Biomedical Engineering Reference
In-Depth Information
and Chen 58 through pgf (probability generation function) method, this al-
ternative approach is equivalent to the above approach but is more powerful
and can get more information. In this section we illustrate how to derive
some basic results by using an extended k-stage multi-event model of car-
cinogenesis with (k2 ) carcinogenesis as an example.
4.1. Stochastic Dierential Equations
For the extended k-stage multi-event model, it is assumed that the number
N 0 (t) = I 0 (t) of normal stem cells are deterministic functions of time since
these numbers are usually very large. Hence, for this model, the state vari-
ables are (I i (t); i = 1; : : : ; k1; T (t)), where I i (t) = number of I i cells at
time t, and T (t) =number of cancer tumors at time t.
To derive stochastic dierential equations for I j (t); j = 1; : : : ; k1, note:
(i) The numbers offI j ; j = 1; : : : ; k1gcells at time t + t derive from
the numbers offI j ; j = 0; 1; : : : ; k1gcells at time t through stochastic
birth and death processes and mutation processes. (ii) The birth-death-
mutation processes during the small interval with length t is equivalent
to multinomial distributions.
Dene for j = 1; : : : ; k1:
B j (t) =Number of new I j cells generated by stochastic cell prolif-
eration (birth) of I j cells during (t; t + t],
D j (t) =Number of death of I j cells during (t; t + dt],
M j (t) =Number of new I j cells arising from I j1 cells by mutation
or some genetic changes during (t; t + t]; j = 1; 2; : : : ; k.
Then, for j = 0; 1; : : : ; k1, the above principle leads to:
M 1 (t)P oissonf 0 (t)tg;
where 0 (t) = I 0 (t) 0 (t), and for j = 1; : : : ; k1,
[B j (t); D j (t); M j+1 (t)]jI j (t)ML[I j (t); b j (t)t; d j (t)t; j (t)t]:
It follows that EM 1 (t) = 0 (t)t and for j = 1; : : : ; k1,
EfB j (t)jI j (t)g= I j (t)b j (t)t;
EfD j (t)jI j (t)g= I j (t)d j (t)t;
EfM j+1 (t)jI j (t)g= I j (t) j (t)t;
By the conservation law,we have then for i = 1; : : : ; k1,
I i (t + t) = I i (t) + M i (t) + B i (t)D i (t):
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