The Filament Based Lamellipodium Model (FBLM) is a two-phase two-dimensional continuum model, describing the dynamcis of two interacting families of locally parallel actin filaments [31]. It contains accounts of the filaments' bending stiffness, of adhesion to the substrate, and of cross-links connecting the two families.An extension of the model is presented with contributions from nucleation of filaments by branching, from capping, from contraction by actin-myosin interaction, and from a pressure-like repulsion between parallel filaments due to Coulomb interaction. The effect of a chemoattractant is described by a simple signal transduction model influencing the polymerization speed. Simulations with the extended model show its potential for describing various moving cell shapes, depending on the signal transduction procedure, and for predicting transients between nonmoving and moving states as well as changes of direction.

We propose a multiscale model of the invasion of the extracellular matrix by two types of cancer cells, the differentiated cancer cells and the cancer stem cells. We assume that the epithelial mesenchymal-like transition between them is driven primarily by the epidermal growth factors. We moreover take into account the transidifferentiation program of the cancer stem cells and the cancer associated fibroblast cells as well as the fibroblast-driven remodelling of the extracellular matrix.The proposed haptotaxis model combines the macroscopic phenomenon of the invasion of the extracellular matrix with the microscopic dynamics of the epidermal growth factors. We analyse our model in a component-wise manner and compare our findings with the literature. We investigate pathological situations regarding the epidermal growth factors and accordingly propose "mathematical-treatment" scenarios to control the aggressiveness of the tumour.

No abstract

We describe in this work the numerical treatment of the Filament Based Lamellipodium Model (FBLM). The model itself is a two-phase two-dimensional continuum model, describing the dynamics of two interacting families of locally parallel F-actin filaments. It includes, among others, the bending stiffness of the filaments, adhesion to the substrate, and the cross-links connecting the two families. The numerical method proposed is a Finite Element Method (FEM) developed specifically for the needs of these problem. It is comprised of composite Lagrange-Hermite two dimensional elements defined over two dimensional space. We present some elements of the FEM and emphasise in the numerical treatment of the more complex terms. We also present novel numerical simulations and compare to in-vitro experiments of moving cells.

The ability to locally degrade the extracellular matrix (ECM) and interact with the tumor microenvironment is a key process distinguishing cancer cells from normal cells, and is a critical step in the metastatic spread of the tumor. The invasion of the surrounding tissue involves the coordinated action of the cancer cells, the ECM, the matrix degrading enzymes, and the epithelial-to-mesenchymal transition. In this paper, we present a mathematical model which describes the transition from an epithelial invasion strategy of the epithelial-like cells (ECs) to an individual invasion strategy for the mesenchymal-like cells (MCs). We achieve this by formulating a genuinely multiscale and hybrid system consisting of partial and stochastic differential equations that describe the evolution of the ECs and the MCs while accounting for the transitions between them. This approach allows one to reproduce, in a very natural way, fundamental qualitative features of the current biomedical understanding of cancer invasion that are not easily captured by classical modelling approaches, for example, the invasion of the ECM by self-generated gradients, and the formation of EC invasion islands outside of the main body of the tumor.

We propose a novel approach to modeling cell migration in an anisotropic environment with biochemical heterogeneity and interspecies interactions, using as a paradigm glioma invasion in brain tissue under the influence of hypoxia-triggered angiogenesis. The multiscale procedure links single-cell and mesoscopic dynamics with population level behavior, leading on the macroscopic scale to flux-limited glioma diffusion and multiple taxis. We verify the non-negativity of regular solutions (provided they exist) to the obtained macroscopic PDE-ODE system and perform numerical simulations to illustrate the solution behavior under several scenarios.

We provide a short review of existing models with multiple taxis performed by (at least) one species and consider a new mathematical model for tumor invasion featuring two mutually exclusive cell phenotypes (migrating and proliferating). The migrating cells perform nonlinear diffusion and two types of taxis in response to non-diffusing cues: away from proliferating cells and up the gradient of surrounding tissue. Transitions between the two cell subpopulations are influenced by subcellular (receptor binding) dynamics, thus conferring the setting a multiscale character. We prove global existence of weak solutions to a simplified model version and perform numerical simulations for the full setting under several phenotype switching and motility scenarios. We also compare (via simulations) this model with the corresponding haptotaxis-chemotaxis one featuring indirect chemorepellent production and provide a discussion about possible model extensions and mathematical challenges.

We consider 3-point numerical schemes for scalar Conservation Laws, that are oscillatory either to their dispersive or anti-diffusive nature. Oscillations are responsible for the increase of the Total Variation (TV); a bound on which is crucial for the stability of the numerical scheme. It has been noticed ([AKM01], [AMT04], [AMS08]) that the use of non-uniform adaptively redefined meshes, that take into account the geometry of the numerical solution itself, is capable of taming oscillations; hence improving the stability properties of the numerical schemes.In this work we provide a model for studying the evolution of the extremes over nonuniform adaptively redefined meshes. Based on this model we prove that proper mesh reconstruction is able to control the oscillations; we provide bounds for the Total Variation (TV) of the numerical solution. We moreover prove under more strict assumptions that the increase of the TV -due to the oscillatory behaviour of the numerical schemes-decreases with time; hence proving that the overall scheme is TV Increase-Decreasing (TVI-D).