Biomedical Engineering Reference
In-Depth Information
As seen in Chapter 2, Section 2.1, HGF/SF is uniformly added to the cul-
ture, diffuses in the ECM, where it decays at a constant rate and is taken up by
ARO cells through Met receptors sited in the membrane compartment. Math-
ematically, the HGF/SF is a continuous object, whose extracellular spatial
profile thus satisfies the following diffusion equation:
@h
@t
= D
h
r
2
h
h
h
|{z}
decay
B(x
;t
;h
)
| {z }
uptake
+ S
;
(5.2)
|{z}
addition
| {z }
diffusion
where h(x;t) denotes the local concentration of the peptide at medium site x
(i.e., (
(
x
)
) = Q). The coecient of diffusivity, D
h
, and of degradation,
h
,
are assumed to be homogeneous throughout the simulated Matrigel and have
been derived from experimental measurements performed for a recent paper of
us [349]. S(x;t) describes the addition of the growth factor at a constant rate
h
outside the cells. The HGF/SF binding and uptake by AROs is defined as
B(x;t;h(x;t)), and is limited to a maximum rate
h
(t) > 0 over the external
surface of the cells:
B(x;t;h) = minf
h
(t);zh(x;t)g;
(5.3)
where (
(
x
)
) = Q and 9x
0
2
0
x
: (
(
x
0
)
) = M. Relation (5.3) is realistic
since the capacity of an ARO cell to locally uptake the growth factor through
its boundary will saturate at a rate limit,
h
, which is the maximal amount
of HGF/SF molecules that can be locally bound and internalized per unit of
time. To compute
h
(t), we have followed the discussion provided in [25] for the
uptake of VEGF molecules by an endothelial cell. In particular, we have here
considered a spatially homogeneous average number of Met receptors per cell
membrane site: this number has been estimated by dividing the total number
of HGF receptors in a generic ARO cell by the actual extension (i.e., at time t)
of the cell membrane. Moreover, we have taken into account an instantaneous
HGF/SF-receptor complex internalization rate of 4.3 10
4
per second [152],
and 95 kDa as the molecular weight for an HGF/SF molecule [91]. The local
number of activated surface Met receptors (i.e., at site x : (
(
x
)
) = M) for
any individual is therefore dened as
X
1
z
B(x
0
;t;h(x
0
;t));
m(x;t) =
(5.4)
x
0
2
0
x
:
(
(
x
0
)
)=Q
where z is the same as in Equation (5.3). In Equation (5.4), we have assumed
a one-to-one interaction between the HGF/SF molecules and their surface
receptors [91].
For each ARO cell , the intracellular events initiating by the activation
of Met receptors regulate the cytosolic biochemical kinetics of PI3K (defined
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