Biomedical Engineering Reference
In-Depth Information
plasma membrane, leading to extracellular calcium entry into the cy-
tosol. The rates of calcium influx are increasing and saturating functions
of AA and NO concentration, respectively [133, 134, 274, 279, 281, 390].
NO release is further mediated by AA itself within cell cytosol [274, 276,
402].
Calcium enhances the rates of cytosolic AA and NO biosynthesis, with
a positive feedback mechanism [236, 274, 276].
calcium ions, which are reversibly buffered to proteins, such as calmod-
ium, or to mitochondria [24, 33, 212, 200], are extruded from the cell by
plasmamembrane ATPase and Ca
2+
-Na
+
exchangers [166, 191, 399].
The evolution of the exogenous VEGF satisfies the following reaction-diffusion
equation:
@v
@t
=
D
v
r
2
v
v
v
B(x;t;v)
| {z }
uptake
+ S
|{z}
addition
;
(6.5)
| {z }
diffusion
|{z}
decay
where v denotes the actual concentration of the VEGF at medium site x (i.e.,
(
(
x
)
) = M). The coecient of diffusivity, D
v
> 0, and of degradation,
v
,
are assumed to be constant and derived from previous experimental deter-
minations [360]. S = S(x;t) describes the input of VEGF at a constant rate
v
per unit of time by a discrete source, whose location and extension will
be discussed in the different sets of simulations. B is the amount of VEGF
molecules per unit of time that are locally sequestered by the cell. As done
in Chapter 5 and following [25, 246], we assume that receptor binding occurs
very rapidly compared to the time-scale of cell migration, and that the local
quantity of VEGF molecules instantly bound by the TEC at time t is equal to
the smallest between the actual available molecular concentration v and the
actual maximal amount of VEGF that can be internalized by the cell surface
receptors, which is defined with
v
(t). Thus, we set:
B(x;t;v) = minf
v
(t);zvg;
(6.6)
where x belongs to the external surface of the cell PM (i.e., (
(
x
)
) = M and
9x
0
2
0
x
: (
(
x
0
)
) = C).
v
(t), the maximal amount of VEGF molecules
that can be locally (i.e., per site) bound and internalized by the TEC for unit
of time has been estimated following [25, 246]. In particular, for the computa-
tion of
v
(t), we have considered a spatially homogeneous average number of
VEGF receptors per cell membrane site, which has been estimated by dividing
the total number of VEGF receptor in a generic endothelial cell, considered
constant and equal to 311.200 (as measured in [405] for human colonic ECs),
for the actual extension of the TEC membrane, which is initially equal to
2700 m
2
[390]. The local number of VEGF receptor thus varies in time: how-
ever, to avoid further overcomplications, we have not taken into account of
other time-dependent phenomena, such as receptor clusterization. Moreover,
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