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.
Similar content being viewed by others
References
Ambrosi D, Mollica F (2002) On the mechanics of a growing tumour. Int J Eng Sci 40: 1297–1316
Ambrosi D, Mollica F (2004) The role of stress in the growth of a multicell spheroid. J Math Biol 48: 477–499
Ambrosi D, Preziosi L (2002) On the closure of mass balance models for tumour growth. Math Mod Methods Appl Sci 12: 737–754
Araujo RP, McElwain DLS (2004) A linear-elastic model of anisotropic tumour growth. Eur J Appl Math 15: 365–384
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: 1261–1284
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: 1285–1299
Basov IV, Shelukhin VV (1999) Generalized solutions to the equations of compressible Bingham flows. ZAMM 49: 185–192
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: 4005–4010
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: 125–152
Breward CJW, Byrne HM, Lewis CE (2003) A multiphase model describing vascular tumour growth. Bull Math Biol 65: 609–640
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: 4249–4260
Byrne HM, King JR, McElwain DLS, Preziosi L (2003) A two-phase model of solid tumour growth. Appl Math Lett 16: 567–573
Byrne HM, Preziosi L (2004) Modeling solid tumour growth using the theory of mixtures. Math Med Biol 20: 341–366
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: 298–303
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: 886–893
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: 197–229
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: 191–220
Cristini V, Lowengrub J, Nie Q (2003) Nonlinear simulation of tumour growth. J Math Biol 46: 191–224
Forgacs G, Foty RA, Shafrir Y, Steinberg MS (1998) Viscoelastic properties of living embryonic tissues: a quantitative study. Biophys J 74: 2227–2234
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: 424–452
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: 277–308
Franks SJ, King JR (2003) Interactions between a uniformly proliferating tumour and its surrounding: uniform material properties. Math Med Biol 20: 47–89
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: 1597–1604
Gibson RF (1994) Principles of Composite Material Mechanics. McGraw-Hill, NY, USA
Green AE, Naghdi PM (1969) On basic equations for mixtures. Quart J Mech Appl Math 22: 427–438
Helmlinger G, Netti PA, Lichtenbeld HC, Melder RJ, Jain RK (1997) Solid stress inhibits the growth of multicellular tumour spheroids. Nature Biotechnol 15: 778–783
Hohenemser K, Prager W (1932) Über die ansätze der mechanik isotroper kontinua. ZAMM 12: 216–226
Holmes NH (1986) Finite deformation of soft tissue: analysis of a mixture model in uni-axial compression. J Biomech Eng 108: 372–381
Joseph DD (1990) Fluid dynamics of viscoelastic liquids. Springer, Berlin
Jones AF, Byrne HM, Gibson JS, Dold JW (2000) A mathematical model of the stress induced during solid tumour growth. J Math Biol 40: 473–499
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: 869–876
Malik WA, Prasad SC, Rajagopal KR, Preziosi L (2008) On the modelling of the viscoelastic response of embryonic tissues. Math Mech Solids 13: 81–91
Macklin P, Lowengrub J (2007) Nonlinear simulation of the effect of the microenvironment on tumour growth. J Theor Biol 245: 677–704
Malvern LE (1969) Introduction of the Mechanics of a Continuous Medium. Prentice Hall Inc., Englewood Cliffs
Netti PA, Jain RK (2003) Interstitial transport in solid tumours. In: Preziosi L (eds) Cancer Modelling and Simulation. CRC Press, Chapman Hall, Boca Raton
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: 66–74
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: 241–254
Preziosi L (1989) On an invariance property of the solution to Stokes’ first problem for viscoelastic fluids. J Non-Newtonian Fluid Mech 33: 225–228
Preziosi L, Joseph DD (1987) Stokes’ first problem for viscoelastic fluids. J Non-Newtonian Fluid Mech 25: 239–259
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
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: 204–212
Rodriguez EK, Hoger A, McCulloch A (1994) Stress dependent finite growth in soft elastic tissues. J Biomech 27: 455–467
Simon BR (1992) Multiphase poroelastic finite element models for soft tissue structures. Appl Mech Rev 45: 191–218
Shelukhin VV (2002) Bingham viscoplastic as a limit of non-Newtonian fluids. J Math Fluid Mech 4: 109–127
Snabre P, Mills P (1996) Rheology of weakly flocculated suspensions of rigid particles. J Phys III France 6: 1811–1834
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: 4320–4329
Volokh KY (2006) Stresses in growing soft tissues. Acta Biomater 2: 493–504
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ambrosi, D., Preziosi, L. Cell adhesion mechanisms and stress relaxation in the mechanics of tumours. Biomech Model Mechanobiol 8, 397–413 (2009). https://doi.org/10.1007/s10237-008-0145-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10237-008-0145-y