Page 79 - 20dynamics of cancer

64 CHAPTER 4

matching the observations. In particular, this theory leads to an ex-
pected incidence of

n n−1
I(t) ≈ (Nu) t /(n − 1)!, (4.1)
where N is the number of cells at risk for transformation, and u is the
transformation rate per cell per unit time; thus, Nu is the rate at which
each transforming step occurs in the tissue.
The multistage theory assumes that changes to a tissue happen se-
quentially. Charles and Luce-Clausen (1942) explicitly discussed and
analyzed quantitatively two sequential mutations to a particular cell;
Muller (1951) discussed in a general way sequential accumulation of
mutations. Nordling (1953) introduced log-log plots of incidence data
to infer the number of steps. Nordling (1953) assumed that the steps
were sequential mutations to a cell lineage, and he suggested that a log-
log slope of n − 1 implied n mutational steps in carcinogenesis. From
data aggregated over various types of cancer, he inferred n ≈ 7.
Stocks (1953) followed Nordling (1953) with a mathematical analysis
to show how sequential accumulation of n changes to a cell leads to
log-log incidence plots with a slope of n − 1. Stocks (1953) had the right
idea, although from a mathematical point of view his analysis was rather
limited because he assumed that changes happened at a constant rate
per year and that at most one change per year occurred.
Armitage and Doll (1954) crystallized multistage theory by extending
the data analysis and mathematical development. With regard to the
data, they examined log-log plots for several distinct cancers rather than
aggregating data over different cancers as had been done by Nordling
(1953). With regard to theory, their mathematical model allowed dif-
ferent rates for different steps; they assumed continuous change rather
than arbitrarily limiting changes to one per year; and they noted that the
stages did not have to be genetic mutations but only had to be sequential
changes to cells. The style of data analysis and mathematical argument
formed the basis for the future development of multistage models.
Armitage and Doll (1954) rejected Fisher and Hollomon’s (1951) mul-
ticell theory in which the changes happen to different cells. Armitage
and Doll argued that if a chemical mutagen caused cancer by causing
mutations to several different cells, then incidence would increase with
dose raised to a high power. For example, in Eq. (4.1), if the mutation