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A continuum treatment of growth in biological tissue: the coupling of mass transport and mechanics. (English) Zbl 1159.74381
J. Mech. Phys. Solids 52, No. 7, 1595-1625 (2004); erratum ibid. 52, No. 12, 2909-2910 (2004).
Summary: Growth (and resorption) of biological tissue is formulated in the continuum setting. The treatment is macroscopic, rather than cellular or sub-cellular. Certain assumptions that are central to classical continuum mechanics are revisited, the theory is reformulated, and consequences for balance laws and constitutive relations are deduced. The treatment incorporates multiple species. Sources and fluxes of mass, and terms for momentum and energy transfer between species are introduced to enhance the classical balance laws. The transported species include: (i) a fluid phase; and (ii) the precursors and byproducts of the reactions that create and break down tissue. A notable feature is that the full extent of coupling between mass transport and mechanics emerges from the thermodynamics. Contributions to fluxes from the concentration gradient, chemical potential gradient, stress gradient, body force and inertia have not emerged in a unified fashion from previous formulations of the problem. The present work demonstrates these effects via a physically consistent treatment. The presence of multiple, interacting species requires that the formulation be consistent with mixture theory. This requirement has far-reaching consequences. A preliminary numerical example is included to demonstrate some aspects of the coupled formulation.

74L15 Biomechanical solid mechanics
92C10 Biomechanics
80A20 Heat and mass transfer, heat flow (MSC2010)
Full Text: DOI
[1] Ambrosi, D.; Mollica, F., On the mechanics of a growing tumor, Int. J. eng. sci., 40, 1297-1316, (2002) · Zbl 1211.74161
[2] Armero, F., Formulation and finite element implementation of a multiplicative model of coupled poro-plasticity at finite strains under fully-saturated conditions, Comput. methods appl. mech. eng., 171, 205-241, (1999) · Zbl 0968.74062
[3] Baar, K., Birla, R., Boluyt, M.D., Borschel, G.H., Arruda, E.M., Dennis, R.G., 2004. Heart muscle by design: self-organization of rat cardiac cells into contractile 3D cardiac tissue to appear in Fed. Amer. Soc. Exp. Biol., submitted for publication.
[4] Bedford, A.; Drumheller, D.S., Recent advancestheories of immiscible and structured mixtures, Int. J. eng. sci., 21, 863-960, (1983) · Zbl 0534.76105
[5] Bilby, B.A., Gardner, L.R.T., Stroh, A.N., 1957. Continuous distribution of dislocations and the theory of plasticity. In: Proceedings of the Nineth International Congress of Applied Mechanics, Brussels, 1956, Université de Bruxelles, pp. 35-44.
[6] Bischoff, J.E.; Arruda, E.M.; Grosh, K., A microstructurally based orthotropic hyperelastic constitutive law, J. appl. mech., 69, 570-579, (2002) · Zbl 1110.74344
[7] Bustamante, C.; Bryant, Z.; Smith, S.B., Ten years of tensionsingle-molecule DNA mechanics, Nature, 421, 423-427, (2003)
[8] Calve, S., Dennis, R.G., Kosnik, P.E., Baar, K., Grosh, K., Arruda, E.M., 2004a. Engineering of functional tendon. Tissue Engineering, to appear.
[9] Calve, S., Baar, K., Grosh, K., Dennis, R.G., Arruda, E.M., 2004b. Biochemical and mechanical characterization of engineered tendon. In: Proceedings of the XIV European Society of Biomechanics Conference, July 4-7, Hertogenbosch, The Netherlands, to appear.
[10] Chadwick, P., Continuum mechanics: concise theory and problems, (1999), Dover Publications Mineola, New York
[11] Cowin, S.C.; Hegedus, D.H., Bone remodeling itheory of adaptive elasticity, J. elasticity, 6, 313-326, (1976) · Zbl 0335.73028
[12] de Boer, R., Theory of porous media: highlights in the historical development and current state, (2000), Springer Berlin · Zbl 0945.74001
[13] de Groot, S.R.; Mazur, P., Nonequilibrium thermodynamics, (1984), Dover New York
[14] Epstein, M.; Maugin, G.A., Thermomechanics of volumetric growth in uniform bodies, Int. J. plasticity, 16, 951-978, (2000) · Zbl 0979.74006
[15] Garikipati, K.; Bassman, L.C.; Deal, M.D., A lattice-based micromechanical continuum formulation for stress-driven mass transport in polycrystalline solids, J. mech. phys. solids, 49, 1209-1237, (2001) · Zbl 1015.74005
[16] Garikipati, K., Narayanan, H., Arruda, E.M., Grosh, K., Calve, S., 2003. Material forces in the context of biotissue remodelling. In: Steinmann, P., Maugin, G.A., (Eds.), Mech. Mater. Forces, Kluwer Academic Publishers, Dordrecht, to appear. e-print available at . · Zbl 1192.74255
[17] Gonzalez, O.C., 1996. Design and analysis of conserving integrators for nonlinear Hamiltonian systems with symmetry. Ph.D. Thesis, Stanford University.
[18] Guyton, A.; Hall, J., Textbook of medical physiology, (1996), W.B. Saunders Company Philadelphia
[19] Harrigan, T.P.; Hamilton, J.J., Finite element simulation of adaptive bone remodellinga stability criterion and a time stepping method, Int. J. numer. methods eng., 36, 837-854, (1993)
[20] Humphrey, J.D., Mechanics of the arterial wallreview and directions, Crit. rev. biomed. eng., 23, 1-164, (1995)
[21] Kuhl, E., Steinmann, P., 2002. Geometrically nonlinear functional adaption of biological microstructures. In: Mang, H., Rammerstorfer, F., Eberhardsteiner, J. (Eds.), Proceedings of the Fifth World Congress on Computational Mechanics. International Association for Computational Mechanics, pp. 1-21.
[22] Lee, E.H., Elastic-plastic deformation at finite strain, J. appl. mech., 36, 1-6, (1969) · Zbl 0179.55603
[23] Lubarda, V.A.; Hoger, A., On the mechanics of solids with a growing mass, Int. J. solids struct., 29, 4627-4664, (2002) · Zbl 1045.74035
[24] Rajagopal, K.R., Wineman, A.S., 1990. Developments in the mechanics of interactions between a fluid and a highly elastic solid. In: Kee, D.D., Kaloni, P.N. (Eds.), Recent Developments in Structured Continua. Vol. II, Longman Scientific and Technical, New York, pp. 249-292.
[25] Rief, M.; Oesterhelt, F.; Heymann, B.; Gaub, H.E., Single molecule force spectroscopy on polysaccharides by atomic force microscopy, Science, 275, 1295-1297, (1997)
[26] Simo, J.C.; Tarnow, N., Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics, Comput. methods appl. mech. eng., 100, 63-116, (1992) · Zbl 0764.73096
[27] Simo, J.C.; Tarnow, N., The discrete energy-momentum method. conserving algorithm for nonlinear elastodynamics, Z. math. phys., 43, 757-793, (1992) · Zbl 0758.73001
[28] Simo, J.C.; Taylor, R.L.; Pister, K.S., Variational and projection methods for the volume constraint in finite deformation elasto-plasticity, Comp. methods appl. mech. eng., 51, 177-208, (1985) · Zbl 0554.73036
[29] Skalak, R.W., Growth as a finite displacement field, (), 347-355
[30] Skalak, R.W.; Zargaryan, S.; Jain, R.K.; Netti, P.A.; Hoger, A., Compatibility and the genesis of residual stress by volumetric growth, J. math. biol., 34, 889-914, (1996) · Zbl 0858.92005
[31] Swartz, M.; Kaipainen, A.; Netti, P.E.; Brekken, C.; Boucher, Y.; Grodzinsky, A.J.; Jain, R.K., Mechanics of interstitial-lymphatic fluid transporttheoretical foundation and experimental validation, J. biomech., 32, 1297-1307, (1999)
[32] Taber, L.A., Biomechanics of growth, remodelling and morphogenesis, Appl. mech. rev., 48, 487-545, (1995)
[33] Taber, L.A.; Humphrey, J.D., Stress-modulated growth, residual stress and vascular heterogeneity, J. biomech. eng., 123, 528-535, (2001)
[34] Taylor, R.L., 1999. FEAP—A Finite Element Analysis Program. University of California at Berkeley, Berkeley, CA, September.
[35] Terzaghi, K., Theoretical soil mechanics, (1943), Wiley New York; Chapman & Hall, London
[36] Truesdell, C., Noll, W., 1965. The Non-linear Field Theories, (Handbuch der Physik, Band III). Springer, Berlin. · Zbl 0779.73004
[37] Truesdell, C., Toupin, R.A., 1960. The Classical Field Theories (Handbuch der Physik, Band I). Springer, Berlin.
[38] Widmaier, E.P.; Raff, H.; Strang, K.T., Vander, sherman and Luciano’s human physiology: the mechanisms of body function, (2003), McGraw-Hill New York
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