## Preface

This book is an introduction to the mathematical modeling of biomedical engineering

systems. In particular, models based on partial differential equations (PDEs) are

presented: antibody binding kinetics, acid-mediated tumor growth, retinal O2 transport,

hemodialyzer dynamics, epidermal wound healing, and polymer matrix drug

delivery.

The numerical solution of the model equations is through a single, well-established

method for PDEs, the method of lines (MOL), in which the spatial derivatives in the

PDEs are replaced with algebraic approximations and a system of ordinary differential

equations (ODEs) in an initial value variable, typically time, follows. The spatial

approximations are finite differences (FDs), although other approximations could easily

be accommodated within the MOL format, e.g., finite elements, finite volumes, spectral

methods, Galerkin methods such as collocation, meshfree methods. The final result is

a set of routines that numerically integrates the ODEs; the format of these routines is

basically the same throughout the book.

To facilitate understanding of the PDE analysis, an introduction to the numerical

methods and associated computational routines is presented in the first chapter. Then each

application is cross referenced to this introduction in each step where some additional

explanation is helpful.

In each example, we follow a combination of the following steps:

• Statement of the model PDE, along with the associated auxiliary (initial and boundary)

conditions. This introduction to the model contains a reference to the original

source and possibly related literature, and includes some discussion of the underlying

biophysics, biochemistry, and physiology.

• Discussion of the numerical methods (algorithms) for the MOL solution of the model

equations, principally by reference to the introduction in Chapter 1.

• List ofMatlab routines based on the MOLnumerical solution of the PDEs, discussed

in some detail, typically a few lines of code at a time. This discussion emphasizes how

the associated mathematics of the model is programmed.

• Discussion of the numerical solution of the model equations, including the origin of

any unusual features of the solution.

• Error analysis to establish if the numerical solution is reliable and has acceptable

accuracy; typically techniques such as varying the MOL grid spacing and order of

approximation are used to infer convergence of the numerical solution to an acceptable

level of accuracy. Also, physical constraints such as conservation of mass energy and

energy are used to evaluate the solution.

• Concluding summary and discussion of extensions of the model and the MOL

algorithms.

Our intention is not to provide a comprehensive treatise, but rather to provide a set of

basic computational procedures that we hope readers can assimilate without becoming

deeply involved in the details of numerical methods forPDEsand computer programming

so that they can concentrate on the problem of interest with reasonable effort. This might

take the form of extending the computer routines provided, or applying and extending

the numerical methods that are presented through examples.

In summary, our intention is to provide a methodology for the PDE analysis of biomedical

engineering systems. This includes the development of numerical methods and

associated computer routines that can be used to study the characteristics and solutions

of the model equations. The approach is not theoretical, e.g., limited theorems and no

proofs; rather, the presentation is based on detailed example applications. The MOL

analysis provides a general framework for the analysis of PDE models that we think can

be broadly applied in biomedical engineering, and which can be applied to all of the

major geometric classes of PDEs (parabolic, hyperbolic, elliptic). All of the Matlab

routines are available (gratis) as a download through a request to wes1@lehigh.edu.

We welcome comments from readers concerning this approach and will be pleased to

answer questions to the extent possible by e-mail.

## Contents

Preface page ix

1 Introduction to partial differential equation integration in

space and time 1

1.1 Introduction 1

1.2 Hyperbolic PDEs 2

1.3 Parabolic PDEs 88

1.4 Hyperbolic-parabolic PDEs 130

2 Antibody binding kinetics 148

2.1 ODE/PDE model equations 148

2.2 Units check 150

2.3 MOL routines 153

2.4 Model output 167

2.5 ODE stiffness 171

2.6 Parameter sensitivity analysis 172

2.7 Spatial derivatives by stagewise differentiation 178

2.8 Spatial derivatives by direct calculation 181

3 Acid-mediated tumor growth 184

3.1 Tumor growth PDE model 184

3.2 MOL routines 189

3.3 Model output 205

3.4 Supplemental output 214

3.5 Extension of the model 219

3.6 Additional references 220

4 Retinal O2 transport 221

4.1 Four-section PDE model 221

4.2 MOL routines 224

4.3 Base case output 241

4.4 Model including photoreceptor cell density 247

4.5 Model including VEGF production 262

5 Hemodialyzer dynamics 284

5.1 1D PDE model 284

5.2 MOL routines 288

5.3 Model output 299

6 Epidermal wound healing 308

6.1 One-PDE model 308

6.2 Two-PDE model 322

6.3 Conclusions concerning space and time discretizations 336

7 Drug distribution from a polymer matrix 339

7.1 Linear model 339

7.2 Variable coefficient model 367

7.3 Nonlinear model 368

Appendix 1 Origin of convection-diffusion-reaction PDEs 388

A1.1 Derivation of a PDE from a conservation principle 388

A1.2 Approximations of derivatives 392

A1.3 Approximation of boundary conditions 394

A1.4 Derivation of a convection-diffusion-reaction PDE 397

Appendix 2 Stiffness and eigenvalue analysis of ODE systems 406

A2.1 Linear algebraic eigenvalue problem 406

A2.2 Exponential solutions 409

A2.3 Laplace transform solutions 412

A2.4 Numerical eigenvalue analysis 414

Index 416