THE INFLUENCE OF MEDIAL-LATERAL CONTACT PAIR CONFORMITY ON CONTACT STRESSES IN TOTAL KNEE REPLACEMENT. Usman and Shyh-Chour Huang

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1 THE INFLUENCE OF MEDIAL-LATERAL CONTACT PAIR CONFORMITY ON CONTACT STRESSES IN TOTAL KNEE REPLACEMENT Usman and Shyh-Chour Huang Department of Mechanical Engineering, National Kaohsiung University of Applied Sciences, Kaohsiung, Taiwan, R.O.C. Received November 2016, Accepted March 2017 No. 16-CSME-134, E.I.C. Accession 4020 ABSTRACT Eleven finite element models representing the medial-lateral conformity level and shape have been analyzed. From the result, we concluded that the flat-on-flat contact model of knee joint replacement was the most vulnerable to damage. Moreover, in the flat-on-flat model, the highest von Mises stress occurred at two points in a model, namely at the medial and lateral edge of the contact area. With respect to the sensitivity of the contact stresses to the conformity (frontal clearance) in the curved-on-curved model, the larger radius of the femoral indenter showed more contact stress sensitivity to the change in conformity. The von Mises stress dropped to almost half of its previous value when the frontal clearance decreased from 10 to 2 mm. Keywords: total knee replacement; finite element analysis; contact mechanics; contact stress; conformity. INFLUENCE DE LA CONFORMITÉ DES PAIRES EN CONTACT MÉDIAL ET LATÉRAL SUR LE STRESS DE CONTACT DANS LE REMPLACEMENT TOTAL DU GENOU RÉSUMÉ On a analysé onze modèles d éléments finis représentant le niveau de conformité interne-externe ainsi que leurs formes. D après les résultats, nous sommes parvenus à la conclusion que le modèle de contact surface sur surface dans le remplacement de l articulation du genou est le plus susceptible d être endommagé. De plus sur ce modèle, le stress Von Mises le plus élevé survient à deux endroits, à savoir sur les bords médial et latéral de la surface de contact. Par rapport à la sensibilité au stress de contact de conformité (dégagement frontal) dans le modèle curved-on-curved, le rayon plus grand de la tête du fémur montre plus de sensibilité au stress de contact lors du changement de conformité. Le stress Von Mises a baissé de près de la moitié de sa valeur précédente quand le dégagement frontal a diminué de 10 à 2 mm. Mots-clés : remplacement total du genou; analyse des éléments finis; mécanique de contact; stress de contact; conformité. Transactions of the Canadian Society for Mechanical Engineering, Vol. 41, No. 2,

2 NOMENCLATURE b half-width of contact (mm) P load per unit contact length (N/mm) R radius of contact body (mm) E modulus of elasticity (MPa) ν Poisson s ratio p(x) contact pressure along x-axis (MPa) r edge radius of tibial indenter (mm) ε pl plastic strain (%) ε true true strain (%) σ true true stress (MPa) Subscripts eq equivalent 1 cylinder 1 2 cylinder 2 f femoral t tibial 1. INTRODUCTION Conformity in the frontal or sagittal plane is one of several parameters in a total knee replacement design. Conformity is defined as the difference between the femoral component radius and the dished-out patch radius of the tibial insert in the frontal or sagittal plane [1]. If we compare the sagittal conformity, the frontal conformity is more important to consider in a total knee replacement design [2] due to the fact that the stresses are the most sensitive to change in the frontal direction [3] and the contact stress in the tibial components is reduced most when the articulating surfaces in the frontal plane have greater conformity [4]. Many researchers have studied the effect of conformity on a total knee replacement performance; hence, many results have been reported. In the curved-on-curved and the flat-on-flat total knee replacement design, the level of conformity of the tibiofemoral contact has a crucial effect on wear and the increase of von Mises stress and contact pressures in the polyethylene insert. Under severest total knee replacement malalignment, the high conformity of the flat-on-flat design produced the greatest contact and von Mises stress, and the curved-on-curved design minimized the polyethylene wear [5]. The low-conforming and increased contact pressure design reduced the surface wear in fixed-bearing knees [6]. Increasing the frontal conformity by modifying the tibial frontal radius may reduce the contact pressure with minimum adverse influence of the implant kinematics [1]. The results show why it is very important to understand the parameter of conformity in total knee replacement design. This is also reinforced by the work of Kuster et al. [7], who found that the small change in conformity (from 0.99 to 0.95) caused a larger increase in shear and surface stress than that caused by a large change of load (from 3000 to 6000 N). Therefore, several pairs of articulating contact mechanisms of contemporary knee design in the frontal plane were studied using finite element analysis (FEA). The Hertz theory of elastic contact was used to validate the models as was previously done by Simpson et al. [8] and van den Heever et al. [9]. For this purpose, we developed several two-dimensional finite element models with various levels of conformity. The models consisted of the main and additional models. The main models geometry was taken from the previous literature, and the additional models were variations of the main models. The additional models were intended to show clearly the effect of conformity against contact stress. The von Mises stress and contact pressure resulted in the analysis of the contact models to determine the behavior of contact stresses against the various levels of conformity occurring in the different models. 302 Transactions of the Canadian Society for Mechanical Engineering, Vol. 41, No. 2, 2017

3 2. METHODS AND MATERIALS 2.1. Elastic Contact Validation As a validation for the finite element model, a two-dimensional contact model between the cylinder and the flat plane was created. The cylinder with a radius of (R 1 ) 40 mm was brought into contact against a flat plane with 300 N/mm of load. The cylinder was a rigid body, and the flat plane was an ultra high molecular weight polyethylene (UHMWPE) deformable body with 550 MPa of modulus of elasticity and 0.44 of Poisson s ratio. First, the contact pressure occurring in the model was calculated by using the Hertz theory of elastic contact. We then compared it with the finite element analysis result to get the well-validated finite element model. The Hertz theory is a popular elastic contact theory usually used by finite element-based researchers to validate their finite element contact model. Since the understanding of this theory is very important in this paper, we will discuss it here briefly. Consider two cylinders 1 and 2 with radii R 1 and R 2 pressed into contact by a load per unit length P as shown in Fig. 1. The half-width of contact b can then be calculated by [10] 4PR eq b =, (1) πe eq where R eq and E eq are the equivalent radius and modulus of elasticity of contact, respectively. The equivalent radius of contact is given by 1 R eq = 1 R R 2. (2) The plus and minus signs are used for the internal and external radii, respectively. If cylinder 1 is brought into contact against the internal radius of cylinder 2, the minus sign is used [11]. The tibial insert of a total knee replacement often has the internal radius. The equivalent modulus of elasticity can be expressed as 1 E eq = (1 ν2 1 ) E 1 + (1 ν2 2 ) E 2, (3) where ν 1 and ν 2 are the Poison s ratio of cylinders 1 and 2, respectively. Then the contact pressure along the x-axis can be expressed by p(x) = 2P (b πb 2 x 2 ). (4) As shown in Fig. 1, x-axis is the horizontal contact counterface between cylinders 1 and 2. But, in this elastic contact validation, the cylinder 2 was a flat plane. Then, the values of contact pressure were calculated using Hertz theory along the positive x-axis starting from the contact center. The values were calculated at every 0.1 mm (from 0 until 4.9 mm). Meanwhile, the values at the same position were also extracted from the finite element model by designing the edge length of elements of 0.1 mm. To get the contact model validation, the contact pressures produced from Hertz theory and FEA were compared Elastic Plastic Finite Element Analysis Eight two-dimensional contact pair models representing the total knee replacement contact in the mediallateral (frontal) plane were created in Abaqus CAE. Three additional models were also created to help provide a clear understanding of the effect of frontal conformity on contact stresses. As we can see in Table 1, Models 1 8 are the main models that were collected from the dimension of total knee replacement Transactions of the Canadian Society for Mechanical Engineering, Vol. 41, No. 2,

4 Fig. 1. Two cylinders are brought into contact by load per unit length. Table 1. Femoral indenter and tibial base geometries in the frontal plane [3, 12]. Model No. Femoral indenter Tibial base radius (R t ) [mm] radius (R t ) [mm] (r = 1, width = 20) 3-1 (r = 2.5, width = 20) 4 (r = 5, width = 20) reported by Rawlinson and Bartel, Sathasivam and Walker [3, 12]. Models 2 1, 2 2 and 3 1 were the additional models. These additional models were modifications of the main models. There are two parts in each model: the femoral indenter and the tibial base. The femoral indenter is a circle representing the central portion of the femoral component in the frontal plane. The tibial base is a rectangle representing the tibial insert as the contact pair of the femoral component in the real total knee replacement. Three categories of contact model were simulated in this study, namely the flat-on-flat, the curved-on-flat, and the curved-on-curved as also previously used by Rawlinson and Bartel [3]. In the flat-on-flat model, the femoral indenter and the tibial base were both flat, but their medial and lateral edges of the femoral indenter were rounded with radius r. The width of the flat-on-flat femoral indenter was 20 mm [3]. The curved-onflat model simulated the contact between the curved femoral indenter with radius R f and the flat tibial base. 304 Transactions of the Canadian Society for Mechanical Engineering, Vol. 41, No. 2, 2017

5 Fig. 2. The contact categories in the frontal plane of the total knee joint replacement for two-dimensional finite element analysis [3]. Last, the curved-on-curved model simulated the contact between the curved femoral indenter with radius R f and the curved tibial base with radius R t. All contact categories are clearly shown in Fig. 2. Table 1 shows the tabulation of the size of the femoral indenter radius R f and the tibial base radius R t for each model. Models 1 3 and 4 represent the frontal profile of contemporary total knee replacement design as previously used by Rawlinson and Bartel [3] in their two-dimensional total knee replacement contact analysis. Meanwhile, Models 5 7 and 8 are the frontal profile size variation of total condylar knees as reported by Sathasivam and Walker [12]. Model 1 was used to study the contact behavior between the curved femoral indenter and the flat tibial base, so the contact category was curved-on-flat for this model. Models 2 7 and 8 were curved-on-curved contact models used to simulate the contact behavior between the curved femoral indenter and the curved tibial base. Meanwhile, Models 3 and 4 were used to conduct the contact simulation between the flat femoral indenter and the flat tibial base. Sathasivam and Walker [12] in their report introduced the term frontal clearance for curved-on-curved contact, namely the difference between the femoral indenter and the tibial base radius (R t R f ). For instance, in Model 2 1, the femoral indenter radius was 20 mm and the tibial base radius was 30 mm. Then, the frontal clearance was 10 mm (30 20). From the definition above, it is clear that the frontal clearance is another term of conformity. In this study, the frontal clearance of 10 and 2 mm was used for the femoral indenter radius of 20 mm (Models 2 1 and 2 2), 30 mm (Models 5 and 6), and 70 mm (Models 7 and 8). This was conducted to determine the sensitiveness of frontal clearance to the contact stresses. As stated before, because of the fact that the stress is most sensitive to the geometry changes in the frontal direction, the analysis in the frontal plane was selected because the anterior-posterior (sagittal) width of the total knee replacement, as observed in three-dimensional total knee replacement studies, could be approximated with a plain-strain analysis [3]. Therefore, in this study, all contact phenomena were only simulated in the frontal plane with two-dimensional analysis by using a plain-strain element. The femoral indenter was assumed to be a rigid body due to the material of the femoral component being a much higher modulus of elasticity than the UHMWPE as the material of the tibial insert. The tibial base was modeled as a deformable body. For the material property of UHMWPE, the nonlinear true stress-true strain of UHMWPE was used in this simulation as depicted in Fig. 3. The modulus of elasticity of UHMWPE was also extracted from Fig. 3 by dividing the first nonzero true stress by the first nonzero true strain. As shown in Fig. 3, the first nonzero true stress was 11 MPa and the corresponding true strain was The modulus of elasticity of UHMWPE equaled 550 MPa. Its Poisson s ratio of 0.44 was used consistent with the work of Willing and Kim [13]. As the input of the plastic property of UHMWPE, Abaqus software requires the true stress versus the plastic strain. The plastic strain can also be extracted from Fig. 3 by using the following equation ε pl = ε true σ true E, (5) Transactions of the Canadian Society for Mechanical Engineering, Vol. 41, No. 2,

6 Fig. 3. Nonlinear true stress-true strain of UHMWPE material model [15]. Table 2. True stress-plastic strain of nonlinear UHMWPE. True Stress (MPa) Plastic Strain where ε pl is plastic strain, ε true is true strain, σ true is true stress, and E is the modulus of elasticity. The true stresses versus plastic strains are tabulated in Table 2. The contact property between the rigid femoral indenter and the UHMWPE tibial base was set as a frictional contact with the coefficient of friction of 0.04 [14]. The surface-to-surface contact and hard contact relationship were also chosen as the contact behavior. A 4-node bilinear plane strain quadrilateral (CPE4) was selected as the tibial base finite element. The area around the contact was finely meshed with the element edge length of 0.1 mm. The representations of three categories of finite element model are shown in Fig. 4, and the number of elements and nodes of each model are tabulated in Table 3. The vertical load of 300 N/mm was applied at the femoral indenter reference point. This load was the in vivo joint load of 2000 N for a 6 7 mm contact width [3]. The femoral indenter was allowed to move only vertically and the tibial base was fixed in all directions. 3. RESULTS AND DISCUSSION 3.1. Elastic Contact Validation Due to the cylinder 1 (Fig. 1) being a rigid body, E 1 =. Then, from Eq. (3), we can obtain E = E 2 /(1 ν 2 2 ). In our case, cylinder 2 was a flat plane. Therefore, R 2 =. Then, from Eq. (2), we can get R = R 1. By using Eq. (4), the contact pressure along the x-axis can be calculated. After that, we can compare it with the result of finite element analysis (FEA). 306 Transactions of the Canadian Society for Mechanical Engineering, Vol. 41, No. 2, 2017

7 Table 3. The number of elements and nodes for all models. Model No. No. elements No.nodes Fig. 4. Finite element model of three contact categories: (a) curved-on-curved, (b) curved-on-flat, and (c) flat-on-flat. In this figure, only a portion of mesh is shown. The maximum contact pressure produced by Hertz theory and FEA were and MPa, respectively (the different level was far below 5%). Both occurred at the contact center. The rest of the contact pressures along x-axis are depicted in Fig. 5. The figure also presented the comparison between the FEA and the Hertz results. The comparison showed that there was a good agreement between FEA and Hertz theory in contact pressure results. Therefore, with this good result, the models created can be decisively used for further simulation Elastic Plastic Finite Element Analysis Figure 6 shows the von Mises stress for all main models. The highest von Mises stress (27.01 MPa) was observed in Model 3. The model was the flat-on-flat category with a medial-lateral edges radius of 1 mm. Transactions of the Canadian Society for Mechanical Engineering, Vol. 41, No. 2,

8 Fig. 5. A good agreement of the contact pressures produced by the FEA and Hertz theory. The contact pressures were recorded along the x-axis from the contact center. R 1 = 40 mm and R 2 = (flat plane). Fig. 6. The highest von Mises stress was observed in Model 3 (flat-on-flat contact category with r = 1 mm), followed by Model 4 (flat-on-flat contact category with r = 5 mm). When the edges radii were increased to 5 mm (Model 4), the von Mises stress decreased by 7.29% to MPa. Then, the von Mises decreased consecutively to 17.32, 12.95, 10.25, 8.19, 6.50 and 3.58 MPa in Models 1, 5, 2, 6, 7 and 8, respectively. From this simulation, we can see that Model 8 produced the lowest von Mises stress of 3.58 MPa. The category of the contact was curved-on-curved with R f = 70 mm and R t = 80 mm (frontal clearance = 10 mm). Figure 7 shows the contact pressures for all models. As with the von Mises stress in Fig. 6, Model 3 experienced the highest contact pressure. The lowest contact stress occurred in Model 8. Nonetheless, 308 Transactions of the Canadian Society for Mechanical Engineering, Vol. 41, No. 2, 2017

9 Fig. 7. The highest contact pressure occurred in Model 3 and the lowest contact pressure occurred in Model 8. Fig. 8. The von Mises stress of flat-on-flat contact with edge radius variation. there was a different trend in contact pressure. Model 1 experienced the second highest contact pressure of MPa and the contact pressure in Model 4 became the third highest with MPa, whereas the von Mises stress in Model 4 was higher than that in Model 1 as shown in Fig. 6. This proved that, in a contact model, the von Mises stress trend was not necessarily the same as the contact pressure trend. This is because the geometry of contact counterface has the different effect on von Mises stress and contact pressure. A similar phenomenon also occurred in the Rawlinson and Bartel s work [3] from which we can conclude that the contact pressure is not the function of von Mises stress in contact mechanism. As already mentioned, to determine a clear effect of the edges radius of the flat-on-flat contact category on von Mises stress, the additional model with an edges radius of 2.5 mm (Model 3 1) had been created. With this model, we can see clearly the influence of the edges radius on von Mises stress. Figure 8 shows the von Mises versus edges radius variation of the flat-on-flat contact model. It can be seen that the von Transactions of the Canadian Society for Mechanical Engineering, Vol. 41, No. 2,

10 Fig. 9. The von Mises stresses in the curved-on-curved contact category with femoral indenter radius of 20, 30 and 70 mm and frontal clearance of 10 and 2 mm. Mises stress for the 2.5 mm edges radius is 25.4 MPa. It decreased by 5.96% from the previous value. It decreased again by 1.42% when the edges radius increased to 5 mm. In the flat-on-flat contact category, if the edges radius increased, the von Mises stress decreased. If we compare the von Mises stress in all models in Fig. 6 with the data of UHMWPE property in Table 2, where the UHMWPE will start yielding at 11 MPa, we can conclude that Models 3, 4, 1, and 5 underwent a permanent strain because the von Mises stress in those models exceeded 11 MPa. The FEA simulation also revealed that the models had a plastic strain. Meanwhile, in Models 2, 6, 7 and 8 the von Mises stresses were below the first yielding stress of UHMWPE. To study the behavior of contact stress in the curved-on-curved contact category with 10 and 2 mm frontal clearance, two additional models were made, namely the femoral indenter with a radius of 20 mm with the frontal clearance 10 and 2 mm (Models 2 1 and 2 2). Meanwhile, the femoral indenter with a radius of 30 mm and 70 mm with a frontal clearance of 10 and 2 mm had been already prepared for the main Models 5, 6, 7 and 8. Figure 9 presents the results of von Mises stress in the curved-on-curved contact category as the effect of frontal clearance. For the 20 mm femoral indenter, von Mises stress decreased by 29.25% from to MPa when the frontal clearance was changed from 10 to 2 mm. For the femoral indenter radius of 30 and 70 mm, the decreasing of the frontal clearance resulted in the decreasing of von Mises stress by and 44.93%, respectively. From these six models, we can conclude that the larger radius of the femoral indenter was more sensitive to the change in conformity. This can be seen in Fig. 9, which shows that the von Mises stress in the model with a femoral indenter radius of 70 mm dropped almost half of its previous value when the frontal clearance decreased from 10 to 2 mm. 4. CONCLUSIONS We conducted simulations of several models of knee joint replacement in the frontal plane using twodimensional finite element analysis. Comparing the models taken from two different resources [3, 12] as well as additional models showed the flat-on-flat contact model of knee joint replacement to be the most vulnerable to damage. Moreover, in the flat-on-flat model, the highest von Mises stress occurred at two 310 Transactions of the Canadian Society for Mechanical Engineering, Vol. 41, No. 2, 2017

11 Fig. 10. (a) The highest von Mises stress occurred in two positions in the flat-on-flat model, in the lateral and medial edge. Meanwhile, (b) in the curved-on-flat and curved-on-curved contact model, the highest von Mises stress occurred only at the contact center. positions in a model as shown in Fig. 10. With respect to the frontal clearance in the curved-on-curved model, we can summarize that the larger radius of the femoral indenter was more sensitive to the change Transactions of the Canadian Society for Mechanical Engineering, Vol. 41, No. 2,

12 of conformity. As we can see in Fig. 9, the von Mises stress dropped by almost half of its previous value when the frontal clearance decreased from 10 to 2 mm. The simulations also showed that in the behavior of contact pressure against von Mises stress, the contact pressure trend was not necessarily the same as the von Mises trend. ACKNOWLEDGEMENT The authors acknowledge and thank the Ministry of Science and Technology of the Republic of China for their partial financial support of this study under Contract Number MOST E REFERENCES 1. Ardestani, M.M., Moazen, M. and Jin, Z., Contribution of geometric design parameters to knee implant performance: Conflicting impact of conformity on kinematics and contact mechanics, The Knee, Vol. 22, No. 3, pp , Dennis, D.A., Komistek, R.D., Walker, S.A., Cheal, E.J. and Stiehl, J.B., Femoral condylar lift-off in vivo in total knee arthroplasty, The Bone and Joint Journal, Vol. 83 B, No. 1, pp , Rawlinson, J.J. and Bartel, D.L., Flat medial-lateral conformity in total knee replacements does not minimize contact stresses, Journal of Biomechanics, Vol. 35, No. 1, pp , Bartel, D.L., Bicknell, V.L. and Wright, T.M., The effect of conformity, thickness, and material on stresses in ultra-high molecular weight components for total joint replacement, The Journal of Bone and Joint Surgery, Vol. 68, No. 7, pp , Liau, J.J., Cheng, C.K., Huang, C.H. and Lo, W.H., The effect of malalignment on stresses in polyethylene component of total knee prostheses-a finite element analysis, Clinical Biomechanics, Vol. 17, No. 2, pp , Galvin, A.L., Kang, L., Udofia, I., Jennings, L.M., McEwen, H.M.J., Jin, Z. and Fisher, J., Effect of conformity and contact stress on wear in fixed-bearing total knee prostheses, Journal of Biomechanics, Vol. 42, No. 12, pp , Kuster, M.S., Horz, S., Spalinger, E., Stachowiak, G.W. and Gächter, A., The effects of conformity and load in total knee replacement, Clinical Orthopaedics and Related Research, Vol. 375, pp , Simpson, D.J., Gray, H., D Lima, D., Murray, D.W. and Gill, H.S., The effect of bearing congruency, thickness and alignment on the stresses in unicompartmental knee replacements, Clinical Biomechanics, Vol. 23, No. 9, pp , van den Heever, D.J., Scheffer, C., Erasmus, P. and Dillon, E., Contact stresses in a patient-specific unicompartmental knee replacement, Clinical Biomechanics, Vol. 26, No. 2,pp , Johson, K.L., Contact Mechanics, Cambridge University Press, Cambridge, Marinescu, I.D. and Rowe, W.B., Tribology of Abrasive Machining Processes, 2nd ed., Elsevier, Sathasivam, S. and Walker, P.S. The conflicting requirements of laxity and conformity in total knee replacement, Journal of Biomechanics, Vol. 32, No. 3, pp , Willing, R. and Kim, I.Y., Three dimensional shape optimization of total knee replacements for reduced wear, Structural and Multidisciplinary Optimization, Vol. 38, No. 4, pp , Godest, A.C., Beaugonin, M., Haug, E. and Taylor, M., Simulation of a knee joint replacement during a gait cycle using explicit finite element analysis, Journal of Biomechanics, Vol. 35, No. 2, pp , Halloran, J.P., Petrella, A.J. and Rullkoetter, P.J., Explicit finite element modeling of total knee replacement mechanics, Journal of Biomechanics, Vol. 38, No. 2, pp , Transactions of the Canadian Society for Mechanical Engineering, Vol. 41, No. 2, 2017