Drs Guo, Sajda, and Liu are inventors of a pending patent derived from this work. All other authors state that they have no conflicts of interest

Research Article

# Complete Volumetric Decomposition of Individual Trabecular Plates and Rods and Its Morphological Correlations With Anisotropic Elastic Moduli in Human Trabecular Bone^{†}^{‡}

Article first published online: 1 OCT 2007

DOI: 10.1359/jbmr.071009

Copyright © 2008 ASBMR

Additional Information

#### How to Cite

Liu, X. S., Sajda, P., Saha, P. K., Wehrli, F. W., Bevill, G., Keaveny, T. M. and Guo, X. E. (2008), Complete Volumetric Decomposition of Individual Trabecular Plates and Rods and Its Morphological Correlations With Anisotropic Elastic Moduli in Human Trabecular Bone. J Bone Miner Res, 23: 223–235. doi: 10.1359/jbmr.071009

^{†}^{‡}Published online on October 1, 2007

#### Publication History

- Issue published online: 4 DEC 2009
- Article first published online: 1 OCT 2007
- Manuscript Accepted: 26 SEP 2007
- Manuscript Revised: 1 SEP 2007
- Manuscript Received: 5 MAY 2007

- Abstract
- Article
- References
- Cited By

### Keywords:

- bone microstructure;
- digital topology;
- elastic moduli;
- μCT imaging;
- trabecular orientation;
- trabecular plate/rod

### Abstract

**Trabecular plates and rods are important microarchitectural features in determining mechanical properties of trabecular bone. A complete volumetric decomposition of individual trabecular plates and rods was used to assess the orientation and morphology of 71 human trabecular bone samples. The ITS-based morphological analyses better characterize microarchitecture and help predict anisotropic mechanical properties of trabecular bone.**

**Introduction:** Standard morphological analyses of trabecular architecture lack explicit segmentations of individual trabecular plates and rods. In this study, a complete volumetric decomposition technique was developed to segment trabecular bone microstructure into individual plates and rods. Contributions of trabecular type-associated morphological parameters to the anisotropic elastic moduli of trabecular bone were studied.

**Materials and Methods:** Seventy-one human trabecular bone samples from the femoral neck (FN), tibia, and vertebral body (VB) were imaged using μCT or serial milling. Complete volumetric decomposition was applied to segment trabecular bone microstructure into individual plates and rods. The orientation of each individual trabecula was determined, and the axial bone volume fractions (aBV/TV), axially aligned bone volume fraction along each orthotropic axis, were correlated with the elastic moduli. The microstructural type-associated morphological parameters were derived and compared with standard morphological parameters. Their contributions to the anisotropic elastic moduli, calculated by finite element analysis (FEA), were evaluated and compared.

**Results:** The distribution of trabecular orientation suggested that longitudinal plates and transverse rods dominate at all three anatomic sites. aBV/TV along each axis, in general, showed a better correlation with the axial elastic modulus (*r*^{2} = 0.95∼0.99) compared with BV/TV (*r*^{2} = 0.93∼0.94). The plate-associated morphological parameters generally showed higher correlations with the corresponding standard morphological parameters than the rod-associated parameters. Multiple linear regression models of six elastic moduli with individual trabeculae segmentation (ITS)-based morphological parameters (adjusted *r*^{2} = 0.95∼0.98) performed equally well as those with standard morphological parameters (adjusted *r*^{2} = 0.94∼0.97) but revealed specific contributions from individual trabecular plates or rods.

**Conclusions:** The ITS-based morphological analyses provide a better characterization of the morphology and trabecular orientation of trabecular bone. The axial loading of trabecular bone is mainly sustained by the axially aligned trabecular bone volume. Results suggest that trabecular plates dominate the overall elastic properties of trabecular bone.

### INTRODUCTION

Osteoporosis is a major public health threat, causing bone fragility and an increased susceptibility to fractures.^{(1)} The major characteristics of osteoporosis are low bone mass and microarchitectural deterioration of trabecular bone, with a dramatic change of trabeculae from plate-like to rod-like.^{(2,3)} With advances in high-resolution μCT and μMRI imaging for trabecular bone, 3D model-independent morphological techniques have been developed during the past 10 yr where the model-independent trabecular thickness (Tb.Th*), trabecular spacing (Tb.Sp*), trabecular number (Tb.N*), and structure model index (SMI) were determined.^{(4–7)} SMI was introduced to estimate, on average, the plate versus rod characteristics of trabecular bone: 0 for an ideal plate and 3 for an ideal rod.^{(2,5)} With this quantitative index, it has been documented that there is a dramatic change of trabeculae from plate-like to rod-like with aging and osteoporosis, confirming qualitative observations.^{(2,3)}

Most of the above studies evaluated the morphology of trabecular bone at a global level without separate analyses for trabecular plates and rods, even though the two types of structure are fundamentally different. Recently, two research groups have independently studied the relative importance of trabecular types (plates and rods) in the architecture and mechanical properties of trabecular bone and developed image processing techniques for volumetrically segmenting 3D trabecular bone microstructure as a collection of trabecular plates and rods.^{(8–10)}

In prior work at the authors' laboratory, the relationship between trabecular microarchitecture and elastic moduli of human trabecular bone was examined using digital topological analysis (DTA) techniques.^{(9)} Rod-like and plate-like trabeculae were separately reconstructed from the skeleton structure with the aid of a newly developed voxel reconstruction technique. The results quantitatively showed that trabecular plates make a far greater contribution than rods to the bone's elastic behavior.^{(9)} Using a similar approach, rod and plate decomposition-based morphological analyses have been reported for human trabecular bone samples.^{(10–12)} In combining new trabecular plate- and rod-based parameters with the standard model independent morphological parameters, they showed that the mean trabecular spacing (Tb.Sp*), the mean rod slenderness (<Ro.SI>), and the relative bone volume fraction of rod (Ro.BV/BV) correlated strongly with the elastic modulus of vertebral trabecular bone.^{(12)}

However, in the above studies, one of the most challenging steps was not accomplished: a segmentation of plate-like components in trabecular bone microstructure into individual elements.^{(9–11)} For example, one or two major plates can span the entire specimen from top to bottom and from one side to the other in plate-dominant trabecular bone such as the femoral head.^{(9–11)} Without the segmentation of these large and curvy trabecular plate components, it is difficult to determine the orientation, thickness, and number of these structurally and mechanically important trabecular plates. Because trabecular bone is an anisotropic material in terms of its microarchitecture and mechanical properties, it is important to volumetrically decompose the plate components into individual elements, which are smaller and have well-defined trabecular orientation. In this manner, the contribution of trabecular plates in different orientations to anisotropic mechanical properties can be quantified.

In this study, a new procedure was developed, based on the DTA-based topology-preserving skeletonization and classification techniques, to fully decompose trabecular bone microstructure into individual rods and individual plates, where the latter was accomplished for the first time. With complete segmentations of both individual rods and individual plates, the distributions of trabecular plate and rod orientations were quantified in high resolution images of human trabecular bone samples from various anatomic sites: proximal femur, proximal tibia, and spine. The orientation dependent contributions of individual trabeculae to the anisotropic elastic moduli, determined by image-based finite element analysis (FEA), were quantified using a newly defined parameter, called “axial bone volume fraction” (aBV/TV). Subsequently, the individual trabeculae segmentation (ITS)-based morphological analyses were applied to these human trabecular bone images to determine several novel morphological parameters such as trabecular plate number and trabecular plate thickness. The anisotropic elastic moduli were correlated to the ITS-based morphological parameters and the standard morphological parameters.^{(5,6)}

### MATERIALS AND METHODS

#### Trabecular bone sample preparation and imaging

Seventy-one cylindrical (8 mm diameter, 20 mm length), on-axis (along the principal trabecular orientation) human trabecular bone samples were obtained from the femoral neck (FN, *n* = 27), proximal tibia (*n* = 23), and lumbar vertebral body (VB, *n* = 21) (Table 1) following a previously published protocol.^{(13–16)} The subjects were screened to exclude metabolic bone disease or bone cancer, and X-ray radiographs were taken to ensure that there was no evidence of damage or other bone pathologies. 3D high-resolution images of central gauge length of 20 mm were obtained for each specimen using μCT systems (μCT 20 or VivaCT 40; SCANCO Medical AG, Bassersdorf, Switzerland) or serial milling.^{(17)} The central ∼4 × 4 × 4-mm^{3} cubic subvolume of each specimen was extracted from the reconstructed image and downsampled to either 21- or 20-μm nominal isotropic resolution by means of a manufacturer supplied program (“scale”; SCANCO Medical AG). A global thresholding technique was applied to binarize grayscale μCT images where the minimum between the bone and bone marrow peaks in the voxel gray value histogram was chosen as the threshold value. The grayscale images resulting from serial milling were binarized using a thresholding technique described elsewhere.^{(17)} Isolated voxels or disconnected voxel-clusters were removed from the largest bone component,^{(18,19)} and the resulting images were used for the following analyses (Fig. 1A).

#### Complete volumetric decomposition of individual trabecular plates and rods

First, digital topological analysis (DTA) including skeletonization^{(20)} and topological classification^{(21)} was performed to transform the trabecular bone image into a representation of a 1-voxel-thick surface and curve skeleton with unique identification of the topological type of each voxel (i.e., surface, curve, surface-curve [plate-rod or P-R], or curve-curve [rod-rod or R-R] junction; Fig. 1B).^{(19,22)} The plate-like and rod-like trabeculae in the original trabecular bone image are referred as “plate” and “rod,” respectively, which corresponds to “surface” and “curve” after skeletonization. By applying DTA-based arc skeletonization,^{(20)} the surface-skeleton was further segmented into a network of curves; we distinguish these curves from the original curves created by first skeletonization and refer to the former as arcs. By applying the DTA-based classification to the arc-skeleton, the arc-arc (also called plate-plate [P-P]) junctions were identified (Fig. 1C). After removing the P-P junctions and the previously identified P-R and R-R junctions, branching points were identified and used to split the skeleton into individual curves and arcs (Fig. 1D). Each branch was labeled with a unique branch identity number and its topological marker as either a curve corresponding to a trabecular rod or an arc indicating a trabecular plate. Small branches <80 μm (the lower bound of the averaged trabecular thickness) were removed.

The final step of the volumetric decomposition process was the volumetric reconstruction of individual trabecular plates and rods. This task was accomplished by an iterative topological marking of nonskeletal voxels starting from the skeletal voxels. The similar algorithm for volumetric reconstruction of plate and rod has been given previously.^{(9)} On completion of the reconstruction process, each bone voxel was labeled as belonging to an individual trabecular plate or rod (Figs. 1E, 1F, and 2). For illustration purpose, volume size 3.2 × 3.2 × 2.1 and 2.1 × 2.1 × 1.5 mm^{3} was chosen for Figs. 1 and 2 respectively. In the study, all analyses were applied to 4 × 4 × 4-mm^{3} trabecular bone images obtained by the method described earlier. The algorithms of the complete volumetric decomposition of individual trabecular plates and rods were programmed in Microsoft Visual C++ (Microsoft, Redmond, WA, USA) and implemented on a Dell XPS PC workstation (Dell, Round Rock, TX, USA).

#### ITS-based morphological analyses

A set of morphological parameters was evaluated directly from the measurements of individual trabeculae to indicate the morphological properties of trabecular plate and rod, respectively. The normal to a trabecular plate was determined as the surface normal to a plane representing surface skeleton using least square fitting.^{(23)} At each voxel of the surface skeleton, a local thickness was calculated along the normal. The thickness of a trabecular plate was evaluated by averaging local thickness of all the voxels on the corresponding surface skeleton. The orientation of a trabecular rod was obtained by using 3D principal component analysis on its corresponding curve skeleton. Local diameters were evaluated for all the voxels of the curve skeleton, and their average was defined as the diameter of the rod. The histograms of plate and rod thickness were plotted for three anatomic sites: FN, tibia, and VB.

By adopting the American Society of Bone and Mineral Research bone histomorphometric nomenclature,^{(24)} the following morphological parameters were reported: bone volume fraction (BV/TV); plate bone volume fraction (pBV/TV), the total volume of plate bone voxels divided by the bulk volume; rod bone volume fraction (rBV/TV), the total volume of rod bone voxels divided by the bulk volume; plate tissue fraction (pBV/BV), the total volume of plate bone voxels divided by the total volume of bone voxels; rod tissue fraction (rBV/BV), the total volume of rod bone voxels divided by the total volume of bone voxels; mean trabecular plate thickness (pTb.Th, mm), the average thickness of trabecular plates; mean trabecular rod thickness (rTb.Th, μm), the average diameter of trabecular rods; trabecular plate and rod numerical density (pTb.N and rTb.N, 1/mm), the cubic root of the total number of trabecular plates or rods divided by the bulk volume; mean trabecular rod length (rTbℓ, μm), the average length of trabecular rods; and mean trabecular plate surface area (pTb.S, μm^{2}), the average surface area of trabecular plates. In addition, the P-R, R-R, and P-P junction density (Junc.D; 1/mm^{3}) were calculated based on the arc-skeleton of the original trabecular bone image as the total number of those junctions normalized by the bulk volume.

#### Standard morphological analyses

The standard morphological parameters such as BV/TV, bone surface to bone volume ratio (BS/BV), trabecular number (Tb.N*), trabecular thickness (Tb.Th*), trabecular spacing (Tb.Sp*), structure model index (SMI), and connectivity density (Conn.D) were evaluated for each specimen using the standard morphological analysis software on a VivaCT 40 system (SCANCO Medical AG).

#### Anisotropic elastic moduli by FEA

A finite element (FE) model was generated from each image by converting each voxel to an eight-node brick element. The trabecular bone tissue was modeled as an isotropic, linear elastic material with a Young's modulus (E_{s}) of 15 GPa and a Poisson's ratio of 0.3 for all the models.^{(25)} Using an element-by-element preconditioned conjugate gradient solver,^{(26)} six FEAs were performed for each model, representing three uniaxial compression tests along three imaging axes (*x*, *y*, and *z*) and three uniaxial shear tests. The general anisotropic stiffness matrix was first determined based on the results from the above analyses, and a new coordinate system of orthotropic axes (*X*_{1}, *X*_{2}, and *X*_{3}) representing the best orthotropic symmetry was calculated by using an optimization procedure^{(27)} (Fig. 3). The transformation of the anisotropic matrix to a new coordinate yielded the full orthotropic stiffness tensor.^{(28)} The elastic constants and stiffness matrix were sorted such that E_{11} was in the direction of the lowest axial modulus and E_{33} was in the direction of the highest axial modulus. The elastic moduli (three Young's moduli, E_{11} < E_{22} < E_{33}, and three shear moduli, G_{23}, G_{31}, G_{12}) were derived from the orthotropic stiffness tensor.

#### Axial bone volume fraction

We introduced an X_{i} axial bone volume fraction, (aBV/TV)_{i}, to characterize the bone volume aligning with the *X*_{i}-axis (i = 1, 2, and 3). For each trabecula (plate or rod), angles Φ_{i} were calculated to indicate its orientation with the reference of the *X*_{i}-axis (i = 1, 2, and 3). For a trabecular plate, an angle Φ_{i} is defined as the angle between the normal of the plate and the axes plane perpendicular to *X*_{i}-axis (Fig. 3, bottom left). For a trabecular rod, the angle Φ_{i} is defined as the angle between the orientation of the rod and *X*_{i}-axis (Fig. 3, bottom right). The X_{i} axial bone volume fraction (aBV/TV)_{i} was evaluated as the volume of axially aligned trabecular plates and rods (0 ≤ Φ_{i} ≤ 30°) in the *X*_{i}-axis (i = 1, 2, and 3) divided by the bulk volume.

Axis *X*_{3} was along the longitudinal direction of tibia, VB, and FN; according to its angle Φ_{3}, orientation of each trabecula was determined (longitudinal, 0° ≤ Φ_{3} ≤ 30°; oblique, 30° < Φ_{3} ≤ 60°; transverse, 60° < Φ_{3} ≤ 90°). The histograms of Φ_{3} for both trabecular plates and rods were evaluated.

#### Statistical analyses

ANOVA and Tukey honestly significant difference (HSD) post hoc tests were used to study any significant difference (*p* < 0.05) of both the ITS-based and standard morphological parameters between FN, tibia, and VB groups. Linear correlations between the standard and ITS-based morphological parameters were also performed.

The forward stepwise multiple linear regression was performed to predict the elastic mechanical properties (E_{11}, E_{22}, E_{33}, G_{23}, G_{31}, and G_{12}) by the ITS-based and standard morphological parameters, respectively. From the ITS-based morphological parameters, pBV/TV, rBV/TV, pTb.N, rTb.N, pTb.Th, rTb.Th, pTb.S, rTb.ℓ, R-R Junc.D, P-R Junc.D, R-R Junc.D, and anatomic sites were used as the independent variables, whereas from the standard morphological parameters, BV/TV, BS/BV, Tb.N*, Tb.Th,* Tb.Sp*, Conn.D, SMI, and anatomic sites were used. At each step of the forward stepwise regression method, the eligible independent variable with the highest statistic strength enters the model. At any subsequent step where two or more independent variables have been selected into the model, the variable with the least statistical strength is removed from the model. The stepping continues until no eligible independent variable exceeds the critical value (*p* < 0.05) for model entry or no independent variable in the model reach the standard (*p* > 0.1) for variable removal. Combinations of critical predictors from either the ITS-based or standard morphological parameters, respectively, will be selected to yield the best prediction of each mechanical constant.

Furthermore, nonlinear correlation analyses by power laws between E_{ii} and BV/TV, pBV/TV, and (aBV/TV)_{i} were performed for all three axes (i = 1, 2, and 3). The Fisher z transformation test^{(29)} with *p* < 0.05 as the standard of statistical significance was used to determine if the correlation coefficients were significantly different.

The stepwise multiple linear regression was performed by SPSS 10.0 software (SPSS, Chicago, IL, USA). All other statistical analyses were performed using KaleidaGraph 3.6 software (Synergy Software, Reading, PA, USA).

### RESULTS

The microarchitecture and mechanical properties of trabecular bone from the three anatomic sites were dramatically different (Tables 2 and 3). Figure 2 shows the results of complete volumetric decomposition of trabecular plates and rods on images of specimens from the FN, tibia, and VB. The FN specimen represented a typical plate-like structure with continuous trabecular plates spanning the whole specimen (Fig. 2A, left). Using the complete volumetric decomposition, all microstructures were successfully decomposed into individual trabecular plates and rods (Fig. 2A, right). In particular, the “major elements,” which could not be segmented by the previous decomposition technique,^{(10)} were segmented according to their topology and geometry into smaller elements with well-defined trabecular orientation.

According to the histograms of Φ_{3} of trabecular plate and rod of three anatomical sites, longitudinal plates were dominant (Fig. 4). The percentage of longitudinal plates was 79.8% in the FN, 86.1% in the tibia, and 76.8% in the VB. On the other hand, more transverse rods existed than oblique and longitudinal ones at all three anatomic sites. The percentage of transverse rods was 55.2% in the FN, 44.2% in the tibia, and 60.2% in the VB.

Histograms of trabecular plate and rod thickness of three anatomical sites, the FN, tibia, and VB, are shown in Fig. 5. The tibia and VB had similar distributions in both plate and rod thickness, and the mean values of plate and rod thickness were similar. In FN, the distributions of plate and rod thickness had a wider range than those of tibia and VB, and the mean value of plate thickness was larger than that of rod thickness.

Significant correlations by power law were found between axial Young's modulus (E_{ii}) and bone volume fraction (BV/TV), plate bone volume fraction (pBV/TV), and X_{i}-axial bone volume fraction (aBV/TV)_{i} for three axial directions (i = 1, 2, and 3; Fig. 6). The correlations of three axial Young's moduli with BV/TV and pBV/TV had the same strength (*p* > 0.05). Although (aBV/TV)_{i} only accounted for 30–70% of BV/TV, correlation coefficients of E_{ii} with (aBV/TV)_{i} (i = 1, 2, and 3) were generally higher than those with BV/TV. It was indicated by the Fisher z transformation test that the differences were statistically significant along the *X*_{1}- and *X*_{3}-axes (*p* < 0.05 and < 0.001, respectively). Shear moduli also correlated strongly with BV/TV, pBV/TV, or (aBV/TV)_{i}; however, no significant difference was found between the correlations (data not shown).

The FN had a higher pBV/TV, pBV/BV, pTb.Th, pTb.N, P-R Junc.D, and P-P Junc.D than the tibia and VB (Table 2). In contrast, the FN showed a lower rBV/BV than both tibia and VB groups. As a parameter indicating the average size of trabecular plates, pTb.S had a similar magnitude in three anatomic sites, suggesting a uniform decomposition of trabecular plates. In terms of rTb.N, rTb.Th, and rTb.ℓ, statistical analyses showed no difference between the FN and tibia groups, whereas VB had a lower rTb.N, thicker rTb.Th, and a longer rTb.ℓ. It should be noted that the ITS-based morphological analysis indicated that the rBV/TV and R-R Junc.D were the same among all anatomic sites. In comparison, the standard morphological analysis revealed that BV/TV was significantly higher in the FN than in the tibia and VB, whereas the BS/BV was significantly lower in the FN than in the tibia and VB. The SMI was significantly lower in the FN than in the tibia and VB. This anatomic difference in the SMI was consistent with the results of pBV/TV (FN > [tibia = VB]), pBV/BV (FN > [tibia = VB]), or rBV/BV (FN < ([tibia = VB]). The findings for standard Tb.Th* paralleled those for pTb.Th (FN > [tibia = VB]), and the results for Tb.N* were consistent with those of pTb.N (FN > [tibia = VB]). However, the standard Tb.Sp* detected significant differences among the anatomic sites (FN < tibia < VB), which were not found for either pTb.S ([FN = VB] > Tibia) or rTb.ℓ ([FN = tibia] < VB). The standard connectivity measure, Conn.D, detected the same anatomic differences as both the P-R Junc.D and P-P Junc.D (FN > [tibia = VB]).

The correlation coefficients of the linear regressions between ITS-based and standard morphological parameters are listed in Table 4. pBV/TV was found to be the best predictor of BV/TV (*r* = 0.99). Except pTb.S, rTb.Th, and R-R Junc.D (*p* > 0.05), all the ITS-based parameters were significantly correlated with BV/TV. In particular, the parameters derived from plates showed generally higher predictive power than those derived from rods: pBV/TV (*r* = 0.99) versus rBV/TV (*r* = −0.35), pTb.Th (*r* = 0.86) versus rTb.Th (*r* = 0.06), pTb.N (*r* = 0.89) versus rTb.N (*r* = 0.38), and P-P Junc.D (*r* = 0.81) versus R-R Junc.D (*r* = 0.11).

The standard morphological parameters showed excellent agreement with the corresponding plate- and rod-associated parameters. For example, significant correlations were found between Tb.Th* and pTb.Th (*r* = 0.98), Tb.N* and pTb.N (*r* = 0.73) and rTb.N (*r* = 0.53), and Conn.D and P-P Junc.D (*r* = 0.95) and P-R Junc.D (*r* = 0.96). The rTb.ℓ showed a significant linear correlation with Tb.Sp* (*r* = 0.64), but pTb.S was not correlated with Tb.Sp*. A significant linear correlation existed between pBV/BV and SMI (*r* = −0.81).

Adjusted correlation coefficients of the multiple linear regressions of six elastic moduli with the ITS-based morphological parameters had the same strength but slightly higher values than those with the standard morphological parameters (Table 5). Among the ITS-based morphological parameters, pBV/TV was found to be the strongest independent predictor of all six elastic moduli, and pTb.Th, pTb.N, and pTb.S were also important predictors to most of the moduli. Among the standard morphological parameters, BV/TV was the strongest independent predictor of all the elastic constants, and BS/BV and Tb.Sp contributed to most of the elastic constants.

### DISCUSSION

In this work, a novel, complete volumetric decomposition technique was developed to segment both individual rods and plates with well-defined trabecular orientation, where the challenging task, decomposition of trabecular plates, was accomplished for the first time. With the completely decomposed individual trabecular elements, the quantitative assessment of trabecular plate or rod thickness and orientation distributions was possible for 3D human trabecular bone images. Histograms of trabecular thickness and orientation have been obtained for trabecular plates and rods of three anatomic sites, and the intersite variance of plate and rod distributions have been quantified from their histograms (Figs. 4 and 5). The distribution of trabecular plate and rod orientation provides quantitative evidence for the majority of trabecular plates being orientated along the principal direction of loading, whereas most of trabecular rods serve as transverse connections between longitudinal plates. Among the three anatomic sites, trabecular plates in the tibia show the best alignment with the principal direction of loading, and most trabecular rods in VB align with the transverse plane of principal loading direction. These observations are consistent with the visual appearance of tibial and vertebral trabecular bone structures.^{(30,31)}

Although only 30∼70% of bone volume was included in the axial bone volume fraction (aBV/TV)_{i}, the correlations of Young's moduli with the axial bone volume fraction along the *X*_{1}- and *X*_{3}-axes were significantly stronger than those with BV/TV. These results imply that the aligned trabeculae play an important role on the axial elastic moduli; thus, we conjectured that the axial loading of trabecular bone is mainly sustained by the axially aligned bone volume.^{(32)} On the other hand, the correlations of shear moduli with either axial or transverse bone volume fraction were not improved. This observation is probably because of the fact that shear involves more complicated mechanisms than axial loading such that the load is more widely distributed among trabeculae with various orientations. In summary, ITS-based orientation analysis provides additional insight into the influence of trabecular orientation on both the axial and shear moduli, and probably also on yield strength.

Independently, Stauber and colleagues^{(10–12)} had developed a similar technique to decompose the trabecular bone microstructure into individual elements according to its plate or rod topology and implemented to assess local morphometry of trabecular bone. The major difference between their studies and ours is the complete volumetric decomposition of trabecular plates. In the paper by Stauber and Muller,^{(10)} the local morphometric technique had been applied to human trabecular bone images from the femoral head, iliac crest, calcaneus, and lumbar spine. A comparison of the results for femoral and vertebral trabecular bone between these two studies showed similar trabecular plate and rod thickness. In contrast, plate tissue fraction (pBV/BV in this study and Pl.BV/BV in reference 10) in vertebral trabecular bone differed (82.5% in this study versus 30∼34% in reference 10). Interestingly, the plate tissue fractions of femoral trabecular bone in both studies are quite close (89.8% in this paper and 83.0% in reference 10). The significant difference in vertebral trabecular bone cannot be reconciled on the basis of biological variations of different human samples. The skeletonization process in this paper and our previous paper^{(9)} follows the rigorous definition of surface (plate) and curve (rod) topologies stated in Saha et al.^{(20)} On the other hand, the modified skeletonization process by Stauber and Muller^{(10)} includes more thinning iterations to reduce the surfaces. Obviously, apparent and major differences exist in the estimates of the plate surface area (pTb.S in this paper and Pl.S in reference 10). The average size of trabecular plates in the other study^{(10)} is roughly 1 mm^{2}, whereas in this study, it was found to be <0.1 mm^{2}. It should be noted that major plate elements (the largest trabecular plate in a specimen) in the other study^{(10)} may have constituted up to 66% of total bone volume in femoral head and 10% in lumbar vertebrae.

Trabecular plate volume estimated by the current technique was found to account for 60∼95% of total bone volume and to dominate the overall properties of trabecular bone structure. Comparing morphological results by ITS-based to those by standard techniques, in general, trabecular plate derived parameters were found to parallel those predicted by the standard morphological analysis (Table 2). However, trabecular rod-associated parameters suggest a behavior different from that of standard morphological analysis. These observations are also supported by the direct correlations between ITS-based and the corresponding standard morphological parameters (Table 4). Plate-derived parameters, in general, strongly correlate with the corresponding standard ones, whereas rod-associated parameters are only weakly correlated or not at all. These observations suggest that the ITS-based morphological analysis not only captures the contribution of the dominant trabecular type (i.e., trabecular plates), it also detects the subtle differences in trabecular rods, which have not been identified by the standard morphological analysis.

Multiple linear regression of anisotropic elastic moduli with the ITS-based morphological parameters result in correlations as those with standard parameters but with higher correlation coefficients. BV/TV was found to be the strongest predictor among the standard morphological parameters of the elastic properties of trabecular bone, whereas pBV/TV was the strongest predictor among the ITS-based parameters. Besides pBV/TV, most of the plate-associated parameters contribute to one or more elastic properties in multiple linear regressions, such as pTb.Th, pTb.N, pTb.S, P-R Junc.D, and P-P Junc.D. In the multiple linear regression model with the standard parameters, BS/BV, Tb.Sp*, and Conn.D contribute to one or more elastic constants. However, Tb.N*, Tb.Th*, and SMI did not add independent predictive power to any of the multiple linear regression models, even though each of these parameters has significant single correlation with all six elastic moduli (*p* < 0.001, data not shown). Thus, we conclude that the plate- and rod-associated morphological parameters, even without direct BV/TV information, predict the elastic properties of human trabecular bone as good as the standard morphological parameters.

Stauber et al.^{(12)} also used their technique to examine the contributions of trabecular plates and rods to the axial elastic modulus of human vertebral trabecular bone. In their study, both standard (global) and plate-rod based (local) morphological parameters were included to construct a multilinear model of elastic modulus of vertebral trabecular bone. Their results suggested that the mean trabecular spacing (Tb.Sp*), the mean slenderness of the rods (<Ro.Sl>), and the relative amount of rod volume to total bone volume (<Ro.BV/BV>, equivalent to rBV/BV in this study) jointly were able to explain 90% of variance in the axial elastic modulus. However, the coefficient of <Ro.BV/BV> in their multiple linear regressions was negative.^{(12)} Given the fact that <Ro.BV/BV> = rBV/BV = 1 − pBV/BV, their results are not necessarily contradictory to the conclusions of this study.

Despite the strengths of this study, certain limitations should be taken into consideration. First, 71 specimens were taken from 54 subjects, and therefore, duplicate specimens could be from the same subject. To examine the sample effect, one specimen from each subject was randomly selected, and all the statistic tests were performed again using the data set of 54 independent specimens. We observed no change of major conclusion after the reduction of sample size compared with those of the original dataset. For this type of methodological study, this approach might be acceptable. Second, two different imaging techniques were used to obtain 3D trabecular bone images. However, both μCT and serial milling are widely used techniques for imaging trabecular bone, and the resolution adopted in this study was sufficiently high to have more than four voxels cross a trabecula.^{(33,34)} We believe the influence introduced by the inconsistency in imaging methods is minor.

The complete volumetric decomposition of trabecular plates and rods has been used to assess the morphology and anisotropic elastic moduli of high-resolution images of human trabecular bone. It is not surprising that bone volume fraction alone accounts for 93–94% of the variation in anisotropic elastic moduli, given the high resolution of the images (∼20 μm) and accuracy of the FE modeling technique, which cannot currently be achieved by in vivo clinical imaging. Although this new technique needs to be tested and validated in a more clinically relevant setting, the preliminary application of the ITS-based morphological analyses of in vivo μMRI images (∼70-μm resolution) indicated that individual trabecular plate- and rod-based morphological analysis is able to detect subtle changes of trabecular bone microarchitecture in response to pharmacological intervention.^{(35)} Recently, high-resolution peripheral quantitative CT (HR-pQCT; XtremeCT; Scanco Medical AG, Bassersdorf, Switzerland) has been shown to permit in vivo assessment of trabecular bone microarchitecture at the distal radius and tibia of patients with an isotropic voxel size of 82 μm.^{(36)} Studies are underway to apply the new ITS-based morphological technique to images of HR-pQCT to examine its usefulness in clinical assessments of osteoporotic fracture risk and efficacy of treatment interventions.

### Acknowledgements

The authors thank X Henry Zhang for contributions in FEA. This work was partially supported by National Institutes of Health Grants AR041443, AR043784, AR051376, and AR053156.

### References

- 1
- 22000 Quantification of age-related changes in the structure model type and trabecular thickness of human tibial cancellous bone. Bone 26: 291–295.,
- 32001 The temporal changes of trabecular architecture in ovariectomized rats assessed by microCT. Osteoporos Int 12: 936–941., , ,
- 41999 Direct three-dimensional morphometric analysis of human cancellous bone: Microstructural data from spine, femur, iliac crest, and calcaneus. J Bone Miner Res 14: 1167–1174., , , ,
- 5 ,
- 61997 A new method for the model independent assessment of thickness in three-dimensional images. J Microsc 185: 67–75.,
- 72002 New model-independent measures of trabecular bone structure applied to in vivo high-resolution MR images. Osteoporos Int 13: 130–136., , ,
- 82006 A 3D morphological analysis of trabecular bone based on individual trabeculae segmentation. Trans Orthop Res Soc 31: 1783., , , ,
- 92006 Quantification of the roles of trabecular microarchitecture and trabecular type in determining the elastic modulus of human trabecular bone. J Bone Miner Res 21: 1608–1617., , , ,
- 102006 Volumetric spatial decomposition of trabecular bone into rods and plates-a new method for local bone morphometry. Bone 38: 475–484.,
- 112006 Age-related changes in trabecular bone microstructures: Global and local morphometry. Osteoporos Int 17: 616–626.,
- 122006 Importance of individual rods and plates in the assessment of bone quality and their contribution to bone stiffness. J Bone Miner Res 21: 586–595., , , ,
- 131994 Trabecular bone exhibits fully linear elastic behavior and yields at low strains. J Biomech 27: 1127–1136., , , ,
- 141998 Yield strain behavior of trabecular bone. J Biomech 31: 601–608.,
- 151999 Uniaxial yield strains for bovine trabecular bone are isotropic and asymmetric. J Orthop Res 17: 582–585., , ,
- 162001 Dependence of yield strain of human trabecular bone on anatomic site. J Biomech 34: 569–577.,
- 171997 Three-dimensional imaging of trabecular bone using the computer numerically controlled milling technique. Bone 21: 281–287., , , , ,
- 181976 Percolation and cluster distribution. I. Cluster multiple labeling technique and critical concentration algorithm. Phys Rev B 14: 3438–3445.,
- 191994 Detection of 3-D simple points for topology preserving. IEEE Trans Pattern Anal Mach Intell 16: 1028–1032.,
- 201997 A new shape preserving parallel thinning algorithm for 3D digital images. Pattern Recogn 30: 1939–1955., ,
- 211996 3D digital topology under binary transformation with applications. Comput Vis Image Underst 63: 418–429.,
- 221994 Topology preservation in 3D digital space. Pattern Recogn 27: 295–300., , ,
- 232003 Topology-based orientation analysis of trabecular bone networks. Med Phys 30: 158–168., ,
- 241987 Bone histomorphometry: Standardization of nomenclature, symbols, and units. Report of the ASBMR Histomorphometry Nomenclature Committee. J Bone Miner Res 2: 595–610., , , , , , ,
- 251997 Is trabecular bone tissue different from cortical bone tissue? Forma 12: 185–196.,
- 261994 A homogenization sampling procedure for calculating trabecular bone effective stiffness and tissue level stress. J Biomech 27: 433–444., ,
- 271992 Numerical Recipes in Fortran, 2nd ed. Cambridge University Press, Cambridge, UK., , ,
- 281996 Direct mechanics assessment of elastic symmetries and properties of trabecular bone architecture. J Biomech 29: 1653–1657., , ,
- 291984 Biostatistical Analysis, 2nd ed. Prentice Hall, Englewood Cliffs, NJ, USA.
- 302007 Spatial autocorrelation and mean intercept length analysis of trabecular bone anisotropy applied to in vivo magnetic resonance imaging. Med Phys 34: 1110–1120., , ,
- 312000 Age-related changes in bone mass, structure, and strength-effects of loading. Z Rheumatol 59(Suppl 1): 1–9.
- 321997 Mechanistic Approaches to Analysis of Trabecular Bone. Forma 12: 267–275.
- 331998 The accuracy of digital image-based finite element models. J Biomech Eng 120: 289–295., ,
- 341999 Convergence behavior of high-resolution finite element models of trabecular bone. J Biomech Eng 121: 629–635., , ,
- 352007 In vivo μMRI based finite element analyses detected the restoration of mechanical properties of tibial trabecular bone in hypogonadal men after testosterone treatment. J Bone Miner Res (in press)., , , , , ,
- 362005 In vivo assessment of trabecular bone microarchitecture by high-resolution peripheral quantitative computed tomography. J Clin Endocrinol Metab 90: 6508–6515., , ,