STOCHASTIC AND STATE SPACE MODELS OF CARCINOGENESIS: SOME NEW MODELING APPROACHES - Quantitative Medical Data Analysis Using Mathematical Tools and Statistical Techniques - page 61

CHAPTER 4
STOCHASTIC AND STATE SPACE MODELS OF
CARCINOGENESIS: SOME NEW MODELING
APPROACHES
W. Y. Tan a and L. J. Zhang
Department of Mathematical Sciences,
The University of Memphis, Memphis, TN 38152-6429, USA
E-mail: a [e-mail]
By surveying recent studies by molecular biologists and cancer geneti-
cists, in this chapter we have proposed general stochastic models of car-
cinogenesis and provided biological evidences for these models. Because
most of these models are quite complicated far beyond the scope of the
MVK two-stage model, the traditional Markov theory approach becomes
too complicated to obtain analytical results. To develop these stochastic
models, in this chapter we thus propose an alternative approach through
stochastic dierential equations. Given observed cancer incidence data,
we further combine these stochastic models with statistical models to de-
velop state space models for carcinogenesis. By using these state space
models, we then develop a generalized Bayesian procedure to estimate
the unknown parameters and to predict state variables via multi-level
Gibbs sampling procedures. In this chapter we have used the multi-event
model as an example to illustrate our modeling approach and some basic
theories.
Keywords: Generalized Bayesian procedures, Observation model,
Multi-event model of carcinogenesis, Multi-level Gibbs sampling pro-
cedures, Multiple pathway model of carcinogenesis, State space model,
Stochastic dierential equations, Stochastic system model.
1. Introduction
It is now universally recognized that carcinogenesis is a multi-stage ran-
dom process involving genetic changes, epigenetic changes and stochastic
proliferation and dierentiation of normal stem cells and genetically altered
stem cells (see [41, 55] and Remark 1). Specically, studies in molecular
biology have conrmed that each cancer tumor develops through stochas-
tic proliferation and dierentiation from a single stem cell which has sus-