Page 112 - 20dynamics of cancer

THEORY I 97

transition may rise when a precancerous cell expands into a large clone,
in which a subsequent change to any one of the clonal cells could cause
progression to the next stage in carcinogenesis. As the clone grows
larger, the target size for a transition increases. Time-varying rates often
cause a rise in acceleration to a midlife peak, followed by a late-life
decline in acceleration.

6.1 Approach

This chapter and the following one develop the theory of progression
dynamics. Most of the sections contain some mathematics. I use the
following structure to make the presentation accessible. A section with
mathematics begins with a précis that highlights the main results. The
mathematical details follow, often with some illustrations to emphasize
the key points. The section ends with a brief statement of the conclu-
sions.
I developed much of the following original theory for this book. Al-
though the overall structure and many of the particular results are new,
my mathematical work grew from a rich and highly developed field. I
gave an overview of the history in Chapter 4. I particularly wish to ac-
knowledge the pioneering contributions of Armitage and Doll, Knudson,
and Moolgavkar, who have been most influential in my own studies.

6.2 Solution with Equal Transition Rates

PR ´ ECIS
I start with the linear chain of stepwise progression illustrated in Fig-
ure 5.2. No type of cancer will always follow the same steps with fixed
transition rates between steps. But a thorough understanding of the
simplest case puts us in a better position to study more realistic as-
sumptions.
In this section, I assume that the transitions between steps happen at
the same rate, u, and that everyone is born in stage 0. Individuals who
progress through the nth stage develop cancer.
With these assumptions, the fraction of the population at age t in each
precancerous stage is given by the Poisson distribution with a mean of
ut. Intuitively, ut would be the average number of transitions passed if
there were unlimited stages, because u is the transition rate per stage