Page 109 - 20dynamics of cancer

94 CHAPTER 5

Not all changes during tumor development limit the rate of progres-
sion. A necessary change may happen very quickly following, for exam-
ple, expansion of a precancerous tumor to a large size. Such a step is
necessary for progression but does not limit the rate of progress, and
so does not determine the ages at which individuals carry tumors of
particular stages. I develop the basic theory under the assumption that
whatever determines a rate-limiting step, tumor progression requires
passing n such steps to develop into cancer. This section follows the
derivations given in Frank (2004a).
I gave a picture of the basic model in Figure 5.2. That picture formally
describes a set of differential equations. Because the picture and the
equations present the same information, one may choose to focus on
either. The equations are

˙ x 0 (t) =−u 0 x 0 (t) (5.2a)
˙ x j (t) = u j−1 x j−1 (t) − u j x j (t) i = j,...,n − 1 (5.2b)
˙ x n (t) = u n−1 x n−1 (t) , (5.2c)
where x i (t) is the fraction of the initial population born at time t = 0
that is in stage i at time t, with time measured in years. Usually, I assume
that when the cohort is born at t = 0, all individuals are in stage 0, that
is, x 0 (0) = 1, and the fraction of individuals in other stages is zero.
As time passes, some individuals move into later stages. The rate of
transition from stage i to stage i + 1is u i . The ˙ x’s are the derivatives of
x with respect to t.

5.6 Technical Definitions of Incidence and Acceleration

Two ways to characterize age-onset patterns play an important role
in analyzing cancer data and studying theories of cancer progression.
Incidence is the rate at which individuals develop cancer at particular
ages. Acceleration is the change in incidence rates. For example, positive
acceleration means that incidence increases with age.
This section provides some technical details for the definitions of
incidence and acceleration. One can get a rough idea of the main results
without these details, so some readers may wish to skip this section and
come back to it later.
Individuals who move into the final, nth stage develop cancer. They
pass into the final stage at the age-specific incidence rate ˙ x n (t), which