Page 115 - 20dynamics of cancer

100 CHAPTER 6

6.3 Parallel Evolution within Each Individual

The model in the previous section assigns each individual in the popu-
lation to a particular stage of progression. Sometimes, it may make more
sense to consider the stage of particular cells or tissue compartments
within a single individual. Different components may be in different
stages of progression.
I described in Chapter 3 how colorectal cancer initiates in individ-
ual crypts, perhaps with mutations that occur to a particular stem cell
within a crypt. So we might choose to focus on different stages of pro-
gression in different crypts or stages of progression in different stem cell
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lineages. The human colon has about 10 crypts, and a slightly higher
number of stem cell lineages, so each individual has many parallel, in-
dependent lines of progression.

PR ´ ECIS

Suppose each individual has L independent lines of progression. We
start by calculating the rate of transition into the final, cancerous stage
for each independent line—the incidence per line. The incidence per
individual is the rate at which one of the L lines moves into the final,
cancerous state. The incidence per individual is simply L multiplied by
the incidence per line: the cancer rate rises linearly with the number of
independent lines that can fail.
If we fix the rate of progression per line, then the number of inde-
pendent lines does not affect log-log acceleration. However, if we wish
to keep constant the overall probability per individual of developing
cancer by a certain age, then as the number of lines increases, the prob-
ability of cancer per line must decline. Interestingly, slower per-line
transformation keeps acceleration higher through later ages, because
slow transformation maintains a high number of stages remaining in
progression.

DETAILS
Let the number of parallel lines of evolution within each individual
be L. We now have to consider progression hierarchically. Within each
individual, cancer arises as soon as one of the L lines progresses to the
nth stage. For each independent line, the probability of progressing to