American Institute of Mathematical Sciences

Advanced
April  2018, 15(2): 361-392. doi: 10.3934/mbe.2018016

A multiscale model for heterogeneous tumor spheroid in vitro

Zhan Chen , and  Yuting Zou

Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA, 30460, USA

* Corresponding author: Zhan Chen (zchen@georgiasouthern.edu).

Received  August 30, 2016 Accepted  April 21, 2017 Published  June 2017

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.

Keywords: Multiscale, 3D, heterogeneous, tumor growth, cell sorting.
Mathematics Subject Classification: Primary: 92B99; Secondary: 35Q92, 34A34.
Citation: Zhan Chen, Yuting Zou. A multiscale model for heterogeneous tumor spheroid in vitro. Mathematical Biosciences & Engineering, 2018, 15 (2) : 361-392. doi: 10.3934/mbe.2018016
References:
[1]

S. Aland, H. Hatzikirou, J. Lowengrub and 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 and 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.  doi: 10.1016/S0168-3659(01)00317-0.

[3]

S. Breslin and L. O'Driscoll, Three-dimensional cell culture: The missing link in drug discovery, Drug Discovery Today, 18 (2013), 240-249.  doi: 10.1016/j.drudis.2012.10.003.

[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 and 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. doi: 10.1002/9781119121534.

[7]

L. L. Campbell and K. Polyak, Breast tumor heterogeneity: Cancer stem cells or clonal evolution?, Cell Cycle, 6 (2007), 2332-2338.  doi: 10.4161/cc.6.19.4914.

[8]

J. Casciari, S. Sotirchos and R. Sutherland, Mathematical modelling of microenvironment and growth in emt6/ro multicellular tumour spheroids, Cell Proliferation, 25 (1992), 1-22.  doi: 10.1111/j.1365-2184.1992.tb01433.x.

[9]

J. Casciari, S. Sotirchos and 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.  doi: 10.1002/jcp.1041510220.

[10]

P. Cirri and P. Chiarugi, Cancer-associated-fibroblasts and tumour cells: A diabolic liaison driving cancer progression, Cancer and Metastasis Reviews, 31 (2012), 195-208.  doi: 10.1007/s10555-011-9340-x.

[11]

J. C. Dallon and H. G. Othmer, How cellular movement determines the collective force generated by the Dictyostelium discoideum slug, J. Theor. Biol., 231 (2004), 203-222.  doi: 10.1016/j.jtbi.2004.06.015.

[12]

T. S. Deisboeck, Z. Wang, P. Macklin and V. Cristini, Multiscale cancer modeling, Ann. Rev. Biomed. Eng., 13 (2011), 127-155. 

[13]

M. J. Dorie, R. F. Kallman and 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.  doi: 10.1016/0014-4827(86)90483-0.

[14]

M. J. Dorie, R. F. Kallman, D. F. Rapacchietta, D. Van Antwerp and Y. R. Huang, Migration and internalization of cells and polystyrene microspheres in tumor cell spheroids, Experimental Cell Research, 141 (1982), 201-209.  doi: 10.1016/0014-4827(82)90082-9.

[15]

D. Drasdo and S. Höhme, A single-cell-based model of tumor growth in vitro: Monolayers and spheroids, Physical Biology, 2 (2005), 133-147.  doi: 10.1088/1478-3975/2/3/001.

[16]

D. Duguay, R. A. Foty and M. S. Steinberg, Cadherin-mediated cell adhesion and tissue segregation: Qualitative and quantitative determinants, Developmental Biology, 253 (2003), 309-323.  doi: 10.1016/S0012-1606(02)00016-7.

[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. doi: 10.1371/journal.pone.0031966.

[18]

E. Evans, Detailed mechanics of membrane-membrane adhesion and separation. ii. discrete kinetically trapped molecular cross-bridges, Biophysical Journal, 48 (1985), 185-192.  doi: 10.1016/S0006-3495(85)83771-1.

[19]

E. A. Evans, Detailed mechanics of membrane-membrane adhesion and separation. i. continuum of molecular cross-bridges, Biophysical Journal, 48 (1985), 175-183.  doi: 10.1016/S0006-3495(85)83770-X.

[20]

E. M. Felipe De Sousa, L. Vermeulen, E. Fessler and 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. doi: 10.1155/2012/762825.

[22]

R. A. Foty and M. S. Steinberg, Cadherin-mediated cell-cell adhesion and tissue segregation in relation to malignancy, International Journal of Developmental Biology, 48 (2004), 397-409.  doi: 10.1387/ijdb.041810rf.

[23]

R. A. Foty and M. S. Steinberg, The differential adhesion hypothesis: A direct evaluation, Developmental Biology, 278 (2005), 255-263.  doi: 10.1016/j.ydbio.2004.11.012.

[24]

R. A. Foty and M. S. Steinberg, Differential adhesion in model systems, Wiley Interdisciplinary Reviews: Developmental Biology, 2 (2013), 631-645.  doi: 10.1002/wdev.104.

[25]

J. Freyer and 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.  doi: 10.1002/jcp.1041240323.

[26]

J. Galle, G. Aust, G. Schaller, T. Beyer and D. Drasdo, Individual cell-based models of the spatial-temporal organization of multicellular systems-achievements and limitations, Cytometry Part A, 69 (2006), 704-710.  doi: 10.1002/cyto.a.20287.

[27]

D. Garrod and M. Steinberg, Tissue-specific sorting-out in two dimensions in relation to contact inhibition of cell movement, Nature, 244 (1973), 568-569.  doi: 10.1038/244568a0.

[28]

P. Gerlee and A. R. Anderson, An evolutionary hybrid cellular automaton model of solid tumour growth, Journal of Theoretical Biology, 246 (2007), 583-603.  doi: 10.1016/j.jtbi.2007.01.027.

[29]

M. Gerlinger, A. J. Rowan, S. Horswell, J. Larkin, D. Endesfelder, E. Gronroos, P. Martinez, N. Matthews, A. Stewart and P. Tarpey, Intratumor heterogeneity and branched evolution revealed by multiregion sequencing, New England Journal of Medicine, 366 (2012), 883-892.  doi: 10.1056/NEJMoa1113205.

[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. doi: 10.1186/1471-2407-12-214.

[31]

J. B. Green, Sophistications of cell sorting, Nature Cell Biology, 10 (2008), 375-377.  doi: 10.1038/ncb0408-375.

[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.  doi: 10.1007/978-1-60761-984-0_1.

[34]

G. Helmlinger, P. A. Netti, H. C. Lichtenbeld, R. J. Melder and R. K. Jain, Solid stress inhibits the growth of multicellular tumor spheroids, Nature Biotechnology, 15 (1997), 778-783.  doi: 10.1038/nbt0897-778.

[35]

F. Hirschhaeuser, H. Menne, C. Dittfeld, J. West, W. Mueller-Klieser and L. A. Kunz-Schughart, Multicellular tumor spheroids: An underestimated tool is catching up again, Journal of Biotechnology, 148 (2010), 3-15.  doi: 10.1016/j.jbiotec.2010.01.012.

[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. doi: 10.1103/PhysRevLett.101.148105.

[37]

J. N. Jennings, A New Computational Model for Multi-cellular Biological Systems PhD thesis, University of Cambridge, 2014.

[38]

Y. Jiang, H. Levine and J. Glazier, Possible cooperation of differential adhesion and chemotaxis in mound formation of dictyostelium, Biophysical Journal, 75 (1998), 2615-2625.  doi: 10.1016/S0006-3495(98)77707-0.

[39]

Y. Jiang, J. Pjesivac-Grbovic, C. Cantrell and J. P. Freyer, A multiscale model for avascular tumor growth, Biophysical journal, 89 (2005), 3884-3894.  doi: 10.1529/biophysj.105.060640.

[40]

K. Kendall, Adhesion: Molecules and mechanics, Science, 263 (1994), 1720-1725.  doi: 10.1126/science.263.5154.1720.

[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. doi: 10.1155/2012/574025.

[42]

Y. Kim, M. Stolarska and H. Othmer, The role of the microenvironment in tumor growth and invasion, Progress in Biophysics and Molecular Biology, 106 (2011), 353-379.  doi: 10.1016/j.pbiomolbio.2011.06.006.

[43]

Y. Kim and H. G. Othmer, A hybrid model of tumor-stromal interactions in breast cancer, Bull. Math. Biol., 75 (2013), 1304-1350.  doi: 10.1007/s11538-012-9787-0.

[44]

Y. KIM and S. ROH, A hybrid model for cell proliferation and migration in glioblastoma, Discrete & Continuous Dynamical Systems-Series B, 18 (2013), 969-1015.  doi: 10.3934/dcdsb.2013.18.969.

[45]

Y. Kim, M. A. Stolarska and 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.  doi: 10.1142/S0218202507002479.

[46]

L. C. Kimlin, G. Casagrande and V. M. Virador, In vitro three-dimensional (3d) models in cancer research: An update, Molecular Carcinogenesis, 52 (2013), 167-182.  doi: 10.1002/mc.21844.

[47]

T. Lecuit and 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.  doi: 10.1038/nrm2222.

[48]

X.-F. Li, S. Carlin, M. Urano, J. Russell, C. C. Ling and J. A. O'Donoghue, Visualization of hypoxia in microscopic tumors by immunofluorescent microscopy, Cancer Research, 67 (2007), 7646-7653.  doi: 10.1158/0008-5472.CAN-06-4353.

[49]

D. Loessner, J. P. Little, G. J. Pettet and 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.  doi: 10.1242/jcs.123836.

[50]

P. Macklin, S. McDougall, A. R. Anderson, M. A. Chaplain, V. Cristini and J. Lowengrub, Multiscale modelling and nonlinear simulation of vascular tumour growth, Journal of Mathematical Biology, 58 (2009), 765-798.  doi: 10.1007/s00285-008-0216-9.

[51]

J.-L. Maître, H. Berthoumieux, S. F. G. Krens, G. Salbreux, F. Jülicher, E. Paluch and 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 and M. Vilela, Multiscale models for the growth of avascular tumors, Physics of Life Reviews, 4 (2007), 128-156.  doi: 10.1016/j.plrev.2007.04.002.

[53]

A. Marusyk, V. Almendro and K. Polyak, Intra-tumour heterogeneity: A looking glass for cancer?, Nature Reviews Cancer, 12 (2012), 323-334.  doi: 10.1038/nrc3261.

[54]

D. McElwain and 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 and C. C. Maley, Cancer as an evolutionary and ecological process, Nature Reviews Cancer, 6 (2006), 924-935.  doi: 10.1038/nrc2013.

[57]

D. Miller, Sugar uptake as a function of cell volume in human erythrocytes, The Journal of Physiology, 170 (1964), 219-225.  doi: 10.1113/jphysiol.1964.sp007325.

[58]

W. F. Mueller-Klieser and 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.  doi: 10.1007/978-1-4684-4895-5_30.

[59]

S. M. Mumenthaler, J. Foo, N. C. Choi, N. Heise, K. Leder, D. B. Agus, W. Pao, F. Michor and 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 and 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.  doi: 10.1021/mp200270v.

[61]

T. J. Newman, Modeling multi-cellular systems using sub-cellular elements, Math. Biosci. Eng., 2 (2005), 613–624, arXiv preprint q-bio/0504028. doi: 10.3934/mbe.2005.2.613.

[62]

H. Ninomiya, R. David, E. W. Damm, F. Fagotto, C. M. Niessen and 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.  doi: 10.1016/S0167-739X(00)00062-5.

[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.  doi: 10.1016/j.jtbi.2008.05.004.

[65]

E. Palsson and 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 and D. McElwain, The migration of cells in multicell tumor spheroids, Bulletin of Mathematical Biology, 63 (2001), 231-257.  doi: 10.1006/bulm.2000.0217.

[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 and A. R. Anderson, Front instabilities and invasiveness of simulated avascular tumors, Bulletin of Mathematical Biology, 71 (2009), 1189-1227.  doi: 10.1007/s11538-009-9399-5.

[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 and A. S. Popel, Multiscale models of angiogenesis, Engineering in Medicine and Biology Magazine, IEEE, 28 (2009), 14-31.  doi: 10.1109/MEMB.2009.931791.

[71]

K. A. Rejniak and R. H. Dillon, A single cell-based model of the ductal tumour microarchitecture, Computational and Mathematical Methods in Medicine, 8 (2007), 51-69.  doi: 10.1080/17486700701303143.

[72]

T. Roose, P. A. Netti, L. L. Munn, Y. Boucher and 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. doi: 10.1103/PhysRevE.71.051910.

[74]

G. Schaller and 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.  doi: 10.1098/rsta.2006.1780.

[75]

E.-M. Schötz, R. D. Burdine, F. Jülicher, M. S. Steinberg, C.-P. Heisenberg and R. A. Foty, Quantitative differences in tissue surface tension influence zebrafish germ layer positioning, HFSP journal, 2 (2008), 42-56. 

[76]

R. Shipley and S. Chapman, Multiscale modelling of fluid and drug transport in vascular tumours, Bulletin of Mathematical Biology, 72 (2010), 1464-1491.  doi: 10.1007/s11538-010-9504-9.

[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. doi: 10.1371/journal.pone.0007190.

[78]

K. Smalley, M. Lioni and 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 and 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.  doi: 10.1016/S0248-4900(03)00051-0.

[80]

M. S. Steinberg, Reconstruction of tissues by dissociated cells, Science, 141 (1963), 401-408.  doi: 10.1126/science.141.3579.401.

[81]

M. S. Steinberg, Adhesion in development: An historical overview, Developmental Biology, 180 (1996), 377-388.  doi: 10.1006/dbio.1996.0312.

[82]

M. Steinberg and 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 and 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.  doi: 10.1098/rsta.2009.0095.

[84]

K. Sung, C. Dong, G. Schmid-Schönbein, S. Chien and R. Skalak, Leukocyte relaxation properties, Biophysical Journal, 54 (1988), 331-336.  doi: 10.1016/S0006-3495(88)82963-1.

[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 and 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.  doi: 10.1083/jcb.105.5.2409.

[87]

K. Thompson and H. Byrne, Modelling the internalization of labelled cells in tumour spheroids, Bulletin of Mathematical Biology, 61 (1999), 601-623.  doi: 10.1006/bulm.1999.0089.

[88]

P. L. Townes and J. Holtfreter, Directed movements and selective adhesion of embryonic amphibian cells, Journal of Experimental Zoology, 128 (1955), 53-120.  doi: 10.1002/jez.1401280105.

[89]

G. Wayne Brodland and 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.  doi: 10.1038/ncb1003-851.

[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. doi: 10.1038/srep19103.

[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. doi: 10.1371/journal.pone.0024999.

[93]

M. Zimmermann, C. Box and S. A. Eccles, Two-dimensional vs. three-dimensional in vitro tumor migration and invasion assays, in Target Identification and Validation in Drug Discovery, Springer, (2013), 227-252.  doi: 10.1007/978-1-62703-311-4_15.

show all references

References:
[1]

S. Aland, H. Hatzikirou, J. Lowengrub and 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 and 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.  doi: 10.1016/S0168-3659(01)00317-0.

[3]

S. Breslin and L. O'Driscoll, Three-dimensional cell culture: The missing link in drug discovery, Drug Discovery Today, 18 (2013), 240-249.  doi: 10.1016/j.drudis.2012.10.003.

[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 and 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. doi: 10.1002/9781119121534.

[7]

L. L. Campbell and K. Polyak, Breast tumor heterogeneity: Cancer stem cells or clonal evolution?, Cell Cycle, 6 (2007), 2332-2338.  doi: 10.4161/cc.6.19.4914.

[8]

J. Casciari, S. Sotirchos and R. Sutherland, Mathematical modelling of microenvironment and growth in emt6/ro multicellular tumour spheroids, Cell Proliferation, 25 (1992), 1-22.  doi: 10.1111/j.1365-2184.1992.tb01433.x.

[9]

J. Casciari, S. Sotirchos and 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.  doi: 10.1002/jcp.1041510220.

[10]

P. Cirri and P. Chiarugi, Cancer-associated-fibroblasts and tumour cells: A diabolic liaison driving cancer progression, Cancer and Metastasis Reviews, 31 (2012), 195-208.  doi: 10.1007/s10555-011-9340-x.

[11]

J. C. Dallon and H. G. Othmer, How cellular movement determines the collective force generated by the Dictyostelium discoideum slug, J. Theor. Biol., 231 (2004), 203-222.  doi: 10.1016/j.jtbi.2004.06.015.

[12]

T. S. Deisboeck, Z. Wang, P. Macklin and V. Cristini, Multiscale cancer modeling, Ann. Rev. Biomed. Eng., 13 (2011), 127-155. 

[13]

M. J. Dorie, R. F. Kallman and 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.  doi: 10.1016/0014-4827(86)90483-0.

[14]

M. J. Dorie, R. F. Kallman, D. F. Rapacchietta, D. Van Antwerp and Y. R. Huang, Migration and internalization of cells and polystyrene microspheres in tumor cell spheroids, Experimental Cell Research, 141 (1982), 201-209.  doi: 10.1016/0014-4827(82)90082-9.

[15]

D. Drasdo and S. Höhme, A single-cell-based model of tumor growth in vitro: Monolayers and spheroids, Physical Biology, 2 (2005), 133-147.  doi: 10.1088/1478-3975/2/3/001.

[16]

D. Duguay, R. A. Foty and M. S. Steinberg, Cadherin-mediated cell adhesion and tissue segregation: Qualitative and quantitative determinants, Developmental Biology, 253 (2003), 309-323.  doi: 10.1016/S0012-1606(02)00016-7.

[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. doi: 10.1371/journal.pone.0031966.

[18]

E. Evans, Detailed mechanics of membrane-membrane adhesion and separation. ii. discrete kinetically trapped molecular cross-bridges, Biophysical Journal, 48 (1985), 185-192.  doi: 10.1016/S0006-3495(85)83771-1.

[19]

E. A. Evans, Detailed mechanics of membrane-membrane adhesion and separation. i. continuum of molecular cross-bridges, Biophysical Journal, 48 (1985), 175-183.  doi: 10.1016/S0006-3495(85)83770-X.

[20]

E. M. Felipe De Sousa, L. Vermeulen, E. Fessler and 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. doi: 10.1155/2012/762825.

[22]

R. A. Foty and M. S. Steinberg, Cadherin-mediated cell-cell adhesion and tissue segregation in relation to malignancy, International Journal of Developmental Biology, 48 (2004), 397-409.  doi: 10.1387/ijdb.041810rf.

[23]

R. A. Foty and M. S. Steinberg, The differential adhesion hypothesis: A direct evaluation, Developmental Biology, 278 (2005), 255-263.  doi: 10.1016/j.ydbio.2004.11.012.

[24]

R. A. Foty and M. S. Steinberg, Differential adhesion in model systems, Wiley Interdisciplinary Reviews: Developmental Biology, 2 (2013), 631-645.  doi: 10.1002/wdev.104.

[25]

J. Freyer and 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.  doi: 10.1002/jcp.1041240323.

[26]

J. Galle, G. Aust, G. Schaller, T. Beyer and D. Drasdo, Individual cell-based models of the spatial-temporal organization of multicellular systems-achievements and limitations, Cytometry Part A, 69 (2006), 704-710.  doi: 10.1002/cyto.a.20287.

[27]

D. Garrod and M. Steinberg, Tissue-specific sorting-out in two dimensions in relation to contact inhibition of cell movement, Nature, 244 (1973), 568-569.  doi: 10.1038/244568a0.

[28]

P. Gerlee and A. R. Anderson, An evolutionary hybrid cellular automaton model of solid tumour growth, Journal of Theoretical Biology, 246 (2007), 583-603.  doi: 10.1016/j.jtbi.2007.01.027.

[29]

M. Gerlinger, A. J. Rowan, S. Horswell, J. Larkin, D. Endesfelder, E. Gronroos, P. Martinez, N. Matthews, A. Stewart and P. Tarpey, Intratumor heterogeneity and branched evolution revealed by multiregion sequencing, New England Journal of Medicine, 366 (2012), 883-892.  doi: 10.1056/NEJMoa1113205.

[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. doi: 10.1186/1471-2407-12-214.

[31]

J. B. Green, Sophistications of cell sorting, Nature Cell Biology, 10 (2008), 375-377.  doi: 10.1038/ncb0408-375.

[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.  doi: 10.1007/978-1-60761-984-0_1.

[34]

G. Helmlinger, P. A. Netti, H. C. Lichtenbeld, R. J. Melder and R. K. Jain, Solid stress inhibits the growth of multicellular tumor spheroids, Nature Biotechnology, 15 (1997), 778-783.  doi: 10.1038/nbt0897-778.

[35]

F. Hirschhaeuser, H. Menne, C. Dittfeld, J. West, W. Mueller-Klieser and L. A. Kunz-Schughart, Multicellular tumor spheroids: An underestimated tool is catching up again, Journal of Biotechnology, 148 (2010), 3-15.  doi: 10.1016/j.jbiotec.2010.01.012.

[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. doi: 10.1103/PhysRevLett.101.148105.

[37]

J. N. Jennings, A New Computational Model for Multi-cellular Biological Systems PhD thesis, University of Cambridge, 2014.

[38]

Y. Jiang, H. Levine and J. Glazier, Possible cooperation of differential adhesion and chemotaxis in mound formation of dictyostelium, Biophysical Journal, 75 (1998), 2615-2625.  doi: 10.1016/S0006-3495(98)77707-0.

[39]

Y. Jiang, J. Pjesivac-Grbovic, C. Cantrell and J. P. Freyer, A multiscale model for avascular tumor growth, Biophysical journal, 89 (2005), 3884-3894.  doi: 10.1529/biophysj.105.060640.

[40]

K. Kendall, Adhesion: Molecules and mechanics, Science, 263 (1994), 1720-1725.  doi: 10.1126/science.263.5154.1720.

[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. doi: 10.1155/2012/574025.

[42]

Y. Kim, M. Stolarska and H. Othmer, The role of the microenvironment in tumor growth and invasion, Progress in Biophysics and Molecular Biology, 106 (2011), 353-379.  doi: 10.1016/j.pbiomolbio.2011.06.006.

[43]

Y. Kim and H. G. Othmer, A hybrid model of tumor-stromal interactions in breast cancer, Bull. Math. Biol., 75 (2013), 1304-1350.  doi: 10.1007/s11538-012-9787-0.

[44]

Y. KIM and S. ROH, A hybrid model for cell proliferation and migration in glioblastoma, Discrete & Continuous Dynamical Systems-Series B, 18 (2013), 969-1015.  doi: 10.3934/dcdsb.2013.18.969.

[45]

Y. Kim, M. A. Stolarska and 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.  doi: 10.1142/S0218202507002479.

[46]

L. C. Kimlin, G. Casagrande and V. M. Virador, In vitro three-dimensional (3d) models in cancer research: An update, Molecular Carcinogenesis, 52 (2013), 167-182.  doi: 10.1002/mc.21844.

[47]

T. Lecuit and 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.  doi: 10.1038/nrm2222.

[48]

X.-F. Li, S. Carlin, M. Urano, J. Russell, C. C. Ling and J. A. O'Donoghue, Visualization of hypoxia in microscopic tumors by immunofluorescent microscopy, Cancer Research, 67 (2007), 7646-7653.  doi: 10.1158/0008-5472.CAN-06-4353.

[49]

D. Loessner, J. P. Little, G. J. Pettet and 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.  doi: 10.1242/jcs.123836.

[50]

P. Macklin, S. McDougall, A. R. Anderson, M. A. Chaplain, V. Cristini and J. Lowengrub, Multiscale modelling and nonlinear simulation of vascular tumour growth, Journal of Mathematical Biology, 58 (2009), 765-798.  doi: 10.1007/s00285-008-0216-9.

[51]

J.-L. Maître, H. Berthoumieux, S. F. G. Krens, G. Salbreux, F. Jülicher, E. Paluch and 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 and M. Vilela, Multiscale models for the growth of avascular tumors, Physics of Life Reviews, 4 (2007), 128-156.  doi: 10.1016/j.plrev.2007.04.002.

[53]

A. Marusyk, V. Almendro and K. Polyak, Intra-tumour heterogeneity: A looking glass for cancer?, Nature Reviews Cancer, 12 (2012), 323-334.  doi: 10.1038/nrc3261.

[54]

D. McElwain and 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 and C. C. Maley, Cancer as an evolutionary and ecological process, Nature Reviews Cancer, 6 (2006), 924-935.  doi: 10.1038/nrc2013.

[57]

D. Miller, Sugar uptake as a function of cell volume in human erythrocytes, The Journal of Physiology, 170 (1964), 219-225.  doi: 10.1113/jphysiol.1964.sp007325.

[58]

W. F. Mueller-Klieser and 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.  doi: 10.1007/978-1-4684-4895-5_30.

[59]

S. M. Mumenthaler, J. Foo, N. C. Choi, N. Heise, K. Leder, D. B. Agus, W. Pao, F. Michor and 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 and 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.  doi: 10.1021/mp200270v.

[61]

T. J. Newman, Modeling multi-cellular systems using sub-cellular elements, Math. Biosci. Eng., 2 (2005), 613–624, arXiv preprint q-bio/0504028. doi: 10.3934/mbe.2005.2.613.

[62]

H. Ninomiya, R. David, E. W. Damm, F. Fagotto, C. M. Niessen and 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.  doi: 10.1016/S0167-739X(00)00062-5.

[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.  doi: 10.1016/j.jtbi.2008.05.004.

[65]

E. Palsson and 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 and D. McElwain, The migration of cells in multicell tumor spheroids, Bulletin of Mathematical Biology, 63 (2001), 231-257.  doi: 10.1006/bulm.2000.0217.

[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 and A. R. Anderson, Front instabilities and invasiveness of simulated avascular tumors, Bulletin of Mathematical Biology, 71 (2009), 1189-1227.  doi: 10.1007/s11538-009-9399-5.

[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 and A. S. Popel, Multiscale models of angiogenesis, Engineering in Medicine and Biology Magazine, IEEE, 28 (2009), 14-31.  doi: 10.1109/MEMB.2009.931791.

[71]

K. A. Rejniak and R. H. Dillon, A single cell-based model of the ductal tumour microarchitecture, Computational and Mathematical Methods in Medicine, 8 (2007), 51-69.  doi: 10.1080/17486700701303143.

[72]

T. Roose, P. A. Netti, L. L. Munn, Y. Boucher and 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. doi: 10.1103/PhysRevE.71.051910.

[74]

G. Schaller and 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.  doi: 10.1098/rsta.2006.1780.

[75]

E.-M. Schötz, R. D. Burdine, F. Jülicher, M. S. Steinberg, C.-P. Heisenberg and R. A. Foty, Quantitative differences in tissue surface tension influence zebrafish germ layer positioning, HFSP journal, 2 (2008), 42-56. 

[76]

R. Shipley and S. Chapman, Multiscale modelling of fluid and drug transport in vascular tumours, Bulletin of Mathematical Biology, 72 (2010), 1464-1491.  doi: 10.1007/s11538-010-9504-9.

[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. doi: 10.1371/journal.pone.0007190.

[78]

K. Smalley, M. Lioni and 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 and 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.  doi: 10.1016/S0248-4900(03)00051-0.

[80]

M. S. Steinberg, Reconstruction of tissues by dissociated cells, Science, 141 (1963), 401-408.  doi: 10.1126/science.141.3579.401.

[81]

M. S. Steinberg, Adhesion in development: An historical overview, Developmental Biology, 180 (1996), 377-388.  doi: 10.1006/dbio.1996.0312.

[82]

M. Steinberg and 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 and 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.  doi: 10.1098/rsta.2009.0095.

[84]

K. Sung, C. Dong, G. Schmid-Schönbein, S. Chien and R. Skalak, Leukocyte relaxation properties, Biophysical Journal, 54 (1988), 331-336.  doi: 10.1016/S0006-3495(88)82963-1.

[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 and 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.  doi: 10.1083/jcb.105.5.2409.

[87]

K. Thompson and H. Byrne, Modelling the internalization of labelled cells in tumour spheroids, Bulletin of Mathematical Biology, 61 (1999), 601-623.  doi: 10.1006/bulm.1999.0089.

[88]

P. L. Townes and J. Holtfreter, Directed movements and selective adhesion of embryonic amphibian cells, Journal of Experimental Zoology, 128 (1955), 53-120.  doi: 10.1002/jez.1401280105.

[89]

G. Wayne Brodland and 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.  doi: 10.1038/ncb1003-851.

[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. doi: 10.1038/srep19103.

[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. doi: 10.1371/journal.pone.0024999.

[93]

M. Zimmermann, C. Box and S. A. Eccles, Two-dimensional vs. three-dimensional in vitro tumor migration and invasion assays, in Target Identification and Validation in Drug Discovery, Springer, (2013), 227-252.  doi: 10.1007/978-1-62703-311-4_15.

Figure 1.  (A): A 3D aggregate in a hanging drop culture; (B): The representation of the Kelvin and growth elements that characterize the internal rheology of each cell, modified from previous papers [45]. Note that in a hanging drop spheroid systems in vitro, the surrounding environment exerts little resistance to growth. As such, it is reasonable to assume that no external force is imposed on in silico spheroids. Here each tumor cell inside a spheroid is modeled as a 3D deformable ellipsoid with three axes a, b, c each of which is represented by a Kelvin element. In the a-axis (similar to b-and c-axis), $u_a$ is the total change of the length, $u_a^0$ and $u_a^g$ are the changes of the length in the a-axis due to the change in the passive and growth elements respectively, $f_2$ is the nonlinear spring force from the spring in parallel, $f_a$ is the magnitude of the force applied to each end, $\mu_a$ is the viscous coefficient of the dash-pot, $k_a$ is the spring constant for the spring in the Maxwell element.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Differential adhesive forces among heterotypical cells lead to various aggregation patterns. Experimental observations of the cross section of a 3D aggregate are listed in (B1), (B2) and (B3). The corresponding numerical patterns, generated by our model after T= 7 hours for an aggregate of 1021 cells, are shown in (A1), (A2) and (A3). The same sorting pattern persists afterward. In particular, the choice of parameters $\alpha_{g,g}:\alpha_{r,r}:\alpha_{g,r}=0.4:1:0.7$ (DA 2) leads to cell sorting in (A2). A cross section of the 3D configuration is shown under its 3D counterpart. Cells do not sort at all when $\alpha_{g,g}:\alpha_{r,r}:\alpha_{g,r}=1:1:1$ in (A1). With $\alpha_{g,g}:\alpha_{r,r}:\alpha_{g,r}=1:1:0.2$ (DA 1), cells separate in (A3). (B1), (B2) and (B3) are the experimental observations from Duguay et al [16] where two L-cell lines express N-cad at different levels. Line N5A expresses about 50% more N-cad than what line N2 does. An aggregate in figure (B1) does not sort, in which both the red-and green-colored cells are from line N5A after 1 day of culture. Yet a similar aggregate in figure (B2) containing a mixture of N5A (red) and N2 (green) cells segregate from one another during 1 day of culture, where higher-expressing N5A cells were completely enveloped by lower-expressing N2 cells. In figure (B3), Aggregates containing equal numbers of L cells lead to mounds of R-cad-expressing cells (red) partially capping a B-cad-expressing mass (green) after being cultured in suspension for 2 days.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  (A) 3D Simulation results of the same engulfment pattern by fragment fusion and sorting shown in a cross section of a 3D aggregate; (B) in vitro observations. In both cases, we set $\alpha_{g,g}:\alpha_{r,r}:\alpha_{g,r}=0.4:1:0.7$. In an aggregate starting with intermixed cells, cells sort by the coalescence of smaller islands to form larger ones (sorting); If two tissues have initial contact, green cells gradually spread over red ones and eventually envelop them (fragment fusion). Our in silico results reproduced the in vitro observations (B) which were taken from Foty's review [24,75] where zebrafish ectoderm and mescendoderm tissues were mixed together or contacted each other. The system reached a stable configuration after 16 h, as the ectoderm occupied the internal position.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Compression forces on cell sorting. In all these simulations, we set $\alpha_{g,g}:\alpha_{r,r}:\alpha_{g,r}=0.4:1:0.7$. Ratio S is calculated by the ratio of the total number of lighter adhesive cells over the total number of the cells in the outer part of the smallest rectangle solid containing the aggregate
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Growth of the tumor spheroid with a pre-existing necrotic core based on the 2D cross section configuration. (a) the oxygen profile which is described in percentage; (b) initial configuration; (c) intermediate state (T= 24 hours); (d): final configuration (T= 48 hours) where green (or blue) is for proliferating cells, red is for quiescent cells and black is for the necrotic core. The unit of color bar in the oxygen is in percentage. One percentage is equal to 0.013 mM. The spatial unit is per 10 $\mu m$.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Plots of the stabilized viable rim (defined by one half of the difference between the diameter of a tumor and the diameter of its necrotic core) in two cases: (a) the evolution of the viable rim inside the tumor spheroid with a pre-existing necrotic core. (b) the evolution of the variable rim inside the tumor spheroid which consists only of initial proliferating cells. Radius is chosen on right plot to enable readers to observe clearly when the necrotic core emerges and how it evolves. Data point at t=0 is not measured since it takes time for an initialized spheroid system to approach a mechanical quasi-equilibrium. In addition, after the stabilization from t=2000 minutes, the same pattern as (a) was observed. But there are slight fluctuations in the simulation due to cellular random walks.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Growth of the tumor spheroid without a pre-existing necrotic core based on the 2D cross section configuration. (a) initial configuration, (b) configuration after T= 44 hours. The spatial unit is per 10 $\mu m$ here.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  (A) Evolution of the frequency of labelled cells in a growing spheroid at three different elapsed time points (2 hours, 24 hours and 48 hours); (B) Evolution of the frequency of labelled cells in a non-growing spheroid. Homotypic labeled cells are initially adhered to the surface of the spheroids. The number of those cells are recorded in different depths of the spheroids.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  First row for a non-growing spheroid where one type of cells experience chemotaxis: (a) initial configuration, (b) final distribution (T= 48 hours). Second row for a growing spheroid where one type of cells experience chemotaxis: (c) initial configuration, (d) final distribution (T= 48 hours). Active forces for both random motion and chemotaxis are 8 nN. The spatial unit is per 10 $\mu m$ here.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 10.  The pathway of cell sorting in a 2D cross section where $\alpha_{r,r}:\alpha_{g,g}:\alpha_{g,r}=0.4:1.5:0.7$. It can be seen that small islands of green cells fuse into large ones.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 11.  An illustration of passive force using Equation 12 Here two standard spherical cells of 10 $\mu m $ diameter are used to carry out the calculations
Figure Options

Download full-size image

Download as PowerPoint slide

Table 2.  Parameters for the cell-based component of the model.
Parameter Description Value Dimensionless in coding Refs.
Adhesion parameters
$\mu_{cell}$ cell-cell adhesiveness 27.0 dyn s/cm 450 [11,45]
$\mu_s$ cell-substrate 27.0 dyn s/cm 450 [11,45]
adhesiveness
$\mu_{f}$ fluid viscosity 2.7 dyn s/cm 450 [11,45]
Rheological parameters
$c^+$ growth function 5.16089$\times 10^{-9}$ mm/(min. nN) 5.16089$\times 10^{-9}$ [45]
$\sigma^+$ growth function 800 nN 800 [45]
$\sigma^-$ growth function -4 nN -4 [45]
$\alpha$ growth function 0.0 nN 0.0 [45]
$k_a$ standard solid 163.8 dyn/cm 163800 [11,45,84]
$k_2$ standard solid 147.5 dyn/cm, 147500 [11,45,84]
$\mu_a$ standard solid 123 dyn min/cm 123000 [11,45,84]
$f_a$ active force 10 nN 10 in this work
Table Options

Download as excel

Parameter Description Value Dimensionless in coding Refs.
Adhesion parameters
$\mu_{cell}$ cell-cell adhesiveness 27.0 dyn s/cm 450 [11,45]
$\mu_s$ cell-substrate 27.0 dyn s/cm 450 [11,45]
adhesiveness
$\mu_{f}$ fluid viscosity 2.7 dyn s/cm 450 [11,45]
Rheological parameters
$c^+$ growth function 5.16089$\times 10^{-9}$ mm/(min. nN) 5.16089$\times 10^{-9}$ [45]
$\sigma^+$ growth function 800 nN 800 [45]
$\sigma^-$ growth function -4 nN -4 [45]
$\alpha$ growth function 0.0 nN 0.0 [45]
$k_a$ standard solid 163.8 dyn/cm 163800 [11,45,84]
$k_2$ standard solid 147.5 dyn/cm, 147500 [11,45,84]
$\mu_a$ standard solid 123 dyn min/cm 123000 [11,45,84]
$f_a$ active force 10 nN 10 in this work
Table 1.  Sorting in the presence of differential adhesion. Various ratios of cohesion can lead to cell sorting in two types of cells, distinguished by green and red in this work. Here $\alpha_{g,g}$, $\alpha_{r,r}$, $\alpha_{g,r}$ represent relative adhesive strengths between like and unlike cells (green and green, red and red, or green and red respectively)
Index $\alpha_{g,g}$ $\alpha_{r,r}$ $\alpha_{g,r}$ Sorting results
1 0.4 1 0.7 green envelops red
2 0.6 1 0.8 green envelops red
3 0.8 1 0.9 green envelops red
4 1 1 1 Not sorting
5 1 1 0.2 green and red separate
Table Options

Download as excel

Index $\alpha_{g,g}$ $\alpha_{r,r}$ $\alpha_{g,r}$ Sorting results
1 0.4 1 0.7 green envelops red
2 0.6 1 0.8 green envelops red
3 0.8 1 0.9 green envelops red
4 1 1 1 Not sorting
5 1 1 0.2 green and red separate
Table 3.  Parameters used in the reaction-diffusion component of the model. We use the cell average packing density carried out $2.01\times10^8\;\;\mbox{cells}/cm^3$ in Casciari $et al$ . [8] to convert uptake parameters $A_{O_2},A_{gl},B_{O_2},B_{gl}$ in this table to rates per unit volume.
P Description Value Dimensionless in coding Refs.
Diffusion Coefficients of oxygen in each region
$D_o^c$ cell based region $1.82\times 10^{-5}\;cm^2/s$ 6.552 [45]
$D_o^q$ continuum region $2.15\times 10^{-6}\; cm^2/s$ 7.74
Diffusion Coefficients of glucose in each region
$D_g^p$ cell based region $ 3.0\times 10^{-6}\; cm^2/s $ 1.08 this work
$D_g^q$ continuum region $ 6.46\times 10^{-6}\; cm^2/s $ 2.3256 [45]
Coefficients in Uptake Functions
$A_{O_2}$ oxygen uptake $1.0642\times 10^{-16}\; \frac{mol}{cell\cdot s}$ 2.01014 [9,45]
$B_{O_2}$ oxygen uptake $6.0202\times 10^{-17}\; \frac{mol\cdot mM}{cell\cdot s}$ 0.0497 [8,9,45]
$A_{gl}$ glucose uptake $1.0642\times 10^{-16}\; \frac{mol}{cell\cdot s}$ 2.01014 [8,9,45]
$B_{gl}$ glucose uptake $1.7879\times 10^{-17}\; \frac{mol\cdot mM}{cell\cdot s}$ 0.0107 [8,25,45]
$k_{O_2} $ critical oxygen concentration $4.640\times 10^{-3}\; mM$ $1.856\times 10^{-4}$ [8,45]
$k_{gl} $ critical glucose concentration $4.0\times 10^{-2}\; mM$ $1.6\times 10^{-3}$ [8,45]
$n_{O_2} $ oxygen uptake $0.55\; mM$ $2.2 \times 10 ^{-2}$ [25,45]
$n_{gl} $ glucose uptake $0.04\; mM$ $1.6 \times 10 ^{-3}$ [25,45]
Table Options

Download as excel

P Description Value Dimensionless in coding Refs.
Diffusion Coefficients of oxygen in each region
$D_o^c$ cell based region $1.82\times 10^{-5}\;cm^2/s$ 6.552 [45]
$D_o^q$ continuum region $2.15\times 10^{-6}\; cm^2/s$ 7.74
Diffusion Coefficients of glucose in each region
$D_g^p$ cell based region $ 3.0\times 10^{-6}\; cm^2/s $ 1.08 this work
$D_g^q$ continuum region $ 6.46\times 10^{-6}\; cm^2/s $ 2.3256 [45]
Coefficients in Uptake Functions
$A_{O_2}$ oxygen uptake $1.0642\times 10^{-16}\; \frac{mol}{cell\cdot s}$ 2.01014 [9,45]
$B_{O_2}$ oxygen uptake $6.0202\times 10^{-17}\; \frac{mol\cdot mM}{cell\cdot s}$ 0.0497 [8,9,45]
$A_{gl}$ glucose uptake $1.0642\times 10^{-16}\; \frac{mol}{cell\cdot s}$ 2.01014 [8,9,45]
$B_{gl}$ glucose uptake $1.7879\times 10^{-17}\; \frac{mol\cdot mM}{cell\cdot s}$ 0.0107 [8,25,45]
$k_{O_2} $ critical oxygen concentration $4.640\times 10^{-3}\; mM$ $1.856\times 10^{-4}$ [8,45]
$k_{gl} $ critical glucose concentration $4.0\times 10^{-2}\; mM$ $1.6\times 10^{-3}$ [8,45]
$n_{O_2} $ oxygen uptake $0.55\; mM$ $2.2 \times 10 ^{-2}$ [25,45]
$n_{gl} $ glucose uptake $0.04\; mM$ $1.6 \times 10 ^{-3}$ [25,45]
[1]

Gülnihal Meral, Christian Stinner, Christina Surulescu. On a multiscale model involving cell contractivity and its effects on tumor invasion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 189-213. doi: 10.3934/dcdsb.2015.20.189
[2]

Yangjin Kim, Hans G. Othmer. Hybrid models of cell and tissue dynamics in tumor growth. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1141-1156. doi: 10.3934/mbe.2015.12.1141
[3]

Thomas Y. Hou, Zuoqiang Shi. Dynamic growth estimates of maximum vorticity for 3D incompressible Euler equations and the SQG model. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1449-1463. doi: 10.3934/dcds.2012.32.1449
[4]

Ahuod Alsheri, Ebraheem O. Alzahrani, Asim Asiri, Mohamed M. El-Dessoky, Yang Kuang. Tumor growth dynamics with nutrient limitation and cell proliferation time delay. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3771-3782. doi: 10.3934/dcdsb.2017189
[5]

Yong Zhou. Remarks on regularities for the 3D MHD equations. Discrete and Continuous Dynamical Systems, 2005, 12 (5) : 881-886. doi: 10.3934/dcds.2005.12.881
[6]

Hyeong-Ohk Bae, Bum Ja Jin. Estimates of the wake for the 3D Oseen equations. Discrete and Continuous Dynamical Systems - B, 2008, 10 (1) : 1-18. doi: 10.3934/dcdsb.2008.10.1
[7]

Indranil SenGupta, Weisheng Jiang, Bo Sun, Maria Christina Mariani. Superradiance problem in a 3D annular domain. Conference Publications, 2011, 2011 (Special) : 1309-1318. doi: 10.3934/proc.2011.2011.1309
[8]

Giovanny Guerrero, José Antonio Langa, Antonio Suárez. Biodiversity and vulnerability in a 3D mutualistic system. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4107-4126. doi: 10.3934/dcds.2014.34.4107
[9]

Jiao He, Rafael Granero-Belinchón. On the dynamics of 3D electrified falling films. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4041-4064. doi: 10.3934/dcds.2021027
[10]

Sadek Gala. A new regularity criterion for the 3D MHD equations in $R^3$. Communications on Pure and Applied Analysis, 2012, 11 (3) : 973-980. doi: 10.3934/cpaa.2012.11.973
[11]

Federica Bubba, Benoit Perthame, Daniele Cerroni, Pasquale Ciarletta, Paolo Zunino. A coupled 3D-1D multiscale Keller-Segel model of chemotaxis and its application to cancer invasion. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2053-2086. doi: 10.3934/dcdss.2022044
[12]

Jian-Guo Liu, Min Tang, Li Wang, Zhennan Zhou. Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3011-3035. doi: 10.3934/dcdsb.2018297
[13]

Jiahong Wu. Regularity results for weak solutions of the 3D MHD equations. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 543-556. doi: 10.3934/dcds.2004.10.543
[14]

Gabriel Deugoue. Approximation of the trajectory attractor of the 3D MHD System. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2119-2144. doi: 10.3934/cpaa.2013.12.2119
[15]

Chongsheng Cao. Sufficient conditions for the regularity to the 3D Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1141-1151. doi: 10.3934/dcds.2010.26.1141
[16]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241
[17]

Alp Eden, Varga K. Kalantarov. 3D convective Cahn--Hilliard equation. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1075-1086. doi: 10.3934/cpaa.2007.6.1075
[18]

Franco Flandoli, Dejun Luo. Euler-Lagrangian approach to 3D stochastic Euler equations. Journal of Geometric Mechanics, 2019, 11 (2) : 153-165. doi: 10.3934/jgm.2019008
[19]

Tomás Caraballo, Antonio M. Márquez-Durán, José Real. Pullback and forward attractors for a 3D LANS$-\alpha$ model with delay. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 559-578. doi: 10.3934/dcds.2006.15.559
[20]

Jianqing Chen. Best constant of 3D Anisotropic Sobolev inequality and its applications. Communications on Pure and Applied Analysis, 2010, 9 (3) : 655-666. doi: 10.3934/cpaa.2010.9.655

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (540)
  • HTML views (281)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy