Biomedical Engineering Reference
In-Depth Information
Hence, the main features of the dynamics include
branching
, due to proliferation
and change of state of cells,
elongation
, due to aggregation and repulsion
phenomena,
remodelling
, and finally
blood circulation
. As already mentioned, here
we do not consider the latter two.
2.2.1
Modelling the Evolution of the Capillary Network and the Coupling
with the Underlying Fields
We again denote by
N
a parameter of scale of the process. We model the network
by tracking the cells building up the vessels. Let be
S =
{
1
,
2
,
3
}
. The considered
individual stochastic processes are
Z
i
X
i
C
i
d
(
t
)=(
(
t
)
,
(
t
))
∈
R
×
S
,
which denote the location and the type of the
i
-th cell at time
t
,forany
i
=
1
is the total number of cells at time
t
. The network
of endothelial cells is described by the union of the trajectories of the tips as in
Eq. (
1
).
,...,
N
(
t
)
,
respectively.
N
(
t
)
Vessel Extension.
We suppose that only type 2 cells are subject to the action of the
underlying fields, while type 1 cells are only subject to a possible randomness and
type 3 cells do not move anymore. Randomness is modelled by additive independent
Wiener processes
W
t
}
t
∈
R
+
. Hence, for
i
T
i
,
{
=
1
,...,
N
(
t
)
and
t
>
d
X
i
X
i
X
i
C
i
d
W
t
,
(
t
)=
β
[
∇
g
(
(
t
)
,
t
)
−
∇
u
(
(
t
)
,
t
)]
δ
C
i
2
d
t
+
σ
(
(
t
))
(22)
(
t
)
,
where
σ
C
i
,
(
t
)=
j
,
for
j
=
1
,
2;
j
C
i
σ
(
(
t
)) =
(23)
C
i
,
(
)=
,
0
t
3
,
σ
∈
R
+
are diffusion coefficients,
β
∈
R
+
σ
δ
j
is the Kroenecker delta, and
i
,
1
2
represents the strength of response to the drift.
Cell Proliferation.
As above, proliferation is described by a branching process,
modelled as a marked counting process, by means of a random measure
Φ
=
ε
(
T
i
,
X
i
,
C
i
)
,
on
B
R
+
×
E
×
S
. Hence, for any measurable set
A
⊆B
R
+
×
R
∑
i
d
×
S
=
∑
i
ε
(
T
i
,
X
i
,
C
i
)
(
A
)=
card
{
i
:
(
T
i
,
X
i
C
i
Φ
(
A
)
:
,
)
∈
A
}
is the random variable which counts those cells which are born in
A
.
The process
Φ
d
is characterized by the following stochastic intensity; for any
(
t
,
x
,
s
)
∈
R
+
×
R
×
S
N
(
t
−
)
i
=
1
ε
(
X
i
(
t
)
,
C
i
(
t
))
(
d
x
×{
s
}
)
d
t
.
Λ
t
(
d
x
×{
s
}
)
d
t
=
h
(
x
,
s
)
(24)
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