Page 122 - 20dynamics of cancer

THEORY I 107

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Figure 6.6 Increasing variation in rates of transition reduces acceleration. In
this example, there are n = 10 steps. The first and last steps are the slowest; the
∗ i
middle steps are the fastest. In particular, u i = u n−1−i = u k for i = 0,..., 4,
with u values chosen so that m/L = 10 −8 . Larger values of k cause greater
variation in rates. Greater rate variation reduces acceleration by concentrating
the limiting transitions onto fewer steps. Here, for the curves from top to bot-
tom, the values are k = 2 and u ∗ = 2.245 × 10 −3 ,2.715 × 10 −4 ,6.85 × 10 −5 ,
2.66 × 10 −5 . The values of accelerations for ages less than 15 were erratic be-
cause of the numerical calculations. At t = 0 the acceleration is n − 1 = 9.

stages 2 and 3, happen relatively quickly and do not limit the flow into
the final, cancerous stage. Only the n s = 3 slow transition rates limit
progression, and so acceleration declines to n s − 1 = 2, as shown in
Figure 6.4.
In the long run, the slowest steps determine acceleration (Moolgavkar
et al. 1999). But the long run may be thousands of years, so we need to
consider how acceleration changes over the course of a typical life when
rates vary. Figure 6.6 shows a different pattern of unequal rates. In
that figure, the first and last transitions happen at the slowest rate, and
the rates rise toward the middle transitions. As one follows the curves
from top to bottom, the variation in rates increases and the accelerations
decline. Figure 6.7 shows the distribution of lines into stages at age 80,
with the panels from top to bottom matching the curves from top to
bottom in Figure 6.6.
Armitage (1953) presented the classical approximation for unequal
rates. However, Moolgavkar (1978) and Pierce and Vaeth (2003) noted