Biomedical Engineering Reference
In-Depth Information
through narrow locations in a brain model in vivo, thereby increasing their
metastatic potential.
9.9 Effect of Matrix Degradation in 3D
In the previous sections, we have demonstrated that cells move within matrix
fibers of varying density and stiffness that act as constraints, deforming both
body and nucleus. As an additional mechanism to overcome limited space,
cells may upregulate proteolytic enzymes that degrade ECM structure (i.e.,
matrix metalloproteinases, MMPs). They act either bound to the cell surface
or when secreted into the extracellular space. Accordingly, cells degrade steric
fiber obstacles either in a cell contact-dependent manner targeting locally
confining fibers, or, in a diffusive manner leading to gradient formation and,
consequently, a more overall weakening of the surrounding tissue structure
[410]. As a result, barrier-free matrix spaces will be created, resulting in longer
traveling distance.
The local concentration of the net proteolytic activity (both surface-bound
or diusive) is dened as m(x;t), and is assumed to evolve following a standard
reaction-diffusion equation:
@m
@t
=
D
m
r
2
m
m
m((
(
x
)
);M)
|
+
P
;
(9.6)
|{z}
production
| {z }
diffusion
{z
}
decay
where ((
(
x
)
);M) = 1 in the interstitial medium M and 0 elsewhere.
m
and D
m
are, respectively, the decay rate and the effective diffusion coecient
of proteolytic enzymes. A low value of D
m
models proteolysis being strongly
localized in regions close to cell membranes, in agreement with experimental
evidence in [336, 408]. P(x;t) models instead the local production of proteases
either at the cell surface or secreted away from the external cell surface, at a
constant rate
m
collagenous component:
8
<
x : (
(
x
)
) = C and 9x
0
2
0
m
if
x
: (
(
x
0
)
) = M;
P(x;t) =
:
0
else;
(9.7)
where we recall that C stands for the cell cytosolic region.
The MMPs are capable of degrading the fibrous component of the matrix:
to reproduce this biological effect, a lattice grid site x belonging to a degraded
collagenous fiber becomes a generalized medium (fluid) site when its local level
of MMPs, m(x;t), is suciently high (in our simulations above 2:5 M). This
change is implemented by changing its type from F (ber) to M (medium),
as in Chapter 3.
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