Biomedical Engineering Reference
In-Depth Information
culture as well as of single individuals by separately analyzing and quanti-
tatively evaluating statistical parameters, such as cell velocity, directionality
of their movement and distribution of their final displacement. In this way,
the invasive capability of the population can be assessed both as a whole
and as a sum of individual behaviors. Indeed, single cells can be specifically
correlated according to their type of movement and grouped in well-defined
subpopulations.
5.2 Mathematical Model
At the mesoscopic cellular level, a compartmentalized CPM represents the
phenomenology of the ARO population, focusing on cell shape, membrane
fluctuations, and adhesive interactions. The internal state of each individual
is then explicitly approached with a continuous model that reproduces the
biochemical signaling pathways activated, via Met receptors, by the HGF/SF,
whose extracellular evolution is in turn described by a standard continuous
equation. Finally, all the levels are inextricably linked, so that the behavior
of the AROs is realistically driven by their microscopic, molecular dynamics.
5.2.1 Cell-Level Model
Since we wish to compare our simulations with experimental wound healing
assays, we use a bidimensional domain R
2
. The AROs are compartmen-
talized individuals of type () = E. In particular, each cell is subdivided in
three units presented at the end of Section 4.2 and in Figure 4.3: the nucleus,
a central cluster of type = N, the surrounding, initially circular, cytosolic
region, of type = C, and the plasmamembrane (PM), of type = M. For
any cell , we dene the state vector of each compartment:
If
is such that (
) = M, s
;
(x;t) = (m(x;t)) 2R
+
, where
m(x;t) is the local concentration of activated surface receptors Met.
If
is such that (
) = C, s
;
(x;t) = (p(x;t);k(x;t);c(x;t);r(x;t)) 2
R
4
+
, where p(x;t) corresponds to the local concentration of PI3K, k(x;t)
of MAPK, c(x;t) of Cdc42, and r(x;t) of Rac.
If
is such that (
) = N, s
;
(x;t) = (c(x;t);r(x;t)) 2R
2
+
.
All these quantities are expressed in units of M. The extracellular environ-
ment, i.e., the experimental Matrigel, is modeled as a special generalized cell
0
of type = Q. As usual, it is assumed to be static, passive, and homoge-
nously distributed throughout the simulation domain, and therefore without
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