Abstract
The annulus fibrosus exhibits complex osmotic and inelastic effects responsible for unusual transversal behavior with a Poisson’s ratio higher than 0.5 in fibers plane and negative (i.e., auxetic) in lamellae plane. In this paper, we present a new chemo-mechanical approach for the intrinsic osmo-inelastic response of the annulus fibrosus in relation to the microstructure of the layered reinforced soft tissue, the biochemical environment and the mechanical loading conditions. The constitutive model introduces the coupling between the deformation-induced inelastic stress in the tangled extracellular matrix and the stress-free swelling due to internal fluid content variation by osmosis. The proposed formulation is implemented into a finite element code, and numerical simulations on annulus specimens, including explicitly lamellae and interlamellar zones, are presented. To illustrate the capability of the approach to capture experimental observations quantitatively, the simulated results are compared to experimental results obtained by monitoring the full-field strain in annulus specimens using digital image correlation method. Some material constants are found by matching the free swelling in a water bath with different salt concentrations, and others are found by matching tensile results in terms of loading–unloading stress–stretch curve and transversal behavior. The constitutive model is found to successfully capture the variations in osmolarity and strain-rate conditions (both statistically significant, p < 0.05) on the intrinsic response and the auxeticity. The stress/strain patterns in the model simulation provide valuable insights into the role of the interlamellar zone in the osmo-inelastic mechanisms.
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Notes
This inter-dependence of the model constants is a consequence of the strong coupling between the chemical response and the mechanical response.
References
Adam C, Rouch P, Skalli W (2015) Inter-lamellar shear resistance confers compressive stiffness in the intervertebral disc: an image-based modelling study on the bovine caudal disc. J Biomech 48:4303–4308
Argoubi M, Shirazi-Adl A (1996) Poroelastic creep response analysis of a lumbar motion segment in compression. J Biomech 29:1331–1339
Ayotte DC, Ito K, Perren SM, Tepic S (2000) Direction-dependent constriction flow in a poroelastic solid: the intervertebral disc valve. J Biomech Eng 122:587–593
Baldit A, Ambard D, Cherblanc F, Royer P (2014) Experimental analysis of the transverse mechanical behaviour of annulus fibrosus tissue. Biomech Model Mechanobiol 13:643–652
Beckstein JC, Sen S, Schaer TP, Vresilovic EJ, Elliott DM (2008) Comparison of animal discs used in disc research to human lumbar disc: axial compression mechanics and glycosaminoglycan content. Spine 33:166–173
Bergstrom JS, Boyce MC (1998) Constitutive modeling of the large strain time-dependent behavior of elastomers. J Mech Phys Solids 46:931–954
Cantournet S, Boyce MC, Tsou AH (2007) Micromechanics and macromechanics of carbon nanotube-enhanced elastomers. J Mech Phys Solids 55:1321–1339
Costi JJ, Stokes IA, Gardner-Morse M, Laible JP, Scoffone HM, Iatridis JC (2007) Direct measurement of intervertebral disc maximum shear strain in six degrees of freedom: motions that place disc tissue at risk of injury. J Biomech 40:2457–2466
Derrouiche A, Zaouali A, Zaïri F, Ismail J, Chaabane M, Qu Z, Zaïri F (2019) Osmo-inelastic response of the intervertebral disc. Proc Inst Mech Eng Part H J Eng Med 233:332–341
Drost MR, Willems P, Snijders H, Huyghe JM, Janssen JD, Huson A (1995) Confined compression of canine annulus fibrosus under chemical and mechanical loading. J Biomech Eng 117:390–396
Ebara S, Iatridis JC, Setton LA, Foster RJ, Mow VC, Weidenbaum M (1996) Tensile properties of nondegenerate human lumbar anulus fibrosus. Spine 21:452–461
Eberlein R, Holzapfel GA, Schulze-Bauer CAJ (2001) An anisotropic model for annulus tissue and enhanced finite element analyses of intact lumbar disc bodies. Comput Methods Biomech Biomed Eng 4:209–229
Ehlers W, Karajan N, Markert B (2009) An extended biphasic model for charged hydrated tissues with application to the intervertebral disc. Biomech Model Mechanobiol 8:233–251
Elliott DM, Setton LA (2001) Anisotropic and inhomogeneous tensile behavior of the human annulus fibrosus: experimental measurement and material model predictions. J Biomech Eng 123:256–263
Emanuel KS, van der Veen AJ, Rustenburg CME, Smit TH, Kingma I (2018) Osmosis and viscoelasticity both contribute to time-dependent behaviour of the intervertebral disc under compressive load: a caprine in vitro study. J Biomech 70:10–15
Eyre DR (1979) Biochemistry of the intervertebral disc. Int Rev Connect Tissue Res 8:227–291
Frijns AJH, Huyghe JM, Janssen JD (1997) A validation of the quadriphasic mixture theory for intervertebral disc tissue. Int J Eng Sci 35:1419–1429
Galbusera F, Mietsch A, Schmidt H, Wilke HJ, Neidlinger-Wilke C (2013) Effect of intervertebral disc degeneration on disc cell viability: a numerical investigation. Comput Methods Biomech Biomed Eng 16:328–337
Gatt R, Wood MV, Gatt A, Zarb F, Formosa C, Azzopardi KM, Casha A, Agius TP, Schembri-Wismayer P, Attard L, Chockalingam N, Grima JN (2015) Negative Poisson’s ratios in tendons: an unexpected mechanical response. Acta Biomater 24:201–208
Gent AN (1996) A new constitutive relation for rubber. Rubber Chem Technol 69:59–61
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 120:169–180
Guerin HL, Elliott DM (2006) Degeneration affects the fiber reorientation of human annulus fibrosus under tensile load. J Biomech 39:1410–1418
Guerin HL, Elliott DM (2007) Quantifying the contributions of structure to annulus fibrosus mechanical function using a nonlinear, anisotropic, hyperelastic model. J Orthop Res 25:508–516
Guo ZY, Peng XQ, Moran B (2006) A composites-based hyperelastic constitutive model for soft tissue with application to the human annulus fibrosus. J Mech Phys Solids 54:1952–1971
Gurtin ME, Anand L (2005) The decomposition F = FeFp, material symmetry, and plastic irrotationality for solids that are isotropic-viscoplastic or amorphous. Int J Plast 21:1686–1719
Han SK, Chen CW, Labus KM, Puttlitz CM, Chen Y, Hsieh AH (2016) Optical coherence tomographic elastography reveals mesoscale shear strain inhomogeneities in the annulus fibrosus. Spine 41:E770–E777
Holzapfel G, Simo J (1996) Entropy elasticity of isotropic rubber-like solids at finite strains. Comput Methods Appl Mech Eng 132:17–44
Holzapfel GA, Gasser TC, Ogden RW (2000) A new constitutive framework for arterial wall mechanics and a comparative study of material models. J Elast Phys Sci Solids 61:1–48
Holzapfel GA, Schulze-Bauer CAJ, Feigl G, Regitnig P (2005) Single lamellar mechanics of the human lumbar anulus fibrosus. Biomech Model Mechanobiol 3:125–140
Huyghe JM, Janssen JD (1997) Quadriphasic mechanics of swelling incompressible porous media. Int J Eng Sci 35:793–802
Iatridis JC, Setton LA, Weidenbaum M, Mow VC (1997) The viscoelastic behavior of the non-degenerate human lumbar nucleus pulposus in shear. J Biomech 30:1005–1013
Iatridis JC, Laible JP, Krag MH (2003) Influence of fixed charge density magnitude and distribution on the intervertebral disc: applications of a poroelastic and chemical electric (PEACE) model. J Biomech Eng 125:12–24
Inoue H, Takeda T (1975) Three-dimensional observation of collagen framework of lumbar intervertebral discs. Acta Orthop 46:949–956
Kemper AR, McNally C, Duma SM (2007) The influence of strain rate on the compressive stiffness properties of human lumbar intervertebral discs. Biomed Sci Instrum 43:176–181
Klisch SM, Lotz JC (2000) A special theory of biphasic mixtures and experimental results for human annulus fibrosus tested in confined compression. J Biomech Eng 122:180–188
Kraemer J (2009) Intervertebral disk diseases: causes, diagnosis, treatment, and prophylaxis. Thieme, New York
Labus KM, Han SK, Hsieh AH, Puttlitz CM (2014) A computational model to describe the regional interlamellar shear of the annulus fibrosus. J Biomech Eng 136:051009
Lai WM, Hou JS, Mow VC (1991) A triphasic theory for the swelling and deformation behaviors of articular cartilage. J Biomech Eng 113:245–258
Lanir Y (1987) Biorheology and fluid flux in swelling tissues. I. Bicomponent theory for small deformations, including concentration effects. Biorheology 24:173–187
Lee EH (1969) Elastic-plastic deformation at finite strains. J Appl Mech 36:1–6
Lees C, Vincent JF, Hillerton JE (1991) Poisson’s ratio in skin. Biomed Mater Eng 1:19–23
Maroudas A (1976) Balance between swelling pressure and collagen tension in normal and degenerate cartilage. Nature 260:808–809
Mengoni M, Luxmoore BJ, Wijayathunga VN, Jones AC, Broom ND, Wilcox RK (2015) Derivation of inter-lamellar behaviour of the intervertebral disc annulus. J Mech Behav Biomed Mater 48:164–172
Michalek AJ, Buckley MR, Bonassar LJ, Cohen I, Iatridis JC (2009) Measurement of local strains in intervertebral disc anulus fibrosus tissue under dynamic shear: contributions of matrix fiber orientation and elastin content. J Biomech 42:2279–2285
Miehe C (1995) Entropic thermoelasticity at finite strains. Aspects of the formulation and numerical implementation. Comput Methods Appl Mech Eng 120:243–269
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 102:73–84
MSC.Marc (2015) MSC. Marc volume D: user subroutines and special routines. MSC Software Corporation
Nerurkar NL, Mauck RL, Elliott DM (2011) Modeling interlamellar interactions in angle-ply biologic laminates for annulus fibrosus tissue engineering. Biomech Model Mechanobiol 10:973–984
Newell N, Grigoriadis G, Christou A, Carpanen D, Masouros SD (2016) Material properties of bovine intervertebral discs across strain rates. J Mech Behav Biomed Mater 65:824–830
Newell N, Little JP, Christou A, Adams MA, Adam CJ, Masouros SD (2017) Biomechanics of the human intervertebral disc: a review of testing techniques and results. J Mech Behav Biomed Mater 69:420–434
O’Connell GD, Guerin HL, Elliott DM (2009) Theoretical and uniaxial experimental evaluation of human annulus fibrosus degeneration. J Biomech Eng 131:1–7
O’Connell GD, Sen S, Elliott DM (2012) Human annulus fibrosus material properties from biaxial testing and constitutive modeling are altered with degeneration. Biomech Model Mechanobiol 11:493–503
Peng XQ, Guo ZY, Moran B (2006) An anisotropic hyperelastic constitutive model with fiber-matrix shear interaction for the human annulus fibrosus. J Appl Mech 73:815–824
Pezowicz CA, Robertson PA, Broom ND (2006) The structural basis of interlamellar cohesion in the intervertebral disc wall. J Anat 208:317–330
Pyrz M, Zaïri F (2007) Identification of viscoplastic parameters of phenomenological constitutive equations for polymers by deterministic and evolutionary approach. Model Simul Mater Sci Eng 15:85–103
Race A, Broom ND, Robertson P (2000) Effect of loading rate and hydration on the mechanical properties of the disc. Spine 25:662–669
Schmidt H, Reitmaier S, Graichen F, Shirazi-Adl A (2016) Review of the fluid flow within intervertebral discs—How could in vitro measurements replicate in vivo? J Biomech 49:3133–3146
Schollum ML, Robertson PA, Broom ND (2009) A microstructural investigation of intervertebral disc lamellar connectivity: detailed analysis of the translamellar bridges. J Anat 214:805–816
Schroeder Y, Sivan S, Wilson W, Merkher Y, Huyghe JM, Maroudas A, Baaijens FPT (2007) Are disc pressure, stress, and osmolarity affected by intra- and extrafibrillar fluid exchange? J Orthop Res 25:1317–1324
Shirazi-Adl A, Ahmed AM, Shrivastava SC (1986) Mechanical response of a lumbar motion segment in axial torque alone and combined with compression. Spine 11:914–927
Simon BR, Wu JS, Carlton MW, Evans JH, Kazarian LE (1985) Structural models for human spinal motion segments based on a poroelastic view of the intervertebral disk. J Biomech Eng 107:327–335
Singha K, Singha M (2012) Biomechanism profile of intervertebral disc’s (IVD): strategies to successful tissue engineering for spinal healing by reinforced composite structure. J Tissue Sci Eng 3:1000118
Skaggs DL, Weidenbaum M, Iatridis JC, Ratcliffe A, Mow VC (1994) Regional variation in tensile properties and biochemical composition of the human lumbar anulus fibrosus. Spine 19:1310–1319
Tavakoli J, Costi JJ (2018) New findings confirm the viscoelastic behaviour of the inter-lamellar matrix of the disc annulus fibrosus in radial and circumferential directions of loading. Acta Biomater 71:411–419
Tavakoli J, Elliott DM, Costi JJ (2016) Structure and mechanical function of the inter-lamellar matrix of the annulus fibrosus in the disc. J Orthop Res 34:1307–1315
Timmins LH, Wu Q, Yeh AT, Moore JE, Greenwald SE (2010) Structural inhomogeneity and fiber orientation in the inner arterial media. Am J Physiol Heart Circ Physiol 298:1537–1545
Vergari C, Mansfield J, Meakin JR, Winlove PC (2016) Lamellar and fibre bundle mechanics of the annulus fibrosus in bovine intervertebral disc. Acta Biomater 37:14–20
Vergroesen PPA, Emanuel KS, Peeters M, Kingma I (2018) Are axial intervertebral disc biomechanics determined by osmosis? J Biomech 70:4–9
Veronda DR, Westmann RA (1970) Mechanical characterization of skin-finite deformations. J Biomech 3:111–124
Wagner DR, Lotz JC (2004) Theoretical model and experimental results for the nonlinear elastic behavior of human annulus fibrosus. J Orthop Res 22:901–909
Wang JL, Parnianpour M, Shirazi-Adl A, Engin AE, Li S, Patwardhan A (1997) Development and validation of a viscoelastic finite element model of an L2/L3 motion segment. Theor Appl Fract Mech 28:81–93
Weiss JA, Maker BN, Govindjee S (1996) Finite element implementation of incompressible, transversely isotropic hyperelasticity. Comput Methods Appl Mech Eng 135:107–128
Williams JL, Lewis JL (1982) Properties and an anisotropic model of cancellous bone from the proximal tibial epiphysis. J Biomech Eng 104:50–56
Yu J, Fairbank JC, Roberts S, Urban JPG (2005) The elastic fiber network of the anulus fibrosus of the normal and scoliotic human intervertebral disc. Spine 30:1815–1820
Yu J, Tirlapur U, Fairbank JC, Handford P, Roberts S, Winlove CP, Cui Z, Urban J (2007) Microfibrils, elastin fibres and collagen fibres in the human intervertebral disc and bovine tail disc. J Anat 210:460–471
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Derrouiche, A., Zaïri, F. & Zaïri, F. A chemo-mechanical model for osmo-inelastic effects in the annulus fibrosus. Biomech Model Mechanobiol 18, 1773–1790 (2019). https://doi.org/10.1007/s10237-019-01176-8
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DOI: https://doi.org/10.1007/s10237-019-01176-8