Biomedical Engineering Reference
In-Depth Information
for any TEC and for (
= C), we set
s
J
ext
;
(x;t) = s
J
ext
;A
(x;t) = (c(x;t));
;
and therefore for each couple and
0
,
E;E
(s
J
ext
;A
(x;t); s
J
ext
;A
J
ext
E;E
((@x 2 @) \ (@x
0
2 @
0
);t) = J
ext
0
;
0
(x
0
;t)) =
;
p
c(x;t)c(x
0
;t)
c
0
(x)c
0
(x
0
)
= g
J
ext
(s
J
ext
;A
(x;t); s
J
ext
;A
0
;
0
(x
0
;t)) = J
0
exp
;
;
where c(x;t) and c(x
0
;t) are the local calcium levels, and c
0
(x) = c
0
(x
0
) = c
0
the basal local concentration of the ion. J
0
represents the typical adhesive
force of resting TECs, estimated by qualitative observations of experimental
cultures (see Appendix C).
The contact energy between the cells and the simulated Matrigel is taken
to be negligible (i.e., J
ext
E;M
= 0). Even though it should obviously be included
in a more detailed model, its exclusion does not strongly influenced the final
outcomes of the approach. The contact interactions between vascular cells and
homogenous and isotropic matrix substrates have been in fact demonstrated
to play a major role in the stabilization of an in vitro vascular network, rather
than in its early formation. In particular, the theoretical works presented
in [10, 215], supported by the biological literature therein, have proved that
the intercellular interactions and their chemotactic and persistent migration
represent the minimal set of biological mechanisms sucient to guarantee the
emerging of the capillary structure.
On the other hand, cell{ECM adhesion only sustains the preservation of its
morphology under shear stresses and perturbations. Consistently, in [391], the
traction/adhesion between the cell population and the gel layer starts acting
when a sucient density is locally obtained, i.e., when cells are no longer iso-
lated but already connected. Indeed, cell-substrate adhesive strengths could
dominate the early stages of vascular patterning only in extreme cases. In
particular, as observed again by the authors of [391], a too-strong cell{ECM
adhesion inhibits the movement of cells, which remain isolated and dispersed
in their initial position. An overly adhesive substrate causes in fact the forma-
tion of clusters of integrin-ligand bounds at cells surfaces, that do not detach,
stopping further cell locomotion. On the contrary, a too weak cell{ECM ad-
hesion results in the packing of cells in big islands, as they try to minimize
the parts of their membrane in contact with the extracellular medium.
Therefore, the exclusion of cell{matrix adhesion from our model is not too
restrictive, since we focused on the initial formation of the network, and not
on its following stabilization, and we implicitly assume that cell{matrix con-
tacts are characterized by an intermediate strength. Following the same line
of thought, the presented approach could obviously be further developed with
the inclusion of other interactions between the cell population and the sub-
strate, that have been analyzed in other computational models. For instance,
Search WWH ::
Custom Search