Microenvironment driven invasion: a multiscale multimodel investigation

  • Alexander R. A. AndersonEmail author
  • Katarzyna A. Rejniak
  • Philip Gerlee
  • Vito Quaranta


Cancer is a complex, multiscale process, in which genetic mutations occurring at a subcellular level manifest themselves as functional and morphological changes at the cellular and tissue scale. The importance of interactions between tumour cells and their microenvironment is currently of great interest in experimental as well as computational modelling. Both the immediate microenvironment (e.g. cell–cell signalling or cell–matrix interactions) and the extended microenvironment (e.g. nutrient supply or a host tissue structure) are thought to play crucial roles in both tumour progression and suppression. In this paper we focus on tumour invasion, as defined by the emergence of a fingering morphology, which has previously been shown to be dependent upon harsh microenvironmental conditions. Using three different modelling approaches at two different spatial scales we examine the impact of nutrient availability as a driving force for invasion. Specifically we investigate how cell metabolism (the intrinsic rate of nutrient consumption and cell resistance to starvation) influences the growing tumour. We also discuss how dynamical changes in genetic makeup and morphological characteristics, of the tumour population, are driven by extreme changes in nutrient supply during tumour development. The simulation results indicate that aggressive phenotypes produce tumour fingering in poor nutrient, but not rich, microenvironments. The implication of these results is that an invasive outcome appears to be co-dependent upon the evolutionary dynamics of the tumour population driven by the microenvironment.

Mathematics Subject Classification (2000)

82B24 92C15 92C17 92B20 92D05 76Z99 92C37 92C50 

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  1. 1.
    Alarcon T, Byrne HM, Maini PK (2005) A multiple scale model for tumor growth. Multiscale Model Simul 3: 440–475zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Albini A, Sporn MB (2007) The tumour microenvironment as a target for chemoprevention. Nat Rev Cancer 7: 139–147CrossRefGoogle Scholar
  3. 3.
    Alexandrova R (2001) Tumour heterogeneity. Exp Pathol Parasitol 4: 57–67Google Scholar
  4. 4.
    Anderson ARA (2003) A hybrid discrete-continuum technique for individual based migration models. In: Alt W, Chaplain M, Griebel M, Lenz J(eds) Polymer and cell dynamics. Birkhauser, BaselGoogle Scholar
  5. 5.
    Anderson ARA (2005) A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion. IMA J Math Med Biol 22: 163–186zbMATHCrossRefGoogle Scholar
  6. 6.
    Anderson ARA (2007) A hybrid multiscale model of tumour invasion: evolution and the microenvironment. In: Anderson ARA, Chaplain MAJ, Rejniak KA(eds) Single-cell-based models in biology and medicine. Birkhauser, BaselCrossRefGoogle Scholar
  7. 7.
    Anderson ARA, Chaplain MAJ (1998) Continuous and discrete mathematical models of tumour-induced angiogenesis angiogenesis. Bull Math Biol 60: 857–899zbMATHCrossRefGoogle Scholar
  8. 8.
    Anderson ARA, Chaplain MAJ, Newman EL, Steele RJC, Thompson AM (2000) Mathematical modelling of tumour invasion and metastasis. J Theor Med 2: 129–154zbMATHGoogle Scholar
  9. 9.
    Anderson ARA, Chaplain MAJ, Rejniak KA (2007) Single-cell-based models in biology and medicine. Birkhauser, BaselCrossRefGoogle Scholar
  10. 10.
    Anderson ARA, Pitcairn A (2003) Application of the hybrid discrete-continuum technique. In: Alt W, Chaplain M, Griebel M, Lenz J(eds) Polymer and cell dynamics. Birkhauser, BaselGoogle Scholar
  11. 11.
    Anderson ARA, Sleeman BD, Young IM, Griffiths BS (1997) Nematode movement along a chemical gradient in a structurally heterogeneous environment. II. Theory. Fundam Appl Nematol 20: 165–172Google Scholar
  12. 12.
    Anderson ARA, Weaver AM, Cummings PT, Quaranta V (2006) Tumor morphology and phenotypic evolution driven by selective pressure from the microenvironment. Cell 127: 905–915CrossRefGoogle Scholar
  13. 13.
    Ao M, Williams K, Bhowmick NA, Hayward SW (2006) Transforming growth factor-β promotes invasion in tumorigenic but not in nontumorigenic human prostatic epithelial cells. Cancer Res 66: 8007–8016CrossRefGoogle Scholar
  14. 14.
    Araujo RP, McElwain DLS (2004) A history of the study of solid tumour growth: the contribution of mathematical modeling. Bull Math Biol 66: 1039–1091CrossRefMathSciNetGoogle Scholar
  15. 15.
    Ben-Jacob E, Cohen I, Levine H (2000) Cooperative self-organization of microorganisms. Adv Phys 49: 395–554CrossRefGoogle Scholar
  16. 16.
    Ben-Jacob E, Garik P (1990) The formation of patterns in non-equilibrium growth. Nature 343: 523–530CrossRefGoogle Scholar
  17. 17.
    Bierie B, Moses HL (2006) Tumour microenvironment: TGF, the molecular Jekyll and Hyde of cancer. Nat Rev Cancer 6: 506–520CrossRefGoogle Scholar
  18. 18.
    Bray D (1990) Intracellular signalling as a parallel distributed process. J Theor Biol 143: 215–231CrossRefGoogle Scholar
  19. 19.
    Brown JM, Wilson WR (2004) Exploring tumour hypoxia in cancer treatment. Nat Rev Cancer 4: 437–447CrossRefGoogle Scholar
  20. 20.
    Byrne HM, Chaplain MAJ (1995) Growth of non-necrotic tumours in the presence and absence of inhibitors. Math Biosci 130: 151–181zbMATHCrossRefGoogle Scholar
  21. 21.
    Byrne HM, Chaplain MAJ (1996) Growth of necrotic tumours in the presence and absence of inhibitors. Math Biosci 135: 187–216zbMATHCrossRefGoogle Scholar
  22. 22.
    Byrne HM, Chaplain MAJ (1996) Modelling the role of cell–cell adhesion in the growth and development of carcinomas. Math Comput Model 24: 1–17zbMATHCrossRefGoogle Scholar
  23. 23.
    Byrne HM, Chaplain MAJ (1998) Free boundary problems arising in models of tumour growth and development. EJAM 8: 639–658MathSciNetGoogle Scholar
  24. 24.
    Casciari JJ, Sotirchos SV, Sutherland RM (1992) Variations in tumor cell growth rates and metabolism with oxygen concentration, glucose concentration, and extracellular pH. J Cell Physiol 151: 386–394CrossRefGoogle Scholar
  25. 25.
    Cavallaro U, Christofori G (2004) Cell adhesion and signaling by cadherins and ig-cams in cancer. Nat Cancer Rev 4: 118–132Google Scholar
  26. 26.
    Chaplain MAJ, Lolas G (2005) Mathematical modelling of cancer cell invasion of tissue: the role of the urokinase plasmino- gen activation system. M3AS 15: 1685–1734zbMATHMathSciNetGoogle Scholar
  27. 27.
    Chaplain MAJ, Graziano L, Preziosi L (2006) Mathematical modelling of the loss of tissue compression resposiveness and its role in solid tumour development. Math Med Biol 23: 197–229zbMATHCrossRefGoogle Scholar
  28. 28.
    Cristini V, Frieboes HB, Gatenby R, Caserta S, Ferrari M, Sinek J (2005) Morphologic instability and cancer invasion. Clin Cancer Res 11: 6772–6779CrossRefGoogle Scholar
  29. 29.
    Daccord G, Nittmann J, Stanley HE (1986) Radial viscous fingers and diffusion-limited aggregation: fractal dimension and growth sites. Phys Rev Lett 56(4): 336–339CrossRefGoogle Scholar
  30. 30.
    Debruyne PR, Bruyneel EA, Karaguni I-M (2002) Bile acids stimulate invasion and haptotaxis in human corectal cancer cells through activation of multiple oncogneic signalling pathways. Oncogene 21: 6740–6750CrossRefGoogle Scholar
  31. 31.
    Dormann S, Deutsch A (2002) Modeling of self-organzied avascular tumor growth with a hybrid cellular automaton. In Silico Biol 2: 393–406Google Scholar
  32. 32.
    Drasdo D, Höhme S (2005) A single-cell-based model of tumor growth in vitro: monolayers and spheroids. Phys Biol 2: 133–147CrossRefGoogle Scholar
  33. 33.
    Düchting W (1990) Tumor growth simulation. Comput. Graphics 14: 505–508CrossRefGoogle Scholar
  34. 34.
    Eden M (1961) A two dimensional growth process. Proc 4th Berkpley Symp Math Stat Prob 4: 223MathSciNetGoogle Scholar
  35. 35.
    Ferreira SC, Martins ML, Vilela MJ (2002) Reaction–diffusion model for the growth of avascular tumor. Phys Rev E 65: 021907CrossRefMathSciNetGoogle Scholar
  36. 36.
    Friedl P, Hegerfeldt Y, Tusch M (2004) Collective cell migration in morphogenesis and cancer. Int J Dev Biol 48: 441–449CrossRefGoogle Scholar
  37. 37.
    Freyer JP, Sutherland RM (1985) A reduction in the in situ rates of oxygen and glucose consumption of cells on EMT6/Ro spheroids during growth. J Cell Physiol 124: 516–524CrossRefGoogle Scholar
  38. 38.
    Freyer JP, Sutherland RM (1986) Regulation of growth saturation and development of necrosis in emt6/ro multicellular spheroids by the glucose and oxygen supply. Cancer Res 46: 3513–3520Google Scholar
  39. 39.
    Gatenby RA, Gawlinski ET (1996) A reaction–diffusion model of cancer invasion. Cancer Res 56: 5745–5753Google Scholar
  40. 40.
    Gerlee P, Anderson ARA (2007) An evolutionary hybrid cellular automaton model of solid tumour growth. J Theor Biol 246(4): 583–603CrossRefMathSciNetGoogle Scholar
  41. 41.
    Gerlee P, Anderson ARA (2007) Stability analysis of a hybrid cellular automaton model of cell colony growth. Phys Rev E 75: 051911CrossRefGoogle Scholar
  42. 42.
    Hanahan D, Weinberg RA (2000) The hallmarks of cancer. Cell 100: 57–70CrossRefGoogle Scholar
  43. 43.
    Haykin S (1999) Neural networks: a comprehensive foundation, 2nd edn. Prentice Hall, New JerseyzbMATHGoogle Scholar
  44. 44.
    Höckel M, Schlenger K, Aral B, Mitze M, Schaffer U, Vaupel P (1996) Association between tumor hypoxia and malignant progression in advanced cancer of the uterine cervix. Cancer Res 56: 4059–4515Google Scholar
  45. 45.
    Jiang Y, Pjesivac-Grbovic JA, Cantrell C, Freyer JP (2005) A multiscale model for avascular tumour growth. Biophys J 89: 3884–3894CrossRefGoogle Scholar
  46. 46.
    Kansal AR, Torquato S, Harsh GR, Chiocca EA, Deisboeck TS (2000) Simulated brain tumor growth using a three-dimensional cellular automaton. J Theor Biol 203: 367–382CrossRefGoogle Scholar
  47. 47.
    Kessler DA, Koplik J, Levine H (1988) Pattern selection in fingered growth phenomena. Adv Phys 37: 255–339CrossRefGoogle Scholar
  48. 48.
    Klominek J, Robert KH, Sundqvist K-G (1993) Chemotaxis and haptotaxis of human malignant mesothelioma cells: effects of fibronectin, laminin, type IV collagen, and an autocrine motility factor-like substance. Cancer Res 53: 4376–4382Google Scholar
  49. 49.
    Lane DP (1994) The regulation of p53 function. Steiner Award Lecture. Int J Cancer 57: 623–627CrossRefGoogle Scholar
  50. 50.
    Liotta LA, Clair T (2000) Checkpoints for invasion. Nature 405: 287–288CrossRefGoogle Scholar
  51. 51.
    Lopez JM, Jensen HJ (2002) Generic model of morphological changes in growing colonies of fungi. Phys Rev E 65(2): 021903CrossRefGoogle Scholar
  52. 52.
    Lowe SW, Lin AW (2000) Apoptosis in cancer. Carcinogenesis 21: 485–495CrossRefGoogle Scholar
  53. 53.
    Macklin P, Lowengrub JS (2005) Evolving interfaces via gradients of geometry-dependent interior Poisson problems: application to tumor growth. J Comput Phys 203(1): 191–220zbMATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    Mallon E, Osin P, Nasiri N, Blain I, Howard B, Gusterson B (2000) The basic pathology of human breast cancer. J Mammary Gland Biol Neoplasia 5: 139–63CrossRefGoogle Scholar
  55. 55.
    Matsushita M, Fujikawa H (1990) Diffusion-limited growth in bacterial colony formation. Physica A 168: 498–506CrossRefGoogle Scholar
  56. 56.
    Matsushita M, Sano M, Hayakawa Y, Honjo H, Sawada Y (1984) Fractal structures of zinc metal leaves grown by electrodeposition. Phys Rev Lett 53(3): 286–289CrossRefGoogle Scholar
  57. 57.
    Matsushita M, Wakita J, Itoh H, Watanabe K, Arai T, Matsuyama T, Sakaguchi H, Mimura M (1999) Formation of colony patterns by a bacterial cell population. Physica A 274: 190–199CrossRefGoogle Scholar
  58. 58.
    Mueller-Klieser WF, Sutherland RM (1982) Oxygen tension in multicellular spheroids of two cell lines. Br J Cancer 45: 256–264Google Scholar
  59. 59.
    Mueller-Klieser W (1987) Multicellular spheroids. A review on cellular aggregates in cancer research. J Cancer Res Clin Oncol 113: 101–122CrossRefGoogle Scholar
  60. 60.
    Mullins WW, Sekerka RF (1963) Morphological stability of a particle growing by diffusion or heat flow. J Appl Phys 34(2): 323–329CrossRefGoogle Scholar
  61. 61.
    Nowell PC (1976) The clonal evolution of tumour cell populations. Science 194: 23–28CrossRefGoogle Scholar
  62. 62.
    Orme ME, Chaplain MAJ (1996) A mathematical model of vascular tumour growth and invasion. Mathl Comp Model 23: 43–60zbMATHCrossRefGoogle Scholar
  63. 63.
    Overall CM, Kleifeld O (2006) Tumour microenvironment ϠOpinion: validating matrix metalloproteinases as drug targets and anti-targets for cancer therapy. Nat Rev Cancer 6: 227–239CrossRefGoogle Scholar
  64. 64.
    Paszek MJ, Zahir N, Johnson KR, Lakins JN, Rozenberg GI, Gefen A, Reinhart-King CA, Margulies SS, Dembo M, Boettiger D, Hammer DA, Weaver VM (2005) Tensional homeostasis and the malignant phenotype. Cancer Cell 8: 241–254CrossRefGoogle Scholar
  65. 65.
    Patel AA, Gawlinski EE, Lemieux SK, Gatenby RA (2001) A cellular automaton model of early tumor growth and invasion: the effects of native tissue vascularity and increased anaerobic tumor metabolism. J Theor Biol 213: 315–331CrossRefMathSciNetGoogle Scholar
  66. 66.
    Peinado H, Olmeda D, Cano A (2007) Snail, ZEB and bHLH factors in tumour progression: an alliance against the epithelial phenotype?. Nat Rev Cancer 7: 415–428CrossRefGoogle Scholar
  67. 67.
    Pennacchietti S, Michieli P, Galluzzo M, Mazzone M, Giordano S, Comoglio PM (2003) Hypoxia promotes invasive growth by transcriptional activation of the met protooncogene. Cancer Cell 3: 347–361CrossRefGoogle Scholar
  68. 68.
    Perumpanani AJ, Sherratt JA, Norbury J, Byrne HM (1996) Biological inferences from a mathematical model of malignant invasion. Invasion Metastases 16: 209–221Google Scholar
  69. 69.
    Peskin CS (1972) Flow patterns around heart valves: a numerical method. J Comput Phys 10: 252–271zbMATHCrossRefMathSciNetGoogle Scholar
  70. 70.
    Peskin CS (1977) Numerical analysis of blood flow in the heart. J Comput Phys 25: 220–252zbMATHCrossRefMathSciNetGoogle Scholar
  71. 71.
    Peskin CS (2002) The immersed boundary method. Acta Numer 11: 479–517zbMATHCrossRefMathSciNetGoogle Scholar
  72. 72.
    Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1996) Numerical recipes in C: the art of scientific computing. Cambridge University, CambridgeGoogle Scholar
  73. 73.
    Preziosi L (2003) Cancer modelling and simulation. CRC Press, West Palm Beach, FLzbMATHGoogle Scholar
  74. 74.
    Qi A, Zheng X, Du C, An B (1993) A Cellular automaton model of cancerous growth. J Theor Biol 161: 1–12CrossRefGoogle Scholar
  75. 75.
    Rejniak KA (2005) A single-cell approach in modeling the dynamics of tumor microregions. Math Biosci Eng 2: 643–655zbMATHMathSciNetGoogle Scholar
  76. 76.
    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: 186–204CrossRefMathSciNetGoogle Scholar
  77. 77.
    Rejniak KA (2007) Modelling the development of complex tissues using individual viscoelastic cells. In: Anderson ARA, Chaplain MAJ, Rejniak KA(eds) Single-cell-based models in biology and medicine. Birkhauser, BaselGoogle Scholar
  78. 78.
    Rejniak KA, Anderson ARA (2008) A computational study of the development of epithelial acini. I. Sufficient conditions for the formation of a hollow structure. Bull Math Biol. 70(3): 677–712zbMATHCrossRefMathSciNetGoogle Scholar
  79. 79.
    Rejniak KA, Anderson ARA (2008) A computational study of the development of epithelial acini. II. Necessary conditions for structure and lumen stability. Bull Math Biol 70(5): 1450–1479zbMATHCrossRefMathSciNetGoogle Scholar
  80. 80.
    Rejniak KA, Dillon RH (2007) A single cell based model of the ductal tumor microarchitecture. Comput Math Methods Med 8(1): 51–69zbMATHCrossRefMathSciNetGoogle Scholar
  81. 81.
    Rejniak KA, Kliman HJ, Fauci LJ (2004) A computational model of the mechanics of growth of the villous trophoblast bilayer. Bull Math Biol 66: 199–232CrossRefMathSciNetGoogle Scholar
  82. 82.
    Scott EL, Britton NF, Glazier JA, Zajac M (1999) Stochastic simulation of benign avascular tumour growth using the Potts Model. Math Computer Model 30: 183–198CrossRefGoogle Scholar
  83. 83.
    Shannon CE (1948) A mathematical theory of information. Bell Syst Tech J 27: 379–423zbMATHMathSciNetGoogle Scholar
  84. 84.
    Sherratt JA, Nowak MA (1992) Oncogenes, anti-oncogenes and the immune response to cancer: a mathematical model. Proc R Soc Lond B 248: 261–271CrossRefGoogle Scholar
  85. 85.
    Smolle J, Stettner H (1993) Computer simulation of tumour cell invasion by a stochastic growth model. J Theor Biol 160: 63–72CrossRefGoogle Scholar
  86. 86.
    Soussi T, Lozano G (2005) p53 mutation heterogeneity in cancer. Biochem Biophys Res Commun 331: 834–842CrossRefGoogle Scholar
  87. 87.
    Swanson KR, Bridge C, Murray JD, Alvord EC Jr (2003) Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion. J Neurol Sci 216: 1–10CrossRefGoogle Scholar
  88. 88.
    Turner S, Sherratt JA (2002) Intracellular adhesion and cancer invasion: a discrete simulation using the extended Potts model. J Theor Biol 216: 85–100CrossRefMathSciNetGoogle Scholar
  89. 89.
    Ward JP, King JR (1999) Mathematical modelling of avascular-tumour growth. II. Modelling growth saturation. IMA J Math Appl Med Biol 16: 171–211zbMATHCrossRefGoogle Scholar
  90. 90.
    Wittekind C, Compton CC, Greene FL, Sobin LH (2002) TNM residual tumor classification revisited. Cancer 94: 2511–2516CrossRefGoogle Scholar
  91. 91.
    Witten TA, Sander LM (1981) Diffusion-limited aggregation, a kinetic critical phenomenon. Phys Rev Lett 47(19): 1400–1403CrossRefGoogle Scholar
  92. 92.
    Zhang L, Athale CA, Deisboeck TS (2007) Development of a three-dimensional multiscale agent-based tumor model: Simulating gene–protein interaction profiles, cell phenotypes and multicellular patterns in brain cancer. J Theor Biol 244: 96–107CrossRefMathSciNetGoogle Scholar
  93. 93.
    Zheng X, Wise SM, Cristini V (2005) Nonlinear simulation of tumour necrosis, neo-vascularization and tissue invasion via an adaptive finite-element/level-set method. Bull Math Biol 67: 211–256CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Alexander R. A. Anderson
    • 1
    • 2
    Email author
  • Katarzyna A. Rejniak
    • 1
    • 2
  • Philip Gerlee
    • 1
    • 3
  • Vito Quaranta
    • 4
  1. 1.Division of MathematicsUniversity of DundeeDundeeScotland, UK
  2. 2.Integrated Mathematical OncologyMoffitt Cancer Center and Research InstituteTampaUSA
  3. 3.Center for Models of LifeNiels Bohr InstituteKøbenhavn ØDenmark
  4. 4.Department of Cancer BiologyVanderbilt University School of MedicineNashvilleUSA

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