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From passive diffusion to active cellular migration in mathematical models of tumour invasion

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Abstract

Mathematical models of tumour invasion appear as interesting tools for connecting the information extracted from medical imaging techniques and the large amount of data collected at the cellular and molecular levels. Most of the recent studies have used stochastic models of cell translocation for the comparison of computer simulations with histological solid tumour sections in order to discriminate and characterise expansive growth and active cell movements during host tissue invasion. This paper describes how a deterministic approach based on reaction-diffusion models and their generalisation in the mechano-chemical framework developed in the study of biological morphogenesis can be an alternative for analysing tumour morphological patterns. We support these considerations by reviewing two studies. In the first example, successful comparison of simulated brain tumour growth with a time sequence of computerised tomography (CT) scans leads to a quantification of the clinical parameters describing the invasion process and the therapy. The second example considers minimal hypotheses relating cell motility and cell traction forces. Using this model, we can simulate the bifurcation from an homogeneous distribution of cells at the tumour surface toward a nonhomogeneous density pattern which could characterise a pre-invasive stage at the tumour-host tissue interface.

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References

  • Ackerman, A.B. (1989). Differentiation of benign from malignant neoplasms by silhouette. Am. J. Dermatopath. 11: 297–300.

    Google Scholar 

  • Adam, J.A. and S.A. Maggelakis (1990). Diffusion regulated growth characteristics of a spherical prevascular carcinoma. Bull. Math. Biol. 52: 549–582.

    Google Scholar 

  • Albini, A., A. Melchiori, A. Garofalo, D.M. Noonan, F. Basolo, G. Taraboletti, G.J. Chader and R. Giavazzi (1992). Matrigel promotes retinoblastoma cell growthin vitro andin vivo. Int. J. Cancer 52: 234–240.

    Google Scholar 

  • Alvord, E.C. and C.M. Shaw (1991). Neoplasms affecting the nervous system of the elderly In: S. Duckett eds., The Pathology of the Aging Human Nervous System, p. 210–286. Philadelphia Lea&Febiger.

    Google Scholar 

  • Arvan, D.A. (1992). Tumor cell heterogeneity: an overview. Clinica Chimica Acta 206: 3–7.

    Google Scholar 

  • Brandley, B.K., J.H. Shaper and R.L. Schnaar (1990). Tumor cell haptotaxis on immobilized N-Acetylglucosamine gradients. Devel. Biol. 140: 161–171.

    Google Scholar 

  • Carter, S.B. (1965). Principles of cell motility: the direction of cell movement and cancer invasion. Nature 208: 1183–1187.

    Google Scholar 

  • Chaplain, M.A.J. and B.D. Sleeman (1992). A mathematical model for the growth and classification of a solid tumor: a new approach via nonlinear elasticity theory using strain-energy functions. Math. Biosc. 11: 169–215.

    Google Scholar 

  • Chaplain, M.A.J. (in press). Reaction-diffusion prepatterning and its potential role in tumour invasion. J. Biol. Syst.

  • Chicoine, M.R. and D.L. Silbergeld (1995). Assessment of brain tumor cell motilityin vivo andin vitro. J. Neurosurg. 82: 615–622.

    Google Scholar 

  • Coldman, A.J. and J.H. Goldie (1985). Role of mathematical modeling in protocol formulation in cancer chemotherapy. Cancer Treatment Reports 69: 1041–1045.

    Google Scholar 

  • Cook, J., P. Tracqui and J.D. Murray (1993). Mechanochemical models and biological morphogenesis—A brief review. Forma 8: 159–178.

    Google Scholar 

  • Cruywagen, G.C., P.K. Maini and J.D. Murray. (1992). Sequential pattern formation in a model for skin morphogenesis. IMA J. Math. Appl. Medic. Biol. 9: 227–248.

    Google Scholar 

  • Dedhar, S. (1993). Integrins and tumor invasion. Bioessays 12: 583–590.

    Google Scholar 

  • Dominguez-Malagon, H. (1993). Proliferative disorders of Myofibroblasts. Ultrastruct. Pathol. 17: 211–220.

    Google Scholar 

  • Düchting, W. and T. Vogelsaenger (1985). Recent progress in modelling and simulation of three-dimensional tumor growth and treatment. BioSystems 18: 79–91.

    Google Scholar 

  • Düchting, W. (1990). Tumor growth simulation. Comp.&Graphics 14: 505–508.

    Google Scholar 

  • Dvorak, H.F. (1986). Tumors: Wounds that do not heal. New England J. Med. 315: 1650–1657.

    Google Scholar 

  • Ermentrout, G.B. and L. Edelstein-Keshet (1993). Cellular automata approaches to biological modelling. J. Theor. Biol. 160: 97–133.

    Google Scholar 

  • Folkman, J. and M. Hochberg (1973). Self-regulation of growth in three dimensions. J. Exp. Med. 138: 745–753.

    Google Scholar 

  • Gabbert, H.E., S. Meier, C.D. Gerharz and G. Hommel (1992). Tumor-cell dissociation at the invasion front: a new prognostic parameter in gastric cancer patients. Int. J. Cancer 50: 202–207.

    Google Scholar 

  • Goldie, J.H. and A.J. Coldman (1979). A mathematical model for relating the drug sensitivity of tumors to their spontaneous mutation rate. Cancer Treatment Reports 63: 1727–1731.

    Google Scholar 

  • Graeff, H., F. Janicke and M. Schmitt (1992). Proteases, matrix degradation and tumor-cell spread. Fibrinolysis 6: 1–2.

    Google Scholar 

  • Grigioni, W.F., A. D'Errico, G. Biagini, A. Mazziotti, L. Bolondi, L.A. Liotta A.M. Mancini and S. Garbisa (1990). The capsule surrounding primary liver tumors: wherefrom its prognostic significance. Int. J. Cancer 45: 63.

    Google Scholar 

  • Harris, A.K. (1987). Cell motility and the problem of anatomical homeostasis. J. Cell Sci. Suppl. 8: 121–140.

    Google Scholar 

  • Harris, A.K., D. Stopak and P. Wild (1981). Fibroblast traction as a mechanism for collagen morphogenesis. Nature 290: 249–251.

    Google Scholar 

  • Honn, K.V. and D.G. Tang (1992). Adhesion molecules and tumor cell interaction with endothelium and subendothelial matrix. Cancer and Metastasis Reviews 11: 353–375.

    Google Scholar 

  • Ingber, D.E. (1991). Integrins as mechanochemical transducers. Curr. Opinion in Cell Biol. 3: 841–848.

    Google Scholar 

  • Kimmel, M. and D.E. Axelrod (1991). Unequal cell division, growth regulation and colony size of mammalian cells: a mathematical model and analysis of experimental data. J. Theor. Biol. 153: 157–180.

    Google Scholar 

  • Klominek, J., K.H. Robert and K.G. Sundqvist (1993). Chemotaxis and haptotaxis of human malignant mesothelial cells: effect of fibronectin, laminin, type IV collagen and an autocrine motility factor-like substance. Cancer Research 53: 4376–4382.

    Google Scholar 

  • Kreth, F.W., P.C. Warnke, R.S. Scheremet and C.B. Ostertag (1993). Surgical resection and radiation therapy versus biopsys and radiation therapy in the treatment of glioblastoma multiforme. J. Neurosurgery 78: 762–766.

    Google Scholar 

  • Kumazawa, H., M. Hess and P. Koldovsky (1993). Morphologic analysis of three-dimensional tumors developed in fibrin matrix and agar culture system. Pathobiology 61: 31–35.

    Google Scholar 

  • Landau, L. and E. Lifchitz (1959). Theory of Elasticity. London, Pergamon Press. (1990) Physique Théorie de l'Elasticité. Paris, Editions MIR.

    Google Scholar 

  • Lazard, D., X. Sastre, M.G. Frid, M.A. Glukhova, J.P. Thiery and V.E. Koteliansky (1993). Expression of smooth muscle-specific proteins in myoepithelium and stromal myofibroblasts of normal and malignant human breast tissue. Proc. Natl. Acad. Sci. USA 90: 999–1003.

    Google Scholar 

  • Leader, W.M., D. Stopak and A.K. Harris (1983). Increased contractile strength and tightened adhesions to the substratum result from reverse transformation of CHO cells by Dibutyryl cyclic adenosine monophosphate. J. Cell Sci. 64: 1–11.

    Google Scholar 

  • Lefever, R., J. Hiernaux and P. Meyers (1989). Evolution of tumours attacked by immune cytotoxic cells: the immune response dilemma. In: A. Goldbeter ed., Cell to Cell Signaling: From Experiments to Theoretical Models, p. 315–333. Academic Press.

  • Leighton, J. and R. Tchao (1984). The propagation of cancer, a process of tissue remodeling. Cancer and Metastasis Reviews 3: 81–97.

    Google Scholar 

  • Leitner, F., I. Marque, S. Lavallée and P. Cinquin (1991). Dynamic segmentation: finding the edge with spline snakes. Proc. Int. Conf. on Curves and Surfaces, p. 279–284. Chamonix, Academic Press.

    Google Scholar 

  • Lester, B.R. and J.B. McCarthy (1992). Tumor cell adhesion to the extracellular matrix and signal transduction mechanisms implicated in tumor cell motility, invasion and metastasis. Cancer and Metastasis Reviews 11: 31–44.

    Google Scholar 

  • Liotta, L.A. (1986). Tumor invasion and metastases—Role of the extracellular matrix. Cancer Res. 46: 1–7.

    Google Scholar 

  • Maini, P.K. (1989). Spatial and spatio-temporal patterns in a cell-haptotaxis model. J. Math. Biol. 27: 507–522.

    Google Scholar 

  • Mareel M.M., F.M. Van Roy, L.M. Messiaen, E.R. Boghaert and E.A. Bruyneel (1987). Qualitative and quantitative analysis of tumour invasionin vivo andin vitro. J. Cell Sci. Suppl. 8: 141–163.

    Google Scholar 

  • Marusic, M., Z. Bajzer, J.P. Freyer and S. Vuk-Pavlovic (1994). Analysis of growth of multicellular tumour spheroids by mathematical models. Cell Prolif. 27: 73–94.

    Google Scholar 

  • McCarthy, J.B., M.L. Basara, S.L. Palm, D.F. Sas and L.T. Furcht (1985). The role of cell adhesion proteins-laminin and fibronectin-in the movement of malignant and metastatic cells. Cancer and Metastasis Reviews 4: 125–152.

    Google Scholar 

  • Michelson, S., B.E. Miller, A.S. Glicksman and J.T. Leith (1987). Tumor micro-ecology and competitive interactions. J. Theor. Biol. 128: 233–246.

    Google Scholar 

  • Mignatti, P. and D.B. Rifkin (1993). Biology and biochemistry of proteinases in tumor invasion. Physiol. Reviews 73: 161–195.

    Google Scholar 

  • Montandon, D., G. D'Andiran and G. Gabbiani (1977). The mechanism of wound contraction and epithelialization. Clin. Plast. Surg. 4: 335–346.

    Google Scholar 

  • Mueller-Klieser, W. (1987). Multicellular spheroids. J. Cancer Res. Clin. Oncol. 113: 101–122.

    Google Scholar 

  • Murray, J.D., P.K. Maini and R.T. Tranquillo (1998). Mechanochemical models for generating biological pattern and form in development. Physics Rep. 171: 60–84.

    Google Scholar 

  • Murray, J.D. (1993). Mathematical Biology. Springer Verlag, 2nd corrected edition.

  • Murray, J.D. and G.F. Oster (1984). Cell tractions models for generating pattern and form in morphogenesis. J. Math. Biol. 19: 265–279.

    Google Scholar 

  • Needham, D. (1991). Possible role of cell cycle-dependent morphology, geometry and mechanical-properties in tumor-cell metastasis. Cell Biophys. 18: 99–121.

    Google Scholar 

  • Ng, I.O.L., M.R.C. Path, E.C.S. Lai, M.M.T. Ng and S.T. Fan (1992). Tumor encapsulation in Hepatocellular carcinoma. Cancer 70: 45–49.

    Google Scholar 

  • Oster, G.F., J.D. Murray and A.K. Harris (1983). Mechanical aspects of mesenchymal morphogenesis. J. Embryol. Exp. Morph. 78: 83–125.

    Google Scholar 

  • Perelson, A.S.K., P.K. Maini, J.D. Murray, J.M. Hyman and G.F. Oster (1986). Nonlinear pattern selection in a mechanical model for morphogenesis. J. Math. Biol. 24: 525–541.

    Google Scholar 

  • Qi, A.S., X. Zheng, C.Y. Du and B.S. An (1993). A cellular automaton model of cancerous growth. J. Theor. Biol. 161: 1–12.

    Google Scholar 

  • Retsky, M.W., D.E. Swartzendruber, R.H. Wardwell and P.D. Bame (1990). Is gompertzian or exponential kinetics a valid description of individual human cancer growth? Medical Hypotheses 33: 95–106.

    Google Scholar 

  • Ruoslahti, E. (1992). Control of cell motility and tumour invasion by extracellular matrix interactions. Br. J. Cancer 66: 239–242.

    Google Scholar 

  • Schwartz, M.A. (1993). Signaling by integrins: implications for tumorigenesis. Cancer Research 53: 1503–1506.

    Google Scholar 

  • Segel, L. (1980). Mathematical Models in Molecular and Cellular Biology. Cambridge University Press.

  • Sherratt, J.A. and M.A. Nowak (1992). Oncogenes, anti-oncogenes and the immune response to cancer: a mathematical model. Proc. R. Soc. Lond. B 248: 261–271.

    Google Scholar 

  • Smolle, J. and H. Stettner (1993). Computer simulation of tumour cell invasion by a stochastic growth model. J. Theor. Biol. 160: 63–72.

    Google Scholar 

  • Smolle, J. and I.A. Grimstad (1992). Tumor-cell motility and invasion within tumors determined by applying computer simulation to histologic patterns. Int. J. Cancer 50: 331–335.

    Google Scholar 

  • Smolle, J., F.M. Smolle-Juettner, H. Stettner and H. Kerl (1992). Relationship of tumor cell motility and morphologic patterns. Part 1. Melanocytic skin tumors. Am. J. Dermatopath. 14: 231–237.

    Google Scholar 

  • Starkey, J.R. (1990). Cell-matrix interactions during tumor invasion. Cancer and Metastasis Reviews, 9: 113–123.

    Google Scholar 

  • Stopak, D. and A.H. Harris (1982). Connective tissue morphogenesis by fibroblast traction. I. Tissue culture observations. Developmental Biol. 90: 383–398.

    Google Scholar 

  • Suh, O. and L. Weiss (1984). The development of a technique for the morphometric analysis of invasion in cancer. J. Theor. Biol. 107: 547–561.

    Google Scholar 

  • Sutherland, R.M. (1988). Cell and environment interactions in tumor microregions: the multicell spheroid model. Science 240: 177–184.

    Google Scholar 

  • Tracqui, P. and M.A.J. Chaplain (in prep.). Tumor invasion and Metastasis: a mechano-chemical formulation.

  • Tracqui, P., G.C. Cruywagen, D.E. Woodward, G.T. Bartoo, J.D. Murray and E.C. Alvord Jr. (1995). A mathematical model of glioma growth: the effect of chemotherapy on spatio-temporal growth. Cell Prolif. 28: 17–31.

    Google Scholar 

  • Volk, T., B. Geiger and A. Raz (1984). Motility and Adhesive properties of High and Low Metastatic Murine neoplastic cells. Cancer Research 44: 811–824.

    Google Scholar 

  • Wheldon, T.E. (1986). Mathematical models in experimental and clinical oncology. In: D. Ingram and R.F. Bloch, eds, Mathematical Methods in Medicine, p. 1–32. J. Wiley&Sons.

  • Woodward, D.E., G.T. Bartoo, P. Tracqui, G.C. Cruywagen, J.D. Murray and E.C. Alvord Jr. (submitted). A mathematical model of glioma growth: the effect of extent of surgical resection.

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  1. Laboratoire TIMC-IMAG, URA CNRS 1618, Faculté de Médecine de Grenoble, Domaine de la Merci, 38700, La Tronche, France

    Philippe Tracqui

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Tracqui, P. From passive diffusion to active cellular migration in mathematical models of tumour invasion. Acta Biotheor 43, 443–464 (1995). https://doi.org/10.1007/BF00713564

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