2015, 12(6): 1173-1187. doi: 10.3934/mbe.2015.12.1173

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

Received  October 2014 Revised  July 2015 Published  August 2015

In this paper, we reformulate the diffuse interface model of the tumor growth (S.M. Wise et al., Three-dimensional multispecies nonlinear tumor growth-I: model and numerical method, J. Theor. Biol. 253 (2008) 524--543). In the new proposed model, we use the conservative second-order Allen--Cahn equation with a space--time dependent Lagrange multiplier instead of using the fourth-order Cahn--Hilliard equation in the original model. To numerically solve the new model, we apply a recently developed hybrid numerical method. We perform various numerical experiments. The computational results demonstrate that the new model is not only fast but also has a good feature such as distributing excess mass from the inside of tumor to its boundary regions.
Citation: Hyun Geun Lee, Yangjin Kim, Junseok Kim. Mathematical model and its fast numerical method for the tumor growth. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1173-1187. doi: 10.3934/mbe.2015.12.1173
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. 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. 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, (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. 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. doi: 10.1006/bulm.1998.0042.

[6]

A. R. A. Anderson and V. Quaranta, Integrative mathematical oncology,, Nat. Rev. Cancer, 8 (2008), 227. 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.

[8]

M. Athanassenas, Volume-preserving mean curvature flow of rotationally symmetric surfaces,, Comment. Math. Helv., 72 (1997), 52. 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. 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. 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. doi: 10.1002/mma.1426.

[12]

W. L. Briggs, A Multigrid Tutorial,, SIAM, (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. doi: 10.1137/S0036141094279279.

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H. M. Byrne, A weakly nonlinear analysis of a model of avascular solid tumour growth,, J. Math. Biol., 39 (1999), 59. 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. 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. 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. 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. doi: 10.1016/S0895-7177(96)00174-4.

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H. M. Byrne and P. Matthews, Asymmetric growth of models of avascular solid tumours: exploiting symmetries,, Math. Med. Biol., 19 (2002), 1. 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. 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.

[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. 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. 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. 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, (2008), 113.

[26]

V. Cristini and J. Lowengrub, Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach,, Cambridge University Press, (2010).

[27]

V. Cristini, J. Lowengrub and Q. Nie, Nonlinear simulation of tumor growth,, J. Math. Biol., 46 (2003), 191. 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. 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. 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.

[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. 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. 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.

[34]

A. Fasano, A. Bertuzzi and A. Gandolfi, Mathematical modelling of tumour growth and treatment,, in Complex Systems in Biomedicine (eds. A. Quarteroni, (2006), 71. 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. 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. 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). doi: 10.1103/PhysRevE.75.051911.

[38]

L. Graziano and L. Preziosi, Mechanics in tumor growth,, in Modeling of Biological Materials (eds. F. Mollica, (2007), 263. 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. 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. 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. doi: 10.1142/S0218202505000960.

[42]

G. Huisken, The volume preserving mean curvature flow,, J. Reine Angew. Math., 382 (1987), 35. 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. 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. 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. doi: 10.1016/j.ijengsci.2014.06.004.

[46]

D.-S. Lee, H. Rieger and K. Bartha, Flow correlated percolation during vascular remodeling in growing tumors,, Phys. Rev. Lett., 96 (2006). doi: 10.1103/PhysRevLett.96.058104.

[47]

I. M. M. van Leeuwen, C. M. Edwards, M. Ilyas and H. M. Byrne, Towards a multiscale model of colorectal cancer,, World J. Gastroentero., 13 (2007), 1399. doi: 10.3748/wjg.v13.i9.1399.

[48]

X. Li, V. Cristini, Q. Nie and J. S. Lowengrub, Nonlinear three-dimensional simulation of solid tumor growth,, Discrete Cont. Dyn-B, 7 (2007), 581. doi: 10.3934/dcdsb.2007.7.581.

[49]

J. S. Lowengrub, H. B. Frieboes, F. Jin, Y.-L. Chuang, X. Li, P. Macklin, S. M. Wise and V. Cristini, Nonlinear modelling of cancer: Bridging the gap between cells and tumours,, Nonlinearity, 23 (2010). doi: 10.1088/0951-7715/23/1/R01.

[50]

Y. Mansury, M. Kimura, J. Lobo and T. S. Deisboeck, Emerging patterns in tumor systems: Simulating the dynamics of multicellular clusters with an agent-based spatial agglomeration model,, J. Theor. Biol., 219 (2002), 343. doi: 10.1006/jtbi.2002.3131.

[51]

U. F. Mayer and G. Simonett, Self-intersections for the surface diffusion and the volume-preserving mean curvature flow,, Differ. Integral Equ., 13 (2000), 1189.

[52]

J. D. Nagy, The ecology and evolutionary biology of cancer: A review of mathematical models of necrosis and tumor cell diversity,, Math. Biosci. Eng., 2 (2005), 381. doi: 10.3934/mbe.2005.2.381.

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S. M. Wise, J. S. Lowengrub and V. Cristini, An adaptive multigrid algorithm for simulating solid tumor growth using mixture models,, Math. Comput. Model., 53 (2011), 1. doi: 10.1016/j.mcm.2010.07.007.

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S. M. Wise, J. S. Lowengrub, H. B. Frieboes and V. Cristini, Three-dimensional multispecies nonlinear tumor growth-I: Model and numerical method,, J. Theor. Biol., 253 (2008), 524. doi: 10.1016/j.jtbi.2008.03.027.

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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. 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. 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, (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. 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. doi: 10.1006/bulm.1998.0042.

[6]

A. R. A. Anderson and V. Quaranta, Integrative mathematical oncology,, Nat. Rev. Cancer, 8 (2008), 227. 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.

[8]

M. Athanassenas, Volume-preserving mean curvature flow of rotationally symmetric surfaces,, Comment. Math. Helv., 72 (1997), 52. 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. 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. 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. doi: 10.1002/mma.1426.

[12]

W. L. Briggs, A Multigrid Tutorial,, SIAM, (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. 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. 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. 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. 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. 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. 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. 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. 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.

[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. 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. 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. 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, (2008), 113.

[26]

V. Cristini and J. Lowengrub, Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach,, Cambridge University Press, (2010).

[27]

V. Cristini, J. Lowengrub and Q. Nie, Nonlinear simulation of tumor growth,, J. Math. Biol., 46 (2003), 191. 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. 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. 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.

[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. 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. 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.

[34]

A. Fasano, A. Bertuzzi and A. Gandolfi, Mathematical modelling of tumour growth and treatment,, in Complex Systems in Biomedicine (eds. A. Quarteroni, (2006), 71. 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. 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. 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). doi: 10.1103/PhysRevE.75.051911.

[38]

L. Graziano and L. Preziosi, Mechanics in tumor growth,, in Modeling of Biological Materials (eds. F. Mollica, (2007), 263. 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. 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. 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. doi: 10.1142/S0218202505000960.

[42]

G. Huisken, The volume preserving mean curvature flow,, J. Reine Angew. Math., 382 (1987), 35. 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. 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. 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. doi: 10.1016/j.ijengsci.2014.06.004.

[46]

D.-S. Lee, H. Rieger and K. Bartha, Flow correlated percolation during vascular remodeling in growing tumors,, Phys. Rev. Lett., 96 (2006). doi: 10.1103/PhysRevLett.96.058104.

[47]

I. M. M. van Leeuwen, C. M. Edwards, M. Ilyas and H. M. Byrne, Towards a multiscale model of colorectal cancer,, World J. Gastroentero., 13 (2007), 1399. doi: 10.3748/wjg.v13.i9.1399.

[48]

X. Li, V. Cristini, Q. Nie and J. S. Lowengrub, Nonlinear three-dimensional simulation of solid tumor growth,, Discrete Cont. Dyn-B, 7 (2007), 581. doi: 10.3934/dcdsb.2007.7.581.

[49]

J. S. Lowengrub, H. B. Frieboes, F. Jin, Y.-L. Chuang, X. Li, P. Macklin, S. M. Wise and V. Cristini, Nonlinear modelling of cancer: Bridging the gap between cells and tumours,, Nonlinearity, 23 (2010). doi: 10.1088/0951-7715/23/1/R01.

[50]

Y. Mansury, M. Kimura, J. Lobo and T. S. Deisboeck, Emerging patterns in tumor systems: Simulating the dynamics of multicellular clusters with an agent-based spatial agglomeration model,, J. Theor. Biol., 219 (2002), 343. doi: 10.1006/jtbi.2002.3131.

[51]

U. F. Mayer and G. Simonett, Self-intersections for the surface diffusion and the volume-preserving mean curvature flow,, Differ. Integral Equ., 13 (2000), 1189.

[52]

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