Multiphasic Intervertebral Disc Mechanics: Theory and Application

  • Nils Karajan


The human spine is the flexible support structure of our body. Its geometric shape is a result of the human evolutionary history, where especially the lumbar spine area (L1–L5) is most at risk of causing discomfort resulting from mechanical stresses. Herein, the intervertebral discs (IVD) between the vertebral bodies are the most susceptible elements, as these avascular structures have to provide the flexibility. It is widely accepted that the IVD are often the trigger for back pain. In the context of biomechanical research, it is therefore important to develop a model for the human lumbar spine with particular focus on the IVD.

The objective of the presented work is to subject the IVD of the lumbar spine to continuum-biomechanical research. Herein, a three-dimensional (3-d) finite-element model is developed that allows to estimate the influence of lumbar spine motion, i.e., bending, torsion and compression, on the resulting stress-field inside the IVD. Following this, the model can be utilised to detect inappropriate or excessive loading of the spine. The theoretical description of the IVD is based on a multi-phase continuum approach in the framework of the well-known “Theory of Porous Media” (TPM). This is a natural choice resulting from the avascular composition of the IVD. In general, IVD tissue is categorised as charged hydrated material with mechanical and electro-chemical internal coupling mechanisms. In order to capture these couplings, the underlying model incorporates an extracellular matrix (ECM) with fixed negative charges, which is saturated by a mixture of a liquid solvent and ions. Following the basic concept of the TPM, a volumetric averaging process is prescribed leading to volume fractions for the pore space and the solid skeleton as well as molar concentrations for the ion species in the pore fluid.

In detail, the IVD exhibits a gelatinous core known as the nucleus pulposus (NP), and an onion-like surrounding structure consisting of anisotropic crosswise fibre-reinforced lamellae, the annulus fibrosus (AF). Both regions are seamlessly merging into each other and consist of mostly collagen fibres of varying strength and direction as well as proteoglycans with adhering negative charges. As a result of these fixed negative charges and the fact that the interstitial fluid carries positively and negatively charged ions, a model is created that describes the mutually coupled behaviour of solid deformation and fluid flow. To illustrate the resulting coupled swelling and shrinkage process, it is sufficient to recall that the size of the human body is reduced by roughly 2 centimetres during the waking phase of the day. This change in height is triggered by mechanical loads stemming from the body weight, which squeezes the interstitial fluid out of the IVD, thereby losing altitude. Simultaneously, an electro-chemical imbalance is generated, which is compensated during the nocturnal resting phase, where the interstitial fluid is driven back into the IVD due to osmotic effects.

In summary, the presented paper provides a review on IVD mechanics and can be understood as a compilation of relevant information giving valuable guidance to researchers starting to work in this challenging field. In this regard, the paper opens with anatomical and chemical fundamentals of IVD tissue with a particular focus on the inherent inhomogeneities. This is followed by an introduction to the continuum-mechanical fundamentals including an overview of the TPM as well as the inelastic non-linear kinematics and the balance equations of the resulting porous continua. Thereafter, the constitutive modelling process is illustrated with a particular focus on the thermo-dynamically consistent modelling process of the intrinsic viscoelasticity of the ECM as well as the inhomogeneous characteristics of the embedded collagen fibres and the viscous pore fluid. The resulting system of coupled partial differential equations is then numerically discretised using the finite-element method which finally allows the simulation of deformation processes of the IVD using either single- or multi-processor machines. Moreover, an automated computation scheme is presented to systematically capture the inhomogeneities of the IVD. The paper is closed with several sample applications, which embrace the capabilities of the presented computational model on the one hand and give advice for further validation in terms of the applied material parameters.



I thank the German Research Foundation (DFG) for funding the research project “Diffusions- und Strömungsprozesse in der anisotropen menschlichen Bandscheibe” under grant number ‘Eh 107/16’.


  1. 1.
    Acartürk A (2009) Simulation of charged hydrated porous media. Dissertation, Bericht Nr. II-18 aus dem Institut für Mechanik (Bauwesen), Universität Stuttgart Google Scholar
  2. 2.
    Ammann M (2005) Parallel finite element simulations of localization phenomena in porous media. Dissertation, Bericht Nr II-11 aus dem Institut für Mechanik (Bauwesen), Universität Stuttgart Google Scholar
  3. 3.
    Antoniou J, Steffen T, Nelson F, Winterbottom N, Hollander AP, Poole RA, Aebi M (1996) The human lumbar intervertebral disc: evidence for changes in the biosynthesis and denaturation of the extracellular matrix with growth. J Clin Invest 98:996–1003 Google Scholar
  4. 4.
    Apel N (2004) Approaches to the description of anisotropic material behaviour at finite elastic and plastic deformations—Theory and numerics. Dissertation, Bericht Nr I-12 aus dem Institut für Mechanik (Bauwesen), Universität Stuttgart Google Scholar
  5. 5.
    Argoubi M, Shirazi-Adl A (1996) Poroelastic creep response analysis of a lumbar motion segment in compression. J Biomech 29:1331–1339 Google Scholar
  6. 6.
    Arnold DN, Brezzi F, Fortin M (1984) A stable finite element for the Stokes equations. Calcolo 21:337–344 MathSciNetzbMATHGoogle Scholar
  7. 7.
    Ayad S, Weiss JB (1987) Biochemistry of the intervertebral disc. In: Jayson MIV (ed) The lumbar spine and back pain, 3rd edn. Churchill Livingstone, New York, pp 100–137 Google Scholar
  8. 8.
    Ball JM (1977) Convexity conditions and existence theorems in nonlinear elasticity. Arch Ration Mech Anal 63:337–403 zbMATHGoogle Scholar
  9. 9.
    Balzani D (2006) Polyconvex anisotropic energies and modeling of damage applied to arterial walls. Dissertation Bericht Nr 2, Fachbereich Bauwissenschaften Google Scholar
  10. 10.
    Balzani D, Neff P, Schröder J, Holzapfel G (2006) A polyconvex framework for soft biological tissues. Adjustment to experimental data. Int J Solids Struct 43:6052–6070 zbMATHGoogle Scholar
  11. 11.
    Bathe KJ (1990) Finite-Elemente-Methoden. Springer, Berlin Google Scholar
  12. 12.
    Benzi M, Golub GH, Liesen J (2005) Numerical solution of saddle point problems. Acta Numer 14:1–137 MathSciNetzbMATHGoogle Scholar
  13. 13.
    Biot MA (1941) General theory of three dimensional consolidation. J Appl Phys 12:155–164 zbMATHGoogle Scholar
  14. 14.
    Bishop AW (1959) The effective stress principle. Tekn Ukebl 39:859–863 Google Scholar
  15. 15.
    Boehler JP (1977) On irreducible representations for isotropic scalar functions. Z Angew Math Mech 57:323–327 MathSciNetzbMATHGoogle Scholar
  16. 16.
    Boehler JP (1979) A simple derivation of representations for non-polynominal constitutive equations in some case of anisotropy. Z Angew Math Mech 59:157–167 MathSciNetzbMATHGoogle Scholar
  17. 17.
    Boehler JP (1987) Introduction of the invariant formulation of anisotropic constitutive equations. In: Boehler JP (ed) Applications of tensor functions in solid mechanics. CISM courses and lectures, vol 292. Springer, Wien, pp 13–30 Google Scholar
  18. 18.
    Bowen RM (1976) Theory of mixtures. In: Eringen AC (ed) Continuum physics, vol III. Academic Press, New York, pp 1–127 Google Scholar
  19. 19.
    Bowen RM (1980) Incompressible porous media models by use of the theory of mixtures. Int J Eng Sci 18:1129–1148 zbMATHGoogle Scholar
  20. 20.
    Brezzi F, Fortin M (1991) Mixed and hybrid finite element methods. Springer, New York zbMATHGoogle Scholar
  21. 21.
    Broberg KB (1993) Slow deformation of intervertebral discs. J Biomech 26:501–512 Google Scholar
  22. 22.
    Chen Y, Chen X, Hisada T (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 zbMATHGoogle Scholar
  23. 23.
    Ciarlet PG (1988) Mathematical elasticity, vol 1: three dimensional elasticity. North-Holland, Amsterdam Google Scholar
  24. 24.
    Coleman BD, Noll W (1963) The thermodynamics of elastic materials with heat conduction and viscosity. Arch Ration Mech Anal 13:167–178 MathSciNetzbMATHGoogle Scholar
  25. 25.
    de Boer R (1982) Vektor und Tensorrechnung für Ingenieure. Springer, Berlin zbMATHGoogle Scholar
  26. 26.
    de Boer R (2000) Theory of porous media. Springer, Berlin zbMATHGoogle Scholar
  27. 27.
    Diebels S (2000) Mikropolare zweiphasenmodelle: formulierung auf basis der theorie poröser medien. Habilitation, Bericht Nr II-4 aus dem Institut für Mechanik (Bauwesen), Universität Stuttgart Google Scholar
  28. 28.
    Diebels S, Ellsiepen P, Ehlers W (1999) Error-controlled Runge-Kutta time integration of a viscoplastic hybrid two-phase model. Tech Mech 19:19–27 Google Scholar
  29. 29.
    Donnan FG (1911) Theorie der Membrangleichgewichte und Membranpotentiale bei Vorhandensein von nicht dialysierenden Elektrolyten. Ein Beitrag zur physikalisch-chemischen Physiologie. Z Elektrochem Angew Phys Chem 17:572–581 Google Scholar
  30. 30.
    Ebara S, Iatridis JC, Setton LA, Foster RJ, Mow C, Weidenbaum M (1996) Tensile properties of nondegenerate human lumbar anulus fibrosus. Spine 21:452–461 Google Scholar
  31. 31.
    Eberlein R, Holzapfel GA, Schulze-Bauer CAJ (2001) An anisotropic model for annulus tissue and enhanced finite element analysis of intact lumbar disc bodies. Comput Methods Biomech Biomed Eng 4:209–229 Google Scholar
  32. 32.
    Eberlein R, Holzapfel GA, Fröhlich M (2004) Multi-segment FEA of the human lumbar spine including the heterogeneity of the anulus fibrosus. Comput Mech 34:147–165 zbMATHGoogle Scholar
  33. 33.
    Effelsberg J (2007) Untersuchung der Diffusions- und Strömungsprozesse in der menschlichen Bandscheibe mittels Magnetresonanztomographie. Diploma Thesis, Bericht Nr 07-II-2 aus dem Institut für Mechanik (Bauwesen), Universität Stuttgart Google Scholar
  34. 34.
    Ehlers W (1989) Poröse Medien—ein kontinuumsmechanisches Modell auf der Basis der Mischungstheorie. Habilitation, Forschungsberichte aus dem Fachbereich Bauwesen, Heft 47, Universität-GH-Essen Google Scholar
  35. 35.
    Ehlers W (1991) Toward finite theories of liquid-saturated elasto-plastic porous media. Int J Plast 7:433–475 zbMATHGoogle Scholar
  36. 36.
    Ehlers W (1993) Constitutive equations for granular materials in geomechanical context. In: Hutter K (ed) Continuum mechanics in environmental sciences and geophysics. CISM courses and lectures, vol 337. Springer, Wien, pp 313–402 Google Scholar
  37. 37.
    Ehlers W (1995–2009) Vector and tensor calculus: An introduction. Lecture notes, Institute of Applied Mechanics (Chair of Continuum Mechanics), Universität Stuttgart,
  38. 38.
    Ehlers W (2002) Foundations of multiphasic and porous materials. In: Ehlers W, Bluhm J (eds) Porous media: theory, experiments and numerical applications. Springer, Berlin, pp 3–86 Google Scholar
  39. 39.
    Ehlers W, Acartürk A (2009) The role of weakly imposed Dirichlet boundary conditions for numerically stable computations of swelling phenomena. Comput Mech 43:545–557 MathSciNetGoogle Scholar
  40. 40.
    Ehlers W, Ellsiepen P (1998) PANDAS: Ein FE-System zur Simulation von Sonderproblemen der Bodenmechanik. In: Wriggers P, Meißner U, Stein E, Wunderlich W (eds) Finite Elemente in der Baupraxis—FEM ’98. Ernst & Sohn, Berlin, pp 391–400 Google Scholar
  41. 41.
    Ehlers W, Ellsiepen P (2001) Theoretical and numerical methods in environmental continuum mechanics based on the Theory of Porous Media. In: Schrefler BA (ed) Environmental geomechanics. CISM courses and lectures, vol 417. Springer, Wien, pp 1–81 Google Scholar
  42. 42.
    Ehlers W, Markert B (2001) A linear viscoelastic biphasic model for soft tissues based on the Theory of Porous Media. J Biomech Eng 123:418–424 Google Scholar
  43. 43.
    Ehlers W, Ellsiepen P, Blome P, Mahnkopf D, Markert B (1999) Theoretische und numerische Studien zur Lösung von Rand- und Anfangswertproblemen in der Theorie Poröser Medien, AbschlußBericht zum DFG-Forschungsvorhaben eh 107/6-2. Bericht Nr 99-II-1 aus dem Institut für Mechanik (Bauwesen), Universität Stuttgart Google Scholar
  44. 44.
    Ehlers W, Markert B, Acartürk A (2005) Swelling phenomena of hydrated porous materials. In: Abousleiman YN, Cheng AHD, Ulm FJ (eds) Poromechanics III, proceedings of the 3rd Biot conference on poromechanics. Balkema at Taylor & Francis, Leiden, pp 781–786 Google Scholar
  45. 45.
    Ehlers W, Karajan N, Markert B (2006) A porous media model describing the inhomogeneous behaviour of the human intervertebral disc. Mater Sci Eng Technol 37:546–551 Google Scholar
  46. 46.
    Ehlers W, Markert B, Karajan N (2006) A coupled FE analysis of the intervertebral disc based on a multiphasic TPM formulation. In: Holzapfel GA, Ogden RW (eds) Mechanics of biological tissue. Springer, Berlin, pp 373–386 Google Scholar
  47. 47.
    Ehlers W, Karajan N, Wieners C (2007) Parallel 3-d simulation of a biphasic porous media model in spine mechanics. In: Ehlers W, Karajan N (eds) Proceedings of the 2nd GAMM seminar on continuum biomechanics, Bericht Nr II-16 aus dem Institut für Mechanik (Bauwesen). Universität Stuttgart, Stuttgart, pp 11–20 Google Scholar
  48. 48.
    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 Google Scholar
  49. 49.
    Ehlers W, Acartürk A, Karajan N (2010) Advances in modelling saturated biological soft tissues and chemically active gels. Arch Appl Mech 80:467–478 Google Scholar
  50. 50.
    Eipper G (1998) Theorie und Numerik finiter elastischer Deformationen in fluidgesättigten Porösen Medien. Dissertation, Bericht Nr II-1 aus dem Institut für Mechanik (Bauwesen), Universität Stuttgart Google Scholar
  51. 51.
    Elliott DA, Setton LA (2000) A linear material model for fiber-induced anisotropy of the anulus fibrosus. J Biomech Eng 122:173–179 Google Scholar
  52. 52.
    Ellsiepen P (1999) Zeit- und ortsadaptive Verfahren angewandt auf Mehrphasenprobleme pröser Medien. Dissertation Bericht Nr II-3 aus dem Institut für Mechanik (Bauwesen), Universität Stuttgart Google Scholar
  53. 53.
    Ellsiepen P, Hartmann S (2001) Remarks on the interpretation of current non-linear finite element analyses as differential-algebraic equations. Int J Numer Methods Eng 51:679–707 zbMATHGoogle Scholar
  54. 54.
    Managing musculoskeletal disorders European Foundation for the Improvement of Living and Working Conditions (2007). Dublin,
  55. 55.
    Eyre DR (1979) Biochemistry of the intervertebral disc. Connect Tissue Res 8:227–291 Google Scholar
  56. 56.
    Eyre DR, Muir H (1977) Quantitative analysis of types I and II collagens in human intervertebral discs at various ages. Biochim Biophys Acta 492:29–42 Google Scholar
  57. 57.
    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 zbMATHGoogle Scholar
  58. 58.
    Frijns AJH, Huyghe JM, Kaasschieter EF, Wijlaars MW (2003) Numerical simulation of deformations and electrical potentials in a cartilage substitute. Biorheology 40:123–131 Google Scholar
  59. 59.
    Fung YC (1981) Biomechanics: mechanical properties of living tissues. Springer, New York Google Scholar
  60. 60.
    Godfrey MD, Hendry DF (1993) The computer as von Neumann planned it. IEEE Ann Hist Comput 15:11–21 MathSciNetzbMATHGoogle Scholar
  61. 61.
    Grace H, Young A (1903) The algebra of invariants. Cambridge University Press, Cambridge zbMATHGoogle Scholar
  62. 62.
    Gu WY, Mao XG, Foster RJ, Weidenbaum M, Mow VC, Rawlins B (1999) The anisotropic hydraulic permeability of human lumbar anulus fibrosus. Spine 24:2449–2455 Google Scholar
  63. 63.
    Gurtin ME, Williams WO (1966) On the inclusion of the complete symmetry group in the unimodular group. Arch Ration Mech Anal 23:163–172 MathSciNetzbMATHGoogle Scholar
  64. 64.
    Hairer E, Wanner G (1991) Solving ordinary differential equations II—stiff and differential-algebraic problems. Springer, Berlin zbMATHGoogle Scholar
  65. 65.
    Hassanizadeh SM, Gray WG (1979) General conservation equations for multi-phase systems: 1. Averaging procedure. Adv Water Resour 2:131–144 Google Scholar
  66. 66.
    Hassanizadeh SM, Gray WG (1987) High velocity flow in porous media. Transp Porous Media 2:521–531 Google Scholar
  67. 67.
    Haupt P (1993) Foundations of continuum mechanics. In: Hutter K (ed) Continuum mechanics in environmental sciences and geophysics. CISM courses and lectures, vol 337. Springer, Wien, pp 1–77 Google Scholar
  68. 68.
    Hayes WC, Bodine AJ (1978) Flow-independent viscoelastic properties of articular cartilage matrix. J Biomech 11:407–419 Google Scholar
  69. 69.
    Holm S, Nachemson A (1983) Variations in the nutrition of the canine intervertebral disc induced by motion. Spine 8:866–974 Google Scholar
  70. 70.
    Holzapfel G, Gasser TC, Ogden RW (2000) A new constitutive framework for arterial wall mechanics and a comparative study of material models. J Elast 61:1–48 MathSciNetzbMATHGoogle Scholar
  71. 71.
    Holzapfel GA, Schulze-Bauer CAJ, Feigl G, Regitnig P (2005) Mono-lamellar mechanics of the human lumbar anulus fibrosus. Biomech Model Mechanobiol 3:125–140 Google Scholar
  72. 72.
    Hsieh AH, Wagner DR, Cheng LY, Lotz JC (2005) Dependence of mechanical behavior of the murine tail disc on regional material properties: a parametric finite element study. J Biomech Eng 127:1158–1167 Google Scholar
  73. 73.
    Hukins DWL (1987) Properties of spinal materials. In: Jayson MIV (ed) The lumbar spine and back pain, 3rd edn. Churchill Livingstone, New York, pp 138–160 Google Scholar
  74. 74.
    Huyghe JM, Houben GB, Drost MR (2003) An ionised/non-ionised dual porosity model of intervertebral disc tissue. Biomech Model Mechanobiol 2:3–19 Google Scholar
  75. 75.
    Iatridis C, Weidenbaum M, Setton LA, Mow VC (1996) Is the nucleus pulposus a solid or a fluid? Mechanical behaviors of the human intervertebral disc. Spine 21:1174–1184 Google Scholar
  76. 76.
    Iatridis JC, Setton A, Weidenbaum M, Mow VC (1997) The viscoelastic behavior of the non-degenerate human lumbar nucleus pulposus in shear. J Biomech 30:1005–1013 Google Scholar
  77. 77.
    Iatridis JC, Setton LA, Foster RJ, Rawlins A, Weidenbaum M, Mow VC (1998) Degeneration affects the anisotropic and nonlinear behaviors of human anulus fibrosus in compression. J Biomech 31:535–544 Google Scholar
  78. 78.
    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 Google Scholar
  79. 79.
    Itskov M, Aksel N (2004) A class of orthotropic and transversely isotropic hyperelastic constitutive models based on a polyconvex strain energy function. Int J Solids Struct 41:3833–3848 MathSciNetzbMATHGoogle Scholar
  80. 80.
    Kaasschieter EF, Frijns AJH, Huyghe JM (2003) Mixed finite element modelling of cartilaginous tissues. Math Comput Simul 61:549–560 MathSciNetzbMATHGoogle Scholar
  81. 81.
    Karajan N, Ehlers W, Röhrle O, Schmitt S (2011) Homogenisation method to capture the non-linear behaviour of intervertebral discs in multi-body systems. Proc. Appl. Math. Mech. 11:95–96 Google Scholar
  82. 82.
    Klawonn A, Pavarino LF (1998) Overlapping Schwarz methods for mixed linear elasticity and Stokes problems. Comput Methods Appl Mech Eng 165:233–245 MathSciNetzbMATHGoogle Scholar
  83. 83.
    Klawonn A, Pavarino LF (2000) A comparison of overlapping Schwarz methods and block preconditioners for saddle point problems. Numer Linear Algebra Appl 7:1–25 MathSciNetzbMATHGoogle Scholar
  84. 84.
    Kleiber M (1975) Kinematics of deformation processes in materials subjected to finite elastic-plastic strains. Int J Eng Sci 13:513–525 zbMATHGoogle Scholar
  85. 85.
    Klisch SM, Lotz JC (1999) Application of a fiber-reinforced continuum theory to multiple deformations of the annulus fibrosus. J Biomech 32:1027–1036 Google Scholar
  86. 86.
    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 Google Scholar
  87. 87.
    Lai WM, Hou JS, Mow VC (1991) A triphasic theory for the swelling and deformation behaviours of articular cartilage. J Biomech Eng 113:245–258 Google Scholar
  88. 88.
    Laible JP, Pflaster DS, Krag MH, Simon BR, Haugh LD (1993) A poroelastic-swelling finite element model with application to the intervertebral disc. Spine 18:659–670 Google Scholar
  89. 89.
    Lambrecht M (2002) Theorie und Numerik von Materialinstabilitäten elastoplastischer Festkörper auf der Grundlage inkrementeller Variationsformulierungen. Dissertation, Bericht Nr I-8 aus dem Institut für Mechanik (Bauwesen), Universität Stuttgart Google Scholar
  90. 90.
    Lanir Y (1987) Biorheology and fluid flux in swelling tissues I. Bicomponent theory for small deformations, including concentration effects. Biorheology 24:173–187 Google Scholar
  91. 91.
    Lee CK, Kim YE, Lee CS, Hong YM, Jung JM, Goel VK (2000) Impact response of the intervertebral disc in a finite-element model. Spine 25:2431–2439 Google Scholar
  92. 92.
    Lee EH (1969) Elastic-plastic deformation at finite strains. J Appl Mech 36:1–6 zbMATHGoogle Scholar
  93. 93.
    Li LP, Shirazi-Adl A, Buschmann MD (2003) Investigation of mechanical behavior of articular cartilage by fibril reinforced poroelastic models. Biorheology 40:227–233 Google Scholar
  94. 94.
    Lide DR (2003) CRC handbook of chemistry and physics. CRC Press, Boca Raton Google Scholar
  95. 95.
    Lis AM, Black M, Korn H, Nordin M (2007) Association between sitting and occupational LBP. Eur Spine J 16:283–298 Google Scholar
  96. 96.
    Liu IS (1972) Method of Lagrange multipliers for exploitation of the entropy principle. Arch Ration Mech Anal 46:131–148 zbMATHGoogle Scholar
  97. 97.
    Liu IS, I M (1984) Thermodynamics of mixtures of fluids. In: Truesdell C (ed) Rational thermodynamics, 2nd edn. Springer, New York, pp 264–285 Google Scholar
  98. 98.
    Ludescher B, Effelsberger J, Martirosian P, Steidle G, Markert B, Claussen C, Schick F (2008) T2- and diffusion-maps reveal diurnal changes of intervertebral disc composition: An in vivo MRI study at 1.5 Tesla. J Magn Reson Imaging 28:252–257 Google Scholar
  99. 99.
    Marchand F, Ahmed AM (1990) Investigation of the laminate structure of the lumbar disc anulus. Spine 15:402–410 Google Scholar
  100. 100.
    Markert B (2005) Porous media viscoelasticity with application to polymeric foams. Dissertation, Bericht Nr II-12 aus dem Institut für Mechanik (Bauwesen), Universität Stuttgart Google Scholar
  101. 101.
    Markert B (2008) A biphasic continuum approach for viscoelastic high-porosity foams: comprehensive theory, numerics, and application. Arch Comput Methods Eng 15:371–446 MathSciNetzbMATHGoogle Scholar
  102. 102.
    Markert B, Ehlers W, Karajan N (2005) A general polyconvex strain-energy function for fiber-reinforced materials. Proc Appl Math Mech 5:245–246 Google Scholar
  103. 103.
    Marsden JE, Hughes TJR (1994) Mathematical foundations of elasticity. Dover, New York. Reprint of Prentice-Hall 1983 Google Scholar
  104. 104.
    Mayer J (2007) A multilevel crout ILU preconditioner with pivoting and row permutation. Numer Linear Algebra Appl, 14, 771–789 MathSciNetzbMATHGoogle Scholar
  105. 105.
    Mayer J (2008) Symmetric permutations for I-matrices to delay and avoid small pivots during factorization. SIAM J Sci Comput 30:982–996 MathSciNetzbMATHGoogle Scholar
  106. 106.
    Mills N (1966) Incompressible mixture of Newtonian fluids. Int J Eng Sci 4:97–112 zbMATHGoogle Scholar
  107. 107.
    Mohr PJ, Taylor BN, Newell DB (eds) (2007) CODATA recommended values of the fundamental physical constants: 2006. National Institute of Standards and Technology, Gaithersburg Google Scholar
  108. 108.
    Mooney M (1940) A theory of large elastic deformation. J Appl Phys 11:582–592 zbMATHGoogle Scholar
  109. 109.
    Morrey CB (1952) Quasi-convexity and the lower semicontinuity of multiple integrals. Pac J Math 2:25–53 MathSciNetzbMATHGoogle Scholar
  110. 110.
    Mow VC, Hayes WC (1997) Basic orthopaedic biomechanics. Lippincott-Raven, New York Google Scholar
  111. 111.
    Mow VC, Ratcliffe A (1997) Structure and function of articular cartilage and meniscus. In: Mow VC, Hayes WC (eds) Basic orthopaedic biomechanics. Lippincott-Raven, New York, pp 113–177 Google Scholar
  112. 112.
    Mow VC, Kuei C, Lai WM, Armstrong CG (1980) Biphasic creep and stress relaxation of articular cartilage in compression: theory and experiments. J Biomech Eng 102:73–84 Google Scholar
  113. 113.
    Mow VC, Gibbs MC, Lai WM, Zhu WB, Athanasiou KA (1989) Biphasic indentation of articular cartilage—II. A numerical algorithm and an experimental study. J Biomech 22:853–861 Google Scholar
  114. 114.
    Mow VC, Ateshian GA, Lai WM, Gu WY (1998) Effects of fixed charges on the stress-relaxation behavior of hydrated soft tissues in a confined compression problem. Int J Solids Struct 35:4945–4962 zbMATHGoogle Scholar
  115. 115.
    Nachemson AL (1987) Lumbar intradiscal pressure. In: Jayson MIV (ed) The lumbar spine and back pain, 3rd edn. Churchill Livingstone, New York, pp 191–203 Google Scholar
  116. 116.
    Naylor A (1962) The biophysical and biochemical aspects of intervertebral disc herniation and degeneration. Ann R Coll Surg Engl 31:91–114 Google Scholar
  117. 117.
    Noll W (1955) On the continuity of the fluid and solid states. J Ration Mech Anal 4:3–81 MathSciNetzbMATHGoogle Scholar
  118. 118.
    Noll W (1958) A mathematical theory of the mechanical behavior of continuous media. Arch Ration Mech Anal 2:197–226 zbMATHGoogle Scholar
  119. 119.
    Ochia RS, Ching RP (2002) Hydraulic resistance and permeability in human lumbar vertebral bodies. J Biomech Eng 124:533–537 Google Scholar
  120. 120.
    Ogden RW (1972) Large deformation isotropic elasticity—on the correlation of theory and experiment for incompressible rubberlike solids. Proc R Soc Lond Ser A, Math Phys Sci 326:565–584 zbMATHGoogle Scholar
  121. 121.
    Parent-Thirion A, Macías EF, Hurley J, Vermeylen G (2007) Fourth European working conditions survey. Report of the European Foundation for the Improvement of Living and Working Conditions. Dublin,
  122. 122.
    Powell MJD (1994) A direct search optimization method that models the objective and constraint functions by linear interpolation. In: Gomez S, Hennart JP (eds) Advances in optimization and numerical analysis. Kluwer Academic, Dordrecht, pp 51–67 Google Scholar
  123. 123.
    Press WH, Teukolsky SA, Vetterling WT, Flannery BP (2007) Numerical recipes: the art of scientific computing, 3rd edn. Cambridge University Press, Cambridge zbMATHGoogle Scholar
  124. 124.
    Reese S, Govindjee S (1998) A theory of finite viscoelasticity and numerical aspects. Int J Solids Struct 35:3455–3482 zbMATHGoogle Scholar
  125. 125.
    Riches PE, Dhillon N, Lotz J, Woods AW, McNally DS (2002) The internal mechanics of the intervertebral disc under cyclic loading. J Biomech 35:1263–1271 Google Scholar
  126. 126.
    Rivlin RS (1948) Large elastic deformations of isotropic materials. Proc R Soc Lond Ser A, Math Phys Sci 241:379–397 MathSciNetzbMATHGoogle Scholar
  127. 127.
    Rivlin RS, Ericksen JL (1955) Stress-deformation relations for isotropic materials. J Ration Mech Anal 4:323–425 MathSciNetzbMATHGoogle Scholar
  128. 128.
    Rohlmann A, Zander T, Schmidt H, Wilke HJ, Bergmann G (2006) Analysis of the influence of disc degeneration on the mechanical behaviour of a lumbar motion segment using the finite element method. J Biomech 39:2484–2490 Google Scholar
  129. 129.
    Saad Y, Schultz MH (1986) GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7:856–869 MathSciNetzbMATHGoogle Scholar
  130. 130.
    Sandhu RS, Wilson EL (1969) Finite-element analysis of seepage in elastic media. J Eng Mech Div 95:641–652 Google Scholar
  131. 131.
    Schenke M (2008) Development of an interface between ABAQUS and PANDAS. Master thesis, Bericht Nr 08-II-12 aus dem Institut für Mechanik (Bauwesen), Universität Stuttgart Google Scholar
  132. 132.
    Schmidt CO, Raspe H, Pfingsten M, Hasenbring M, Basler HD, Eich W, Kohlmann T (2007) Back pain in the German adult population. Prevalence, severity, and sociodemographic correlations in a multiregional survey. Spine 32:2005–2011 Google Scholar
  133. 133.
    Schmidt H, Heuer F, Simon U, Kettler A, Rohlmann A, Claes L, Wilke HJ (2006) Application of a new calibration method for a three-dimensional finite element model of a human lumbar annulus fibrosus. Clin Biomech 21:337–344 Google Scholar
  134. 134.
    Schmidt H, Heuer F, Drumm J, Klezl Z, Claes L, Wilke HJ (2007) Application of a calibration method provides more realistic results for a finite element model of a lumbar spinal segment. Clin Biomech 22:377–384 Google Scholar
  135. 135.
    Schröder J (1996) Theoretische und algorithmische Konzepte zur phänomenologischen Beschreibung anisotropen Materialverhaltens. Dissertation, Bericht Nr I-1 aus dem Institut für Mechanik (Bauwesen), Universität Stuttgart Google Scholar
  136. 136.
    Schröder J (2000) Homogenisierungsmethoden der nichtlinearen Kontinuumsmechanik unter Beachtung von Stabilitätsproblemen. Habilitation, Bericht Nr I-7 aus dem Institut für Mechanik (Bauwesen), Universität Stuttgart Google Scholar
  137. 137.
    Schröder J, Neff P (2003) Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions. Int J Solids Struct 40:401–445 zbMATHGoogle Scholar
  138. 138.
    Schröder 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 Google Scholar
  139. 139.
    Schwarz HR (1991) Methode der finiten Elemente. Teubner, Stuttgart Google Scholar
  140. 140.
    Shirazi-Adl A (1994) Nonlinear stress analysis of the whole lumbar spine in torsion-mechanics of facet articulation. J Biomech 27:289–299 Google Scholar
  141. 141.
    Shirazi-Adl A (2006) Analysis of large compression loads on lumbar spine in flexion and torsion using a novel wrapping element. J Biomech 39:267–275 Google Scholar
  142. 142.
    Shirazi-Adl A, Ahmed AM, Shrivastava SC (1986) A finite element study of a lumbar motion segment subjected to pure sagittal plane moments. J Biomech 19:331–350 Google Scholar
  143. 143.
    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 Google Scholar
  144. 144.
    Šilhavý M (1997) The mechanics and thermodynamics of continuous media. Springer, Berlin zbMATHGoogle Scholar
  145. 145.
    Simo JC, Taylor RL (1985) Consistent tangent operators for rate-independent elastoplasticity. Comput Methods Appl Mech Eng 48:101–118 zbMATHGoogle Scholar
  146. 146.
    Skaggs L, Weidenbaum M, Iatridis C, Ratcliffe A, Mow VC (1994) Regional variation in tensile properties and biochemical composition of the human lumbar anulus fibrosus. Spine 19:1310–1319 Google Scholar
  147. 147.
    Skempton AW (1960) Significance of Terzaghi’s concept of effective stress (Terzaghi’s discovery of effective stress). In: Bjerrum L, Casagrande A, Peck RB, Skempton AW (eds) From theory to practice in soil mechanics. Wiley, New York, pp 42–53 Google Scholar
  148. 148.
    Smith GF (1971) On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors. Int J Eng Sci 9:899–916 zbMATHGoogle Scholar
  149. 149.
    Snijders H, Huyghe JM, Janssen JD (1995) Triphasic finite element model for swelling porous media. Int J Numer Methods Fluids 20:1039–1046 zbMATHGoogle Scholar
  150. 150.
    Spencer AJM (1971) Theory of invariants. In: Eringen AC (ed) Continuum physics, vol 1. Academic Press, New York, pp 239–353 Google Scholar
  151. 151.
    Spencer AJM (1982) The formulation of constitutive equations for anisotropic solids. In: Boehler JP (ed) Mechanical behavior of anisotropic solids. Proceedings of the Euromech colloquium, vol 115. Martinus Nijhoff, The Haque, pp 2–26 Google Scholar
  152. 152.
    Spencer AJM (1984) Constitutive theory for strongly anisotropic solids. In: Spencer AJM (ed) Continuum theory of the mechanics of fibre reinforced composites. CISM courses and lectures, vol 282. Springer, Wien, pp 1–32 Google Scholar
  153. 153.
    Spencer AJM (1987) Isotropic polynominal invariants and tensor functions. In: Boehler JP (ed) Applications of tensor functions in solid mechanics. CISM courses and lectures, vol 292. Springer, Wien, pp 141–169 Google Scholar
  154. 154.
    Spencer AJM, Rivlin RS (1962) Isotropic integrity bases for vectors and second-order tensors. Arch Ration Mech Anal 9:45–63 MathSciNetzbMATHGoogle Scholar
  155. 155.
    Sten-Knudsen O (2002) Biological membranes: theory of transport, potentials and electric impulses. Cambridge University Press, Cambridge Google Scholar
  156. 156.
    Svendson B, Hutter K (1995) On the thermodynamics of a mixture of isotropic materials with constraints. Int J Eng Sci 33:2021–2054 Google Scholar
  157. 157.
    Szirmai JA (1970) Structure of the intervertebral disc. In: Balazs EA (ed) Chemistry and molecular biology of the intercellular matrix, vol 3. Academic Press, London, pp 1279–1308 Google Scholar
  158. 158.
    Taylor C, Hood P (1973) A numerical solution of the Navier-Stokes equations using the finite element technique. Comput Fluids 1:73–100 MathSciNetzbMATHGoogle Scholar
  159. 159.
    Treloar LRG (1975) The physics of rubber elasticity, 3rd edn. Clarendon Press, Oxford Google Scholar
  160. 160.
    Truesdell C (1949) A new definition of a fluid, II. The Maxwellian fluid. Tech Rep P-3553, §19, US Naval Research Laboratory Google Scholar
  161. 161.
    Truesdell C (1984) Thermodynamics of diffusion. In: Truesdell C (ed) Rational thermodynamics, 2nd edn. Springer, New York, pp 219–236 Google Scholar
  162. 162.
    Truesdell C, Noll W (1965) The nonlinear field theories of mechanics. In: Flügge S (ed) Handbuch der physik, vol III/3. Springer, Berlin Google Scholar
  163. 163.
    Truesdell C, Toupin RA (1960) The classical field theories. In: Flügge S (ed) Handbuch der physik, vol III/1. Springer, Berlin Google Scholar
  164. 164.
    Tsuji H, Hirano N, Ohshima H, Ishihara H, Terahata N, Motoe T (1993) Structural variation of the anterior and posterior anulus fibrosus in the development of human lumbar intervertebral disc. A risk factor for intervertebral disc rupture. Spine 18:204–210 Google Scholar
  165. 165.
    Tyrrell AR, Reilly T, Troup JDG (1985) Circadian variation in stature and the effects of spinal loading. Spine 10:161–164 Google Scholar
  166. 166.
    Urban JPG, Holm S (1986) Intervertebral disc nutrition as related to spinal movements and fusion. In: Hargens AR (ed) Tissue nutrition and viability. Springer, Berlin, pp 101–119 Google Scholar
  167. 167.
    Urban G, Maroudas A (1979) The measurement of fixed charged density in the intervertebral disc. Biochim Biophys Acta 586:166–178 Google Scholar
  168. 168.
    Urban JPG, Roberts S (1996) Intervertebral disc. In: Comper WD (ed) Extracellular matrix, vol 1: tissue function. Harwood Academic, Amsterdam, pp 203–233 Google Scholar
  169. 169.
    Urban JPG, Holm S, Maroudas A, Nachemson A (1982) Nutrition of the intervertebral disc: effect of fluid flow on solute transport. Clin Orthop 170:296–302 Google Scholar
  170. 170.
    Vanharanta H, Guyer RD, Ohnmeiss DD, Stith WJ, Sachs BL, Aprill C, Spivey M, Rashbaum RF, Hochschuler SH, Videman T, Selby DK, Terry A, Mooney V (1988) Disc deterioration in low-back syndromes. A prospective, multi-center CT/discography study. Spine 13:1349–1351 Google Scholar
  171. 171.
    van Loon R, Huyghe FM, Wijlaars MW, Baaijens FPT (2003) 3D FE implementation of an incompressible quadriphasic mixture model. Int J Numer Methods Eng 57:1243–1258 zbMATHGoogle Scholar
  172. 172.
    Varga OH (1966) Stress-strain behavior of elastic materials. Interscience, New York zbMATHGoogle Scholar
  173. 173.
    Walker DW, Dongarra JJ (1996) MPI: a standard message passing interface. Supercomputer 12:56–68 Google Scholar
  174. 174.
    Wall WA (1999) Fluid-Struktur-Interaktion mit stabilisierten finiten elementen. Dissertation, Bericht Nr 31 aus dem Institut für Baustatik, Universität Stuttgart Google Scholar
  175. 175.
    Wang CC (1969) On representations for isotropic functions, part I and II. Arch Ration Mech Anal 33:249–287 zbMATHGoogle Scholar
  176. 176.
    Wang CC (1970) A new representation theorem for isotropic functions: an answer to Professor G.F. Smith’s criticism of my papers on representations for isotropic functions, part I and II. Arch Ration Mech Anal 36:166–223 zbMATHGoogle Scholar
  177. 177.
    Wang C, Truesdell C (1973) Introduction to rational elasticity. Noordhoff International, Leyden zbMATHGoogle Scholar
  178. 178.
    Weyl H (1946) The classical groups, their invariants and representation. Princeton University Press, Princeton Google Scholar
  179. 179.
    White AA, Panjabi MM (1990) Clinical biomechanics of the spine, 2nd edn. Lippincott Williams, Philadelphia Google Scholar
  180. 180.
    Wieners C (2003) Taylor-Hood elements in 3D. In: Wendland WL, Efendiev M (eds) Analysis and simulation of multifield problems. Springer, Berlin, pp 189–196 Google Scholar
  181. 181.
    Wieners C (2004) Distributed Point Objects. A new concept for parallel finite elements. In: Kornhuber R, Hoppe R, Périaux J, Pironneau O, Widlund O, Xu J (eds) Domain decomposition methods in science and engineering. Lecture notes in computational science and engineering, vol 40. Springer, Berlin, pp 175–183 Google Scholar
  182. 182.
    Wieners C, Ammann M, Ehlers W, Graf T (2005) Parallel Krylov methods and the application to 3-d simulations of a tri-phasic porous media model in soil mechanics. Comput Mech 36:409–420 zbMATHGoogle Scholar
  183. 183.
    Wieners C, Ehlers W, Ammann M, Karajan N, Markert B (2005) Parallel solution methods for porous media models in biomechanics. Proc Appl Math Mech 5:35–38 Google Scholar
  184. 184.
    Wieners C, Ammann M, Ehlers W (2006) Distributed Point Objects: a new concept for parallel finite elements applied to a geomechanical problem. Future Gener Comput Syst 22, 532–545 Google Scholar
  185. 185.
    Wilke HJ, Claes LE (eds) (1999) Die traumatische und degenerative Bandscheibe. Hefte zur Zeitschrift, Der Unfallchirurg, vol 271. Springer, Berlin Google Scholar
  186. 186.
    Wilke HJ, Neef P, Caimi M, Hoogland T, Claes LE (1999) New in vivo measurements of pressures in the intervertebral disc in daily life. Spine 24:755–762 Google Scholar
  187. 187.
    Wilson W, van Donkelar CC, Huyghe JM (2005) A comparison between mechano-electrochemical and biphasic swelling theories for soft hydrated tissues. J Biomech Eng 127:158–165 Google Scholar
  188. 188.
    Wriggers P (1988) Konsistente Linearisierung in der Kontinuumsmechanik und ihre Anwendung auf die Finite-Elemente-Methode. Habilitation, Technischer Bericht F88/4, Forschungs und Seminarberichte aus dem Bereich der Mechanik der Universität Hannover Google Scholar
  189. 189.
    Wu JSS, Chen JH (1996) Clarification of the mechanical behavior of spinal motion segments through a three-dimensional poroelastic mixed finite element model. Med Eng Phys 18:215–224 Google Scholar
  190. 190.
    Zheng QS (1994) Theory of representations for tensor functions—a unified invariant approach to constitutive equations. Appl Mech Rev 47:545–587 Google Scholar
  191. 191.
    Zheng QS, Boehler JP (1994) The description, classification, and reality of material and physical symmetries. Acta Mech 102:73–89 MathSciNetzbMATHGoogle Scholar
  192. 192.
    Zheng QS, Spencer AJM (1993) Tensors which characterize anisotropies. Int J Eng Sci 31:679–693 MathSciNetzbMATHGoogle Scholar
  193. 193.
    Zienkiewicz OC, Taylor RL (2005) The finite element method for solid and structural mechanics, vol 2, 6th edn. Butterworth–Heinemann, Oxford zbMATHGoogle Scholar
  194. 194.
    Zienkiewicz OC, Taylor RL (2005) The finite element method. The basis, vol 1, 6th edn. Butterworth–Heinemann, Oxford Google Scholar
  195. 195.
    Zienkiewicz OC, Taylor RL, Sherwin SJ, Peiró J (2003) On discontinuous Galerkin methods. Int J Numer Methods Eng 58:1119–1148 zbMATHGoogle Scholar

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© CIMNE, Barcelona, Spain 2012

Authors and Affiliations

  1. 1.Institute of Applied Mechanics (CE)Universität StuttgartStuttgartGermany

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