A hyperelastic biphasic fibre-reinforced model of articular cartilage considering distributed collagen fibre orientations: Continuum basis, computational aspects and applications

Computer Methods in Biomechanics and Biomedical Engineering

Volumn 16, Issue 12, 2013, Pages 1344-1361

  Pierce, David M ^a   Ricken, Tim ^b   Holzapfel, Gerhard A ^a,c  

^a GRAZ UNIVERSITY OF TECHNOLOGY   (Austria)

^b TU DORTMUND UNIVERSITY   (Germany)

^c ROYAL INSTITUTE OF TECHNOLOGY   (Sweden)

Author keywords

biphasic; cartilage; collagen fibre; constitutive modelling; finite element simulation; homogenisation

Indexed keywords

ARTICULAR CARTILAGES; BIPHASIC; COLLAGEN FIBRES; COMPUTATIONAL ASPECTS; FINITE ELEMENT IMPLEMENTATION; FINITE ELEMENT SIMULATIONS; ISOTROPIC DIFFUSIVITY; MULTIPHASE MATERIALS;

CARTILAGE; COLLAGEN; CONSTITUTIVE MODELS; DIFFUSION IN LIQUIDS; GLYCOPROTEINS; HOMOGENIZATION METHOD; REINFORCEMENT; TENSILE STRENGTH;

FIBERS;

COLLAGEN FIBER; PROTEOGLYCAN;

ARTICLE; ARTICULAR CARTILAGE; COMPUTER MODEL; HUMAN; HYPERELASTIC BIPHASIC FIBRE REINFORCED MODEL; MATHEMATICAL MODEL; MEMBRANE PERMEABILITY; NONLINEAR SYSTEM; PRIORITY JOURNAL; TENSION; TISSUE PRESSURE;

ANISOTROPY; CARTILAGE, ARTICULAR; COLLAGEN; COMPUTATIONAL BIOLOGY; ELASTICITY; FINITE ELEMENT ANALYSIS; HUMANS; MODELS, BIOLOGICAL; WEIGHT-BEARING;

EID: 84877308884     PISSN: 10255842     EISSN: 14768259     Source Type: Journal    
DOI: 10.1080/10255842.2012.670854     Document Type: Article

References (106)

1. AbdullahOM, OthmanSF, ZhouXJ, MaginRL. 2007. Diffusion tensor imaging as an early marker for osteoarthritis. Proc Int Soc Magn Reson Med. 15:814.
2. AbousleimanY, ChengAHD, JiangC, RoegiersJC. 1996. Poroviscoelastic analysis of borehole and cylinder problems. Acta Mech. 109(1-4):199-219.
3. AndersonAE, EllisBJ, MaasSA, PetersCL, WeissJA. 2008. Validation of finite element predictions of cartilage contact pressure in the human hip joint. J Biomech Eng. 130:051008.
4. AndersonAE, EllisBJ, MaasSA, WeissJA. 2010. Effects of idealized joint geometry on finite element predictions of cartilage contact stresses in the hip. J Biomech. 43:1351-1357.
5. AteshianGA, ChahineNO, BasaloIM, HungCT. 2004. The correspondence between equilibrium biphasic and triphasic material properties in mixture models of articular cartilage. J Biomech. 37:391-400.
6. AteshianGA, EllisBJ, WeissJA. 2007. Equivalence between short-time biphasic and incompressible elastic material responses. J Biomech Eng. 129:405-412.
7. AteshianGA, RajanV, ChahineNO, CanalCE, HungCT. 2009. Modeling the matrix of articular cartilage using a continuous fiber angular distribution predicts many observed phenomena. J Biomech Eng. 131:61003.
8. AteshianGA, WardenWH, KimJJ, GrelsamerRP, MowVC. 1997. Finite deformation biphasic material properties of bovine articular cartilage from confined compression experiments. J Biomech. 30:1157-1164.
9. AteshianGA, WeissJA. 2010. Anisotropic hydraulic permeability under finite deformation. J Biomech Eng. 132:111004.
10. AthanasiouKA, DarlingEM, HuJC2010. Articular cartilage tissue engineering. San Rafael: Morgan & Claypool.
11. AthanasiouKA, RosenwasserMP, BuckwalterJA, MalininTI, MowVC. 1991. Interspecies comparisons of in situ intrinsic mechanical properties of distal femoral cartilage. J Orthop Res. 9:330-340.
12. BachrachNM, MowVC, GuilakF. 1998. Incompressibility of the solid matrix of articular cartilage under high hydrostatic pressures. J Biomech. 31:445-451.
13. BettenJ. 1987. Formulation of anisotropic constitutive equations. In: BoehlerJP, editor. Applications of tensor functions in solid mechanics. CISM courses and lectures no. 292. Wien: Springer-Verlag. p. 227-250.
14. BishopAW. 1959. The principle of effective stress. Tek Ukebl. 39:859-863.
15. BluhmJ. 2002. Modelling of saturated thermo-elastic porous solids with different phase temperatures. In: EhlersW, BluhmJ, editors. Porous media: theory, experiments and numerical applications. Berlin, Heidelberg, New York: Springer-Verlag. p. 87-118.
16. BoehlerJP. 1987. Introduction to the invariant formulation of anisotropic constitutive equations. In: BoehlerJP, editor. Applications of tensor functions in solid mechanics. CISM courses and lectures no. 292. Wien: Springer-Verlag. p. 13-30.
17. BowenRM. 1980. Incompressible pourous media models by use of theory of mixture. Int J Eng Sci. 18:1129-1148.
18. BowenRM. 1982. Compressible porous media models by use of the theory of mixtures. Int J Eng Sci. 20:697-735.
19. BuckleyMR, GleghornJP, CohenI, BonassarLJ. 2007. Depth dependence of shear properties in articular cartilage. In: Transactions of the 53rd annual meeting of the orthopaedic research society. San Diego, CA.
20. ChahineNO, ChenFH, HungCT, AteshianGA. 2005. Direct measurement of osmotic pressure of glycosaminoglycan solutions by membrane osmometry at room temperature. Biophys J. 89:1543-1550.
21. ChahineNO, WangCC, HungCT, AteshianGA. 2004. Anisotropic strain-dependent material properties of bovine articular cartilage in the transitional range from tension to compression. J Biomech. 37:1251-1261.
22. ChapelleD, GerbeauJF, Sainte-MarieJ, Vignon-ClementelIE. 2010. A poroelastic model valid in large strains with applications to perfusion in cardiac modeling. J Vasc Interv Radiol. 14:1427-1432.
23. ChenAC, BaeWC, SchinaglRM, SahRL. 2001. Depth- and strain-dependent mechanical and electromechanical properties of full-thickness bovine articular cartilage in confined compression. J Biomech. 34:1-12.
24. ChenY, ChenX, HisadaT. 2006. Non-linear finite element analysis of mechanical electrochemical phenomena in hydrated soft tissues based on triphasic theory. Int J Numer Methods Eng. 65:147-173.
25. ChenAHD, DetournayE. 1998. On singular integral equations and fundamental solutions of poroelasticity. Int J Solids Struct. 35(34/35):4521-4555.
26. CoussyO. 2004. Poromechanics. Chichester: John Wiley & Sons.
27. de BoerR. 2000. Theory of porous media. Highlights in the historical development and current state. Heidelberg: Springer-Verlag.
28. DelfinoA, StergiopulosN, MooreJE, Jr, MeisterJJ. 1997. Residual strain effects on the stress field in a thick wall finite element model of the human carotid bifurcation. J Biomech. 30:777-786.
29. DemirayH. 1972. A note on the elasticity of soft biological tissues. J Biomech. 5:309-311.
30. deVisserSK, BowdenJC, Wentrup-BryneE, RintoulL, BostromT, PopeJM, MomotKI. 2008a. Anisotropy of collagen fibre alignment in bovine cartilage: comparison of polarised light microscopy and spatially resolved diffusion-tensor measurements. Osteoarthritis Cartilage. 16:689-697.
31. deVisserSK, CrawfordRW, PopeJM. 2008b. Structural adaptations in compressed articular cartilage measured by diffusion tensor imaging. Osteoarthritis Cartilage. 16:83-89.
32. EhlersW. 1989. Poröse Medien - ein kontinuumsmechanisches Modell auf der Basis der MischungstheorieUniversität GH Essen. Forschungsbericht aus dem Fachbereich Bauwesen. 47.
33. EhlersW. 1993. Constitutive equations for granular materials in geomechanical context. In: HutterK, editor. Continuum mechanics in environmental sciences and geophysics. CISM courses and lectures no. 337. Wien: Springer-Verlag. p. 313-402.
34. EhlersW. 2002. Foundations of multiphasic and porous materials. In: EhlersW, BluhmJ, editors. Porous media: theory, experiments and numerical applications. Berlin: Springer-Verlag. p. 3-86.
35. EhlersW, EipperG. 1997. Finite elastizität bei uidgesättigten hochporösen festkörpern. Z Angew Math Mech. 77:79-80.
36. EhlersW, EipperG. 1999. Finite elastic deformations in liquid-saturated and empty porous solids. Transport Porous Med. 34:179-191.
37. EhlersW, KarajanN, MarkertB. 2009. An extended biphasic model for charged hydrated tissues with application to the intervertebral disc. Biomech Model Mechanobiol. 8:233-251.
38. Eipper G. 1998. Theorie und Numerik finiter elastischer Deformationen in fluidgesättigten porösen FestkörpernUniversität Stuttgart. Bericht Nr. II-1 aus dem Institut für Mechanik (Bauwesen).
39. ElliottDM, NarmonevaDA, SettonLA. 2002. Direct measurement of the Poisson's ratio of human patella cartilage in tension. J Biomech Eng. 124:223-228.
40. ErneOK, ReidJB, EhmkeLW, SommersMB, MadeySM, BottlangM. 2005. Depth-dependent strain of patellofemoral articular cartilage in unconfined compression. J Biomech. 38:667-672.
41. FedericoS, GasserTC. 2010. Nonlinear elasticity of biological tissues with statistical fibre orientation. J R Soc Interface. 7:955-966.
42. FedericoS, GrilloA. 2012. Elasticity and permeability of porous fibre reinforced materials under large deformations. Mech Mater. 44:58-71.
43. FedericoS, HerzogW. 2008. On the anisotropy and inhomogeneity of permeability in articular cartilage. Biomech Model Mechanobiol. 7:367-378.
44. FilidoroL, DietrichO, WeberJ, RauchE, OetherT, WickM, ReiserMF, GlaserC. 2005. High-resolution diffusion tensor imaging of human patellar cartilage: feasibility and preliminary findings. Magn Reson Med. 53:993-998.
45. GarcíaJJ, CortésDH. 2007. A biphasic viscohyperelastic fibril-reinforced model for articular cartilage: formulation and comparison with experimental data. J Biomech. 40:1737-1744.
46. GasserTC, OgdenRW, HolzapfelGA. 2006. Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. J R Soc Interface. 3:15-35.
47. Goreham-VossCM, McKinleyTO, BrownTD. 2007. A finite element exploration of cartilage stress near an articular incongruity during unstable motion. J Biomech. 40:3438-3447.
48. GuKB, LiLP. 2011. A human knee joint model considering fluid pressure and fiber orientation in cartilages and menisci. Med Eng Phys. 33:497-503.
49. HanSK, FedericoS, EpsteinM, HerzogW. 2005. An articular cartilage contact model based on real surface geometry. J Biomech. 38:179-184.
50. HassanizadehSM, GrayWG. 1979a. General conservation equations for multi-phase systems: 1. Averaging procedure. Adv Water Resour. 2:131-144.
51. HassanizadehSM, GrayWG. 1979b. General conservation equations for multi-phase systems: 2. Mass, momenta, energy, and entropy equations. Adv Water Resour. 2:191-208.
52. HassanizadehSM, GrayWG. 1980. General conservation equations for multi-phase systems: 3. Constitutive theory for porous media flow. Adv Water Resour. 3:25-40.
53. HassanizadehSM, GrayWG. 1990. Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries. Adv Water Resour. 13:169-186.
54. HolmesMH, LaiWM, MowVC. 1985. Singular perturbation analysis of the nonlinear, flow-dependent compressive stress relaxation behavior of articular cartilage. J Biomed Eng. 107:206-218.
55. HolzapfelGA, SommerG, GasserCT, RegitnigP. 2005a. Determination of the layer-specific mechanical properties of human coronary arteries with non-atherosclerotic intimal thickening, and related constitutive modelling. Am J Physiol Heart Circ Physiol. 289:H2048-H2058.
56. HolzapfelGA, StadlerM, GasserTC. 2005b. Changes in the mechanical environment of stenotic arteries during interaction with stents: computational assessment of parametric stent design. J Biomech Eng. 127:166-180.
57. HuangCY, MowVC, AteshianGA. 2001. The role of flow-independent viscoelasticity in the biphasic tensile and compressive responses of articular cartilage. J Biomech Eng. 123:410-417.
58. HuangCY, SoltzMA, KopaczM, MowVC, AteshianGA. 2003. Experimental verification of the roles of intrinsic matrix viscoelasticity and tension-compression nonlinearity in the biphasic response of cartilage. J Biomech Eng. 125:84-93.
59. HuangCY, StankiewiczA, AteshianGA, MowVC. 2005. Anisotropy, inhomogeneity, and tension-compression nonlinearity of human glenohumeral cartilage in finite deformation. J Biomech. 38:799-809.
60. HumphreyJD. 2002. Cardiovascular solid mechanics. Cells, tissues, and organs. New York: Springer-Verlag.
61. JoshiMD, SuhJK, MaruiT, WooSLY. 1995. Interspecies variation of compressive biomechanical properties of the meniscus. J Biomed Mater Res. 29:823-828.
62. JulkunenP, KivirantaP, WilsonW, JurvelinJS, KorhonenRK. 2007. Characterization of articular cartilage by combining microscopic analysis with a fibril-reinforced finite-element model. J Biomech. 40:1862-1870.
63. JulkunenP, WilsonW, JurvelinJS, RieppoJ, QuCJ, LammiMJ, KorhonenRK. 2008. Stress-relaxation of human patellar articular cartilage in unconfined compression: prediction of mechanical response by tissue composition and structure. J Biomech. 41:1978-1986.
64. JurvelinJS, BuschmannMD, HunzikerEB. 1997. Optical and mechanical determination of Poisson's ratio of adult bovine humeral articular cartilage. J Biomech. 30:235-241.
65. KorhonenRK, LaasanenMS, TöyräsJ, LappalainenR, HelminenHJ, JurvelinJS. 2003. Fibril reinforced poroelastic model predicts specifically mechanical behavior of normal, proteoglycan depleted and collagen degraded articular cartilage. J Biomech. 36:1373-1379.
66. LaiWM, HouJS, MowVC. 1991. A triphasic theory for the swelling and deformation behaviours of articular cartilage. J Biomech Eng. 113:245-258.
67. LiLP, BuschmannMD, Shirazi-AdlA. 2000. A fibril reinforced nonhomogeneous poroelastic model for articular cartilage: inhomogeneous response in unconfined compression. J Biomech. 33:1533-1541.
68. LiLP, CheungJTM, HerzogW. 2009. Three-dimensional fibril-reinforced finite element model of articular cartilage. Med Biol Eng Comput. 47:607-615.
69. LiLP, HerzogW. 2004. The role of viscoelasticity of collagen fibers in articular cartilage: theory and numerical formulation. Biorheology. 41:181-194.
70. LiG, LopezO, RubashH. 2001. Variability of a three-dimensional finite element model constructed using magnetic resonance images of a knee for joint contact stress analysis. J Biomech Eng. 123:341-346.
71. LilledahlMB, PierceDM, RickenT, HolzapfelGA, de Lange DaviesC. 2011. Structural analysis of articular cartilage using multiphoton microscopy: input for biomechanical modeling. IEEE Trans Med Imaging. 30(9):1635-1648.
72. MandaK, RydL, ErikssonA. 2011. Finite element simulations of a focal knee resurfacing implant applied to localized cartilage defects in a sheep model. J Biomech. 44:794-801.
73. MansourJM. 2008. Biomechanics of cartilage. In: OatisCA, editor. Kinesiology: The Mechanics and Pathomechanics of Human Movement, 2nd Ed., Philadelphia: Lippincott Williams and Wilkins. p. 69-83.
74. MarkertB. 2007. A constitutive approach to 3-D nonlinear fluid flow through finite deformable porous continua. Transport Porous Med. 70:427-450.
75. MederR, deVisserSK, BowdenJC, BostromT, PopeJM. 2006. Diffusion tensor imaging of articular cartilage as a measure of tissue microstructure. Osteoarthritis Cartilage. 14:875-881.
76. MononenM, MikkolaM, JulkunenP, OjalaR, NieminenM, JurvelinJ, KorhonenR. 2012. Effect of superficial collagen patterns and fibrillation of femoral articular cartilage on knee joint mechanics - a 3D finite element analysis. J Biomech. 45(3):579-587.
77. MowVC, GibbsMC, LaiWM, ZhuWB, AthanasiouKA. 1989. Biphasic indentation of articular cartilage-II. A numerical algorithm and an experimental study. J Biomech. 22:853-861.
78. MowVC, GuWY, ChenFH. 2005. Structure and function of articular cartilage and meniscus. In: MowVC, HuiskesR, editors. Basic orthopaedic biomechanics & mechano-biology. 3rd ed. Philadelphia: Lippincott Williams & Wilkins. p. 181-258.
79. ParkS, KrishnanR, NicollSB, AteshianGA. 2003. Cartilage interstitial fluid load support in unconfined compression. J Biomech. 36:1785-1796.
80. PeñaE, CalvoB, MartínezMA, PalancaD, DoblaréM. 2005. Finite element analysis of the effect of meniscal tears and meniscectomies on human knee biomechanics. Clin Biomech. p. 498-507.
81. PierceDM, TrobinW, RayaJG, TrattnigS, BischofH, GlaserC, HolzapfelGA. 2010. DT-MRI based computation of collagen fiber deformation in human articular cartilage: a feasibility study. Ann Biomed Eng. 38:2447-2463.
82. PierceDM, TrobinW, TrattnigS, BischofH, HolzapfelGA. 2009. A phenomenological approach toward patient-specific computational modeling of articular cartilage including collagen fiber tracking. J Biomech Eng. 131:091006.
83. PotterHG, BlackBR, ChongLR. 2009. New techniques in articular cartilage imaging. Clin Sports Med. 28:77-94.
84. ResponteDJ, NatoliRM, AthanasiouKA. 2007. Collagens of articular cartilage: structure, function, and importance in tissue engineering. Crit Rev Biomed Eng. 35:363-411.
85. ReynaudB, QuinnTM. 2006. Anisotropic hydraulic permeability in compressed articular cartilage. J Biomech. 39:131-137.
86. RickenT, BluhmJ. 2009. Evolutional growth and remodeling in multiphase living tissue. Comp Mater Sci. 45:806-811.
87. RickenT, BluhmJ. 2010. Remodeling and growth of living tissue: a multiphase theory. Arch Appl Mech. 80:453-465.
88. RickenT, DahmenU, DirschO. 2010. A biphasic model for sinusoidal liver perfusion remodeling after outflow obstruction. Biomech Model Mechanobiol. 9:435-450.
89. SimoJC, PisterKS. 1984. Remarks on rate constitutive equations for finite deformation problems: computational implications. Comput Methods Appl Mech Engrg. 46:201-215.
90. SkemptonAW. 1960. Terzaghi's discovery of effective stress. In: BjerrumL, CasagrandeA, PeckRB, SkemptonAW, editors. From theory to practice in soil mechanics. New York, London: John Wiley. p. 42-53.
91. SoltzMA, AteshianGA. 1998. Experimental verification and theoretical prediction of cartilage interstitial fluid pressurization at an impermeable contact interface in confined compression. J Biomech. 31:927-934.
92. SoltzMA, AteshianGA. 2000. A conewise linear elasticity mixture model for the analysis of tension-compression nonlineartiy in articular cartilage. J Biomech Eng. 122:576-586.
93. SpencerAJM. 1971. Part III. Theory of invariants. In: EringenAC, editor. Continuum physics. New York: Academic Press. p. 239-353.
94. SunW, ChaikofEL, LevenstonME. 2008. Numerical approximation of tangent moduli for finite element implementations of nonlinear hyperelastic material models. J Biomech Eng. 130:061003.
95. TongJ, CohnertT, RegitnigP, HolzapfelGA. 2011. Effects of age on the elastic properties of the intraluminal thrombus and the thrombus-covered wall in abdominal aortic aneurysms: biaxial extension behavior and material modeling. Eur J Vasc Endovasc Surg. 42:207-219.
96. van LoonR, HuygheJ, WijlaarsM, BaaijensF. 2003. 3D FE implementation of an incompressible quadriphasic mixture model. Int J Numer Methods Eng. 57:1243-1258.
97. WangL, FrittonSP, WeinbaumS, CowinSC. 2003. On bone adaptation due to venous stasis. J Biomech. 36:1439-1451.
98. WhitakerS. 1977. Simultaneous heat, mass, and momentum transfer in porous media: a theory of drying. Adv Heat Transfer. 13:119-203.
99. WilsonW, HuygheJM, van DonkelaarCC. 2007. Depth-dependent compressive equilibrium properties of articular cartilage explained by its composition. Biomech Model Mechanobiol. p. 43-53.
100. WilsonW, van DonkelaarCC, van RietbergenB, HuiskesR. 2005. A fibril-reinforced poroviscoelastic swelling model for articular cartilage. J Biomech. 38:1195-1204.
101. WilsonW, van DonkelaarCC, van RietbergenB, ItoK, HuiskesR. 2004. Stresses in the local collagen network of articular cartilage: a poroviscoelastic fibril-reinforced finite element study. J Biomech. 37:357-366.
102. WongBL, BaeWC, ChunJ, GratzKR, SahRL. 2007. Micro-mechanics of cartilage articulation: effect of degeneration on shear deformation. In: Transactions of the 53rd annual meeting of the orthopaedic research society. San Diego, CA.
103. WongM, PonticielloM, KovanenV, JurvelinJS. 2000. Volumetric changes of articular cartilage during stress relaxation in unconfined compression. J Biomech. 33:1049-1054.
104. ZhengQS, SpencerAJM. 1993a. On the canonical representations for Kronecker powers of orthogonal tensors with application to material symmetry problems. Int J Eng Sci. 4:617-635.
105. ZhengQS, SpencerAJM. 1993b. Tensors which characterize anisotropies. Int J Eng Sci. 5:679-693.
106. ZipfelWR, WilliamsRM, WebbWW. 2003. Nonlinear magic: multiphoton microscopy in the biosciences. Nat Biotechnol. 21:1368-1376.