Page 223 - 20dynamics of cancer

208 CHAPTER 10

Several theories of age-specific mortality have been based on multiple
stages or multiple states of progression. Specific models almost always
derive from reliability theory—the engineering field that evaluates time
to failure for manufactured devices (Gavrilov and Gavrilova 2001).
In engineering, components of a device that protect against failure
may be arranged in various pathways. Serial protection means that sys-
tem failure follows a pathway in which first one component fails, fol-
lowed by a second component, and so on; the probability of failure of
later components in the sequence occurs conditionally on the failure of
earlier components in the sequence. Parallel protection describes func-
tional redundancy, in which any single functioning component keeps the
system going; failure occurs only after all redundant components fail;
and component failures occur independently. Various combinations of
serial and parallel pathways may be designed.
Reliability theory calculates time to failure (mortality) based on as-
sumptions about component failure rates and pathways by which com-
ponents are related. Obviously, the multistage theory I developed ear-
lier forms a branch of reliability theory. However, the reliability theory
found in texts focuses on engineering problems, and those problems
rarely match the particular biological scenarios for cancer progression.
So, although the principles exist in reliability texts, many of the specific
results in my theory chapters are new.
Gavrilov and Gavrilova (2001) provided a nice review of reliability the-
ory applied to human mortality. They note that when system failure
depends on the simultaneous failure of several components, the accel-
eration of age-specific mortality declines later in life. I have already
discussed the idea several times. If system failure requires failure of n
components, then log-log acceleration (LLA) is n − 1. As systems age
and components fail, say a have failed, then LLA tends to drop toward
n − a − 1. Details vary, but the idea holds widely. Vaupel (2003) gives
a good, intuitive description of how multicomponent reliability may ex-
plain the late-life mortality plateau.
In light of reliability theory, we can state more generally an expla-
nation for the late-life decline in the acceleration of mortality (Frank
2004a). Suppose a measurable disease outcome, such as death, occurs
only after several different rate-limiting events have occurred. Each
event has at least some aspect of its time course that is independent
of other events. If so, then the dynamics of onset will not follow the