Biomedical Engineering Reference
In-Depth Information
first model, or the type of the cell, in the second one. We model sprout extension by
tracking the random network
X
, given by all the trajectories
X
i
(
t
)
(
t
)
, i.e.,
N
(
t
)
X
i
T
d
)
.
T
b
≤
(
)=
(
)
,
≤
(
,
X
t
s
s
min
t
(1)
i
=
1
Thus, the random network of endothelial cells is the union of the random trajectories
described by all existing cells.
The
Lagrangian description
of the system may be also given in terms of
measures. Precisely, if we denote by
d
D
the range
R
×
I
, we might describe the
state of the
k
-th cell by
ε
Z
k
)
=
ε
(
X
k
))
∈M
(D)
,
Y
k
(
t
(
t
)
,
(
t
X
k
Y
k
which is the
Dirac measure
localized at
(
(
t
)
,
(
t
))
. This is such that for any
B
1
×
B
2
∈B
D
,
1
X
k
Y
k
,
(
t
)
∈
B
1
,
(
t
)
∈
B
2
,
ε
Z
k
)
(
B
1
×
B
2
)=
(
t
,
,
0
otherwise
d
so that, for any sufficiently smooth
g
:
R
×
S
→
R
,
g
X
k
Y
k
g
(
x
,
y
)
ε
(
X
k
(
t
)
,
Y
k
(
t
))
(
d
y
×
d
y
)=
(
t
)
,
(
t
)
.
d
R
I
According to an
Eulerian discrete approach
, the system can be described via a
space-type distribution of cells by considering the global random empirical measure
Q
N
of the process, such that, for any
t
∈
R
+
,
N
)
i
=
1
ε
(
X
k
(
t
)
,
Y
k
(
t
))
∈M
(D)
.
(
t
1
N
Q
N
(
t
)=
(2)
Q
N
as
We define the marginal measure
N
)
i
=
1
ε
X
k
(
t
)
∈M
(R
(
t
1
N
Q
N
(
d
t
)=
Q
N
(
t
)(
·,
I)=
)
.
(3)
A key question concerns the modelling of the
interaction
; interaction among
cells may be direct or indirect. Here we consider two cases of indirect interaction,
i.e., the force exerted on each cell depends on an underlying field whose evolution
depends on the distribution of the entire population; as a consequence the depen-
dence of the evolution of the spatial distribution of a single individual upon the
spatial distribution of the whole population is mediated by the underlying field.
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