Method of Lines PDE Analysis in Biomedical Science and Engineering

Preface xi About the Companion Website xiii 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 6.4.2 Adsorption unit AU1, 177 6.4.3 Adsorption unit AU2, 178 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