Page 150 - 20dynamics of cancer

THEORY II 135

The fraction of the population that has not yet progressed to cancer
is
b = 1 − p (t, u) f (u) du, (7.7)
which is one minus the average probability of progression per individual.
With these expressions, incidence is I(t) = a/b, and log-log accelera-
tion is
d log (I)
˙ a ˙
b
LLA (t) = = t ˙ I/I = t − . (7.8)
d log (t) a b
˙
Because b =−a, we can also write

˙
˙ a b ˙ a a ˙ a
LLA (t) = t − = t + = t + I . (7.9)
a b a b a
To make calculations, we need to express a and ˙ a in terms of x i , for
which we have explicit solutions. First, to expand a, we need ˙ p = L˙ x n (1−
x n ) L−1 , with ˙ x n = ux n−1 (see Eqs. 6.1). Second, ˙ a = ¨ p(t, u)f(u)du,
2
with ¨ p = L[¨ x n (1 − x n ) L−1 − ˙ x (L − 1)(1 − x n ) L−2 ] and ¨ x n = u˙ x n−1 =
n
2
u (x n−2 − x n−1 ).
CONCLUSIONS

Increasing heterogeneity causes a strong decline in the acceleration
of cancer. Heterogeneity could, for example, cause a cancer with n = 10
stages to have acceleration values below 5 that decline with age. Thus,
low values of acceleration (slopes of incidences curves) do not imply a
limited number of stages in progression. Heterogeneity must be nearly
universal in natural populations, so heterogeneity should be analyzed
when trying to understand differences in epidemiological patterns be-
tween populations.
Heterogeneity in progression rates causes cancer to be a form of trun-
cation selection, in which those above a threshold almost certainly de-
velop cancer and those below a threshold rarely develop cancer. Under
truncation selection, the amount of variation in progression rates will
play a more important role than the average rate of progression in de-
termining what fraction of the population develops cancer and at what
ages they do so. To understand the distribution of cancer, it may be
more important to measure heterogeneity than to measure the average
value of processes that determine rates of progression.