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

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2015, 12(6): 1157-1172. doi: 10.3934/mbe.2015.12.1157

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

Tracy L. Stepien ^1, , Erica M. Rutter ^1, and Yang Kuang ^2,

1.	School of Mathematical & Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, United States, United States
2.	School of Mathematics and Statistical Sciences, Arizona State University, Tempe, AZ 85281

Received October 2014 Revised March 2015 Published August 2015

Abstract
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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.

Keywords: traveling waves, glioblastoma, Biomathematical modeling, tumor growth simulation..

Mathematics Subject Classification: Primary: 92C50, 35C07; Secondary: 35K5.

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

References:

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[4]	A. Q. Cai, K. A. Landman and B. D. Hughes, Multi-scale modeling of a wound-healing cell migration assay, J. Theor. Biol., 245 (2007), 576-594. doi: 10.1016/j.jtbi.2006.10.024.
[5]	C. Chicone, Ordinary Differential Equations with Applications, vol. 34 of Texts in Applied Mathematics, 2nd edition, Springer, 2006.
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[8]	T. Harko and M. K. Mak, Traveling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach, Math. Biosci. Eng., 12 (2015), 41-69. doi: 10.3934/mbe.2015.12.41.
[9]	S. Harris, Fisher equation with density-dependent diffusion: Special solutions, J. Phys. A-Math. Gen., 37 (2004), 6267-6268. doi: 10.1088/0305-4470/37/24/005.
[10]	S. Jbabdi, E. Mandonnet, H. Duffau, L. Capelle, K. R. Swanson, M. Pélégrini-Issac, R. Guillevin and H. Benali, Simulation of anisotropic growth of low-grade gliomas using diffusion tensor imaging, Magnetic Resonance in Medicine, 54 (2005), 616-624. doi: 10.1002/mrm.20625.
[11]	E. Kengne, M. Saudé, F. B. Hamouda and A. Lakhssassi, Traveling wave solutions of density-dependent nonlinear reaction-diffusion equation via the extended generalized Riccati equation mapping method, Eur. Phys. J. Plus, 128 (2013), p136. doi: 10.1140/epjp/i2013-13136-7.
[12]	J. C. Lagarias, J. A. Reeds, M. H. Wright and P. E. Wright, Convergence properties of the Nelder-Mead simplex method in low dimensions, SIAM J. Optimiz., 9 (1999), 112-147. doi: 10.1137/S1052623496303470.
[13]	P. K. Maini, L. Malaguti, C. Marcelli and S. Matucci, Diffusion-aggregation processes with mono-stable reaction terms, Discrete Cont. Dyn.-B, 6 (2006), 1175-1189. doi: 10.3934/dcdsb.2006.6.1175.
[14]	L. Malaguti and C. Marcelli, Travelling Wavefronts in Reaction-Diffusion Equations with Convection Effects and Non-Regular Terms, Math. Nachr., 242 (2002), 148-164. doi: 10.1002/1522-2616(200207)242:1<148::AID-MANA148>3.0.CO;2-J.
[15]	L. Malaguti, C. Marcelli and S. Matucci, Continuous dependence in front propagation of convective reaction-diffusion equations, Commun. Pur. Appl. Anal., 9 (2010), 1083-1098. doi: 10.3934/cpaa.2010.9.1083.
[16]	L. Malaguti, C. Marcelli and S. Matucci, Continuous dependence in front propagation for convective reaction-diffusion models with aggregative movements, Abstr. Appl. Anal., 2011 (2011), 1-22. doi: 10.1155/2011/986738.
[17]	N. L. Martirosyan, E. M. Rutter, W. L. Ramey, E. J. Kostelich, Y. Kuang and M. C. Preul, Mathematically modeling the biological properties of gliomas: A review, Math. Biosci. Eng., 12 (2015), 879-905. doi: 10.3934/mbe.2015.12.879.
[18]	J. D. Murray, Mathematical Biology: I: An Introduction, vol. 17 of Interdisciplinary Applied Mathematics, 3rd edition, Springer, 2002.
[19]	W. Ngamsaad and K. Khompurngson, Self-similar solutions to a density-dependent reaction-diffusion model, Phys. Rev. E, 85 (2012), 066120. doi: 10.1103/PhysRevE.85.066120.
[20]	A. D. Norden and P. Y. Wen, Glioma therapy in adults, Neurologist, 12 (2006), 279-292. doi: 10.1097/01.nrl.0000250928.26044.47.
[21]	K. J. Painter and T. Hillen, Mathematical modelling of glioma growth: The use of Diffusion Tensor Imaging (DTI) data to predict the anisotropic pathways of cancer invasion, J. Theor. Biol., 323 (2013), 25-39. doi: 10.1016/j.jtbi.2013.01.014.
[22]	M. G. Pedersen, Wave speeds of density dependent Nagumo diffusion equations - inspired by oscillating gap-junction conductance in the islets of Langerhans, J. Math. Biol., 50 (2005), 683-698. doi: 10.1007/s00285-004-0304-4.
[23]	V. M. Pérez-García, G. F. Calvo, J. Belmonte-Beitia, D. Diego and L. Pérez-Romasanta, Bright solitary waves in malignant gliomas, Phys. Rev. E, 84 (2011), 021921. doi: 10.1103/PhysRevE.84.021921.
[24]	F. Sánchez-Garduño and P. K. Maini, Traveling wave phenomena in some degenerate reaction-diffusion equations, J. Differ. Equations, 117 (1995), 281-319. doi: 10.1006/jdeq.1995.1055.
[25]	F. Sánchez-Garduño, P. K. Maini and J. Pérez-Velásquez, A non-linear degenerate equation for direct aggregation and traveling wave dynamics, Discrete Cont. Dyn.-B, 138 (2010), 455-487. doi: 10.3934/dcdsb.2010.13.455.
[26]	R. D. Skeel and M. Berzins, A method for the spatial discretization of parabolic equations in one space variable, SIAM J. Sci. Stat. Comp., 11 (1990), 1-32. doi: 10.1137/0911001.
[27]	A. M. Stein, T. Demuth, D. Mobley, M. Berens and L. M. Sander, A mathematical model of glioblastoma tumor spheroid invasion in a three-dimensional in vitro experiment, Biophy. J., 92 (2007), 356-365. doi: 10.1529/biophysj.106.093468.
[28]	A. M. Stein, D. A. Vader, L. M. Sander and D. A. Weitz, A stochastic model of glioblastoma invasion, 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. doi: 10.1007/978-0-8176-4558-8_19.
[29]	K. R. Swanson, C. Bridge, J. Murray and E. C. Alvord Jr, Virtual and real brain tumors: Using mathematical modeling to quantify glioma growth and invasion, J. Neurol. Sci., 216 (2003), 1-10. doi: 10.1016/j.jns.2003.06.001.
[30]	P. Tracqui, G. Cruywagen, D. Woodward, G. Bartoo, J. Murray and E. Alvord, A mathematical model of glioma growth: The effect of chemotherapy on spatio-temporal growth, Cell Proliferat., 28 (1995), 17-31. doi: 10.1111/j.1365-2184.1995.tb00036.x.
[31]	T. P. Witelski, An asymptotic solution for traveling waves of a nonlinear-diffusion Fisher's equation, J. Math. Biol., 33 (1994), 1-16. doi: 10.1007/BF00160171.

show all references

References:

[1]	N. J. Armstrong, K. J. Painter and J. A. Sherratt, A continuum approach to modelling cell-cell adhesion, J. Theor. Biol., 243 (2006), 98-113. doi: 10.1016/j.jtbi.2006.05.030.
[2]	C. Atkinson, G. E. H. Reuter and C. J. Ridler-Rowe, Traveling wave solutions for some nonlinear diffusion equations, SIAM J. Math. Anal., 12 (1981), 880-892. doi: 10.1137/0512074.
[3]	P.-Y. Bondiau, O. Clatz, M. Sermesant, P.-Y. Marcy, H. Delingette, M. Frenay and N. Ayache, Biocomputing: Numerical simulation of glioblastoma growth using diffusion tensor imaging, Physics in Medicine and Biology, 53 (2008), p879. doi: 10.1088/0031-9155/53/4/004.
[4]	A. Q. Cai, K. A. Landman and B. D. Hughes, Multi-scale modeling of a wound-healing cell migration assay, J. Theor. Biol., 245 (2007), 576-594. doi: 10.1016/j.jtbi.2006.10.024.
[5]	C. Chicone, Ordinary Differential Equations with Applications, vol. 34 of Texts in Applied Mathematics, 2nd edition, Springer, 2006.
[6]	J. Doke, GRABIT, MATLAB Central File Exchange, http://www.mathworks.com/matlabcentral/fileexchange/7173-grabit, (2005), Retrieved July 1, 2014.
[7]	B. H. Gilding and R. Kersner, A Fisher/KPP-type equation with density-dependent diffusion and convection: travelling-wave solutions, J. Phys. A-Math. Gen., 38 (2005), 3367-3379. doi: 10.1088/0305-4470/38/15/009.
[8]	T. Harko and M. K. Mak, Traveling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach, Math. Biosci. Eng., 12 (2015), 41-69. doi: 10.3934/mbe.2015.12.41.
[9]	S. Harris, Fisher equation with density-dependent diffusion: Special solutions, J. Phys. A-Math. Gen., 37 (2004), 6267-6268. doi: 10.1088/0305-4470/37/24/005.
[10]	S. Jbabdi, E. Mandonnet, H. Duffau, L. Capelle, K. R. Swanson, M. Pélégrini-Issac, R. Guillevin and H. Benali, Simulation of anisotropic growth of low-grade gliomas using diffusion tensor imaging, Magnetic Resonance in Medicine, 54 (2005), 616-624. doi: 10.1002/mrm.20625.
[11]	E. Kengne, M. Saudé, F. B. Hamouda and A. Lakhssassi, Traveling wave solutions of density-dependent nonlinear reaction-diffusion equation via the extended generalized Riccati equation mapping method, Eur. Phys. J. Plus, 128 (2013), p136. doi: 10.1140/epjp/i2013-13136-7.
[12]	J. C. Lagarias, J. A. Reeds, M. H. Wright and P. E. Wright, Convergence properties of the Nelder-Mead simplex method in low dimensions, SIAM J. Optimiz., 9 (1999), 112-147. doi: 10.1137/S1052623496303470.
[13]	P. K. Maini, L. Malaguti, C. Marcelli and S. Matucci, Diffusion-aggregation processes with mono-stable reaction terms, Discrete Cont. Dyn.-B, 6 (2006), 1175-1189. doi: 10.3934/dcdsb.2006.6.1175.
[14]	L. Malaguti and C. Marcelli, Travelling Wavefronts in Reaction-Diffusion Equations with Convection Effects and Non-Regular Terms, Math. Nachr., 242 (2002), 148-164. doi: 10.1002/1522-2616(200207)242:1<148::AID-MANA148>3.0.CO;2-J.
[15]	L. Malaguti, C. Marcelli and S. Matucci, Continuous dependence in front propagation of convective reaction-diffusion equations, Commun. Pur. Appl. Anal., 9 (2010), 1083-1098. doi: 10.3934/cpaa.2010.9.1083.
[16]	L. Malaguti, C. Marcelli and S. Matucci, Continuous dependence in front propagation for convective reaction-diffusion models with aggregative movements, Abstr. Appl. Anal., 2011 (2011), 1-22. doi: 10.1155/2011/986738.
[17]	N. L. Martirosyan, E. M. Rutter, W. L. Ramey, E. J. Kostelich, Y. Kuang and M. C. Preul, Mathematically modeling the biological properties of gliomas: A review, Math. Biosci. Eng., 12 (2015), 879-905. doi: 10.3934/mbe.2015.12.879.
[18]	J. D. Murray, Mathematical Biology: I: An Introduction, vol. 17 of Interdisciplinary Applied Mathematics, 3rd edition, Springer, 2002.
[19]	W. Ngamsaad and K. Khompurngson, Self-similar solutions to a density-dependent reaction-diffusion model, Phys. Rev. E, 85 (2012), 066120. doi: 10.1103/PhysRevE.85.066120.
[20]	A. D. Norden and P. Y. Wen, Glioma therapy in adults, Neurologist, 12 (2006), 279-292. doi: 10.1097/01.nrl.0000250928.26044.47.
[21]	K. J. Painter and T. Hillen, Mathematical modelling of glioma growth: The use of Diffusion Tensor Imaging (DTI) data to predict the anisotropic pathways of cancer invasion, J. Theor. Biol., 323 (2013), 25-39. doi: 10.1016/j.jtbi.2013.01.014.
[22]	M. G. Pedersen, Wave speeds of density dependent Nagumo diffusion equations - inspired by oscillating gap-junction conductance in the islets of Langerhans, J. Math. Biol., 50 (2005), 683-698. doi: 10.1007/s00285-004-0304-4.
[23]	V. M. Pérez-García, G. F. Calvo, J. Belmonte-Beitia, D. Diego and L. Pérez-Romasanta, Bright solitary waves in malignant gliomas, Phys. Rev. E, 84 (2011), 021921. doi: 10.1103/PhysRevE.84.021921.
[24]	F. Sánchez-Garduño and P. K. Maini, Traveling wave phenomena in some degenerate reaction-diffusion equations, J. Differ. Equations, 117 (1995), 281-319. doi: 10.1006/jdeq.1995.1055.
[25]	F. Sánchez-Garduño, P. K. Maini and J. Pérez-Velásquez, A non-linear degenerate equation for direct aggregation and traveling wave dynamics, Discrete Cont. Dyn.-B, 138 (2010), 455-487. doi: 10.3934/dcdsb.2010.13.455.
[26]	R. D. Skeel and M. Berzins, A method for the spatial discretization of parabolic equations in one space variable, SIAM J. Sci. Stat. Comp., 11 (1990), 1-32. doi: 10.1137/0911001.
[27]	A. M. Stein, T. Demuth, D. Mobley, M. Berens and L. M. Sander, A mathematical model of glioblastoma tumor spheroid invasion in a three-dimensional in vitro experiment, Biophy. J., 92 (2007), 356-365. doi: 10.1529/biophysj.106.093468.
[28]	A. M. Stein, D. A. Vader, L. M. Sander and D. A. Weitz, A stochastic model of glioblastoma invasion, 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. doi: 10.1007/978-0-8176-4558-8_19.
[29]	K. R. Swanson, C. Bridge, J. Murray and E. C. Alvord Jr, Virtual and real brain tumors: Using mathematical modeling to quantify glioma growth and invasion, J. Neurol. Sci., 216 (2003), 1-10. doi: 10.1016/j.jns.2003.06.001.
[30]	P. Tracqui, G. Cruywagen, D. Woodward, G. Bartoo, J. Murray and E. Alvord, A mathematical model of glioma growth: The effect of chemotherapy on spatio-temporal growth, Cell Proliferat., 28 (1995), 17-31. doi: 10.1111/j.1365-2184.1995.tb00036.x.
[31]	T. P. Witelski, An asymptotic solution for traveling waves of a nonlinear-diffusion Fisher's equation, J. Math. Biol., 33 (1994), 1-16. doi: 10.1007/BF00160171.

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