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 ofPDEs, 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.
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 and 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 trial-and-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).

William E. Schiesser

Bethlehem, PA

February 2014