Tumor Development Under Combination Treatments with Anti-angiogenic Therapies - Mathematical Methods and Models in Biomedicine - page 301

their survival in a hostile environment. Summarizing, cancer is a disease with at
times anti-intuitive behavior whose macroscopic time course reflects intra-cellular
and inter-cellular phenomena that are strongly nonlinear and time varying.
In this framework, methods of modern mathematics, such as the theory of
finite and infinite dimensional dynamical systems, can play an important role in
better understanding, preventing, and treating the family of bio-physical phenomena
collectively called cancer. As Bellomo and Maini stressed in [ 3 ]: “the heuristic
experimental approach, which is the traditional investigative method in the bio-
logical sciences,” and in medicine, “should be complemented by the mathematical
modeling approach. The latter can be used as a hypothesis-testing and indeed,
hypothesis-generating tool which can help to direct experimental research. In turn,
the results of experiments help to refine the modeling. The goal of this research
is that, by iterating back and forth between experiment and theory, we eventually
arrive at a deeper conceptual understanding of how the highly nonlinear processes
in biology interact. The ultimate goal in the clinical setting is to use mathematical
models to help design therapeutic strategies [our emphasis]”.
In this chapter, we describe some of the fundamental principles that underlie
the mathematical modeling of the evolution of a tumor that need to be taken
into account in any treatment approach. Clearly, no attempt can be made to be
comprehensive in a short chapter and for this reason we focus on one particular
topic, combination therapies that involve anti-angiogenic treatments. We start with
a brief discussion of various mathematical models for tumor growth which form
the basis on which any kind of treatment needs to be imposed. Of these, by far the
most important ones are chemo- and radiotherapies and we discuss their benefits
and shortcomings. Anti-angiogenic therapies target the vasculature of a developing
tumor and in combination with these traditional treatment approaches provide a
two-pronged attack on both the cancer cells and the vasculature that supports
them. Starting with general models that capture the characteristic features of these
treatment approaches, we lead over to more detailed models as they are needed
to optimize treatment protocols and discuss the implications of our mathematical
analysis.

2 Phenomenological Models of Tumor Growth

A phenomenological model that describes the growth of a population of cells may
be written in the general form

p = pR ( p ) , (1)

where p is the size (measured as volume, number of cells, density of cells, etc.)
of the population and R models the net proliferation rate, i.e., the difference

between the proliferation rate of the tumor cells,

( · )

Π ( p ) , and their death and

apoptosis rate M . Ideally, these rates are constant and then the growth law is

a pure exponential [ 68 ]. In the biological reality, however, the rate R

( p )

( p ) normally