A multiscale model for heterogeneous tumor spheroid in vitro

  • Received: 30 August 2016 Accepted: 21 April 2017 Published: 01 April 2018
  • MSC : Primary: 92B99; Secondary: 35Q92, 34A34

  • In this paper, a novel multiscale method is proposed for the study of heterogeneous tumor spheroid growth in vitro. The entire tumor spheroid is described by an ellipsoid-based model while nutrient and other environmental factors are treated as continua. The ellipsoid-based discrete component is capable of incorporating mechanical effects and deformability, while keeping a minimum set of free variables to describe complex shape variations. Moreover, our purely cell-based description of tumor avoids the complex mutual conversion between a cell-based model and continuum model within a tumor, such as force and mass transformation. This advantage makes it highly suitable for the study of tumor spheroids in vitro whose size are normally less than 800 $μ m$ in diameter. In addition, our numerical scheme provides two computational options depending on tumor size. For a small or medium tumor spheroid, a three-dimensional (3D) numerical model can be directly applied. For a large spheroid, we suggest the use of a 3D-adapted 2D cross section configuration, which has not yet been explored in the literature, as an alternative for the theoretical investigation to bridge the gap between the 2D and 3D models. Our model and its implementations have been validated and applied to various studies given in the paper. The simulation results fit corresponding in vitro experimental observations very well.

    Citation: Zhan Chen, Yuting Zou. A multiscale model for heterogeneous tumor spheroid in vitro[J]. Mathematical Biosciences and Engineering, 2018, 15(2): 361-392. doi: 10.3934/mbe.2018016

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  • In this paper, a novel multiscale method is proposed for the study of heterogeneous tumor spheroid growth in vitro. The entire tumor spheroid is described by an ellipsoid-based model while nutrient and other environmental factors are treated as continua. The ellipsoid-based discrete component is capable of incorporating mechanical effects and deformability, while keeping a minimum set of free variables to describe complex shape variations. Moreover, our purely cell-based description of tumor avoids the complex mutual conversion between a cell-based model and continuum model within a tumor, such as force and mass transformation. This advantage makes it highly suitable for the study of tumor spheroids in vitro whose size are normally less than 800 $μ m$ in diameter. In addition, our numerical scheme provides two computational options depending on tumor size. For a small or medium tumor spheroid, a three-dimensional (3D) numerical model can be directly applied. For a large spheroid, we suggest the use of a 3D-adapted 2D cross section configuration, which has not yet been explored in the literature, as an alternative for the theoretical investigation to bridge the gap between the 2D and 3D models. Our model and its implementations have been validated and applied to various studies given in the paper. The simulation results fit corresponding in vitro experimental observations very well.


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    [1] [ S. Aland,H. Hatzikirou,J. Lowengrub,A. Voigt, A mechanistic collective cell model for epithelial colony growth and contact inhibition, Biophysical Journal, 109 (2015): 1347-1357.
    [2] [ R. K. Banerjee,W. W. van Osdol,P. M. Bungay,C. Sung,R. L. Dedrick, Finite element model of antibody penetration in a prevascular tumor nodule embedded in normal tissue, Journal of Controlled Release, 74 (2001): 193-202.
    [3] [ S. Breslin,L. O'Driscoll, Three-dimensional cell culture: The missing link in drug discovery, Drug Discovery Today, 18 (2013): 240-249.
    [4] [ G. W. Brodland, Computational modeling of cell sorting, tissue engulfment, and related phenomena: A review, Applied Mechanics Reviews, 57 (2004): 47-76.
    [5] [ G. W. Brodland,D. Viens,J. H. Veldhuis, A new cell-based fe model for the mechanics of embryonic epithelia, Computer Methods in Biomechanics and Biomedical Engineering, 10 (2007): 121-128.
    [6] [ J. C. Butcher, Numerical Methods for Ordinary Differential Equations John Wiley & Sons, 2016.
    [7] [ L. L. Campbell,K. Polyak, Breast tumor heterogeneity: Cancer stem cells or clonal evolution?, Cell Cycle, 6 (2007): 2332-2338.
    [8] [ J. Casciari,S. Sotirchos,R. Sutherland, Mathematical modelling of microenvironment and growth in emt6/ro multicellular tumour spheroids, Cell Proliferation, 25 (1992): 1-22.
    [9] [ J. Casciari,S. Sotirchos,R. Sutherland, Variations in tumor cell growth rates and metabolism with oxygen concentration, glucose concentration, and extracellular ph, Journal of Cellular Physiology, 151 (1992): 386-394.
    [10] [ P. Cirri,P. Chiarugi, Cancer-associated-fibroblasts and tumour cells: A diabolic liaison driving cancer progression, Cancer and Metastasis Reviews, 31 (2012): 195-208.
    [11] [ J. C. Dallon,H. G. Othmer, How cellular movement determines the collective force generated by the Dictyostelium discoideum slug, J. Theor. Biol., 231 (2004): 203-222.
    [12] [ T. S. Deisboeck,Z. Wang,P. Macklin,V. Cristini, Multiscale cancer modeling, Ann. Rev. Biomed. Eng., 13 (2011): 127-155.
    [13] [ M. J. Dorie,R. F. Kallman,M. A. Coyne, Effect of cytochalasin b, nocodazole and irradiation on migration and internalization of cells and microspheres in tumor cell spheroids, Experimental Cell Research, 166 (1986): 370-378.
    [14] [ M. J. Dorie,R. F. Kallman,D. F. Rapacchietta,D. Van Antwerp,Y. R. Huang, Migration and internalization of cells and polystyrene microspheres in tumor cell spheroids, Experimental Cell Research, 141 (1982): 201-209.
    [15] [ D. Drasdo,S. Höhme, A single-cell-based model of tumor growth in vitro: Monolayers and spheroids, Physical Biology, 2 (2005): 133-147.
    [16] [ D. Duguay,R. A. Foty,M. S. Steinberg, Cadherin-mediated cell adhesion and tissue segregation: Qualitative and quantitative determinants, Developmental Biology, 253 (2003): 309-323.
    [17] [ K. Erbertseder, J. Reichold, B. Flemisch, P. Jenny and R. Helmig, A coupled discrete/continuum model for describing cancer-therapeutic transport in the lung PloS One 7 (2012), e31966.
    [18] [ E. Evans, Detailed mechanics of membrane-membrane adhesion and separation. ii. discrete kinetically trapped molecular cross-bridges, Biophysical Journal, 48 (1985): 185-192.
    [19] [ E. A. Evans, Detailed mechanics of membrane-membrane adhesion and separation. i. continuum of molecular cross-bridges, Biophysical Journal, 48 (1985): 175-183.
    [20] [ E. M. Felipe De Sousa,L. Vermeulen,E. Fessler,J. P. Medema, Cancer heterogeneity-a multifaceted view, EMBO Reports, 14 (2013): 686-695.
    [21] [ T. Fiaschi and P. Chiarugi, Oxidative stress, tumor microenvironment, and metabolic reprogramming: A diabolic liaison International Journal of Cell Biology 2012 (2012), Article ID 762825, 8pp.
    [22] [ R. A. Foty,M. S. Steinberg, Cadherin-mediated cell-cell adhesion and tissue segregation in relation to malignancy, International Journal of Developmental Biology, 48 (2004): 397-409.
    [23] [ R. A. Foty,M. S. Steinberg, The differential adhesion hypothesis: A direct evaluation, Developmental Biology, 278 (2005): 255-263.
    [24] [ R. A. Foty,M. S. Steinberg, Differential adhesion in model systems, Wiley Interdisciplinary Reviews: Developmental Biology, 2 (2013): 631-645.
    [25] [ J. Freyer,R. Sutherland, A reduction in the in situ rates of oxygen and glucose consumption of cells in emt6/ro spheroids during growth, Journal of Cellular Physiology, 124 (1985): 516-524.
    [26] [ J. Galle,G. Aust,G. Schaller,T. Beyer,D. Drasdo, Individual cell-based models of the spatial-temporal organization of multicellular systems-achievements and limitations, Cytometry Part A, 69 (2006): 704-710.
    [27] [ D. Garrod,M. Steinberg, Tissue-specific sorting-out in two dimensions in relation to contact inhibition of cell movement, Nature, 244 (1973): 568-569.
    [28] [ P. Gerlee,A. R. Anderson, An evolutionary hybrid cellular automaton model of solid tumour growth, Journal of Theoretical Biology, 246 (2007): 583-603.
    [29] [ M. Gerlinger,A. J. Rowan,S. Horswell,J. Larkin,D. Endesfelder,E. Gronroos,P. Martinez,N. Matthews,A. Stewart,P. Tarpey, Intratumor heterogeneity and branched evolution revealed by multiregion sequencing, New England Journal of Medicine, 366 (2012): 883-892.
    [30] [ R. H. Grantab and I. F. Tannock, Penetration of anticancer drugs through tumour tissue as a function of cellular packing density and interstitial fluid pressure and its modification by bortezomib BMC Cancer 12 (2012), 214.
    [31] [ J. B. Green, Sophistications of cell sorting, Nature Cell Biology, 10 (2008): 375-377.
    [32] [ E. Hairer, S. Norsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Second edition. Springer Series in Computational Mathematics, 8. Springer-Verlag, Berlin, 1993.
    [33] [ J. W. Haycock, 3d cell culture: A review of current approaches and techniques, 3D Cell Culture, 695 (2010): 1-15.
    [34] [ G. Helmlinger,P. A. Netti,H. C. Lichtenbeld,R. J. Melder,R. K. Jain, Solid stress inhibits the growth of multicellular tumor spheroids, Nature Biotechnology, 15 (1997): 778-783.
    [35] [ F. Hirschhaeuser,H. Menne,C. Dittfeld,J. West,W. Mueller-Klieser,L. A. Kunz-Schughart, Multicellular tumor spheroids: An underestimated tool is catching up again, Journal of Biotechnology, 148 (2010): 3-15.
    [36] [ M. S. Hutson, G. W. Brodland, J. Yang and D. Viens, Cell sorting in three dimensions: Topology, fluctuations, and fluidlike instabilities Physical Review Letters 101 (2008), 148105.
    [37] [ J. N. Jennings, A New Computational Model for Multi-cellular Biological Systems PhD thesis, University of Cambridge, 2014.
    [38] [ Y. Jiang,H. Levine,J. Glazier, Possible cooperation of differential adhesion and chemotaxis in mound formation of dictyostelium, Biophysical Journal, 75 (1998): 2615-2625.
    [39] [ Y. Jiang,J. Pjesivac-Grbovic,C. Cantrell,J. P. Freyer, A multiscale model for avascular tumor growth, Biophysical journal, 89 (2005): 3884-3894.
    [40] [ K. Kendall, Adhesion: Molecules and mechanics, Science, 263 (1994): 1720-1725.
    [41] [ Z. I. Khamis, Z. J. Sahab and Q. X. A. Sang, Active roles of tumor stroma in breast cancer metastasis International Journal of Breast Cancer 2012 (2012), Article ID 574025, 10pp.
    [42] [ Y. Kim,M. Stolarska,H. Othmer, The role of the microenvironment in tumor growth and invasion, Progress in Biophysics and Molecular Biology, 106 (2011): 353-379.
    [43] [ Y. Kim,H. G. Othmer, A hybrid model of tumor-stromal interactions in breast cancer, Bull. Math. Biol., 75 (2013): 1304-1350.
    [44] [ Y. KIM,S. ROH, A hybrid model for cell proliferation and migration in glioblastoma, Discrete & Continuous Dynamical Systems-Series B, 18 (2013): 969-1015.
    [45] [ Y. Kim,M. A. Stolarska,H. G. Othmer, A hybrid model for tumor spheroid growth in vitro i: Theoretical development and early results, Mathematical Models and Methods in Applied Sciences, 17 (2007): 1773-1798.
    [46] [ L. C. Kimlin,G. Casagrande,V. M. Virador, In vitro three-dimensional (3d) models in cancer research: An update, Molecular Carcinogenesis, 52 (2013): 167-182.
    [47] [ T. Lecuit,P.-F. Lenne, Cell surface mechanics and the control of cell shape, tissue patterns and morphogenesis, Nature Reviews Molecular Cell Biology, 8 (2007): 633-644.
    [48] [ X.-F. Li,S. Carlin,M. Urano,J. Russell,C. C. Ling,J. A. O'Donoghue, Visualization of hypoxia in microscopic tumors by immunofluorescent microscopy, Cancer Research, 67 (2007): 7646-7653.
    [49] [ D. Loessner,J. P. Little,G. J. Pettet,D. W. Hutmacher, A multiscale road map of cancer spheroids-incorporating experimental and mathematical modelling to understand cancer progression, J Cell Sci, 126 (2013): 2761-2771.
    [50] [ P. Macklin,S. McDougall,A. R. Anderson,M. A. Chaplain,V. Cristini,J. Lowengrub, Multiscale modelling and nonlinear simulation of vascular tumour growth, Journal of Mathematical Biology, 58 (2009): 765-798.
    [51] [ J.-L. Maître,H. Berthoumieux,S. F. G. Krens,G. Salbreux,F. Jülicher,E. Paluch,C.-P. Heisenberg, Adhesion functions in cell sorting by mechanically coupling the cortices of adhering cells, Science, 338 (2012): 253-256.
    [52] [ M. Martins,S. Ferreira,M. Vilela, Multiscale models for the growth of avascular tumors, Physics of Life Reviews, 4 (2007): 128-156.
    [53] [ A. Marusyk,V. Almendro,K. Polyak, Intra-tumour heterogeneity: A looking glass for cancer?, Nature Reviews Cancer, 12 (2012): 323-334.
    [54] [ D. McElwain,G. Pettet, Cell migration in multicell spheroids: Swimming against the tide, Bulletin of Mathematical Biology, 55 (1993): 655-674.
    [55] [ E. Méhes, E. Mones, V. Németh and T. Vicsek, Collective motion of cells mediates segregation and pattern formation in co-cultures, PloS One 7.
    [56] [ L. M. F. Merlo,J. W. Pepper,B. J. Reid,C. C. Maley, Cancer as an evolutionary and ecological process, Nature Reviews Cancer, 6 (2006): 924-935.
    [57] [ D. Miller, Sugar uptake as a function of cell volume in human erythrocytes, The Journal of Physiology, 170 (1964): 219-225.
    [58] [ W. F. Mueller-Klieser,R. M. Sutherland, Oxygen consumption and oxygen diffusion properties of multicellular spheroids from two different cell lines, in Oxygen Transport to Tissue-VI , Springer, 180 (1984): 311-321.
    [59] [ S. M. Mumenthaler,J. Foo,N. C. Choi,N. Heise,K. Leder,D. B. Agus,W. Pao,F. Michor,P. Mallick, The impact of microenvironmental heterogeneity on the evolution of drug resistance in cancer cells, Cancer Informatics, 14 (2015): 19-31.
    [60] [ S. Mumenthaler,J. Foo,K. Leder,N. Choi,D. Agus,W. Pao,P. Mallick,F. Michor, Evolutionary modeling of combination treatment strategies to overcome resistance to tyrosine kinase inhibitors in non-small cell lung cancer, Molecular Pharmaceutics, 8 (2011): 2069-2079.
    [61] [ T. J. Newman, Modeling multi-cellular systems using sub-cellular elements, Math. Biosci. Eng., 2 (2005), 613–624, arXiv preprint q-bio/0504028.
    [62] [ H. Ninomiya,R. David,E. W. Damm,F. Fagotto,C. M. Niessen,R. Winklbauer, Cadherin-dependent differential cell adhesion in xenopus causes cell sorting in vitro but not in the embryo, Journal of Cell Science, 125 (2012): 1877-1883.
    [63] [ E. Palsson, A three-dimensional model of cell movement in multicellular systems, Future Generation Computer Systems, 17 (2001): 835-852.
    [64] [ E. Palsson, A 3-d model used to explore how cell adhesion and stiffness affect cell sorting and movement in multicellular systems, Journal of Theoretical Biology, 254 (2008): 1-13.
    [65] [ E. Palsson,H. G. Othmer, A model for individual and collective cell movement in dictyostelium discoideum, Proceedings of the National Academy of Sciences, 97 (2000): 10448-10453.
    [66] [ G. Pettet,C. Please,M. Tindall,D. McElwain, The migration of cells in multicell tumor spheroids, Bulletin of Mathematical Biology, 63 (2001): 231-257.
    [67] [ K. Polyak, Heterogeneity in breast cancer, The Journal of Clinical Investigation 121 (2011), 3786.
    [68] [ N. J. Poplawski,U. Agero,J. S. Gens,M. Swat,J. A. Glazier,A. R. Anderson, Front instabilities and invasiveness of simulated avascular tumors, Bulletin of Mathematical Biology, 71 (2009): 1189-1227.
    [69] [ A. Quarteroni, R. Sacco and F. Saleri, Matematica Numerica Springer Science & Business Media, 1998.
    [70] [ A. A. Qutub,F. M. Gabhann,E. D. Karagiannis,P. Vempati,A. S. Popel, Multiscale models of angiogenesis, Engineering in Medicine and Biology Magazine, IEEE, 28 (2009): 14-31.
    [71] [ K. A. Rejniak,R. H. Dillon, A single cell-based model of the ductal tumour microarchitecture, Computational and Mathematical Methods in Medicine, 8 (2007): 51-69.
    [72] [ T. Roose,P. A. Netti,L. L. Munn,Y. Boucher,R. K. Jain, Solid stress generated by spheroid growth estimated using a linear poroelastisity model, Microvascular Research, 66 (2003): 204-212.
    [73] [ G. Schaller and M. Meyer-Hermann, Multicellular tumor spheroid in an off-lattice voronoi-delaunay cell model Physical Review E 71 (2005), 051910, 16pp.
    [74] [ G. Schaller,M. Meyer-Hermann, Continuum versus discrete model: a comparison for multicellular tumour spheroids, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 364 (2006): 1443-1464.
    [75] [ E.-M. Schötz,R. D. Burdine,F. Jülicher,M. S. Steinberg,C.-P. Heisenberg,R. A. Foty, Quantitative differences in tissue surface tension influence zebrafish germ layer positioning, HFSP journal, 2 (2008): 42-56.
    [76] [ R. Shipley,S. Chapman, Multiscale modelling of fluid and drug transport in vascular tumours, Bulletin of Mathematical Biology, 72 (2010): 1464-1491.
    [77] [ A. Shirinifard, J. S. Gens, B. L. Zaitlen, N. J. Poplawski, M. Swat and J. A. Glazier, 3d multi-cell simulation of tumor growth and angiogenesis PloS One 4 (2009), e7190.
    [78] [ K. Smalley,M. Lioni,M. Herlyn, Life ins't flat: Taking cancer biology to the next dimension, In Vitro Cellular & Developmental Biology-Animal, 42 (2006): 242-247.
    [79] [ A. Starzec,D. Briane,M. Kraemer,J.-C. Kouyoumdjian,J.-L. Moretti,R. Beaupain,O. Oudar, Spatial organization of three-dimensional cocultures of adriamycin-sensitive and-resistant human breast cancer mcf-7 cells, Biology of the Cell, 95 (2003): 257-264.
    [80] [ M. S. Steinberg, Reconstruction of tissues by dissociated cells, Science, 141 (1963): 401-408.
    [81] [ M. S. Steinberg, Adhesion in development: An historical overview, Developmental Biology, 180 (1996): 377-388.
    [82] [ M. Steinberg,D. Garrod, Observations on the sorting-out of embryonic cells in monolayer culture, Journal of Cell Science, 18 (1975): 385-403.
    [83] [ M. A. Stolarska,Y. Kim,H. G. Othmer, Multi-scale models of cell and tissue dynamics, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 367 (2009): 3525-3553.
    [84] [ K. Sung,C. Dong,G. Schmid-Schönbein,S. Chien,R. Skalak, Leukocyte relaxation properties, Biophysical Journal, 54 (1988): 331-336.
    [85] [ M. H. Swat, S. D. Hester, R. W. Heiland, B. L. Zaitlen, J. A. Glazier and A. Shirinifard, Compucell3d manual and tutorial version 3. 5. 0.
    [86] [ G. Taraboletti,D. D. Roberts,L. A. Liotta, Thrombospondin-induced tumor cell migration: Haptotaxis and chemotaxis are mediated by different molecular domains, The Journal of Cell Biology, 105 (1987): 2409-2415.
    [87] [ K. Thompson,H. Byrne, Modelling the internalization of labelled cells in tumour spheroids, Bulletin of Mathematical Biology, 61 (1999): 601-623.
    [88] [ P. L. Townes,J. Holtfreter, Directed movements and selective adhesion of embryonic amphibian cells, Journal of Experimental Zoology, 128 (1955): 53-120.
    [89] [ G. Wayne Brodland,H. H. Chen, The mechanics of cell sorting and envelopment, Journal of Biomechanics, 33 (2000): 845-851.
    [90] [ D. G. Wilkinson, How attraction turns to repulsion, Nature Cell Biology, 5 (2003): 851-853.
    [91] [ M. Zanoni, F. Piccinini, C. Arienti, A. Zamagni, S. Santi, R. Polico, A. Bevilacqua and A. Tesei, 3d tumor spheroid models for in vitro therapeutic screening: A systematic approach to enhance the biological relevance of data obtained Scientific Reports 6 (2016), 19103.
    [92] [ Y. Zhang, G. Thomas, M. Swat, A. Shirinifard and J. Glazier, Computer simulations of cell sorting due to differential adhesion PloS One 6 (2011), e24999.
    [93] [ M. Zimmermann,C. Box,S. A. Eccles, Two-dimensional vs. three-dimensional in vitro tumor migration and invasion assays, in Target Identification and Validation in Drug Discovery, Springer, null (2013): 227-252.
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