Page 114 - 20dynamics of cancer

THEORY I 99

i
e −ut (ut) /i! for i = 0,...,n − 1, with the initial condition that x 0 (0) = 1
and x i (0) = 0 for i> 0. Note that the x i (t) follow the Poisson distribu-
tion for the probability of observing i events when the expected number
of events is ut.
In the multistage model above, the derivative of x n (t) is given by
˙ x n (t) = ux n−1 (t). From the solution for x n−1 (t), we have ˙ x n (t) =
ue −ut (ut) n−1 /n − 1!. Age-specific incidence is
n−1
˙ x n (t) ˙ x n (t) u(ut) /n − 1!
I(t) = = n−1 = n−1 i , (6.2)
1 − x n (t) x i (t) (ut) /i!
i=0 i=0
and log-log acceleration from Eq. (5.3) is
dI (t) /I (t)
LLA (t) = = n − 1 − ut (S n−2 /S n−1 ), (6.3)
dt/t
k i
where S k = i=0 (ut) /i!.
The total fraction of the population that has suffered cancer by age
t—the cumulative probability—is

x n (t) = 1 − e −ut S n−1 . (6.4)
This analysis does not explicitly follow causes of mortality other than
cancer. Frank (2004a) analyzed the case in which each stage has a con-
stant transition rate to the next stage, u, as above, and also a constant
mortality rate from other causes, d. With constant mortality, d, the only
i
change in the solution arises in the expression x i (t) = e −(u+d)t (ut) /i!
for i = 0,...,n − 1, in particular, with extrinsic mortality, we must
use e −(u+d)t in the solution rather than e −ut . Because these exponen-
tial terms arise in both the numerator and denominator of the expres-
sion for incidence and so cancel out, extrinsic mortality does not affect
the incidence and acceleration solutions given here. The classes x i for
i = 0,...,n − 1 can be interpreted as those individuals alive and tumor-
less at different stages in progression.

CONCLUSIONS
This simple model shows the tendency of incidence to increase with
age in an approximately linear way on log-log scales. The increase in
incidence with age occurs because individuals progress through multi-
ple precancerous stages. Many processes cause departures from log-log
linearity. The following sections explore some of the ways in which pro-
gression affects the shape of the age-incidence curve.