This work was aimed at developing a first approach to produce structural models of the Annulus Fibrosus, i.e. constitutive models of individual lamellae. Parameters of Holzapfel’s model were fitted using literature data on the behaviour of tissue components. Used on sample-specific or simplified geometries, the effect on the whole annulus deformation of the fibre angles, interlamellar behaviour, and incompressibility was analysed to assess their role in the disc mechanics

Jones, AC

Mengoni, M

Wijayathunga, VN

Wilcox, RK

White Rose Research Online

promoting access to White Rose research papers    White Rose Research Online eprints@whiterose.ac.uk    Universities of Leeds, Sheffield and York http://eprints.whiterose.ac.uk/    This is an author produced version of a paper published in 3rd International Conference on Mathematical and Computational Biomedical Engineering  White Rose Research Online URL for this paper:  http://eprints.whiterose.ac.uk/76931/     Paper: Mengoni, M, Wijayathunga, VN, Jones, AC and Wilcox, RK (2013) Structural modelling of the annulus fibrosus - an anisotropic hyperelastic model approach at the lamellar level. In: Nithiarasu, P and Löhner, R, (eds.) 3rd International Conference on Mathematical and Computational Biomedical Engineering.   3rd International Conference on Mathematical and Computational Biomedical Engineering – CMBE201316-18 December 2013, Hong-KongP. Nithiarasu and R. Löhner (eds)Structural modelling of the annulus fibrosus - an anisotropic hyperelastic modelapproach at the lamellar levelMarlène Mengoni, Vithanage N. Wijayathunga, Alison C. Jones and Ruth K. WilcoxInstitute of Medical and Biological Engineering, University of Leeds, Leeds LS29JT, UK, m.mengoni@leeds.ac.ukSUMMARYThis work was aimed at developing a first approach to produce structural models of the AnnulusFibrosus, i.e. constitutive models of individual lamellae. Parameters of Holzapfel’s model werefitted using literature data on the behaviour of tissue components. Used on sample-specific orsimplified geometries, the effect on the whole annulus deformation of the fibre angles, inter-lamellar behaviour, and incompressibility was analysed to assess their role in the disc mechanics.Key Words: annulus fibrosus lamellae, anisotropic hyperelasticity, finite element analysis.1. INTRODUCTIONThe annulus fibrosus is the fibrous avascular outer part of the intervertebral disc (IVD). It iscomposed of a series of circumferential layers called the lamellae [1]. Each lamella is composedof collagen fibre bundles embedded into a ground matrix. Secondary structures such as bridgesacross the lamellae or surrounding elastin walls are also present. In each lamella, the fibres arequasi-uniformly oriented in one direction at an angle of 20 to 50 with the transverse plane. Thisangle alternates between adjacent lamellae. The inner part of the IVD, surrounded by the annulus,is a gel-like substance called the nucleus pulposus.The IVD is a permanently loaded environment. It acts as a joint between two adjacent vertebraeand gives the spine its flexibility. It is subjected to compressive loads, bending and shear. Inparticular, the annulus withstands internal pressure due to the nucleus hydrostatic compressivestress. As the annulus bulges, the tissue is extended, making tension an important mode ofloading for the annulus fibrosus [1]. A realistic virtual model of the IVD can give insight into thesignificance of deformation patterns and is a help for the design of novel treatment approaches.As such the understanding of the mechanical properties of the different constituent tissues andtheir role in the IVD mechanics is of greater importance. In the last decade, many constitutivemodels of the annulus fibrosus have been proposed, most of these considering an annulusconstituted of two sets of fibres with different orientations and an isotropic elastic matrix [2,3],with some accounting for the cross-links between adjacent lamellae [4]. To our knowledge noprevious model has accounted for fibres and matrix of individual lamella, the inter-lamellarbehaviour and the cross-bridges. This study took the first step towards such a model. Theapproach was focused on the structural aspects and did not account for viscous or damage effects.2. METHODSConsidering the nonlinear behaviour of the collagen fibres and the structural aspect the model hasto represent, an anisotropic hyperelastic model was chosen for the lamellar behaviour. Such amodel allows consideration of separate energy density functions for the matrix and the fibres,including their specific orientation. It also allows the consideration of incompressibility for thelamellae. In particular, Holzapfel’s constitutive model [5], developed mainly for arterial walls,was used in this work. It assumes a linear shear behaviour of the ground matrix (neo-Hookeanmodel) and a non-linear exponential behaviour of oriented fibres, that bear load only in tension.First, 1D analytical stress-strain curves for each component of Holzapfel’s model were fittedagainst experimental data for a single lamellae extracted from the literature. Details of mechanicaltests on lamellar samples prepared in such a way that the fibres would not bear loads, and thusrepresentative of the matrix behaviour, are available in [6]. Data from shear tests were used tocharacterise the ground matrix mechanical behaviour. Single lamellar tension tests performedalong the fibre direction [3,7,8] were used to characterise the fibre behaviour.The constitutive model with the fitted parameters was then used in a sample-specific finiteelement (FE) model. The geometry was acquired from a micrograph of a sample cut through theannulus width in the radial direction [9]. Segmentation and tetrahedral mesh generation wereperformed using ScanIP 5.1 (Simpleware Ltd). This FE model was used to assess the relevance ofan anisotropic hyperelastic model versus a neo-Hookean model with equivalent shear stiffness.Finally, the constitutive model was used in a finite element model of the whole annulus fibrosusrepresented as a simplified cylindrical geometry. This model consisted of 12 concentric perfectlycircular lamellae subjected to internal pressure. It was used to analyse the effect of inter-lamellarbehaviour on the annulus bulge as well as the effect of the site-dependency of the fibrousbehaviour, the fibre orientation, and the incompressibility.All finite element analyses were performed using the implicit finite element software Abaqus6.12 (Dassault Systèmes) with linear tetrahedral or hexahedral elements.3. RESULTSData available in [6] clearly shows the ground matrix shear behaviour is non-linear. However alinear model that characterises the shear behaviour of the neo-Hookean model can be fitted(r2=0.982) up to 30% shear strains (Figure 1, left). The non-linear behaviour of the fibresdepends on the fibre location around the annulus (Figure 1, right - dots). An exponentialbehaviour as in Holzapfel’s model can be fitted (Figure 1, right - lines) for each experimentaltest (r2 between 0.986 and 0.999) over the whole available stretch range.The longitudinal force (radial in terms of the IVD geometry) needed to stretch a radial cutthrough the annulus, for two material models (Holzapfel’s model or a neo-Hookean model), isshown on Figure 2 (right). As the elongation is perpendicular to the fibre stretch, the neo-Hookean model, being isotropic, necessitates a higher force (20% greater at 20% strain).Figure 1- Results of a parameter fit: Left: a neo-Hookean model of the ground matrix (exp. data from [6],linear fit in red); Right: an exponential behaviour of the collagen fibres (exp. data from [7,8]).Figure 2 – Left: FE model of 20% applied strain to a radial cut - white: lamellae cut along the fibredirection, red lamellae with sectioned fibres, green: cross-bridges; Right: Force vs. strain for a neo-Hookean model (in orange) and a Holzapfel one (in blue).Figure 3 – Results from an FE model of cylindrical annulus under pressure - Left: Change in bulge vs.internal pressure as a function of the fibre angle. Right: Change in bulge vs. internal pressure as a functionof the inter-lamellar interaction model.Finally, the bulge of a cylindrical annulus under pressure is presented in Figure 3. It shows adependence on the assumed fibre angle (adjacent lamellae have supplementary angles from thetransverse plane). It also shows a strong dependence on the assumed inter-lamellar behaviour.4. DISCUSION / CONCLUSIONSThis work, focussing on a structural model of the annulus fibrosus, showed that the use ofHolzapfel’s constitutive model with one fibre orientation to represent each lamellae showspromising results. The exponential model of the fibre mechanics is adequate on the whole testedrange available in the literature. The linear model of the matrix gives relevant results up to 30%shear strains. This is a limitation at very large strain but not essentially for clinically relevantdeformations. A radial stretching model of an actual sample showed that using an anisotropic lawrather than a neo-Hookean law is worth the extra-computational cost. When used on a cylindricalrepresentation of the annulus, the model shows not only a sensitivity to the fibre angle but also tothe assumed inter-lamellar behaviour. Future work will propose experimental tests to furthervalidate the model of the lamellar components, including the incompressibility and the possiblepre-stress. The use of a validated lamellar constitutive model in conjunction with whole annulusexperimental data could give insight to the inter-lamellar behaviour.AcknowledgmentThis work was funded through WELMEC, a Centre of Excellence in Medical Engineering fundedby the Wellcome Trust and EPSRC, under grant number 088908/Z/09/ZREFERENCES[1] J.J. Cassidy, A. Hiltner & E. Baer, Hierarchical structure of the intervertebral-disc, ConnectTissue Res, 23, 75-88, 1989[2] D.M. Elliott & L.A. Setton, A Linear Material Model for Fiber-Induced Anisotropy of theAnulus Fibrosus, J Biomech Eng, 122, 173-179, 2000[3] R. Eberlein, G.A. Holzapfel & C.A.J. Schulze-Bauer, An Anisotropic Model for AnnulusTissue and Enhanced Finite Element Analyses of Intact Lumbar Disc Bodies, CMBBE, 4,209-229, 2001[4] H.L. Guerin & D.M. Elliott, Quantifying the Contributions of Structure to Annulus FibrosusMechanical Function Using a Nonlinear, Anisotropic, Hyperelastic Model, J. of OrthopaedicResearch, 25, 508-516, 2007[5] G.A. Holzapfel, T.C. Gasser & R.W. Ogden, A New Constitutive Framework for ArterialWall Mechanics and a Comparative Study of Material Models, J Elasticity 61, 1-48, 2000[6] J.P. Little, M.J. Pearcy, G. Tevelen, J.H. Evans, G. Pettet & C.J. Adam, The mechanicalresponse of the ovine lumbar anulus fibrosus to uniaxial, biaxial and shear loads, J MechBehav Biomed Mater, 3, 146-157, 2010[7] G.A. Holzapfel, C.A.J. Schulze-Bauer, G. Feigl & P. Regitnig, Single lamellar mechanics ofthe human lumbar anulus fibrosus, Biomech Model Mechanobiol, 3, 125-40, 2005[8] D.L. Skaggs, M. Weidenbaum, J.C. Iatridis, A. Ratcliffe & V.C,. Mow Regional Variation inTensile Properties and Biochemical Composition of the Human Anulus Fibrosus, Spine, 19,1310-1319, 1994[9] B.J. Luxmoore, Computational Simulation of the Intervertebral Disc, PhD Thesis, Universityof Leeds, School of Mechanical Engineering, 2013

Structural modelling of the annulus fibrosus - an anisotropic hyperelastic model approach at the lamellar level

Mengoni, Marlène

Wijayathunga, V. N.

Jones, Alison C.

Wilcox, Ruth K.

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Structural Modelling of the Annulus Fibrosus - an Anisotropic Hyperelastic Model Approach at the Lamellar Level

Structural modelling of the annulus fibrosus - an anisotropic hyperelastic model approach at the lamellar level

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