univ of minnesota pr mathematical models in the health sciences a computer aided approach nov 1979

However, forone interested in quantitative approaches to biology and medicine most ofthe text should prove useful.Mathematical Models in the Health Sciences may also be of value to gradu

IN THE HEALTH SCIENCES

A Computer-Aided Approach

Lael Cranmer Gatewood, Ph.D.

Associate Professor and Director

Health Computer Sciences

University of Minnesota

UNIVERSITY OF M I N N E S O T A PRESS D M I N N E A P O L I S

Published by the University Minnesota Press,

2037 University Avenue Southeast, Minneapolis, Minnesota 55455

Printed in the United States of America

at North Central Publishing Company, St Paul

Library of Congress Cataloging in Publication Data

2 Medicine—Data processing I Gatewood,

Lael Cranmer, joint author II Title.

R858.A36 610'.28'54 79-9481

ISBN 0-8166-0864-4

The University of Minnesota is an equal-opportunity educator and employer.

Mathematical techniques have long been employed in the biological,medical, and related health disciplines Within the past few decades, thefrequency of such applications has increased significantly, as can be seen

by scanning current literature in a field such as physiology This changehas been made possible by the availability of electronic aids to computa-tion and by the development of appropriate numeric and graphicmethodologies

The most ubiquitous mathematical techniques as applied to biomedicalareas have been grouped together under the title biostatistics Probablyall quantitative studies incorporate statistical methodology, at least to alimited degree Numerous textbooks have been written about biostatis-tics, its subdisciplines, and its applications to the health sciences On theother hand, there exist a variety of mathematical techniques that areemployed in the health sciences but that are not primarily statistical innature These are called mathematical modeling and form the basis for thevarious topics discussed in this book

Computer technology has made possible many of the applications ofmathematics to biology and medicine Accordingly, computer programs,graphics and tabular output, and block diagrams are included in the illus-trative material throughout the text It is assumed that the reader has hadprevious exposure to scientific computing, but specific knowledge of aprogramming language is not required Thus the text is concernedexplicitly with selected topics from the biological and health sciences forwhich computers have been a natural tool for analysis

v

One of the first reactions that a knowledgeable reader may have whenlooking at the table of contents is a sense of the incompleteness of thetopics covered The pedagogic technique followed here is sometimes re-ferred to as a block-and-gap method The entire field of mathematicalmodeling is divided into a group of blocks with intervening gaps Theblocks are discussed as fully as space permits; the topics in the gaps aresimply omitted It is the intention to emphasize in this fashion the generalphilosophic approach as well as to present specific methodologies andapplications whose importance will not fade too rapidly Such a selection

is clearly a compromise, but one that proves useful to a variety of types ofstudents

A text concerned with biomedical applications of mathematics mustperforce refer to a variety of areas of biology and medicine It seemsunreasonable to assume that all readers will be equally familiar with all ofthe areas included If the book is to be more than a collection of recipes,some knowledge of the significance and implications of the areas of appli-cation is necessary References are given to allow the interested reader topursue each study more thoroughly However, it is hoped that the sup-plemental material presented with each example is adequate in itself formany readers

The book has been written with the hope that it will be used as a text forcourses at the graduate level The emphasis has been placed on themathematical techniques rather than on detailed derivations The latterare the logical justification for the techniques discussed On the otherhand, a text on biomedical applications must assume that the interestedreader will have mathematical books available that develop the underly-ing proofs to the degree of rigor that is desired Such knowledge willaugment the understanding of readers with more mathematical interests,but others whose training and research emphasize practical applicationsshould find the methodologies as presented here to be sufficient in them-selves

The primary audience for whom this text has been written are students

in the program of Biometry and Health Information Systems at the versity of Minnesota who are working toward an M.S or Ph.D degree.They have had graduate courses in biostatistics, biomedical computing,and at least one area of biology, as well as an interest in quantitative,analytical approaches to biomedical studies For such students this courseprovides an introduction to a different set of mathematical and computermethodologies applied to the health sciences

Uni-The book should also prove useful for those working in other related and biomedical sciences Essentially, what is required as prereq-

health-uisites are mathematics through calculus and advanced training in somehealth science or biomedical field A knowledge of biostatistics and com-puter programming may be useful in following some of the detailedexamples Readers may find some parts of the text overly simplified andredundant, other parts too far from their area of interest However, forone interested in quantitative approaches to biology and medicine most ofthe text should prove useful.

Mathematical Models in the Health Sciences may also be of value to

graduate and postdoctoral students in mathematics, computer science,the physical sciences, and engineering They may have been exposed tothorough developments of mathematical and computer techniques butmay find their biological background requires more supplementation than

is provided in this text Nonetheless, if they wish to expand their edge of biomedical applications of mathematics, this book and its refer-ences should help to meet their needs

knowl-All of the types of students described in the preceding paragraphs havebeen included in courses entitled "Mathematical Biology" and taught aspart of the graduate program at the University of Minnesota Each timethe course has been offered student preparation and interests have beendifferent Attempts were made to vary the content and even the emphasis

of the course to meet the perceived needs of the class as well as to includesome of the current interests of the instructors

In addition to the formal lectures, the course at the University of nesota included individual reports and homework assignments Thesereports, presented both orally and in writing, have encouraged greaterlibrary utilization The other out-of-class assignments have includedcomputer-based problems that increased familiarity with the locally avail-able computer resources Topics for reports and problems were obtainedfrom current references similar to those presented at the end of eachchapter

Min-These classes in Mathematical Biology have resulted in extensive dent participation and interaction Although this varied from one person

stu-to the next, all contributed in some fashion stu-to the selection of applicationsand examples The authors gratefully acknowledge their help and advice.Numerous of the authors' colleagues have also provided assistance in onefashion or another Particularly deserving of acknowledgment is Dr.Lynda Ellis, who originally suggested including the material in Chapter

13, leading to a major revision in the selected chapters

Several groups have supported in part the preparation of this text.These include the Northwest Area Foundation as well as the Biotechnol-ogy Research Resource Facility, the College of Pharmacy and the De-

partment of Laboratory Medicine and Pathology of the University ofMinnesota In order to complete this text, the senior author spent a year

on sabbatical leave at the University of Washington's Department of oratory Medicine The help of the latter faculty is also gratefully acknowl-edged The text would not have been possible without the typing andeditorial support provided by Mrs Margie Henry, Ms Kathy Seidl, and

Lab-Dr Margaret Ewing

E.A

L.C.G

Preface v

I N T R O D U C T I O N 1Chapter 1 Models and Goals 3

A Origins and Definitions 3

B Automated Computational Aids 5

C Deterministic and Stochastic Models 6

F Computer Simulation 38

G Non-Linear Parameter Estimation 41

H Model Selection and Validation 45

I Summary 49Chapter 3 Modified Compartmental Analysis 53

A Extensions of Compartmental Analysis 53

B Blood Glucose Regulation 54

C Ceruloplasmin Synthesis 64

D Dye Dilution Curves 68

E Lung Models 69

F Summary 72Chapter 4 Enzyme Kinetics 76

A Enzymes and Biology 76

B Proteins and Amino Acids 77

C Prosthetic Groups, Cofactors, and Coenzymes 80

D Molecular Conformation and ChemicalReactions 82

E Michaelis-Menten Kinetics 85

F Estimation of Michaelis-Menten Parameters 88

G Catalase and Peroxidase Reactions 92

H Enzyme Kinetics and Mathematical Biology 96Chapter 5 Enzyme Systems 99

A Introduction 125

B Analog to Digital Signal Conversion 126

C Fourier Transforms 128

D Discrete Fourier Transforms 138

E Fast Fourier Transforms 142

F Laplace Transforms 148

G Sampling Theorems 150

H Summary 155Chapter 7 Transforms and Transfer Functions 157

A Transfer Functions 157

B Convolution Integrals 159

C Compartmental Analysis 164

D Dye Dilution Curves 169

E Fast Walsh Transforms 172

F Applications 175Chapter 8 Electrocardiographic Interpretation 178

A Physiological Basis 178

B EKG Characteristics 182

C VKG Patterns 185

D Abnormalities 189

E Simulation and the Inverse Problem 191

F Automated Interpretation of the EKG 197

G Automated Aids to Clinical Diagnosis 200

H Summary 202Chapter 9 Electroencephalographic Analyses 206

A Central Nervous System 206

H Evoked Response Averages 227

I Automation and the EEC 229

I N F O R M A T I O N A N D

S I M U L A T I O N 2 3 3

Chapter 10 Information Theory 235

G Health Sciences Applications 248

Chapter 11 Genetic Transfer of Information 250

A Genes and Chromosomes 250

B Cell Replication and Division 252

C Molecular Basis of Genetics 253

D Information Content of DNA 255

E Types of Genes 259

F RNA and Protein Synthesis 262

G Information Theory and Evolution 265

H Genetic Models and Evolution 267

Chapter 12 Simulation of Epidemics 271

A Epidemics and Epidemic Theory 271

B Simulation of Stochastic Models 274

C Simplest Stochastic Models 276

D Competition and Vaccination 282

E Structured Populations 289

F Influenza Epidemic Model 293

G Overview 300

Chapter 13 Population, Ecology, and the World System 304

A Introduction: Population Models 304

B Exponential Growth 306

C Logistic Growth 309

D Competition and Predator-Prey Interactions 312

E Other Ecology Models 317

F World Systems Models 320

G Simulation and Prediction 325

H Summary 330

O V E R V I E W 3 3 3

Chapter 14 Mathematical Models in the Health Sciences 335

A Summary of Text 335

B Other Areas of Mathematical Biology 337

C Other Health Science Applications 339

D Health Computer Sciences 341

E Future Implications 342

Index 347

Chapter 1 on models and goals provides an overview of the philosophic approach taken in the text It is hoped that this chapter will be read first and then reread several times while the text is being used The scientific setting of the text, references to the biomedical litera- ture, and an explanation of the notational scheme used throughout the text are pre- sented here.

CHAPTER 1

Models and Goals

A Origins and Definitions

For centuries scientists have used mathematical functions to describethe observable world, but the early records of applications of mathematics

to biological phenomena are difficult to find The types of applicationsselected for presentation in this text have been developed since thenineteenth century by a diverse group of scientists working in manyfields As recently as 1850 it was possible for one person to acquire theskills of a physician, surgeon, physicist, and mathematician as exemplified

by von Helmholtz Until the introduction of digital computers, thestudies of these scientists, individually and in groups, were usually in theareas now called biophysics Examples include von Helmholtz's andRayleigh's studies of hearing and Einthoven's analyses of electrocardio-grams Rashevsky's group at the University of Chicago chose the termmathematical biophysics for their studies of diffusion, permeability,growth, metabolism, and neurobiology From perhaps 1900 activities ofthis nature grew at an exponential rate but with a long time constant.Many biologists and most clinicians regarded this growth as an oddity,having little to do with biology or medicine However, a discipline de-scribed as mathematical biology began to emerge as a separate field ofstudy and research although frequently as part of programs still calledbiostatistics or biophysics

The introduction of the digital computer and the consequent ical developments such as operating systems, high-level programming

technolog-3

languages, and special simulation languages, caused a rapid change in theuse of mathematical models for all health sciences In the 1970s, thequestion of whether a separate or integrated discipline devoted tomathematical modeling exists is competitively discussed and debated.This text discusses selected applications of mathematics to biology, tomedicine, and to other health-related disciplines in which the analysesare neither overly simplistic nor primarily biostatistical These qualifiersimply considerable personal judgment by the authors as influenced bytheir colleagues and students.

The use of quantitative analytic techniques including mathematicalmodels in biology and medicine is often termed mathematical biology.However, many different concepts or relationships are suggested by thisterm Mathematical biology and biostatistics are often combined andcalled biomathematics, and if biomedical computing is incorporated, thecombination is sometimes called biometry Some reserve the last word forbiostatistics per se Mathematical modeling as presented in this text can

be considered an essential part of a program in health computer sciences.The modeling techniques included in mathematical biology are inti-mately involved in many other interdisciplinary areas, such as physiology,biophysics, biochemistry, medical physics, and biomedical engineering.Many of the topics discussed in the following chapters are included incourses in these disciplines In addition models have been used in manyother health-related areas, including epidemiology, basic health sciences,and health services Many hospitals and clinics use techniques derivedfrom modeling studies in laboratory instruments, radiological treatmentplanning, resource allocation and scheduling, and other facets of healthcare delivery

Quantitation in the health sciences is dependent on the use ofmathematical models This approach is natural to the physicist, thechemist, and the engineer; they often do not note the extent to whichthey use models or abstractions of reality The biological and health sci-ences have been so dominated by descriptive methodologies that the use

of mathematics requires the explicit definition of a model Biomedicalscientists, often unfamiliar with this approach, sometimes tend to expectfar too much or to accept far too little of what a study based on amathematical model can offer Consequently it is important in mathemat-ical modeling to define the uses, goals, and validation of models Theremainder of this chapter is a general discussion of various types of mod-els, as these bear on the goals of mathematical modeling in the healthsciences

B Automated Computational Aids

Before the introduction of computer technology, it was necessary inworking with mathematical models of biomedical systems either to over-simplify and approximate to an unacceptable degree or to perform labori-ous numerical calculations by hand or with a desk calculator; the laborcost was often prohibitively high Thus computer representation has be-come a necessary part of many mathematical models The following dis-cussion explains this relationship in more detail by considering howmathematical models are used

First, a quantitative representation is hypothesized for the relationshipamong variables within the model The internal variables may involve, forexample, concentrations and their time derivatives or factory output andpollution indices Customarily the model is then solved to describe rela-tionships that can be observed experimentally, such as the plasma con-centration of one or more tracers as a function of time or age-specificattack rates during an epidemic These examples of such use are discussed

in other chapters The solution may involve integrating differential tions, but, depending on the model, need not be of that form

equa-Given specific details for the mathematical model, the solutions that areobtained can generally be represented as tables of numbers People find itdifficult to recognize the information contained in such lists of numbers,whereas they can quickly grasp the form and message of a well-con-structed graph If many solutions for different forms of the model anddifferent initial values of conditions are desired, numerous graphs may beneeded The computer allows the preparation of graphic displays of data

in a form that is easier to modify and is far less expensive than a drawn presentation

hand-However, the frequent use of numeric calculations creates a basic needfor automated computational techniques The models with which it issimplest to deal, namely, those that permit a closed solution, neverthelessrequire calculations to express the solution in a form that can be comparedwith experimental results If solutions for several different sets of initialvalues or for several sets of pseudorandom numbers are desired, themanual calculation task may become prohibitively expensive In someapplications the model system can be solved only by numeric techniques

In others it may prove more convenient to solve the model by numericanalysis than to derive and use a closed-form solution

Both analog computers that deal with continuous signals and digitalcomputers that deal with discrete numbers have been used to aid innumeric computation In the early 1950s many scientists preferred the

analog computer because of its speed and accuracy, which was similar tothat of experimental methodology Subsequent experience and develop-ment of the digital computer have proved that the latter is easier to use formost purposes Special digital computer languages that mimic analogcomputers have made the advantages of both types of computers available

in one Analog computer techniques are still used to preprocess ous signals from biological systems Except for that role, the digital com-puter is today the necessary and essential apparatus for a health scientist

continu-C Deterministic and Stochastic Models

The models used in the health sciences can be classified in severalfashions One system differentiates between deterministic and stochasticmodels A deterministic model is one that has, given the initial condi-tions, an exact, determined solution that relates the dependent variables

of the model to each other and to the independent variable (or variables)

In contrast, a stochastic model and its solution involve probablistic siderations

con-Classical physics and chemistry dealt almost exclusively with ministic models This type of model is also popular in biomedical studies.Most uses of tracers are based on an explicit or implicit deterministicmodel Enzyme kinetic models, hydrodynamic models of the cardiovascu-lar system, and other physiological models using physical and engineeringanalogies are, by and large, deterministic Models of medical diagnosisthat have a dendritic pattern with definitive decisions at each node are alsodeterministic

deter-On the other hand modern quantum physics and chemistry haveturned to models that are stochastic and provide only the probability of anevent occurring rather than a statement that it will or will not occur.Biostatistical models are by definition stochastic, and information theory,another tool of the health scientist, deals with stochastic processes To-day's approaches to epidemic simulation and to analysis of electrocardio-grams also contain major stochastic elements Thus both deterministic andstochastic models are used in applying mathematics in the health sci-ences

Although the dichotomy between deterministic and stochastic models

is intellectually pleasing, in actual practice it is simplistic All tic models that are intended to represent real, measurable quantitiesmust be used recognizing the limits of precision of the measurements.These limits introduce an uncertainty and hence a probabilistic element,

determinis-into both the initial conditions used in the model and the values of theobservables predicted by the model.

Stochastic models may be reduced in a trivial fashion to deterministicones under some circumstances For example, if the number of molecules

or persons involved is so large that the random stochastic events cannot

be observed, the model leads to deterministic predictions even thoughthe underlying process is stochastic In addition many stochastic models,perhaps all, contain some deterministic elements

Because the distinction between these models, as defined, is not alwaysclear, a revised definition is perhaps needed Models are deterministic iftheir principal features lead to definitive predictions, albeit modulated byrecognized uncertainties On the other hand, models are stochastic iftheir more important parts depend on probabilistic or chance consid-erations, even though the model also contains deterministic elements

D Inverse Solutions

There is frequently a major difference between model applications inthe physical and the engineering sciences on one hand and the biomedicaldisciplines on the other The physicist and engineer often can design andbuild systems to predetermined specifications Accordingly they often use

a model to predict how a given system will behave This type of solution ofthe mathematical model, whether performed analytically or numerically,

is referred to as a direct or forward solution The design of health caredelivery systems also may involve such forward solutions of mathematicalmodels

By contrast the biomedical scientist usually cannot design the system to

be studied but can observe the behavior of the system In this case a goal

of model study is often to find characteristics by which the system can bedescribed For this purpose the model's forward solution is comparedwith observed behavior and some form of an objective function is com-puted The objective function provides a suitably weighted measure ofthe agreement (or lack thereof) between the forward solution and theactual system's behavior It is then possible to seek parameters that willoptimize this agreement These parameters are referred to as the inversesolution, which can then be used to characterize the individual system.Engineering technology often faces a similar problem Suppose a trialsystem has been designed, a suitable mathematical model described, and

a forward solution found If this system is to perform a preassigned task,one may ask how well the model predicts that these objectives will be

met To answer this question quantitatively an objective function isneeded The technologist then must seek alternate forms for the model orperhaps alternate parameters within the model, which will be used tobring the performance of the system closer to its objectives Such a designprocess is called system optimization.

The objectives of a biomedical scientist in seeking an inverse solutionmay differ from those of an engineer attempting to optimize a system.Nonetheless the mathematical and computer-based techniques are quitesimilar Therefore, some biomedical scientists adopt engineering ter-minology and speak of system optimization as though it were equivalent

to finding an inverse solution

E Model Conformation and Parameter Estimation

In one area of the physical sciences, namely, X-ray crystallography,inverse solutions of the type used in the health sciences are essential.Given a set of X-ray diffraction spots (an X-ray diffraction pattern), theproblem is to select locations and bond angles for the atoms or atomicgroups within the crystal The solution of this problem is particularlyimportant in studying crystals of large molecules such as occur in biologi-cal systems The process is closely analogous to the system optimization ofthe engineer although different computer and mathematical techniquesare used Crystallographers call their process refinement; in effect it con-sists of iteratively selecting the atom locations, bond angles, and arrange-ments to find forward solutions that conform increasingly well to therequirements of the X-ray diffraction pattern In mathematical modelingthe iterative process of refining an inverse solution is sometimes calledmodel conformation

Inverse solutions are often developed by biostatisticians who call thisprocess parameter estimation Unbiased estimates are sought that willprovide closer correspondence to reality as more data are examined Byand large the biostatistician seeks estimates that in some sense optimize

an objective function Some measure of uncertainty of these estimates isdesirable This procedure works best when the parameters to be esti-mated appear in a linear fashion in the solution of the model Linearparameter estimation is discussed in statistical texts on linear models andlinear regression analysis

It is well to note that most models discussed in this text are nonlinear

by the biostatistician's definition In other words, the parameters to beestimated do not appear in a linear fashion in the analytical solution to the

model The word nonlinear is the source of much confusion because it isoften used in two different fashions by scientists and technologists Essen-tially, technologists use linearity to refer to the differential (or other)relationships between the variables in the model rather than to the oc-currence of the parameters to be estimated in the analytical solution.

In the succeeding chapters most of the examples presented are related

to specific biomedical applications However, to emphasize the two senses

in which linear is used, four abstract examples are presented in an companying table Mathematical models are presented in the table both

ac-as differential equations and ac-as their analytic solutions Arbitrary decisionsconcerning integration constants have been introduced The variables

are labeled y and t, and the parameters to be estimated as a, b, and c The

notation is explained in Section G of this chapter

Linear for Linear for Differential Analytical

Biostatistician? Engineer? Equation Solution

Yes Yes d 2 y/dt 2 = a y = a-t L > /2 + b - t + c (1-1)

No Yes dy/dt = - a-y y - b-exp (- a-t) (1-3)

No No dy/dt = a - y - b - y 2 y = c/[b-c/a + exp (- a-t)] (1-1)The first example (Equation 1—1) has been chosen to emphasize that eventhough the differential equation may be linear and the parameters to beestimated may appear only in linear fashions, the resultant analyticalsolution need not be the equation of a straight line The second examplehas been included for completeness only However, models similar intheir linearity to Equations 1—3 and 1-4 form the bases for several modelsdiscussed in this text The specific example in Equation 1-3 is used inChapter 2 and the one illustrated in Equation 1-4 appeals in Chapter 13.Although all real biological systems can be shown to be nonlinear in theengineering sense, nonetheless many can be adequately approximated bymodels that are based on linear differential relationships between thevariables but involve parameters in a nonlinear fashion in their solution.One property of linear differential equations should be noted, namely,

if there are two or more solutions known, the sum of these solutions orany linear combination thereof is also a solution This is sometimes re-ferred to as the superposition theorem It implies that in a model withseveral inputs (or initial conditions), one may solve repeatedly allowingonly one input (or initial condition) at a time to be nonzero and then addthese partial solutions to find the general solution By the same reasoningmultiplying all the inputs and initial conditions by a fixed constant results

in multiplying the general solution by the same constant In some cases it

is convenient to use experimental tests of the superposition theorem to

judge whether a mathematical model is linear in the engineering sense.

No matter what the decision, however, finding inverse solutions to themodel usually involves nonlinear parameter estimates

Sometimes nonlinear estimation can be avoided by transforming theanalytical solution into a form in which new parameters can be definedthat are linear in the biostatistical sense Thus taking the logarithm of bothsides of the solution to Equation 1-3 leads to

When a transformation of this nature is possible, statisticians call solutions

of the form of Equation 1—3 pseudononlinear It should be noted that in

most cases estimates of a and b based on Equation 1-5 differ from ones

based directly on Equation 1-3

The problems of nonlinear parameter estimation are far more cated than of linear parameter estimation The latter can be done exactly,whereas nonlinear parameter estimation always requires an iterative,trial and retrial approach Various schemes have been developed for au-tomated computation of nonlinear parameter estimates Many computercenters have several packaged programs for this purpose because no oneprogram is ideal for all models

compli-Nonlinear parameter estimation is also difficult in a number of otherways Usually there are not suitable data to determine whether the esti-mate is biased Worse, there is usually a large coupling between differentparameters, which some biostatisticians describe as very large covarianceterms Accordingly estimates of uncertainty in the nonlinearly estimatedparameters become questionable in meaning A better approach seems to

be to seek combinations of parameters that are relatively insensitive toexperimental error (See Chapter 2 for further discussion.)

Use of many nonlinear parameter estimation routines requires edge of the numerical values of partial derivatives By and large methodsthat do not require derivatives are easier to use because analytical speci-fication of the partial derivatives of the objective function is not needed.Some so-called derivative-free methods actually approximate the deriva-tives numerically within the routines, whereas others use directly thevalues of the function itself at various trial points

knowl-Any iterative method of parameter estimation may end at a localminimum of the objective function There is no way to guard against thiseventuality Moreover, in a search for the best set of parameters thewhere

global minimum need not necessarily be the best The existence of localminima introduces a level of uncertainty into nonlinear parameter estima-tion which is overlooked by most statistical estimates of confidence inter-vals for the estimated parameters Therefore, it might be best to avoidsuch representations except if supported by repeated experiments with avariety of subjects Even then it is not possible to reject the hypothesisthat the iterative method chosen has introduced bias into the parameterestimates (See Chapter 2 for further discussion.)

F Health-related Goals of Mathematical Modeling

It is necessary to specify clearly the goals to be met by a given model.This results in part from the difference between the experience ofbiomedical scientists and of engineers and physical scientists in represent-ing and analyzing various phenomena In addition, the original goals oftenare lost sight of in the search for inverse solutions, thus the goals should

be defined explicitly

In subsequent chapters goals will be described in concrete terms, asthey apply to the model under discussion For the moment, considerthese goals in a more global fashion Different methods of classifying thegoals of using mathematical models exist The following grouping hasproved convenient and is used throughout this text

Goals of Mathematical Model UseData Description

Diagnostic ClassificationHypothesis TestingExperimental DesignIsomorphic MappingData Description Even if one uses a model only to reduce a mass ofdata to a small number of constants, which are more easily discussed, thenthe model serves a real purpose; the model need not have any direct orimplied relationship to the underlying biological processes This use issometimes referred to as curve fitting Although it does not explain un-derlying mechanisms, it can reduce the data to a useful form An example

is the frequency analysis of electroencephalographic signals

Diagnostic Classification The estimated parameters of the model may

be used for diagnostic classification This goal, like data description, isindependent of the other goals listed in the table If the estimatedparameters can be used to separate normal from abnormal or to help

characterize quantitatively different disease states or responses to tion, then the model need not even produce an acceptable description ofthe empirical data Certain electrocardiographic models meet this goalalone.

medica-Hypothesis Testing If several hypotheses appear conceptually able, then it may be possible by model studies to eliminate some of thealternatives It may also be possible using a single model to test questionssuch as if proposition A occurs, what results are to be expected? Forexample, some model studies of virus epidemics have investigated possi-ble public health interventions

accept-Experimental Design Certain scientists regard the design of new periments as the most important or the only important goal in usingmathematical models This seems to the authors to be an unnecessarilyextreme position Nonetheless, a major use of model studies is to designnew, critical experiments These are most gratifying if they confirm theinvestigator's preconceived notions but yield more information if theydemonstrate that the scheme embodied in the mathematical model isuntenable Many tracer studies following the course of a labeled metabo-lite over time are used to delineate hypotheses for further examination.Isomorphic Mapping This is regarded by some as the ultimate goal ofmathematical modeling Although intuitively appealing, this goal cannever be fully realized, for no matter how much detail is included thereremain other details that are excluded This goal is not consonant with theother four listed, which tend to simplify just as most models in the physi-cal sciences do In contrast, this last goal requires greater and greatercomplexity as models become more isomorphic with the real world Ingeneral a compromise seems necessary, and the demand for true isomor-phic mapping may be self-defeating

ex-In addition to choosing one or more goals, it is necessary to establishcriteria for model selection and acceptance The first and most importantcriterion is that the model meets the goal(s) chosen It is also customary toask that the solution (or simulation) of the model adequately describe theobserved data This, unfortunately, is a somewhat subjective criterion Ifthe model predictions are continuous curves for variables that are ex-perimentally known to be continuous and if the predicted curve differs atall observed points by an amount less than the experimental error, thenthe model may adequately describe the data There may, however, beconsistent trends in the deviations of the predicted curve from observeddata that would lead to rejecting the model

Another very important criterion for model selection is that the modelinclude as much knowledge as possible about the structure and interrela-

tionships of the biomedical system However, the search for such a modelmust be tempered by an attempt to find a model including as fewparameters and as few unknown constants as possible The latter demandfor simplicity is sometimes referred to as Ockham's razor Each time anew constant or an additional relationship with its own characteristicparameters is introduced, one can more closely approximate the data.Moreover, the difficulties in assigning uncertainties to nonlinear para-meters decrease the clarity in the decision rule for model selection.Some type of heuristic rule must be selected, such as demanding a fiftypercent reduction in the cumulative deviations between model predictionand experimental observation for each new relationship added.

In order to validate the model further, it is desirable to ask if it torily conforms to the data from new experiments on other individuals orunder new conditions As this process continues, it should be expectedthat some limits will be found beyond which the model proves unsatisfac-tory This limitation should not be regarded as a failure of the model but,more correctly, as the range of applicability of the mathematical model toachieve one or more of the modeling goals

satisfac-G Notation Used in Text

This text emphasizes the application of mathematics to the healthsciences using digital computer processing Accordingly the notationadopted facilitates coding the equations in the text in high-level computerlanguages To further support this orientation program segments havebeen incorporated into some of the chapters

However, the book is about mathematical applications and is not a text

of computer use Therefore, the notation retains several standard matical conventions that the authors believe facilitate reading, par-ticularly for the mathematically adept Thus several conventions shared

mathe-by many scientific computational languages, such as the use of capitalletters only, have been disregarded in the mathematical equations.Specific steps to employ a notation close to that of computer languagesinclude the restriction of equations to one line wherever possible and theexplicit use of two operators, "•" and "/", to indicate multiplication anddivision respectively (The operator "*" is reserved to denote convolution,and the operator "°" indicates the dot product for vector multiplication.)Where convenient, functional expressions are used, the arguments alwaysbeing included with a set of parentheses A partial list of the functionabbreviations follows:

exp (x) = exponential function of x

logn (x) = natural (base e) logarithm of x

Iog2 (x) = base 2 logarithm ol'.v

On the other hand, subscripts, superscripts, and powers are sented by numbers or by single letters using standard mathematical nota-tion Integrals, summations, and products are indicated by the mathemat-ical operators "J ", " ^ ", and "II" respectively Limits are shown above andbelow these operators if appropriate, in some cases actually involvingseparate relationships to specify the limits The representations noted inthis paragraph are thus oriented to the convenience of an interdisciplinarygroup of readers

in the health sciences are explored in this overview

The remainder of the text is divided into three parts, which develop in

a more quantitative fashion and by example the topics reviewed in thisintroduction No attempt is made to exhaustively treat all possiblemathematical or computer techniques nor to include all health-scienceapplications of mathematics mentioned in the introduction Rather, cer-tain applications have been selected, as mentioned in the preface, and aretreated in greater detail

The first of the three major parts of the text is entitled DeterministicModels Two general types of deterministic models are considered, thosebased on compartmental models and variants thereof and those based on

models of enzyme reactions and of systems of enzymes Not only are allthese models deterministic, but they can also be represented by simul-taneous first-order differential equations.

The second major part deals with biological time series The techniquesand methodologies of analyzing such data are explored The examplespresented are based on both electrocardiography and electroencephalog-raphy These models are primarily deterministic, but stochastic consid-erations play a greater role than in models presented in the earlier chap-ters

The third part deals with information theory and simulation of tions and of the world in which we live In these chapters the role ofstochastic elements in the models grows These models have been in-cluded in this text because they fail to fit the usual pattern of biostatisticsand because of the crucial role of simulation in these studies

popula-The final chapter is an overview of mathematical modeling in the healthsciences It is the companion to this introduction, albeit considerablyshorter It should prove more meaningful after the intervening chaptersare read

The reader who desires to read further is urged to investigate thereferences at the end of this chapter Mathematical models also appear innumerous journals, some, dedicated to mathematical biology per se,others, concerned with biomedical computing and health services re-search or with simulation alone Physiology, ecology, bioengineering,epidemiology, and biochemistry journals also include many articles based

on the use of a mathematical model All these provide a better view ofcurrent areas of research activity than can a textbook On the other hand,

it is hoped that this textbook provides more perspective and a more nearlyglobal overview of health-related mathematical models by consideringselected health science applications of mathematics with emphasis on theuse of computer technologies

Selected References

General Biophysics and Bioengineering

Ackerman, E., L Ellis, and L Williams 1979 Biophysical science 2nd ed Englewood

Cliffs, New Jersey: Prentice-Hall 634 pp.

Ray, C D., ed 1974 Medical engineering Chicago: Year Book Medical Publishers 1256

pp.

General Mathematical Biology

Bailey, N T J 1967 The mathematical approach to biology and medicine London: Wiley.

296 pp.

Rosen, R., eel Foundations of mathematical biology New York: Academic Press.

1972 Vol 1, 287 pp.

1973 Vol 2, ,348 pp.

1973 Vol 3, 412 pp.

Rubinovv, S I 1975 Introduction to mathematical biology New York: Wiley 386 pp.

Specific Mathematical Models

Bailey, N T J., Bl Sendov, and R Tsanev, eds 1974 Mathematical models in biology and medicine Amsterdam: North-Holland 152 pp.

Caspari, E W., and W J Horvath, eds 1970 Systems principles in biology: A symposium.

Bch Sci 15:1-117 (entire January issue).

Heinmetz, F., ed 1969 Concepts and models of biomathematics: Simulation techniques and methods New York: Marcel Dekker 287 pp.

In a sense all the models considered in this text are deterministic, and all involve some stochastic elements However, in the follow- ing four chapters the models considered are called deterministic because they lend them- selves to representation in terms of differential equations Two selected types of deterministic models are described The first type is solved

by a variety of techniques called tal analysis, which is particularly appropriate for models of tracer and metabolic studies The simplest form of such models is discussed

compartmen-in Chapter 2; a few of the less restrictive iants of compartmental analysis are presented

var-in Chapter 3 The second type of determvar-inistic- model to be considered, one for enzyme kinet- ics, is discussed in Chapter 4 The final chap- ter of this part, Chapter 5, deals with models

deterministic-of systems deterministic-of enzymes; these models are closely related to compartmental models of metabolic systems but are based on the sim- pler methodology of enzyme and chemical kinetics.

as an arrow and, as discussed later, may be assigned a rate parameter

often designated as k.

Some examples are introduced to indicate in a narrative fashion howcompartmental analysis can be used and what its limitations are Com-partmental analysis is a valuable tool of the mathematical modeler, andnumerous examples could be cited The choice of the following three ispurely arbitrary and is made for illustrative purposes only

1 Thyroxine and the Liver

The first compartmental model to be considered involves the fate of thyroxine, also known as 3,5,3',5'-tetraiodothyronine or T4 Thismolecule is formed in normal humans in an endocrine gland called the

metabolic-19

thyroid and plays a critical role in regulating metabolism An excesscauses hyperactivity, loss of body weight, and manic behavior amongother conditions A deficiency of thyroxine leads to lowered metabolicrates, sluggishness, and decrease in mental activity Although thyroxine isformed in the thyroid, it affects metabolic rates in many target organs and

is metabolized at least in part by the liver

A structural formula for thyroxine is shown in Figure 2.1 Duringmetabolism successive iodines are removed, and compounds such as T3,T2, and Tl with respectively 3, 2, and 1 iodine atoms per molecule areformed Although these are physiologically active in a fashion analogous toT4, all the other derivatives are more rapidly broken down than T4 ThusT4 appears to be the longest lasting form of thyroxine It exists in theplasma, partially but reversibly bound to certain proteins (Physiologicalevidence indicates that T3 may actually be the active form The concen-tration of T3, however, is strongly dependent in normal individuals on theT4 level.)

2.1 Thyroxine molecule (3,5,3',5'-tetraiodo thyronine).

Various models have been proposed to represent iodine and thyroxinemetabolism in the intact animal One of these is shown in Figure 2.2.Although this model is greatly oversimplified if compared with the realanimal, it is sufficiently complex that it is not clear whether there areanatomic correlates of the various compartments included Accordingly,Flock and her colleagues (see Ackerman, Hazelrig, and Gatewood, 1967)considered the simpler system shown in Figure 2.3 of an isolated rat liver,perfused with rat blood into which radio-iodine-labeled T4 was introduced

In this preparation the radioactivity of the liver and blood are tored as a function of time Cumulative radioactivity of bile collected fromthe canulated bile duct was also measured Thus the fate of the radio-labeled iodine was followed Because some inorganic iodine is returned tothe blood after metabolism by the liver, it is necessary to take into accountthe concentration of inorganic iodine as a function of time as well aschanges in the perfusate as aliquots are removed periodically for analyses

moni-In this fashion a quantitative compartmental analysis was carried outusing a variety of experiments These studies were highly successful in

2.2 Block diagram of iodine kinetics model The circles represent ments and the connecting arrows, flow pathways The thyroid is composed of three compartments: a rapid turn-over compartment, a delay phase organized

compart-as a chain of compartments to simulate a lag, and a storage compartment Note that the numerals in subscripts identifying rate constants are ordered oppo- sitely from those in Equation 2-19, i.e., \2i is identical to kl 2 - (Adapted with permission from M Berman, "The iodine pool," in Compartments Pools and Spaces in Medical Physiology, eds P Bergner, C Lushbaugh, and E Ander-

son, CONF-661010, AEC Symposium Series, Oak Ridge, 1967, p 356.)

showing that the preconceived compartmental system diagrammed inFigure 2.3 was too simple, and predictions based on its use were unac-ceptably different from the experimental values This system is reex-amined in Section E of this chapter

2 Glucose Distribution

Glucose is a six-carbon sugar (hexose) widely used in metabolism in ous tissues including liver, muscle, and brain Glucose may enter thecirculatory system by absorption in the small intestine, by production inthe liver (gluconeogenesis), and by conversion of glycogen in liver (or

vari-2.3 Two-compartment open system depicting liver metabolism of thyroxine Asterisk represents radio-thyroxine input (Adapted with permission from E Ackerman and J Hazelrig, "Computer applications to the evaluation of

dynamic biological processes," in Dynamic Clinical Studies with Radio-isotopes, ed R Knisely, R Tauxe, and E Anderson, TID-768; AEC

Symposium Series, Oak Ridge, 1964, p 50.)

muscle) to glucose (glycolysis) It can be metabolized in most tissues andcan be stored after conversion to glycogen in liver or muscle

Because the liver plays a major role in the metabolism, storage, andrelease of glucose, a separate representation of the hepatic portal circula-tion is included in the model shown in Figure 2.4 Using a model of thissort it is possible to develop expressions for rates of release and removal ofblood glucose from the systemic circulation Fluxes, concentrations, andrate constants are used in these expressions

Although this model appears quite complex, portions of it can be used

to quantitatively study the distribution of glucose Using tracer niques it is possible to estimate the amount of glucose in the variouscompartments, and these numbers are called pool sizes It is also possible

tech-to estimate rates of turnover and hence of introduction of glucose, as well

as to distinguish gluconeogenesis from glycolysis

However, this compartmental model has numerous shortcomings forother types of determinations If it is desired, for example, to measurecarbon dioxide and bicarbonate production, metabolism must be includedexplicitly If studies are conducted over periods of physiological change,then it is found that any change in blood glucose level induces change inhormonal levels The latter in turn alters all the rates of synthesis, release,and removal of glucose shown in Figure 2.4 A model to take account ofthese interactions is discussed in Chapter 3

3 Pulmonary Function

The final example was developed to quantitate the performance of thelung Two different parameters are used One is the perfusion or bloodflow per unit volume through the lung The other is the ventilation or

inspired gas flow per unit volume through the lung If the subjectbreathes a radioactive gas such as radio-xenon 133 and the concentrations

in the inspired and expired gas and plasma are monitored, then one canuse a compartmental scheme such as is shown in Figure 2.5 to derive thedesired parameters On a gross basis this method is quite successful.For a more detailed analysis of pulmonary performance a somewhatmore complicated scheme is needed Three separate lung regions can beconsidered, dependent on whether the alveoli (air-sacs) are open always,only during part of the breathing and heart cycles, or never Currentlythe data obtained with gamma cameras, which can be used to examineradioactivity in many parts of the lung, allow modeling the lung withmany more compartments This subject and others, such as effects of

2.4 Isomorphic-type model of the blood-glucose regulatory system The

squares represent blood concentrations; the solid lines, exchange between compartments; and the dashed lines, modification of transfer rates (Adapted with permission from E Ackerman, L Gatewood, J Rosevear, and G Mol-

nar, "Blood Glucose Regulation and Diabetes," in Concepts and Models of Biomathematics: Simulation Techniques and Methods, ed F Heinmets, Mar-

cel Dekker, Inc., New York, 1969 Reprinted from page 132 by courtesy of

Marcel Dekker, Inc.)

2.5 Exchange system for lung, blood, and tissue CO 2 The inputs to the system designate CO 2 entering from inspired air and tissue metabolism (Adapted with permission from F Grodins, J Gray, K Schroeder, A Norris, and R Jones, "Respiratory responses to CO 2 inhalation A theoretical study of

a nonlinear biological regulator,"/ Appl Phijsiol 7:286, 1954).

breathing, of the heart cycle, and of the respiratory dead space are sidered in Chapter 3

con-B Compartmental Analysis

The general aim in Compartmental analysis is to replace a system that iscontinuous and, by and large, nonhomogeneous with a system of discretecompartments that can be considered to be homogeneous The mathe-matical representation of the model thereby becomes much sim-pler, especially if partial differential equations can be replaced by ordinarydifferential equations To define such a model not only the compartments,but also the relationships of fluxes of substances between the variouscompartments must be specified A biological system that may be far toocomplex to describe in all details, even with the aid of a computer, canthus be reduced to a form amenable to tracer studies and computer-basedanalysis

It is then possible to examine a particular Compartmental analysis interms of the five goals enumerated in the last chapter The first goal refers

to the representation of large masses of data in terms of a relatively smallnumber of constants It will be shown that, following the postulatespresented subsequently in this section of this chapter, Compartmentalanalysis predicts that the concentrations (or amounts) of the substancesunder consideration can be represented in each compartment as a sum ofexponential terms These have the form

Ci = concentration or amount of substance in ith compartment

i = compartment number (or designation)

\j = jth exponential decay constant

t = time

Aij.= amplitude of the jth exponential in the ith compartment

N = number of decay terms

If the total number of parameters of the forms A(J and A.J is small compared

to the number of experimental points, then the first goal will be satisfiedeven though none of the others described in Chapter 1 are

Whether or not there is substantial data reduction, it may be that theestimated parameters or some combination thereof discriminate betweennormal and pathological states better than do the raw data If severalconceptually acceptable hypotheses exist and can be expressed as com-partmental models, it is also possible to predict for each the form of curves

of concentrations versus time This is a useful approach if some of themodels are sufficiently at variance with the data to be rejected

Simulation of a compartmental model can permit selecting tal conditions that will most critically test the range of validity of themodel and the hypotheses that it represents Finally, one may ask to whatextent do the compartments correspond to real anatomic structures? Tothe extent that they do, the model can be considered isomorphic with thesystem However, it should be noted that this isomorphism exists only at

experimen-a mexperimen-acro level If questions experimen-are rexperimen-aised experimen-about finer experimen-anexperimen-atomic or biochemicexperimen-aldetail it is seen that the isomorphism does not exist at that level

In many applications more than one of these goals is important partmental analysis as a general term is used to describe a variety ofmodels of different complexity In this chapter, a simple, perhapsclassical, form of compartmental analysis is considered It may be defined

Com-by a series of five postulates, which must be valid if this form of analysis is

to be used

Postulate 1: Existence This states that it is possible in some way to

represent the system under study as consisting of one or more regionscalled compartments These need not be related in a straightforwardfashion to anatomic structures but may be purely logical constructs How-ever, it is intuitively more pleasing to deal with compartments corre-sponding to real, biological subsystems Some biological systems do not