Selected Topics in Cancer Modeling

A major challenge in the modeling and simulation of tumor growth is the mathematical description of living matter, which is far more complex than a mathematical description of inert matter. One critical piece of this challenge is creating multiscale models that take into account subcellular, cellular, and macroscopic levels of cancer. The complexity of these different levels requires the development of new mathematical methods and ideas, which are examined in this work.Written by first-rate researchers in the field of mathematical biology, this collection of selected chapters offers a comprehensive overview of state-of-the-art mathematical methods and tools for modeling and analyzing cancer phenomena.Topics covered include:* Genetic and epigenetic pathways to colon cancer* A game theoretical perspective on the somatic evolution of cancer* Nonlinear modeling and simulation of tumor growth* Tumor cords and their response to anticancer agents* Modeling diffusely invading brain tumors* Multiphase models of tumor growth* Mathematical modeling of breast carcinogenesis* Predictive models in tumor immunology* Multiscale modeling of solid tumor growthSelected Topics in Cancer Modeling is an excellent reference for researchers, practitioners, and graduate students in applied mathematics, mathematical biology, and related fields. The book has an overall aim of quantitative, predictive mathematical modeling of solid tumor growth at all scales, from genetics all the way through to treatment therapy for patients.

Preface	p. xi
List of Contributors	p. xiii
Genetic and Epigenetic Pathways to Colon Cancer: Relating Experimental Evidence with Modeling	p. 1
Introduction	p. 1
Genomic Instability in CRC: The Three Molecular Phenotypes	p. 2
Mathematical Modeling of Pathways to Cancer	p. 8
Modeling Sporadic and Familial CRC Initiation	p. 10
Discussion	p. 20
References	p. 23
From Kinetic Theory for Active Particles to Modelling Immune Competition	p. 31
Introduction	p. 31
Mathematical Frameworks and a General Model	p. 32
Qualitative Analysis	p. 36
Simulations	p. 40
Looking Forward	p. 42
Appendix	p. 44
References	p. 46
Towards Microscopic and Nonlocal Models of Tumour Invasion of Tissue	p. 49
Introduction	p. 49
A Simple Example	p. 51
Macroscopic Models of Tumour Invasion	p. 53
Microscopic Nonlocal Models of Tumour Invasion	p. 55
Mesoscopic Model	p. 58
Conclusions	p. 61
References	p. 62
Nonlinear Renewal Equations	p. 65
Introduction	p. 65
Examples of Age-Structured Models	p. 67
Assumptions and Eigenelements	p. 71
Existence of Solutions to the Linear Problem	p. 73
Nonlinear Problem, Uniform Bounds	p. 74
A Case with Asymptotic Decoupling	p. 77
A Concave Nonlinearity on the Birth Term	p. 80
Stability for Nonlinearities Reducing to ODE Systems	p. 82
References	p. 94
A Game Theoretical Perspective on the Somatic Evolution of Cancer	p. 97
Introduction	p. 97
Tumourigenesis	p. 99
Angiogenesis	p. 102
Motility/Invasion	p. 105
Outlook	p. 108
References	p. 111
Nonlinear Modeling and Simulation of Tumor Growth	p. 113
Introduction	p. 113
Solid Tumor Growth Using a Boundary Integral Method	p. 115
Vascularized Tumor Growth Using an Adaptive Finite Element/Level Set Method	p. 130
Solid Tumor Growth Using a Ghost Cell/Level Set Method	p. 146
Vascularized 3-D Tumor Growth Using a Diffuse Interface Approach	p. 160
Conclusion	p. 168
References	p. 169
Tumour Cords and Their Response to Anticancer Agents	p. 183
Introduction	p. 183
The Basic Model for the Evolution of Tumour Cords	p. 187
Response of Cords to Treatments	p. 193
Conclusions	p. 203
References	p. 204
Modeling Diffusely Invading Brain Tumors: An Individualized Approach to Quantifying Glioma Evolution and Response to Therapy	p. 207
Introduction	p. 207
Growth Dynamics of Gliomas in Vivo	p. 208
Tissue Heterogeneity: Grey and White Matter	p. 213
Modeling Glioma Response to Therapy	p. 213
Improvements in Medical Imaging and Tumor Evolution	p. 217
Conclusion	p. 217
References	p. 218
Multiphase Models of Tumour Growth	p. 223
Introduction	p. 223
Mass Balance	p. 224
Force Balance	p. 229
Interaction Forces	p. 232
Stress Tensors	p. 237
Tumour Growth in a Rigid ECM	p. 239
Tumour Growth in a Remodelling ECM	p. 243
Vascularized Tumour Growth	p. 244
Final Remarks and Open Problems	p. 248
References	p. 250
The Lymphatic Vascular System in Lymphangiogenesis, Invasion and Metastasis: A Mathematical Approach	p. 255
Introduction	p. 255
A Mathematical Model	p. 258
Numerical Results	p. 266
Conclusions	p. 269
References	p. 272
M for Invasion: Morphology, Mutation and the Microenvironment	p. 277
Introduction	p. 277
Hybrid Discrete-Continuum Technique	p. 281
The Continuum Model	p. 282
The Discrete Model	p. 283
Morphology and Metabolism	p. 287
Mutation and the Microenvironment	p. 290
Conclusion	p. 293
Discussion	p. 294
References	p. 295
Methods of Stochastic Geometry, and Related Statistical Problems in the Analysis and Therapy of Tumour Growth and Tumour-Driven Angiogenesis	p. 299
Introduction	p. 299
Stochastic Geometry	p. 303
The Hazard Function	p. 304
Generalized Densities	p. 307
Approximation of Mean Densities	p. 311
Mean Densities of Stochastic Tessellations	p. 313
Birth-and-Growth Processes	p. 315
Interaction with Underlying Fields	p. 317
Statistical Methods for Fibre Systems	p. 325
Application of the Estimators to Simulated Angiogenic Processes	p. 329
References	p. 332
Mathematical Modelling of Breast Carcinogenesis, Treatment with Surgery and Radiotherapy, and Local Recurrence	p. 337
Introduction	p. 337
A Continuous Mathematical Model of Breast Cancer Initiation and Progression	p. 342
Modelling Surgery and Radiotherapy	p. 350
Discussion	p. 354
References	p. 357
Predictive Models in Tumor Immunology	p. 363
Introduction	p. 363
Tumor Immunology	p. 364
Cancer Immunoprevention	p. 368
The Model	p. 372
The Optimal Scheduling Problem	p. 375
Conclusion	p. 381
References	p. 383
Dynamically Adaptive Tumour-Induced Angiogenesis: The Impact of Flow on the Developing Capillary Plexus	p. 385
Introduction	p. 385
Angiogenesis and Perfusion: A Brief Review	p. 387
Dynamic Adaptive Tumour-Induced Angiogenesis (DATIA)	p. 391
Chemotherapy Delivery via Dynamically Adaptive Capillary Beds	p. 400
Discussion and Conclusions	p. 404
Capillary Growth Details	p. 406
Adaptive Flow Modelling Details	p. 410
References	p. 413
Dynamic Irregular Patterns and Invasive Wavefronts: The Control of Tumour Growth by Cytotoxic T-Lymphocytes	p. 419
Introduction	p. 419
The Mathematical Model	p. 420
Numerical Simulation Results	p. 425
Linear Stability Analysis	p. 431
Bifurcation Analysis	p. 434
Travelling Waves: Simulation and Analysis	p. 435
Finite Element Method Simulations	p. 441
Discussion and Conclusions	p. 443
References	p. 445
Multiscale Modelling of Solid Tumour Growth	p. 449
Introduction	p. 449
Metabolic Changes During Carcinogenesis	p. 451
Vascular Tumour Growth	p. 454
Colorectal Cancer	p. 463
Discussion	p. 469
References	p. 470
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Selected Topics in Cancer Modeling

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