Page 139 - 20dynamics of cancer

124 CHAPTER 7

the retinoblastoma (Rb) gene (Section 8.1). I will also compare colon
cancer incidence between normal individuals and those who carry a mu-
tation to the APC gene. In both cases, the ratio of age-specific incidences
between normal and mutant individuals follows roughly along the curve
predicted by multistage theory if the mutants begin life one stage further
along in progression than do normal individuals (Frank 2005). Here, I
develop the theory for predicting the ratio of incidences between normal
and mutant genotypes.
Assume a simple model of progression, with n stages and a constant
rate of transition between stages, u. Mutant individuals begin life in
stage j, and so have n − j stages to progress to cancer. The results of
Section 6.2 provide the age-specific incidence for progression through n
stages, I n , so the ratio of incidences of normal and mutant individuals
is
j
(ut) n − j − 1 ! S n−j−1
R = I n /I n−j = , (7.3)
(n − 1)! S n−1
j i
where S j = i=0 (ut) /i!. When j = 1, then R ≈ ut/(n − 1) is often a
good approximation (Frank 2005).
When comparing the incidences between two genotypes, it may often
be useful to look at the slope of log(R) versus log(t), which is

d log (R) d log (I n ) − d log I n−j
ΔLLA = =
d log (t) d log (t)
= LLA n − LLA n−j

S n−2 S n−j−2
= j − ut − , (7.4)
S n−1 S n−j−1
where LLA k , the log-log acceleration for a cancer with k stages, is given
in Eq. (6.3). The slope of log(R) versus log(t) is equal to the difference
in LLA, so I will sometimes refer to this slope as ΔLLA.
When progression causes acceleration to drop at later ages, then the
slope of log(R) tends to decline with age. For example, in Figure 7.3,
cancer develops through a single line of progression, L = 1. Often, a
small number of progression lines tends to cause acceleration to drop at
later ages. By contrast, in Figure 7.4, cancer develops through many lines
8
of progression, L = 10 , which keeps acceleration nearly constant across
all ages. Consequently, the ratio of incidences has a constant slope equal