copy to come

Chapman & Hall/CRCMathematical and Computational Biology Series

Chapman & Hall/CRCMathematical and Computational Biology Series

C990X

w w w . c r c p r e s s . c o m

INTRODUCTION TO MATHEMATICAL ONCOLOGY

YANG KUANGJOHN D. NAGY

STEFFEN E. EIKENBERRY

KUANG • NAGYEIKENBERRY

INTRODUCTION TO MATHEMATICAL ONCOLOGY

C990X_cover.indd 1 12/15/15 12:16 PM

Yang Kuang, John D. Nagy and Steffen E. Eikenberryc©2015

Introduction toMathematical Oncology[December 16, 2015]

CRC PRESS

Boca Raton London New York Washington, D.C.

Contents

Preface ix

1 Introduction to Theory in Medicine 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Disease . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 A brief survey of trends in health and disease . . . . . . . . . 41.4 The scientific basis of medicine . . . . . . . . . . . . . . . . . 101.5 Aspects of the medical art . . . . . . . . . . . . . . . . . . . 111.6 Key scientific concepts in mathematical medicine . . . . . . . 12

1.6.1 Genetics . . . . . . . . . . . . . . . . . . . . . . . . . . 131.6.2 Evolution . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.7 Pathology—where science and art meet . . . . . . . . . . . . 21References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2 Introduction to Cancer Modeling 252.1 Introduction to cancer dynamics . . . . . . . . . . . . . . . . 252.2 Historical roots . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.1 The von Bertalanffy growth model . . . . . . . . . . . 262.2.2 Gompertzian growth . . . . . . . . . . . . . . . . . . . 28

2.3 Applications of Gompertz and von Bertalanffy models . . . . 332.4 A more general approach . . . . . . . . . . . . . . . . . . . . 392.5 Mechanistic insights from simple tumor models . . . . . . . . 422.6 Sequencing of chemotherapeutic and surgical treatments . . 442.7 Stability of steady states for ODEs . . . . . . . . . . . . . . 482.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.9 Projects and open questions . . . . . . . . . . . . . . . . . . 58

2.9.1 Mathematical open questions . . . . . . . . . . . . . . 602.9.2 Tumor growth with a time delay . . . . . . . . . . . . 612.9.3 Tumor growth with cell diffusion . . . . . . . . . . . . 61

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3 Spatially Structured Tumor Growth 653.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.2 The simplest spatially structured tumor model . . . . . . . . 66

3.2.1 Model formulation . . . . . . . . . . . . . . . . . . . . 663.2.2 Equilibrium nutrient profile with no necrosis . . . . . 693.2.3 Size of the necrotic core . . . . . . . . . . . . . . . . . 71

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3.3 Spheroid dynamics and equilibrium size . . . . . . . . . . . . 723.4 Greenspan’s seminal model . . . . . . . . . . . . . . . . . . . 77

3.4.1 The Greenspan model . . . . . . . . . . . . . . . . . . 783.4.2 Threshold for quiescence . . . . . . . . . . . . . . . . . 803.4.3 Growth dynamics of the Greenspan model . . . . . . . 81

3.5 Testing Greenspan’s model . . . . . . . . . . . . . . . . . . . 833.6 Sherratt-Chaplain model for avascular tumor growth . . . . 84

3.6.1 MATLABR© file for Figure 3.6 . . . . . . . . . . . . . . 863.6.2 Minimum wave speed . . . . . . . . . . . . . . . . . . 88

3.7 A model of in vitro glioblastoma growth . . . . . . . . . . . 893.7.1 Model formulation . . . . . . . . . . . . . . . . . . . . 893.7.2 Traveling wave system properties . . . . . . . . . . . . 913.7.3 Existence of traveling wave solutions . . . . . . . . . . 92

3.8 Derivation of one-dimensional conservation equation . . . . . 963.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983.10 Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

3.10.1 Nutrient limitation induced quiescence . . . . . . . . . 1033.10.2 Inhibitor generated by living cells . . . . . . . . . . . . 1033.10.3 Glioblastoma growth in a Petri dish or in vivo . . . . 1033.10.4 A simple model of tumor-host interface . . . . . . . . 103

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4 Physiologically Structured Tumor Growth 1074.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.2 Construction of the cell-size structured model . . . . . . . . 1084.3 No quiescence, some intuition . . . . . . . . . . . . . . . . . 1124.4 Basic behavior of the model . . . . . . . . . . . . . . . . . . 1144.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5 Prostate Cancer: PSA, AR, and ADT Dynamics 1235.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235.2 Models of PSA kinetics . . . . . . . . . . . . . . . . . . . . . 124

5.2.1 Vollmer et al. model . . . . . . . . . . . . . . . . . . . 1255.2.2 Prostate cancer volume . . . . . . . . . . . . . . . . . 125

5.3 Dynamical models . . . . . . . . . . . . . . . . . . . . . . . . 1275.3.1 Swanson et al. model . . . . . . . . . . . . . . . . . . 1275.3.2 Vollmer and Humphrey model . . . . . . . . . . . . . 1305.3.3 PSA kinetic parameters: Conclusions from dynamical

models . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.4 Androgens and the evolution of prostate cancer . . . . . . . 136

5.4.1 Evolutionary role . . . . . . . . . . . . . . . . . . . . . 1375.4.2 Intracellular AR kinetics model . . . . . . . . . . . . . 1385.4.3 Basic dynamics of the AR kinetics model . . . . . . . 140

5.5 Prostate growth mediated by androgens . . . . . . . . . . . . 141

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5.6 Evolution and selection for elevated AR expression . . . . . . 146

5.6.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

5.6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 147

5.7 Jackson ADT model . . . . . . . . . . . . . . . . . . . . . . . 148

5.8 The Ideta et al. ADT model . . . . . . . . . . . . . . . . . . 152

5.9 Predictions and limitations of current ADT models . . . . . 155

5.10 An immunotherapy model for advanced prostate cancer . . . 156

5.11 Other prostate models . . . . . . . . . . . . . . . . . . . . . . 161

5.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

5.13 Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

5.13.1 The epithelial-vascular interface and serum PSA . . . 167

5.13.2 A clinical algorithm based on a dynamical model . . . 168

5.13.3 An extension of Vollmer and Humphrey’s model . . . 168

5.13.4 Androgens positively regulating AI cell proliferation . 169

5.13.5 Combining androgen ablation with other therapies . . 169

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

6 Resource Competition and Cell Quota in Cancer Models 175

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

6.2 A cell-quota based population growth model . . . . . . . . . 176

6.3 From Droop cell-quota model to logistic equation . . . . . . 180

6.4 Cell-quota models for prostate cancer hormone treatment . . 183

6.4.1 Preliminary model . . . . . . . . . . . . . . . . . . . . 183

6.4.2 Final model . . . . . . . . . . . . . . . . . . . . . . . . 184

6.4.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . 185

6.4.4 Predictions . . . . . . . . . . . . . . . . . . . . . . . . 187

6.5 Other cell-quota models for prostate cancer hormone treatment 188

6.5.1 Basic model . . . . . . . . . . . . . . . . . . . . . . . . 188

6.5.2 Long-term competition in the basic model . . . . . . . 189

6.5.3 Intermittent androgen deprivation . . . . . . . . . . . 190

6.5.4 Cell quota with chemical kinetics . . . . . . . . . . . . 192

6.6 Stoichiometry and competition in cancer . . . . . . . . . . . 193

6.6.1 KNE model . . . . . . . . . . . . . . . . . . . . . . . . 194

6.6.2 Predictions . . . . . . . . . . . . . . . . . . . . . . . . 196

6.7 Mathematical analysis of a simplified KNE model . . . . . . 198

6.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

6.9 Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

6.9.1 Beyond the KNE model . . . . . . . . . . . . . . . . . 207

6.9.2 Iodine and thyroid cancer . . . . . . . . . . . . . . . . 208

6.9.3 Iron and microbes . . . . . . . . . . . . . . . . . . . . 208

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

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7 Natural History of Clinical Cancer 2137.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2137.2 Conceptual models for the natural history of breast cancer:

Halsted vs. Fisher . . . . . . . . . . . . . . . . . . . . . . . . 2147.2.1 Surgery and the Halsted model . . . . . . . . . . . . . 2157.2.2 Systemic chemotherapy and the Fisher model . . . . . 2177.2.3 Integration of Halsted and Fisher concepts: Surgery

with adjuvant chemotherapy . . . . . . . . . . . . . . 2187.3 A simple model for breast cancer growth kinetics . . . . . . . 219

7.3.1 Speer model: Irregular Gompertzian growth . . . . . . 2207.3.2 Calibration and predictions of the Speer model . . . . 2217.3.3 Limitations of the Speer approach . . . . . . . . . . . 222

7.4 Metastatic spread and distant recurrence . . . . . . . . . . . 2237.4.1 The Yorke et al. model . . . . . . . . . . . . . . . . . 2237.4.2 Parametrization and predictions of the Yorke model . 2267.4.3 Limitations of the Yorke approach . . . . . . . . . . . 2277.4.4 Iwata model . . . . . . . . . . . . . . . . . . . . . . . . 2277.4.5 Thames model . . . . . . . . . . . . . . . . . . . . . . 2317.4.6 Other models . . . . . . . . . . . . . . . . . . . . . . . 231

7.5 Tumor dormancy hypothesis . . . . . . . . . . . . . . . . . . 2317.6 The hormonal environment and cancer progression . . . . . . 2357.7 The natural history of breast cancer and screening protocols 236

7.7.1 Pre-clinical breast cancer and DCIS . . . . . . . . . . 2387.7.2 CISNET program . . . . . . . . . . . . . . . . . . . . 2387.7.3 Continuous growth models . . . . . . . . . . . . . . . 2407.7.4 Conclusions and optimal screening strategies . . . . . 245

7.8 Cancer progression and incidence curves . . . . . . . . . . . . 2467.8.1 Basic multi-hit model . . . . . . . . . . . . . . . . . . 2467.8.2 Two-hit models . . . . . . . . . . . . . . . . . . . . . . 2487.8.3 The case of colorectal cancer . . . . . . . . . . . . . . 2517.8.4 Multiple clonal expansions . . . . . . . . . . . . . . . . 2547.8.5 Smoking and lung cancer incidence . . . . . . . . . . . 2547.8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 255

7.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

8 Evolutionary Ecology of Cancer 2658.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2658.2 Necrosis: What causes the tumor ecosystem to collapse? . . 266

8.2.1 Necrosis in multicell spheroids . . . . . . . . . . . . . 2688.2.2 Necrosis in tumor cords . . . . . . . . . . . . . . . . . 2708.2.3 Diffusion limitation in ductal carcinoma in situ . . . . 2728.2.4 Necrosis caused by mechanical disruption of cells . . . 2738.2.5 Necrosis from local acidosis . . . . . . . . . . . . . . . 2768.2.6 Necrosis due to local ischemia . . . . . . . . . . . . . . 277

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8.3 What causes cell diversity within malignant neoplasia? . . . 2808.3.1 Causes of Type I diversity . . . . . . . . . . . . . . . . 2808.3.2 Causes of Type II diversity . . . . . . . . . . . . . . . 286

8.4 Synthesis: Competition, natural selection and necrosis . . . . 2958.5 Necrosis and the evolutionary dynamics of metastatic disease 297

8.5.1 Pre-metastatic selection hypothesis . . . . . . . . . . . 2988.5.2 Reproductive fitness and export probability . . . . . . 3008.5.3 Tumor self-seeding . . . . . . . . . . . . . . . . . . . . 301

8.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3028.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

9 Models of Chemotherapy 3139.1 Dose-response curves in chemotherapy . . . . . . . . . . . . . 314

9.1.1 Simple models . . . . . . . . . . . . . . . . . . . . . . 3149.1.2 Concentration, time, and cyotoxicity plateaus . . . . . 3179.1.3 Shoulder region . . . . . . . . . . . . . . . . . . . . . . 3179.1.4 Pharmacodynamics for antimicrobials . . . . . . . . . 318

9.2 Models for in vitro drug uptake and cytotoxicity . . . . . . . 3189.2.1 Models for cistplatin uptake and intracellular pharma-

cokinetics . . . . . . . . . . . . . . . . . . . . . . . . . 3199.2.2 Paclitaxel uptake and intracellular pharmacokinetics . 320

9.3 Pharmacokinetics . . . . . . . . . . . . . . . . . . . . . . . . 3239.4 The Norton-Simon hypothesis and the Gompertz model . . . 327

9.4.1 Gompertzian model for human breast cancer growth . 3289.4.2 The Norton-Simon hypothesis and dose-density . . . . 3289.4.3 Formal Norton-Simon model . . . . . . . . . . . . . . 3299.4.4 Intensification and maintenance regimens . . . . . . . 3319.4.5 Clinical implications and results . . . . . . . . . . . . 3319.4.6 Depletion of the growth fraction . . . . . . . . . . . . 332

9.5 Modeling the development of drug resistance . . . . . . . . . 3339.5.1 Luria-Delbruck fluctuation analysis . . . . . . . . . . . 3339.5.2 The Goldie-Coldman model . . . . . . . . . . . . . . . 3389.5.3 Extensions of Goldie-Coldman and alternating therapy 3409.5.4 The Monro-Gaffney model and palliative therapy . . . 3479.5.5 The role of host physiology . . . . . . . . . . . . . . . 350

9.6 Heterogeneous populations: The cell cycle . . . . . . . . . . 3519.6.1 The Smith-Martin conceptual model . . . . . . . . . . 3519.6.2 A delay differential model of the cell cycle . . . . . . . 3539.6.3 Age-structured models for the cell cycle . . . . . . . . 3559.6.4 More general sensitivity and resistance . . . . . . . . . 358

9.7 Drug transport and the spatial tumor environment . . . . . . 3599.7.1 Solute transport across tumor capillaries . . . . . . . . 3599.7.2 Fluid flow in tumors . . . . . . . . . . . . . . . . . . . 3619.7.3 Tumor spheroid . . . . . . . . . . . . . . . . . . . . . . 362

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9.7.4 Tumor cord framework . . . . . . . . . . . . . . . . . . 3629.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

10 Major Anticancer Chemotherapies 37110.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 37110.2 Alkylating and alkalating-like agents . . . . . . . . . . . . . . 372

10.2.1 Nitrogen mustards . . . . . . . . . . . . . . . . . . . . 37310.2.2 Platinum-based drugs . . . . . . . . . . . . . . . . . . 37510.2.3 Nitrosoureas . . . . . . . . . . . . . . . . . . . . . . . 37610.2.4 Methylating agents . . . . . . . . . . . . . . . . . . . . 377

10.3 Antitumor antibiotics . . . . . . . . . . . . . . . . . . . . . . 37710.3.1 Anthracyclines . . . . . . . . . . . . . . . . . . . . . . 37810.3.2 Mitomycin-C . . . . . . . . . . . . . . . . . . . . . . . 37810.3.3 Bleomycins . . . . . . . . . . . . . . . . . . . . . . . . 378

10.4 Antimetabolites . . . . . . . . . . . . . . . . . . . . . . . . . 38010.5 Mitotic inhibitors . . . . . . . . . . . . . . . . . . . . . . . . 380

10.5.1 Taxanes . . . . . . . . . . . . . . . . . . . . . . . . . . 38010.5.2 Vinca alkaloids . . . . . . . . . . . . . . . . . . . . . . 382

10.6 Non-cytotoxic and targeted therapies . . . . . . . . . . . . . 383References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

11 Radiation Therapy 38911.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 38911.2 Molecular mechanisms . . . . . . . . . . . . . . . . . . . . . . 393

11.2.1 Ions and radical reactions . . . . . . . . . . . . . . . . 39411.2.2 Oxygen status . . . . . . . . . . . . . . . . . . . . . . 39711.2.3 The four R’s . . . . . . . . . . . . . . . . . . . . . . . 398

11.3 Classical target-hit theory . . . . . . . . . . . . . . . . . . . 39811.4 Lethal DNA misrepair . . . . . . . . . . . . . . . . . . . . . . 400

11.4.1 Repair-misrepair model . . . . . . . . . . . . . . . . . 40011.4.2 Lethal-potentially lethal model . . . . . . . . . . . . . 40611.4.3 Parametrization . . . . . . . . . . . . . . . . . . . . . 407

11.5 Saturable and enzymatic repair . . . . . . . . . . . . . . . . 40811.5.1 Haynes model . . . . . . . . . . . . . . . . . . . . . . . 40911.5.2 Goodhead model . . . . . . . . . . . . . . . . . . . . . 41011.5.3 General saturable-repair model . . . . . . . . . . . . . 411

11.6 Kinetics of damage repair . . . . . . . . . . . . . . . . . . . . 41111.7 The LQ model and dose fractionation . . . . . . . . . . . . . 41411.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

11.8.1 Tumor cure probability . . . . . . . . . . . . . . . . . 42011.8.2 Regrowth . . . . . . . . . . . . . . . . . . . . . . . . . 42011.8.3 Hypoxia . . . . . . . . . . . . . . . . . . . . . . . . . . 42311.8.4 Radiation with chemotherapy . . . . . . . . . . . . . . 423

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

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12 Chemical Kinetics 42912.1 Introduction and the law of mass action . . . . . . . . . . . . 429

12.1.1 Dissociation constant . . . . . . . . . . . . . . . . . . 43212.2 Enzyme kinetics . . . . . . . . . . . . . . . . . . . . . . . . . 432

12.2.1 Equilibrium approximation . . . . . . . . . . . . . . . 43312.3 Quasi-steady-state approximation . . . . . . . . . . . . . . . 434

12.3.1 Turnover number . . . . . . . . . . . . . . . . . . . . . 43512.3.2 Specificity constant . . . . . . . . . . . . . . . . . . . . 43512.3.3 Lineweaver-Burk equation . . . . . . . . . . . . . . . . 435

12.4 Enzyme inhibition . . . . . . . . . . . . . . . . . . . . . . . . 43712.4.1 Competitive inhibition . . . . . . . . . . . . . . . . . . 43712.4.2 Allosteric inhibition . . . . . . . . . . . . . . . . . . . 438

12.5 Hemoglobin and the Hill equation . . . . . . . . . . . . . . . 44012.6 Monod-Wyman-Changeux model . . . . . . . . . . . . . . . . 442References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

13 Epilogue: Toward a Quantitative Theory of Oncology 447References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

Index 454