Cell Migration in Extracellular Matrices - Cellular Potts Models: Multiscale Extensions and Biological Applications - page 170

we distribute 3 10 5 flat collagen-like fibers in each x and y-direction of the 3.5
cm-side length dish, yielding a density of 500 fibers/mm2; 2 ; see left top panel of
Figure 9.1. This corresponds to a concentration of nearly 5 10 3 collagen-like
molecules/m 2 , since, as seen, each thread is here assumed to be formed by
10 6 collagen-like molecules [7].
The analogous isotropic 3D scaffold consists of a regular cubic mesh of col-
lagen bers creating a uniform pore distribution 10 m wide (i.e., the same or-
der of magnitude of the initial cell diameter; see Figure 9.1, left bottom panel).
This gives a fibrous concentration of nearly 1.5 mg/mL, as each collagen-like
molecule has a molecular weight of 285 kDa [7]. We simulate a regular fibrous
network to avoid the minor heterogeneities often experienced in experimental
matrices, where the distribution of the threads and the relative pore diameters
is only roughly constant [242, 294, 409].
As shown in the wind-rose graphs (right panels of Figure 9.1), when cells
migrate on both 2D and in 3D matrices, the selected cell paths display a ran-
dom walk, without any preferred direction, in the absence of biasing chemical
gradients or matrix anisotropies.
Such migratory path structures and quantitative parameters are consistent
with experimental results for both 2D and 3D porous ECMs, such as for human
adult vascular smooth muscle cells (HSMCs) plated on flat type IV collagen
(CnIV) substrates of similar concentrations [106] or human glioma cells plated
on polyacrylamide ECMs [397], and for different fibroblastic and cancerous cell
lines migrating within 3D fibrous matrices of similar geometrical and struc-
tural properties, i.e., NR6 mouse fibroblasts in collagen-glycosaminoglycan
matrices [177], or human melanoma cells in collagen lattices [143]. Indeed,
these comparisons provide confidence in the choice of parameters describing
the biophysical and mechanical properties of the simulated cell{ECM system.

9.4 Anisotropic 2D and 3D Matrices
Next, we analyze the migratory characteristics of a cell population in the case
of anisotropic matrices. In particular, we keep fixed the quantity of fibers
as displayed in Figure 9.1, but progressively change their distribution by in-
creasing their number along the same x-direction, leaving the remaining fibers
disposed in their standard direction. The alignment of the matrix is quantified
by evaluating a proper index that can be called alignment index, given by

1
d 1

d n x N align =

n tot 1

; (9.5)

where d is the dimension of the domain, n x the number of threads along the
x-direction, and n tot their overall number. This quantity scales the percentage
of fibers aligned along the x-direction, so that it is zero in the case of isotropic