# American Institute of Mathematical Sciences

2015, 12(1): 41-69. doi: 10.3934/mbe.2015.12.41

## 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

Received  May 2014 Revised  September 2014 Published  December 2014

We consider quasi-stationary (travelling wave type) solutions to a nonlinear reaction-diffusion equation with arbitrary, autonomous coefficients, describing the evolution of glioblastomas, aggressive primary brain tumors that are characterized by extensive infiltration into the brain and are highly resistant to treatment. The second order nonlinear equation describing the glioblastoma growth through travelling waves can be reduced to a first order Abel type equation. By using the integrability conditions for the Abel equation several classes of exact travelling wave solutions of the general reaction-diffusion equation that describes glioblastoma growth are obtained, corresponding to different forms of the product of the diffusion and reaction functions. The solutions are obtained by using the Chiellini lemma and the Lemke transformation, respectively, and the corresponding equations represent generalizations of the classical Fisher--Kolmogorov equation. The biological implications of two classes of solutions are also investigated by using both numerical and semi-analytical methods for realistic values of the biological parameters.
Citation: Tiberiu Harko, Man Kwong Mak. Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach. Mathematical Biosciences & Engineering, 2015, 12 (1) : 41-69. doi: 10.3934/mbe.2015.12.41
##### References:
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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. 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##### 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. 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