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An Anisotropic Diffusion Model for Brain Tumour Spread Open Access


Other title
Anisotropic Diffusion
Mathematical Biology
Type of item
Degree grantor
University of Alberta
Author or creator
Swan, Amanda C
Supervisor and department
Hillen, Thomas (Mathematical and Statistical Sciences)
Bowman, John (Mathematical and Statistical Sciences)
Examining committee member and department
Lewis, Mark (Mathematical and Statistical Sciences)
de Vries, Gerda (Mathematical and Statistical Sciences)
Murray, David (Oncology)
Swanson, Kristin (Mayo Clinic)
Department of Mathematical and Statistical Sciences
Applied Mathematics
Date accepted
Graduation date
2016-06:Fall 2016
Doctor of Philosophy
Degree level
Gliomas, tumours arising from the glial cells of the nervous system, are some of the most difficult tumours to treat. In particular, glioblastoma are a particularly aggressive glioma subtype carrying a life expectancy of only 14 months. Typically, treatment combines surgery, radiation and chemotherapy where a key component of treatment planning is determining an appropriate treatment region over which to administer radiation therapy. Because gliomas are diffuse, only the main tumour mass shows up using imaging, while many undetectable cancer cells infiltrate the surrounding brain tissue. To account for this, treatment regions typically extend the visible tumour mass by a uniform $2$ $\centi \meter$ margin. We propose that a mathematical model for glioma cell density could help by modelling the spread of cancer cells and contribute to treatment plans that target the largest densities of these undetectable cells. In this thesis, we focus on the Painter-Hillen model for glioma spread, which uses anisotropic diffusion, allowing the rate of spread of the cells to vary with direction. This is meant to simulate the biological phenomenon where cancer cells spread preferentially along the fibrous white matter tracts within the brain, resulting in tumours having irregular shapes and projections. We establish the utility of this model by implementing it using data from ten patients, in both two and three dimensions. For comparison, a previous isotropic glioma model, the Swanson model, is used, as it has been applied successfully in a clinical setting. The results of our simulations indicate that the inclusion of anisotropy offers an advantage over the previous model. Finally, we develop two extensions to the Painter-Hillen model. In Chapter~6, we explore the derivation of a ``mass effect'' model, using a multiphase model framework. The mass effect model includes the forces induced by the growing mass, introducing a mechanical component to the model. This effect becomes important where a tumour is growing in close proximity to the skull, where the growth in this direction will be impeded by the pressure generated by the increased density. In Chapter 7, we discuss an extension of the Painter-Hillen model, using a transport model framework. The generalization considered allows the turning rate to vary with the direction that a cell is travelling in relation to the underlying structure, since a cell travelling along a fibre will turn less frequently than one travelling perpendicularly. Both of these model extensions have potential for further exploration.
This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for the purpose of private, scholarly or scientific research. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.
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