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767 Elastohydrodynamic lubrication analysis of hip implants with ultra high molecular weight polyethylene cups under transient conditions D Jalali-Vahid1, Z M Jin2* and D Dowson3 1 D epartment of M aterial Engineering, Sahand U niversity of Technology, Tabriz, Iran 2 M edical Engineering, School of Engineering, D esign and Technology, U niversity of Bradford, West Yorkshire, U K 3 ‘R yedale’, Adel, Leeds, U K Abstract: The transient variation of both the load and speed experienced during walking in an elastohydrodynamic lubrication (EH L) analysis for arti cial hip joints employing an ultra high molecular weight polyethylene (U H M WPE) acetabular cup against either a metallic or ceramic femoral head was considered in this study. A general numerical procedure to solve the transient EH L problem in spherical ball-in-socket coordinates, developed in a previous study by Jalali-Vahid and Jin in 2002, was applied under three speci c conditions experienced during typical gait cycles, including speed reversal, a sudden load increase and a sudden load decrease. The predicted minimum lm thickness was found to stay remarkably constant and similar to that prior to the change in either the load or the angular velocity, despite a large change in these operating conditions. This was attributed to the remarkably effective squeeze- lm action of preserving and maintaining the lubricating lm developed before the transient variation in either the load or speed. It is therefore possible to neglect the effect of these speci c transient variations of load and speed under physiological walking conditions considered in the present study on the predicted lm thickness in hip implants with U H M WPE cups. Keywords: transient elastohydrodynamic lubrication (EH L), arti cial hip joint replacement, speed reversal, squeeze lm NOTATION x , y, z linear coordinates c radial clearance ˆ R 2 ¡ R 1 d elastic deformation of the U H M WPE d cup wall thickness ˆ R 3 ¡ R 2 liner de ned in equation (3) ex , ey , ez eccentricities between the centres of the ex , ey , ez non-dimensional eccentricity ratios femoral head and the acetabular socket ˆ ex =c, ey =c, ez =c E modulus of elasticity for U H M WPE Z viscosity of synovial uid fx , fy , fz calculated load components de ned in n Poisson’ s ratio equation (4) f, y angular coordinates in the entraining and h total lm thickness side-leakage directions respectively p pressure o angular velocity R1 femoral head radius R2 cup radius R3 outside radius of the cup t time from the change of load or speed w 1 INTRODUCTION applied load in the y direction N atural synovial joints such as hips and knees are T he M S was received on 4 M arch 2003 and was accepted after revision remarkable bearings. These bearings are expected to for publication on 3 A pril 2003. function in the human body for a lifetime while * Corresponding author: M edical Engineering, S chool of Engineering, Design and T echnology, University of Bradford, Bradford, W est transmitting large dynamic loads and yet accommodat- Y orkshire BD7 1DP, UK. ing a wide range of movements. H owever, diseases such C03903 # IM echE 2003 Proc. Instn M ech. Engrs Vol. 217 Part C: J. M echanical Engineering Science 768 D JALALI-VAH ID, Z M JIN AN D D DOWSON as osteoarthritis and rheumatoid arthritis and trauma joint replacements. F urthermore, it has been pointed out sometimes require these natural bearings to be replaced that the average transient minimum lm thickness by arti cial ones. Total joint replacement has been the predicted throughout one cycle is very close to that most successful surgical treatment for hip joint diseases under quasi-static conditions based upon the average in the last forty years. Currently, more than 800 000 hip angular velocity and load. D ue to the convergence joint replacements are carried out worldwide every year. problem of the numerical method commonly experi- The majority of current arti cial hip joints utilize a enced with EH L solutions, only a cyclic sinusoidal speed material combination of ultra high molecular weight with a constant load and a cyclic sinusoidal load with a polyethylene (U H M WPE) articulating against either a constant speed were considered. Although it is generally metallic or ceramic component. These man-made desirable to consider the complete cyclic nature of gait bearings can sometimes last 20 years in the body cycles, the load in the swing phase is generally much without failure. H owever, problems such as osteolysis smaller and less well de ned, and often considered to be and loosening of the prosthesis have recently been less important compared with the stance phase load. In identi ed as the main factor limiting the in vivo many studies, the load in the swing phase is not even performance of implants, particularly with the increas- given. M ore recently, it has been demonstrated that in ing use of these devices in younger patients with life vivo microseparation can occur between the femoral expectancies after surgery in excess of 25 years. head and the acetabular cup during gait. Average Osteolysis and loosening are usually caused by an separations in subjects with U H M WPE-on-metal hip adverse tissue reaction to wear particles of U H M WPE prostheses have been shown to be 2 mm [14, 15]. This [1–3]. microseparation occurs during the swing phase in the Tribological studies of arti cial hip joints with direction along the axis of the cup, and when a load is U H WM PE acetabular cups play a major role in the applied in the stance phase, the femoral head contacts long-term success of these implants. The wear particles, the superior rim of the cup because of the load vector which can cause adverse biological reactions, are mainly imposed on the hip joint at this instant of the walking generated at the articulating bearing surfa ces because of cycle. The implication of the microseparation on the a mixed or boundary lubrication regime experienced in swing phase load is still debatable [16], but the the majority of these hip implants [4, 5]. The generation magnitude of the swing phase load must be very small. of wear particles can be greatly reduced by improving Although a small load generally favours the numerical the wear resistance of the bearing materials. H owever, it solution to EH L problems, a sudden decrease in load may also be possible to reduce the proportion of the after the stance phase may pose the problem of total load carried by asperity contact by promoting uid numerical convergence. Similar problems could also be lm lubrication in these man-made bearings, since the encountered just before the stance phase when the load two articulating surfaces are then partially separated. It is suddenly increased. F urthermore, the angular speed is is therefore important to predict the magnitude of the reversed during the stance phase, which has not been lubricating lm thickness in hip joint replacements in considered in the previous transient EH L analysis of hip order to assess the lubrication regime and to improve implants [13]. Therefore, it is important to analyse the the contribution of uid lm lubrication to the lubricating lm thickness under these speci c transient performance of these implants. periods during the gait cycle. The purpose of this study Quasi-static entraining motion has been considered in was to apply the general transient EH L methodology a number of previous elastohydrodynamic lubrication developed in a previous study [13] to an example of hip studies of arti cial hip joints with U H M WPE cups [5–7]. implants with U H M WPE cups under three transient In addition, a number of squeeze- lm lubrication periods during a typical gait cycle: (a) angular speed analyses have also been carried out [8–11]. It is well reversal, (b) a sudden increase in load and (c) a sudden known that neither the load nor the speed experienced in decrease in load. hip joints are constant during walking cycles [12]. R ecently, a general numerical procedure to solve the transient elastohydrodynamic lubrication (EH L) pro- blem in spherical ball-in-socket coordinates has been 2 LUBRICATION MODEL, GOVERNING developed by Jalali-Vahid and Jin [13] and applied to an EQUATIONS AND NUMERICAL ANALYSIS example of hip implants with U H M WPE cups under transient cyclic variations of load and speed. The A simple ball-in-socket con guration was considered in predicted minimum lm thickness was shown to stay this study for EH L analysis of arti cial hip joints with remarkably constant, despite a large change in the U H M WPE cups, as shown in F ig. 1. The important angular velocity and the load. This has been attributed parameters required for the EH L analysis are the radius to the combined effect of entraining and squeeze- lm of the femoral head …R 1†, the radius of the acetabular actions in generating, replenishing and maintaining load cup …R 2† or the radial clearance …c ˆ R 2 ¡ R 1 †, and the support by means of a lubricating lm in arti cial hip thickness of the cup wall …d†, the elastic modulus …E† Proc. Instn M ech. Engrs Vol. 217 Part C: J. M echanical Engineering Science C03903 # IMechE 2003 EHL AN ALYSIS OF H IP IM PLANTS WITH UH M WPE CUPS UNDER TR AN SIENT CON DITIONS 769 Case b: sudden increase in load, as shown in Fig. 2b. This sudden increase in load commences just before heel strike and is completed shortly afterwards. The load increases signi cantly from a small fraction of body- weight, while the speed remains relatively constant. F or this case, the load was assumed to increase linearly from 200 to 2100 N in 0.05 s, under a constant angular speed of 1 rad/s. Case c: sudden decrease in load, as shown in Fig. 2c. The load decreases rapidly just before toe off. F or this case, the load was assumed to decrease from 2100 to 200 N in 0.05 s. Although the speed reverses during Fig. 1 A simple ball-in-socket model for the transient EHL this stage of the walking cycle, a constant speed of analysis of hip joint replacements with U H M WPE 1 rad/s was assumed in the present analysis. cups The governing equation for the pressure generation …p† for the present EH L problem is the R eynolds and Poisson’s ratio …n† for the U H M WPE cup. A typical equation in spherical coordinates (shown in F ig. 3), as hip joint replacement was considered with the following detailed below [13, 17]: parameters: ´ ´ q qp q qp sin y h3 sin y ‡ h3 (a) femoral head radius …R 1† of 14 mm, qy qy qf qf (b) cup radius …R 2† of 14.1 mm, ´ qh qh (c) cup thickness …d† of 7 mm, ˆ 6ZR 2 sin 2 y o 2 ‡2 …1† qf qt (d) elastic modulus …E† of 1 G Pa, (e) Poisson’s ratio …n) of 0.4. Other equations include the geometrical representation of lm thickness …h†: The femoral head articulating with the U H M WPE cup is usually metallic or ceramic. The modulus of these ¡ ¢ h ˆ c 1 ¡ ex sin y cos f ¡ ey sin y sin f ¡ ez cos y ‡ d hard materials is at least two orders of magnitude greater than that of U H M WPE and therefore the …2† femoral head was assumed to be rigid. F urthermore, where the elastic deformation of the U IH M WPE cup …d† the lubricant present in arti cial hip implants, synovial was calculated, based on a simple constrained column uid, was assumed to be N ewtonian and iso-viscous for model: the purpose of lubrication analysis in the present study h i [5]. A relatively high viscosity of 0.01 Pa s was adopted R 2 …R 3 =R 2 †3 ¡1 in order to facilitate convergence of numerical solutions dˆ n op …3† [13]. E 1=…1 ¡ 2n† ‡ ‰2=…1 ‡ n†Š…R 3 =R 2 †3 The load and speed experienced in hip joints during walking are generally three-dimensional, but the main F inally, the integration of the pressure distribution must load component is in the vertical direction and the main be balanced by the external load applied: motion is exion and extension. Therefore, only the load …p …p in the y direction …w† and the angular velocity around 2 fx ˆ R 2 p sin y cos f sin y dy df ˆ 0 the z axis …o† were considered. F igure 2 shows the variation of both the vertical load and the angular speed …0 …0 p p ( exion/extension) during one walking cycle [17]. Three fy ˆ R 22 p sin y sin f sin y dy df ˆ w 0 0 speci c transient periods during the walking cycles were …p …p considered: fz ˆ R 22 p cos y sin y dy df ˆ 0 0 0 Case a: speed reversal during the stance phase, as shown …4† in Fig. 2a. This period occurs just after the heel strike. The load increases signi cantly and stays relatively The following boundary conditions for the present constant, while the direction of the speed is reversed. lubrication problem were adopted: The load was assumed to be 2100 N (three times (a) zero pressure at the edge of the cup in the plane …x z†, bodyweight for a person with 70 kg mass). The (b) cavitation on the outlet boundary. angular speed was assumed to decrease linearly from 1.7 rad/s to zero in 0.12 s, and then to increase from 0 The speed reversal was considered in the present study to 1.7 rad/s in the opposite direction in 0.12 s. by simply changing the inlet and the outlet boundaries. C03903 # IM echE 2003 Proc. Instn M ech. Engrs Vol. 217 Part C: J. M echanical Engineering Science 770 D JALALI-VAH ID, Z M JIN AN D D DOWSON Fig. 2 Transient variation of the load and speed for simulating (a) speed reversal during the stance phase (case a), (b) a sudden load increase before the heel strike (case b) and (c) a sudden load decrease after the toe off (case c) Proc. Instn M ech. Engrs Vol. 217 Part C: J. M echanical Engineering Science C03903 # IMechE 2003 EHL AN ALYSIS OF H IP IM PLANTS WITH UH M WPE CUPS UNDER TR AN SIENT CON DITIONS 771 F igure 6a shows the minimum and central lm thicknesses as a function of time after the load is decreased from 2100 to 200 N (case c). The lm pro le and the pressure distribution are shown in F igs 6b and c respectively at various instants during the load change. The lm pro les after the load is kept constant at 200 N for a further 0.5 s are shown in F ig. 6d. 4 DISCUS SION It can be seen from F ig. 4a that both the central and the minimum lm thicknesses decrease as the speed decreases, up to the instant when the direction of Fig. 3 De nition of spherical coordinates …f, y† motion is reversed …t ˆ 0.12 s†. H owever, the reduction in the lm thickness is both gentle and small. N either the central nor the minimum lm thickness fall to zero at the The effect of cavitation on the location of the inlet instant of speed reversal when the speed is reduced to boundary and ow continuity during speed reversal zero due to squeeze- lm action. After the speed reversal, were not considered in the present analyses. The effect the minimum lm thickness quickly increases by a small of these assumptions on the predicted lubricant lm amount and stays relatively constant, while the central thickness has been shown to be negligible in an EH L lm thickness continues to decrease at a very slow rate. analysis considering speed reversal, particularly under F or example, at the end of the simulation period of heavily loaded conditions [18]. 0.72 s from the start of the speed change, the central lm The nite difference method was employed to solve thickness is decreased by only 12 per cent, while the the governing equations (1) to (4), subject to the load minimum lm thickness is almost identical to that and speed variations shown in F igs 2a, b and c. The established before the speed change. The detailed details of the numerical method have been given changes of the lm thickness at different time instants elsewhere [13]. The number of mesh grids used in the are shown in F ig. 4b. It is evident that the effect of speed present study was between 4806480 and 9616961. The reversal upon the lm pro les is most pronounced in the time step was chosen between 0.05/300 and 0.24/300 s. inlet region, both before and after the speed reversal. F urthermore, numerical solutions to the lm pro le and Before the speed reversal …t < 0.12 s†, the inlet region is the pressure distribution were computed only for the on the left of the contact in F ig. 4b and the convergent speci c transient periods of 0.72 s for case (a) and 0.55 s lm pro les can be seen from left (inlet) to right (outlet). for both cases (b) and (c). After the speed reversal …t > 0.12 s†, the inlet region moves to the right of the contact. It is interesting to note that the lm thicknesses begin to increase on the right of the conjunction (inlet), while the lm thicknesses on the 3 RESULTS left (outlet) decrease. H owever, both the central and the minimum lm thicknesses stay relatively constant, as The predicted minimum and central lm thicknesses as a shown in F ig. 4a. It should be pointed out that at 0.72 s, function of time after the rapid speed change at constant the lubricating lm thickness is convergent between load (case a) are shown in F ig. 4a. The corresponding f ˆ 1358 and 1228 and divergent from f ˆ 1228 to 458 lm shape and thickness and pressure distributions within the contact conjunction. D espite relatively large along the centre-line in the entraining direction …f† are changes of the lm pro les, the change in the pressure shown in F igs 4b and c respectively for different time distribution is negligible due to the assumption of a instants. constant load (F ig. 4c). The time history of the predicted central and The effect of a sudden load increase from 200 to minimum lm thicknesses when the load is increased 2100 N on the predicted minimum and central lm from 200 to 2100 N in 0.05 s, and then kept constant for thicknesses is quite small, as shown in F ig. 5a. A slight another 0.5 s (case b), is shown in F ig. 5a. The changes increase in both the minimum and the central lm in lm shape and thickness and pressure distributions at thicknesses around t ˆ 0.05 s is probably due to the large different time instants during the ramp period of 0.05 s squeeze- lm velocity at the outlet region. This becomes are shown in F igs 5b and c. F igure 5d shows how the more evident when the lm pro les at different time lm shape and thickness changes during and after the instants shown in F ig. 5b are considered. It is clear that load increase from 200 to 2100 N . the load increase leads to an increase in the width of the C03903 # IM echE 2003 Proc. Instn M ech. Engrs Vol. 217 Part C: J. M echanical Engineering Science 772 D JALALI-VAH ID, Z M JIN AN D D DOWSON Fig. 4 (a) Prediction of the central and minimum lm thicknesses as a function of time from the speed change (case a); (b) lm pro les at different instants after the speed change (case a); (c) pressure distributions at different instants after the speed change (case a) Proc. Instn M ech. Engrs Vol. 217 Part C: J. M echanical Engineering Science C03903 # IMechE 2003 EHL AN ALYSIS OF H IP IM PLANTS WITH UH M WPE CUPS UNDER TR AN SIENT CON DITIONS 773 Fig. 5 (continued over) C03903 # IM echE 2003 Proc. Instn M ech. Engrs Vol. 217 Part C: J. M echanical Engineering Science 774 D JALALI-VAH ID, Z M JIN AN D D DOWSON Fig. 5 (a) Prediction of the central and minimum lm thicknesses as a function of time from the load increase from 200 to 2100 N (case b); (b) lm pro les at different instants during the load increase from 200 to 2100 N (case b); (c) pressure distributions at different instants during the load increase from 200 to 2100 N (case b); (d) lm pro les at different instants after the load increase from 200 to 2100 N (case b) contact conjunction and a large decrease in the lm 2100 N prior to the load change but still considerably thicknesses in both the inlet and the outlet regions. smaller than the steady state prediction of 0.267 mm at a H owever, the lm thickness in the central part of the load of 200 N , as shown in F ig. 6d. The maximum contact region between f ˆ 648 and 1148, mainly pressure decreases rapidly, from 29 to 8 M Pa as shown corresponding to the contact region of 200 N , stays in F ig. 6c, as the load is decreased from 2100 to 200 N . remarkably constant, due to the squeeze- lm action. It has been pointed out that it is important to estimate Consequently, this leads to almost constant minimum the lubricating lm thickness under general transient and central lm thicknesses, as shown in F ig. 5a. It is conditions. It has been shown previously that under also interesting to note that even after 0.55 s (or 0.50 s cyclic variations of both the load and speed, the use of a after the maximum load of 2100 N has been achieved), steady state formula based on the average load, speed the transient lm thickness predicted in the present and viscosity gives a good estimate of the transient lm study does not resemble that under the steady state thickness. It is clear from the present study that under condition shown in F ig. 5d. The minimum lm thickness physiological walking conditions imposing a relatively predicted at 0.55 s was 0.254 mm, as compared with short cycle of time, the effect of speed reversal has a 0.267 mm at the onset of the load change and 0.180 mm at negligible effect upon the predicted lm thickness. This a steady state load of 2100 N . This once again implies is also true when either a sudden load increase or a the importance of the squeeze- lm action in preserving sudden load decrease occurs. D ue to the powerful the lubricating lm thickness developed under low load. squeeze- lm action, the lubricant lm developed prior to Although the pressure builds up rapidly in both the inlet the transient condition stays relatively unchanged for and the outlet regions as the load increases, the quite a long time of 0.5 s, when either the load is magnitude, located within the centre of the contact, is increased or decreased or the direction of motion is largely determined by the load imposed as shown in reversed. These observations are consistent with the F ig. 5c. persistence of a relatively constant lubricant lm The effect of a sudden load decrease on the predicted thickness during walking cycles reported from previous minimum and central lm thicknesses is shown in theoretical studies of both synovial joints and their F ig. 6a. U nder transient conditions, a decrease in load replacements by D owson and Jin [19], Jin et al. [20], actually results in a decrease in the predicted minimum Chan et al. [21] and Jalali-Vahid and Jin [13]. It is and the central lm thicknesses, throughout two thirds reassuring that the major load increase experienced in of the duration of the load change. After that, both the the stance phase has a very small effect on the predicted central and minimum lm thicknesses begin to increase, transient lm thickness and the load decrease in the as also shown in F ig. 6b. H owever, at the end of the swing phase considered in the present study is also simulation (0.55 s) after the load has remained constant unlikely to lead to a large increase in the lubricating lm. at 200 N for 0.5 s, the predicted minimum lm thickness It should be pointed out that the main purpose of the was 0.167 mm, similar to that of 0.180 mm at a load of present study was to assess the effect on uid lm Proc. Instn M ech. Engrs Vol. 217 Part C: J. M echanical Engineering Science C03903 # IMechE 2003 EHL AN ALYSIS OF H IP IM PLANTS WITH UH M WPE CUPS UNDER TR AN SIENT CON DITIONS 775 Fig. 6 (continued over) C03903 # IM echE 2003 Proc. Instn M ech. Engrs Vol. 217 Part C: J. M echanical Engineering Science 776 D JALALI-VAH ID, Z M JIN AN D D DOWSON Fig. 6 (a) Prediction of the central and minimum lm thicknesses as a function of time from the load decrease from 2100 to 200 N (case c); (b) lm pro les at different instants during the load decrease from 2100 to 200 N (case c); (c) pressure distribution at different instants during the load decrease from 2100 to 200 N (case c); (d) lm pro les at different instants after the load decrease from 2100 to 200 N (case c) lubrication of the transient variation of the load and average load, viscosity and average speed under the speed experienced in hip joints during walking for a cyclic conditions considered in the present study. typical hip joint replacement with an U H M WPE cup. It is generally known that the lubricating lm thickness developed in these hip implants is not suf ciently large REFERENCES to separate completely the two bearing surfaces [5]. The assumption of full uid lm lubrication is not valid for 1 Willert, H. G. and Semlitsch, M. R eaction of the articular these hip implants and the mode of lubrication is capsule to wear products of arti cial joint prostheses. therefore boundary or mixed. H owever, the general J. Biomed. M ater. R es., 1977, 11, 157–164. ndings from this transient elastohydrodynamic lubri- 2 Amstutz, H. C., Campell, P., Kossovsky, N. and Clark, I. C. cation analysis for hip joint replacements with M echanism and clinical signi cance of wear debris-induced U H M WPE cups may be equally applicable to other osteolysis. Clin. Orthop. R elated R es., 1991, 276, 7–17. forms of hip prostheses such as metal-on-metal material 3 Ingham, E. and Fisher, J. Biological reactions to wear combinations [22]. debris in total joint replacement. 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