Biomedical Engineering Reference
In-Depth Information
(
,
)
[
]
,
,
=
,
,
underlying field
C
t
x
. By specializing operators
F
C
Op
i
i
1
2
3, we may
discriminate different dynamics.
For any
g
C
2
,
1
b
Z
k
∈
(D
×
R
+
)
,Ito's formula applied to
g
(
(
)
,
))
=
t
t
,forany
k
,···,
(
)
1
, leads to the
Eulerian (discrete) description
via an evolution equation
for the empirical measure
Q
N
N
t
(
)
t
[
6
,
27
]; indeed,
t
(
,
)
(
)(
)=
(
,
)
(
)(
)+
(
(
)
,
(
·,
)
,
(
·,
))
g
z
t
Q
N
t
d
z
g
z
0
Q
N
0
d
z
Op
1
Q
N
s
C
s
g
s
d
s
D
D
0
+
M
N
[
Z
,
W
](
t
)
,
(7)
where
M
N
[
is a zero mean martingale.
So far, we have shown how the models (
5
)-(
6
) may provide a general Lagrangian
description of the evolution of the processes
Z
k
, coupled with the evolution of a field
C
. On the other hand Eq. (
7
), again coupled with Eq. (
6
), is an Eulerian (discrete)
description of the system, where single identities of cells are lost. Both equations
may be coupled with other stochastic processes too, due for example to the evolution
of
N
Z
,
W
](
t
)
.
In the next two sections we provide two examples of models for angiogenesis, in
which we specify system (
5
)-(
6
), and then Eq. (
7
).
(
t
)
2.1
Tumor-Induced Angiogenesis
An interesting example of formation of patterns may be found in the process of
tumor growth and in particular in angiogenesis. Tumor-induced angiogenesis is
believed to occur when normal tissue vasculature is no longer able to support the
growth of an avascular tumor. A well-known model in literature is based on the
idea that endothelial cells proliferate and migrate in response to different signalling
cues; in particular they move though a gap in the basement membrane and into
extra cellular matrix (ECM). They secrete proteolytic enzymes, which also degrade
the ECM. Migration is through to be controlled by chemotaxis, the directed cell
movement up the gradient of a diffusible substance [
17
], a growth factor emitted by
the tumor [here tumour angiogenetic factor (TAF)], and by haptotaxis, the directed
cell movement along a nondiffusible substance, an adhesive gradient (here the
fibronectin). TAF and fibronectin bind to specific membrane receptors of endothelial
cells, activating cell migration machinery. Then cells produce a matrix degrading
enzyme (MDE), which improves the attachment of the cells to fibronectin contained
in the extracellular matrix. As a consequence, endothelial cells are able to exert
the traction forces needed for migration. ECs subsequently respond to the TAF
concentration gradients by forming sprouts, dividing and migrating towards the
tumor. So, at an individual level, cells interact and perform a branching process
coupled with elongation, under the stimulus of a chemical field produced by a tumor.
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