Contact mechanics and wear simulations of hip resurfacing devices using computational methods

Transcription

1 Ata of Bioengineering and Biomehanis Vol. 16, No. 2, 2014 Original paper DOI: /abb Contat mehanis and wear simulations of hip resurfaing devies using omputational methods MURAT ALI*, KEN MAO Shool of Engineering, University of Warwik, Coventry, UK. The development of omputational and numerial methods provides the option to study the ontat mehanis and wear of hip resurfaing devies. The importane of these tehniques is justified by the extensive amount of testing and experimental work required to verify and improve urrent orthopaedi implant devies. As the demands for devie longevity is inreasing, it is as important as ever to study tehniques for providing muh needed orthopaedi hip implant solutions. Through the use of advaned omputer aided design and the finite element method, ontat analysis of hip resurfaing devies was arried out by developing both three-dimensional and twodimensional axisymmetri models whilst onsidering the effets of loading onditions and material properties on the ontat stresses. Following on from this, the three-dimensional model was used in ombination with a unique programme to develop wear simulations and obtain umulative wear for both the aetabular up and femoral head simultaneously. Key words: arhard wear law, biotribology, ontat, finite element analysis, wear 1. Introdution The introdution of hip replaements inspired the development and prodution of hip resurfaing devies, whih are designed to redue the volumetri wear rates of the implant devies and inrease the longevity of the implantation. These produts are also aimed to be suitable for younger patients who require a hip joint implant. The study of wear and surfae damage has long been of a high importane for hip implant researh, development and manufature. In the past, the majority of researh was mainly foused on the use of experimental in vitro testing simulators to predit the mehanial wear of the implant devies using international standards. With the inreasing demands on orthopaedi devie ompanies to produe validated and reliable implants whih are safe for patients, the neessity for new and improved tehniques to be developed and support the engineering ativities of suh devies is imperative. In addition to arrying out experimental simulator testing, a viable alternative method for studying the mehanial wear of hip resurfaing devies is through the development of omputational ontat and wear models whih an predit the wear of hip resurfaing devies [1]. The omplexity of wear mehanisms [2] makes modelling suh problems more hallenging than traditional stress and strain analysis using the finite element method. To redue the omplexity of modelling wear mehanisms, a number of wear models have been proposed in the past and the most widely used is the Arhard wear model [3], whih is still used today as shown by reent literature by Rońda and Wojnarowski [4]. The objetive of this study was to model the ontat mehanis of hip resurfaing devies under normal, edge loading and serve loading onditions, speifially assessing the ontat stresses of the hip implant. Obtaining the ontat stress results along with arefully assessing the boundary onditions and material properties applied, formed the basis for developing wear models whih ould be applied to as- * Corresponding author: Murat Ali, Shool of Engineering, University of Warwik, Coventry, CV4 7AL, UK. Tel: , fax: , murat.ali@warwik.a.uk. Reeived: July 19th, 2013 Aepted for publiation: November 14th, 2013

2 104 M. ALI, K. MAO sess the volumetri wear of both the femoral head and aetabular up simultaneously, whih has not been onsidered in previous literature. 2. Materials and methods A three-dimensional segmented hip resurfaing devie model and two-dimensional axisymmetri model were developed based on the models used for ontat analysis by Ali and Mao [5], [6], with a up and head bearing diameter of 50 mm and diametral learane of 80 µm based geometrially on the models proposed by Udofia et al. [7]. The segmented model allowed for the use of DICOM (Digital Imaging and Communiations in Mediine) bone sans to be prepared for an assembly with hip resurfaing implant models using Solidworks (Dassault Systèmes Solid- Works Corporation, Waltham, Massahusetts, USA) Fig. 1. Segmented three-dimensional hip resurfaing finite element model Fig. 2. Two-dimensional axisymmetri hip resurfaing finite element model omputer aided design. Through an assoiative interfae with the ommerially and aademially available finite element analysis software Simulia ABAQUS (Version , Dassault Systèmes SolidWorks Corporation, Waltham, Massahusetts, USA), finite element assembly models were developed as shown in Fig. 1 and Fig. 2. A ombination of hexagonal (C3D8I) and wedge (C3D6) finite elements were used for modelling the aetabular up and femoral head omponents. These elements were seleted based on their suitability for studying the ontat mehanis of hip implant devies. During the model meshing proess, seeding tehniques were used inluding definitions of part seed sizes and edge seeding with diretional bias to define nodal points in areas of high ontat stresses suh as edge loading and rim ontat. Due to the geometrial omplexity of the femur and pelvi bone models, an automati meshing algorithm was utilised and based on tetrahedral elements. A sliding ontat interation algorithm was defined with a nominal value of frition oeffiient between the femoral head and aetabular up of 0.16, whih was based on the frition fator of CoCrMo on CoCrMo (obalt hromium molybdenum) in both bovine serum and synovial fluid [8]. Following on from the development of the threedimensional model, the two-dimensional axisymmetri model allowed for a omputational effiient omparison of ontat pressures and stresses of three different vertial loading onditions. The ISO vertial loading of 3000 N (F I ) and vertial load of 3900 N (F y ) were applied based on the peak load expeted to our during the walking yle. In addition to this, a stumbling load of N (F s ) was also onsidered, as these high vertial loads have been highlighted to our in vivo [9]. Both impliit and expliit finite element solvers were used to study the ontat pressure during edge loading onditions and rim ontat between the femoral head and aetabular up. The use of the impliit solver allowed for an assessment of rim ontat to be made under lateral sliding of the femoral head leading to edge loading rim ontat. The expliit solver provided a suitable numerial method for studying the ontat mehanis at the rim of the aetabular up aused by inferior and lateral miroseparation leading to edge loading. This type of edge loading ontat ours during the re-engagement of the femoral head within the aetabular up as a result of vertial loading at heal strike whih is referred to as pure miroseparation in this study. The differenes between both impliit and expliit tehniques for engineering prob-

3 Contat mehanis and wear simulations of hip resurfaing devies using omputational methods 105 lem solving has been onisely explained by Harewood and MHugh [10]. For an impliit analysis, a system of equations are required to be solved for eah analysis and displaement inrementation. The global stiffness matrix K an be inverted to solve for inremental displaements, however, this is very often a omputationally intensive proess, espeially for non-linear quasi-stati problems involving ontat. Modelling normal ontat onditions and lateral displaement edge loading between the femoral head and aetabular up was ahieved by reahing inremental equilibrium using the impliit method based on the Newton Raphson solver. For solving the finite element problem under laxity based edge loading the expliit method was utilised. For laxity based edge loading a miroseparation distane between the femoral head and aetabular up was first established, as this is expeted to our during swing phase of the human walking gait yle. Following on from this, ontat would our under the ation of the vertial load during the stane phase. This is a omplex ontat ondition to model, therefore, a forward Euler integration sheme was used, whih an be used to solve omplex ontat problems. Overall, the expliit analysis is less omputational intensive than impliit modelling as large stiffness matries are not required to be inverted. In order to simulate wear based on the Arhard Wear model, the magnitude of linear wear depth h I given by n h I = kw pi ( si si 1) (1) i= 1 was defined at eah node as being linearly proportional to ontat pressure p, sliding distane s and wear oeffiient value k w at eah analysis inrement i. The number of quasi-stati finite element iterations were defined from i = 1 to n. Wear oeffiients were assumed to be onstant during bedding-in and steady state phases as observed from experimental hip simulator studies, with the same approah applied within previous wear simulation models [1]. The gait loading boundary onditions have been applied based on the ISO (International Organisation of Standardization) standards developed for in vitro testing, whih simulate typial walking gait loading profiles as disussed by Bergmann [11]. This auses ontat to our between the hip implant bearing omponents. The sliding distane is determined by the physial sliding between the bearing omponents in ontat, whih our through the relative angular displaement of the bearing. Therefore, ISO standard flexion-extension and inward-outward rotations are modelled as kinemati boundary onditions. To alulate the sliding distane between the implant devies, two numerial methods were used through a ombination of the finite element output database and user defined interfaes. One of the methods involved alulating the magnitude of the relative sliding distane d between the two ontat surfaes and the other was based on the hange in oordinate position of the femoral head during the angular displaement yle d i = ( ui ui 1 ) + ( i ν i 1) + ( w wi 1) ν (2) where u, v and w are the displaements in the x, y and z diretion. For both methods, the finite element iteration defined the intervals at whih the sliding distane was alulated. Although the abdution-addution angular displaement was not inluded in the ontat sliding model due to modelling onstraints, literature has shown the wear rates from the assessments of in vivo retrieval implants math wear rates from two-axis experimental simulator studies more losely than studies using threeaxis simulators (three motion rig) [12]. All rotations have been applied about the entre point of the femoral head (Fig. 3), whih was defined as the entre of rotation. Fig. 3. Hip resurfaing ontat and wear simulation model with angular displaements The wear simulation method was developed using a Python sripting interfae along with the finite element model. Following the alulation of linear wear at eah inrement, the total wear h T is multiplied by the number of yles k before finite element remeshing ours k h T = h I. (3) i= 1 The total umulative nodal linear and volumetri wear was alulated and the results presented. More detail on the wear simulation proess is provided in Fig. 4. This is an alternative method to that previ-

4 106 M. ALI, K. MAO ously developed by other researhers using subroutines to simulate wear. The material models and methodology for both the metal-on-metal CoCrMo hip bearing implants and bone models are disussed within a previous study by Ali and Mao [5]. A summary of the nominal materials model values are provided in Table 1, where E F and E P are the equivalent elasti modulus values applied to the femur and pelvi bone models, respetively. is given by µ x. The density and Young s modulus values are determined from definition of the user defined oeffiients a j, b j, j and d j where j denotes the subsripts a, b and ν a = E a b ρ = a b GS, (5) a + bb + db 0 a 0 b + d 1 ρ ρ b. (6) To determine if obtaining an equivalent modulus values for eah patient speifi sanned femur model and pelvi model should be repeated, a parametri study was arried out by varying bone elasti modulus values. Firstly, the same elasti modulus was applied to both the femur and pelvis, this was varied between 3 GPa to 25 GPa. Then a separate segmented bone model was assembled with the femoral head and aetabular up omponents, the elasti modulus was varied independently between values of 3 GPa and 30 GPa. This provided an assessment for the relative material properties of the femur and pelvi bone models. By arrying out this analysis, the effet of bone material properties on the ontat pressures and stresses were assessed. Fig. 4. Python sripting wear simulation proedure Table 1. Material properties for finite element modelling Material Elasti modulus (GPa) Poisson s ratio Density (kg/m 3 ) E F E P CoCrMo In addition to establishing isotropi and homogeneous material properties for the pelvi and femur bone setions, more omplex orthotropi and Hounsfield based density greysale material models were also modelled. For the greysale model, eah finite element was defined with a value of density ( ρ). Therefore, Poisson s ratio (ν) and Young s modulus (E) were defined as a funtion of density. The Hounsfield unit (GS) is determined from μ μ x water GS = 1000 (4) μwater where the linear attenuation of water is given by µ water and the oeffiient of the substane being measured 3. Results 3.1. Contat mehanis of normal, edge loading and severe loading onditions The variation of maximum ontat pressure, von Mises stress and prinipal stress against the femur and pelvi bone elasti modulus is provided in Fig. 5. A relative bone elasti modulus was applied between the femur and pelvis, the ontat pressure and von Mises stresses are shown in Fig. 6. Along with hange of ontat pressure aused by different values of elasti modulus, any differene in ontat pressure magnitude between the two graphs was also due to the development of the segmented model being repeated to hek for any model errors, and assess the sensitivity of the geometry and assembly differenes on the ontat pressure and stress results. Although both threedimensional segmented hip implant models were based on the same DICOM bone model data, loading onditions and implant devies, there was a differene

5 Contat mehanis and wear simulations of hip resurfaing devies using omputational methods of 15º anteversion angle between the models, however, it was observed that the differene in ontat pressure distribution for eah of the models was negligible, therefore detailed pressure and stress distribution plots are not provided. ning to the end of one omplete gait yle (Fig. 8), as required for the implant volumetri wear analysis. Fig. 7. Relative elasti modulus differene against relative ontat pressure differene Fig. 5. Maximum ontat pressure and stress against bone elasti modulus Fig. 8. Aetabular up maximum prinipal stress, maximum von Mises stress and maximum shear stress with surfae frition oeffiient of 0.16 Fig. 6. Maximum ontat pressure and von Mises stress against relative bone elasti modulus By applying an orthotropi bone model, an assessment was onduted to obtain the ontat pressure and von Mises stresses based on the material properties defined by Couteau et al. [13]. The perentage ontat pressure differene as the relative elasti modulus was varied is shown in Fig. 7. The range of elasti modulus as presented from the study by Rho et al. [14] is highlighted on the graph. When applying the flexionextension and inward-outward angular rotation to the three-dimensional finite element model along with the ISO gait loading onditions, the maximum von Mises stress, prinipal stress and three omponents of shear stress S12, S23 and S13 were plotted from the begin- 107 Fig. 9. Maximum ontat pressure during loading yle with 200 μm and 500 μm medial-lateral displaement

6 108 M. ALI, K. MAO The differene in ontat pressure between applying the vertial loading magnitudes of F y and F I was observed to be 18%, however the differene inreased further when lateral edge loading boundary onditions were applied. To be onsistent with experimental simulator testing methods, the range of lateral displaements were applied between the ranges of 200 μm and 500 μm, the results in Fig. 9 show the maximum ontat pressure during a single ISO gait loading yle with lateral miroseparation applied throughout the loading yle. Results disussed above were based on a nominal up inlination angle of 45. By hanging the inlination angle and anteversion angle, the effet on ontat pressure and von Mises stress was assessed using the three-dimensional head and up models without the baking of the pelvi and femur bones to obtain results with inreased omputational effiieny. The maximum ontat pressure and von Mises stress at varying up inlination angles and anteversion angles are provided in Fig. 10. From the two-dimensional axisymmetri model, the maximum subsurfae stresses were obtained for both the aetabular up and femoral head omponents under F I, F y and F s vertial loading onditions. The maximum stresses ourred below the surfae of ontat in all but one of the ases. It was only when a stumbling load was applied that the maximum stress ourred at the base of the head omponent. The maximum values of shear stress τ xy and maximum σ x, σ y prinipal stress are provided in Table 2 and Table 3. Table 2. Aetabular up maximum stresses under vertial loads Load Max. shear stress Prinipal stress Prinipal stress (τ xy ) MPa (σ x ) MPa (σ y ) MPa F I F y F s Table 3. Femoral head maximum stresses under vertial loads Load Max. shear stress Prinipal stress Prinipal stress (τ xy ) MPa (σ x ) MPa (σ y ) MPa F I F y F s Wear simulations Fig. 10. Maximum ontat pressure and von Mises stress against varying up inlination angle and anteversion angle The kinematis of pure miroseparation was modelled to study laxity based edge loading of hip implants under ontat. Following hip bearing miroseparation, laxity of the hip joint is present as disussed in Setion 2. By running an expliit model and applying a vertial load of 3900 N, reloation of the hip joint was observed whilst being able to assess the ontat during this reloation analysis step. At a up inlination angle of 45 ontat initially ourred at the rim of the aetabular up. The maximum ontat pressure observed was 341 MPa and no ontat was observed below the rim of the aetabular up, i.e., the rim radius. When the up inlination angle inreased to 60 the maximum ontat pressure inreased to 646 MPa. The volumetri material loss due to mehanial wear under flexion-extension, internal-external rotation and ISO gait loading onditions were reorded for both the aetabular up and femoral head simultaneously. By using the sliding distane algorithm method to determine the ontat sliding distane and when applying a wear oeffiient value ( mm 3 /Nm) to the wear model, the volumetri wear rates were 1.5 mm 3 per million yles for the femoral head and 2.6 mm 3 per million yles for the aetabular up. A omparison between using the sliding distane output methodology and the oordinate output methodology was arried out. The differene in wear rates between using eah method for the same segmented hip resurfaing model under angular displaement was 3.5% over one million yles. For a wear simulation developed for 10 million yles, the wear ode was adapted to simulate and reflet long term wear, whilst onsidering the ourrene and trend of the wear observed experimentally. Ahieving this result required the setup and development of a numerially stable model to manage the altering finite element mesh ourring during the wear simulations. A umulative volumetri wear loss of material for both the femoral head and aetabular up is presented in Fig. 11, as well as the

7 Contat mehanis and wear simulations of hip resurfaing devies using omputational methods 109 presentation of the volumetri wear loss of the femoral head against wear oeffiient value over one million yles (Fig. 12). Fig. 11. Femoral head and aetabular up volumetri wear over 10 million yles Fig. 12. Volumetri wear of the femoral head at different wear oeffiient values 4. Disussion The three-dimensional and two-dimensional hip resurfaing models have both provided unique and speifi models for assessing the ontat mehanis using a range of materials models and boundary onditions. By modelling bone material properties as isotropi, orthotropi and density based, it was determined that the ontat pressure was insensitive to the intriaies assoiated with omplex material models and relative differenes in material models between the femur and pelvi bone setions. For low values of bone elasti modulus the prinipal stresses and von Mises stresses varied more greatly than the stress values at an elasti modulus greater than 5 GPa. The angular displaements and ISO loading onditions applied to the three-dimensional segmented model provided the option of alulating the sliding distanes along with the appliation of variable loading using omputational methods. The ontat pressure between the femoral head and aetabular up, as well as the shear stresses of these ontat loadings mathed the trend and profile of the input time dependant loading urve applied during the walking yle. By using the finite element method, the full range of lateral displaements typially applied during miroseparation based hip simulator tests was modelled. The differene in maximum ontat pressure observed when applying 200 μm and 500 μm was shown. Although ISO loading was applied, the ontat between the bearing surfaes was dominated by edge loading due to the lateral displaement of the femoral head. This is shown by the ontat pressure profile throughout the yle not onforming the ISO gait loading profile. The two-dimensional axisymmetri model allowed for an effiient assessment of the ontat pressure and stresses at the surfae and sub-surfae of the bone and hip resurfaing implant devies. By applying the F I, F y and F s vertial loads, inrease in stresses were observed for both the femoral head and aetabular up implant omponents as the vertial load inreased. A lose omparison of sliding distane was obtained by using the two different sliding distane alulation methodologies, whih provide an inreased level of onfidene in the validity of the wear results. Both methods provided a valid platform for assessing the volumetri wear between hip resurfaing devies using the finite element method with a ustom sripting interfae. The three-dimensional hip implant wear model provided volumetri wear results for both the aetabular up and femoral head simultaneously, whih has not been onduted before in previous studies. Although not ompared extensively in previous studies, the results show the volumetri wear of the aetabular up to be higher than for the femoral head whih is also observed experimentally from ylially simulated hip implant omponents [15]. A linear inrease in volumetri wear was observed as the wear oeffiient value inreased, whih shows the effet of the wear fator on the loss of material and highlights the importane for further study. The advantages of using the sripting method for developing hip implant wear simulations over previously published

8 110 M. ALI, K. MAO methods using subroutine based simulations inlude less restritions on element types and ontat algorithms, as well as allowing an option for applying the wear model to both the aetabular up and femoral head omponents simultaneously. Overall, this study had demonstrated the use of omputational and numerial methods for studying the ontat mehanis of three-dimensional and twodimensional axisymmetri hip resurfaing models. This was then extended to develop wear simulations to obtain the volumetri wear rates of hip implant devies under ontat, angular displaement and variable loading onditions. This study has shown the possibility of using a Python sripting interfae to simulate the volumetri wear loss by applying the Arhard wear model to hip implants, and therefore model the long term performane of implant devies beyond urrent ISO standards for experimental simulator testing. This has been ahieved whilst establishing the appropriate boundary onditions and material properties to be applied to the omputational implant models. Further work inludes longer term omputational wear simulations with the inlusion of edge loading during the loading yle. Aknowledgements This work was supported by the EPSRC (Engineering and Physial Sienes Researh Counil). Referenes [1] LIU F., LESLIE I., WILLIAMS S., FISHER J., JIN Z., Development of omputational wear simulation of metal-on-metal hip resurfaing replaements, Journal of Biomehanis, 2008, 41 (3), [2] MENG H., LUDEMA K., Wear models and preditive equations: their form and ontent, Wear, 1995, (2), [3] ARCHARD J.F., Contat and rubbing of flat surfaes, Journal of Applied Physis, 1953, 24(8), [4] ROŃDA J., WOJNAROWSKI P., Analysis of wear of polyethylene hip joint up related to its positioning in patient s body, Ata of Bioengineering and Biomehanis, 2013, 15(1). [5] ALI M., MAO K., Contat analysis of hip resurfaing devies under normal and edge loading onditions, IAENG Speial Issues Journal, 2012, 20(4), [6] ALI M., MAO K., Modelling of hip resurfaing devie ontat under entral and edge loading onditions, Leture Notes in Engineering and Computer Siene Proeedings of The World Congress on Engineering, 2012, 20(4), [7] UDOFIA I.J., YEW A., JIN Z.M., Contat mehanis analysis of metal-on-metal hip resurfaing prostheses, Proeedings of the Institution of Mehanial Engineers Part H Journal of Engineering in Mediine, 2004, 218, [8] SCHOLES S.C., UNSWORTH A., GOLDSMITH A.A.J., A fritional study of total hip joint replaements, Physis in Mediine and Biology, 2000, 45(12), [9] BERGMANN G., GRAICHEN F., ROHLMANN A., BENDER A., HEINLEIN B., DUDA G., HELLER M., MORLOCK M., Realisti loads for testing hip implants, Bio-medial Materials & Engineering, 2010, 20(2), [10] HAREWOOD F., MCHUGH P., Comparison of the impliit and expliit finite element methods using rystal plastiity, Computational Materials Siene, 2007, 39(2), [11] BERGMANN G., DEURETZBACHER G., HELLER M., GRAICHEN F., ROHLMANN A., STRAUSS J., DUDA G., Hip ontat fores and gait patterns from routine ativities, Journal of Biomehanis, 2001, 34(7), [12] FIRKINS P., TIPPER J., INGHAM E., STONE M., FARRAR R., FISHER J., Influene of simulator kinematis on the wear of metal-on-metal hip prostheses, Proeedings of the Institution of Mehanial Engineers, Part H, 2001, 215(1), [13] COUTEAU B., LABEY L., HOBATHO M.C., VANDER SLOTEN J., ARLAUD J.Y., BRIGNOLA J.C., Validation of a three dimensional finite element model of a femur with a ustomized hip implant, Computer Methods in Biomeh. & Biomed. Eng., 1998, 1(1), [14] RHO J.Y., ASHMAN R.B., TURNER C.H., Young s modulus of trabeular and ortial bone material: Ultrasoni and mirotensile measurements, Journal of Biomehanis, 1993, 26(2), [15] MANAKA M., CLARKE I.C., YAMAMOTO K., SHISHIDO T., GUSTAFSON A., IMAKIIRE A., Stripe wear rates in alumina THR Comparison of miroseparation simulator study with retrieved implants, Journal of Biomedial Materials Researh Part B Applied Biomaterials, 2004, 69(B),