Biomedical Engineering Reference
In-Depth Information
after 15000 MCS, so that they reproduce a time-lapse of nearly 4 days. As
represented in Figure 8.3, the initial conditions of all simulations consist of a
layer of tumor cells, whose initial dimensions correspond to their target mea-
sures given in the Appendix. This configuration is consistent with a spatially
extended cancer mass which is invading from an epithelial cell lining down its
basement membrane into the surrounding stroma.
The boundary conditions for both growth factors, n, and proteolytic en-
zymes, m, are periodic at the left and the right sides of the domain (i.e., at
x = 0 and x = 500) and zero flux at the bottom (i.e., at y = 0). In par-
ticular, the lateral periodic conditions are set since the simulated malignant
population is a section of a much larger lesion, that could not be reasonably
modeled here. The zero fluxes at y = 0 are set assuming that this part of the
tissue is far enough from the front of the tumor mass. For the same reason,
we set a no flux condition for the MMP field at the top of the domain (i.e.,
at y = 500). The upper border of the grid represents instead an extended,
planar source for the environmental growth factors, which are supplied by
the host stroma throughout the basement membrane: consequently, we set
n(@(x;y = 500) 2 @;t) =
n
, where n
@
x
is the outward normal to @x.
The parameter describing the biophysical properties of the cells, such as
their basal motility, elasticity and adhesive strengths, have been evaluated,
consistently with the biological considerations provided in the previous sec-
tion, through preliminary simulations, that showed the model consistency in
a wide range of values. The intracellular basal level of chemical factors, n
0
,
is suciently high to avoid that cells irreversibly enter the apoptotic state.
The extracellular environment is initially seeded with a saturating level of
proteins, p
ext;0
, while there are no growth factors or secreted MMPs in the
extracellular medium.
To quantify the effects of the different model parameters on the tumor
phenotype, we concentrate on the final depth of invasion of the mass at the
end of the simulations (i.e., at t = 15000 MCS 4 days), given by:
X
1
N
d
f
=
(d
i
d
0
) ;
(8.8)
i=1
where d
i
represents the final distance between a tumor cell and the bottom
border of the domain and d
0
is the initial width of malignant mass. The
average of N = 10 randomly chosen values is used to avoid biases towards
accounting for outlier individuals. df
f
has a clear clinical relevance, since it
quantitatively estimates the severity of the disease and characterizes the pres-
ence of metastasis delocalized with respect to the main body of the tumor.
We first simulate the evolution of the tumor with the standard param-
eter setting. The malignant mass is observed to have an overall movement
biased toward the extracellular environment, which is significantly invaded,
as d
f
200 m; see Figure 8.4(B). Consistently with the experimental ev-
idences provided in the biological introduction (see also Figure 8.1), the in-
vasive phenotype is largely mediated by the aggressive behavior of the more
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