Cross-diffusion in chemotaxis and tumor growth models

Cross-diffusion models are typically systems of strongly nonlinear equations in divergence form, whose diffusion matrix is nondiagonal. Such models naturally appear in, for instance, biology, chemistry, and physics in the study of multi-species systems. Cross diffusion may cause segregation of populations or granular materials, pattern formation in reaction-diffusion systems, or aggregation in chemotaxis models. In this project, we intend to analyze certain cross-diffusion models from biology describing chemotaxis or tumor growth. Our studies will include the well-posedness of the initial-boundary-value problems, the qualitative behavior of the solutions, and their numerical approximation. This project is not concerned with the modeling of the biological and chemical phenomena. Instead, the project is concerned with the understanding of the mathematical structure of cross-diffusion models and with the development of refined mathematical tools for their analysis. In the first part of the project, we will study cross-diffusion Keller-Segel-type models for chemotaxis. Chemotaxis is a biological phenomenon describing the change of motion of a population of species (amoebae, bacteria, cells etc.), which reacts in response to an external chemical stimulus spread in the environment of the population. A possible aggregation of the cells is reflected by the fact that the solutions to the multi-dimensional Keller-Segel model may blow up in finite time. We suggest a new extension of the parabolic-parabolic Keller-Segel model possessing weak solutions for all time and for any initial mass. The global existence result is based on new a priori estimates, which are a consequence of a characterization of more general cross-diffusion problems. We specify conditions under which such general problems possess a priori estimates, which we call entropy estimates. These conditions are related to the equivalence between the existence of a Lyapunov functional (mathematical entropy) for the system and the existence of a "symmetrizing" transformation of variables. Furthermore, the classical two-dimensional Keller-Segel model is numerically solved before and after blow up, employing a combination of a particle method (for the singular solution part) and a finite-element moving-mesh method (for the smooth part). In the second part of the project, we will analyze cross-diffusion tumor-growth models. Typically, the first stage of a tumor is the avascular growth (i.e. without blood vessels) which is mostly governed by the proliferation of tumor cells. We analyze two macroscopic avascular growth models from the literature: the model of Casciari et al. (1992), which is based on a drift-diffusion approach for the molar fluxes of the chemical species, and the model of Jackson and Byrne (2002), which describes the evolution of the volume fractions of the tumor cells and the extracellular matrix (i.e. the material surrounding the cells). The understanding of the mathematical structure of these cross- diffusion models is challenging since the diffusion matrices may be nonsymmetric and not positive definite. This problem will be overcome by suitable transformations of variables and corresponding entropy estimates. The objective is to prove the well-posedness of weak solutions and to develop mass-conserving and positivity- preserving numerical schemes.

The main aim of this project was the analysis of cross-diffusion models arising in biological applications like chemotaxis, tumor growth, and multicomponent fluids. Whereas diffusion is expected to smooth out the solution, cross diffusion may lead to a number of counterintuitive effects like up-hill diffusion and pattern formation. The mathematical difficulty is that the diffusion matrix of such systems often does not possess the properties needed for a mathematical analysis (symmetry, positive definiteness). The project was intended to contribute to the understanding of cross-diffusion effects. This goal has been achieved by exploiting a particular structure of the equations (called formal gradient-flow or entropy structure). Inspired by non-equilibrium thermodynamics, we have introduced so-called entropy variables which allow us to formulate the models in such a way that the diffusion matrix becomes (symmetric and) positive definite, hence overcoming the mathematical difficulties. Another advantage of this transformation is that it may be used to design stable numerical schemes which may be employed in biological applications.