
1 t 1=2 5

τ¼8h e
K E τ ¼ e
ð0:1386Þ:8 ¼ e
ð1:1088Þ ¼ 0:3299  0:33 At the steady-state    
 x 1 150 1 ¼ 8:95 mg=L cp max ¼ 0 ¼
K τ 25 1
0:33 vd 1
e E  
 x 1 cp min ¼ 0 :e
K E τ ¼ 8:95  0:33 ¼ 2:95 mg=L vd 1
e
K E τ     1 1 or lading dose ðxÞ ¼ x0 ¼ 223:88 ¼ 224 mg ¼ x 0 1
0:33 1
e
K E τ 7. A drug that undergoes elimination following nonlinear kinetics has been administered to a patient with two different doses, 500 and 650 mg, in two different situations. The values for Km and vmax of the drug in the patient are 120 mg and 75 mg/h. Calculate the half-life of the drug on both occasions in the patient. Solution Here K m ¼ 120 mg; vmax ¼ 75 mg=h; D0 ¼ 500 mg For half-life, D ¼ D20 ¼ 500 2 ¼ 250 mg

292

14

Pharmacokinetic Numerical Problems with Solutions

D K m ln ð D0 ÞþðD0
DÞ We know, t ¼ (where, the amount of drug, dose, is D0 and vmax D is the amount of drug present   in blood at time t).

K m ln

D0 D0 2

þðD0
20 Þ D

¼ 120 ln ð2Þþ75ð500
250Þ ¼ 1200:6931þ250 Therefore, t 50% ¼ vmax 75 Therefore, t50% ¼ 4.44 h On the second occasion, when D0 ¼ 650 mg, For half-life, D ¼ D20 ¼ 650 ¼ 325 mg 2  D D K m ln D0 þðD0
20 Þ 0 2 Therefore, t 50% ¼ ¼ 120 ln ð2Þþ75ð650
325Þ ¼ 1200:6931þ325 vmax 75 Therefore, t50% ¼ 5.44 h 8. A drug has been administered to a patient by intravenous infusion with two different daily doses, 200 and 350 mg, on two separate occasions. They achieved steady-state plasma concentration of the drug 14 mg/L and 27 mg/L, respectively. Find out the values of Km and vmax of the drug in the patient. Calculate the dose that can provide steady-state plasma concentration of the drug 22 mg/L in the patient. Solution We know that, D
D1 350
200 150 150  ¼
200 350 ¼ Km ¼  2 ¼ ¼ 113:6 mg=L 1:32 D1 D2 ð 14:285
12:96 Þ

14 27 C1 C2 v ðdoseÞ ¼ 200 mg=day vmax ¼

vðK m þ cÞ 200ð113:6 þ 14Þ ¼ ¼ 1822:8  1822 mg=day c 14

Alternatively, v ðdoseÞ ¼ 350 mg=day vmax ¼

vðK m þ cÞ 350ð113:6 þ 27Þ ¼ ¼ 1822:5  1822 mg=day c 27

Alternatively, it can be done graphically also We know that, v ðDoseÞ ¼
K m : vc þ vmax Therefore, if we plot, v (Dose) on the y
axis and vc on the x
axis, the slope of the line gives the value of
Km, and the intercept on the y
axis provides the value of vmax as shown in the figure given underneath.

14

Pharmacokinetic Numerical Problems with Solutions

293

Again, c v ¼ ðKvmax (when c is the steady-state plasma concentration of the drug) m þcÞ

v¼

ð1822Þð22Þ 40, 084 mg ¼ 295:60  300 mg=day ¼ 135:6 day ð113:6 þ 22Þ

9. In a 60-year-old patient with body weight of 75 kg, a drug has been administered by intravenous infusion with an elimination half-life of 4 h. The apparent volume of the dose distribution is 0.6 L/kg, and the minimum plasma concentration of the drug in the patient at the steady-state is 8 mg/L. The drug is available in 5-mL with a drug concentration of 150 mg/mL. Find out the rate of infusion of the drug in the patient mg/h and mL/h. Calculate the loading dose of the drug for the patient. Solution vd ¼ 0:6  75 ¼ 45 L; K E ¼

0:693 ¼ 0:1732 h
1 ; css ¼ 8 mg=L 4

or k 0 ¼ css vd K E ¼ 8  45  0:1732 ¼ 62:35 mg=h When k0 is expressed in mL/h, the value of k0 is ¼

62:35 ¼ 0:4156 mL=h 150

Loading dose ðxÞ ¼

k0 62:35 ¼ ¼ 359:98 ¼ 360 mg K E 0:1732

294

14

Pharmacokinetic Numerical Problems with Solutions

 css ¼ x ¼ css vd ¼

x0 τ vd K E

x0 k ¼ 0 τ KE KE

  x ,k0 ¼ 0  τ

10. Phenytoin sodium that undergoes elimination following nonlinear kinetics has been administered to a patient (380 mg/day) to achieve a steady-state plasma concentration of 12 mg/L and 500 mg/day to achieve a steady-state plasma concentration of 22 mg/L in two different situations. Calculate Km and vmax, respective clearances, and a daily dose of the drug to achieve a steady-state plasma concentration of 18.5 mg/L of the drug in the patient. (Molecular weight of phenytoin sodium is 274.25 g/mole.) Solution The molecular weight of phenytoin sodium is 274.25 g/mole. Then, the molecular weight of phenytoin is 252.27 g/mole. We know, Weight ðmgÞ of phenytoin ¼ weight ðmgÞ of phenytoin sodium 

molecular weight of phenytoin the molecular weight of phenytoin sodium

Therefore, 380 mg of phenytoin sodium ¼ 349.54 mg of phenytoin and 500 mg of phenytoin sodium ¼ 459:92 mg of phenytoin Then, for 349.54 mg drug, desired steady-state plasma concentration (css) is 12 mg/L and for 459.92 mg drug, desired steady-state plasma concentration is 22 mg/L. We know that, Clearance ¼

Daily dose css

Therefore, clearance for the dose 380 mg/day ¼ 349:54 12 ¼ 29:12  29 L/day ¼ 20:9  21 L/day and clearance for the dose 500 mg/day ¼ 459:92 22 D
D1 459:92
349:54 110:38 110:38  ¼
349:54 459:92 ¼ ¼ ¼ 13:42 mg=L Km ¼  2 8:22 D1 D2 ð 29:12
20:9 Þ 12
22 C1
C2 v ðdoseÞ ¼ 380 mg=day

14

Pharmacokinetic Numerical Problems with Solutions

vmax ¼

295

vðK m þ cÞ 349:54ð13:42 þ 12Þ ¼ ¼ 740:44  740:5 mg=day c 12

Alternatively, v ðdoseÞ ¼ 500 mg=day vmax ¼

vðK m þ cÞ 459:92ð13:42 þ 22Þ ¼ ¼ 740:47  740:5 mg=day c 22

Alternatively, The values of Km and vmax can be determined by graphical method. Therefore, if we plot, v (Dose) on the y
axis and vc on x
axis, the slope of the line gives the value of
Km, and the intercept on the y
axis provides the value of vmax as shown in the figure given underneath.

Again, daily dose to achieve steady-state plasma concentration 18.5 mg/L c v ðdaily doseÞ ¼ ðKvmax (when c is the steady-state plasma concentration of m þcÞ the drug) v ðdaily doseÞ ¼

vmax c 740:5  18:5 13, 699:25 ¼ ¼ 31:92 ðK m þ cÞ 13:42 þ 18:5

¼ 429:17 mg of phenytoin=day Therefore, the daily dose is ¼ 429.17  274:25 252:27 ¼ 466.56 mg of phenytoin sodium. 11. A bi-exponential series represents the plasma concentration of a drug at time t in a two-compartmental model in the form of cp ¼ Ae
αt + Be
βt, where A ¼ x0 ðα
K 21 Þ vc ðα
βÞ


βÞ and B ¼ x0 ðvKc ð21α
β Þ : The value of vc ¼ 30 L, the dose of the drug is

296

14

Pharmacokinetic Numerical Problems with Solutions

400 mg, KE is 0.18 h
1, and the values of K21 and K12 are 1.3 and 1.9 h
1, respectively. Find α, β and A and B. Solution Here K10 is the elimination rate constant, KE which is equal to 0.18 h
1 Dose ¼ 400 mg, K 21 ¼ 1:3 h
1 and K 12 ¼ 1:9 h
1 : We know that, K10 + K12 + K21 ¼ α + β and K21K10 ¼ αβ Then, K 10 þ K 12 þ K 21 ¼ α þ β or 0:18 þ 1:9 þ 1:3 ¼ α þ β or α þ β ¼ 3:38

ð14:1Þ

Again, αβ ¼ K 21 K 10 or αβ ¼ 1:3  0:18 ¼ 0:234

ð14:2Þ

or, α ¼ 0:234 β , by replacing the value to the Eq. (14.1), we get, or

0:234 þ β ¼ 3:38 β

or

β þ 0:234 ¼ 3:38 β

2

2

or β þ 0:234 ¼ 3:38 β 2

Therefore, β
3:38 β þ 0:234 ¼ 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
3:38Þ  ð
3:38Þ2
4:1:0:234 Therefore, β ¼ 2:1 Therefore, β ¼ 0:075 or 3:305 (since for the equation ax + bx + c ¼ 0, the solution of x ¼
b 2

Again, α ¼

0:234 , β

Therefore, α ¼ 3:12 or 0:708

pffiffiffiffiffiffiffiffiffiffiffi b2
4ac 2a

)

14

Pharmacokinetic Numerical Problems with Solutions

Now, A ¼ B¼

297

x0 ðα
K 21 Þ 400:ð3:12
1:3Þ 728 ¼ 7:96 ¼ ¼ vc ð α
β Þ 30 ð3:12
0:075Þ 91:35

x0 ðK 21
βÞ 400:ð1:3
0:075Þ 490 ¼ 5:363: ¼ ¼ vc ðα
βÞ 30 ð3:12
0:075Þ 91:35

By putting the other set of values of alpha and beta, we get, A ¼ 3:039 and B ¼ 10:29 12. A bi-exponential series represents the plasma concentration of a drug at time t in a two-compartmental model in the form of cp ¼ Ae
αt + Be
βt, where A ¼ 28 and B ¼ 7.5 and α ¼ 1.8 and β ¼ 0.32. The dose of the drug is 600 mg. Find K21, K12, KE and vc. Solution Here K10 is the elimination rate constant, KE We know that K10 + K12 + K21 ¼ α + β and K21K10 ¼ αβ Or, K 10 þ K 12 þ K 21 ¼ 1:8 þ 0:32 ¼ 2:12

ð14:3Þ

Again, K 21 K 10 ¼ αβ or K 21 K 10 ¼ 1:8  0:32 ¼ 0:576 We know, A ¼

x0 ðα
K 21 Þ x ðK
βÞ and B ¼ 0 21 : vc ðα
βÞ vc ð α
β Þ

Dividing A by B, we get x0 ðα
K 21 Þ

ðα
K 21 Þ A v ðα
βÞ ¼ c ¼ B x0 ðK 21
βÞ ðK 21
βÞ vc ðα
βÞ

or

ð1:8
K 21 Þ 28 ¼ 7:5 ðK 21
0:32Þ

or 28:ðK 21
0:32Þ ¼ 7:5ð1:8
K 21 Þ or 28:ðK 21 Þ
28:ð0:32Þ ¼ 7:5ð1:8Þ
7:5ðK 21 Þ or ð28 þ 7:5ÞK 21 ¼ 7:5ð1:8Þ þ 28:ð0:32Þ or K 21 ¼

22:46 ¼ 0:63 h
1 35:5

Now A ¼

x0 ðα
K 21 Þ vc ð α
β Þ

ð14:4Þ

298

14

or 28 ¼ or 28 ¼

Pharmacokinetic Numerical Problems with Solutions

600ð1:8
0:63Þ vc ð1:8
0:32Þ

600ð1:8
0:63Þ 702 ¼ 1:48 vc vc ð1:8
0:32Þ

or 28  1:48 vc ¼ 702 Therefore, vc ¼ 16:94  17 L From Eq. (14.4), we know, K 21 K 10 ¼ 0:576 or K 21 K E ¼ 0:576 ∴K E ¼

0:576 0:576 ¼ 0:91 h
1 ð,K 10 ¼ K E Þ ¼ K 21 0:63

From Eq. (14.3), we get, K 10 þ K 12 þ K 21 ¼ 2:12 Or, 0:91 þ K 12 þ 0:63 ¼ 2:12 ∴K 12 ¼ 0:58 h
1 13. A patient with body weight of 65 kg received multiple intravenous doses of a drug to maintain drug plasma concentration between 11.5 and 1.8 mg/mL. The apparent volume of distribution of the dose is 0.8 L/kg. The rate of elimination is 0.12 h
1. Find out the volume of distribution of the drug and its dosing interval. Solution The volume of distribution (vd) of the drug in the patient ¼ 65  0.8 ¼ 52 L
 cp min
 ¼ e
K E τ cp max ∴e
K E τ ¼

1:8 ¼ 0:156 or e
ð0:12Þτ ¼ 0:156 11:5

Taking natural logarithm ðlnÞ, we get, ln ð0:156Þ ¼
ð0:12Þ τ Or,
1:857 ¼
ð0:12Þ τ

14

Pharmacokinetic Numerical Problems with Solutions

299

Therefore, τ ¼ 15:47 h 14. A patient received multiple intravenous doses (each dose 150 mg) every 8 h to maintain drug plasma concentration between 15 and 40 mg/mL. Calculate the loading dose of the drug. Solution
 cp min
 ¼ e
K E τ cp max or

15 ¼ e
K E τ 40

∴e
K E τ ¼ 0:375     1 1 Loading dose ¼ Maintenance dose  ¼ 150  1
0:375 1
e
K E τ   1 ¼ 150  ¼ 240 mg 0:625 15. Several doses of a drug that is eliminated through the liver following nonlinear kinetics were investigated on a patient. After administration of 450 mg of the drug daily on the 4th day, steady-state plasma concentration reaches 8 mg/L in the patient. The dose was continued. On the 10th day and on, he started receiving 250 mg of dose twice daily, and on the 21st day, he had a steadystate plasma concentration of 13.5 mg/L, and on the 60th day from the very first day of treatment, he had the steady-state plasma concentration of 24 mg/L. Calculate Km and vmax, maximum clearance, and a daily dose of the drug required to achieve steady-state plasma concentration 17 mg/L of the drug in the patient. When the daily dose is 350 mg of the drug in the patient, calculate the plasma concentration of the drug. Solution Here D1 ¼ 450 mg daily and D2 ¼ 250 mg  2 ¼ 500 mg daily We know that, D
D1 500
450 50 50  ¼
450 500  ¼ Km ¼  2 ¼ ¼ 2:6 mg=L 19:22 D1 D2 ð 56:25
37:03 Þ 8
13:5 C1
C2 v ðdoseÞ ¼ 450 mg=day vmax ¼

vðK m þ cÞ 450ð2:6 þ 8Þ ¼ ¼ 596:25 mg=day c 8

300

14

Pharmacokinetic Numerical Problems with Solutions

Clearance ¼

Daily dose css

Clearance on day 4 ¼

450 ¼ 56:25 L=day 8

Clearance on day 10 ¼

500 ¼ 37:03 L=day 13:5

Clearance on day 60 ¼

500 ¼ 20:83 L=day 24

Therefore, the maximum clearance is 56.25 L/day on day 4 among the given values c v ðdaily doseÞ ¼ ðKvmax (when c is the steady-state plasma concentration of m þcÞ the drug) Then, a daily dose of the drug that is required to achieve a steady-state plasma concentration of 17 mg/L of the drug in the patient v ðdaily doseÞ ¼

vmax c 596:25  17 10, 136:25 ¼ ¼ 2:6 þ 17 19:6 ð K m þ cÞ

¼ 517:15 mg of drug=day When the daily dose is 350 mg of the drug in the patient, plasma concentration of the drug will be v ðdaily doseÞ ¼ Then, 350 ¼

vmax c ð K m þ cÞ

596:25c ð2:6 þ cÞ

or 350:ð2:6 þ cÞ ¼ 596:25c or 910 þ 350c ¼ 596:25c or 596:25c
350c ¼ 910 or 246:25c ¼ 910 ∴c ¼ 3:695 mg=L 16. The number of doses of a drug that is eliminated through kidneys following nonlinear kinetics was investigated on a patient. Daily dose and steady-state plasma concentration data are given below. Calculate Km and vmax, maximum clearance, and a daily dose of the drug required to achieve steady-state plasma

14

Pharmacokinetic Numerical Problems with Solutions

301

concentration 24 mg/L of the drug in the patient. When the daily dose is 75 mg of the drug in the patient, calculate the drug steady-state plasma concentration. Daily dose (mg) 10 20 30 40 50 60

Steady-state plasma concentration (mg/mL) 1.8 4.2 9.8 16.3 25.5 34.2

Solution We know that, Clearance ¼ (Daily dose)/Steady state plasma drug concentration, Steady-state plasma concentration (mg/mL) 1.8 4.2 9.8 16.3 25.5 34.2

Daily dose (mg) 10 20 30 40 50 60

Clearance (L/day) 5.55 4.76 3.06 2.45 1.96 1.75

D
D1 60
50 10 10  ¼
50 ¼ Km ¼  2 ¼ ¼ 47:6 mg=L 60 0:21 D1 D2 ð 1:96
1:75 Þ
25:5 34:2 C1
C 2 v ðdoseÞ ¼ 50 mg=day, For 50 mg daily dose vmax ¼

vðK m þ cÞ 50ð47:6 þ 25:5Þ ¼ ¼ 143:33 mg=day c 25:5 Clearance ¼

Daily dose css

Therefore, the maximum clearance among the given doses is 5.55 L/day c v ðdaily doseÞ ¼ ðKvmax (when c is the steady-state plasma concentration of m þcÞ the drug) Then, a daily dose of the drug that is required to achieve a steady-state plasma concentration 24 mg/L of the drug in the patient v ðdaily doseÞ ¼

vmax c 143:33x24 3439:92 ¼ ¼ 48:04 mg of drug=day ¼ 71:6 ðK m þ cÞ 47:6 þ 24

302

14

Pharmacokinetic Numerical Problems with Solutions

When the daily dose is 75 mg of the drug in the patient, the plasma concentration of the drug will be v ðdaily doseÞ ¼ Then, 75 ¼

vmax c ð K m þ cÞ

143:33:c ð47:6 þ cÞ

or 75:ð47:6 þ cÞ ¼ 143:33:c or 3570 þ 75:c ¼ 143:33:c or 143:33:c
75:c ¼ 3570 or 68:33:c ¼ 3570 ∴c ¼ 52:24 mg=L 17. In a three-compartment model system, the drug has been administered by the intravenous route to the central compartment and is eliminated from the central compartment also. The amount of drug in the central compartment is represented 2 by Laplace transform as ðsþ1sÞðþ6sþ9 sþ2Þðsþ4Þ. Without using the Laplace table directly, calculate the amount of drug present in the central compartment at time t. Solution 2 To arrange the equation ðsþ1sÞðþ6sþ9 sþ2Þðsþ4Þ in summation of parts, Let us consider s2 þ 6s þ 9 A B C ¼ þ þ ð s þ 1Þ ð s þ 2Þ ð s þ 4Þ ð s þ 1Þ ð s þ 2Þ ð s þ 4Þ Where A, B, and C are constants ∴s2 þ 6s þ 9 ¼ Aðs þ 2Þðs þ 4Þ þ Bðs þ 1Þðs þ 4Þ þ Cðs þ 1Þðs þ 2Þ Comparing coefficients of powers of s on both sides of the equality, we know that three (3) parts on the right sides of the equation are equivalent to a system of three equations with the three unknowns, A, B, and C, on the left-hand side. But there is a shortcut to determine the three unknowns. If we set, s ¼
1, and put the value in s2 + 6s + 9 ¼ A(s + 2)(s + 4), we obtain, 4 ¼ A ð 1Þ ð 3Þ

14

Pharmacokinetic Numerical Problems with Solutions

or A ¼

303

4 3

Similarly, if we set s ¼
2, and put the value in s2 + 6s + 9 ¼ B(s + 1)(s + 4), we obtain, 1 ¼ B ð
1Þð2Þ or B ¼


1 2

If we set, s ¼
4, and put the value in s2 + 6s + 9 ¼ C(s + 1)(s + 2), we obtain, 1 ¼ C ð
3Þð
2Þ or C ¼

1 6

Now we need to calculate the inverse Laplace value of the equation. That is,
1 

4  1
2 s2 þ 6s þ 9
1 3 L ¼L þ þ 6 ð s þ 1Þ ð s þ 2Þ ð s þ 4Þ ð s þ 1Þ ð s þ 2Þ ð s þ 4Þ





 4
1 1 1
1 1 1
1 1 ¼ L
L þ L 3 2 6 ð s þ 1Þ ðs þ 2Þ ðs þ 4Þ

   4 1 1 1 ¼ e
t
e
2t þ e
4t ,L
1 ¼ e
αt 3 2 6 ðs þ αÞ
1



Therefore, the amount of drug in the central compartment at time t is given by 4
t 1
2t 1
4t e
e þ e 3 2 6 18. The half-life of a drug in a patient with body weight of 68 kg is 4 h. The volume of distribution of the drug is 0.65 L/kg. Calculate the clearance of the drug in the patient. Solution Volume of distribution of the drug is 68  0.65 ¼ 44.2 L The half-life of the drug ¼ 4 h Then, K E ¼

0:693 ¼ 0:173 h
1 4

304

14

Pharmacokinetic Numerical Problems with Solutions

Therefore, clearance ¼ V d: K E ¼ 0:173  44:2 ¼ 7:64 L=h 19. A drug undergoes degradation following zero-order kinetics. The amounts of the drug became 450 mg at 6 months and 441 mg in a year. Find the rate of degradation. Solution: At 6 months, amount of drug was 450 mg, and at 12 months, amount of drug became 441 mg. 9 As per zero-order kinetics, rate (K) ¼ xt12

xt12 ¼ 450
441 12
6 ¼ 6 ¼ 1:5 mg=month 20. Following zero-order degradation kinetics, 150 mg of a drug became 145 mg in 9 months. Find the time required by the drug to become 138 mg. Solution: Zero-order kinetic equation is xt ¼ x0
kt or 145 ¼ 150
k:9 Or, k ¼

5 ¼ 0:55 mg=month 9

Again, xt ¼ x0
kt or 138 ¼ 150
0:55 t Therefore, t ¼

12 ¼ 21:818 months 0:55

21. A drug that undergoes degradation following zero-order kinetics has its dose 100 mg and becomes ineffective upon 5% degradation. The drug becomes 98 mg at 6 months. Find its expiry date. Solution: Zero-order kinetic equation is xt ¼ x0
kt or 98 ¼ 100
k:6 or 6:k ¼ 100
98 ¼ 2 or k ¼

2 ¼ 0:33 mg=month 6

Upon 5% degradation, drug becomes 95 mg.

14

Pharmacokinetic Numerical Problems with Solutions

305

Therefore, 95 ¼ 100
0:33:t or 0:33:t ¼ 100
95 ¼ 5 or t ¼

5 ¼ 15:15 months 0:33

Therefore, the expiry date should be 15.15 months from its preparation. 22. Plasma drug concentration relationship of an intravenous bolus dose in a one-compartment model is given by ct ¼ 5.12 e
0.15t, where ct is a concentration of drug (μg/mL) at t hours. Determine bioavailability of the drug till 10 h by trapezoid rule. Take the reading every half an hour time interval. Solution: Area of a trapezoid ¼ (0.5)  perpendicular height between the two parallel sides  (sum of the two parallel sides) Since every half an hour (i.e., 0.5 h), the reading is taken. Hence, the distance between the two time subsequent points is 0.5 h (which is the perpendicular height between the two subsequent parallel sides). Therefore, area of a trapezoid ¼ (0.5)  (0.5)  (sum of the two parallel sides) ¼ 0.25  (sum of the two parallel sides). ct values measure the lengths of the parallel sides. Time “t” (h) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

ct, plasma concentration at time t (μg/ mL) 0 4.75 4.41 4.10 3.79 3.51 3.26 3.02 2.80 2.61 2.42 2.24 2.08 1.93 1.79 1.66 1.54 1.43 1.33 1.23 1.14

Trapezoid area computation 0 0.25 (0 + 4.75) 0.25 (4.75 + 4.41) 0.25 (4.41 + 4.10) 0.25 (4.10 + 3.79) 0.25 (3.79 + 3.51) 0.25 (3.51 + 3.26) 0.25 (3.26 + 3.02) 0.25 (3.02 + 2.80) 0.25 (2.80 + 2.61) 0.25 (2.61 + 2.42) 0.25 (2.42 + 2.24) 0.25 (2.24 + 2.08) 0.25 (2.08 + 1.93) 0.25 (1.93 + 1.79) 0.25 (1.79 + 1.66) 0.25 (1.66 + 1.54) 0.25 (1.54 + 1.43) 0.25 (1.43 + 1.33) 0.25 (1.33 + 1.23) 0.25 (1.23 + 1.14)

Therefore, the total trapezoid area ¼ 25.18 Therefore, the AUC of the drug up to first 10 h is 25.18 μg.h/mL.

Trapezoid area 0 1.18 2.29 2.12 1.97 1.82 1.69 1.57 1.45 1.35 1.25 1.16 1.08 1.00 0.93 0.86 0.80 0.74 0.69 0.64 0.59

Questions, and Questions and Answers for Practice

15.1

15

The Practice of Questions and Answers on Topics Based on Pharmacokinetics

(Against all the questions, heading numbers of section/subsection are mentioned in the brackets where the responses of the questions are available.) Chapter 1 1. What is meant by pharmacokinetics? What is pharmacodynamics? What is the minimum effective drug concentration? What is called the minimum safe concentration of drugs? What is called the therapeutic window? What are Cmax, tmax, and lag phase? (Sects. 1.1 and 1.1.1) 2. Draw drug absorption curve as plasma drug concentration against time. Show MEC, MSC, and therapeutic window. Show AUC, Cmax, and tmax. (Sect. 1.1.1) 3. Explain the term onset of drug action, and differentiate the onset of drug action, and lag phase of a dose, of a drug (Sect. 1.1.1). 4. What is zero-order kinetics? Deduce zero-order kinetics, and determine the halflife of a drug (Sect. 1.1.1). 5. Deduce first-order kinetics, and determine the half-life of a drug. Deduce the equation for the amount of a drug at any time t. (Sect. 1.1.1) 6. What is the volume distribution of a drug? (Sect. 1.1.3) 7. What is called Krüger-Thiemer “pharmacokinetic factor”? (Sect. 1.1.6) 8. What is called the Krüger-Thiemer dose ratio? Write its importance. (Sect. 1. 1.7) 9. In a multiple dosing regimen, determine the maximum and the minimum plasma concentrations of a drug after nth dose. (Sect. 1.1.2) 10. What do you mean by steady-state plasma concentration? Give its significances. Deduce the equation of steady-state plasma concentration. (Sect. 1.1.4) 11. What is accumulation factor? Deduce its value in a multiple dosing regimen. Give its significances. (Sect. 1.1.5) # The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B. Mukherjee, Pharmacokinetics: Basics to Applications, https://doi.org/10.1007/978-981-16-8950-5_15

307

308

15

Questions, and Questions and Answers for Practice

Chapter 2 1. “To determine drug absorption rate constant, plotting the data of the percentage of drug remaining to be absorbed versus time is truly rational”—justify the statement. (Sect. 2.1) 2. Write Dominguez equation and give its importance. (Sect. 2.1.1) 3. Deduce Wagner Nelson equation for determination of drug absorption rate constant. Give the importance of the equation. Write the pre-assumptions of the equation. (Sect. 2.1.2) 4. Deduce drug absorption rate constant from urinary excretion data using the Wagner-Nelson equation. (Sect. 2.1.3) 5. Using urinary excretion data, show that x1 u /x0 ¼ ke/kE [Sect. 2.1.3, Eq. (2.3)] 6. Deduce the equation for determination of rate of drug absorption by Loo-Riegelman method. Give the importance of the method. (Sect. 2.1.6) 7. Explain with example flip flop phenomena in pharmacokinetics. (Sect. 2.1.8) 8. How will you determine the absorption rate constant by the residual method? Explain it. Why is it called the feathering or stripping, or peeling method? Using graph, explain the importance of lag time. (Sect. 2.1.7) Chapter 3 1. What do you mean by bioavailability of a drug? Define absolute bioavailability and relative bioavailability. What do you mean by renal clearance? (Sects. 3.1 and 3.1.1) 2. Determine absolute bioavailability using the rate of urinary excretion. (Sect. 3. 1.3) 3. Determine absolute bioavailability in terms of the biological half-life of a drug. (Sect. 3.1.2) 4. What is called bioequivalence? What are generic and brand drugs? (Sect. 3.1.4) 5. Explain the terms chemical equivalent, pharmaceutical equivalent, bioequivalent, and therapeutic equivalent. (Sects. 3.1.4.13.1.4.4) 6. Give the importance of drug protein binding. (Sect. 3.1.5) Deduce the equation for Klotz reciprocal plot. Write the lacuna of the reciprocal plot (Sect. 3.1.6). What is Scatchard plot? (Sect. 3.1.7) Write its importance. Deduce the importance of Scatchard plot. (Sect. 3.1.7) 7. Deduce the equation of Sandberg plot. Give its importance. (Sect. 3.1.8) Chapter 4 1. Write a short note on Physiological pharmacokinetic model, blood flow limited model or perfusion model, drug-protein binding physiological model, diffusion-limited model, and model based on the statistical moment (Sects. 4.1.14.1.5) 2. What is drug extraction ratio? [Sect. 4.1.2, portion next to Eq. (4.2)] What do you mean by intrinsic clearance? [Sect. 4.1.2, portion next to Eq. (4.2)] What do you mean by statistical moments? (Sect. 4.1.5) What do you mean by zero moment? (Sect. 4.1.5) Name the moments that are called variance, skewness, and kurtosis. (Sect. 4.1.5)

15.1

The Practice of Questions and Answers on Topics Based on Pharmacokinetics

309

3. What do you mean by one-compartmental open model (Sect. 4.1.6.1) and closed model (Sect. 4.1.6.2)? What do you mean by catenary model, cyclic model, and mamillary model? (Sects. 4.1.6.34.1.6.5) What do you mean by Laplace operator? (Sect. 4.1.7.2) 4. What do you mean by two-compartment open models? (Sect. 4.1.9.2) Write the assumptions of two-compartment models. (Sect. 4.1.8) 5. For intravenous infusion, determine steady-state drug level. (Sect. 4.1.9.1.2) 6. In one-compartment open model, determine overall elimination rate constant during the time t0 elapsed after the stop of the infusion at the steady-state level of a drug in a patient. (Sect. 4.1.9.1.3) 7. In one-compartment open model, determine overall elimination rate constant when the infusion has been stopped before the plasma drug concentration reaches the steady-state level in a patient. (Sect. 4.1.9.1.4) 8. In a two-compartment open model, determine plasma drug concentration and amount of drug at the peripheral compartment, following an intravenous bolus dose. (Sect. 4.1.9.2) 9. In a three-compartment open model, establish plasma drug concentration-time relationship. (Sect. 4.1.9.3) Chapter 5 1. Give the importance of drug metabolism. (Sect. 5.1) What do you mean by the hepatic first-pass metabolism effect? (Sect. 5.1.1) Write the role of phase I and phase II enzymes in hepatic drug metabolism. (Sect. 5.1.1) 2. Using a two-compartment open model, show how the total bioavailability of drugs by the oral route is invariably low compared to that by intravenous drug administration. (Sect. 5.1.2) 3. What is drug metabolism? (Sect. 5.1) Using the compartmental model, deduce the equation of estimation of blood level of drug-metabolite excreted unchanged through urine. (Sect. 5.1.3) Chapter 6 1. What do you mean by nonlinear kinetics? (Sect. 6.1.1) What is the LineweaverBurk plot? Give its importance. (Sect. 6.1.2) 2. Give various linear plots of the Michaelis-Menten equation. (Sect. 6.1.2) 3. What do you mean by capacity-limited process? Give examples. (Sect. 6.1.1) 4. Give a time concentration relationship for a capacity-limited process. (Sect. 6.1.3) 5. Give time concentration relationship for a more than one capacity-limited process. (Sect. 6.1.5) 6. Deduce the equation for the time concentration relationship of capacity-limited process to establish a relationship of dose and Michaelis-Menten constant. (Sect. 6.1.4) 7. Deduce equation for sigma minus method. Why is the method named so? Explain how you determine the rate of elimination from it. (Sect. 6.1.6)

310

15

Questions, and Questions and Answers for Practice

8. For an orally administered drug that is eliminated unchanged through urine, develop bi-exponential absorptionelimination relationship using a compartment model. (Sect. 6.1.7) 9. Explain how you will determine “Ke” using the excretion rate method. (Sect. 6. 1.8) Chapter 7 1. What is called pharmacokinetic drug-drug interaction? (Sect. 7.1) Give its importance. (Sect. 7.1.1) What are the categories of pharmacokinetic drug-drug interaction? (Sect. 7.1.2) 2. Give an example of the pharmacokinetic drug-drug interaction of the following categories: absorption, distribution, metabolism, and elimination. (Sect. 7.1.3) 3. Give an example of the pharmacodynamic drug-drug interaction of the following categories: (a) involving receptor activation (b) modulating physiological regulatory function (c) by agonistic function (d) by antagonistic function. (Sect. 7.1.4) 4. Write the various data sources of drug-drug interaction. (Sect. 7.1.5) Chapter 8 1. What do you mean by therapeutic drug monitoring? (Sect. 8.1) List the areas where therapeutic drug monitoring is essential. (Sect. 8.1.1.3) Give its significance. (Sect. 8.1.1.3) 2. Differentiate plasma and serum. (Sects. 8.1.1.1 and 8.1.1.2) 3. Describe the impact of physiological processes in drug absorption, distribution, metabolisms, and elimination in children. (Sects. 8.1.28.1.2.5) 4. Describe the impact of physiological processes in drug absorption, distribution, metabolisms, and elimination in adults. (Sects. 8.1.38.1.3.4) 5. Give the various formulas for dose calculation in children and adults. (Sects. 8.1. 2.6 and 8.1.3.5) 6. What is called obese? Write the importance of dose calculation in obese patients. (Sect. 8.1.4) 7. How will you calculate the dose of a drug for patients suffering from renal insufficiency? (Sect. 8.1.5) 8. Describe the parameters for designing a pharmacokinetic preclinical study. (Sects. 8.1.68.1.6.1.11) 9. Describe the parameters for designing a pharmacokinetic clinical study. (Sects. 8.1.68.1.6.1.11) 10. Write stepwise the method of analysis of data for various pharmacokinetic models. (Sects. 8.1.78.1.7.5) Chapter 9 1. What is called pharmacokinetic sampling? What are called primary sampling and secondary sampling? (Sect. 9.1.1) 2. What is phlebotomy? (Sect. 9.1.2) How is it commonly done? (Sect. 9.1.2) How are the blood samples processed immediately after the collection? (Sect. 9.1.2.1)

15.1

The Practice of Questions and Answers on Topics Based on Pharmacokinetics

311

3. What is a tourniquet? (Sect. 9.1.2) Why is it required? (Sect. 9.1.2) What is hematoma? (Sect. 9.1.2) How can we prevent hematoma that may occur during blood withdrawal? (Sect. 9.1.2) How can you prevent hemolysis of blood samples? (Sect. 9.1.2.1) What are anticoagulants? Give their examples and role. (Sect. 9.1.2.1) 4. Write the site of collection of blood from small animals. (Sect. 9.1.2.2) 5. Write the site of collection of blood from babies, and answer the following questions (Sect. 9.1.2) (A) Write the procedure for collection and fixation of tissue samples for analysis. (Sects. 9.1.49.1.5) (B) How is the urine sample processed? (Sect. 9.1.3) (C) Explain the processing of a fecal sample till before analysis. (Sect. 9.1.6) 6. How is urine sample collected from small animals? (Sect. 9.1.3) 7. What is called extraction? How is a drug extracted from tissue for analysis? (Sect. 9.1.7) 8. Give the mechanism of extraction with the relevant equations. (Sect. 9.1.7) 9. Explain Noyes Whitney equation, Nernst law of distribution and partition coefficient regarding drug extraction from tissue samples. (Sect. 9.1.7) Chapter 10 1. Write the working principle of LC-MS/MS. (Sect. 10.1.1.1) What is a quadrupleassembly? Describe its functions. (Sect. 10.1.1.2.7) What is the interface in LC-MS/MS? Give its importance. (Sect. 10.1.1.2.4) 2. Describe with a neat diagram one LC-MS/MS system. (Sects. 10.1.110.1.1.2.7) 3. Describe stepwise method optimization by LC-MS/MS. (Sects. 10.1.1.3.110.1. 1.3.5) 4. Write the working principle of HPLC. (Sect. 10.1.2.1) Name various types of HPLC. (Sects. 10.1.2.210.1.2.2.4) Write about them briefly. (Sects. 10.1.2.2.1 10.1.2.2.4) 5. With a neat diagram, describe an HPLC system. (Sects. 10.1.2.310.1.2.3.7) 6. How will you optimize HPLC methods? (Sects. 10.1.2.510.1.2.5.9) 7. What is a tailing factor? Give its significance. (Sect. 10.1.2.4) Chapter 11 1. Define the term statistics. What do you mean by biometry? Write the importance of biometry. (Sect. 11.1) 2. What is called “central tendency”? Give some examples of central tendencies for a set of data. (Sect. 11.1.1) 3. What is called statistical mean? How many types of arithmetic means are there? What are they? Explain them. (Sects. 11.1.1.111.1.1.1.3) 4. Give the direct method and shortcut method of simple arithmetic mean. (Sect. 11.1.1.1.1) 5. Describe the direct method, shortcut method, and step deviation method of weighted arithmetic mean. (Sect. 11.1.1.1.2) 6. What do you understand by the arithmetic mean of a composite group? (Sect. 11.1.1.1.3)

312

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Questions, and Questions and Answers for Practice

7. What is median? Explain it. (Sect. 11.1.1.3) 8. What is called mode? Explain it. (Sect. 11.1.1.4) 9. Give the empirical relationship among mean, median, and mode. (Sect. 11.1. 1.4) What is called range? (Sect. 11.1.2.1.1) What is quartile deviation? (Sect. 11.1.2.1.2) What is mean deviation? (Sect. 11.1.2.1.3) 10. What is called standard deviation? Explain it. (Sect. 11.1.2.1.4) What is called variance? (Sect. 11.1.2.1.4) What is called standard error of means or SEM? (Sect. 11.1.1.2) 11. What is called probability? What is called an event? How many types of events are there? Explain them. (Sect. 11.1.3) 12. What is an event space or sample space? (Sect. 11.1.3.7) What are called sample points? (Sect. 11.1.3.8) 13. Define the following terms: Continuous data, discontinuous data, parametric data, nonparametric data, normal distribution, homogeneous and heterogeneous population, skewness, and kurtosis (Sects. 11.1.4–11.1.4.10) 14. What is a statistical decision tree? Give its importance. (Sect. 11.1.4.11) 15. Write the importance of Bartlett’s test of homogeneity (Sect. 11.1.8.1), Cochran test (Sect. 11.1.8.2), and Student’s t-test (Sect. 11.1.8.5). 16. Give the equation to identify homogeneity of variance by F-test. (Sect. 11.1.8.3) 17. Describe the methods of analysis by ANOVA and ANCOVA. (Sects. 11.1.8.6 and 11.1.8.11) 18. What is a scattergram? (Sect. 11.1.9) How will you calculate correlation coefficient (r) in a linear regression model? (Sect. 11.1.9.2) 19. Write the types of data analyzed by the following tests. Give their methods of analysis in brief. Scheffe’s test, Dunnett’s t-test, Tukey’s test, Williams’ test, Duncan’s multiple range test, Fisher’s exact test, Chi-square test, Mann-Whitney U-test, and Kruskal-Wallis test (Sect. 11.1.7, please see the respective method) Chapter 12 1. What is software? (Sect. 12.1.1) How many types of software do you know? Give examples. (Sect. 12.1.2) 2. Give the importance of software in pharmacokinetics. (Sect. 12.1.3) 3. Name some pharmacokinetic software tools and their application. (Sect. 12.1.4)

15.2

Multiple Choice Questions with Answers for Various Competitive Examinations

Q.1. In a pharmacokinetic model depicted in the following scheme, what is the halflife of the drug if the apparent volume of distribution of the drug is 25 L?

15.2

Multiple Choice Questions with Answers for Various Competitive Examinations

313

250 mg i.v

0.173/hr

(A) (B) (C) (D)

1.7 h 2h 4h 3h

(Answer C)

Q.2. What will be the maintenance dose of a sustained release 12-h formulation of a drug exhibiting one compartment kinetics with a half-life of 6 h, plasma concentration (steady-state) 6 mg/ml, volume of distribution 30 L, and an oral bioavailability of 80%? (A) 249.48 (B) 225.48 mg (C) 311.85 mg (Answer C) (D) 281.85 mg Q.3. Which is NOT applicable to protein binding? (A) Klotz reciprocal plot (B) Sandberg modified equation (Answer D) (C) Blanchard equation (D) Detli plot Q.4. According to USP, the speed regulating device of the dissolution apparatus should be capable of maintaining the speed within limits of what % of the selected speed? (A) 1% (B) 2% (Answer C) (C) 4% (D) 5% Q.5. Which of the following parameters from plasma concentration time profile study gives indication of the rate of drug absorption? (A) Cmax (B) Tmax (Answer B) (C) AUC (D) tl/2 Q.6. If C is the concentration of dissolved drug and Cs is the saturation concentration, in which case the sink conditions are said to be maintained? (A) C < 20% of Cs (B) C > 20% of Cs (Answer C) (C) C < 10% of Cs (D) C > 10% of Cs

314

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Questions, and Questions and Answers for Practice

Q.7. Which condition does not apply as per Indian law while conducting single-dose bioavailability study of an immediate release product? (A) Sampling period should be at least three times of drug tl/2. (B) Sampling should represent pre-exposure, peak exposure, and postexposure phases. (C) There should be at least four sampling points during elimination phase. (Answer D) (D) Sampling should be continued till measured AUC is at least equal to 80% of AUC. Q.7. A drug is administered to a 65-kg patient as 500 mg tablets every 4 h. Half-life of the drug is 3 h, volume of distribution is 2 L/kg and oral bioavailability of the drug is 0.85. Calculate the steady-state concentration of the drug? (A) 5.05 mcg/mL (B) 4.50 mcg/mL (Answer C) (C) 3.53 mcg/mL (D) 3.00 mcg/mL Q.8. A drug whose solubility is 1 g/L in water, when given orally at a dose of 500 mg is absorbed up to 95% of the administered dose. The drug belongs to which class according to the BCS classification? (A) Class I (B) Class II (C) Class III (Answer B) (D) Class IV Q.9. What will be the dose required to maintain therapeutic concentration of 20 μg/ mL for 24 h of a drug exhibiting total clearance of 2 L/h? (A) 96 mg (B) 480 mg (C) 960 mg (Answer C) (D) 48 mg Q.10. What will be the urine to plasma ratio of a weakly acidic drug having pKa of 5? [urine (pH ¼ 5) plasma (pH ¼ 7)] (A) 1 : 101 (B) 1 : 201 (Answer B) (C) 2 : 101 (D) 1 : 202 Q.11. The characteristic of nonlinear pharmacokinetics includes: (A) Area under the curve is proportional to the dose. (B) Elimination half-life remains constant. (Answer C) (C) Area under the curve is not proportional to the dose. (D) Amount of drug excreted through remains constant. Q.12. Measurement of inulin renal clearance is a measure for: (A) Effective renal blood flow (B) Renal drug excretion rate (Answer D) (C) Active renal secretion (D) Glomerular filtration rate

15.2

Multiple Choice Questions with Answers for Various Competitive Examinations

315

Q.13. The applicability of Noyes-Whitney equation is to describe: (A) First-order kinetics (B) Zero-order kinetics (Answer D) (C) Mixed-order kinetics (D) Dissolution rate Q.14. Drugs following one-compartment open model pharmacokinetics eliminate: (A) Bi-exponentially (B) Tri-exponentially (C) Non-exponentially (Answer D) (D) Mono-exponentially Q.15. Drugs in suspensions and semisolid formulations always degrade by: (A) First-order kinetics (B) Second-order kinetics (C) Zero-order kinetics (Answer C) (D) Nonlinear kinetics Q.16. A drug (200 mg dose) administered in tablet form and as intravenous injection (50 mg dose) showed AUC of 100 and 200 μg h/mL, respectively. The absolute availability of the drug through oral administration is: (A) 125% (B) 250% (Answer C) (C) 12.5% (D) 1.25% Q.17. The volume of distribution of a drug administered at a dose of 300 mg and exhibiting 30 μg/mL instantaneous concentration in plasma shall be: (A) 10 L (B) 100 L (Answer A) (C) 1.0 L (D) 0.10 L Q.18. It is required to maintain a therapeutic concentration of 10 μg/mL for 12 h of a drug having half-life of 1.386 h and Vd of 5 L. The dose required in a sustained release product will be: (A) 600 mg (B) 300 mg (Answer A) (C) 30 mg (D) 60 mg Q.19. A 250 mg dose of a drug was administered to a patient by rapid i.v. injection. The initial plasma concentration was 2.50 μg/mL. After 4 h, the plasma concentration was 1.89 μg/mL. Assuming that the drug was eliminated by a pseudo firstorder process and the body behaves as one-compartment model: 19(i). Kel is (A) 0.0699 h1 (Answer A) (B) 0.0349 h1 (C) 1.623 h1 (D) 0.699 h1

316

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Questions, and Questions and Answers for Practice

19 (ii). Biological half-life is: (A) 4.95 h (B) 19.82 h (Answer D) (C) 99.1 h (D) 9.91 h Q.20. In crossover bioavailability studies, in which the subjects must be rested for sufficient time between each drug administration to ensure that “washout” is complete practically, or “washout” is deemed to be complete when (P) 95% is washed out. (Q) 100% is washed out. (R) Five biological half-lives have elapsed. (S) Two biological half-lives have elapsed. (A) P, R (B) P, S (Answer A) (C) Q, R (D) Q, S Q.21. A drug was administered to 30 subjects as a tablet (30 mg), an oral aqueous solution (30 mg) and as an intravenous infusion (0.3 mg). Mean AUC’s (ng.h/ mL), dose normalized to 1 mg, for tablet, oral solution, and IV infusion were 0.91, 0.87, and 103.0, respectively. Calculate the relative bioavailability of the drug in tablet compared to the oral solution and the absolute bioavailability of tablet form: (A) 104.6%, 0.883% (B) 81%, 5.6% (Answer A) (C) 10.46%, 8.83% (D) 19%, 56% Q.22. Mode is the ____________ measure of central tendency (select correct answer to fill up). (a) First (b) Second (c) Third (d) Fourth (Answer C) Q.23. In a data distribution, mean is 12, median is 10, what is the value of mode? (a) 12 (b) 8 (c) 9 (d) 6 (Answer D) Q.24. “Survey of a like or dislike of lipstick” is an example of which of the following data type? (a) Parametric (b) Nonparametric (c) Continuous (d) Discontinuous (Answer B)

15.2

Multiple Choice Questions with Answers for Various Competitive Examinations

317

Q.25. Gaussian distribution is: (a) Normal distribution (b) Non-normal distribution (c) Skewed distribution (d) Abnormal distribution (Answer A) Q.26. Fisher’s exact test is done for: (a) Quantitative data (b) Qualitative data (c) Quantitative data with small samples (d) Qualitative data with small samples (Answer D) Q.27. Gastric acid output in neonatal life is: (a) 2 mEq/10 kg/h (b) 20 mEq/10 kg/h (c) 15 mEq/10 kg/h (d) 0.15 mEq/10 kg/h (Answer D) Q.28. A drug has maximum plasma concentration 30 mg/L and minimum plasma concentration 10 mg/L. What is the value of ekE T ? (T is dosing interval) (a) 0.33 (b) 3 (c) 0.03 (d) 300 (Answer A) Q.29. Wagner Nelson equation is: (a) Absorption kinetics (b) Elimination kinetics (c) Mixed kinetics (d) Lineweaver-Burke kinetics (Answer A) Q.30. What will be the AUC values of lidocaine if the administered dose is 0.2 g and the total body clearance is 45 L/h? (a) 4.44 h.mg/L (b) 0.0044 h.mg/L (Answer A) (c) 9.00 h.gm/L (d) 9000 h.mg/L Q.31. If a drug is known to be distributed into total body water, how many milligrams are needed to obtain an initial plasma level of 5 mg/L in a patient weighing 70 kg? (a) 210 (b) 150 (Answer A) (c) 50 (Since 70 kg man has a standard body water content 42 L) (d) 35 Q.32. Which method is not suitable to calculate the area under the curve? (a) Least square method (b) Weighing and platometry (c) Trapezoid rule (Answer A) (d) Integration of curve

318

15

Questions, and Questions and Answers for Practice

Q.33. Phase solubility analysis curve is not a good tool for: (a) Complex formation (b) Bioavailability determination (c) Polymorph detection (Answer D) (d) Impurity detection Q.34. The following techniques is/are used to determine the amount of the drug bound to protein? (a) Equilibrium dialysis (b) Solubility (c) pH analysis (Answer A) (d) Distribution method Q.35. For drug substances with highly variable pharmacokinetic characteristics, the following bioequivalence study design is used: (a) Parallel design (b) Non-replicate design (Answer D) (c) Non-parallel design (d) Replicate design Q.36. Volume of blood that flows per unit time per unit volume of the tissue is: (a) Residence time (b) Elimination rate (c) Gastric emptying rate (Answer D) (d) Perfusion rate Q.37. In case of one-compartment open model intravenous infusion, CSS (steadystate plasma concentration) is equal to: (a) [Plasma concentration][infusion rate]/clearance (b) [Cmax][infusion rate]/clearance (c) [tmax][infusion rate]/clearance (Answer D) (d) Infusion rate/clearance Q.38. IVIVC utilizes the principles of statistical moment analysis: (a) Level A (b) Level B (Answer B) (c) Level C (d) Level D Q.39. During determination of absorption rate constant by method of residual, flipflop phenomenon occurs when (Ka absorption rate constant and KE overall elimination rate constant): (a) KE/Ka  3 (b) Ka/KE  3 (Answer A) (c) KE/Ka  3 (d) Ka/KE  3 Q.40. Apparent volume of distribution will be highest in the case of the drug with % plasma protein binding: (a) 10% (b) 89% (c) 50% (Answer A) (d) 68%

15.2

Multiple Choice Questions with Answers for Various Competitive Examinations

319

Q.41. For a particular drug, the rate of absorption but not the extent of the absorption of GIT is affected by the presence of food in GIT then taking the drug with food will result in: (a) Smaller area under the plasma drug concentration time curve (b) Smaller maximal plasma drug concentration (Answer B) (c) Smaller time at which the maximal plasma drug concentration occurs (d) Smaller fractional bioavailability and total clearance Q.42. Characteristics of drug-protein binding: (P) Often parallels drug lipid solubility. (Q) Drug-plasma albumin binding tends to be relatively nonselective. (R) Acidic drugs bind to albumin while basic drug bind to glycoproteins. (S) In rheumatoid arthritis patients, increased alpha1-acidic glycoprotein tends to promote increased lidocaine protein binding. (a) P and Q (b) P, Q, and R (c) P, Q, R, and S (Answer C) (d) P and R. Q.43. Which of the following method is an example of facilitated diffusion? (a) Passive diffusion (b) Endocytosis (c) Carrier-mediated diffusion (Answer C) (d) Active transport Q.44. Conc v/s time curve drown from single oral dose, which parameter can be calculated: (a) Elimination constant (b) Rate constant (Answer D) (c) Absorption peak (d) Plasma concentration Q.45. Bioavailability differences among drug’s oral formulations are more likely to occur if it: (a) Is freely water-soluble (b) Is incompletely absorbed (Answer B) (c) Is completely absorbed (d) Undergoes little first-pass metabolism Q.46. Creatinine clearance is used as a measurement of: (a) Passive renal absorption (b) Glomerular filtration rate (Answer B) (c) Renal excretion rate (d) All Q.47. What is the renal clearance of substance, if its concentration in plasma is 10 mg/mL, concentration in urine is 100 mg/L and urine flow 2 mL/min? (a) 0.02 mL/min (b) 0.2 mL/min (c) 2 mL/min (Answer A) (d) 20 mL/min

320

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Questions, and Questions and Answers for Practice

Q.48. For first-order reactions, the rate constant, k, has the units as: (A) M s-1 (B) M-1 s-1 (Answer D) (C) M-2 s-1 (D) s-1 Q.48. If the given drug is absorbed by passive diffusion what will be its absorption kinetics? (a) Zero order (b) First order (Answer A) (c) Second order (d) Pseudo-zero order Q.49. Which of the following parameters is/are important in determining bioequivalence? (A) Tmax (B) Cmax (Answer C) (C) AUC and C (D) None of these Q.50. Which of the following may be used to assess the relative bioavailability of two chemically equivalent drug products in a crossover study? (A) Dissolution test (B) Peak concentration (C) Time-to-peak concentration (Answer D) (D) Area under the plasma level time curve Q.51. Regarding two-compartment pharmacokinetics all are true EXCEPT: (A) A drug is always removed from the peripheral compartment. (Answer A) (B) A drug with a high volume of distribution is likely to be lipophillic. (C) A drug can have a short duration of action while being eliminated very slowly. (D) Most anesthetic drugs are modeled well with a two-compartment model. Q.52. The Vd for phenytoin is 70 L and half-life is 1.5 h. What is the total clearance of phenytoin? (A) 34.32 L/h (B) 32.34 L/h (Answer B) (C) 151.5 L/h (D) 51.51 L/h Q.53. What will be the approximate Tmax of a drug exhibiting Ka of 2 h1 and K of 0.2 h1? (A) 1.2 h (B) 2.4 h (Answer A) (C) 4.8 h (D) 2.0 h

15.3

15.3

Selected Questions from the University Examinations

321

Selected Questions from the University Examinations

1. (a) Determine Ka by Loo-Riegelman method. Write the advantages of this method over other methods for determination of Ka. (b) Deduce the time-concentration relationship equations for more than one capacity limited elimination process. Write its significance. (c) A drug follows nonlinear kinetics in its elimination process in the body. It has Km and Vmax 150 mg and 75 mg/h. If the drug is administered 500 and 400 mg to a patient on different occasions, determine t1/2 at the different doses. 5 þ 2 þ 7 þ 6 ¼ 20 2. (a) Define loading dose and accumulation factor. Give their significances. Deduce their equations. (b) From the loading dose-accumulation factor relationship, show that KE ¼ 2:303=t log ½ðCss  CÞ=Css  (c) A patient is 55 years old, and the body weight is 78 kg is administered an iv infusion. Elimination half-life is 3 h, Vd is 0.75 L/kg and effective plasma concentration is 12 mg/L. Drug is supplied in 5 mL ampoules in a concentration 250 mg/mL. Calculate the starting infusion rate in mg/h and mL/h. 4 þ 2 þ 5 þ 4 þ 5 ¼ 20 3. (a) Define compartmental model and write its advantages. Give its assumptions. A drug (dose X0) is administered by iv infusion to a patient and eliminated from both the compartments. Draw a two-compartmental model, and develop its equations, and determine plasma concentration using the compartmental model. (b) By repetitive iv injection, a dose of 600 mg is given to a patient every 8 h. The drug has t1/2 value 3 h in the patient. Vd is 22 L. Calculate minimum, maximum, and average plasma levels of the drug, loading dose, and maintenance dose in the patient. 2 þ 2 þ 3 þ 7 þ 6 ¼ 20 4. (a) Deduce the drug levels in a two-compartmental model in both the compartments, when the drug undergoes hepatic first-pass metabolism, and show that bioavailability of a drug is invariably more by i.v. route of drug administration.

322

15

Questions, and Questions and Answers for Practice

(b) A patient received a drug infusion 5 mg/h for 12 h. K is 0.04 h1, Vd ¼ 24 L Determine the concentration of the drug in the body 3 h after the drug cessation. 16 þ 4 ¼ 20 5. (a) Determine Ka by Wagner-Nelson method. How will you determine it when the drug is excreted unchanged through urine? Show by compartmental model how you will determine plasma level of drug metabolite. (b) A patient is administered a drug with an iv infusion rate 28 mg/h and blood levels of the drug were found to be 10 mg/mL and 13 mg/mL at 10 and 30 h, respectively. Reported elimination half-life of the drug in patients is 35 h. Determine Css, elimination half-life of the drug in the patient and Vd. 5 þ 5 þ 5 þ 5 ¼ 20 6. Write short notes on: (a) Flip-flop model (b) Peeling method (c) Dose calculation for obese patients, patients with renal insufficiency, and old patients (d) Apparent and absolute bioavailability and their relationship with t1/2 values 5  4 ¼ 20 7. What is Dominguez equation? Give its importance. Deduce Wagner-Nelson equation to determine Ka. Why is “% of drug remaining to be absorbed” instead of “% of drug absorbed” plotted against time in case of Wagner-Nelson equation? (Justify). What is the principal demerit of Wagner-Nelson equation? 2 þ 2 þ 5 þ 4 þ 2 þ 5 ¼ 20 8. In a multiple dosing regimen, determine the maximum and minimum plasma concentrations of a drug after the nth dose. What do you mean by steady-state plasma concentration? Give its significances. Deduce the equation of steady-state plasma concentration? What is accumulation factor? Deduce its value in a multiple dosing regimen. Give its significances.

15.3

Selected Questions from the University Examinations

323

6 þ 1 þ 2 þ 3 þ 2 þ 4 þ 2 ¼ 20 9. (a) What do you mean by absolute bioavailability? How is it different from relative bioavailability? Determine absolute bioavailability in terms of half-life. How can you determine it using urinary excretion data (deduce it)? What is Krügner-Thiemer pharmacokinetic factor? Give its significance. (b) Infusion of a drug was administered to a patient at 30 mg/h. Blood drug concentrations were found to be 9.5 mg/mL at 8 h and 13 mg/mL at 28 h. The drug has average half-life in common population is 24 h. Determine Css and Vd. 1 þ 1 þ 4 þ 4 þ 2 þ 2 þ 6 ¼ 20 10. (a) Determine, by developing equation, time-concentration relationship for more than one capacity limited processes. (b) Using compartment model, determine the concentration of metabolite of a drug in plasma. (c) A drug eliminated by nonlinear kinetics has Km 125 mg and Vmax 65 mg/h. If 450 and 360 mg of drug have been administered to a patient on different occasions, calculate t1/2 of the drug at those dose levels. 10 þ 5 þ 5 ¼ 20 11. Define first-pass metabolism and its impact on drug bioavailability. Using compartmental model, show that bioavailability of a drug is always more when the drug is administered by intravenous route than by oral route. 3 þ 17 ¼ 20 12. Write short notes on: (a) Flip-flop model (b) Residual method (c) Deduce equation for absolute bioavailability using urinary excretion data (d) Dose calculation for patients with renal insufficiency and for elderly patients

324

15

Questions, and Questions and Answers for Practice

5 þ 6 þ 6 þ 3 ¼ 20 13. Define compartment model. Write its assumptions. Draw the following compartment model, and develop the equations for each compartment. A dose (x0 amount) of a drug is administered into the central compartment, and from there it has been distributed to lungs, kidney, and brain and eliminated from kidney. 2 þ 4 þ 6 ¼ 12 14. Deduce Wagner-Nelson equation and graph for the determination of ka. Write its significances. 5þ3¼8 15. What do you mean by loading dose? Deduce the equation for loading dose. 2þ3¼5 16. Write the significance of steady-state plasma concentration. Find out the dose of an infusion of a drug in a patient to achieve the steady-state plasma concentration 2 mg/mL. Volume of distribution of the drug is 20 L, and elimination rate is 0.15/h. 2þ3¼5

Index

A Absolute bioavailability, 37, 39–42, 308, 315, 316, 322, 323 Absorption, 1, 21, 37, 51, 111, 137, 145, 158, 197, 228, 250, 307 Absorption, distribution, metabolism, and elimination (ADME), vii, 1, 5, 33, 51, 158, 229, 240 Accuracy, 2, 85, 158, 164, 168, 169, 191, 194, 197, 199, 200, 211, 222, 228, 233, 282 Acetaminophen, 42, 147, 149, 265 Acetonitrile, 182, 187, 191, 196, 243, 244, 248, 268, 270–273, 275–278 Acetylation, 113, 162 Acetylcholine, 150, 152 Acidic drugs, 151, 162, 314, 319 Aciloc, 269 acslXtreme, 231 Active pharmaceutical ingredient (API), 42, 277 Active reabsorption, 125 Active secretion, 125, 162 Active transport, 45, 319 ADAPT 5, 231 Adobe, 227 Adverse drug reactions, 51, 145 Agonist, 150, 152 Agonistic, 145, 150, 152, 310 Akaike information criterion, 172 Albumin, 45, 58, 148, 151, 160, 162, 237–247, 273, 319 Alcohol dehydrogenase, 112, 247 Aldehyde dehydrogenase, 112 Algorithm, 171, 231 Alpha (α)-acid-glycoprotein, 45 Alpha one (α1)-acid glycoprotein, 45, 151, 162, 319 Alternate hypothesis, 210, 220, 222, 224

Aluminum/magnesium hydroxide, 147, 150, 151 Alzheimer’s disease, 152 Ampicillin, 147, 151 Analysis of covariance (ANCOVA), 223, 312 Analysis of variance (ANOVA), 199, 211, 217– 221, 223, 232, 312 Antagonistic, 145, 150, 152, 310 Antecubital fossa, 176, 177 Anticoagulants, 175–178, 235–236, 268, 270, 271, 278, 311 Antioxidant, 175 Apparent volume of distribution, 12, 87, 96, 117, 119, 163, 287, 291, 293, 298, 312, 318 Application software, 227, 228 Area under the curve (AUC), 3, 14, 23, 24, 38, 44, 60, 97, 168–170, 237, 267, 305, 307, 313, 314, 316, 317, 320 Area under the first moment curve (AUMC), 97, 169, 170, 237 Arithmetic mean, 200–202, 205, 311 Aryl hydrocarbon hydroxylase, 112, 247 Aspartate aminotransferase (AST), 178 Assessment of efficacy, clinical trails, 283 Assessment of safety, clinical trails, 283 Assumptions of compartment models, 199, 324 Asymmetric peak, 195 Atazanavir, 147, 151 Atenolol, 146, 147, 151, 243, 244 Atmospheric pressure ionization, 188, 189 Atorvastatin, 274–276 Atropine, 147, 151 Autolysis, 180

B Bartlett’s test, 211–213, 312

# The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B. Mukherjee, Pharmacokinetics: Basics to Applications, https://doi.org/10.1007/978-981-16-8950-5

325

326 Basic drugs, 151, 162, 319 Bayesian designs, 229 Bayesian information criterion, 172 Benztropine mesylate, 147, 151 BestDose, 229 Bi-exponential equation, 31, 98, 122 Billiary elimination rate, 86 Bioavailability, vii, 3, 4, 23, 24, 31, 37–49, 97, 117, 119, 120, 137, 146, 147, 149, 159, 168, 237, 269–272, 305, 308, 309, 313, 314, 316, 319–323 Bioequivalence, vii, 37–49, 182, 269–270, 308, 318, 320 Bioequivalents, 43, 44, 269, 308 Bioinformatics, 229 Biological half-life, 3, 12, 33, 45, 89, 162, 169, 308, 316 Biometry, 199–225, 311 Biopsy, 179 Bioptome, 179 Biostatistics, 199 Biotransformation, 111, 149, 151, 248 Bipolar disorder, 150, 152 Bisoprolol, 146, 147 Blinding (masking) process, 166 Blood-brain barrier, 229 Blood clotting, 157, 176, 177, 235 Blood compartment, 21, 28, 61, 84, 85, 92, 98– 100, 121 Blood flow-limited model, 55, 56, 59, 60, 308 Bolus injection, 2, 3, 167 Boomer, 230 Bouin’s solution, 180 Boundary layer, 182, 183 Bound drug %, 242, 244 Brand name, 42, 269, 276 Buffer, 180, 187, 188, 194, 196, 234–235, 238– 240, 242, 244–246, 248, 260, 261, 263, 268, 272, 273 Buffer compartment, 239, 240, 244

C Cadaver, 263, 265 Calcitriol, 248 Canaglifozin (CFZ), 276–278 Capacity-limited process, 125, 127, 129–135, 309, 323 Carbamazepine, 271 Carnoy’s solution, 180 Categorical data, 208, 215 Catenary model, 62, 63, 309 C18 column, 196, 270, 273

Index Celecoxib, 149, 152 Cell culture, 247, 248, 261 Cell lysis buffer, 235 Cellophane, 263–265 Cellular uptake, 51, 53, 260–263 Central compartment, 28, 61, 63, 84, 85, 92–94, 96, 97, 99–104, 115, 117–121, 138, 253, 255, 256, 302, 303, 324 Central tendency, 200–203, 311, 316 Chang liver cells, 247 Channels matching (CM) algorithm, 171 Characteristic, 2, 33, 44, 45, 60, 73, 78, 191, 196, 197, 208, 314, 318, 319 Chemically equivalent, 43, 320 CHL-CYP2C18, 247 Chlorothiazide, 149, 152 Chromatographic conditions, 195 Chromatography, 185–197, 266, 273 Ciprofloxacin, 147, 151 Citrate, 147, 176, 177, 235, 236 Clarithromycin, 146, 147, 151 Clark’s formula, 161 Clearance, vii, 18, 37–49, 55, 56, 97, 148, 152, 162, 163, 170, 172, 173, 230, 231, 236, 237, 253, 256, 267, 269–271, 277, 287, 294, 299–301, 303, 304, 308, 314, 317– 320 Clidinium, 147, 151 Clinical examination, 164 Clinical pharmacy, vii, 279 Clinical trial protocol, 279–283 Clinical trials, 161, 164–168, 172, 228, 229, 279, 282, 283 Clopidogrel, 147, 149, 151 Cmax, 2–4, 8, 44, 168, 169, 237, 269, 270, 272, 277, 278, 307, 313, 318, 320 Cochrane library, 153 Cochran test, 211, 213–215, 312 Cohort group, 166, 279 Cohort study, 166, 279 Collection of patient data, clinical trails, 279– 281 Collision energy optimization, 191 Collision gas pressure, 278 Collision-induced dissociation, 189 Common logarithm, 19, 84, 87, 99, 130, 236 Compartmental data analysis, 170, 171 Complementary events, 207 Composite events, 206 Concentration gradient, 59, 183 Confocal microscopy, 260 Conjugation, 112, 113, 160, 247, 261 Constant-head water reservoir, 248–254, 257

Index Continuous data, 208, 211–213, 215, 217, 222, 312 Continuous random variable, 200 Control group, 165, 166, 211, 219, 262 Correlation coefficient, 224–225, 312 Covariates, 170, 173, 223 Cowling’s formula, 161 C++, Python, 228 Cramer’s rule, 63, 65–69, 104, 107 Critical value, 210, 216, 217, 219–221 Cross-over design, 44, 166, 271 Crystalluria, 111 Cumulative frequency curves, 200 Curtain gas pressure, 278 Cyclic model, 62–63, 309 Cyclohexane, 271, 272 Cyclopentolate, 147, 151 CYP2A6, 151 CYP2B6, 151 CYP2C18, 247 CYP2C19, 151 CYP3A, 151 Cytochrome P-450 enzymes, 112 Cytochrome P-450 monooxygenases, 112, 272 Cytosolic fraction, 179, 181

D DAPI, 261, 262 Darifenacin, 147, 151 Data analysis, vii, 169–171, 173, 199, 200, 228, 229, 243 pharmacokinetic/pharmacodynamic model, 170, 172, 228 physiological/physiology-based pharmacokinetic model, 170, 172 Data editing, 169–170 Data handling, 169, 283 DC voltage, 189, 190 Deamination, 112 Degree of freedom, 210, 211, 213, 217–222 Denominator determinant, 65, 66, 68 Descriptive statistics, 199 Design–Expert, 232 Desolvation, 188, 274, 276 Detection limit, 196, 197, 268 Determinant, 63–69, 95–98, 105–107, 116, 118 Dialysis chamber system, 238, 243 Dialysis tubing cellulose membrane, 242 Diazepam, 147, 151 Diclofenac, 148, 149, 151, 152, 242, 245, 276 Dicyclomine, 147, 151

327 Diffusion, 51, 52, 59–60, 151, 175, 182, 183, 193, 241, 248, 251, 263, 264, 308, 319 Diffusion cell, 263, 264 Diffusion coefficient, 183 Diffusion-limited model, 51, 59–60, 308 Digoxin, 146, 147, 151, 162 Dihydroxycholecalciferol, 248 Dimethyl sulfoxide (DMSO), 243, 244, 273 Dimethylthiazolyl-2, 5-diphenyltetrazolium bromide, 236 3-(4, 5-Dimethylthiazol-2-yl)-2, 5-diphenyltetrazolium bromide (MTT) assay, 236, 261, 262 Direct method, 201, 311 Discontinuous data, 208, 221, 312 Discrete data, 208 Discrete random variable, 200, 202, 203 Distribution, 1, 21, 45, 51, 113, 131, 145, 158, 183, 191, 199, 229, 237, 287, 307 Distribution constant, 183 Distribution properties, 200 Dominguez equation, 22, 308, 322 Donepezil, 150, 152 Donnan effect, 240 Donor compartment, 263, 265 Dopamine, 150, 152 Dose formula children, 157, 159–161 elderly patients, 157, 159–163 infants, 157, 159–161 neonates, 157, 159–161 obese patients, 157, 159–161 patients with liver and kidney insufficiencies, 157, 159–161, 163 Dose interval, 9, 12, 14–16, 18, 97, 163, 168, 287, 288, 291, 298, 317 Dose-plasma drug concentration relationship, 131–132, 305 Double-blind process, 166 Doxorubicin, 146, 147, 151, 261 Driver software, 228 Drug absorption, vii, 2–5, 8, 13, 21–34, 37–49, 57, 99, 101, 111, 139, 140, 142, 146, 147, 150, 151, 159, 162, 170, 171, 228, 258–260, 269, 271, 307, 308, 310, 313 Drug absorption rate, 5, 21–25, 28–32, 101, 138, 143, 308 Drug accumulation factor, 15–18 Drug dissolution rate, 183 Drug distribution, 5, 12, 21, 22, 45, 51–109, 159, 160, 162, 199, 230, 237, 249, 251, 254, 256, 258

328 Drug-drug interaction (DDI), vii, 145–153, 158, 161, 281, 310 Drug elimination, 5, 16, 21, 24, 39, 52, 53, 62, 70, 86, 89, 94, 97, 99, 101, 123–143, 163, 170, 171, 254 Drug Interaction Checker, 153 Drug metabolism, 5, 21, 45, 59, 111–124, 160, 162, 247–248, 258, 309 Drug-plasma protein binding, 58, 151, 162, 199, 237–247, 318 Drug-protein binding, 2, 5, 45, 48, 49, 56–59, 237–247, 258, 308, 319 Duccan’s multiple range test, 211 Dulbecco’s Modified Eagle Medium (DMEM), 261, 262 Dunnett difference, 219 Dunnett’s t-test, 211, 219–220, 312 Duration of action, 3, 4, 320

E Eddy’s diffusion, 183 Edema, 267 Edsim++, 229 Electrochemical detector, 194, 197 Electron microscopy, 180 Electrospray ionization (ESI), 186, 188–190, 274, 278 Elementary events, 206 Elimination, 1, 21, 38, 51, 111, 125, 145, 158, 249, 287, 309 Elimination half-life, 44, 249, 252, 254, 257, 289, 291–293, 314, 321, 322 Elimination rate/elimination rate constant, 7, 9, 16, 18–19, 25, 26, 33, 45, 85, 86, 88, 90–92, 101, 115, 118, 121–124, 135– 139, 142, 143, 170, 250, 253, 287, 289, 296, 309, 318, 324 Embase, 153 Endoscopy, 179 Endpoints, 165, 166, 168, 169, 279, 282 Energy optimization, 190–191 Enteric-coated, 1 Enzyme saturation kinetic, 132 Equally likely event, 207 Equilibrium dialysis, 49, 237–240, 242, 318 Erythromycin, 147, 151 Esterase, 112, 247 Ethical clearance, 236, 267, 271, 277 Ethyl acetate, 276–278 Ethylenediaminetetraacetic acid (EDTA), 235, 236, 268, 270, 271, 278 Etoricoxib, 148, 149, 152

Index European Medicines Agency (EMA), 168 Events, 60, 200, 206, 207, 219, 279, 312 Event space, 207, 312 Everted ileum, 258, 259 Excretion, 25–27, 37, 39, 41–42, 60, 121, 125, 142–143, 146, 150, 237, 308, 310, 314, 319, 323 Excretion rate method, 142–143, 310 Exhaustive events, 207 Expected frequency, 222 Extent of drug absorption, vii, 2, 37–49, 269, 319 Extract, 182, 270, 275, 278 Extraction, 55, 56, 169, 181–183, 196, 275, 278, 308, 311 Extraction ratio, 55, 56, 308 Ezetimibe, 274–276

F Famotidine, 147, 151 F-distribution, 172, 217 Feathering method, 32 Fecal sample, 181–182, 185, 311 Fesoterodine, 147, 151 Financing policy, clinical trails, 283 First-order kinetics, 5, 6, 24, 31, 86, 121, 126, 127, 132, 139, 248, 249, 251, 254, 256, 257, 307, 315, 320 First-pass metabolism, 112, 309, 319, 321, 323 Fisher’s exact test, 211, 221–222, 312, 317 FITC, 260–262 Flip-flop disposition, 33 Flip-flop phenomenon, 32–33 Flow cytometer (FACS), 260, 262 Flow model, 51–53 Fluconazole, 146, 147, 151 Fluorescence detector, 194, 197 Fluorescent microscopy, 260 Forced degradation, 197 Formalin, 180 Formic acid, 273, 275–278 Fractional multiple, of dose, 40, 41 Fraction of free drug concentration, 58 Fraction of tissue free drug, 58 Fraction unbound drug, 239, 240 Fragmented ions, 191 Free-drug, 239, 241, 246 Fried’s rule for infants, 161 Fronting, 195 Frusemide, 267, 268 F-test, 172, 211, 215–216, 223 F-value, 217–219, 223

Index G Gastric ulcer, 270 Gaussian curve, 195 Gendre’s solution, 180 Generalized least-squares (GLS), 171 Generic medicine, 42, 44 Generic name, 42, 281 Genetic algorithm, 171 Glass separatory funnel, 179 Glibenclamide, 147, 151 Glomerular filtration, 125, 160, 162, 314, 319 Glucuronidation, 113, 160, 162 Glutathione S-transferase, 112, 247 Glycine conjugation, 113 Goodness of fit, 172 Grand mean, 210, 218 Granulated polymers, 187, 194 GraphPad Prism, 229–230 Group design, 165–166

H Half-lives, 12, 44, 89, 162, 253, 316 Halothane, 178 HCT116 cells, 262 Hematoma, 176, 177, 311 Hemolysis, 177, 178, 311 Heparin, 176, 177, 236 Hepatic drug-metabolizing enzymes, 247 Hepatocyte, 112, 247 Hepatoportal route, 111, 112, 247 Heterogeneous data, 211, 213, 216 Heterogeneous populations, 209 Heteroscedastic data, 171 High-performance liquid chromatography (HPLC), 169, 185, 187, 192–197, 239, 241, 243, 244, 249, 260, 261, 263, 265– 273, 275, 277, 278 High plasma protein binding drug, 237–247 Hirudin, 236 Histac, 269 Histamine-2 receptor blocker, 269 Homogeneity of variance, 211, 215, 312 Homogeneous data, 218 Homogenous populations, 209 Homoscedastic, 171 Honestly significant difference (HSD) test, 220 Humanized mouse, 247 Hydrochlorothiazide, 149, 152 Hydrolysis, 112, 247

I Ibuprofen, 148, 149, 152 ICH GCP guidelines, clinical trails, 282

329 ICH guidelines, clinical trails, 282 Impossible events, 207 Indapamide, 149, 152 Individual mean, 210 Indomethacin, 148, 149, 152 Inferential statistics, 199 Initial estimates, 171 Insurance, clinical trails, 283 Interface, 186, 188, 230, 274, 311 Internal standard (IS), 244, 248, 268, 270–278 Intravenous bolus injection, 3, 167 Intravenous infusion, 3, 15, 88–91, 132, 167, 251–253, 255–258, 289, 292, 293, 316, 318 Intrinsic clearance, 56 In vitro, 199, 247, 258–260, 263–265 Ion-exchange HPLC, 192, 193 Ionization, 2, 186–91, 274, 278 Isoflurane, 178 Isopropanol, 187 Itraconazole, 146, 147, 151

J JavaPK, 230 Java SE, 230 JGuiB, 230 JPKD, 230

K Ketamine, 178 Ketoconazole, 146, 147, 273 Kinetica, 231 Kinetica Streaming Data Warehouse, 231 Klotz reciprocal plot, 46–48, 313 Krüger-Thiemer dose ratio, 17–19 Krüger-Thiemer’s “factor”/Krüger-Thiemer’s “pharmacokinetic factor”, 17 Kruskal–Wallis test, 211, 223–224 Kurtosis, 61, 209–210

L Laboratory Applied Pharmacokinetics and Bioinformatics (LAPKB), 229 Laboratory work, 234–237 Lactate dehydrogenase (LDH), 178 Lag time, 3, 32, 33, 167 Lambda maxima, λmax, 250, 252, 255 Lansoprazole, 146–148, 150, 151 Laplace–Gauss distribution, 208 Laplace operator, 70 Laplace table, 70, 107, 302

330 Laplace transforms, 63, 70–73, 86, 88, 90, 91, 94, 96–98, 101, 104, 107, 115, 116, 118, 119, 121, 302 Least-squares method, 172 Leukopenia, 149, 152 Level 1 drug-drug interaction, 145 Levenberg–Marquardt algorithm (LM), 171 Linearity, 197 Linear kinetics, vii, 125–143, 228, 229, 291, 294, 300, 315, 321, 323 Linear regression model, 223–225 Lineweaver-Burk plot, 127, 128, 317 Liquid Chromatography with Tandem Mass Spectroscopy (LC-MS/MS), 168, 169, 182, 185–191, 239, 241, 244, 245, 260–263, 272, 274, 276–278 Lithium, 149, 152 Liver microsomes, 235, 248, 272–274 Lixoft, 229 Loading dose, 17, 18, 158, 287, 288, 291, 293, 299, 321, 324 Loo-Riegelman method, 28–30, 32, 321 Low plasma protein binding drug, 237–247

M Macintosh, 230 Mammillary model, 63 Mann–Whitney U test, 222–223 Mann–Whitney–Wilcoxon, 222–223 Mantissa, 73, 78, 84 Mass spectra detectors, 186, 187, 190 Mass spectrometry (MS), 185–188, 190, 273, 275–277 Mass to charge ratio (m/z), 186, 187, 189, 190, 274, 276, 278 Mass transfer, 46, 54, 182, 183, 187 Mathematical models, 3, 51, 61 MATLAB, 231 Matrix, 63–69, 95–98, 104–107, 116, 118, 263 Maximum likelihood of tests, 171 Maximum mutual information (MMI), 171 Maximum plasma drug concentration, 2, 13 Maximum safe drug concentration (MSC), 2–4, 8, 158 Maximum safe drug level, 3 Maxsim2, 231–232 Mean, 61, 73–76, 78–82, 84, 97, 172, 173, 199–205, 208, 210, 211, 216–221, 223, 230, 237, 316, 322–324 Mean deviation, 204–25 Mean residence time (MRT), 61, 97, 230, 237

Index Median, 176, 199–200, 202–204, 211, 216, 316 Medline, 153 Mefenamic acid, 148, 149, 152 Membrane-limited model, 59–60 Metabolic elimination rate, 86 Metabolism, vii, 5, 6, 21, 45, 51, 59, 111–125, 145, 146, 151, 158–160, 162, 199, 229, 247–248, 258, 272–274, 319, 321, 323 Metabolites, 1, 5, 38, 44, 52, 53, 57, 85, 111, 113, 120–125, 151, 172, 179, 181–183, 185, 196, 247, 248, 272, 274, 322, 323 Metformin (MET), 150, 276–278 Methacarn, 180 Methanol, 181, 187, 191, 192, 196, 243, 248, 266, 271, 273–278 Method of residual, 31–32, 137, 318 Methotrexate, 147, 148, 152 Methyltransferase, 112 Metolazone, 149, 152 Metoprolol, 42, 147–148, 151, 152 Michaelis–Menten constant, 128, 129, 131–132 Michaelis–Menten equation, 125, 127–129, 131 Microsoft Excel, 169, 230 Microsoft Windows, 228 Microsomal fraction, 179 Minimum effective concentration (MEC), 1–4, 8, 9, 13, 21, 23, 24, 37, 158 Minimum inhibitory concentration (MIC), 1 Minimum plasma drug amount, 258 Minimum plasma drug concentration multipledose regimen, 7–11, 17 Minimum toxic concentration (MTC), 2–4, 8, 13 Mobile phase, 187, 188, 191–197, 263, 266–268, 270, 272, 273, 275, 276, 278 Model selection, 171, 172 Mode of drug administration, 165, 167–168 Modes, 165, 167–168, 189, 190, 199, 200, 203, 230, 274, 278, 282, 316 Molar concentration, 46–48, 196, 234 Molecular diffusion, 182 Monoamine oxidase, 112 Monolix, 229 Morphine, 111, 113, 148, 152 Mosteller’s equation, 160–161 MS/MS operation and parameters, 190 Multicentered trial, 283 Multi-compartment model, 61, 84 Multiple-dose regimen, 7–11, 17 Multiple modes, 167 Mutually exclusive events, 207

Index N N-acetyl transferase, 247 NADPH, 248 Naloxone, 148, 152 Naproxen, 149, 152 Natural logarithm, 7, 9, 19, 84, 87, 99, 130, 236 N-desethyl oxybutynin, 272–274 Nebulization, 188 Needle biopsy, 179 Negative lag time, 32, 33 Nelder-Mead algorithm, 171 Nelson equation, 27–28 Nernst law of distribution, 183, 311 N-(4-hydroxyphenyl)acetamide, 265 N-(4-hydroxyphenyl)ethanamide, 265 N-octadecyl, 187 Noncompartmental data analysis, 170–172, 228 Nonlinear kinetics, vii, 125–143, 228, 229, 291, 294, 300, 315, 321, 323 Nonlinear regression model, 224–225 Non-normal distribution, 208, 316 Nonparametric data, 208, 211 Nonparametric rank statistics, 222 Nonparametric test, 211, 213 Normal distribution, 208, 212, 216, 317 Normal-phase HPLC, 192 Noyes-Whitney equation, 183, 315 Null hypothesis, 172, 210, 216–218, 220, 222, 223 Numerator determinant, 65–69

O Ogive, 200 Omeprazole, 146–151, 270–272 One-compartment closed model, 61 One-compartment open model, 62, 84, 86–92, 171, 249, 251, 309, 315, 318 One-way ANOVA, 217, 219, 223 Onset of action, 2–4, 167, 307 Optimization, 190–191, 195–197, 229, 230, 232, 311 Orange book, 44 Ordinary least-squares (OLS), 171 Osmium tetroxide, 180 Outlier, 170, 208 Overall elimination rate, 86, 88, 90–92, 121, 309, 318 Overall mean, 210 Oxidation, 112, 162, 175, 247 Oxidative dealkylation, 112 Oxybutynin, 272–274

331 P Paired test, 216 Pantoprazole, 146–148, 150, 151 Paracetamol, 42, 160, 246–247, 265–267 Paraformaldehyde, 180 Parallel design, 44, 166, 282, 318 Parametric data, 208, 212, 312, 316 Parametric test, 211, 212, 216, 221 Parkinson’s disease, 150, 152 Paroxetine, 147, 151 Partition coefficient, 2, 52, 54, 55, 58, 183, 195, 311 Passive reabsorption, 125 Pasteur pipette, 250, 253, 255, 257 Patch, 263–265 Patient Manager CSC, 153 PCModfit, 230 PCModfit V7.0, 230 Peak time, 4, 194 Peeling method, 32, 308, 322 Penicillamine, 147, 151 Percentage of drug remaining to be absorbed, 24, 25, 27, 28, 30, 258, 260, 308, 322 Perchloric acid, 181, 236, 261, 263 Perfusion model, 51, 54–57, 59, 308 Peripheral compartment, 28, 29, 61, 63, 84, 85, 92–94, 97–104, 109, 113, 115, 116, 118, 253–256, 258, 309, 320 p-glycoprotein (p-gp/p-gp 120), 151 Pharmaceutical equivalents, 43, 44, 308 Pharmacodynamic action, 1 Pharmacokinetic models, 51–109, 170–173, 228, 231, 308, 312 Pharmacokinetic report, 170, 173 Pharmacokinetics, vii, viii, 1–19, 31, 33, 38, 51–109, 113–120, 127, 145–153, 157–173, 175–183, 185–197, 199–225, 227–283, 287–305, 307–312, 314, 315, 318, 320, 323 Pharmacokinetic sampling, 175, 310 Pharmacokinetic software, vii, 227–232, 312 Pharmacovigilance, 279 Phase I clinical trial, 164, 167, 168, 230, 279 Phase I type hepatic drug-metabolizing reactions, 112 Phase II clinical trial, 164, 167, 168, 230 Phase II type hepatic drug-metabolizing reactions, 112 Phase III clinical trial, 164, 167, 168, 229 Phase “0” trial, 168 Phlebotomy, 175–178, 310 Phosphate buffer, 180, 196, 234, 242, 245, 246, 248, 260, 263, 272, 273

332 Phosphate buffer saline, 180, 234, 260–263 Physiological effects on pharmacokinetic drug parameters, 159–161, 163 Physiological pharmacokinetic model, 51–57, 172, 308 Physiological pharmacokinetic model with protein binding, 58–59 Pkanalix, 229 PKQuest, 228 PKSolver 2.0, 230 Placebo, 166, 282 Plasma, 2, 22, 37, 57, 117, 129, 145, 157, 177, 185, 199, 230, 237, 287, 307 Plasma compartment, 29, 239, 240, 244 Pmetrics, 229 Pneumatic intensifier, 188 Polyethylene glycol 400, 263 Poly-therapy, 146, 151, 152 Population, 161, 162, 164, 173, 199, 208–210, 217, 218, 220, 228, 229, 231, 282, 312, 323 Population-based pharmacokinetic data analysis, 170, 173 Post hoc tests, 218–221 Precision, 44, 197, 231 Preclinical investigations, 164, 165, 167, 172 Pre-considerations of compartment models, 85 Preservatives, 175, 179, 243 Primary sampling, 175, 310 Primates, 247 Priming, 223 Prism, 229–230 Probability, 60, 172, 206–208, 217, 222, 312 Programmer software, 227, 228 Propranolol, 146, 147, 151, 243, 244, 277, 278 Prospective study, 166, 279 Protein-binding, 21, 46, 47, 51, 53, 57, 58, 151, 162, 199, 237–247, 258, 308, 313, 318, 319 Protein precipitation, 236 Publication policy, clinical trails, 283 PubMed, 153 Pump, 187, 188, 193, 194, 255, 264, 266 Purity %, 267 Putrefaction, 180

Q Quadrupole, 186, 187, 189–190, 273, 275, 277 Quality assurance, clinical trails, 283 Quality control, clinical trails, 283 Quantal data, 211 Quantification limit, 268, 270, 272

Index Quartile deviation, 204, 312 Quetiapine, 150, 152

R Raltegravir, 147, 151 Randomization design, 166 Random variable, 200, 202, 203 Range, 44, 188, 190, 191, 196, 197, 200, 204, 211, 218, 220, 221, 231, 242, 268–270, 272, 275, 312 Ranitidine, 147, 151, 269 Rantac, 269 Rapid equilibrium dialysis (RED), 243–244 Rate of drug absorption, 5, 21, 25, 28, 34, 43, 57, 139, 258, 260, 308, 313 Receptor compartment, 264, 265 Reciprocating pump, 188 Record-keeping, clinical trails, 283 Recovery %, 240 Reduction, 112, 147–152, 162, 163, 247, 249, 250, 252–255, 257, 258 Refractive index detector, 197 Regression analysis, 172, 199, 223, 224 Regression coefficient, 265 Relative bioavailability, 37, 38, 308, 316, 320, 323 Renal clearance, 38–39, 41, 162, 237, 308, 314, 319 Repeated dose regimen, 8, 9, 16, 18 Residual method, 31, 32, 308, 323 Retention time, 194, 196, 197 Retrospective data, 279 Retrospective study, 279 Reversed-phase HPLC, 192, 196, 197, 267–269 Revolution per minute and g relationship, 235 RF amplitude, 190 Rifampicin, 149, 151 Robustness, 197 Route of administration, 3, 167, 281 Roxithromycin, 146, 147, 151 RXC Chi-square test, 211, 222

S Saline sodium citrate (SSC) buffer, 235 Salt formation, 113 Sample injector, 188, 266 Sample points, 207–208 Sample space, 207, 208, 312 Sample, statistical, 199, 218, 219, 222 Sampling time points, 168 Sandberg plot, 48, 49, 308

Index Scatchard plot, 47–48, 242, 243, 308 Scattergram, 211, 224–225, 312 Scheffe’s test, 218–219, 312 Schizophrenia, 150, 152 Schwarz Criterion, 172 Scintigraphy, 261 Secondary sampling, 175, 310 Selected ion monitoring (SIM), 190 Selected ion recording (SIR), 189 Selection of subjects, clinical trails, 282, 283 Separating column, 187 Serotonin, 150, 152 Serum, 157, 162, 169, 177, 182, 185, 236, 261, 262, 310 Short-cut method, 201, 202 Sigma minus method, 135–138, 309 Silica particles, 192, 196, 268 SimBiology, 231 Simple arithmetic mean, 200, 201, 311 Simulux, 229 Single compartment, 22, 24, 27, 28, 32, 61, 84 Single-compartment model, 32, 61 Size-exclusion HPLC, 192 Skewness, 61, 209, 309, 312 Skin, 52, 55, 176, 263–265 Software or a programming tool, 227, 229–231 Spectrofluorimetry, 260 Spectrophotometer, 239, 242, 244, 246, 249, 250, 252–255, 257 SRM mode, 274 Stability, 197 Stabilizers, 175 Standard deviation (SD), 197, 199, 202, 204–206, 216, 217, 220, 225, 312 Standard error of the mean/measurement (SEM), 202, 312 Stat-Ease Inc., 232 Stationary phase, 187, 191–197, 266, 275 Statistical decision tree, 211–215, 312 Statistical dispersion, 203 Statistical equation modelling, 199 Statistical moment, 51, 60, 308, 318 Statistical moment models, 51 Statistics, viii, 199–225, 283, 311 Statrum corneum, 263, 265 Steady-state average plasma concentration, 12–14, 16 Steady-state plasma concentration, 5, 12–15, 17–19, 89–91, 170, 289–295, 299–301, 307, 313, 324 Step derivation method, 201, 202 Steripette, 250, 253, 255, 257 Stripping method, 32

333 Student’s t-test, 199, 211, 216–217 Sulfotransferase, 112 Sum of squares of errors (SSE), 217 Sum of squares regression (SSR), 217 Sum of squares total (SST), 217 Supplements, clinical trails, 173, 283 Sure or a certain event, 207 SwissADME, 229 Symmetric peak, 195 Synergistic additive, 145 Syringe pumps, 188 System software, 228 System suitability, 197

T Tadalafil, 277, 278 Tailing factor, 195, 197 Tandem mass spectroscopy, 185–187 Teamwork, 233 Tetracycline, 147, 151, 159 Thaw, 179, 181, 244, 268 Theophylline, 112, 146, 147, 151, 162 Theoretical plates, 197 Therapeutically effective drug level, 1, 2, 9 Therapeutic drug monitoring, sectors, 158 Therapeutic equivalence, 43–44 Therapeutic window, 2–4, 158, 307 Thiazides, 152 Three-compartment open model, 86, 100–109, 256, 302, 309 Thrombocytopenia, 149, 152 Tissue compartment, 29, 84, 85, 92, 97–100 Tissue fixation, 180, 181, 311 Tissue homogenate, 179, 235, 236 Tissue homogenizer, 179, 235 Tmax, 2–4, 8, 44, 45, 168, 169, 237, 269, 270, 272, 277, 278, 307, 313, 318, 320 Tolbutamide, 112, 147, 151, 162 Total operative body clearance, 237, 269, 270 Tourniquet, 176, 311 Toxic concentration, 3 Toxline, 153 Transdermal drug delivery systems, 263 Transgenic, 247 Trapezoidal rule, 170, 237 Trial design, 282 Trichloroacetic acid (TCA), 181, 236, 246, 261, 263 Triple-blind process, 166, 282 Tris-EDTA (TE) buffer, 235 Tubular secretion, 125 Tukey’s test, 218, 219, 312

334 Two-compartment open model, 33, 92–101, 113, 114, 253–255, 295, 297, 309, 320, 321 Two-tailed test, 216, 222

U Ultracentrifugation, 49, 179, 237, 241–242, 246–247 Ultrafiltration, 237, 240, 241, 245 Ultraviolet detector, 194, 197 Unchanged drug excreted through urine, 23, 25–27, 39, 41, 122, 136, 138, 142 United States Food and Drug Administration (USFDA), 42–44, 168 U points, 222 Uridine diphosphate (UDP)-glucuronyl transferase, 112 Urinary elimination rate, 38, 86 Urinary excretion data, 25, 35, 37, 60, 308, 323 Urinary excretion rate, 142, 237, 308 Urine specimen, 179 Use of antilogarithm table, 73–84 Use of log tables, 63 U values, 223

V Validation, 171, 195, 197 Vapor-fix, 180 Variance parameters, 173 Variation, 158–160, 163, 164, 204, 211 Variation coefficient, 204 Venepuncture, 176 Visual Basic 6, 230 Vitamin B12, 147, 151 Volume of distribution, 11, 12, 18, 22, 29, 57, 61, 62, 87, 88, 96, 117, 119, 122–124, 131, 135, 163, 231, 237, 287–289, 291, 298, 303, 312–315, 318, 320, 324

Index Volunteer, 44, 161, 165, 167, 168, 176, 178, 179, 236, 267–272, 277–280, 282

W Wagner-Nelson method, 22, 24, 32, 258, 259, 322 Warfarin, 112, 146–149, 151 Water, 111, 152, 157, 159, 162, 163, 176, 181, 187, 191, 192, 237, 240, 242–244, 246–260, 264–267, 269, 271, 273–275, 314, 317, 319 Weighted arithmetic mean, 200–202, 311 Weighted least-square (WLS), 171 Weights, 151, 158–160, 162, 163, 166, 171–173, 189, 192, 212, 234, 238, 242, 246, 266, 275, 281, 287, 293, 294, 298, 303, 321 Wilcoxon-Mann-Whitney test, 222–223 Wilcoxon rank-sum test, 222–223 Williams’ test, 220, 312 Withdrawal of subjects, clinical trails, 282 Working principle HPLC, 192 LC-MS/MS, 186–187 WRL-68 cells, 247

X Xenobiotics, 38, 247 Xylazine, 178

Y Young’s formula, 161

Z Zero-order kinetics, 3, 5, 88, 126, 304, 307, 315