Page 134 - 20dynamics of cancer

THEORY II 119

less than 0.2, then we have m/L ≈ z(T), and the cumulative probability
of cancer at age t is p(t) ≈ z(t)L.
The transitions between stages are u ji (t), the rate of flow in the jth
pathway from stage i to stage i + 1. The transition rates may change
with time. These distinct, time-varying rates provide the most general
formulation. It is easy enough to keep the analysis at this level of gen-
erality, but then we have so many parameters and specific assumptions
for each case that it becomes hard to see what novel contributions are
made by having multiple pathways. To keep the emphasis on multiple
pathways for this section, I assume that all transitions in each pathway
are the same, u j , that transition rates do not vary over time, and that
distinct pathways indexed by j may have different transition rates.
Incidence at age t is
˙ z
I = ,
1 − z
where I is the incidence at age t; the numerator, ˙ z, is the total flow into
terminal stages at age t; and the denominator, 1 − z, is proportional to
the number of pathways that remain at risk at age t.
The rate of progression for a line is

k k

˙ x jn j
˙ z = ˙ x jn j 1 − x in i = (1 − z) .
j=1 i=j j=1 1 − x jn j
), so the previous two
The incidence per pathway is I j = ˙ x jn j /(1 − x jn j
equations can be combined to give

k k

˙ x jn j
I = I j = ,
j=1 j=1 1 − x jn j
in words, the total incidence per line is the sum of the incidences for
each pathway. Differentiating I yields
k

¨ x jn j 2
˙ I = + I j .
j=1 1 − x jn j
Earlier, I showed that log-log acceleration is LLA(t) = t ˙ I/I, which can be
expanded from the previous expressions.
Using this formula for LLA to make calculations requires applying the
=
pieces from earlier sections. In particular, ˙ x jn j = u j x jn j −1 and ¨ x jn j