## CONTENTS

Preface ix

1. Introduction to Partial Differentiation Equation

Analysis: Chemotaxis 1

2. Pattern Formation 43

3. Belousov–Zhabotinskii Reaction System 103

4. Hodgkin–Huxley and Fitzhugh–Nagumo Models 127

5. Anesthesia Spatiotemporal Distribution 163

6. Influenza with Vaccination and Diffusion 207

7. Drug Release Tracking 243

8. Temperature Distributions in Cryosurgery 287

Index 323

## PREFACE

This book focuses on the rapidly expanding development and use of

computer-based mathematical models in the life sciences, designated

here as biomedical science and engineering (BMSE). The mathematical

models are stated as systems of partial differential equations

(PDEs) and generally come from papers in the current research literature

that typically include the following steps:

1. The model is presented as a system of PDEs that explain associated

chemistry, physics, biology, and physiology.

2. A numerical solution to the model equations is presented, particularly

a discussion of the important features of the solution.

What is missing in this two-step approach are the details of how

the solution was computed, particularly the details of the numerical

algorithms. Also, because of the limited length of a research paper,

the computer code used to produce the numerical solution is not

provided. Thus, for the reader to reproduce (confirm) the solution

and extend it is virtually impossible with reasonable effort.

The intent of this book is to fill in the steps for selected example

applications that will give the reader the knowledge to reproduce

and possibly extend the numerical solutions with reasonable effort.

Specifically, the numerical algorithms are discussed in some detail,

with additional background references, so that the reader will have

some understanding of how the calculations were performed, and a

set of transportable routines in R that the reader can study and execute

to produce and extend the solutions is provided.1

Thus, the typical format of a chapter includes the following steps:

1. The model is presented as a system of PDEs that explain

associated chemistry, physics, biology, and physiology. The

requirements of a well-posed set of equations such as the

number of dependent variables, the number of PDEs, algebraic

equations used to calculate intermediate variables, and the

initial and boundary conditions for the PDEs are included

(which is often not the case in research papers so that all of

the details of the model are not included or known to the

reader).

2. The features of the model that determine the selection of numerical

algorithms are discussed; for example, how spatial derivatives

are approximated, whether the MOL ODEs are nonstiff or

stiff, and therefore, whether an explicit or implicit integration

algorithm has been used. The computational requirements of the

particular selected algorithms are identified such as the solution

of nonlinear equations, banded matrix processing, or sparse

matrix processing.

3. The routines that are the programming of the PDEs and numerical

algorithms are completely listed and then each section of

code is explained, including referral to the mathematical model

and the algorithms. Thus, all of the computational details for

producing a numerical solution are in one place. Reference to

another source for the software, possibly with little or no documentation,

is thereby avoided.

1R is a quality open source scientific programming system that can be easily

downloaded from the Internet (http://www.R-project.org/). In particular, R

has (i) vector-matrix operations that facilitate the programming of linear algebra,

(ii) a library of quality ODE integrators that can also be applied to PDE

and ODE/PDE systems through the numerical method of lines (MOL), and (iii)

graphical utilities for the presentation of the numerical solutions. All of these

features and utilities are demonstrated through the applications in this book.

4. A numerical solution to the model equations is presented,

particularly a discussion of the important features of the

solution.

5. The accuracy of the computed solution is inferred using established

methods such as h and p refinement. Alternative algorithms

and computational details are considered, particularly to

extend the model and the numerical solution.

In this way, a complete picture of the model and its computer

implementation is provided without having to try to fill in the details

of the numerical analysis, algorithms, and computer programming

(often a time-consuming procedure that leads to an incomplete and

unsatisfactory result). The presentation is not heavily mathematical,

for example, no theorems and proofs, but rather the presentation is

in terms of detailed examples of BMSE applications.

End of the chapter problems have not been provided. Rather, the

instructor can readily construct problems and assignments that will be

in accordance with the interests and objectives of the instructor. This

can be done in several ways by developing variations and extensions

of the applications discussed in the chapters. The following are a few

examples.

1. Parameters in the model equations can be varied, and the effects

on the computed solutions can be observed and explained.

Exploratory questions can be posed such as whether the changes

in the solutions are as expected. In addition, the terms in the

right-hand sides (RHSs) of the PDEs (without the derivatives

in the initial-value independent variable, usually time) can be

computed and displayed numerically and graphically to explain

in detail why the parameter changes had the observed effect.

The computation and display of PDE RHS terms is illustrated

in selected chapters to serve as a guide.

2. Additional terms can be added to the PDE RHSs to model physical,

chemical, and biological effects that might be significant

in determining the characteristics of the problem system. These

additional terms can be computed and displayed along with the

original terms to observe which terms have a significant effect

on the computed model solution.

3. One or more PDEs can be added to an existing model to include

additional phenomena that are considered possibly relevant to

the analysis and understanding of the problem system. Also,

ODEs can be added, typically as boundary conditions.

4. An entirely new model can be proposed and programmed for

comparison with an existing model. The existing routines might

serve as a starting point, for example, as a template.

These suggested problem formats are in the order of increasing

generality to encourage the reader to explore new directions, including

the revision of an existing model and the creation of a new model.

This process is facilitated through the availability of existing routines

for a model that can first be executed and then modified. The trialand-

error development of a model can be explored, particularly if

experimental data that can be used as the basis for model development

are provided, starting from parameter estimation based on a

comparison of experimentally measured data and computed solutions

from an existing model, up to the development of a new model to

interpret the data.

The focus of this book is primarily on models expressed as systems

of PDEs that generally result from including spatial effects so

that the dependent variables of the PDEs, for example, concentrations,

are functions of space and time, which is a basic distinguishing

characteristic of PDEs (ODEs have only one independent variable,

typically time). The spatial derivatives require boundary conditions

for a complete specification of the PDE model and several boundary

condition types are discussed in the example applications.

In summary, my intention is to provide a set of basic computational

procedures for ODE/PDE models that readers can use with

modest effort without becoming deeply involved in the details of

numerical methods for ODE/PDEs and computer programming. All

of the R routines discussed in this PDE volume and the companion

ODE volume Differential Equation Analysis in Biomedical Science

and Engineering: Ordinary Differential Equation Applications with R

are available from a software download site, booksupport.wiley.com,

which requires the ISBN: 9781118705483 for the ODE volume or

9781118705186 for this volume. I welcome comments and will be

pleased to respond to questions to the extent possible by e-mail

(wes1@lehigh.edu).