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Bone. Author manuscript; available in PMC 2010 February 1.

Published in final edited form as:

Published online 2008 October 25. doi: 10.1016/j.bone.2008.10.039

PMCID: PMC2669651

NIHMSID: NIHMS96216

Elise F. Morgan,^{*}^{†} Zachary D. Mason,^{*}^{†} Karen B. Chien,^{*} Anthony J. Pfeiffer,^{*} George L. Barnes,^{†} Thomas A. Einhorn,^{†} and Louis C. Gerstenfeld^{†}

Corresponding author: Elise F. Morgan, Department of Mechanical Engineering, Boston University, 110 Cummington Street, Boston, MA 02215, p: (617) 353-2791, f: (617) 353-5866, e: Email: ude.ub@nagromfe

Non-invasive characterization of fracture callus structure and composition may facilitate development of surrogate measures of the regain of mechanical function. As such, quantitative computed tomography- (CT-) based analyses of fracture calluses could enable more reliable clinical assessments of bone healing. Although previous studies have used CT to quantify and predict fracture healing, it is unclear which of the many CT-derived metrics of callus structure and composition are the most predictive of callus mechanical properties. The goal of this study was to identify the changes in fracture callus structure and composition that occur over time and that are most closely related to the regain of mechanical function. Micro-computed tomography (μCT) imaging and torsion testing were performed on murine fracture calluses (n=188) at multiple post-fracture timepoints and under different experimental conditions that alter fracture healing. Total callus volume (TV), mineralized callus volume (BV), callus mineralized volume fraction (BV/TV), bone mineral content (BMC), tissue mineral density (TMD), standard deviation of mineral density (σ_{TMD}), effective polar moment of inertia (J_{eff}), torsional strength, and torsional rigidity were quantified. Multivariate statistical analyses, including multivariate analysis of variance, principal components analysis, and stepwise regression were used to identify differences in callus structure and composition among experimental groups and to determine which of the μCT outcome measures were the strongest predictors of mechanical properties. Although calluses varied greatly in the absolute and relative amounts of mineralized tissue (BV, BMC, and BV/TV), differences among timepoints were most strongly associated with changes in tissue mineral density. Torsional strength and rigidity were dependent on mineral density as well as the amount of mineralized tissue: TMD, BV, and σ_{TMD} explained 62% of the variation in torsional strength (p<0.001); and TMD, BMC, BV/TV, and σ_{TMD} explained 70% of the variation in torsional rigidity (p<0.001). These results indicate that fracture callus mechanical properties can be predicted by several μCT-derived measures of callus structure and composition. These findings form the basis for developing non-invasive assessments of fracture healing and for identifying biological and biomechanical mechanisms that lead to impaired or enhanced healing.

Approximately 5–10% of the more than six million bone fractures that occur each year in the United States show delayed or impaired healing and require subsequent surgical treatment [1]. The regain of bone strength and stiffness are the fundamental qualities that define healing; however, they are difficult if not impossible to measure directly in the clinical setting. At present, clinical assessments rely on poorly defined, qualitative metrics such as range of motion, discomfort, and regain of structural integrity based on plain film X-ray. Radiographs are inherently a two-dimensional assessment of the three-dimensional callus structure. In comparison, computed tomography (CT) provides numerous, quantitative, and three-dimensional measurements of the structure and mineralization of the fracture callus, and these measurements could potentially be related to callus stiffness and strength. Consequently, quantitative, CT-based analyses of callus structure and composition may lead to the development of reliable, non-invasive metrics of healing. Characterization of the temporal changes in callus structure and mineralization under differing physiological conditions will also be of tremendous value in relating the biological healing processes to the regain of strength, thereby establishing structure-function relationships that synthesize the biology and biomechanics of fracture healing.

A number of pre-clinical studies have used quantitative computed tomography (QCT) or peripheral quantitative computed tomography (pQCT) to assess fracture healing [2–11], healing of unicortical defects [10, 12, 13], and bone formation in distraction osteogenesis [14, 15]. Several of these studies have demonstrated good agreement among CT-, radiograph-, and histology-derived measures of healing. For example, callus size measured from radiographs was shown to be correlated with callus cross-sectional area and bone mineral content (BMC) measured using pQCT [6]. Bone mineral density (BMD) measured by pQCT was found to be strongly associated with histological measurements of the percentage of the osteotomy gap occupied by mineralized tissue [7]. Direct comparisons of CT and standard radiographic analyses have indicated that the former can yield comparable or better predictions of callus compressive strength [6] and torsional strength and stiffness [4, 7] and more definitive diagnoses of healing progression [2] and of non-unions [16]. However, no consensus currently exists as to which CT-derived measures, or combinations of measures, best predict callus strength and stiffness.

Micro-computed tomography (μCT) provides superior resolution to pQCT and QCT that is of benefit when studying bone healing in small animals. Several studies have used μCT to measure quantities such as bone volume, bone volume fraction, and mineral density in the fracture callus [17–23]. However, contradictory results have been reported regarding how well these quantities predict callus mechanical properties [17, 20, 22]. This discrepancy may have arisen because these studies chose different outcome measures and different timepoints for assessment. Given that μCT can provide a host of measures of callus structure and composition, the challenge remains to determine how these measures can be used to characterize healing in a manner that is relevant to callus mechanical properties and to the underlying biological healing mechanisms over the full timecourse of repair.

The overall goal of this study was to characterize the changes in fracture callus structure and composition that occur during the healing process with the intent of relating these changes to regain of mechanical function. Fracture healing from three different types of experiments was analyzed. These experiments focused on an anabolic therapeutic (parathyroid hormone (PTH)), two different types of anti-resorptive agents (alendronate and denosumab), and a catabolic pathology (a lupus-like syndrome). These experiments provided substantial diversity in callus structure, stiffness, and strength, thus enabling a broad investigation of relationships among healing processes, callus structure, and mechanical function. The specific objectives of this study were: 1) to determine which measures of callus structure and composition discriminate most clearly among experimental groups; 2) to determine which measures best describe the variability in callus structure and composition among specimens; and 3) to determine which measures are most predictive of callus stiffness and strength. The results were used to identify CT-based indices of healing that also reflect key aspects of fracture repair biology.

Animal research was conducted in conformity with all federal and USDA guidelines, as well as IACUC-approved protocols. Unilateral, mid-diaphyseal, pinned femoral fractures were produced in mice as previous described [24] using a modification of the same blunt guillotine procedure as developed for rat by Bonnarens and Einhorn [25]. Radiographic assessment of the fractures was performed immediately after fracture and at the time of euthanasia. Fractures that did not occur in the mid-diaphysis or were excessively comminuted were not used in the study. Animals were euthanized by CO_{2} asphyxiation, and the fractured femora were harvested and carefully cleaned of muscle and soft connective tissue.

The analyses in this study were performed on fracture calluses from three different fracture healing experiments (Table 1). The timepoints used in each experiment were chosen based on joint consideration of the timecourse of murine fracture healing and the expected effects of the biological perturbation investigated in each experiment. In the first experiment, 72 male C57BL/6 mice eight to ten weeks of age were enrolled. Animals were given daily subcutaneous injections of PTH (1–34) (30 μg/kg, n=36) or saline (n=36) beginning on postoperative day 1. The fractured femora were harvested on post-operative days 14, 21, and 28. The second experiment compared two different anti-resorptive therapies: denosumab and alendronate. Male C57BL/6 human RANKL knock-in mice eight to 17 weeks of age were provided through a materials transfer agreement with Amgen Inc. (Thousand Oaks, CA) and were shipped directly from breeding colonies (Charles River Laboratory, San Diego, CA). The transgenic mice were created so the fifth exon of the RANKL gene was exchanged for the human exon, resulting in expression of a chimeric form of RANKL. The humanized RANKL gene remained under the control of normal endogenous regulatory elements of the murine RANKL and was capable of maintaining normal bone resorption. These mice have no obvious skeletal phenotype, except for slightly higher bone mass, lower bone resorption parameters, and the unique ability to respond to denosumab [26]. Following production of the femoral fracture, animals received a subcutaneous injection twice weekly of denosumab (Dmab, 10 mg/kg, n=27), alendronate (Aln, 0.1 mg/kg, n=29), or saline (Control, 0.1 mL, n=29). Saline was also used as the carrier for both denosumab and alendronate. Fractured femora were harvested on postoperative days 21 and 42. The third experiment compared male C57BL/6 mice eight to ten weeks of age (n=17) to age-matched mice (n=20) that were carrying an inactivating mutation within the Fas receptor gene, leading to lymphoproliferation (lpr) disorder. These B6.MRL/FAS^{lpr} mice are a well established model of systemic lupus erythematosus [27]. Fractured femora were harvested on post-operative days 21 and 35.

Femora were scanned using an isotropic voxel size of 12 μm (70 kVp, 114 mA; μCT 40, Scanco Medical, Brüttisellen, Switzerland). On each 2D tomogram located between the proximal and distal boundaries of the callus, semi-automated segmentation was used to define the outer boundary of the callus as well as the area enclosed by the periosteal surface of the pre-existing cortical bone (Figure 1A). The volume enclosed by these two surfaces was the volume of interest (Figure 1B). The segmentation routine was included in the system software and requires the user to supply an initial guess for the segmentation boundary. This guess is then refined by automated detection of an edge gradient with additional constraints regarding the distance from the initial guess and the smoothness of the boundary.

A fixed, global threshold of 25% of the maximum grayvalue (0.25•2^{16}/2=8192), which corresponded to a mineral density of 641.9 mg HA/ccm, was used to distinguish mineralized tissue from unmineralized and poorly mineralized tissue. This threshold value corresponded approximately to 45% of the attenuation of mature cortical bone in the cohorts of specimens examined in this study. This value was chosen based on visual inspection of the tomograms and qualitative comparison of these tomograms with paired, decalcified histological sections from additional specimens from Experiment 1.

The following measures of callus structure and composition were evaluated from the μCT image data for each specimen: total callus volume (TV); mineralized callus volume (BV); callus mineralized volume fraction (BV/TV); tissue mineral density (TMD); standard deviation of mineral density (σ_{TMD}); and bone mineral content (BMC, defined as the callus BV multiplied by TMD). Prior to computing the values of each of these outcome measures, a Gaussian filter (sigma = 0.8, support = 1.0) was applied for noise reduction. Calculation of TMD, σ_{TMD}, and BMC was made possible by density calibration data obtained from scans of a hydroxyapatite (HA) phantom provided by the system manufacturer. This phantom consisted of four 6-mm-diameter, HA cylinders of known density (100, 200, 400, and 800 mgHA/cc). Although the scan data for the phantom were not filtered, the attenuation value used for each cylinder was the average over all voxels within the cylinder and thus was very stable. Both TV and BV were normalized by the length of the callus (mean ± std length = 6.29 ± 0.98 mm). Both TMD and σ_{TMD} were calculated using only the voxels whose intensity exceeded the threshold, and two voxels were removed from all bone surfaces prior to computing these two quantities in order to reduce partial volume effects. We verified through an inter-observer repeatability analysis performed on sixteen of the specimens used in this study that the image segmentation method described above introduced minimal observer bias. The mean difference and mean absolute deviation for all outcome measures were less than 2.5% and 8.0%, respectively, and a paired *t*-test indicated no systematic difference among observers (p>0.05).

In addition to the outcome measures listed above, the effective polar moment of inertia of each callus was calculated (custom script, Matlab R2007a, The Mathworks, Natick, MA). For a homogeneous, prismatic bar, the torsional rigidity is equal to the product of the shear modulus and the torsional constant. For circular bars, the torsional constant is equal to the polar moment of inertia. If the bar is not prismatic but rather has a cross-sectional shape that varies along the length, an effective torsional constant can be derived under the assumption that the shear modulus is constant throughout the bar [28]. In the present study, the porosity of the callus prevented closed-form calculation of the torsional constant. Hence, we relied on the approximate circular shape of the callus cross-sections and computed the effective polar moment of inertia *J _{eff}* as an estimate of the effective torsional constant of the callus.

$${J}_{i}=\sum _{k=1}^{N}{r}_{k}^{2}A$$

(1)

where the summation was performed over all *N* voxels whose attenuation exceeded the previously specified threshold. *r _{k}* is the distance between voxel

$${J}_{\mathit{eff}}=L{\left(\underset{0}{\overset{L}{\int}}\frac{1}{J(z)}dz\right)}^{-1}$$

(2)

where *L* is the length of the callus and *J(z)* is the polar moment of inertia *J _{i}* of the callus cross-section located at the longitudinal position

Following μCT scanning, the proximal and distal ends of the femora were potted using polymethyl methacrylate (PMMA) in casings consisting of 10mm lengths of square aluminum tubing (10×10 mm^{2}). Care was taken to keep the specimens hydrated with saline during preparation, to center the bones within each casing, and to align the casings with the diaphyseal axis. The gage length was defined as the length of the specimen that spanned the distance between the two casings and was calculated as the average of four measurements taken at 90° increments around the specimen circumference. Specimens were mounted in the testing system (MT55, Instron, Norwood, MA) by securing the casings in the system grips with thumb screws. Angular displacement (twist) was applied at the rate of 0.5°/second until failure. This rotation was applied such that the distal end of the femur rotated inward relative to the proximal end. Torque was measured with a 2.25-Nm transducer and angular displacement with an optical encoder. Torsional strength was defined as the maximum torque sustained by the specimen. Torsional rigidity was calculated as the slope of the linear portion of the torque-twist curve, where twist was normalized by the gage length.

Three types of statistical analyses were used (JMP 6.0, SAS Institute, Cary, NC): 1) multivariate analyses of variance (MANOVA) to test for differences in callus structure and composition among groups within a given experiment; 2) principal components analysis (PCA) to describe variability in callus structure and composition among all specimens, independent of group or experiment; and 3) regression analyses to test for the dependence of callus mechanical properties on the μCT outcome measures. Prior to these analyses, log transformations were performed where necessary so that each of the mechanical and μCT outcome measures was approximately normally distributed.

Both MANOVA and PCA are multivariate statistical analyses that can account for mutually correlated independent variables. MANOVA tests for differences among groups in the multivariate space defined by the original set of outcome measures (TV, BV, BV/TV, TMD, σ_{TMD}, and BMC). In this space, each of the *k* experimental groups is described in terms of a vector of means, rather than a single mean value. MANOVA is more appropriate and statistically powerful than multiple, univariate ANOVAs when moderate correlations exist among the outcome measures [29]. MANOVA defines a set of *k*-1 “canonical variates”, or composite variables, that are mutually orthogonal linear combinations of the original outcome measures. These linear combinations are constructed such that the separation among groups is maximized, and the coefficients of these linear combinations indicate the relative importance of each original outcome measure in a given canonical variate. The canonical variates are ordered in terms of the amount of variation among groups that each explains, with the first variate accounting for the greatest variation. A two-factor MANOVA was performed for each experiment separately with time and treatment (Experiments 1 and 2) or time and genotype (Experiment 3) as the factors. When a factor was found to be significant, a Tukey *post hoc* test was performed on the first canonical variate corresponding to that factor in order to determine pairwise differences. Examination the coefficients in these factor-specific canonical variates also indicated the relative importance of the original outcome measures in discriminating among the levels of a given factor.

PCA identifies the linear combinations of original outcome measures, called “principal components”, that best describe the variation among specimens. A single principal component analysis was performed on the data pooled from all three experiments. For the regression analyses, stepwise regression was used to determine the dependence of maximum torque and torsional rigidity on the μCT outcome measures. Given the large degree of collinearity between BV and BMC (r=0.95), these analyses were performed twice: once with BV excluded and once with BMC excluded. For torsional rigidity, these two reduced sets of outcome measures also included *J _{eff}* or J

$${\mathit{error}}_{CV}=\sqrt{\frac{1}{N}{\sum}_{i=1}^{N}{({Y}_{i}-{\widehat{Y}}_{i})}^{2}}$$

(3)

where *Y _{i}* and

As illustrated qualitatively by three-dimensional reconstructions of representative specimens (Figure 2) and quantitatively by the MANOVA for each experiment (Figure 3, Table 2), callus structure and composition differed between treatments (Experiments 1 and 2, p<0.001) or genotypes (Experiment 3, p=0.01) and with time (p<0.001). A significant effect of the interaction between time and treatment or time and genotype was also found for Experiments 2 and 3 (p≤0.002). Tissue mineral density (TMD) was the measure that discriminated most strongly among timepoints in all three experiments. Clear separation among timepoints was observed in the first canonical variate, and this variate was almost exclusively associated with TMD (Figure 3, Table 2). In contrast, the μCT outcome measures that discriminated among treatments or genotypes differed depending on the experiment (Table 2). In Experiment 1, differences between PTH-treated and control groups were associated with primarily with differences in BV/TV, and, to a lesser extent, σ_{TMD}, BV, and BMC. In Experiment 2, differences among treatments increased with time and were most closely associated with TMD, TV, and BV. Lastly, in Experiment 3, discrimination between genotypes also increased with time, and these genotypic differences were most closely associated with BV/TV and TMD.

Longitudinal, cut-away views of 3-D renderings of representative calluses from each of the three experiments: these specimens correspond to the median BV/TV for their respective experimental groups. The calluses were defined through image segmentation **...**

MANOVA results depicting the first and second canonical variates for each of the three experiments. The circles denote 95% confidence intervals, and the centers of the circles represent the group means. The axis that defines each original outcome measure **...**

In MANOVA, the coefficients in the linear combinations of μCT outcome measures that define a canonical variate (CV) indicate the relative importance of each of the outcome measures in discriminating among experimental groups. The highest coefficients **...**

When specimens from all three experiments were pooled, the variation in callus structure and composition among specimens was captured largely by BV, BMC, and BV/TV. These measures contributed heavily to the first principal component, and this component explained 51.2% of the total variation (Table 3). TV and TMD contributed most strongly to the second principal component, while the third principal component was almost completely weighted towards σ_{TMD}. In total, 92% of the variability among specimens in callus structure was captured by the first three principal components.

Results of the principal component analysis (PCA). The first principal component (PC1) explained 51.2% of the variation in callus structure among specimens. Together, the first three principal components explained 92% of the variation. Also shown are **...**

The stepwise regression analyses resulted in regression models that explained 62% (p<0.001) and 70% (p<0.001) of the variation in maximum torque and torsional rigidity, respectively (Table 4). For maximum torque, this regression model consisted of TMD, BV, and σ_{TMD} as the independent variables (Figure 4). For torsional rigidity, TMD, BMC, BV/TV, and σ_{TMD} were the independent variables. Notably, the regression model consisting of TMD, BMC, and σ_{TMD} as the independent variables also provided moderately good predictions of maximum torque (R^{2}=0.61, p<0.001). Stepwise regression performed on random subsets of the data produced the above models nine out of ten times and eight out of ten times for maximum torque and torsional rigidity, respectively. In the one or two cases in which a different model resulted, the resulting set of predictors were a subset of the predictors identified from the stepwise regressions performed on the full data set. Cross validation estimates of the prediction error ranged 10.9–13.2% of the mean maximum torque or torsional rigidity and were only 0.2–16.8% higher than the root mean square errors corresponding to the above regression models performed on the full data set.

Results of the stepwise regression analysis for maximum torque: Measured, log-transformed values of maximum torque plotted against the values of maximum torque predicted from TMD, BV, and σ_{TMD} for all three experiments. The inset table lists the **...**

Results of the stepwise regression analyses using the μCT outcome measures as independent variables and results of the standard regression analyses using each of the principal components as independent variables. For the stepwise regression analyses, **...**

Regressions of maximum torque and torsional rigidity against the principal components indicated that the first and second principal components were significant predictors of these callus mechanical properties. In addition, *J _{eff,w}* and, to a lesser extent,

This study characterized the changes in fracture callus structure and composition that occur over time and with pharmacologic or genotypic modulation in order to relate these changes to the gradual restoration of bone stiffness and strength. The MANOVA results indicate that the temporal progression of healing in all three experiments was strongly linked to an increase in tissue mineral density (TMD) in the callus. Although temporal changes in the other measures of callus structure and composition were more subtle in comparison, these other measures did discriminate among treatments and genotypes. For example, BV/TV was strongly associated with differences between treatments or genotypes in Experiments 1 and 3, respectively.

The results of the principal components analysis indicate that the majority of the variability in callus structure and composition among all specimens, irrespective of experimental group, was captured by the outcome measures that quantify the absolute and relative amounts of mineralized tissue in the callus, *i.e.* BV, BMC, and BV/TV. Comparatively less variability was found among specimens in total callus size (TV), tissue mineral density (TMD), and the standard deviation in mineral density (σ_{TMD}). Thus, despite the importance of TMD in describing the time-dependent changes in the calluses, the greatest amount of diversity among the calluses was in the quantity of mineralized tissue present.

Importantly, however, the combination of μCT outcome measures that resulted in the best predictions of torsional strength and rigidity included descriptors of both the quantity *and* mineral density of the mineralized tissue. Given that the maximum torque and torsional rigidity are extrinsic measures of strength and stiffness, this finding reflects the importance of both geometry and material properties for the mechanical behavior of the callus. Although significant relationships between callus mechanical properties and mineral density and between callus mechanical properties and the quantity of mineralized tissue have been shown previously [4, 6, 7], the current results are notable in that the use of multiple regression analyses establishes the independent and respective contributions of each of these two factors. When viewed in conjunction with the MANOVA and PCA results, these regression results indicate that while the regain of bone strength and stiffness over time is due largely to a time-dependent increase in mineral density, this relationship between mechanical properties and mineral density can be modulated by factors that alter geometry.

Several aspects of the experimental design strengthen the findings of this study. First, multivariate statistical analyses were used in order to account for mutual correlations among the μCT outcome measures. For the specimens included in this study, Pearson correlation coefficients (*r*) for pairs of the outcome measures ranged −0.64 to 0.95. Whereas the results of univariate comparisons of callus structure and composition among experimental groups would be confounded by these correlations, the multivariate analyses control for them such that the results identify the fundamental and most salient differences in callus structure and composition among experimental groups. Second, this study used a large sample size (188 bones) that provided not only a broad range of callus structure and composition but also greater power in the statistical analyses. The results of PCA can be sensitive to sample size when the ratio of the number of samples to number of variables is less than ten [29]. Third, cross-validation was performed in conjunction with the stepwise regression analyses in order to test the validity of the resulting set of predictors for maximum torque and torsional rigidity. The robustness of the resulting set of predictors and the low cross- validation prediction errors indicated that meaningful regression models were obtained for the full data set. Given that stepwise regression is confounded by collinearity among the predictor variables, these analyses were performed excluding either BV or BMC, as these two variables were highly correlated (*r*=0.95). With either of these two variables excluded from the set of candidate predictors, the maximum correlation coefficient was 0.75. Fourth, because the assessment of callus structure and composition was performed using non-invasive imaging techniques, the results have direct bearing on development of diagnostic indices of fracture healing. The parameters found to be most relevant to callus strength—BV (or BMC), TMD and σ_{TMD} —can all be quantified *in vivo* with μCT in small animals and with QCT in humans. Thus, these parameters could potentially be used in pre-clinical and clinical studies as surrogate measures of callus mechanical properties. This in turn would facilitate non-invasive and also longitudinal assessments of the extent and rate of healing. Although it remains to be seen whether the current results also hold for the lower resolution imaging afforded by QCT as compared to μCT, these results certainly indicate that such an investigation is warranted.

A final strength of this study is that specimens from three murine fracture healing experiments were included in order to investigate the effects of several, clinically relevant, biological perturbations on callus structure and mechanical properties. Intermittent PTH treatment has previously been shown to increase callus size [30, 31], largely by enhancing the endochondral phase of repair [32]. The MANOVA results for the PTH experiment (Experiment 1) revealed that the effects of PTH on callus structure and composition at the peak of the endochondral phase (Day 14) and beyond were most strongly associated with an increase in BV/TV. In Experiment 2, the separation among treatment groups was greatest during the remodeling phase of repair (Day 42, Figure 3) and the treatment effect was due primarily to differences in TMD and callus size (TV and BV). The differences in callus size, particularly at late timepoints, are consistent with known effects of anti-resorptive treatments on osteoclast differentiation and function [33–36]. Experiment 3, which investigated the consequences of a catabolic pathology on healing, also showed the greatest differences during callus remodeling (Day 35, Figure 3), and these differences were due primarily to lower BV/TV in the B6.MRL/FAS^{lpr} calluses as compared to the C57BL/6 calluses. This difference in callus mineralized volume fraction most likely reflects the increased osteoclast count found in the B6.MRL/FAS^{lpr} calluses as compared to controls [37].

Some limitations of this study also warrant discussion. For example, because different genotypes and timepoints were used in the three experiments, it was not possible to make direct, quantitative comparisons among the different biological perturbations. Analyses of callus structure and composition in studies such as those investigating the combined effects of osteogenic and anti-resorptive agents on fracture healing [38] are needed in order to make such comparisons. An additional limitation of this study is that the effective torsional constant was not computed and was instead estimated by the effective polar moment of inertia. Further, in relating the effective polar moment of inertia to the measured torsional rigidity, it was implicitly assumed that the tissue shear modulus was either constant throughout the callus (in the case of *J _{eff}*) or was proportional to X-ray attenuation (in the case of

The distribution of grayvalues throughout the fracture callus presents difficulties in devising an automated method for identifying a suitable threshold for distinguishing mineralized tissue from unmineralized tissue. **(A)** Histogram for a specimen from **...**

Interestingly, both maximum torque and torsional rigidity were found to depend on the standard deviation in mineral density (σ_{TMD}), with larger values of σ_{TMD} associated with higher strength and rigidity. Although the mechanistic basis of this relationship is not clear at this time, there are several possible explanations. First, this result may be a consequence of differences in tissue composition among genotypes. Larger values of σ_{TMD}, strength, and rigidity were observed in the specimens from Experiment 2, which was conducted on C57BL/6 human RANKL knock-in transgenic mice, as compared to the other two experiments that were conducted on C57BL/6 wild type mice. Second, larger values of σ_{TMD} could occur when both mineralized cartilage and mineralized bone tissue are present in the callus, as the former is expected to have somewhat lower mineral density than the latter. Both of these types of mineralized tissue are present in significant amounts towards the end of the endochondral phase, which typically coincides with an increase in callus stiffness and strength. Further investigation of the specific X-ray attenuation values for mineralized cartilage vs. mineralized bone tissue and of the spatial distribution of mineral density throughout the callus are needed to understand more fully the contribution of σ_{TMD} to callus mechanical properties.

Given that several of the results of this study indicate the importance of TMD and σ_{TMD} for callus mechanical properties and for describing the variation among calluses, it is necessary to consider factors such as beam hardening and partial volume effects that can confound μCT measurements of tissue mineralization. Although two-voxel “peeling” was used when calculating TMD and σ_{TMD}, it was not possible to ensure that all partial volume effects were excluded. With respect to beam hardening, a correction algorithm based on a 200-mgHA/cc wedge phantom was used in reconstructing the μCT image data in this study. Recent reports have indicated that the use of a 1200-mgHA/cc correction algorithm can reduce the beam hardening artifact, particularly for specimens of high bone volume fraction (BV/TV) [41, 42]. For the experiment and timepoint that contained specimens with the highest BV/TV and greatest variations in callus size (Experiment 3, day 42), we repeated the μCT analyses on image data reconstructed using the higher density correction algorithm and also repeated the multivariate statistical analyses. The values of TMD and σ_{TMD} changed by 9.7% and 16.1%, respectively (paired -tests, p<0.001), *t* indicating that additional work is needed to fully validate measurements of tissue mineralization obtained by μCT. However, the conclusions drawn from the MANOVA, PCA, and regression analyses were unchanged, leading us to conclude that the outcomes of this study were not specific to the beam hardening correction used.

In summary, the results of this study have identified subsets of μCT-derived metrics that describe the time-dependent changes in callus structure and composition and that are significant predictors of callus mechanical properties. These results were obtained through analyses of a diverse set of specimens representing investigations of specific anabolic and catabolic perturbations on healing, and as such, the findings generate a broad picture of how alterations to different biological processes can modulate callus structure, composition, and mechanical function. Taken together, the results demonstrate the use of μCT-based assessments to relate the biology of fracture healing to the gradual regain of bone stiffness and strength and to identify potential surrogate measures of healing.

Funding for this study was provided in part by NIH AR049920 (TAE), NIH AR047045 (LCG), the Amgen Corporation, and the Boston University Undergraduate Research Opportunities Program. The authors would like to thank Masia Al Sebaei, Stephanie Stapleton, Daniel Bellin, Daniel Sacks, Lincoln Miara, Megan Pelis, and Daniel Tobin for their technical assistance.

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