Page 116 - 20dynamics of cancer

THEORY I 101

the final malignant state by time t is x n (t). The cumulative probability
of cancer is the probability that at least one of the L lines has progressed
to the malignant state. This cumulative probability of cancer by age t is

L
p(t) = 1 − [1 − x n (t)] . (6.5)

For large L and small x n (t), the Poisson approximation is very accurate,
p(t) ≈ 1 − e −x n (t)L . The Poisson distribution with mean x n (t)L gives the
distribution of the number of independent tumors per individual at age
t.
Incidence is the rate of new cases divided by the fraction of the pop-
ulation at risk. Using the definition for p(t) in Eq. (6.5) and dropping t
from the notation,
L−1
˙ p L˙ x n (1 − x n ) L˙ x n
I = = L = .
1 − p (1 − x n ) 1 − x n
Comparing this result with Eq. (6.2) shows that having L independent
lines of progression within an individual simply increases incidence by
a constant value L. Log-log acceleration is independent of constant mul-
tiples of incidence, as shown in Eq. (5.3), so log-log acceleration is inde-
pendent of L and is given by Eq. (6.3).
What does change is the value of u that one must assume in order for a
certain total fraction of the population to have cancer by a particular age,
T. If the fraction of the population with cancer is m, then u is obtained by
solving m = p(T) for u, using Eq. (6.5) for p(T) and Eq. (6.4) for x n (T).
As the number of independent lines, L, increases, slower transitions
must be assumed to give the same overall incidence. This reduction of
u causes each line to progress more slowly, but, by chance, one of the
many separate lines within an individual progresses to the final stage
with probability p(T).
If, under the assumptions of this model, individuals rarely have more
than one independent tumor, then the per-line probability of progres-
sion is approximately m/L, the total probability of progression per in-
dividual, m, divided by the number of lines, L. It is often most informa-
tive to evaluate progression on a per-line basis and to present results
for particular levels of m/L ≈ x n (T). In this model, multiple tumors per
individual are rare when m = p(T) < 0.2.