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Cell adhesion mechanisms and stress relaxation in the mechanics of tumours

  • Davide Ambrosi
  • Luigi PreziosiEmail author
Original Paper

Abstract

Tumour cells usually live in an environment formed by other host cells, extra-cellular matrix and extra-cellular liquid. Cells duplicate, reorganise and deform while binding each other due to adhesion molecules exerting forces of measurable strength. In this paper, a macroscopic mechanical model of solid tumour is investigated which takes such adhesion mechanisms into account. The extracellular matrix is treated as an elastic compressible material, while, in order to define the relationship between stress and strain for the cellular constituents, the deformation gradient is decomposed in a multiplicative way distinguishing the contribution due to growth, to cell rearrangement and to elastic deformation. On the basis of experimental results at a cellular level, it is proposed that at a macroscopic level there exists a yield condition separating the elastic and dissipative regimes. Previously proposed models are obtained as limit cases, e.g. fluid-like models are obtained in the limit of fast cell reorganisation and negligible yield stress. A numerical test case shows that the model is able to account for several complex interactions: how tumour growth can be influenced by stress, how and where it can generate cell reorganisation to release the stress level, how it can lead to capsule formation and compression of the surrounding tissue.

Keywords

Stress Relaxation Deformation Gradient Math Biol Multicellular Spheroid Solid Tumour Growth 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Ambrosi D, Mollica F (2002) On the mechanics of a growing tumour. Int J Eng Sci 40: 1297–1316CrossRefMathSciNetGoogle Scholar
  2. Ambrosi D, Mollica F (2004) The role of stress in the growth of a multicell spheroid. J Math Biol 48: 477–499zbMATHCrossRefMathSciNetGoogle Scholar
  3. Ambrosi D, Preziosi L (2002) On the closure of mass balance models for tumour growth. Math Mod Methods Appl Sci 12: 737–754zbMATHCrossRefMathSciNetGoogle Scholar
  4. Araujo RP, McElwain DLS (2004) A linear-elastic model of anisotropic tumour growth. Eur J Appl Math 15: 365–384zbMATHCrossRefMathSciNetGoogle Scholar
  5. Araujo RP, McElwain DLS (2005a) A mixture theory for the genesis of residual stresses in growing tissues. I. A general formulation. SIAM J Appl Math 65: 1261–1284zbMATHCrossRefMathSciNetGoogle Scholar
  6. Araujo RP, McElwain DLS (2005b) 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 65: 1285–1299CrossRefMathSciNetGoogle Scholar
  7. Basov IV, Shelukhin VV (1999) Generalized solutions to the equations of compressible Bingham flows. ZAMM 49: 185–192CrossRefMathSciNetGoogle Scholar
  8. Baumgartner W, Hinterdorfer P, Ness W, Raab A, Vestweber D, Schindler H, Drenckhahn D (2000) Cadherin interaction probed by atomic force microscopy. Proc Natl Acad Sci USA 97: 4005–4010CrossRefGoogle Scholar
  9. 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: 125–152zbMATHCrossRefMathSciNetGoogle Scholar
  10. Breward CJW, Byrne HM, Lewis CE (2003) A multiphase model describing vascular tumour growth. Bull Math Biol 65: 609–640CrossRefGoogle Scholar
  11. Buscall R, Mills PDA, Goodwin JW, Lawson DW (1988) Scaling behaviour of the rheology of aggregate networks formed from colloidal particles. J Chem Soc Faraday Trans 84: 4249–4260CrossRefGoogle Scholar
  12. Byrne HM, King JR, McElwain DLS, Preziosi L (2003) A two-phase model of solid tumour growth. Appl Math Lett 16: 567–573zbMATHCrossRefMathSciNetGoogle Scholar
  13. Byrne HM, Preziosi L (2004) Modeling solid tumour growth using the theory of mixtures. Math Med Biol 20: 341–366CrossRefGoogle Scholar
  14. Canetta E, Duperray A, Leyrat A, Verdier C (2005) Measuring cell viscoelastic properties using a force-spectrometer: Influence of the protein–cytoplasm interactions. Biorheology 42: 298–303Google Scholar
  15. Caveda L, Martin-Padura I, Navarro P, Breviario F, Corada M, Gulino D, Lampugnani MG, Dejana E (1996) Inhibition of cultured cell growth by vascular endothelial cadherin (cadherin-5/VE-cadherin). J Clin Invest 98: 886–893CrossRefGoogle Scholar
  16. Chaplain MAJ, 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
  17. Chen CY, Byrne HM, King JR (2001) The influence of growth-induced stress from the surrounding medium on the development of multicell spheroids. J Math Biol 43: 191–220zbMATHCrossRefMathSciNetGoogle Scholar
  18. Cristini V, Lowengrub J, Nie Q (2003) Nonlinear simulation of tumour growth. J Math Biol 46: 191–224zbMATHCrossRefMathSciNetGoogle Scholar
  19. Forgacs G, Foty RA, Shafrir Y, Steinberg MS (1998) Viscoelastic properties of living embryonic tissues: a quantitative study. Biophys J 74: 2227–2234CrossRefGoogle Scholar
  20. Franks SJ, Byrne HM, King JR, Underwood JCE, Lewis CE (2003a) Modelling the early growth of ductal carcinoma in situ of the breast. J Math Biol 47: 424–452zbMATHCrossRefMathSciNetGoogle Scholar
  21. Franks SJ, Byrne HM, Mudhar HS, Underwood JCE, Lewis CE (2003b) Mathematical modelling of comedo ductal carcinoma in situ of the breast. Math Med Biol 20: 277–308zbMATHCrossRefGoogle Scholar
  22. Franks SJ, King JR (2003) Interactions between a uniformly proliferating tumour and its surrounding: uniform material properties. Math Med Biol 20: 47–89zbMATHCrossRefGoogle Scholar
  23. Frieboes H, Zheng X, Sun C-H, Tromberg B, Gatenby R, Cristini V (2006) An integrated computational/experimental model of tumour invasion. Cancer Res 66: 1597–1604CrossRefGoogle Scholar
  24. Gibson RF (1994) Principles of Composite Material Mechanics. McGraw-Hill, NY, USAGoogle Scholar
  25. Green AE, Naghdi PM (1969) On basic equations for mixtures. Quart J Mech Appl Math 22: 427–438zbMATHCrossRefGoogle Scholar
  26. Helmlinger G, Netti PA, Lichtenbeld HC, Melder RJ, Jain RK (1997) Solid stress inhibits the growth of multicellular tumour spheroids. Nature Biotechnol 15: 778–783CrossRefGoogle Scholar
  27. Hohenemser K, Prager W (1932) Über die ansätze der mechanik isotroper kontinua. ZAMM 12: 216–226CrossRefGoogle Scholar
  28. Holmes NH (1986) Finite deformation of soft tissue: analysis of a mixture model in uni-axial compression. J Biomech Eng 108: 372–381CrossRefGoogle Scholar
  29. Joseph DD (1990) Fluid dynamics of viscoelastic liquids. Springer, BerlinzbMATHGoogle Scholar
  30. Jones AF, Byrne HM, Gibson JS, Dold JW (2000) A mathematical model of the stress induced during solid tumour growth. J Math Biol 40: 473–499zbMATHCrossRefMathSciNetGoogle Scholar
  31. Levenberg S, Yarden A, Kam Z, Geiger B (1999) p27 is involved in N-cadherin-mediated contact inhibition of cell growth and S-phase entry. Oncogene 18: 869–876CrossRefGoogle Scholar
  32. Malik WA, Prasad SC, Rajagopal KR, Preziosi L (2008) On the modelling of the viscoelastic response of embryonic tissues. Math Mech Solids 13: 81–91zbMATHCrossRefMathSciNetGoogle Scholar
  33. Macklin P, Lowengrub J (2007) Nonlinear simulation of the effect of the microenvironment on tumour growth. J Theor Biol 245: 677–704CrossRefMathSciNetGoogle Scholar
  34. Malvern LE (1969) Introduction of the Mechanics of a Continuous Medium. Prentice Hall Inc., Englewood CliffsGoogle Scholar
  35. Netti PA, Jain RK (2003) Interstitial transport in solid tumours. In: Preziosi L (eds) Cancer Modelling and Simulation. CRC Press, Chapman Hall, Boca RatonGoogle Scholar
  36. Panorchan P, Thompson MS, Davis KJ, Tseng Y, Konstantopoulos K, Wirtz D (2006) Single-molecule analysis of cadherin-mediated cell–cell adhesion. J Cell Sci 119: 66–74CrossRefGoogle Scholar
  37. 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
  38. Preziosi L (1989) On an invariance property of the solution to Stokes’ first problem for viscoelastic fluids. J Non-Newtonian Fluid Mech 33: 225–228zbMATHCrossRefGoogle Scholar
  39. Preziosi L, Joseph DD (1987) Stokes’ first problem for viscoelastic fluids. J Non-Newtonian Fluid Mech 25: 239–259zbMATHCrossRefGoogle Scholar
  40. Preziosi L, Tosin A (2009) Multiphase modeling of tumour growth and extracellular matrix interaction: mathematical tools and applications. J Math Biol.. doi: 10.1007/s00285-008-0218-7 MathSciNetGoogle Scholar
  41. Roose T, Netti PA, Munn LL, Boucher Y, Jain RK (2003) Solid stress generated by spheroid growth estimated using a linear poroelasticity model. Microvasc Res 66: 204–212CrossRefGoogle Scholar
  42. Rodriguez EK, Hoger A, McCulloch A (1994) Stress dependent finite growth in soft elastic tissues. J Biomech 27: 455–467CrossRefGoogle Scholar
  43. Simon BR (1992) Multiphase poroelastic finite element models for soft tissue structures. Appl Mech Rev 45: 191–218CrossRefGoogle Scholar
  44. Shelukhin VV (2002) Bingham viscoplastic as a limit of non-Newtonian fluids. J Math Fluid Mech 4: 109–127zbMATHCrossRefMathSciNetGoogle Scholar
  45. Snabre P, Mills P (1996) Rheology of weakly flocculated suspensions of rigid particles. J Phys III France 6: 1811–1834CrossRefGoogle Scholar
  46. 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
  47. Volokh KY (2006) Stresses in growing soft tissues. Acta Biomater 2: 493–504CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Dipartimento di MatematicaPolitecnico di TorinoTurinItaly

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