AN INTRODUCTION TO MATHEMATICAL PHYSIOLOGY AND BIOLOGY
J. Mazumdar, Cambridge University Press, 1999, pp: 224, ISBN 0-521-64675-8 (pbk), 0-521-64110-1 (hc); Price: $29.95 (pbk), $74.95 (hc).

Practitioners of mathematical biology come from a variety of backgrounds: mathematics and biology, obviously, but also the physical sciences, engineering and computer science. Despite the variety of knowledge and experience we bring to our studies, we have a few things in common: (1) we share an enthusiasm for a rapidly developing field which we are very eager to share with students and colleagues, and (2) we all have a great deal to learn. Very few of us studied both biology and mathematics as undergraduates. Mathematical biology is therefore in need of textbooks. Each group of students entering the field needs textbooks adapted to their particular background, which brings us to the subject of this review, a textbook for a one-term course in mathematical biology oriented toward senior mathematics students.

Mazumdar's approach is ambitious: In one small book, he aims to give students a broad appreciation of mathematical biology. The book includes chapters on such disparate topics as population biology, pharmacokinetics, and the fluid mechanics of blood flow. There are also two chapters on epidemiology and immunology written by K.B. Naidu. I wish I could say that this textbook was a success because I think the survey-course approach is very much the right one for the intended audience. Unfortunately, the early chapters use mostly trivial mathematics, and some of the later chapters, on blood flow and related issues, can't properly be appreciated without a fairly extensive background in anatomy and mechanics. Moreover, very few sections adequately discuss the interplay of data and models, without which the subtle beauty of the subject can't properly be appreciated.

The book starts off well enough. Chapters 1 and 2, respectively on dimensional analysis and on diffusion, lay down some basic ideas, which I thought would form the basis for the rest of the book. Unfortunately, the themes introduced in these chapters seldom recur in the rest of the book, and never very effectively. Chapter 3 introduces population modeling and Chapter 4 covers biogeography. I would have thought that a chapter on biogeography following chapters on diffusion and population models would have discussed diffusion models of species dispersion, if only qualitatively. However, Chapter 4 limits itself almost entirely to a study of the equilibrium solutions of ODE models of migration of species to and from islands. While this is an interesting introductory topic, it is mathematically far too easy for undergraduate mathematics students. In fact, with some changes in notation, the material covered by Mazumdar in this chapter could easily be used in a high school course.

The chapter on biogeography is typical of most of this book. One feels in many places a peculiar reluctance on the part of the author to use the full power of the mathematical techniques with which senior mathematics students should be acquainted. To me, the most glaring example of this is the almost complete avoidance of linear algebra throughout the book, even when this would have considerably simplified the discussion. In Chapter 3, where the stability of solutions of systems of ODEs is discussed, the connection to properties of the Jacobian matrix is not introduced early enough. This is done only after a great deal of tedious algebraic manipulation has been deployed to answer questions which could have been resolved much more simply by a judicious use of linear algebra. The appropriate theorems are never stated, and the author never returns to the ideas presented in this chapter. Indeed, the author sometimes goes to the most remarkable lengths to avoid the use of matrix techniques. In the chapter on pharmacokinetics for instance (Chapter 5), one encounters a variable transformation whose effect is to diagonalize a system of linear differential equations. Showing that this transformation arises from the diagonalization of the coefficient matrix would have brought out the structure of the problem. Simply presenting students with the transformation teaches them nothing.

The level of mathematics in the later chapters (on blood flow and related issues, the author's research area) is much more appropriate. Unfortunately, in these chapters, Mazumdar's enthusiasm sometimes gets the better of him. Biomedical terminology is sometimes used with little explanation. There are few diagrams to help a student with little previous background in the area. Also, the author seems to assume that everyone has taken a course in the mechanics of deformable materials. While many engineering-based applied mathematics programs would include such a course, in most cases a lecturer would have to fill in a good deal of background to make this material intelligible to the average student.

To compound these difficulties, there is a very rough quality to the writing in several of the chapters in this book. I sometimes felt that I was reading someone's lecture notes rather than a polished textbook, which surprised me greatly given that this is a second edition. Material that distracts rather than informs is sometimes included. Terms are sometimes used on one page and not defined until the next. And, as mentioned earlier in this review, there is not much continuity from one chapter to the next.

Despite these criticisms, there are some very interesting ideas in this book. Chapter 1 on dimensional analysis introduces a number of important themes that few textbooks in applied mathematics even touch on. In chapter 3, the time-T solution map of a differential equation is introduced. This leads to a number of mathematically interesting topics not covered in this book, but which a lecturer with appropriate interests could discuss with the class. In the chapter on pharmacokinetics (Chapter 5), the parameter identifiability problem is briefly discussed. This is an extremely important and mathematically subtle problem which, again, I would have liked to have seen fleshed out with a detailed example or two. However, the author should be commended for raising these issues at all since they are not often mentioned in applied mathematics courses. K.B. Naidu's chapters on epidemiology and immunology include some beautiful and detailed examples of mathematical model building in a biological context. Again, I would have liked to see some discussion of methods for estimating the parameters from experimental data, but that is at least partly a matter of taste. Finally, the chapters on blood flow, ventricular mechanics and heart valve vibration which end the book are a rewarding study, despite my misgivings about the author's assumptions regarding the students' backgrounds.

If you teach a course in mathematical biology, or perhaps even a more general course in mathematical modeling, buy yourself a copy of this book. I came away from reading Mazumdar's book with many ideas for new material to be incorporated into my own mathematical modeling course. However, I wouldn't recommend it as a textbook. The level of mathematics is generally not high enough for mathematically trained students (including physicists), and the biology is not sufficiently developed in most chapters to really get the excitement of the field across to students.

Marc R. Roussel
Department of Chemistry and Biochemistry
University of Lethbridge