Page 152 - 20dynamics of cancer

THEORY II 137

situations, so they use the model to fit data and reduce pattern descrip-
tion to a few simple parameters. Various forms of the Weibull model
exist. A simple and widely applied form can be written as

β
W(t) − W(0) = αt ,

where W(t) is the Weibull failure rate at age t, W(0) is the baseline
failure rate, and α and β are parameters that describe how failure rate
increases with age.
The simple model of multistage progression with equal transition
rates, given in Eq. (6.2), can be rewritten as

β
I(t) = αt /S n−1
≈ αt β if S n−1 ≈ 1

n
where α = u /(n−1)!, the exponent β = n−1, and S n−1 is the probability
that a particular line of progression has not reached the final disease
state by age t.
β
If I(t) ≈ αt is a good approximation of the observed pattern of age-
specific incidence, then multistage progression dynamics approximately
follows the Weibull model. On a log-log scale, the relation is

log (I) ≈ log (α) + β log (t) .

With this form of the model expressed on a log-log scale, estimates for
the height of the line, log(α), and the slope, β, provide a full description
of the relation between incidence and age. The log-log acceleration for
this pattern of incidence is β, the slope of the line.
Whenever log-log acceleration remains constant with age, the multi-
stage and Weibull models will be similar. The previous sections dis-
cussed the assumptions under which log-log acceleration remains con-
stant with age.
The Weibull model simply describes pattern, and so cannot be used
to develop testable predictions about the processes that control age-
specific rates. With multistage models of progression, we can predict
how incidence will change in individuals with inherited mutations com-
pared with normal individuals, or how incidences of different diseases
compare based on the number of stages of progression, the number of