Page 153 - 20dynamics of cancer

138 CHAPTER 7

independent lines of progression, the variation in transition rates be-
tween stages, and the temporal changes in transition rates over a life-
time.
The Gompertz model provides a widely used alternative description
of mortality rates. Let G(t) be the age-specific mortality rate of a Gom-
pertz model, and let a dot denote the derivative with respect to t. The
Gompertz model assumes that the mortality rate increases at a constant
rate γ with age:
˙ G = γG.
Solving this simple differential equation yields

γt
G(t) = ae ,
where a = G(0). From the differential equation, we can also write

˙ G dln (G)
= = γ,
G dt
which shows that the slope of the logarithm of mortality rate with re-
spect to time is the constant γ. Horiuchi and Wilmoth (1997, 1998)
defined d ln(G)/dt as the life table aging rate.
The Gompertz model arises when one assumes a constant life table
aging rate. As with the Weibull model, the Gompertz model describes
the pattern that follows from a simple assumption about age-related
changes in failure rates. Neither model provides insight into the pro-
cesses that influence age-related changes in disease. However, these
models can be useful when analyzing certain kinds of data. For exam-
ple, the observed age-specific incidence curves may be based on rela-
tively few observations. With relatively few data, it may be best to esti-
mate only the slope and intercept for the incidence curves and not try
to estimate nonlinearities.
When fitting a straight line on a log-log scale, one is estimating Weibull
parameters. Similarly, fitting a straight line of incidence versus time on
a log-linear scale estimates parameters from a Gompertz model. The
Weibull distribution may be the better choice because it provides a lin-
ear approximation to an underlying model of multistage progression
dynamics.