Biomedical Engineering Reference
In-Depth Information
(see [2, 3, 7, 14, 17, 30, 32, 40, 43, 49, 64]).
(2). The estimates of the mutation rates
j
(i) =
j
for j = 1; 2; 3 are
in general independent of the dose level x
i
. These estimates are of
order 10
5
and do not dier signicantly from one another.
(3). The results in Table 3 indicate that for non-smokers, the death rates
d
j
(0) (j = 1; 2; 3) are slightly greater than the birth rates b
j
(0) so
that the proliferation rates
j
(0) = b
j
(0)d
j
(0) are negative. For
smokers, however, the proliferation rates
j
(i) = b
j
(i)d
j
(i)(i > 0)
are positive and increases as dose level increases (the only exception
is
2
(3)). This is not surprising since most of the genes are tumor
suppressor genes which are involved in cell dierentiation and cell
proliferation and apoptosis (e.g., p53) (see [20, 44, 67]).
(4). From Table 3, we observed that the estimates of
3
(i) are of or-
der 10
2
which are considerably greater that the estimates of
1
(i)
respectively. The estimates of
1
(i) are of order 10
3
and are con-
siderably greater than the estimates of
2
(i) respectively. The esti-
mates of
2
(i) are of order 10
4
and
2
(3) assumed negative value.
One may explain these observations by noting the results: (1) Sig-
nicant cell proliferation may trickle apoptosis leading to increased
cell death unless the apoptosis gene (p53) has been inactivated and
(2) the inactivation of the apoptosis gene (p53) occurred in the very
last stage; see [67].
7. Conclusions and Summary
Based on most recent biological studies, in this chapter we have presented
some stochastic models for carcinogenesis. To develop mathematical analy-
sis for these models, the traditional approach based on theories of Markov
process is extremely dicult and has some serious drawbacks. To get around
these diculties, in this chapter we have proposed an alternative approach
through stochastic dierential equations and state space models for car-
cinogenesis. This provides an unique approach to combine information from
both stochastic models and statistical models of carcinogenesis. By using
state space models, we have developed a general procedure via multiple
Gibbs sampling method to estimate the unknown parameters. In this pa-
per we have used the multi-event model as an example to illustrate the
basic approach and our new modeling ideas.
To illustrate some applications of results of this chapter, we have applied
the model and method to the British physician data on lung cancer and
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