Med. & Biol. Eng. & Comput., 1979, 17, 155-160
Spherometer measurements of human hip-joint geometry lan C. Clarke
C h u c k Fiske
Harlan C. A m s t u t z
Biomechanics Research Section and Division of Orthopaedics, Centre for the Health Sciences, University of California, Los Angeles, Calif. 90024, USA
Abstract--The accuracy of sphericity measurements on human hip joints was studied with two types o f 3-legged micrometer gauges. Initial calibration work on metal cylinders ( 3 0 - 6 5 mm diameter) achieved an error range o f 4-8%. Correction was then made for this degree of error on a nomogram which reduced the errors in subsequent measurements to 0 . 8 - 1 . 5 % . The problem was that the percentage error in the micrometer reading was directly reflected in the percentage error in diametrical calculation. To achieve an acceptable accuracy o f 4- 1%, the micrometer readings had to be accurate to within +_ (15-23)/tm, i.e, O. 0 0 0 6 - 0 . 0 0 0 9 in. (average of five measurements). This was only achieved in the metal calibration studies 4- (0-15)/~m, but not for the hip j o i n t or its replicas, 4- (7-109)/~m. It was concluded that 3-legged gauges were a very unreliable method of taking diametrical measurements from human j o i n t surfaces, K e y w o r d s - - H u m a n - h i p , Femoral geometry, Acetabular geometry, Dimensional studies
l Introduction
THE 3-dimensional configuration of our joints has obvious importance in understanding and modelling both the macro- and micro-behaviour of joint mechanics. In particular, the sphericity or otherwise of the human hip joint has intrigued many investigators over the last 113 years, following on from an early study by A~BV (1863). Such studies generally compared the cross-sectional contours of either the natural joint or its wax or plaster replicas (AEBY 1863; SCHMID,1876; HELWIG, 1912; WALMSEY,1928). These authors concluded that this ball and socket joint was distinctly aspherical. HAMMOND and CHARNLEV (1967) re-examined these methods in c o m b i n a t i o n with other techniques, using dye, vernier caliper and profile-projection techniques. They asserted that the hip was remarkably spherical and therefore congruent, thereby discounting all the previously published data (GooDsm, 1868; MACCONEILL, 1946; BARNETTet al., 1961). Table 1. Comparison of "out-of-round" dimensions for the hip joint derived from linear regression data (Bullough et al., 1988), using hip joint diameters in the 40 to 60 mm range
Site
Out-of-round
BULLOUGH et al. (1968) used a 3-legged contour gauge (spherometer) with micrometer attachment to characterise the joint contours. The diametrical out-of-round values ( D l m a x - - D t m i n ) shown in Table 1 were derived from their linear-regression data. Two values of 40 and 60 mm were used for the sup.
po~. ~
~nt.
.pest.
ant,
inf.
a
~upen
b
Ms
ant' / Mp post.
(Dmax-Dmin) femoral head (20-80yrs)
mm 2- 4
acetabulum (20 yrs) (80 yrs)
9-13 5-8
Firstreceived 16th Decemberandinfinalform 25th March 1978
inf. Fig. 1 Definition of orientation of measurement sites and symbols used to describe geometry o f Femoral head, a Lateral view b Superior view c Corresponding acetabular geometry
9 IFMBE: 1979 0140-0118/79/0201 55 + 06 $01 950/0
Medical & Biological Engineering & Computing
March 1979
155
average diameter values as representative of the range of femoral head sizes (CLARKE and AMSTUTZ, 1975). Their 3-legged gauge study therefore indicated that the femoral head was 2 to 4 mm out-of-round, while the acetabulum was much more aspherical, being 5 to 13 mm out-of-round. However, this data conflicted with that from a study by DAVIS and FRYMOYER (1969), who used a similar type of gauge, but with four legs. The authors commented that 'the nonarthritic femoral head in all age groups is spherical to within 0" 07 mm.' Hence the conception of the hip had twice swung away from and returned to aspherical and was now back to spherical again. It was difficult to resolve the differences between the 3- and 4-legged spherometer studies because there was little or no methodology or experimental data presented. Particularly important in these studies was the apparent omission of any calibration work to determine the validity of the methodology. The objective of this study was therefore to investigate the use of 3-legged gauges on human hip-joint contours and thereby resolve some of the conflicting data on hip-joint configurations. 2 Method Six freshly dissected adult human hip joints with little or no visible degeneration were used in this study. Silicone and acrylic media were used for either 1- or 2-stage replication procedures for the acetabular and femoral components as required. The orientation of the measurement sites followed the equatorial and meridian concepts (Fig. 1) as described by the early anatomists (AEBY, 1863; SCH~ID, 1876). The 3-legged gauge (t.l.g.) was selected for this purpose rather than the 4-legged design, which would only provide an average value virtually insensitive to orientation. Examination of the hip-joint cartilage showed that the acetabular articular surfaces were more limited than on the
femoral side (Fig. 1) and measured approximately 70 to 90 mm in circumference and 20 to 30 mm wide, tapering to 12 mm or less at the extremities. Two gauges were therefore assembled, one (t.l.g.-50) with 12' 7 mm span (0.5 in) for the smaller cartilage dimensions and one (t.l.g.-75) with 19"1 mm span (0"75 in) for comparison with the gauge used by BULLOUOH et al. (1968). The gauges (Fig. 2) were built around two vernier-equipped micrometers reading to 2.5 pm (0" 0091 in) per division with nonrotating spindle barrels (Mitutoyo Manufacturing Co. Ltd., Los Angeles). The various features of the gauge geometry were measured in an optical comparator (Schem Tumico Inc., Los Angeles) at x 20 magnification. The gauges were calibrated on 14 specially machined aluminium cylinders for the 30 to 65 mm diameter range (1"2 to 2.5 in) in approximately 2" 5 mm (0" 1 in) increments. Acrylic replicas were made of these cylinders to provide concave calibration standards when measurements of the internal radii of acetabulae were required. For the 3-legged gauge geometry, where it is assumed that the legs have negligible thickness (see Appendix), the equation relating the joint radius R to the micrometer value x is given by a 2
R~ =
2x
X
+ -2
.
.
.
.
.
.
.
(1)
.
where 2a represents the chord of the circle across the contact points of the gauge legs. This formula for the 3-legged gauges applies equally well for the chord of a circle on either the cylindrical calibration specimens or the spherical hip-joint geometry. Previous studies with the 3-legged gauge used the expression a 2
R
......
2x
.
.
.
.
.
.
.
.
.
.
(2)
ignoring the x/2 term (GREENWALD, 1970). When the geometry of the gauge legs is included in the model, eqn. 1 becomes (al) 2
R =
x + -- .
. . . .
(3)
where r is the radius at tip of gauge legs and a 1 is defined by a
=
01 - - - -
1+
Fig. 2 S p h e r o m e t e r s (t.L g. - 50, t, L g, - 75) a n d a l u m i n i u m calibration cylinders
156
.
.
.
.
.
.
.
.
.
(4)
R
The values of a and a 1 were determined for each micrometer gauge (see Appendix) and provided in tabular form for ease of calculation.
Medical & Biological Engineering & Computing
March 1979
3 Results The initial calibration with the gauges on the aluminum cylinders produced errors of 4 to 8 ~ , using eqns. 1 and 2. It was hoped to achieve an accuracy of at least _+1~ , which still represented a possible error of 0.5 mm on a 50ram diameter 9femoral head. These larger errors were thought to be caused by omitting the geometry of the gauge feet.
C12 2 90 6 2 0 "
-(a') = 0 . 0 6 2 5
.0600-
TLG-50
.05BO HUMAN HIP
//,
9O5OO '0480
The values a and a ~were then compared (Fig. 3A) and used in eqn. 3, but the same magnitude of error still remained. The data obtained with eqn. 1 was then plotted in a nomogram (Fig. 3B), i.e. ignoring for the moment thx/2 term and the effects of the gauge feet. The error E~oo in the calculated values then represented a correction factor for use with the nominal diameter magnitude D. This showed that when working on convex surfaces the correction factors were virtually constant (approximately 4 ~ ) for t.l.g.-50 in the 45 to 65 mm range and 35 to 65 mm range for t.l.g.-75. This correction was then incorporated into the tabular work sheets and thereafter provided an accuracy of 0 . 4 to 0-5~o with t,l.g.-75 and up to 1"5~o for t.l.g.-50 as used on the calibration standards by both operators. DIA mm
I I t I
I, I [ I
/
li
Ill
0440
L,
2'i08
,
04
06
,
I-0
30 ,,I 1.2
,
t4
40, , 1.6
,
1-8
50 ' , 2"0
in
6
4
-2
=" . =
1.2
-4 -6 -8 -10 -12 -14
-16.
-18,
1'4
. 4 0I
. i
1"6
1'8
e
, 5 '-0 2'0
,
60
mrn.,, _ u 2:2 ' 2.4 ' 2 : 6 in. r
~
~ TLG-50
outside dia. o
TLG - 7 5
o u t s i d e dia.
inside dia, 9
inside dia.
9
Fig. 3B Calibration graph for gauges t.Lg.-50 and t./.g.-75, showing the errors E% in calculating the diameters D of the concave and convex surfaces o f the calibration jigs, using the equation a2
2Rg --
/
/
/
2-00
L
~
1.96 /
1.92 9
1.88
II R
8
30
i1
~
--II---9 "" 2417
L
DIA 2!~4
/
48
10
2
CARTILAGE /|
,50- ,~1,~,.
Fig. 3,4 Calibration graph for use with 3 - l e g g e d gauge geometry (t.l g,-50), Parameter (M ) represents the halfspan of the gauge legs (here a 1 = 0 . 2 5 in, (al)2----0.0625). When the real geometry for the contact points of the gauge legs is considered, parameter "a" varies with respect to the j o i n t radius R, the gauge-leg radius r and the gauge halfspan a i
0
F
51
49 -0460
JOINT NO 5f T LG-.50 RADIAL
X
Medical & Biologica! Engineering & Computing
4
7
46.
:
1
"
.
8
4
~
.1-80
-
45
#
1.76
SITE NO
Fig. 4 Comparison o f calculated diameters for a femoral head at five meridian sites M immediately f o l l o w i n g dissection, and 17 and 24 h later. Data derived using gauge t.l.g.-50 with each data p o i n t representing the average of five consecutive measurements
The typical pattern of diameter measurements as measured on five meridian sites (M, Fig. 1) are shown for a fresh femoral head (Fig. 4). The measurements were made with t.l.g.-50 following excision, then 17 and 24 h later at the same marked sites, the head being re-equilibrated in buffered saline between measurements. The apparent out-of-roundness (D,,,,,-D,,i,,) varied from 1 to 4 mm in a seemingly unpredictable manner, depending on which set of tests were taken. Similar profiles are shown for another femoral head as measured by two operators using both gauges (Fig. 5). To check that there were no cartilage changes during the tests, a negative silicone replica was made and then a positive acrylic replica, the latter providing a dimensionally-stable polished rigid surface for easier measurement. Again the puzzling inconsistencies were apparent. Ideally, all,~he values at each of the sites (numbers 1 to 5) would be closely grouped together. The acrylic replica
March 1979
157
of the femoral head was then measured in a simple profilimeter (CLARKE and AMSTUTZ, 1975). The resulting diametrical variation is shown in Fig. 5 and is very much smaller ( 0 . 4 m m ) than that predicted by the 3-1egged gauges (1.4 to 4.4 mm). Table 2, Comparison o f theoretical errors f in radii calculations R, using eqns. 1 and 2 with predetermined errors S for gauge t,I. g. - 75 82
2R = - - + x X
where
Selected error S in gauge value x I in
_+S { (/~m) t%
R=l.0in a21.o-= 0 . 1 2 0 4 x l . o -- 0.0621 in Theoretical error E in radius R in
+ E l (ram) -
t%
0-0001 (2.5) 0.2
0'002 (0.05) 0.2
0.0005 (12.5) 0-8
0.009 (0.22) 0.9
0.0010 (25-4) 1.6
0.016 (0.40) 1 -6
0- 0020 (50.8) 3.2
0. 032 (0.81) 3.2
It was obvious from the foregoing results that the spherometer data taken from cartilage surfaces was much in error. The effect of possible measurement errors was investigated theoretically by providing varying errors + s in the gauge parameter x for the various equations. This showed (Table 2) that the errors E in the calculated radii varied directly as the errors in x. Hence, a 1~ accuracy in the calculated radius (approximately +0.25 mm or 0.0l in) required a 1~ accuracy in the micrometer readings. Theoretically, this required an accuracy of at least 10 ltm (0-0004 in) for t.l.g.-50 and 23 pm (0.0007 in) for t.l.g.-75 (Table 3) in the required diameter range. Table 3. Data shows maximum theoretical errors + S in micrometer parameter x for both spherometers to achieve an error in the 1% range for the calculated radii R
Joint diameter
158
M a x i m u m permissible micrometer error _+S t.l.g.-50
t.l.g.-75
in (mm) 1 95748 (40)
in (/zm) 0" 0004 (1 0)
in (Fro) 0- 0009 (23)
1 95748 (40)
0" 0004 (1 0)
0. 0009 (23)
A summary of the experimental data (Table 4) showed that this was virtually achieved for the aluminum and acrylic standards but not for the cartilage or its replicas. Fifty consecutive readings were then taken with t.l.g.-75 on the same marked spot of a femoral head (cartilage in excellent condition) to determine how much scatter was likely to be present under optimal experimental conditions. Rather than rely on the operator's 'feel' or the micrometer ratchet, the gauge was mounted with the femoral head on a jig and viewed at x 20 magnification in the optical comparator for 'kissing' contact of gauge legs against cartilage. The variation (2s) in each of the ten consecutive sets of five measurements varied from 16 to 66/zm (0.0006-0"0026in). As indicated in Table 2, this range corresponded to an error of 1 to 4 ~ , i.e. a possible error of 2 mm when measuring a 50 mm diameter ball. The study was terminated at this point. 4 Discussion and conclusions Although spherometers: are commonly used in industry for detecting variations in surface curvature, the data in this study indicated that such devices were of little value in quantifying the actual diameter magnitudes. The initial analytical calibration methods which were intended to compensate for the geometry of the gauge feet still resulted in unacceptably DIA,
mm
58 57 58
2.30
,,,', ",,.
t'~ / .,,-- t
~ 2.18
55
2,14 54 53.
2.10
2.06
52
2
i
4
;
SITE NO
Fig. 5 Comparison of calculated femoral head diameters for t w o operators, using both gauges on the natural head and its acrylic replica. The shaded band represents the diametrical variations of the same replica as measured by profilimeter for equatorialA sites Specimen Operator Gauge Graph
cartilage cartilage replica replica replica replica
2 1 1 2 1 2
t.l.g.-50 t.l.g.-75 tl.g.-50 t.l.g.-50 t.l.g.-75 t.l.g.-75
Medical & Biological Engineering & Computing
A E B C D F
March 1979
large errors ( > 4 ~ ) . By tabulating these errors and correcting for them individually, it was possible to reduce the measurement errors on the metal calibration cylinders to less than 1 ~ , but only for the larger gauge (t.l.g.-75). This was thought to be acceptable for joint measurements. However, when used on either cartilage surfaces or their replicas, the results again became erratic, this time due to inconsistencies in the micrometer measurements. The theoretical error analysis showed that the errors in the radii calculations were directly related to the errors in the micrometer readings. Therefore, to achieve an accuracy of + 1 ~ (possible error + 0" 5 mm average) on femoral heads with a known diameter range of 40 to 60 ram, required a consistency of micrometer measurement within +15 /~m (0' 0006) in) for t.l.g.-75 Tables 2 and 3). Comparison of the data in Table 4 demonstrates that this was just not achieved for either the cartilage surfaces or their replicas. It must be concluded, therefore, that the spherometer gauges are only marginally acceptable for measuring diameters of machined components, and likely to be very much in error when assessing the more irregular nature of cartilaginous joint surfaces. Hence the validity of previous spherometer studies and their arguments for and against asphericity of the human hip joint (BULLOUGH et al., 1968; DAVIS and FRYMOYER, 1969) are open to question. Fable 4. Variations (2S) in sets of 5 micrometer values (x) from I 630 measurements on calibration jigs, and replicas cartilage Specimen
aluminium standard acrylic standard articular cartilage cartilage replicas
Number of sets
56 41 133 96 326
Error range (2S) in x parameter in (/~m) 0 - 0 . 0006 (0-15) 0-0" 0005 (0-13) O. 003-0. 0074 (7-190) 0-0" 0041 (0-104)
BLOWERS, D. H., ELSON, R. and KORLEY, E. (1972) An investigation of the sphericity of the human femoral head. Med. & Biol. Eng. 10, 762-774. Britar-nica Roundtable (1972): Antique instruments: Elegance from the past 1, 4, 20-23. ]~ULLOUGH)P., GOODFELLOW,J., GREENWALD,A. S. and O'CONNOR, J. (1968) Incongruent surfaces in the human hip joint. Nature 217, 1290. CATHCART. R. F. (1973) New ideas in the design and function of the Austin-Moore prosthesis. Orthop. Rev. 2, 15-22. CLARKE, I. C. and AMSTUTZ,H. C. (1975): Human hip joint geometry and hemi-arthroplasty selection. Proceedings of the hip society, C. V. Mosby Co., St. Louis, 63-92. DAVIS,P. H. and FRYMOYER,J. W. (1969) A femoral head measuring device. J. Bone Joint Surg. 51A, 1663-1664. GOODFELLOW, J. W. and BULLOUGH,P. G. (1969) The interpretation of the radiological joint space. Buech. Orthop. 4, 52-58. GOODSIR, J. (1968) On the curvatures and movements of the acting facets of articular surfaces. In TURNER, W. (Ed.) Anatomical memoirs of J. Goodsir, London, A. & C. Black Ltd., 246-248. GREENWALD, A. S. (1970) The transmission of forces through animal joints. D. Phil. thesis, Oxford University. HAMMOND,B. T. and CHARNLEY,J. (1967) The sphericity of the femoral head. Med. & Biol. Eng. 5, 445-453. HELWIG, R. (1912) Uber die Form des Huftgelenkes des Menschen, Kiev (quoted by Walmsey). MACCONAILL, M. A. (1961) Joint surfaces. In BARNETT, C. H., DAVIES,D. V. and MACCONAILL,M.A., Synovial joints, Longmans, London. SCHMID, F. (1876) Uber Form und Mechanik des Huftgelenkes. Dtsch. Z. Chit. 5, 1-24. SCHMID, F. (1874) ibid. 3. WALMSEY, T. (1928) The articular mechanism of the diarthoses. J. Bone Joint Surg. 10, 40-45.
Appendix For the simple geometry shown in Fig. 6, the spherometer can be considered applied to either the inside or outside of the circle of radius R at points A, D and C. Therefore AB'BC ~ DB'BE Therefore a2 ~
a2
2R -- - - + x . . . . . . . . . . x
326 X 5 = 1630
Acknowledgments--This work was financed by the Deutsch Foundation and Blalock Foundation (H.S. Rinker) and by National Institutes of Health Grant No. 5 PO1 AMi 6120-03AMP. Thanks are also due to the Health Sciences Computing Facility of UCLA for computing assistance supported by NIH Special Resources Grant RR-3 and to E. Morgan for machining expertise.
x(2R--x)
A
~B
(5)
~
References AEBY, C. (1963) Die Spharoidgelenke der Extremitatengurtel. Z. Med. 17 (quoted by Schmid). BARNETT, C. H., DAVIES,D. V. and MACCONAILL,M. A. (1961) In Synovial Joints, Longmans, London, 170-179.
Fig. 6 Simple geometry
Medical & Biological Engineering & Computing
March 1979
159
T h i s e q u a t i o n was abbreviated by
GREENWALD
(1970) to
a 2
2R9 =
-- 9
.
X
.
.
.
.
.
.
.
.
.
W h e n the s p h e r o m e t e r is applied to a concave surface this becomes
(6) a 1 a -
F o r the real situation s h o w n in Fig. 7 with a r a d i u s r on the tip o n t h e gauge legs a=a
1-AF=a
1---
l----
r
R
ar
R
Therefore, the final soluti on for radius R is given by
Therefore 2R --
a 1
(al)2
a--
I+--
+ x
. . . . . .
(7)
F
R Table 5.
Spherometer
r
aI
a2(R = 1 -0 in)
x(R = 1 "0 in)
in t.l.g.-50
O. 072
in O- 249
O. 0544
O" 0276
t.l.g.-75
O. 072
O- 872
O. 1204
O" 0621
G a u g e p a r a m e t e r ' x ' represents the difference between two m e a s u r e m e n t s , one being the zero reading Xo w h e n the gauge legs are aligned in the s a m e plane a n d the o t h e r the experimental reading xe o n a curved surface, i.e. x ~ x ~ - - X o . T h e d i m e n s i o n s o f the 3-legged g e o m e t r y were m e a s u r e d in an optical c o m p a r a t o r ( • 20 m a g n i t u d e ) to provide the p a r a m e t e r s in Table 5. C o m p a r i n g eqns. 1 a n d 2 ~
Rg--R
,a
R
rF Fig. 7
160
Real situation
x
= -- - 2
where R = 1 . 0 i n
U s i n g the c o r r e s p o n d i n g x values given in Table 5, the theoretical error in using eqn. 2 for radii Ra calculations is a p p r o x i m a t e l y 1.5~/o for t.l.g.-50 a n d 3 % for t.l.g.-75, i.e. e q u i v a l e n t t o u n d e r e s t i m a t e s o f 3/4 a n d 1"5 r a m , respectively, o n the diameter m e a s u r e m e n t s .
Medical & Biological Engineering & Computing
March 1979
Spherometer measurements of human hip-joint geometry lan C. Clarke
C h u c k Fiske
Harlan C. A m s t u t z
Biomechanics Research Section and Division of Orthopaedics, Centre for the Health Sciences, University of California, Los Angeles, Calif. 90024, USA
Abstract--The accuracy of sphericity measurements on human hip joints was studied with two types o f 3-legged micrometer gauges. Initial calibration work on metal cylinders ( 3 0 - 6 5 mm diameter) achieved an error range o f 4-8%. Correction was then made for this degree of error on a nomogram which reduced the errors in subsequent measurements to 0 . 8 - 1 . 5 % . The problem was that the percentage error in the micrometer reading was directly reflected in the percentage error in diametrical calculation. To achieve an acceptable accuracy o f 4- 1%, the micrometer readings had to be accurate to within +_ (15-23)/tm, i.e, O. 0 0 0 6 - 0 . 0 0 0 9 in. (average of five measurements). This was only achieved in the metal calibration studies 4- (0-15)/~m, but not for the hip j o i n t or its replicas, 4- (7-109)/~m. It was concluded that 3-legged gauges were a very unreliable method of taking diametrical measurements from human j o i n t surfaces, K e y w o r d s - - H u m a n - h i p , Femoral geometry, Acetabular geometry, Dimensional studies
l Introduction
THE 3-dimensional configuration of our joints has obvious importance in understanding and modelling both the macro- and micro-behaviour of joint mechanics. In particular, the sphericity or otherwise of the human hip joint has intrigued many investigators over the last 113 years, following on from an early study by A~BV (1863). Such studies generally compared the cross-sectional contours of either the natural joint or its wax or plaster replicas (AEBY 1863; SCHMID,1876; HELWIG, 1912; WALMSEY,1928). These authors concluded that this ball and socket joint was distinctly aspherical. HAMMOND and CHARNLEV (1967) re-examined these methods in c o m b i n a t i o n with other techniques, using dye, vernier caliper and profile-projection techniques. They asserted that the hip was remarkably spherical and therefore congruent, thereby discounting all the previously published data (GooDsm, 1868; MACCONEILL, 1946; BARNETTet al., 1961). Table 1. Comparison of "out-of-round" dimensions for the hip joint derived from linear regression data (Bullough et al., 1988), using hip joint diameters in the 40 to 60 mm range
Site
Out-of-round
BULLOUGH et al. (1968) used a 3-legged contour gauge (spherometer) with micrometer attachment to characterise the joint contours. The diametrical out-of-round values ( D l m a x - - D t m i n ) shown in Table 1 were derived from their linear-regression data. Two values of 40 and 60 mm were used for the sup.
po~. ~
~nt.
.pest.
ant,
inf.
a
~upen
b
Ms
ant' / Mp post.
(Dmax-Dmin) femoral head (20-80yrs)
mm 2- 4
acetabulum (20 yrs) (80 yrs)
9-13 5-8
Firstreceived 16th Decemberandinfinalform 25th March 1978
inf. Fig. 1 Definition of orientation of measurement sites and symbols used to describe geometry o f Femoral head, a Lateral view b Superior view c Corresponding acetabular geometry
9 IFMBE: 1979 0140-0118/79/0201 55 + 06 $01 950/0
Medical & Biological Engineering & Computing
March 1979
155
average diameter values as representative of the range of femoral head sizes (CLARKE and AMSTUTZ, 1975). Their 3-legged gauge study therefore indicated that the femoral head was 2 to 4 mm out-of-round, while the acetabulum was much more aspherical, being 5 to 13 mm out-of-round. However, this data conflicted with that from a study by DAVIS and FRYMOYER (1969), who used a similar type of gauge, but with four legs. The authors commented that 'the nonarthritic femoral head in all age groups is spherical to within 0" 07 mm.' Hence the conception of the hip had twice swung away from and returned to aspherical and was now back to spherical again. It was difficult to resolve the differences between the 3- and 4-legged spherometer studies because there was little or no methodology or experimental data presented. Particularly important in these studies was the apparent omission of any calibration work to determine the validity of the methodology. The objective of this study was therefore to investigate the use of 3-legged gauges on human hip-joint contours and thereby resolve some of the conflicting data on hip-joint configurations. 2 Method Six freshly dissected adult human hip joints with little or no visible degeneration were used in this study. Silicone and acrylic media were used for either 1- or 2-stage replication procedures for the acetabular and femoral components as required. The orientation of the measurement sites followed the equatorial and meridian concepts (Fig. 1) as described by the early anatomists (AEBY, 1863; SCH~ID, 1876). The 3-legged gauge (t.l.g.) was selected for this purpose rather than the 4-legged design, which would only provide an average value virtually insensitive to orientation. Examination of the hip-joint cartilage showed that the acetabular articular surfaces were more limited than on the
femoral side (Fig. 1) and measured approximately 70 to 90 mm in circumference and 20 to 30 mm wide, tapering to 12 mm or less at the extremities. Two gauges were therefore assembled, one (t.l.g.-50) with 12' 7 mm span (0.5 in) for the smaller cartilage dimensions and one (t.l.g.-75) with 19"1 mm span (0"75 in) for comparison with the gauge used by BULLOUOH et al. (1968). The gauges (Fig. 2) were built around two vernier-equipped micrometers reading to 2.5 pm (0" 0091 in) per division with nonrotating spindle barrels (Mitutoyo Manufacturing Co. Ltd., Los Angeles). The various features of the gauge geometry were measured in an optical comparator (Schem Tumico Inc., Los Angeles) at x 20 magnification. The gauges were calibrated on 14 specially machined aluminium cylinders for the 30 to 65 mm diameter range (1"2 to 2.5 in) in approximately 2" 5 mm (0" 1 in) increments. Acrylic replicas were made of these cylinders to provide concave calibration standards when measurements of the internal radii of acetabulae were required. For the 3-legged gauge geometry, where it is assumed that the legs have negligible thickness (see Appendix), the equation relating the joint radius R to the micrometer value x is given by a 2
R~ =
2x
X
+ -2
.
.
.
.
.
.
.
(1)
.
where 2a represents the chord of the circle across the contact points of the gauge legs. This formula for the 3-legged gauges applies equally well for the chord of a circle on either the cylindrical calibration specimens or the spherical hip-joint geometry. Previous studies with the 3-legged gauge used the expression a 2
R
......
2x
.
.
.
.
.
.
.
.
.
.
(2)
ignoring the x/2 term (GREENWALD, 1970). When the geometry of the gauge legs is included in the model, eqn. 1 becomes (al) 2
R =
x + -- .
. . . .
(3)
where r is the radius at tip of gauge legs and a 1 is defined by a
=
01 - - - -
1+
Fig. 2 S p h e r o m e t e r s (t.L g. - 50, t, L g, - 75) a n d a l u m i n i u m calibration cylinders
156
.
.
.
.
.
.
.
.
.
(4)
R
The values of a and a 1 were determined for each micrometer gauge (see Appendix) and provided in tabular form for ease of calculation.
Medical & Biological Engineering & Computing
March 1979
3 Results The initial calibration with the gauges on the aluminum cylinders produced errors of 4 to 8 ~ , using eqns. 1 and 2. It was hoped to achieve an accuracy of at least _+1~ , which still represented a possible error of 0.5 mm on a 50ram diameter 9femoral head. These larger errors were thought to be caused by omitting the geometry of the gauge feet.
C12 2 90 6 2 0 "
-(a') = 0 . 0 6 2 5
.0600-
TLG-50
.05BO HUMAN HIP
//,
9O5OO '0480
The values a and a ~were then compared (Fig. 3A) and used in eqn. 3, but the same magnitude of error still remained. The data obtained with eqn. 1 was then plotted in a nomogram (Fig. 3B), i.e. ignoring for the moment thx/2 term and the effects of the gauge feet. The error E~oo in the calculated values then represented a correction factor for use with the nominal diameter magnitude D. This showed that when working on convex surfaces the correction factors were virtually constant (approximately 4 ~ ) for t.l.g.-50 in the 45 to 65 mm range and 35 to 65 mm range for t.l.g.-75. This correction was then incorporated into the tabular work sheets and thereafter provided an accuracy of 0 . 4 to 0-5~o with t,l.g.-75 and up to 1"5~o for t.l.g.-50 as used on the calibration standards by both operators. DIA mm
I I t I
I, I [ I
/
li
Ill
0440
L,
2'i08
,
04
06
,
I-0
30 ,,I 1.2
,
t4
40, , 1.6
,
1-8
50 ' , 2"0
in
6
4
-2
=" . =
1.2
-4 -6 -8 -10 -12 -14
-16.
-18,
1'4
. 4 0I
. i
1"6
1'8
e
, 5 '-0 2'0
,
60
mrn.,, _ u 2:2 ' 2.4 ' 2 : 6 in. r
~
~ TLG-50
outside dia. o
TLG - 7 5
o u t s i d e dia.
inside dia, 9
inside dia.
9
Fig. 3B Calibration graph for gauges t.Lg.-50 and t./.g.-75, showing the errors E% in calculating the diameters D of the concave and convex surfaces o f the calibration jigs, using the equation a2
2Rg --
/
/
/
2-00
L
~
1.96 /
1.92 9
1.88
II R
8
30
i1
~
--II---9 "" 2417
L
DIA 2!~4
/
48
10
2
CARTILAGE /|
,50- ,~1,~,.
Fig. 3,4 Calibration graph for use with 3 - l e g g e d gauge geometry (t.l g,-50), Parameter (M ) represents the halfspan of the gauge legs (here a 1 = 0 . 2 5 in, (al)2----0.0625). When the real geometry for the contact points of the gauge legs is considered, parameter "a" varies with respect to the j o i n t radius R, the gauge-leg radius r and the gauge halfspan a i
0
F
51
49 -0460
JOINT NO 5f T LG-.50 RADIAL
X
Medical & Biologica! Engineering & Computing
4
7
46.
:
1
"
.
8
4
~
.1-80
-
45
#
1.76
SITE NO
Fig. 4 Comparison o f calculated diameters for a femoral head at five meridian sites M immediately f o l l o w i n g dissection, and 17 and 24 h later. Data derived using gauge t.l.g.-50 with each data p o i n t representing the average of five consecutive measurements
The typical pattern of diameter measurements as measured on five meridian sites (M, Fig. 1) are shown for a fresh femoral head (Fig. 4). The measurements were made with t.l.g.-50 following excision, then 17 and 24 h later at the same marked sites, the head being re-equilibrated in buffered saline between measurements. The apparent out-of-roundness (D,,,,,-D,,i,,) varied from 1 to 4 mm in a seemingly unpredictable manner, depending on which set of tests were taken. Similar profiles are shown for another femoral head as measured by two operators using both gauges (Fig. 5). To check that there were no cartilage changes during the tests, a negative silicone replica was made and then a positive acrylic replica, the latter providing a dimensionally-stable polished rigid surface for easier measurement. Again the puzzling inconsistencies were apparent. Ideally, all,~he values at each of the sites (numbers 1 to 5) would be closely grouped together. The acrylic replica
March 1979
157
of the femoral head was then measured in a simple profilimeter (CLARKE and AMSTUTZ, 1975). The resulting diametrical variation is shown in Fig. 5 and is very much smaller ( 0 . 4 m m ) than that predicted by the 3-1egged gauges (1.4 to 4.4 mm). Table 2, Comparison o f theoretical errors f in radii calculations R, using eqns. 1 and 2 with predetermined errors S for gauge t,I. g. - 75 82
2R = - - + x X
where
Selected error S in gauge value x I in
_+S { (/~m) t%
R=l.0in a21.o-= 0 . 1 2 0 4 x l . o -- 0.0621 in Theoretical error E in radius R in
+ E l (ram) -
t%
0-0001 (2.5) 0.2
0'002 (0.05) 0.2
0.0005 (12.5) 0-8
0.009 (0.22) 0.9
0.0010 (25-4) 1.6
0.016 (0.40) 1 -6
0- 0020 (50.8) 3.2
0. 032 (0.81) 3.2
It was obvious from the foregoing results that the spherometer data taken from cartilage surfaces was much in error. The effect of possible measurement errors was investigated theoretically by providing varying errors + s in the gauge parameter x for the various equations. This showed (Table 2) that the errors E in the calculated radii varied directly as the errors in x. Hence, a 1~ accuracy in the calculated radius (approximately +0.25 mm or 0.0l in) required a 1~ accuracy in the micrometer readings. Theoretically, this required an accuracy of at least 10 ltm (0-0004 in) for t.l.g.-50 and 23 pm (0.0007 in) for t.l.g.-75 (Table 3) in the required diameter range. Table 3. Data shows maximum theoretical errors + S in micrometer parameter x for both spherometers to achieve an error in the 1% range for the calculated radii R
Joint diameter
158
M a x i m u m permissible micrometer error _+S t.l.g.-50
t.l.g.-75
in (mm) 1 95748 (40)
in (/zm) 0" 0004 (1 0)
in (Fro) 0- 0009 (23)
1 95748 (40)
0" 0004 (1 0)
0. 0009 (23)
A summary of the experimental data (Table 4) showed that this was virtually achieved for the aluminum and acrylic standards but not for the cartilage or its replicas. Fifty consecutive readings were then taken with t.l.g.-75 on the same marked spot of a femoral head (cartilage in excellent condition) to determine how much scatter was likely to be present under optimal experimental conditions. Rather than rely on the operator's 'feel' or the micrometer ratchet, the gauge was mounted with the femoral head on a jig and viewed at x 20 magnification in the optical comparator for 'kissing' contact of gauge legs against cartilage. The variation (2s) in each of the ten consecutive sets of five measurements varied from 16 to 66/zm (0.0006-0"0026in). As indicated in Table 2, this range corresponded to an error of 1 to 4 ~ , i.e. a possible error of 2 mm when measuring a 50 mm diameter ball. The study was terminated at this point. 4 Discussion and conclusions Although spherometers: are commonly used in industry for detecting variations in surface curvature, the data in this study indicated that such devices were of little value in quantifying the actual diameter magnitudes. The initial analytical calibration methods which were intended to compensate for the geometry of the gauge feet still resulted in unacceptably DIA,
mm
58 57 58
2.30
,,,', ",,.
t'~ / .,,-- t
~ 2.18
55
2,14 54 53.
2.10
2.06
52
2
i
4
;
SITE NO
Fig. 5 Comparison of calculated femoral head diameters for t w o operators, using both gauges on the natural head and its acrylic replica. The shaded band represents the diametrical variations of the same replica as measured by profilimeter for equatorialA sites Specimen Operator Gauge Graph
cartilage cartilage replica replica replica replica
2 1 1 2 1 2
t.l.g.-50 t.l.g.-75 tl.g.-50 t.l.g.-50 t.l.g.-75 t.l.g.-75
Medical & Biological Engineering & Computing
A E B C D F
March 1979
large errors ( > 4 ~ ) . By tabulating these errors and correcting for them individually, it was possible to reduce the measurement errors on the metal calibration cylinders to less than 1 ~ , but only for the larger gauge (t.l.g.-75). This was thought to be acceptable for joint measurements. However, when used on either cartilage surfaces or their replicas, the results again became erratic, this time due to inconsistencies in the micrometer measurements. The theoretical error analysis showed that the errors in the radii calculations were directly related to the errors in the micrometer readings. Therefore, to achieve an accuracy of + 1 ~ (possible error + 0" 5 mm average) on femoral heads with a known diameter range of 40 to 60 ram, required a consistency of micrometer measurement within +15 /~m (0' 0006) in) for t.l.g.-75 Tables 2 and 3). Comparison of the data in Table 4 demonstrates that this was just not achieved for either the cartilage surfaces or their replicas. It must be concluded, therefore, that the spherometer gauges are only marginally acceptable for measuring diameters of machined components, and likely to be very much in error when assessing the more irregular nature of cartilaginous joint surfaces. Hence the validity of previous spherometer studies and their arguments for and against asphericity of the human hip joint (BULLOUGH et al., 1968; DAVIS and FRYMOYER, 1969) are open to question. Fable 4. Variations (2S) in sets of 5 micrometer values (x) from I 630 measurements on calibration jigs, and replicas cartilage Specimen
aluminium standard acrylic standard articular cartilage cartilage replicas
Number of sets
56 41 133 96 326
Error range (2S) in x parameter in (/~m) 0 - 0 . 0006 (0-15) 0-0" 0005 (0-13) O. 003-0. 0074 (7-190) 0-0" 0041 (0-104)
BLOWERS, D. H., ELSON, R. and KORLEY, E. (1972) An investigation of the sphericity of the human femoral head. Med. & Biol. Eng. 10, 762-774. Britar-nica Roundtable (1972): Antique instruments: Elegance from the past 1, 4, 20-23. ]~ULLOUGH)P., GOODFELLOW,J., GREENWALD,A. S. and O'CONNOR, J. (1968) Incongruent surfaces in the human hip joint. Nature 217, 1290. CATHCART. R. F. (1973) New ideas in the design and function of the Austin-Moore prosthesis. Orthop. Rev. 2, 15-22. CLARKE, I. C. and AMSTUTZ,H. C. (1975): Human hip joint geometry and hemi-arthroplasty selection. Proceedings of the hip society, C. V. Mosby Co., St. Louis, 63-92. DAVIS,P. H. and FRYMOYER,J. W. (1969) A femoral head measuring device. J. Bone Joint Surg. 51A, 1663-1664. GOODFELLOW, J. W. and BULLOUGH,P. G. (1969) The interpretation of the radiological joint space. Buech. Orthop. 4, 52-58. GOODSIR, J. (1968) On the curvatures and movements of the acting facets of articular surfaces. In TURNER, W. (Ed.) Anatomical memoirs of J. Goodsir, London, A. & C. Black Ltd., 246-248. GREENWALD, A. S. (1970) The transmission of forces through animal joints. D. Phil. thesis, Oxford University. HAMMOND,B. T. and CHARNLEY,J. (1967) The sphericity of the femoral head. Med. & Biol. Eng. 5, 445-453. HELWIG, R. (1912) Uber die Form des Huftgelenkes des Menschen, Kiev (quoted by Walmsey). MACCONAILL, M. A. (1961) Joint surfaces. In BARNETT, C. H., DAVIES,D. V. and MACCONAILL,M.A., Synovial joints, Longmans, London. SCHMID, F. (1876) Uber Form und Mechanik des Huftgelenkes. Dtsch. Z. Chit. 5, 1-24. SCHMID, F. (1874) ibid. 3. WALMSEY, T. (1928) The articular mechanism of the diarthoses. J. Bone Joint Surg. 10, 40-45.
Appendix For the simple geometry shown in Fig. 6, the spherometer can be considered applied to either the inside or outside of the circle of radius R at points A, D and C. Therefore AB'BC ~ DB'BE Therefore a2 ~
a2
2R -- - - + x . . . . . . . . . . x
326 X 5 = 1630
Acknowledgments--This work was financed by the Deutsch Foundation and Blalock Foundation (H.S. Rinker) and by National Institutes of Health Grant No. 5 PO1 AMi 6120-03AMP. Thanks are also due to the Health Sciences Computing Facility of UCLA for computing assistance supported by NIH Special Resources Grant RR-3 and to E. Morgan for machining expertise.
x(2R--x)
A
~B
(5)
~
References AEBY, C. (1963) Die Spharoidgelenke der Extremitatengurtel. Z. Med. 17 (quoted by Schmid). BARNETT, C. H., DAVIES,D. V. and MACCONAILL,M. A. (1961) In Synovial Joints, Longmans, London, 170-179.
Fig. 6 Simple geometry
Medical & Biological Engineering & Computing
March 1979
159
T h i s e q u a t i o n was abbreviated by
GREENWALD
(1970) to
a 2
2R9 =
-- 9
.
X
.
.
.
.
.
.
.
.
.
W h e n the s p h e r o m e t e r is applied to a concave surface this becomes
(6) a 1 a -
F o r the real situation s h o w n in Fig. 7 with a r a d i u s r on the tip o n t h e gauge legs a=a
1-AF=a
1---
l----
r
R
ar
R
Therefore, the final soluti on for radius R is given by
Therefore 2R --
a 1
(al)2
a--
I+--
+ x
. . . . . .
(7)
F
R Table 5.
Spherometer
r
aI
a2(R = 1 -0 in)
x(R = 1 "0 in)
in t.l.g.-50
O. 072
in O- 249
O. 0544
O" 0276
t.l.g.-75
O. 072
O- 872
O. 1204
O" 0621
G a u g e p a r a m e t e r ' x ' represents the difference between two m e a s u r e m e n t s , one being the zero reading Xo w h e n the gauge legs are aligned in the s a m e plane a n d the o t h e r the experimental reading xe o n a curved surface, i.e. x ~ x ~ - - X o . T h e d i m e n s i o n s o f the 3-legged g e o m e t r y were m e a s u r e d in an optical c o m p a r a t o r ( • 20 m a g n i t u d e ) to provide the p a r a m e t e r s in Table 5. C o m p a r i n g eqns. 1 a n d 2 ~
Rg--R
,a
R
rF Fig. 7
160
Real situation
x
= -- - 2
where R = 1 . 0 i n
U s i n g the c o r r e s p o n d i n g x values given in Table 5, the theoretical error in using eqn. 2 for radii Ra calculations is a p p r o x i m a t e l y 1.5~/o for t.l.g.-50 a n d 3 % for t.l.g.-75, i.e. e q u i v a l e n t t o u n d e r e s t i m a t e s o f 3/4 a n d 1"5 r a m , respectively, o n the diameter m e a s u r e m e n t s .
Medical & Biological Engineering & Computing
March 1979
