Page 126 - 20dynamics of cancer

THEORY I 111

A transition rate might increase rapidly and then not change further.
This sudden increase in a transition rate would be similar to a sudden
abrogation of a rate-limiting step. Apart from a very brief burst in ac-
celeration, the main effect of a sudden knockout would be a decline in
acceleration because fewer limiting steps would remain.

DETAILS

In the first model, transition rates increase with advancing age (Frank
2004a). Let u j (t) = uf(t), where f is a function that describes changes
in transition rates over different ages. We will usually want f to be a
nondecreasing function that changes little in early life, rises in midlife,
and perhaps levels off late in life. In numerical work, one commonly
uses the cumulative distribution function (CDF) of the beta distribution
to obtain various curve shapes that have these characteristics. Following
this tradition, I use

t/T

Γ (a + b) a−1 b−1
β(t) = x (1 − x) dx,
0 Γ (a) Γ (b)
where T is maximum age so that t/T varies over the interval [0, 1], and
the parameters a and b control the shape of the curve. The value of β(t)
varies from zero at age t = 0 to one at age t = T.
We need f to vary over [1,F], where the lower bound arises when
f has no effect, and F sets the upper bound. So, let f(t) = 1 + (F −
1)β(t). Figure 6.8 shows examples of how increasing transition rates
affect acceleration.
In the second model, the transition rate between certain stages may
rise with clonal expansion. Models of clonal expansion have been stud-
ied extensively in the past (Armitage and Doll 1957; Fisher 1958; Mool-
gavkar and Venzon 1979; Moolgavkar and Knudson 1981; Luebeck and
Moolgavkar 2002). I describe the particular assumptions used in Frank
(2004b), which allow for multiple rounds of clonal expansion. Multiple
clonal expansions would be consistent with multistage tumorigenesis
being caused by progressive loss of control of cellular birth and death,
ultimately leading to excessive cellular proliferation.
I use the following strategy to study clonal expansion. First, assume
that all lines start in stage 0 at birth, t = 0, and use the initial condition
x 0 (0) = 1 so that x i (t) is the probability of a line being in stage i at
age t. Second, describe the value of x i (t) by summing all the influx into