Mathematical modelling of chemotaxis (the movement of biological cells or organisms in response to chemical gradients) has developed into a large and diverse discipline, whose aspects include its mechanistic basis, the modelling of specific systems and the mathematical behaviour of the underlying equations. The Keller-Segel model of chemotaxis (Keller and Segel in J Theor Biol 26:399-415, 1970; 30:225-234, 1971) has provided a cornerstone for much of this work, its success being a consequence of its intuitive simplicity, analytical tractability and capacity to replicate key behaviour of chemotactic populations. One such property, the ability to display "auto-aggregation", has led to its prominence as a mechanism for self-organisation of biological systems. This phenomenon has been shown to lead to finite-time blow-up under certain formulations of the model, and a large body of work has been devoted to determining when blow-up occurs or whether globally existing solutions exist. In this paper, we explore in detail a number of variations of the original Keller-Segel model. We review their formulation from a biological perspective, contrast their patterning properties, summarise key results on their analytical properties and classify their solution form. We conclude with a brief discussion and expand on some of the outstanding issues revealed as a result of this work.

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In this paper mesoscopic (individual based) and macroscopic (population based) models for mesenchymal motion of cells in fibre networks are developed. Mesenchymal motion is a form of cellular movement that occurs in three-dimensions through tissues formed from fibre networks, for example the invasion of tumor metastases through collagen networks. The movement of cells is guided by the directionality of the network and in addition, the network is degraded by proteases.The main results of this paper are derivations of mesoscopic and macroscopic models for mesenchymal motion in a timely varying network tissue. The mesoscopic model is based on a transport equation for correlated random walk and the macroscopic model has the form of a drift-diffusion equation where the mean drift velocity is given by the mean orientation of the tissue and the diffusion tensor is given by the variance-covariance matrix of the tissue orientations. The transport equation as well as the drift-diffusion limit are coupled to a differential equation that describes the tissue changes explicitly, where we distinguish the cases of directed and undirected tissues. As a result the drift velocity and the diffusion tensor are timely varying. We discuss relations to existing models and possible applications.

In this paper we study a version of the Keller-Segel model where the chemotactic cross-diffusion depends on both the external signal and the local population density. A parabolic quasi-linear strongly coupled system follows. By incorporation of a population-sensing (or "quorum-sensing") mechanism, we assume that the chemotactic response is switched off at high cell densities. The response to high population densities prevents overcrowding, and we prove local and global existence in time of classical solutions. Numerical simulations show interesting phenomena of pattern formation and formation of stable aggregates. We discuss the results with respect to previous analytical results on the Keller-Segel model.

Gliomas are a class of rarely curable tumors arising from abnormal glia cells in the human brain. The understanding of glioma spreading patterns is essential for both radiological therapy as well as surgical treatment. Diffusion tensor imaging (DTI) allows to infer the white matter fibre structure of the brain in a noninvasive way. Painter and Hillen (J Theor Biol 323:25-39, 2013) used a kinetic partial differential equation to include DTI data into a class of anisotropic diffusion models for glioma spread. Here we extend this model to explicitly include adhesion mechanisms between glioma cells and the extracellular matrix components which are associated to white matter tracts. The mathematical modelling follows the multiscale approach proposed by Kelkel and Surulescu (Math Models Methods Appl Sci 23(3), 2012). We use scaling arguments to deduce a macroscopic advection-diffusion model for this process. The tumor diffusion tensor and the tumor drift velocity depend on both, the directions of the white matter tracts as well as the binding dynamics of the adhesion molecules. The advanced computational platform DUNE enables us to accurately solve our macroscopic model. It turns out that the inclusion of cell binding dynamics on the microlevel is an important factor to explain finger-like spread of glioma.