Dr Orwoll serves as a consultant for Eli Lilly and Company, Merck & Co., Novartis, Procter & Gamble, and TAP Pharmaceutical Products, Inc. All other authors have no conflict of interest.

Dennis R Carter

Bone and Joint Center, VA Palo Alto Health Care System, Palo Alto, California, USA

The influence of fall orientation on femur strength has important implications for understanding hip fracture risk. A new image analysis technique showed that the strength of the femoral neck in 37 males varied significantly along the neck axis and that bending strength varied by a factor of up to 2.8 for different loading directions.

Introduction: Osteoporosis is associated with decreased BMD and increased hip fracture risk, but it is unclear whether specific osteoporotic changes in the proximal femur lead to a more vulnerable overall structure. Nonhomogeneous beam theory, which is used to determine the mechanical response of composite structures to applied loads, can be used along with QCT to estimate the resistance of the femoral neck to axial forces and bending moments.

Materials and Methods: The bending moment {M_{y}(θ)} sufficient to induce yielding within femoral neck sections was estimated for a range of bending orientations (θ) using in vivo QCT images of 37 male (mean age, 73 years; range, 65–87 years) femora. Volumetric BMD, axial stiffness, average moment at yield (M_{y,avg}), maximum and minimum moment at yield (M_{y,max} and M_{y,min}), bone strength index (BSI), stress-strain index (SSI), and density-weighted moments of resistance (R_{x} and R_{y}) were also computed. Differences among the proximal, mid-, and distal neck regions were detected using ANOVA.

Results: M_{y}(θ) was found to vary by as much as a factor of 2.8 for different bending directions. Axial stiffness, M_{y,avg}, M_{y,max}, M_{y,min}, BSI, and R_{x} differed significantly between all femoral neck regions, with an overall trend of increasing axial stiffness and bending strength when moving from the proximal neck to the distal neck. Mean axial stiffness increased 62% between the proximal and distal neck, and mean M_{y,avg} increased 53% between the proximal and distal neck.

Conclusions: The results of this study show that femoral neck strength strongly depends on both fall orientation and location along the neck axis. Compressive yielding in the superior portion of the femoral neck is expected to initiate fracture in a fall to the side.

OSTEOPOROSIS IS ASSOCIATED with decreased BMD and increased hip fracture risk, but it is unclear how specific changes in the material and sectional properties of the proximal femur interact to produce a more vulnerable overall structure. As with other physical structures, the geometry and material properties of the proximal femur can be used to determine whether a fracture will occur under a load of specific type and magnitude. Because of the complexity of femoral geometry and the heterogeneous distribution of material properties within the proximal femur, it is difficult to accurately estimate hip strength. In addition, other factors such as areal BMD, risk of fall, the amount of soft tissue surrounding the hip, age, race, gender, and trabecular architecture have been shown to affect fracture risk. (^{1–6})

Images obtained using QCT provide a 3D description of both femoral geometry and bone material distribution. In recent years, QCT has been used to calculate the areas, second area moments of inertia, density-weighted moments of resistance (R_{x} and R_{y}), bone strength indices (BSI), and polar stress-strain indices (SSI) of cross-sections of the humerus, radius, ulna, tibia, fibula, femoral shaft, and femoral neck. (^{7–17}) Nonhomogeneous beam theory, which accounts for the distribution of material properties within bone sections, has been used to accurately predict failure loads using QCT images of thoracic and lumbar vertebrae, trabecular bone cylinders, and rabbit humeri. (^{18–20})

This study aims to provide a detailed analysis of femoral neck structure to determine how strongly in vivo QCT-derived estimates of axial strength and bending strength of the femoral neck depend on measurement location and fall direction. The method presented here follows nonhomogeneous beam theory, which takes into account the geometry of bone sections and the variation of mechanical properties within bone sections. Using this method along with yield strain failure criteria provides estimates of the location and type (either compressive or tensile) of yielding that will occur in each section under axial and bending loads.

MATERIALS AND METHODS

Analysis approach

During normal daily activities and in a fall to the side, the femoral neck experiences a combination of bending and axial compression. (^{12}, ^{21}, ^{22}) Nonhomogeneous beam theory provides a method for estimating the amount of deformation that will occur in a bone section subjected to these two types of loads. The analysis method presented below is performed for sections located along the femoral neck axis, but a similar analysis can be applied to any long bone section, provided that the image of the bone is sliced along the bone axis. Because all of the calculated properties for bone sections may vary along the length of a bone (e.g., along the z-axis of the femoral neck shown in Fig. 1), the dependence of all calculated properties on the z-location of the section is implicit in the analysis. Table 1 contains the definitions and SI units of the symbols used in the analysis, and Fig. 1 provides a visual reference for the coordinate axes and applied loads to which the text refers.

Table Table 1.. Symbols Used in Analysis

When a bone is deformed under a pure bending moment, the normal strains within a given cross-section vary linearly from tensile strain (stretching; positive sign convention) on one side to compressive strain (shortening; negative sign convention) on the other side. The transition from tensile strain to compressive strain occurs along the neutral axis of the section. The bone tissue located along the neutral axis in a bone section under pure bending is therefore not deformed, because the strain at all points along the neutral axis is zero. The orientation of the neutral axis changes for different bending directions, but the neutral axis always passes through a point in the section called the effective centroid. The location of the effective centroid can be different than both the location of the section's area centroid and the location of the section's center of mass, because the effective centroid is determined both by section geometry and the distribution of material properties within the section. For a section in a QCT image with isometric voxels, the effective centroid (x,y) is obtained using

(1)

where E_{i} is the axial elastic modulus (a measure of bone stiffness) of the voxel at location (x_{i},y_{i}), and ∑ indicates summation over all of the voxels in a section.

Four biomechanical properties of a bone section are needed to obtain the strain distribution under axial and bending loads. (^{23}) The flexural rigidity about the x-axis (I_{xx}), the flexural rigidity about the y-axis (I_{yy}), the generalized product of inertia (I_{xy}), and the axial stiffness (A) of the section are calculated using

(2)

(3)

(4)

and

(5)

where a = voxel area. The a^{2}/12 term in each of the flexural rigidity expressions accounts for the fact that voxels are of finite size (parallel axis theorem). These are the four basic sectional properties that characterize the stiffness of a bone section.

Using the sectional properties described above, the generalized flexural formula for normal strains at any location (x,y) in the section is

(6)

where P is an axial load oriented in the z-direction and passing through the effective centroid of the section, and M is a bending moment applied in a direction θ measured from the x-axis. (^{24})

The portion of the strain at all points in the section due only to the axial load P is

(7)

Because the yield strain (the amount of deformation that begins to cause damage) of bone within a particular anatomical region can be considered uniform, maximum strain can be used as a failure criterion for a bone section. (^{25}) Because Adetermines the magnitude of the strain in the section because of P, Ais a measure of the bone's relative axial strength at the z-location of the section.

From Eq. 6, the strain at a particular location (x,y) in the section caused only by the moment M oriented at an angle θ is

(8)

where

(9)

and

(10)

Note that C_{x}(x,y) and C_{y}(x,y) are constant at any given location in the section.

The dependence of ε_{b} on θ shows that, if the orientation of the bending moment changes, the strain distribution within the section will also change. For the coordinate system used in this analysis, a bending angle of 0° corresponds to downward bending of the femoral neck about the anterior-posterior (A-P) axis. For example, a force pushing directly downward on the top surface of the femoral head while standing will produce a bending moment near θ = 0°. A force pushing upward on the inferior surface of the femoral head during a fall to the side will induce a bending moment near θ = 180°.

The type of strain, either compressive or tensile, that occurs at locations throughout a femoral neck section can be used to provide a failure criterion based on yielding. Recent experimental evidence suggests that cancellous bone in the femoral neck has a yield strain of ∼0.85% in compression and 0.61% in tension and that cortical bone has a yield strain of ∼0.86% in compression and 0.57% in tension. (^{25}, ^{26}) The ratio of compressive yield strain to tensile yield strain is ∼1.4-1.5 for bone in the femoral neck. The maximum (most positive) tensile strain {ε_{t,max}(θ)} and the maximum (most negative) compressive strain {ε_{c,max}(θ)} in a section caused by a bending moment of magnitude 1 N · m oriented at an angle θ can be calculated by applying Eq. 8 with M = 1 N · m at each location within the section. If the maximum strain ratio {r(θ)}, given by

(11)

is ⩽1.4, yielding is likely to first occur in tension, and bone tissue will be pulled apart. If r(θ) is >1.4, yielding is likely to first occur in compression, and bone tissue will be crushed. Using Eq. 7 along with the known r(θ) and the tensile and compressive yield strains of bone in the femoral neck, the moment (M_{y}) that will induce tensile yielding is

(12)

and the M_{y} that will induce compressive yielding is

(13)

Note that the yield strains for femoral neck cancellous bone are used in Eqs. 12 and 13. Because the yield strains for cortical bone and femoral neck cancellous bone are very similar, cancellous bone yield strains were used for all bone in this study. Other site-specific values may be more appropriate when analyzing other bone regions. (^{25})

Phantom tests

To show that QCT images can be used to predict bending strains in a composite beam section, a two-material phantom was scanned, analyzed, and mechanically tested. An elliptical phantom (length along major axis = 40 mm, length along minor axis = 30 mm) machined out of glass-filled phenolic (PLENCO 06990; Plastics Engineering Corp., Sheboygan, WI, USA) and containing a circular core (diameter = 24 mm) of acrylic was scanned at 80 kVp and 300 mAs using a 3-mm slice thickness, a 48-cm field of view (FOV), and the standard GE reconstruction algorithm, resulting in a voxel size of 0.94 × 0.94 × 3 mm^{3} (Fig. 2). The center of the phantom's circular core was offset from the center of the ellipse by 4 mm along the major axis, resulting in a minimum thickness of 2 mm for the phenolic “cortex.” A single slice of the CT image was segmented using a region-growing algorithm, (^{10}) and a computer program called StrainMax was written to apply the analysis method described above to segmented QCT scans. Elastic moduli were assigned to each voxel in the section based on a linear calibration between the average Hounsfield numbers of phenolic (1390 HU) and acrylic (91 HU) and their previously measured material properties (elastic modulus = 14.2 GPa for the phenolic, 3.7 GPa for the acrylic). Voxel locations and estimated elastic moduli were used to compute A, I_{xx}, I_{yy}, and I_{xy}for the section, and these sectional properties were used to predict the strains in the section caused by applied bending loads. Four strain gages (EA-13-250BG-120; Vishay MicroMeasurements, Raleigh, NC, USA) were attached to the surface of the phantom at the location of the analyzed section, and loads from 0 to 80N in the four directions corresponding to the strain gage locations were applied to the end of the phantom. The strain gages were located 81.7 mm from the point of loading, resulting in an applied bending moment ranging from 0 to 6.5 N · m at the position of the analyzed section. Using linear regression, the strains predicted by StrainMax for the range of applied bending moments were compared with the strains measured during mechanical testing.

In vivo application

Images of 37 male (mean age, 73 years; range, 65–87 years) proximal left femora obtained in the Osteoporotic Fractures in Men (MrOS) Study were analyzed using StrainMax. All subjects were scanned along with a solid hydroxyapatite calibration phantom (Image Analysis, Columbia, KY, USA) at 80 kVp and 300 mAs using a 3-mm slice thickness, a 48-cm FOV, and the standard reconstruction algorithm. The images were segmented and resliced along the neck z-axis using previously described software. (^{10}) The data were interpolated to produce cubic voxels with 0.94-mm sides, and the equivalent hydroxyapatite concentration of each voxel was determined using the calibration phantom. Voxels with an equivalent hydroxyapatite concentration below 0 mg/cm^{3} were excluded from the analysis. A total of 521 femoral neck slices were included in the final analysis.

The elastic modulus of each voxel was estimated using a series of three relationships. The equivalent solid calcium hydroxyapatite concentration (ρ_{HA, S}, in mg/cm^{3}) of each voxel was converted to equivalent liquid K_{2}HPO_{4} concentration (ρ_{HA, L})(^{27}) using

(14)

Then ρ_{HA, L} was converted to wet apparent density (ρ)(^{28}) using

(15)

and final estimates of elastic modulus (E) were obtained using a bilinear power relationship(^{29}):

(16)

(17)

The distance (z) of each section from the center of the femoral head was measured using a single coronal slice from each analyzed femur. The neck axis length (NAL), defined as the distance in the z-direction from the center of the femoral head to the lateral periosteal boundary of the proximal femur, was also measured using a single coronal slice for each femur. The relative location of each section along the neck axis was expressed as % NAL = (z/NAL × 100%).

The total BMC and bone volume of each section were used to compute volumetric BMD (vBMD), and the locations and QCT-derived elastic moduli of the bone voxels were used to compute A, I_{xx}, I_{yy}, and I_{xy}for each section along the femoral neck z-axis. The moment at yield M_{y}(θ) of each section was computed at integer angles from 0° to 360° measured from the A-P axis. The average moment at yield (M_{y,avg}) and the maximum and minimum moments at yield (M_{y,max} and M_{y,min}) were recorded along with their respective bending angles (θ_{max} and θ_{min}) for each section along the neck z-axis. The yield moment ratio r_{M} = M_{y,max}/M_{y,min} was recorded for each section as an indicator of the extent to which the bending strength changed with bending direction.

Several density-weighted bone strength measures that have been applied in the literature were also computed for each femoral neck section. Polar SSI (a density-weighted measure of bone torsional strength), BSI (a measure of cortex bending strength), and the density-weighted moments of resistance about the x-axis (R_{x}) and the y-axis (R_{y}; measures of bone bending strength) were calculated using custom software. (^{8}, ^{14}, ^{15}) SSI and moments of resistance were computed using all voxels with an equivalent hydroxyapatite concentration >0 mg/cm^{3}, and a threshold of 400 mg/cm^{3} was used to separate the cortex and dense subcortical bone from the marrow cavity for the BSI calculation. (^{11})

The various femoral neck strength estimates described above offer three different choices for the location of the coordinate system origin: the area centroid (x_{a},y_{a}), the center of mass (x_{m},y_{m}), and the effective centroid (x,y). The location of the area centroid is determined using only the geometry of the bone section, whereas the location of the center of mass is obtained using the geometry and the BMD distribution. The location of the effective centroid, as explained above, is determined using the geometry and the distribution of bone material properties. It has been shown that a shift in the effective centroid toward the inferior aspect of the femoral neck will lead to higher stresses and strains in the superior aspect of the femoral neck. (^{30}) To determine whether the relative locations of these origins change along the femoral neck axis, the y-locations of the area centroid (y_{a}), center of mass (y_{m}), and effective centroid (y) were recorded for each section. The location of yrelative to y_{m} and y_{a} was computed for each section using y− y_{m} and y− y_{a}.

To compare the mean sectional property values for different anatomical regions, femoral neck sections were divided into three groups based on relative location along the neck axis. Sections in the proximal femoral neck (n = 149) were located at 15–24% NAL, sections in the mid-femoral neck (n = 256) were located at 25–34% NAL, and sections in the distal femoral neck (n = 116) were located at 35–44% NAL (Fig. 3). ANOVA was used to detect any significant (p < 0.05) differences in the sample mean values of vBMD, A, M_{y,avg}, M_{y,min}, M_{y,max}, r_{M}, y− y_{m}, y− y_{a}, SSI, BSI, R_{x}, and R_{y} among the three regions. Fourteen sections from each hip on average were analyzed. Because it was possible to have multiple sections from a given subject included in the calculation of mean properties in a hip region, the subject from which each structural parameter was calculated was included as a random factor in the ANOVA. The inclusion of the subject as a random factor provided a conservative approach to detecting differences among groups. In cases where ANOVA indicated the presence of significant differences between regions, Games-Howell posthoc tests (α = 0.05) were used to compare the mean sectional property values among the three regions. All statistical analyses were performed using SSPS software (SSPS, Chicago, IL, USA).

RESULTS

For the elliptical phantom, the maximum tensile strains predicted by StrainMax showed excellent agreement with the values measured directly from the strain gages during mechanical testing. The linear regression equation for measured strain (ε_{meas}) versus predicted strain (ε_{pred}), both in units of microstrain, was ε_{meas} = 1.03(ε_{pred}) + 0.01 (r^{2} = 0.998). Thus, the results indicate that the StrainMax program can provide accurate predictions of the strains that will occur in sections of a composite structure under bending loads in different directions.

Mean QCT-derived sectional properties and their SDs for the three femoral neck regions in the in vivo scans are given in Table 2. Also provided in Table 2 are the percent increases in the mean values when moving from the proximal femoral neck to the distal femoral neck. Substantial variability was observed in the sectional properties, as evidenced by the relatively large SDs. In the distal femoral neck, for example, the SD for Awas 24% of the mean, the SD for M_{y,max} was 28% of the mean, and the SD of BSI was 34% of the mean.

Table Table 2.. Mean QCT-Derived Sectional Properties and Their SDs for Proximal Femoral Neck (PFN), Mid-Femoral Neck (MFN), and Distal Femoral Neck (DFN) Sections

All computed properties except vBMD, R_{y}, SSI, and r_{M} increased significantly between each region when moving from the proximal neck to the distal neck. The mean values of y− y_{m} and y− y_{a} were negative for all three groups, indicating that the effective centroid was located in a more inferior position than both the area centroid and the center of mass. These differences in y location increased (i.e., became more negative) significantly between each femoral neck group when moving from the proximal femoral neck to the distal femoral neck. Thus, the effective centroid was shown to shift to a more inferior position when moving from the proximal femoral neck to the distal femoral neck.

For a given hip, the maximum strain ratio r(θ) and the yield moment M_{y}(θ) varied widely for different bending directions (Fig. 4). A region of compressive yielding, as indicated by a maximum strain ratio r(θ) > 1.4, was typically observed for a subset of bending directions within the bending angle range of ∼135° to 235°. For these bending angles, the superior aspect of the femoral neck experiences compressive strains, whereas the inferior aspect experiences tensile strains. M_{y,max} was greater than M_{y,min} by as much as a factor of 2.8 (i.e., the maximum r_{M} observed in the femoral neck was 2.8), indicating that the magnitude of the moment that would induce yielding in the femoral neck is strongly affected by bending direction. The mean yield moment ratio r_{M} was the highest for the mid-femoral neck, indicating that the largest relative change in femoral neck strength with bending direction occurs in the mid-neck region of elderly male femora.

DISCUSSION

The results of the femoral neck analysis show that the relative bending strengths of femoral neck sections and the type of yielding expected in a fall strongly depend on fall orientation. For some femoral neck sections, the value of M_{y} varied by as much as a factor of 2.8 for different bending directions, indicating that some hips may be up to 2.8 times stronger under a bending moment applied in the θ_{max} direction as opposed to the θ_{min} direction. To determine the clinical significance of these results, we narrowed our focus to the loading directions expected in a fall. A fall to the side resulting in impact on the lateral aspect of the greater trochanter is thought to produce the highest risk of hip fracture. (^{31–33}) A fall directly to the side is expected to produce a combination of axial compression and bending in a direction near θ = 180°, and the strength of the proximal femur has been found in mechanical tests and numerical models to decrease when a posteriorly oriented bending component is introduced. (^{34}, ^{35}) The addition of a posteriorly oriented bending load is consistent with the fall configuration observed by van den Kroonenberg et al. (^{36}) in a study of fall dynamics. Falls to the side with a posterior bending component correspond to a bending angle slightly <180° in Fig. 4. For the femoral neck section characterized in Fig. 4, a fracture resulting from fall directly to the side (θ = 180°) would be expected to begin with compressive yielding in the superior portion of the cortex. For a fall to the side and slightly backward (θ = 160°) instead of directly to the side, compressive yielding in the posterosuperior cortex is predicted, and the estimated bending strength of the section decreases by 21%. If the posterior bending component is increased even further (i.e., θ drops below 160°), initial tensile yielding in the anteroinferior cortex is expected, and the estimated bending strength continues to decrease.

The estimated location of the effective centroid has important implications for the type of yielding expected in a fall to the side. In this study, the effective centroid was located an average of 2–5 mm inferior to the center of mass and 4–10 mm inferior to the area centroid. The use of the effective centroid, as opposed to the area centroid or center of mass, results in an increase in estimated strain in the superior aspect of the femoral neck. Compressive strain estimates in the superior aspect of the femoral neck are subsequently increased for bending angles near 180°. When strains caused by axial compression are superimposed on those caused by bending, as is expected in a fall to the side, the compressive strains predicted for the superior portion of the femoral neck are increased even further.

Compressive failure in the superior and posterosuperior aspects of the femoral neck and tensile failure in the inferior and anteroinferior aspects of the femoral neck are observed clinically in subcapital and transcervical fractures and have been produced in ex vivo mechanical tests in a fall loading configuration. (^{22}, ^{34}, ^{37}) However, in clinical fracture cases, it is not possible to determine whether compressive yielding or tensile yielding occurred first. The prediction of initial compressive yielding in the superior aspect of the femoral neck is consistent with the results of a study using finite element models to study the effects of fall loading direction on femoral neck strength. (^{35}) It is important to differentiate this type of yielding from the buckling failure mechanism proposed in studies of the effects of aging and altered load on femoral neck structure. (^{38}, ^{39}) Nonhomogeneous beam analysis takes into account all of the bone in a given cross-section, including both cortical and trabecular bone compartments. Using this method, yielding in the superior aspect of the femoral neck is predicted to occur when the local compressive strain surpasses the assumed bone yield strain. Buckling instability is based on the assumption that the femoral neck is equivalent to a hollow, thin-walled tube that becomes more prone to failure as the cortex becomes thinner in relation to the neck width. Because well over one-half of the volume of the femoral neck is filled with trabecular bone, (^{40}) in most cases nonhomogenous beam theory provides a more appropriate characterization of femoral neck structure. The thin-walled tube model may be applicable for pathological situations in which extensive cortical thinning has occurred and almost no trabecular bone remains within the femoral neck.

The results of this study also show that the sectional properties of the femoral neck strongly depend on measurement location. Because the bending moment arm for hip joint reaction forces increases with distance from the femoral head, M_{y,max}, M_{y,min}, M_{y,avg}, R_{x}, R_{y}, and BSI, which are estimates of relative sectional bending strength and stiffness, were expected to increase with distance from the femoral head. The results of the femoral neck analysis thus agreed with the expected trends for all but one of these bending strength measures: the value of R_{y} did not vary significantly between the proximal and distal neck regions. A, which is a measure of relative axial strength, was expected to remain relatively constant along the length of the femoral neck, because the total axial force remains constant from section to section. However, mean Adiffered significantly between all groups and increased by 62% when moving from the proximal neck to the distal neck.

Although our results suggest a general pattern of increasing femoral neck strength when moving distally along the femoral neck, some of the individual hips analyzed in this study did not follow this trend. For example, one hip exhibited a slight decrease in mean M_{y,avg} when moving from the mid-femoral neck (M_{y,avg} = 33 N · mm) to the distal femoral neck (M_{y,avg} = 31 N · mm). The observed decrease in average bending strength between the mid- and distal neck regions is important, because the magnitude of the applied bending moment is expected to increase with distance from the femoral head in both normal and fall loading configurations. Therefore, the structure of the distal femoral neck in this particular hip may not be well suited for supporting the elevated bending loads expected in a fall. The structure of this hip reveals the value of analyzing multiple locations in a given subject as opposed to a single section. Including additional sections may help to identify outliers that would not be noticed when focusing on a single measurement location.

The dependence of femoral neck sectional properties on location along the neck axis has implications for studies that use QCT. Sectional properties can vary greatly along the femoral neck axis of a given hip (Fig. 5). Because of concerns with radiation dose and clinical feasiblilty, only a single bone slice in a particular region is typically analyzed in studies using QCT. The results of this study indicate that the bending strength of the femoral neck increases with distance from the center of the femoral head. However, whereas the proximal neck region may have a lower bending strength, this region is also expected to experience a lower applied bending moment resulting from the hip joint reaction force. When choosing a single site within the femoral neck to provide a strength estimate, it is important to consider the changes in sectional properties that occur along the femoral neck axis as well as the expected in vivo loading conditions. For analyses of overall femoral neck structure, it may be most prudent to analyze three characteristic sections: one from the proximal neck, one from the mid-neck, and one from the distal neck. A plot such as the one shown in Fig. 5 can be created and used to study the change in femoral neck strength along the neck axis and with changing bending direction.

The relatively high SDs observed for mean sectional properties in this study are similar to those reported in the literature for both in vitro and in vivo QCT-derived estimates of femoral neck vBMD and cross-sectional area. (^{10}, ^{11}, ^{17}, ^{40}) Some previously reported SDs for femoral neck sectional properties are considerably higher: Lochmüller et al. (^{11}) reported a SD of 67% for area moment of inertia and 70% for SSI computed from 42 in vitro QCT scans of the male femoral neck. These consistently high SDs suggest that there is a high degree of variability in QCT-derived hip strength parameters within the elderly male population. It is possible that this variability may contribute to interindividual differences in hip fracture risk.

The advantage of this method compared with density-weighted bone strength measures such as SSI, BSI, and R_{x} from QCT is that Aand M_{y}(θ) are quantities that can be used to directly estimate the relative response of complex femoral neck sections to a variety of applied loads. In addition, the series of calculations used to obtain M_{y}(θ) accounts for the difference between compressive and tensile strains. When bone fails in compression, it is still possible for the bone to continue to sustain compressive loads. However, if bone fails in tension and the tensile load is not removed, the bone can no longer remain intact. (^{41}) Thus, it is important to account for the relative magnitudes of the tensile and compressive strains a bone experiences under a given load.

Although StrainMax accurately predicted strains for the elliptical phantom, there are some issues that must be addressed when applying the same analysis to the femoral neck. The use of beam theory may not be entirely appropriate in the femoral neck, because the cross-sectional geometry of the neck and the distribution of bone mass within sections change rapidly along its length. In addition, the length of the femoral neck is relatively small compared with its height and depth, resulting in a “stubby beam” geometry. Nonhomogeneous beam theory nevertheless provides a method to characterize the relative bending and axial strengths of femoral neck sections. In effect, the analysis method presented above takes a single section of the femoral neck and estimates the equivalent bending and axial strengths of a hypothetical long beam whose cross-sections are all identical to the single section being analyzed. Thus, although the femoral neck itself does not fit the description of a long beam, the relative axial strength and bending strength of each individual section can be estimated and compared with those of other sections.

The elastic moduli of the phenolic and acrylic used in the elliptical phantom were determined through mechanical testing, but the calculation of mechanical properties of bone in femoral neck sections is more problematic. At clinical imaging resolutions, the calibration process used in QCT only offers information on the overall amount of bone mineral contained in a voxel. Therefore, material properties must be estimated without quantitative information about the organic portion of the bone matrix or the trabecular architecture. The equations to estimate elastic modulus were chosen for this study because they span the entire range of observed densities and have been used by our laboratory in past simulations. (^{29}) Analyses performed on bones from other locations may require the use of site-specific relationships. (^{42}) Because the sectional properties are directly weighted by the elastic modulus, the analysis method presented here is sensitive to the choice of density/elastic modulus relationship. A relationship that provides estimates of elastic modulus twice as high as those in this study would give estimates of sectional properties that are also twice as high. To compare sectional properties within a population, it is therefore crucial to use a consistent density/elastic modulus relationship that is well defined within the range of densities observed. Extrapolating values from relationships obtained using a smaller range of bone densities could introduce additional errors in the analysis.

The image resolution used in this study was chosen based on obtaining the best possible image quality while keeping radiation dose at a clinically feasible level. The resulting in vivo images are somewhat coarse, and “stepping” artifacts along the femoral neck length evident in Fig. 5 exist because of sampling errors resulting from the digital rotation and reformatting process. Images obtained at higher resolution would allow more accurate geometric measurements and help to moderate the effects of partial volume averaging on estimates of BMD and material properties. However, provided that a consistent image resolution, density calibration technique, and density/elastic modulus relationship are used for a given set of data, the sectional properties described in this study can be used to compare the relative axial and bending strengths for different locations within a given bone and to confidently draw conclusions about the structure of bones in a population.

This study developed and tested a QCT-based technique to provide a detailed analysis of femoral neck structure and the response of the femoral neck to axial loads and a variety of bending loads. This technique assesses important biomechanical parameters in vivo and can be used to study the influence of factors such as aging, osteoporosis, and pharmacological treatments on the relative strength of femoral neck sections. The results of this study suggest that there is a large degree of variability of femoral neck sectional properties within the elderly male population, and this high degree of variability is consistent with previously reported values for BMD and geometric properties measured in the femoral neck using QCT. The orientation of the bending moment applied to the femoral neck in falling was shown to have a strong effect on estimates of femoral neck bending strength, and the estimated bending and axial strengths of the femoral neck were shown to depend strongly on measurement location. The influence of fall orientation on femoral neck strength and the population variation in femoral neck structural properties have important implications in understanding fracture risk.

Acknowledgements

The authors thank Derek Lindsey, Eric Topp, Jim Anderson, Ben Chan, Lynn Marshall, and Laura Logan. This project was funded by VA Grant A2424P, NIH Grants UO1 AG18197-02, UO1 AR45580-02, UO1 AR45614, UO1 AR45632, UO1 AR45647, UO1 AR45654, UO1 AR45583, and MO1 RR00334, and a Stanford University Office of Technology Licensing Fellowship.