
-
Previous Article
Effect of rotational grazing on plant and animal production
- MBE Home
- This Issue
-
Next Article
The potential impact of a prophylactic vaccine for Ebola in Sierra Leone
A multiscale model for heterogeneous tumor spheroid in vitro
Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA, 30460, USA |
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.
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. |











Parameter | Description | Value | Dimensionless in coding | Refs. |
Adhesion parameters | ||||
|
cell-cell adhesiveness | 27.0 dyn s/cm | 450 | [11,45] |
|
cell-substrate | 27.0 dyn s/cm | 450 | [11,45] |
adhesiveness | ||||
|
fluid viscosity | 2.7 dyn s/cm | 450 | [11,45] |
Rheological parameters | ||||
|
growth function | 5.16089 |
5.16089 |
[45] |
growth function | 800 nN | 800 | [45] | |
|
growth function | -4 nN | -4 | [45] |
|
growth function | 0.0 nN | 0.0 | [45] |
|
standard solid | 163.8 dyn/cm | 163800 | [11,45,84] |
|
standard solid | 147.5 dyn/cm, | 147500 | [11,45,84] |
|
standard solid | 123 dyn min/cm | 123000 | [11,45,84] |
|
active force | 10 nN | 10 | in this work |
Parameter | Description | Value | Dimensionless in coding | Refs. |
Adhesion parameters | ||||
|
cell-cell adhesiveness | 27.0 dyn s/cm | 450 | [11,45] |
|
cell-substrate | 27.0 dyn s/cm | 450 | [11,45] |
adhesiveness | ||||
|
fluid viscosity | 2.7 dyn s/cm | 450 | [11,45] |
Rheological parameters | ||||
|
growth function | 5.16089 |
5.16089 |
[45] |
growth function | 800 nN | 800 | [45] | |
|
growth function | -4 nN | -4 | [45] |
|
growth function | 0.0 nN | 0.0 | [45] |
|
standard solid | 163.8 dyn/cm | 163800 | [11,45,84] |
|
standard solid | 147.5 dyn/cm, | 147500 | [11,45,84] |
|
standard solid | 123 dyn min/cm | 123000 | [11,45,84] |
|
active force | 10 nN | 10 | in this work |
Index | |
|
|
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 |
Index | |
|
|
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 |
P | Description | Value | Dimensionless in coding | Refs. |
Diffusion Coefficients of oxygen in each region | ||||
|
cell based region | |
6.552 | [45] |
|
continuum region | |
7.74 | |
Diffusion Coefficients of glucose in each region | ||||
|
cell based region | |
1.08 | this work |
|
continuum region | |
2.3256 | [45] |
Coefficients in Uptake Functions | ||||
|
oxygen uptake | |
2.01014 | [9,45] |
|
oxygen uptake | |
0.0497 | [8,9,45] |
|
glucose uptake | |
2.01014 | [8,9,45] |
|
glucose uptake | |
0.0107 | [8,25,45] |
|
critical oxygen concentration | |
|
[8,45] |
critical glucose concentration | |
|
[8,45] | |
|
oxygen uptake | |
|
[25,45] |
|
glucose uptake | |
|
[25,45] |
P | Description | Value | Dimensionless in coding | Refs. |
Diffusion Coefficients of oxygen in each region | ||||
|
cell based region | |
6.552 | [45] |
|
continuum region | |
7.74 | |
Diffusion Coefficients of glucose in each region | ||||
|
cell based region | |
1.08 | this work |
|
continuum region | |
2.3256 | [45] |
Coefficients in Uptake Functions | ||||
|
oxygen uptake | |
2.01014 | [9,45] |
|
oxygen uptake | |
0.0497 | [8,9,45] |
|
glucose uptake | |
2.01014 | [8,9,45] |
|
glucose uptake | |
0.0107 | [8,25,45] |
|
critical oxygen concentration | |
|
[8,45] |
critical glucose concentration | |
|
[8,45] | |
|
oxygen uptake | |
|
[25,45] |
|
glucose uptake | |
|
[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
Other articles
by authors
[Back to Top]