Biomechanics and Modeling in Mechanobiology

, Volume 18, Issue 6, pp 1773–1790 | Cite as

A chemo-mechanical model for osmo-inelastic effects in the annulus fibrosus

  • Amil Derrouiche
  • Fahmi ZaïriEmail author
  • Fahed Zaïri
Original Paper


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.


Annulus fibrosus Constitutive modeling Osmo-inelastic coupling Transversal behavior Finite element analysis 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 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 48:4303–4308CrossRefGoogle Scholar
  2. Argoubi M, Shirazi-Adl A (1996) Poroelastic creep response analysis of a lumbar motion segment in compression. J Biomech 29:1331–1339CrossRefGoogle Scholar
  3. Ayotte DC, Ito K, Perren SM, Tepic S (2000) Direction-dependent constriction flow in a poroelastic solid: the intervertebral disc valve. J Biomech Eng 122:587–593CrossRefGoogle Scholar
  4. Baldit A, Ambard D, Cherblanc F, Royer P (2014) Experimental analysis of the transverse mechanical behaviour of annulus fibrosus tissue. Biomech Model Mechanobiol 13:643–652CrossRefGoogle Scholar
  5. 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 33:166–173CrossRefGoogle Scholar
  6. Bergstrom JS, Boyce MC (1998) Constitutive modeling of the large strain time-dependent behavior of elastomers. J Mech Phys Solids 46:931–954zbMATHCrossRefGoogle Scholar
  7. Cantournet S, Boyce MC, Tsou AH (2007) Micromechanics and macromechanics of carbon nanotube-enhanced elastomers. J Mech Phys Solids 55:1321–1339zbMATHCrossRefGoogle Scholar
  8. 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 40:2457–2466CrossRefGoogle Scholar
  9. 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 233:332–341CrossRefGoogle Scholar
  10. 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 117:390–396CrossRefGoogle Scholar
  11. Ebara S, Iatridis JC, Setton LA, Foster RJ, Mow VC, Weidenbaum M (1996) Tensile properties of nondegenerate human lumbar anulus fibrosus. Spine 21:452–461CrossRefGoogle Scholar
  12. 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 4:209–229CrossRefGoogle Scholar
  13. Ehlers W, Karajan N, Markert B (2009) An extended biphasic model for charged hydrated tissues with application to the intervertebral disc. Biomech Model Mechanobiol 8:233–251CrossRefGoogle Scholar
  14. Elliott DM, Setton LA (2001) Anisotropic and inhomogeneous tensile behavior of the human annulus fibrosus: experimental measurement and material model predictions. J Biomech Eng 123:256–263CrossRefGoogle Scholar
  15. 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 70:10–15CrossRefGoogle Scholar
  16. Eyre DR (1979) Biochemistry of the intervertebral disc. Int Rev Connect Tissue Res 8:227–291CrossRefGoogle Scholar
  17. Frijns AJH, Huyghe JM, Janssen JD (1997) A validation of the quadriphasic mixture theory for intervertebral disc tissue. Int J Eng Sci 35:1419–1429zbMATHCrossRefGoogle Scholar
  18. 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 16:328–337CrossRefGoogle Scholar
  19. 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 24:201–208CrossRefGoogle Scholar
  20. Gent AN (1996) A new constitutive relation for rubber. Rubber Chem Technol 69:59–61CrossRefGoogle Scholar
  21. 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 120:169–180CrossRefGoogle Scholar
  22. Guerin HL, Elliott DM (2006) Degeneration affects the fiber reorientation of human annulus fibrosus under tensile load. J Biomech 39:1410–1418CrossRefGoogle Scholar
  23. Guerin HL, Elliott DM (2007) Quantifying the contributions of structure to annulus fibrosus mechanical function using a nonlinear, anisotropic, hyperelastic model. J Orthop Res 25:508–516CrossRefGoogle Scholar
  24. 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 54:1952–1971zbMATHCrossRefGoogle Scholar
  25. 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 21:1686–1719zbMATHCrossRefGoogle Scholar
  26. 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–E777CrossRefGoogle Scholar
  27. Holzapfel G, Simo J (1996) Entropy elasticity of isotropic rubber-like solids at finite strains. Comput Methods Appl Mech Eng 132:17–44MathSciNetzbMATHCrossRefGoogle Scholar
  28. 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 61:1–48MathSciNetzbMATHCrossRefGoogle Scholar
  29. Holzapfel GA, Schulze-Bauer CAJ, Feigl G, Regitnig P (2005) Single lamellar mechanics of the human lumbar anulus fibrosus. Biomech Model Mechanobiol 3:125–140CrossRefGoogle Scholar
  30. Huyghe JM, Janssen JD (1997) Quadriphasic mechanics of swelling incompressible porous media. Int J Eng Sci 35:793–802zbMATHCrossRefGoogle Scholar
  31. Iatridis JC, Setton LA, Weidenbaum M, Mow VC (1997) The viscoelastic behavior of the non-degenerate human lumbar nucleus pulposus in shear. J Biomech 30:1005–1013CrossRefGoogle Scholar
  32. 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 125:12–24CrossRefGoogle Scholar
  33. Inoue H, Takeda T (1975) Three-dimensional observation of collagen framework of lumbar intervertebral discs. Acta Orthop 46:949–956CrossRefGoogle Scholar
  34. 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 43:176–181Google Scholar
  35. 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 122:180–188CrossRefGoogle Scholar
  36. Kraemer J (2009) Intervertebral disk diseases: causes, diagnosis, treatment, and prophylaxis. Thieme, New YorkCrossRefGoogle Scholar
  37. 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 136:051009CrossRefGoogle Scholar
  38. Lai WM, Hou JS, Mow VC (1991) A triphasic theory for the swelling and deformation behaviors of articular cartilage. J Biomech Eng 113:245–258CrossRefGoogle Scholar
  39. Lanir Y (1987) Biorheology and fluid flux in swelling tissues. I. Bicomponent theory for small deformations, including concentration effects. Biorheology 24:173–187CrossRefGoogle Scholar
  40. Lee EH (1969) Elastic-plastic deformation at finite strains. J Appl Mech 36:1–6zbMATHCrossRefGoogle Scholar
  41. Lees C, Vincent JF, Hillerton JE (1991) Poisson’s ratio in skin. Biomed Mater Eng 1:19–23Google Scholar
  42. Maroudas A (1976) Balance between swelling pressure and collagen tension in normal and degenerate cartilage. Nature 260:808–809CrossRefGoogle Scholar
  43. 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 48:164–172CrossRefGoogle Scholar
  44. 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 42:2279–2285CrossRefGoogle Scholar
  45. Miehe C (1995) Entropic thermoelasticity at finite strains. Aspects of the formulation and numerical implementation. Comput Methods Appl Mech Eng 120:243–269MathSciNetzbMATHCrossRefGoogle Scholar
  46. 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 102:73–84CrossRefGoogle Scholar
  47. MSC.Marc (2015) MSC. Marc volume D: user subroutines and special routines. MSC Software CorporationGoogle Scholar
  48. Nerurkar NL, Mauck RL, Elliott DM (2011) Modeling interlamellar interactions in angle-ply biologic laminates for annulus fibrosus tissue engineering. Biomech Model Mechanobiol 10:973–984CrossRefGoogle Scholar
  49. 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 65:824–830CrossRefGoogle Scholar
  50. 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 69:420–434CrossRefGoogle Scholar
  51. O’Connell GD, Guerin HL, Elliott DM (2009) Theoretical and uniaxial experimental evaluation of human annulus fibrosus degeneration. J Biomech Eng 131:1–7CrossRefGoogle Scholar
  52. 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 11:493–503CrossRefGoogle Scholar
  53. 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 73:815–824zbMATHCrossRefGoogle Scholar
  54. Pezowicz CA, Robertson PA, Broom ND (2006) The structural basis of interlamellar cohesion in the intervertebral disc wall. J Anat 208:317–330CrossRefGoogle Scholar
  55. 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 15:85–103CrossRefGoogle Scholar
  56. Race A, Broom ND, Robertson P (2000) Effect of loading rate and hydration on the mechanical properties of the disc. Spine 25:662–669CrossRefGoogle Scholar
  57. 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 49:3133–3146CrossRefGoogle Scholar
  58. Schollum ML, Robertson PA, Broom ND (2009) A microstructural investigation of intervertebral disc lamellar connectivity: detailed analysis of the translamellar bridges. J Anat 214:805–816CrossRefGoogle Scholar
  59. 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 25:1317–1324CrossRefGoogle Scholar
  60. Shirazi-Adl A, Ahmed AM, Shrivastava SC (1986) Mechanical response of a lumbar motion segment in axial torque alone and combined with compression. Spine 11:914–927CrossRefGoogle Scholar
  61. 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 107:327–335CrossRefGoogle Scholar
  62. 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 3:1000118MathSciNetCrossRefGoogle Scholar
  63. 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 19:1310–1319CrossRefGoogle Scholar
  64. 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 71:411–419CrossRefGoogle Scholar
  65. 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 34:1307–1315CrossRefGoogle Scholar
  66. 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 298:1537–1545CrossRefGoogle Scholar
  67. Vergari C, Mansfield J, Meakin JR, Winlove PC (2016) Lamellar and fibre bundle mechanics of the annulus fibrosus in bovine intervertebral disc. Acta Biomater 37:14–20CrossRefGoogle Scholar
  68. Vergroesen PPA, Emanuel KS, Peeters M, Kingma I (2018) Are axial intervertebral disc biomechanics determined by osmosis? J Biomech 70:4–9CrossRefGoogle Scholar
  69. Veronda DR, Westmann RA (1970) Mechanical characterization of skin-finite deformations. J Biomech 3:111–124CrossRefGoogle Scholar
  70. Wagner DR, Lotz JC (2004) Theoretical model and experimental results for the nonlinear elastic behavior of human annulus fibrosus. J Orthop Res 22:901–909CrossRefGoogle Scholar
  71. 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 28:81–93CrossRefGoogle Scholar
  72. Weiss JA, Maker BN, Govindjee S (1996) Finite element implementation of incompressible, transversely isotropic hyperelasticity. Comput Methods Appl Mech Eng 135:107–128zbMATHCrossRefGoogle Scholar
  73. Williams JL, Lewis JL (1982) Properties and an anisotropic model of cancellous bone from the proximal tibial epiphysis. J Biomech Eng 104:50–56CrossRefGoogle Scholar
  74. 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 30:1815–1820CrossRefGoogle Scholar
  75. 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 210:460–471CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Civil Engineering and geo-Environmental Laboratory (EA 4515 LGCgE)Lille UniversityLilleFrance
  2. 2.Ramsay Générale de SantéHôpital privé Le BoisLilleFrance

Personalised recommendations