Page 316 - 20dynamics of cancer

CELL LINEAGE HISTORY 301

predicts that new cases will be very rare, and incidence at age 65 should
be near zero.
Let us suppose, for the moment, that it is possible to use mitotic age to
obtain an approximation for incidence along the lines followed by Pike.
How would we proceed to get the correct formulation? Start by assuming
that the rate of mitosis determines the rate of transition between stages
in a multistage model. Let the rate of mitosis at age t be u(t). Then
mitotic age at age t is the cumulative number of mitoses at that age,
t
m(t) = 0 u(x)dx, where the integral simply means the summing up of
all the mitoses between ages 0 and t.
This measure of mitotic age describes the cumulative number of mi-
toses, so we need to work with cumulative incidence to keep cause and
effect on the same scale. Cumulative incidence is the summing up of
t
incidence between ages 0 and t, which is notationally CI(t) = 0 I(x)dx.
Then the widely used approximation that Pike wished to analyze is

n
CI (t) ≈ c[m (t)] .

Age-specific incidence, I(t), is the rate of additional cases at age t, which
is the derivative of cumulative incidence with respect to t. Taking the
derivative of both sides of the expression for cumulative incidence yields

I(t) ≈ c[m (t)] n−1 u(t) , (14.2)

which is the correct formula that follows from Pike’s logic instead of
Eq. (14.1). This correct formula can be read as: the rate of cancer at age
t depends on mitotic age, m(t), raised to the n − 1st power, multiplied
by the age-specific rate of mitosis at age t, u(t). Mitotic age raised to the
n−1st power is approximately proportional to the number of individuals
that have progressed through the first n−1 stages of carcinogenesis and
need only one additional step to be transformed into a case of cancer.
The rate of mitosis at age t, u(t), is the rate at which those individuals
in stage n−1 pass the final step and are transformed. If the age-specific
rate of mitosis, u(t), drops significantly at menopause, then the inci-
dence would also decline significantly, and the slope of the incidence
curve would be negative. Pierce and Vaeth (2003) developed this sort of
formulation properly and extensively for a period of carcinogen expo-
sure followed by cessation of exposure. In that formulation, incidence