Cell Migration in Extracellular Matrices - Cellular Potts Models: Multiscale Extensions and Biological Applications - page 165

cytosol, limits the migration capacity of the entire cell. Finally, we will in-
clude ECM-directed proteolysis, resulting in enhanced migration in restricted
environments.

9.2 Mathematical Model
The simulation domain is a d-dimensional regular lattice R d . In partic-
ular, d = 2 describes the case of cells moving on a gel and d = 3 the case
of cells moving in a 3D ECM. In both cases, the simulated cells are defined
as bi-compartmentalized units of type = E, composed, as usual, of the
nucleus of type = N and the surrounding cytosol, = C. It is useful to
emphasize that in this model we do not consider the molecular state of moving
cells: therefore, it is not necessary to define their internal state vector.
The cell population resides in an extracellular matrix, which is differenti-
ated in a medium-like state, = M, and a collagen-like state, = F. The
medium-like state reproduces the mixture of soluble components (among oth-
ers, proteoglycans and glycoproteins in water), which compose the interstitial
fluid. It is assumed to be isotropically distributed throughout the simulation
domain, forming no large-scale structures.
The collagen state represents instead a network of insoluble macro-
molecules, such as collagen, that associates into first-order fibrils and second-
order fibers and displays the most abundant structure in mammalian
tissues. Each fibrous component is treated as CPM standard and non-
compartmentalized CPM objects . The dimensions, density, and distri-
bution of the fibrous structures will be specified in next sections and will
reproduce 2D and 3D matrix types, respectively, typically employed for in
vitro assays.
The inclusion of an explicit two-component matrix environment, already
present in some other CPM applications [25, 162, 246, 334] and in Chapter
3, is a fundamental aspect of this work: it allows an accurate analysis of how
cells migratory behavior is influenced by the heterogeneous fibrillar extracel-
lular environment and therefore by the ECMs' specic biophysical and biome-
chanical properties while they glide in medium of constant and homogeneous
physical properties.
The system Hamiltonian is

H(t) = H shape (t) + H adhesion (t): (9.1)

In this case, H shape models the geometrical attributes of both the cells and the
matrix threads, which are written as nondimensional relative deformations in
the standard quadratic form of Equation (4.6). Trivially, in the 2D case, we
have constraints on the cell surface and perimeter, while in the 3D case, we