Biomedical Engineering Reference
In-Depth Information
tumor cells are typically deformable, as they are able to significantly remodel
to invade their surroundings more eciently, for any individual
, we use a
suciently low
surface
=
surfac
C
.
Since, the cancer cells are uncompartmentalized objects, H
adhesion
has the
classical form given in Equation (1.5). In particular, J
C;C
represents the adhe-
sive strength between the membranes of two nearby cells, as usual a measure
of the quantity of active and exposed cadherins, while J
C;M
evaluates the het-
erophilic adhesive bonds between the integrins on the cell surface and suitable
ligands in the extracellular matrix. As in the bidimensional set-up, we set
constant and homogeneous values for both J
C;C
and J
C;M
. However, tumor
cells within spheroid masses prefer to adhere to one another rather than to the
host (the strong homotypic interactions within solid tumor masses are typi-
cally regulated by the over-expression of E-cadherins and by other intercellular
mechanisms): therefore J
C;C
has to be lower than J
C;M
. In particular, as given
in Table C.9, we set J
C;C
= 1=2 J
C;M
: this is an arbitrary choice, however
a complete screening of the role played by the cell{cell adhesive strength in
determining the tumor phenotype will be performed in the Results section (cf.
Figures 8.16 and 8.17).
The other major change is due to the fact that, in the three-dimensional
set-up, the production of diusing growth factors, n(x;t), is set (at a constant
rate per unit of time) at the entire border of the domain. Again the matrix
soluble proteins, p(x;t), are assumed to have a uniform distribution at the
beginning of each simulation and are degraded upon contact by the metallo-
proteinases (MMPs) secreted by malignant cells m(x;t), that, in turn, diuse
throughout the tissue and undergo some form of decay (passive, active, or due
to the neutralization by endogenous matrix inhibitors).
8.10 Simulations
The simulation domain is a cubic lattice of 350 350 350 sites. The
characteristic length of each site is 1 m, and therefore represents a tissue
with a volume of 0.04 mm
3
. One MCS is set to correspond to 20 sec: the
overall simulations stop after 56250 MCS, so that they reproduce a time-lapse
of nearly 15 days. The PDE for the evolution of nutrients is numerically solved
with a nite dierence scheme on a grid with the same spatial resolution as ,
characterized by 30 diffusion steps per MCS. This temporal scale is suciently
small to guarantee the stability of the numerical method.
As represented in Figure 8.12, we start all simulations with a cluster of 24
cells in the center of the lattice. Each cell is initially a sphere whose volume
is consistent with the average dimensions of human glioma cells [50]. The
specific initial configuration reproduces an avascular solid tumor spheroid,
which is invading the surrounding host tissue.
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