Page 138 - 20dynamics of cancer

THEORY II 123

From these parts, we can write the total age-specific incidence in the
population as
n−1 n−j−1
˙ z u j=0 p j (ut) /n − j − 1!
I = = n−1 n−i−1 j ,
1 − z (ut) /j!
i=0 p i j=0
and the log-log acceleration as

LLA = t ˙ I/I
⎛ ⎞
n−2 n−j−2 n−2 n−i−2 j
j=0 p j (ut) /n − j − 2! i=0 p i j=0 (ut) /j!
= ut ⎝ ⎠ .
n−1 n−j−1 − n−1 n−i−1 j

j=0 p j (ut) /n − j − 1! i=0 p i j=0 (ut) /j!
Figure 7.2 shows that genetic heterogeneity will typically have little
effect on aggregate patterns of cancer. That figure assumes a common
genotype with n = 10 steps and a rare mutant genotype with n−j steps,
where j is the number of stages in progression by which the mutant
genotype is advanced at birth. If the mutant advances only by j = 1,
then the patterns differ little between the genotypes. If, however, n is
small, as for retinoblastoma, then advancing one step, j = 1, can have a
significant effect (not shown). Mutants are usually thought to advance
progression by just one stage (Knudson 2001; Frank 2005), although
relatively little direct evidence exists.
If mutants advance progression by j = 4 stages, then the mutants
can have a significant impact on aggregate patterns, as shown in the
middle column of Figure 7.2 in which the mutant occurs at a frequency
of 0.01. However, the mutant must not be too rare—the right column of
Figure 7.2 shows that genetic heterogeneity has little effect for j = 4, if
the mutant occurs at a frequency of 0.001.

COMPARISON BETWEEN GENOTYPES: RATE-LIMITING STEPS
Mutant genotypes may often have little effect on aggregate pattern, as
shown in the previous section. However, if one can track the incidence
patterns separately for different genotypes, then much can be learned by
comparison of incidence patterns between genotypes. Indeed, relative
incidence patterns between genotypes may be the most powerful way
to learn about cancer progression and the link between particular genes
and cancer risk (Knudson 1993, 2001; Frank 2005).
In the next chapter, I will compare retinoblastoma incidence in hu-
mans between normal individuals and those who carry a mutation to