Page 155 - 20dynamics of cancer

140 CHAPTER 7

Note that median time to failure, m,is

S(m) = 0.5 = e −μ(m)

and so
ln (0.5) =−μ(m) .
Age-specific incidence, I(t), is the instantaneous decrease in survival
divided by the fraction of the original population still surviving, thus

I(t) =−S /S =−dln (S) /dt = λ(t) ,
so the instantaneous failure rate from the nonstationary Poisson process
is also the age-specific incidence rate.
Cumulative incidence sums up the age-specific incidences; cumulative
incidence measures the total failure intensity over the total time period,
thus

t t
CI (t) = I (x) dx = −dln (S (x)) /dx
0 0
t

=− ln (S (t)) = λ (x) dx = μ(t) .
0
This background provides the details needed to decipher the rather
cryptic analysis in Peto et al. (1991) on the Weibull distribution and the
Druckrey equation.
To start, assume that cumulative failure follows the Weibull distribu-
tion
n
μ(t) =− ln (S) = bt .
Then the median time to failure is

μ(m) =− ln (0.5) = bm n

and so
b =− ln (0.5)/m n

and
n
n
CI = μ(t) = bt =− ln (0.5)(t/m) .
Thus, the median, m, and the exponent, n, completely determine the
course of survival, time to failure, and incidence.