Page 182 - 20dynamics of cancer

CARCINOGENS 167

tions between stages that must be passed before cancer, u is the rate of
transition between stages, and t is age. Suppose a carcinogen increases
the rate of transition between some of the stages to u(1 + bd), where d
is dosage and b scales dose level into an increment in transition rate.
n
If the carcinogen affects r of the transitions, then I(t) ≈ ku (1 +
r n−1
bd) t . Two further changes to this equation provide a more useful
formula for studies of dosage and duration.
First, in examples such as cigarette smoking, the onset of carcinogen
exposure does not begin at birth but at some age t 0 at which smoking
starts, so the duration of exposure is t − t 0 = τ.
Second, in empirical studies, one cannot directly estimate u, the base-
n
line transition rate between stages, so the term ku = c enters in anal-
ysis only as a single constant, c. In different formulations, there will
be different combinations of factors that together would be estimated
as a single constant from data. I will use c to denote such constants,
although the particular aggregate of factors subsumed by c may change
from case to case.
With these assumptions, one may begin an analysis of dosage, d, and
duration, τ, with an expression such as
r n−1
I(τ) ≈ c(1 + bd) τ (9.1)
or a suitably modified equation to match the particular problem.
If, as often assumed, moderate to large doses significantly increase
transitions, then bd is much larger than one, and the transition rate
becomes u(1 + bd) ≈ ubd. Incidence is then

r
I(τ) ≈ cd τ n−1 (9.2)
r
with incidence rising as the rth power of dose, d and the n − 1st power
n r
of duration, τ n−1 . Here, c = ku b , representing a single constant that
may be varied or estimated from data.
Sometimes it is useful to study cumulative incidence, the summing
up (integration) of incidence rates over the duration of exposure. This
leads to the simple expression for cumulative incidence
n
r
CI (τ) ≈ cd τ . (9.3)
Here, c differs from above but remains an arbitrary constant to vary or
estimate from data.