Introduction to Mathematical Oncology

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:
Introduction to Theory in Medicine Introduction Disease A brief survey of trends in health and disease The scientific basis of medicine Aspects of the medical art Key scientific concepts in mathematical medicine Pathology—where science and art meet Introduction to Cancer Modeling Introduction to cancer dynamics Historical roots Applications of Gompertz and von Bertalanffy models A more general approach Mechanistic insights from simple tumor models Sequencing of chemotherapeutic and surgical treatments Stability of steady states for ODEs Exercises Projects and open questions Spatially Structured Tumor Growth Introduction The simplest spatially structured tumor model Spheroid dynamics and equilibrium size Greenspan's seminal model Testing Greenspan's model Sharratt–Chaplain model for avascular tumor growth A model of in vitro glioblastoma growth Derivation of one dimensional balance equation Exercises Projects Physiologically Structured Tumor Growth Introduction Construction of the cell-size structured model No quiescence, some intuition Basic behavior of the model Exercises Prostate Cancer: PSA, AR, and ADT Dynamics Introduction Models of PSA kinetics Dynamical models Androgens and the evolution of prostate cancer Prostate growth mediated by androgens Evolution and selection for elevated AR expression Jackson ADT model The Ideta et al. ADT model Predictions and limitations of current ADT models An immunotherapy model for advanced prostate cancer Other prostate models Exercises Projects Resource Competition and Cell Quota in Cancer Models Introduction A cell-quota based population growth model From Droop cell-quota model to logistic equation Cell-quota models for prostate cancer hormone treatment Other cell-quota models for prostate cancer hormone treatment Stoichiometry and competition in cancer Mathematical analysis of a simplified KNE model Exercises Projects Natural History of Clinical Cancer Introduction Conceptual models for the natural history of breast cancer: Halsted vs. Fisher A simple model for breast cancer growth kinetics Metastatic spread and distant recurrence Tumor dormancy hypothesis The hormonal environment and cancer progression The natural history of breast cancer and screening protocols Cancer progression and incidence curves Exercises Evolutionary Ecology of Cancer Introduction Necrosis: What causes the tumor ecosystem to collapse? What causes cell diversity within malignant neoplasia? Synthesis: Competition, natural selection and necrosis Necrosis and the evolutionary dynamics of metastatic disease Conclusion Exercises Models of Chemotherapy Dose-response curves in chemotherapy Models for in vitro drug uptake and cytotoxicity Pharmacokinetics The Norton-Simon hypothesis and the Gompertz model Modeling the development of drug resistance Heterogeneous populations: the cell cycle Drug transport and the spatial tumor environment Exercises Major Anti-Cancer Chemotherapies Introduction Alkylating and alkalating-like agents Anti-tumor antibiotics Anti-metabolites Mitotic inhibitors Non-cytotoxic and targeted therapies Radiation Therapy Introduction Molecular mechanisms Classical target-hit theory Lethal DNA misrepair Saturable and enzymatic repair Kinetics of damage repair The LQ model and dose fractionation Applications Chemical Kinetics Introduction and the law of mass action Enzyme kinetics Quasi-steady-state approximation Enzyme inhibition Hemoglobin and the Hill equation Monod–Wyman–Changeux model Epilogue: Toward a Quantitative Theory of Oncology References appear at the end of each chapter.