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Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach
1. | Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom |
2. | Department of Computing and Information Management, Hong Kong Institute of Vocational Education, Chai Wan, Hong Kong, China |
References:
[1] |
M. S. Ahluwalia, J. de Groot, W. M. Liu and C. L. Gladson, Targeting SRC in glioblastoma tumors and brain metastases: Rationale and preclinical studies, Cancer Letters, 298 (2010), 139-149.
doi: 10.1016/j.canlet.2010.08.014. |
[2] |
E. C. Alvord, Jr., Patterns of growth of gliomas, American Journal of Neuroradiology, 16 (1995), 1013-1017. |
[3] |
M. Bellini, R. Deza and N. Giovanbattista, Exact travelling annular waves in generalized reaction-diffusion equations, Physics Letters A, 232 (1997), 200-206.
doi: 10.1016/S0375-9601(97)00360-5. |
[4] |
T. Brugarino, Exact solutions of nonlinear differential equations using the Abelian equation of the first type, Il Nuovo Cimento, 119 (2004), 975-982. |
[5] |
A. Chiellini, Sull'integrazione dell'equazione differenziale $y^{'} + Py^ 2 +Qy^ 3 =0$, Bollettino dell Unione Matematica Italiana, 10 (1931), 301-307. |
[6] |
T. Demuth and M. E. Berens, Molecular mechanisms of glioma cell migration and invasion, Journal of Neuro-Oncology, 70 (2004), 217-228.
doi: 10.1007/s11060-004-2751-6. |
[7] |
S. E. Eikenberry, T. Sankar, M. C. Preul, E. J. Kostelich, C. J. Thalhauser and Y. Kuang, Virtual glioblastoma: Growth, migration and treatment in a three-dimensional mathematical model, 2009, Cell Prolif., 42 (2009), 511-528. |
[8] |
R. A. Fisher, The Genetical Theory of Natural Selection, Oxford, Oxford University Press, 1999. |
[9] |
R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369.
doi: 10.1111/j.1469-1809.1937.tb02153.x. |
[10] |
N. Follain, J.-M. Valleton, L. Lebrun, B. Alexandre, P. Schaetzel, M. Metayer and S. Marais, Simulation of kinetic curves in mass transfer phenomena for a concentration-dependent diffusion coefficient in polymer membranes, Journal of Membrane Science, 349 (2010), 195-207.
doi: 10.1016/j.memsci.2009.11.044. |
[11] |
P. Gerlee and S. Nelander, The Impact of Phenotypic Switching on Glioblastoma Growth and Invasion, PLOS Computational Biology, 8 (2012), 1002556, 12pp.
doi: 10.1371/journal.pcbi.1002556. |
[12] |
P. Gerlee and S. Nelander, Travelling wave analysis of a mathematical model of glioblastoma growth, preprint, arXiv:1305.5036. |
[13] |
S. G. Giatili and G. S. Stamatakos, A detailed numerical treatment of the boundary conditions imposed by the skull on a diffusion-reaction model of glioma tumor growth. Clinical validation aspects, Applied Mathematics and Computation, 218 (2012), 8779-8799.
doi: 10.1016/j.amc.2012.02.036. |
[14] |
B. H. Gilding and R. Kersner, The Characterization of Reaction-Convection-Diffusion Processes by Travelling Waves, Journal of differential equations, 124 (1996), 27-79.
doi: 10.1006/jdeq.1996.0002. |
[15] |
B. H. Gilding and R. Kersner, Travelling Waves in Nonlinear Diffusion Convection Reaction, Birkhäuser, 2004.
doi: 10.1007/978-3-0348-7964-4. |
[16] |
V. V. Golubev, Vorlesungen über Differentialgleichungen im Komplexen, Deutsch. Verlag Wissenschaft., Berlin, 1958. |
[17] |
T. Harko and M. K. Mak, Relativistic dissipative cosmological models and Abel differential equation, Computers and Mathematics with Applications, 46 (2003), 849-853.
doi: 10.1016/S0898-1221(03)90147-7. |
[18] |
T. Harko, F. S. N. Lobo and M. K. Mak, Exact analytical solutions of the Susceptible-Infected- Recovered (SIR) epidemic model and of the SIR model with equal deaths and births, Applied Mathematics and Computation, 236 (2014), 184-194.
doi: 10.1016/j.amc.2014.03.030. |
[19] |
T. Harko, F. S. N. Lobo and M. K. Mak, A Chiellini type integrability condition for the generalized first kind Abel differential equation, Universal Journal of Applied Mathematics, 1 (2013), 101-104. |
[20] |
T. Harko, F. S. N. Lobo and M. K. Mak, A class of exact solutions of the Liénard type ordinary non-linear differential equation, Journal of Engineering Mathematics, accepted for publication, arXiv:1302.0836. |
[21] |
H. L. P. Harpold, E. C. Alvord, Jr. and K. R. Swanson, The Evolution of Mathematical Modeling of Glioma Proliferation and Invasion, J Neuropathol Exp Neurol, 66 (2007), 1-9.
doi: 10.1097/nen.0b013e31802d9000. |
[22] |
H. Hatzikirou, L. Brusch, K. Schaller, M. Simon and A. Deutsch, Prediction of traveling front behavior in a lattice-gas cellular automaton model for tumor invasion, Computers and Mathematics with Applications, 59 (2010), 2326-2339.
doi: 10.1016/j.camwa.2009.08.041. |
[23] |
H. Hatzikirou, D. Basanta, M. Simon, K. Schaller and A. Deutsch, "Go or Grow": the key to the emergence of invasion in tumour progression?, Mathematical Medicine and Biology, 29 (2012), 49-65.
doi: 10.1093/imammb/dqq011. |
[24] |
P. Hinow, P. Gerlee, L. J. McCawley, V. Quaranta, M. Ciobanu, S. Wang, J. M. Graham, B. P. Ayati, J. Claridge, K. R. Swanson, M. Loveless and A. R. A. Anderson, A Spatial Model of Tumor-Host Interaction: Application of Chemotherapy, Math. Biosci. Eng., 6 (2009), 521-546.
doi: 10.3934/mbe.2009.6.521. |
[25] |
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. |
[26] |
S. Joannès, L. Mazé and A. R. Bunsell, A concentration-dependent diffusion coefficient model for water sorption in composite, Composite Structures, 108 (2014), 111-118. |
[27] |
D. R. Johnson and B. P. O'Neill, Glioblastoma survival in the United States before and during the temozolomide era, Journal of Neurooncology, 107 (2012), 359-364.
doi: 10.1007/s11060-011-0749-4. |
[28] |
M. Kinoshita, N. Hashimoto, T. Goto, N. Kagawa, H. Kishima, S. Izumoto, H. Tanaka, N. Fujita and T. Yoshimine, NeuroImage, 43 (2008), 29-35. |
[29] |
E. Kamke, Differentialgleichungen: Lösungsmethoden und Lösungen, Chelsea, New York, 1959. |
[30] |
A. Kolmogorov, I. Petrovskii and N. Piscounov, Étude de l'équation de la diffusion avec croissance de la quantité de matiére et son application a un problème biologique, Bulletin de l'Université d'État à Moscou, Série Internationale, Section A Mathématiques et Mécanique, 1 (1937), 1-25. |
[31] |
H. Lemke, Über eine von R. Liouville untersuchte Differentialgleichung erster Ordnung, Sitzungs. Berl. Math. Ges., 18 (1920), 26-31. |
[32] |
C. K. N. Li, The glucose distribution in 9L rat brain multicell tumour spheroids and its effect on cell necrosis, Cancer, 50 (1982), 2066-2073. |
[33] |
D. N. Louis, H. Ohgaki, O. D. Wiestler, W. K. Cavenee, P. C. Burger, A. Jouvet, B. W. Scheithauer and P. Kleihues, The 2007 WHO Classification of Tumours of the Central Nervous System, Acta Neuropathol., 114 (2007), 97-109.
doi: 10.1007/s00401-007-0243-4. |
[34] |
M. K. Mak, H. W. Chan and T. Harko, Solutions generating technique for Abel type non-linear ordinary differential equations, Computers and Mathematics with Applications, 41 (2001), 1395-1401.
doi: 10.1016/S0898-1221(01)00104-3. |
[35] |
M. K. Mak and T. Harko, New method for generating general solution of Abel differential equations, Computers and Mathematics with Applications, 43 (2002), 91-94.
doi: 10.1016/S0898-1221(01)00274-7. |
[36] |
S. C. Mancas and H. C. Rosu, Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations, Physics Letters, A 377 (2013), 1234-1238.
doi: 10.1016/j.physleta.2013.04.024. |
[37] |
S. C. Mancas and H. C. Rosu, Integrable Ermakov-Pinney equations with nonlinear Chiellini 'damping', preprint, arXiv:1301.3567. |
[38] |
S. C. Mancas, G. Spradlin and H. Khanal, Weierstrass traveling wave solutions for dissipative Benjamin, Bona, and Mahony (BBM) equation, Journal of Mathematical Physics, 54 (2013), 081502, 15pp.
doi: 10.1063/1.4817342. |
[39] |
E. Mandonnet, J.-Y. Delattre, M.-L. Tanguy, K. R. Swanson, A. F. Carpentier, H. Duffau, P. Cornu, R. Van Effenterre, E. C. Alvord. Jr. and L. Capelle, Continuous growth of mean tumor diameter in a subset of grade II gliomas, Annals of Neurology, 53 (2003), 524-528. |
[40] |
D. C. Markham, M. J. Simpson, P. K. Maini, E. A. Gaffney and R. E. Baker, Comparing methods for modelling spreading cell fronts, Journal of Theoretical Biology, 353 (2014), 95-103.
doi: 10.1016/j.jtbi.2014.02.023. |
[41] |
A. Martínez-González, G. F. Calvo, L. A. Pérez Romasanta and V. M. Pérez-García, Hypoxic Cell Waves Around Necrotic Cores in Glioblastoma: A Biomathematical Model and Its Therapeutic Implications, Bulletin of Mathematical Biology, 74 (2012), 2875-2896.
doi: 10.1007/s11538-012-9786-1. |
[42] |
M. Marusic, Z. Bajzer, J. P. Freyer and S. Vuk-Palovic, Analysis of growth of multicellular tumour spheroids by mathematical models, Cell Prolif, 27 (1994), 73-94. |
[43] |
A. Matzavinos and M. A. J. Chaplain, Traveling-wave analysis of a model of the immune response to cancer, 2004, C. R. Biologies, 327 (2004), 995-1008. |
[44] |
E. Mehrara, E. Forssell-Aronsson, H. Ahlman and P. Bernhardt, Quantitative analysis of tumor growth rate and changes in tumor marker level: Specific growth rate versus doubling time, Acta Oncologica, 48 (2009), 591-597.
doi: 10.1080/02841860802616736. |
[45] |
J. D. Murray, On travelling wave solutions in a model for the Belousov-Zhabotinskii reaction, Journal of Theoretical Biology, 56 (1976), 329-353.
doi: 10.1016/S0022-5193(76)80078-1. |
[46] |
J. D. Murray, Mathematical Biology, 3rd ed, New York, NY: Springer- Verlag, 2002.
doi: 10.1007/b98869. |
[47] |
B. Perthame, M. Tang and N. Vauchelet, Traveling wave solution of the Hele-Shaw model of tumor growth with nutrient, Math. Models Methods Appl. Sci., 24 (2014), 2601-2626. arXiv:1401.3649.
doi: 10.1142/S0218202514500316. |
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show all references
References:
[1] |
M. S. Ahluwalia, J. de Groot, W. M. Liu and C. L. Gladson, Targeting SRC in glioblastoma tumors and brain metastases: Rationale and preclinical studies, Cancer Letters, 298 (2010), 139-149.
doi: 10.1016/j.canlet.2010.08.014. |
[2] |
E. C. Alvord, Jr., Patterns of growth of gliomas, American Journal of Neuroradiology, 16 (1995), 1013-1017. |
[3] |
M. Bellini, R. Deza and N. Giovanbattista, Exact travelling annular waves in generalized reaction-diffusion equations, Physics Letters A, 232 (1997), 200-206.
doi: 10.1016/S0375-9601(97)00360-5. |
[4] |
T. Brugarino, Exact solutions of nonlinear differential equations using the Abelian equation of the first type, Il Nuovo Cimento, 119 (2004), 975-982. |
[5] |
A. Chiellini, Sull'integrazione dell'equazione differenziale $y^{'} + Py^ 2 +Qy^ 3 =0$, Bollettino dell Unione Matematica Italiana, 10 (1931), 301-307. |
[6] |
T. Demuth and M. E. Berens, Molecular mechanisms of glioma cell migration and invasion, Journal of Neuro-Oncology, 70 (2004), 217-228.
doi: 10.1007/s11060-004-2751-6. |
[7] |
S. E. Eikenberry, T. Sankar, M. C. Preul, E. J. Kostelich, C. J. Thalhauser and Y. Kuang, Virtual glioblastoma: Growth, migration and treatment in a three-dimensional mathematical model, 2009, Cell Prolif., 42 (2009), 511-528. |
[8] |
R. A. Fisher, The Genetical Theory of Natural Selection, Oxford, Oxford University Press, 1999. |
[9] |
R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369.
doi: 10.1111/j.1469-1809.1937.tb02153.x. |
[10] |
N. Follain, J.-M. Valleton, L. Lebrun, B. Alexandre, P. Schaetzel, M. Metayer and S. Marais, Simulation of kinetic curves in mass transfer phenomena for a concentration-dependent diffusion coefficient in polymer membranes, Journal of Membrane Science, 349 (2010), 195-207.
doi: 10.1016/j.memsci.2009.11.044. |
[11] |
P. Gerlee and S. Nelander, The Impact of Phenotypic Switching on Glioblastoma Growth and Invasion, PLOS Computational Biology, 8 (2012), 1002556, 12pp.
doi: 10.1371/journal.pcbi.1002556. |
[12] |
P. Gerlee and S. Nelander, Travelling wave analysis of a mathematical model of glioblastoma growth, preprint, arXiv:1305.5036. |
[13] |
S. G. Giatili and G. S. Stamatakos, A detailed numerical treatment of the boundary conditions imposed by the skull on a diffusion-reaction model of glioma tumor growth. Clinical validation aspects, Applied Mathematics and Computation, 218 (2012), 8779-8799.
doi: 10.1016/j.amc.2012.02.036. |
[14] |
B. H. Gilding and R. Kersner, The Characterization of Reaction-Convection-Diffusion Processes by Travelling Waves, Journal of differential equations, 124 (1996), 27-79.
doi: 10.1006/jdeq.1996.0002. |
[15] |
B. H. Gilding and R. Kersner, Travelling Waves in Nonlinear Diffusion Convection Reaction, Birkhäuser, 2004.
doi: 10.1007/978-3-0348-7964-4. |
[16] |
V. V. Golubev, Vorlesungen über Differentialgleichungen im Komplexen, Deutsch. Verlag Wissenschaft., Berlin, 1958. |
[17] |
T. Harko and M. K. Mak, Relativistic dissipative cosmological models and Abel differential equation, Computers and Mathematics with Applications, 46 (2003), 849-853.
doi: 10.1016/S0898-1221(03)90147-7. |
[18] |
T. Harko, F. S. N. Lobo and M. K. Mak, Exact analytical solutions of the Susceptible-Infected- Recovered (SIR) epidemic model and of the SIR model with equal deaths and births, Applied Mathematics and Computation, 236 (2014), 184-194.
doi: 10.1016/j.amc.2014.03.030. |
[19] |
T. Harko, F. S. N. Lobo and M. K. Mak, A Chiellini type integrability condition for the generalized first kind Abel differential equation, Universal Journal of Applied Mathematics, 1 (2013), 101-104. |
[20] |
T. Harko, F. S. N. Lobo and M. K. Mak, A class of exact solutions of the Liénard type ordinary non-linear differential equation, Journal of Engineering Mathematics, accepted for publication, arXiv:1302.0836. |
[21] |
H. L. P. Harpold, E. C. Alvord, Jr. and K. R. Swanson, The Evolution of Mathematical Modeling of Glioma Proliferation and Invasion, J Neuropathol Exp Neurol, 66 (2007), 1-9.
doi: 10.1097/nen.0b013e31802d9000. |
[22] |
H. Hatzikirou, L. Brusch, K. Schaller, M. Simon and A. Deutsch, Prediction of traveling front behavior in a lattice-gas cellular automaton model for tumor invasion, Computers and Mathematics with Applications, 59 (2010), 2326-2339.
doi: 10.1016/j.camwa.2009.08.041. |
[23] |
H. Hatzikirou, D. Basanta, M. Simon, K. Schaller and A. Deutsch, "Go or Grow": the key to the emergence of invasion in tumour progression?, Mathematical Medicine and Biology, 29 (2012), 49-65.
doi: 10.1093/imammb/dqq011. |
[24] |
P. Hinow, P. Gerlee, L. J. McCawley, V. Quaranta, M. Ciobanu, S. Wang, J. M. Graham, B. P. Ayati, J. Claridge, K. R. Swanson, M. Loveless and A. R. A. Anderson, A Spatial Model of Tumor-Host Interaction: Application of Chemotherapy, Math. Biosci. Eng., 6 (2009), 521-546.
doi: 10.3934/mbe.2009.6.521. |
[25] |
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. |
[26] |
S. Joannès, L. Mazé and A. R. Bunsell, A concentration-dependent diffusion coefficient model for water sorption in composite, Composite Structures, 108 (2014), 111-118. |
[27] |
D. R. Johnson and B. P. O'Neill, Glioblastoma survival in the United States before and during the temozolomide era, Journal of Neurooncology, 107 (2012), 359-364.
doi: 10.1007/s11060-011-0749-4. |
[28] |
M. Kinoshita, N. Hashimoto, T. Goto, N. Kagawa, H. Kishima, S. Izumoto, H. Tanaka, N. Fujita and T. Yoshimine, NeuroImage, 43 (2008), 29-35. |
[29] |
E. Kamke, Differentialgleichungen: Lösungsmethoden und Lösungen, Chelsea, New York, 1959. |
[30] |
A. Kolmogorov, I. Petrovskii and N. Piscounov, Étude de l'équation de la diffusion avec croissance de la quantité de matiére et son application a un problème biologique, Bulletin de l'Université d'État à Moscou, Série Internationale, Section A Mathématiques et Mécanique, 1 (1937), 1-25. |
[31] |
H. Lemke, Über eine von R. Liouville untersuchte Differentialgleichung erster Ordnung, Sitzungs. Berl. Math. Ges., 18 (1920), 26-31. |
[32] |
C. K. N. Li, The glucose distribution in 9L rat brain multicell tumour spheroids and its effect on cell necrosis, Cancer, 50 (1982), 2066-2073. |
[33] |
D. N. Louis, H. Ohgaki, O. D. Wiestler, W. K. Cavenee, P. C. Burger, A. Jouvet, B. W. Scheithauer and P. Kleihues, The 2007 WHO Classification of Tumours of the Central Nervous System, Acta Neuropathol., 114 (2007), 97-109.
doi: 10.1007/s00401-007-0243-4. |
[34] |
M. K. Mak, H. W. Chan and T. Harko, Solutions generating technique for Abel type non-linear ordinary differential equations, Computers and Mathematics with Applications, 41 (2001), 1395-1401.
doi: 10.1016/S0898-1221(01)00104-3. |
[35] |
M. K. Mak and T. Harko, New method for generating general solution of Abel differential equations, Computers and Mathematics with Applications, 43 (2002), 91-94.
doi: 10.1016/S0898-1221(01)00274-7. |
[36] |
S. C. Mancas and H. C. Rosu, Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations, Physics Letters, A 377 (2013), 1234-1238.
doi: 10.1016/j.physleta.2013.04.024. |
[37] |
S. C. Mancas and H. C. Rosu, Integrable Ermakov-Pinney equations with nonlinear Chiellini 'damping', preprint, arXiv:1301.3567. |
[38] |
S. C. Mancas, G. Spradlin and H. Khanal, Weierstrass traveling wave solutions for dissipative Benjamin, Bona, and Mahony (BBM) equation, Journal of Mathematical Physics, 54 (2013), 081502, 15pp.
doi: 10.1063/1.4817342. |
[39] |
E. Mandonnet, J.-Y. Delattre, M.-L. Tanguy, K. R. Swanson, A. F. Carpentier, H. Duffau, P. Cornu, R. Van Effenterre, E. C. Alvord. Jr. and L. Capelle, Continuous growth of mean tumor diameter in a subset of grade II gliomas, Annals of Neurology, 53 (2003), 524-528. |
[40] |
D. C. Markham, M. J. Simpson, P. K. Maini, E. A. Gaffney and R. E. Baker, Comparing methods for modelling spreading cell fronts, Journal of Theoretical Biology, 353 (2014), 95-103.
doi: 10.1016/j.jtbi.2014.02.023. |
[41] |
A. Martínez-González, G. F. Calvo, L. A. Pérez Romasanta and V. M. Pérez-García, Hypoxic Cell Waves Around Necrotic Cores in Glioblastoma: A Biomathematical Model and Its Therapeutic Implications, Bulletin of Mathematical Biology, 74 (2012), 2875-2896.
doi: 10.1007/s11538-012-9786-1. |
[42] |
M. Marusic, Z. Bajzer, J. P. Freyer and S. Vuk-Palovic, Analysis of growth of multicellular tumour spheroids by mathematical models, Cell Prolif, 27 (1994), 73-94. |
[43] |
A. Matzavinos and M. A. J. Chaplain, Traveling-wave analysis of a model of the immune response to cancer, 2004, C. R. Biologies, 327 (2004), 995-1008. |
[44] |
E. Mehrara, E. Forssell-Aronsson, H. Ahlman and P. Bernhardt, Quantitative analysis of tumor growth rate and changes in tumor marker level: Specific growth rate versus doubling time, Acta Oncologica, 48 (2009), 591-597.
doi: 10.1080/02841860802616736. |
[45] |
J. D. Murray, On travelling wave solutions in a model for the Belousov-Zhabotinskii reaction, Journal of Theoretical Biology, 56 (1976), 329-353.
doi: 10.1016/S0022-5193(76)80078-1. |
[46] |
J. D. Murray, Mathematical Biology, 3rd ed, New York, NY: Springer- Verlag, 2002.
doi: 10.1007/b98869. |
[47] |
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