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Computational and Mathematical Methods in Medicine
Volume 2016, Article ID 1239861, 11 pages
Research Article

Modeling the Treatment of Glioblastoma Multiforme and Cancer Stem Cells with Ordinary Differential Equations

1Department of Mathematics, Winthrop University, Rock Hill, SC 29733, USA
2Department of Mathematics, Vassar College, Poughkeepsie, NY 12604, USA

Received 3 September 2015; Revised 12 November 2015; Accepted 22 December 2015

Academic Editor: Chung-Min Liao

Copyright © 2016 Kristen Abernathy and Jeremy Burke. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Despite improvements in cancer therapy and treatments, tumor recurrence is a common event in cancer patients. One explanation of recurrence is that cancer therapy focuses on treatment of tumor cells and does not eradicate cancer stem cells (CSCs). CSCs are postulated to behave similar to normal stem cells in that their role is to maintain homeostasis. That is, when the population of tumor cells is reduced or depleted by treatment, CSCs will repopulate the tumor, causing recurrence. In this paper, we study the application of the CSC Hypothesis to the treatment of glioblastoma multiforme by immunotherapy. We extend the work of Kogan et al. (2008) to incorporate the dynamics of CSCs, prove the existence of a recurrence state, and provide an analysis of possible cancerous states and their dependence on treatment levels.