Page 111 - 20dynamics of cancer

6 Theory I

To test hypotheses about how particular biochemical processes affect
cancer, we need quantitative predictions for how biochemical changes
alter the age of cancer onset. This chapter develops the quantitative
theory of progression dynamics.
The first section outlines my strategy for presentation. I divide each
quantitative analysis into a précis that gives the main points, a mathe-
matical presentation of the analytical details, and a set of conclusions.
The second section solves the basic model of multistage progression
dynamics. In that model, individuals progress through a series of stages
with the same constant transition rate from each stage to the next. That
model follows the classical analysis of multistage progression, leading
to the conclusion that a log-log plot of cancer incidence versus age is
approximately linear with a slope of n − 1, where n is the number of
rate-limiting steps in progression. The slope of n − 1 measures the ac-
celeration of cancer with age. I present an exact solution for the model,
which shows that, under some conditions, the incidence curve flattens
late in life and drops below the linear approximation, causing a late-life
decline in acceleration.
The third section analyzes parallel lines of progression within indi-
viduals. The models follow the stages of cells or tissue compartments,
in which different cells or compartments may be in different stages of
progression within the same tissue. The greater the number of indepen-
dent lines of progression, the slower progression must be in each line to
keep the overall incidence from rising to very high levels. The smaller
the number of lines, the more strongly acceleration tends to decline later
in life.
The fourth section discusses how incidence changes when the rates
of transition vary between different stages in progression. The greater
the variation in rates of transition, the more strongly the acceleration of
cancer tends to decline with advancing age.
The fifth section studies what happens when rates of transition vary
with age. Rates may increase with age if DNA repair capacity or other
checks on cell-cycle integrity decline with age. Alternatively, rates of