Annals of Biomedical Engineering

, Volume 21, Issue 6, pp 679–697 | Cite as

Quantitative characterization of cell invasionIn vitro: Formulation and validation of a mathematical model of the collagen gel invasion assay

  • Richard B. Dickinson
  • James B. McCarthy
  • Robert T. Tranquillo


Anin vitro assay proposed to systematically characterize and compare cell invasion under different conditions is the collagen gel invasion assay where cells, initially seeded onto the surface of a type I collagen gel, penetrate the surface and migrate within the gel over time. Using simplifying assumptions about cell transport across the gel surface and migration within the gel, we formulate and solve a mathematical model of this assay which predicts the resulting cell distribution based on three phenomenological parameters characterizing the ability of cells to penetrate the gel surface interface, migrate randomly within the gel, and return to the gel surface. An index of cell invasiveness is defined based on these parameters that reflects the overall ability of cells to transport across the gel surface interface, that is, invade the gel. Cell concentration profiles predicted by the model correspond well to measured profiles for murine melanoma cells invading gels supplemented with extracellular matrix proteins fibronectin and type IV collagen as well as unsupplemented gels, allowing these parameters to be estimated by a nonlinear regression fit of the model solution to the measured profiles. Our analysis suggests that type IV collagen and fibronectin primarily modulate cell transport across the gel surface interface rather than migration within the gel. Further, we validate the key model assumptions and obtain independent, direct estimates of model parameters by time-lapse video microscopy and digital image analysis of cell penetration of the gel surface and migration within the gel during the assay.


Invasion Collagen gel Mathematical model Metastasis Migration Cell tracking 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abramowitz, M.; Stegun, I.A., eds. Handbook of mathematical functions. Washington, D.C.: Department of Commerce; 1972.Google Scholar
  2. 2.
    Bissell, M.J.; Aggeler, J. Dynamic reciprocity: How do extracellular matrix and hormones direct gene expression? Prog. Clin. Biol. Res. 249:251–262; 1987.PubMedGoogle Scholar
  3. 3.
    Brown, A.F. Neutrophil granulocytes: Adhesion and locomotion on collagen substrata and in collagen matrices. J. Cell Sci. 58:455–467; 1982.PubMedGoogle Scholar
  4. 4.
    Crank, J. The mathematics of diffusion. Oxford: Clarendon Press; 1975.Google Scholar
  5. 5.
    Dickinson, R.B.; Tranquillo, R.T. Optimal estimation of cell migration indices from the statistical analysis of cell tracking data. AIChE J. 39(12):1995–2010; 1993.CrossRefGoogle Scholar
  6. 6.
    Dunn, G.A. Characterizing a kinesis response: Time averaged measures of cell speed and directional persistence. Agents Act. Suppl. 12:14–33; 1983.Google Scholar
  7. 7.
    Erkell, L.J.; Schirrmacher, V. Quantitative in vitro assay for tumor cell invasion through extracellular matrix or into protein gels. Cancer Res. 48(23):6933–6937; 1988.PubMedGoogle Scholar
  8. 8.
    Faassen, A.E.; Schrager, J.A.; Klein, D.J.; Oegema, T.R.; Couchman, J.R.; McCarthy, J.B. A cell surface chondroitin sulfate proteoglycan, immunologically related to CD44, is involved in type I collagen-mediated melanoma cell motility and invasion. J. Cell Biol. 116(2):521–531; 1992.CrossRefPubMedGoogle Scholar
  9. 9.
    Furukawa, M.; Kono, T.; Tanii, T.; Ishii, M.; Hamada, T.; Shibata, T. Proliferative potential of murine melanoma cells cultured in or on collagen gel. J. Dermatol. 17:297–302; 1990.PubMedGoogle Scholar
  10. 10.
    Goldfarb, R.H.; Liotta, L.A. Proteolytic enzymes in cancer invasion and metastasis. Semin. Thromb. Hemost. 12(4):294–307; 1986.PubMedGoogle Scholar
  11. 11.
    Hendrix, M.J.; Seftor, E.A.; Seftor, R.E.; Misiorowski, R.L.; Saba, P.Z.; Sundareshan, P.; Welch, D.R. Comparison of tumor cell invasion assays: human amnion versus reconstituted basement membrane barriers. Invas. Metast. 9(5):278–297; 1989.Google Scholar
  12. 12.
    Herbst, T.; McCarthy, J.B.; Tsilibary, E.C.; Furcht, L.T. Differential effects of laminin, intact type IV collagen, and specific domains of type IV collagen on endothelial cell adhesion and migration. J. Cell Biol. 106:1365–1373; 1988.CrossRefPubMedGoogle Scholar
  13. 13.
    Islam, L.N.; McKay, I.C.; Wilkinson, P.C. The use of collagen or fibrin gels for the assay of human neutrophil chemotaxis. J. Immunol. Methods 85(1):137–151; 1985.CrossRefPubMedGoogle Scholar
  14. 14.
    Kleinman, H.K.; McGarvey, M.L.; Liotta, L.A.; Geron Robey, P.; Trygvasson, K.; Martin, G.R. Isolation and characterization type IV procollagen, laminin, and heparan sulfate proteoglycan from the EHS sarcoma. Biochem. 21:6188–6193; 1982.Google Scholar
  15. 15.
    Lackie, J.M.; Chaabane, N.; Crocket, K.V. A critique of the methods used to assess leucocyte behaviour. Biomed. Pharmacother. 41(6):265–278; 1987.PubMedGoogle Scholar
  16. 16.
    Lawless, J.F. Statistical models and methods for lifetime data. New York: Wiley; 1982.Google Scholar
  17. 17.
    Liotta, L.; Schiffmann, E. Tumor autocrine motility factors. Important Adv. Oncol. 17–30; 1988.Google Scholar
  18. 18.
    Liotta, L.; Rao, C.N.; Barsky, S.H. Tumor invasion and the extracellular matrix. Lab Invest. 49(6):636–649; 1983.PubMedGoogle Scholar
  19. 19.
    McCarthy, J.B.; Chelberg, M.K.; Mickelson, D.J.; Furcht, L.T. Localization and chemical synthesis of fibronectin peptides with melanoma adhesion and heparin binding activities. Biochemistry 27(4):1380–1388; 1988.CrossRefPubMedGoogle Scholar
  20. 20.
    McCarthy, J.B.; Sas, D.F.; Furcht, L.T. Mechanisms of parenchymal cell migration into wounds. In: Clark, R.A.F., Henson, P.M. eds. The molecular and cellular biology of wound repair. New York: Plenum Press; 1988. pp. 281–308.Google Scholar
  21. 21.
    Montesano, R.; Pepper, M.S.; Vassalli, J.D.; Orci, L. Phorbol ester induces cultured endothelial cells to invade a fibrin matrix in the presence of fibrinolytic inhibitors. J. Cell Physiol. 132(3):509–516; 1987.CrossRefPubMedGoogle Scholar
  22. 22.
    Mooradian, D.L.; McCarthy, J.B.; Komanduri, K.V.; Furcht, L.T. Effects of transforming growth factor-beta 1 on human pulmonary adenocarcinoma cell adhesion, motility, and invasion in vitro. J. Natl. Cancer Inst. 84(7):523–527; 1992.PubMedGoogle Scholar
  23. 23.
    Othmer, H.G.; Dunbar, S.R.; Alt, W. Models of dispersal in biological systems. J. Math. Biol. 26(3):263–298; 1988.CrossRefPubMedGoogle Scholar
  24. 24.
    Runyan, R.B.; Markwald, R.R. Invasion of mesenchyme into three-dimensional collagen gels: A regional and temporal analysis of interaction in embryonic heart tissue. Dev. Biol. 95(1):108–114; 1983.CrossRefPubMedGoogle Scholar
  25. 25.
    Russo, R.G.; Thorgeirsson, U.; Liotta, L.A. In vitro quantitative assay of invasion using human amnion. In: Liotta, L.A., Hart, I.R., eds. Tumor invasion and metastasis. Boston: Nijhoff; 1982. pp. 175–187.Google Scholar
  26. 26.
    Schor, S.L.; Allen, T.D.; Harrison, C.J. Cell migration through three-dimensional gels of native collagen fibres: Collagenolytic activity is not required for the migration of two permanent cell lines. J. Cell Sci. 41:159–175; 1980.PubMedGoogle Scholar
  27. 27.
    Schor, S.L.; Allen, T.D.; Winn, B. Lymphocyte migration into three-dimensional collagen matrices: A quantitative study. J. Cell Biol. 96(4):1089–1096; 1983.CrossRefPubMedGoogle Scholar
  28. 28.
    Schor, S.L.; Schor, A.M.; Winn, B.; Rushton, G. The use of three-dimensional collagen gels for the study of tumour cell invasion in vitro: Experimental parameters influencing cell migration into the gel matrix. Int. J. Cancer. 29(1):57–62; 1982.PubMedGoogle Scholar
  29. 29.
    Seber, G.A.F.; Wild, C.J. Nonlinear regression. New York: John Wiley & Sons; 1989.Google Scholar
  30. 30.
    Smith, C.W.; Anderson, D.C. PMN adhesion and extravasation as a paradigm for tumor cell dissemination. Cancer Metast. Rev. 10(1):61–78; 1991.CrossRefGoogle Scholar
  31. 31.
    Trygvasson, K.; Hoyhtya, M.; Salo, T. Proteolytic degradation of extracellular matrix in tumor invasion. Biochim. Biophys. Acta. 907(3):191–217; 1987.Google Scholar

Copyright information

© Pergamon Press Ltd. 1993

Authors and Affiliations

  • Richard B. Dickinson
    • 1
  • James B. McCarthy
    • 2
  • Robert T. Tranquillo
    • 1
  1. 1.Department of Chemical Engineering and Materials ScienceUniversity of MinnesotaMinneapolis
  2. 2.Department of Laboratory Medicine and PathologyUniversity of MinnesotaMinneapolis

Personalised recommendations