# Mathematical Models in the Health Sciences: A Computer-Aided Approach

Eugene Ackerman
Laël Cranmer Gatewood
Edition: NED - New edition
Pages: 372
https://www.jstor.org/stable/10.5749/j.ctttt5gn

1. Front Matter
(pp. i-iv)
2. Preface
(pp. v-viii)
E.A. and L.C.G.
(pp. ix-xiv)
4. ### INTRODUCTION

• CHAPTER 1 Models and Goals
(pp. 3-16)

For centuries scientists have used mathematical functions to describe the observable world, but the early records of applications of mathematics to biological phenomena are difficult to find. The types of applications selected for presentation in this text have been developed since the nineteenth century by a diverse group of scientists working in many fields. As recently as 1850 it was possible for one person to acquire the skills of a physician, surgeon, physicist, and mathematician as exemplified by von Helmholtz. Until the introduction of digital computers, the studies of these scientists, individually and in groups, were usually in the areas...

5. ### DETERMINISTIC MODELS

• CHAPTER 2 Compartmental Analysis
(pp. 19-52)

In this chapter only the simplest compartmental models are included; they all can be described as obeying a set of postulates presented in Section B of the chapter. Numerous monographs and a few textbooks have been written about models containing one or more compartments. (Use of more than one compartment is sometimes described as multicompartmental analysis.) These models all share the feature that the real biological system is treated by analogy as though it consisted of a series of relatively independent chambers or compartments, which can be represented graphically as boxes. Some exchange of the tracer or molecule of interest...

• CHAPTER 3 Modified Compartmental Analysis
(pp. 53-75)

In the previous chapter, it was noted that Compartmental analysis generates models that are distinguished by simplicity and ease of manipulation. This type of analysis provides an excellent first approximation. If it meets the goals sought, Compartmental analysis is, in itself, sufficient. However, Compartmental analysis of the type discussed in the previous chapter is not always adequate; a few examples of such cases are presented in this chapter.

The only models considered in the previous chapter are ones that obey all five given postulates. These postulates can be summarized as follows: The existence (the first) postulate states that a Compartmental...

• CHAPTER 4 Enzyme Kinetics
(pp. 76-98)

Deterministic models suitable for describing enzyme reactions are presented in this chapter and are extended to models of enzyme systems in Chapter 5. The goals discussed in Chapter 1 for health-related modeling apply equally well to this area. Mathematical models of enzyme kinetics depend heavily on automated computation. A discussion of these models presupposes a knowledge of enzyme function, thus a brief overview follows. Readers who are familiar with biochemistry are encouraged to either scan briefly or omit altogether the remainder of this section as well as Sections B, C, and D in this chapter.

Enzymes are often defined as...

• CHAPTER 5 Enzyme Systems
(pp. 99-122)

Enzymes usually occur as part of multienzyme systems in the living cell. This chapter develops various models and mathematical techniques that apply to such enzyme systems. In the previous chapter, it was shown that when an enzyme is mixed with its substrates, the intermediate complexes rise rapidly to a maximum. Thereafter, the intermediate complexes change slowly relative to the rate of substrate utilization. In a system in which a supply of new substrate is provided at a rate comparable to its removal, then the relative amount of the enzyme, as an intermediate complex, tends to be constant; the enzyme is...

6. ### TIME SERIES

• CHAPTER 6 Discrete Time Series
(pp. 125-156)

Variables that are functions of time play a central role in quantitative measurements for biology and medicine. In tracer studies of the kinetic behavior of physiological systems and in studies of enzyme kinetics, the empirical observations that are simulated or modeled are functions of time. Other models and model parameters deal with steady-state concentrations and pool sizes, but even these may be estimated from tracer measurements or from perturbation data recorded initially as a function of time. Many physiological variables such as blood pressure, hormonal levels, and nerve conduction of information are observed as functions of the independent variable time....

• CHAPTER 7 Transforms and Transfer Functions
(pp. 157-177)

In any deterministic system that in some sense can be described as having an input and an output these two signals must be uniquely related. The input can be dye inserted into the vascular system, a sound wave impinging on the ear, or tracer entering a given compartment. The respective outputs might be the dye concentration farther along the vascular system, the neuronal pulses in the auditory nerve, and the tracer leaving the compartment. By and large, the time relationships cannot be expressed as simple proportionalities even if the system is linear in the engineering sense.

For linear systems, it...

• CHAPTER 8 Electrocardiographic Interpretation
(pp. 178-205)

Probably every type of continuous signal that is associated with the living human has at one time or another been recorded as or transduced into an electrical form and then converted into a discrete time series. Among the signals whose time variations have primary importance, none has been studied more thoroughly using a variety of computer and mathematical techniques than the electrocardiogram. It and its usual adjectives are abbreviated as EKG in this test. (The German abbreviation EKG rather than the English ECG is used to limit confusion with the electroencephalogram, which is abbreviated EEG.) EKG signals can be detected...

• CHAPTER 9 Electroencephalographic Analyses
(pp. 206-232)

In Chapter 8 the time series resulting from the digitization of electrocardiographic potentials are discussed. Other physiological processes also give rise to time-variant electrical potentials that can be detected at the body surface. Like the EKG, these other surface potentials provide an indication of underlying physiological phenomena and can be observed noninvasively, causing minimal damage or discomfort to the subject. The potentials associated with the brain have received special attention and study. These are called electroencephalographic potentials and are referred to throughout this chapter as the EEG. This abbreviation is used indiscriminately as a noun and an adjective to refer...

7. ### INFORMATION AND SIMULATION

• CHAPTER 10 Information Theory
(pp. 235-249)

Information and its transfer have been alluded to and used implicitly in several places in this text. These concepts are developed and expanded in this and the next chapter, which focus on an approach to quantification of information called information theory. An early formulation was developed by Fisher in 1925; however, information theory was introduced to the scientific and engineering community primarily as the result of the work of Claude Shannon (see Shannon and Weaver, 1949). Information theory appeared and became widely used in the period of initial development and introduction of electronic digital computer technology. The representation of information...

• CHAPTER 11 Genetic Transfer of Information
(pp. 250-270)

The genetic transfer of information represents one of the major applications of information theory and the use of computer-based simulations. Before introducing these topics in this chapter, a brief review of genetics and its molecular basis is included. Rather than attempting a historic review or a detailed account of the supporting evidence, only those concepts felt by the authors to be most important for mathematical biology are included.

Every individual, human or otherwise, represents the combined results of both environmental and inherited factors. The latter are called genetic. Generally speaking, an individual’s inheritance is built upon pairs of factors; one...

• CHAPTER 12 Simulation of Epidemics
(pp. 271-303)

In the previous chapter some types of models introduced dealt with evolution of genetically inherited characteristics within populations. This chapter continues with a more general discussion of these types of models and several specific examples selected from epidemic theory, a part of epidemiology. Before further discussing simulation and the models used in epidemic theory, this section presents an extremely brief introduction to epidemiology.

Originally, epidemiology referred to studies of epidemics. An epidemic may be thought of as an occurrence of a particular disease in a particular location at a rate that is significantly greater than the usual, average rate of...

• CHAPTER 13 Population, Ecology, and the World System
(pp. 304-332)

Mathematical models of populations and simulations to characterize the dynamic behavior inherent within such models are major components of modeling in the health sciences. This general approach is included in Chapter 11, where genetic and evolutionary models were introduced. Those models enable simulation of processes that would require many human lifetimes to observe. Other population models, of epidemics, form the basis for Chapter 12. Such models made it possible to test hypotheses that cannot be tested on human populations. This introduction to Chapter 13 recapitulates the general classifications of population models and outlines the specific characteristics of the models discussed...

8. ### OVERVIEW

• CHAPTER 14 Mathematical Models in the Health Sciences
(pp. 335-344)

A preview of the text is included in the first chapter. In addition, most chapters contain their own introductions with more detailed outlines of the topics under discussion. Most chapters conclude with a section summarizing the material of that chapter, often relating it to preceding and following ones and placing the chapter in the perspective of a broader, biomedical context. It is the function of this chapter to provide a final perspective for the entire text. Section A is concerned with the topics of the text per se. Succeeding sections discuss the place of the topics selected within broad areas...

9. Index
(pp. 347-357)