MAT 598----Tentative Contents----Spring, 2007

Mathematical Models in Medicine

Medicine is the science and art dealing with the maintenance of health and the prevention,
alleviation, or cure of disease. In this course, we explore how mathematics can be applied to
improve the practice and art of medicine. In particular, we focus on the role mathematics
can play in understanding the etiology, epidemiology, pathogenesis and morphology of
disease, as well as improving treatment.  The course will try to cover three types of diseases-
malignant neoplasia, viral infections, and dynamical diseases.

Instructors:  Yang Kuang,     E-mail: kuang@asu.edu
                      John Nagy,   E-mail: john.nagy@sccmail.maricopa.edu

Course Home page: http://math.asu.edu/~kuang/class/mmm.html. Details on potential
topics can be found there.

Section:  62370; CDS 321, 1:40pm--2:55pm, Tu, Th.            
Office: PSA 429,   Phone: 965-6915     Office hours: 12pm-1pm, T, W, Th,  and by appointment.

PREREQUISITES: MAT 475, or MAT 494 in Mathematical Biology in fall, 2006, or the
approval of the instructors.

Introduction:

While modern medicine has provided vaccines and antibiotics for most common viral
and bacterial diseases, many of these microbes still threaten human health. Bacterial pathogens tend
to evolve resistance to antibiotics, and cheap mass transportation systems and massive overpopulation
have greatly increased opportunities for  new and potentially fatal viral agents, like SARS virus and
ebola, for which we have developed no effective vaccines, to spread throughout the world. Global
poverty, which bars much of humanity from effective protection against even well-known and curable
diseases, also threatens human health in both developing and wealthy developed nations alike. Even
in the wealthiest nations, where nearly everyone is living longer and enjoying seemingly unlimited food
and material wealth, we face unprecedented rates of cancer, heart disease, diabetes and neurological
disorders.

Fortunately, the recent breathtaking and explosive growth in modern sciences, especially cell biology,
genetics and computer sciences, promises to provide tools with which to take on the enormous
challenges facing modern medicine. To understand and effectively deal with diseases of all forms is
clearly a pressing and profound undertaking as the human populations in developed countries continues
 to age, and those in developing countries face increasing devastation from widespread viral and bacterial
diseases.

In the 1970s and 1980s, hope abounded that a cure for the most common diseases, especially cancer,
was not far off. However, almost 35 years and hundreds of billions of dollars—not to mention many
millions of lives—after President Nixon signed the National Cancer Act into law on December 23, 1971,
we are still  far from a complete understanding of even the most common cancers. Much the same can
be said for HIV disease, although we have witnessed more progress here. What went wrong? A
possible answer to this difficult question may be the dominant reductionistic method used in the
conduct of scientific research. Reductionism holds that the nature of complex entitities can always
be understood by breaking them down into simpler or more fundamental components. Although this
awareness spurred a great number of advances in medicine, many recognized that another approach
was needed for complex diseases like cancer.  This recognition sparked a new trend, called holism,
in disease research. Holism, contrary to reductionism, takes the view that the whole can be more
 than the sum of its parts, and hence a full understanding can only be gained in a integrative way.
The holistic view of disease gives birth to the emerging area of mathematical medicine.

Mathematical medicine is deeply rooted in systems biology and mathematical ecology. Systems
biology provides a sound scientific framework and structure to the study of disease, and
mathematical ecology renders proven mathematical models and effective techniques to simulate
disease progression and evaluate details of reatment efficacy. Mathematical medicine began with
Kermack and McKendrick’s study of the advent of  plague in Bombay in 1927, and was put
on firm theoretical foundations by Anderson and May in the 1970's. In the last two decades,
we have seen an explosive interest in mathematical modeling of various diseases,  especially viral
diseases and cancer.

This course will be largely self-contained. It will focus on three categories of disease: cancer, viral
diseases  and dynamical diseases.

Part 1: Mathematical Oncology.

Cancer kills more people around the world than any other disease except coronary artery
disease. Because of its importance, cancer has attracted the attention of clinical, mathematical,
molecular and cellular biologists for many years. Besides elucidating the complexities of this
disease, basic cancer research has motivated, almost as a byproduct, many fundamental
contributions to our understanding of basic cell biology. It also has grown into an important
area of applied mathematics. In this part, we address a number of fundamental issues in
oncology from a mathematical perspective. This part is organized into three major chapters:
etiology, pathogenesis and treatment. In the etiology chapter we explore models of malignant
oncogenesis, starting with leukemia and moving to more complex issues, including the
complicated, and still somewhat mysterious, carcinogenesis of solid epithelial tumors,
and the indirect but critical process of angiogenesis. We begin the pathology chapter with
a tour of basic tumor growth models before moving to other fundamental aspects of the
pathology: pleomorphism and tumor natural history, idiosyncratic tumor metabolism and
its consequences, necrosis, invasion and metastasis. Throughout this chapter we focus
on how one uses models to predict the clinical course of disease and define prognosis.
This part ends with a short tour of treatment models, including  chemo-, radiation, and
immune therapies.

Chapter 1.  Etiology

1.1 Leukemogenesis. Leukemia is characterized by a particularly simple etiology. In this
section we study that etiology with one- and two-hit mutation models. We start with simple
ordinary differential equation (ODE) models and show how these relate to more realistic
stochastic models. As a case study we model chromosomal translocations known to
produce leukemia.

1.2 The “One Renegade Cell” Theory of Solid Tumor Oncogenesis. This section begins
with an extension of the leukemia models into multiple-hit scenarios. We also introduce
models of mutations in particular genes—most notably p53—and their effect on
carcinogenesis. We show how such models may help explain the variation in morphological
and histological patterns seen in tumors originating in different tissues.

1.3 Complete Carcinogenesis. Malignant transformation cannot be explained entirely
by genomic changes alone. Ecological and evolutionary factors must also be considered,
since carcinogenesis is essentially natural selection acting on cell types within an ecology
determined by the host's physiology. This understanding forms the core of this chapter.
Here we introduce this concept using models of angiogenesis, which is not only an
important ecological phenomenon, but also bridges the gap between etiology and
pathogenesis.

Chapter 2.  Pathogenesis

2.1 Nonspatial Tumor Growth. Malignant neoplasia is chiefly characterized by uncontrolled
cell proliferation. A number of models from population biology—primarily ODEs—have
been adapted to this application. Here we look at two such models: one of a classic form,
and one that includes angiogenesis.

2.2 Spatial Tumor Growth. Tumors exist in a complex three dimensional environment.
Describing tumor growth in such an environment has long interested mathematical biologists.
The simplest such models are abstracted from a tissue culture model of cancer called multicell
spheroids. We introduce PDE (partial differential equations) formalisms describing such
systems before moving to ductal carcinoma in situ as an in vivo case study.

2.3 Malignant Metabolism.  Most malignant tumors acidify their local environments, largely
because malignant cells rely on glycolysis and lactic acid fermentation as their core metabolic
process. In this section we introduce and explore models, due primarily to Gatenby and his
coworkers, of the causes and consequences of this switch from aerobic to anaerobic
metabolism in neoplasia.

2.4 Necrosis.  Solid tumor morphology is often characterized by necrosis—regions of the
tumor that suffer a complete ecological collapse. The prevalence of necrosis varies by tissue
of origin and histological type. In this section we use mathematical models to analyze
competing hypotheses explaining the cause of necrosis—local tumor acidosis, explored in
the previous section, and  ischemia due to blood vessel collapse and other disruptions of tumor
blood flow.

2.5 Invasion.  We finish this section with an exploration of the two signature properties of
malignant neoplasia-invasion and metastasis. We begin with simple PDE descriptions of
invasion that produce propagating waves, before we move to two more realistic models of
invasion that include penetration of the basement membrane, chemotaxis and haptotaxis.

2.6 Metastasis.  In this section we take on two problems. First, how do cancers metastasize?
Second, why do they do so? After introducing a simple metapopulation-like model of
metastasis, we then adapt it to deal with the evolutionary question, under what circumstances
do the benefits of metastasis for individual cells—primarily escape from competition-outweigh
the costs of exposure to the immune system?  Finally we touch on how these insights can help
us understand the variation in metastatic potential and pattern among different types of tumors.

Chapter 3. Treatment

3.1 Radiation Therapy. Chemo- and radiation therapy remain principal modalities in the
treatment of malignant neoplasia. In this section we introduce radiation therapy to the
simpler growth models of the previous section and explore how modeling may be used
to improve treatment schedules.

3.2 Chemotherapy. This section begins in the same way as the previous one—with an
introduction of chemotherapy to previous models as simple killing terms. Then we
construct more realistic models that include the immune attack on tumors and
chemotherapy’s impact on the immune system. With these models we explore
how one can improve treatment dosage and scheduling.

3.3 Immune Therapy.  Attempts to introduce immune therapy as a major cancer
treatment modality have met with difficulty. However, in some cases, immune therapy
has been used successfully as an adjuvant to classical treatment. Here we use the
previously introduced immune models to explore why the difficulties arise and the most
effect adjuvant use of immune therapy.

3.4 Treatment Failure.  The most common cause of cancer treatment failure is natural
selection. Both chemo- and radiation therapy select for tumor cell types that are resistant
to treatment. If such cells survive and repopulate the tumor, subsequent treatment often fails.
This explains why oncologists tend to prefer multiple treatment modalities for certain tumor
types, because this approach limits the directions evolution can take. Here we explore natural
selection’s role in treatment failure by synthesizing many of the models presented previously.

Part 2: Mathematical Immunology and Virology

Researchers have recently found impressive applications for differential equations in the study of
viral dynamics and viral disease treatment. They are increasingly being used to explore the logical
consequences of theoretical hypotheses and interpret experimental data.  This situation contrasts
sharply with cancer, where difficulty in obtaining suitable human in vivo tumor growth and treatment
data has slowed the application of cancer models for practical use. Among the most noteworthy
successful applications of viral models are the many striking and practical findings obtained by a
team of researchers led by Alan Perelson, a mathematical immunologist at Los Alamos National
Laboratory. Their results have lead clinicians to rethink their standards for drug therapies (Cipra,
Will Viruses Succumb to Mathematical Models? 32, 1999, SIAM News). in this part of the book,
we will address such questions as: How does a healthy human immune system clear or control
influenza virus? What is the quantitative relationship between virus load and immune response?
Why do viral infections often fail to grow uncontrollably when immune response is low? How
can the multiple viral steady states at low immune levels be explained? Given a set of treatment
options, which is best under which conditions? We will use differential equation models to answer
all these questions.

Chapter 1.  A Primer of Viruses and Immunity

(a) Introduction to Influenza. This group of sections will cover the replication of influenza, influenza
epidemiology and ecology, and antiviral drug treatments.

(b) Immune Response. This group of sections will introduce the reader to basic human immunity and
immune response to viral infection.

(c) Introduction to HIV. This section will cover the replication of HIV, HIV epidemiology and ecology,
immune response to HIV and antiviral drug treatments.

Chapter 2.  Influenza Models

(a) Influenza Models. This group of sections will selectively cover some simple ODE and DDE (delay
differential equations) influenza models with explicit immune response dynamics, but no treatment.
Most such models will be simulated with realistic parameter values.

(b) Influenza Models with Treatment. This group of sections will selectively cover some simple ODE
and DDE influenza models with explicit immune response dynamics and antiviral drug treatment.
Most such models will be simulated with realistic parameter values and occasionally with experiment
data.

Chapter 3.  HIV Models

(a) HIV Models. This group of sections will selectively cover some simple ODE and DDE HIV models
with explicit immune response dynamics, but no treatment. Most such models will be simulated with
realistic parameter values.

(b) HIV Models withTreatment. This group of sections will selectively cover some simple ODE and
DDE HIV models with explicit immune response dynamics and antiviral drug treatment. These models
will be challenged with experimental data.

Part 3: Dynamic Diseases

Diseases can be distinguished by their onset (acute versus sub-acute) and by their subsequent clinical
course (self-limiting, relapsing-remitting, cyclic, chronic progressive). Dynamical diseases (also called
periodic diseases) refer to those that exhibit abnormal temporal profiles, or mathematically, those with
pathologies that are caused by a qualitative change in dynamics (bifurcation). In other words, the
hallmark of dynamical disease is a sudden qualitative change in the temporal pattern of some
physiological variables. Hence, there is a natural connection between the nonlinear dynamics and
dynamical diseases. A solid understanding of such models may allow us to develop therapeutic
strategies to ensure that those physiological variables remain within normal ranges. Indeed, recognition
of the temporal rhythms, combined with knowledge that certain temporal rhythms respond best to
dynamic therapies, has driven the development of a growing number of implantable devices that treat
disease dynamically through their effects on physiological control mechanisms. Examples include
cardiac and brain defibrillators, deep brain stimulators, drug delivery systems, ‘smart’ insulin pumps,
artificial kidneys and pancreas, and the therapeutic benefits of noise for balance control and hearing
aids. These tremendous advances have occurred mostly over the last decade and have been sparked
by direct comparisons between model predictions and experimental observation that have recently
become possible. In this chapter, we illustrate this point through analyses of dynamical models of
some simple biological oscillators, including disorders of nervous system such as Parkinson disease,
and feedback control mechanisms, illustrated in particular with the insulin-glucose interaction.

Chapter 1.  Dynamical Diseases, Oscillations and Chaos

(a) Introduction to Dynamical Diseases. We will start with an overview of rhythms of life, introduce
examples of dynamical diseases and discuss their identification and possible treatment strategies.
(b) Oscillations and Chaos in Physiological Systems. This group of sections will be a primer on the
applications of mathematical theory of oscillation and chaos in the context of physiological processes.
Chapter 2.  Biological Oscillators

(a) Mathematical Models of Biological Oscillators. This group of sections will selectively cover some
simple ODE and DDE models that generate physiological rhythms.

(b) Periodic Stimulations. Periodic stimulation is a popular strategy in controlling some dynamical d
iseases. This group of sections will introduce mathematical concepts and methods needed to
understand the complex interplay of the periodic stimulations and the intrinsic oscillatory/chaotic
dynamics of the modeled biological oscillators.

Chapter 3.  Models of Dynamical Diseases

(a) Parkinson Disease as a Dynamical Disease. We will introduce a model of tremor developed
by Beuter and Vasilakos (1994), based on negative feedback with time delay. Application of
this model to Parkinson disease will be presented.

(b) Models of Ultradian Insulin Oscillations and Insulin Therapies. This group of sections will
model the observed human ultradian insulin oscillations and some current insulin therapies with
several ODE and DDE models. These models are challenged with experimental data.

(c) Acute Myelogenous Leukemia. We will cover the work of Louise Kold-Andersen and M.C.
Mackey on "Resonance in periodic chemotherapy: A case study of acute myelogenous leukemia",
J. Theor. Biol. (2001), 209, 113-130.

Appendices (about 40 pages) will contain review material/crash courses for linear stability theory
and Liapunov method for ordinary, partial and delay differential equations.

References will be placed at the end of each chapter.

The key references of part 1 include John Nagy’s 2005 cancer model review paper, other cancer
review articles and the many edited volumes on cancer modeling, including the recent DCDS-B
special issue on cancer modeling that contains our first cancer modeling work.

The key references of part 2 include several classical and current paper of Alan Perelson, such as
the review paper by him and Patrick Nelson in 1999, and the 2000 book by Nowak and May.

The key references of part 3 include the book of L. Glass and M.C. Mackey (1988, From Clocks
to Chaos: The Rhythms of Life, Princeton University Press) and the information-rich website of
John Milton (http://faculty.jsd.claremont.edu/jmilton/dd.htm).