Page 315 - 20dynamics of cancer

300 CHAPTER 14

acceleration with age. Those declining LLA curves fit Pike’s prediction.
However, notice in Figure A.8 that cancers of the kidney, esophagus, and
larynx have declining patterns of LLA that closely match the pattern of
decline for ovarian cancer.
The slowing of mitosis with age in the female reproductive tissues
may very well reduce the LLA of those tissues. But the fact that non-
reproductive tissues show similar declines suggests that ubiquitous as-
pects of aging may dominate the patterns of incidence.

MATHEMATICAL ANALYSIS OF MITOTIC AGE AND INCIDENCE
Pike developed a mathematical expression to link mitotic age to inci-
dence (Pike et al. 1983, 2004). That formulation arises from the correct
notion that the age-specific rate of mitosis may influence age-specific
incidence. However, Pike’s particular formulation incorrectly expresses
the relation between mitosis and incidence. In this section, I show Pike’s
formulation, explain why it is wrong, and discuss the correct way to an-
alyze the problem.
I begin by following Pike’s formulation, but I modify the notation to
match mine. Pike began with the widely used approximation for inci-
dence
I(t) ≈ ct n−1 ,
where t is time since birth, or, equivalently, age, and c is a constant
that absorbs all terms independent of age. This formulation assumes
that the risk factors driving cancer happen at the same constant rate
throughout life. If mitosis is the main risk factor, and the rate of mitosis
varies with age, then instead of measuring the accumulation of time by
t, one should measure the accumulation of mitoses over time, or mitotic
age, m(t), where m is the cumulative number of mitoses at age t.
Pike therefore substituted mitotic age for age and presented the for-
mula
n−1
I(t) ≈ c[m (t)] . (14.1)
This formulation is incorrect. For example, suppose that the age-specific
rate of mitosis slows to near zero at age 65. The cumulative number
of mitoses since birth at age 65, m(65), may be a large number, and
so according to Pike, the incidence will be high at age 65. However,
incidence at age 65 is the rate of new cases at that age. If mitoses have
slowed to almost zero, then this particular form of multistage theory