The Mathematical Approach to Physiological Problems: A Critical Primer

With its formulations derived entirely from algebra and introductory calculus, this primer takes as its central purpose the encouragement of high standards of mathematical conduct. And yet the author is not so interested in fostering a formal elegance or abstract rigor as in imparting to the student a care for the definition of the problem, the choice of units, the avoidance of ambiguities and inner contradictions; and as in instilling in him a feeling for dimensionality, an awareness of the pitfalls of various techniques, an appreciation for the habitual checking of derivations through numerical substitution. Riggs adduces a number of examples from the open literature in which biologists of the highest standards of experimental exactness are shown presenting their careful data in mathematically inexact form. As the mathematization of the biological sciences proceeds and accelerates, it becomes even more important that definitions and derivations be as closely scrutinized as tissue sections and radiation counters.

In most cases, the various chapters develop their material in a general mathematical form, with applications to particular biological problems developed in thought-provoking exercises. Many of these are based on published papers, and the student is asked to find mistaken assumptions and procedures. A lengthy appendix presents (and where necessary explains step by step) the derivation of the solutions.

Other appendices concern approximations, the use of logarithms, and units and symbols. The text includes treatment of the following topics: Definitions, assumptions, and models—Symbols, dimensions, and units—Aids to mathematical work: analog and digital computers; graphical methods; the method of least squares; linear transformations—Constants, variables, and functional relationships: correlations; symbol-and-arrow diagrams—Feedback relationships: feedback loops; positive and negative feedback; homeostasis—Exponential growth and disappearance—Transfer of substances between biological compartments: steady states and equilibrium states; simple diffusion; general kinetics; membrane permeability—Further kinetic problems: fluid flow; metabolic transformations; various biological examples—The law of mass action: effects of volume and temperature; collision theory; chemical equilibria; ionization—Substrate-enzyme and drug-receptor interactions—The derivation of equations: a general method—The derivation of equations: a detailed example (anesthetic gases)—Methods for checking the validity of equations.