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

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

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

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. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1157-1172. doi: 10.3934/mbe.2015.12.1157
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. doi: 10.1016/j.jtbi.2006.05.030. Google Scholar

[2]

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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). doi: 10.1088/0031-9155/53/4/004. Google Scholar

[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. doi: 10.1016/j.jtbi.2006.10.024. Google Scholar

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C. Chicone, Ordinary Differential Equations with Applications, vol. 34 of Texts in Applied Mathematics,, 2nd edition, (2006). Google Scholar

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J. Doke, GRABIT, MATLAB Central File Exchange,, , (2005). Google Scholar

[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. doi: 10.1088/0305-4470/38/15/009. Google Scholar

[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. doi: 10.3934/mbe.2015.12.41. Google Scholar

[9]

S. Harris, Fisher equation with density-dependent diffusion: Special solutions,, J. Phys. A-Math. Gen., 37 (2004), 6267. doi: 10.1088/0305-4470/37/24/005. Google Scholar

[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. doi: 10.1002/mrm.20625. Google Scholar

[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). doi: 10.1140/epjp/i2013-13136-7. Google Scholar

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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. doi: 10.1137/S1052623496303470. Google Scholar

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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. doi: 10.3934/dcdsb.2006.6.1175. Google Scholar

[14]

L. Malaguti and C. Marcelli, Travelling Wavefronts in Reaction-Diffusion Equations with Convection Effects and Non-Regular Terms,, Math. Nachr., 242 (2002), 148. doi: 10.1002/1522-2616(200207)242:1<148::AID-MANA148>3.0.CO;2-J. Google Scholar

[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. doi: 10.3934/cpaa.2010.9.1083. Google Scholar

[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. doi: 10.1155/2011/986738. Google Scholar

[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. doi: 10.3934/mbe.2015.12.879. Google Scholar

[18]

J. D. Murray, Mathematical Biology: I: An Introduction, vol. 17 of Interdisciplinary Applied Mathematics,, 3rd edition, (2002). Google Scholar

[19]

W. Ngamsaad and K. Khompurngson, Self-similar solutions to a density-dependent reaction-diffusion model,, Phys. Rev. E, 85 (2012). doi: 10.1103/PhysRevE.85.066120. Google Scholar

[20]

A. D. Norden and P. Y. Wen, Glioma therapy in adults,, Neurologist, 12 (2006), 279. doi: 10.1097/01.nrl.0000250928.26044.47. Google Scholar

[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. doi: 10.1016/j.jtbi.2013.01.014. Google Scholar

[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. doi: 10.1007/s00285-004-0304-4. Google Scholar

[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). doi: 10.1103/PhysRevE.84.021921. Google Scholar

[24]

F. Sánchez-Garduño and P. K. Maini, Traveling wave phenomena in some degenerate reaction-diffusion equations,, J. Differ. Equations, 117 (1995), 281. doi: 10.1006/jdeq.1995.1055. Google Scholar

[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. doi: 10.3934/dcdsb.2010.13.455. Google Scholar

[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. doi: 10.1137/0911001. Google Scholar

[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. doi: 10.1529/biophysj.106.093468. Google Scholar

[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, (2007), 217. doi: 10.1007/978-0-8176-4558-8_19. Google Scholar

[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. doi: 10.1016/j.jns.2003.06.001. Google Scholar

[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. doi: 10.1111/j.1365-2184.1995.tb00036.x. Google Scholar

[31]

T. P. Witelski, An asymptotic solution for traveling waves of a nonlinear-diffusion Fisher's equation,, J. Math. Biol., 33 (1994), 1. doi: 10.1007/BF00160171. Google Scholar

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. doi: 10.1016/j.jtbi.2006.05.030. Google Scholar

[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. doi: 10.1137/0512074. Google Scholar

[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). doi: 10.1088/0031-9155/53/4/004. Google Scholar

[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. doi: 10.1016/j.jtbi.2006.10.024. Google Scholar

[5]

C. Chicone, Ordinary Differential Equations with Applications, vol. 34 of Texts in Applied Mathematics,, 2nd edition, (2006). Google Scholar

[6]

J. Doke, GRABIT, MATLAB Central File Exchange,, , (2005). Google Scholar

[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. doi: 10.1088/0305-4470/38/15/009. Google Scholar

[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. doi: 10.3934/mbe.2015.12.41. Google Scholar

[9]

S. Harris, Fisher equation with density-dependent diffusion: Special solutions,, J. Phys. A-Math. Gen., 37 (2004), 6267. doi: 10.1088/0305-4470/37/24/005. Google Scholar

[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. doi: 10.1002/mrm.20625. Google Scholar

[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). doi: 10.1140/epjp/i2013-13136-7. Google Scholar

[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. doi: 10.1137/S1052623496303470. Google Scholar

[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. doi: 10.3934/dcdsb.2006.6.1175. Google Scholar

[14]

L. Malaguti and C. Marcelli, Travelling Wavefronts in Reaction-Diffusion Equations with Convection Effects and Non-Regular Terms,, Math. Nachr., 242 (2002), 148. doi: 10.1002/1522-2616(200207)242:1<148::AID-MANA148>3.0.CO;2-J. Google Scholar

[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. doi: 10.3934/cpaa.2010.9.1083. Google Scholar

[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. doi: 10.1155/2011/986738. Google Scholar

[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. doi: 10.3934/mbe.2015.12.879. Google Scholar

[18]

J. D. Murray, Mathematical Biology: I: An Introduction, vol. 17 of Interdisciplinary Applied Mathematics,, 3rd edition, (2002). Google Scholar

[19]

W. Ngamsaad and K. Khompurngson, Self-similar solutions to a density-dependent reaction-diffusion model,, Phys. Rev. E, 85 (2012). doi: 10.1103/PhysRevE.85.066120. Google Scholar

[20]

A. D. Norden and P. Y. Wen, Glioma therapy in adults,, Neurologist, 12 (2006), 279. doi: 10.1097/01.nrl.0000250928.26044.47. Google Scholar

[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. doi: 10.1016/j.jtbi.2013.01.014. Google Scholar

[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. doi: 10.1007/s00285-004-0304-4. Google Scholar

[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). doi: 10.1103/PhysRevE.84.021921. Google Scholar

[24]

F. Sánchez-Garduño and P. K. Maini, Traveling wave phenomena in some degenerate reaction-diffusion equations,, J. Differ. Equations, 117 (1995), 281. doi: 10.1006/jdeq.1995.1055. Google Scholar

[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. doi: 10.3934/dcdsb.2010.13.455. Google Scholar

[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. doi: 10.1137/0911001. Google Scholar

[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. doi: 10.1529/biophysj.106.093468. Google Scholar

[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, (2007), 217. doi: 10.1007/978-0-8176-4558-8_19. Google Scholar

[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. doi: 10.1016/j.jns.2003.06.001. Google Scholar

[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. doi: 10.1111/j.1365-2184.1995.tb00036.x. Google Scholar

[31]

T. P. Witelski, An asymptotic solution for traveling waves of a nonlinear-diffusion Fisher's equation,, J. Math. Biol., 33 (1994), 1. doi: 10.1007/BF00160171. Google Scholar

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