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.

We prove the global existence, along with some basic boundedness properties, of weak solutions to a PDE-ODE system modeling the multiscale invasion of tumor cells through the surrounding tissue matrix. The model has been proposed in [22] and accounts on the macroscopic level for the evolution of cell and tissue densities, along with the concentration of a chemoattractant, while on the subcellular level it involves the binding of integrins to soluble and insoluble components of the peritumoral region. The connection between the two scales is realized with the aid of a contractivity function characterizing the ability of the tumor cells to adapt their motility behavior to their subcellular dynamics.The resulting system, consisting of three partial and three ordinary differential equations including a temporal delay, in particular involves chemotactic and haptotactic cross-diffusion. In order to overcome technical obstacles stemming from the corresponding highest-order interaction terms, we base our analysis on a certain functional, inter alia involving the cell and tissue densities in the diffusion and haptotaxis terms respectively, which is shown to enjoy a quasi-dissipative property. This will be used as a starting point for the derivation of a series of integral estimates finally allowing for the construction of a generalized solution as the limit of solutions to suitably regularized problems.

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.

We investigate a PDE-ODE system describing cancer cell invasion in a tissue network. The model is an extension of the multiscale setting in [28,40], by considering two subpopulations of tumor cells interacting mutually and with the surrounding tissue. According to the go-or-grow hypothesis, these subpopulations consist of moving and proliferating cells, respectively. The mathematical setting also accommodates the effects of some therapy approaches. We prove the global existence of weak solutions to this model and perform numerical simulations to illustrate its behavior for different therapy strategies.

We propose and study a strongly coupled PDE-ODE system with tissue-dependent degenerate diffusion and haptotaxis that can serve as a model prototype for cancer cell invasion through the extracellular matrix. We prove the global existence of weak solutions and illustrate the model behaviour by numerical simulations for a two-dimensional setting.Comment: 25 page

We derive a multiscale model for tumor cell migration allowing to account for the receptor-mediated movement of the cells, the degradation of tissue fibers and the subsequent production of a soluble ligand whose concentration gradient then acts together with the distribution of tissue fibers as a directional cue for the cells. For this model we present a result on the local existence and uniqueness of a solution in all biologically relevant space dimensions.

We consider the multiscale model for glioma growth introduced in [19] and extend it to account for therapy effects. Thereby, three treatment strategies involving surgical resection, radio-, and chemotherapy are compared for their efficiency. The chemotherapy relies on inhibiting the binding of cell surface receptors to the surrounding tissue, which impairs both migration and proliferation.