Biomedical Engineering Reference
In-Depth Information
In this way we obtain an approximation of the driving fields which are determin-
istic at the macroscale. These fields now drive, at the microscale, a simply stochastic
evolution for the single cells. More specifically, a typical cell
k
will satisfy the
following system of stochastic differential equations, for any
k
=
1
,...,
N
(
t
)
:
d
Y
k
Y
k
Y
k
C
k
dW
t
d
t
(
t
)=
β
[
∇
g
(
(
t
)
,
t
)
−
∇
u
(
(
t
)
,
t
)]
δ
C
k
2
d
t
+
σ
(
(
t
))
,
(59)
(
t
)
,
Y
k
C
k
subject to a change of state governed by the matrix
M
(
(
t
)
,
(
t
))
, with prolifera-
tion rate
h
1
Y
k
C
k
(
(
t
)
,
(
t
))
=
λ
1
δ
C
k
+
λ
2
))
δ
C
k
,
(60)
(
t
)
,
1
(
t
)
,
2
Y
k
1
+
N
(
p
2
(
·,
t
)
∗
K
)(
(
t
and coupled with systems (
54
)-(
55
)fortheVEGF
g
and the nutrient
u
.
In conclusion, we may describe this retinal angiogenic process in two different
ways; either by a fully stochastic system, involving the individual cell processes,
and the associated empirical measure process, strongly coupled with the evolution
of relevant underlying fields, or as a simply stochastic system coupled with the
evolution of the deterministic mean underlying field obtained upon a limit of large
numbers of the empirical measure
Q
N
. Whenever applicable, this last approach
makes the model computationally more affordable.
4
Conclusions and Discussion
The following question arises; do we lose any important feature of the dynamics
of the system by considering the hybrid model, i.e., by eliminating the stochastic
variability of the underlying fields? We have addressed this question by means of
numerical simulations.
In this section we compare the simulation results (a) for the fully deterministic
system of spatial densities of both cells and underlying fields, (b) for the hybrid
system, i.e., stochastic birth and growth processes (at the microscale (cells)) driven
by deterministic averaged underlying fields (at the macroscale), and (c) for the fully
stochastic model of retinal anagenesis.
The Deterministic System.
First of all, we consider system (
54
)-(
58
). From Fig.
6
(thanks to spherical symmetry it is sufficient to show 2D sections of the relevant
distributions) it is clear that:
1. Each profile is smooth and symmetric with respect to the origin.
2. Type 2 cells show a higher density in the external regions, where the capillaries
are built (up-left).
3. Type 1 cell density decreases from the origin to the frontier (up-right).
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