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Prediction of lubricating film thickness in UHMWPE hip joint replacements
Journal of Biomechanics 34 (2001) 261}266
Technical note
Prediction of lubricating "lm thickness in UHMWPE hip joint replacements D. Jalali-Vahid , M. Jagatia , Z.M. Jin *, D. Dowson Department of Mechanical and Medical Engineering, University of Bradford, Bradford DB7 1DP, UK School of Mechanical Engineering, University of Leeds, Leeds LS2 9JT, UK Accepted 20 August 2000
Abstract An elastohydrodynamic lubrication model developed for a ball-in-socket con"guration in a previous studies by the present authors (Jalali-Vahid et al., Thinning "lms and tribological interfaces, 26th Leeds}Lyon Symposium on Tribology, 2000, pp. 329}339) was applied to analyse the lubrication problem of a typical arti"cial hip joint replacement, consisting of an ultra-high molecular weight polyethylene (UHMWPE) acetabular cup against a metallic or ceramic femoral head. The cup was assumed to be stationary whilst the ball was assumed to rotate at a steady angular velocity and under a constant load. A wide range of main design parameters were considered. It has been found that the predicted lubricating "lm thickness increases with a decrease in the radial clearance, an increase in the femoral head radius, an increase in UHMWPE thickness and a decrease in UHMWPE modulus. However, the predicted lubricating "lm thicknesses are not found to be su$ciently large in relation to the surface roughness of the cup and head to indicate separation of the two articulating surfaces. It should also be noted that if the design features are unable to secure full #uid "lm lubrication, it may be preferable to select them for minimum wear rather than maximum "lm thickness. For example, an increase in head radius will enhance the "lm thickness, but it will also increase the sliding distance and hence wear in mixed or boundary lubrication conditions. Furthermore, it is pointed out that an increase in the predicted lubricant "lm thickness is usually associated with an increase in the contact area, and this may cause lubricant starvation and stress concentration at the edge of the cup, and adversely a!ect the tribological performance of the implant. The e!ect of running-in process on the lubrication in UHMWPE hip joint replacements is also discussed. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Elastohydrodynamic lubrication; Arti"cial hip joint replacements; UHMWPE
1. Introduction The main problem associated with the long-term survival of current hip joint replacements is loosening caused by the adverse tissue reaction (osteolysis) to wear particles (Willert and Semlitsch, 1977; Livermore et al., 1990; Amstutz et al., 1991; Jasty and Smith, 1992; Ingham and Fisher, 2000). The majority of wear particles are generated at the articulating surfaces because of asperity contact in a mixed or boundary lubrication regime experienced in current hip joint replacements. Therefore, it is very important to consider lubrication in the design of such implants. Under ideal conditions where the two articulating surfaces are completely separated by a continuous lubricant "lm, minimum wear can be
* Corresponding author.
expected. However, in current total hip joint replacements employing ultra-high molecular weight polyethylene (UHMWPE) cups, the lubricant "lm thickness is generally thought to be much smaller than the surface roughness of the UHMWPE bearing surface, consequently leading to a mixed or boundary lubrication regime and wear of the bearing surface and generation of wear particles (Unsworth et al., 1975; Fisher and Dowson, 1991). The purpose of this study was to analyse elastohydrodynamic lubrication "lm thickness and to compare the theoretical prediction of the lubrication regime in current UHMWPE hip joint replacements with the previous experimental observation. It should be pointed out that the comparison of the predicted lubricating "lm thickness with the surface roughness of the bearing surfaces may indicate the degree of asperity contacts and hence likely severity of wear. Furthermore, such a lubrication model can be extended to other forms of hip prostheses where #uid "lm lubrication may be very
0021-9290/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 9 2 9 0 ( 0 0 ) 0 0 1 8 1 - 0
262
D. Jalali-Vahid et al. / Journal of Biomechanics 34 (2001) 261}266
R u
Nomenclature c d e ,e ,e V W X E E F ,F ,F V W X h h E h
H
H*
M p R R R
radial clearance, R !R cup thickness, R !R eccentricities modulus of elasticity for UHMWPE equivalent modulus of elasticity de"ned in Eq. (5) calculated load components de"ned in Eq. (A.6) total "lm thickness de"ned in Eq. (A.4) rigid "lm thickness de"ned in Eq. (A.2) minimum lubricant "lm thickness non-dimensional "lm thickness, h /R
Moes "lm thickness parameter de"ned in Eq. (4) Moes load parameter de"ned in Eq. (4) pressure e!ective radius for the ball-on-plane model, R"R R /c femoral head radius cup radius
important, for example metal-on-metal and ceramic-onceramic material combinations (Chan et al., 1999; Nevelos et al., 1999). Few theoretical studies of the lubrication of arti"cial hip joint replacements have been reported in the literature. In most studies of the lubrication of ball-insocket con"gurations, it has been assumed that the bearing surfaces were rigid, making the analysis particularly relevant to the lubrication of metal-onmetal or ceramic-on-ceramic hip joint replacements (Ai and Cheng, 1996; Kothari et al., 1995; Jin and Dowson, 1999). There are only a few studies reported in the literature where the elastic deformation of the bearing surfaces was considered in the lubrication analysis. Jin et al. (1997) applied the well-known Hamrock and Dowson (1978) formulae for isoviscous lubricants and semi-in"nite solids to both metal-on-metal and metal-on-UHMWPE hip joint replacements, while a number of other studies examined the squeeze-"lm lubrication problem (Sasada and Mabuchi, 1985; Mabuchi and Sasada, 1990; Wang et al., 1990). The e!ect of entraining action has been recently considered by Jalali-Vahid et al. (2000) in the development of an elastohydrodynamic lubrication analysis and a successful numerical procedure based upon the Newton}Raphson method has been developed for ball-in-socket UHMWPE hip joint replacements. The speci"c objectives of this
; w = x, y, z d e ,e ,e V W X
,h
g m x
outside radius of the cup entraining velocity for the ball-on-plane model, u"uR /2 non-dimensional velocity for the ballon-plane model de"ned in Eq. (5) applied load in the y direction non-dimensional load for the ball-onplane model de"ned in Eq. (5) coordinates used in Fig. 1a elastic deformation of UHMWPE liner de"ned in Eq. (A.3) eccentricity ratios, e /c, e /c, e /c V W X angular coordinates in the entraining and side-leakage directions, respectively half-contact angle (in degrees) viscosity of synovial #uid Poisson's ratio angular velocity
Subscripts b-p b-s
ball-on-plane ball-in-socket
study were: (a) To apply the elastohydrodynamic lubrication model developed by Jalali-Vahid et al. (2000) to a wide range of conditions experienced in current hip joint replacements with UHMWPE sockets. (b) To investigate the e!ect of important design parameters such as the radial clearance, the femoral head radius, UHMWPE thickness and modulus on the formation of lubricating "lm thickness in current UHMWPE hip joint replacements.
2. Elastohydrodynamic lubrication model In this study, an elastohydrodynamic lubrication analysis developed by Jalali-Vahid et al. (2000) for the ballin-socket con"guration shown in Fig. 1a was used to investigate the formation of the lubricant "lm in UHMWPE hip joint replacements under steady-state conditions. It has previously been shown that such a steady-state analysis can provide a reasonable estimate of the lubricating "lm thickness in the natural hip joint, even under the dynamic conditions experienced during steady walking (Jin et al., 1993). Only one load component in the y direction (w) and one angular velocity about the z}axis (u) were considered. The important design parameters to be investigated were the radii of the
D. Jalali-Vahid et al. / Journal of Biomechanics 34 (2001) 261}266
263
the numerical results thus obtained were used to curve-"t the following equation, to predict the minimum lubricant "lm thickness for the present ball-in-socket con"guration, (h ) } ;
(h ) }
"1.0191 !2.3153 #0.8151 (h ) } 90 90
;
90
!0.1133
#1, 90
r"0.9909
(2)
where (h ) } is the corresponding minimum lubricant
"lm thickness for the ball-on-plane model considered by Wang (1994) and is the half-contact angle, which can be determined from a contact mechanics analysis (Jin et al., 1999). The minimum lubricant "lm thickness for the ball-onplane model shown in Fig. 1b is (HH ) } "1.47M+\ > +\ ,
(3)
where (H* ) } and M denote the Moes "lm and load
parameters, respectively, de"ned as follows: = M" , (HH ) } "(H ) } /;
; Fig. 1. (a) Ball-in-socket model for lubrication analysis of UHMWPE arti"cial hip joint replacements. (b) Equivalent ball-on-plane model for lubrication analysis of UHMWPE arti"cial hip joint replacements.
and
(1!2l) d =" (1!l) R E "E/(1!l)
femoral head (R ) and the cup (R ), or the radial clear ance (c"R !R ), the elastic modulus (E), Poisson's ratio (l) and the thickness (d) of the UHMWPE layer. The prediction of the lubricant "lm thickness for the ball-in-socket model will be compared with the corresponding ball-on-plane model as shown in Fig. 1b considered by Wang (1994), with an equivalent radius (R) determined by the following equation: R R R" c
(1)
The main governing equations for the present lubrication model are the Reynolds equation, which governs the #uid "lm lubrication between the two bearing surfaces, and the elasticity equation. The details of these governing equations are presented in the appendix and the numerical method to solve these equations can be found in a previous study by Jalali-Vahid et al. (2000). A wide range of the main design parameters were considered and
(4)
w (1!2l) d , ;" (1!l) R E R
gu , E R (5)
The general "lm thickness formula shown in Eq. (2) was used in a parametric study to investigate the e!ect of a wide range of conditions experienced in current UHMWPE hip joint replacements on the prediction of lubricant "lm thickness. The following parameters were chosen for the lubrication analysis: The angular velocity (u), 2 rad/s The Poisson's ratio for UHMWPE (l), 0.4 Load (w), 2500 N The viscosity of synovial #uid (g), 0.005 Pa s Other important design parameters were varied to cover a wide range of conditions experienced in current UHMWPE hip joint replacements: Radial clearance (c), 75}200 lm Femoral head radius (R ), 11}16 mm UHMWPE elastic modulus (E), 250}2000 MPa UHMWPE thickness (d), 5}15 mm For example, consider a 28 mm diameter femoral head and a polyethylene cup of thickness 10 mm and elastic
264
D. Jalali-Vahid et al. / Journal of Biomechanics 34 (2001) 261}266
radial modulus of 1 GPa, with a radial clearance of 120 lm, the dimensionless parameters used in Eqs. (4) and (5) became
Figs. 2a,b and 3a,b show the e!ect of the femoral head radius, the radial clearance, the UHMWPE thickness and UHMWPE elastic modulus, respectively, on the predicted minimum "lm thickness from Eq. (2). It is generally noticed that an increase in the femoral head radius, a decrease in the radial clearance, an increase in the UHMWPE thickness and a decrease in the
UHMWPE elastic modulus all result in an increase in the predicted minimum "lm thickness. However, the effect of the radial clearance does not appear to be important within the range considered, unlike the "ndings for metal-on-metal hip joint replacements (Jin et al., 1997). A similar conclusion can also be reached for the UHMWPE thickness, particularly when a realistic value is used. The variation of the predicted lubricating "lm thickness is relatively small, despite of a wide range of design parameters considered. Furthermore, the magnitude of the predicted lubricating "lm thickness is much smaller than the surface roughness of the UHMWPE bearing surface which is often quoted as 1 lm (Jin et al., 1997). Therefore, a mixed or boundary lubrication is predicted for all the conditions considered in the present study and this is consistent with the previous experimental observation reported in the literature. The "lm thickness predicted in the present study is based upon the assumption of smooth bearing surfaces. However, as already mentioned above, in none of the cases considered is the predicted "lm thickness large enough to indicate that the two bearing surfaces can be separated by a #uid "lm. Therefore, the roughness of
Fig. 2. (a) E!ect of femoral head radius on the predicted minimum "lm thickness (u"2 rad/s; w"2500 N; d"10 mm; g"0.005 Pa s; E"1000 MPa; c"120 lm), (b) E!ect of radial clearance on the predicted minimum "lm thickness (u"2 rad/s; w"2500 N; d"10 mm; g"0.005 Pa s; E"1000 MPa; R "14 mm).
Fig. 3. (a) E!ect of UHMWPE thickness on the predicted minimum "lm thickness (u"2 rad/s; w"2500 N; g"0.005 Pa s; c"120 lm; E"1000 MPa; R "14 mm). (b) E!ect of elastic modulus of UHMWPE on the predicted minimum "lm thickness (u"2 rad/s; w"2500 N; d"10 mm; g"0.005 Pa s; c"120 lm; R "14 mm).
E "1.19 Gpa, M"14200.
;"1.2;10\,
="2.6;10\,
From Eq. (3), this yields a dimensionless Moes "lm thickness parameter (H* ) } of 0.285. Hence, the ball-on plane analysis indicates a minimum "lm thickness of 0.20 lm. If the half-contact angle is predicted to be 48.463 (Jin et al., 1994; 1999), the "lm thickness will be reduced by 10%, or 0.18 lm as predicted from Eq. (2).
3. Results and discussion
D. Jalali-Vahid et al. / Journal of Biomechanics 34 (2001) 261}266
the UHMWPE bearing surface should be taken into account in future analyses of the elastohydrodynamic lubrication of total replacement hip joints. Furthermore, the consequence of mixed or boundary lubrication must be considered in the selection of the design parameters. For example, although an increase in the femoral head radius leads to an increase in the e!ective radius as well as the entraining velocity and consequently lubricant "lm thickness, the e!ect of increasing sliding distance upon wear must also be taken into consideration. Furthermore, the lubricant starvation due to an increase in the contact area must be addressed. For example, although a decrease in the elastic modulus from 2000 to 250 MPa results in almost a twofold increase in the predicted minimum "lm thickness, the half-contact angle is also increased from 41.35 to 64.093, which exceeds the maximum available angle when the cup is placed anatomically in the body. It is clear that design of UHMWPE hip joint replacements from the lubrication point view requires a careful and balanced approach. It should also be pointed out that all the design parameters considered in the present study were assumed to remain constant. However, this may not be true for some parameters such as the radial clearance since the femoral head usually wears linearly into the polyethylene socket. Furthermore, the retrieved polyethylene cups are often described as polished and the e!ective surface roughness must therefore be reduced. For example, Wang et al. (1995) reported that the initial centreline average roughness of 2 lm was reduced to 0.04 lm after 2 million cycles tested in simulators, while El"ck et al. (1998) found the r.m.s. surface roughness was decreased from 0.39 to 0.07 lm in retrieved polyethylene cups. All these factors, will help to promote #uid "lm lubrication, in UHMWPE hip joint replacements and should be taken into consideration in further lubrication studies in order to provide a complete understanding of the tribology at the bearing surfaces of arti"cial hip joint replacements.
4. Conclusion A ball-in-socket elastohydrodynamic lubrication model developed in a previous study by the present authors has been applied to current UHMWPE arti"cial hip joint replacements. A general formula for the minimum lubricant "lm thickness as functions of the halfcontact angle has been developed and applied to current UHMWPE hip joint replacements under realistic conditions. It has been shown that despite an increase in the femoral head radius, a decrease in the radial clearance, an increase in UHMWPE thickness and a decrease in UHMWPE elastic modulus all lead to an increase in the predicted minimum "lm thickness, the magnitude of the "lm thickness predicted is signi"cantly less than the initial machined surface roughness of the UHMWPE bear-
265
ing surface. It is recognized that the UHMWPE surface is smoothed during articulation and `running-ina, but there is a paucity of data on the resulting steady-state surface "nish on the polyethylene, as well as the modi"ed radial clearance. It has been shown theoretically that all current forms of metal-on-polyethylene total replacement hip joints operate in the mixed lubrication regime, with load transmission being e!ected partly by asperity contact and partly by hydrodynamic action. If the roughness of the polyethylene decreases as the joint articulates, the proportion of the load carried by the asperities and the associated wear will decrease. Lubrication studies of total joint replacements can thus indicate likely severity of wear of given combinations of implant materials.
Acknowledgements This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) and DePuy International Ltd. * a Johnson & Johnson Co.
Appendix A. Governing equations for elastohydrodynamic lubrication analysis of UHMWPE hip joint replacements The governing equation for the #uid #ow between the two articulating surfaces is Reynolds equation, which takes the following form in spherical polar co-ordinates for the present lubrication model shown in Fig. 1a (Jin and Dowson, 1999); sin h
* *p * *p *h h sin h # h "6gR u sin h *h *h *
*
*
(A.1)
The rigid "lm thickness between the undeformed cup and the ball is given by h "c(1!e sin h cos !e sin h sin ) E V W
(A.2)
The elastic deformation of the bearing surfaces is mainly due to the UHMWPE acetabular cup, since the elastic modulus for UHMWPE is much lower than that for either metallic or ceramic femoral components. In the present study, an approximate method based upon the constrained column model was used to calculate the elastic deformation for the UHMWPE cup (Jin et al., 1999);
R !1 R d" p. 1 2 R E # 1!2t 1#t R R
(A.3)
266
D. Jalali-Vahid et al. / Journal of Biomechanics 34 (2001) 261}266
Therefore, the total "lm thickness is given as follows h"h #d. (A.4) E The boundary conditions for the hydrodynamic pressure are p"0 at the edge of the cup in the plane (xz) and the Swift}Steiber}Reynolds cavitation boundary is adopted on the outlet boundary, where; *p *p " "0. * *h
(A.5)
Furthermore, the integration of the hydrodynamic pressure distribution components must be equal to the load imposed on the joint as shown in Fig. 1a.
F "R W F "R X
F "R V
0
0
0
p sin h cos sin h dh d "0 p sin h sin sin h dh d "w
(A.6)
p cos h sin h dh d "0
The numerical solution to the above governing equations was based upon the Newton-Raphson method. The Reynolds equation was discretized using the half-point "nite di!erence scheme for the left hand terms, and the backward di!erence scheme for the right-hand term in order to balance the accuracy and stability of the numerical method (Wang, 1994). The details of the numerical procedure have been given in a previous study by the present authors (Jalali-Vahid et al., 2000).
References Ai, C., Cheng, H.S., 1996. Hydrodynamic lubrication analysis of metallic hip joint. Tribology Transactions 39 (1), 103}111. Amstutz, H.C., Campell, P., Kossovsky, N., Clark, I.C., 1991. Mechanism and clinical signi"cance of wear debris-induced osteolysis. Clinical Orthopaedics and Related Research 276, 7}17. Chan, F.W., Bobyn, J.D., Medley, J.B., Krygier, J.J., Tanzer, M., 1999. The Otto Aufranc Award * Wear and lubrication of metal-onmetal hip implants. Clinical Orthopaedics and Related Research 369, 10}24. El"ck, A.P., Hall, R.M., Unsworth, A., Pinder, I.M., 1998. Surface topography of retrieved PCA acetabular liners: proposal of a novel wear mechanism. Journal of Materials Science Letters 17 (13), 1085}1088. Fisher, J., Dowson, D., 1991. Tribology of Arti"cial Joints. Journal of Engineering in Medicine, Proceedings of the Institution of Mechanical Engineers 205(H2), 73}79. Hamrock, B.J., Dowson, D., 1978. Elastohydrodynamic lubrication of elliptical contacts for materials of low elastic modulus: 1 fully #ooded conjunction. Trans ASME, Journal of Lubrication Technology 100, 236}245.
Ingham, E., Fisher, J., 2000. Biological reactions to wear debris in total joint replacement. Journal of Engineering in Medicine, Proceedings of the Institution of Mechanical Engineers 214(H1), 21}37. Jalali-Vahid, D., Jagatia, M., Jin, Z.M., Dowson, D., 2000. Elastohydrodynamic lubrication analysis of UHMWPE hip joint replacements. In: Dowson, D. et al. (Eds.), Thinning Films and Tribological Interfaces. Elsevier Science, Amsterdam, pp. 329}339. Jasty, M., Smith, E., 1992. Wear particles of total joint replacements and their role in periprosthetic osteolysis. Current Opinions in Rheumatology 4, 204}209. Jin, Z.M., Dowson, D., 1999. A full numerical analysis of hydrodynamic lubrication in arti"cial hip joint replacements constructed from hard materials. Journal of Mechanical Engineering Science, Proceedings of the Institution of Mechanical Engineers 213, 355}370. Jin, Z.M., Dowson, D., Fisher, J., 1993. Fluid "lm lubrication in natural hip joints In: Dowson, D., Taylor, C.M., Godet, M. (Eds.), Thin Films in Tribology, Elsevier, Amsterdam, pp. 545}555. Jin, Z.M., Dowson, D., Fisher, J., 1994. A parametric analysis of the contact stress in ultra-high-molecular-weight polyethylene acetabular cups. Medical Engineering and Physics 16 (5), 398}405. Jin, Z.M., Dowson, D., Fisher, J., 1997 Analysis of #uid "lm lubrication in arti"cial hip joint replacements with surfaces of high elastic modulus. Journal of Engineering in Medicine, Proceedings of the Institution of Mechanical Engineers 211(H3), 247}256. Jin, Z.M., Heng, S.M., Ng, H.W., Auger, D.D., 1999. An axisymmetric contact model of ultra high molecular weight polyethylene cups against metallic femoral heads for arti"cial hip joint replacements. Journal of Engineering in Medicine, Proceedings of the Institution of Mechanical Engineers 213(H4), 317}327. Kothari, M., Booker, J.F., Bartel, D.L., 1995. Analysis of arti"cial hip joints as spherical bearings. Proceedings of the 21st Leeds}Lyon Symposium on Tribology 93}98, Elsevier, Amsterdam. Livermore, J., Ilstrup, Morrey, B., 1990. E!ect of femoral head size on the wear of the polyethylene acetabular component. Journal of Bone and Joint Surgery 72A, 518}528. Mabuchi, K., Sasada, T., 1990. Numerical analysis of elastohydrodynamic squeeze "lm lubrication of total hip prostheses. Wear 140, 1}16. Nevelos, J.E., Ingham, E., Doyle, C., Fisher, J., Nevelos, A.B., 1999. Analysis of retrieved alumina ceramic components from Mittelmeier total hip prostheses. Biomaterials 20 (19), 1833}1840. Sasada, T., Mabuchi, K., 1985. Elastohydrodynamic lubrication of total hip prosthesis. Proceedings of the JSLE International Tribology Conference, Tokyo, Japan, July 8}10. Unsworth, A., Dowson, D., Wright, V., 1975. The frictional behaviour of human synovial joints * Part II. arti"cial joints. Trans ASME, Journal of Lubrication Technology 97(3), 377}382. Wang, A., Stark, C., Dumbleton, J.H., 1995. Role of cyclic plastic deformation in the wear of UHMWPE acetabular cups. Wear 29, 619}626. Wang, C.T., Wang, Y.L., Chen, Q.L., Yang, M.R., 1990. Calculation of elastohydrodynamic lubrication "lm thickness for hip prostheses during normal walking. Tribology Transactions 33, 239}245. Wang, D., 1994. Elastohydrodynamic lubrication of point contacts for layers of &soft' solids and monolithic &hard' materials in the transient bouncing ball problem. Ph.D. Thesis, University of Leeds, England. Willert, H. G., Semlitsch, 1977. Reactions of the articular capsule to wear products of arti"cial joint prostheses. Journal of Biomedical Materials Research 11, 157}164.
Technical note
Prediction of lubricating "lm thickness in UHMWPE hip joint replacements D. Jalali-Vahid , M. Jagatia , Z.M. Jin *, D. Dowson Department of Mechanical and Medical Engineering, University of Bradford, Bradford DB7 1DP, UK School of Mechanical Engineering, University of Leeds, Leeds LS2 9JT, UK Accepted 20 August 2000
Abstract An elastohydrodynamic lubrication model developed for a ball-in-socket con"guration in a previous studies by the present authors (Jalali-Vahid et al., Thinning "lms and tribological interfaces, 26th Leeds}Lyon Symposium on Tribology, 2000, pp. 329}339) was applied to analyse the lubrication problem of a typical arti"cial hip joint replacement, consisting of an ultra-high molecular weight polyethylene (UHMWPE) acetabular cup against a metallic or ceramic femoral head. The cup was assumed to be stationary whilst the ball was assumed to rotate at a steady angular velocity and under a constant load. A wide range of main design parameters were considered. It has been found that the predicted lubricating "lm thickness increases with a decrease in the radial clearance, an increase in the femoral head radius, an increase in UHMWPE thickness and a decrease in UHMWPE modulus. However, the predicted lubricating "lm thicknesses are not found to be su$ciently large in relation to the surface roughness of the cup and head to indicate separation of the two articulating surfaces. It should also be noted that if the design features are unable to secure full #uid "lm lubrication, it may be preferable to select them for minimum wear rather than maximum "lm thickness. For example, an increase in head radius will enhance the "lm thickness, but it will also increase the sliding distance and hence wear in mixed or boundary lubrication conditions. Furthermore, it is pointed out that an increase in the predicted lubricant "lm thickness is usually associated with an increase in the contact area, and this may cause lubricant starvation and stress concentration at the edge of the cup, and adversely a!ect the tribological performance of the implant. The e!ect of running-in process on the lubrication in UHMWPE hip joint replacements is also discussed. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Elastohydrodynamic lubrication; Arti"cial hip joint replacements; UHMWPE
1. Introduction The main problem associated with the long-term survival of current hip joint replacements is loosening caused by the adverse tissue reaction (osteolysis) to wear particles (Willert and Semlitsch, 1977; Livermore et al., 1990; Amstutz et al., 1991; Jasty and Smith, 1992; Ingham and Fisher, 2000). The majority of wear particles are generated at the articulating surfaces because of asperity contact in a mixed or boundary lubrication regime experienced in current hip joint replacements. Therefore, it is very important to consider lubrication in the design of such implants. Under ideal conditions where the two articulating surfaces are completely separated by a continuous lubricant "lm, minimum wear can be
* Corresponding author.
expected. However, in current total hip joint replacements employing ultra-high molecular weight polyethylene (UHMWPE) cups, the lubricant "lm thickness is generally thought to be much smaller than the surface roughness of the UHMWPE bearing surface, consequently leading to a mixed or boundary lubrication regime and wear of the bearing surface and generation of wear particles (Unsworth et al., 1975; Fisher and Dowson, 1991). The purpose of this study was to analyse elastohydrodynamic lubrication "lm thickness and to compare the theoretical prediction of the lubrication regime in current UHMWPE hip joint replacements with the previous experimental observation. It should be pointed out that the comparison of the predicted lubricating "lm thickness with the surface roughness of the bearing surfaces may indicate the degree of asperity contacts and hence likely severity of wear. Furthermore, such a lubrication model can be extended to other forms of hip prostheses where #uid "lm lubrication may be very
0021-9290/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 9 2 9 0 ( 0 0 ) 0 0 1 8 1 - 0
262
D. Jalali-Vahid et al. / Journal of Biomechanics 34 (2001) 261}266
R u
Nomenclature c d e ,e ,e V W X E E F ,F ,F V W X h h E h
H
H*
M p R R R
radial clearance, R !R cup thickness, R !R eccentricities modulus of elasticity for UHMWPE equivalent modulus of elasticity de"ned in Eq. (5) calculated load components de"ned in Eq. (A.6) total "lm thickness de"ned in Eq. (A.4) rigid "lm thickness de"ned in Eq. (A.2) minimum lubricant "lm thickness non-dimensional "lm thickness, h /R
Moes "lm thickness parameter de"ned in Eq. (4) Moes load parameter de"ned in Eq. (4) pressure e!ective radius for the ball-on-plane model, R"R R /c femoral head radius cup radius
important, for example metal-on-metal and ceramic-onceramic material combinations (Chan et al., 1999; Nevelos et al., 1999). Few theoretical studies of the lubrication of arti"cial hip joint replacements have been reported in the literature. In most studies of the lubrication of ball-insocket con"gurations, it has been assumed that the bearing surfaces were rigid, making the analysis particularly relevant to the lubrication of metal-onmetal or ceramic-on-ceramic hip joint replacements (Ai and Cheng, 1996; Kothari et al., 1995; Jin and Dowson, 1999). There are only a few studies reported in the literature where the elastic deformation of the bearing surfaces was considered in the lubrication analysis. Jin et al. (1997) applied the well-known Hamrock and Dowson (1978) formulae for isoviscous lubricants and semi-in"nite solids to both metal-on-metal and metal-on-UHMWPE hip joint replacements, while a number of other studies examined the squeeze-"lm lubrication problem (Sasada and Mabuchi, 1985; Mabuchi and Sasada, 1990; Wang et al., 1990). The e!ect of entraining action has been recently considered by Jalali-Vahid et al. (2000) in the development of an elastohydrodynamic lubrication analysis and a successful numerical procedure based upon the Newton}Raphson method has been developed for ball-in-socket UHMWPE hip joint replacements. The speci"c objectives of this
; w = x, y, z d e ,e ,e V W X
,h
g m x
outside radius of the cup entraining velocity for the ball-on-plane model, u"uR /2 non-dimensional velocity for the ballon-plane model de"ned in Eq. (5) applied load in the y direction non-dimensional load for the ball-onplane model de"ned in Eq. (5) coordinates used in Fig. 1a elastic deformation of UHMWPE liner de"ned in Eq. (A.3) eccentricity ratios, e /c, e /c, e /c V W X angular coordinates in the entraining and side-leakage directions, respectively half-contact angle (in degrees) viscosity of synovial #uid Poisson's ratio angular velocity
Subscripts b-p b-s
ball-on-plane ball-in-socket
study were: (a) To apply the elastohydrodynamic lubrication model developed by Jalali-Vahid et al. (2000) to a wide range of conditions experienced in current hip joint replacements with UHMWPE sockets. (b) To investigate the e!ect of important design parameters such as the radial clearance, the femoral head radius, UHMWPE thickness and modulus on the formation of lubricating "lm thickness in current UHMWPE hip joint replacements.
2. Elastohydrodynamic lubrication model In this study, an elastohydrodynamic lubrication analysis developed by Jalali-Vahid et al. (2000) for the ballin-socket con"guration shown in Fig. 1a was used to investigate the formation of the lubricant "lm in UHMWPE hip joint replacements under steady-state conditions. It has previously been shown that such a steady-state analysis can provide a reasonable estimate of the lubricating "lm thickness in the natural hip joint, even under the dynamic conditions experienced during steady walking (Jin et al., 1993). Only one load component in the y direction (w) and one angular velocity about the z}axis (u) were considered. The important design parameters to be investigated were the radii of the
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263
the numerical results thus obtained were used to curve-"t the following equation, to predict the minimum lubricant "lm thickness for the present ball-in-socket con"guration, (h ) } ;
(h ) }
"1.0191 !2.3153 #0.8151 (h ) } 90 90
;
90
!0.1133
#1, 90
r"0.9909
(2)
where (h ) } is the corresponding minimum lubricant
"lm thickness for the ball-on-plane model considered by Wang (1994) and is the half-contact angle, which can be determined from a contact mechanics analysis (Jin et al., 1999). The minimum lubricant "lm thickness for the ball-onplane model shown in Fig. 1b is (HH ) } "1.47M+\ > +\ ,
(3)
where (H* ) } and M denote the Moes "lm and load
parameters, respectively, de"ned as follows: = M" , (HH ) } "(H ) } /;
; Fig. 1. (a) Ball-in-socket model for lubrication analysis of UHMWPE arti"cial hip joint replacements. (b) Equivalent ball-on-plane model for lubrication analysis of UHMWPE arti"cial hip joint replacements.
and
(1!2l) d =" (1!l) R E "E/(1!l)
femoral head (R ) and the cup (R ), or the radial clear ance (c"R !R ), the elastic modulus (E), Poisson's ratio (l) and the thickness (d) of the UHMWPE layer. The prediction of the lubricant "lm thickness for the ball-in-socket model will be compared with the corresponding ball-on-plane model as shown in Fig. 1b considered by Wang (1994), with an equivalent radius (R) determined by the following equation: R R R" c
(1)
The main governing equations for the present lubrication model are the Reynolds equation, which governs the #uid "lm lubrication between the two bearing surfaces, and the elasticity equation. The details of these governing equations are presented in the appendix and the numerical method to solve these equations can be found in a previous study by Jalali-Vahid et al. (2000). A wide range of the main design parameters were considered and
(4)
w (1!2l) d , ;" (1!l) R E R
gu , E R (5)
The general "lm thickness formula shown in Eq. (2) was used in a parametric study to investigate the e!ect of a wide range of conditions experienced in current UHMWPE hip joint replacements on the prediction of lubricant "lm thickness. The following parameters were chosen for the lubrication analysis: The angular velocity (u), 2 rad/s The Poisson's ratio for UHMWPE (l), 0.4 Load (w), 2500 N The viscosity of synovial #uid (g), 0.005 Pa s Other important design parameters were varied to cover a wide range of conditions experienced in current UHMWPE hip joint replacements: Radial clearance (c), 75}200 lm Femoral head radius (R ), 11}16 mm UHMWPE elastic modulus (E), 250}2000 MPa UHMWPE thickness (d), 5}15 mm For example, consider a 28 mm diameter femoral head and a polyethylene cup of thickness 10 mm and elastic
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radial modulus of 1 GPa, with a radial clearance of 120 lm, the dimensionless parameters used in Eqs. (4) and (5) became
Figs. 2a,b and 3a,b show the e!ect of the femoral head radius, the radial clearance, the UHMWPE thickness and UHMWPE elastic modulus, respectively, on the predicted minimum "lm thickness from Eq. (2). It is generally noticed that an increase in the femoral head radius, a decrease in the radial clearance, an increase in the UHMWPE thickness and a decrease in the
UHMWPE elastic modulus all result in an increase in the predicted minimum "lm thickness. However, the effect of the radial clearance does not appear to be important within the range considered, unlike the "ndings for metal-on-metal hip joint replacements (Jin et al., 1997). A similar conclusion can also be reached for the UHMWPE thickness, particularly when a realistic value is used. The variation of the predicted lubricating "lm thickness is relatively small, despite of a wide range of design parameters considered. Furthermore, the magnitude of the predicted lubricating "lm thickness is much smaller than the surface roughness of the UHMWPE bearing surface which is often quoted as 1 lm (Jin et al., 1997). Therefore, a mixed or boundary lubrication is predicted for all the conditions considered in the present study and this is consistent with the previous experimental observation reported in the literature. The "lm thickness predicted in the present study is based upon the assumption of smooth bearing surfaces. However, as already mentioned above, in none of the cases considered is the predicted "lm thickness large enough to indicate that the two bearing surfaces can be separated by a #uid "lm. Therefore, the roughness of
Fig. 2. (a) E!ect of femoral head radius on the predicted minimum "lm thickness (u"2 rad/s; w"2500 N; d"10 mm; g"0.005 Pa s; E"1000 MPa; c"120 lm), (b) E!ect of radial clearance on the predicted minimum "lm thickness (u"2 rad/s; w"2500 N; d"10 mm; g"0.005 Pa s; E"1000 MPa; R "14 mm).
Fig. 3. (a) E!ect of UHMWPE thickness on the predicted minimum "lm thickness (u"2 rad/s; w"2500 N; g"0.005 Pa s; c"120 lm; E"1000 MPa; R "14 mm). (b) E!ect of elastic modulus of UHMWPE on the predicted minimum "lm thickness (u"2 rad/s; w"2500 N; d"10 mm; g"0.005 Pa s; c"120 lm; R "14 mm).
E "1.19 Gpa, M"14200.
;"1.2;10\,
="2.6;10\,
From Eq. (3), this yields a dimensionless Moes "lm thickness parameter (H* ) } of 0.285. Hence, the ball-on plane analysis indicates a minimum "lm thickness of 0.20 lm. If the half-contact angle is predicted to be 48.463 (Jin et al., 1994; 1999), the "lm thickness will be reduced by 10%, or 0.18 lm as predicted from Eq. (2).
3. Results and discussion
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the UHMWPE bearing surface should be taken into account in future analyses of the elastohydrodynamic lubrication of total replacement hip joints. Furthermore, the consequence of mixed or boundary lubrication must be considered in the selection of the design parameters. For example, although an increase in the femoral head radius leads to an increase in the e!ective radius as well as the entraining velocity and consequently lubricant "lm thickness, the e!ect of increasing sliding distance upon wear must also be taken into consideration. Furthermore, the lubricant starvation due to an increase in the contact area must be addressed. For example, although a decrease in the elastic modulus from 2000 to 250 MPa results in almost a twofold increase in the predicted minimum "lm thickness, the half-contact angle is also increased from 41.35 to 64.093, which exceeds the maximum available angle when the cup is placed anatomically in the body. It is clear that design of UHMWPE hip joint replacements from the lubrication point view requires a careful and balanced approach. It should also be pointed out that all the design parameters considered in the present study were assumed to remain constant. However, this may not be true for some parameters such as the radial clearance since the femoral head usually wears linearly into the polyethylene socket. Furthermore, the retrieved polyethylene cups are often described as polished and the e!ective surface roughness must therefore be reduced. For example, Wang et al. (1995) reported that the initial centreline average roughness of 2 lm was reduced to 0.04 lm after 2 million cycles tested in simulators, while El"ck et al. (1998) found the r.m.s. surface roughness was decreased from 0.39 to 0.07 lm in retrieved polyethylene cups. All these factors, will help to promote #uid "lm lubrication, in UHMWPE hip joint replacements and should be taken into consideration in further lubrication studies in order to provide a complete understanding of the tribology at the bearing surfaces of arti"cial hip joint replacements.
4. Conclusion A ball-in-socket elastohydrodynamic lubrication model developed in a previous study by the present authors has been applied to current UHMWPE arti"cial hip joint replacements. A general formula for the minimum lubricant "lm thickness as functions of the halfcontact angle has been developed and applied to current UHMWPE hip joint replacements under realistic conditions. It has been shown that despite an increase in the femoral head radius, a decrease in the radial clearance, an increase in UHMWPE thickness and a decrease in UHMWPE elastic modulus all lead to an increase in the predicted minimum "lm thickness, the magnitude of the "lm thickness predicted is signi"cantly less than the initial machined surface roughness of the UHMWPE bear-
265
ing surface. It is recognized that the UHMWPE surface is smoothed during articulation and `running-ina, but there is a paucity of data on the resulting steady-state surface "nish on the polyethylene, as well as the modi"ed radial clearance. It has been shown theoretically that all current forms of metal-on-polyethylene total replacement hip joints operate in the mixed lubrication regime, with load transmission being e!ected partly by asperity contact and partly by hydrodynamic action. If the roughness of the polyethylene decreases as the joint articulates, the proportion of the load carried by the asperities and the associated wear will decrease. Lubrication studies of total joint replacements can thus indicate likely severity of wear of given combinations of implant materials.
Acknowledgements This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) and DePuy International Ltd. * a Johnson & Johnson Co.
Appendix A. Governing equations for elastohydrodynamic lubrication analysis of UHMWPE hip joint replacements The governing equation for the #uid #ow between the two articulating surfaces is Reynolds equation, which takes the following form in spherical polar co-ordinates for the present lubrication model shown in Fig. 1a (Jin and Dowson, 1999); sin h
* *p * *p *h h sin h # h "6gR u sin h *h *h *
*
*
(A.1)
The rigid "lm thickness between the undeformed cup and the ball is given by h "c(1!e sin h cos !e sin h sin ) E V W
(A.2)
The elastic deformation of the bearing surfaces is mainly due to the UHMWPE acetabular cup, since the elastic modulus for UHMWPE is much lower than that for either metallic or ceramic femoral components. In the present study, an approximate method based upon the constrained column model was used to calculate the elastic deformation for the UHMWPE cup (Jin et al., 1999);
R !1 R d" p. 1 2 R E # 1!2t 1#t R R
(A.3)
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Therefore, the total "lm thickness is given as follows h"h #d. (A.4) E The boundary conditions for the hydrodynamic pressure are p"0 at the edge of the cup in the plane (xz) and the Swift}Steiber}Reynolds cavitation boundary is adopted on the outlet boundary, where; *p *p " "0. * *h
(A.5)
Furthermore, the integration of the hydrodynamic pressure distribution components must be equal to the load imposed on the joint as shown in Fig. 1a.
F "R W F "R X
F "R V
0
0
0
p sin h cos sin h dh d "0 p sin h sin sin h dh d "w
(A.6)
p cos h sin h dh d "0
The numerical solution to the above governing equations was based upon the Newton-Raphson method. The Reynolds equation was discretized using the half-point "nite di!erence scheme for the left hand terms, and the backward di!erence scheme for the right-hand term in order to balance the accuracy and stability of the numerical method (Wang, 1994). The details of the numerical procedure have been given in a previous study by the present authors (Jalali-Vahid et al., 2000).
References Ai, C., Cheng, H.S., 1996. Hydrodynamic lubrication analysis of metallic hip joint. Tribology Transactions 39 (1), 103}111. Amstutz, H.C., Campell, P., Kossovsky, N., Clark, I.C., 1991. Mechanism and clinical signi"cance of wear debris-induced osteolysis. Clinical Orthopaedics and Related Research 276, 7}17. Chan, F.W., Bobyn, J.D., Medley, J.B., Krygier, J.J., Tanzer, M., 1999. The Otto Aufranc Award * Wear and lubrication of metal-onmetal hip implants. Clinical Orthopaedics and Related Research 369, 10}24. El"ck, A.P., Hall, R.M., Unsworth, A., Pinder, I.M., 1998. Surface topography of retrieved PCA acetabular liners: proposal of a novel wear mechanism. Journal of Materials Science Letters 17 (13), 1085}1088. Fisher, J., Dowson, D., 1991. Tribology of Arti"cial Joints. Journal of Engineering in Medicine, Proceedings of the Institution of Mechanical Engineers 205(H2), 73}79. Hamrock, B.J., Dowson, D., 1978. Elastohydrodynamic lubrication of elliptical contacts for materials of low elastic modulus: 1 fully #ooded conjunction. Trans ASME, Journal of Lubrication Technology 100, 236}245.
Ingham, E., Fisher, J., 2000. Biological reactions to wear debris in total joint replacement. Journal of Engineering in Medicine, Proceedings of the Institution of Mechanical Engineers 214(H1), 21}37. Jalali-Vahid, D., Jagatia, M., Jin, Z.M., Dowson, D., 2000. Elastohydrodynamic lubrication analysis of UHMWPE hip joint replacements. In: Dowson, D. et al. (Eds.), Thinning Films and Tribological Interfaces. Elsevier Science, Amsterdam, pp. 329}339. Jasty, M., Smith, E., 1992. Wear particles of total joint replacements and their role in periprosthetic osteolysis. Current Opinions in Rheumatology 4, 204}209. Jin, Z.M., Dowson, D., 1999. A full numerical analysis of hydrodynamic lubrication in arti"cial hip joint replacements constructed from hard materials. Journal of Mechanical Engineering Science, Proceedings of the Institution of Mechanical Engineers 213, 355}370. Jin, Z.M., Dowson, D., Fisher, J., 1993. Fluid "lm lubrication in natural hip joints In: Dowson, D., Taylor, C.M., Godet, M. (Eds.), Thin Films in Tribology, Elsevier, Amsterdam, pp. 545}555. Jin, Z.M., Dowson, D., Fisher, J., 1994. A parametric analysis of the contact stress in ultra-high-molecular-weight polyethylene acetabular cups. Medical Engineering and Physics 16 (5), 398}405. Jin, Z.M., Dowson, D., Fisher, J., 1997 Analysis of #uid "lm lubrication in arti"cial hip joint replacements with surfaces of high elastic modulus. Journal of Engineering in Medicine, Proceedings of the Institution of Mechanical Engineers 211(H3), 247}256. Jin, Z.M., Heng, S.M., Ng, H.W., Auger, D.D., 1999. An axisymmetric contact model of ultra high molecular weight polyethylene cups against metallic femoral heads for arti"cial hip joint replacements. Journal of Engineering in Medicine, Proceedings of the Institution of Mechanical Engineers 213(H4), 317}327. Kothari, M., Booker, J.F., Bartel, D.L., 1995. Analysis of arti"cial hip joints as spherical bearings. Proceedings of the 21st Leeds}Lyon Symposium on Tribology 93}98, Elsevier, Amsterdam. Livermore, J., Ilstrup, Morrey, B., 1990. E!ect of femoral head size on the wear of the polyethylene acetabular component. Journal of Bone and Joint Surgery 72A, 518}528. Mabuchi, K., Sasada, T., 1990. Numerical analysis of elastohydrodynamic squeeze "lm lubrication of total hip prostheses. Wear 140, 1}16. Nevelos, J.E., Ingham, E., Doyle, C., Fisher, J., Nevelos, A.B., 1999. Analysis of retrieved alumina ceramic components from Mittelmeier total hip prostheses. Biomaterials 20 (19), 1833}1840. Sasada, T., Mabuchi, K., 1985. Elastohydrodynamic lubrication of total hip prosthesis. Proceedings of the JSLE International Tribology Conference, Tokyo, Japan, July 8}10. Unsworth, A., Dowson, D., Wright, V., 1975. The frictional behaviour of human synovial joints * Part II. arti"cial joints. Trans ASME, Journal of Lubrication Technology 97(3), 377}382. Wang, A., Stark, C., Dumbleton, J.H., 1995. Role of cyclic plastic deformation in the wear of UHMWPE acetabular cups. Wear 29, 619}626. Wang, C.T., Wang, Y.L., Chen, Q.L., Yang, M.R., 1990. Calculation of elastohydrodynamic lubrication "lm thickness for hip prostheses during normal walking. Tribology Transactions 33, 239}245. Wang, D., 1994. Elastohydrodynamic lubrication of point contacts for layers of &soft' solids and monolithic &hard' materials in the transient bouncing ball problem. Ph.D. Thesis, University of Leeds, England. Willert, H. G., Semlitsch, 1977. Reactions of the articular capsule to wear products of arti"cial joint prostheses. Journal of Biomedical Materials Research 11, 157}164.