Mathematical Biology: Modeling and Analysis

This short monograph considers biological processes that can be described by systems of partial differential equations (PDEs). It focuses on modeling those processes and analyzing the results. It also presents detailed mathematical analysis of the models, or simplified versions of them, and suggests several open problems.

The pattern throughout the book is to develop a model to address a biological question, show that the model agrees with what is known of the biology, and then provide a solution to the original question. The author was the founding director of the Mathematical Biosciences Institute at Ohio State. He has collected and selected examples that he believes would be useful to students and researchers who may have little background in biology.

The author begins with a helpful short background chapter on biology that captures both terminology and essential concepts. Many topics are touched on through the monograph but three subjects are explored in some detail. These are models of cancer, atherosclerosis, and chronic wound healing.

To introduce the practice of mathematical modeling in biology the author illustrates the process by developing models for enzyme dynamics. He then provides an example that uses an advection-diffusion equation with chemotaxis (movement in the direction of a chemical gradient) to model the role of angiogenesis (formation of new blood vessels) in enabling the growth of cancerous tumors.

Models relating to cancer appear at a few different places in the book. One place is the example just described. Another is a population model that examines whether T cells from the immune system are able to eradicate cancer cells in a tumor. A third set of models investigates the ability of certain drugs to suppress or eliminate a tumor. Simulation results of a simplified version of one of these models are also presented.

The atherosclerosis model investigates plaque formation in blood vessels. It focuses specifically on the role played by concentrations of low and high density lipoproteins (LDL and HDL). The author describes the variables that influence growth of the plaque, identifies the network of relationships among these variables, and then assembles a corresponding system of PDEs. The model leads to a contour map that estimates the risk level for growth of plaque based on LDL and HDL levels.

Modeling of chronic wound healing incorporates the effect of angiogenesis and depends heavily on the amount of oxygen available to tissue surrounding the wound. The model describes the progression of wound healing as the extracellular matrix, supported by plate-derived and vascular endothelial growth factors, expands to cover wounded tissue. The PDE involved has both fixed and free boundary conditions.

All three major subjects (cancer, atherosclerosis and wound healing models) have separate accompanying chapters on mathematical analysis of the models (or simplified versions thereof). These generally state and prove theorems about the existence and properties of solutions of the relevant PDEs and their stability. The PDEs are typically elliptic or parabolic, often nonlinear and with a variety of boundary conditions.

The author says that no prior experience with PDEs is assumed, and he provides an appendix to fill in the necessary background. This appendix is an admirably complete and concise summary, but is probably insufficient for novices to make good use of this book. Overall this monograph is most suitable for supplemental reading in a mathematical biology course for those with at least a modest acquaintance with PDEs.

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Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.