Biomedical Engineering Reference
In-Depth Information
function, that is, when
s
ch
;
(
x
)
(x;t) = s
ch
;A
;
(
x
)
(x;t) = (c(x;t));
c
0
+ c(x;t)
c
0
+ jc(x;t)
ch
;
(
x
)
(t) =
ch
;C
(s
ch
;A
;
(
x
)
(x;t)) = f
ch
(s
ch
;A
;
(
x
)
(x;t)) =
c
0
;
where
c
0
is a basal chemotactic response. With this relation, we model the
experimental evidence that the local redistribution of VEGF receptors from
the Golgi apparatus to the plasmamembrane, and their following activation,
is caused by a local elevation of cytosolic calcium ions [264].
In (6.2), q measures the local extracellular VEGF concentration sensed by
the moving membrane site:
q(x;V ) =
X
x
0
2
0
x
v(x
0
;t);
(6.3)
where x 2fx
source
; x
target
g, and the x
0
are all the matrix first-nearest neigh-
bors of x. Such a spatial approximation is in agreement with experimental
results given in [157], where a direct chemotactic guidance by VEGF has been
provided to require the existence of precisely shaped extracellular gradients
in the close proximity of cell surfaces.
The local characterization of cell chemical responses is a key advantage
of the proposed CPM extensions with respect to similar published works
[259, 260, 261]. Therein, the cells experience in fact a homogeneous chemo-
tactic strength over the entire membrane, hiding relevant microscopic in-
homogeneities such as the clusterization of VEGF receptors or their local
agonist-induced activation which, as we will see in next section, is fundamental
for polarization mechanisms. Moreover, in classical descriptions, the chemical
strengths are constant over time, without adapting during the simulated pro-
cess. This is an unplausible situation, since real cells constantly change their
biophysical and biomechanical properties as a consequence of continuous in-
ternal and external stimuli: this can be obtained from the modeling point of
view connecting the value of the Potts parameters to the internal concentra-
tion of chemical factors and activation of related pathways, as commented in
Chapter 4. It is also useful to underline that the chemotactic response of TEC
individuals is similar, from a mechanical view point, to that of \normal" ECs,
except from an enhanced magnitude, due to the overexpression to VEGF re-
ceptors [55, 170]. This feature is taken into account in the above relation by
setting an asymptotic chemical sensitivity which is twice its basal level.
Besides the directional chemotactic migration, real tumor vascular cells,
as well as their normal counterparts, are seen to have a persistent motion
dictated by their longer axes. It is a consequence of their polarization, i.e., their
differentiation in a leading and a trailing edge. This inertial, shape-dependent
motion can be modeled with an energy term of type (2.4), which coherently
takes into account of the running mean over the cell past movements [20, 349].
Search WWH ::
Custom Search