### Mathematical Biosciences and Engineering

2015, Issue 6: 1157-1172. doi: 10.3934/mbe.2015.12.1157

# A data-motivated density-dependent diffusion model of in vitro glioblastoma growth

• Received: 01 October 2014 Accepted: 29 June 2018 Published: 01 August 2015
• MSC : Primary: 92C50, 35C07; Secondary: 35K57.

• Glioblastoma multiforme is an aggressive brain cancer that is extremely fatal. It is characterized by both proliferation and large amounts of migration, which contributes to the difficulty of treatment. Previous models of this type of cancer growth often include two separate equations to model proliferation or migration. We propose a single equation which uses density-dependent diffusion to capture the behavior of both proliferation and migration. We analyze the model to determine the existence of traveling wave solutions. To prove the viability of the density-dependent diffusion function chosen, we compare our model with well-known in vitro experimental data.

Citation: Tracy L. Stepien, Erica M. Rutter, Yang Kuang. A data-motivated density-dependent diffusion model of in vitro glioblastoma growth[J]. Mathematical Biosciences and Engineering, 2015, 12(6): 1157-1172. doi: 10.3934/mbe.2015.12.1157

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• Glioblastoma multiforme is an aggressive brain cancer that is extremely fatal. It is characterized by both proliferation and large amounts of migration, which contributes to the difficulty of treatment. Previous models of this type of cancer growth often include two separate equations to model proliferation or migration. We propose a single equation which uses density-dependent diffusion to capture the behavior of both proliferation and migration. We analyze the model to determine the existence of traveling wave solutions. To prove the viability of the density-dependent diffusion function chosen, we compare our model with well-known in vitro experimental data.

 [1] J. Theor. Biol., 243 (2006), 98-113. [2] SIAM J. Math. Anal., 12 (1981), 880-892. [3] Physics in Medicine and Biology, 53 (2008), p879. [4] J. Theor. Biol., 245 (2007), 576-594. [5] 2nd edition, Springer, 2006. [6] http://www.mathworks.com/matlabcentral/fileexchange/7173-grabit, (2005), Retrieved July 1, 2014. [7] J. Phys. A-Math. Gen., 38 (2005), 3367-3379. [8] Math. Biosci. Eng., 12 (2015), 41-69. [9] J. Phys. A-Math. Gen., 37 (2004), 6267-6268. [10] Magnetic Resonance in Medicine, 54 (2005), 616-624. [11] Eur. Phys. J. Plus, 128 (2013), p136. [12] SIAM J. Optimiz., 9 (1999), 112-147. [13] Discrete Cont. Dyn.-B, 6 (2006), 1175-1189. [14] Math. Nachr., 242 (2002), 148-164. [15] Commun. Pur. Appl. Anal., 9 (2010), 1083-1098. [16] Abstr. Appl. Anal., 2011 (2011), 1-22. [17] Math. Biosci. Eng., 12 (2015), 879-905. [18] 3rd edition, Springer, 2002. [19] Phys. Rev. E, 85 (2012), 066120. [20] Neurologist, 12 (2006), 279-292. [21] J. Theor. Biol., 323 (2013), 25-39. [22] J. Math. Biol., 50 (2005), 683-698. [23] Phys. Rev. E, 84 (2011), 021921. [24] J. Differ. Equations, 117 (1995), 281-319. [25] Discrete Cont. Dyn.-B, 138 (2010), 455-487. [26] SIAM J. Sci. Stat. Comp., 11 (1990), 1-32. [27] Biophy. J., 92 (2007), 356-365. [28] in Mathematical Modeling of Biological Systems (eds. A. Deutsch, L. Brusch, H. Byrne, G. Vries and H. Herzel), vol. I of Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, 2007, 217-224. [29] J. Neurol. Sci., 216 (2003), 1-10. [30] Cell Proliferat., 28 (1995), 17-31. [31] J. Math. Biol., 33 (1994), 1-16.
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