Mathematics for Biomedical Applications [Reprint 2020 ed.] 9780520323094

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MATHEMATICS FOR BIOMEDICAL APPLICATIONS

MATHEMATICS FOR BIOMEDICAL APPLICATIONS

S T A N T O N A. G L A N T Z

UNIVERSITY OF CALIFORNIA PRESS BERKELEY • LOS ANGELES • LONDON

University of California Press Berkeley and Los Angeles, California University of California Press, Ltd. London, England Copyright © 1979 by The Regents of the University of California ISBN 0-520-03599-2 Library of Congress Catalog Card Number: 77-20320 Printed in the United States of America 123456789

To FRIEDA and LOUIS GLANTZ

CONTENTS

PREFACE

xi

1.

INTRODUCTION

1

2.

WRITING DIFFERENTIAL EQUATIONS Solutions of ordinary differential equations

5 8

One-compartment drug distribution 11 The function which is its own derivative: The exponential The natural logarithm function 17 Time constants and half-times 19 Metabolic turnover

23

One-compartment drug distribution with constant infusion Two-compartment drug distribution 27 Fluid-filled catheter dynamics 30 Indicator-dilution methods to measure physiological f l o w Formula for computing valve areas in the heart 41 The Michaelis-Menten mechanism for an enzyme-catalyzed reaction 45 Additional readings 47

3.

13

25

34

DIRECT METHODS TO SOLVE DIFFERENTIAL EQUATIONS 48 Separable first-order equations

48

One-compartment drug distribution with constant infusion: The formal solution 51 The operative principle of the pacemaker 52

viii

CONTENTS Exact first-order equations and integrating factors 55 Linear transformations and their properties 61 Linear differential equations are linear transformations 69 Other functions used to solve differential equations: The sine and cosine 72 Linear differential equations with constant coefficients: Finding eigenvalues and fundamental solutions 81 Linear differential equations with constant coefficients: Finding a particular solution 88 Linear differential equations: Finding the complete solution 95 Two-compartment drug distribution following a bolus injection or constant infusion and its application to lidocaine 98 Explaining age-related differences in ouabain pharmacokinetics 110 Response of a fluid-filled catheter to step and sinusoidal pressure changes 116 Approaches to nonlinear differential equations 1 27 Pseudo-steady-state solution for the Michaelis-Menten enzymecatalyzed reaction 133 Additional readings 136

4.

L A P L A C E T R A N S F O R M M E T H O D S FOR S O L V I N G L I N E A R DIFFERENTIAL EQUATIONS 137 Basic properties of the Laplace transform 138 Important Laplace transform pairs 142 Partial fractions 150 The unit step function 157 The unit impulse 161 The direct solution of systems of differential equations 165 Drug distribution following an oral or intramuscular dose 168

5.

W A Y S TO C H A R A C T E R I Z E L I N E A R P R O C E S S E S 175 Why take all initial conditions equal to zero? 177 Impulse response 178 Step response 185 Frequency response 190 Frequency response of fluid-filled catheters 200 Transfer functions 205 The convolution integral and the equivalence of the impulse response, step response, and transfer function 208 Additional readings 216

6.

F O U R I E R A N A L Y S I S , SPECTRA, A N D F I L T E R S 217 The Fourier series represents periodic functions 218 The complex variable form of the Fourier series 230

CONTENTS

ix

A comprehensive example 234 The Fourier integral represents nonperiodic functions 239 Fourier transforms of periodic functions 244 Derivation of the Laplace transform 252 Some common Fourier transforms and their symmetries 254 Basic properties of the Fourier transform 258 Frequency response is the Fourier transform of the impulse response 269 The ideal transducer 272 Filters 277 Spectral analysis of blood pressure waveforms 282 Additional readings 288 7.

DIGITAL COMPUTERS AND NUMERICAL METHODS Floating point numbers are not real 293 The Taylor series and truncation error 297 Integration 302 Differential equations 310 Curve fitting 322 Summary 333 Additional readings 334 APPENDICES 335 A. Review of calculus 335 B. Table of integrals 374 C. Basic properties of complex numbers and functions D. Table of Laplace transform pairs 387 E. Solutions to problems 391 INDEX

411

292

382

PREFACE

This text has its origins in an applied mathematics course I developed in 1973 while a postdoctoral fellow in the Stanford University Cardiology Division. Like that course, this text builds on the student's (perhaps distant) exposure to introductory calculus to develop applied differential equations and linear systems theory motivated by biomedical problems. I have sought to include enough mathematical theory to provide a sense of the structure and unity that mathematics can bring to other sciences, while always remembering

that a biomedical audience is most interested in

learning how to use mathematics to solve practical problems. Because this audience is traditionally skeptical about the value of mathematics, the first two chapters formulate more-or-less real problems in terms of differential equations, but defer solving these equations until Chapter 3. Chapter 3 summarizes direct methods to solve linear ordinary differential equations and works the problems formulated in Chapter 2. Chapter 2 also constructs the exponential function from a power series solution to a first-order differential equation to show the reader why this function appears so often. Chapter 3 does the same thing for sines and cosines. My students have found this approach a useful complement to their original exposure to these functions in introductory courses. The next three chapters on the Laplace transform, properties o f linear systems, and Fourier analysis introduce impulse and step functions with applications to biological signal processing. In many cases, I re-solve problems that were solved by direct methods in Chapter 3, so the reader can

xii

PREFACE

compare the relative difficulty of the different methods. Finally, Chapter 7 introduces numerical techniques for integration, solving differential equations, and curve fitting. The emphasis is on providing a context for reading more advanced texts or using library subroutines; no attempt is made to make the reader an expert computer programmer. Chapter 7 shows that one can use what would be perfectly acceptable logic in the algebra of real numbers to produce unreliable algorithms in floating point arithmetic. To help the rusty reader remember the relevant portions of calculus, Appendix A contains a brief review of these topics. Appendix C presents the properties of complex numbers which are necessary to understand the Fourier transform (Chapter 6) and a few other topics. Appendices B and D contain tables of integrals and Laplace transforms to save the reader the trouble of evaluating complicated integrals. In a formal course meeting three hours per week, one can reasonably expect to cover a brief review of calculus and the first three chapters in one quarter and the remainder of the book in a second quarter. I thank Donald Harrison, Chief of the Cardiology Division at Stanford University, for providing the resources to start this project, and William Parmley, Chief of the Cardiology Division at the University of California at San Francisco, for providing the resources to finish it. I completed the manuscript while a Senior Research Fellow of the San Francisco Bay Area Heart Research Committee and put the finishing touches on it while holding a Research Career Development Award from the National Institutes of Health. A. Lawrence Spitz wrote the computer programs to generate the power spectra of human cardiac pressure waves included in Chapter 6. I am especially grateful to Gail Hayes, Margaret Tidd, Kathleen Hecker, and Marilyn Gruen for typing the original manuscript, David Toy, Mary Helen Stull, and Norma Riffle for preparing the illustrations, Anne Holly for helping get the manuscript ready to send to the publisher, Douglas Bullis for editing it, and Michael Bass for doing the book design and layout. Harry Miller, who helped overcome many administrative difficulties at Stanford, and Grant Barnes, my sponsor at U.C. Press, deserve special thanks. Most of all, I am grateful to Robert Goldman, Peter Renz, and my students for providing insightful criticism which helped me tailor this work for its intended audience. S.A.G. San Francisco, California November, 1977

1 INTRODUCTION

A qualitative understanding o f physiological processes comes from the integration of knowledge of several factors which span many orders of magnitude in space, time, and energy. These factors include behavior exhibited by the entire body, performance of organ systems, histologic structure of the organs, and biochemical reactions inside cells. When one integrates these different perspectives on a problem in an attempt to understand a biological process or to reach a practical decision about how to manage a disease, one often slips from one set of assumptions to another, even though taken as a whole these assumptions might not be logically consistent. In contrast, a quantitative understanding o f biological processes, based on mathematics, requires logical consistency above all. Although theoretically it may be possible to consider several factors which span many orders of magnitude, in practice it is often difficult to obtain numerical solutions when the variables which quantify these factors differ by more than a few orders of magnitude. Thus, successful mathematical analysis generally requires a consistent scale of events. This requirement sometimes restricts use of prior knowledge, but it can help avoid unimportant details. One's prior knowledge of the structure of a process under consideration determines the questions one asks and the mathematical tools one applies to answer them. With little prior knowledge, one is limited to the description of general patterns of available observations. Quantitative methods are used to answer relatively

simple questions, such as the

estimation o f a parameter's value or the probability that two samples were

2

INTRODUCTION

drawn from the same population. We use statistical tools to answer these questions. In contrast, if one knows (or can hypothesize) the underlying structure and mechanisms which produce the observed patterns, one can use the deterministic mathematics developed in this text to predict behavior and to draw deductive conclusions from mathematical manipulation. For example, cardiologists often prescribe drugs from the digitalis family, which improve the heart's pumping action in patients with heart failure or slow the heart rate in patients with abnormal rhythms. T o study why children require larger doses of these drugs to achieve the same serum concentrations as adults, my colleagues and I administered the same dose (per kilogram of body weight) of ouabain to adult dogs and puppies and

TIME,

t (mini

FIGURE 1.1: After an ouabain dose of 0.05 mg/kg, serum ouabain concentrations in adult dogs remained significantly higher than in puppies given the same dose, although the curves for both groups had the same shape. (Adapted with permission of the American Heart Association, Inc., from S. A. Glantz, R. Kernoff, and R. H. Goldman, Age-Related Changes in Ouabain Pharmacology, Circ. Res. 39 (1976): 407.)

INTRODUCTION

3

measured the resulting concentrations* (Fig. 1.1). With no prior knowledge—or hypotheses-about ouabain distribution, we applied statistical tests and found that, for the same dose, puppies had significantly lower serum ouabain concentrations than adult dogs. This observation was interesting but provided no direct insight into the underlying mechanism which produced it. The mathematical tools this text develops will permit us, among other things, to propose and verify a theoretical description for ouabain distribution. Suppose that some of the ouabain present in the body remains dissolved in the plasma, some is reversibly bound in the body, and some is irreversibly removed (Fig. 1.2). The constants kl2, fc2i> a n < i describe how quickly ouabain is bound (kl2) and released (k2,) from reversible binding sites or removed from the body (fc 1 0 ). The equations which describe this system show that these constants determine the shape of the curve for serum concentration versus time, whereas the dose and the volume in which the drug is distributed determine the magnitude of the concentration but not the way in which it changes with time. (This fact follows from the important mathematical property of linearity, which we will discuss at length.) Since both adult dogs and puppies received the same dose and since their curves had the same shape, we concluded that

BOLUS DOSE 1r k

12

t

VOLUME OF DISTRIBUTION (PLASMA VOLUME, etc)

REVERSIBLY BOUND OUABAIN

k21

k

10

METABOLISM AND EXCRETION F I G U R E 1.2: One way to picture the distribution of ouabain in a dog. This picture, called a two-compartment model, will lead to equations that describe how ouabain concentration changes with time. We will see that the constants that describe how the drug is reversibly bound ( / r 1 2 and / r 2 i ) and eliminated from the body

determine the shape, but not the

magnitude, of the concentration versus time curve. * Chapter 3 treats this problem in detail.

4

INTRODUCTION

the only difference between puppies and adult dogs was that the effective volume in which the drug was distributed was larger in puppies. This mathematical result suggested additional experiments, which confirmed that the puppies had larger physiological fluid spaces per kilogram of body weight than the adult dogs, a finding which accounts for the greater volume of distribution in the puppies. Thus, the techniques this text develops permitted us to draw much stronger conclusions from our experiments than would have been possible if we had been limited to a statistical approach. Sometimes, however, one should spurn a mathematical approach. The value of any analysis depends heavily on the initial assumptions, and often one cannot translate them into convenient mathematical expressions. Some workers, with glib disregard for the violence done to known underlying physiology, recast their assumptions to make the problem workable. Perhaps the most common such offense is the assumption, often implicit, of linearity. Linear equations are relatively easy to solve; indeed, often one cannot solve nonlinear equations analytically. But since biological systems often exhibit nonlinear behavior, assumptions of linearity require careful scrutiny and, if necessary, rejection. Rejection may preclude solving the problem and restrict one to qualitative arguments. The question then arises whether to trust precise analysis based on imprecise assumptions or imprecise analysis based on precise assumptions. The answer depends both on one's knowledge of the physiology involved and on one's ability to apply appropriate mathematical tools. When mathematical analysis predicts a nonintuitive conclusion, one must be secure enough in knowledge of both physiology and mathematics to answer two questions: Are the initial assumptions reasonable? Is the analysis correctly executed? Affirmative answers to these questions require one to accept new conclusions and reorient his or her intuition. As the reader comes to understand the common underlying structure in many problems, illuminated by analogous equations, his or her developing mathematical intuition will improve how one designs experiments and views physiology.

WRITING DIFFERENTIAL EQUATIONS

Most problems in physiology and medicine concern dynamic processes which involve the rates of change of several variables, as well as their values. With derivatives to describe a variable's rate of change, we will describe dynamic processes with equations containing derivatives, which are called differential equations. After understanding differentiation, translating physiological statements into mathematical statements provides the key to writing useful differential equations. The formulation and solution of a dynamic problem includes three distinct phases. First, one must use knowledge and intuition to write one or more differential equations and an equal number of uniqueness conditions which describe the physiological situation. Second, one must manipulate these mathematical expressions to solve the differential equations. Finally, one must interpret the mathematical solution in physiological terms. Typical examples best illustrate this process, so the bulk of this chapter consists of examples which highlight different approaches to the translation of physiological statements into differential equations. Most cases present and verify the solutions, but we defer discussion of the formal procedures for solving differential equations to Chapters 3, 4, and 7. Before we study these examples, however, we must address the theoretical question: What is a solution t o a differential equation and when does it exist? We begin with four definitions. Differential equation: an equation containing one or more ordinary or partial derivatives.

6

WRITING EQUATIONS

Ordinary differential equation: an equation containing only ordinary derivatives, all with respect to the same variable. Partial differential equation: an equation which contains one or more partial derivatives. (It may also contain ordinary derivatives.) The order of a differential equation is the order of the highest derivative that appears in the equation. For example,

F =m

d2x

m = constant

dt2

(2.1)

and q+q+q

+ 1=0

(2.2)

are second-order ordinary differential equations. In contrast, - kx

(2.3)

is a first-order partial differential equation. Partial differential equations are much harder to solve than ordinary differential equations. Luckily, ordinary differential equations suffice for most applications we will encounter, so we will concentrate on writing and solving them. Many applications, such as multiple-compartment descriptions for drug distribution, give rise to a set of first-order ordinary differential equations. Such a set of n first-order differential equations is equivalent to a single nth-order differential equation. Suppose a problem led to the two first-order ordinary differential equations (2.4) and ¿2=*!

(2.5)

To convert (2.4) and (2.5) to a single equivalent second-order differential equation, differentiate (2.4) with respect to time, =i2

(2-6)

WRITING EQUATIONS

7

and eliminate x 2 from ( 2 . 6 ) with ( 2 . 5 ) : (2-7) Thus, we have converted the set o f two first-order differential equations in the variables xl

and x2

into a single equivalent second-order differential

equation in x t . One could use ( 2 . 4 ) and ( 2 . 5 ) simultaneously to find x l and x2

or

find

result to get x2.

from the single equation ( 2 . 7 ) , then differentiate the This process o f differentiation and substitution permits

the conversion o f any set o f n first-order ordinary differential equations into a single nth-order differential equation. Therefore, a set of n order

differential

equations

is nth order.

first-

Conversely, an nth-order ordinary

differential equation may be transformed into a set o f n

first-order

differential equations by reversing this process.

P R O B L E M S E T 2.1 What order are these differential equations?

dx 2. y +y

x

~ ZQ z 3. = +z ox 4. x1=x}+x2

x2=x2+x3

5.xl=x1+x2

x2 =x2

x3=x] +x3

x3 = x\

Write each system o f differential equations as a single differential equation in terms o f the first variable. 6. xl

=xl

*,