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
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
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
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
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
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