Page 154 - 20dynamics of cancer

THEORY II 139

CONCLUSIONS
Weibull and Gompertz models provide useful tools to reduce data
to a small number of estimated parameters. However, I prefer to begin
with an explicit model of progression dynamics and derive the predicted
shape of the incidence curve. Explicit dynamical models allow one to test
comparative hypotheses about the processes that influence progression.

7.5 Weibull Analysis of
Carcinogen Dose-Response Curves

PR ´ ECIS

Peto et al. (1991) provided the most comprehensive experiment and
analysis of carcinogen dose-response curves. In their analysis, they com-
pared the observed age-specific incidence of cancer (the response) over
varying dosage levels. They described the incidence curves by fitting the
data to the Weibull distribution. They also related the Weibull incidence
pattern to the classic Druckrey formula for carcinogen dose-response
relations. The Druckrey formula summarizes the many carcinogen ex-
periments that give linear dose-response curves when plotting the me-
dian time to tumor onset versus dosage of the carcinogen on log-log
scales (Druckrey 1967).
I discussed the Druckrey equation, the data from Peto et al.’s study,
and some experimental results from other carcinogen experiments in
Section 2.5. Here, I summarize the theory that ties the Weibull approxi-
mation for incidence curves to the Druckrey equation between carcino-
gens and tumor incidence.

DETAILS

Define the instantaneous failure rate as λ(t). Cumulative failure in-
t

tensity is μ(t) = λ(x)dx. Then, from the nonstationary Poisson pro-
0
cess, the probability of survival (nonfailure) to age t is
S(t) = e −μ(t)

and failure is 1 − S.