Page 108 - 20dynamics of cancer

PROGRESSION DYNAMICS 93

5.4 Why Study Quantitative Theories?

An ordered, linear sequence leaves out many of the complexities of
carcinogenesis. However, it pays to begin with this simple model, to
understand all of its logical consequences, and to study how well that
model can predict changes in incidence. Following on the simple model,
we can begin to explore alternatives, such as parallel lines of progres-
sion in different cellular lineages or incidence aggregated over different
pathways.
After I have analyzed the basic model, I will explore a range of more
complex assumptions, because we need to understand the possible al-
ternative explanations for observed patterns. Without broad conceptual
understanding, there is a tendency to latch onto the first available expla-
nation that fits the data without full consideration of reasonable alter-
natives. The theory I develop will run ahead of empirical understanding,
but if used properly, this is exactly what theory must do.
Another issue concerns the definition of stages and rate-limiting steps.
To address this issue, we must consider what we wish to accomplish with
mathematical models. The models are tools, so we need be concerned
only about defining stages and rate-limiting steps in ways that help us
to achieve particular goals for particular problems.
Sometimes we may formulate a model in a very abstract, nonbiological
way, for example, to study how variation in rates of transition between
stages influences age-onset patterns. In this case, stages remain abstract
notions that we manipulate in a mathematical model in order to under-
stand the logical consequences of various assumptions. In other cases,
we may try to match the definition of stages and rates to the biological
details of a particular cancer. A stage may, for example, be an adenoma
of a particular size, histology, and genetic makeup. A transition between
stages may occur at the rate of a somatic mutation to a particular gene.

5.5 The Basic Model

Assume that cancer progression requires passage through n rate-
limiting steps, each step moving through the sequence of tumor pro-
gression to the next stage. A step could, for example, be mutation to
APC or p53, as in colorectal cancer progression. But for now, I just
assume that such steps must be passed.