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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