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On the theory of reactive mixtures for modeling biological growth

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Biomechanics and Modeling in Mechanobiology Aims and scope Submit manuscript

An Erratum to this article was published on 25 July 2007

Abstract

Mixture theory, which can combine continuum theories for the motion and deformation of solids and fluids with general principles of chemistry, is well suited for modeling the complex responses of biological tissues, including tissue growth and remodeling, tissue engineering, mechanobiology of cells and a variety of other active processes. A comprehensive presentation of the equations of reactive mixtures of charged solid and fluid constituents is lacking in the biomechanics literature. This study provides the conservation laws and entropy inequality, as well as interface jump conditions, for reactive mixtures consisting of a constrained solid mixture and multiple fluid constituents. The constituents are intrinsically incompressible and may carry an electrical charge. The interface jump condition on the mass flux of individual constituents is shown to define a surface growth equation, which predicts deposition or removal of material points from the solid matrix, complementing the description of volume growth described by the conservation of mass. A formulation is proposed for the reference configuration of a body whose material point set varies with time. State variables are defined which can account for solid matrix volume growth and remodeling. Constitutive constraints are provided on the stresses and momentum supplies of the various constituents, as well as the interface jump conditions for the electrochemical potential of the fluids. Simplifications appropriate for biological tissues are also proposed, which help reduce the governing equations into a more practical format. It is shown that explicit mechanisms of growth-induced residual stresses can be predicted in this framework.

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References

  • Alberts B, Johnson A, Lewis J, Raff M, Roberts K, Walter P (2002) Molecular biology of the cell. Garland Science, Oxford

    Google Scholar 

  • Ateshian GA, Chahine NO, Basalo IM, Hung CT (2004) The correspondence between equilibrium biphasic and triphasic material properties in mixture models of articular cartilage. J Biomech 37(3):391–400

    Article  Google Scholar 

  • Ateshian GA, Likhitpanichkul M, Hung CT (2006) A mixture theory analysis for passive transport in osmotic loading of cells. J Biomech 39(3):464–475

    Google Scholar 

  • Bachrach NM, Mow VC, Guilak F (1998) Incompressibility of the solid matrix of articular cartilage under high hydrostatic pressures. J Biomech 31(5):445–451

    Article  Google Scholar 

  • Baek S, Rajagopal KR, Humphrey JD (2006) A theoretical model of enlarging intracranial fusiform aneurysms. J Biomech Eng 128(1):142–149

    Article  Google Scholar 

  • Basser PJ, Schneiderman R, Bank RA, Wachtel E, Maroudas A (1998) Mechanical properties of the collagen network in human articular cartilage as measured by osmotic stress technique. Arch Biochem Biophys 351(2):207–219

    Article  Google Scholar 

  • Bedford A, Drumheller DS (1983) Recent advances theories of immiscible and structured mixtures. Int J Eng Sci 21(8): 863–960

    Article  MATH  MathSciNet  Google Scholar 

  • Biot M (1962) Mechanics of deformation and acoustic propagation in porous media. J Appl Phys 33(4):1482–1498

    Article  MATH  MathSciNet  Google Scholar 

  • Biot MA (1972) Theory of finite deformations of porous solids. Indiana U Math J 21(7):597–620

    Article  MathSciNet  Google Scholar 

  • Bowen RM (1968) Thermochemistry of reacting materials. J Chem Phys 49(4):1625–1637

    Article  Google Scholar 

  • Bowen RM (1969) The thermochemistry of a reacting mixture of elastic materials with diffusion. Arch Ration Mech An 34(2):97–127

    Article  MATH  MathSciNet  Google Scholar 

  • Bowen RM (1976) Theory of mixtures. In: Eringen AE (ed) Continuum physics, vol 3. Academic, New York, pp 1–127

    Google Scholar 

  • Bowen RM (1980) Incompressible porous media models by use of the theory of mixtures. Int J Eng Sci 18(9):1129–1148

    Article  MATH  Google Scholar 

  • Bowen RM (1982) Compressible porous media models by use of the theory of mixtures. Int J Eng Sci 20(6):697–735

    Article  MATH  Google Scholar 

  • Chahine NO, Wang CC, Hung CT, Ateshian GA (2004) Anisotropic strain-dependent material properties of bovine articular cartilage in the transitional range from tension to compression. J Biomech 37(8):1251–1261

    Article  Google Scholar 

  • Chen YC, Hoger A (2000) Constitutive functions of elastic materials in finite growth and deformation. J Elast 59: 175–193

    Article  MATH  Google Scholar 

  • Cowin SC (1983) The mechanical and stress adaptive properties of bone. Ann Biomed Eng 11:263–295

    Article  Google Scholar 

  • Cowin SC, Hegedus DH (1976) Bone remodeling-1, theory of adaptive elasticity. J Elast 6(3):313–326

    Article  MATH  MathSciNet  Google Scholar 

  • Curnier A, He QC, Zysset P (1995) Conewise linear elastic materials. J Elast 37:1–38

    Article  MATH  MathSciNet  Google Scholar 

  • Dugdale D (1993) Essentials of electromagnetism. American Insitute of Physics MacMillan, New York

  • Eisenberg SR, Grodzinsky AJ (1987) The kinetics of chemically induced nonequilibrium swelling of articular cartilage and corneal stroma. J Biomech Eng 109(1):79–89

    Google Scholar 

  • Eringen AE (1998) A mixture theory of electromagnetism and superconductivity. Int J Eng Sci 36(5/6):525–543

    Article  MathSciNet  Google Scholar 

  • Eringen A, Ingram J (1965) Continuum theory of chemically reacting media-1. Int J Eng Sci 3:197–212

    Article  Google Scholar 

  • Fung YC (1990) Biomechanics: motion, flow, stress, and growth. Springer, Heidelberg, New York

    MATH  Google Scholar 

  • Garikipati K, Arruda E, Grosh K, Narayanan H, Calve S (2004) A continuum treatment of growth in biological tissue: the coupling of mass transport and mechanics. J Mech Phys Solids 52(7):1595–1625

    Article  MATH  MathSciNet  Google Scholar 

  • Grimshaw PE, Grodzinsky AJ, Yarmush ML, Yarmush DM (1989) Dynamic membranes for protein transport: chemical and electrical control. Chem Eng Sci 44(4):827–840

    Article  Google Scholar 

  • Grimshaw PE, Nussbaum JH, Grodzinsky AJ, Yarmush ML (1990) Kinetics of electrically and chemically induced swelling in polyelectrolyte gels. J Chem Phys 93(6):4462–4472

    Article  Google Scholar 

  • Grodzinsky AJ (1983) Electromechanical and physicochemical properties of connective tissue. Crit Rev Biomed Eng 9(2):133–199

    Google Scholar 

  • Gu WY, Lai WM, Mow VC (1993) Transport of fluid and ions through a porous-permeable charged-hydrated tissue, and streaming potential data on normal bovine articular cartilage. J Biomech 26(6):709–723

    Article  Google Scholar 

  • Gu WY, Lai WM, Mow VC (1997) A triphasic analysis of negative osmotic flows through charged hydrated soft tissues. J Biomech 30(1):71–78

    Article  Google Scholar 

  • Gu WY, Lai WM, Mow VC (1998) A mixture theory for charged-hydrated soft tissues containing multi-electrolytes: passive transport and swelling behaviors. J Biomech Eng 120(2): 169–180

    Google Scholar 

  • Guillou A, Ogden RW (2006) Growth in soft biological tissue and residual stress development. In: Holzapfel GA, Ogden RW (eds) Mechanics of biological tissue. Springer, Heidelberg, pp 47–62

    Chapter  Google Scholar 

  • van Holde KE, Johnson WC, Ho PS (1998) Principles of physical biochemistry. Prentice-Hall, USA

    Google Scholar 

  • Holmes MH, Mow VC (1990) The nonlinear characteristics of soft gels and hydrated connective tissues in ultrafiltration. J Biomech 23(11):1145–1156

    Article  Google Scholar 

  • Hou JS, Holmes MH, Lai WM, Mow VC (1989) Boundary conditions at the cartilage-synovial fluid interface for joint lubrication and theoretical verifications. J Biomech Eng 111(1):78–87

    Google Scholar 

  • Hsu FH (1968) The influences of mechanical loads on the form of a growing elastic body. J Biomech 1(4):303–311

    Article  Google Scholar 

  • Humphrey JD (2002) Cardiovascular solid mechanics: cells, tissues, and organs. Springer, Heidelberg

    Google Scholar 

  • Humphrey JD, Rajagopal KR (2002) A constrained mixture model for growth and remodeling of soft tissues. Math Mod Meth Appl S 12(3):407–430

    Article  MATH  MathSciNet  Google Scholar 

  • Humphrey JD, Rajagopal KR (2003) A constrained mixture model for arterial adaptations to a sustained step change in blood flow. Biomech Model Mechanobiol 2(2):109–126

    Article  Google Scholar 

  • Huyghe JM, Janssen JD (1997) Quadriphasic mechanics of swelling incompressible porous media. Int J Eng Sci 35(8): 793–802

    Article  MATH  Google Scholar 

  • Incropera FP, DeWitt DP (1981) Fundamentals of heat transfer. Wiley, New York

    Google Scholar 

  • Katzir-Katchalsky A, Curran PF (1965) Nonequilibrium thermodynamics in biophysics. Harvard books in biophysics; no. 1. Harvard University Press, Cambridge

  • Kenyon DE (1976) Transient filtration in a porous elastic cylinder. J Appl Mech 43 Ser E(4):594–598

  • Klisch SM, Chen SS, Sah RL, Hoger A (2003) A growth mixture theory for cartilage with application to growth-related experiments on cartilage explants. J Biomech Eng 125(2):169–179

    Article  Google Scholar 

  • Lai WM, Hou JS, Mow VC (1991) A triphasic theory for the swelling and deformation behaviors of articular cartilage. J Biomech Eng 113(3):245–158

    Google Scholar 

  • Lee RT, Yamamoto C, Feng Y, Potter-Perigo S, Briggs WH, Landschulz KT, Turi TG, Thompson JF, Libby P, Wight TN (2001) Mechanical strain induces specific changes in the synthesis and organization of proteoglycans by vascular smooth muscle cells. J Biol Chem 276(17):13847–13851

    Google Scholar 

  • Lemon G, King JR, Byrne HM, Jensen OE, Shakesheff KM (2006) Mathematical modelling of engineered tissue growth using a multiphase porous flow mixture theory. J Math Biol 52(5):571–594

    Article  MATH  MathSciNet  Google Scholar 

  • Liu IS (1980) On chemical potential and incompressible porous media. J Mec 19(2):327–342

    Google Scholar 

  • Maroudas AI (1976) Balance between swelling pressure and collagen tension in normal and degenerate cartilage. Nature 260(5554):808–809

    Article  Google Scholar 

  • Meerveld JV, Molenaar MM, Huyghe JM, Baaijens FPT (2003) Analytical solution of compression, free swelling and electrical loading of saturated charged porous media. Tranport Porous Med 50:111–126

    Article  Google Scholar 

  • Mow VC, Lai M (1990) Biorheology of swelling tissue. Biorheology 27(1):110–119

    Google Scholar 

  • Mow VC, Kuei SC, Lai WM, Armstrong CG (1980) Biphasic creep and stress relaxation of articular cartilage in compression: theory and experiments. J Biomech Eng 102(1):73–84

    Article  Google Scholar 

  • Mueller I (1975) Thermodynamics of mixtures of fluids. J Mec 14(2):267–303

    MATH  Google Scholar 

  • Narmoneva DA, Wang JY, Setton LA (1999) Nonuniform swelling-induced residual strains in articular cartilage. J Biomech 32(4):401–408

    Article  Google Scholar 

  • Overbeek JT (1956) The donnan equilibrium. Prog Biophys Biophys Chem 6:57–84

    Google Scholar 

  • Porterfield SP, Calhoon TB, Weiss HS (1968) Changes in connective tissue colloidal charge density with atherosclerosis and age. Am J Physiol 215(2):324–329

    Google Scholar 

  • Rodriguez EK, Hoger A, McCulloch AD (1994) Stress-dependent finite growth in soft elastic tissues. J Biomech 27(4):455–467

    Article  Google Scholar 

  • Setton LA, Tohyama H, Mow VC (1998) Swelling and curling behaviors of articular cartilage. J Biomech Eng 120(3):355–361

    Google Scholar 

  • Skalak R, Dasgupta G, Moss M, Otten E, Dullumeijer P, Vilmann H (1982) Analytical description of growth. J Theor Biol 94(3):555–577

    Article  Google Scholar 

  • Tinoco Jr I, Sauer K, Wang JC (1995) Physical chemistry: principles and applications in biological sciences. Prentice-Hall, USA

    Google Scholar 

  • Truesdell C (1984) Rational thermodynamics, 2nd edn., Chapt. 7, Springer, Heidelberg, pp 365–395

  • Truesdell C, Noll W (1992) The non-linear field theories of mechanics, 2nd edn. Springer, Heidelberg

    MATH  Google Scholar 

  • Truesdell C, Toupin R (1960) The classical field theories. In: Flugge S(ed) Handbuch der physik, vol III/1. Springer, Heidelberg

    Google Scholar 

  • Weiss JA, Gardiner JC (2001) Computational modeling of ligament mechanics. Crit Rev Biomed Eng 29(3):303–371

    Google Scholar 

  • Wight TN, Ross R (1975) Proteoglycans in primate arteries. i. ultrastructural localization and distribution in the intima. J Cell Biol 67(3):660–674

    Article  Google Scholar 

  • Yao LY, Moody C, Schonherr E, Wight TN, Sandell LJ (1994) Identification of the proteoglycan versican in aorta and smooth muscle cells by dna sequence analysis, in situ hybridization and immunohistochemistry. Matrix Biol 14(3):213–225

    Article  Google Scholar 

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  1. Department of Mechanical Engineering, Columbia University, 500 West 120th St., MC4703, 220 S.W. Mudd, New York, NY, 10027, USA

    Gerard A. Ateshian

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  1. Gerard A. Ateshian
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Correspondence to Gerard A. Ateshian.

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An erratum to this article can be found at http://dx.doi.org/10.1007/s10237-007-0092-z

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Ateshian, G.A. On the theory of reactive mixtures for modeling biological growth. Biomech Model Mechanobiol 6, 423–445 (2007). https://doi.org/10.1007/s10237-006-0070-x

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  • DOI: https://doi.org/10.1007/s10237-006-0070-x

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