Advertisement

Multiphase modelling of tumour growth and extracellular matrix interaction: mathematical tools and applications

  • Luigi PreziosiEmail author
  • Andrea Tosin
Article

Abstract

Resorting to a multiphase modelling framework, tumours are described here as a mixture of tumour and host cells within a porous structure constituted by a remodelling extracellular matrix (ECM), which is wet by a physiological extracellular fluid. The model presented in this article focuses mainly on the description of mechanical interactions of the growing tumour with the host tissue, their influence on tumour growth, and the attachment/detachment mechanisms between cells and ECM. Starting from some recent experimental evidences, we propose to describe the interaction forces involving the extracellular matrix via some concepts coming from viscoplasticity. We then apply the model to the description of the growth of tumour cords and the formation of fibrosis.

Mathematics Subject Classification (2000)

92C10 92C50 

Supplementary material

ESM 1 (AVI 1041 kb)

ESM 2 (AVI 2483 kb)

ESM 3 (AVI 3213 kb)

ESM 4 (AVI 1661 kb)

ESM 5 (AVI 1627 kb)

References

  1. 1.
    Ambrosi D, Guana F (2007) Stress modulated growth. Math Mech Solids 12(3): 319–342zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ambrosi D, Mollica F (2002) On the mechanics of a growing tumor. Int J Eng Sci 40(12): 1297–1316CrossRefMathSciNetGoogle Scholar
  3. 3.
    Ambrosi D, Mollica F (2004) The role of stress in the growth of a multicell spheroid. J Math Biol 48(5): 477–499zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Ambrosi D, Preziosi L (2008) Cell adhesion mechanisms and elasto-viscoplastic mechanics of tumours. Biomech Model Mechanobiol (to appear)Google Scholar
  5. 5.
    Ambrosi D, Preziosi L (2002) On the closure of mass balance models for tumor growth. Math Models Methods Appl Sci 12(5): 737–754zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Araujo RP, McElwain DLS (2004) A history of the study of solid tumour growth: the contribution of mathematical modelling. Bull Math Biol 66(5): 1039–1091CrossRefMathSciNetGoogle Scholar
  7. 7.
    Araujo RP, McElwain DLS (2004) A linear-elastic model of anisotropic tumour growth. Eur J Appl Math 15(3): 365–384zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Araujo RP, McElwain DLS (2005) A mixture theory for the genesis of residual stresses in growing tissues. I. A general formulation. SIAM J Appl Math 65(4): 1261–1284 (electronic)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Araujo RP, McElwain DLS (2005) A mixture theory for the genesis of residual stresses in growing tissues. II. Solutions to the biphasic equations for a multicell spheroid. SIAM J Appl Math 66(2): 447–467 (electronic)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Astanin S, Preziosi L (2007) Multiphase models of tumour growth. In: Bellomo N, Chaplain M, DeAngelis E (eds) Selected topics on cancer modelling: genesis—evolution—immune competition—therapy. Birkhäuser, BaselGoogle Scholar
  11. 11.
    Baumgartner W, Hinterdorfer P, Ness W, Raab A, Vestweber D, Schindler H, Drenckhahn D (2000) Cadherin interaction probed by atomic force microscopy. Proc Nat Acad Sci USA 97: 4005–4010CrossRefGoogle Scholar
  12. 12.
    Behravesh E, Timmer MD, Lemoine JJ, Liebschner MA, Mikos AG (2002) Evaluation of the in vitro degradation of macroporous hydrogels using gravimetry, confined compression testing, and microcomputed tomography. Biomacromolecules 3: 1263–1270CrossRefGoogle Scholar
  13. 13.
    Bertuzzi A, Fasano A, Gandolfi A (2004/2005) A free boundary problem with unilateral constraints describing the evolution of a tumor cord under the influence of cell killing agents. SIAM J Math Anal 36(3): 882–915 (electronic)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Bertuzzi A, Fasano A, Gandolfi A (2005) A mathematical model for tumor cords incorporating the flow of interstitial fluid. Math Models Methods Appl Sci 15(11): 1735–1777zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Breward CJW, Byrne HM, Lewis CE (2001) Modelling the interactions between tumour cells and a blood vessel in a microenvironment within a vascular tumour. Eur J Appl Math 12(5): 529–556zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Breward CJW, Byrne HM, Lewis CE (2002) The role of cell–cell interactions in a two-phase model for avascular tumour growth. J Math Biol 45(2): 125–152zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Breward CJW, Byrne HM, Lewis CE (2003) A multiphase model describing vascular tumour growth. Bull Math Biol 65: 609–640CrossRefGoogle Scholar
  18. 18.
    Byrne HM, King JR, McElwain DLS, Preziosi L (2003) A two-phase model of solid tumour growth. Appl Math Lett 16(4): 567–573zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Byrne HM, Preziosi L (2004) Modeling solid tumor growth using the theory of mixtures. Math Med Biol 20: 341–366CrossRefGoogle Scholar
  20. 20.
    Canetta E, Leyrat A, Verdier C, Duperray A (2005) Measuring cell viscoelastic properties using a force-spectrometer: influence of the protein–cytoplasm interactions. Biorheology 42(5): 321–333Google Scholar
  21. 21.
    Chaplain M, Graziano L, Preziosi L (2006) Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour development. Math Med Biol 23: 197–229zbMATHCrossRefGoogle Scholar
  22. 22.
    De Masi A, Luckhaus S, Presutti E (2007) Two scales hydrodynamic limit for a model of malignant tumour cells. Ann IHP Prob Stat 43(3): 257–279zbMATHGoogle Scholar
  23. 23.
    DiMilla PA, Kenneth B, Lauffenburger DA (1991) Mathematical model for the effects of adhesion and mechanics on cell migration speed. Biophys J 60(1): 15–37CrossRefGoogle Scholar
  24. 24.
    DiMilla PA, Stone JA, Quinn JA, Albelda SA, Lauffenburger DA (1993) Maximal migration of human smooth muscle cells fibronectin and type iv collagen occurs at an intermediate attachment strength. J Cell Biol 122: 729–737CrossRefGoogle Scholar
  25. 25.
    Forgacs G, Foty RA, Shafrir Y, Steinberg MS (1998) Viscoelastic properties of living embryonic tissues: a quantitative study. Biophys J 74: 2227–2234CrossRefGoogle Scholar
  26. 26.
    Franks SJ, Byrne HM, King JR, Underwood JCE, Lewis CE (2003) Modelling the early growth of ductal carcinoma in situ of the breast. J Math Biol 47(5): 424–452zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Franks SJ, Byrne HM, Mudhar HS, Underwood JCE, Lewis CE (2003) Mathematical modelling of comedo ductal carcinoma in situ of the breast. Math Med Biol 20: 277–308zbMATHCrossRefGoogle Scholar
  28. 28.
    Franks SJ, King JR (2003) Interactions between a uniformly proliferating tumor and its surrounding. Uniform material properties. Math Med Biol 20: 47–89zbMATHCrossRefGoogle Scholar
  29. 29.
    Fung YC (1993) Biomechanics: mechanical Properties of living tissues. Springer, HeidelbergGoogle Scholar
  30. 30.
    Graziano L, Preziosi L (2007) Mechanics in tumour growth. In: Mollica F, Preziosi L, Rajagopal KR (eds) Modeling of biological materials. Birkhäuser, Basel, pp 267–328Google Scholar
  31. 31.
    Greenspan HP (1976) On the growth and stability of cell-cultures and solid tumours. J Theor Biol 56: 229–242CrossRefMathSciNetGoogle Scholar
  32. 32.
    Gu WY, Yao H, Huang CY, Cheung HS (2003) New insight into deformation-dependent hydraulic permeability of gels and cartilage, and dynamic behavior of agarose gels in confined compression. J Biomech 36: 593–598CrossRefGoogle Scholar
  33. 33.
    Humphrey JD, Rajagopal KR (2002) A constrained mixture model for growth and remodeling of soft tissues. Math Mod Meth Appl Sci 12: 407–430zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Humphrey JD, Rajagopal KR (2003) A constrained mixture model for arterial adaptations to a sustained step-change in blood flow. Biomech Model Mechanobiol 2: 109–126CrossRefGoogle Scholar
  35. 35.
    Iordan A, Duperray A, Verdier C (2008) A fractal approach to the rheology of concentrated cell suspensions. Phys Rev E 77: 011911CrossRefGoogle Scholar
  36. 36.
    Iredale JP (2007) Models of liver fibrosis: exploring the dynamic nature of inflammationa and repair in a solid organ. J Clin Invest 117(3): 539–548CrossRefGoogle Scholar
  37. 37.
    Johnson PRA (2001) Two scales hydrodynamic limit for a model of malignant tumour cells. Clin Exp Pharm Physiol 28: 233–236CrossRefGoogle Scholar
  38. 38.
    Lanza V, Ambrosi D, Preziosi L (2006) Exogenous control of vascular network formation in vitro: a mathematical model. Netw Heterog Media 1(4): 621–637 (electronic)zbMATHMathSciNetGoogle Scholar
  39. 39.
    Liotta LA, Kohn EC (2007) The microenvironment of the tumor–host interface. Nature 411: 375–379CrossRefGoogle Scholar
  40. 40.
    Newby AC, Zaltsman AB (2000) Molecular mechanisms in intimal hyperplasia. J Pathol 190: 300–309CrossRefGoogle Scholar
  41. 41.
    Oxlund H, Andreassen TT (1980) The roles of hyaluronic acid, collagen and elastin in the mechanical properties of connective tissues. J Anat 131(4): 611–620Google Scholar
  42. 42.
    Pujuguet PP, Hammann A, Moutet M, Samuel JL, Martin F, Martin M (1996) Expression of fibronectin eda+ and edb+ isoforms by human and experimental colorectal cancer. Am J Patho 148: 579–592Google Scholar
  43. 43.
    Palecek SP, Loftus JC, Ginsberg MH, Lauffenburger DA, Horwitz AF (1997) Integrin-ligand binding properties govern cell migration speed through cell-substratum adhesiveness. Nature 385: 537–540CrossRefGoogle Scholar
  44. 44.
    Preziosi L (ed) (2003) Cancer modelling and simulation. Chapman & Hall/CRC Mathematical Biology and Medicine Series, Boca Raton/FLGoogle Scholar
  45. 45.
    Preziosi L, Farina A (2001) On Darcy’s law for growing porous media. Int J Nonlinear Mech 37: 485–491CrossRefGoogle Scholar
  46. 46.
    Rao IJ, Humphrey JD, Rajagopal KR (2003) Biological growth and remodeling: a uniaxial example with possible application to tendons and ligaments. CMES 4: 439–455zbMATHGoogle Scholar
  47. 47.
    Sun M, Graham JS, Hegedus B, Marga F, Zhang Y, Forgacs G, Grandbois M (2005) Multiple membrane tethers probed by atomic force microscopy. Biophys J 89: 4320–4329CrossRefGoogle Scholar
  48. 48.
    Tosin A (2008) Multiphase modeling and qualitative analysis of the growth of tumor cords. Netw Heterog Media 3: 43–83zbMATHMathSciNetGoogle Scholar
  49. 49.
    Truskey GA, Yuan F, Katz DF (2004) Transport phenomena in biological systems. Prentice Hall, Englewood CliffsGoogle Scholar
  50. 50.
    Winters BS, Shepard SR, Foty RA (2005) Biophysical measurement of brain tumor cohesion. Int J Cancer 114: 371–379CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsPolitecnico di TorinoTorinoItaly

Personalised recommendations