Numerical resolution of a reinforced random walk model arising in haptotaxis

In this paper we study the numerical resolution of a reinforced random walk model arising in haptotaxis and the stabilization of solutions. The model consists of a system of two differential equations, one parabolic equation with a second order non-linear term (haptotaxis term) coupled to an ODE in a bounded two dimensional domain. We assume radial symmetry of the solutions. The scheme of resolution is based on the application of the characteristics method together with a finite element one. We present some numerical simulations which illustrate some features of the numerical stabilization of solutions

Computer modelling of haematopoietic stem cells migration

AbstractA mathematical model for migration of haematopoietic stem cells towards their niche in the bone marrow has been proposed in the literature. It consists of a chemotaxis system of partial differential equations with nonhomogeneous boundary conditions and an additional ordinary differential equation on a part of the computational boundary. The aim of the current work is to extend appropriately a second order positivity preserving central upwind scheme, originally proposed for a chemotaxis system with zero-flux boundary conditions and to apply it for the numerical solution of the considered problem. This paper introduces a first glance of such modification and outlines open questions in the handling of the nonlinear boundary conditions in a way that preserves the positivity of the solution. The presented numerical tests illustrate the need of the development of new specialized schemes for more complex chemotaxis systems

An unconditionally energy stable and positive upwind DG scheme for the Keller-Segel model

The well-suited discretization of the Keller-Segel equations for chemotaxis has become a very challenging problem due to the convective nature inherent to them. This paper aims to introduce a new upwind, mass-conservative, positive and energy-dissipative discontinuous Galerkin scheme for the Keller-Segel model. This approach is based on the gradient-flow structure of the equations. In addition, we show some numerical experiments in accordance with the aforementioned properties of the discretization. The numerical results obtained emphasize the really good behaviour of the approximation in the case of chemotactic collapse, where very steep gradients appear.Comment: 24 pages, 17 figures, 4 table

A mathematical insight in the epithelial-mesenchymal-like transition in cancer cells and its effect in the invasion of the extracellular matrix

Current biological knowledge supports the existence of a secondary group of cancer cells within the body of the tumour that exhibits stem cell-like properties. These cells are termed Cancer Stem Cells (CSCs}, and as opposed to the more usual Differentiated Cancer Cells (DCCs), they exhibit higher motility, they are more resilient to therapy, and are able to metastasize to secondary locations within the organism and produce new tumours. The origin of the CSCs is not completely clear; they seem to stem from the DCCs via a transition process related to the Epithelial-Mesenchymal Transition (EMT) that can also be found in normal tissue. In the current work we model and numerically study the transition between these two types of cancer cells, and the resulting "ensemble" invasion of the extracellular matrix. This leads to the derivation and numerical simulation of two systems: an algebraic-elliptic system for the transition and an advection-reaction-diffusion system of Keller-Segel taxis type for the invasion

Finite Difference Approximation with ADI Scheme for Two-dimensional Keller-Segel Equations

Keller-Segel systems are a set of nonlinear partial differential equations used to model chemotaxis in biology. In this paper, we propose two alternating direction implicit (ADI) schemes to solve the 2D Keller-Segel systems directly with minimal computational cost, while preserving positivity, energy dissipation law and mass conservation. One scheme unconditionally preserves positivity, while the other does so conditionally. Both schemes achieve second-order accuracy in space, with the former being first-order accuracy in time and the latter second-order accuracy in time. Besides, the former scheme preserves the energy dissipation law asymptotically. We validate these results through numerical experiments, and also compare the efficiency of our schemes with the standard five-point scheme, demonstrating that our approaches effectively reduce computational costs.Comment: 29 page