Skip to main content
SpringerLink
Log in

From single cells to tissue architecture—a bottom-up approach to modelling the spatio-temporal organisation of complex multi-cellular systems

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

Collective phenomena in multi-cellular assemblies can be approached on different levels of complexity. Here, we discuss a number of mathematical models which consider the dynamics of each individual cell, so-called agent-based or individual-based models (IBMs). As a special feature, these models allow to account for intracellular decision processes which are triggered by biomechanical cell–cell or cell–matrix interactions. We discuss their impact on the growth and homeostasis of multi-cellular systems as simulated by lattice-free models. Our results demonstrate that cell polarisation subsequent to cell–cell contact formation can be a source of stability in epithelial monolayers. Stroma contact-dependent regulation of tumour cell proliferation and migration is shown to result in invasion dynamics in accordance with the migrating cancer stem cell hypothesis. However, we demonstrate that different regulation mechanisms can equally well comply with present experimental results. Thus, we suggest a panel of experimental studies for the in-depth validation of the model assumptions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ingber, D.E.: Cellular mechanotransduction: putting all the pieces together again. FASEB J. 20(7), 811–27 (2006)

    Article  Google Scholar 

  2. Chen, C.S., Mrksich, M., Huang, S., Whitesides, G.M., Ingber, D.E.: Geometric control of cell life and death. Science 276, 1425–1428 (1997)

    Article  Google Scholar 

  3. Gloushankova, N.A., Alieva, N.A., Krendel, M.F., Bonder, E.M., Feder, H.H., Vasiliev, J.M., Gelfand, I.M.: Cell–cell contact changes the dynamics of lamellar activity in nontransformed epitheliocytes but not in their ras-transformed descendants. Proc. Natl. Acad. Sci. USA 94, 879–883 (1997)

    Article  Google Scholar 

  4. Grant, M.R., Mostov, K.E., Tlsty, T.D., Hunt, C.A.: Simulating properties of in vitro epithelial cell morphogenesis. PLoS Comput. Biol. 6; 2(10), e129 (2006)

  5. Meineke, F.A., Potten, C.S., Loeffler, M.: Cell migration and organisation in the intestinal crypt using a lattice-free model. Cell. Prolif. 34, 253–266 (2001)

    Article  Google Scholar 

  6. Hogeweg, P.: Evolving mechanisms of morphogenesis: on the interplay between differential adhesion and cell differentiation. J. Theor. Biol. 203, 317–333 (2000)

    Article  Google Scholar 

  7. Bauer, A.L., Jackson, T.L., Jiang, Y.: A cell-based model exhibiting branching and anastomosis during tumor-induced angiogenesis. Biophys. J. 92(9), 3105–3121 (2007)

    Article  Google Scholar 

  8. Anderson, A.R.: A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion. Math. Med. Biol. 22(2), 163–186 (2005)

    Article  MATH  Google Scholar 

  9. Dormann, S., Deutsch, A.: Modelling of self-organized avascular tumor growth with a hybrid cellular ’automaton. In Silico Biol. 2(3), 393–406 (2002)

    Google Scholar 

  10. Graner, F., Glazier, J.A.: Simulation of biological cell sorting using a two-dimensional extended Potts model. Phys. Rev. Lett. 69(13), 2013–2016 (1993)

    Article  Google Scholar 

  11. Loeffler, M., Potten, C.S., Wichmann, H.E.: Epidermal cell proliferation. Virchows Arch. B 53, 286–300 (1987)

    Article  Google Scholar 

  12. Schaller, G., Meyer-Hermann, M.: Multicellular tumor spheroid in an lattice-free Voronoi-Delaunay cell model. Phys. Rev. E 71, 051910 (2005)

    Article  MathSciNet  Google Scholar 

  13. Honda, H., Tanemura, M., Nagai, T.: A three-dimensional vertex dynamics cell model of space-filling polyhedra simulating cell behavior in a cell aggregate. J. Theor. Biol. 226(4), 439–453 (2004)

    Article  MathSciNet  Google Scholar 

  14. Brodland, G.W., Veldhuis, J.H.: Computer simulations of mitosis and interdependencies between orientation, cell shape and epithelia reshaping. J. Biomech. 35, 673–681 (2002)

    Article  Google Scholar 

  15. Galle, J., Loeffler, M., Drasdo, D.: Modelling the effect of deregulated proliferation and apoptosis on the growth dynamics of epithelial cell populations in vitro. Biophys. J. 88(1), 62–75 (2005)

    Article  Google Scholar 

  16. Dallon, J., Othmer, H.G.: How cellular movement determines the collective force generated by the Dictyostelium discoideum slug. J. Theor. Biol. 231(2), 203–222 (2004)

    Article  MathSciNet  Google Scholar 

  17. Drasdo, D., Höhme, S.: A single-cell-based model of tumor growth in vitro: monolayers and spheroids. Phys. Biol. 2, 133–147 (2005)

    Article  Google Scholar 

  18. Kreft, J.U., Picioreanu, C., Wimpenny, J.W., van Loosdrecht, M.C.: Individual-based modelling of biofilms. Microbiology 147, 2897–2912 (2001)

    Google Scholar 

  19. Palsson, E., Othmer, H.G.: A model for individual and collective cell movement in Dictyostelium discoideum. Proc. Natl. Acad. Sci. USA 97(19), 10448–10453 (2000)

    Article  Google Scholar 

  20. Drasdo, D., Kree, R., McCaskill, J.S.: Monte Carlo approach to tissue-cell populations. Phys. Rev. E 52(6), 6635–6657 (1995)

    Article  Google Scholar 

  21. Schaller, G., Meyer-Hermann, M.: A modelling approach towards epidermal homoeostasis control. J. Theor. Biol. 247(3), 554–573 (2007)

    Article  MathSciNet  Google Scholar 

  22. Hoehme, S., Hengstler, J.G., Brulport, M., Schafer, M., Bauer, A., Gebhardt, R., Drasdo, D.: Mathematical modelling of liver regeneration after intoxication with CCl(4). Chem. Biol. Interact. 20 168(1), 74–93 (2007)

    Article  Google Scholar 

  23. Beyer, T., Meyer-Hermann, M.: Modeling emergent tissue organization involving high-speed migrating cells in a flow equilibrium. Phys. Rev. E: Stat. Nonlin. Soft Matter Phys. 76(2), 021929 (2007)

    Google Scholar 

  24. Galle, J., Sittig, D., Hanisch, I., Wobus, M., Wandel, E., Loeffler, M., Aust, G.: Individual cell-based models of tumor–environment interactions: multiple effects of CD97 on tumor invasion. Am. J. Pathol. 169(5), 1802–1811 (2006)

    Article  Google Scholar 

  25. Engler, A., Bacakova, L., Newman, C., Hategan, A., Griffin, M., Discher, D.: Substrate compliance versus ligand density in cell on gel responses. Biophys. J. 86, 617–628 (2004)

    Article  Google Scholar 

  26. Friedl, P.: Prespecification and plasticity: shifting mechanisms of cell migration. Curr. Opin. Cell Biol. 16(1), 14–23 (2004)

    Article  Google Scholar 

  27. Sahai, E., Marshall, C.J.: Differing modes of tumor cell invasion have distinct requirements for Rho/ROCK signalling and extracellular proteolysis. Nat. Cell Biol. 5(8), 711–719 (2003)

    Article  Google Scholar 

  28. Grunert, S., Jechlinger, M., Beug, H.: Diverse cellular and molecular mechanisms contribute to epithelial plasticity and metastasis. Nat. Rev. Mol. Cell Biol. 4(8), 657–665 (2003)

    Article  Google Scholar 

  29. Hahn, G.M.: State vector description of the proliferation of mammalian cells in tissue culture. I. Exponential growth. Biophys. J. 6(3), 275–290 (1966)

    Article  Google Scholar 

  30. Ingber, D.E.: Tensegrity I. Cell structure and hierarchical systems biology. J. Cell Sci. 116, 1157–173 and Tensegrity II. How structural networks influence cellular information processing networks. 1397–408 (2003)

    Google Scholar 

  31. Mahaffy, R.E., Park, S., Gerde, E., Kas, J., Shih, C.K.: Quantitative analysis of the viscoelastic properties of thin regions of fibroblasts using atomic force microscopy. Biophys. J. 86(3), 1777–1793 (2004)

    Article  Google Scholar 

  32. Charras, G.T., Horton, M.A.: Determination of cellular strains by combined atomic force microscopy and finite element modelling. Biophys. J. 83, 858–879 (2002)

    Article  Google Scholar 

  33. Schwarz, U.S., Balaban, N.Q., Riveline, D., Bershadsky, A., Geiger, B., Safran, S.A.: Calculation of forces at focal adhesions from elastic substrate data: the effect of localized force and the need for regularization. Biophys. J. 83, 1380–1394 (2002)

    Article  Google Scholar 

  34. Guck, J., Ananthakrishnan, R., Mahmood, H., Moon, T.J., Cunningham, C.C., Kas, J.: The optical stretcher: a novel laser tool to micromanipulate cells. Biophys. J. 81(2), 767–784 (2001)

    Article  Google Scholar 

  35. Benoit, M., Gabriel, D., Gerisch, G., Gaub, H.E.: Discrete interactions in cell adhesion measured by single-molecule force spectroscopy. Nat. Cell Biol. 2, 313–317 (2000)

    Article  Google Scholar 

  36. Benoit, M., Gaub, H.E.: Measuring cell adhesion forces with the atomic force microscope at the molecular level. Cells Tissues Organs 172(3), 174–189 (2002)

    Article  Google Scholar 

  37. Beysens, D.A., Forgacs, G., Glazier, J.A.: Cell sorting is analogous to phase ordering in fluids. Proc. Natl. Acad. Sci. USA 97(17), 9467–9471 (2000)

    Article  Google Scholar 

  38. Drasdo, D., Hoehme, S., Block, M.: On the role of physics in the growth and pattern formation of multi-cellular systems: what can we learn from individual-cell based models?. J. Stat. Phys. 128, 287–345 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  39. Galle, J., Aust, G., Schaller, G., Beyer, T., Drasdo, D.: Individual cell-based models of the spatial-temporal organization of multicellular systems—achievements and limitations. Cytometry A. 69(7), 704–710 (2006)

    Google Scholar 

  40. Aplin, A.E., Howe, A.K., Juliano, R.L.: Cell adhesion molecules, signal transduction and cell growth. Curr. Opin. Cell Biol. 11, 737–744 (1999)

    Article  Google Scholar 

  41. Mollard, R., Dziadek, M.: A correlation between epithelial proliferation rates, basement membrane component localization patterns, and morphogenetic potential in the embryonic mouse lung. Am. J. Respir. Cell Mol. Biol. 19(1), 71–82 (1998)

    Google Scholar 

  42. Pullan, S., Wilson, J., Metcalfe, A., Edwards, G.M., Goberdhan, N., Tilly, J., Hickman, J.A., Dive, C., Streuli, C.H.: Requirement of basement membrane for the suppression of programmed cell death in mammary epithelium. J. Cell Sci. 109, 631–642 (1996)

    Google Scholar 

  43. Pirone, D.M., Liu, W.F., Ruiz, S.A., Gao, L., Raghavan, S., Lemmon, C.A., Romer, L.H., Chen, C.S.: An inhibitory role for FAK in regulating proliferation: a link between limited adhesion and RhoA-ROCK signaling. J. Cell Biol. 174(2), 277–288 (2006)

    Article  Google Scholar 

  44. DeMali, K.A., Wennerberg, K., Burridge, K.: Integrin signaling to the actin cytoskeleton. Curr. Opin. Cell Biol. 15(5), 572–582 (2003)

    Article  Google Scholar 

  45. Brabletz, T., Spaderna, S., Kolb, J., Hlubek, F., Faller, G., Bruns, C.J., Jung, A., Nentwich, J., Duluc, I., Domon-Dell, C., Kirchner, T., Freund, J.N.: Down-regulation of the homeodomain factor Cdx2 in colorectal cancer by collagen type I: an active role for the tumor environment in malignant tumor progression. Cancer Res. 64(19), 6973–6977 (2004)

    Article  Google Scholar 

  46. Brabletz, T., Jung, A., Reu, S., Porzner, M., Hlubek, F., Kunz-Schughart, L.A., Knuechel, R., Kirchner, T.: Variable beta-catenin expression in colorectal cancers indicates tumor progression driven by the tumor environment. Proc. Natl. Acad. Sci. USA 98(18), 10356–10361 (2001)

    Article  Google Scholar 

  47. Bru, A., Albertos, S., Luis Subiza, J., Garcia-Asenjo, J.L., Bru, I.: The universal dynamics of tumor growth. Biophys. J. 85(5), 2948–2961 (2003)

    Article  Google Scholar 

  48. Helmlinger, G., Netti, P.A., Lichtenbeld, H.C., Melder, R.J., Jain, R.K.: Solid stress inhibits the growth of multicellular tumor spheroids. Nat. Biotechnol. 15(8), 778–783 (1997)

    Article  Google Scholar 

  49. Long, H., Han, H., Yang, B., Wang, Z.: Opposite cell density-dependence between spontaneous and oxidative stress-induced apoptosis in mouse fibroblast L-cells. Cell Physiol. Biochem. 13(6), 401–414 (2003)

    Article  Google Scholar 

  50. Fiore, M., Degrassi, F.: Dimethyl sulfoxide restores contact inhibition-induced growth arrest and inhibits cell density-dependent apoptosis in hamster cells. Exp. Cell Res. 251(1), 102–110 (1999)

    Article  Google Scholar 

  51. Maniotis, A.J., Chen, C.S., Ingber, D.E.: Demonstration of mechanical connections between integrins, cytoskeletal filaments, and nucleoplasm that stabilize nuclear structure. Proc. Natl. Acad. Sci. USA 94(3), 849–854 (1997)

    Article  Google Scholar 

  52. McBeath, R., Pirone, D.M., Nelson, C.M., Bhadriraju, K., Chen, C.S.: Cell shape, cytoskeletal tension, and RhoA regulate stem cell lineage commitment. Dev. Cell 6(4), 483–495 (2004)

    Article  Google Scholar 

  53. Nelson, C.M., Jean, R.P., Tan, J.L., Liu, W.F., Sniadecki, N.J., Spector, A.A., Chen, C.S.: Emergent patterns of growth controlled by multicellular form and mechanics. Proc. Natl. Acad. Sci. U.S.A 102(33), 11594–11599 (2005)

    Article  Google Scholar 

  54. Benaud, C.M., Dickson, R.B.: Adhesion-regulated G1 cell cycle arrest in epithelial cells requires the downregulation of c-Myc. Oncogene 20(33), 4554–4567 (2001)

    Article  Google Scholar 

  55. Ren, X.D., Kiosses, W.B., Sieg, D.J., Otey, C.A., Schlaepfer, D.D., Schwartz, M.A.: Focal adhesion kinase suppresses Rho activity to promote focal adhesion turnover. J. Cell Sci. 113, 3673–3678 (2000)

    Google Scholar 

  56. Steinert, M., Wobus, M., Boltze, C., Schutz, A., Wahlbuhl, M., Hamann, J., Aust, G.: Expression and regulation of CD97 in colorectal carcinoma cell lines and tumor tissues. Am. J. Pathol. 161(5), 1657–1667 (2002)

    Google Scholar 

  57. Friedl, P., Wolf, K.: Tumor-cell invasion and migration: diversity and escape mechanisms. Nat. Rev. Cancer 3(5), 362–374 (2003)

    Article  Google Scholar 

  58. Wang, W., Goswami, S., Sahai, E., Wyckoff, J.B., Segall, J.E., Condeelis, JS.: Tumor cells caught in the act of invading: their strategy for enhanced cell motility. Trends Cell Biol. 15(3), 138–145 (2005)

    Article  Google Scholar 

  59. Chaplain, M.A., Graziano, L., Preziosi, L.: Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour development. Math. Med. Biol. 23(3), 197–229 (2006)

    Article  MATH  Google Scholar 

  60. Thiery, J.P.: Epithelial–mesenchymal transitions in tumor progression. Nat. Rev. Cancer 2(6), 442–454 (2002)

    Article  Google Scholar 

  61. Brabletz, T., Jung, A., Spaderna, S., Hlubek, F., Kirchner, T.: Opinion: migrating cancer stem cells—an integrated concept of malignant tumour progression. Nat. Rev. Cancer 5(9), 744–749 (2005)

    Article  Google Scholar 

  62. Loeffler, M., Potten, C.S.: Stem cells and cellular pedigrees—a conceptual introduction. In: Potten, C.S.(eds) Stem Cells, pp. 1–27. Academic Press, London (1997) doi:10.1016/B978-012563455-7/50002-7

    Chapter  Google Scholar 

  63. Quesenberry, P.J., Abedi, M., Aliotta, J., Colvin, G., Demers, D., Dooner, M., Greer, D., Hebert, H., Menon, M.K., Pimentel, J., Paggioli, D.: Stem cell plasticity: an overview. Blood Cells Mol. Dis. 32(1), 1–4 (2004)

    Article  Google Scholar 

  64. Wagers, A.J., Weissman, I.L.: Plasticity of adult stem cells. Cell 116(5), 639–48 (2004)

    Article  Google Scholar 

  65. Loeffler, M., Roeder, I.: Tissue stem cells: definition, plasticity, heterogeneity, self organization and models—a conceptual approach. Cells Tissues Organs 171(1), 8–26 (2002)

    Article  Google Scholar 

  66. Kirkland, M.A.: A phase space model of hemopoiesis and the concept of stem cell renewal. Exp. Hematol. 32(6), 511–9 (2004)

    Article  Google Scholar 

  67. Glauche, I., Cross, M., Loeffler, M., Roeder, I.: Lineage specification of hematopoietic stem cells: mathematical modeling and biological implications. Stem Cells 25(7), 1791–1799 (2007)

    Article  Google Scholar 

  68. Stockholm, D., Benchaouir, R., Picot, J., Rameau, P., Neildez, T.M., Landini, G., Laplace-Builhe, C., Paldi, A.: The origin of phenotypic heterogeneity in a clonal cell population in vitro. PLoS ONE 2, e394 (2007)

  69. Palmqvist, R., Rutegard, J.N., Bozoky, B., Landberg, G., Stenling, R.: Human colorectal cancers with an intact p16/cyclin D1/pRb pathway have up-regulated p16 expression and decreased proliferation in small invasive tumor clusters. Am. J. Pathol. 157(6), 1947–1953 (2000)

    Google Scholar 

  70. Balkwill, F.: Cancer and the chemokine network. Nat. Rev. Cancer 4(7), 540–550 (2004)

    Article  Google Scholar 

  71. Locke, M., Heywood, M., Fawell, S., Mackenzie, I.C.: Retention of intrinsic stem cell hierarchies in carcinoma-derived cell lines. Cancer Res. 65(19), 8944–8950 (2005)

    Article  Google Scholar 

  72. Ichinose, S., Tagami, M., Muneta, T., Sekiya, I.: Morphological examination during in vitro cartilage formation by human mesenchymal stem cells. Cell Tissue Res. 322(2), 217–226 (2005)

    Article  Google Scholar 

  73. Teixeira, A.I., Abrams, G.A., Bertics, P.J., Murphy, C.J., Nealey, P.F.: Epitheilial contact guidance on well-defined micro- and nanostructured substrates. J. Cell Sci. 116, 1881–1892 (2003)

    Article  Google Scholar 

  74. Izaguirre, J.A., Chaturvedi, R., Huang, C., Cickovski, T., Coffland, J., Thomas, G., Forgacs, G., Alber, M., Hentschel, G., Newman, S.A., Glazier, J.A.: CompuCell, a multi-model framework for simulation of morphogenesis. Bioinformatics 20(7), 1129–1137 (2004)

    Article  Google Scholar 

  75. Teller, I.C., Beaulieu, J.F.: Interactions between laminin and epithelial cells in intestinal health and disease. Exp. Rev. Mol. Med. 28(09), 1–16 (2001)

    Google Scholar 

  76. Preston, S.L., Wong, W.M., Chan, A.O., Poulsom, R., Jeffery, R., Goodlad, R.A., Mandir, N., Elia, G., Novelli, M., Bodmer, W.F., Tomlinson, I.P., Wright, N.A.: Bottom-up histogenesis of colorectal adenomas: origin in the monocryptal adenoma and initial expansion by crypt fission. Cancer Res. 63(13), 3819–3825 (2003)

    Google Scholar 

  77. Roeder, I., Glauche, I.: Towards an understanding of lineage specification in hematopoietic stem cells: a mathematical model for the interaction of transcription factors GATA-1 and PU.1. J. Theor. Biol. 241(4), 852–865 (2006)

    MathSciNet  Google Scholar 

  78. Pribyl, M., Muratov, C.B., Shvartsman, S.Y.: Discrete models of autocrine cell communication in epithelial layers. Biophys. J. 84(6), 3624–3635 (2003)

    Article  Google Scholar 

  79. Roose, T., Chapman, S.J., Maini, P.K.: Mathematical models of avascular cancer. SIAM Rev. 49(2), 179–208 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  80. Frisch, T., Thoumine, O.: Predicting the kinetics of cell spreading. Biomechanics 35(8), 1137–1141 (2002)

    Article  Google Scholar 

  81. Grossmann, J., Walther, K., Artinger, M., Kiessling, S., Schölmerich, J.: Apoptotic signaling during initiation of detachment-induced apoptosis (“anoikis– of primary human intestinal epithelial cells. Cell Growth Differ. 12(3), 147–155 (2001)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Interdisciplinary Center for Bioinformatics, University Leipzig, Leipzig, Germany

    J. Galle & M. Hoffmann

  2. Centre of Surgery, Research Laboratories, University Leipzig, Leipzig, Germany

    G. Aust

Authors
  1. J. Galle
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Hoffmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. Aust
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Galle.

Electronic Supplementary material

Below are the Electronic Supplementary materials.

285_2008_172_MOESM1_ESM.mpg

Movies 1a, 1b (related to Fig.5 of the article): Population growth dynamics. Comparison of population growth dynamics according to scenarios: a) R1 and b) R2. Top views on growing colonies. Colour saturation indicates imminent cell division. b) Substrate adapted cells are shown in yellow. The two colonies regulated according to R1 and to R2 show nearly identical spreading. This is achieved by assuming different cell-substrate friction coefficients. For parameter sets ‘R1 vs. R2’, please, see Appendix.

285_2008_172_MOESM2_ESM.mpg

Movies 1a, 1b (related to Fig.5 of the article): Population growth dynamics. Comparison of population growth dynamics according to scenarios: a) R1 and b) R2. Top views on growing colonies. Colour saturation indicates imminent cell division. b) Substrate adapted cells are shown in yellow. The two colonies regulated according to R1 and to R2 show nearly identical spreading. This is achieved by assuming different cell-substrate friction coefficients. For parameter sets ‘R1 vs. R2’, please, see Appendix.

285_2008_172_MOESM3_ESM.mpg

Movies 2a, 2b (related to Fig.7 of the article): Cancer stem cell organisation. Top views on growing tumours (blue: migratory inactive, red: migratory active) within stroma (yellow). a) Environmentally regulated proliferation according to the plasticity scenario R3: Colour saturation indicates stem cells. The migratory cells at the tumour front (dark red) are considered to be quiescent stem cells. b) Intrinsically regulated proliferation according to the pedigree scenario R4: Colour saturation indicates differentiated cells. Most of the stem cells become confined in the tumour bulk. Stem cells located at the tumour periphery stay there and induce fast invasion. For parameter sets ‘R3 vs. R4’, please, see Appendix.

285_2008_172_MOESM4_ESM.mpg

Movies 2a, 2b (related to Fig.7 of the article): Cancer stem cell organisation. Top views on growing tumours (blue: migratory inactive, red: migratory active) within stroma (yellow). a) Environmentally regulated proliferation according to the plasticity scenario R3: Colour saturation indicates stem cells. The migratory cells at the tumour front (dark red) are considered to be quiescent stem cells. b) Intrinsically regulated proliferation according to the pedigree scenario R4: Colour saturation indicates differentiated cells. Most of the stem cells become confined in the tumour bulk. Stem cells located at the tumour periphery stay there and induce fast invasion. For parameter sets ‘R3 vs. R4’, please, see Appendix.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Galle, J., Hoffmann, M. & Aust, G. From single cells to tissue architecture—a bottom-up approach to modelling the spatio-temporal organisation of complex multi-cellular systems. J. Math. Biol. 58, 261–283 (2009). https://doi.org/10.1007/s00285-008-0172-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-008-0172-4

Keywords

Mathematics Subject Classification (2000)

Search

Navigation