Page 110 - 20dynamics of cancer

PROGRESSION DYNAMICS 95

is roughly the probability of developing cancer per year at age t. The
age-specific incidence is the fraction of all individuals in the cohort who
develop cancer for the first time at age t, which is the probability of
developing cancer at age t divided by the fraction of individuals, S(t),
who have not yet developed cancer by that age. In symbols, we write
that the age-specific incidence is I(t) = ˙ x n (t)/S(t).
The incidence, I(t), is the rate at which cancer cases accumulate at a
particular age. I frequently refer to the acceleration of cancer, which is
how fast the rate, I(t), changes at a particular age, t. The most useful
measure of acceleration in multistage models scales incidence and time
logarithmically (Frank 2004a, 2004b).
Use of logarithms provides a scale-free measure of change. In other
words, differences on a logarithmic scale summarize percentage change
in a variable independently of the value of the variable. This can be seen
by examining the derivative of the logarithm for a variable x, which is
dx
d log (x) = .
x
The right side is the change in x divided by x, which measures the frac-
tional change in x independently of how large or small x is.
For example, if we wanted to measure the percentage increase in the
age-specific incidence for a given percentage increase in age, then we
need to measure in a scale-free way changes in both age-specific inci-
dence and age. We obtain a scale-free measure by defining the log-log
acceleration (LLA) at age t as
dI (t) /I (t) d log (I (t))
LLA (t) = = . (5.3)
dt/t d log (t)
The derivative of incidence, dI(t)/dt, is the age-specific acceleration,
so LLA is just a normalized (nondimensional) measure of age-specific
acceleration.

5.7 Summary
This chapter introduced the quantitative tools needed to build mod-
els of cancer progression. Such models make predictions about how
particular genetic or physiological changes alter age-specific incidence.
The ability to make such predictions successfully defines a causal under-
standing of cancer. The next chapter begins my mathematical analysis
of the ways in which particular causes affect age-specific incidence.