Bulletin of Mathematical Biology

, Volume 80, Issue 5, pp 1259–1291 | Cite as

A Patient-Specific Anisotropic Diffusion Model for Brain Tumour Spread

  • Amanda Swan
  • Thomas Hillen
  • John C. Bowman
  • Albert D. Murtha
Special Issue : Mathematical Oncology


Gliomas are primary brain tumours arising from the glial cells of the nervous system. The diffuse nature of spread, coupled with proximity to critical brain structures, makes treatment a challenge. Pathological analysis confirms that the extent of glioma spread exceeds the extent of the grossly visible mass, seen on conventional magnetic resonance imaging (MRI) scans. Gliomas show faster spread along white matter tracts than in grey matter, leading to irregular patterns of spread. We propose a mathematical model based on Diffusion Tensor Imaging, a new MRI imaging technique that offers a methodology to delineate the major white matter tracts in the brain. We apply the anisotropic diffusion model of Painter and Hillen (J Thoer Biol 323:25–39, 2013) to data from 10 patients with gliomas. Moreover, we compare the anisotropic model to the state-of-the-art Proliferation–Infiltration (PI) model of Swanson et al. (Cell Prolif 33:317–329, 2000). We find that the anisotropic model offers a slight improvement over the standard PI model. For tumours with low anisotropy, the predictions of the two models are virtually identical, but for patients whose tumours show higher anisotropy, the results differ. We also suggest using the data from the contralateral hemisphere to further improve the model fit. Finally, we discuss the potential use of this model in clinical treatment planning.


Mathematical medicine Gliomas Partial differential equations Mathematical modelling Anisotropic diffusion 



AS would like to acknowledge funding from an NSERC CGS D3 scholarship and an Alberta Innovates Technology Futures Graduate Student Scholarship. Both TH and JB are supported through NSERC discovery Grants. The DTI images were acquired through an Alberta Cancer Foundation sponsored grant aimed at using DTI imaging to help predict the pattern of glioma growth.


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Copyright information

© Society for Mathematical Biology 2017

Authors and Affiliations

  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada
  2. 2.Department of Mathematical and Statistical Sciences, Centre for Mathematical BiologyUniversity of AlbertaEdmontonCanada
  3. 3.Cross Cancer InstituteEdmontonCanada

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