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Hybrid models of cell and tissue dynamics in tumor growth
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 |
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. |
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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