Abstract
The dawn of the new era in precise diagnostic devices and personalized cancer treatment is intertwined with computational models capable of integrating biochemical factors and biophysical processes to simulate and predict cancer progression. In the last decade, thanks to the increase in the computational power, the development of more sophisticated and realistic models has gained attention in cancer research. These computational models can advance our fundamental understandings of the complicated processes involved in metastasis, as a challenging stage for cancer treatment, and can provide new means for developing novel tools for predicting cancer progression. Considering the potential of these models and the plethora of recent computational models of different steps of the metastasis process, this timely review provides an up-to-date outlook of novel approaches, identifies research gaps and suggests future research directions. This review focuses on physics-based approaches for modeling metastasis process and covers recent computational models and frameworks related to all steps of metastasis from primary tumor growth to secondary tumor formation. In this review, computational models and simulations are classified based on their targeted step of metastasis. Tumor growth and tumor-induced angiogenesis together are considered as a necessary primary step for the metastasis cascade that has four steps: the intravasation of an individual tumor cell into circulatory system, circulation of the cell, arrest and extravasation, and eventually colonization and formation of a metastatic tumor.
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References
Chaffer CL, Weinberg RA (2011) A perspective on cancer cell metastasis. Science 331:1559–1564. https://doi.org/10.1126/science.1203543
Bray F, Ferlay J, Soerjomataram I et al (2018) Global cancer statistics 2018: GLOBOCAN estimates of incidence and mortality worldwide for 36 cancers in 185 countries. CA Cancer J Clin 68:394–424. https://doi.org/10.3322/caac.21492
Sugarbaker EV (1979) Cancer metastasis: a product of tumor-host interactions. Curr Probl Cancer 3:1–59. https://doi.org/10.1016/S0147-0272(79)80008-2
Pope EL (1938) Metastasis and metastases*. Can Med Assoc J 38:244–249
Lambert AW, Pattabiraman DR, Weinberg RA (2017) Emerging biological principles of metastasis. Cell 168:670–691. https://doi.org/10.1016/j.cell.2016.11.037
Rejniak KA, McCawley LJ (2010) Current trends in mathematical modeling of tumor–microenvironment interactions: a survey of tools and applications. Exp Biol Med. https://doi.org/10.1258/ebm.2009.009230
Wirtz D, Konstantopoulos K, Searson PC (2011) The physics of cancer: the role of physical interactions and mechanical forces in metastasis. Nat Rev Cancer 11:512–522. https://doi.org/10.1038/nrc3080
Malandrino A, Kamm RD, Moeendarbary E (2018) In vitro modeling of mechanics in cancer metastasis. ACS Biomate Sci Eng 4:294–301. https://doi.org/10.1021/acsbiomaterials.7b00041
Prospective Outlook of Mechanics in Oncology | Physical Sciences in Oncology. https://physics.cancer.gov/report/workshop6.aspx. Accessed 22 Apr 2019
Jain RK, Batista A (2018) A physical view of cancer. Trends in Cancer 4:257. https://doi.org/10.1016/j.trecan.2018.03.001
Hanahan D, Weinberg RA (2011) Hallmarks of cancer: the next generation. Cell 144:646–674. https://doi.org/10.1016/j.cell.2011.02.013
Egeblad M, Nakasone ES, Werb Z (2010) Tumors as organs: complex tissues that interface with the entire organism. Dev Cell 18:884–901. https://doi.org/10.1016/j.devcel.2010.05.012
Stetler-Stevenson WG, Aznavoorian S, Liotta LA. Tumor cell interactions with the extracellular matrix during invasion and metastasis. 35
Gravitz L (2012) Physical scientists take on cancer. Nature 491:S49. https://doi.org/10.1038/491S49a
Newton PK, Mason J, Bethel K et al (2012) A stochastic markov chain model to describe lung cancer growth and metastasis. PLoS ONE. https://doi.org/10.1371/journal.pone.0034637
Cook LM, Araujo A, Pow-Sang JM et al (2016) Predictive computational modeling to define effective treatment strategies for bone metastatic prostate cancer. Sci Rep. https://doi.org/10.1038/srep29384
Altrock PM, Liu LL, Michor F (2015) The mathematics of cancer: integrating quantitative models. Nat Rev Cancer 15:730–745. https://doi.org/10.1038/nrc4029
Newton PK, Mason J, Bethel K et al (2013) Spreaders and sponges define metastasis in lung cancer: a markov chain monte carlo mathematical model. Cancer Res 73:2760–2769. https://doi.org/10.1158/0008-5472.CAN-12-4488
Cristini V, Lowengrub J (2010) Multiscale modeling of cancer: an integrated experimental and mathematical modeling approach. Cambridge University Press, Cambridge
Basanta D, Hatzikirou H, Deutsch A (2008) Studying the emergence of invasiveness in tumours using game theory. Eur Phys J B 63:393–397. https://doi.org/10.1140/epjb/e2008-00249-y
Gerisch A, Chaplain MAJ (2008) Mathematical modelling of cancer cell invasion of tissue: local and non-local models and the effect of adhesion. J Theor Biol 250:684–704. https://doi.org/10.1016/j.jtbi.2007.10.026
Katira P, Bonnecaze RT, Zaman MH (2013) Modeling the mechanics of cancer: effect of changes in cellular and extra-cellular mechanical properties. Front Oncol. https://doi.org/10.3389/fonc.2013.00145
Edelman LB, Eddy JA, Price ND (2010) In silico models of cancer. Wiley Interdiscip Rev Syst Biol Med 2:438–459. https://doi.org/10.1002/wsbm.75
Kolev M, Zubik-Kowal B (2011) Numerical solutions for a model of tissue invasion and migration of tumour cells. In: Computational and Mathematical Methods in Medicine. https://www.hindawi.com/journals/cmmm/2011/452320/. Accessed 30 Jan 2019
Dallon JC (2000) Numerical aspects of discrete and continuum hybrid models in cell biology. Appl Numer Math 32:137–159. https://doi.org/10.1016/S0168-9274(99)00021-5
Metzcar J, Wang Y, Heiland R, Macklin P (2019) A review of cell-based computational modeling in cancer biology. JCO Clin Cancer Inform. https://doi.org/10.1200/CCI.18.00069
De Matteis G, Graudenzi A, Antoniotti M (2013) A review of spatial computational models for multi-cellular systems, with regard to intestinal crypts and colorectal cancer development. J Math Biol 66:1409–1462. https://doi.org/10.1007/s00285-012-0539-4
Sanga S, Frieboes HB, Zheng X et al (2007) Predictive oncology: multidisciplinary, multi-scale in-silico modeling linking phenotype, morphology and growth. Neuroimage 37:S120–S134. https://doi.org/10.1016/j.neuroimage.2007.05.043
Karolak A, Markov DA, McCawley LJ, Rejniak KA (2018) Towards personalized computational oncology: from spatial models of tumour spheroids, to organoids, to tissues. J R Soc Interface 15:20170703. https://doi.org/10.1098/rsif.2017.0703
Chwalek K, Bray LJ, Werner C (2014) Tissue-engineered 3D tumor angiogenesis models: potential technologies for anti-cancer drug discovery. Adv Drug Deliv Rev 79–80:30–39. https://doi.org/10.1016/j.addr.2014.05.006
Cobelli C, Carson E (2008) Introduction to modeling in physiology and medicine. Elsevier, Amsterdam
Gatenby RA, Gawlinski ET (1996) A reaction-diffusion model of cancer invasion. Cancer Res 56:5745–5753
Jeon J, Quaranta V, Cummings PT (2010) An off-lattice hybrid discrete-continuum model of tumor growth and invasion. Biophys J 98:37–47. https://doi.org/10.1016/j.bpj.2009.10.002
Anderson ARA, Chaplain MAJ, Newman EL et al (2000) Mathematical modelling of tumour invasion and metastasis. J Theo Med 2:129–154. https://doi.org/10.1080/10273660008833042
Lolas G, Chaplain MAJ (2006) Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity. Netw Heterogeneous Media 1:399–439. https://doi.org/10.3934/nhm.2006.1.399
Meral G (2019) DRBEM-FDM solution of a chemotaxis–haptotaxis model for cancer invasion. J Comput Appl Math 354:299–309. https://doi.org/10.1016/j.cam.2018.04.047
Waldeland JO, Evje S (2018) A multiphase model for exploring tumor cell migration driven by autologous chemotaxis. Chem Eng Sci 191:268–287. https://doi.org/10.1016/j.ces.2018.06.076
Preziosi L (2003) Cancer modelling and simulation. CRC Press, Boca Raton
Deisboeck TS, Wang Z, Macklin P, Cristini V (2011) Multiscale cancer modeling. Annu Rev Biomed Eng 13:127–155. https://doi.org/10.1146/annurev-bioeng-071910-124729
Alder BJ, Wainwright TE (1959) Studies in molecular dynamics: I: general method. J Chem Phys 31:459–466. https://doi.org/10.1063/1.1730376
Bellomo N, Li NK, Maini PK (2008) On the foundations of cancer modelling: selected topics, speculations, and perspectives. Math Models Methods Appl Sci 18:593–646
Stéphanou A, Volpert V (2016) Hybrid modelling in biology: a classification review. Math Model Nat Phenom 11:37–48. https://doi.org/10.1051/mmnp/201611103
Schaller G, Meyer-Hermann M (2006) Continuum versus discrete model: a comparison for multicellular tumour spheroids. Philos Trans R Soc Math Phys Eng Sci 364:1443–1464. https://doi.org/10.1098/rsta.2006.1780
Harding JH (1997) Mesoscopic modelling. Curr Opin Solid State Mater Sci 2:728–732. https://doi.org/10.1016/S1359-0286(97)80017-4
Wolfram S (1983) Statistical mechanics of cellular automata. Rev Mod Phys 55:601–644. https://doi.org/10.1103/RevModPhys.55.601
Mak M, Kim T, Zaman MH, Kamm RD (2015) Multiscale mechanobiology: computational models for integrating molecules to multicellular systems. Integr Biol 7:1093–1108. https://doi.org/10.1039/c5ib00043b
Wolfram S (1984) Universality and complexity in cellular automata. Phys D 10:1–35. https://doi.org/10.1016/0167-2789(84)90245-8
Voss-Böhme A (2012) Multi-scale modeling in morphogenesis: a critical analysis of the cellular potts model. PLoS ONE 7:e42852. https://doi.org/10.1371/journal.pone.0042852
Swat MH, Thomas GL, Belmonte JM et al (2012) Chapter 13: multi-scale modeling of tissues using compucell3D. In: Asthagiri AR, Arkin AP (eds) Methods in cell biology. Academic Press, Cambridge, pp 325–366
Graner F, Glazier JA (1992) Simulation of biological cell sorting using a two-dimensional extended Potts model. Phys Rev Lett 69:2013–2016. https://doi.org/10.1103/PhysRevLett.69.2013
Teschner T-R, Könözsy L, Jenkins KW (2016) Progress in particle-based multiscale and hybrid methods for flow applications. Microfluid Nanofluid 20:68. https://doi.org/10.1007/s10404-016-1729-y
Bhui R, Hayenga HN (2017) An agent-based model of leukocyte transendothelial migration during atherogenesis. PLoS Comput Biol 13:e1005523. https://doi.org/10.1371/journal.pcbi.1005523
B. Liu M, Liu GR, W. Zhou L, Z. Chang J, (2014) Dissipative particle dynamics (DPD): an overview and recent developments. Arch Comput Methods Eng. https://doi.org/10.1007/s11831-014-9124-x
Español P, Warren PB (2017) Perspective: dissipative particle dynamics. J Chem Phys 146:150901. https://doi.org/10.1063/1.4979514
Chopard B, Ouared R, Deutsch A et al (2010) Lattice-gas cellular automaton models for biology: from fluids to cells. Acta Biotheor 58:329–340. https://doi.org/10.1007/s10441-010-9118-5
Moeendarbary E, Ng TY, Zangeneh M (2009) Dissipative particle dynamics: introduction, methodology and complex fluid applications—a review. Int J Appl Mech 01:737–763. https://doi.org/10.1142/S1758825109000381
Basan M, Prost J, Joanny J-F, Elgeti J (2011) Dissipative particle dynamics simulations for biological tissues: rheology and competition. Phys Biol 8:026014. https://doi.org/10.1088/1478-3975/8/2/026014
Friedman R, Boye K, Flatmark K (2013) Molecular modelling and simulations in cancer research. Biochim Biophys Acta (BBA) Rev Cancer 1836:1–14. https://doi.org/10.1016/j.bbcan.2013.02.001
Hoogerbrugge PJ, Koelman JMVA (1992) Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics. EPL Lett J Explor Front Phys 19:155–160. https://doi.org/10.1209/0295-5075/19/3/001
Irving JH, Kirkwood JG (1950) The statistical mechanical theory of transport processes: IV: the equations of hydrodynamics. J Chem Phys 18:817–829. https://doi.org/10.1063/1.1747782
Franssen LC, Lorenzi T, Burgess AEF, Chaplain MAJ (2019) A Mathematical framework for modelling the metastatic spread of cancer. Bull Math Biol 81:1965–2010. https://doi.org/10.1007/s11538-019-00597-x
Bielenberg DR, Zetter BR (2015) The contribution of angiogenesis to the process of metastasis. Cancer J 21:267–273. https://doi.org/10.1097/PPO.0000000000000138
Tien Y-W, Chang K-J, Jeng Y-M et al (2001) Tumor angiogenesis and its possible role in intravasation of colorectal epithelial cells. Clin Cancer Res 7:1627–1632
Liotta LA, Kleinerman J, Saldel GM (1976) The significance of hematogenous tumor cell clumps in the metastatic process. Cancer Res 36:889–894
Wittekind C, Neid M (2005) Cancer invasion and metastasis. Oncology 69(Suppl 1):14–16. https://doi.org/10.1159/000086626
Zetter BR (1998) Angiogenesis and tumor metastasis. Annu Rev Med 49:407–424. https://doi.org/10.1146/annurev.med.49.1.407
Frieboes HB, Jin F, Chuang Y-L et al (2010) Three-dimensional multispecies nonlinear tumor growth—II: tumor invasion and angiogenesis. J Theor Biol 264:1254–1278. https://doi.org/10.1016/j.jtbi.2010.02.036
Bearer EL, Lowengrub JS, Frieboes HB et al (2009) Multiparameter computational modeling of tumor invasion. Cancer Res 69:4493–4501. https://doi.org/10.1158/0008-5472.CAN-08-3834
Ayati BP, Webb GF, Anderson ARA (2006) Computational methods and results for structured multiscale models of tumor invasion. Multiscale Model Simul 5:1–20. https://doi.org/10.1137/050629215
Bresch D, Colin T, Grenier E et al (2010) Computational modeling of solid tumor growth: the avascular stage. SIAM J Sci Comput 32:2321–2344. https://doi.org/10.1137/070708895
Ambrosi D, Preziosi L (2002) On the closure of mass balance models for tumor growth. Math Models Methods Appl Sci 12:737–754. https://doi.org/10.1142/S0218202502001878
Anderson ARA, Chaplain MAJ, McDougall S (2012) A hybrid discrete-continuum model of tumour induced angiogenesis. In: Jackson TL (ed) Modeling tumor vasculature: molecular, cellular, and tissue level aspects and implications. Springer, New York, pp 105–133
Anderson ARA, Chaplain MAJ (1998) Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull Math Biol 60:857–899. https://doi.org/10.1006/bulm.1998.0042
Araujo R (2004) A history of the study of solid tumour growth: the contribution of mathematical modelling. Bull Math Biol 66:1039–1091. https://doi.org/10.1016/j.bulm.2003.11.002
Byrne HM (2010) Dissecting cancer through mathematics: from the cell to the animal model. Nat Rev Cancer 10:221–230. https://doi.org/10.1038/nrc2808
Roose T, Chapman SJ, Maini PK (2007) Mathematical models of avascular tumor growth. SIAM Rev 49:179–208. https://doi.org/10.1137/S0036144504446291
Cristini V, Frieboes HB, Gatenby R et al (2005) Morphologic instability and cancer invasion. Clin Cancer Res 11:6772–6779. https://doi.org/10.1158/1078-0432.CCR-05-0852
Soltani M, Chen P (2013) Numerical modeling of interstitial fluid flow coupled with blood flow through a remodeled solid tumor microvascular network. PLoS ONE 8:e67025. https://doi.org/10.1371/journal.pone.0067025
Salavati H, Soltani M, Amanpour S (2018) The pivotal role of angiogenesis in a multi-scale modeling of tumor growth exhibiting the avascular and vascular phases. Microvasc Res 119:105–116. https://doi.org/10.1016/j.mvr.2018.05.001
Hanahan D, Weinberg RA (2000) The hallmarks of cancer. Cell 100:57–70. https://doi.org/10.1016/S0092-8674(00)81683-9
Santos J, Monteagudo Á (2012) Study of cancer hallmarks relevance using a cellular automaton tumor growth model. In: Coello CAC, Cutello V, Deb K et al (eds) Parallel problem solving from nature: PPSN XII. Springer, Berlin, pp 489–499
Butler J, Mackay F, Denniston C, Daley M (2016) Halting the hallmarks: a cellular automaton model of early cancer growth inhibition. Nat Comput 15:15–30. https://doi.org/10.1007/s11047-015-9508-3
Gödde R, Kurz H (2001) Structural and biophysical simulation of angiogenesis and vascular remodeling. Dev Dyn 220:387–401. https://doi.org/10.1002/dvdy.1118
Fredrich T, Welter M, Rieger H (2017) Tumorcode - A framework to simulate vascularized tumors. https://doi.org/10.1101/216903
Cytowski M, Szymanska Z (2015) Large-scale parallel simulations of 3D cell colony dynamics: the cellular environment. Comput Sci Eng 17:44–48. https://doi.org/10.1109/MCSE.2015.66
Cytowski M, Szymanska Z (2014) Large-scale parallel simulations of 3D cell colony dynamics. Comput Sci Eng 16:86–95. https://doi.org/10.1109/MCSE.2014.2
Cytowski M (2014) Large scale computational modelling of cellular biosystems
Izaguirre JA, Chaturvedi R, Huang C et al (2004) COMPUCELL, a multi-model framework for simulation of morphogenesis. Bioinformatics 20:1129–1137. https://doi.org/10.1093/bioinformatics/bth050
Cickovski TM, Chaturvedi and R, Glimm T, et al (2005) A framework for three-dimensional simulation of morphogenesis. IEEE/ACM Trans Comput Biol Bioinf 2:273–288. https://doi.org/10.1109/TCBB.2005.46
Shirinifard A, Gens JS, Zaitlen BL et al (2009) 3D multi-cell simulation of tumor growth and angiogenesis. PLoS ONE 4:e7190. https://doi.org/10.1371/journal.pone.0007190
Jeanquartier F, Jean-Quartier C, Cemernek D, Holzinger A (2016) In silico modeling for tumor growth visualization. BMC Syst Biol 10:59. https://doi.org/10.1186/s12918-016-0318-8
Mirams GR, Arthurs CJ, Bernabeu MO et al (2013) Chaste: an open source C++ library for computational physiology and biology. PLoS Comput Biol 9:e1002970. https://doi.org/10.1371/journal.pcbi.1002970
Tanaka S, Sichau D, Iber D (2015) LBIBCell: a cell-based simulation environment for morphogenetic problems. Bioinformatics 31:2340–2347. https://doi.org/10.1093/bioinformatics/btv147
Lee H, Smelser A, Low J et al (2017) Mechanical properties of normal breast cells and metastatic cancer cells in co-culture. Biophys J 112:124a. https://doi.org/10.1016/j.bpj.2016.11.693
Yu H, Mouw JK, Weaver VM (2011) Forcing form and function: biomechanical regulation of tumor evolution. Trends Cell Biol 21:47–56. https://doi.org/10.1016/j.tcb.2010.08.015
Suresh S (2007) Biomechanics and biophysics of cancer cells. Acta Biomater 3:413–438. https://doi.org/10.1016/j.actbio.2007.04.002
Cross SE, Jin Y-S, Tondre J et al (2008) AFM-based analysis of human metastatic cancer cells. Nanotechnology 19:384003. https://doi.org/10.1088/0957-4484/19/38/384003
Thiery JP, Sleeman JP (2006) Complex networks orchestrate epithelial–mesenchymal transitions. Nat Rev Mol Cell Biol 7:131–142. https://doi.org/10.1038/nrm1835
Rejniak KA (2005) A single-cell approach in modeling the dynamics of tumor microregions. Math Biosci Eng 2:643–655
Rejniak KA, Wang SE, Bryce NS et al (2010) Linking changes in epithelial morphogenesis to cancer mutations using computational modeling. PLoS Comput Biol 6:e1000900. https://doi.org/10.1371/journal.pcbi.1000900
Rejniak KA (2016) Circulating tumor cells: when a solid tumor meets a fluid microenvironment. In: Rejniak KA (ed) Systems biology of tumor microenvironment. Springer, Cham, pp 93–106
Moraru II, Schaff JC, Slepchenko BM et al (2008) Virtual cell modelling and simulation software environment. IET Syst Biol 2:352–362. https://doi.org/10.1049/iet-syb:20080102
Slepchenko BM, Schaff JC, Macara I, Loew LM (2003) Quantitative cell biology with the Virtual Cellq. 7
Akhurst RJ, Derynck R (2001) TGF-β signaling in cancer: a double-edged sword. Trends Cell Biol 11:S44–S51. https://doi.org/10.1016/S0962-8924(01)02130-4
Price JT, Wilson HM, Haites NE (1996) Epidermal growth factor (EGF) Increases the in vitro invasion, motility and adhesion interactions of the primary renal carcinoma cell line, A704. Eur J Cancer 32:1977–1982. https://doi.org/10.1016/0959-8049(96)00207-9
Wang Z, Birch CM, Sagotsky J, Deisboeck TS (2009) Cross-scale, cross-pathway evaluation using an agent-based non-small cell lung cancer model. Bioinformatics 25:2389–2396. https://doi.org/10.1093/bioinformatics/btp416
Wang Z, Zhang L, Sagotsky J, Deisboeck TS (2007) Simulating non-small cell lung cancer with a multiscale agent-based model. Theo Biol Med Model 4:50. https://doi.org/10.1186/1742-4682-4-50
Lee P, Wolgemuth CW (2016) Physical mechanisms of cancer in the transition to metastasis. Biophys J 111:256–266. https://doi.org/10.1016/j.bpj.2016.05.046
Brodland GW, Veldhuis JH (2012) The mechanics of metastasis: insights from a computational model. PLoS ONE 7:e44281. https://doi.org/10.1371/journal.pone.0044281
Viens D, Brodland GW (2007) A three-dimensional finite element model for the mechanics of cell-cell interactions. J Biomech Eng 129:651–657. https://doi.org/10.1115/1.2768375
Ramis-Conde I, Chaplain MAJ, Anderson ARA, Drasdo D (2009) Multi-scale modelling of cancer cell intravasation: the role of cadherins in metastasis. Phys Biol 6:016008. https://doi.org/10.1088/1478-3975/6/1/016008
Mitchell MJ, King MR (2013) Computational and experimental models of cancer cell response to fluid shear stress. Front Oncol. https://doi.org/10.3389/fonc.2013.00044
King MR, Phillips KG, Mitrugno A et al (2015) A physical sciences network characterization of circulating tumor cell aggregate transport. Am J Physiol-Cell Physiol 308:C792–C802. https://doi.org/10.1152/ajpcell.00346.2014
Xiao LL, Liu Y, Chen S, Fu BM (2017) Effects of flowing RBCs on adhesion of a circulating tumor cell in microvessels. Biomech Model Mechanobiol 16:597–610. https://doi.org/10.1007/s10237-016-0839-5
Takeishi N, Imai Y, Yamaguchi T, Ishikawa T (2015) Flow of a circulating tumor cell and red blood cells in microvessels. Phys Rev E. https://doi.org/10.1103/PhysRevE.92.063011
Rejniak KA (2012) Investigating dynamical deformations of tumor cells in circulation: predictions from a theoretical model. Front Oncol. https://doi.org/10.3389/fonc.2012.00111
Cross SE, Jin Y-S, Rao J, Gimzewski JK (2007) Nanomechanical analysis of cells from cancer patients. Nat Nanotechnol 2:780–783. https://doi.org/10.1038/nnano.2007.388
Liu Y, Liu WK (2006) Rheology of red blood cell aggregation by computer simulation. J Comput Phys 220:139–154. https://doi.org/10.1016/j.jcp.2006.05.010
Zhang L, Gerstenberger A, Wang X, Liu WK (2004) Immersed finite element method. Comput Methods Appl Mech Eng 193:2051–2067. https://doi.org/10.1016/j.cma.2003.12.044
Zhang LT, Gay M (2007) Immersed finite element method for fluid-structure interactions. J Fluids Struct 23:839–857. https://doi.org/10.1016/j.jfluidstructs.2007.01.001
Morse PM (1929) Diatomic molecules according to the wave mechanics. ii. vibrational levels. Phys Rev 34:57–64. https://doi.org/10.1103/PhysRev.34.57
Slater NB (1957) Classical motion under a morse potential. Nature 180:1352. https://doi.org/10.1038/1801352a0
Gusenbauer M, Cimrak I, Bance S, et al A tunable cancer cell filter using magnetic beads: cellular and fluid dynamic simulations. 11
Rejniak KA (2007) An immersed boundary framework for modelling the growth of individual cells: an application to the early tumour development. J Theor Biol 247:186–204. https://doi.org/10.1016/j.jtbi.2007.02.019
Reymond N, d’Água BB, Ridley AJ (2013) Crossing the endothelial barrier during metastasis. Nat Rev Cancer 13:858–870. https://doi.org/10.1038/nrc3628
Paget S (1889) The distribution of secondary growths in cancer of the breast. The Lancet 133:571–573. https://doi.org/10.1016/S0140-6736(00)49915-0
Ewing J (1919) Neoplastic diseases, A Text-book On Tumors. W.B. Saunders Company, London
Fidler IJ (2003) The pathogenesis of cancer metastasis: the “seed and soil” hypothesis revisited. Nat Rev Cancer 3:453–458. https://doi.org/10.1038/nrc1098
Coman DR, Delong RP. Studies on the mechanisms of metastasis. The Distribution of Tumors in Various Organs in Relation to the Distribution of Arterial Emboli. 8
Weiss L (1992) Comments on hematogenous metastatic patterns in humans as revealed by autopsy. Clin Exp Metast 10:191–199. https://doi.org/10.1007/BF00132751
Dembo M, Bell GI (1987) The thermodynamics of cell adhesion. In: Bronner F, Klausner RD, Kempf C, van Renswoude J (eds) Current topics in membranes and transport. Academic Press, Cambridge, pp 71–89
Bell G (1978) Models for the specific adhesion of cells to cells. Science 200:618–627. https://doi.org/10.1126/science.347575
Bell GI, Dembo M, Bongrand P (1984) Cell adhesion. Competition between nonspecific repulsion and specific bonding. Biophys J 45:1051–1064. https://doi.org/10.1016/S0006-3495(84)84252-6
Hammer DA, Apte SM (1992) Simulation of cell rolling and adhesion on surfaces in shear flow: general results and analysis of selectin-mediated neutrophil adhesion. Biophys J 63:35–57
Yan WW, Liu Y, Fu BM (2010) Effects of curvature and cell–cell interaction on cell adhesion in microvessels. Biomech Model Mechanobiol 9:629–640. https://doi.org/10.1007/s10237-010-0202-1
Xiao LL, Yan WW, Liu Y et al (2018) Modeling cell adhesion and extravasation in microvascular system. In: Fu BM, Wright NT (eds) Molecular, cellular, and tissue engineering of the vascular system. Springer, Cham, pp 219–234
Yan WW, Cai B, Liu Y, Fu BM (2012) Effects of wall shear stress and its gradient on tumor cell adhesion in curved microvessels. Biomech Model Mechanobiol 11:641–653. https://doi.org/10.1007/s10237-011-0339-6
Takeishi N, Imai Y, Ishida S et al (2016) Cell adhesion during bullet motion in capillaries. Am J Physiol-Heart Circul Physiol 311:H395–H403. https://doi.org/10.1152/ajpheart.00241.2016
Follain G, Osmani N, Azevedo AS et al (2018) Hemodynamic forces tune the arrest, adhesion, and extravasation of circulating tumor cells. Dev Cell 45:33-52.e12. https://doi.org/10.1016/j.devcel.2018.02.015
Angio TK (2016). In: Cemosis. http://www.cemosis.fr/projects/angiotk/. Accessed 5 Apr 2019
Hou J-M, Krebs M, Ward T et al (2011) Circulating tumor cells as a window on metastasis biology in lung cancer. Am J Pathol 178:989–996. https://doi.org/10.1016/j.ajpath.2010.12.003
Anderson KJ, de Guillebon A, Hughes AD et al (2017) Effect of circulating tumor cell aggregate configuration on hemodynamic transport and wall contact. Math Biosci 294:181–194. https://doi.org/10.1016/j.mbs.2017.10.002
Fidler IJ, Gersten DM, Riggs CW (1977) Relationship of host immune status to tumor cell arrest, distribution, and survival in experimental metastasis. Cancer 40:46–55. https://doi.org/10.1002/1097-0142(197707)40:1%3c46::AID-CNCR2820400110%3e3.0.CO;2-T
Giuliano M, Shaikh A, Lo HC et al (2018) Perspective on circulating tumor cell clusters: why it takes a village to metastasize. Cancer Res 78:845–852. https://doi.org/10.1158/0008-5472.CAN-17-2748
Friedl P, Locker J, Sahai E, Segall JE (2012) Classifying collective cancer cell invasion. Nat Cell Biol 14:777–783. https://doi.org/10.1038/ncb2548
Phillips KG, Lee AM, Tormoen GW et al (2015) The thrombotic potential of circulating tumor microemboli: computational modeling of circulating tumor cell-induced coagulation. Am J Physiol-Cell Physiol 308:C229–C236. https://doi.org/10.1152/ajpcell.00315.2014
Guo P, Cai B, Lei M et al (2014) Differential arrest and adhesion of tumor cells and microbeads in the microvasculature. Biomech Model Mechanobiol 13:537–550. https://doi.org/10.1007/s10237-013-0515-y
Gomez-Garcia MJ, Doiron AL, Steele RRM et al (2018) Nanoparticle localization in blood vessels: dependence on fluid shear stress, flow disturbances, and flow-induced changes in endothelial physiology. Nanoscale 10:15249–15261. https://doi.org/10.1039/C8NR03440K
Groot RD, Warren PB (1997) Dissipative particle dynamics: Bridging the gap between atomistic and mesoscopic simulation. J Chem Phys 107:4423–4435. https://doi.org/10.1063/1.474784
Li Y, Lian Y, Zhang LT et al (2016) Cell and nanoparticle transport in tumour microvasculature: the role of size, shape and surface functionality of nanoparticles. Interface Focus. https://doi.org/10.1098/rsfs.2015.0086
Xiao LL, Liu Y, Chen S, Fu BM (2016) Numerical simulation of a single cell passing through a narrow slit. Biomech Model Mechanobiol 15:1655–1667. https://doi.org/10.1007/s10237-016-0789-y
Yingling M, O’Neill T, Skalak TC, Peirce-Cottler S (2005) A cellular automata model of circulating cell adhesion and transmigration in the microvaculature. In: 2005 IEEE design symposium, systems and information engineering, pp 356–361
NetLogo Home Page. https://ccl.northwestern.edu/netlogo/. Accessed 19 Apr 2019
Cao X, Moeendarbary E, Isermann P et al (2016) A chemomechanical model for nuclear morphology and stresses during cell transendothelial migration. Biophys J 111:1541–1552. https://doi.org/10.1016/j.bpj.2016.08.011
Chen LL, Blumm N, Christakis NA et al (2009) Cancer metastasis networks and the prediction of progression patterns. Br J Cancer 101:749–758. https://doi.org/10.1038/sj.bjc.6605214
Anderson ARA (2005) A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion. Math Med Biol 22:163–186. https://doi.org/10.1093/imammb/dqi005
Chaplain M, a. J, Lolas G, (2005) Mathematical modelling of cancer cell invasion of tissue: the role of the urokinase plasminogen activation system. Math Models Methods Appl Sci 15:1685–1734. https://doi.org/10.1142/S0218202505000947
Bitsouni V, Trucu D, Chaplain MAJ, Eftimie R (2018) Aggregation and travelling wave dynamics in a two-population model of cancer cell growth and invasion. Math Med Biol J IMA. https://doi.org/10.1093/imammb/dqx019
Annila A, Annila E (2008) Why did life emerge? Int J Astrobiol 7:293–300. https://doi.org/10.1017/S1473550408004308
Lucia U (2013) Thermodynamics and cancer stationary states. Phys A 392:3648–3653. https://doi.org/10.1016/j.physa.2013.04.033
Lucia U (2014) Transport processes in biological systems: Tumoral cells and human brain. Phys A 393:327–336. https://doi.org/10.1016/j.physa.2013.08.066
Lucia U (2013) Different chemical reaction times between normal and solid cancer cells. Med Hypotheses 81:58–61. https://doi.org/10.1016/j.mehy.2013.04.007
Luo L (2009) Entropy production in a cell and reversal of entropy flow as an anticancer therapy. Front Phys China 4:122. https://doi.org/10.1007/s11467-009-0007-9
Kam Y, Rejniak KA, Anderson ARA (2012) Cellular modeling of cancer invasion: Integration of in silico and in vitro approaches. J Cell Physiol 227:431–438. https://doi.org/10.1002/jcp.22766
Koch TM, Münster S, Bonakdar N et al (2012) 3D traction forces in cancer cell invasion. PLoS ONE 7:e33476. https://doi.org/10.1371/journal.pone.0033476
Magjarevic R, Fabry B, Koch TM et al (2009) Contractile forces during cancer cell invasion. In: Dössel O, Schlegel WC (eds) World congress on medical physics and biomedical engineering, September 7–12, 2009, Munich, Germany. Springer, Berlin, pp 85–86
Ghaffarizadeh A, Friedman SH, Macklin P (2016) BioFVM: an efficient, parallelized diffusive transport solver for 3-D biological simulations. Bioinformatics 32:1256–1258. https://doi.org/10.1093/bioinformatics/btv730
Ghaffarizadeh A, Heiland R, Friedman SH et al (2018) PhysiCell: An open source physics-based cell simulator for 3-D multicellular systems. PLoS Comput Biol 14:e1005991. https://doi.org/10.1371/journal.pcbi.1005991
Stoll G, Caron B, Viara E et al (2017) MaBoSS 2.0: an environment for stochastic Boolean modeling. Bioinformatics 33:2226–2228. https://doi.org/10.1093/bioinformatics/btx123
Stoll G, Viara E, Barillot E, Calzone L (2012) Continuous time boolean modeling for biological signaling: application of Gillespie algorithm. BMC Syst Biol 6:116. https://doi.org/10.1186/1752-0509-6-116
Letort G, Montagud A, Stoll G et al (2019) PhysiBoSS: a multi-scale agent-based modelling framework integrating physical dimension and cell signalling. Bioinformatics. https://doi.org/10.1093/bioinformatics/bty766
Madsen CD, Sahai E (2010) Cancer dissemination—lessons from leukocytes. Dev Cell 19:13–26. https://doi.org/10.1016/j.devcel.2010.06.013
Carey SP, Rahman A, Kraning-Rush CM et al (2015) Comparative mechanisms of cancer cell migration through 3D matrix and physiological microtracks. Am J Physiol-Cell Physiol 308:C436–C447. https://doi.org/10.1152/ajpcell.00225.2014
Zaman MH, Kamm RD, Matsudaira P, Lauffenburger DA (2005) Computational model for cell migration in three-dimensional matrices. Biophys J 89:1389–1397. https://doi.org/10.1529/biophysj.105.060723
Friedl P, Wolf K (2003) Tumour-cell invasion and migration: diversity and escape mechanisms. Nat Rev Cancer 3:362–374. https://doi.org/10.1038/nrc1075
Taking Aim at Moving Targets in Computational Cell Migration | Elsevier Enhanced Reader. https://reader.elsevier.com/reader/sd/pii/S0962892415001658?token=31BC4C2F2A756E07ABB6313B4E5A7F300A5A4FCF5373C5D9DB85D496DF5798FC5DA0305A7DA23BBFABBD865F71934984. Accessed 5 Apr 2019
Iwata K, Kawasaki K, Shigesada N (2000) A dynamical model for the growth and size distribution of multiple metastatic tumors. J Theor Biol 203:177–186. https://doi.org/10.1006/jtbi.2000.1075
Baratchart E, Benzekry S, Bikfalvi A et al (2015) Computational modelling of metastasis development in renal cell carcinoma. PLoS Comput Biol 11:e1004626. https://doi.org/10.1371/journal.pcbi.1004626
Audigier C, Mansi T, Delingette H et al (2015) Efficient lattice boltzmann solver for patient-specific radiofrequency ablation of hepatic tumors. IEEE Trans Med Imaging 34:1576–1589. https://doi.org/10.1109/TMI.2015.2406575
Karpatkin S (1981) Role of platelets in tumor cell metastases. Ann Intern Med 95:6
Buettner R (2018) Platelets promoting tumor metastasis: Culprits or victims? J Thorac Dis 10:550–553. https://doi.org/10.21037/jtd.2017.12.24
Leblanc R, Peyruchaud O (2016) Metastasis: new functional implications of platelets and megakaryocytes. Blood 128:24–31. https://doi.org/10.1182/blood-2016-01-636399
Mahalingam M, Ugen KE, Kao K-J, Klein PA (1988) Functional role of platelets in experimental metastasis studied with cloned murine fibrosarcoma cell variants. Cancer Res 48:1460–1464
Zaman MH (2013) The role of engineering approaches in analysing cancer invasion and metastasis. Nat Rev Cancer 13:596–603. https://doi.org/10.1038/nrc3564
Lowe CP (1999) An alternative approach to dissipative particle dynamics. EPL 47:145. https://doi.org/10.1209/epl/i1999-00365-x
August DA, Sugarbaker PH, Schneider PD (1985) Lymphatic dissemination of hepatic metastases. Implications for the follow-up and treatment of patients with colorectal cancer. Cancer 55:1490–1494
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This study was supported by a grant from the Centre for Quantitative Analysis and Modelling of The Fields Institute for Research in Mathematical Sciences.
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Anvari, S., Nambiar, S., Pang, J. et al. Computational Models and Simulations of Cancer Metastasis. Arch Computat Methods Eng 28, 4837–4859 (2021). https://doi.org/10.1007/s11831-021-09554-1
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DOI: https://doi.org/10.1007/s11831-021-09554-1