The book aims to provide an introduction to mathematical models that describe the dynamics of tumor growth and the evolution of tumor cells. It can be used as a textbook for advanced undergraduate or graduate courses, and also serves as a reference book for researchers. The book has a strong evolutionary component and reflects the viewpoint that cancer can be understood rationally through a combination of mathematical and biological tools. It can be used both by mathematicians and biologists. Mathematically, the book starts with relatively simple ordinary differential equation models, and subsequently explores more complex stochastic and spatial models. Biologically, the book starts with explorations of the basic dynamics of tumor growth, including competitive interactions among cells, and subsequently moves on to the evolutionary dynamics of cancer cells, including scenarios of cancer initiation, progression, and treatment. The book finishes with a discussion of advanced topics, which describe how some of the mathematical concepts can be used to gain insights into a variety of questions, such as epigenetics, telomeres, gene therapy, and social interactions of cancer cells.

Preface vii

1 Teaching guide 1

1.1 How to use this book 1

1.2 A sample syllabus for a Mathematics course 2

1.3 A sample syllabus for a Biology course 3

2 Cancer and somatic evolution 5

2.1 What is cancer? 5

2.2 Basic cancer genetics 6

2.3 Multi-stage carcinogenesis and colon cancer 8

2.4 Genetic instability 10

2.5 Barriers to cancer progression: importance of the microenvironment 12

2.6 Cellular hierarchies in cancer 15

2.7 Genetic and epigenetic changes 15

2.8 Evolutionary theory and Darwinian selection 17

3 Mathematical modeling of tumorigenesis 19

3.1 Ordinary differential equations 20

3.2 Extensions of ODE modeling 22

      3.2.1 Optimal control 22

      3.2.2 ODEs and cancer epidemiology 23

3.3 Partial differential equations 23

3.4 Stochastic modeling 25

3.5 Cellular automaton models 28

3.6 Hybrid and multiscale modeling 30

Basic growth dynamics and deterministic models 33

4 Single species growth 35

4.1 Exponential growth 35

4.2 Surface growth 37

4.3 Sigmoidal growth 39

      4.3.1 Logistic growth 39

      4.3.2 Other sigmoidal laws 41

4.4 Atypical growth 43

4.5 Multistep growth 44

4.6 Conclusions 44

5 Two-species competition dynamics 47

5.1 Logistic growth of two species and the basic dynamics of competition 47

5.2 Two-species dynamics: the axiomatic approach 50

5.3 Summary 55

6 Competition between genetically stable and unstable cells 57

6.1 Competition dynamics 58

6.2 Competition dynamics and cancer evolution 63

      6.2.1 A quasispecies model 63

      6.2.2 Strong apoptosis 71

      6.2.3 Weak apoptosis 74

6.3 Overview of the insights obtained so far 76

6.4 Can competition be reversed by chemotherapy? 77

6.5 Summary 79

7 Chromosomal instability and tumor growth 81

7.1 The effect of chromosome loss on the generation of cancer 82

7.2 Calculating the optimal rate of chromosome loss 84

7.3 The optimal rate of LOH: a time-dependent problem 89

      7.3.1 Formulation of the time-dependent problem 91

      7.3.2 Mathematical apparatus 94

      7.3.3 The optimal strategy for cancer 98

7.4 The bigger picture 100

      7.4.1 Does cancer solve an optimization problem? 102

      7.4.2 Summary 102

8 Angiogenesis, inhibitors, promoters, and spatial growth 105

8.1 Model 1: Angiogenesis inhibition induces cell death 107

8.2 Model 2: Angiogenesis inhibition prevents tumor cell division 112

      8.2.1 Linear stability analysis of the ODEs 113

      8.2.2 Conclusions from the linear analysis 115

8.3 Spread of tumors across space 115

      8.3.1 Turing stability analysis 116

      8.3.2 Stationary periodic solutions 119

      8.3.3 Biological implications and numerical simulations 120

8.4 Somatic cancer evolution and progression 121

8.5 Summary and clinical implications 127

Evolutionary dynamics and stochastic models 131

9 Evolutionary dynamics of tumor initiation through oncogenes: the gain-of-function model 133

9.1 Introduction 133

9.2 Mutation-selection diagrams and the stochastic Moran process 135

9.3 Analysis 137

      9.3.1 The method of differential equations 138

      9.3.2 The probability of absorption 139

9.4 Probability and timing of mutant fixation 140

      9.4.1 The approximation of “almost absorbing” states and the growth of mutants 143

      9.4.2 Nearly-deterministic regime 144

9.5 Summary 145

10 Evolutionary dynamics of tumor initiation through tumor-suppressor genes: the loss-of-function model and stochastic tunneling 147

10.1 Introduction 147

10.2 Process description and the mutation-selection diagram 148

10.3 Three regimes: a two-step process, stochastic tunneling, and a nearly-deterministic regime 150

10.4 The transition matrix 151

10.5 Mathematical theory 152

      10.5.1 The Kolmogorov forward equation in the absence of intermediate mutant fixation 152

      10.5.2 The probability generating function 153

      10.5.3 The method of characteristics and the Riccati equation 154

      10.5.4 Tunneling for disadvantageous, neutral, and advantageous intermediate mutants 156

      10.5.5 Genuine two-step process vs tunneling 157

      10.5.6 Time-scales of the process 157

      10.5.7 Neutral intermediate mutants 158

      10.5.8 Disadvantageous intermediate mutants 160

      10.5.9 Advantageous intermediate mutants 161

10.6 Dynamics of loss-of-function mutations 162

      10.6.1 The genuine two-step processes 162

      10.6.2 Tunneling 163

      10.6.3 Nearly deterministic regime 165

      10.6.4 Disadvantageous, neutral and advantageous intermediate mutants 165

      10.6.5 The role of the population size 166

10.7 Summary 168

11 Microsatellite and chromosomal instability in sporadic and familial colorectal cancers 171

11.1 Some biological facts about genetic instability in colon cancer 173

11.2 A model for the initiation of sporadic colorectal cancers 173

      11.2.1 The first model of the APC gene inactivation: no instabilities 173

      11.2.2 Colorectal cancer and chromosomal instability 179

11.3 Sporadic colorectal cancers, CIN and MSI 184

11.4 FAP 189

11.5 HNPCC 191

11.6 Summary 192

12 Evolutionary dynamics in hierarchical populations 197

12.1 Introduction 197

12.2 Types of stem cells divisions 198

12.3 The set-up 200

12.4 Methodology 202

      12.4.1 Analysis of the Moran process 202

      12.4.2 Numerical simulations 208

12.5 Generation of mutations in a hierarchical population 210

      12.5.1 Tunneling rates 210

      12.5.2 Double-hit mutants are produced slower under symmetric compared to asymmetric divisions 211

      12.5.3 Comparison with the homogeneous model 213

      12.5.4 The optimal fraction of stem cells 214

      12.5.5 Do mutations in TA cells produce double-mutants? 216

12.6 Biological discussion 218

      12.6.1 Symmetric divisions can have a cancer-delaying effect 219

      12.6.2 Can TA cells create double-hit mutants? 221

      12.6.3 Cancer stem cell hypothesis 222

12.7 Summary 223

13 Spatial evolutionary dynamics of tumor initiation 225

13.1 Introduction 225

13.2 1D spatial Moran process 226

13.3 Two-species dynamics 228

      13.3.1 Preliminaries 228

      13.3.2 Probability of mutant fixation 229

13.4 Three-species dynamics 231

      13.4.1 Calculating the tunneling rate by the doublystochastic approximation 231

      13.4.2 Limiting cases and the tunneling rate approximations 234

      13.4.3 When is tunneling important? 236

13.5 Dynamics of mutant generation 238

      13.5.1 Gain-of-function mutations: a two-species problem 238

      13.5.2 Loss-of-function mutations: a three-species problem 239

      13.5.3 Definition of neutrality 241

      13.5.4 Three-species dynamic: a comparison with the space-free model 241

13.6 Outlook 244

14 Complex tumor dynamics in space 247

14.1 Introduction 247

14.2 Complex traits and fitness valleys 248

14.3 The Moran process 249

      14.3.1 Spatial restriction accelerates evolution 250

      14.3.2 Dependence on parameters 252

14.4 The contact process 258

      14.4.1 The steady-state density of cells 259

      14.4.2 Complex effects of spatial restriction 263

      14.4.3 Parameter dependencies 264

14.5 Advantageous intermediate mutants 265

14.6 Summary and discussion 268

15 Stochastic modeling of cancer growth, treatment, and resistance generation 275

15.1 Introduction 275

15.2 The basic model of cancer growth and generation of mutations 276

      15.2.1 The concept: a birth-death process with mutations 276

      15.2.2 Summary of all the probabilities 277

      15.2.3 Stochastic description: the example of one mutation 278

      15.2.4 The probability generating function description 280

      15.2.5 The method of characteristics 281

15.3 Application to cancer treatment and generation of resistance 282

      15.3.1 The framework 283

      15.3.2 Treatment regimes 285

      15.3.3 Probability of extinction and treatment success 286

      15.3.4 Symmetric coefficients 287

15.4 Example: the case of two drugs 288

      15.4.1 Equations for the moments 289

      15.4.2 Equations for the characteristics 290

15.5 Mutant production before and during treatment 291

      15.5.1 General theory 291

      15.5.2 The case of one drug 294

      15.5.3 The case of two drugs 297

15.6 Outlook 299

16 Evolutionary dynamics of drug resistance in chronic myeloid leukemia 301

16.1 Biology of CML 302

16.2 Therapy and targeted small molecule inhibitors 302

16.3 The computational framework 305

16.4 When do resistant cells emerge? 307

16.5 Cancer turnover and the evolution of resistance 308

16.6 Combination therapy and the prevention of resistance 309

16.7 Parameters and CML 312

16.8 Tumor architecture and tumor stem cells 314

16.9 Short-term versus long-term treatment strategies 317

16.10 Cross-resistance and combination therapy 319

16.11 Combination versus cyclic sequential treatment 324

16.12 Summary 328

Advanced topics 331

17 Evolutionary dynamics of stem-cell driven tumor growth 333

17.1 The model 334

17.2 Evolutionary dynamics in ODE models 336

17.3 Evolutionary dynamics in a stochastic, spatial model 339

17.4 Predicted versus observed tumor growth patterns 340

17.5 The order of phenotypic transitions 342

17.6 Summary 345

18 Tumor growth kinetics and disease progression 347

18.1 Cell death and mutant generation 349

18.2 Does PCD protect against cancer? 353

18.3 Cell turnover and pathology 356

18.4 Conclusions 357

19 Epigenetic changes and the rate of DNA methylation 359

19.1 De novo methylation kinetics in CIMP and non-CIMP cells following demethylation 361

19.2 Quantifying the de novo methylation kinetics 363

19.3 Interpreting the results with the help of a mathematical model 366

19.4 De novo methylation kinetics in highly methylated cells 371

19.5 Importance of experimental verification 372

19.6 Summary 372

20 Telomeres and cancer protection 375

20.1 Lineages and replication limits 377

20.2 Model analysis 380

      20.2.1 Population turnover and replication capacity: analytical results 380

      20.2.2 Agent-based model 387

      20.2.3 Decrease in the replication capacity of stem cells 388

20.3 Tissue architecture and the development of cancer 389

20.4 Theory and observed tissue architecture 398

20.5 Summary 401

21 Gene therapy and oncolytic virus therapy 403

21.1 A basic ordinary differential equation model 404

      21.1.1 Non-replicating viruses 407

      21.1.2 Replicating viruses 408

21.2 Different mathematical formulations and the robustness of results 411

21.3 A spatially explicit model of oncolytic virus dynamics 412

      21.3.1 Initial virus growth patterns 413

      21.3.2 Growth patterns and the extinction of cells 415

21.4 Experimentally observed patterns of virus spread 424

21.5 Conclusions 428

22 Immune responses, tumor growth, and therapy 431

22.1 Some facts about immune responses 433

22.2 The model 435

22.3 Properties of equilibria and parameter dependencies 439

22.4 Immunity versus tolerance 442

22.5 Cancer initiation 443

22.6 Tumor dormancy, evolution, and progression 443

22.7 Immunotherapy against cancers 446

22.8 Case study: immune responses and the treatment for chronic myeloid leukemia 449

22.9 Role of immunity and resistance in driving treatment dynamics 452

22.10 Possible role of immune stimulation for long-term remission 456

22.11 Summary 457

23 Towards higher complexities: social interactions 459

23.1 Microenvironment 459

23.2 Cooperation and division of labor 460

23.3 Conclusion 462

Bibliography 463

Index 511

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