Page 113 - 20dynamics of cancer

98 CHAPTER 6

and t is the time that has elapsed. So the probability of i transitions
among the precancerous stages follows the standard Poisson process.
If cancer remains uncommon by age t, then incidence is I(t) ≈ kt n−1 ,
n
where k = u /(n − 1)!. On log-log scales,
log (I (t)) ≈ log (k) + (n − 1) log (t) .

The log-log acceleration is

LLA (t) ≈ n − 1.
This is the classical result that log-log plots of incidence versus age will
be approximately linear with a slope of n − 1 (Armitage and Doll 1954).
When a significant fraction of individuals develops cancer, the log-log
incidence plot tends to accelerate more slowly at later ages, causing the
curve to flatten late in life and drop below the linear approximation. The
following details provide an exact solution for this simple model. The
exact solution shows how acceleration declines with age.

DETAILS

I introduced the basic model in Eqs. (5.2) of the previous chapter. I
repeat those equations here to provide the starting point for further
analysis

˙ x 0 (t) =−u 0 x 0 (t) (6.1a)
˙ x j (t) = u j−1 x j−1 (t) − u j x j (t) i = j,...,n − 1 (6.1b)
˙ x n (t) = u n−1 x n−1 (t) , (6.1c)

where x i (t) is the fraction of the initial population born at time t = 0
that is in stage i at time t, with time measured in years. Usually, I assume
that when the cohort is born at t = 0, all individuals are in stage 0, that
is, x 0 (0) = 1, and the fraction of individuals in other stages is zero.
As time passes, some individuals move into later stages. The rate of
transition from stage i to stage i + 1is u i . The ˙ x’s are the derivatives of
x with respect to t.
If the transition rates are constant and equal, u j = u for all j, then we
can obtain an explicit solution for the multistage model (Frank 2004a).
This provides a special case that helps to interpret more complex as-
sumptions that must be evaluated numerically. The solution is x i (t) =