Biomedical Engineering Reference
In-Depth Information
We consider pairwise interaction. Thus the interaction between two locations
x
,
y
∈
d
is modelled by a symmetric reference kernel
K
1
, depending on their distance.
We assume that, in a population of
N
cells, the interaction of two cells, out of
N
,
located in
x
and
y
, respectively, is modelled by
R
1
N
K
N
N
η
K
1
N
η
/
d
z
(
−
)
,
(
)=
(
)
,
x
y
where
K
N
z
(4)
which expresses the rescaling of
K
1
with respect to the total member
N
of cells, in
terms of a scaling coefficient
d
, the interaction with
η
∈
[
0
,
1
]
. At a position
x
∈
R
the population of cells at time
t
∈
R
+
is given by a mollified version of the marginal
spatial random distribution Eq. (
3
)
Q
N
K
N
N
)
k
=
1
K
N
(
x
−
X
k
(
t
1
N
(
)) =
(
)
∗
(
)
.
t
t
x
The choice of
in Eq. (
4
) determines the range and the strength of the influence
of neighboring cells; indeed, the number of cells influencing the underlying field
at a point
x
η
is of the order
N
1
−
η
. In particular, for
N
increasing to infinity,
d
∈
R
1let
N
1
−
η
tend to infinity, allowing the applicability of a local law
of large numbers. From a mathematical point of view, the use of mollifiers reduces
analytical complexity; from a modelling point of view this might correspond to the
range of a nonlocal interaction between cells and the relevant underlying fields.
Another possible interpretation would be that a cell is not exactly a point, but it may
extend as a spot in the relevant spatial domain. Altogether the parameter
the choice
η
<
η
defines
the order of what we call here a mesoscale, i.e.,
N
−
η
[
6
,
7
,
28
].
A general model may then appear of the following form, for any
t
≥
0,
d
Z
k
Z
k
d
W
k
(
t
)=
F
[
C
(
·,
t
)] (
(
t
))
d
t
+
σ
(
t
)
,
k
=
1
,...,
N
(
t
)
;
(5)
Op
3
K
N
∂
∂
Q
N
(
d
t
C
(
x
,
t
)=
Op
2
(
C
(
·,
t
))(
x
)+
t
)
∗
(
x
)
,
x
∈
R
.
(6)
System (
5
)-(
6
) say that the stochastic evolution of an individual state
Z
k
(
t
)
is driven
by an underlying field
C
(
x
,
t
)
(such as nutrient, growth factor, and alike [
18
,
19
])
via the operator
F
depending on the field and acting on each individual;
on the other hand the evolution equation of the field
C
[
C
(
·,
t
)]
depends itself upon
the structure of the system of individuals by means of an operator
Op
3
,which
depends on the convolution
(
x
,
t
)
Q
N
(
K
N
of individuals, acting at a spatial location
x
. In our concrete examples, reported in Sects.
2.1
and
2.2
, the underlying fields
are better specified as suitable biochemical substances. For simplicity, the diffusion
coefficient
t
)
∗
in Eq. (
5
) is assumed to be a constant, in time and space, diagonal
matrix with entries in
σ
R
+
. Randomness is modelled via a diagonal matrix with
diagonal entries given by two independent Wiener processes
W
i
(
t
)
,
i
=
1
,
2. Note
that also the evolution of the stochastic process
{
N
(
t
)
}
t
∈
R
+
may depend upon the
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