Page 120 - 20dynamics of cancer

THEORY I 105

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Figure 6.4 Increasing variation in rates of transition reduces acceleration. In
this example, there are n = 10 steps. Three steps have relatively slow transition
rates, u 0 = u 3 = u 7 = s, and the other seven steps have fast rates, f. The lifetime
7
risk per line, m/L, was set to 10 −8 for all curves, so if L = 10 , then the lifetime
∗
risk per individual is 0.1. The slow and fast rates are calculated by s = u /d 2
and f = u d. For the curves, from top to bottom, u ∗ = 0.00962, 0.00963,
∗
√
0.0119, 0.0238, 0.0516, and d =1, 5, 10, 20, 100. In all cases, the ratio of fast
3
to slow rates is f/s = d ; the lower the curve, the greater the variation in rates.
in Figure 6.4 shows the nearly constant acceleration with age when tran-
sition rates do not differ and m/L = 10 −8 . As the variation in transition
rates rises, the curves in Figure 6.4 drop to lower accelerations. (I nu-
merically evaluated Eqs. (6.1) for all calculations in this section.)
Figure 6.5 shows the distribution of lines in different stages at age 80,
where the panels from top to bottom match the increasing variation in
rates for the curves from top to bottom in Figure 6.4.
Why does rate variation cause a drop in acceleration with age? At
birth, all individuals are in stage 0, and there are n = 10 steps to pass
to get to the final cancerous stage of progression. So, the acceleration
is n − 1 = 9, independently of the variation in rates, because each of the
n steps remains a barrier.
The bottom panel of Figure 6.5 shows the consequences of high vari-
ation in rates for the distribution of lines into stages at age 80. The
probability peaks for stages 0, 3, and 7 arise because transitions out
of those stages are relatively slow compared to all other transitions.
The fast transitions between, for example, stages 1 and 2, and between