STOCHASTIC AND STATE SPACE MODELS OF CARCINOGENESIS: SOME NEW MODELING APPROACHES - Quantitative Medical Data Analysis Using Mathematical Tools and Statistical Techniques - page 62

tained a series of irreversible genetic changes. Furthermore, the number of
stages and the number of pathways of the carcinogenesis process are signif-
icantly inuenced by environmental factors underlying the individual 41;55 .
Recently, it has been demonstrated that carcinogenesis is an evolution pro-
cess in cell populations referred to as a micro-evolution process; and each
cancer tumor is the outcome of growth of a most tted genetically altered
stem cell 8;28 .
In this chapter we will summarize recent results from cancer biology
and propose general stochastic models of carcinogenesis. For these models,
mathematical results by the classical methods are very dicult even under
some simplifying assumptions which may not be realistic in the real world;
see Remark 2. It follows that except possibly for the simplest two stage
model, analytical mathematical results remain to be developed and pub-
lished. In order to derive analytical mathematical results and to relax some
unrealistic assumptions, in this chapter we will provide new approaches
through stochastic differential equations to analyze these models.
For combining information from dierent sources and for easing problems
of identiability, we will combine these stochastic models with statistical
models to develop state space models for carcinogenesis. By using these
state space models, we will develop generalized Bayesian method and pre-
dictive inference procedures to estimate the unknown parameters and to
predict the state variables.
In Section 2, we will summarize recent results from cancer biology. Based
on these cancer biology, in Section 3, we will propose general stochastic
models of carcinogenesis. To derive analytical results and to extend the
models, in Section 4, we will propose an alternative approach to analyze
these stochastic models through stochastic dierential equations. For com-
bining information from stochastic models and statistical models and for
tting the models to cancer data, in Section 5, we will proceed to develop
state space models for the process of carcinogenesis. In Section 6, we
will illustrate the application of the models and methods by analyzing the
British data from physician's lung cancer and smoking. Finally in Section
6, we will discuss some possible applications of these models and methods.
Remark 1: The number of cells increases through somatic cell division
by entering into cell division cycle and complete the cell division cycle
giving rise to daughter cells. This has been referred to as cell proliferation.
When a cell enters into cell division cycle, there is also a chance that this
cell would dierentiate to become a dierentiated cell without completing
the cell division cycle. This is referred to as cell dierentiation. In terms