Research Papers

An Anisotropic Multiphysics Model for Intervertebral Disk

[+] Author and Article Information
Xin Gao

Department of Mechanical and
Aerospace Engineering,
University of Miami,
Coral Gables, FL 33146
e-mail: x.gao3@umiami.edu

Qiaoqiao Zhu

Department of Biomedical Engineering,
University of Miami,
Coral Gables, FL 33146
e-mail: q.zhu3@umiami.edu

Weiyong Gu

Fellow ASME
Department of Mechanical and
Aerospace Engineering,
University of Miami,
Coral Gables, FL 33146;
Department of Biomedical Engineering,
University of Miami,
Coral Gables, FL 33146
e-mail: wgu@miami.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 1, 2015; final manuscript received October 12, 2015; published online November 9, 2015. Editor: Yonggang Huang.

J. Appl. Mech 83(2), 021011 (Nov 09, 2015) (8 pages) Paper No: JAM-15-1461; doi: 10.1115/1.4031793 History: Received September 01, 2015; Revised October 12, 2015

Intervertebral disk (IVD) is the largest avascular structure in human body, consisting of three types of charged hydrated soft tissues. Its mechanical behavior is nonlinear and anisotropic, due mainly to nonlinear interactions among different constituents within tissues. In this study, a more realistic anisotropic multiphysics model was developed based on the continuum mixture theory and employed to characterize the couplings of multiple physical fields in the IVD. Numerical simulations demonstrate that this model is capable of systematically predicting the mechanical and electrochemical signals within the disk under various loading conditions, which is essential in understanding the mechanobiology of IVD.

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Urban, J. P. G. , and Roberts, S. , 2003, “ Degeneration of the Intervertebral Disc,” Arthritis Res. Ther., 5(3), pp. 120–130. [CrossRef] [PubMed]
Antoniou, J. , Steffen, T. , Nelson, F. , Winterbottom, N. , Hollander, A. P. , Poole, R. A. , Aebi, M. , and Alini, M. , 1996, “ The Human Lumbar Intervertebral Disc—Evidence for Changes in the Biosynthesis and Denaturation of the Extracellular Matrix With Growth, Maturation, Ageing, and Degeneration,” J. Clin. Invest., 98(4), pp. 996–1003. [CrossRef] [PubMed]
Urban, J. P. , and Maroudas, A. , 1981, “ Swelling of the Intervertebral Disc In Vitro,” Connect Tissue Res., 9(1), pp. 1–10. [CrossRef] [PubMed]
Lai, W. M. , Hou, J. S. , and Mow, V. C. , 1991, “ A Triphasic Theory for the Swelling and Deformation Behaviors of Articular Cartilage,” ASME J. Biomech. Eng., 113(3), pp. 245–258. [CrossRef]
Donnan, F. G. , 1924, “ The Theory of Membrane Equilibria,” Chem. Rev., 1(1), pp. 73–90. [CrossRef]
Iatridis, J. C. , Weidenbaum, M. , Setton, L. A. , and Mow, V. C. , 1996, “ Is the Nucleus Pulposus a Solid or a Fluid? Mechanical Behaviors of the Nucleus Pulposus of the Human Intervertebral Disc,” Spine, 21(10), pp. 1174–1184. [CrossRef] [PubMed]
Vos, T. , Flaxman, A. , and Naghavi, M. , 2014, “ Years Lived With Disability (YLDs) for 1160 Sequelae of 289 Diseases and Injuries 1990–2010: A Systematic Analysis for the Global Burden of Disease Study 2010 (Vol 380, pg 2163, 2012),” Lancet, 384(9943), p. 582. [CrossRef] [PubMed]
Schmidt, H. , Galbusera, F. , Rohlmann, A. , and Shirazi-Adl, A. , 2013, “ What Have We Learned From Finite Element Model Studies of Lumbar Intervertebral Discs in the Past Four Decades?,” J. Biomech., 46(14), pp. 2342–2355. [CrossRef] [PubMed]
Belytschko, T. , Kulak, R. F. , Schultz, A. B. , and Galante, J. O. , 1974, “ Finite Element Stress Analysis of an Intervertebral Disc,” J. Biomech., 7(3), pp. 277–285. [CrossRef] [PubMed]
Schmidt, H. , Heuer, F. , Drumm, J. , Klezl, Z. , Claes, L. , and Wilke, H. J. , 2007, “ Application of a Calibration Method Provides More Realistic Results for a Finite Element Model of a Lumbar Spinal Segment,” Clin. Biomech. (Bristol, Avon), 22(4), pp. 377–384. [CrossRef] [PubMed]
Eberlein, R. , Holzapfel, G. A. , and Schulze-Bauer, C. A. , 2001, “ An Anisotropic Model for Annulus Tissue and Enhanced Finite Element Analyses of Intact Lumbar Disc Bodies,” Comput. Methods Biomech. Biomed. Eng., 4(3), pp. 209–229. [CrossRef]
Laible, J. P. , Pflaster, D. S. , Krag, M. H. , Simon, B. R. , and Haugh, L. D. , 1993, “ A Poroelastic-Swelling Finite-Element Model With Application to the Intervertebral Disc,” Spine, 18(5), pp. 659–670. [CrossRef] [PubMed]
Argoubi, M. , and ShiraziAdl, A. , 1996, “ Poroelastic Creep Response Analysis of a Lumbar Motion Segment in Compression,” J. Biomech., 29(10), pp. 1331–1339. [CrossRef] [PubMed]
Schroeder, Y. , Huyghe, J. M. , van Donkelaar, C. C. , and Ito, K. , 2010, “ A Biochemical/Biophysical 3D FE Intervertebral Disc Model,” Biomech. Model. Mechanobiol., 9(5), pp. 641–650. [CrossRef] [PubMed]
Karajan, N. , 2012, “ Multiphasic Intervertebral Disc Mechanics: Theory and Application,” Arch. Comput. Methods Eng., 19(2), pp. 261–339. [CrossRef]
Jacobs, N. T. , Cortes, D. H. , Peloquin, J. M. , Vresilovic, E. J. , and Elliott, D. M. , 2014, “ Validation and Application of an Intervertebral Disc Finite Element Model Utilizing Independently Constructed Tissue-Level Constitutive Formulations that are Nonlinear, Anisotropic, and Time-Dependent,” J. Biomech., 47(11), pp. 2540–2546. [CrossRef] [PubMed]
Ehlers, W. , Karajan, N. , and Markert, B. , 2009, “ An Extended Biphasic Model for Charged Hydrated Tissues With Application to the Intervertebral Disc,” Biomech. Model. Mechanobiol., 8(3), pp. 233–251. [CrossRef] [PubMed]
Yao, H. , and Gu, W. Y. , 2006, “ Physical Signals and Solute Transport in Human Intervertebral Disc During Compressive Stress Relaxation: 3D Finite Element Analysis,” Biorheology, 43(3–4), pp. 323–335. [PubMed]
Iatridis, J. C. , Laible, J. P. , and Krag, M. H. , 2003, “ Influence of Fixed Charge Density Magnitude and Distribution on the Intervertebral Disc: Applications of a Poroelastic and Chemical Electric (PEACE) Model,” ASME J. Biomech. Eng., 125(1), pp. 12–24. [CrossRef]
Sun, D. D. N. , and Leong, K. W. , 2004, “ A Nonlinear Hyperelastic Mixture Theory Model for Anisotropy, Transport, and Swelling of Annulus Fibrosus,” Ann. Biomed. Eng., 32(1), pp. 92–102. [CrossRef] [PubMed]
Lanir, Y. , 1987, “ Biorheology and Fluid Flux in Swelling Tissues.1. Bicomponent Theory for Small Deformations, Including Concentration Effects,” Biorheology, 24(2), pp. 173–187. [PubMed]
Wilson, W. , van Donkelaar, C. C. , and Huyghe, J. M. , 2005, “ A Comparison Between Mechano-Electrochemical and Biphasic Swelling Theories for Soft Hydrated Tissues,” ASME J. Biomech. Eng., 127(1), pp. 158–165. [CrossRef]
Gao, X. , and Gu, W. , 2014, “ A New Constitutive Model for Hydration-Dependent Mechanical Properties in Biological Soft Tissues and Hydrogels,” J. Biomech., 47(12), pp. 3196–3200. [CrossRef] [PubMed]
Perie, D. , Korda, D. , and Iatridis, J. C. , 2005, “ Confined Compression Experiments on Bovine Nucleus Pulposus and Annulus Fibrosus: Sensitivity of the Experiment in the Determination of Compressive Modulus and Hydraulic Permeability,” J. Biomech., 38(11), pp. 2164–2171. [CrossRef] [PubMed]
Iatridis, J. C. , Setton, L. A. , Foster, R. J. , Rawlins, B. A. , Weidenbaum, M. , and Mow, V. , 1998, “ Degeneration Affects the Anisotropic and Nonlinear Behaviors of Human Anulus Fibrosus in Compression,” J. Biomech., 31(6), pp. 535–544. [CrossRef] [PubMed]
Elliott, D. M. , and Setton, L. A. , 2001, “ Anisotropic and Inhomogeneous Tensile Behavior of the Human Anulus Fibrosus—Experimental Measurement and Material Model Predictions,” ASME J. Biomech. Eng., 123(3), pp. 256–263. [CrossRef]
Gu, W. Y. , Mao, X. G. , Foster, R. J. , Weidenbaum, M. , Mow, V. C. , and Rawlins, B. A. , 1999, “ The Anisotropic Hydraulic Permeability of Human Lumbar Anulus Fibrosus—Influence of Age, Degeneration, Direction, and Water Content,” Spine, 24(23), pp. 2449–2455. [CrossRef] [PubMed]
Jackson, A. R. , Yuan, T. Y. , Huang, C. Y. , Brown, M. D. , and Gu, W. Y. , 2012, “ Nutrient Transport in Human Annulus Fibrosus is Affected by Compressive Strain and Anisotropy,” Ann. Biomed. Eng., 40(12), pp. 2551–2558. [CrossRef] [PubMed]
Gu, W. Y. , Lai, W. M. , and Mow, V. C. , 1998, “ A Mixture Theory for Charged-Hydrated Soft Tissues Containing Multi-Electrolytes: Passive Transport and Swelling Behaviors,” ASME J. Biomech. Eng., 120(2), pp. 169–180. [CrossRef]
Frijns, A. J. H. , Huyghe, J. M. , and Janssen, J. D. , 1997, “ A Validation of the Quadriphasic Mixture Theory for Intervertebral Disc Tissue,” Int. J. Eng. Sci., 35(15), pp. 1419–1429. [CrossRef]
Ateshian, G. A. , 2007, “ On the Theory of Reactive Mixtures for Modeling Biological Growth,” Biomech. Model. Mechanobiol., 6(6), p. 447. [CrossRef]
Sun, D. N. , Gu, W. Y. , Guo, X. E. , Lai, W. M. , and Mow, V. C. , 1999, “ A Mixed Finite Element Formulation of Triphasic Mechano-Electrochemical Theory for Charged, Hydrated Biological Soft Tissues,” Int. J. Numer. Methods Eng., 45(10), pp. 1375–1402. [CrossRef]
Yao, H. , and Gu, W. Y. , 2007, “ Convection and Diffusion in Charged Hydrated Soft Tissues: A Mixture Theory Approach,” Biomech. Model. Mechanobiol., 6(1–2), pp. 63–72. [CrossRef] [PubMed]
Ehlers, W. , and Eipper, G. , 1999, “ Finite Elastic Deformations in Liquid-Saturated and Empty Porous Solids,” Transp. Porous Med., 34(1–3), pp. 179–191. [CrossRef]
Holzapfel, G. A. , Gasser, T. C. , and Ogden, R. W. , 2000, “ A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models,” J. Elasticity, 61(1–3), pp. 1–48. [CrossRef]
Federico, S. , and Herzog, W. , 2008, “ On the Permeability of Fibre-Reinforced Porous Materials,” Int. J. Solids Struct., 45(7–8), pp. 2160–2172. [CrossRef]
Gu, W. Y. , Yao, H. , Huang, C. Y. , and Cheung, H. S. , 2003, “ New Insight Into Deformation-Dependent Hydraulic Permeability of Gels and Cartilage, and Dynamic Behavior of Agarose Gels in Confined Compression,” J. Biomech., 36(4), pp. 593–598. [CrossRef] [PubMed]
Gu, W. Y. , Yao, H. , Vega, A. L. , and Flagler, D. , 2004, “ Diffusivity of Ions in Agarose Gels and Intervertebral Disc: Effect of Porosity,” Ann. Biomed. Eng., 32(12), pp. 1710–1717. [CrossRef] [PubMed]
Broberg, K. B. , 1983, “ On the Mechanical-Behavior of Intervertebral Disks,” Spine, 8(2), pp. 151–165. [CrossRef] [PubMed]
O’Connell, G. D. , Vresilovic, E. J. , and Elliott, D. M. , 2007, “ Comparison of Animals Used in Disc Research to Human Lumbar Disc Geometry,” Spine, 32(3), pp. 328–333. [CrossRef] [PubMed]
Urban, J. P. G. , and Maroudas, A. , 1979, “ Measurement of Fixed Charge-Density in the Intervertebral-Disk,” Biochim. Biophys. Acta, 586(1), pp. 166–178. [CrossRef]
Iatridis, J. C. , MacLean, J. J. , O’Brien, M. , and Stokes, I. A. F. , 2007, “ Measurements of Proteoglycan and Water Content Distribution in Human Lumbar Intervertebral Discs,” Spine, 32(14), pp. 1493–1497. [CrossRef] [PubMed]
Ruiz, C. , Noailly, J. , and Lacroix, D. , 2013, “ Material Property Discontinuities in Intervertebral Disc Porohyperelastic Finite Element Models Generate Numerical Instabilities Due to Volumetric Strain Variations,” J. Mech. Behav. Biomed., 26, pp. 1–10. [CrossRef]
Cortes, D. H. , Jacobs, N. T. , DeLucca, J. F. , and Elliott, D. M. , 2014, “ Elastic, Permeability and Swelling Properties of Human Intervertebral Disc Tissues: A Benchmark for Tissue Engineering,” J. Biomech., 47(9), pp. 2088–2094. [CrossRef] [PubMed]
Iatridis, J. C. , Setton, L. A. , Weidenbaum, M. , and Mow, V . C. , 1997, “ Alterations in the Mechanical Behavior of the Human Lumbar Nucleus Pulposus With Degeneration and Aging,” J. Orthopaed. Res., 15(2), pp. 318–322. [CrossRef]
Farrell, M. D. , and Riches, P. E. , 2013, “ On the Poisson’s Ratio of the Nucleus Pulposus,” ASME J. Biomech. Eng., 135(10), p. 104501. [CrossRef]
Iatridis, J. C. , Kumar, S. , Foster, R. J. , Weidenbaum, M. , and Mow, V . C. , 1999, “ Shear Mechanical Properties of Human Lumbar Annulus Fibrosus,” J. Orthopaed. Res., 17(5), pp. 732–737. [CrossRef]
Best, B. A. , Guilak, F. , Setton, L. A. , Zhu, W. B. , Saednejad, F. , Ratcliffe, A. , Weidenbaum, M. , and Mow, V. C. , 1994, “ Compressive Mechanical-Properties of the Human Anulus Fibrosus and Their Relationship to Biochemical-Composition,” Spine, 19(2), pp. 212–221. [CrossRef] [PubMed]
Ebara, S. , Iatridis, J. C. , Setton, L. A. , Foster, R. J. , Mow, V. C. , and Weidenbaum, M. , 1996, “ Tensile Properties of Nondegenerate Human Lumbar Anulus Fibrosus,” Spine, 21(4), pp. 452–461. [CrossRef] [PubMed]
Yao, H. , and Gu, W. Y. , 2004, “ Physical Signals and Solute Transport in Cartilage Under Dynamic Unconfined Compression: Finite Element Analysis,” Ann. Biomed. Eng., 32(3), pp. 380–390. [CrossRef] [PubMed]
Lu, Y. M. , Hutton, W. C. , and Gharpuray, V. M. , 1996, “ Do Bending, Twisting, and Diurnal Fluid Changes in the Disc Affect the Propensity to Prolapse? A Viscoelastic Finite Element Model,” Spine, 21(22), pp. 2570–2579. [CrossRef] [PubMed]
O’Connell, G. D. , Jacobs, N. T. , Sen, S. , Vresilovic, E. J. , and Elliott, D. M. , 2011, “ Axial Creep Loading and Unloaded Recovery of the Human Intervertebral Disc and the Effect of Degeneration,” J. Mech. Behav. Biomed., 4(7), pp. 933–942. [CrossRef]
Mcnally, D. S. , and Arridge, R. G. C. , 1995, “ An Analytical Model of Intervertebral Disc Mechanics,” J. Biomech., 28(1), pp. 53–68. [CrossRef] [PubMed]
Heuer, F. , Schmidt, H. , Klezl, Z. , Claes, L. , and Wilke, H. J. , 2007, “ Stepwise Reduction of Functional Spinal Structures Increase Range of Motion and Change Lordosis Angle,” J. Biomech., 40(2), pp. 271–280. [CrossRef] [PubMed]
Skaggs, D. L. , Weidenbaum, M. , Iatridis, J. C. , Ratcliffe, A. , and Mow, V. C. , 1994, “ Regional Variation in Tensile Properties and Biochemical-Composition of the Human Lumbar Anulus Fibrosus,” Spine, 19(12), pp. 1310–1319. [CrossRef] [PubMed]
Adams, M. A. , and Roughley, P. J. , 2006, “ What is Intervertebral Disc Degeneration, and What Causes it?,” Spine, 31(18), pp. 2151–2161. [CrossRef] [PubMed]
Stokes, I. A. F. , Laible, J. P. , Gardner-Morse, M. G. , Costi, J. J. , and Iatridis, J. C. , 2011, “ Refinement of Elastic, Poroelastic, and Osmotic Tissue Properties of Intervertebral Disks to Analyze Behavior in Compression,” Ann. Biomed. Eng., 39(1), pp. 122–131. [CrossRef] [PubMed]
Setton, L. A. , and Chen, J. , 2006, “ Mechanobiology of the Intervertebral Disc and Relevance to Disc Degeneration,” J. Bone Jt. Surg. Am., 88A, pp. 52–57. [CrossRef]
Iatridis, J. C. , Furukawa, M. , Stokes, I. A. F. , Gardner-Morse, M. G. , and Laible, J. P. , 2009, “ Spatially Resolved Streaming Potentials of Human Intervertebral Disk Motion Segments Under Dynamic Axial Compression,” ASME J. Biomech. Eng., 131(3), p. 031006. [CrossRef]


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Fig. 2

Schematic diagrams for experimental protocols used in simulations in this study. (a) Creep test, (b) bending and torsion tests, (c) osmotical loading test, and (d) dynamic loading test.

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Fig. 3

Simulated disk height loss during creep test, and compared with experimental results [52]

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Fig. 1

A schematic of the IVD showing the AF, NP, and CEPs. The dimensions are in millimeter.

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Fig. 4

Fluid pressure during creep test. (a) Before load applied, (b) right after load applied, (c) after 2 hrs of creep, and (d) after 4 hrs of creep.

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Fig. 5

Simulated range of motion of disk under (a) 2.5 N·m and (b) 5.0 N·m bending momentums, and compared with experimental results (minimum, median, and maximum) [54]

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Fig. 6

Distributions of fiber stretch under 5.0 N·m bending momentum in (a) flexion, (b) extension, (c) lateral bending, and (d) axial rotation. Only one family of fibers are shown.

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Fig. 7

Simulated mechanical response of disk to the change of bath saline solution, from 0.15 M to 1.50 M, and compared with experimental results (mean ± SD) [57]

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Fig. 8

Simulated distribution of electrical potential under dynamical loading at midsagittal line, and compared with experimental results (mean ± SEM) [59]



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