Cell migration on and through extracellular matrix is fundamental ina wide variety of physiological and pathological phenomena, and isexploited in scaffold-based tissue engineering. Migration isregulated by a number of extracellular matrix- or cell-derivedbiophysical parameters, such as matrix fiber orientation, pore size,and elasticity, or cell deformation, proteolysis, and adhesion. Wehere present an extended Cellular Potts Model (CPM) able toqualitatively and quantitatively describe cell migrationefficiencies and phenotypes both on two-dimensional substrates andwithin three-dimensional matrices, close to experimental evidence.As distinct features of our approach, cells are modeled ascompartmentalized discrete objects, differentiated into nucleusand cytosolic region, while the extracellular matrix iscomposed of a fibrous mesh and a homogeneous fluid. Our modelprovides a strong correlation of the directionality of migrationwith the topological extracellular matrix distribution and abiphasic dependence of migration on the matrix structure, density,adhesion, and stiffness, and, moreover, simulates that celllocomotion in highly constrained fibrillar obstacles requires thedeformation of the cell's nucleus and/or the activity ofcell-derived proteolysis. In conclusion, we here propose amathematical modeling approach that serves to characterize cellmigration as a biological phenomenon in healthy and diseased tissuesand in engineering applications.

Multiscale problems are ubiquitous and fundamental in all biological phenomena that emerge naturally from the complex interaction of processes which occur at various levels. A number of both discrete and continuous mathematical models and methods have been developed to address such an intricate network of organization. One of the most suitable individual cell-based model for this purpose is the well-known cellular Potts model (CPM). The CPM is a discrete, lattice-based, flexible technique that is able to accurately identify and describe the phenomenological mechanisms which are responsible for innumerable biological (and nonbiological) phenomena. In this work, we first give a brief overview of its biophysical basis and discuss its main limitations. We then propose some innovative extensions, focusing on ways of integrating the basic mesoscopic CPM with accurate continuous models of microscopic dynamics of individuals. The aim is to create a multiscale hybrid framework that is able to deal with the typical multilevel organization of biological development, where the behavior of the simulated individuals is realistically driven by their internal state. Our CPM extensions are then tested with sample applications that show a qualitative and quantitative agreement with experimental data. Finally, we conclude by discussing further possible developments of the method.

The description of the cell spatial pattern and characteristic distances is fundamental in a wide range of physio-pathological biological phenomena, from morphogenesis to cancer growth. Discrete particle models are widely used in this field, since they are focused on the cell-level of abstraction and are able to preserve the identity of single individuals reproducing their behavior. In particular, a fundamental role in determining the usefulness and the realism of a particle mathematical approach is played by the choice of the intercellular pairwise interaction kernel and by the estimate of its parameters. The aim of the paper is to demonstrate how the concept of H-stability, deriving from statistical mechanics, can have important implications in this respect. For any given interaction kernel, it in fact allows to a priori predict the regions of the free parameter space that result in stable configurations of the system characterized by a finite and strictly positive minimal interparticle distance, which is fundamental when dealing with biological phenomena. The proposed analytical arguments are indeed able to restrict the range of possible variations of selected model coefficients, whose exact estimate however requires further investigations (e.g., fitting with empirical data), as illustrated in this paper by series of representative simulations dealing with cell colony reorganization, sorting phenomena and zebrafish embryonic development.