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Estimation of the wear volume after total hip replacement
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Medical Engineering & Physics 30 (2008) 373–379
Estimation of the wear volume after total hip replacement A simple access to geometrical concepts T. Ilchmann a,∗,1 , M. Reimold b,1 , W. M¨uller-Schauenburg b a
Department of Orthopaedics, Kantonsspital Rheinstr. 26, CH-4410 Liestal, Switzerland b Department of Nuclear Medicine, University of T¨ ubingen, Germany
Received 1 August 2006; received in revised form 21 March 2007; accepted 5 April 2007
Abstract Various formulas have been proposed to calculate the volume of prosthetic wear from the penetration depth of the head as assessed on plain radiographs, based on idealized, three-dimensional geometrical models of a prosthetic hip. However, for most published formulas no (or no simple) derivation is available and not all of them are correct. We describe a simple geometrical model that allows for transparent derivation of equations for various components of prosthetic wear volume and compare the calculated volumes with those obtained from published equations. These components are: (1) a right generalized cylinder resulting from a linear shift of a half spherical part of the prosthetic head into the hemispherical cup, (2) an additional wedge that is “cut” from the cup and (3) the wear from an optional additional cylindrical portion of the cup. We emphasize that calculation of a three-dimensional wear volume from linear penetration depth should be based on a geometrical concept that is transparent and simple enough for clinical research, such as the one presented. The incorrect formula of Kabo et al. should be completely abandoned. © 2007 IPEM. Published by Elsevier Ltd. All rights reserved. Keywords: Hip; Arthroplasty; Wear; Volume; Calculation
1. Introduction Wear is seen as a reason for osteolysis and loosening after hip replacement. The harm of the wear depends on the size and the volume of the particles and the overall volume of debris is of importance. There is general agreement that the volume of wear debris should be kept small [1–4] and efforts were made to reduce the volume of wear by introducing ultra high crosslinked polyethylene, metal on metal or ceramic on ceramic bearings [5]. Linear wear and the direction of wear can be assessed from plain radiographs [6–9]. As the biological effect of wear products is a function of the overall volume of deliberated particles, there is an interest to estimate the wear volume from measurements obtained from radiographs. Several formulas have been proposed for calculating the volumetric ∗ 1
Corresponding author. Tel.: +41 61 9252236; fax: +41 61 9252808. E-mail address: [email protected] (T. Ilchmann). These authors contributed equally.
wear starting from Charnley et al. [10]. The established formulas of Kabo et al. [11] and Hall et al. [12] are incorrect with errors up to 45% [13,14]. Derbyshire [13] recognized this error and illustrates the (wrong) geometrical concept that underlies Kabo’s equation. These illustrations permit to reproduce Kabo’s wrong wear formula surprisingly fast. We developed an unified geometric way to calculate a wide spectrum of wear volumes from clinical radiographs, without any complicated integration procedure, taking the direction of wear and the anteversion of the cup into account.
2. Theory 2.1. Wear perpendicular to the entrance plane (β = 90◦ ) The simplest calculation of the volume of wear is the circular cylinder approach by Charnley et al. [10]. The femoral head is assumed to bore a linear trace perpendicular (β = 90◦ ,
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Fig. 1. Circular cylinder [9]. Volume calculation for wear perpendicular to the entrance plane (β = 90◦ ). Note that both dotted areas are equal in size, therefore the wear volume (curved dotted area) corresponds to a cylinder (rectangular dotted area) with a circular cross-section (Fig. 4) and the height d (Vcyl = πr2 d).
Fig. 1) to the entrance plane of the cup. The cross-section of that trace is a full circle (Fig. 4). Since each point of the displaced head has the same displacement length d, the product of the circular cross-section and the displacement d represents the wear volume. This product also describes the volume of a right circular cylinder. This correspondence of a worn curved volume to a simple rectified form like such a circular cylinder will be used several times in the following text to obtain wear volumes without mathematical integration. 2.2. Wear parallel to the entrance plane (β = 0◦ ) For β = 0◦ , half of the prosthetic head is in contact with the cup (Fig. 2). The cross-section of the trace is therefore a half-circle (Fig. 4) and the related part of the wear volume corresponds to the product of this half-
circle and the displacement d, represented by a half-circular cylinder. Some cups additionally have a cylindrical portion [12], which must be considered when calculating the total volume of wear. The volume that is worn from this additional portion will be denoted as Vadd (Figs. 2, 3 and 5, grey rectangle). Whereas Vadd is zero for wear perpendicular to the entrance plane, it is maximal for parallel wear (β = 0◦ ). For simplifying the calculation of Vadd we assume that the equator of the head cuts straight down towards and perpendicular to the entrance plane instead of a curved cut with the shape of the head. By this assumption this additional wear gets a rectangular cross-section with the height f multiplied by the wear displacement d. Since the spatial extension of the worn volume in the third dimension (i.e. perpendicular to the rectangular cross-section) is 2r (Fig. 5), the additional wear for β = 0◦ amounts to Vadd = 2rfd. Again we have calculated a curved wear volume by rectification (here: to a cuboid). 2.3. Oblique wear (0◦ < β < 90◦ ) Most worn cups show an oblique direction of wear (0◦ < β < 90◦ ). Three partial volumes contribute to the total wear volume: (1) The main part of the wear volume (Figs. 3 and 4, dotted area) originates from a shift of the supraequatorial part of the prosthetic head (i.e. of a half-sphere). We denote this volume Vcyl as, mathematically speaking, it corresponds to a generalized right cylinder (not necessarily a circular cylinder). Its cross-section consists of a half-circle and a half-ellipse. The small axis of the ellipse depends on the direction of wear and is given by r sin(β) (Fig. 4). The volume of this deformed cylinder is the product of its cross-section and its height: Vcyl = (1⁄2πr2 + 1⁄2πr2 sinβ)d. (2) Below this wear volume, an additional wedge is worn by the infraequatorial part of the head. As with the wear of the additional socket described above, we assume a
Fig. 2. Half cylinder. As in Fig. 1, the wear volume for β = 0◦ can be calculated by rectification. For β = 0◦ the wear volume (dotted curved area) corresponds to a half-circular cylinder (rectangular dotted area) with the volume being the product of the half-circular cross-section (Fig. 4) and the penetration depth d (Vcyl = 1⁄2πr2 d). In addition, an optional cylindrical part of height f was attached to the spherical cup. For β = 0◦ the additional wear volume is maximal (Vadd = 2rdf, grey rectangle), for β = 90◦ it is zero.
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Fig. 3. Three partial volumes contribute to the total wear volume: (1) A cylindrical volume (dotted area). Its cross-section (Fig. 4) depends on β. The volume is calculated as a product of the cross-section and the displacement d: Vcyl = (1⁄2πr2 + 1⁄2πr2 sinβ) × d. (2) The wedge, corresponding to the hatched triangle EFE (area = 1⁄2d cosβ × d sinβ, for derivation of the corresponding volume, Fig. 5). (3) The contribution of an optional cylindrical part (grey rectangle, area = d cosβ × f, Fig. 5).
Fig. 4. Cross-sections of the cylindrical wear Vcyl (dotted area). The geometry can be compared with the illuminated moon as seen from a distant observer. For β = 90◦ (Fig. 1) it is a circle (“full moon”), for β = 0◦ (Fig. 2) a half-circle (“half moon”). For 0◦ < β < 90◦ (Fig. 3) it corresponds to the “3/4 moon”. It consists of a half-circle (radius r, area 1⁄2πr2 ) and a half-ellipse (small axis r sinβ, area 1⁄2πr2 sinβ).
straight instead of a curved delineation (Figs. 3 and 5, line FE ) and obtain a triangular cross-section (Figs. 1 and 3, triangle EFE ) which is maximal for β = 45◦ and zero for β = 0◦ and β = 90◦ . The volume corresponding to the triangle EFE again is calculated by rectification (Fig. 5): Vwedge = 1⁄2d sinβ × d cosβ × 2r (Table 1). (3) For cups with an additional cylindrical portion, the additional wear is calculated as described above (β = 0◦ ), with d being replaced by d cosβ (Figs. 3 and 5): Vadd = fd cosβ × 2r.
2.4. Estimation of β from a plain radiograph Measures of length refer to true patient space and direct measurements on the radiograph have to be corrected for radiological magnification. If the cup has no anteversion, the penetration angle βrx on the radiograph reflects the (true) penetration angle β between the entrance plane and the direction of penetration. In case
Fig. 5. Wedge and additional cylinder. Because of the parallel movement of all parts of the tracing half-sphere, all (hatched) triangle cross-sections of the wedge are identical. The same is assumed for the wear of an additional socket (grey rectangle). The corresponding volumes are calculated by multiplication of the area of the cross-section (Fig. 3) with the spatial extent in the third dimension (2r): Vwedge = 1⁄2d cosβ × d sinβ × 2r and Vadd = fd cosβ × 2r.
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Table 1 Summary of wear volumes derived in this paper Volume β = 90◦
1. Circular cylinder for 2. Half cylinder for β = 0◦ 3. Generalized cylinder for any β 4. Wedge 5. Additional cylindrical part 6. Total volume
Figures
Principle
Formula
Figs. 1 and 4 Figs. 2 and 4 Figs. 3 and 4 Figs. 3 and 5 Figs. 2, 3 and 5
Circle × d Half-circle × d (Half-circle + half-ellipse) × d Triangle EFE × 2r Square × 2r
Vcyl = πr2 d Vcyl = 1⁄2πr2 d Vcyl = 1⁄2πr2 d + 1⁄2dr2 π sinβ = 1⁄2πr2 d(1 + sinβ) Vwedge = 1⁄2d cosβ × d sinβ × 2r = 1⁄2rd2 sin(2β) Vadd = fd cosβ × 2r
Vcyl + Vwedge + Vadd
V = 1⁄2πr2 d(1 + sinβ) + 1⁄2rd2 sin(2β) + fd cosβ × 2r
of anteversion, the angle of anteversion α can be calculated from the ratio of the long versus the short axis of the elliptic projection of the entrance plane [15] or other methods [16,17]. Then, the true β can be calculated from α and βrx , with βrx being the apparent penetration angle on the radiograph, that is, the angle between the direction of maximal wear and the long axis of the ellipse: β = arcsin (sin βrx cosα) (Fig. 6).
Fig. 6. Estimation of the penetration angle from planar radiographs in case of anteversion α > 0. The ellipse represents the entrance plane of the cup (see Fig. 8). We now imagine the cup and the vector of penetration being rotated by the amount of α around the long-axis ‘a’ so that the former ellipse coincides with ‘a’. Accordingly, the penetration vector d rotates out of the film plane and the component that is parallel to the short axis of the ellipse ‘b’, i.e. db = d sinβrx , appears shortened by a factor cosα, corresponding with a shift of its tip from D0 to D1 (dotted arrow). In a second step, we imagine the cup being rotated within the entrance plane, so that d moves back into the film plane from D1 to D2 , restoring its original length and preserving its component db = d sinβrx cosα, which is now identified as d sinβ, where β is the “true” penetration angle used in this paper: sinβ = sinβrx cosα.
3. Results All equations derived in this paper are listed in Table 1. The main part of the wear volume corresponds to a generalized right cylinder (Vcyl ), whose volume depends on the direction of wear β (Fig. 7). For sake of simplicity of Fig. 7 we normalized Vcyl (β) by πr2 d, resulting in a normalized volume = 1 at β = 90◦ (the “reference point” reflecting Charnley’s circular cylinder) and 0.5 at β = 0◦ . Unlike Vcyl , which is maximal at β = 90◦ , Vwedge has its maximum at 45◦ . As Vwedge is proportional to rd2 but not to r2 d, we took a fixed ratio d/r to depict Vwedge in the same diagram as Vcyl . In Fig. 6, Vwedge is plotted for d/r = 0.25 and for d/r = 0.1. Without accounting for the optional socket Vadd , the total wear corresponds to the sum of Vcyl and Vwedge (dashed lines above Vcyl , Fig. 7). Due to the contribution of the wedge this volume has its maximum slightly below 90◦ .
Fig. 7. Angular dependency of calculated wear volumes. The angular dependency of calculated wear volumes V(β) is plotted as a ratio to Charnley’s circular cylinder (wear at 90◦ = πr2 d). The volumes of the wedge as well as the wear volume according to Kabo depend on the ratio d/r (d = penetration depth, r = radius of the head). They are displayed for the samples d/r = 0.1 and d/r = 0.25. For these samples, the sum of Vcyl and Vwedge (dashed lines) lies mostly between Vcyl (thick dotted line) and VHashimoto (thick dotted line), characterizing VHashimoto as an acceptable approach, in contrast to VKabo which is mostly either close to 0.5 or undefined, corresponding to the true volume of wear only at β = 0◦ (Table 2).
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4. Discussion The circular cylinder approach of Charnley et al. [10] has been used to estimate the volume of prosthetic wear from biplane measurements from radiographs for a long time. It has been shown that the direction of wear is primarily cranial and rather lateral than medial [19,20]. Therefore, the wear volume is systematically overestimated with Charnley’s formula. The formula of Kabo et al. [11] was the first taking the direction of wear in account. Hall et al. [12] noted that the formula of Kabo et al. underestimated the true wear volume. But they accepted the (wrong) concept of Kabo et al. and added a mathematically incorrect term that was meant to account for a cylindrical bore in the entrance plane of some cups. Derbyshire [13] showed that Kabo et al. used a wrong geometrical concept of wear and recognized that Hall’s “correction term” [12] was wrong (Table 2). Furthermore, he investigated the effect from a discrepancy of radii between head and cup, which is not scope of this paper. A numerical integration to solve the problem for various shapes of the cup and directions of wear was proposed [13]. It permits calculation of wear volumes even in case of exceptional shapes of the entrance plane of the studied cups but it appears less transparent for the clinician, performing wear studies on the long term of larger patient populations. Hashimoto and co-workers have proposed [18] has proposed another equation for the wear volume (Table 2). We do not see an explanation why this geometry should reflect wear, as the cut of the head in the cup is a curved line for β > 0◦ and why wear should be estimated by an equation with a term that depends linearly on β. The vicinity of Hashimoto’s volume to Vcyl + Vwedge in Fig. 7 suggests to calculate for each β a ratio d/r where they would coincide. In general Hashimoto’s volume slightly overestimates wear for realistic low ratios d/r. Kosak et al. [21] developed a new formula for calculating wear volumes and found that their formula came closer to the fluid displacement measurements of retrieved cups as
Fig. 8. Example of a worn cup after 18 years. The penetration d was 1.8 mm, the direction of penetration in the film plane βrx was 66◦ and the anteversion 25◦ , as measured with the EBRA method [8]. According to Charnley’s equation [10], the calculated volume was 1448 mm3 . Taking the direction of wear and the anteversion into account, the true direction of wear β was 56◦ and the calculated volume 1323 mm3 according to Kosak et al. [21], 1387 mm3 with the equation of Hashimoto and co-workers [18] and 1347 mm3 with the formula presented here, respectively. The cup was retrieved due to aseptic loosening. The directly measured volume of wear was 1356 mm3 (sum of volume segments as a product of surface segments and their corresponding radial deviation; Mitutoyo® , Strato 9106, accuracy 1.7 m).
We compared the wear according to our equations with that from the equations of Kabo et al. [11] and Hashimoto and co-workers [18]. Hashimoto’s equation yields slightly larger values than those given by V = Vcyl + Vwedge for d/r = 0.25. Kabo’s equation yields values that are in a wide range of β either fairly close to 0.5 or not defined (Fig. 7). The effect of anteversion has to be taken into account before calculating volumetric wear. It leads to a diminishing of the calculated volume of wear (Figs. 6 and 8). Table 2 Available formulas for the calculation of the wear volume Charnley et al. [10]a
V = πr2 d
Kabo et al. [11]b
VKabo = πr2 d − r2 d cos−1
Hall et al.
[12]c
V = VKabo + 2
d tan(β) −
r
r2 tan2 (β)
− d2 +
Vcyl = r 2d (π + π sin β) 2 VHashimoto = r 2d (π + 2β + sin(2β)) 2 V = r 2d π + π sin β + dr sin(2β) = Vcyl + Vwedge
This paperg
V =
−
r3 3 tan(β)
1−
d 2 tan2 (β) r2
3/2
−1
2rfd cos β
Kosak et al. [21]d Hashimoto and co-workers [18]e Kadaba and Ramakrishnanf
r2 d 2
r tan(β)
π + π sin β +
d r
sin(2β) + 2rfd cos β = Vcyl + Vwedge + Vadd
Only correct for wear perpendicular to the entrance plane (β = 90◦ ). Only correct for wear parallel to the entrance plane (β = 0◦ ). c The (wrong) concept of Kabo was accepted, an additional cylindrical part added. In the published equation, the factor cosβ was put erroneously in the denominator (obviously wrong, e.g. for β = 90◦ : the formula predicts infinite instead of zero wear). d Linear wear, derived by integration. The provided formula is identical to our V cyl (Figs. 1–3). The wedge is not taken into account. e Hashimoto’s volume corresponds to V ◦ ◦ ◦ ◦ cyl for β = 0 and β = 90 . For 0 < β < 90 , Hashimoto’s equation yields slightly higher values [13] (Figs. 7 and 8). f Presented on a meeting, not published and no derivation available. It corresponds to our V cyl + Vwedge . g Total wear as proposed in this paper. a
b
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compared to calculations with the Kabo formula. Their formula was derived with an integration and corresponds to our Vcyl for any β (Table 1) but disregards the wedge, which is maximal for β = 45◦ , the most frequent direction of wear (Fig. 7). The formula of Kadaba and Ramakrishnan (congress communication, Table 2) corresponds to our formula Vcyl + Vwedge but it is not published and there is no derivation available. There are considerable errors in measuring linear head penetration on a.p. radiographs [8]. Sophisticated methods have improved the measurement accuracy for the linear wear [6,22–24]. There is little information about the accuracy of assessment of the direction of wear. For wear less than 1 mm, the error of direction measurement might be relatively high, in case of higher wear, the direction might be determined more accurate [14]. The orientation of the cup has to be taken into account for any volume calculations out of measurements from plain radiographs. Disregarding, the effect of anteversion will lead to a systematic overestimation of the volumetric wear. Most of the wear occurs in the film plane and wear perpendicular to the film plane (sagittal wear) is difficult to assess [25]. Two-dimensional measurements might be sufficient in clinical studies [26]. In case of measurement of sagittal wear, further calculations are needed to get the amount and direction of the true wear. A curved direction of wear [27], creep penetration and plastic deformation processes [28], variations in the geometry of the entrance plane of the cup like asymmetric bores [12] and discrepancies in head and cup diameters [13] cannot be detected on plain radiographs and might add further errors in the assessment of wear volume. These errors have to be taken in account for an accurate description of the wearing in process of a cup and for linear wear below 1 mm by sophisticated methods of measurement [29] and are not scope of this paper. The accuracy of wear volume estimations from twodimensional measurements should mainly depend on the methods of measurement but not on the mathematical equation used. Thus we believe that our simplifying assumption of a straight cut of the femoral head from the wedge and the optional cylinder is acceptable. The main contributing volumes of wear, the cylindrical part, the wedge and the part of the optional additional cylinder are easily calculated by converting curved wear volumes into simple rectified geometric objects. Another issue is the clinical benefit from transforming a linear penetration depth into an estimated three-dimensional volume. It can only be investigated with data from a large sample of patients and analysis of retrieved cups. Concerning clinical relevance, we do not provide new equations which have not been at least mentioned somewhere in a conference presentation, but we provide a set of transparent and simple straight forward derivations for formula previously deduced only by integration or lacking a derivation; yet we strongly emphasize that if one chooses to calculate volumetric wear, the employed geometrical concept should be mathematically
correct, transparent and as simple as possible to be suitable for clinical research. Acknowledgements We thank Hans Wingstrand, Department of Orthopedics, University Lund, Sweden, for the hint, that Kabo’s formula might be wrong and for pointing out that our equation for Vcyl + Vwedge has already been presented by Kadaba and Ramakrishnan on a meeting. Furthermore, we thank Martin L¨uem, Department of Orthopedics, Kantonsspital Liestal, Switzerland for providing the example and analysing the data.
References [1] Dumbleton JH, Manley MT, Edidin AA. A literature review of the association between wear rate and osteolysis in total hip arthroplasty. J Arthroplasty 2002;17:649–61. [2] Harris WH. Wear and periprosthetic osteolysis: the problem. Clin Orthop 2001;393:66–70. [3] Maloney WJ, Galante JO, Anderson M, Goldberg V, Harris WH, Jacobs J, et al. Fixation, polyethylene wear, and pelvic osteolysis in primary total hip replacement. Clin Orthop 1999;369:157–64. [4] Oparaugo PC, Clarke IC, Malchau H, Herberts P. Correlation of wear debris-induced osteolysis and revision with volumetric wear-rates of polyethylene: a survey of 8 reports in the literature. Acta Orthop Scand 2001;72:22–8. [5] Heisel C, Silva M, Schmalzried TP. Bearing surface options for total hip replacement in young patients. J Bone Joint Surg Am 2003;85A:1366–79. [6] Devane PA, Bourne RB, Rorabeck CH, Hardie RM, Horne JG. Measurement of polyethylene wear in metal-backed acetabular cups. I. Three-dimensional technique. Clin Orthop 1995;319:303–16. [7] Griffith MJ, Seidenstein MK, Williams D, Charnley J. Socket wear in Charnley low friction arthroplasty of the hip. Clin Orthop 1978;137:37–47. [8] Ilchmann T, Mjoberg B, Wingstrand H. Measurement accuracy in acetabular cup wear. Three retrospective methods compared with Roentgen stereophotogrammetry. J Arthroplasty 1995;10:636–42. [9] Martell JM, Leopold SS, Liu X. The effect of joint loading on acetabular wear measurement in total hip arthroplasty. J Arthroplasty 2000;15:512–8. [10] Charnley J, Kamangar A, Longfield MD. The optimum size of prosthetic heads in relation to the wear of plastic sockets in total replacement of the hip. Med Biol Eng 1969;7:31–9. [11] Kabo JM, Gebhard JS, Loren G, Amstutz HC. In vivo wear of polyethylene acetabular components. J Bone Joint Surg Br 1993;75:254–8. [12] Hall RM, Unsworth A, Craig PS, Hardaker C, Siney P, Wroblewski BM. Measurement of wear in retrieved acetabular sockets. Proc Inst Mech Eng 1995;209:233–42. [13] Derbyshire B. The estimation of acetabular cup wear volume from twodimensional measurements: a comprehensive analysis. Proc Inst Mech Eng 1998;212:281–91. [14] Mizoue T, Yamamoto K, Masaoka T, Imakiire A, Akagi M, Clarke IC. Validation of acetabular cup wear volume based on direct and two-dimensional measurements: hip simulator analysis. J Orthop Sci 2003;8:491–9. [15] Seradge H, Nagle KR, Miller RJ. Analysis of version in the acetabular cup. Clin Orthop Relat Res 1982;166:152–7. [16] Hassan DM, Johnston GH, Dust WN, Watson LG, Cassidy D. Radiographic calculation of anteversion in acetabular prostheses. J Arthroplasty 1995;10:369–72.
T. Ilchmann et al. / Medical Engineering & Physics 30 (2008) 373–379 [17] Krismer M, Bauer R, Tschupik J, Mayrhofer P. EBRA: a method to measure migration of acetabular components. J Biomech 1995;28:1225–36. [18] Sychterz CJ, Moon KH, Hashimoto Y, Terefenko KM, Engh Jr CA, Bauer TW. Wear of polyethylene cups in total hip arthroplasty. A study of specimens retrieved post mortem. J Bone Joint Surg Am 1996;78:1193–200. [19] Hall RM, Siney P, Wroblewski BM, Unsworth A. Observations on the direction of wear in Charnley sockets retrieved at revision. J Bone Joint Surg Br 1998;80:1067–72. [20] Wroblewski BM. Direction and rate of socket wear in Charnley lowfriction arthroplasty. J Bone Joint Surg Br 1985;67:757–61. [21] Kosak R, Antolic V, Pavlovcic V, Kralj-Iglic V, Milosev I, Vidmar G, et al. Polyethylene wear in total hip prostheses: the influence of direction of linear wear on volumetric wear determined from radiographic data. Skeletal Radiol 2003;32:679–86. [22] Ilchmann T. Radiographic assessment of cup migration and wear after hip replacement. Acta Orthop Scand Suppl 1997;276:1–26. [23] Martell JM, Berdia S. Determination of polyethylene wear in total hip replacements with use of digital radiographs. J Bone Joint Surg Am 1997;79:1635–41.
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[24] Shaver SM, Brown TD, Hillis SL, Callaghan JJ. Digital edge-detection measurement of polyethylene wear after total hip arthroplasty. J Bone Joint Surg Am 1997;79:690–700. [25] Wan Z, Boutary M, Dorr LD. Precision and limitation of measuring two-dimensional wear on clinical radiographs. Clin Orthop Relat Res 2006;449:267–74. [26] Sychterz CJ, Yang AM, McAuley JP, Engh CA. Two-dimensional versus three-dimensional radiographic measurements of polyethylene wear. Clin Orthop Relat Res 1999;365:117–23. [27] Yamaguchi M, Bauer TW, Hashimoto Y. Three-dimensional analysis of multiple wear vectors in retrieved acetabular cups. J Bone Joint Surg Am 1997;79:1539–44. [28] Isaac GH, Dowson D, Wroblewski BM. An investigation into the origins of time-dependent variation in penetration rates with Charnley acetabular cups—wear, creep or degradation? Proc Inst Mech Eng 1996;210:209–16. [29] Bragdon CR, Martell JM, Greene ME, Estok DM, Thanner J, Karrholm J, et al. Comparison of femoral head penetration using RSA and the Martell method. Clin Orthop Relat Res 2006;448:52–7.
Medical Engineering & Physics 30 (2008) 373–379
Estimation of the wear volume after total hip replacement A simple access to geometrical concepts T. Ilchmann a,∗,1 , M. Reimold b,1 , W. M¨uller-Schauenburg b a
Department of Orthopaedics, Kantonsspital Rheinstr. 26, CH-4410 Liestal, Switzerland b Department of Nuclear Medicine, University of T¨ ubingen, Germany
Received 1 August 2006; received in revised form 21 March 2007; accepted 5 April 2007
Abstract Various formulas have been proposed to calculate the volume of prosthetic wear from the penetration depth of the head as assessed on plain radiographs, based on idealized, three-dimensional geometrical models of a prosthetic hip. However, for most published formulas no (or no simple) derivation is available and not all of them are correct. We describe a simple geometrical model that allows for transparent derivation of equations for various components of prosthetic wear volume and compare the calculated volumes with those obtained from published equations. These components are: (1) a right generalized cylinder resulting from a linear shift of a half spherical part of the prosthetic head into the hemispherical cup, (2) an additional wedge that is “cut” from the cup and (3) the wear from an optional additional cylindrical portion of the cup. We emphasize that calculation of a three-dimensional wear volume from linear penetration depth should be based on a geometrical concept that is transparent and simple enough for clinical research, such as the one presented. The incorrect formula of Kabo et al. should be completely abandoned. © 2007 IPEM. Published by Elsevier Ltd. All rights reserved. Keywords: Hip; Arthroplasty; Wear; Volume; Calculation
1. Introduction Wear is seen as a reason for osteolysis and loosening after hip replacement. The harm of the wear depends on the size and the volume of the particles and the overall volume of debris is of importance. There is general agreement that the volume of wear debris should be kept small [1–4] and efforts were made to reduce the volume of wear by introducing ultra high crosslinked polyethylene, metal on metal or ceramic on ceramic bearings [5]. Linear wear and the direction of wear can be assessed from plain radiographs [6–9]. As the biological effect of wear products is a function of the overall volume of deliberated particles, there is an interest to estimate the wear volume from measurements obtained from radiographs. Several formulas have been proposed for calculating the volumetric ∗ 1
Corresponding author. Tel.: +41 61 9252236; fax: +41 61 9252808. E-mail address: [email protected] (T. Ilchmann). These authors contributed equally.
wear starting from Charnley et al. [10]. The established formulas of Kabo et al. [11] and Hall et al. [12] are incorrect with errors up to 45% [13,14]. Derbyshire [13] recognized this error and illustrates the (wrong) geometrical concept that underlies Kabo’s equation. These illustrations permit to reproduce Kabo’s wrong wear formula surprisingly fast. We developed an unified geometric way to calculate a wide spectrum of wear volumes from clinical radiographs, without any complicated integration procedure, taking the direction of wear and the anteversion of the cup into account.
2. Theory 2.1. Wear perpendicular to the entrance plane (β = 90◦ ) The simplest calculation of the volume of wear is the circular cylinder approach by Charnley et al. [10]. The femoral head is assumed to bore a linear trace perpendicular (β = 90◦ ,
1350-4533/$ – see front matter © 2007 IPEM. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.medengphy.2007.04.003
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Fig. 1. Circular cylinder [9]. Volume calculation for wear perpendicular to the entrance plane (β = 90◦ ). Note that both dotted areas are equal in size, therefore the wear volume (curved dotted area) corresponds to a cylinder (rectangular dotted area) with a circular cross-section (Fig. 4) and the height d (Vcyl = πr2 d).
Fig. 1) to the entrance plane of the cup. The cross-section of that trace is a full circle (Fig. 4). Since each point of the displaced head has the same displacement length d, the product of the circular cross-section and the displacement d represents the wear volume. This product also describes the volume of a right circular cylinder. This correspondence of a worn curved volume to a simple rectified form like such a circular cylinder will be used several times in the following text to obtain wear volumes without mathematical integration. 2.2. Wear parallel to the entrance plane (β = 0◦ ) For β = 0◦ , half of the prosthetic head is in contact with the cup (Fig. 2). The cross-section of the trace is therefore a half-circle (Fig. 4) and the related part of the wear volume corresponds to the product of this half-
circle and the displacement d, represented by a half-circular cylinder. Some cups additionally have a cylindrical portion [12], which must be considered when calculating the total volume of wear. The volume that is worn from this additional portion will be denoted as Vadd (Figs. 2, 3 and 5, grey rectangle). Whereas Vadd is zero for wear perpendicular to the entrance plane, it is maximal for parallel wear (β = 0◦ ). For simplifying the calculation of Vadd we assume that the equator of the head cuts straight down towards and perpendicular to the entrance plane instead of a curved cut with the shape of the head. By this assumption this additional wear gets a rectangular cross-section with the height f multiplied by the wear displacement d. Since the spatial extension of the worn volume in the third dimension (i.e. perpendicular to the rectangular cross-section) is 2r (Fig. 5), the additional wear for β = 0◦ amounts to Vadd = 2rfd. Again we have calculated a curved wear volume by rectification (here: to a cuboid). 2.3. Oblique wear (0◦ < β < 90◦ ) Most worn cups show an oblique direction of wear (0◦ < β < 90◦ ). Three partial volumes contribute to the total wear volume: (1) The main part of the wear volume (Figs. 3 and 4, dotted area) originates from a shift of the supraequatorial part of the prosthetic head (i.e. of a half-sphere). We denote this volume Vcyl as, mathematically speaking, it corresponds to a generalized right cylinder (not necessarily a circular cylinder). Its cross-section consists of a half-circle and a half-ellipse. The small axis of the ellipse depends on the direction of wear and is given by r sin(β) (Fig. 4). The volume of this deformed cylinder is the product of its cross-section and its height: Vcyl = (1⁄2πr2 + 1⁄2πr2 sinβ)d. (2) Below this wear volume, an additional wedge is worn by the infraequatorial part of the head. As with the wear of the additional socket described above, we assume a
Fig. 2. Half cylinder. As in Fig. 1, the wear volume for β = 0◦ can be calculated by rectification. For β = 0◦ the wear volume (dotted curved area) corresponds to a half-circular cylinder (rectangular dotted area) with the volume being the product of the half-circular cross-section (Fig. 4) and the penetration depth d (Vcyl = 1⁄2πr2 d). In addition, an optional cylindrical part of height f was attached to the spherical cup. For β = 0◦ the additional wear volume is maximal (Vadd = 2rdf, grey rectangle), for β = 90◦ it is zero.
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Fig. 3. Three partial volumes contribute to the total wear volume: (1) A cylindrical volume (dotted area). Its cross-section (Fig. 4) depends on β. The volume is calculated as a product of the cross-section and the displacement d: Vcyl = (1⁄2πr2 + 1⁄2πr2 sinβ) × d. (2) The wedge, corresponding to the hatched triangle EFE (area = 1⁄2d cosβ × d sinβ, for derivation of the corresponding volume, Fig. 5). (3) The contribution of an optional cylindrical part (grey rectangle, area = d cosβ × f, Fig. 5).
Fig. 4. Cross-sections of the cylindrical wear Vcyl (dotted area). The geometry can be compared with the illuminated moon as seen from a distant observer. For β = 90◦ (Fig. 1) it is a circle (“full moon”), for β = 0◦ (Fig. 2) a half-circle (“half moon”). For 0◦ < β < 90◦ (Fig. 3) it corresponds to the “3/4 moon”. It consists of a half-circle (radius r, area 1⁄2πr2 ) and a half-ellipse (small axis r sinβ, area 1⁄2πr2 sinβ).
straight instead of a curved delineation (Figs. 3 and 5, line FE ) and obtain a triangular cross-section (Figs. 1 and 3, triangle EFE ) which is maximal for β = 45◦ and zero for β = 0◦ and β = 90◦ . The volume corresponding to the triangle EFE again is calculated by rectification (Fig. 5): Vwedge = 1⁄2d sinβ × d cosβ × 2r (Table 1). (3) For cups with an additional cylindrical portion, the additional wear is calculated as described above (β = 0◦ ), with d being replaced by d cosβ (Figs. 3 and 5): Vadd = fd cosβ × 2r.
2.4. Estimation of β from a plain radiograph Measures of length refer to true patient space and direct measurements on the radiograph have to be corrected for radiological magnification. If the cup has no anteversion, the penetration angle βrx on the radiograph reflects the (true) penetration angle β between the entrance plane and the direction of penetration. In case
Fig. 5. Wedge and additional cylinder. Because of the parallel movement of all parts of the tracing half-sphere, all (hatched) triangle cross-sections of the wedge are identical. The same is assumed for the wear of an additional socket (grey rectangle). The corresponding volumes are calculated by multiplication of the area of the cross-section (Fig. 3) with the spatial extent in the third dimension (2r): Vwedge = 1⁄2d cosβ × d sinβ × 2r and Vadd = fd cosβ × 2r.
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Table 1 Summary of wear volumes derived in this paper Volume β = 90◦
1. Circular cylinder for 2. Half cylinder for β = 0◦ 3. Generalized cylinder for any β 4. Wedge 5. Additional cylindrical part 6. Total volume
Figures
Principle
Formula
Figs. 1 and 4 Figs. 2 and 4 Figs. 3 and 4 Figs. 3 and 5 Figs. 2, 3 and 5
Circle × d Half-circle × d (Half-circle + half-ellipse) × d Triangle EFE × 2r Square × 2r
Vcyl = πr2 d Vcyl = 1⁄2πr2 d Vcyl = 1⁄2πr2 d + 1⁄2dr2 π sinβ = 1⁄2πr2 d(1 + sinβ) Vwedge = 1⁄2d cosβ × d sinβ × 2r = 1⁄2rd2 sin(2β) Vadd = fd cosβ × 2r
Vcyl + Vwedge + Vadd
V = 1⁄2πr2 d(1 + sinβ) + 1⁄2rd2 sin(2β) + fd cosβ × 2r
of anteversion, the angle of anteversion α can be calculated from the ratio of the long versus the short axis of the elliptic projection of the entrance plane [15] or other methods [16,17]. Then, the true β can be calculated from α and βrx , with βrx being the apparent penetration angle on the radiograph, that is, the angle between the direction of maximal wear and the long axis of the ellipse: β = arcsin (sin βrx cosα) (Fig. 6).
Fig. 6. Estimation of the penetration angle from planar radiographs in case of anteversion α > 0. The ellipse represents the entrance plane of the cup (see Fig. 8). We now imagine the cup and the vector of penetration being rotated by the amount of α around the long-axis ‘a’ so that the former ellipse coincides with ‘a’. Accordingly, the penetration vector d rotates out of the film plane and the component that is parallel to the short axis of the ellipse ‘b’, i.e. db = d sinβrx , appears shortened by a factor cosα, corresponding with a shift of its tip from D0 to D1 (dotted arrow). In a second step, we imagine the cup being rotated within the entrance plane, so that d moves back into the film plane from D1 to D2 , restoring its original length and preserving its component db = d sinβrx cosα, which is now identified as d sinβ, where β is the “true” penetration angle used in this paper: sinβ = sinβrx cosα.
3. Results All equations derived in this paper are listed in Table 1. The main part of the wear volume corresponds to a generalized right cylinder (Vcyl ), whose volume depends on the direction of wear β (Fig. 7). For sake of simplicity of Fig. 7 we normalized Vcyl (β) by πr2 d, resulting in a normalized volume = 1 at β = 90◦ (the “reference point” reflecting Charnley’s circular cylinder) and 0.5 at β = 0◦ . Unlike Vcyl , which is maximal at β = 90◦ , Vwedge has its maximum at 45◦ . As Vwedge is proportional to rd2 but not to r2 d, we took a fixed ratio d/r to depict Vwedge in the same diagram as Vcyl . In Fig. 6, Vwedge is plotted for d/r = 0.25 and for d/r = 0.1. Without accounting for the optional socket Vadd , the total wear corresponds to the sum of Vcyl and Vwedge (dashed lines above Vcyl , Fig. 7). Due to the contribution of the wedge this volume has its maximum slightly below 90◦ .
Fig. 7. Angular dependency of calculated wear volumes. The angular dependency of calculated wear volumes V(β) is plotted as a ratio to Charnley’s circular cylinder (wear at 90◦ = πr2 d). The volumes of the wedge as well as the wear volume according to Kabo depend on the ratio d/r (d = penetration depth, r = radius of the head). They are displayed for the samples d/r = 0.1 and d/r = 0.25. For these samples, the sum of Vcyl and Vwedge (dashed lines) lies mostly between Vcyl (thick dotted line) and VHashimoto (thick dotted line), characterizing VHashimoto as an acceptable approach, in contrast to VKabo which is mostly either close to 0.5 or undefined, corresponding to the true volume of wear only at β = 0◦ (Table 2).
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4. Discussion The circular cylinder approach of Charnley et al. [10] has been used to estimate the volume of prosthetic wear from biplane measurements from radiographs for a long time. It has been shown that the direction of wear is primarily cranial and rather lateral than medial [19,20]. Therefore, the wear volume is systematically overestimated with Charnley’s formula. The formula of Kabo et al. [11] was the first taking the direction of wear in account. Hall et al. [12] noted that the formula of Kabo et al. underestimated the true wear volume. But they accepted the (wrong) concept of Kabo et al. and added a mathematically incorrect term that was meant to account for a cylindrical bore in the entrance plane of some cups. Derbyshire [13] showed that Kabo et al. used a wrong geometrical concept of wear and recognized that Hall’s “correction term” [12] was wrong (Table 2). Furthermore, he investigated the effect from a discrepancy of radii between head and cup, which is not scope of this paper. A numerical integration to solve the problem for various shapes of the cup and directions of wear was proposed [13]. It permits calculation of wear volumes even in case of exceptional shapes of the entrance plane of the studied cups but it appears less transparent for the clinician, performing wear studies on the long term of larger patient populations. Hashimoto and co-workers have proposed [18] has proposed another equation for the wear volume (Table 2). We do not see an explanation why this geometry should reflect wear, as the cut of the head in the cup is a curved line for β > 0◦ and why wear should be estimated by an equation with a term that depends linearly on β. The vicinity of Hashimoto’s volume to Vcyl + Vwedge in Fig. 7 suggests to calculate for each β a ratio d/r where they would coincide. In general Hashimoto’s volume slightly overestimates wear for realistic low ratios d/r. Kosak et al. [21] developed a new formula for calculating wear volumes and found that their formula came closer to the fluid displacement measurements of retrieved cups as
Fig. 8. Example of a worn cup after 18 years. The penetration d was 1.8 mm, the direction of penetration in the film plane βrx was 66◦ and the anteversion 25◦ , as measured with the EBRA method [8]. According to Charnley’s equation [10], the calculated volume was 1448 mm3 . Taking the direction of wear and the anteversion into account, the true direction of wear β was 56◦ and the calculated volume 1323 mm3 according to Kosak et al. [21], 1387 mm3 with the equation of Hashimoto and co-workers [18] and 1347 mm3 with the formula presented here, respectively. The cup was retrieved due to aseptic loosening. The directly measured volume of wear was 1356 mm3 (sum of volume segments as a product of surface segments and their corresponding radial deviation; Mitutoyo® , Strato 9106, accuracy 1.7 m).
We compared the wear according to our equations with that from the equations of Kabo et al. [11] and Hashimoto and co-workers [18]. Hashimoto’s equation yields slightly larger values than those given by V = Vcyl + Vwedge for d/r = 0.25. Kabo’s equation yields values that are in a wide range of β either fairly close to 0.5 or not defined (Fig. 7). The effect of anteversion has to be taken into account before calculating volumetric wear. It leads to a diminishing of the calculated volume of wear (Figs. 6 and 8). Table 2 Available formulas for the calculation of the wear volume Charnley et al. [10]a
V = πr2 d
Kabo et al. [11]b
VKabo = πr2 d − r2 d cos−1
Hall et al.
[12]c
V = VKabo + 2
d tan(β) −
r
r2 tan2 (β)
− d2 +
Vcyl = r 2d (π + π sin β) 2 VHashimoto = r 2d (π + 2β + sin(2β)) 2 V = r 2d π + π sin β + dr sin(2β) = Vcyl + Vwedge
This paperg
V =
−
r3 3 tan(β)
1−
d 2 tan2 (β) r2
3/2
−1
2rfd cos β
Kosak et al. [21]d Hashimoto and co-workers [18]e Kadaba and Ramakrishnanf
r2 d 2
r tan(β)
π + π sin β +
d r
sin(2β) + 2rfd cos β = Vcyl + Vwedge + Vadd
Only correct for wear perpendicular to the entrance plane (β = 90◦ ). Only correct for wear parallel to the entrance plane (β = 0◦ ). c The (wrong) concept of Kabo was accepted, an additional cylindrical part added. In the published equation, the factor cosβ was put erroneously in the denominator (obviously wrong, e.g. for β = 90◦ : the formula predicts infinite instead of zero wear). d Linear wear, derived by integration. The provided formula is identical to our V cyl (Figs. 1–3). The wedge is not taken into account. e Hashimoto’s volume corresponds to V ◦ ◦ ◦ ◦ cyl for β = 0 and β = 90 . For 0 < β < 90 , Hashimoto’s equation yields slightly higher values [13] (Figs. 7 and 8). f Presented on a meeting, not published and no derivation available. It corresponds to our V cyl + Vwedge . g Total wear as proposed in this paper. a
b
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compared to calculations with the Kabo formula. Their formula was derived with an integration and corresponds to our Vcyl for any β (Table 1) but disregards the wedge, which is maximal for β = 45◦ , the most frequent direction of wear (Fig. 7). The formula of Kadaba and Ramakrishnan (congress communication, Table 2) corresponds to our formula Vcyl + Vwedge but it is not published and there is no derivation available. There are considerable errors in measuring linear head penetration on a.p. radiographs [8]. Sophisticated methods have improved the measurement accuracy for the linear wear [6,22–24]. There is little information about the accuracy of assessment of the direction of wear. For wear less than 1 mm, the error of direction measurement might be relatively high, in case of higher wear, the direction might be determined more accurate [14]. The orientation of the cup has to be taken into account for any volume calculations out of measurements from plain radiographs. Disregarding, the effect of anteversion will lead to a systematic overestimation of the volumetric wear. Most of the wear occurs in the film plane and wear perpendicular to the film plane (sagittal wear) is difficult to assess [25]. Two-dimensional measurements might be sufficient in clinical studies [26]. In case of measurement of sagittal wear, further calculations are needed to get the amount and direction of the true wear. A curved direction of wear [27], creep penetration and plastic deformation processes [28], variations in the geometry of the entrance plane of the cup like asymmetric bores [12] and discrepancies in head and cup diameters [13] cannot be detected on plain radiographs and might add further errors in the assessment of wear volume. These errors have to be taken in account for an accurate description of the wearing in process of a cup and for linear wear below 1 mm by sophisticated methods of measurement [29] and are not scope of this paper. The accuracy of wear volume estimations from twodimensional measurements should mainly depend on the methods of measurement but not on the mathematical equation used. Thus we believe that our simplifying assumption of a straight cut of the femoral head from the wedge and the optional cylinder is acceptable. The main contributing volumes of wear, the cylindrical part, the wedge and the part of the optional additional cylinder are easily calculated by converting curved wear volumes into simple rectified geometric objects. Another issue is the clinical benefit from transforming a linear penetration depth into an estimated three-dimensional volume. It can only be investigated with data from a large sample of patients and analysis of retrieved cups. Concerning clinical relevance, we do not provide new equations which have not been at least mentioned somewhere in a conference presentation, but we provide a set of transparent and simple straight forward derivations for formula previously deduced only by integration or lacking a derivation; yet we strongly emphasize that if one chooses to calculate volumetric wear, the employed geometrical concept should be mathematically
correct, transparent and as simple as possible to be suitable for clinical research. Acknowledgements We thank Hans Wingstrand, Department of Orthopedics, University Lund, Sweden, for the hint, that Kabo’s formula might be wrong and for pointing out that our equation for Vcyl + Vwedge has already been presented by Kadaba and Ramakrishnan on a meeting. Furthermore, we thank Martin L¨uem, Department of Orthopedics, Kantonsspital Liestal, Switzerland for providing the example and analysing the data.
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