Introduction to Mathematical Oncology

Yang Kuang, John D. Nagy, Steffen E. Eikenberry

• Publisher: Chapman and Hall/CRC
• Year: 2018
• Print ISBN: 9781584889908
• eBook ISBN: 9781498752978
• Edition: 1
• Topic: Mathématiques
• Subtopic: Mathématiques appliquées

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.

Chapter 1

Introduction to Theory in Medicine

1.1 Introduction

On September 5, 1976, a man named Mabalo Lokela was admitted to the Yambuku Mission Hospital in what is now the Democratic Republic of the Congo [8, Ch. 5]. Gravely ill, Lokela was suffering an intense fever, headache, chest pain and nausea. He vomited blood and had bloody diarrhea. Medical personnel recognized the signs of a hemorrhagic fever, but they were still largely in the dark. A variety of pathogens cause hemorrhagic fever, and there was no time to determine which one was causing this particular case. Lokela was clearly in serious trouble. Unfortunately, his health care workers failed to recognize that they were also in serious trouble.

Roughly speaking, Lokela’s tissues, including his skin, were melting away, causing massive internal bleeding. After a few days he had exhausted the clotting factors in his blood, and he began “bleeding out.” By then, the hospital staff could do little more than watch him die.

Ominously, just after Lokela’s burial, a number of his friends and family began experiencing similar symptoms. Eventually, 20 of them contracted the same disease. Two survived. While these 20 people suffered, the hospital in Yambuku started admitting patient after patient with the same sickness. Eventually, hospital staff, too, began to fall ill. The epidemic spread like wildfire. Within weeks of the outbreak, astonished, and frankly terrified, scientists and medical professionals around the world scrambled to understand what was happening in Yambuku. Initially they focused on two questions: what was the pathogen, and how did it pass from person to person? Surprisingly quickly they discovered that the pathogen was unknown to science (at that time) and that it jumped between hosts via body fluids. Unaware of this latter point early in the outbreak, many Yambuku medical workers contracted the disease because they failed to protect themselves from their patients’ blood. The hospital became an amplifier of the epidemic as the pathogen spread from patient to patient, carried at times by the medical workers themselves. Quickly, however, hospitals throughout Sub-Saharan Africa were made aware of the disease and taught how to handle it. The epidemic died out nearly as rapidly as it flared.

In this way the world learned of Ebola hemorrhagic fever, also called Ebola virus disease (EVD). Since then the world has seen a number of other Ebola outbreaks. In 2014 an Ebola epidemic spread more extensively to West Africa, specifically to Sierra Leone, Liberia and Guinea. This extremely long lasting outbreak was probably started by a 2-year-old boy died in December 2013 in the village of Meliandou, Guinea. It caught the world off guard because, unlike previous outbreaks, this one did not flare and rapidly die out on its own, largely due to the poor health infrastructure and the lack of standard practices to prevent the outbreak in the affected countries. As of November 1, 2015, over 28,607 cases have been reported, of which over 11,314 patients have died, making it the deadliest Ebola epidemic thus far according to WHO Ebola situation reports [19]. Despite this tragic human suffering and loss, the dynamics of this epidemic can be reasonably predicted by simple mathematical models correctly modeling the human behavior change dynamics after some initial period of time [2].

Against the background of the tragedies in Yambuku and West Africa we see the functioning of modern medicine. Here we use the word medicine in its most general sense—it means “the art of preventing or curing disease” and “the science concerned with disease in all its relations” [15].¹ In Yambuku, the two “arms” of medicine, curative and public health care, complemented each other beautifully. As its name implies, curative medicine focuses on cures or treatments for diseases—the Yambuku Mission Hospital staff trying to keep Mabalo Lokela alive, for example. In contrast, the goal of public health is to prevent disease. In Yambuku, public health professionals (perhaps) slowed the epidemic by identifying the pathogen and recommending techniques to prevent infection among hospitals in the region. When these two arms of medicine work together, the effectiveness of medical intervention is maximized. Although popular culture tends to focus on curative medicine— nearly all health-related movies and shows are set in hospitals with physicians and surgeons treating individual patients, and our experience tends to suggest that most young students entering so-called “pre-med” undergraduate studies are unaware of the existence of public health as an arm of medicine—a strong argument can be made that public health is primary. As the developing world shows us daily, inadequate public health care makes curative medicine superfluous. Without clean water, for example, the number of cases of infectious disease simply swamps curative efforts (see Section 1.3 below).

The definition above also suggests that medicine is both art and science. The art of medicine typically refers to clinical practice.² In the clinic, medical professionals work with individual patients out of necessity—each patient presents a unique case. In contrast, medical science seeks broad patterns and causative relationships within the chaos of individual cases. These patterns exist both within and among patients.

As the tragedies of Yambuku and West Africa show, medical science informs, or should inform, the practice of the medical art. Health professionals in the clinic rely on discoveries made by their scientific colleagues. At least, they should. When they do, we refer to the practice as evidence-based medicine. Our goal in this book is to explore how mathematics, dynamical models in particular, have in the past and can in the future advance the practice of evidence-based medicine, specifically as it applies to oncology, the science and art of studying and treating tumors.

1.2 Disease

Central to all of medicine, and its founding scientific discipline of physiology (see below), is the concept of homeostasis, a concept that includes both equilibrium and disequilibrium. For example, we say that mammals homeostatically regulate body temperature because they maintain a constant body temperature in disequilibrium with the environment. A dead mammal in a thermally invariant environment will maintain a constant body temperature, but not homeostasis.

Antithetical to homeostasis is the concept of disease. By the standard definition [15], disease is “an interruption, cessation, or disorder of body function, system or organ.” A more modern outlook would take this down to the level of cells and even molecules. Since almost all organs, systems, cells and molecules work to maintain homeostasis, it might be tempting to define disease as a threat to homeostasis. However, this definition would not apply to diseases of the reproductive system, which functions not to maintain homeostasis but to perpetuate the genes. Nevertheless, homeostatic mechanisms exist, ultimately, in support of the reproductive system in metazoans (multicellular animals).

Like the word medicine, “disease” can be used with subtly different meanings. The word also applies to a sickness with “at least two of these criteria: recognized [causative] agent(s), identifiable group of signs and symptoms, or consistent anatomic alterations” [15]. A symptom is something a patient feels that indicates disease, whereas a sign is an outward, objective manifestation of disease. According to these definitions, sore throat is a common symptom of a cold, whereas fever is a common sign of bacterial infection. A collection of signs and symptoms characteristic of disease is called a syndrome. For example, HIV infection is a disease characterized by acquired immunodeficiency syndrome (AIDS), signs of which include loss of certain types of immune cells and the presence of various opportunistic infections and cancers, like Pneumocystis carinii pneumonia and lymphoma, among others.

1.3 A brief survey of trends in health and disease

We adopt the view that any mathematical model of disease dynamics must connect in some way to the clinic. Otherwise the exercise is either pure mathematics, in which case its origin as a model of disease is hardly relevant, or it reduces to a triviality. This viewpoint justifies our decision to start with a survey of biomedical science and not mathematical techniques. Our goal is to help both mathematics and science students interested in theory to develop the skills and understanding necessary to contribute in a meaningful way to pathology—literally, the study of suffering—with the ultimate goal of alleviating some of that suffering. It is a daunting task that no one should enter without a clear understanding of what they are up against. Therefore, we include here a survey of the patterns of disease around the world.

About half the people on the planet will die from infection, cancer, coronary artery disease or cerebrovascular disease (primarily strokes).³ However, simple lists like this are misleading because patterns of disease are strongly influenced by socioeconomics at all levels, from individuals to na-tions (Fig. 1.1). Speaking generally, poor countries contend primarily with infectious diseases, particularly pulmonary (including tuberculosis), diarrheal, HIV, malaria and neonatal (among infants) infections. Infectious diseases are less significant in wealthier countries, where ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 Introduction to Theory in Medicine
• 2 Introduction to Cancer Modeling
• 3 Spatially Structured Tumor Growth
• 4 Physiologically Structured Tumor Growth
• 5 Prostate Cancer: PSA, AR, and ADT Dynamics
• 6 Resource Competition and Cell Quota in Cancer Models
• 7 Natural History of Clinical Cancer
• 8 Evolutionary Ecology of Cancer
• 9 Models of Chemotherapy
• 10 Major Anticancer Chemotherapies
• 11 Radiation Therapy
• 12 Chemical Kinetics
• 13 Epilogue: Toward a Quantitative Theory of Oncology
• Index