Journal of Orthopaedic Research 9674-682 Raven Press, Ltd., New York 0 1991 Orthopaedic Research Society

Evaluation of Orthogonal Mechanical Properties and Density of Human Trabecular Bone From the Major Metaphyseal Regions with Materials Testing and Computed Tomography M. J. Ciarelli, S. A. Goldstein, J. L. Kuhn, *D. D. Cody, and TM. B. Brown Orthopaedic Research Laboratories, Section of Orthopaedic Surgery and ?School of Public Health, Biostatistics, University of Michigan, Ann Arbor; and *Medical Image Research Laboratory, Department of Radiology, Henry Ford Hospital, Detroit, MI U.S.A.

Summary: We evaluated the orthogonal mechanical properties of human trabecular bone from the major metaphyseal regions with materials testing and quantitative computed tomography (CT). The proximal tibia, distal femur, proximal femur, distal radius, and proximal humerus from fresh cadaver specimens between the ages of 55 and 70 years were excised and prepared for experimentation. The bones were embedded and scanned at 1 or 1.5 mm intervals on a Technicare HPS 1440 and GE 9800 CT scanner. After scanning, the bones were sectioned, producing 8-mm cubes of trabecular bone which were mechanically tested in uniaxial compression at a strain rate of l%/s. The testing sequence consisted of preyield tests in two of the three orthogonal directions and failure in the third. After testing, the cubes were evaluated for apparent density and ash weight. The results of the study show that the strength and stiffness of trabecular bone varies significantly within metaphyseal regions and from metaphysis to metaphysis. The power and significance of relationships between density and modulus varied as a function of metaphyseal location. Both linear and nonlinear models were significant, suggesting that trabecular deformation occurs in response to both axial and bending loads. Finally, the need for architectural measures of trabecular bone to predict mechanical properties is emphasized. Key Words: Trabecular bone-Computed tomography density-Modulus-Orthogonal properties-Anatomic variation.

erties and anisotropy is necessary to support experimental and analytical studies related to normal or abnormal joint function, aging phenomena, mechanically induced remodeling, and response to total ioint arthroplasty. Received November 7, 1990; accepted January 23, 1991. Previous investigators have shown that the mePresented in part at the 32nd Annual Meeting of the Orthopaeproperties of trabecular bone are similar to dic Research Society, February 17-20, 1986, New orleans, L ~ ~ chanical those of fluid-filled porous engineering isiana. - materials (4) Address correspondence and reprint requests to Dr. Steven A. and that its porosity and structural orientation conGoldstein at Orthopaedic Research Laboratories, University of Michigan, G-0161 400 N. Ingalls, Ann Arbor, MI 48109-0486, tribute to its anisotropy (9,10,24,26). The strength U.S.A. and stiffness of trabecular bone has also been

Characterization of the mechanical properties of human metaphyseal trabecular bone remains an important endeavor in musculoskeletal research. Quantifying the distribution of its mechanical prop-

6 74

MECHANICAL PROPERTIES OF TRABECULAR BONE

shown to be proportional to its mineral density and orientation (1,4,9,15,20,22,24). It is the variability of these parameters within trabecular bone that accounts for its wide range of mechanical properties. The phenomenon known as Wolff s Law, which assumes that mechanical stresses resulting from normal joint function influence the orientation, structure, and strength of bone (27), has been shown to explain the tremendous diversity in trabecular bone mechanical properties as a function of anatomic location (3,12,14,18). Trabecular bone specimens machined from a variety of anatomic locations, including the proximal and distal femur, proximal tibia, and vertebral bodies, have been mechanically tested by many investigators (1,3,4,8,9,12,14,15,18,22,25,28).Brown and Ferguson (3) determined the orthogonal mechanical properties of 5-mm cubes of trabecular bone from the proximal femur. They showed that the variations in stiffness correlated well with radiographic trabeculation patterns and that stiffness and yield strength were linearly proportional. Experimental determination of the modulus and strength distribution in the proximal tibia1 metaphysis (12) and epiphysis (14) has also been reported. Consistent patterns of increased stiffness arose from the medial and lateral cortices and extended inward beneath the contact regions of the femoral condyles, thus supporting the hypothesis of functional remodeling. Although these and other studies have provided more accurate data for incorporation in analytic models of joint function or replacement, to our knowledge no study has attempted to catalogue mechanical property distributions of multiple metaphyseal regions tested under similar conditions. We wished to determine the orthogonal mechanical properties of trabecular bone of multiple human metaphyseal locations. In addition, this study was designed to evaluate the use of computed tomography (CT) as a noninvasive method of estimating the lo-

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cal mechanical properties of trabecular bone. The results of this study will enable direct comparisons of mechanical properties and the degree of mechanical anisotropy within selected anatomic regions. MATERIALS A N D METHODS

Bone specimens from the metaphyseal regions of four autopsy subjects between the ages of 55 and 70 years were obtained within 48 h of donation and immediately stored at - 10°C in sealed bags. None of the specimens exhibited severe arthritic pathology or pathologic osteopenia on gross examination. All subjects appeared to be of average weight and stature, with no previous medical conditions or treatments that would have affected bone mineralization. The left and right proximal and distal femurs, proximal humeri, and distal radii of a 70year-old male, the left and right distal femurs of a 62-year-old male, and the right proximal tibias of a 55-year-old female and a 69-year-old male were excised and cleaned of soft tissue (Table 1). The diaphysis of each bone was embedded in an aluminum block using a plaster mix (Vel-Mix Stone, Kerr, Chicago, IL, U.S.A.). Positioning fixtures standardized the orientation of the bone specimens in an anatomic position with the anterior-posterior (AP), medial-lateral (ML), and inferior-superior (IS) directions forming the basis of the reference coordinate system. Under constant irrigation, each bone specimen was sectioned on a computer-controlled numerical milling machine (Series I CNC, Bridgeport Machines, Troy, MI, U.S.A.) equipped with a thin stainless-steel blade. The initial cut on each metaphysis removed the articular surface and subchondral plate, exposing a plane of trabecular bone. If necessary, additional parallel cuts were made which removed incremental amounts of trabecular bone until this first layer was large enough to ac-

TABLE 1. Number and distribution of trabecular bone cubes tested Metaphyseal regions tested Specimen 70-Year-old male 62-Year-old male 55-Year-old female 69-Year-old male Totals

Proximal tibia 42 (R) 53 (R) 95

Distal femur

Proximal femur

Proximal humerus

Distal radius

Totals

220 (R and L) 270 (R and L)

54 (R and L) -

74 (R and L)

10 (R and L) -

358 270 42 53

54

74

10

723

-

-

-

490

The designations R and L correspond to the origin of the cubes; the right and left bones of the autopsy subjects. The uneven distribution is due to the relative differences in size between metaphyseal regions.

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commodate the appropriate number of bone cubes. In all cases, the total amount of trabecular bone removed was 1-4 mm and was standardized within each metaphyseal region to maximize the number of bone cubes in the first layer. Each region was then sectioned in two perpendicular directions producing 8 X 8-mm columns of bone referenced to the natural anatomic axes (Fig. 1A). The proximal and distal femurs, proximal humeri, and distal radii were placed in a water-filled cylindrical Plexiglas chamber to simulate soft tissue mass and examined on a Technicare HPS 1440 CT scanner (Technicare, Cleveland, OH, U.S.A.). The scanning parameters were 130 kV and 100 mA with a 4-s scan time. A subset of specimens was also analyzed on a General Electric 9800 CT scanner (General Electric Medical Systems, Waukesha, WI, U.S.A.). Each metaphyseal region was positioned alone or in pairs and scanned to include the entire presectioned volume. The CT slices (1 or 1.5 mm thick) were contiguous, with standard beam hardening corrections applied. The sectioning and scanning protocol provided a method for direct comparisons of volumes of trabecular bone which were scanned by CT and subsequently mechanically tested (Fig. 1B). From each axial image, the mean and SD of the CT Hounsfield number, measured in Hounsfield units (HU), was calculated for each specimen region. The mean Hounsfield numbers from sequential CT slices were averaged to determine an average CT density, subsequently referred to as CT density, for each corresponding 8-mm bone cube. The CT scale is based on relative attenuation of x-rays by a scanned body as compared with attenuation by water. In general, zero HU equals the density of water and - 1,000 HU corresponds to the relative density of air. Cortical bone had CT values greater than + 1,000HU whereas trabecular bone had values ranging from - 25 to 714 HU. An average CT

value of water was determined for each CT scan and used to adjust the trabecular bone values for scanner drift. After scanning, the bones were remounted in the computer-operated milling machine and sectioned along the third plane to produce a series of 8-mm cubes of trabecular bone. Each cube was stored in Ringers’ solution, and its location and orientation was documented. Cubes containing any cortical bone remnants were removed from the study. The three-dimensional control of the numerical milling machine insured planoparallel sides on all our trabecular specimens. All sectioning was done at constant room temperature and under constant irrigation. Care was taken to expose all specimens to approximately the same environmental conditions such as thawing time to mechanical testing. The defined sectioning matrices for each metaphyseal region were standardized so that the results of the experiment could be averaged across our sample population. The number and location of extracted specimens are summarized in Table 1. Due to the relatively large size of the distal femur, most cubes tested were from the two pairs of distal femurs. Each cube was mechanically tested on an Instron Materials Testing System (Model 1000, Instron, Canton, MA, U.S.A.) in the three orthogonal directions corresponding to the anatomic axes of AP, ML, and IS. The testing sequence, as described by Kuhn et al. (16), consisted of uniaxial compressive loading to incipient yield in two directions (preyield) followed by loading to failure in the third. Incipient yield was determined by monitoring the load-deflection curve on a storage oscilloscope. The order of testing was alternated so that the failure direction was equally distributed between the three orthogonal directions of AP, ML, and IS. The load data were acquired directly by a Tektronix 4054 computer system (Tektronix, Beaverton, OR,

FIG. 1. First layer of trabecular bone of a distal femur with saw marks defining 8-mm columns of bone (left, A). Computed tomography (CT) scan of the same femur (right, B). The saw marks are clearly discernible and provide a guide for matching CT regions accurately with their corresponding bone cubes.

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677

MECHANICAL PROPERTIES OF TRABECULAR BONE AP Modulus vs CT #

.

y=

- 97.448 + 1.6201~Rz= 0.682

800 -

lo00

600-

a

<

I

.

200

0

.

I

.

.

I

400 600 CT #, Hounsfield

.

.

800

ML Modulus vs CT# y= -

d

I

60.822+ 1.0976~ 2 = 0.427

-

0

200

400 CT #, Hounsfield

600

IS Modulus vs CT #

The specimens were tested moist at constant room temperature in uniaxial compressive stress at a strain rate of -l%/s. Before loading, each specimen was preloaded between three and six times to -4040% of the ultimate load, as necessary, until a repeatable load-displacement curve was produced. Each cube was then tested in preyield in two directions and failure in the third. The tangent elastic modulus was calculated from each stress-strain curve by performing a linear regression on the most linear portion of the response curve. After mechanical testing, the specimens were evaluated for apparent density according to the combined protocols of Arnold (29), Galante et al. (9), and Carter and Hayes (4). Apparent density was defined as wet weight (in grams) divided by' bulk volume (in cubic centimeters). Wet weight was calculated by removing the marrow with a highpressure water spray, degreasing the specimens in ethanol, centrifuging them at 9,000 rpm for 15 min, and weighing them. Finally, the specimens were dried at 100°C for 4 h, ashed at 500°C for 60 h, and weighed for ash weight density calculations. Linear and nonlinear regression analyses were performed on the density and mechanical data using the least-squares method. The nonlinear analysis involved logarithmic transformations of both the mechanical and density measures. The variation of these parameters as a function of metaphyseal location was then evaluated with a repeated measures analysis of variance.

1200 lo00 -

y = 39.069 + 1.3988~ R2=0.577

RESULTS

$

E

800-

j.

d

1z

400-

0

200

400 CT #, Hounsfield

600

800

FIG. 2. Relationships between the anterior-posterior, rnediallateral, and inferior-superior moduli (MPa) and computed tomography density are illustrated above. This analysis represents pooled data for all metaphyseal regions.

U.S.A.) through an analog-to-digital converter. The displacement data were calculated using cross-head displacement with a correction for system compliance.

The orthogonal mechanical properties (tangent elastic moduli), failure stresses, and density measures of trabecular bone cubes from the proximal humerus, distal radius, proximal tibia, proximal femur (femoral head), and distal femur of four autopsy subjects (Table 1) were determined. These results indicate that the stiffnesses and anisotropy, quantified by the degree of differences between the three orthogonal moduli, vary significantly within the individual metaphyseal regions and from metaphysis to metaphysis (p < 0.0001). For all regions, the IS modulus was significantly greater than both the AP and ML moduli. The AP moduli were either greater than or not significantly different from the ML moduli depending on the region. The overall geometric mean of the IS moduli was 2.50 times greater than that of the AP moduli, which was 1.45 times greater than that of the ML moduli. The maximal moduli were found in the distal femoral meJ Orthop Res, Vol. 9, No. 5 , 1991

M . J . CIARELLI ET AL.

678

taphysis, and the minimal values were from the distal radius. To evaluate correlations between density measures and mechanical parameters, the data were analyzed for each individual metaphyseal region as well as grouped together over the entire population. A statistically significant relationship was noted between the tangent moduli and the CT densities. When the CT density was used to predict the modulus in any one orthogonal direction on the pooled data, the unexplained variance was very large, up to 60% (Fig. 2). Because the three orthogonal moduli were highly correlated for each cube (? = 0.340.80), the values were averaged to produce a simple gross estimate of the overall stiffness for that cube. The relationship for the pooled data between CT density and mean modulus was excellent (Fig. 3), explaining 79% of the variance. A linear relationship between apparent density and CT density was also significant (? = 0.82), a finding that supports the use of CT as an excellent predictor of local in vivo bone density (Fig. 4). Ash weight density varied linearly with both apparent density (? = 0.89) and CT density (? = 0.84) and was within normal ranges of published values for trabecular bone. In addition, all CT density measurements were nearly identical on both the General Electric and Technicare scanners (? = 0.98). The relationships between the tangent moduli of the trabecular bone specimens and the density determined either by CT or physical measures were analyzed by both linear and nonlinear regression techniques. As shown in Table 2, significant correlations were observed for both the linear and power models. In addition, these results indicate that the ----

0

400

81 0

600

CT #, Hounsfreld FIG. 4. A statistically significant relationship was noted between apparent density (g/cm3) and computed tomography (CT)density, supporting the use of CT as a predictor of local in vivo bone density.

amount of variance explained, the coefficients of the regression equations, and the power of the relationship are different for different testing directions as well as different metaphyseal regions. The order of the power for the nonlinear models ranged between 0.95 and 3.55. As expected, the best relationships were those between the mean modulus and density. To explain the differences in the regression relationships between CT density and modulus and apparent density and modulus, we performed a nonlinear regression on CT density and apparent density. The nonlinear model showed that CT density was related to apparent density raised to the 1.32 power (? = 0.79). Although the amount of variance explained is slightly less than a linear model of CT 20

I

y = - 38.644+ 1.3665~Rz=0.791

200

15

h 10

5

0

200

400

600

800

CT #, Hounsfield FIG. 3. Average of individual directional moduli correlated very well with computed tomography density. Density, which is a scalar quantity, was better correlated to an average stiffness because it is representative of the entire bone specimen.

J Orthop Res, Vol. 9, N o . 5 , 1991

0 0

200

400

600 800 Failure Modulus, MPa

lo00

1200

FIG. 5 Failure stress versus failure modulus. Similar to results of previous studies, the failure stress (MPa) was linearly related to the tangent moduli (MPa). This analysis represents pooled data for all metaphyseal regions.

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MECHANICAL PROPERTIES OF TRABECULAR BONE

TABLE 2. Relationship between CT density (Hounsfield units), apparent density (glcm3),ash density (glcm3), and orthogonal moduli (MPa) Ei

=

Ash density

Power

3

Power

?

Power

r.'

0.50 0.31 0.48 0.54

53 53 53 53

2,046 1,192 2,739 1,992

0.55 0.29 0.46 0.54

1.37 1.10 0.99 1.10

0.84 0.57 0.54 0.87

1.94 1.38 2.17 1.80

0.55 0.40 0.48 0.57

1.49 0.96 1.43 1.25

0.56 0.34 0.37 0.49

1,231 957 883 ,019

0.65 0.55 0.38 0.77

278 279 280 277

2,311 1,801 1,656 1,921

0.68 0.58 0.40 0.80

1.44 1.62 0.88 1.21

0.59 0.58 0.46 0.79

1.78 2.15 0.95 1.45

0.65 0.68 0.33 0.80

1.90 2.31 1.00 1.57

0.63 0.67 0.32 0.78

95 95 94 94

343 520 ,217 693

0.41 0.69 0.35 0.52

95 95 94 94

574 829 2,068 1,157

0.45 0.68 0.39 0.56

-

-

1.54 2.79 2.12 2.05

0.40 0.67 0.61 0.68

1.61 2.75 2.18 2.10

0.46 0.68 0.67 0.74

0.32 0.46 0.25 0.46

46 45 46 45

712 708 761

0.65 0.62 0.43 0.72

36 36 36 36

1,280 1,139 1,562 1,327

0.67 0.51 0.45 0.75

0.70 0.91 0.69 0.75

0.39 0.48 0.33 0.61

2.19 2.32 2.04 2.06

0.66 0.66 0.55 0.83

1.08 1.04 0.79 0.91

0.71 0.56 0.33 0.73

0.52 0.74 0.33 0.92

10 10 10

403 1,453 1,202 1,019

0.17 0.73 0.52 0.88

10

10 10

0.850 1.789 1.163 1.267

10 10

742 2,349 1,936 1,676

0.23 0.73 0.52 0.91

1.24 2.23 0.65 1.14

0.38 0.52 0.44 0.90

1.17 3.55 1.18 1.80

0.13 0.52 0.57 0.89

1.34 3.50 1.22 1.86

0.17 0.49 0.59 0.91

402 404 404 402

1.620 1.098 1.399 1.367

0.68 0.43 0.58 0.79

484 484 485 481

1,162 865 1,077 1,033

0.72 0.61 0.54 0.81

472 473 473 470

1,944 1,405 1,819 1,719

0.67 0.51 0.52 0.76

1.13 1.13 0.96 1.02

0.61 0.53 0.58 0.79

2.13 2.27

0.76 0.75 0.63 0.85

1.53 1.57 1.23 1.35

0.60 0.54 0.55 0.68

n

Slope

?

Mean

27 27 27 27

1.751 1.149 1.801 1.567

0.84 0.50 0.52 0.84

54 54 54 54

1,241 791 1,797 1,276

Distal femur

AP ML IS Mean

305 306 306 305

1.803 1.283 1.302 1.456

0.65 0.41 0.47 0.77

279 280 281 278

Proximal tibia

AP ML

-

-

Mean

-

-

-

AP ML IS Mean

60 61 61 60

0.783 0.733 0.680 0.738

AP ML IS Mean

10

Proximal humerus

Distal radius

Overall population

AP ML

IS Mean

Ash density

3

3

IS

Apparent density

Slope

Slope

IS

CT density

n

n AP ML

+ error

Ei = A(density)'

Apparent density

CT density

Proximal femur

Power function

Linear model A(density) + error

10

10

858

10

1.64

1.87

Two regression models were formulated for both a linear and power function. The overall population represents the pooled raw data for all regions tested.

versus apparent density (? = 0.82), the significance of both regressions may contribute to the variability observed in the density-modulus relationships. Finally, as in previous studies, the failure stress was linearly related to the tangent moduli, and when analyzed for differences between metaphysea1 regions, showed small differences. In the AP and IS directions, the failure relationships were relatively similar (p > 0.05), but the ML direction demonstrated significantly different relationships between metaphyseal regions (p < 0.005). The combined data (AP, ML, IS) for all regions is shown in Fig. 5 . DISCUSSION Many investigators have studied the relationship between the density of trabecular bone and its mechanical properties. The results of these studies have been inconsistent, with various investigators reporting linear relationships (1,6,9,15,18,19,23,26) and power functions (4,15,24). The results of this current study suggest that a significant amount of the variation in previously reported results may

have been due to anatomic location, direction of testing, and testing protocol. We examined trabecular bone from different metaphyseal locations using one standard testing protocol. In this manner, differences in mechanical properties between regions reflected true differences and not differences resulting from variations in the testing procedure. The metaphyseal regions tested were from four autopsy subjects, three men and one woman, with ages ranging from 55 to 70 years. Only specific regions, as specified in Table 1 , were tested for each donor. With this limited sample size, conclusions regarding the distribution patterns of trabecular bone mechanical properties between metaphyseal regions cannot be made. What the results of this study do provide are general relationships concerning trabecular bone, its properties, and its density measures. For the specimens tested in this study, the mechanical properties fell within the range of previously published data (13). Variations in the mechanical properties also seemed to correspond to the functional demands of the specific joint involved; e.g., while bone in a central plane of the

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M . J . CIARELLI ET AL.

femoral head tended toward isotropy, the distal femur appeared to be transversely isotropic and the radius was very anisotropic. In the distal femur, the greatest stiffnesses were in the AP and IS directions. In the most distal plane, the results appeared to demonstrate transversely isotropic properties in a sagittal plane. This correlates well with the need of the distal femur to withstand large contact stresses over an extended range of flexion angles. As the metaphysis extends proximally, the pattern of increased stiffness tended to diverge toward the diaphyseal walls. This relationship between joint function and the observed pattern of trabecular stiffnesses was apparent for all the metaphyseal regions tested. Similarly, the relationship between the ultimate stress and the tangent modulus was in good agreement with previously reported correlations (1,3,4,6,12). This linear correlation between strength and modulus serves as partial verification that within the limits of our testing analysis the preyield tests in the three orthogonal directions had no detectable effect on the failure tests. Another study within our laboratory evaluated the effect of this testing protocol on canine trabecular bone specimens. Overall, this study showed average differences between preyield and failure moduli on the same cube and in the same direction to be <8%, but for cubes with moduli >170 MPa the differences were <4% (16). Essentially all IS moduli in the current study were >170 MPa, and most AP and ML moduli were also >170 MPa. Therefore, the preyield tests provided excellent predictions of the tangent moduli for all three orthogonal directions. Density measures were also performed as a means of estimating the mechanical parameters. The linear relationship found between apparent density and CT density (Fig. 4) was in good agreement with results of previous studies (7,15,17,19). Independent of the scanning and calibration protocols used, in each study linear relationships between apparent density and CT density were significant, with 12 values ranging between 0.60 and 0.87. The inability in some studies to match analyzed regions exactly using CT and physical apparent density measures may have been a source of some of this unexplained variance. The sectioning protocol used in our study exactly matched analyzed regions within the CT scans to their corresponding bone cubes and thus resulted in a 3 value of 0.82. Correlations between density measures of trabec-

J Orthop Res, Vol. 9, No. 5 , 1991

ular bone and its mechanical properties are shown in Table 2. These results indicate that the coefficients, as well as the power of the relationship between the modulus and density, were very dependent on the anatomic distribution and therefore on the inherent architecture of the tested specimens. Also demonstrated, was a variability in which model, linear or power, could explain more of the variance in the density-modulus relationships. When compared with results of previous studies on individual metaphyseal regions, the correlations shown in Table 2 fall within published values. Lotz et al. (17), testing trabecular bone specimens from the proximal femur, reported significant power relationships when the compressive moduli were correlated with apparent density (1.4 power) and CT density (1.2 power). Because their loading axis was parallel to the femoral neck, direct comparison with our data is not possible. Their power relationships are similar, however, to those derived in our study for the proximal femur tested in the ML direction. For these specimens, the compressive moduli was related to the apparent density raised to the 1.38 power and the CT density raised to the 1.10 power. Similar relationships were noted for the other metaphyseal regions when comparisons were made with results of previous studies. Another observation made in this and numerous other studies was that although density measures continued to provide significant correlations to mechanical properties, 20 to 60% of unexplained variance remained. This unexplained variance was not completely unexpected with use of the scalar quantity density to describe the orthogonal mechanical properties of trabecular bone. As a scalar, density lacks any directional information and at best can only model averaged cube properties. Ducheyne et al. (6) reported that of trabecular specimens with the same density but different strengths stronger specimens had thicker and more branched trabeculae than weaker specimens. Thus, for any given density, there can be many different architectural schemes, each resulting in different mechanical properties. Additional architectural parameters are needed to account for the many types of architecture noted throughout the different metaphyseal regions. Another study in our laboratory is currently investigating age-related effects concerning trabecular bone mechanical properties and architectural parameters within and between different metaphyseal regions. The importance of architectural parameters was

MECHANICAL PROPER TIES OF TRABECULAR BONE evident even in studies that attempted to limit the amount of architectural variation. Hvid et al. (15) tested trabecular bone specimens taken from the condylar areas of the tibia, thus avoiding the weaker central tibial bone. Their results comparing density measures to moduli values showed between 34 and 45% unexplained variance. In comparison, for cubes tested in the IS direction from the proximal tibia in the current study, unexplained variance was between 33 and 65%. The larger amounts of unexplained variance in our study may be due in part to the wider range of architecture observed when the entire proximal tibia is evaluated. The effect of limiting the analyzed regions to the condylar areas may have reduced this architectural variation, but the remaining unexplained variance suggests that architectural changes over even relatively small areas in trabecular bone can have substantial effects on its mechanical properties. Therefore, parameters quantifying these architectural changes are needed to characterize trabecular bone mechanical properties further. Statistical evaluation of the relationship between apparent density and mechanical properties of trabecular bone was evaluated by Rice et al. (21). The results of their analysis indicated that for the data analyzed Young’s modulus and strength were related to the square of the apparent density. These results appear to be consistent with the analytical predictions of Gibson and Ashby (1l), Christensen (9,and Gibson (lo), who have suggested that deformation of the trabeculi occurs primarily in response to bending. The results of the present study are in general agreement with these findings but also provide evidence that axial loading of trabeculi may occur, as reported earlier by Williams and Lewis (26). Axial loading conditions explain linear relationships between modulus and apparent density. Limits of our testing protocol may have masked any significant differences between the linear and power models, however. Although density models alone cannot fully model the mechanical characteristics of trabecular bone, they can provide initial estimates of its mechanical integrity. The excellent correlations between CT and density measures supports its application in mechanical evaluation of trabecular bone. Use of quantitative CT to determine relative bone density nondestructively and noninvasively and thus infer mechanical strength has been demonstrated in this and other studies (2,7,12,15,17). In conclusion, the results from this study have

681

confirmed our previous appreciation for the tremendous variation in trabecular bone mechanical properties. The differences in the orthogonal mechanical properties for the major metaphyseal regions demonstrate that the mechanical property distribution and mechanical anisotropy varied up to three orders of magnitude from bone to bone and within each metaphyseal region. The patterns of mechanical properties and anisotropy correlated well with the functional demands of each anatomic region. In additon, because quantitative CT scans correlated very well with apparent density, ash weight, and mechanical properties, CT is a reasonable tool for noninvasive estimation of trabecular bone mechanical integrity. Finally, density alone cannot completely explain the variance of the mechanical property results; quantification of trabecular architecture is necessary to predict local bone properties accurately. Acknowledgment: This study was supported by Grant No. AR34399 from the National Institutes of Health. We thank Dennis Kayner, Mark Capper, and Jeff Stanley for their technical assistance.

REFERENCES 1. Behrens JC, Walker PS, Shoji H: Variation in strength and structure of cancellous bone at the knee. J Biomech 7:201207, 1974 2. Bentzen SM, Hvid I, Jorgensen J: Mechanical strength and tibial trabecular bone evaluated by x-ray computed tomography. J Biomech 20:743-752, 1987 3. Brown TD, Ferguson AB: Mechanical property distribution in the cancellous bone of the human proximal femur. Acta Orthop Scand 51:429437, 1980 4. Carter DR, Hayes WC: The compressive behavior of bone as a two-phase porous structure. J Bone Joint Surg [Am] 59:954962, 1977 5. Christensen RM: Mechanics of load density materials. J Mech Phys Solids 34563-578, 1986 6. Ducheyne P, Heymans L, Martens M, Aemoudt E, Meester PD, Mulier JC: The mechanical behavior of intracondylar cancellous bone of the femur at different loading rates. J Biomech 10:747-762, 1977 7. Esses, SI, Lotz JF, Hayes WC: Biomechanical properties of the proximal femur determined in vitro by single-energy quantitative computed tomography. J Bone Min Res 4:715722, 1989 8 Evans FG, King AL: Regional differences in some physical properties of human spongy bone. In: Biomechanical Studies of the Musculoskeletal System, ed. by FG Evans, Springfield, Charles C Thomas, 1961, pp 49-67 9 Galante J, Rostoker W, Ray RD: Physical properties of trabecular bone. Calcif Tissue Res 5:236246, 1970 10. Gibson LJ: The mechanical behavior of cancellous bone. J Biomech 18:317-328, 1985. 11. Gibson LJ, Ashby MS: The mechanics of three-dimensional cellular materials. Proc R Soc London A382:43-59, 1982 12. Goldstein SA, Wilson DL, Sonstegard DA, Matthews LS: The mechanical properties of human tibial trabecular bone

J

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