Accuracy of methods for calculating volumetric wear from coordinate measuring machine data of retrieved metal-on-metal hip joint implants Zhen Lu and Harry A McKellop Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine 2014 228: 237 originally published online 14 February 2014 DOI: 10.1177/0954411914524188 The online version of this article can be found at: http://pih.sagepub.com/content/228/3/237

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Original Article

Accuracy of methods for calculating volumetric wear from coordinate measuring machine data of retrieved metal-on-metal hip joint implants

Proc IMechE Part H: J Engineering in Medicine 2014, Vol. 228(3) 237–249 Ó IMechE 2014 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0954411914524188 pih.sagepub.com

Zhen Lu and Harry A McKellop

Abstract This study compared the accuracy and sensitivity of several numerical methods employing spherical or plane triangles for calculating the volumetric wear of retrieved metal-on-metal hip joint implants from coordinate measuring machine measurements. Five methods, one using spherical triangles and four using plane triangles to represent the bearing and the best-fit surfaces, were assessed and compared on a perfect hemisphere model and a hemi-ellipsoid model (i.e. unworn models), computer-generated wear models and wear-tested femoral balls, with point spacings of 0.5, 1, 2 and 3 mm. The results showed that the algorithm (Method 1) employing spherical triangles to represent the bearing surface and to scale the mesh to the best-fit surfaces produced adequate accuracy for the wear volume with point spacings of 0.5, 1, 2 and 3 mm. The algorithms (Methods 2–4) using plane triangles to represent the bearing surface and to scale the mesh to the best-fit surface also produced accuracies that were comparable to that with spherical triangles. In contrast, if the bearing surface was represented with a mesh of plane triangles and the best-fit surface was taken as a smooth surface without discretization (Method 5), the algorithm produced much lower accuracy with a point spacing of 0.5 mm than Methods 1–4 with a point spacing of 3 mm.

Keywords Hip prosthesis, metal on metal, volumetric wear, coordinate measuring machine measurement, explant

Date received: 12 May 2013; accepted: 22 January 2014

Introduction Excessive wear plays an important role in the failure of metal-on-metal (MOM) hip joint replacements, particularly with some MOM designs.1–7 In prior studies, the total volumetric wear of the retrieved MOM femoral balls and acetabular cups usually has been evaluated in three steps: (1) the geometry of the bearing surface is measured using a coordinate measuring machine (CMM); (2) the resultant CMM data, that is, the ‘point cloud’, are processed to generate the wear map and (3) the volumetric wear is calculated based on the wear map. To determine the wear map, it is ideal to compare the surface geometry of the retrieved ball and cup to their preoperative condition. Unfortunately, preoperative CMM measurements are seldom available. An alternative, widely used method is to fit an analytical surface to the CMM points in the unworn areas of the implant using a least-square method, and the wear depth is taken as the distance between each CMM point in the worn area and the best-fit surface, that is, the

radial deviation of each CMM point from the best-fit surface. A number of studies assumed that the unworn area of a retrieved ball or cup was part of a nearly perfect sphere.8–16 This assumption can be justified first by the specifications of the ASTM F2033 standard, which requires that the maximum eccentricity of new metal or ceramic balls and cups should be not more than 5 mm. Second, most of the deformation of a cup during pressfit fixation is elastic and is recovered due to subsequent creep of the surrounding bone.17 For the retrieved cups whose eccentricity due to manufacturing tolerance and/

J. Vernon Luck Orthopaedic Research Center, UCLA & Orthopaedic Institute for Children Department of Orthopaedic Surgery, Los Angeles, CA, USA Corresponding author: Zhen Lu, J. Vernon Luck Orthopaedic Research Center, UCLA & Orthopaedic Institute for Children, Department of Orthopaedic Surgery, 403 West Adams Boulevard, Los Angeles, CA 90007, USA. Email: [email protected]

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Table 1. Comparison of the five numerical methods.

Method 1 Method 2 Method 3 Method 4 Method 5

Developed by

Surface discretized

Triangle

Approximation

Accuracy

Complexity

Present study Present study Published study Published study Published study

Bearing and fitting Bearing and fitting Bearing Bearing and fitting Bearing

Spherical Plane + tetrahedron Plane Plane Plane

Second order Derived from Method 1 Derived from Method 2 Variation of Method 3 First order

Highest Adequate Adequate Adequate Lowest

Most Moderate Least Moderate Moderate

or permanent deformation is not negligible, several methods have been suggested to represent a best-fit surface with eccentricity.6,7,18,19 For example, Morlock et al.18 proposed using an elliptical fitting surface for a deformed cup. Conceptually, the volumetric wear can be calculated by subtracting the volume enclosed by the bearing surface of a ball or cup from the volume enclosed by the corresponding portion of the best-fit surface, which can be calculated analytically. However, since the worn surface is a general curved surface, the wear volume must be calculated numerically. A number of studies have employed different algorithms to calculate the wear volume of MOM hip implants. McKellop et al.10 meshed the surface with quadrilaterals with a CMM point at the centre of each quadrilateral. The wear depth in each element was assumed uniform and taken from its centre point. The wear volume was calculated by multiplying the wear depths by the areas of the quadrilaterals, and the total wear volume was obtained by summing over all of the surface elements. Similarly, Lord et al.11 divided the CMM-measured surface into quadrilaterals and calculated the volumetric wear at each element by multiplying the area of the quadrilateral by the mean wear depth of the four corners. The total volumetric wear was obtained by summing the wear volume over the entire surface. Morlock et al.13 triangulated the CMMmeasured surface and calculated the volumetric wear at each triangle by multiplying the area of the triangle by the mean wear depth of the three corners. Bills et al.12 triangulated the CMM-measured surface and compared the volume enclosed by the triangles with a perfect sphere when evaluating the sensitivity of CMM point spacing on the volume calculation. The study by Underwood et al.20 provided a detailed comparison of evaluations of volumetric wear among different research institutes. Although some of the details on the triangles or quadrilaterals used with discretization in the previous studies are not available, it appears that in each case, plane triangles or quadrilaterals were used to represent the curved bearing surface. However, it is well known in numerical calculations that plane (i.e. linear) elements produce less accuracy than higher order elements with the same mesh density or they require a denser mesh than higher order elements to provide the same accuracy. The study by Bills et al.12 showed that with plane triangles, the point spacing had to be 0.3 mm or

smaller to achieve an accuracy with the error lower than 1 mm3 for calculating the volume of a 50-mm-diameter hemisphere. The effect of using second-order elements, such as spherical triangles, to represent the surface of MOM balls and cups on the accuracy of the calculation of the wear volume and the required density of the point mesh has not been determined. This study employed spherical triangles and plane triangles in discretizing the surface of femoral balls and acetabular cups. Three questions were addressed: (1) Do spherical triangles produce a more accurate estimate of the volumetric wear than plane triangles? (2) Which point mesh density is required with spherical triangles for acceptable accuracy in calculating the volumetric wear? (3) Under which conditions can plane triangles provide the wear volume with an accuracy that is comparable to that provided by spherical triangles? Five numerical methods (Table 1), one with spherical triangles and four with plane triangles, were evaluated and compared on a perfect hemisphere model and a hemi-ellipsoid model (unworn models), on computergenerated wear models and on a wear-tested model with four femoral balls. Among the five alternatives (Table 1), Method 1 with spherical triangles was developed in this study. Method 2 with plane triangles and tetrahedrons was derived from Method 1 with some simplifications. Methods 3–5 were similar to those used in previously published studies.10–13 Since a quadrilateral could always be divided into two triangles, only triangles were included in this study.

Materials and methods As shown in Figure 1, after triangulation of the CMMmeasured surface of a ball or a cup, each set of three adjacent CMM points formed a plane triangle DABC(p) or a curved triangle DABC(c). By scaling the points A, B and C to the best-fit surface, the triangulation also was performed on the best-fit surface. Similarly, the corresponding points D, E and F on the best-fit surface formed a plane triangle DDEF(p) or a curved triangle DDEF(c). The right side of Figure 1 illustrates a femoral ball with the worn surface being below the best-fit surface. For an acetabular cup, the worn surface was above the best-fit surface. Theoretically, the volumetric wear at one element is the volume enclosed by the curved triangle DABC(c) on the bearing surface and the curved triangle DDEF(c) on the best-fit surface. By connecting the curved triangles

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Figure 1. A surface meshed with triangles.

DABC(c) and DDEF(c) to the centre (Point O) of the best-fit surface, tetrahedrons tetABCO(c) and tetDEFO(c) were created, respectively. The volume enclosed by the curved triangles DABC(c) and DDEF(c) was calculated as the volumetric difference between the tetrahedrons tetABCO(c) and tetDEFO(c). The total wear volume was the summation over the worn area X Wear volume = (VtetDEFO(c) VtetABCO(c)) worn

ð1Þ

where VtetABCO(c) and VtetDEFO(c) are the volume of the tetrahedrons tetABCO(c) and tetDEFO(c), respectively. The volumes of VtetABCO(c) and VtetDEFO(c) consisted of two parts (Figure 1). Connecting the plane triangles DABC(p) and DDEF(p) to the centre (Point O) of the best-fit surface created regular tetrahedrons tetABCO(p) and tetDEFO(p), respectively. The volume of VtetABCO(c) contained the volume of tetABCO(p) and the volume enclosed by the curved triangle DABC(c) and plane triangle DABC(p). Similarly, the volume of VtetDEFO(c) contained the volume of tetDEFO(p) and the volume enclosed by the curved triangle DDEF(c) and plane triangle DDEF(p). Consequently, equation (1) could be rewritten equivalently as X Wear volume = (VtetDEFO(c) VtetABCO(c)) worn

=

X

(VtetDEFO(p) + VArcDEFðcpÞÞ

worn

VtetABCO(p) VArcABCðcpÞ

ð2Þ

where VtetDEFO(p) and VtetABCO(p) are the volume of the regular tetrahedrons tetDEFO(p) and tetABCO(p), respectively; VArcDEF(cp) is the volume enclosed by the curved triangle DDEF(c) and plane triangle DDEF(p) and VArcABC(cp) is the volume enclosed by the curved triangle DABC(c) and plane triangle DABC(p) (Figures 1 and 2). Since the worn surface is a general curved surface, there are no analytical solutions for the curved triangle DABC(c). In order to calculate the wear volume enclosed by triangles DABC(c) and DDEF(c), the curved triangle DABC(c) could be simplified as a plane triangle

Figure 2. Volume errors of VArcABC and VArcDEF due to using plane triangles to represent the bearing surface and the best-fit surface.

DABC(p) or a spherical triangle DABC(s) or any other element that made the calculations feasible. Similarly, the curved triangle DDEF(c) on the best-fit surface could be simplified as a plane triangle DDEF(p) or a spherical triangle DDEF(s) or any other element. If a spherical surface is used as the best-fit surface, spherical triangles are accurate representations of the best-fit surface. Thus, by replacing the curved triangles with spherical triangles, equation (1) can be approximated as X Wear volume = (VtetDEFO(s) VtetABCO(s)) worn

ð3Þ

and equation (2) can be approximated as X Wear volume = (VtetDEFO(s) VtetABCO(s)) worn

=

X

(VtetDEFO(p) + VArcDEF

worn

VtetABCO(p) VArcABC)

ð4Þ

where VArcDEF was the volume enclosed by the spherical triangle DDEF(s) and plane triangle DDEF(p), and VArcABC was the volume enclosed by the spherical triangle DABC(s) and plane triangle DABC(p). In this study, each of the five algorithms (Table 1) was deduced from equations (3) and (4). Methods 1 and 2 were developed in this study. While Methods 3–5 were, in principle, similar to those used by previous studies,10–13 the details of calculation procedures might not be the same as those studies, and a more mathematical basis for Methods 3–5 was given in this study.

Method 1 This method meshed both the bearing and best-fit surfaces with spherical triangles. The volumetric wear was calculated using equation (3). As noted above, since, in general, the worn area and the best-fit surface were not spherical, the radius of each spherical triangle DABC(s) or DDEF(s) was

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approximated as the average of the distances of the three points A, B and C or points D, E and F to the centre (Point O) of the best-fit surface (Figure 1). The spherical triangles DABC(s) and DDEF(s) at different locations had different radii, depending on the wear depths at the locations or the best-fit surface used. Thus, using spherical triangles was a second-order approximation for the bearing surface and best-fit surface. If a perfect spherical surface is used as the best-fit surface, spherical triangles are an accurate expression for the best-fit surface. The main steps of the calculation are as follows: 1. 2.

3.

4.

Determining the worn areas, the best-fit surface and the wear map. Triangulating the CMM points with the spherical triangles DABC(s) and generating the tetrahedrons tetABCO(s) by connecting points A, B and C and the centre (Point O) of the best-fit sphere. Scaling the CMM points to the best-fit surface and generating the corresponding spherical triangle DDEF(s) and tetrahedron tetDEFO(s). Calculating the areas of the two spherical triangles DABC(s) and DDEF(s) using Girard’s formula21 S = R2 (a + b + g p)

7. Calculating the total wear volume using equation (3).

Method 2 This method was derived from Method 1 with a simplification and meshed the bearing and best-fit surfaces with plane triangles. If it was assumed that the difference between the volume VArcABC and the volume VArcDEF (Figure 2) was negligible due to the small distance between the worn and best-fit surfaces, equation (4) could be simplified as X Wear volume = (VtetDEFO(p) + VArcDEF worn

VtetABCO(p) VArcABC) X ’ (VtetDEFO(p) VtetABCO(p)) worn

ð9Þ

ð5Þ

where S is the area of the spherical triangle, R is the radius of the sphere and a, b and g are the three angles of the spherical triangle. 5. Calculating the volume of tetrahedron tetABCO(s) using the relation SDABC(s) VtetABCO(s) = Sabc Vabco

where Sdef is the area of the entire spherical surface and Vdefo is the volume of the entire sphere. Since each spherical triangle DDEF(s) had a different radius, the values of Sdef and Vdefo varied over the elements.

Thus, the volumetric wear in one element was approximated as the volumetric difference between the two regular tetrahedrons, tetABCO(p) and tetDEFO(p), that were generated from the plane triangles DABC(p) and DDEF(p). The main steps of the calculation are as follows: 1.

ð6Þ

2.

where SDABC(s) is the area of the spherical triangle DABC(s), Sabc is the area of the entire spherical surface, VtetABCO(s) is the volume of the tetrahedron tetABCO(s) and Vabco is the volume of the entire sphere. Since each spherical triangle DABC(s) in the worn area had a different radius, the values of Sabc and Vabco varied over the elements.

3.

4. 6. Calculating the volume of tetrahedron tetDEFO(s), using the relation when a perfect spherical surface was used as the best-fit surface SDDEF(s) VtetDEFO(s) = S V

ð7Þ

where SDDEF(s) is the area of the spherical triangle DDEF(s), S is the area of the entire best-fit spherical surface, VtetDEFO(s) is the volume of the tetrahedron tetABCO(s) and V is the volume of the entire best-fit sphere. Calculating the volume of tetrahedron tetDEFO(s), using the relation when the best-fit surface was not spherical (e.g. elliptical) SDDEF(s) VtetDEFO(s) = Sdef Vdefo

ð8Þ

Determining the worn areas, the best-fit surface and the wear map. Triangulating the CMM points with plane triangles DABC(p) and generating the regular tetrahedrons tetABCO(p) by connecting points A, B, C and O. Scaling the CMM points to the best-fit surface and generating the corresponding plane triangles DDEF(p) and regular tetrahedrons tetDEFO(p) by connecting points D, E, F and O. Calculating the volume of the two regular tetrahedrons using the Cayley–Menger determinant21 0 a2 21 1 2 det a231 V = 288 2 a41 1

a212 0 a232 a242 1

a213 a223 0 a243 1

a214 a224 a234 0 1

1 1 1 1 0

ð10Þ

where V is the volume of the regular tetrahedron and aij are the lengths of the six edges of the tetrahedron with aij = aji, i = 1–4 and j = 1–4, representing the four vertices of a tetrahedron. 5. Calculating the total wear volume using equation (9).

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Method 3

4.

This method was a simplification of Method 2 (equation (9)) and similar to those used in the previous studies.10,11,13 The wear volume calculated from equation (9) or Method 2 was the volume of a general wedge wABCDEF enclosed by the plane triangles DABC(p) and DDEF(p), whose two triangles were oblique (Figure 1). If it was assumed that the variations of the edges AD, BE and CF, or the wear depths at points A, B and C, were very small, and the difference between the areas of the triangles DABC(p) and DDEF(p) were negligible due to the small distance between the worn and best-fit surfaces, the volume of the wedge wABCDEF or the wear volume could be approximated as X Wear volume = (WearDepth(average) worn ð11Þ 3 areaDABC(p)) where WearDepth(average) is the average of the wear depths of the three CMM points A, B and C and areaDABC(p) is the area of the plane triangle DABC(p). Equation (11) calculated the volume of a regular wedge, whose two triangles DABC(p) and DDEF(p) were approximated as parallel. The main steps of the calculation are as follows: 1. 2. 3.

4.

Determining the worn areas, the best-fit sphere and the wear map. Triangulating the CMM points with plane triangles DABC(p). Calculating the average of the wear depth at points A, B and C and the area of the plane triangle DABC(p). Calculating the volumetric wear using equation (11).

Method 4

The main calculation steps are as follows:

2. 3.

Method 5 The volumetric wear was obtained by subtracting the volume enclosed by the bearing surface that was represented by the plane triangles DABC(p) from the corresponding partial volume of the best-fit sphere. This method was equivalent to ignoring the term VArcABC in equation (4), that is, the volume enclosed by the curved triangle DABC(c) and plane triangle DABC(p) (Figure 2), such that X Wear volume = (VtetDEFO(p) + VArcDEF VtetABCO(p)) = Partial volume of best-fit sphere X VtetABCO(p) ð13Þ

Method 5 was similar to that used by Bills et al.12 to determine the optimal point spacing for CMM measurements of MOM hip implants. The main calculation steps are as follows: 1. 2.

3.

4.

5.

This method was a variation of Method 3. The area of the plane triangle DABC(p) on the bearing surface was replaced with the area of the plane triangle DDEF(p) on the best-fit surface in equation (11) of Method 3, that is X Wear volume = (WearDepth(average) worn ð12Þ 3areaDDEF(p))

1.

5.

Determining the worn areas, the best-fit sphere and the wear map. Triangulating the CMM points with plane triangles DABC(p). Scaling CMM point to the best-fit surface and generating the corresponding plane triangles DDEF(p).

Calculating the average of the wear depth at points A, B and C and the area of the plane triangles DDEF(p). Calculating the volumetric wear using equation (12).

6.

Determining the worn areas and the best-fit sphere. Triangulating the CMM points with plane triangles DABC(p) and generating the regular four-point tetrahedrons tetABCO(p) by connecting points A, B, C and O. Calculating the volume of the regular tetrahedron using the Cayley–Menger determinant21 (equation (10)). Calculating the total volume enclosed by the plane triangle mesh by summing the volume of all of the tetrahedrons. Calculating the corresponding partial volume of the best-fit sphere using the analytical solution. Calculating the volumetric wear using equation (13).

All of the calculations were carried out with usercompiled programs using MATLAB (MathWorks Inc., Natick, MA, USA) and Visual Basic (Microsoft Corp., Redmond, WA, USA). Three models were generated to evaluate and compare the five methods: (1) a perfect hemisphere model and a hemi-ellipsoid model, representing unworn surfaces without and with eccentricity, respectively; (2) computer-generated wear models of a ball and cup, whose wear volume had analytical solutions and (3) four wear-tested femoral balls, whose wear volume was calculated from the weight loss that was measured during the wear test. The perfect hemisphere model, hemi-ellipsoid model and computer-generated wear models could exclude the common uncertainties in the calculation of volumetric

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wear due to the unknown eccentricity of the surface, the accuracy of the CMM and the fitting process of a perfect analytical surface that involved individual experience.12,18,20 In addition, the models’ analytical solutions were available for comparison. These features were appropriate for this study to focus on assessing the effect of using spherical or plane triangles on the accuracy of the calculation of wear volume. The perfect hemisphere model had a diameter of 46 mm. The hemi-ellipsoid model simulated that a cup with a diameter of 46 mm had a radial deformation of 20.1 mm at its equator with the formula x2 y2 z2 + + =1 2 2 (22:9) (23:08) (23:02)2

where x- and y-axes were in the plane of cup’s equator and z-axis was along the centre to the apex of the cup. The two models were generated using MATLAB program and meshed with point spacings of 0.3, 0.5, 1, 2 or 3 mm. The data were processed using usercompiled programs that were the same as those used for processing the CMM data. These unworn models served to (1) examine whether the five methods satisfied the initial condition that the wear volume should be zero for an unworn surface, or when the best-fit surface was the same as bearing surface, and (2) compare the volume of the hemisphere and the hemi-ellipsoid calculated with the spherical triangles (Method 1) and the plane triangles (Method 5) to the analytical solution at different point spacings. For the hemisphere model, Method 1 was expected to produce the same volume of a perfect sphere as the analytical solution, except for minor rounding errors due to discretization, and its results should not be affected by the point spacing. For the ellipsoid model, Method 1 was expected to produce negligible deviation in volume from the analytical solution, and its results should be affected by the point spacing slightly. In contrast, the accuracy with Method 5 was expected to be strongly affected by the point spacing, due to using plane triangles to represent the curved surface. The computer-generated wear models (Figure 3) were created using the Patran finite element program (MSC Software Corp., Santa Ana, CA, USA). The femoral ball was modelled as a hemisphere with a nominal diameter of 46 mm. A worn area around the pole was created using two concentric hemispheres with diameters of 46 and 46.4 mm. The larger hemisphere was moved down along the axis that passed through the centre and was perpendicular to the plane going through the equator, such that the two hemispheres intersected each other. Each hemisphere was broken into two parts at the intersection: a spherical cap and a lower partial sphere. The wear model was generated by combining the spherical cap of the 46.4 mm hemisphere and the lower partial sphere of the 46 mm hemisphere. By moving the 46.4 mm hemisphere down 0.3 and 0.22 mm, wear models with maximum wear depths of 0.1 and 0.02 mm were produced, respectively.

Figure 3. Computer-generated wear models of a femoral ball and an acetabular cup with a maximum wear depth of 0.1 mm.

Analytical solutions for the volumetric ‘wear’ were calculated from the volume of the spherical cap of the 46.4 mm hemisphere and the lower partial portion of the hemisphere. Wear models for an acetabular cup with a maximum wear depth of 0.1 or 0.02 mm near the edge were created using the same method (Figure 3). Thus, total of four wear models were generated: two for the ball and two for the cup. Each model was meshed with point spacings of 0.5, 1, 2 and 3 mm. The five methods were used to calculate the wear volume for each condition and compared with the analytical solutions. The actual wear specimens were four 44-mm-diameter femoral balls of cobalt–chromium–molybdenum alloy (Wright Medical Technology, Inc., Arlington, TN, USA) that had been previously wear-tested against acetabular cups of the same material for a total of 5 million cycles on a hip joint wear simulator. Before and after the test, the balls were weighed on a Mettler AT621 Microbalance (Mettler-Toledo Inc., Columbus, OH, USA) with a resolution of 60.1 mg. The volumetric wear corresponding to the loss in weight was

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Table 2. Volume of a perfect hemisphere and a hemi-ellipsoid calculated with Methods 1 and 5. Point spacing (mm)

Method 1

Method 5 3

Hemisphere 0.3 0.5 1 2 3 Hemi-ellipsoid 0.3 0.5 1 2 3

3

Volume (mm )

Error (mm )

Volume (mm3)

Error (mm3)

25,482.51 25,482.51 25,482.51 25,482.51 25,482.51

0.00 0.00 0.00 0.00 0.00

25,480.91 25,478.12 25,465.06 25,413.89 25,325.87

1.60 4.39 17.45 68.62 156.63

25,482.10 25,482.10 25,482.09 25,482.07 25,482.02

0.00 0.00 0.01 0.03 0.08

25,480.31 25,477.28 25,462.96 25,408.37 25,315.77

1.79 4.82 19.14 73.73 166.33

The analytical volume is 25,482.51 and 25,482.10 mm3 for the hemisphere and hemi-ellipsoid, respectively.

calculated using the nominal density of cobalt– chromium alloy (8.28 g/cm3). The dimensions of the balls were measured using a Mitutoyo Legex 322 CMM (Mitutoyo America Corp., Aurora, IL) fitted with a Renishaw SP25M scanning probe and a 3-mm ruby stylus. The CMM had a resolution of 0.01 mm and an accuracy of 0.8 + 2L/1000 mm. The balls were measured with point spacings of 0.5, 1, 2 and 3 mm. A perfect sphere was fit to the unworn areas with a leastsquare algorithm, using the Geomagic Qualify 12 program (Geomagic, Morrisville, NC, USA). The depth in the worn area was taken as the distance from each CMM point inside the wear area to the surface of the best-fit sphere. Once the distribution of the wear depth was determined, the wear volume was calculated using each of the five methods, with the same user-compiled programs as noted above.

0.3 mm. Similarly, the volume of the hemi-ellipsoid calculated using Method 5 had an error of 4.82 mm3 compared to the analytical solution with a point spacing of 0.5 mm and an error of 166.33 mm3 with a point spacing of 3 mm (Table 2). The error volume was still 1.79 mm3 when the point spacing was reduced to 0.3 mm. It was observed during the trial calculation with Method 1 that the volume error exhibited slight increase unexpectedly as the point spacing was reduced. The problem stemmed from the fact that initially, the constant p was represented to seven decimal places. However, calculations with the smaller point spacings, or more spherical triangles, required using p more times, which, in turn, resulted in more accumulated rounding errors. The problem was solved by setting p to 14 decimal places.

Results

Computer-generated wear models

Perfect hemisphere and hemi-ellipsoid models

As shown in Tables 3 and 4, Method 1 produced the most accurate results among the five methods with the fine and coarse point meshes, and Methods 2, 3 and 4 also produced results with accuracies comparable to Method 1. For example, for the ball model (Table 3 and Figure 4) with a maximum wear depth of 0.1 mm and an analytical wear volume of 55.88 mm3, the minimum volume error of 0.02 mm3 occurred with Method 1 with a point spacing of 0.5 mm, and the maximum volume error of 0.97 mm3 occurred with Method 2 with a point spacing of 3 mm. Similarly, for the model with a maximum wear depth of 0.02 mm and an analytical wear volume of 3.05 mm3, the minimum volume error of 0.00 mm3 occurred with Methods 1, 2 and 4 with a point spacing of 0.5 mm, and the maximum volume error of 0.15 mm3 occurred with Method 2 with a point spacing of 3 mm. The cup model showed similar results (Table 4 and Figure 5). For example, for the model with a maximum wear depth of 0.1 mm and an analytical wear volume of

As expected, for the perfect hemisphere and hemiellipsoid models (unworn models), Methods 1–4 satisfied the initial condition, giving zero wear volume. Table 2 shows that for the volume of the perfect hemisphere calculated using Method 1, the maximum deviations from the analytical solutions or volume error were 0.00 mm3 with point spacings of 0.5, 1, 2 and 3 mm. For the volume of the hemi-ellipsoid calculated using Method 1, the maximum deviations from the analytical solutions or volume error were 0.00 and 0.08 mm3 with point spacings of 0.5 and 3 mm, respectively. In contrast, Method 5 was unable to produce zero wear volume on the unworn surface, even at the smallest point spacing. The volume of the hemisphere calculated using Method 5 had an error of 4.39 mm3 compared to the analytical solution at a point spacing of 0.5 mm and an error of 156.63 mm3 at a point spacing of 3 mm (Table 2). The error volume was still 1.60 mm3 when the point spacing was reduced to

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Table 3. Wear volume of the computer-generated wear model of a ball calculated with Methods 1–5. Depth (mm)

Point spacing (mm) 0.5

1

Point spacing (mm) 2

3

0.5

Wear volume (mm3) Method 1 0.1 0.02 Method 2 0.1 0.02 Method 3 0.1 0.02 Method 4 0.1 0.02 Method 5 0.1 0.02

1

2

3

Volume error (mm3)

55.86 3.04

55.81 3.03

55.64 2.99

55.28 2.92

20.02 0.00

20.06 20.01

20.24 20.06

20.60 20.13

55.85 3.04

55.77 3.03

55.50 2.98

54.91 2.90

20.03 0.00

20.10 20.02

20.38 20.07

20.97 20.15

55.69 3.04

55.63 3.03

55.40 2.98

54.92 2.91

20.18 20.01

20.25 20.02

20.47 20.06

20.96 20.14

56.02 3.05

55.95 3.03

55.72 2.99

55.23 2.91

0.14 0.00

0.08 20.01

20.15 20.06

20.64 20.14

60.25 7.49

73.29 21.05

128.17 76.94

227.45 158.55

4.37 4.44

17.41 18.01

72.29 73.89

171.57 155.50

The analytical volume is 55.88 and 3.05 mm3 for the maximum wear depth of 0.1 and 0.02 mm, respectively.

Figure 4. Wear volume of the computer-generated wear model of a ball calculated with Methods 1–5 with point spacings of 0.5, 1, 2 and 3 mm. The maximum wear depth is 0.1 mm, and the analytical volume is 55.88 mm3.

Figure 5. Wear volume of the computer-generated wear model of a cup calculated with Methods 1–5 with point spacings of 0.5, 1, 2 and 3 mm. The maximum wear depth is 0.1 mm, and the analytical volume is 20.50 mm3.

20.50 mm3, the minimum volume error of 0.01 mm3 occurred with Methods 1 and 2 with a point spacing of 0.5 mm, and the maximum volume error of 0.38 mm3 occurred with Methods 2 and 4 with a point spacing of 3 mm. Similarly, for a maximum wear depth of 0.02 mm and an analytical wear volume of 1.03 mm3, the minimum volume error of 0.00 mm3 occurred with Methods 1–4 with a point spacing of 0.5 mm, and the maximum volume error of 0.07 mm3 occurred with Methods 1–4 with a point spacing of 3 mm. In contrast, Method 5 produced much greater volume errors at the four-point densities than Methods 1– 4 (Tables 3 and 4 and Figures 4 and 5). For example, for the ball model with a maximum wear depth of 0.02 mm and an analytical wear volume of 3.05 mm3, Method 5 produced volume errors of 4.44 and

155.5 mm3 with point spacings of 0.5 and 3 mm, respectively. Similarly, for the cup model with a maximum wear depth of 0.02 mm and an analytical wear volume of 1.03 mm3, Method 5 produced volume errors of 4.46 and 170.01 mm3 with point spacings of 0.5 and 3 mm, respectively. Similarly, Method 5 produced much greater volume errors with a point spacing of 0.5 mm than did Methods 1–4 with a point spacing of 3 mm (Tables 3 and 4 and Figures 4 and 5). A point spacing of 0.2 mm had to be used for Method 5 to reduce the volume error to magnitudes comparable to those produced by Methods 1–4 at a point spacing of 3 mm. The results (Tables 5 and 6) confirmed the assumption made with Methods 2–4 that there was negligible difference between the volume VArcABC enclosed by

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Table 4. Wear volume of the computer-generated wear model of a cup calculated with Methods 1–5. Depth (mm)

Point spacing (mm) 0.5

1

Point spacing (mm) 2

3

0.5

Wear volume (mm3) Method 1 0.1 0.02 Method 2 0.1 0.02 Method 3 0.1 0.02 Method 4 0.1 0.02 Method 5 0.1 0.02

1

2

3

Volume error (mm3)

20.49 1.02

20.47 1.02

20.37 1.00

20.23 0.96

20.01 0.00

20.03 20.01

20.14 20.03

20.27 20.07

20.49 1.02

20.46 1.02

20.31 0.99

20.12 0.95

20.01 0.00

20.04 20.01

20.19 20.03

20.38 20.07

20.55 1.02

20.52 1.02

20.39 0.99

20.23 0.96

0.05 0.00

0.02 20.01

20.11 20.03

20.27 20.07

20.43 1.02

20.40 1.02

20.28 0.99

20.12 0.96

20.07 0.00

20.10 20.01

20.23 20.03

20.38 20.07

16.07 23.43

2.52 216.69

251.22 270.52

2142.42 2168.98

24.44 24.46

217.98 217.72

271.72 271.54

2162.93 2170.01

The analytical volume is 20.50 and 1.03 mm3 for the maximum wear depth of 0.1 and 0.02 mm, respectively.

Table 5. Total volume of VArcABC and VArcDEF over the entire surface due to using plane triangles to represent the curved surface and their difference with the point spacing from 0.5 to 3 mm calculated from the computer-generated wear model of a ball with Methods 1 and 2. Depth (mm)

Point spacing (mm)

P

P VArcDEF (mm3)

P (VArcABC 2 VArcDEF) (mm3)

0.1

0.5 1 2 3 0.5 1 2 3

4.39 17.47 72.53 172.17 4.45 18.02 73.95 155.62

4.40 17.51 72.67 172.53 4.45 18.02 73.96 155.64

0.01 0.04 0.14 0.37 0.00 0.00 0.01 0.02

0.02

VArcABC (mm3)

Table 6. Total volume of VArcABC and VArcDEF over the entire surface due to using plane triangles to represent the curved surface and their difference with the point spacing from 0.5 to 3 mm calculated from the computer-generated wear model of a cup with Methods 1 and 2. Depth (mm)

Point spacing (mm)

P

P VArcDEF (mm3)

P (VArcABC 2 VArcDEF) (mm3)

0.1

0.5 1 2 3 0.5 1 2 3

4.43 17.95 71.59 162.66 4.46 17.71 71.51 169.95

4.43 17.94 71.53 162.55 4.46 17.71 71.51 169.94

0.00 20.01 20.06 20.11 0.00 0.00 0.00 20.01

0.02

VArcABC (mm3)

the plane triangle DABC(p) and the spherical triangle DABC(s) on the bearing surface and the volume VArcDEF enclosed by the plane triangle DDEF(p) and the spherical triangle DDEF(s) on the best-fit surface (equation (9)). For example, for the ball model with a maximum wear depth of 0.1 mm and an analytical wear volume of 55.88 mm3 (Table 5), while the total volume of VArcABC over the entire hemisphere surface increased from 4.39 to 172.17 mm3 as the point spacing

increased from 0.5 to 3 mm, the total volume of VArcDEF over the entire fitting surface also increased from 4.40 to 172.53 mm3. This produced differences of 0.01, 0.04, 0.14 and 0.37 mm3 between the total volumes of VArcABC and VArcDEF for the point spacing of 0.5, 1, 2 and 3 mm, respectively. Similarly, for the cup model (Table 6) with a maximum wear depth of 0.02 mm and an analytical volumetric wear of 1.03 mm3, the difference was 0.00 mm3 for point

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Table 7. Comparison of wear volumes measured from the wear test and calculated with Methods 1–5 with point spacings of 0.5, 1, 2 and 3 mm. Specimen

Point spacing (mm) 0.5

1

Point spacing (mm) 2

3

Wear volume (mm3) Ball 1

Ball 2

Ball 3

Ball 4

Method 1 Method 2 Method 3 Method 4 Method 5 Method 1 Method 2 Method 3 Method 4 Method 5 Method 1 Method 2 Method 3 Method 4 Method 5 Method 1 Method 2 Method 3 Method 4 Method 5

1.72 1.72 1.72 1.73 6.24 1.25 1.25 1.25 1.25 5.95 1.66 1.66 1.65 1.65 6.27 1.40 1.40 1.37 1.37 5.67

1.78 1.78 1.79 1.79 21.87 1.28 1.28 1.28 1.28 21.43 1.66 1.66 1.62 1.62 22.62 1.41 1.41 1.43 1.43 23.61

0.5

1

2

3

20.10 20.10 20.11 20.11 88.44 0.09 0.09 0.05 0.05 94.48 0.02 0.01 0.00 0.00 94.41 0.08 0.07 0.11 0.11 95.87

20.02 20.02 20.04 20.04 208.89 20.01 20.02 20.03 20.03 209.42 0.10 0.09 0.09 0.10 209.19 0.06 0.05 0.08 0.08 210.29

Volume error (mm3) 1.72 1.72 1.71 1.71 90.26 1.28 1.28 1.24 1.24 95.67 1.64 1.63 1.62 1.62 96.03 1.33 1.32 1.36 1.36 97.12

1.80 1.80 1.78 1.78 210.71 1.18 1.17 1.16 1.16 210.61 1.72 1.71 1.71 1.72 210.81 1.31 1.30 1.33 1.33 211.54

20.10 20.10 20.10 20.09 4.42 0.06 0.06 0.06 0.06 4.76 0.04 0.04 0.03 0.03 4.65 0.15 0.15 0.12 0.12 4.42

20.04 20.04 20.03 20.03 20.05 0.09 0.09 0.09 0.09 20.24 0.04 0.04 0.00 0.00 21.00 0.16 0.16 0.18 0.18 22.36

The wear volume converted from the weight loss measured at 5 million cycles was Ball 1 = 1.82 mm3, Ball 2 = 1.19 mm3, Ball 3 = 1.62 mm3 and Ball 4 = 1.25 mm3.

spacings of 0.5, 1 and 2 mm and was 0.01 mm3 for a point spacing of 3 mm. The large errors with Method 5 stemmed from ignoring the volume VArcABC, instead of the difference between the volumes of VArcABC and VArcDEF in equation (4). For example, the total volumes of VArcABC for the ball model with a maximum wear depth of 0.1 mm were 4.39 and 172.17 mm3 at point spacings of 0.5 and 3 mm, respectively, producing volume errors of 4.39 and 172.17 mm3, respectively (Table 5). Similarly, the total of volumes VArcABC for the cup model with a maximum wear depth of 0.1 mm were 4.43 and 162.66 mm3 at point spacings of 0.5 and 3 mm, respectively, producing volume errors of 4.43 and 162.66 mm3, respectively (Table 6). Due to ignoring the volume VArcABC, Method 5 overestimated the volumetric wear for the ball model (Table 3) and underestimated it for the cup model (Table 4). It even produced volume gain for the cup model shown as negative wear volume in Table 4 when the volume error VArcABC exceeded the wear volume. As shown in Tables 3 and 4, while Method 3 had higher volume errors than Method 4 for a ball and Method 4 had higher volume errors than Method 3 for a cup, the difference between the two methods was negligible.

Wear-tested model The pretest measurements showed that Balls 1–3 had negligible apparent ‘wear’ volumes due to the initial eccentricities, whereas Ball 4 had an initial ‘wear’

Figure 6. Wear volume measured from the wear test and calculated with Methods 1–5 with point spacings of 0.5, 1, 2 and 3 mm. The wear volume of Ball 1 converted from the weight loss measured at 5 million cycles was 1.82 mm3.

volume of 2.77 mm3. Therefore, the wear volume calculated post test based on the CMM data for Ball 4 was corrected by 2.77 mm3. As shown in Table 7 and Figure 6, Methods 1–4 produced results with adequate accuracy at the four CMM point densities for the four balls tested on a hip joint wear simulator for 5 million cycles. For example, the maximum volume errors were 0.11, 0.09, 0.10 and 0.18 mm3 compared with the weight-loss-converted wear volumes of 1.82, 1.19, 1.62 and 1.25 mm3 for the wear-tested Balls 1–4, respectively. In contrast, Method 5 produced ‘wear’ volumes that were several times greater than the weightloss-converted wear (Table 7 and Figure 6), even at the

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point spacing of only 0.5 mm, with volume errors of 4.42, 4.76, 4.65 and 4.42 mm3 at a point spacing of 0.5 mm, compared with the weight-loss-converted wear volumes of 1.82, 1.19, 1.62 and 1.25 mm3 for the weartested Balls 1–4, respectively.

Discussion This study evaluated the effects of representing the surfaces of MOM femoral balls and acetabular cups with meshes of spherical or plane triangles on the accuracy of the estimates of the volumetric wear based on the CMM measurements. Using different point spacings, five numerical methods were assessed, with one using spherical triangles and four using plane triangles, on a perfect hemisphere model and a hemi-ellipsoid model, on computer-generated wear models and on four femoral balls that were wear-tested. Method 1, which employed spherical triangles to represent the bearing and best-fit surfaces, produced the most accurate results in calculating the volumetric wear. The accuracy with this method was adequate with point spacings of 0.5, 1, 2 and 3 mm. When the bearing surface was represented by plane triangles, and the plane triangle mesh was scaled to the best-fit surface (Methods 2–4), the algorithms also produced accuracies that were comparable to that with spherical triangles. In contrast, if the bearing surface was represented with a mesh of plane triangles and the best-fit surface was taken as a smooth surface without discretization, such as a sphere or ellipsoid (Method 5), the accuracy was low even with the point spacing of only 0.5 mm. The advantages of using spherical triangles were evident from the results calculated by Method 1 on the perfect hemisphere model, the hemi-ellipsoid model, the computer-generated wear models and the weartested femoral balls. One main simplification with this method was to use piece-wise spherical patches (triangles) to represent the curved bearing surface and bestfit surface, which required that the radius was constant within one spherical triangle and varied among the triangles. In Method 1, the average radius of the three points of a spherical triangle was assigned as the constant radius for that triangle. The results showed that this simplification had negligible influence on the accuracy of the results (Tables 2–4) and that the method produced an accurate calculation of the wear volume as the point spacing varied from 0.5 to 3 mm. For the best-fit sphere, this simplification was not applied, and the volume calculated with the spherical triangle was the same as the analytical solution. While Method 2 represented the best-fit and bearing surfaces with meshes of plane triangles, it was derived from Method 1 with a simplification. The difference between Methods 1 and 2 was that the volumetric difference of (VArcDEF 2 VArcABC) (Figures 1 and 2) contained in equation (4) was assumed negligible and removed in Method 2 (equation (9)). In fact, the

volumes of VArcDEF and VArcABC (Figures 1 and 2) were the volume errors due to using plane triangles to discretize the best-fit and bearing surfaces, respectively. Removing both VArcDEF and VArcABC was equivalent to using the two volume errors to cancel out each other and altered the spherical triangles on the best-fit and bearing surfaces to plane triangles. The results (Tables 5 and 6) of the computer-generated wear models confirmed this assumption. Thus, the volume errors due to using plane triangles to discretize the curved surface were effectively minimized in Method 2 by removing both volumes of VArcDEF and VArcABC or by representing both the bearing and the best-fit surfaces with meshes of plane triangles. Methods 3 and 4 were essentially the same as Method 2 in terms of minimizing the volume error due to using plane triangles to represent a curved surface. However, the two methods made some additional simplifications for calculating the volume of the wedge wABCDEF enclosed by the plane triangles DABC(p) on the bearing surface and DDEF(p) on the best-fit surface (Figure 1). The wedge wABCDEF calculated with Method 2 was a general wedge, whose two plane triangles DABC(p) and DDEF(p) were oblique or the wear depth varied over the element. With Methods 3 and 4, the wedge was simplified as a regular wedge, whose two plane triangles DABC(p) and DDEF(p) were parallel. This was equivalent to assigning a constant wear depth over the plane triangle DABC(p) or DDEF(p), which was the average of the wear depth of the three points A, B and C on the triangles in Methods 3 and 4. Therefore, although Method 3 did not include the calculation steps to mesh the best-fit surface with plane triangles explicitly, it was virtually equivalent to meshing both the best-fit and the bearing surfaces with plane triangles as with Method 2, which was derived from meshing both surfaces with spherical triangles with the simplification. This made Method 3 the simplest algorithm among Methods 1–4. The algorithms reported by some previous studies10,11,13 were similar to Methods 3 and 4. In those studies, the CMM-measured surfaces of femoral balls or acetabular cups were discretized with plane triangles or quadrilaterals. The wear volume for each element was calculated by multiplying the area of the triangle or the quadrilateral by the average wear depth over the element. Satisfactory results were reported for point spacings from 1 to 5 mm and were consistent with the wear volume calculated on the computer-generated wear model and the wear-tested femoral balls using Methods 3 and 4 with varying point spacing in this study. Apparently, the reported acceptable results that were calculated from a relative coarse mesh10,11,13 and the results of Methods 2–4 in this study were inconsistent with the study by Bills et al.,12 which showed that the volume error was 4.57 mm3 at a point spacing about 0.5 mm when using plane triangles to mesh a 50-mmdiameter hemisphere. However, this inconsistency was anticipated. The method used by Bills et al., or Method

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5 in this study, represented the worst situation in discretizing a curved surface. With this method, while the best-fit sphere was treated as a sphere, the bearing surfaces of the balls or cups were represented by plane triangles. Thus, the volume of VArcABC was removed from equation (4) and became the volume error. This was a significant difference from Methods 2 to 4, where the volumetric difference (VArcDEF 2 VArcABC) was removed from equation (4) and became the volume error. As shown in Tables 5 and 6, while the total volume of (VArcDEF 2 VArcABC) was minimal and changed little with varying point spacing, the total volume of VArcABC was very large with a coarse point mesh and still a few cubic millimetres with a point spacing of 0.5 mm. Therefore, Method 5 required a very fine point mesh to achieve acceptable accuracy in the calculation of wear volume, which was consistent with the study by Bills et al.12 An appropriate numerical method is expected to at least satisfy the initial and boundary conditions. In the calculation of wear volume, the initial condition is that when there is no wear on the balls or cups, the calculated wear volume should be zero. The boundary condition is that when wear occurs, the wear volume in the unworn area should be zero. Apparently, Methods 1–4 satisfy both of these conditions, whereas Method 5 satisfies neither. While the best-fit surface of the computer-generated wear models and the wear-tested model in this study was assumed as a perfect spherical surface, apparently, Methods 1–4 did not require the best-fit surface to be spherical, elliptical or any specific shape. As shown in the calculation steps, the algorithms of Methods 1–4 were established within a single triangle element rather than the overall shape of a cup or ball. Therefore, there are no restrictions for Methods 1–4 in selecting the best-fit surface to represent the unworn areas of the bearing surface with either negligible eccentricity or significant eccentricity. However, since the calculation of volumetric wear was based on the distributions of the wear depth, its accuracy was affected by the accuracy of the wear map, which depended on selecting an appropriate best-fit surface. Previous studies8–15 have shown that a perfect best-fit spherical surface could produce wear distributions with acceptable accuracy in many cases, where the initial eccentricity due to manufacturing tolerance and the eccentricity due to permanent deformation were small. For a ball or cup with relative large eccentricity, several methods have been suggested by previous studies6,7,18,19 to take into account the eccentricity. In this study, prior to the wear test, one wear-tested ball had an apparent ‘wear’ volume of 2.77 mm3 due to its eccentricity, and its actual wear volume was corrected by the pretest volume. However, the initial eccentricity of balls and cups that are retrieved from patients typically is unknown. One approach to reduce the uncertainty due to the initial eccentricity is to perform the calculation over the worn area only.10,16 Although Methods 1–4

could be performed over the entire surface, in this study, they were programmed to do the volume calculation only in the worn area for this purpose. However, this approach might be effective only when the initial eccentricity is randomly distributed over the surface. If the eccentricity is primarily in a single location and overlaps the worn area, it would be very difficult to distinguish the apparent ‘wear’ volume due to the initial eccentricity from the actual wear volume.

Conclusion Volumetric wear is an important factor for investigating failure of MOM hip implants. Five numerical methods with spherical or plane triangles for calculating volumetric wear were evaluated for their accuracy and sensitivity to point spacing. The results are as follows: 1.

2.

3.

4.

5.

Method 1 with spherical triangles to mesh the bearing and best-fit surfaces produced the most accurate results among the five methods. The wear volume calculated with Method 1 had an acceptable accuracy from point spacing of 0.5– 3 mm. Methods 2–4 with plane triangles were actually derived from Method 1 with simplifications. The discretization errors due to using plane triangles to represent curved surfaces were minimized by scaling the mesh from the bearing surface to the bestfit surface. As a result, Methods 2–4 produced the wear volume with an accuracy comparable to that of Method 1. Method 5 discretized the bearing surface with plane triangles and kept the best-fit surface as a smooth surface without discretization. It calculated the wear volume with much lower accuracy with a point spacing of 0.5 mm than Methods 1–4 with a point spacing of 3 mm. Among Methods 1–4, Method 1 was the most complicated in calculation due to inclusion of second-order approximation or spherical triangles. Method 3 is the simplest.

Declaration of conflicting interests The authors declare that there is no conflict of interest. Funding This research was supported by the Los Angeles Orthopaedic Hospital Foundation. References 1. Catelas I and Wimmer MA. New insights into wear and biological effects of metal-on-metal bearings. J Bone Joint Surg Am 2011; 93(Suppl. 2): 76–83. 2. Ebramzadeh E, Campbell PA, Takamura KM, et al. Failure modes of 433 metal-on-metal hip implants: how,

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