Biomedical Engineering Reference
In-Depth Information
y ij given n i (j) and given the state variables, letfI k1 (t; i; r) be the number
of I k1 (t) cells in the rth individual who was exposed to the carcinogen
with dose level s i and let k1 (i) be the rate of the transition I k1 !I k
under exposure to the carcinogen with dose level s i . Among people who have
been exposed to the carcinogen with dose level s i , let P r (i; j) denote the
conditional probability given the state variables that the rth individual
would develop cancer during the j-th age group. Then, as shown in Tan 56 ,
Chapter 8, P r (i; j) is given by:
t j1 1
X
I k1 (t; i; r) k1 (i)g(1e R(i;j;r) k1 (i) );
P r (i; j) = expf
t=0
P
t j 1
where R(i; j; r) =
t=t j1 I k1 (t; i; r).
From the above it follows that the conditional probability density of
y ij given n i (j) and given the state variables I k1 (i; j) =fI k1 (t; i; r); t
t j ; r = 1; : : : ; n i (j)gis
y ij
n i (j)
Y
Y
n i (j)
y ij
Pfy ij jn i (j); I k1 (i; j)g=
P r (i; j)
[1P u (i; j)]: (6)
r=1
u=y ij +1
Let I k1 (t; i) denote the number of I k1 cells at time t under dose level
s i . When n i (j) and n i (j)y i (j) are very large and when n i (j)P r (i; j) are
nite for all r, the above probability is closely approximated by:
y ij
Y
1
y ij ! expf i (j)g
Pfy ij jn i (j); I k1 (i; j)g=
[n i (j)P r (i; j)];
(7)
r=1
where i (j) = n i (j)EP T (i; j) = n i (j)EP r (i; j); r = 1; : : : ; n i (j) and where
t j1 1
X
I k1 (t; i) k1 (i) + log(1e fR(i;j) k1 (i)g )g;
P T (i; j) = expf
t=0
P
t j 1
t=t j1 I k1 (t; i). (For Proof, see [63])
From equation (7), the conditional likelihood of the parameters given
data Y
with R(i; j) =
=fy ij ; i = 1; : : : ; m; j = 1; : : : ; ngand given the state variables is
m
n
Y
Y
LfjY ; I k1 g=
Pfy ij jn i (j); I k1 (i; j)g;
(8)
i=1
j=1
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