The fixed cortical fraction assumption in Hip Structure Analysis produces systematic, sex- and age- dependent errors in femoral neck morphology estimates

Introduction

Osteoporotic hip fractures were estimated at 14.2 million globally in 2019, with annual increases in the age-standardized rate (1, 2), resulting in 2.9 million years lived with disability (2) and increased mortality rates, particularly for males and older individuals (3). Monitoring hip strength is critical for identifying individuals at increased fracture risk who may benefit from preventive treatment. Because hip strength cannot be measured directly in living humans, bone health assessments must rely on estimates of bone mass and structure derived from noninvasive imaging technologies as surrogate measures of strength (4–6). Age-related declines in hip strength arise in part from an imbalance between bone resorption and formation leading to increased porosity, cortical thinning, and loss of trabecular bone volume fraction (BV/TV). Recent work reported that age-related changes in femoral neck structure vary significantly among individuals depending on their external bone size (7). We propose that assessments of bone strength and strength decline should account for structural heterogeneity across the population and variation in biomechanical pathways increasing fracture risk. Hip Structure Analysis (HSA) is widely used to estimate femoral neck (FN) structure from hip DXA images (Figure 1); however, it is unclear whether the simplifying assumptions used in HSA introduce systematic errors that could lead to bias (8–10), particularly for parameters that depend on assumptions about cortical bone distribution (Figure 1).

HSA has been widely used to assess hip structural changes associated with aging (11, 12) kidney transplantation (13), and type 1 diabetes (14), demonstrating its value for characterizing structural aspects of bone strength in diverse clinical applications. HSA estimates FN structural measures including bone width, cross-sectional area (CSA), cross-sectional moment of inertia (CSMI), and section modulus (Z) by integrating mineral X-ray attenuation profiles (11, 15, 16). CSA reflects the total load-bearing area of bone at a given cross-section, CSMI quantifies resistance to bending about the neutral axis, and Z = CSMI/c provides an estimate of bending strength (where c is the distance from the neutral axis to the outer surface). These structural measures correlate well with those measured from corresponding regions of computed tomography images (9). HSA also estimates R_inner (endocortical diameter/2), buckling ratio (BR), and cortical area (Ct.Ar) using a suite of algorithms along with simplifying assumptions (11, 15, 16). For these parameters, HSA assumes the femoral neck has a circular cross-sectional shape with a uniformly thick cortical shell that accounts for 60% of the femoral neck bone mineral content (BMC). Accurate estimation of these parameters is necessary because BR and Ct.Ar directly indicate cortical instability and load-bearing capacity, respectively, and estimation errors could lead to mischaracterizing fracture risk. The extent to which these assumptions are associated with systematic errors when estimating R_inner, Ct.Ar, and BR has not been evaluated.

The ratio of cortical to total mineral content, designated as f_c in the HSA algorithms, is assumed to be 0.6 at the narrow neck regardless of factors such as sex, race/ethnicity, age, or disease condition (11). However, we reported that the proportion of cortical to total bone mineral content is not fixed but varies with sex (17). Given known age-related changes in bone structure (7), f_c would also be expected to vary with age. The goals of the current study were three-fold: first, to assess how the ratio of cortical to total mineral content (f_c) varies with sex, age, and body size characteristics using high-resolution nano-computed tomography (nanoCT) as the reference standard; second, to assess the errors associated with using a fixed f_c (f_c = 0.6) when estimating femoral neck morphology compared to structural measures determined from nanoCT data; and third, to determine if using the measured f_c would reduce the errors associated with HSA estimates compared to using a fixed f_c.

Methods

Human tissue samples

Cadaveric femora from 32 White female and 30 White male donors were obtained from Science Care (Phoenix, AZ, USA), Anatomy Gifts Registry (Hanover, MD, USA), and the University of Michigan Anatomical Donations program (Ann Arbor, MI, USA). Samples were excluded if they had known or suspected musculoskeletal pathologies. Human tissue use and handling were approved by the Institutional Biosafety Committee (IBC), and the study was exempted by the Institutional Review Board (IRB). All samples were cleaned of soft tissue and prepared for high-resolution imaging by sectioning the femoral shaft at a location 16.5 cm distal to the superior aspect of the femoral head. The proximal femur samples were embedded in a 5 cm square aluminum channel filled with acrylic resin (OrthoJet BCA, Lang Dental, Wheeling IL, USA) with the femoral necks oriented at 15 degrees of internal rotation relative to the embedding block faces. Alignment consistency across samples was accomplished using a custom-made alignment fixture. The samples used in this study were a subset of those examined previously (17), limiting the analytic sample to those with body weight data.

NanoComputed tomography images

Whole proximal femora were scanned using nanoCT at 27 µm voxel size, 110 kV, 200 µA, and a 0.762-mm aluminum filter, as described previously (17). A calibration phantom containing air, water, and a hydroxyapatite mimicker (1.69 mg/cc; Gammex, Middleton, WI, USA) was included in each scan to convert attenuation grayscale values to Hounsfield units. Image volumes were reconstructed using datos|x software (GE Inspection Technologies, LP, Skaneateles, NY, USA). Each volume was digitally rotated 15 degrees to account for anteversion using Dragonfly software (Comet Technologies, Montreal, QC, Canada). Two fully convolutional neural networks (FCNNs) were used; one to segment bone from background (Dice = 0.983 ± 0.016) and another to segment cortical from trabecular tissue (Dice = 0.977 ± 0.023), as validated previously (17). Both FCNNs were trained and validated on a sample of 30 white female cadaveric donors ranging in age from 36 to 89+ years, spanning the full spectrum of age-related cortical structural changes including superior cortical thinning to the width of a single trabecula. Each segmented image was manually reviewed for thresholding quality and cortical-trabecular segmentation accuracy, with the boundary defined by established protocols, and manual corrections were made for any false-positive or false-negative voxels. The delineation between cortical and trabecular bone was based on the approach described by Bell et al. (18, 19); the cortical:trabecular FCNN was trained to segment dense, low-porosity compact regions of cortical bone and higher-porosity endocortical regions that, while visually similar to trabecular bone, are anatomically distinct and reflect the progressive increase in resorption within the Haversian canals adjacent to the endocortical surface (20, 21) (Figure 1). The cortical:trabecular FCNN was not trained to segment based on a specific porosity value but rather on the pattern and continuity of similar structural features characteristic of tissue circumscribing the cross-section. The FCNN was trained to segment cortical tissue in the superior region that is as thin as a single trabecula and to simultaneously segment the thick cortical tissue in the inferior region.

Three femoral neck volumes of interest (FN VOIs) were generated (Figure 1): a 15 mm wide VOI located at the base of the FN, a 15 mm wide VOI centered at the narrow neck region, and a 3 mm wide VOI centered at the narrow neck region. The narrow neck site was identified as the location within the femoral neck having the smallest width perpendicular to the hip axis length. The two 15 mm wide VOIs were anatomically located to be similar to those of FN regions of interest defined by two widely used commercial manufacturers. The 3 mm wide VOI was consistent with the size and anatomical location of the region used in the HSA algorithm (16). Reliability of the FN VOI extraction process was confirmed with an intraclass correlation coefficient (ICC) of 0.9997 for area and 0.988 for bone mineral content (BMC) (17). Average cross-sectional cortical area (Ct.Ar), total subperiosteal area (Tt.Ar), trabecular BV/TV, superior-inferior outer diameter, and superior-inferior internal diameter were determined for the 3 mm VOI (Figure 1). The fraction of total bone mass (BMC) attributed to cortical and trabecular tissues was calculated for the three VOIs for each femur. Cross-sectional circularity was calculated for the narrow neck site as the ratio of the superior-inferior to the anterior-posterior periosteal diameter.

PseudoDXA images

PseudoDXA images were generated for each of the three FN VOIs described above using a custom written Dragonfly plug-in. Briefly, a two-dimensional bone density map (pseudoDXA image) was generated by projecting all bone voxels onto a planar region of interest (ROI) with the proximal femur anteverted 15 degrees, consistent with standard positioning during a routine DXA image acquisition. PseudoDXA images were created for cortical bone, trabecular bone, and the combination of both. PseudoDXA parameters, including the projected ROI area (pseudoDXA area), total number of bone voxels (pseudoDXA BMC), and the total number of bone voxels per area (pseudoDXA aBMD), were calculated. Average and minimum FN widths (FNW) were measured directly from the pseudoDXA images. PseudoDXA parameters were significantly correlated with corresponding parameters measured from a commercial Hologic DXA system in a validation sample of 19 White female donors (22): BMC (R² = 0.95, p<0.0001), areal BMD (R² = 0.92, p<0.0001), and ROI area (R² = 0.58, p<0.0001).

HSA algorithms

The hip structure analysis (HSA) estimates of FN morphology were calculated at the narrow neck location using the algorithms and methods described previously (16). The HSA structural measures included the following:

Router=FNW/2.

A=∑(ti ps)

Rinner=(Router2− fc(A / π))12

BR=Router/(Router− Rinner)

Ct.Ar=π (Router2− Rinner2)

Ct.Th=Router− Rinner

where FNW is the femoral neck width, R_outer is the outer radius of the femoral neck cross-section, t_j is the linear attenuation value at pixel j along the mass profile, p_s is the pixel size, A is the cross-sectional area of bone minus voids, spaces, and marrow cavity (equivalent to CSA), BR is the buckling ratio, Ct.Ar is cortical area, R_inner is the endocortical radius, and Ct.Th is cortical thickness. It is important to distinguish between cross-sectional area (CSA) in the HSA program and cortical area (Ct.Ar). For purposes of this investigation and to be consistent with the HSA algorithms, CSA is derived from the X-ray attenuation profile without simplifying assumptions and represents the total area occupied by bone tissue at a given cross-section, excluding voids, marrow spaces, and other cavities that do not contribute meaningfully to load-bearing capacity and CSA and A are used interchangeably (16). Ct.Ar is the cross-sectional area occupied by cortical bone and is calculated assuming a circular profile with a uniformly thick cortex.

The HSA structural measures were calculated in two ways: first, using a fixed proportion of BMC attributable to cortical bone (fixed f_c = 0.6) and second, using the proportion of BMC attributable to cortical bone measured from the 3 mm nanoCT volume centered at the narrow neck site (measured f_c). For comparison to actual structural values, Ct.Ar, R_inner, and the buckling ratio were measured directly from the 3 mm nanoCT volume located at the narrow neck site (Figure 1).

Analysis

Data were expressed as mean ± standard deviation (SD). Normality was confirmed using the D’Agostino and Pearson test. Data were reported for females and males separately. Independent t-tests were used to compare traits between female and male donors. Data were adjusted to the average age of the female and male donors (61.6 years) when FN traits were directly compared between sexes. For all analyses, the 3 mm VOI was used for the HSA algorithms to be consistent with the protocols described previously (16). Likewise, comparisons to the actual structural measures were conducted using traits measured from the 3 mm VOI located at the narrow neck site. Values measured directly from the nanoCT images were the standard against which HSA parameters calculated using fixed f_c and measured f_c were compared. All statistical analyses were conducted using Prism (version 10, GraphPad Software, LLC, Boston, MA, USA), except as noted.

The first analysis (f_c Variation) determined the extent to which f_c varied among individuals, with body size, and with age. All analyses were conducted separately for females and males. To test how the measured f_c values varied along the femoral neck, a linear regression analysis was conducted between the f_c measured for the 3 mm VOI and the f_c measured for the 15 mm VOIs corresponding to two commercially available DXA systems (Figure 1). A repeated measures (RM) one-way ANOVA with Geisser-Greenhouse correction was conducted separately for females and males to assess whether f_c differed across the three VOI locations. A correlation matrix was established among f_c, age, weight, pseudoDXA BMC, and FNW to assess the associations among these parameters; f_c and pseudoDXA BMC were assessed for the 3 mm VOI. We examined associations with age, weight, BMC, and FNW because these parameters are available clinically. Pairwise correlations between f_c and demographic and imaging variables (Table 1) were computed as an exploratory screening step to inform variable selection for multivariable regression modeling. To account for multiple comparisons within this screening step, p-values were adjusted using the Benjamini-Hochberg false discovery rate (FDR) procedure applied separately within each sex (15 pairwise tests per sex) using p.adjust() in R (v 4.6.0, The R Foundation for Statistical Computing, Vienna, Austria). Associations with FDR-adjusted q<0.05 were considered statistically significant for the purpose of variable selection. Finally, a multivariable linear regression analysis was conducted between f_c and age, weight, BMC, and FNW to identify significant predictors of f_c. Variance inflation factors (VIFs) were calculated to assess the extent to which multicollinearity inflated the variance of the estimated regression coefficients.

Table 1. Demographic characteristics and femoral neck morphology for female and male donors.

Values are mean ± SD. BMI, body mass index; FNW, femoral neck width; f_c, cortical fraction; SI, superior-inferior; AP, anterior-posterior. ^aIndependent t-test for unadjusted comparisons; independent t-test on age-adjusted values.

The second analysis (Effects of fixed versus measured f_c on morphology estimates) determined the extent to which assuming a fixed f_c affected estimates of morphology using the HSA algorithms compared to using the measured f_c. First, a Bland-Altman analysis was conducted to assess the agreement between the HSA estimates using the fixed f_c versus the measured f_c values. The bias and 95% limits of agreement were calculated for this analysis. Second, a linear regression analysis was conducted comparing the structural values estimated from HSA using fixed f_c and measured f_c with those measured directly from the corresponding nanoCT images. The slopes and elevations of these two regressions were compared using the ‘Compare lines’ function in Prism, which tests for differences using an F-test and is mathematically equivalent to ANCOVA. For each regression, we also calculated Lin’s concordance correlation coefficients, which assess the agreement (concordance) between two measures of a variable (i.e., HSA estimate versus nanoCT value) (R v 4.6.0). If the HSA estimates for R_inner, Ct.Ar, and BR were identical to those measured from the nanoCT image, then the regression would approximate the identity line (i.e., x = y) and Lin’s concordance correlation coefficient would be 1. Differences in concordance between the regression of the HSA estimates using fixed f_c versus nanoCT-derived values and the regression of the HSA estimates using measured f_c versus nanoCT-derived values were considered significant when the 95% confidence intervals did not overlap.

The third analysis (Age-related errors in morphology estimates) assessed how the errors associated with using fixed f_c and measured f_c to estimate R_inner, Ct.Ar, and BR differed as a function of age. Female and male samples were sorted into younger (≤65 years of age) and older (>65 years of age) age groups. Sorting the samples using 65 years of age split the samples into relatively similar-sized subgroups and coincided with the age when bone health monitoring tends to begin. The structural estimates were compared to the traits measured from nanoCT to assess the errors when estimating the parameters using fixed f_c and measured f_c. The errors were compared between younger and older subgroups using a t-test to determine if the errors differed between age groups. The errors were also compared using paired t-tests for each of the age subgroups to determine if using the measured f_c reduced the errors compared to using a fixed f_c. Analyses were conducted for females and males separately.

The fourth analysis (VOI mismatch error estimation) estimated the additional error introduced when applying a 3 mm narrow neck VOI-derived f_c correction to commercial HSA systems that average over a 15 mm region. HSA structural parameters (R_inner, Ct.Th, Ct.Ar, and BR) were computed for each specimen under three f_c scenarios: f_c measured at the 3 mm narrow neck VOI, f_c measured at the 15 mm narrow neck VOI, and f_c measured at the 15 mm base of FN VOI, using the equations described in the HSA Algorithms section. Percent errors for each 15 mm scenario were computed relative to estimates derived using f_c from the 3 mm narrow neck VOI, which served as the reference standard consistent with Analyses 2 and 3. Mismatch errors are additionally reported as a proportion of the fixed f_c error magnitude to contextualize their clinical relevance. Differences in f_c between the 3 mm narrow neck VOI and each 15 mm VOI were tested with one-sample t-tests against a null hypothesis of zero difference, conducted separately for females and males.

Results

Analysis 1: f_c variation

The demographic data, FNW, measured f_c values, and circularity are shown in Table 1. Female and male donors spanned 18 to over 89 years of age with group averages of 65.5 ± 20.6 years and 57.6 ± 21.4 years for females and males (p=0.144, t-test), respectively. Males were heavier (p=0.007, t-test) and taller (p < 0.0001) but showed no difference in BMI (p=0.474, t-test) compared to females. Males had wider FNW (p<0.001, t-test) compared to females, as expected. Mean values for f_c were 0.48 ± 0.09 for females and 0.42 ± 0.08 for males, which differed significantly between sexes (p=0.010, t-test) even after adjusting for age (p=0.024, t-test). The circularity, which was calculated as the ratio of periosteal border diameters in the superior-inferior and anterior-posterior directions, was 1.00 ± 0.09 for females and 1.00 ± 0.06 for males, which did not differ between sexes after adjusting for age (p=0.826, t-test).

The f_c values calculated for the 3 mm NN region correlated significantly with the f_c values calculated for two 15 mm wide VOIs (Figure 2), though f_c differed significantly across the three VOI locations for both females and males (Females: F(1.12, 34.73)=59.63, p<0.0001; Males: F(1.19, 34.39)=46.48, p<0.0001; RM one-way ANOVA with Geisser-Greenhouse correction). The linear regressions showed significant differences in the elevations between the base and narrow neck VOIs, with the f_c values calculated for the FN base location being larger than those calculated for the narrow neck and 3 mm VOIs for both females and males (p<0.0001, paired t-test).

The correlation matrix showed associations between f_c and clinically available parameters including age, weight, BMC, and FNW (Table 2). For females, significant associations were found between f_c and age (p=0.004). For males, significant associations were found between f_c and weight (p<0.001). After Benjamini-Hochberg FDR correction (15 pairwise tests per sex, α = 0.05), the two primary associations remained significant: f_c vs. Age in females (raw p=0.004, q = 0.010) and f_c vs. Weight in males (raw p=0.0004, q=0.006). In females, no additional f_c correlations were lost, as all correlations significant at the uncorrected threshold survived FDR correction. In males, f_c vs. pseudoDXA BMC was nominally significant at the uncorrected threshold (p=0.044) but did not survive FDR correction (q=0.165); this variable did not enter the final male multivariable model, and no downstream conclusions are affected by this correction. A multivariable linear regression analysis was performed separately for females and males to identify significant independent predictors of f_c (Table 3). For females, age and BMC were significant predictors with adjusted R² = 0.361 (p=0.003) for the overall regression. For males, weight remained the only significant predictor in the multivariable regression model with adjusted R² = 0.402 (p=0.002) for the overall regression. Variance inflation factors (VIFs) were less than 5 confirming little to moderate correlation among predictor variables.

Table 2. Associations among measured f_c and clinically available parameters.

Bold font indicates significant associations (p<0.05). f_c, cortical fraction; R_outer, outer radius = FNW/2; FNW, femoral neck width; pseudoDXA BMC, bone mineral content derived from simulated DXA images generated by projection of nanoCT data. Pairwise correlations were computed as an exploratory screen (15 tests per sex) with Benjamini-Hochberg FDR correction applied. Bold font indicates associations surviving FDR correction (q< 0.05). In males, the f_c vs. pseudoDXA BMC correlation was significant at the uncorrected threshold (p = 0.044) but did not survive FDR correction (q = 0.165) and is not bolded; this variable did not enter the final male multivariable model.

Table 3. Multivariable linear regression analysis between measured f_c and clinically available parameters.

Bold font indicates significant predictors (p<0.05). BMC: bone mineral content; FNW: femoral neck width; VIF: variance inflation factor; t: t-statistic (Estimate/Standard Error).

To better understand the sex-specific associations among f_c, age, and weight observed in the multivariable models, follow-up linear regression analyses were conducted using the amount of cortical and trabecular bone versus age and weight. Cortical and trabecular tissues can be segmented in pseudoDXA images, and the amount of each tissue type is measured as the number of voxels. The total number of bone voxels was not included in the ANCOVA regression comparisons but is shown for reference purposes and to provide context for how f_c varies with age and weight (Figures 3A, B, D, E). For the age associations (Figures 3A–C), both females and males showed significant negative associations between the number of trabecular voxels and age (R² = 0.525, p<0.0001 females; R² = 0.175, p=0.022 males) and total voxels and age (R²=0.329, p=0.0006 females; R²=0.218, p=0.009 males), but the associations between the number of cortical voxels and age were not significant (R² = 0.028, p=0.356 females; R² = 0.036, p=0.312 males). The resulting f_c values showed a positive association with age for females (R² = 0.243, p=0.004) but not males (R² = 0.042, p=0.280). To understand the association between f_c and weight (Figures 3D–F), we first adjusted the number of cortical and trabecular voxels for age by computing residuals from a linear regression of voxel counts on age, separately for females and males. For females, the number of age-adjusted cortical (R²=0.273, p=0.002), trabecular (R²=0.314, p=0.0009), and total voxels (R²=0.452, p<0.0001) all showed significant positive associations with weight, with a significant difference in elevations (p=0.020, ANCOVA) but no difference in slopes (p=0.702, ANCOVA). In contrast, the number of age-adjusted cortical and trabecular voxels showed divergent associations with weight for males (difference in slopes: p=0.0004, ANCOVA), with the number of cortical voxels showing a significant negative association with weight (R² = 0.144, p=0.042) and the number of trabecular voxels showing a positive association with weight (R² = 0.263, p=0.005), while total voxels did not show a significant association with weight (R²=0.077, p=0.146). Given the sex-specific regressions between cortical and trabecular voxels versus weight, it was not surprising that males (R² = 0.384, p=0.0003) but not females (R² = 0.036, p=0.298) showed a significant association between f_c and weight (slopes, p=0.002, ANCOVA).

Analysis 2: effects of fixed versus measured f_c on morphology estimates

In the second analysis, we tested whether using the measured f_c in the HSA algorithms improved estimates of FN structural parameters compared to using a fixed f_c. The Bland-Altman analysis showed that substituting fixed f_c for measured f_c systematically underestimated R_inner and BR and overestimated Ct.Ar in both females and males, with larger bias in males and wider limits of agreement or R_inner and Ct.Ar in males (Table 4). Linear regression analysis showed significant correlations between measured R_inner and both HSA estimates for both females and males (Figure 4). A comparison of the linear regressions showed significantly lower elevations for the fixed f_c regression for females (p=0.007, ANCOVA) and males (p<0.0001, ANCOVA). Both BR estimates correlated significantly with measured BR for females and males. The slopes differed significantly between estimates (females: p=0.002; males: p=0.007, ANCOVA), with fixed f_c estimates generally falling below those using measured f_c. Both Ct.Ar estimates correlated significantly with measured Ct.Ar for females and males. The elevations differed significantly between estimates (females: p<0.0001; males: p<0.0001, ANCOVA), with fixed f_c estimates generally falling well above those using the measured f_c. Lin’s concordance correlation coefficients were generally improved when the measured f_c values were used to estimate R_inner, Ct.Ar, and BR (Table 5). Although using measured f_c generally produced estimates more closely aligned with nanoCT-measured values, a significantly greater improvement in concordance based on non-overlapping 95% confidence intervals for Lin’s concordance correlation coefficients was observed for R_inner and Ct.Ar in males.

Table 4. Bland-Altman bias and 95% limits of agreement comparing HSA estimates using fixed f_c versus measured f_c.

Bias = fixed f_c estimate minus measured f_c estimate. Positive values indicate overestimation by fixed f_c; negative values indicate underestimation. LOA, limits of agreement.

Table 5. Lin’s concordance correlation coefficients (CCC) for HSA estimates using fixed f_c and measured f_c compared to nanoCT-measured values.

CCC, concordance correlation coefficient; CI, confidence interval. Differences in concordance were considered significant when 95% CIs did not overlap. A significantly greater improvement for concordance with measured f_c was observed for R_inner and Ct.Ar in males.

Analysis 3: age-related errors in morphology estimates

The third analysis tested if using a fixed f_c to estimate structural parameters showed greater errors in older (>65) individuals compared to younger (≤65) individuals and whether these errors were reduced when using measured f_c values (Table 6). For R_inner, the percent error differed significantly between older and younger females but not males. Using the measured f_c significantly reduced the errors for males but not females. The errors for BR were substantial (16-37%) for females whether using fixed or measured f_c values. In contrast, the large errors (23-28%) associated with estimating BR using fixed f_c were significantly reduced to 9-10% when using measured f_c values (p<0.0001, paired t-test) for males. Finally, using fixed f_c values to estimate Ct.Ar resulted in large errors for both females (13-36%) and males (31-36%), and in both sexes these errors were significantly reduced to 5-10% when using measured f_c values (p≤0.001, paired t-test). Errors associated with estimating structural parameters using fixed f_c values were significantly different between younger and older women. R_inner and BR were underestimated for younger women and overestimated for younger women. Ct.Ar was overestimated for both age groups, but the error was significantly larger for younger women. No significant difference in errors between younger and older women were found when using the measured f_c to estimate structure. The errors in estimating structural parameters using fixed f_c and the measured f_c did not differ between younger and older men (t-test); however, post-hoc power analyses revealed low achieved power for these comparisons (5–27%), indicating that these null results should be interpreted cautiously and are most likely attributable to insufficient sample size rather than a true absence of age-related differences.

Table 6. Comparison of errors for females and males sorted by age into younger (≤65 years) and older (>65 years) subgroups.

Data are shown as the mean % error ± SD, where % error = 100 × (estimated − measured)/measured. Bold font indicates significant differences (p<0.05). Age-effect: comparison between younger and older subgroups using an independent t-test.

Analysis 4: VOI mismatch error estimation

The mean f_c at the 3 mm narrow neck VOI did not differ significantly from f_c at the 15 mm narrow neck VOI in either females (+0.003 ± 0.020, t (31)=0.780, p=0.441, one-sample t-test) or males (+0.005 ± 0.027, t(29)=1.036, p=0.309, one-sample t-test). Consequently, the additional error introduced by applying a 3 mm-derived f_c correction to a commercial narrow neck system was negligible, representing 8–15% of the fixed f_c error magnitude across structural parameters and both sexes. In contrast, f_c at the 3 mm narrow neck VOI was significantly lower than f_c at the 15 mm base of FN VOI in both females (−0.095 ± 0.070, t(31)=7.674, p<0.0001, one-sample t-test) and males (−0.095 ± 0.074, t(29)=6.979, p<0.0001, one-sample t-test). Applying a 3 mm-derived f_c correction to a base of FN commercial system introduced mismatch errors representing 47–83% of the fixed f_c error magnitude, indicating that only a portion of the available improvement would be recovered in that context.

Discussion

The analyses described in the current study expand on our previous study by focusing on how the proportion of cortical and trabecular tissue mass varies among individuals and how this varying proportion affects algorithms used to estimate structural properties (7). The goal of this study was to critically evaluate HSA estimates of femoral neck structural measures for systematic errors. DXA images are a 2D projection of a 3D structure, and simplifying assumptions are needed to extract meaningful structural information. The HSA algorithms assume the femoral neck has a circular cross-section with a uniformly thick cortex that accounts for 60% of the total bone mineral content to estimate R_inner, BR, and Ct.Ar. Using high-resolution nanoCT data as a means of comparison, we 1) investigated whether the ratio of cortical to total bone mineral content (f_c) varied with sex, age, and body size; 2) determined errors in estimating femoral neck morphology resulting from the assumption of a fixed f_c; and 3) determined errors in estimating femoral neck morphology with measured f_c determined from nanoCT data. The primary source of error was the assumption that cortical bone accounts for 60% of the total mineral content. Analysis of proximal femur nanoCT images confirmed that f_c measured at the narrow neck site is not fixed, but varies significantly with factors such as sex, age, weight, and anatomical site along the femoral neck. When compared to structural measures assessed from nanoCT volumes, the errors associated with using a fixed f_c reached up to 36% for Ct.Ar in both sexes. Replacing fixed f_c with measured f_c significantly reduced errors for Ct.Ar and R_inner in males and Ct.Ar in females. Thus, we showed that the fixed f_c assumption is associated with significant errors in structural estimates that are age- and sex-specific.

One of the clearest findings from our data is that f_c varies considerably between individuals, and that the mean narrow neck values of 0.48 for females and 0.42 for males, with individual values ranging from 0.23 to 0.67, are well below the 0.6 value that HSA assumes. The femoral neck is not a uniform cortical structure; rather, its composition varies depending on location along the neck axis. The cross-section is predominantly cortical near the femoral shaft, but the proportion of trabecular bone increases progressively toward the femoral head (23). A single fixed f_c value will therefore fail to capture both the spatial heterogeneity of femoral neck composition and the considerable interindividual variation in cortical fraction that exists within a population. The fact that our mean values fall below 0.6 in both sexes likely reflects the mixed cortical-trabecular character of the mid-neck region where the narrow neck measurement is taken, rather than a fundamental disagreement with prior estimates made at more shaft-adjacent locations along the femoral neck.

The origin of the f_c = 0.6 value itself warrants examination. In Beck et al. (11), the assumption that 60% of measured mineral mass at the narrow neck is cortical is attributed to two in vitro studies: Kuiper et al. (24) and Bell et al. (19). However, neither study directly supports this value as applied to the narrow neck ROI used in HSA. Kuiper et al. used quantitative computed tomography (QCT) in 20 embalmed cadaveric femoral necks and reported that at the smallest cross-sectional slice, corresponding most closely to the HSA narrow neck ROI, trabecular bone constituted 33% of total bone mass, implying a cortical fraction of 0.67, not 0.6 (24). A cortical fraction closer to 0.6 in the Kuiper data corresponds to a location approximately 8 millimeters proximal to the narrow neck, toward the femoral head. Bell et al. used histomorphometry in cadaveric femoral neck biopsies to characterize regional cortical and cancellous bone distribution by octant, reporting geometric area fractions rather than mineral mass fractions (19). Because cortical bone is considerably denser than trabecular bone, the same geometric area of cortical bone contributes disproportionately more to total mineral mass than trabecular bone of equal area, meaning the mineral mass fraction of cortical bone will always exceed its geometric area fraction. Bell’s histomorphometric data cannot therefore be directly used to establish the mineral mass fraction that HSA requires without additional density assumptions that Bell et al. did not provide. Neither study was designed specifically to establish a mineral mass fraction for DXA-based structural analysis at the narrow neck, and neither used measurements at precisely the 3 mm narrow neck VOI that the HSA software analyzes. The empirical basis of the f_c = 0.6 assumption, as applied to the 3 mm narrow neck ROI used in HSA, does not appear to be well established by the studies cited as its source. To our knowledge, the present study provides the first systematic evaluation of this assumption using high-resolution nanoCT data across a large cadaveric sample spanning a wide age range and finds mean narrow neck f_c values of 0.48 for females and 0.42 for males, both substantially below 0.6.

Differences in segmentation approach, specifically our use of density-based FCNN segmentation at 27 µm resolution versus the QCT thresholding used by Kuiper et al. and the histomorphometric area fractions reported by Bell et al., may partially account for this discrepancy (19, 24). The circularity of the narrow neck cross-section was confirmed at 1.00 ± 0.09 for females and 1.00 ± 0.06 for males, confirming that the circular cross-section assumption held while the f_c assumption did not. The Kuiper et al. data show that f_c changes approximately 2.5% per mm along the femoral neck axis, and our analyses confirm that this spatial variation is detectable even within the narrow neck region itself, with f_c differing significantly across the three VOI locations in both females and males (Females: F(1.12, 34.73)=59.63, p<0.0001; Males: F(1.19, 34.39)=46.48, p<0.0001; RM one-way ANOVA with Geisser-Greenhouse correction). Examination of cortical and trabecular voxels separately, afforded by pseudoDXA images and not available in clinical DXA, provided a deeper understanding of why assuming a fixed f_c is problematic for the femoral neck. For the age associations (Figures 3A–C), both females and males showed significant negative associations between the number of trabecular voxels and age, while cortical voxels showed no significant age association in either sex. However, females appear to have proportionally less trabecular bone than males at all ages, so as trabecular bone declines with age, the cortical fraction rises, producing a significant positive association between f_c and age in females (R²=0.243, p=0.004) that was not observed in males (R²=0.042, p=0.280). Population-based QCT data show that trabecular vBMD at the femoral neck declines by more than 50% over life in women compared with approximately 45% in men, with declines beginning before midlife, while cortical area at the femoral neck is relatively preserved in both sexes (25). Longitudinal QCT data further show that women lose cortical thickness and cortical vBMD at the femoral neck more rapidly than men, yet these losses are only weakly reflected in standard DXA-derived femoral neck aBMD (26), which helps explain why HSA algorithms relying on a fixed f_c assumption cannot account for the age-related shifts in cortical fraction documented in the present study. This age association was not seen for males, where similar cortical and trabecular declines produce no net association between f_c and age, consistent with more gradual, non-event-driven bone loss in men (27).

This study provided new insight into sex-specific differences in weight associations (Figures 3D–F). For females, larger body weights were accommodated with both greater trabecular and cortical bone, which explains why f_c does not show a significant association with weight for females. However, for males, larger body weights were supported by an increase in trabecular bone but not cortical bone, which explains why f_c decreased with weight for males. Heavier men had a proportionally lower percentage of bone mineral content occupied by cortical bone, suggesting different sex-specific skeletal loading strategies to accommodate higher body weight, a novel observation enabled by the pseudoDXA approach used here. Note that analyses were conducted within the observed weight range and do not include obese individuals.

Errors were large when using fixed f_c. For females, errors in Ct.Ar ranged from 12.6% in older to 36.3% in younger females, and BR errors ranged from 17.4% in older to 20.8% in younger females, with younger females showing underestimation and older females showing overestimation of BR. For males, errors in Ct.Ar ranged from 31.0–35.9% and BR errors ranged from 22.6–27.9%. The Bland-Altman analysis showed that substituting fixed f_c for measured f_c systematically underestimated R_inner and BR and overestimated Ct.Ar in both females and males, with larger bias in males and wider limits of agreement for R_inner and Ct.Ar in males (Table 4). Errors for R_inner were smaller but compound into Ct.Ar estimates via squared terms in the algorithm, amplifying their effect on downstream structural parameters.

Using measured f_c significantly reduced errors for Ct.Ar in both sexes and for BR in males, but BR errors remained large in females (15.7–37.1%) even with measured f_c. Errors associated with using a fixed f_c differed significantly between younger and older females but not males (Table 6), with younger females most affected, as Ct.Ar was overestimated by 36.3% because their relatively high trabecular content pulls measured f_c well below 0.6. Using measured f_c eliminated the age effect on errors in females and substantially reduced errors for all traits in males. Lin’s concordance correlation coefficients showed significantly greater improvement in concordance, based on non-overlapping 95% confidence intervals, for Ct.Ar and R_inner in males (Table 5). The persistence of large BR errors in females despite using the correct f_c value, compared with BR errors of 9-10% in males with measured f_c (Table 6) suggests that f_c is not the only source of systematic error in the BR algorithm for females, and that a second structural assumption requires examination.

The BR calculation assumes not only a fixed f_c but also that the cortex is uniformly thick around the circumference of the femoral neck. This uniform cortical thickness assumption is less accurate in females than in males. Using CT in 77 proximal femurs spanning ages 20–95 years, Mayhew et al. showed that the femoral neck cortex is markedly asymmetric, with the inferior cortex substantially thicker than the superolateral cortex at all ages, and that this asymmetry increases with age more rapidly in females than males with superior and superoposterior cortical thickness declining 6.4% per decade in women compared with significantly smaller effects in men (28). Similarly, Bell et al. demonstrated using histomorphometry that the inferior cortex is disproportionately thick relative to the superior cortex, with a significant sex difference in cortical bone proportion specifically in the inferoanterior region, and Carpenter et al. confirmed using QCT in the AGES-Reykjavik cohort that females have a relatively thicker inferior cortex despite having smaller femoral necks overall (19, 27). When the inferior cortex is disproportionately thick, the actual BR, which reflects the true distribution of bone around the neutral axis, will be lower than the algorithm predicts, meaning the femoral neck is stronger in bending than HSA estimates even when f_c is correctly specified. The HSA algorithm cannot account for this circumferential asymmetry in cortical thickness because it derives BR from a concentric circular model, and this limitation is more consequential in females than in males. These prior findings support the hypothesis that the uniform cortical thickness assumption is less valid in females than in males, though this was not directly tested in the present dataset.

In males, the more uniform circumferential distribution of cortical bone means that the uniform Ct.Th assumption holds more closely, and correcting f_c alone is therefore sufficient to substantially reduce errors. The residual BR errors in females are not a failure of the f_c correction but are consistent with the hypothesis that the uniform Ct.Th assumption introduces additional error in females; quantifying this contribution directly and developing methods to account for it using clinically available imaging represents an important direction for future work, particularly for female fracture risk assessment.

Beyond the residual female BR errors, the age-stratified analysis revealed additional clinically important patterns in the errors associated with fixed f_c. Errors associated with using a fixed f_c were significantly greater in younger females than in older females for both Ct.Ar and BR (Table 6), while no significant age effect was observed in males. This pattern is mechanistically consistent with the f_c–age association described above: younger females have proportionally more trabecular bone, pulling their measured f_c further below 0.6 and therefore producing larger errors when the fixed value is applied. The practical consequence is that younger women in whom the f_c divergence from 0.6 is greatest are also the individuals for whom fixed f_c HSA estimates are most systematically biased. Using measured f_c eliminated the significant age effect on errors in females, suggesting that individualized f_c estimation could substantially improve the clinical utility of HSA in this population.

HSA has become a widely used tool in osteoporosis research because it offers geometric information about the proximal femur that standard BMD cannot provide. Beck et al. (29) described how the software derives cross-sectional area, cross-sectional moment of inertia, section modulus, cortical thickness, and buckling ratio from the distribution of mineral mass across pixel profiles at the narrow neck, intertrochanter, and shaft, parameters that reflect how bone resists bending and compressive forces during a fall. Beck et al. (29) acknowledged key limitations: the method reconstructs a three-dimensional structure from a two-dimensional image, is sensitive to leg rotation during scanning, and relies on a fixed assumption about the fraction of mineral mass residing in the cortical compartment. Prevrhal et al. previously compared HSA estimates to volumetric QCT and found reasonable agreement for BMD and CSMI, but acknowledged that no true gold standard existed for the cortical parameters that depend on the f_c assumption (30), leaving this as an open question in the literature that the present study directly addresses.

This limitation is particularly consequential because the cortical parameters most affected by the f_c assumption are also the ones that carry the strongest clinical signal for fracture risk. Kaptoge et al. demonstrated this in what remains the largest prospective validation of HSA to date, following 7,474 women over thirteen years and capturing 635 incident hip fractures (12). Their analysis found that cortical thickness, cross-sectional area, and buckling ratio each predicted hip fracture at least as well as BMD alone at the narrow neck, and that combining geometric variables with age improved discrimination beyond BMD alone. Ha et al. reinforced this from a different angle with a cross-sectional propensity score matched study, finding that hip fracture patients had significantly lower femoral neck cross-sectional area and cortical thickness than matched controls, with HSA variables achieving sensitivities of 82–90% for identifying fracture cases (31). The case for buckling ratio specifically was made most forcefully by Anderson et al., whose twenty-year follow-up of nearly 1,000 Australian women showed that higher shaft buckling ratio remained independently associated with low-trauma fracture even after adjusting for total hip BMD, meaning the information buckling ratio carries about fracture risk is not redundant with BMD and errors in its estimation cannot simply be absorbed by adjusting for BMD in a clinical model (32).

The present findings connect directly to these clinical stakes. The errors associated with fixed f_c documented here are not random but systematic, age-dependent, and sex-specific, meaning they will distort group comparisons across age and sex in precisely the populations where HSA is most commonly applied. The reversal in the direction of BR error between younger and older females is particularly concerning, as younger women are underestimated as stronger than they are while older women are overestimated as weaker than they are, both of which carry implications for clinical decision-making. Replacing fixed f_c with measured f_c significantly reduced errors in Ct.Ar for both sexes and in BR for males, demonstrating that the f_c assumption is the primary driver of these errors and that they are at least partially correctable. The broader implication is that the HSA parameters most relied upon in clinical fracture risk assessment are also the ones most distorted by an assumption whose empirical basis, as demonstrated above, is uncertain.

The finding that f_c varies systematically with age and BMC in females and with weight in males raises the question of whether f_c could be estimated from clinically available parameters and used to improve HSA accuracy without requiring nanoCT imaging. The multivariable regression models explained 36.1% and 40.2% of the variance in f_c for females and males respectively (Table 3), and both models were statistically significant. While these R² values confirm that the identified predictors capture meaningful variation in f_c, they also indicate that a substantial proportion of the variance remains unexplained, meaning that individual f_c cannot be reliably predicted from age, BMC, and weight alone with sufficient precision for clinical application. The sex-specific nature of the predictors adds a further layer of complexity, as a single prediction equation applicable to both sexes is not supported by the data. Nevertheless, these findings demonstrate proof of concept that individualized f_c estimation from clinical measurements is feasible and worth pursuing in larger, more diverse samples. An important next step would be to develop and validate f_c prediction models in a larger cohort with concurrent DXA and QCT or nanoCT data, where the additional structural detail available from volumetric imaging could be used to identify additional predictors and improve model accuracy. Ultimately, a clinically deployable f_c estimation approach, even one that reduces rather than eliminates error, would represent a meaningful improvement over the current fixed assumption and could be implemented within existing HSA software frameworks without requiring additional imaging.

While this study sought to understand how assumptions in HSA calculations lead to systematic error, not every assumption was investigated. The current study was limited to examining errors caused by using a constant f_c versus measured f_c and did not assess methods that estimate CSA directly from the mineral density profile, nor the impact of a non-concentric cortex on HSA parameters. Using 3D nanoCT volumes we found that the narrow neck cross-section is circular but the cortex is not concentric, which is well known (19, 28) and represents an important avenue for future investigation. The current study used a non-Hispanic White donor population spanning a wider age range than the cadaveric samples used in early HSA validation work (15,11. Generalizability across racial and ethnic groups is an important limitation that warrants explicit acknowledgment. Hip fracture rates differ substantially across racial and ethnic groups, with rates approximately 50% lower in African American and Asian women than in White women, and intermediate rates in Hispanic and Native American women (33, 34). Significant differences in femoral neck geometry and composite strength indices have been documented among non-Hispanic White, African American, Mexican American, and Native American women using HSA (35–37), and these structural differences are consistent with the known variation in hip fracture risk across groups. Because the patterns of cortical bone distribution that determine f_c at the narrow neck are likely to differ across racial and ethnic groups in ways that would affect both the magnitude and direction of HSA errors documented here, future studies should characterize f_c in racially and ethnically diverse samples to determine whether population-specific f_c values or correction models are needed for accurate HSA-based fracture risk assessment.

The bone-background and cortical-trabecular FCNNs achieved Dice coefficients of 0.983 ± 0.016 and 0.977 ± 0.023, respectively, in a training cohort of 30 white female donors ranging in age from 36 to 89+ years; Dice was not assessed separately by age group, and future studies should confirm that segmentation accuracy is maintained uniformly across the full age range. Additionally, the training cohort was all-female; whether segmentation performance is equivalent in male specimens, which have a more uniformly distributed cortex and different cortical-trabecular boundary characteristics, has not been formally evaluated. Further, we used pseudoDXA images from nanoCT for actual measurements rather than commercial DXA images, which HSA is generally run on; commercial DXA images of the cadaveric proximal femurs were not available. The discrepancies between fixed and measured f_c are large enough to justify repeating this study on a larger cohort with commercial DXA and nanoCT pseudoDXA for comparison.

Additional limitations warrant acknowledgment. Sample sizes in the male age subgroups were modest (younger: n=18, older: n=12), and post-hoc power analyses conducted in R (v 4.6.0, pwr::pwr.t2n.test, α=0.05, two-tailed) revealed very low achieved power for the null age-group comparisons of BR and Ct.Ar: BR under fixed f_c (Cohen’s d=0.20, power=8.2%), BR under measured f_c (d=0.04, power=5.1%), Ct.Ar under fixed f_c (d=0.26, power=10.3%), and Ct.Ar under measured f_c (d=0.26, power=10.3%). These results indicate that the absence of significant age effects in males is most likely attributable to insufficient statistical power rather than a true biological difference from the pattern observed in females, and that a substantially larger male sample would be required to detect age-related differences of the magnitude observed here. The present study computed f_c as the ratio of cortical to total bone voxels at 27 μm resolution, which is a volumetric fraction rather than the mineral mass fraction that the HSA algorithm implicitly requires. Quantitative backscattered electron imaging (qBEI) studies demonstrate that femoral cortical bone exhibits a distinctively higher peak calcium content (Ca_PEAK) than the reference trabecular bone mineral density distribution (BMDD) in healthy adults (38–40). Because cortical bone tissue therefore has a higher tissue-level mineral density than trabecular bone, the mineral mass fraction of cortical bone will be systematically higher than its volumetric fraction. Denoting the mean tissue mineral densities of cortical and trabecular bone as ρ_c and ρ_t, and voxel counts as V_c and V_t, the mineral-mass-based f_c,mass that the HSA algorithm requires is:

fc,mass= (ρc×Vc)/(ρc×Vc+ρt×Vt).

Since ρ_c > ρ_t, f_c,mass exceeds the measured volumetric f_c,vol by a factor between ρ_c/ρ_t. Lerebours et al. (40) reported mean calcium contents of 25.05 ± 0.28 weight% for femoral cortical bone and 22.20 ± 0.45 weight% for trabecular bone from qBEI measurements, yielding a ρ_c/ρ_t ratio of approximately 1.13. Applying this correction to mean volumetric f_c values of 0.48 (females) and 0.42 (males) suggests mineral-mass-based f_c values of approximately 0.51 (females) and 0.45 (males), both of which remain well below the assumed 0.6 and do not alter the direction or significance of the sex, age, and weight associations reported here. This systematic upward shift would modestly reduce, but not eliminate, the gap between measured and assumed f_c. Additionally, because the volumetric f_c was substituted into an algorithm expecting f_c,mass, the error reductions reported for the “measured f_c” condition represent a lower bound on the improvement achievable with a true mineral-mass-based measurement. Future studies incorporating voxel-level attenuation calibration to convert volumetric fractions to mineral mass fractions would enable a more complete evaluation of this assumption and may yield further improvements in HSA accuracy. Each nanoCT acquisition included a hydroxyapatite calibration phantom, which in principle could be used to estimate tissue-level mineral density directly from the study’s own calibrated attenuation values, providing an internally derived ρ_c/ρ_t ratio specific to this sample rather than one imported from an independent population. Extraction of per-voxel calibrated attenuation values separately for cortical and trabecular compartments was not feasible within the scope of the present study; this represents a meaningful direction for future work that could replace the externally derived correction factor with a sample-specific estimate. The f_c measured from the 3 mm narrow neck VOI used in the present study is not directly comparable to f_c derived from the 15 mm VOIs used in commercial HSA software, given that f_c varies significantly across VOI locations as shown here, and further study is needed to characterize how VOI-specific f_c differences propagate into HSA algorithm errors. Finally, the weight analyses were conducted within the observed non-obese weight range and may not generalize to individuals with higher body weight.

In conclusion, this study found that the proportion of cortical bone at the narrow neck, represented by f_c, is not a constant 0.6 as used in HSA estimates. f_c differs significantly between females and males and varies with age and pseudoDXA BMC in females and with weight in males. Fixed f_c = 0.6 produces large, systematic errors in HSA-derived Ct.Ar and BR that are both sex-specific and age-specific, reaching up to 36% for Ct.Ar in both sexes. Using measured f_c substantially reduces errors, with significantly greater improvement in concordance for Ct.Ar and R_inner in males. However, systematic errors remained, particularly for BR in females, which suggests that a fundamental structural difference, likely the non-uniform cortical thickness distribution, limits the effectiveness of the f_c correction in females and represents the next assumption requiring investigation. In the present study, nanoCT served exclusively as a reference standard for quantifying error and does not constitute a clinically viable pathway for individualized f_c estimation given its radiation dose and limited availability. The practical need arising from this study is the development and validation of models that predict f_c from variables already available on the DXA scan itself, such as attenuation profile, neck geometry, areal density, age, sex, and weight, without requiring additional imaging. The multivariable models presented here, which explained 36.1% and 40.2% of f_c variance in females and males respectively, demonstrate proof of concept for this approach but are insufficient for individual clinical use. These findings motivate further development of DXA-derived f_c prediction models and re-evaluation of HSA-derived fracture risk metrics, particularly in younger women and heavier men.

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Footnotes

Edited by: Nico Sollmann, Ulm University Medical Center, Germany

Reviewed by: Youngjun Lee, Indiana University Purdue University, United States

Marcelo Eduardo Berli, Universidad Nacional de Entre Ríos, Argentina