Page 186 - 20dynamics of cancer

CARCINOGENS 171

discrimination, it does provide a good point of departure for figuring
out how to construct informative comparative tests. I delay discussion
of tests until later in this chapter.

CARCINOGENS AFFECT SOME STAGES BUT NOT OTHERS
Suppose, as discussed above, that a carcinogen increases the rate of
transition between stages to u(1 + bd), where d is dosage and b trans-
lates dose level into an increment in transition rate. If, for certain stages
in progression, moderate to large doses significantly increase the tran-
sition rate, then bd is much larger than one, and the transition rate
becomes u(1 + bd) ≈ ubd. For other stages not much affected by the
carcinogen, bd is small, and u(1 + bd) ≈ u.
If a large increase in transition rate occurs for r of the stages, and
the carcinogen has little effect on the other n − r transitions, then as I
showed in Eq. (9.3) above,
r
n
CI (τ) ≈ cd τ ,
r
with the cumulative incidence rate rising as the rth power of dose, d ,
n
and the nth power of duration, τ .
This explanation easily fits any case in which incidence increases ex-
ponentially with dosage and duration. However, the mathematics of
curves provides no reason to believe that the number of steps affected
by carcinogens can be inferred by measuring the empirical fit to the
exponent on dosage.

THE MATHEMATICS OF CURVES: CARCINOGENS AFFECT ALL STAGES
Consider the most famous dose-response study: smoking among Brit-
ish doctors. Figure 9.1 shows the fit given by Doll and Peto (1978), in
which the highest exponent of dose is two. From that fit, many authors
have stated that lung cancer depends on the second power of dose, and
thus the carcinogens in cigarette smoke affect only two stages in lung
cancer progression. Against that explanation, the dashed curve in Fig-
ure 9.1 illustrates my calculation that a nearly equivalent fit for incidence
can be obtained with a higher power of dose, in this case proportional
5
to (1 + d/46) .
The fact that one can fit a higher power of dose to those lung cancer
data certainly does not mean that the carcinogens in cigarette smoke
affect five stages of carcinogenesis rather than two. It does mean that