Page 141 - 20dynamics of cancer

126 CHAPTER 7

R ! R !



6




7PSTI

%KI

Figure 7.4 Ratio of incidence rates between normal and mutant genotypes
8
when there are multiple lines of progression. For these plots, L = 10 . To keep
the cumulative probability at 0.1 for the normal genotype at age 80, u = 0.00052
for n = 5, and u = 0.00753 for n = 10. All other aspects match Figure 7.3.

u, and the transition rate for the mutant genotype be v = δu, with δ> 1.
As in Eq. (7.4), I calculate the log-log slope of the ratio of incidences, in
this case taking the ratio of mutant to normal genotypes, R. The solution
follows from Eq. (6.3):
v u
δS n−2 S n−2
ΔLLA = LLA u − LLA v = ut v − u , (7.6)
S n−1 S n−1
α j i
where S = i=0 (αt) /i!
j
Figure 7.5 illustrates this theory. The left column shows the stan-
dard log-log incidence curves. The bottom curve plots the wild-type in-
cidence; the curves above show incidence for mutants with higher tran-
sition rates. The right column plots the difference in the slopes of the
incidence curves, ΔLLA, between the wild-type and the various mutant
genotypes.
8
The bottom right panel, Figure 7.5h, uses L = 10 independent lines
of progression within the tissue under study. With large L, almost all