STOCHASTIC AND STATE SPACE MODELS OF CARCINOGENESIS: SOME NEW MODELING APPROACHES - Quantitative Medical Data Analysis Using Mathematical Tools and Statistical Techniques - page 73

derived by the FAP and/or the HNPCC pathways while for the age groups
after 50 years old, most of the colon cancer cases would derive from the LOH
pathway. To reect the real situation, we will thus assume that ! i (j) =
! i P i (j)=f

P
5
u=1 ! u P u (j)g, where P i (j) is the conditional probability that
the individual will develop colon cancer during the jth age group given
that this person develops colon cancer by following the ith pathway.

4. Some New Approaches for Analyzing Stochastic Models
of Carcinogenesis

To develop mathematical theories for the stochastic models of carcino-
genesis, the traditional approach in the literature 55 is by way of Markov
theories. The basic approach along this line consists of the following four
basic steps: (a) Deriving the probability generating function (PGF) of the
number of cancer tumors, (b) deriving the incidence function of cancer
tumors, (c) deriving the probability distribution of time to tumor onset,
and (d) deriving the probabilities of the number of cancer tumors. This
approach has been described and illustrated in detail in [55].
Using the above approach, theoretically one may derive some useful
information for stochastic models of carcinogenesis. However, a careful
scrutiny would reveal that the above approach suers from several draw-
backs: (1) The process may not be Markov so that the above approach is
not applicable. For example, if one can not ignore cancer progression, then
the number of cancer tumors is not Markov since it depends on the time
when the last stage initiated cell is generated (see [54, 58, 60, 61, 63]) ; see
also Remark 2. (2) As illustrated in [55], it is mathematically manageable
only for a two stage model under very restrictive assumptions as described
in Remark 2; further many of these assumptions have signicant impacts
on cancer incidence (see [54, 55, 58, 60, 61, 63, 68]). (3) It is extremely di-
cult, if not impossible, to t and to adapt to cancer data, especially beyond
the simplest MVK two stage model. (4) The cancer stages and many of the
parameters are not identiable when three or more stages are involved. In
fact, as shown by Hanin and Yakovlev 24 , even for the simple homogeneous
two-stage MVK model, it is not possible to estimatefb 1 ; d 1 ; 0 gand 1 sep-
arately; hence, not all parameters are estimable by using the above Markov
approach unless some other data and some further external information
about the parameters is available.
Because of the above diculties, we have developed an alternative ap-
proach to developed stochastic models of carcinogenesis. As shown by Tan