Page 190 - 20dynamics of cancer

CARCINOGENS 175

a

This expression is, on a log-log scale, log(m) = k −log(1+bd ), and may
often fit the data well. For example, in the large carcinogen study shown
in Figure 9.2, if we use Peto’s (1977) suggested value of a = r/n = 1/2,
with fitted values for two parameters of k = 1.01 and b = 16, we obtain

a line that is almost exactly equivalent to the fit of the Druckrey formula
shown in the figure.
The match of this diminishing effect theory to the observed relation
in Figure 9.2 shows that the data fit equally well to a model in which the
carcinogen affects only r< n of the stages in progression or a model
in which the effects of carcinogen dose rise at a diminishing rate with
increasing dose.
Diminishing effects of carcinogens with dose readily explain the ob-
servation that r< n. At present, little information exists about how
widespread such diminishing effects may be. Carcinogenic acceleration
of mitogenesis provides a plausible mechanism by which diminishing
effects may arise, but additional mechanisms probably occur.

HETEROGENEITY
Individuals vary in their susceptibility to carcinogens. Heterogene-
ity in susceptibility arises from both genetic and environmental factors.
Lutz (1999) suggested that heterogeneity may tend to linearize the dose-
response curve, that is, to reduce the exponent on dosage in such curves.
Lutz based his argument on a graph that illustrated how the aggregate
dose-response curve may form when summed over individuals with dif-
ferent susceptibilities. To evaluate this idea, I describe a few specific
quantitative models. These models suggest that heterogeneity can in-
fluence the dose-response curves, but heterogeneity does not provide a
convincing explanation for the widely observed low exponent on dose.
Consider the following rough calculation to illustrate the effect of
heterogeneity on the dose-response curve. Suppose a carcinogen affects
the relative risk of cancer, S. Let S depend on bd, where d is the dose,
and b is a factor that scales the effect of dose on relative risk.
Heterogeneity in individual susceptibility enters the analysis through
individual variability in b, the scaling factor that translates dose into an
increment in transition rate between stages of progression. We need the
value of S averaged over the different individual susceptibilities in the
population. Let the probability distribution for the values of b among
individuals be f(b). The value of S for each level of susceptibility, b,