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A data-motivated density-dependent diffusion model of in vitro glioblastoma growth
Mathematical model and its fast numerical method for the tumor growth
1. | Institute of Mathematical Sciences, Ewha Womans University, Seoul 120-750, South Korea |
2. | Department of Mathematics, Konkuk University, Seoul 143-701 |
3. | Department of Mathematics, Korea University, Seoul 136-713 |
References:
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N. D. Alikakos, P. W. Bates and X. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model, Arch. Rational Mech. Anal., 128 (1994), 165-205.
doi: 10.1007/BF00375025. |
[2] |
T. Alarcón, H. M. Byrne and P. K. Maini, A cellular automaton model for tumour growth in inhomogeneous environment, J. Theor. Biol., 225 (2003), 257-274.
doi: 10.1016/S0022-5193(03)00244-3. |
[3] |
B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter, Molecular Biology of the Cell, $5^{th}$ edition, Garland Science, New York, 2007. |
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S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Mater., 27 (1979), 1085-1095.
doi: 10.1016/0001-6160(79)90196-2. |
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A. R. A. Anderson and M. A. J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis, Bull. Math. Biol., 60 (1998), 857-899.
doi: 10.1006/bulm.1998.0042. |
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A. R. A. Anderson and V. Quaranta, Integrative mathematical oncology, Nat. Rev. Cancer, 8 (2008), 227-234.
doi: 10.1038/nrc2329. |
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C. Athale, Y. Mansury and T. S. Deisboeck, Simulating the impact of a molecular 'decision-process' on cellular phenotype and multicellular patterns in brain tumors, J. Theor. Biol., 233 (2005), 469-481. |
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M. Athanassenas, Volume-preserving mean curvature flow of rotationally symmetric surfaces, Comment. Math. Helv., 72 (1997), 52-66.
doi: 10.1007/PL00000366. |
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K. Bartha and H. Rieger, Vascular network remodeling via vessel cooption, regression and growth in tumors, J. Theor. Biol., 241 (2006), 903-918.
doi: 10.1016/j.jtbi.2006.01.022. |
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N. Bellomo, N. K. Li and P. K. Maini, On the foundations of cancer modeling: Selected topics, speculations, and perspective, Math. Models Methods Appl. Sci., 18 (2008), 593-646.
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M. Brassel and E. Bretin, A modified phase field approximation for mean curvature flow with conservation of the volume, Math. Meth. Appl. Sci., 34 (2011), 1157-1180.
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J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. |
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E. A. Carlen, M. C. Carvalho and E. Orlandi, Approximate solutions of the Cahn-Hilliard equation via corrections to the Mullins-Sekerka motion, Arch. Rational Mech. Anal., 178 (2005), 1-55.
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Y. Chen, S. M. Wise, V. B. Shenoy and J. S. Lowengrub, A stable scheme for a nonlinear, multiphase tumor growth model with an elastic membrane, Int. J. Numer. Meth. Biomed. Engng., 30 (2014), 726-754.
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T. S. Deisboeck, L. Zhang, J. Yoon and J. Costa, In silico cancer modeling: Is it ready for prime time?, Nat. Clin. Pract. Oncol., 6 (2009), 34-42.
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S. Dormann and A. Deutsch, Modeling of self-organized avascular tumor growth with a hybrid cellular automaton, In Silico Biol., 2 (2002), 393-406. |
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D. Drasdo, S. Hohme and M. Block, On the role of physics in the growth and pattern formation of multi-cellular systems: what can we learn from individual-cell based models?, J. Stat. Phys., 128 (2007), 287-345.
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show all references
References:
[1] |
N. D. Alikakos, P. W. Bates and X. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model, Arch. Rational Mech. Anal., 128 (1994), 165-205.
doi: 10.1007/BF00375025. |
[2] |
T. Alarcón, H. M. Byrne and P. K. Maini, A cellular automaton model for tumour growth in inhomogeneous environment, J. Theor. Biol., 225 (2003), 257-274.
doi: 10.1016/S0022-5193(03)00244-3. |
[3] |
B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter, Molecular Biology of the Cell, $5^{th}$ edition, Garland Science, New York, 2007. |
[4] |
S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Mater., 27 (1979), 1085-1095.
doi: 10.1016/0001-6160(79)90196-2. |
[5] |
A. R. A. Anderson and M. A. J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis, Bull. Math. Biol., 60 (1998), 857-899.
doi: 10.1006/bulm.1998.0042. |
[6] |
A. R. A. Anderson and V. Quaranta, Integrative mathematical oncology, Nat. Rev. Cancer, 8 (2008), 227-234.
doi: 10.1038/nrc2329. |
[7] |
C. Athale, Y. Mansury and T. S. Deisboeck, Simulating the impact of a molecular 'decision-process' on cellular phenotype and multicellular patterns in brain tumors, J. Theor. Biol., 233 (2005), 469-481. |
[8] |
M. Athanassenas, Volume-preserving mean curvature flow of rotationally symmetric surfaces, Comment. Math. Helv., 72 (1997), 52-66.
doi: 10.1007/PL00000366. |
[9] |
K. Bartha and H. Rieger, Vascular network remodeling via vessel cooption, regression and growth in tumors, J. Theor. Biol., 241 (2006), 903-918.
doi: 10.1016/j.jtbi.2006.01.022. |
[10] |
N. Bellomo, N. K. Li and P. K. Maini, On the foundations of cancer modeling: Selected topics, speculations, and perspective, Math. Models Methods Appl. Sci., 18 (2008), 593-646.
doi: 10.1142/S0218202508002796. |
[11] |
M. Brassel and E. Bretin, A modified phase field approximation for mean curvature flow with conservation of the volume, Math. Meth. Appl. Sci., 34 (2011), 1157-1180.
doi: 10.1002/mma.1426. |
[12] |
W. L. Briggs, A Multigrid Tutorial, SIAM, Philadelphia, 1987. |
[13] |
L. Bronsard and B. Stoth, Volume-preserving mean curvature flow as a limit of a nonlocal Ginzburg-Landau equation, SIAM J. Math. Anal., 28 (1997), 769-807.
doi: 10.1137/S0036141094279279. |
[14] |
H. M. Byrne, A weakly nonlinear analysis of a model of avascular solid tumour growth, J. Math. Biol., 39 (1999), 59-89.
doi: 10.1007/s002850050163. |
[15] |
H. M. Byrne, T. Alarcon, M. R. Owen, S. D. Webb and P. K. Maini, Modelling aspects of cancer dynamics: A review, Phil. Trans. R. Soc. A, 364 (2006), 1563-1578.
doi: 10.1098/rsta.2006.1786. |
[16] |
H. M. Byrne and M. A. J. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors, Math. Biosci., 130 (1995), 151-181.
doi: 10.1016/0025-5564(94)00117-3. |
[17] |
H. M. Byrne and M. A. J. Chaplain, Growth of necrotic tumors in the presence and absence of inhibitors, Math. Biosci., 135 (1996), 187-216.
doi: 10.1016/0025-5564(96)00023-5. |
[18] |
H. M. Byrne and M. A. J. Chaplain, Modelling the role of cell-cell adhesion in the growth and development of carcinomas, Math. Comput. Model., 24 (1996), 1-17.
doi: 10.1016/S0895-7177(96)00174-4. |
[19] |
H. M. Byrne and P. Matthews, Asymmetric growth of models of avascular solid tumours: exploiting symmetries, Math. Med. Biol., 19 (2002), 1-29.
doi: 10.1093/imammb/19.1.1. |
[20] |
J. W. Cahn, C. M. Elliott and A. Novick-Cohen, The Cahn-Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature, Eur. J. Appl. Math., 7 (1996), 287-301.
doi: 10.1017/S0956792500002369. |
[21] |
J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. |
[22] |
E. A. Carlen, M. C. Carvalho and E. Orlandi, Approximate solutions of the Cahn-Hilliard equation via corrections to the Mullins-Sekerka motion, Arch. Rational Mech. Anal., 178 (2005), 1-55.
doi: 10.1007/s00205-005-0366-5. |
[23] |
X. Chen, The Hele-Shaw problem and area-preserving curve-shortening motions, Arch. Rational Mech. Anal., 123 (1993), 117-151.
doi: 10.1007/BF00695274. |
[24] |
Y. Chen, S. M. Wise, V. B. Shenoy and J. S. Lowengrub, A stable scheme for a nonlinear, multiphase tumor growth model with an elastic membrane, Int. J. Numer. Meth. Biomed. Engng., 30 (2014), 726-754.
doi: 10.1002/cnm.2624. |
[25] |
V. Cristini, H. B. Frieboes, X. Li, J. S. Lowengrub, P. Macklin, S. Sanga, S. M. Wise and X. Zheng, Nonlinear modeling and simulation of tumor growth, in Selected Topics in Cancer Modeling: Genesis, Evolution, Immune Competition, and Therapy (eds. N. Bellomo, M. Chaplain and E. de Angelis), Birkhäuser, (2008), 113-181. |
[26] |
V. Cristini and J. Lowengrub, Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, Cambridge University Press, Cambridge, 2010. |
[27] |
V. Cristini, J. Lowengrub and Q. Nie, Nonlinear simulation of tumor growth, J. Math. Biol., 46 (2003), 191-224.
doi: 10.1007/s00285-002-0174-6. |
[28] |
S. Dai and Q. Du, Motion of interfaces governed by the Cahn-Hilliard equation with highly disparate diffusion mobility, SIAM J. Appl. Math., 72 (2012), 1818-1841.
doi: 10.1137/120862582. |
[29] |
T. S. Deisboeck, L. Zhang, J. Yoon and J. Costa, In silico cancer modeling: Is it ready for prime time?, Nat. Clin. Pract. Oncol., 6 (2009), 34-42.
doi: 10.1038/ncponc1237. |
[30] |
S. Dormann and A. Deutsch, Modeling of self-organized avascular tumor growth with a hybrid cellular automaton, In Silico Biol., 2 (2002), 393-406. |
[31] |
D. Drasdo, S. Hohme and M. Block, On the role of physics in the growth and pattern formation of multi-cellular systems: what can we learn from individual-cell based models?, J. Stat. Phys., 128 (2007), 287-345.
doi: 10.1007/s10955-007-9289-x. |
[32] |
J. Escher, U. F. Mayer and G. Simonett, The surface diffusion flow for immersed hypersurfaces, SIAM J. Math. Anal., 29 (1998), 1419-1433.
doi: 10.1137/S0036141097320675. |
[33] |
J. Escher and G. Simonett, Classical solutions for Hele-Shaw models with surface tension, Adv. Differ. Equ., 2 (1997), 619-642. |
[34] |
A. Fasano, A. Bertuzzi and A. Gandolfi, Mathematical modelling of tumour growth and treatment, in Complex Systems in Biomedicine (eds. A. Quarteroni, L. Formaggia and A. Veneziani), Springer, (2006), 71-108.
doi: 10.1007/88-470-0396-2_3. |
[35] |
H. B. Frieboes, F. Jin, Y.-L. Chuang, S. M. Wise, J. S. Lowengrub and V. Cristini, Three-dimensional multispecies nonlinear tumor growth-II: tumor invasion and angiogenesis, J. Theor. Biol., 264 (2010), 1254-1278.
doi: 10.1016/j.jtbi.2010.02.036. |
[36] |
A. Friedman, Mathematical analysis and challenges arising from models of tumor growth, Math. Models Methods Appl. Sci., 17 (2007), 1751-1772.
doi: 10.1142/S0218202507002467. |
[37] |
P. Gerlee and A. R. A. Anderson, Stability analysis of a hybrid cellular automaton model of cell colony growth, Phys. Rev. E, 75 (2007), 051911.
doi: 10.1103/PhysRevE.75.051911. |
[38] |
L. Graziano and L. Preziosi, Mechanics in tumor growth, in Modeling of Biological Materials (eds. F. Mollica, L. Preziosi and K.R. Rajagopal), Birkhäuser, (2007), 263-321.
doi: 10.1007/978-0-8176-4411-6_7. |
[39] |
H. P. Greenspan, On the growth and stability of cell cultures and solid tumors, J. Theor. Biol., 56 (1976), 229-242.
doi: 10.1016/S0022-5193(76)80054-9. |
[40] |
H. L. P. Harpold, E. C. Alvord and K. R. Swanson, The evolution of mathematical modeling of glioma proliferation and invasion, J. Neuropath. Exp. Neur., 66 (2007), 1-9.
doi: 10.1097/nen.0b013e31802d9000. |
[41] |
H. Hatzikirou, A. Deutsch, C. Schaller, M. Simon and K. Swanson, Mathematical modelling of glioblastoma tumour development: A review, Math. Models Methods Appl. Sci., 15 (2005), 1779-1794.
doi: 10.1142/S0218202505000960. |
[42] |
G. Huisken, The volume preserving mean curvature flow, J. Reine Angew. Math., 382 (1987), 35-48.
doi: 10.1515/crll.1987.382.35. |
[43] |
Y. Jiang, J. Pjesivac-Grbovic, C. Cantrell and J. P. Freyer, A multiscale model for avascular tumor growth, Biophys. J., 89 (2005), 3884-3894.
doi: 10.1529/biophysj.105.060640. |
[44] |
A. R. Kansal, S. Torquato, G. R. Harsh IV, E. A. Chiocca and T. S. Deisboeck, Simulated brain tumor growth dynamics using a three-dimensional cellular automaton, J. Theor. Biol., 203 (2000), 367-382.
doi: 10.1006/jtbi.2000.2000. |
[45] |
J. Kim, S. Lee and Y. Choi, A conservative Allen-Cahn equation with a space-time dependent Lagrange multiplier, Int. J. Eng. Sci., 84 (2014), 11-17.
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