Biomedical Engineering Reference
In-Depth Information
Equation (
45
) may be seen as the weak form of the following partial differential
equation for the density
p
(
t
,
x
,
v
)
:
∂
∂
(
C
t
p
(
t
,
x
,
v
)=
−
v
·
∇
x
p
(
t
,
x
,
v
)+
k
∇
v
·
(
vp
(
t
,
x
,
v
))+
α
h
(
t
,
x
))
p
(
t
,
x
,
v
0
)
F
p
2
2
Δ
v
p
+
σ
C
)
,
f
−
∇
v
·
(
t
,
x
(
t
,
x
)
(
t
,
x
,
v
)
(
t
,
x
,
v
)
.
(49)
Given the limit system (
45
)-(
48
), one may recover the dynamics of the individu-
als coupling such a system with
dX
i
v
i
(
t
)=
(
t
)
,
[
C
,
f
d
v
i
kv
i
d
W
i
(
)=
−
(
)+
]
+
(
)
.
t
t
F
d
t
t
3.2
Retinal Angiogenesis
As considered above, let us perform a heuristic derivation of a continuum dynamics.
Note that also in this case we may state that the martingale Eq. (
33
)vanishesin
probability. Let us assume that, as
N
→
∞
, the measure
Q
N
(
t
)
admits a limit measure
Q
(
t
)
with a density
p
(
x
,
s
,
t
)
, such that, for any
A
⊂
S
,
)=
s
∈
A
p
(
x
,
s
,
t
)
d
x
.
Q
(
t
)(
d
x
×
A
This is equivalent to say that for any
s
∈
S
, the empirical measure related to the cell
of type
s
,
Q
[
s
N
(
given by Eq. (
31
) converges to a measure
Q
[
s
]
(
t
)
t
)
such that
Q
[
s
]
(
t
)(
d
x
)=
p
(
x
,
s
,
t
)
d
x
=
:
p
s
(
x
,
t
)
d
x
.
(50)
From Eqs. (
32
)and(
50
), together with Eqs. (
29
)and(
30
), a weak form for the
time evolution of the continuum densities
p
s
(
x
,
t
)
,
s
∈
S
, is obtained
s
∈
S
=
s
∈
S
f
(
x
,
s
)
p
s
(
x
,
t
)
d
x
f
(
x
,
s
)
p
s
(
x
,
0
)
d
x
B
B
σ
2
1
t
+
2
Δ
x
f
(
x
,
1
)
p
1
(
x
,
t
)+
h
(
x
,
1
)
f
(
x
,
1
)
p
1
(
x
,
t
)
0
B
−
m
12
f
(
x
,
1
)
p
1
(
x
,
t
)
−
m
13
f
(
x
,
1
)
p
1
(
x
,
t
)
d
x
d
t
+
m
21
f
(
x
,
2
)
p
2
(
x
,
t
)
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