Analysis of Reaction-Diffusion Models with the Taxis Mechanism

This open access book deals with a rich variety of taxis-type cross-diffusive equations. Particularly, it intends to show the key role played by quasi-energy inequality in the derivation of some necessary a priori estimates. This book addresses applied mathematics and all researchers interested in mathematical development of reaction-diffusion theory and its application and can be a basis for a graduate course in applied mathematics

Analysis of Reaction-Diffusion Models with the Taxis Mechanism

This open access book deals with a rich variety of taxis-type cross-diffusive equations. Particularly, it intends to show the key role played by quasi-energy inequality in the derivation of some necessary a priori estimates. This book addresses applied mathematics and all researchers interested in mathematical development of reaction-diffusion theory and its application and can be a basis for a graduate course in applied mathematics

Convergence of a cancer invasion model to a logistic chemotaxis model

A characteristic feature of tumor invasion is the destruction of the healthy tissue surrounding it. Open space is generated, which invasive tumor cells can move into. One such mechanism is the urokinase plasminogen system (uPS), which is found in many processes of tissue reorganization. Lolas, Chaplain and collaborators have developed a series of mathematical models for the uPS and tumor invasion. These models are based upon degradation of the extracellular material through plasmid plus chemotaxis and haptotaxis. In this paper we consider the uPS invasion models in one-space dimension and we identify a condition under which this cancer invasion model converges to a chemotaxis model with logistic growth. This condition assumes that the density of the extracellular material is not too large. Our result shows that the complicated spatio-temporal patterns, which were observed by Lolas and Chaplain et al. are organized by the chaotic attractor of the logistic chemotaxis system. Our methods are based on energy estimates, where, for convergence, we needed to find lower estimates in Lγ for 0 < γ < 1. This is a new method for these types of PDE. © 2013 World Scientific Publishing Company

Global existence of classical solutions and numerical simulations of a cancer invasion model

In this paper, we study a cancer invasion model both theoretically and numerically. The model is a nonstationary, nonlinear system of three coupled partial differential equations modeling the motion of cancer cells, degradation of the extracellular matrix, and certain enzymes. We first establish existence of global classical solutions in both two- and three-dimensional bounded domains, despite the lack of diffusion of the matrix-degrading enzymes and corresponding regularizing effects in the analytical treatment. Next, we give a weak formulation and apply finite differences in time and a Galerkin finite element scheme for spatial discretization. The overall algorithm is based on a fixed-point iteration scheme. Our theory and numerical developments are accompanied by some simulations in two and three spatial dimensions

Mini-Workshop: Mathematical Models for Cancer Cell Migration

Tumour cell invasion is an essential hallmark in the progression of malignant cancer. Thereby, cancer cells migrate through the surrounding tissue (normal cells, extracellular matrix, interstitial fluid) towards blood or lymph vessels which they penetrate and thus access the blood flow. They are carried by blood circulation to distant locations where they extravasate and develop new tumours, a phenomenon known as metastasis. The invasive spread of cancer cells is highly complex – it involves several mechanisms, like diffusion, chemotaxis and haptotaxis; these in turn are conditioned by and influence the subcellular dynamics. Mathematical models offer a powerful tool to gain insight into the complicated biological processess connected to tumour invasion and have also stimulated advanced mathematical research. Some of the new developments in the field of biomedical oncology were inspired by such models. A significant challenge arises due to the interactions of cancer cells with a complicated and structured microenvironment of healthy tissue. Many of the models of cancer cell migration are based on partial differential equations (PDEs) including spatial heterogeneity, orientational tissue structure, tissue stiffness and deformability. Specific settings relate to reaction-diffusion equations, transport equations, continuum equations, and to their multi-scale analysis, to local and global existence and uniqueness, to pattern formation, blow-ups and invasions. A further approach involves agent-based models providing a characterisation of cell migration by way of simulating the (inter)actions of autonomous agents (individual cells, collective dynamics) and aiming for assessing their effects on the entire system. In this meeting we covered the full spectrum between macroscopic PDE models and microscopic individual based models with the common goal of modelling cancer cell migration. Of particular interest was the derivation of macroscopic properties from microscopic details. Similar multiscale models have been used in other contexts (such as chemotaxis for example), and we gained some significant insight from the collaborations in this workshop. In this one week meeting we posted nine open ended problems (outlined below), which will form the seed for new collaborations going far beyond this workshop