Page 90 - 20dynamics of cancer

HISTORY OF THEORIES 75

Armitage and Doll (1957) developed a two-stage mathematical the-
ory in which the first hit causes proliferation of the altered cell, and
the second hit causes progression to cancer. They developed this the-
ory to explain two observations. First, prior experimental studies of
carcinogens had emphasized only two distinct stages in carcinogenesis.
Second, many common cancers increased in incidence with about the
fifth or sixth power of age.
Previously, Armitage and Doll (1954) showed that a simple multistage
model could explain the increase of incidence with age based on six
or seven hits, the number of hits being the exponent on age plus one.
However, given the two-stage interpretation of experimental carcinogen
studies, they sought in their 1957 paper an alternative theory to fit the
data. Their new clonal expansion theory could be fit to the observed rise
of incidence with a high power of age. The rapid increase in incidence
with age occurs because, given the first hit, the rate of transformation
by the second hit increases with time as the clone of initiated cells grows
and raises the number of cells at risk for obtaining the second hit.
Starting with Fisher (1958), many others have given variant mathe-
matical treatments of clonal expansion. They all come down to the same
process: increasing the number of target cells with i − 1 hits raises the
rate at which the ith hit occurs. This increase in the rate of transition
between stages raises the slope of the incidence curve (acceleration), al-
lowing a model with a small number of hits to generate incidence curves
with high acceleration.
I develop some of the technical details of clonal expansion models
in the mathematical chapters of this book. For example, if the clone
expands rapidly, the next hit comes so quickly that it is not rate limit-
ing in progression. Once a clone approaches in size the inverse of the
mutation rate, the next hit comes inevitably and does not limit the rate
of progression. So, these models depend on slow clonal expansion over
many years to provide a fit to observed incidence curves.
Recent molecular studies implicate several key genetic changes in pro-
gression for many cancers. Because of those studies, the two-stage mod-
els of clonal expansion have given way to more sophisticated multistage
models that include one or more stages of clonal proliferation (Luebeck
and Moolgavkar 2002).
Many models can provide a moderately good fit to the data for com-
mon cancers. Thus, the data do not strongly discriminate between the