J. Lliomechanics Printed

ox!-9290/91

Vol. 24, No. 5. pp. 317 329, 1991.

in Great

Pergamon

Bntain

MECHANICAL

PROPERTIES OF METAPHYSEAL THE PROXIMAL FEMUR

$3.00+.00 Press

plc

BONE IN

JEFFREY C. LOTZ,* TOBIN N. GERHART and WILSON C. HAYES? Orthopaedic Biomechanics Laboratory, Department of Orthopaedic Surgery, Charles A. Dana Research Institute. Beth Israel Hospital and Harvard Medical School, Boston, MA 02215, U.S.A. Abstract-We used a three-point bending test to investigate the structural behavior of 123 rectangular flat plate specimens harvested from the metaphyseal shell of the cervical and intertrochanteric regions of five fresh/frozen human proximal remora. For comparison purposes, 36 specimens of similar geometry were also fabricated from bone of the femoral diaphysis. All specimens were oriented in either the local longitudinal or transverse directions. The mean longitudinal elastic modulus was 9650+2410(SD) MPa and demonstrated a 24% decrease from that measured for the diaphysis (12500+ 2140 MPa) using the same testing technique. However, the transverse elastic moduli did not differ significantly between the proximal (547Ok 1720 MPa) and diaphyseal(5990 & 1520 MPa) specimens. The globally averaged values for the ultimate tensile strengths of the metaphyseal shell were 101+26 MPa in the longitudinal and 50+ 12 MPa in the transverse directions. These compared with diaphyseal values of 128 + 16 MPa and 47 & 12 MPa, respectively. While these differences were largely due to the reduced density of the proximal specimens, a slight decrease in transverse anisotropy for the proximal specimens was also noted by comparing the ratio of longitudinal to transverse moduli (1.76) and tensile strength (2.02) to the diaphyseal values (2.09 and 2.71, respectively). Use of these data should lead to improved performance of analytical models for the proximal femur, and thus help focus increased attention on the structural contributions of trabecular bone to the strength and rigidity of the proximal femur.

INTRODUCTION The elastic and ultimate properties of femoral cortical bone have been measured by many investigators

(Cowin, 1988; Lipson and Katz, 1984; Reilly and Burstein, 1975). These efforts have focused primarily on diaphyseal bone, where the relatively thick cortex allows harvesting of specimens of sufficient size for standard tension and compression tests. However, the mechanical properties of the thin cortical shell at the metaphyses of long bones are less well understood. While previous analytic studies have assumed that this thin cortex has similar mechanical properties to those of the diaphysis, recent investigations of subchondral bone in the femoral head (Brown and Vrahas, 1984) and metaphyseal (Murray et al., 1984) and subchondral (Choi et al., 1989) bone in the proximal tibia have suggested that this shell may have markedly reduced mechanical properties in comparison to diaphyseal bone. If this is true, the contribution of cortical bone to the strength of these regions has been overestimated in many previous analytic studies. In addition, since osteoporosis is thought to affect trabecular and cortical bone at different rates and at different times (Riggs et al.. 1982), this issue becomes important when

Received infinal form 12 November 1990. *Current address: Failure Analysis Associates, 149 Commonwealth Dr., Menlo Park, CA 94025, U.S.A. tAddress correspondence to: Wilson C. Hayes, Orthopaedic Biomechanics Laboratory, Beth Israel Hospital, 330 Brookline Avenue, Boston, MA 02215, U.S.A. Tel.: (617) 735-2940.

investigating the influence of osteoporosis on the load carrying capacity of the proximal femur. Overestimating the contribution of the metaphyseal shell would lead to an underestimate of the osteoporosis-linked decrease in strength of the proximal femur, which is thought to be due primarily to a reduction of the more metabolically active trabecular bone. The thin geometry of the metaphyseal shell in the proximal femur precludes the fabrication of standard specimens for tensile and compression tests. Other investigators have overcome this difficulty by using non-standard specimen geometries. To determine the material properties of subchondral bone in the femoral head, Brown and Vrahas (1984) fabricated spherical caps ranging in thickness from 1.0 to 4.0 mm, which were tested under polar point loads. The material properties were inferred from corresponding analytical shell solutions. Murray et al. (1984) fabricated thin disks ranging in thickness from 0.1 to 1.0 mm from the proximal tibia that were tested as centrally loaded circular plates with plate theory used in determining the mechanical properties. Choi et al. (1989) machined small rectangular beam specimens ranging in thickness from 0.1 to 0.2 mm and used beam theory to determine mechanical properties from three-point bending tests. The irregular geometry of the cortical shell in the proximal femur also necessitates the use of very thin specimens. Therefore, flat, rectangular prismatic plates, fabricated from the metaphyseal shell of the proximal femur, were tested in three-point bending and the elastic modulus and ultimate tensile strength inferred from comparisons between experimental re-

317

J. C. LOTZ et al.

318

suits and the corresponding analytical thin plate solutions. Thus, the objectives of this study were: (1) to validate the formulations for deriving the elastic modulus and ultimate tensile strength from threepoint bending loaddeflection data using specimens of wood and acrylic (plexiglass); and (2) to determine if there exists a significant difference between the modulus and strength of the thin cortical shell within the proximal femur and that of diaphyseal bone of the femoral shaft.

MATERIALS

AND METHODS

Specimens

A total of 123 rectangular metaphyseal cortical shell specimens were machined from the left femur of five fresh/frozen human femora, harvested from cadavers ranging in age from 28 to 90yr. The contralateral femur was reserved for intact bone fracture studies reported elsewhere (Lotz and Hayes, 1990). The specimens were fabricated from bone collected from the distal femoral neck and intertrochanteric regions. Proximal to these regions, the metaphyseal shell within the femoral neck was too thin to allow specimen harvesting with the following protocol and hence was excluded from this study. Since the cortical shell within the distal neck and intertrochanteric regions demonstrated large variations in thickness (with the inferomedial cortex exhibiting typical thicknesses of 3.5 mm and the superolateral cortex thicknesses of 1.0 mm), the specimens were separated into two groups: inferomedial and superolateral. In addition, 36 cortical specimens were fabricated from bone taken from the femoral diaphysis. The major dimension of the rectangular specimens was aligned in either the longitudinal or circumferential direction of the bone. Within the femoral neck, the longitudinal direction aligned with the femoral neck axis. The bone specimens were prepared by first thawing the frozen bones and sectioning them at several locations. Within the distal femoral neck, the sections were cut perpendicular to the neck axis. Within the diaphysis, the sections were cut parallel to the diaphyseal axis, and within the intertrochanteric region, one cut was made which bisected the cervico-diaphyseal angle. Next, portions of the cortex along with the underlying trabecular bone were removed from each section using a hand saw. The thickness of the cortical shell was then measured using metric calipers and the trabecular bone carefully ground away using a metallurgical grinder, first on 300 and then 600 grit abrasive paper. After achieving a planar endosteal surface free of defects, the specimen was inverted and a minimum of material ground from the periosteal surface until a thin region of constant thickness was obtained (Murray et al, 1984). The local architecture was observed by transmission light microscopy at low magnification (10 x). Where a definite Haversian architecture was observed, an effort was made to align

the axes of the specimen with the material axes. The resulting plate thicknesses ranged from 0.18 to 0.4 mm. The specimen widths were held near 5.0 mm while the lengths were maintained at 7.0 mm. All bone samples were kept moist with normal saline during all phases of fabrication and testing. Mechanical

testing

The specimens were tested using an electro-hydraulic materials testing system (Model 1331, Instron Corp., Canton, MA) with a miniature fixture specially designed to apply the three-point bending loads. Within this fixture, each end of the specimen rested on a 1.5 mm diameter stainless steel cylinder. The two cylinders were parallel to each other and perpendicular to the major axis of the bone specimen. A third cylinder, which was attached to the moving cross-head of the testing system and was parallel to the first two, applied the bending loads so as to produce a strain rate of 0.05 s-l at the location of maximum stress at the bottom surface of the plate. The load was measured by a calibrated load cell (Model 41, Sensotec, Columbus, OH) and specimen deformation was measured by the displacement transducer of the test system linear actuator. The resulting data were collected using a personal computer (Model AT, IBM Corp., Boca Raton, FL) with commercially available software (Labtech Notebook, Wilmington, MA) and analyzed using in-house software. A fast Fourier transform was performed to determine the frequency content of the signal and the appropriate cut-off frequency for a low-pass filter to eliminate superimposed noise. Based on the determined cut-off frequency, the appropriate constants for a five-point low-pass filter were determined and the data filtered. Material properties

After testing, the specimens were dried and weighed on an analytical balance (Mettler No. HSlAR, Hightstown, NJ) to an accuracy of 1 mg. Densities were calculated by dividing specimen volume (as determined from moist specimen dimensions) by the specimen dry weight. Since typical specimens had a small thickness as compared to their width and length, the analytic relationship used to estimate the elastic modulus from the load-deflection data was based on the theory of thin plates (Timoshenko and Woinowsky-Krieger, 1959) E=PL’(l-vLr-vrL) 48dl

’

(1)

where E is the Young’s modulus (MPa), vLT is the . longttudmal Potsson ratto, vTLis the transverse Poisson ratio, P is the applied load (N), L is the specimen length (mm), d is the specimen deflection at mid-span (mm), and I is the cross-sectional moment of inertia (mm4). This relationship is dependent on both the elastic modulus E and the Poisson ratios vLTand vTL. The values of vLT and vrL were not measured, but

Mechanical properties of metaphyseal bone rather their product (subsequently referred to as v’) was assigned a value of 0.09 (Brown and Vrahas, 1984). A sensitivity analysis of the influence of variations of vz (from between 0.04 and 0.16) on the calculated modulus E was performed and demonstrated that errors resulting from variation of v were small (less than 3%). Additional refinement to equation (1) was made to account for the concentrated nature of the applied loads (Timoshenko and Goodier, 1970) by setting E=--

(2)

319

and P is equal to the highest load (P,,,) carried by the specimen before fracture (Timoshenko and Goodier, 1970). This calculated beam breaking strength can be related to the material ultimate tensile strength by the ‘rupture factor’ (Roark and Young, 1975), defined as the calculated maximum tensile stress in the outer fibers of the beam at failure divided by the ultimate tensile stress of the material. This parameter indicates that a finite portion of the specimen cross-section must fail before structural failure results. For brittle materials, this factor is reported to be between 1.60 and 1.75. Therefore, the material ultimate tensile strength can be related to the calculated maximum tensile stress by

where h is the specimen thickness. Assuming v* = 0.09 gives

s,=

y [

1 )

(6)

where S, is the material ultimate tensile strength and R is the rupture factor. The slope of the 1oadAeflection curve (AP/Ad) used in equation (3) was determined by calculating a best fit regression line from the data. This regression line was calculated in a moving cell fashion for a series of n consecutive points starting at the beginning of the data and stepping up to the peak load, where n equals the number of data points to the peak load divided by four. For example, where there were 250 data points to the peak load, n = 63 and regression lines were determined for points 062, l-63, 2-64 . 187-250. The regression line with the best coefficient of determination (R*) was used to represent the slope of the loadcleflection curve in its most linear region. This method gave repeatable results which corresponded well with values obtained by hand from recordings of the load-deflection data. The travel of the center cylinder was assumed to equal the plate mid-span deflection. Therefore, any local deformations present at the three points of load application would lead to an overestimation of the true beam deflection. The magnitude of this local plastic deformation could be expected to vary for different specimen materials and indentor geometries and thus was determined experimentally and incorporated into a modified form of equation (3) (4) where K is the constant multiplier of equation Appendix). The expression used to estimate ultimate strength assumed brittle behavior, with no occurring in the material prior to failure. situation the stress varies linearly from the axis, and beam theory, with a correction concentrated nature of the applied loads, giving ,,.=0.125~]-0.266[$],

(3) (see tensile yielding In this neutral for the applies,

(5)

where h is the beam thickness, w is the specimen width

Error analysis

Given the number of measurements performed for each calculation of modulus and strength, some estimate of the maximum absolute error inherent in the results is appropriate. This error was determined given the magnitude of error present in the measurement of the specimen dimensions and the assumption of a Poisson ratio product of 0.09. The bounds for the calculated parameter AF (modulus or strength) was determined given the bounds for the error of each of the variables Aa, (specimen length 1,width w, thickness h, and Poisson ratio product v*). Using elementary calculus of errors based on a first-order Taylor series expansion (Bronshtein and Semendyayev, 1985) AF=CAaiIf,i(ai,.

. ., a,Jl

(7)

wherefXi represents the partial derivative of the function with respect to each of the variables. For the calculation of the elastic modulus, using equations (1) and (7), the resulting expression for the error bounds is

AE=~~Z~z2~~v2)]+Aw~~] +Ah[3I’;;‘*)]+Av/i]}.

(8)

For the calculation of the bending strength, using equations (3) and (7), the resulting expression for the error bounds is AS=;b,[$]+Aw[--&]+Ah[$]}.(9) To verify the accuracy of the above formulations for determining the elastic modulus and ultimate tensile strength of the metaphyseal shell of the proximal femur, specimens fabricated from materials with known properties (wood and acrylic) were tested. Rectangular beams of varying thickness (0.4, 0.6, 0.8, 1.0 mm) were fabricated from bulk materials, with widths and lengths similar to those of the bone

J. C. LOTZet al.

320

samples. In addition, a series of acrylic tensile specimens were fabricated to determine the true elastic modulus and tensile strength. These cylindrical specimens were 4.5 mm in diameter and 26 mm long. They were tested to failure using the electro-hydraulic system, with the load measured using the calibrated load cell of the test system and the specimen deformation measured using a contact extensometer (Model No. 2620-532, Instron Corp., Canton, MA) attached directly to the specimen over a known gage length (12.5 mm)

22000

TT

Statistical analysis

Thickness

All statistical analyses were performed using the SAS statistical software system (SAS, Cary, NC) on an IBM personal computer. Standard analysis of variance (ANOVA) and analysis of covariance (ANCOVA) procedures were performed to compare specimen group means and to estimate the effect of the specimen variables (parent bone, harvest location and harvest orientation entered as categorical predictors, and density, entered as a continuous predictor) on the measured parameters of interest (strength and modulus). Within the ANOVA and ANCOVA procedures, each data point was weighted by its error estimate, as calculated with equation (8) or (9), in order to account for the variable precision associated with each measurement.

Fig. l(a). Uncorrected wood modulus vs specimen thickness.

30000

25000 -z z 2

20000

I s 15000

10000_ 0.4

RESULTS Wood

The influence of the local deformations in the calculation of E is demonstrated in Fig. l(a). Without the correction for local deformations (Appendix), the estimated value of E was highly dependent on the material thickness, with a value of approximately 20,000 MPa for a thickness of 0.04 mm and 10,400 MPa for a thickness of 1.0 mm (p = 0.0001, R2 = 0.84). When the correction for the local deformation was included [Fig. l(b)], this dependency of the elastic modulus on thickness was reduced. The mean value of E for the corrected data was 19,950 f 3755 [SD] MPa. The maximum fiber stress of the oak specimens demonstrated no significant correlation with thickness (p = 0.36, R2 = 0.3), with the mean bending strength calculated to be 277+ 22 MPa. Using an assumed rupture factor for wood of 1.84 (Roark and Young, 1975), the calculated ultimate tensile strength was estimated to be 150 MPa. The data compare with a range of literature values for the static bending modulus and strength of oak from 11,000 to 13,000 MPa for flexural modulus and of 82-127 MPa for bending strength (Forest Products Laboratory, 1955). Acrylic

Uniaxial tensile tests of six acrylic specimens (fabricated from the same bulk material as the beam

(mm)

0.5

0.6

07 Thickness

08

09

1.0

(mm)

Fig. l(b). Corrected wood modulus vs specimen thickness.

specimens) were performed to determine the true elastic modulus and ultimate tensile strength. These tests resulted in an average modulus of 2980 +330 MPa and average tensile strength of 72 +4 MPa both of which are within the published values of 2410-3100 MPa and 55-75 MPa respectively (Modern Plastics Encyclopedia, 1982). Three-point bending tests were performed with 16 acrylic specimens ranging in thickness from 0.4 to 1.2 mm. The trend of decreasing modulus with thickness that was apparent for the wood specimens was not observed for acrylic [p = 0.5, R2 = 0.02; Fig. 21. The resulting mean modulus was 2890f270 MPa. The mean outer fiber tensile strength was 126 + 12 MPa. Given the true tensile strength of 72 MPa, the bending strength and ultimate tensile stress were related by a rupture factor of 1.75. The elastic modulus as determined from the bending tests did not differ significantly from the values obtained from the tensile tests (t-test; t = 0.62, p = 0.5). Therefore equation (3), which was used to determine the elastic modulus, appears to give accurate results. Also, since the value of the rupture factor (1.75) is within the expected range, the three-point bending test also appears to be an accurate method for

Mechanical properties of metaphyseal bone 40001

1 2000t -~+ -+- -~03 04 05 06

07 Thickness

,

08

09

10

II

12

(mm)

Fig. 2. Uncorrected acrylic modulus vs specimen thickness.

the determination of the ultimate tensile strength. Since the geometry of the bone specimens was similar to that of acrylic, this empirical rupture factor of 1.75 was also applied to the cortical shell data. Bone--elastic

modulus

Typical diaphyseal bone loadcleflection curves for longitudinal and transverse directions are shown in Fig. 3. The averaged values for modulus at the three harvest locations is presented in Table 1. Both the modulus and density measured for specimens obtained from the diaphyseal region are on the average

321

greater than those measured for the proximal specimens. In the longitudinal direction, the mean value of the elastic modulus of all the proximal specimens was 9560 + 2410 or 23% less than that measured within the diaphyseal region. The average density of the proximal specimens (1.59 g cc- ‘) was 8% less than that measured for the diaphyseal specimens (1.72 g cc- I). The results of an ANOVA (Table 2) demonstrated that the difference between longitudinal modulus measured for the proximal specimens and that measured in the diaphysis is significant (p = O.OOOl), while no significant difference exists between the longitudinal modulus measured for the two proximal groups (p = 0.353). For the transverse direction, the mean value of the elastic modulus of all the proximal specimens (5450 f 1720 MPa) was less than that of the diaphyseal specimens (5990 + 1520 MPa). However, the difference was not significant (p > 0.05, Table 2). The density of the proximal specimens (1.65 gee- ‘) was 5% less than that measured for the diaphyseal specimens (1.73 gee-1). The elastic modulus data for the entire specimen population (proximal plus diaphyseal) is shown plotted against specimen density in Fig. 4. An ANCOVA was performed to determine whether a significant difference exists between the modulus/density relationship for specimens tested in either the longitudinal

Table 1. Specimen data by orientation and location Orientation

Longitudinal Longitudinal Longitudinal Transverse Transverse Transverse

Location

Diaphyseal Superolateral Inferomedial Diaphyseal Superolateral Inferomedial

”

19 31 29 17 37 26

E

~m,x

P

MPa

MPa

gcmm3

12500 (2140) 9280 (2000) 9860 (2800) 5990 (1520) 5380 (1610) 5550 (1900)

225 (28) 177 (44) 177 (47) 83 (21) 86 (22) 88 (22)

1.72 (0.10) 1.59 (0.12) 1.59 (0.13) 1.73 (0.07) 1.65 (0.11) 1.64 (0.13)

Table 2. Weighted ANOVA for differences in modulus and stress Orientation

Parameter*

t

P

- 2920 (665) - 3420 (663) 50.5 (541)

-4.39 -5.16 0.93

0.0001 0.0001 0.353

-53.9 (12) -58.4 (12) 4.45 (10)

- 4.45 -4.82 0.45

0.0001 0.0001 0.656

(566) (539) (472)

-0.50 - 1.18 0.74

0.617 0.243 0.459

(7) (6) (6)

0.14 0.70 -0.59

0.887 0.488 0.554

Value

MPa Longitudinal

E,-E, Es-E, E,-Es

Longitudinal

cl-@D us*0 ~1-~s

Transverse

E,-E, Es-E, &Es

Transverse

Ul-UD US-(TD ut-us

-284 -635 351 0.9 4.3 -3.4

*D refers to diaphyseal, S to superolateral, I to inferomedial, E to elastic modulus and u refers to maximum fiber tensile stress.

322

J. C. LOTZ

et al.

-pa*

-2%

0.0

-0 2

-02

00

02

Stroke (mm)

1.0

1.2

Stroke (mm)

Fig. 3(a). Diaphyseal bone load-deflection

curve for speci-

mens fabricated in the longitudinal direction.

Modulus

Fig. 3(b). Diaphyseal bone load-deflection curve for speci mens fabricated in the transverse direction.

vs. Density

by Orientation (Diaphyseal

and

Proximal

Combined)

20000

18000

t

Ic

0

-

Longitudlnel;

0

-

Transverse.

modulus modulus

= -13430+1426l(density) = -3122+4979(density)

16000 14000

0

a

3 3

12000

2 z

10000

4 6000

6000

I

1.50 Density

(pm/cc)

Fig. 4. Longitudinal and transverse modulus vs specimen apparent density.

or transverse directions. In addition to density, harvest orientation and a random effect for parent bone, a cross-term of density with orientation was included in the model to account for a varying effect of density on the elastic modulus for different specimen harvest orientations. This model fit the data well (R2 = 0.67) and indicated that the modulus had a different dependence on density for specimens tested in either the

longitudinal or transverse direction as demonstrated by the significance of the model parameter associated with the above-mentioned cross-term (p = 0.0001). Since, in addition to density, the specimen elastic modulus appeared dependent on harvest iocation, harvest orientation and parent bone, an additional ANCOVA was performed using all these parameters. Also, cross-terms of density with orientation and

Mechanical properties of metaphyseal bone

Stress

323

vs. Density

by Orientation (Diaphyseal

1 00

and Proximal

Cl -

Longltudlnal.

stress

= -180+223(dens1ty)

0

Transverse,

stress

=

-

Combined)

-4t3+78(density)

1 25

I.50 Density

1.75

2 00

(gm//cc)

Fig. 5. Longitudinal and transverse maximum fiber stress vs specimen apparent density

orientation with location were included in the model to account for a varying effect of density on modulus for different specimen orientations and a different effect of orientation on modulus for different specimen harvest locations. The results of this ANCOVA indicate that the specimen density (p = 0.0001) and parent bone (p = 0.020) significantly affected the elastic modulus. In addition, the variable effect of density with orientation was also significant (p = 0.0035). Once these variables were included in the statistical model, there was no significant residual effect of specimen harvest location on the elastic modulus (p = 0.09 19).

Bone-strength

The maximum fiber tensile stress measured for specimens obtained from the diaphyseal region were in general greater than those measured for the proximal specimens. In the longitudinal direction, the mean value of the maximum fiber tensile stress of all the proximal specimens was 101 MPa (177/1.75) or 21% less than that measured within the diaphysis. This proximal population also demonstrated a reduction in density of 8% when compared to the diaphyseal specimens. The results of an ANOVA (Table 2) demonstrates that a significant difference does exist between the longitudinal maximum fiber

tensile stress measured for the proximal specimens and that measured in the diaphysis (p < O.OOl),while no significant difference exists between the logitudinal maximum fiber tensile stress measured for proximal groups (p = 0.656). In the transverse direction, the mean value of the tensile strength of the proximal specimens (50 MPa or 87/1.75) was not significantly different than that of the diaphyseal specimens (47 MPa or 83j1.75) (p > 0.05, Tables 1 and 2). The maximum fiber tensile stress data for the entire specimen population (proximal plus diaphyseal) is shown plotted against specimen density in Fig. 5. To determine whether a significant difference exists between the stress/density relationship for specimens tested in either the longitudinal or transverse directions an ANOVA was performed as with the elastic modulus data. This model fit the data well (R’ = 0.78) and indicated that the maximum fiber tensile stress had a different dependence on density for specimens tested in either the longitudinal or transverse direction as demonstrated by the significance of the density/orientation cross-term (p = 0.0001). As with the modulus data, a statistical model was created to determine which parameters in addition to density were statistically significant for the maximum fiber tensile stress. The results of this ANCOVA indicate that the specimen density (p = O.OOOl),the

324

J. C. LOTZ

variable effect of density with orientation (p = 0.011) and the variable effect of orientation with location (p = 0.003) all significantly affected the maximum fiber tensile stress. Once these variables were included in the statistical model, there was no residual effect of the specimen harvest location (p = 0.1817) or the parent bone (p = 0.4659).

DISCUSSION

Metuphyseal vs diaphyseal properties

The elastic properties of diaphyseal bone have been widely characterized, supporting a transversely isotropic model. The elastic constants for femoral diaphyseal bone in tension have been presented by Reilly and Burstein (1975) as 17,000 MPa in the longitudinal direction and 11,500 MPa in the transverse direction. The results from the three-point bending tests presented here differ from these values. For diaphyseal bone, the mean longitudinal modulus was 12,500 MPa or 26% lower than that reported above. However, the average diaphyseal specimen density of 1.72 gee-’ is well below that which would be expected for young healthy bone (1.95 g cc- ‘). Using the linear weighted regression derived from our data (Fig. 4), specimens with a density of 1.95 gee- ’ would have an estimated modulus of 14,400 MPa, which is closer to the published value. In addition, the calculated transverse modulus of 5990 MPa was also less (48%) than that reported above. Performing a similar extrapolation to 1.95gcc-’ using the transverse specimen regression (Fig. 4) results in a value of 6590 MPa. The residual differences may be due to an increased sensitivity of the bending test to local defects. The diaphyseal specimens tested here exhibited an Haversian architecture, the canals of which could potentially occupy a large portion of the cross-section. This would have the effect of decreasing the effective moment of inertia and lead to an underestimate of the material elastic modulus based on surface dimensions alone. For instance, based on an average canal diameter of 90 pm and average specimen dimensions, the effective moment of inertia would be overestimated by 50% (for specimens tested in the transverse direction). The specimens fabricated from the proximal femur demonstrated significantly decreased mechanical properties from those of the diaphysis. Within the proximal femur, the mean value of the longitudinal elastic modulus was 9650 MPa or 23% less than that measured for the diaphysis (12,500 MPa). However, the transverse elastic moduli did not differ significantly between the diaphyseal (5990 MPa) and proximal (5470 MPa) populations despite the significant density difference. These observations were partially explained by noting the variable effect of density on modulus for specimens of different orientations. Surprisingly, once one allows for a different slope for modulus vs density for specimens of different harvest orientation (Fig. 4), specimen harvest orientation

et al.

alone is no longer significant for predicting specimen elastic modulus. This result suggests that as density changes, the specimen architecture changes at a different rate in the two testing directions, and consequently, architectural as well as density differences exist between the proximal and diaphyseal regions. This is further supported by the measured difference in the ratio of longitudinal to transverse moduli for the diaphyseal(2.09) and proximal (1.76) groups. Figure 6 shows transmission light micrographs (20 x ) of specimens taken from the diaphyseal and proximal regions. Architectural differences are apparent, although a detailed analysis of micrographs of all specimens was not performed. For the cortical shell within the proximal femur, our overall mean values for the elastic properties were Erongitudinar ~9650 MPa E tranSYerSe = 5470 MPa

p= 1.62 gcmw3. These calculated values of the elastic modulus E, and E, are associated with standard deviations of 2400 and 1760 MPa respectively, which are near those reported by other investigators for specimens tested in pure tension or compression (Burstein et al., 1976; Currey and Brear, 1974; Dickenson et al., 1981). The ultimate tensile strength of femoral diaphyseal bone has also been widely investigated. Reilly and Burstein (1975) found the values of the ultimate tensile strengths to be Q,,,= 133 MPa (longitudinal) and cult=51 MPa (transverse). For our data, the use of a rupture factor of 1.75 resulted in the diaphyseal ultimate tensile strength for the longitudinal (128 MPa) and transverse (47 MPa) specimens which did not differ significantly from those values presented above (p > 0.05). This was observed despite the density differences as suggested by the modulus data between our population and that of Reilly and Burstein. Since the flexural rigidity, from which our modulus was estimated, is sensitive to defects throughout the entire structure, differences between our diaphyseal modulus data and that of others are reasonable. However, the strength is dependent only on the material where the failure occurs, presumably at the center of the beam where the bending moment is the greatest (for threepoint bending). The use of four-point bending would produce a region of constant bending stress in which failure would occur at the weakest location. Therefore, if four-point bending were used, it could be expected that the strength data would demonstrate similar sensitivities to heterogeneous porosity as the modulus data and as a result similar differences from previously reported values. The proximal specimens demonstrated a 21% decrease in longitudinal ultimate tensile strength compared with diaphyseal bone. As with the elastic modulus, this difference was explained by differences in density and a variable effect of density with orientation. Additionally, there was a varying effect of ori-

Fig. 6. Transmission

light micrographs

(20 x ) of diaphyseal beam specimen (above) and proximal specimen (below).

325

beam

Mechanical properties of metaphyseal bone entation with harvest location. This latter result indicates that there was a large orientation effect (in excess of that explained by density) at one location (diaphyseal) and a small orientation effect at the others (proximal). Together, these data support the hypothesis that there exists a slight decrease in transverse isotropy for the proximal specimens. The globally averaged values for the ultimate tensile strengths of the cortical shell within the proximal femur were CT,,~ = 101 MPa longitudinal o,,, = 50 MPa transverse. These calculated values of the ultimate tensile strength were associated with standard deviations of approximately 25%. The characteristics of the load-deflection data demonstrate additional differences between the longitudinal and transverse populations. The specimens fabricated in the transverse direction failed in a more brittle fashion, with the strain at failure in the longitudinal direction being approximately four to five times that in the transverse direction (Fig. 3). This also agrees with the observations of others (Reilly and Burstein, 1975). The use of three-point bending experiments potentially results in greater uncertainty in the estimation of specimen material properties than standard tensile or compression tests. The three primary sources of this additional uncertainty are: (1) the inability of the analytic formulations to completely account for the complex stress state which includes regions of compression, tension and plastic deformation; (2) inclusion of errors inherent in the added dimensional measurements which must be made for each experiment; and (3) the small specimen geometries, resulting in an increased sensitivity to material defects. The validation studies performed with acrylic demonstrate that the determination of elastic modulus and ultimate tensile strength can be made with good accuracy for defect free material. In addition, the experiments performed with the oak specimens demonstrate the need to consider the magnitude of the local deformation at the site of load application. However, for small specimen thickness (co.5 mm) this local deformation resulted in less than 1% error and hence could be neglected. Comparison with previous metaphyseal and subchondral bone data

Other investigators studying the mechanical properties of the metaphyseal shell or subchondral bone have reported values for elastic modulus and strength which are far less than those we report here. Murray et al. (1984) estimated the elastic modulus and strength of the metaphyseal shell from the proximal tibia by testing thin, circular disks of 7.5 mm diameter and 0.14.4 mm thickness. Modulus and strength values were inferred from thin plate theory with no

321

corrections applied for local deformations at points of load application. Good agreement with literature values for aluminum was reported when using their specimen configuration over this range of thicknesses. The resulting modulus values ranged from about 1500 MPa at the most proximal metaphysis to about 7000 MPa in the diaphysis. Calculated values for ultimate strength ranged from 90 to 310 MPa in the shell and diaphysis. The authors did not report property variations with local apparent density nor did they suggest reasons for the markedly reduced modulus and strength values observed. Brown and Vrahas (1984) measured the elastic modulus of the subchondral bone from the human femoral head using an ingenious spherical shell specimen with thicknesses ranging from 1 to 4 mm. Modulus was estimated from shell theory, with no corrections applied for localized deformation at the points of load application. Excellent agreement with literature data for acrylic was observed for shell thicknesses of about 1.5 mm, with shells of 4.0 mm thickness yielding modulus values about 40% below literature data. They reported apparent modulus values for subchondral bone which depended on thickness, ranging from about 1500 MPa (for 1 mm thick specimens) to about 500 MPa (for 4 mm thick shells). While apparent density data were not reported, the authors attributed the markedly reduced modulus values of the subchondral shell to the presence of histologically evident marrow spaces which extended nearly to the specimen surfaces. Recently, Choi et al. (1989) reported modulus values for subchondral and cortical bone from the human proximal tibia. These were determined using small, rectangular beam specimens with cross-sectional dimensions of between 0.1 and 0.2 mm and length of about 1.0 mm. The flexural modulus was calculated using beam theory with the same corrections for shear effects that are reflected in our equation (3). They reported good agreement with the literature values for acrylic and for larger specimens (1 x I x 10 mm) but did not report validation data for acrylic specimens over the specimen size ranges used for bone. For cortical and subchondral bone, they reported modulus values of 5440 and 1150 MPa, respectively. Apparent density was not reported but a microradiographic estimate of mineral density was significantly correlated with the cortical property measurement. Size effects were also found to be important, with cortical specimens of cross-sectional dimensions of about 1 x 1 mm yielding modulus values in the range of previous literature (about 15,000 MPa). Cortical microspecimens in the range of those used for subchondral bone (0.1 x 0.1 mm) yielded modulus estimated of about 5500 MPa. The authors suggested the effects of local structural defects to be the reason for these reductions. In contrast to these earlier studies, our results suggest that the thin cortical shell surrounding the proximal femur is characterized by mechanical

328

J. C. LOT2

properties which are only moderately reduced from those of the diaphysis with most of the reduction accounted for by differences in density. Since we did not measure properties for subchondral bone in these experiments, direct comparisons to the data reported by Brown and Vrahas (1984) and Choi et al. (1989) are not possible. We suspect, however, that apparent density (along with the possibility oflocal deformation effects at the load points for the thicker shell specimens of Brown and Vrahas) would largely account for the low modulus values they report for subchondral bone. Our results indicate that analytical models of the proximal femur should incorporate values for the elastic modulus and strength of the metaphyseal shell in the femoral neck and intertrochanteric regions different from those used for the femoral diaphysis. Specifically, the metaphyseal elastic modulus and strength should be reduced by approximately 23 and 21% respectively. Anisotropy should also be considered with the ratio of major to minor properties being 1.8 for the elastic modulus and 2.0 for strength. The direction of these principle axes should be aligned with the cervical axis when within the femoral neck and with the diaphyseal axis when within the femoral diaphysis. Within the intertrochanteric region, the orientation of the principle material property direction lies approximately between that of the femoral neck and diaphysis. The incorporation of these reduced metaphyseal material properties should lead to improved performance of analytical models of the proximal femur and thus increased attention on the structural contributions of trabecular bone within this region. Acknowledgement-This

study was supported by grants from the National Institutes of Health (CA 41295) and from the Centers of Disease Control (CR 102550)and by the Maurice E. Mueller Professorship of Biomechanics at the Harvard Medical School (WCH). We thank Jeanine Goodwin and Steve Stern for aHsistan&ein manuscript preparation. REFERENCES

Bronshtein, I. N. and Semendyayev, K. A. (1985)Handbook of Mathematics. Van Nostrand-Reinhold,.New York. _ Brown. T. D. and Vrahas. M. S. (19841The aeuarent elastic modulus of the juxtarticular ‘subchondrai ‘bone of the femoral head. J. Ortbop. Res. 2(l), 32-38. Burstein, A. H., Reilly, D. T. and Martens, M. (1976) Aging of bone tissue: mechanical properties. J. Bone Jt Surg. (Am) 58, 82-86.

Choi, K., Kuhn, J. L., Ciarelli, M. J. and Goldstein, S. A. (1989) The elastic modulus of trabecular, subchondral, and cortical bone tissue. Trans. 35th ORS 14, 102. Cowin, S. C. (1988) The mechanical properties of cortical bone tissue. Bone Mechanics (Edited by Cowin, S. C.), pp. 97-127. CRC Press, Boca Raton, FL. Currey, J. D. and Brear, K. (1974) Tensile yield in bone. Calc$ Tissue Res. 15, 173-179. Dickenson, R. P., Hutton, W. C. and Stott, J. R. R. (1981) The mechanical properties of bone in osteoporosis. J. Bone Jr Surg. (Br) 63(2), 233-238. Forest Products Laboratory (1955) Basic information on wood as a material of construction with data for its use in

et al.

design and specification. In U.S. Forest Service Wood Handbook.

Lipson, S. F. and Katz, J. L. (1984) The relationship between elastic properties and microstructure of bovine cortical bone. J. Biomechanics 17, 231-240. Lotz, J. C. and Hayes, W. C. (1990) Estimates of hip fracture risk from falls using quantitative computed tomography. J. Bone Jt Surg. (Am) 72, 689-700. Modern Plastics Encyclopedia (1982) McGraw-Hill, New York. Murray, R. P., Hayes, W. C., Edwards, W. T. and Harry, J. D. (1984) Mechanical properties of the subchondral plate and the metaphyseal shell. Trans. 30th ORS 9, 197. Reilly, D. T. and Burstein, A. H. (1975) The elastic and ultimate properties of compact bone tissue. J. Biomechanits 8, 393405.

Riggs, B. L., Wahner, H. W., Seeman, E., Offord, K. P., Dunn, W. L., Mazess, R. B., Johnson, K. A. and Melton, L. J. III (1982) Changes in bone mineral density of the proximal femur and spine with aging. Differences between the postmenopausal and senile osteoporosis syndromes. J. Clin. Invest. 70, 716723. Roark, R. J. and Young, W. C. (1975) Formulasfor Stress and Strain (5th Edn). McGraw-Hill, New York. Timoshenko, S. P. and Goodier, J. N. (1970) Theory of Elasticity. McGraw-Hill, New York. Timoshenko, S. and Woinowsky-Krieger, S. (1959) Theory of Plates and Shells. McGraw-Hill, New York.

APPENDIX

The magnitude of the local deformation at the point of load application during three-point bending was determined experimentally using the three-point test center cylinder (attached to the moving cross-head of the test system) and the particular specimen material of interest. This test cylinder was pressed into a rigidly backed block of the specimen bulk material. We assumed that for this testing configuration the deformation is a function of the applied load, the specimen width, and the specimen thickness d =f (P, b, h),

(AI)

where P is the applied load, b is the specimen width, and h is the thickness. To determine the nature of this expression, experiments were performed with oak, acrylic and bone in which a concentrated load was applied to a slab of material while recording the resulting force/displacement relationships. The results demonstrated an initial nonlinear region at the initiation of contact, followed by a highly linear zone. This linear region was used to determine the relationship between the contact load and the local deformation [equation (A.l)]. Slab tests were performed for a series of specimen widths and thicknesses. We assumed a linear relationship of the form (A.2) where l/Q is the slope of the linear portion of the load/deflection curve, w is the specimen width and h is the specimen thickness. By measuring l/Q from the results of the slab tests of several samples of varying widths and thicknesses, the constants K,, K, and K, were determined for each material. Thus, given the specimen width, the value of Cpwas determined for each sample, and the deflection due only to the local deformation (2AP@) was subtracted from the total beam deflection d. This gives a modified form of equation (3)

Mechanical properties of metaphyseal bone

329

Table A 1. Constants relating local deformation to sample size Material (Ntib)

White oak Acrylic

140 680

I