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Introduction to Mathematical Oncology

Book Description

Introduction to Mathematical Oncology presents biologically well-motivated and mathematically tractable models that facilitate both a deep understanding of cancer biology and better cancer treatment designs. It covers the medical and biological background of the diseases, modeling issues, and existing methods and their limitations. The authors introduce mathematical and programming tools, along with analytical and numerical studies of the models. They also develop new mathematical tools and look to future improvements on dynamical models.

After introducing the general theory of medicine and exploring how mathematics can be essential in its understanding, the text describes well-known, practical, and insightful mathematical models of avascular tumor growth and mathematically tractable treatment models based on ordinary differential equations. It continues the topic of avascular tumor growth in the context of partial differential equation models by incorporating the spatial structure and physiological structure, such as cell size. The book then focuses on the recent active multi-scale modeling efforts on prostate cancer growth and treatment dynamics. It also examines more mechanistically formulated models, including cell quota-based population growth models, with applications to real tumors and validation using clinical data. The remainder of the text presents abundant additional historical, biological, and medical background materials for advanced and specific treatment modeling efforts.

Extensively classroom-tested in undergraduate and graduate courses, this self-contained book allows instructors to emphasize specific topics relevant to clinical cancer biology and treatment. It can be used in a variety of ways, including a single-semester undergraduate course, a more ambitious graduate course, or a full-year sequence on mathematical oncology.

Table of Contents

  1. Preliminaries
  2. Preface
  3. Chapter 1 Introduction to Theory in Medicine
    1. 1.1 Introduction
    2. 1.2 Disease
    3. 1.3 A brief survey of trends in health and disease
    4. 1.4 The scientific basis of medicine
    5. 1.5 Aspects of the medical art
    6. 1.6 Key scientific concepts in mathematical medicine
      1. 1.6.1 Genetics
      2. 1.6.2 Evolution
    7. 1.7 Pathology—where science and art meet
    8. References
      1. Figure 1.1
      2. Figure 1.2
      3. Figure 1.3
      4. Figure 1.4
      5. Figure 1.5
  4. Chapter 2 Introduction to Cancer Modeling
    1. 2.1 Introduction to cancer dynamics
    2. 2.2 Historical roots
      1. 2.2.1 The von Bertalanffy growth model
      2. 2.2.2 Gompertzian growth
    3. 2.3 Applications of Gompertz and von Bertalanffy models
    4. 2.4 Amore general approach
    5. 2.5 Mechanistic insights from simple tumor models
    6. 2.6 Sequencing of chemotherapeutic and surgical treatments
    7. 2.7 Stability of steady states for ODEs
    8. 2.8 Exercises
    9. 2.9 Projects and open questions
      1. 2.9.1 Mathematical open questions
      2. 2.9.2 Tumor growth with a time delay
      3. 2.9.3 Tumor growth with cell diffusion
    10. References
      1. Figure 2.1
      2. Figure 2.2
      3. Figure 2.3
      4. Figure 2.4
      5. Figure 2.5
      1. Table 2.1
  5. Chapter 3 Spatially Structured Tumor Growth
    1. 3.1 Introduction
    2. 3.2 The simplest spatially structured tumor model
      1. 3.2.1 Model formulation
      2. 3.2.2 Equilibrium nutrient profile with no necrosis
      3. 3.2.3 Size of the necrotic core
    3. 3.3 Spheroid dynamics and equilibrium size
    4. 3.4 Greenspan’s seminal model
      1. 3.4.1 The Greenspan model
      2. 3.4.2 Threshold for quiescence
      3. 3.4.3 Growth dynamics of the Greenspan model
    5. 3.5 Testing Greenspan’s model
    6. 3.6 Sherratt-Chaplain model for avascular tumor growth
      1. 3.6.1 MATLAB® file for Figure 3.6
      2. 3.6.2 Minimum wave speed
    7. 3.7 A model of in vitro glioblastoma growth
      1. 3.7.1 Model formulation
      2. 3.7.2 Traveling wave system properties
      3. 3.7.3 Existence of traveling wave solutions
    8. 3.8 Derivation of one-dimensional conservation equation
    9. 3.9 Exercises
    10. 3.10 Projects
      1. 3.10.1 Nutrient limitation induced quiescence
      2. 3.10.2 Inhibitor generated by living cells
      3. 3.10.3 Glioblastoma growth in a Petri dish or in vivo
      4. 3.10.4 A simple model of tumor-host interface
    11. References
      1. Figure 3.1
      2. Figure 3.2
      3. Figure 3.3
      4. Figure 3.4
      5. Figure 3.5
      6. Figure 3.6
      7. Figure 3.7
      8. Figure 3.8
      9. Figure 3.9
  6. Chapter 4 Physiologically Structured Tumor Growth
    1. 4.1 Introduction
    2. 4.2 Construction of the cell-size structured model
    3. 4.3 No quiescence, some intuition
    4. 4.4 Basic behavior of the model
    5. 4.5 Exercises
    6. References
      1. Figure 4.1
      2. Figure 4.2
      3. Figure 4.3
  7. Chapter 5 Prostate Cancer: PSA, AR, and ADT Dynamics
    1. 5.1 Introduction
    2. 5.2 Models of PSA kinetics
      1. 5.2.1 Vollmer et al. model
      2. 5.2.2 Prostate cancer volume
    3. 5.3 Dynamical models
      1. 5.3.1 Swanson et al. model
      2. 5.3.2 Vollmer and Humphrey model
      3. 5.3.3 PSA kinetic parameters: Conclusions from dynamical models
    4. 5.4 Androgens and the evolution of prostate cancer
      1. 5.4.1 Evolutionary role
      2. 5.4.2 Intracellular AR kinetics model
      3. 5.4.3 Basic dynamics of the AR kinetics model
    5. 5.5 Prostate growth mediated by androgens
    6. 5.6 Evolution and selection for elevated AR expression
      1. 5.6.1 Model
      2. 5.6.2 Results
    7. 5.7 Jackson ADT model
    8. 5.8 The Ideta et al. ADT model
    9. 5.9 Predictions and limitations of current ADT models
    10. 5.10 An immunotherapy model for advanced prostate cancer
    11. 5.11 Other prostate models
    12. 5.12 Exercises
    13. 5.13 Projects
      1. 5.13.1 The epithelial-vascular interface and serum PSA
      2. 5.13.2 A clinical algorithm based on a dynamical model
      3. 5.13.3 An extension of Vollmer and Humphrey’s model
      4. 5.13.4 Androgens positively regulating AI cell proliferation
      5. 5.13.5 Combining androgen ablation with other therapies
    14. References
      1. Figure 5.1
      2. Figure 5.2
      3. Figure 5.3
      4. Figure 5.4
      5. Figure 5.5
      1. Table 5.1
      2. Table 5.2
      3. Table 5.3
      4. Table 5.4
      5. Table 5.5
      6. Table 5.6
      7. Table 5.7
  8. Chapter 6 Resource Competition and Cell Quota in Cancer Models
    1. 6.1 Introduction
    2. 6.2 A cell-quota based population growth model
    3. 6.3 From Droop cell-quota model to logistic equation
    4. 6.4 Cell-quota models for prostate cancer hormone treatment
      1. 6.4.1 Preliminary model
      2. 6.4.2 Final model
      3. 6.4.3 Simulation
      4. 6.4.4 Predictions
    5. 6.5 Other cell-quota models for prostate cancer hormonetreatment
      1. 6.5.1 Basic model
      2. 6.5.2 Long-term competition in the basic model
      3. 6.5.3 Intermittent androgen deprivation
      4. 6.5.4 Cell quota with chemical kinetics
    6. 6.6 Stoichiometry and competition in cancer
      1. 6.6.1 KNE model
      2. 6.6.2 Predictions
    7. 6.7 Mathematical analysis of a simplified KNE model
    8. 6.8 Exercises
    9. 6.9 Projects
      1. 6.9.1 Beyond the KNE model
        1. 6.9.1.1 Phosphate homeostasis
        2. 6.9.1.2 Intracellular phosphate: A Droop approach?
        3. 6.9.1.3 Tumor lysis syndrome
      2. 6.9.2 Iodine and thyroid cancer
      3. 6.9.3 Iron and microbes
        1. 6.9.3.1 Salmonella infection
        2. 6.9.3.2 Malaria
    10. References
      1. Figure 6.1
      2. Figure 6.2
      3. Figure 6.3
      4. Figure 6.4
      5. Figure 6.5
      6. Figure 6.6
      7. Figure 6.7
      8. Figure 6.8
      9. Figure 6.9
      1. Table 6.1
      2. Table 6.2
  9. Chapter 7 Natural History of Clinical Cancer
    1. 7.1 Introduction
    2. 7.2 Conceptual models for the natural history of breast cancer: Halsted vs. Fisher
      1. 7.2.1 Surgery and the Halsted model
      2. 7.2.2 Systemic chemotherapy and the Fisher model
      3. 7.2.3 Integration of Halsted and Fisher concepts: Surgery with adjuvant chemotherapy
    3. 7.3 A simple model for breast cancer growth kinetics
      1. 7.3.1 Speer model: Irregular Gompertzian growth
      2. 7.3.2 Calibration and predictions of the Speer model
      3. 7.3.3 Limitations of the Speer approach
    4. 7.4 Metastatic spread and distant recurrence
      1. 7.4.1 The Yorke et al. model
      2. 7.4.2 Parametrization and predictions of the Yorke model
      3. 7.4.3 Limitations of the Yorke approach
      4. 7.4.4 Iwata model
      5. 7.4.5 Thames model
      6. 7.4.6 Other models
    5. 7.5 Tumor dormancy hypothesis
    6. 7.6 The hormonal environment and cancer progression
    7. 7.7 The natural history of breast cancer and screening protocols
      1. 7.7.1 Pre-clinical breast cancer and DCIS
      2. 7.7.2 CISNET program
      3. 7.7.3 Continuous growth models
      4. 7.7.4 Conclusions and optimal screening strategies
    8. 7.8 Cancer progression and incidence curves
      1. 7.8.1 Basic multi-hit model
      2. 7.8.2 Two-hit models
      3. 7.8.3 The case of colorectal cancer
      4. 7.8.4 Multiple clonal expansions
      5. 7.8.5 Smoking and lung cancer incidence
      6. 7.8.6 Summary
    9. 7.9 Exercises
    10. References
      1. Figure 7.1
      2. Figure 7.2
      3. Figure 7.3
      4. Figure 7.4
      5. Figure 7.5
      6. Figure 7.6
      7. Figure 7.7
      8. Figure 7.8
  10. Chapter 8 Evolutionary Ecology of Cancer
    1. 8.1 Introduction
    2. 8.2 Necrosis: What causes the tumor ecosystem to collapse?
      1. 8.2.1 Necrosis in multicell spheroids
      2. 8.2.2 Necrosis in tumor cords
      3. 8.2.3 Diffusion limitation in ductal carcinoma in situ
      4. 8.2.4 Necrosis caused by mechanical disruption of cells
      5. 8.2.5 Necrosis from local acidosis
      6. 8.2.6 Necrosis due to local ischemia
    3. 8.3 What causes cell diversity within malignant neopla-sia?
      1. 8.3.1 Causes of Type I diversity
        1. 8.3.1.1 Incomplete competitive exclusion
        2. 8.3.1.2 Fungiform invasion
        3. 8.3.1.3 Turing instabilities
      2. 8.3.2 Causes of Type II diversity
        1. 8.3.2.1 Natural selection favoring proliferation rate and efficient nutrient use
        2. 8.3.2.2 Natural selection favoring insensitivity to hypoxia
        3. 8.3.2.3 Natural selection favoring fungiform invasion
    4. 8.4 Synthesis: Competition, natural selection and necrosis
    5. 8.5 Necrosis and the evolutionary dynamics of metastatic disease
      1. 8.5.1 Pre-metastatic selection hypothesis
      2. 8.5.2 Reproductive fitness and export probability
      3. 8.5.3 Tumor self-seeding
    6. 8.6 Conclusion
    7. 8.7 Exercises
    8. References
      1. Figure 8.1
      2. Figure 8.2
      3. Figure 8.3
      1. Table 8.1
  11. Chapter 9 Models of Chemotherapy
    1. 9.1 Dose-response curves in chemotherapy
      1. 9.1.1 Simple models
      2. 9.1.2 Concentration, time, and cyotoxicity plateaus
      3. 9.1.3 Shoulder region
      4. 9.1.4 Pharmacodynamics for antimicrobials
    2. 9.2 Models for in vitro drug uptake and cytotoxicity
      1. 9.2.1 Models for cistplatin uptake and intracellular pharma-cokinetics
      2. 9.2.2 Paclitaxel uptake and intracellular pharmacokinetics
    3. 9.3 Pharmacokinetics
    4. 9.4 The Norton-Simon hypothesis and the Gompertz model
      1. 9.4.1 Gompertzian model for human breast cancer growth
      2. 9.4.2 The Norton-Simon hypothesis and dose-density
      3. 9.4.3 Formal Norton-Simon model
      4. 9.4.4 Intensification and maintenance regimens
      5. 9.4.5 Clinical implications and results
      6. 9.4.6 Depletion of the growth fraction
    5. 9.5 Modeling the development of drug resistance
      1. 9.5.1 Luria-Delbrück fluctuation analysis
        1. 9.5.1.1 Step 1: Expected number of mutations and resistant cells
        2. 9.5.1.2 Step 2: Test the mutation-selection hypothesis
        3. 9.5.1.3 Later work
      2. 9.5.2 The Goldie-Coldman model
      3. 9.5.3 Extensions of Goldie-Coldman and alternating therapy
        1. 9.5.3.1 Two-drug Goldie-Coldman model
        2. 9.5.3.2 Stochastic two-drug Goldie-Coldman model
        3. 9.5.3.3 Relaxation of the “symmetry assumption”
        4. 9.5.3.4 Goldie-Coldman contra Norton-Simon
      4. 9.5.4 The Monro-Gaffney model and palliative therapy
        1. 9.5.4.1 The model
        2. 9.5.4.2 Predictions of the Monro-Gaffney model
        3. 9.5.4.3 Summary
      5. 9.5.5 The role of host physiology
    6. 9.6 Heterogeneous populations: The cell cycle
      1. 9.6.1 The Smith-Martin conceptual model
      2. 9.6.2 A delay differential model of the cell cycle
      3. 9.6.3 Age-structured models for the cell cycle
      4. 9.6.4 More general sensitivity and resistance
    7. 9.7 Drug transport and the spatial tumor environment
      1. 9.7.1 Solute transport across tumor capillaries
      2. 9.7.2 Fluid flow in tumors
      3. 9.7.3 Tumor spheroid
      4. 9.7.4 Tumor cord framework
    8. 9.8 Exercises
    9. References
      1. Figure 9.1
      2. Figure 9.2
      3. Figure 9.3
      4. Figure 9.4
      5. Figure 9.5
      6. Figure 9.6
      7. Figure 9.7
      8. Figure 9.8
      9. Figure 9.9
      10. Figure 9.10
      11. Figure 9.11
      12. Figure 9.12
      13. Figure 9.13
      14. Figure 9.14
      15. Figure 9.15
      16. Figure 9.16
  12. Chapter 10 Major Anticancer Chemotherapies
    1. 10.1 Introduction
    2. 10.2 Alkylating and alkalating-like agents
      1. 10.2.1 Nitrogen mustards
      2. 10.2.2 Platinum-based drugs
      3. 10.2.3 Nitrosoureas
      4. 10.2.4 Methylating agents
    3. 10.3 Antitumor antibiotics
      1. 10.3.1 Anthracyclines
      2. 10.3.2 Mitomycin-C
      3. 10.3.3 Bleomycins
    4. 10.4 Antimetabolites
    5. 10.5 Mitotic inhibitors
      1. 10.5.1 Taxanes
      2. 10.5.2 Vinca alkaloids
    6. 10.6 Non-cytotoxic and targeted therapies
    7. References
      1. Figure 10.1
      2. Figure 10.2
  13. Chapter 11 Radiation Therapy
    1. 11.1 Introduction
    2. 11.2 Molecular mechanisms
      1. 11.2.1 Ions and radical reactions
        1. 11.2.1.1 DSB repair mechanisms
        2. 11.2.1.2 Radical detoxification mechanisms
      2. 11.2.2 Oxygen status
      3. 11.2.3 The four R’s
    3. 11.3 Classical target-hit theory
    4. 11.4 Lethal DNA misrepair
      1. 11.4.1 Repair-misrepair model
      2. 11.4.2 Lethal-potentially lethal model
      3. 11.4.3 Parametrization
    5. 11.5 Saturable and enzymatic repair
      1. 11.5.1 Haynes model
      2. 11.5.2 Goodhead model
      3. 11.5.3 General saturable-repair model
    6. 11.6 Kinetics of damage repair
    7. 11.7 The LQ model and dose fractionation
    8. 11.8 Applications
      1. 11.8.1 Tumor cure probability
      2. 11.8.2 Regrowth
      3. 11.8.3 Hypoxia
      4. 11.8.4 Radiation with chemotherapy
    9. References
      1. Figure 11.1
      2. Figure 11.2
      3. Figure 11.3
      4. Figure 11.4
      5. Figure 11.5
      6. Figure 11.6
      7. Figure 11.7
      8. Figure 11.8
      9. Figure 11.9
  14. Chapter 12 Chemical Kinetics
    1. 12.1 Introduction and the law of mass action
      1. 12.1.1 Dissociation constant
    2. 12.2 Enzyme kinetics
      1. 12.2.1 Equilibrium approximation
    3. 12.3 Quasi-steady-state approximation
      1. 12.3.1 Turnover number
      2. 12.3.2 Specificity constant
      3. 12.3.3 Lineweaver-Burk equation
    4. 12.4 Enzyme inhibition
      1. 12.4.1 Competitive inhibition
      2. 12.4.2 Allosteric inhibition
    5. 12.5 Hemoglobin and the Hill equation
    6. 12.6 Monod-Wyman-Changeux model
    7. References
      1. Figure 12.1
      2. Figure 12.2
      3. Figure 12.3
  15. Chapter 13 Epilogue: Toward a Quantitative Theory of Oncology
    1. References