The authors have no conflict of interest.

Research-Article

# Three-Dimensional-Line Skeleton Graph Analysis of High-Resolution Magnetic Resonance Images: A Validation Study From 34-μm-Resolution Microcomputed Tomography^{†}

Article first published online: 1 OCT 2002

DOI: 10.1359/jbmr.2002.17.10.1883

Copyright © 2002 ASBMR

Additional Information

#### How to Cite

Pothuaud, L., Laib, A., Levitz, P., Benhamou, C. L. and Majumdar, S. (2002), Three-Dimensional-Line Skeleton Graph Analysis of High-Resolution Magnetic Resonance Images: A Validation Study From 34-μm-Resolution Microcomputed Tomography. J Bone Miner Res, 17: 1883–1895. doi: 10.1359/jbmr.2002.17.10.1883

^{†}

#### Publication History

- Issue published online: 2 DEC 2009
- Article first published online: 1 OCT 2002
- Manuscript Accepted: 10 APR 2002
- Manuscript Revised: 22 MAR 2002
- Manuscript Received: 31 OCT 2001

- Abstract
- Article
- References
- Cited By

### Keywords:

- trabecular bone structure;
- magnetic resonance imaging;
- topological parameters;
- three-dimensional-line skeleton graph analysis;
- skeleton graph smoothing

### Abstract

- Top of page
- Abstract
- INTRODUCTION
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- Acknowledgements
- REFERENCES
- APPENDIX

The resolution achievable in vivo by magnetic resonance imaging (MRI) techniques is not sufficient to depict precisely individual trabeculae and, thus, does not permit the quantification of the “true” trabecular bone morphology and topology. Nevertheless, the characterization of the “apparent” trabecular bone network derived from high-resolution MR images (MRIs) and their potential to provide information in addition to bone mineral density (BMD) alone has been established in studies of osteoporosis. The aim of this work was to show the ability of the three-dimensional-line skeleton graph analysis (3D-LSGA) to characterize high-resolution MRIs of trabecular bone structure. Fifteen trabecular bone samples of the distal radius were imaged using the high-resolution MRI (156 × 156 × 300 μm^{3}) and microcomputed tomography (μCT; 34 × 34 × 34 μm^{3}). After thresholding, the 3D skeleton graph of each binary image was obtained. To remove the assimilated-noise branches of the skeleton graph and smooth this skeleton graph before it was analyzed, we defined a smoothing length criterion (*l*_{c}), such that all “termini” branches having a length lower than *l*_{c} were removed. Local topological and morphological LSGA measurements were performed from MRIs and μCT images of the same samples. The correlations between these two sets of measurements were dependent on the smoothing criterion *l*_{c}, reaching *R*^{2} = 0.85 for topological measurements and *R*^{2} = 0.57–0.64 for morphological measurements. 3D-LSGA technique could be applied to in vivo high-resolution MRIs of trabecular bone structure, giving an indirect characterization of the microtrabecular bone network.

### INTRODUCTION

- Top of page
- Abstract
- INTRODUCTION
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- Acknowledgements
- REFERENCES
- APPENDIX

The in vivo evaluation of trabecular bone structure could be useful in the diagnosis of osteoporosis for the characterization of therapeutic response and understanding the role of parameters other than bone mineral density (BMD) in defining skeletal status. High-resolution imaging based on computed tomography (CT) or magnetic resonance imaging (MR) have been used to obtain in vivo images of peripheral bone sites such as phalanges,^{(1,2)} distal radius,^{(3–7)} and calcaneus.^{(8–10)}

Thus, although in vivo images do not depict individual trabeculae, their validity and usefulness in osteoporosis studies have been established. Majumdar et al.^{(11)} have shown that the “apparent” structural measurements determined from in vivo high-resolution MR images (MRIs) of the distal radius and calcaneus may be used to differentiate postmenopausal women with and without osteoporotic fractures of the proximal femur. Furthermore, the combination of the “apparent” structural parameters and bone mineral density measurements improve the discrimination between osteoporotic subjects and age-matched controls. Link et al.^{(9)} used high-resolution MRIs of the calcaneus and showed that MR-derived parameters of trabecular bone structure differentiated patients with and without osteoporotic hip fractures. Wehrli et al. found the same trend^{(7)} from MRIs of the distal radius in differentiating women with and without osteoporotic vertebral fractures, showing that MR-derived parameters were predictive of vertebral deformity. Newitt et al.^{(12)} developed an automatic efficiency and reproducible process for the characterization of trabecular bone structure from high-resolution MRIs. This process has been applied to longitudinal studies with interesting first results.^{(12–14)} Although some authors have shown the potential of peripheral CT images to discriminate patients with and without osteoporotic fractures,^{(15,16)} the longitudinal in vivo studies performed with peripheral CT imaging are at this time more limited.

Recently, we have developed a three-dimensional-line skeleton graph analysis (3D-LSGA) technique.^{(17)} The skeleton graph is used as a template to model the trabecular bone network, each branch of the graph being associated to a trabecula and each vertex of the graph to a connection between several trabeculae. Some local topological and morphological parameters are deduced from this modeling. 3D-LSGA has been validated on a set of 12 femoral head samples by high-resolution MRI at 78 × 78 × 78 μm^{3}.^{(17)} A recent study performed in vitro on spine bone samples at a resolution of 117 × 117 × 300 μm^{3} showed the potential of LSGA topological parameters to predict bone strength.^{(18,19)}

Recognizing that the resolution achievable in vivo is limited, the purpose of our current work was to relate 3D-LSGA measurements of high-resolution MRI (156 × 156 × 300 μm^{3}) to those obtained from micro-CT (μCT; 34 × 34 × 34 μm^{3}) for a set of 15 trabecular bone samples.^{(20,21)} MR and μCT LSGA-parameters such as the branches and vertices densities as well as the mean length and mean volume of the trabeculae were compared. The mean ratios between MR and μCT LSGA measurements were evaluated, as well as the mean ratios of the global parameters: bone volume fraction (bone volume/total volume [BV/TV]), mean chord length of the solid phase^{(22)} (*l*_{s}), and Euler-Poincaré characteristic^{(23)} (EPC). In parallel, we have studied the impact of smoothing of the skeleton graph on the MR/μCT 3D-LSGA comparison.

### MATERIALS AND METHODS

- Top of page
- Abstract
- INTRODUCTION
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- Acknowledgements
- REFERENCES
- APPENDIX

#### Bone samples

Fifteen trabecular bone samples were obtained from the distal radius of 15 cadavers (age range, 49–92 years). The samples were cut following a cubic core of a 12-mm mean side length at a 9.75-mm mean location from the joint line of the radius. These samples have been used in previous studies analyzing and assessing different processing methodologies.^{(20,21)} The trabecular bone samples, with intact marrow, were placed in formaldehyde solution.

#### μCT imaging

3D images with an isotropic resolution of 34 × 34 × 34 μm^{3} were obtained using μCT (Scanco 20; Scanco Medical AG, Zurich, Switzerland).^{(24)} The gray level images were filtered with a low-pass Gaussian filter to reduce the gray level noise, and then a global threshold was used to separate bone (or solid) and marrow (or assimilated void) phases.^{(25)} The 3D binary images (Fig. 1A) were enlarged to insure a 5 pixels void border following the three axes. Average size of the enlarged binary images was 255 × 255 × 302 pixels^{3}, with an average size region of interest (ROI) of 245 × 245 × 292 pixels^{3} or 8.3 × 8.3 × 9.9 mm^{3}.

#### MRI

Bone marrow was removed and the samples were immersed in gadolinium-diethylenetriaminepentaacetic acid (DTPA)-doped saline. They were placed into a vacuum to remove air bubbles. 3D MRIs with an anisotropic resolution of 156 × 156 × 300 μm^{3} were obtained (GE 1.5 T; General Electric Medical Systems, Waukesha, WI, USA), using a 3D fast gradient echo sequence with partial echo time^{(39)}: TE (echo time) = 3.7 ms; TR (repetition time) = 30 ms; 30° flip angle; ±15.6-kHz bandwidth.

The gray level images were segmented into bone (solid) and void phases from a global threshold evaluated using a dual reference limit approach.^{(26)} The 3D binary images (Fig. 1B) were enlarged to insure a 5-pixel void border following the three axes. Average size of the enlarged binary images was 61 × 61 × 36 pixels,^{(3)} with an average size ROI of 51 × 51 × 26 pixels^{(3)} or 8.0 × 8.0 × 7.8 mm^{3}.

#### Clustering analysis

After thresholding, some small isolated clusters of the solid and void phases remain in the binary images. To determine that the experimental condition was met and that there was only one cluster of solid (zero-order Betti number, β_{0} = 1) and no internal surface (second-order Betti number, β_{2} = 0), we applied a clustering analysis to both solid and void phases. The Hoshen and Kopelman algorithm^{(27)} was applied in 26-connexity to the solid phase to identify each isolated cluster of solid, and in 6-connexity to the complementary phase to identify each isolated cluster of void. Only the larger cluster of each phase was maintained, the deleted clusters corresponding to 4.90% and 0.88% of the larger cluster for the solid phase for the MRIs and μCT images, respectively, and to 0.0012% and 0.0003% of the larger cluster for the void phase.

#### BV/TV

BV/TV is defined as the ratio of BV and TV. It was evaluated by pixel counting in the ROI.

#### Connectivity number

Connectivity number (χ) is a topological parameter expressing the number of loops of the trabecular bone network. It was evaluated from the Euler-Poincaré relation^{(28)}: χ = β_{0} + β_{2} − EPC. EPC was evaluated in a global way from an algorithm described by Vogel.^{(23)} The previous conditions imposed by the clustering analysis (β_{0} = 1 and β_{2} = 0) induce a simplified relation between χ and EPC: χ = 1 − EPC.

#### Mean chord length of the solid phase

We used the mean chord length^{(22)} of the solid phase (*l*_{s}) as length indicator of the trabecular bone network. A chord of solid is defined as the intersection between the solid phase and a random line, plotted from a random initial point of the void phase and following a random direction in the 3D space. *l*_{s} was calculated as the mean length of several chords of solid.

#### 3D skeleton graph analysis

##### 3D skeleton graph

The 3D skeleton graph was obtained from a constrained thinning algorithm applied to the solid phase.^{(17)} Topological constraints based on the topological invariants β_{0}, β_{2}, and EPC were used to check the conservation of the topology between the initial image and its skeleton graph (Fig. 2). Morphological constraints were used to check the conservation of the trabecular elongation between the initial trabecular bone network and its skeleton graph.

##### Classification of the skeleton graph points

Each point of the skeleton graph was classified into one of the classes (isolate, end, begin, final, node, particular, or connect) defined in the Appendix. This classification was performed taking into consideration both the number of neighbors of the considered point and the neighboring configuration of these neighbors.^{(17)}

##### Topological and morphological parameters

The 3D skeleton graph can be described as branches (*B*) and vertices (*V*) and was used as a template of the trabecular bone network to evaluate both topological and morphological characteristics.^{(17)} The principle of this modeling is to associate each branch of the skeleton graph to a trabecula of the bone network and each vertex of the skeleton graph to a connection between several trabeculae.

We classified as “connection” topological parameters the vertices (*V*_{c}) connecting several branches together and the branches (*B*_{cc}) linking two *V*_{c} vertices. We classified as “termini” topological parameters the free-end vertices (*V*_{t}) and the branches (*B*_{ct}) linking one *V*_{c} and one *V*_{t} vertices together. The relation between *B*, *V*, *B*_{cc}, *V*_{c}, *B*_{ct}, *V*_{t} and the {isolate, end, begin, final, node, particular, connect} numbers are reported in the Appendix. All topological parameters (*K*) were normalized by the total volume of analysis (*V*_{0}) of the ROI, giving topological densities (numbers per unit of volume): *K*_{v} = *K*/*V*_{0}.

A branch of the skeleton graph is identified as the set of points of the skeleton graph belonging to a path between two vertices, this path being explored in 26-connexity. As all branches are identified, a dilation algorithm is applied conditionally to the initial solid phase and each reconstructed trabecula is identified from the corresponding branch identification. Local morphological parameters were evaluated as trabecular length (distance between the two extremities of the corresponding branch) and trabecular volume (number of pixels belonging to the identified trabecula). *L*_{cc} and *V*_{cc} are, respectively, the length and volume of the connection trabeculae stemmed from the connection branches, and *L*_{ct} and *V*_{ct} are relative to the termini trabeculae stemmed from the termini branches.

##### Skeleton graph smoothing

The thinning algorithm is sensitive to the irregularities of the digitized trabecular surface, and some “noise-termini” branches appear because of these irregularities, being not related to any significant pattern. These noise-termini branches are mixed with the “true-termini” branches related to the trabecular bone state. They disturb the interpretation of the skeleton graph, and, consequently, disturb the topological and morphological characterization of the trabecular bone network.

To limit this effect, we used a length criterion (*l*_{c}) to remove some of the termini branches: all termini branches having a length lower than *l*_{c} were considered as noise-termini branches and were removed. In a previous study,^{(17)} we used an a priori criterion of *l*_{c} = 2 × *l*_{s}, where *l*_{s} was the mean length chord of the solid phase.^{(22)} In this study, we used *l*_{c} = *k* × *l*_{s}, with *k* = 0, 0.5, 1, 1.5, and 2, to check the effect of the selection of the criterion value on the local topological and morphological parameters. For each of these criteria, the smoothing was performed from the initial skeleton graph.

After removing a termini branch, it is necessary to check the reclassification of connection and termini branches that were linked to the removed branch. In some cases, the two branches that were linked to the removed branch become a single branch, and the point that was a vertex linking the old branches together becomes a “current” point belonging to the new single branch (Fig. 3).

#### Statistical analysis

Linear regressions were performed using JMP software (SAS Institute, Inc., Cary, NC, USA).

### RESULTS

- Top of page
- Abstract
- INTRODUCTION
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- Acknowledgements
- REFERENCES
- APPENDIX

#### Global evaluation

BV/TV, mean chord length (*l*_{s}), and connectivity density (χ) were evaluated for the μCT images and MRIs. The linear correlations of these parameters between μCT images and MRIs of the same samples were evaluated; the *R*^{2} values were *R*^{2} = 0.89 for BV/TV, *R*^{2} = 0.71 for *l*_{s}, and *R*^{2} = 0.79 for χ.

The mean ratio (*r*_{0}) of a global parameter (*G*) between μCT and MR characterizations was calculated as the mean of the *n* = 15 individual ratios (*r*_{i}) obtained for each sample:

- (1)

The mean ratios evaluated for the global parameters BV/TV, *l*_{s}, and χ were *r*_{0} = 0.40 ± 0.05, *r*_{0} = 0.45 ± 0.03, and *r*_{0} = 2.95 ± 0.94, respectively.

#### 3D skeleton graph

3D-LSGA was applied to the two sets of μCT images and MRIs. The classification of the points of the skeleton graph into branches (*B*) and vertices (*V*) was validated following a topological criterion (relative error) previously reported,^{(17)} based on the EPC. This criterion was applied to the two sets of images. The mean relative error was ε_{mean} = 0.02% (0% ≤ ε ≤ 0.11%) for the 15 μCT images, and ε_{mean} = 0.59% (0% ≤ ε ≤ 1.35%) for the 15 MRIs.

#### Skeleton graph smoothing

The smoothing of the skeleton graph induced dramatic changes of both topological and morphological parameters. Figure 4 shows the changes of some mean topological densities (*B*, *V*, and χ) with increasing *l*_{c} for the μCT images (Fig. 4A) and MRIs (Fig. 4B).

From *l*_{c} = 0 (initial skeleton graph) to *l*_{c} = 2 × *l*_{s}, the decrease of *B* and *V* represents 77% of their initial values for μCT images (Fig. 4A) and 17% for MRIs (Fig. 4B), *B* and *V* decreasing with the same proportion. Nevertheless, the connectivity density χ′ evaluating from *B* and *V* densities (χ′ = *B* − *V*) remains constant, checking that the topology remains constant during the smoothing of the initial graph (the conditions β_{0} = 1 and β_{2} = 0 remaining satisfied).

Figures 5 and 6 show the changes of the mean morphological parameters (trabecular length and volume, respectively) with increasing *l*_{c} for both μCT images and MRIs. A slight increase of the length (and volume) of the connection trabeculae and, on the contrary, a dramatic increase of the length (and volume) of the termini trabeculae are observed. These trends are more pronounced for the μCT images than for the MRIs.

#### Comparisons between μCT and MR local measurements

##### Topological parameters

The mean ratios of the topological densities (*K*_{v}) were evaluated using Eq. (1). These mean ratios (*r*_{0}) as well as the associated SDs (σ) are reported in Table 1 for different values of the smoothing length criterion *l*_{c} (*l*_{c} = *k* × *l*_{s}, *k* = 0, …, 2). Linear correlation coefficients (*R*^{2}) between μCT and MR characterizations are reported in each section of the Table 1.

The section *l*_{c} = 0 corresponds to the comparison between μCT images and MRIs without smoothing of the skeleton graph. In this case, the correlations are nonstatistically significant for all of the topological densities (0.11 ≤ *R*^{2} ≤ 0.51), with high mean ratios (10.88 ≤ *r*_{0} ≤ 29.90).

For the first threshold value (*l*_{c} = 0.5 × *l*_{s}), there is good correlation between most of the topological densities (*R*^{2} ≥ 0.88), except for the termini topological densities (*R*^{2} = 0.09). Although the correlations between termini topological densities increase from *R*^{2} = 0.09 to *R*^{2} = 0.54 (*R* = 0.73) for *l*_{c} = 2 × *l*_{s}, the correlations between the other topological densities slightly decrease with values remaining higher than *R*^{2} = 0.85. In parallel, the mean ratios (*r*_{0}) decrease to reach *r*_{0} = 3.85 for the termini topological densities, and reach values lower than 3.5 (3.30 ≤ *r*_{0} ≤ 3.48) for the connection topological densities.

##### Morphological parameters

The mean ratios of the morphological parameters were evaluated using Eq. (1). These mean ratios (*r*_{0}) as well as the associated SDs (σ) are reported in Table 2 for different values of the smoothing length criterion *l*_{c} (*l*_{c} = *k* × *l*_{s}, *k* = 0, …, 2). Linear correlation coefficients (*R*^{2}) between μCT and MR characterizations are reported in each section of the Table 2. Globally, when *l*_{c} increases, *R*^{2} values increase with increasing mean ratio *r*_{0}.

The higher correlations were obtained for *l*_{c} = 2 × *l*_{s}. Although the correlations between the connection parameters reach *R*^{2} = 0.64 (*R* = 0.80) for *L*_{cc} and *R*^{2} = 0.57 (*R* = 0.75) for *V*_{cc}, the correlations between the termini parameters remain low, with *R*^{2} ≤ 0.18 for *V*_{ct} and *R*^{2} ≤ 0.53 (*R* ≤ 0.73) for *L*_{ct}. In parallel, the mean ratios reach values between 0.42 and 0.5 for *l*_{c} = 2 × *l*_{s}, while for *l*_{c} = 0 the same mean ratios are in the range of 0.11–0.23.

### DISCUSSION

- Top of page
- Abstract
- INTRODUCTION
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- Acknowledgements
- REFERENCES
- APPENDIX

In this study, we have compared 3D-LSGA measurements of high-resolution MRIs of trabecular bone samples with those of micro-high-resolution CT images of the same samples. The correlations between μCT and MR measurements were dependent on the smoothing criterion (*l*_{c}) of the skeleton graph, the smoothing consisting of deletion of all termini branches of the skeleton graph having a length lower than *l*_{c}. Without smoothing (*l*_{c} = 0), no correlations were found between MR and μCT LSGA measurements. A criterion *l*_{c} = 0.5 × *l*_{s} (*l*_{s}, mean chord length of the solid phase) improved the correlations between μCT and MR LSGA topological measurements (*R*^{2} ∼ 0.90). For LSGA morphological measurements, a criterion of *l*_{c} = 2 × *l*_{s} increased the correlations of the connection morphological measurements (*R*^{2} = 0.64 for the mean length of the connection trabeculae; *R*^{2} = 0.57 for the mean volume of the connection trabeculae).

With recent advances in hardware and software developments, different MR methods have emerged as a means of measuring trabecular bone structure.^{(29)} Small-bore high-field magnets have been used widely for imaging specimens of human^{(30,31)} or animal^{(32–34)} trabecular bone. Because the potential strength and role of the MRI lies in its human in vivo applications, a large number of in vitro studies also have been performed using clinical scanners. Such an approach has been used for the study of resolution dependency^{(5,10,21,35,36)} or for the study of the prediction of bone strength.^{(5,35,37,38)} In this study, we have used the high-resolution MRI of ultradistal radius human specimens performed on a 1.5 T clinical scanner, because our main objective was to validate the use of the 3D-LSGA technique with in vivo-like MRI settings.

Several authors have studied the resolution dependency of the trabecular bone network measurements obtained from the MRI ^{(5,10,21,26,35,36,39)} or peripheral quantitative CT.^{(40)} It was observed that the error of the standard morphological measurements rises rapidly with decreasing resolution^{(39,40)} as well as with slice thickness in the case of anisotropic resolution.^{(39)} A decrease of the resolution leads to an increase of the BV/TV and trabecular thickness (Tb.Th) and a decrease of the trabecular spacing (Tb.Sp) and trabecular number (Tb.N). The increase of Tb.Th and indirectly of BV/TV have been attributed to the partial volume effect induced in the gray level images because of the structure averaging, and the same averaging induces a decrease of Tb.Sp (dual measurement of Tb.Th) and a decrease of Tb.N. Laib and Ruegsegger^{(41)} have compared morphological measurements of two sets of images having the same isotropic resolution of 165 μm, one being artificially obtained from scaled high-resolution images and inherently unblurred and noise free and the other one being obtained from peripheral quantitative CT following clinical protocol imaging. The results showed that the error measurement of the morphological parameters was higher with the real images than with the artificial noise-free images, suggesting a nonnegligible noise dependency on the in vivo measurement accuracy. This noise dependency also is likely to be image modality dependent, hence different for CT-based techniques compared with MR.

The over- or underestimation factors of the trabecular bone network measurements have been evaluated by different authors.^{(5,20)} For example, Majumdar et al.^{(5)} found an overestimation of BV/TV of a factor of 3 between high-resolution MRI and X-ray tomographic microscopy of human radius samples. Furthermore, Tb.Th was overestimated by a factor of 3. In this study, the overestimation of BV/TV between MR and μCT measurements was a factor of 2.5 (1/*r*_{0}, *r*_{0} = 0.40), and the mean chord length of the solid phase, used as a morphological indicator, was overestimated by a factor of 2.2 (1/*r*_{0}, *r*_{0} = 0.45). Although the resolution dependency of the morphological parameters was well studied, the resolution dependency of the topological parameters has not been researched. In this study, we found a global underestimation of the connectivity density of a factor of 2.9. From this result, we hypothesized that the in vivo resolution does not allow depiction of all the loops of the trabecular bone network; therefore, some loops disappear due to the structure averaging. Evaluated from a global analysis, the mean ratios *r*_{0}(*G*) were considered as references: *r*_{0}(BV/TV) = 0.40 ± 0.05 referencing the volume change between the two sets of images, *r*_{0}(*l*_{s}) = 0.45 ± 0.03 referencing the length change, and *r*_{0}(χ_{v}) = 2.95 ± 0.94 referencing the topology change. Because of the linear dependence between the connectivity density (χ_{v}) and the vertex and branch densities (see Appendix), we hypothesized *r*_{0}(*K*) = *r*_{0}(χ) and defined a normalized mean ratio *r*_{0}′(*K*) = *r*_{0}(*K*)/*r*_{0}(χ). Figures 7A and 7B show the evolution of the normalized mean ratio *r*_{0}′(*K*) with increasing *l*_{c}, respectively, for the branch densities (*B*, *B*_{cc}, and *B*_{ct}) and for the vertex densities (*V*, *V*_{c}, and *V*_{t}). An increase of *l*_{c} is associated with a decrease of the normalized mean ratios *r*_{0}′(*K*), these ratios reaching a limit that is very close to the suggested hypothesis *r*_{0}(*K*) = *r*_{0}(χ), or *r*_{0}′(*K*) = 1. Hypothesizing that the volume change of the trabeculae was explained by the change of BV/TV, we defined normalized mean ratios as follows: *r*_{0}′(*V*_{cc,ct}) = *r*_{0}(*V*_{cc,ct})/*r*_{0}(BV/TV). Likewise, we hypothesized that the length change of the trabeculae was explained by the change of the mean chord length of the solid phase (*l*_{s}), and we defined *r*_{0}′(*L*_{cc,ct}) = *r*_{0}(*L*_{cc,ct})/*r*_{0}(*l*_{s}). Figures 8A and 8B show the evolution of the normalized mean ratio *r*_{0}′ with increasing *l*_{c}, respectively, for the trabecula length (*L*_{cc} and *L*_{ct}) and for the trabecula volume (*V*_{cc} and *V*_{ct}). An increase of *l*_{c} is associated with an increase of the normalized mean ratios *r*_{0}′(IK), these ratios reaching a limit that is very close to the suggested hypotheses *r*_{0}(*L*) = *r*_{0}(*l*_{s}) and *r*_{0}(*V*) = *r*_{0}(BV/TV) for the connection trabeculae (*L*_{cc} and *V*_{cc}) and more distant for the termini trabeculae (*L*_{ct} and *V*_{ct}).

The over- or underestimation factors could be used as calibration factors to recover the true value directly from the estimated or apparent one.^{(20,21)} Nevertheless, this approach leads to a hypothesized linear relation between true and apparent measurements as well as high reproducibility of the calibration factors. These hypotheses are not confirmed at this time. Furthermore, calibration procedure between apparent and true measurements necessitates a reference imaging modality at high resolution, which can be performed only from in vitro experiments that are free of confounding in vivo patient examination conditions. Another approach is to use apparent measurements as a reflection of the trabecular bone structure, noting that in this case the apparent measurements are not identical to those obtained by histological evaluation but are relatively well correlated to the true measurements. For example, Laib et al.^{(21)} compared standard morphological parameters evaluated following plate-model assumption, using the same samples as this study. The correlation coefficients were *R*^{2} = 0.83 for Tb.N, *R*^{2} = 0.60 for Tb.Th, and *R*^{2} = 0.81 for Tb.Sp. Laib and Ruegsegger^{(20)} compared peripheral quantitative CT measurements of the same samples with high-resolution μCT measurements. The correlation coefficients were *R*^{2} = 0.81 for Tb.N, *R*^{2} = 0.92 for Tb.Th, and *R*^{2} = 0.81 for Tb.Sp. The accuracy of Tb.Th was better in this second study, noting that it was performed with an in vivo isotropic resolution of 165 μm. In this study, we have checked the correlations of both topological and morphological LSGA measurements. These correlations were dependent on the criterion (*l*_{c}) used for the smoothing of the skeleton graph. Without smoothing, no correlation was found between MR and μCT measurements; however, a criterion *l*_{c} = 0.5 × *l*_{s} was sufficient to obtain high correlations (*R*^{2} ∼ 0.90) with the topological parameters, and the best correlations with the morphological parameters were obtained with a criterion *l*_{c} = 2 × *l*_{s} (*R*^{2} = 0.64 for the mean length of the connection trabeculae). 3D-LSGA allows morphological characterization of each trabecula.^{(17)} Nevertheless, our results have showed that such local morphological measurements are not adequate to in vivo resolution MRIs, leading to low correlations with high-resolution μCT measurements. When the length criterion applied to the smoothing of the skeleton graph increases, favoring in this case the longer trabeculae, the inaccuracy of the morphological MR measurements could be reduced. This trend could be explained by the fact that the partial volume effect, linked to the low resolution, affects the longer trabecular patterns less than the shorter patterns. 3D-LSGA is well adapted to the topological characterization of the trabecular bone network, and the application to high-resolution MRIs leads to good correlations with the true measurements, except for the termini parameters. Nevertheless, the termini parameters are not pertinent in terms of trabecular network connection, and, by extrapolation, could not be pertinent in terms of bone strength.

In this study, we used a smoothing criterion based on the length of the solid phase (*l*_{s}), which is a morphological indicator of the solid phase. Nevertheless, it has been found that this parameter was evaluated with only medium accuracy (*R*^{2} = 0.71 between MR and μCT measurements), and it would have been difficult to ascertain that the smoothing of the skeleton graph was applied in a comparable way on MRIs and μCT images. The smoothing of the skeleton graph is of the utmost importance, particularly at low resolution because of the irregularities of the surface of the solid phase. To limit the effect of these irregularities, it is current practice to apply a low-pass filter directly on the initial gray level images. Nevertheless, this low-pass filtering is not adequate for the high-resolution MRIs of trabecular bone structure, where the size of the solid phase is comparable with the pixel size, and in this case a pretreatment low-pass filtering could considerably increase the inaccuracy of the solid phase measurements. Both approaches, which apply the length criterion or pretreatment low-pass filter, are delicate because the extent of the smoothing can be controlled only by defining an experimental criterion (length for the first approach and filter coefficient for the second one).^{(42)} Recent skeletonization techniques based on a veinerization process^{(43)} allow a better control of the noise-termini branches induced by the skeletonization step. Such techniques could be tested on high-resolution MRIs of the trabecular bone structure.

Several authors have used connectivity as a topological indicator of the trabecular bone network.^{(44–46)} Garrahan et al.^{(47)} were the first to apply a “strut analysis” to 2D trabecular bone images. This technique allows evaluation of topological parameters as the numbers of nodes and vertices.^{(47–49)} Nevertheless, this topological approach is theoretically debatable because of its 2D evaluation and the fact that real topology can be evaluated only from a 3D approach.^{(50,51)} 3D-LSGA has been applied to trabecular bone structure in vitro.^{(17–19,31,52,53)} This technique has been optimized to capture the topology of 3D digitized images with very high accuracy.^{(17)} A similar approach, based on a “surface” skeleton graph analysis has been developed by Saha et al.^{(54)} and applied to trabecular bone images.^{(55,56)} Recently, 3D-LSGA topological parameters have been confronted to biomechanical properties of the trabecular bone structure of spine samples, the results showing very high correlations (*R*^{2} > 0.95) between 3D-LSGA topological parameters and biomechanical parameters evaluated from microfinite element analysis.^{(53)}

This study has showed that 3D-LSGA technique could be applied to high-resolution MRIs, given indirect topological information of the true trabecular bone network with good accuracy. This technique is now envisaged for transversal and longitudinal in vivo studies to check the potential of the 3D-LSGA measurements in the in vivo MR evaluation of osteoporosis and bone treatment effects.

### Acknowledgements

- Top of page
- Abstract
- INTRODUCTION
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- Acknowledgements
- REFERENCES
- APPENDIX

The authors thank Dr. H. Cousart for her invaluable help during study preparation and acknowledge Dr. Thomas Link for assistance in procuring these specimens and Ahi Sema Issever for preparing them. This study was supported by grants from the School of Medicine Research Evaluation and Allocation Committee (REAC) and the Academic Senate-Committee on Research (COR). We acknowledge the Regional Council of Region Centre (France) for its financial help during this work.

### REFERENCES

- Top of page
- Abstract
- INTRODUCTION
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- Acknowledgements
- REFERENCES
- APPENDIX

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### APPENDIX

- Top of page
- Abstract
- INTRODUCTION
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- Acknowledgements
- REFERENCES
- APPENDIX

##### 3D-LSGA nomenclature^{17}

current = point of the graph that has only two neighbors;

node = vertex of the graph that has more than two neighbors;

end = vertex of the graph that has only one neighbor;

begin = current point of the graph for which one and only one of its two neighbors is a node vertex;

final = end vertex for which the corresponding branch has only one point—a final vertex is topologically equivalent to a begin point plus an end vertex;

isolate = point of the graph that has zero neighbors—an isolate point is topologically equivalent to two end vertices plus one branch;

connect = point of the graph that has only two neighbors that are two node vertices;

particular = defined by a nil length connection between two adjacent node vertices;

V = total number of vertices, including both connection (

*V*_{c}) and termini (*V*_{t}) vertices;B = total number of branches, including both connection (

*B*_{cc}) and termini (*B*_{ct}) branches;*V*_{c}= total number of connection vertices;*V*_{t}= total number of termini vertices;*B*_{cc}= total number of connection branches, linking two connection vertices;*B*_{ct}= total number of termini branches, linking a connection vertex and a termini vertex;*V*= node + end + final + 2 · isolate*B*= (begin + end)/2 + final + connect + particular + isolate*V*_{c}= node*V*_{t}= end + final + 2 · isolate*B*_{cc}= (begin -end)/2 + connect + particular*B*_{ct}= end + final + isolateβ

_{0}= (zero-order Betti number) number of connected solid clusters;β

_{1}= (first-order Betti number) number of loops or connectivity;β

_{2}= (second-order Betti number) number of internal surfaces;EPC = Euler-Poincaré characteristic;

EPC′ = Euler-Poincaré characteristic estimated from the vertices and branches counting

EPC = β

_{0}− β_{1}+ β_{2}EPC′ = V − B + β

_{2}.