Page 201 - 20dynamics of cancer

186 CHAPTER 9

a five-stage model: the six stages of progression minus one stage that
is not rate limiting in the presence of the carcinogen. After cessation,
progression follows the full six rate-limiting stages, and so incidence
instantly drops to the rate for a six-stage model.
If the carcinogen affects only the penultimate transition, i = 4, then
during carcinogen application, individuals move very rapidly from stage
4 to stage 5, where they await the final transforming event at the nor-
mal, background rate. By essentially skipping a stage during carcinogen
application, the incidence follows a five-stage model. After cessation,
almost all new cancers arise from the pool of individuals in stage 5 who
await the final transition. When transformation occurs by a single ran-
dom event, the incidence rate remains flat over time. The final event is
as likely to happen this year as next year or a later year. If the carcino-
gen affected only the third transition, i = 3, then after cessation most
cancers would arise in the pool of individuals that require two further
steps, causing incidence to increase only slowly with time as in a model
with only two stages.
In Figure 9.10c,d, the carcinogen is applied only between age 25 and
age 80. The carcinogen has relatively little effect when it increases the
earliest transition, i = 0, because that transition has already occurred
by age 25 in many of the individuals who ultimately progress to cancer.
For the next transition, i = 1, fewer would have passed that step by age
25, and so more will be affected by the carcinogen. For the later steps,
almost no one would have passed those steps by age 25, and so the
carcinogen increases incidence equally for all of the later transitions.
In Figure 9.10e,f, the carcinogen is applied only between age 25 and
age 60, after which carcinogen application ceases. This case matches the
problem of cessation smoking, with onset of smoking in the first third of
life and cessation in the last third of life. The patterns can be understood
from the previous cases. If smoking affects only an early stage, then the
earlier the stage, the less the effect, because the earliest stages are more
likely to have been passed already before the onset of smoking and the
acceleration of that stage. If smoking affects only a later transition, i,
then after cessation, the pool of individuals most susceptible has n − i
steps remaining; if smoking affects the final transition, no excess pool
of susceptibles exists, and incidence reverts to the background rate.
The first theoretical studies of smoking cessation considered models
in which smoking affected only one stage (Whittemore 1977; Day and