Page 204 - 20dynamics of cancer

CARCINOGENS 189

I then used the following crude procedure to fit the model to the
data. I set the cumulative lifetime risk of lung cancer for nonsmokers
to 0.005 to match the lowest curve in Figure 9.8, which shows data for
nonsmokers. I then fit the transition rate between stages per year, u,
needed to match that nonsmoker incidence curve, resulting in the esti-
mate u = 7.24 × 10 −4 . Given this value for the baseline transition rate,
I next assumed that during exposure to smoke carcinogens, all transi-
tions between stages rise to u(1 + bd), where d is dose, and bd is the
increase in the transition rate caused by carcinogens. The value of b
sets a proportionality constant for the effect of a given dose; without
loss of generality, I used b = 1, because all calculations depend only on
the product bd and not on the separate values of the two parameters.
I estimated the value of d = 1.187 to match the top curve, in order
to obtain a lifetime cumulative risk for continuing smokers of 0.158.
Finally, I assumed that, upon cessation of smoking, carcinogenic effects
decay with a half-life of 5 years; this assumption prevents an unrealistic
instantaneous decline in incidence immediately upon cessation.
This fitting procedure required estimation of only two parameters, u
and d. The other values came from prior studies or common assump-
tions. The fit shown in Figure 9.8b provides a reasonable qualitative
match to the observed patterns in Figure 9.8a; some deviation occurs
at age 80—a few observations at this point cause some of the incidence
curves to rise late in life. Better fit could be obtained by optimizing
the fit procedure and by using additional parameters. But my point is
simply that the basic multistage model gives a nice match to the data
without the need for any special adjustment or refined fitting.
Originally, Armitage, Peto, and others rejected a model in which car-
cinogens affect all stages because the estimated exponent of the dose-
response curve is between one and two. Does the model I used, with all
stages affected, also match that observed dose-response relation?
To test the model fit to the observed dose-response curve, I focused on
the estimated value of d, which in the standard models is proportional
to dose. At the maximum age measured, in this case 80 years, I varied
the cumulative lifetime risk for continuing smokers between the value
for nonsmokers, 0.005, and the approximate observed value for lifetime
smokers of 0.158. For each cumulative risk value (the response), I fit the
d value (the dose) needed to match the cumulative risk. I then calculated
the log-log slope of the dose-response curve, which turned out to be