Biomedical Engineering Reference
In-Depth Information
The type 1 cell proliferation rate is constant, while the type 2 cell proliferation
rate decreases with their local density number. As a consequence the nonnegative
function
h
d
(
.,
s
)
∈
C
b
(R
)
,for
s
∈
S
, may be modelled in the following way:
h
(
))
=
λ
1
δ
C
i
1
Z
i
X
i
C
i
h
(
(
t
)) =
(
t
)
,
(
t
+
λ
2
δ
C
i
,
(25)
(
t
)
,
1
(
t
)
,
2
Q
[
2
]
N
+
(
∗
)(
X
i
(
))
1
N
K
t
where
λ
1
λ
2
and
K
is just a regularizing kernel.
The counting process
N
λ
1
,
λ
2
∈
R
+
, with
d
(
t
)=
Φ
([
0
,
t
]
×
R
×
S)
has intensity given by
N
)
i
=
1
h
(
Z
i
(
t
d
ν
(
)=
Λ
(R
×
S)
=
(
))
d
t
d
t
t
d
t
t
Cell State Evolution.
The change of state of each cell is described via a continuous
time Markov chain. The associated time-dependent stochastic matrix for the
i
-th cell
is given by the following:
⎡
⎤
−
(
m
12
+
m
13
)
m
12
m
13
Z
i
⎣
⎦
,
M
(
(
t
)
,
t
)=
m
21
−
(
m
21
+
m
23
)
m
23
,
(26)
0
0
0
where
C
i
C
i
prob
{
(
t
+
Δ
t
)=
k
|
(
t
)=
h
}
Z
i
m
hk
:
=
m
hk
(
(
t
)) =
lim
Δ
Δ
t
t
→
0
are given by
)=
λ
12
u
λ
13
m
12
(
x
,
1
)
,
m
13
(
x
,
1
)=
,
(27)
(
,
Q
[
1
]
N
x
t
(
∗
K
)(
x
)
Q
[
2
N
∗
m
21
(
x
,
2
)=
λ
21
(
K
)(
x
)
,
m
23
(
x
,
2
)=
λ
23
,
(28)
for
i
,
j
=
1
,
2
,
3,
λ
ij
>
0.
Evolution of the Fields.
VEGF diffuses and naturally degrades; furthermore, as a
matter of modelling simplification, we assume that VEGF is produced at the front
of type 2 cells [
19
]. So, given a region of interest
E
d
⊂
R
=
(for
d
2 or 3), for
d
, the concentration of VEGF
g
(
,
)
∈
R
+
×
R
(
,
)
any
t
x
x
t
is subject to the following
evolution equation:
j
=
1
ε
(
·,
C
j
(
t
))
(
x
,
2
)
∗
K
N
(
X
j
N
(
t
)
∂
g
(
x
,
t
)
)+
α
g
1
N
=
−
d
g
g
(
x
,
t
)+
D
g
Δ
g
(
x
,
t
(
t
))
.
(29)
∂
t
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