Multiphase and multicomponent models

The project aims to develop models for tissue and blood vessel growth and to examine them mathematically. Models in cell biology are usually based on heuristic derivations, which can, however, lead to physical incompatibilities and thus to an incorrect description of biological phenomena. A particular focus is here on models that simulate tumor growth and the directed movement of cells based on chemical concentrations (chemotaxis). The project`s approach is to use so-called multiphase models. It is assumed that the temporal evolution of the biological components (e.g., cells, fibers, nutrients, water) can be described by fluid-dynamical equations in which the stress forces of the mixture are balanced by the forces between the components and by friction forces. This leads to highly complex so-called cross-diffusion equations, which are to be analyzed and numerically solved in this project. In addition to their complexity, the challenge posed by these equations lies in the fact that the diffusion matrices do not fulfill the necessary mathematical requirements (positive definiteness). This problem can be overcome by the inherently present physical energy or entropy structure, which underscores the physical consistency and can be translated to mathematical methods. The analytical part of the project will investigate the biological conditions under which the models are actually physically consistent, and the qualitative behavior of the model solutions will be studied. Structure-preserving finite volume methods and higher-order time discretizations allow quantitative calculations of the solutions and the comparison with biologically expected results. These goals are to be achieved with cooperation partners in Italy, France, Croatia, and Germany. The fundamental objectives of the project are to substantially refine mathematical entropy methods and to establish thermodynamic principles in cell biology in order to improve our understanding of the occurring phenomena.