Page 203 - 20dynamics of cancer

188 CHAPTER 9

fit closely to data such as in Figure 9.8. Second, later studies of dose-
response favored a quadratic fit to the data, leading many to suppose
that smoking affects two stages in progression.
One can see from Figure 9.10e,f that a combination of the earliest
transition, i = 0, and the penultimate transition, i = n − 2 = 4, provides
the shapes needed to fit the data in Figure 9.8, and with two transitions
affected, the overall incidence would be higher. Various authors fit the
data in this way, sometimes weighting the role of those two stages dif-
ferently (Day and Brown 1980; Brown and Chu 1987; Whittemore 1988).
Those fitted models based on two affected stages match the data rea-
sonably well for both dose-response and incidence. In particular, one
can easily explain the flattening of the incidence curves upon cessation
by the penultimate transition and the later rise in incidence several years
after cessation by the earliest transition.
The data and matching models tell a pleasing empirical and logical
story. However, other plausible models also fit nicely to the data. The
next section provides an example.

ALL STAGES AFFECTED
Armitage’s quote shows that the linear or perhaps quadratic dose-
response curve motivated the initial models in which smoke carcino-
gens affect only one or two stages of progression. Those assumptions
about number of stages affected may over-interpret the data: one cannot
draw firm biological conclusions about a molecular mechanism from a
fitted exponent of a dose-response curve. In addition, the mathematical
analyses of progression have in the past typically used approximations;
those approximations do not capture key aspects of incidence curves
and dose-response curves.
I decided to analyze how well the standard model of multistage pro-
gression fits the data, in which the carcinogens affect equally all n stages.
I first fit the data in Figure 9.8a, giving the fitted curves shown in Fig-
ure 9.8b. To obtain those fitted curves, I began with the basic multistage
model described earlier in the theory chapters. I took the following pa-
rameters as given based on previous studies or on common assump-
tions: the number of stages, n = 6; the number of independent cell
8
lineages at risk, L = 10 ; the age at which smoking starts, 25 years; and
the maximum age of the analysis, 80 years. Those parameters were not
fit to the data but instead derived from extrinsic considerations.