Biomedical Engineering Reference
In-Depth Information
computational cost rapidly increases with the number of cells modeled, which
makes it dicult to simulate lesions greater than one millimeter.
Following the recent literature [13, 158, 159], we here use the multi-level
modeling framework introduced in Chapter 4 to reproduce and quantitatively
analyze the first phase of tumor invasion. In particular, the main focus is
on the emergence of different morphologies at the front of tumor invasion,
which result both from cell-based processes (such as cell elasticity, adhesive
properties and motility) and from subcellular molecular dynamics (such as
growth factor internalization, ECM protein digestion and MMP secretion).
The resulting model is able to characterize the morphology of the invasive
tumor and to quantify its malignancy in term of invasive depth in several
different conditions. Furthermore, the proposed approach has the potential to
make clear the relevance of the various mechanisms involved and to suggest
possible intervention strategies able to reduce the aggressiveness of the lesion
by controlling its morphological stability, i.e., by enforcing its compactness so
that it could in principle be more easily resectable.
8.2 Mathematical Model
The malignant cells are represented at a mesoscopic level by compartmental-
ized objects, that locally interact with each other and with the microenviron-
ment through their membranes. The molecular biology is instead incorporated
with a macroscopic description of the evolution of nutrients, ECM proteins
and tumor matrix metalloproteinasis (continuous CPM objects), see Figure
8.2 for a schematic representation. As a key feature of our discrete-continuum
composite approach, the different scales affects each other, as the distribu-
tion of nutrients and ECM proteins in the extracellular environment influence
cells' properties and phenomenology (with carefully-calibrated constitutive
relations).
8.2.1 Cell-Level Model
The simulation domain is a planar square domain R
2
. The cancer cells
are bicompartmental individuals of type = T, composed of a central
cell nucleus ( = N) and the surrounding cytosol, = C, as depicted in
Figure 4.3. The tumor mass lives on an extracellular matrix, = M, which
is assumed to be isotropically distributed throughout the simulation domain,
forming no large-scale structures. It reproduces in fact the mixture of soluble
components (among others, long carbohydrate polymers, and nonproteoglycan
polysaccharides), which, together with the water solvent, compose the so-
called interstitial medium. For any cell , we dene the state vector of each
compartment:
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