Abstract
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.
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Notes
We obtained preprocessed DTI data by courtesy of Carsten Wolters and Felix Lucka, Institute for Biomagnetism und Biosignalanalysis, University of Münster. This includes segmentation data, the apparent water diffusion tensors as well as a brainmask for each voxel of the brain of a healthy male student.
the larger the value of \(\kappa \), the higher the concentration of distribution around the mean direction.
MATLAB Release 2012b, The MathWorks, Inc., Natick, Massachusetts, United States.
We assumed that attachment and detachment of receptors to tissue components take place with similar rates. However, variations in the choice of the binding rate \(k^+\) (we tried with up to \(50\,\%\)) do not seem to influence the dynamics in a notable way. The choice of \(\lambda _1\) is rather imprecise, as there are no data or references available for this parameter. We took it to be rather small, in order to endorse the positivity of the turning rate and not to exaggerate the influence of the microscale dynamics. Larger values of \(\lambda _1\) would mean higher advection, hence a more pronounced anisotropic behavior.
at least qualitatively, till data become available for an adequate validation.
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Acknowledgments
We thank Carsten Wolters and Felix Lucka (Institute of Biomagnetism and Biosignalanalysis, University of Münster) for a lot of preprocessed DTI data and much advice with the visualization software SciRun. We are grateful for supportive and challenging remarks by Andreas Deutsch (TU Dresden) and for enlightening discussions with Katarina Wolf (Radboud University Nijmegen) concerning the biology of glioma invasion.
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Engwer, C., Hillen, T., Knappitsch, M. et al. Glioma follow white matter tracts: a multiscale DTI-based model. J. Math. Biol. 71, 551–582 (2015). https://doi.org/10.1007/s00285-014-0822-7
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DOI: https://doi.org/10.1007/s00285-014-0822-7