Biomedical Engineering Reference
In-Depth Information
)+
α
u
Q
[
1
N
∗
K
N
(
,
)
∂
u
x
t
=
−
d
u
u
(
x
,
t
)+
D
u
Δ
u
(
x
,
t
(
x
)
.
(35)
∂
t
System (
32
)-(
35
) gives an Eulerian description of the fully stochastic retinal
angiogenic process.
3
Towards Hybrid Models
Let us go back to the general Lagrangian system (
5
)-(
6
) and to the Eulerian version
givenbyEq.(
7
). The analysis and the computation of the above system require the
knowledge of the evolution of all individuals up to time
t
. In both examples we have
considered, in the detailed models, either the Lagrangian one or the Eulerian one,
the evolution of the stochastic processes of branching and extension is driven by
parameters which depend upon the underlying fields; since the evolution of these
ones is vice versa coupled with the above stochastic processes, they are themselves
stochastic; as a consequence, we are dealing at the microscale with a doubly
stochastic system. A major difficulty, both analytical and computational, derives
from the fact that indeed the parameters are
{F
t
−
}
—stochastic, i.e., their values at
F
t
of the whole system up to time
t
−
.
The strong coupling with the fields is a source of complexity, as already discussed.
Under these circumstances, a possible way to reduce complexity, which has been
suggested by the authors [
7
] and by a large literature [
35
,
40
], is to apply suitable
laws of large numbers at the mesoscale, i.e., in suitably scaled neighborhoods of
any relevant point
x
time
t
>
0 depend upon the actual history
d
, such that, at that scale we may approximate
Q
N
(
∈
R
t
)
by
a deterministic measure (the so called mean field approximation)
Q
(
t
)
,
possibly
d
,i.e.,
having a density
ρ
(
·,
t
)
with respect to the usual Lebesgue measure
ν
d
Q
(
t
)=
ρ
(
x
,
y
,
t
)
ν
.
(36)
Given that Eq. (
36
) is the limit measure of the sequence of the empirical measures
{
Q
N
(
t
)
}
t
∈
R
+
, we may additionally observe the following. From Eq. (
4
), it is clear
that
lim
N
→
+
∞
K
N
=
δ
0
,
0
is Dirac's delta function; as a consequence, the measure
Q
N
where
δ
(
t
)
is such that
its regularized measure, defined, for any
s
∈
S
,as
1
N
1
−
η
k
)=(
Q
N
(
N
η
/
d
X
k
h
N
(
x
,
t
t
)
∗
K
N
)(
x
)=
K
1
(
(
x
−
(
t
)))
,
(37)
is such that
)=
ρ
(
lim
N
→
+
∞
h
N
(
x
,
t
x
,
t
)=
I
ρ
(
x
,
y
,
t
)
d
y
.
(38)
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