Page 136 - 20dynamics of cancer

THEORY II 121

of cases, aggregate acceleration may be dominated by the lower accel-
eration associated with mutants, which have fewer steps in progression
than do normal genotypes. Late in life, aggregate acceleration will be
dominated by the normal genotype, which has more steps and a higher
acceleration. The net effect may be low acceleration early when domi-
nated by the mutants, a rise to a midlife peak as dominance switches to
the normal individuals, and a late-life decline in acceleration following
the trend set by the normal genotype (Figure 7.2).
In the second case, one can distinguish between mutant and normal
genotypes. This is an important case, because it allows one to test di-
rectly the role of particular genes by comparison of mutant and normal
patterns of incidence and acceleration. I show that the ratio, R, of nor-
mal to mutant incidence provides a good way to compare genotypes. The
change in this ratio with age on log-log scales is the difference in acceler-
ation between the normal and mutant genotype. Under simple models of
progression dynamics, the observed difference in acceleration provides
an estimate for the difference in the number of rate-limiting stages in
progression.

DETAILS
I assume a single pathway of progression in each line and a single
line of progression per tissue, that is, k = L = 1. Extensions for multiple
pathways and lines can be obtained by following the methods in prior
sections. I assume the pathway of progression has n rate-limiting steps,
with the transition rate between stages, u. Here, u is the same between
all stages and does not vary with time.
A fraction of the population, p j , has mutations that start them j steps
along the pathway of progression; in other words, those individuals have
n − j steps remaining before cancer. I refer to individuals that start j
steps along as members of class j or as being born in the jth stage of
progression.
AGGREGATE PATTERNS
If different genotypes cannot be distinguished, then all measurements
on cancer incidence will combine the incidences for the different geno-
types. The aggregate rate of transition into the final, cancerous state is
n−1
˙ z = j=0 p j ˙ x jn−j , where x ji is the probability that an individual born
in the jth stage has progressed a further i stages. The population-wide