Biomedical Engineering Reference
In-Depth Information
Note that from Eq. (
6
), the interaction of the underlying field
C
with the
neighboring cells is realized via the function
h
N
. The size of the spatial region of
interaction is a consequence of the choice of our mesoscale, related to the value of
η
<
1. Indeed, Eq. (
37
) represents an averaging over a large number of cells, at the
level of the mesoscale (given by the size of
η
); furthermore
K
N
converges to a Dirac
δ
-function sufficiently slow, so that we may apply a “law of large numbers” in such
a way that
h
N
(
approaches the limit Eq. (
38
). We need to be sure that the sum in
Eq. (
37
) is performed over a sufficiently large number of cells, which is equivalent
to say that the range of
K
N
has not to shrink to zero too fast with respect to the
increase of the number of cells.
In the limit the relevant density will satisfy a deterministic evolution equation of
the type
x
,
t
)
Op
1
d
s
t
)
,
C
(
,
)
ρ
(
,
)
=
(
,
)
ρ
(
,
)
+
ρ
(
·,
(
·,
)
,
(
·,
)
.
g
z
t
z
t
d
z
g
z
0
z
0
d
z
s
s
g
s
(39)
D
D
0
This approximated measure then substitutes the random measure
Q
N
(
in the
evolution equations for the underlying fields at the macroscale, so that they are now
completely deterministic
t
)
Op
3
∂
∂
t
C
Op
2
[
C
C
(
x
,
t
)=
(
·,
t
)](
x
)+
(
·,
t
)
∗
ρ
(
·,
t
)
(
x
)
;
(40)
they may be used to drive the kinetic parameters of the cell processes at the
microscale, thus leading to simple stochastic processes
F
C
d
Z
k
Z
k
d
W
k
(
t
)=
(
·,
t
)
(
(
t
))
d
t
+
σ
(
t
)
,,
k
=
1
,...,
N
(
t
)
.
(41)
These kind of models are known as
hybrid models
, since they are stochastic at
the level of individuals Eq. (
41
) and deterministic at the level of the field Eq. (
40
).
3.1
Tumor-Induced Angiogenesis
Given the evolution equation (
16
) for the empirical measure
Q
N
(
, let us observe
that due to the Doob's inequality and Eq. (
21
), the quadratic variation of the zero
mean martingale Eq. (
17
) is such that
t
)
sup
t
)
|
F
0
2
2
F
0
M
N
(
M
N
(
E
T
|
t
≤
4
C
E
|
T
)
|
≤
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