Biomedical Engineering Reference
In-Depth Information
uniformly distributed over the objects' surfaces and therefore the localiza-
tion of adhesive proteins is homogenous. However, as will be described in
detail in the next section, the Js may vary during the simulation, i.e.,
J
(
);(
0
)
= J
(
);(
0
)
(t).
H
shape
models the geometrical properties of the simulated individuals:
given as in (1.6), the initial dimensions of discrete objects are assumed to be
their target measures. In specific cases, the growth (respectively, death) of
individuals is modeled by allowing their target measures to increase (respec-
tively, decrease) with time, as we will explain below.
Finally, the chemotactic motion of malignant individuals along concen-
tration gradients of signal molecules is implemented by relation (1.9), where
che
C
is the effective potential related to tumor cells and c represents the local
level of the chemical substances.
The chemotactic factors are released in the interstitial/peritoneal fluid
by both the mesothelial cells and the ECM molecules in order to direct the
motion of tumor cells. Indeed, their spatiotemporal evolution is controlled by
the following PDE:
@c
@t
=
D
c
r
2
c
c
c
+
c
;
(3.2)
| {z }
diffusion
|{z}
decay
|{z}
secretion
where the coecients of diffusion D
c
and decay
c
are assumed constant
in time and homogeneous throughout the extracellular fluid.
c
=
c
(x;t)
describes the secretion of the chemoattractant:
8
<
if (
(
x
)
) 2fM;Eg and 9x
0
2
0
c
x
: (
(
x
0
)
) = F;
c
(x;t) =
:
0
else:
(3.3)
As stated in Section 3.1, the tumor cells are able to activate and release
different types of MMPs, m, whose evolution is described by
@m
@t
=
D
m
r
2
m
m
m
|{z}
decay
+
m
|{z}
release
;
(3.4)
| {z }
diffusion
where, as before, D
m
and
m
are, respectively, the diffusion and degradation
rate, constant in time and homogeneous in space.
m
=
m
(x;t) models
instead the secretion of proteinases over the external surface of the cancer in
the interstitial fluid when meeting an ECM fiber, i.e.,
8
<
if (
(
x
)
) = C and 9x
0
2
0
m
x
: (
(
x
0
)
) = E;
m
(x;t) =
:
0
else:
(3.5)
During the invasion process, tumor MMPs are capable of degrading both
the pericellular and the submesothelial matrix components: to model this
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