Biomedical Engineering Reference
In-Depth Information
thus inherit their randomness. The advantage of using averaged quantities at the
larger scales is anyway convenient, both from a theoretical point of view and for
being computational affordable; the message is: do it when suitably supported by
applicability of laws of large numbers.
For the time being, we have performed a heuristic derivation of a law of large
numbers as the number of cells increases, showing that, when the number of cells
is sufficiently large, the empirical measure
Q
N
(
t
)
can be approximated by its limit
measure
Q
[
6
,
7
,
27
] which admits a spatial density with respect to the usual
Lebesgue measure. We also provide the evolution equations for the relevant spatial
densities associated with the limit measure. Indeed, one should check that the hybrid
system is compatible with a rigorous derivation of the evolution for the vessel densi-
ties. Nonlinearities in the full model are a big difficulty in this direction. In literature,
examples of derivation of limit systems from a stochastic particle dynamics may be
found also in [
10
,
29
,
39
], but to the best of the knowledge of the authors, the kind
of stochastic hybrid models considered here has not been studied yet.
In Sect.
2
we discuss the mathematical modelling of the stochastic interacting
population when the number of individuals is finite. We consider both Lagrangian
and Eulerian (discrete) descriptions. In Sects.
2.1
and
2.2
, we analyze two specific
cases. The first case refers to a model for stochastic tumor-induced angiogenesis.
There is a widespread literature on the subject [
1
,
3
,
11
-
13
,
15
,
23
,
24
,
32
-
34
,
37
]. The interested reader may refer to the introduction of [
7
] for a detailed
description. The second model considered here is a model for retinal angiogenesis.
Both models are examples of stochastic fiber processes, coupled with the continuum
underlying field of a chemoattractor (indirect interaction). In Sect.
3
we study the
derivation of the corresponding hybrid models, for the two working examples.
Finally, in Sect.
4
we discuss the role of randomness in the proposed model via
simulations.
(
t
)
2
Cells, Interactions and Evolution
We denote by
N
a parameter of scale of the process; from a modelling point of view,
it may well represent the total number of cells involved in the phenomenon, in a way
or another. Let
N
be the random number of cells entered in the dynamics by
time
t
,
T
b
∈
R
+
, the random time of birth of the
i
-th cell;
T
d
∈
R
+
, the random
time of death of the
i
-th cell. Here, cell states are described by a bivariate stochastic
process
Z
i
(
t
)
∈
N
Z
i
=
{
(
t
)
}
t
∈
[
0
,
T
]
, such that
Z
i
X
i
Y
i
(
t
)=(
(
t
)
,
(
t
))
,
i
=
1
,...,
N
(
t
)
,
where
X
i
d
∈
R
+
,and
Y
i
(
t
)
∈
R
is the random location of the
i
-th cell at time
t
(
t
)
∈
I
may represent a possible random characterization of the
i
-th cell at time
t
∈
R
+
.
Later on, we will see that
Y
i
(
t
)
will represent the velocity of the
i
-th cell, in the
Search WWH ::
Custom Search