zbMATH — the first resource for mathematics

An adaptive multigrid algorithm for simulating solid tumor growth using mixture models. (English) Zbl 1211.65123
Summary: In this paper we give the details of the numerical solution of a three-dimensional multispecies diffuse interface model of tumor growth, which was derived in earlier work. The model has a thermodynamic basis, is related to recently developed mixture models, and is capable of providing a detailed description of tumor progression. It utilizes a diffuse interface approach, whereby sharp tumor boundaries are replaced by narrow transition layers that arise due to differential adhesive forces among the cell-species. The model consists of fourth-order nonlinear advection-reaction-diffusion equations (of Cahn-Hilliard-type) for the cell-species coupled with reaction-diffusion equations for the substrate components. Computing numerical solutions of the model is challenging because the equations are coupled, highly nonlinear, and numerically stiff. In this paper we describe a fully adaptive, nonlinear multigrid/finite difference method for efficiently solving the equations. We demonstrate the convergence of the algorithm and we present simulations of tumor growth in 2D and 3D that demonstrate the capabilities of the algorithm in accurately and efficiently simulating the progression of tumors with complex morphologies.

65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
92C15 Developmental biology, pattern formation
92C50 Medical applications (general)
Full Text: DOI
[1] Alberts, B.; Johnson, A.; Lewis, J.; Raff, M.; Roberts, K.; Walter, P., Molecular biology of the cell, (2002), Garland Science New York
[2] Ribba, B.; Alarcón, T.; Marron, K.; Maini, P.K.; Agur, Z., The use of hybrid cellular automaton models for improving cancer therapy, (), 444-453 · Zbl 1116.92314
[3] Quaranta, V.; Weaver, A.M.; Cummings, P.T.; Anderson, A.R.A., Mathematical modeling fo cancer: the future of prognosis and treatment, Clini. chim. acta, 357, 173-179, (2005)
[4] Hatzikirou, H.; Deutsch, A.; Schaller, C.; Simon, M.; Swanson, K., Mathematical modeling of glioblastoma tumour development: a review, Math. models methods appl. sci., 15, 1779-1794, (2005) · Zbl 1077.92032
[5] Nagy, J.D., The ecology and evolutionary biology of cancer: a review of mathematical models of necrosis and tumor cell diversity, Math. biosci. eng., 2, 381-418, (2005) · Zbl 1070.92026
[6] Byrne, H.M.; Alarcón, T.; Owen, M.R.; Webb, S.W.; Maini, P.K., Modeling aspects of cancer dynamics: a review, Philos. trans. R. soc A, 364, 1563-1578, (2006)
[7] Fasano, A.; Bertuzzi, A.; Gandolfi, A., Mathematical modelling of tumour growth and treatment, (), 71-108 · Zbl 1387.92050
[8] van Leeuwen, I.M.M.; Edwards, C.M.; Ilyas, M.; Byrne, H.M., Towards a multiscale model of colorectal cancer, World J. gastroenterol., 13, 1399-1407, (2007)
[9] Roose, T.; Chapman, S.J.; Maini, P.K., Mathematical models of avascular tumor growth, SIAM rev., 49, 179-208, (2007) · Zbl 1117.93011
[10] Graziano, L.; Preziosi, L., Mechanics in tumor growth, (), 263-321
[11] Harpold, H.L.; Alvord, E.C.; Swanson, K.R., The evolution of mathematical modeling of glioma proliferation and invasion, J. neuropathol. exp. neurol., 66, 1-9, (2007)
[12] Drasdo, D.; Höhme, S., On the role of physics in the growth and pattern formation of multicellular systems: what we learn from individual-cell based models?, J. stat. phys., 128, 287-345, (2007) · Zbl 1118.82033
[13] Friedman, A.; Bellomo, N.; Maini, P.K., Mathematical analysis and challenges arising from models of tumor growth, Math. models methods appl. sci., 17, 1751-1772, (2007) · Zbl 1135.92013
[14] Sanga, S.; Frieboes, H.B.; Zheng, X.; Gatenby, R.; Bearer, E.L.; Cristini, V., Predictive oncology: a review of multidisciplinary, multiscale in silico modeling linking phenotype, morphology and growth, Neuroimage, 37, S120-S134, (2007)
[15] Deisboeck, T.S.; Zhang, L.; Yoon, J.; Costa, J., In silico cancer modeling: is it ready for prime time?, Nat. clin. pract. oncol., 6, 34-42, (2009)
[16] Anderson, A.R.A.; Quaranta, V., Integrative mathematical oncology, Nature rev. cancer, 8, 227-244, (2008)
[17] Bellomo, N.; Li, N.K.; Maini, P.K., On the foundations of cancer modeling: selected topics, speculations, and perspectives, Math. models methods appl. sci., 4, 593-646, (2008) · Zbl 1151.92014
[18] Cristini, V.; Frieboes, H.B.; Li, X.; Lowengrub, J.S.; Macklin, P.; Sanga, S.; Wise, S.M.; Zheng, X., Nonlinear modeling and simulation of tumor growth, () · Zbl 1181.92046
[19] Lowengrub, J.S.; Frieboes, H.B.; Jin, F.; Chuang, Y.-L.; Li, X.; Macklin, P.; Wise, S.M.; Cristini, V., Nonlinear modeling of cancer: bridging the gap between cells and tumors, Nonlinearity, 23, R1-R91, (2010) · Zbl 1181.92046
[20] Anderson, A.R.A.; Weaver, A.M.; Commmings, P.T.; Quaranta, V., Tumor morphology and phenotypic evolution driven by selective presure from the microenvironment, Cell, 127, 905-915, (2006)
[21] Bearer, E.L.; Lowengrub, J.S.; Chuang, Y.L.; Frieboes, H.B.; Jin, F.; Wise, S.M.; Ferrari, M.; Agus, D.B.; Cristini, V., Multiparameter computational modeling of tumor invasion, Cancer res., 69, 4493-4501, (2009)
[22] Cristini, V.; Frieboes, H.B.; Gatenby, R.; Caserta, S.; Ferrari, M.; Sinek, J., Morphologic instability and cancer invasion, Clin. cancer res., 11, 6772-6779, (2005)
[23] Cristini, V.; Lowengrub, J.S.; Nie, Q., Nonlinear simulation of tumor growth, J. math. biol., 46, 191-224, (2003) · Zbl 1023.92013
[24] Frieboes, H.B.; Lowengrub, J.S.; Wise, S.M.; Zheng, X.; Macklin, P.; Bearer, E.L.; Cristini, V., Computer simulation of glioma growth and morphology, Neuroimage, 37, S59-S70, (2007)
[25] Gatenby, R.A.; Smallbone, K.; Maini, P.K.; Rose, F.; Averill, J.; Nagle, R.B.; Worrall, L.; Gillies, R.J., Cellular adaptations to hypoxia and acidosis during somatic evolution of breast cancer, Br. J. cancer, 97, 646-653, (2007)
[26] Macklin, P.; Lowengrub, J.S., An improved geometry-aware curvature discretization for level set methods: application to tumor growth, J. comput. phys., 215, 392-401, (2006) · Zbl 1089.92024
[27] Wise, S.M.; Lowengrub, J.S.; Frieboes, H.B.; Cristini, V., Three-dimensional multispecies nonlinear tumor growth—I: model and numerical method, J. theoret. biol., 253, 524-543, (2008)
[28] Frieboes, H.B.; Jin, F.; Chuang, Y.L.; Wise, S.M.; Lowengrub, J.S.; Cristini, V., Three-dimensional multispecies nonlinear tumor growth—II: tissue invasion and angiogenesis, J. theoret. biol., 264, 1254-1278, (2010)
[29] Cahn, J.W.; Hilliard, J.E., Free energy of a nonuniform system. I. interfacial free energy, J. chem. phys., 28, 258, (1958)
[30] Khain, E.; Sander, L.M., Generalized cahn – hilliard equation for biological applications, Phys. rev. E, 77, 051129, (2008)
[31] Armstrong, N.J.; Painter, K.J.; Sherratt, J.A., A continuum approach to modeling cell – cell adhesion, J. theoret. biol., 243, 98-113, (2006)
[32] Gerisch, A.; Chaplain, M.A., Mathematical modelling of cancer cell invasion of tissue: local and non-local models and the effect of adhesion, J. theoret. biol., 250, 684-704, (2008)
[33] Ambrosi, D.; Preziosi, L., On the closure of mass balance models for tumor growth, Math. models methods appl. sci., 12, 737-754, (2002) · Zbl 1016.92016
[34] Ambrosi, D.; Preziosi, L., Cell adhesion mechanisms and stress relaxation in the mechanics of tumours, Biomech. model. mechanobiol., 8, 397-413, (2009)
[35] Araujo, R.P.; McElwain, D.L.S., A mixture theory for the genesis of residual stresses in growing tissues I: a general formulation, SIAM J. appl. math., 65, 1261-1284, (2005) · Zbl 1074.74043
[36] Breward, C.J.W.; Byrne, H.M.; Lewis, C.E., The role of cell – cell interactions in a two-phase model for avascular tumour growth, J. math. biol., 45, 125-152, (2002) · Zbl 1012.92017
[37] Breward, C.J.W.; Byrne, H.M.; Lewis, C.E., A multiphase model describing vascular tumour growth, Bull. math. biol., 65, 609-640, (2003) · Zbl 1334.92190
[38] Byrne, H.M.; King, J.R.; McElwain, D.L.S; Preziosi, L., A two-phase model of solid tumour growth, Appl. math. lett., 16, 567-573, (2003) · Zbl 1040.92015
[39] Byrne, H.M.; Preziosi, L., Modelling solid tumour growth using the theory of mixtures, Math. med. biol., 20, 341-366, (2003) · Zbl 1046.92023
[40] Chaplain, M.A.J.; Graziano, L.; Preziosi, L., Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumor development, Math. med. biol., 23, 192-229, (2006) · Zbl 1098.92037
[41] Cristini, V.; Li, X.; Lowengrub, J.S.; Wise, S.M., Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching, J. math. biol., 58, 723-763, (2009) · Zbl 1311.92039
[42] Franks, S.J.; Byrne, H.M.; King, J.R.; Underwood, J.C.E.; Lewis, C.E., Modelling the early growth of ductal carcinoma in situ of the breast, J. math. biol., 47, 424-452, (2003) · Zbl 1050.92030
[43] Franks, S.J.; Byrne, H.M.; Mudhar, H.S.; Underwood, J.C.E.; Lewis, C.E., Mathematical modelling of comedo ductal carcinoma in situ of the breast, Math. med. biol., 20, 277-308, (2003) · Zbl 1039.92021
[44] Franks, S.J.; King, J.R., Interactions between a uniformly proliferating tumor and its surrounding. uniform material properties, Math. med. biol., 20, 47-89, (2003) · Zbl 1044.92032
[45] Tosin, A., Multiphase modeling and qualitative analysis of the growth of tumor cords, Netw. heterog. media, 3, 43-84, (2008) · Zbl 1144.35476
[46] Byrne, H.M.; Chaplain, M.A.J., Growth of nonnecrotic tumors in the presence and absence of inhibitors, Math. biosci., 130, 151-181, (1995) · Zbl 0836.92011
[47] Byrne, H.M.; Chaplain, M.A.J., Growth of necrotic tumors in the presence and absence of inhibitors, Math. biosci., 135, 187-216, (1996) · Zbl 0856.92010
[48] Chaplain, M.A.J., Avascular growth, angiogenesis and vascular growth in solid tumours: the mathematical modelling of the stages of tumour development, Math. comput. model., 23, 47-87, (1996) · Zbl 0859.92012
[49] Friedman, A.; Reitich, F., Analysis of a mathematical model for the growth of tumors, J. math. biol., 38, 262-284, (1999) · Zbl 0944.92018
[50] Greenspan, H.P., On the growth and stability of cell cultures and solid tumors, J. theoret. biol., 56, 229-242, (1976)
[51] Hogea, C.S.; Murray, B.T.; Sethian, J.A., Simulating complex tumor dynamics from avascular to vascular growth using a general level-set method, J. math. biol., 53, 86-134, (2006) · Zbl 1100.92029
[52] Macklin, P.; Lowengrub, J.S., A new ghost cell/level set method for moving boundary problems: application to tumor growth, J. sci. comput., 35, 266-299, (2008) · Zbl 1203.65144
[53] Macklin, P.; McDougall, S.; Anderson, A.R.A.; Chaplain, M.A.J.; Cristini, V.; Lowengrub, J., Multiscale modelling and simulation of vascular tumour growth, J. math. biol., 58, 765-798, (2008) · Zbl 1311.92040
[54] Zheng, X.; Wise, S.M.; Cristini, V., Nonlinear simulation of tumor necrosis, neo-vascularization and tissue invasion via an adaptive finite-element/level-set method, Bull. math. biol., 67, 211-259, (2005) · Zbl 1334.92214
[55] Abbott, R.G.; Forrest, S.; Pienta, K.J., Simulating the hallmarks of cancer, Artif. life, 12, 617-634, (2006)
[56] Alarcón, T.; Byrne, H.M.; Maini, P.K., A cellular automaton model for tumour growth in inhomogeneous environment, J. theoret. biol., 225, 257-274, (2003)
[57] Anderson, A.R.A., A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion, Math. med. biol., 22, 163-186, (2005) · Zbl 1073.92013
[58] Bartha, K.; Rieger, H., Vascular network remodeling via vessel cooption, regression and growth in tumors, J. theoret. biol., 241, 903-918, (2006)
[59] Bauer, A.L.; Jackson, T.L.; Jiang, Y., A cell-based model exhibiting branching and anastomosis during tumor-induced angiogenesis, Biophys. J., 92, 3105-3121, (2007)
[60] Byrne, H.M.; Drasdo, D., Individual-based and continuum models of growing cell populations: a comparison, J. math. biol., 58, 657-687, (2009) · Zbl 1311.92060
[61] Drasdo, D.; Höhme, S., Individual-based approaches to birth and death in avascular tumors, Math. comput. modelling, 37, 1163-1175, (2003) · Zbl 1047.92023
[62] Drasdo, D.; Höhme, S., A single-scale-based model of tumor growth in vitro: monolayers and spheroids, Phys. biol., 2, 133-147, (2005)
[63] Gerlee, P.; Anderson, A.R.A., An evolutionary hybrid cellular automaton model of solid tumor growth, J. theoret. biol., 246, 583-603, (2007)
[64] Gerlee, P.; Anderson, A.R.A., A hybrid cellular automaton model of clonal evolution in cancer: the emergence of the glycolytic phenotype, J. theoret. biol., 250, 705-722, (2008)
[65] Jiang, Y.; Pjesivac-Grbovic, J.; Cantrell, C.; Freyer, J.P., A multiscale model for avascular tumor growth, Biophys. J., 89, 3884-3894, (2005)
[66] Lee, D.S.; Rieger, H., Flow correlated percolation during vascular remodeling in growing tumors, Phys. rev. lett., 96, 058104, (2006)
[67] Ramis-Conde, I.; Drasdo, D.; Anderson, A.R.A.; Chaplain, M.A.J., Modeling the influence of the E-cadherin-beta-catenin pathway in cancer cell invasion: a multiscale approach, Biophys. J., 95, 155-165, (2008)
[68] Rejniak, K., A single-cell approach in modeling the dynamics of tumor microregions, Math. biosci. eng., 2, 643-655, (2005) · Zbl 1079.92046
[69] Rejniak, K., An immersed boundary framework for modeling the growth of individual cells: an application to the early tumour development, J. theoret. biol., 247, 186-204, (2007)
[70] Li, X.; Cristini, V.; Nie, Q.; Lowengrub, J.S., Nonlinear three-dimensional simulation of solid tumor growth, Discrete contin. dyn. syst. B, 7, 581-604, (2007) · Zbl 1124.92022
[71] Macklin, P.; Lowengrub, J.S., Nonlinear simulation of the effect of microenvironment on tumor growth, J. theoret. biol., 245, 677-704, (2007)
[72] Sinek, J.; Frieboes, H.B.; Zheng, X.; Cristini, V., Two-dimensional chemotherapy simulations demonstrate fundamental transport and tumor response limitations involving nanoparticles, Biomed. microdev., 6, 297-309, (2004)
[73] Preziosi, L.; Tosin, A., Multiphase modeling of tumor growth and extracellular matrix interaction: mathematical tools and applications, J. math. biol., 58, 625-656, (2009) · Zbl 1311.92029
[74] Kim, Y.; Stolarska, M.A.; Othmer, H.G., A hybrid model for tumor spheroid growth in vitro I: theoretical development and early results, Math. models methods appl. sci., 17, 1773-1798, (2007) · Zbl 1135.92016
[75] Stolarska, M.A.; Kim, Y.; Othmer, H.G., Multi-scale models of cell and tissue dynamics, Phil. trans. R. soc. A, 367, 3525-3553, (2009) · Zbl 1185.74066
[76] Wise, S.M.; Kim, J.S.; Lowengrub, J.S., Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method, J. comput. phys., 226, 414-446, (2007) · Zbl 1310.82044
[77] Briggs, W.L.; Henson, V.E.; McCormick, S.F., A multigrid tutorial, (2000), SIAM Philadelphia · Zbl 0958.65128
[78] Trottenberg, U.; Oosterlee, C.W.; Schüller, A., Multigrid, (2001), Academic Press New York
[79] Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. comput. phys., 126, 202-228, (1996) · Zbl 0877.65065
[80] Kim, J.S.; Kang, K.; Lowengrub, J.S., Conservative multigrid methods for ternary Cahn-Hilliard systems, Commun. math. sci., 2, 53-77, (2004) · Zbl 1085.65093
[81] Morton, K.W.; Mayers, D.F., Numerical solution of partial differential equations, (2005), Cambridge University Press Cambridge · Zbl 0811.65063
[82] Frieboes, H.B.; Zheng, X.; Sun, C.-H.; Tromberg, B.J.; Gatenby, R.; Cristini, V., An integrated computational/experimental model of tumor invasion, Cancer res., 66, 1597-1604, (2006)
[83] Hu, Z.; Wise, S.M.; Wang, C.; Lowengrub, J.S., Stable and efficient finite-difference nonlinear-multigrid schemes for the phase-field crystal equation, J. comput. phys., 228, 5323-5339, (2009) · Zbl 1171.82015
[84] Wise, S.M.; Wang, C.; Lowengrub, J.S., An energy stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. numer. anal., 47, 2269-2288, (2009) · Zbl 1201.35027
[85] Wise, S.M., Unconditionally stable finite difference, nonlinear multigrid simulation of the cahn – hilliard – hele – shaw system of equations, J. sci. comput., 44, 38-68, (2010) · Zbl 1203.76153
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.