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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.

Glioma is a common type of primary brain tumour, with a strongly invasive potential, often exhibiting non-uniform, highly irregular growth. This makes it difficult to assess the degree of extent of the tumour, hence bringing about a supplementary challenge for the treatment. It is therefore necessary to understand the migratory behaviour of glioma in greater detail. In this paper, we propose a multiscale model for glioma growth and migration. Our model couples the microscale dynamics (reduced to the binding of surface receptors to the surrounding tissue) with a kinetic transport equation for the cell density on the mesoscopic level of individual cells. On the latter scale, we also include the proliferation of tumour cells via effects of interaction with the tissue. An adequate parabolic scaling yields a convection-diffusion-reaction equation, for which the coefficients can be explicitly determined from the information about the tissue obtained by diffusion tensor imaging (DTI). Numerical simulations relying on DTI measurements confirm the biological findings that glioma spread along white matter tracts.

Glioma is a broad class of brain and spinal cord tumors arising from glia cells, which are the main brain cells that can develop into neoplasms. They are highly invasive and lead to irregular tumor margins which are not precisely identifiable by medical imaging, thus rendering a precise enough resection very difficult. The understanding of glioma spread patterns is hence essential for both radiological therapy as well as surgical treatment. In this paper we propose a multiscale model for glioma growth including interactions of the cells with the underlying tissue network, along with proliferative effects. Our current accounting for two subpopulations of cells to accomodate proliferation according to the go-or-grow dichtomoty is an extension of the setting in [16]. As in that paper, we assume that cancer cells use neuronal fiber tracts as invasive pathways. Hence, the individual structure of brain tissue seems to be decisive for the tumor spread. Diffusion tensor imaging (DTI) is able to provide such information, thus opening the way for patient specific modeling of glioma invasion. Starting from a multiscale model involving subcellular (microscopic) and individual (mesoscale) cell dynamics, we perform a parabolic scaling to obtain an approximating reaction-diffusion-transport equation on the macroscale of the tumor cell population. Numerical simulations based on DTI data are carried out in order to assess the performance of our modeling approach.

SUMMARYIn this paper we present a new approach to simulations on complex-shaped domains. The method is based on a discontinuous Galerkin (DG) method, using trial and test functions defined on a structured grid. Essential boundary conditions are imposed weakly via the DG formulation. This method offers a discretization where the number of unknowns is independent of the complexity of the domain.We will show numerical computations for an elliptic scalar model problem in R 2 and R 3 . Convergence rates for different polynomial degrees are studied.

In order to perform electroencephalography (EEG) source reconstruction, i.e., to localize the sources underlying a measured EEG, the electric potential distribution at the electrodes generated by a dipolar current source in the brain has to be simulated, which is the so-called EEG forward problem. To solve it accurately, it is necessary to apply numerical methods that are able to take the individual geometry and conductivity distribution of the subject's head into account. In this context, the finite element method (FEM) has shown high numerical accuracy with the possibility to model complex geometries and conductive features, e.g., white matter conductivity anisotropy. In this article, we introduce and analyze the application of a discontinuous Galerkin (DG) method, a finite element method that includes features of the finite volume framework, to the EEG forward problem. The DG-FEM approach fulfills the conservation property of electric charge also in the discrete case, making it attractive for a variety of applications. Furthermore, as we show, this approach can alleviate modeling inaccuracies that might occur in head geometries when using classical FE methods, e.g., so-called "skull leakage effects", which may occur in areas where the thickness of the skull is in the range of the mesh resolution. Therefore, we derive a DG formulation of the FEM subtraction approach for the EEG forward problem and present numerical results that highlight the advantageous features and the potential benefits of the proposed approach.

SUMMARY A discontinuous Galerkin method for the solution of the immiscible and incompressible two‐phase flow problem based on the nonsymmetric interior penalty method is presented. Therefore, the incompressible Navier–Stokes equation is solved for a domain decomposed into two subdomains with different values of viscosity and density as well as a singular surface tension force. On the basis of a piecewise linear approximation of the interface, meshes for both phases are cut out of a structured mesh. The discontinuous finite elements are defined on the resulting Cartesian cut‐cell mesh and may therefore approximate the discontinuities of the pressure and the velocity derivatives across the interface with high accuracy. As the mesh resolves the interface, regularization of the density and viscosity jumps across the interface is not required. This preserves the local conservation property of the velocity field even in the vicinity of the interface and constitutes a significant advantage compared with standard methods that require regularization of these discontinuities and cannot represent the jumps and kinks in pressure and velocity. A powerful subtessellation algorithm is incorporated to allow the usage of standard time integrators (such as Crank–Nicholson) on the time‐dependent mesh. The presented discretization is applicable to both the two‐dimensional and three‐dimensional cases. The performance of our approach is demonstrated by application to a two‐dimensional benchmark problem, allowing for a thorough comparison with other numerical methods. Copyright © 2012 John Wiley & Sons, Ltd.