Page 132 - 20dynamics of cancer

THEORY II 117

progression over all pathways in a tissue. In this section, I analyze inci-
dence and acceleration when aggregated over multiple underlying path-
ways of progression.
If one pathway progresses rapidly and another slowly, then incidence
and acceleration will shift with age from dominance by the early pathway
to dominance by the late pathway. For example, the early pathway may
have few steps and low acceleration, whereas the late pathway may have
many steps and high acceleration. Early in life, most cases arise from
the early, low-acceleration pathway; late in life, most cases arise from
the late, high-acceleration pathway.
In this example, the aggregate acceleration curve may be low early in
life, rise to a peak in midlife when dominated by the later pathway, and
then decline as the acceleration of the later pathway decays with ad-
vancing age. Aggregated pathways provide an alternative explanation
for midlife peaks in acceleration. In the Conclusions at the end of this
section, Figure 7.1 illustrates the main points and provides an intuitive
sense of how multiple pathways affect incidence and acceleration. (Var-
ious multipathway models are scattered throughout the literature. See
the references in Mao et al. (1998)).

DETAILS

For a particular tissue, I assume k distinct pathways to cancer indexed
by j = 1,...,k. Each pathway has n j transitions and i = 0,...,n j states.
The probability of being in state i of pathway j at age t is x ji (t). A tissue
is subdivided into L distinct lines of progression. A line might be a stem
cell lineage, a compartment of the tissue, or some other architecturally
defined component. Each line is an independent replicate of the system
with all k distinct pathways.
Cancer arises if any of the Lk distinct pathways has reached its final
state. All pathways begin in state 0 such that x j0 (0) = 1 and x ji (0) = 0
for all i> 0. I interpret x ji (t) as the probability that pathway j is in
state i at time t.
The probability that a particular line progresses to malignancy is the
probability that at least one pathway in that line has progressed to the
final state,
k

z(t) = 1 − 1 − x jn j (t) . (7.1)
j=1