DEVELOPMENT AND VALIDATION OF A SUBJECT-SPECIFIC FINITE ELEMENT MODEL OF THE PELVIS: ASSESSMENT OF MODEL SENSITVITY

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1 DEVELOPMENT AND VALIDATION OF A SUBJECT-SPECIFIC FINITE ELEMENT MODEL OF THE PELVIS: ASSESSMENT OF MODEL SENSITVITY A.E. Aderso 1, C.L. Peters 2, B.D. Tuttle 3, ad J.A. Weiss 4,5 ABSTRACT The objectives of this study were to: 1) develop ad validate a realistic FE model of the pelvis usig subject-specific boe geometry, locatio-depedet cortical thickess ad trabecular boe elastic modulus, ad 2) assess the sesitivity of FE predictios to assumptios i both assumed ad measured model iputs. A fiite elemet model of a cadaveric pelvis was created usig subject-specific CT image data. The pelvis was loaded experimetally usig a prosthetic femoral stem i a fashio that could be easily duplicated i the computatioal model. Cortical boe strais were moitored usig rosette strai gauges i te locatios throughout the left hemi-pelvis. FE predictios were compared directly with experimetal results for purposes of validatio. Overall, FE predictios were cosistet with experimetal results. I additio, the results of the sesitivity study suggest that chages to cortical boe thickess had the largest effect o cortical boe strais. Deviatios to all other material parameters had little effect o cortical strais. INTRODUCTION The pelvic joit is oe of the most importat weight bearig structures i the huma body. Despite its efficiet structure, the pelvis has little tolerace for abormal loads [1]. Although the major problems associated with the pelvis are largely attributed to altered biomechaics, there is very little quatitative iformatio cocerig the i-vivo stress distributio i the pelvis. A i vivo approach to the study of pelvic stresses ad strais may provide valuable iformatio that could lead to improved implat desigs, surgical approaches, diagosis, ad may provide the framework ecessary for preoperative surgical plaig. The FE method has ofte bee utilized to ivestigate various aspects of orthopedic medicie. The FE method is preferred over experimetal aalyses whe it is ecessary to model cliical abormalities. This becomes clear whe oe cosiders how difficult (if ot impossible) it would be to assemble a populatio of cadaveric tissue that exhibits a specific pathology. Nevertheless, the process to create subject-specific models remais questioable due to the fact that direct validatio has ot bee performed. Furthermore, a complete sesitivity aalysis has ot bee doe to uderstad the mechaical sigificace of both assumed ad measured iputs. Keywords: pelvis, fiite elemet model, biomechaics, strai gauges 1 Ph.D. Studet, 3 Research Assistat, 4 Correspodig Author, Associate Professor, Departmet of Bioegieerig, Uiversity. of Utah, 50 S. Cetral Campus Dr., Rm 2460 Salt Lake City, Utah, USA 2 Associate Professor, 5 Adjuct Associate Professor, Departmet of Orthopedics, Uiversity. of Utah, 3B165 SOM, Salt Lake City, Utah, USA

2 METHODS A combied experimetal ad computatioal protocol was used to develop ad validate a subject-specific three-dimesioal model of a 68 y/o female pelvis. Figure 1: Model idicatig the locatios of Experimetal Study. The sacroiliac joit ad the te rosette strai gauges that were attached all soft tissues, with the exceptio of articular to the left cadaveric hemi-pelvis. cartilage, were removed. A registratio block ad wires were attached to the iliac crest. The block allowed for spatial registratio of experimetal ad FE coordiate systems, while the wires served as a guide to reproduce the boudary C A B coditios used i the experimetal model [2]. A volumetric CT sca (512x512 acquisitio matrix, FOV=225 mm, 354 D slices, slice thickess=0.6 mm) was obtaied i a superior to iferior fashio. E The moutig ad loadig of the pelvis followed the protocol of Dalstra et al. [3]. The iliac crests were submerged i a moutig pa of catalyzed polymer resi (Bodo, Mar-Hyde, Atlata, GA) to the depth defied by the iliac guide wires. Te three-elemet rectagular rosette strai gauges (WA-060WR- 120, Vishay Measuremets Group, Raleigh, NC), represetig 30 chaels of data, were attached to the left hemi-pelvis at locatios aroud the acetabulum, pubis, ischium, ad ilium (Fig. 1). Vertically orieted loads (0.25, 0.50, 0.75, ad 1.0 BW) were applied to the acetabulum via a femoral prosthesis attached to a liear actuator, while strais were recorded cotiuously (Fig. 2). 3D coordiates of Aterior the strai gauges ad registratio block were determied with respect to the Block laboratory referece frame usig a electromagetic digitizer (Immersio Corp., Sa Jose, CA). F Figure 2: Fixture for loadig pelvis via femoral compoet. A) load cell, B) ball joit, C) femoral compoet, D) pelvis, E) moutig pa for embeddig pelvis, ad F) X-Y table. Geometry Extractio, Mesh Geeratio. Cotours for the outer cortex ad the boudary of the cortical ad trabecular boe, registratio block, ad guide wire were extracted from the CT data for the left hemi-pelvis via maual segmetatio (Fig. 3). The poits comprisig the Posterior Figure 3: CT image slice at the ilium showig the radio-opaque registratio block ad the distict boudary betwee cortical ad trabecular boe (left). The origial cortical boe surface was recostructed by stackig the segmeted cotours (middle). This surface was smoothed ad decimated to create the fial surface (right). cotours were triagulated to form the origial surface, which was the decimated ad smoothed to form the fial surface usig the Visualizatio Toolkit (VTK). The triagles composig the fial surface were used to defie a FE mesh of shell elemets to represet cortical boe. A volumetric tetrahedral mesh was created usig the FE surface mesh ad the mesh geeratio program CUBIT (Sadia Natioal Labs, Albuquerque, NM). The fial FE model cosisted of 210,000 tetrahedral solid elemets for trabecular boe ad 30,000 triagular shell elemets for cortical boe (Fig. 4). Acetabular

3 cartilage was represeted with 500 shell elemets with a costat thickess of 2 mm, determied by averagig the distace betwee the implat ad acetabulum i the eutral kiematic positio. Cortical Thickess. A algorithm was developed to determie the spatially varyig thickess of the cortex based o the distaces betwee the polygoal surfaces represetig the outer cortex ad the boudary betwee the cortical ad trabecular boe. Vectors were costructed betwee each ode o the cortical surface ad the 100 earest odes o the surface defiig the corticaltrabecular boudary (Fig 5, left). Cortical thickess was determied by miimizig both the distace betwee the odes of the surfaces ad the agle of the dot product betwee the surface ormal of the cortical surface with that of each correspodig trabecular vector. I areas of high curvature (such as the acetabular rim), special cosideratio of thickess was Figure 4: FE mesh of the pelvis (left) with a close-up view of the mesh at the acetabulum (right). θ L = = = betwee the odes was take as the cortical thickess. ecessary (Fig. 5, right). Whe the above-described algorithm reported a thickess value that exceeded 1.5 times the smallest distace betwee the odes, the smallest distace betwee odes o the two surfaces was used to determie the cortical thickess. The miimum value of odal thickess was assumed to be 0.44 mm or the width of oe pixel (FOV= 225, FOV/512 = 0.44 mm/pixel). The algorithm was tested for accuracy usig polygoal surfaces represetig parallel plaes, cocetric spheres, ad layered boxes with varyig mesh desities. Material Properties, Boudary Coditios, & Aalysis. The femoral implat was represeted as rigid while cortical ad trabecular boe were represeted as isotropic hypoelastic. Iitial material properties for cortical boe were E=17 GPa ad ν=0.29 [3]. Relatioships betwee CT Housfield uit, apparet desity ad elastic modulus were used to assig a desity-depedet modulus for each tetrahedral elemet [4,5]. Acetabular articular cartilage was represeted as a hyperelastic Mooey-Rivli material. Coefficiets C 1 ad C 2 for articular cartilage were assumed to be 4.1 MPa ad 0.41 MPa, respectively, with ν=0.4 [6]. A FE coordiate system was created from the polygoal surface of the recostructed registratio block. A correspodig coordiate system was established for the experimetal measuremets usig the digitized coordiates of the registratio block [2]. Nodes superior to the iliac guide wires ad odes alog the pubis sythesis joit were costraied to simulate the boudary coditios i the experimet. Frictioless cotact was eforced betwee the femoral implat ad cartilage. Load was applied to the implat usig the same magitude ad θ L dot product ode ormal legth t Figure 5: Schematics illustratig the cortical thickess algorithm with cortical odes (ope circles) ad earest trabecular odes (filled circles). Left - the legths of the earest trabecular odes ad dot product were aalyzed. Middle - the smallest agle of the dot product betwee the cortical ode ad earest trabecular ode eighbor yields the desired thickess measuremet (t). Right - the ormal vector from the cortical ode does ot itersect the trabecular surface. I this situatio a weightig scheme was applied such that the smallest distace t

4 directio measured experimetally. All aalyses were performed with the implicit time itegratio capabilities of LS-DYNA (Livermore Software Techology Corporatio, Livermore, CA). FE predictios of cortical pricipal strais were averaged over elemets that were located beeath each strai gauge. Best-fit lies ad r 2 values were reported for each model at all loads for purposes of FE validatio ad sesitivity aalysis. Sesitivity studies. Sesitivity studies were performed to assess the effects of variatios i assumed ad measured material properties ad cortical thickess o the predicted cortical surface strais (Table 1). Table 1: Models studied for FE sesitivity aalysis. Stadard deviatios (SD) i material properties ad cortical thickess were take from experimetally measured values () as well as deviatios ad rages of values reported i the literature. Sesitivty Models Aalyzed Referece Costat cortical boe thickess = ± 0, 1 SD (0.44 mm) Costat trabecular boe modulus, E = ± 0, 1 SD (80 MPa) Costat cortical thickess & trabecular modulus, 1.0 mm, E = 160 MPa Trabecular boe Poisso s ratio, ν=0.2, ν=0.39 [7] Cortical boe Poisso s ratio, ν=0.29 [8] Cortical boe modulus, E = ± 1 SD (1.62 GPa) [9] Cartilage thickess = 0.0, 4.0 mm (Mi/Max) Cartilage modulus, E = 1.36, 7.79 MPa (Mi/Max) [10] RESULTS Cortical Thickess, Trabecular Elastic Modulus. Usig boxes, spheres, ad parallel plaes with kow thickess, it 2.5 mm was determied that the maximum RMS error of the thickess algorithm was ±2%. Cortical boe thickess Aterior 0.4 mm Medial raged from 0.44 to 3.83 mm (mea Figure 6: Cotours of positio depedet cortical boe 1.0 ± 0.4 mm) for the cadaveric pelvis thickess. Cortical thickess was highest alog the iliac (Fig. 6). Usig empirical relatioships crest, the ascedig pubis ramus, at the gluteal surface betwee CT Houdsfield uit, apparet ad aroud the acetabular rim. Cortical boe was miimal at the acetabular cup, the ischial tuberosity, the desity ad elastic modulus, it was iliac fossa ad the area surroudig the pubic tubercle. determied that trabecular elastic modulus raged from MPa (mea 160±80 MPa). FE Model Predictios. The subject-specific FE model predictios of pricipal strais showed strog correlatio (r 2 =0.776) with experimetal measuremets (Fig. 7, top pael). Similar correlatio coefficiets were obtaied for the models examied i the sesitivity studies. Models represetig stadard deviatios i average trabecular elastic modulus did ot alter strais cosiderably (Fig 7, middle pael). I cotrast, chages i the thickess of the cortical boe had a substatial effect o cortical strais (Fig 7, bottom pael). Nevertheless, the model with average cortical thickess predicted strais that were cosistet with subject-specific model results (data ot show). Alteratios to all other material properties did ot alter cortical strais sigificatly (data ot show).

5 DISCUSSION A subject-specific fiite elemet model of the pelvis was developed ad validated. Accurate FE model predictios were obtaied whe positio-depedet cortical thickess ad elastic modulus were used. The FE models were most sesitive to alteratios i cortical boe thickess. Accurate predictios of cortical strais were obtaied whe a average cortical thickess ad modulus was used; thus, the subjectspecific model was ot ay more accurate tha those that assume average properties. However, the average cortical shell thickess i areas where strai gauges were preset was very close to the average thickess for the etire model but deviated much less (mea 1.07±0.2 mm). This suggests that accurate measuremet of cortical thickess should be cosidered for future models sice the model predictios were very sesitive to deviatios i this iput. Experimetal Mi/Max Pricipal Strai (µstrai) Subject-Specific Subject-Specific Exp = FE Cost. elastic modulus + 0 SD Cost. elastic modulus -1 SD Cost. elastic modulus +1SD + 0 SD - 1 SD + 1 SD Exp = FE Cost. thickess -1 SD Cost. thickess +1 SD Cost. thickess +0 SD Three-dimesioal FE models of the pelvis have bee studied o several occasios. Most of these models used simplified pelvic geometry, average material properties ad did ot validate FE predictios of stress ad strai. To our kowledge, the work of Dalstra et al. was the first ad oly attempt to develop ad validate a three-dimesioal FE model of the pelvis usig subjectspecific geometry ad material properties [3]. The FE model was validated usig - 1 SD + 1 SD + 0 SD Exp = FE FE Mi/Max Pricipal Strai (µstrai) Figure 7: FE predicted vs. experimetal cortical boe pricipal strais. Top pael- subjectspecific, middle pael- costat trabecular modulus, bottom pael- costat cortical thickess. experimetal measures of strai i the peri-acetabular regio of a cadaveric pelvis. However, validatio by direct compariso with subject-specific experimetal measuremets was ot performed. Differet cadaveric specimes were used for FE mesh geeratio ad experimetal tests. The results of the sesitivity studies demostrated the relative importace of both assumed ad measured parameters for FE modelig of the pelvis. This study showed that the predictio of cortical boe strais was most sesitive to chages i cortical thickess. Alteratios to both boe Poisso s ratios, cartilage elastic modulus, ad cartilage thickess did ot alter cortical strais. However, chages to these iputs did have a effect o overall model displacemet ad cartilage/trabecular boe mechaics. Therefore, model sesitivity should ot be approached as all or oe ; the required accuracy of idividual material iputs/parameters will deped o the ature of the study

6 Although the results of the sesitivity studies suggest that errors i material property values will ot likely to produce sigificat chages i the stress ad strai gradiets i the cortical boe, it is likely that strais would be more sesitive to chages i the boudary coditios ad applied loadig coditios. For this reaso, it was ot the itet of this study to replicate physiological loadig coditios. By applyig a welldefied experimetal loadig cofiguratio, the applied loads were accurately replicated i the FE model with miimal error. It is cocluded that the presetly developed approach for subject-specific FE modelig of the pelvis has the ability to predict cortical boe strais accurately. The techiques validated i this study ca be applied to the aalysis of the detailed pelvic geometry of idividual patiets based o volumetric CT scas. This will provide a meas to examie the mechaics of the pelvis for cases whe detailed subject-specific geometry should be cosidered. ACKNOWLEDGMENTS Fiacial support from the Uiversity of Utah Seed Grat # ad the Departmet of Orthopedics, Uiversity of Utah is gratefully ackowledged. REFERENCES 1. Cooperma, D. R., Walleste, R., ad Stulberg, S. D., 1983, "Acetabular Dysplasia i the Adult," Cli Orthop, pp Fischer, K. J., Maso, T. T., Pfaeffle, H. J., Tomaio, M. M., ad Woo, S. L., 2001, "A Method for Measurig Joit Kiematics Desiged for Accurate Registratio of Kiematic Data to Models Costructed from Ct Data," J Biomech, 34, pp Dalstra, M., Huiskes, R., ad va Erig, L., 1995, "Developmet ad Validatio of a Three-Dimesioal Fiite Elemet Model of the Pelvic Boe," J Biomech Eg, 117, pp Ploeg, H., "Fiite Elemet Aalysis ad Boe Remodelig Followig Total Hip Replacemet," preseted at NAFEMS World Cogress, Dalstra, M., Huiskes, R., Odgaard, A., ad va Erig, L., 1993, "Mechaical ad Textural Properties of Pelvic Trabecular Boe," J Biomech, 26, pp Little, R. B., Wevers, H. W., Siu, D., ad Cooke, T. D., 1986, "A Three- Dimesioal Fiite Elemet Aalysis of the Upper Tibia," J Biomech Eg, 108, pp Lappi, V. G., Kig, M. S., ad Lemay, I., 1979, "Determiatio of Elastic Costats for Huma Femurs," J Biomech Eg, 101, pp Ooishi, H., Isha, H., ad Hasegawa, T., 1983, "Mechaical Aalysis of the Huma Pelvis ad Its Applicatio to the Artificial Hip Joit--by Meas of the Three Dimesioal Fiite Elemet Method," J Biomech, 16, pp Syder, S. M. ad Scheider, E., 1991, "Estimatio of Mechaical Properties of Cortical Boe by Computed Tomography," J Orthop Res, 9, pp Armstrog, C. G., Bahrai, A. S., ad Garder, D. L., 1979, "I Vitro Measuremet of Articular Cartilage Deformatios i the Itact Huma Hip Joit uder Load," J Boe Joit Surg Am, 61, pp