An extended biphasic model for charged hydrated tissues with application to the intervertebral disc

Biomechanics and Modeling in Mechanobiology

Volumn 8, Issue 3, 2009, Pages 233-251

  Ehlers, W ^a   Karajan, N ^a   Markert, B ^a  

^a UNIVERSITY OF STUTTGART   (Germany)

Author keywords

Anisotropy; Coupled problems; Finite viscoelasticity; Inhomogeneity; Intervertebral disc (IVD); Mixed FEM; Swelling; Theory of Porous Media (TPM)

Indexed keywords

ANISOTROPIC MEDIA; FINITE ELEMENT METHOD; FLOW OF FLUIDS; HYDRATION; OSMOSIS; POROUS MATERIALS; TISSUE; VISCOELASTICITY;

COUPLED PROBLEMS; FINITE VISCOELASTICITIES; HYDRATED TISSUES; INHOMOGENEITIES; INTERVERTEBRAL DISK; MIXED-FEM; SOFT BIOLOGICAL TISSUE; THEORY OF POROUS MEDIAS; THEORY OF POROUS MEDIUM;

ANISOTROPY;

WATER;

ARTICLE; BIOLOGICAL MODEL; FINITE ELEMENT ANALYSIS; HUMAN; INTERVERTEBRAL DISK; OSMOTIC PRESSURE; PHYSIOLOGY; POROSITY;

FINITE ELEMENT ANALYSIS; HUMANS; INTERVERTEBRAL DISK; MODELS, BIOLOGICAL; OSMOTIC PRESSURE; POROSITY; WATER;

EID: 67349201415     PISSN: 16177959     EISSN: 16822978     Source Type: Journal    
DOI: 10.1007/s10237-008-0129-y     Document Type: Article

References (103)

1. M Argoubi A Shirazi-Adl 1996 Poroelastic creep response analysis of a lumbar motion segment in compression J Biomech 29 1331 1339 (Pubitemid 26302378)
2. DN Arnold F Brezzi M Fortin 1984 A stable finite element for the stokes equations Calcolo 21 337 344
3. Ayad S, Weiss JB (1987) Biochemistry of the intervertebral disc. In: Jayson MIV(eds) The lumbar spine and back pain, 3rd edn. Churchill Livingstone, New York, pp 100-137
4. DC Ayotte K Ito SM Perren S Tepic 2000 Direction-dependent constriction flow in a poroelastic solid: The intervertebral disc valve ASME J Biomech Eng 122 587 593 (Pubitemid 32047020)
5. NM Bachrach VC Mow F Guilak 1998 Incompressibility of the solid matrix of articular cartilage under high hydrostatic pressures J Biomech 31 445 451 (Pubitemid 28408343)
6. D Balzani P Neff J Schröder G Holzapfel 2005 A polyconvex framework for soft biological tissues. Adjustment to experimental data Int J Solids Struct 43 6052 6070
7. MA Biot 1941 General theory of three dimensional consolidation J Appl Phys 12 155 164
8. AW Bishop 1959 The effective stress principle Teknisk Ukeblad 39 859 863
9. Boehler JP (1987) Introduction of the invariant formulation of anisotropic constitutive equations. In: Boehler JP(eds) Applications of tensor functions in solid mechanics, CISM courses and lectures No. 292. Springer, Wien, pp 13-30
10. de Boer R (2000) Theory of Porous Media. Springer, Berlin
11. Bowen RM (1976) Theory of mixtures. In: Eringen AC(eds) Continuum physics, vol III. Academic Press, New York, pp 1-127
12. RM Bowen 1980 Incompressible porous media models by use of the theory of mixtures Int J Eng Sci 18 1129 1148
13. Braess D (1997) Finite elemente. Springer, Berlin
14. Brezzi F, Fortin M (1991) Mixed and hybrid finite element methods. Springer, New York
15. S Diebels P Ellsiepen W Ehlers 1999 Error-controlled Runge-Kutta time integration of a viscoplastic hybrid two-phase model Technische Mechanik 19 19 27
16. FG Donnan 1911 Theorie der Membrangleichgewichte und Membranpotentiale bei Vorhandensein von nicht dialysierenden Elektrolyten. Ein Beitrag zur physikalisch-chemischen Physiologie Zeitschrift für Elektrochemie und angewandte physikalische Chemie 17 572 581
17. S Ebara JC Iatridis LA Setton RJ Foster VC Mow M Weidenbaum 1996 Tensile properties of nondegenerate human lumbar anulus fibrosus Spine 21 452 461 (Pubitemid 26067854)
18. R Eberlein GA Holzapfel CAJ Schulze-Bauer 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
19. R Eberlein GA Holzapfel M Fröhlich 2004 Multi-segment FEA of the human lumbar spine including the heterogeneity of the anulus fibrosus Comput Mech 34 147 165
20. Ehlers W (1989) Poröse Medien-ein kontinuumsmechanisches Modell auf der Basis der Mischungstheorie. Habilitation, Forschungsberichte aus dem Fachbereich Bauwesen, Heft 47, Universität-GH-Essen
21. W Ehlers 1991 Toward finite theories of liquid-saturated elasto-plastic Porous Media Int J Plast 7 433 475 (Pubitemid 21713600)
22. Ehlers W (1993) Constitutive equations for granular materials in geomechanical context. In: Hutter K(eds) Continuum mechanics in environmental sciences and geophysics, CISM courses and lectures No. 337. Springer, Wien, pp 313-402
23. 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
24. Ehlers W, Acartürk A (2007) The role of weakly imposed Dirichlet boundary conditions for numerically stable computations of swelling phenomena. Comput Mech (submitted)
25. W Ehlers B Markert 2001 A linear viscoelastic biphasic model for soft tissues based on the Theory of Porous Media ASME J Biomech Eng 123 418 424 (Pubitemid 32938939)
26. 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. Nr. 99-II-1 aus dem Institut für Mechanik (Bauwesen), Universität Stuttgart
27. Ehlers W, Markert B, Acartürk A (2002) Large strain viscoelastic swelling of charged hydrated porous media. In: Auriault JL, Geindreau C, Royer P, Bloch JF, Boutin C, Lewandowska J (eds) Poromechanics II, Proceedings of the 2nd Biot conference on poromechanics, Swets & Zeitlinger, Lisse (Netherlands), pp 185-191
28. 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 Publishers, pp 781-786
29. W Ehlers N Karajan B Markert 2006 A porous media model describing the inhomogeneous behaviour of the human intervertebral disc Mater Sci Eng Technol 37 546 551 (Pubitemid 44012822)
30. Ehlers W, Markert B, Karajan N (2006b) 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
31. Ehlers W, Karajan N, Wieners C (2007) Parallel 3-d simulations of a biphasic porous media model in spine mechanics. In: Ehlers W, Karajan N (eds) Proceedings of the 2nd GAMM Seminar on Continuum Biomechanics, Report No. II-16 of the Institute of Applied Mechanics (CE), Universität Stuttgart, Germany, pp 11-20
32. 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
33. DA Elliott LA Setton 2000 A linear material model for fiber-induced anisotropy of the anulus fibrosus ASME J Biomech Eng 122 173 179 (Pubitemid 30318342)
34. 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
35. P Ellsiepen S Hartmann 2001 Remarks on the interpretation of current non-linear finite element analyses as differential-algebraic equations Int J Numer Methods Eng 51 679 707
36. AJH Frijns JM Huyghe JD Janssen 1997 A validation of the quadriphasic mixture theory for intervertebral disc tissue Int J Eng Sci 35 1419 1429 (Pubitemid 127400432)
37. AJH Frijns JM Huyghe EF Kaasschieter MW Wijlaars 2003 Numerical simulation of deformations and electrical potentials in a cartilage substitute Biorheology 40 123 131
38. WY Gu XG Mao RJ Foster M Weidenbaum VC Mow B Rawlins 1999 The anisotropic hydraulic permeability of human lumbar anulus fibrosus Spine 24 2449 2455
39. SM Hassanizadeh WG Gray 1987 High velocity flow in porous media Transp Porous Media 2 521 531 (Pubitemid 18581951)
40. WC Hayes AJ Bodine 1978 Flow-independent viscoelastic properties of articular cartilage matrix J Biomech 11 407 419 (Pubitemid 9043631)
41. S Holm A Nachemson 1983 Variations in the nutrition of the canine intervertebral disc induced by motion Spine 8 866 974
42. GA Holzapfel TC Gasser 2001 A viscoelastic model for fiber-reinforced composites at finite strains: Continuum basis, computational aspects and applications Comput Methods Appl Mech Eng 190 4379 4403 (Pubitemid 32541190)
43. GA Holzapfel TC Gasser RW Ogden 2004 Comparison of a multi-layer structural model for arterial walls with a Fung-type model, and issues of material stability ASME J Biomech Eng 126 264 275 (Pubitemid 38655934)
44. GA Holzapfel CAJ Schulze-Bauer G Feigl P Regitnig 2005 Mono-lamellar mechanics of the human lumbar anulus fibrosus Biomech Model Mechanobiol 3 125 140 (Pubitemid 40454312)
45. AH Hsieh DR Wagner LY Cheng JC Lotz 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 (Pubitemid 43067271)
46. JM Huyghe GB Houben MR Drost 2003 An ionised/non-ionised dual porosity model of intervertebral disc tissue Biomech Model Mechanobiol 2 3 19
47. JC Iatridis M Weidenbaum LA Setton VC Mow 1996 Is the nucleus pulposus a solid or a fluid? Mechanical behaviors of the human intervertebral disc Spine 21 1174 1184 (Pubitemid 26161382)
48. JC Iatridis LA Setton M Weidenbaum VC Mow 1997 The viscoelastic behavior of the non-degenerate human lumbar nucleus pulposus in shear J Biomech 30 1005 1013 (Pubitemid 27490229)
49. JC Iatridis LA Setton RJ Foster BA Rawlins M Weidenbaum VC Mow 1998 Degeneration affects the anisotropic and nonlinear behaviors of human anulus fibrosus in compression J Biomech 31 535 544 (Pubitemid 28434469)
50. JC Iatridis JP Laible MH Krag 2003 Influence of fixed charge density magnitude and distribution on the intervertebral disc: Applications of a Poroelastic and Chemical Electric (PEACE) model Trans ASME 125 12 24 (Pubitemid 36297224)
51. EF Kaasschieter AJH Frijns JM Huyghe 2003 Mixed finite element modelling of cartilaginous tissues Math Comput Simul 61 549 560
52. M Kleiber 1975 Kinematics of deformation processes in materials subjected to finite elastic-plastic strains Int J Eng Sci 13 513 525
53. SM Klisch JC Lotz 1999 Application of a fiber-reinforced continuum theory to multiple deformations of the annulus fibrosus J Biomech 32 1027 1036 (Pubitemid 29356042)
54. SM Klisch JC Lotz 2000 A special theory of biphasic mixtures and experimental results for human annulus fibrosus tested in confined compression ASME J Biomech Eng 122 180 188 (Pubitemid 30318343)
55. WM Lai JS Hou VC Mow 1991 A triphasic theory for the swelling and deformation behaviors of articular cartilage ASME J Biomech Eng 113 245 258
56. JP Laible DS Pflaster MH Krag BR Simon LD Haugh 1993 A poroelastic-swelling finite element model with application to the intervertebral disc Spine 18 659 670 (Pubitemid 23125036)
57. Y Lanir 1987 Biorheology and fluid flux in swelling tissues. I. Bicomponent theory for small deformations, including concentration effects Biorheology 24 173 187
58. CK Lee YE Kim CS Lee YM Hong JM Jung VK Goel 2000 Impact response of the intervertebral disc in a Finite-Element Model Spine 25 2431 2439
59. EH Lee 1969 Elastic-plastic deformation at finite strains J Appl Mech 36 1 6
60. LP Li J Soulhat MD Buschmann A Shirazi-Adl 1999 Nonlinear analysis of cartilage in unconfined ramp compression using a fibril reinforced poroelastic model Clin Biomech 14 673 682 (Pubitemid 29410485)
61. LP Li A Shirazi-Adl MD Buschmann 2003 Investigation of mechanical behavior of articular cartilage by fibril reinforced poroelastic models Biorheology 40 227 233
62. TH Lim JH Hong 2000 Poroelastic properties of bovine vertebral trabecular bone J Orthop Res 18 671 677
63. F Marchand AM Ahmed 1990 Investigation of the laminate structure of the lumbar disc anulus Spine 15 402 410 (Pubitemid 20188042)
64. Markert B (2005) Porous media viscoelasticity with application to polymeric foams. Dissertation, Bericht Nr. II-12 aus dem Institut für Mechanik (Bauwesen), Universität Stuttgart
65. B Markert 2007 A constitutive approach to 3-d nonlinear fluid flow through finite deformable porous continua with application to a high-porosity polyurethane foam Transp Porous Media 70 427 450
66. B Markert W Ehlers N Karajan 2005 A general polyconvex strain-energy function for fiber-reinforced materials Proc Appl Math Mech 5 245 246
67. M Mooney 1940 A theory of large elastic deformation J Appl Phys 11 582 592
68. Mow VC, Hayes WC (1997) Basic orthopaedic biomechanics. Lippincott-Raven, New York
69. VC Mow SC Kuei WM Lai CG Armstrong 1980 Biphasic creep and stress relaxation of articular cartilage in compression: theory and experiments ASME J Biomech Eng 102 73 84 (Pubitemid 11082099)
70. VC Mow MC Gibbs WM Lai WB Zhu KA Athanasiou 1989 Biphasic indentation of articular cartilage. II. A numerical algorithm and an experimental study J Biomech 22 853 861 (Pubitemid 20087386)
71. W Noll 1958 A mathematical theory of the mechanical behavior of continous media Arch Rat Mech Anal 2 197 226
72. RS Ochia RP Ching 2002 Hydraulic resistance and permeability in human lumbar vertebral bodies J Biomech Eng 124 533 537 (Pubitemid 35239601)
73. Parent-Thirion A, Macías EF, Hurley J, Vermeylen G (2007) Fourth European Working Conditions Survey. European Foundation for the Improvement of Living and Working Conditions, Dublin
74. H Raspe A Hueppe H Neuhauser 2008 Back pain, a communicable disease? Int J Epidemiol 37 69 74
75. S Reese S Govindjee 1998 A theory of finite viscoelasticity and numerical aspects Int J Solids Struct 35 3455 3482
76. PE Riches N Dhillon J Lotz AW Woods DS McNally 2002 The internal mechanics of the intervertebral disc under cyclic loading J Biomech 35 1263 1271
77. RS Rivlin 1948 Large elastic deformations of isotropic materials Proc R Soc Lond Ser A 241 379 397
78. RS Sandhu EL Wilson 1969 Finite-element analysis of seepage in elastic media ASCE J Eng Mech Div 95 641 652
79. M Schanz S Diebels 2003 A comperative study of biot's theory and the linear Theory of Porous Media for wave propagation problems Acta Mech 161 213 235
80. CO Schmidt H Raspe M Pfingsten M Hasenbring HD Basler W Eich T Kohlmann 2007 Back pain in the german adult population. Prevalence, severity, and sociodemographic correlations in a multiregional survey Spine 32 2005 2011 (Pubitemid 47301005)
81. J Schröder P Neff 2003 Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions Int J Solids Struct 40 401 445 (Pubitemid 36128780)
82. Y Schröder S Sivan W Wilson Y Merkher JM Huyghe A Maroudas FPT Baaijens 2007 Are disc pressure, stress and osmolarity affected by intra- and extrafibrillar fluid exchange J Orthop Res 25 1317 1324 (Pubitemid 47523361)
83. A Shirazi-Adl 1994 Nonlinear stress analysis of the whole lumbar spine in torsion-mechanics of facet articulation J Biomech 27 289 299 (Pubitemid 24061724)
84. A Shirazi-Adl 2006 Analysis of large compression loads on lumbar spine in felxion and torsion using a novel wrapping element J Biomech 39 267 275 (Pubitemid 41729000)
85. A Shirazi-Adl AM Ahmed SC Shrivastava 1986 A finite element study of a lumbar motion segment subjected to pure sagittal plane moments J Biomech 19 331 350 (Pubitemid 16080328)
86. A Shirazi-Adl AM Ahmed SC Shrivastava 1986 Mechanical response of a lumbar motion segment in axial torque alone and combined with compression Spine 11 914 927 (Pubitemid 17012170)
87. DL Skaggs M Weidenbaum JC Iatridis A Ratcliffe VC Mow 1994 Regional variation in tensile properties and biochemical composition of the human lumbar anulus fibrosus S 19 1310 1319 (Pubitemid 24219932)
88. 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
89. Spencer AJM (1972) Deformations of fiber-reinforced materials. Oxford University Press, NY, USA
90. Spencer AJM (1982) The formulation of constitutive equations for anisotropic solids. In: Boehler JP(eds) Mechanical behavior of anisotropic solids, Proceedings of the Euromech Colloquium, vol 115. Martinus Nijhoff Publishers, The Haque, pp 2-26
91. Spencer AJM (1984) Constitutive theory for strongly anisotropic solids. In: Spencer AJM(eds) Continuum theory of the mechanics of fibre reinforced composites, CISM Courses and Lectures No. 282. Springer, Wien, pp 1-32
92. DN Sun WY Gu XE Guo WM Lai VC Mow 1999 A mixed finite element formulation of triphasic mechano-electrochemical theory for charged, hydrated biological soft tissues Int J Numer Methods Eng 45 1375 1402 (Pubitemid 129540467)
93. Treloar LRG (1975) The physics of rubber elasticity, 3rd edn. Clarendon Press, Oxford
94. Truesdell C (1949) A new Definition of a Fluid, II. The Maxwellian fluid. Tech. Rep. P-3553, § 19, US Naval Research Laboratory
95. Urban J, Holm S (1986) Intervertebral disc nutrition as related to spinal movements and fusion. In: Hargens AR(eds) Tissue nutrition and viability. Springer, Berlin, pp 101-119
96. Urban JPG, Roberts S (1996) Intervertebral disc. In: Comper WD(eds) Extracellular matrix, vol 1, Tissue function. Harwood Academic Publishers, GmbH, pp 203-233
97. R van Loon FM Huyghe MW Wijlaars FPT Baaijens 2003 3D FE implementation of an incompressible quadriphasic mixture model Int J Numer Methods Eng 57 1243 1258 (Pubitemid 36906173)
98. Varga OH (1966) Stress-strain behavior of elastic materials. Interscience, New York
99. Wieners C (2003) Taylor-Hood elements in 3D. In: Wendland WL, Efendiev M(eds) Analysis and simulation of multified problems. Springer, Berlin, pp 189-196
100. C Wieners W Ehlers M Ammann N Karajan B Markert 2005 Parallel solution methods for porous media models in biomechanics Proc Appl Math Mech 5 35 38
101. W Wilson CC van Donkelar JM Huyghe 2005 A comparison between mechano-electrochemical and biphasic swelling theories for soft hydrated tissues ASME J Biomech Eng 127 158 165 (Pubitemid 40463490)
102. JSS Wu JH Chen 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 (Pubitemid 26147344)
103. Zienkiewicz OC, Taylor RL (2000) The finite element method: the basis, vol 1, 5th edn. Butterworth Heinemann, Oxford