STOCHASTIC AND STATE SPACE MODELS OF CARCINOGENESIS: SOME NEW MODELING APPROACHES - Quantitative Medical Data Analysis Using Mathematical Tools and Statistical Techniques - page 70

3.1. The Extended Multi-Event Model of Carcinogenesis

The most general model for a single pathway is the extended k-stage
(k2) multi-event model proposed by Tan and co-workers 58;63 . This is
an extension of the multi-event model rst proposed by Chu 11 and studied
by Tan 55 and Little 36 . It views carcinogenesis as the end point of k (k2)
discrete, heritable and irreversible events (mutations, genetic changes or
epigenetic changes) with intermediate cells subjected to stochastic prolifer-
ation and dierentiation. It takes into account cancer progression by follow-
ing Yang and Chen 69 to postulate that cancer tumors develop from primary
I k cells by clonal expansion (i.e. stochastic birth-death process), where a
primary I k cell is an I k cell which arise directly from an I k1 cell.
Let N denote normal stem cells, T the cancer tumors and I j the jth
stage initiated cells arising from the (j1)th stage initiated cells (j =
1; : : : ; k) by mutation or some genetic changes. Then the model assumes
N!I 1 !I 2 !!I k with the N cells and the I j cells subject
to stochastic proliferation and dierentiation. The cancer tumors develop
from primary I k cells by clonal expansion.

Ras Src Second Copy of p53

N I 1 I 2 I 3 I 4 I 5 I 6

Myc

APC in 5q Second Copy

of APC

Second Copy
of Smad4

Smad4 in 18q P53 in 17P

Carcinomas

Fig. 1. The APC--Catenin-TCF-Myc Pathway of Human Colon Cancer. Here, N =
Normal stem cell, I j = The jth-stage initiated cell in the LOH pathway,#denotes
mutation, inactivation or loss of suppressor genes (APC, Smad, p53), and&denotes
mutation or activation of oncogenes (Ras, Src, Myc).