# Method of Lines PDE Analysis in Biomedical Science and Engineering

## Books

Presents the methodology and applications of ODE and PDE models within biomedical science and engineering

With an emphasis on the method of lines (MOL) for partial differential equation (PDE) numerical integration, Method of Lines PDE Analysis in Biomedical Science and Engineering demonstrates the use of numerical methods for the computer solution of PDEs as applied to biomedical science and engineering (BMSE). Written by a well-known researcher in the field, the book provides an introduction to basic numerical methods for initial/boundary value PDEs before moving on to specific BMSE applications of PDEs.

Featuring a straightforward approach, the book’s chapters follow a consistent and comprehensive format. First, each chapter begins by presenting the model as an ordinary differential equation (ODE)/PDE system, including the initial and boundary conditions.  Next, the programming of the model equations is introduced through a series of R routines that primarily implement MOL for PDEs. Subsequently, the resulting numerical and graphical solution is discussed and interpreted with respect to the model equations. Finally, each chapter concludes with a review of the numerical algorithm performance, general observations and results, and possible extensions of the model. Method of Lines PDE Analysis in Biomedical Science and Engineering also includes:

• Examples of MOL analysis of PDEs, including BMSE applications in wave front resolution in chromatography, VEGF angiogenesis, thermographic tumor location, blood-tissue transport, two fluid and membrane mass transfer, artificial liver support system, cross diffusion epidemiology, oncolytic virotherapy, tumor cell density in glioblastomas, and variable grids
• Discussions on the use of R software, which facilitates immediate solutions to differential equation problems without having to first learn the basic concepts of numerical analysis for PDEs and the programming of PDE algorithms
• A companion website that provides source code for the R routines
Method of Lines PDE Analysis in Biomedical Science and Engineering is an introductory reference for researchers, scientists, clinicians, medical researchers, mathematicians, statisticians, chemical engineers, epidemiologists, and pharmacokineticists as well as anyone interested in clinical applications and the interpretation of experimental data with differential equation models. The book is also an ideal textbook for graduate-level courses in applied mathematics, BMSE, biology, biophysics, biochemistry, medicine, and engineering.

Preface xi

1 An Introduction to MOL Analysis of PDEs: Wave Front Resolution in Chromatography 1
1.1 1D 2-PDE model, 2
1.2 MOL routines, 7
1.2.1 Main program, 7
1.2.2 MOL/ODE routine, 16
1.2.3 Subordinate routines, 20
1.3 Model output, single component chromatography, 21
1.3.1 FDs, step BC, 21
1.3.2 Flux limiters, step BC, 39
1.3.3 FDs, pulse BC, 48
1.3.4 Flux limiters, pulse BC, 50
1.4 Multi component model, 53
1.5 MOL routines, 54
1.5.1 Main program, 54
1.5.2 MOL/ODE routine, 62
1.6 Model output, multi component chromatography, 67
References, 68

2 Wave Front Resolution in VEGF Angiogenesis 69
2.1 1D 2-PDE model, 70
2.2 MOL routines, 72
2.2.1 Main program, 72
2.2.2 MOL/ODE routine, 81
2.2.3 Subordinate routines, 85
2.3 Model output, 86
2.3.1 Comparison of numerical and analytical solutions, 86
2.3.2 Effect of diffusion on the traveling-wave solution, 88
2.4 Conclusions, 88
References, 89

3 Thermographic Tumor Location 91
3.1 2D, 1-PDE model, 92
3.2 MOL analysis, 94
3.2.1 ODE routine, 94
3.2.2 Main program, 100
3.3 Model output, 105
3.4 Summary and conclusions, 110
References, 111

4 Blood-Tissue Transport 113
4.1 1D 2-PDE model, 114
4.2 MOL routines, 115
4.2.1 MOL/ODE routine, 115
4.2.2 Main program, 119
4.2.3 Bessel function routine, 128
4.3 Model output, 129
4.4 Model extensions, 133
4.5 Conclusions and summary, 142
References, 143

5 Two-Fluid/Membrane Model 145
5.1 2D, 3-PDE model, 146
5.2 MOL analysis, 147
5.2.1 MOL/ODE routine, 148
5.2.2 Main program, 153
5.3 Model output, 160
5.4 Summary and conclusions, 162

6 Liver Support Systems 165
6.1 2-ODE patient model, 166
6.2 Patient ODE model routines, 167
6.2.1 Main program, 167
6.2.2 ODE routine, 172
6.3 Model output, 174
6.4 8-PDE ALSS model, 176
6.4.1 Membrane unit MU1, 177
6.4.4 Membrane unit MU2, 179
6.5 Patient-ALSS ODE/PDE model routines, 180
6.5.1 Main program, 180
6.5.2 ODE routine, 188
6.6 Model output, 195
6.7 Summary and conclusions, 196
Appendix - Derivation of PDEs for Membrane and Adsorption Units, 200
A.1 PDEs for Membrane Units, 200
A.2 PDEs for Adsorption Units, 202
References, 203

7 Cross Diffusion Epidemiology Model 205
7.1 2-PDE model, 205
7.2 Model routines, 207
7.2.1 Main program, 207
7.2.2 ODE routine, 215
7.3 Model output, 218
7.3.1 ncase = 1, time-invariant solution, 218
7.3.2 ncase = 2, transient solution, no cross diffusion, 220
7.3.3 ncase = 3, transient solution with cross diffusion, 222
7.4 Summary and conclusions, 224
Reference, 225

8 Oncolytic Virotherapy 227
8.1 1D 4-PDE model, 228
8.2 MOL routines, 229
8.2.1 Main program, 230
8.2.2 MOL/ODE routine, 240
8.2.3 Subordinate routine, 245
8.3 Model output, 246
8.4 Summary and conclusions, 273
Reference, 274

9 Tumor Cell Density in Glioblastomas 275
9.1 1D PDE model, 276
9.2 MOL routines, 277
9.2.1 Main program, 277
9.2.2 MOL/ODE routine, 286
9.3 Model output, 289
9.3.1 Output for ncase = 1, linear, 290
9.3.2 Output for ncase = 2, logistic, 295
9.3.3 Output for ncase = 3, Gompertz, 296
9.4 p-refinement error analysis, 299
9.5 Summary and conclusions, 301
References, 301

10 MOL Analysis with a Variable Grid: Antigen-Antibody Binding Kinetics 303
10.1 ODE/PDE model, 303
10.2 MOL routines, 306
10.2.1 Main program, 306
10.2.2 MOL/ODE routine, 314
10.3 Model output, 318
10.3.1 Uniform grid, 318
10.3.2 Variable grid, 321
10.4 Summary and conclusions, 325
Appendix: Variable Grid Analysis, 327
A.1 Derivation of numerical differentiators, 327
A.2 Testing of numerical differentiators, 331
A.2.1 Differentiation matrix, 331
A.2.2 Test functions, 332
References, 340

Appendices
Appendix A Derivation of Convection-Diffusion-Reaction
Partial Differential Equations 341
Appendix B Functions dss012, dss004, dss020, vanl 345
Index 351

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