Biomedical Engineering Reference
In-Depth Information
CPM: Multiscale Developments and Biological Applications
where the quantity n
(t) =
P
x
2
n(x;t) is the total amount of growth fac-
tors inside cell , as n(x;t) is their local intracellular concentration, dened
in Equation (8.5), while N
0
=
P
x
2
n
0
is instead their overall basal level. T
0
therefore corresponds to the basal motility of tumor cells, while T
0
=h is their
asymptotic motility for saturating concentrations of chemicals. The above hy-
pothesis are in agreement with experimental observations providing, through
classical wound healing experiments, that high concentrations of growth fac-
tors stimulate the migratory capacity of different tumor cell lines (see, for
example, [101, 104] for the hepatocyte growth factor, [38, 383] for the vas-
cular endothelial growth factor families, and [375] for the fibroblast growth
factor).
8.2.2 Molecular-Level Model
Available growth factors are supplied to the medium, diffuse and decay at a
constant rate, and are consumed by tumor cells. Their spatial prole, n(x;t),
therefore satisfies the following equation:
@n
@t
=
D
n
r
2
n
n
n((
(
x
)
);M)
|
+
| {z }
diffusion
{z
}
decay
minfn
max
;
n
ng(1 ((
(
x
)
);M))
|
+
S
|{z}
production
;
(8.5)
{z
}
uptake
where ((
(
x
)
);M) = 1 in the extracellular environment and 0 within cells.
D
n
is the characteristic diffusion coecient, homogeneous throughout domain
, and
n
is the decay rate in the ECM. The third term in (8.5) models the
local uptake by tumor cells, which follows a piecewise-linear approximation
of a Michaelis{Menten law. In particular,
n
n
, as we assume that the
nutrient natural decay is negligible compared to the uptake by tumor cells.
S describes the production (or input) of chemical factors at a constant rate
n
per unit of time by a planar source, whose location and extension will be
discussed in the next section.
The substrate contains matrix soluble proteins with concentration p(x;t)
(i.e., we neglect their production, assuming a uniform distribution at the be-
ginning of each simulation, see next section for more details), that naturally
decay and that are degraded by the metalloproteinases (MMPs) secreted by
malignant cells, m(x;t). The change in the local amount of ECM proteins is
therefore described by:
@p
@t
=
p
p
p
pm
| {z }
degradation
;
(8.6)
|{z}
decay
where
p
and
p
are, respectively, nonnegative decay and degradation rates,
constant within the entire simulated substrate. In particular, we assume
p
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