Biomedical Engineering Reference
In-Depth Information
model aims at reproducing the formation of the overall tumor capillary struc-
ture and at characterizing its topological properties, with a close comparison
with published experimental observations. In particular, through different sets
of numerical realizations, we show that the initial density of the overall cul-
ture, as well as selected biophysical properties of single individuals, such as
their calcium-dependent chemical and adhesive strength and the persistent
component of their migration, play a fundamental role in determining the
dynamic and the final configuration of the overall network formation. If, on
one hand, the systematical analysis of the TEC pattern ability confirms the
eciency of some currently available anti-angiogenic therapies, on the other
hand it may suggest novel and experimentally testable strategies that have
the potential to disrupt TEC capillary formation.
7.2 Mathematical Model
The model for the overall TEC tubulogenesis contains exactly the same in-
gredients as that for the single cell motion, with some slightly modifications
regulating intercellular interactions that will be clarified below.
7.2.1 Cell-Level Model
First, since we wish to compare simulations with experimental cultures, where
the vascular patterns are essentially monolayers, we turn to use a bidimen-
sional domain R
2
+
. Each simulated TEC, , is therefore dened as a
planar bi-compartmental unit, composed of the cell nucleus, in this case a
round cluster of type = N and the surrounding cytosol, = C (see Fig-
ure 4.3). Indeed, given the bidimensional approach,
volume
and
surface
now
correspond to, respectively,
surface
and
perimeter
. The TECs reside again
in a generalized substrate, a further discrete object of type = M, which
represents the experimental Matrigel. The internal state vector of each sub-
cellular compartment is the same as in the case of the single cell, while the
Hamiltonian functional H reads again as in (6.1).
However, with respect to the model for the single TEC, H
adhesion
has
to consider also the effective adhesion between membranes of different cells,
given by (4.3). In particular, J
ext
E;E
, where () = (
0
) = E, represents the
local adhesive strength acting between neighboring cells, a measure from a
statistical view point, of the probability of the formation of local intercellular
cadherin-cadherin complexes, which depends on the quantity of active exposed
molecules on either sides of the interface. Indeed, since VE-cadherin activity
is enhanced by calcium ions, which generate clusters of activated molecules,
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