Application of the Scaling Index Method to µ-CT images of human trabecular bone for the characterization of biomechanical strength Ernst
Rummenyb ,
Roberto A. Monettia , Jan Bauerb,a , Dirk M¨ ullerb , c d Maiko Matsuura , Felix Eckstein , Thomas Linke , and Christoph R¨atha
a
Max-Planck-Institut fuer extraterrestrische Physik, Giessenbachstr. 1, 85748, Garching, Germany b Department of Radiology, Technische Universitaet Muenchen, Ismaninger Str. 22, 81675 Munich, Germany c Department of Anatomy, Ludwig-Maximilian-University, Pettenkoferstr 11 80336 Munich, Germany d Institute of Anatomy and Musculoskeletal Research, Paracelsus Private Medical University, 5020, Salzburg, Austria. e Magnetic Resonance Science Center, Department of Radiology, UCSF, San Francisco, CA 94143, USA ABSTRACT Osteoporosis is a metabolic bone disorder characterized by the loss of bone mineral density (BMD) and the deterioration of the bone micro-architecture. Rarefied bone structures are more susceptible to fractures which are the worst complications of osteoporosis. Here, we apply a structure characterization method, namely the Scaling Index Method, to micro-computed tomographic (µ-CT) images of the distal radius and extract 3D nonlinear structure measures to assess the biomechanical properties of trabecular bone. Biomechanical properties were quantified by the maximum compressive strength (MCS) obtained in a biomechanical test and bone mineral density (BMD) was calculated using dual X-ray absorptiometry (DXA). µ-CT images allow for the application of two different modalities of the SIM which differ in the dimensional embedding of the image. Both representations lead to similar correlation coefficients with MCS which are significantly better than the ones obtained using standard 3D morphometric parameters and comparable to the result given by BMD. The analysis of µ-CT images based on the SIM allows for a sharp distinction of the different structural elements which compose the trabecular bone network. Keywords: Osteoporosis, µ-CT Imaging, Scaling Index Method, Bone Structure Analysis, Structure Analysis
1. INTRODUCTION Osteoporosis has been defined as a systemic skeletal disease, characterized by a decrease of bone density and quality, leading to a reduction in bone strength and increased susceptibility to fracture.1 Among the relevant characteristics of bone quality, its architecture, i.e. microstructure has been suggested, as well as bone turnover, damage accumulation and mineralization. The status of trabecular bone microstructure in the human skeleton of subjects at advanced age has, however, so far not been well characterized, and the impact of gender and site on trabecular microstructure remains ill defined. Amling et al2 used histomorphometry to study trabecular microarchitecture at the lumbar vertebral bodies, the iliac crest, the femur, and the calcaneus of 12 healthy autopsy cases aged 28-84 years. They reported a high degree of heterogeneity of bone microstructure with the trabecular bone volume fraction (BV/TV, %) ranging from 8.3% in the lumbar spine to 15.8% in the femoral neck. Hildebrand et al.3 used µCT imaging to derive three-dimensional measures of trabecular microstructure Further author information: (Send correspondence to R.A.M.) R.A.M.: E-mail:
[email protected], Telephone: +49 89 30000 3345, Fax: +49-89-30000-3950 Medical Imaging 2007: Image Processing, edited by Josien P. W. Pluim, Joseph M. Reinhardt, Proc. of SPIE Vol. 6512, 65124H, (2007) · 1605-7422/07/$18 · doi: 10.1117/12.709195
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Figure 1. Left: P (α) histogram for two different specimens obtained using a 3D image embedding. Right: Idem for 4D image embedding.
from 5 skeletal sites (femoral head, lumbar vertebral bodies (L) 2 and 4, iliac crest and calcaneus) from 52 donors and also found microstructural properties to vary substantially throughout the human skeleton. Recently, the Scaling Index Method (SIM) has been successfully applied to magnetic resonance images for the characterization of bone trabecular structure. It has been shown that texture measures extracted using the SIM are able to predict both the fracture status in vivo4, 5 or the biomechanical properties of trabecular bone in vitro.6–10 In this work, we apply for the first time the SIM to µ-CT images of trabecular bone in vitro and derive non-linear structure measures to be used for the characterization of biomechanical properties of the bone expressed by the MCS. Our purpose is to compare our results with the prediction of bone strength given by the well-established standard 3D morphometric parameters and BMD. In µ-CT images the mineral trabecular net can easily be segmented by thresholding the grey-level distribution. This allows us to apply and compare two different approaches of the SIM, namely 3D and 4D image embedding representations.
2. MATERIAL AND METHODS 2.1. Specimens 17 cylindrical specimens were harvested from the distal radius using diamond trephines (Salzmann, Munich, Germany) as described previously,11 with an 8 mm inner diameter drill. The sample is not age-matched and heterogeneous, i.e. bones harvested from men and women cadaver were considered. The length was set to 14 mm. The samples were stored in a solution of 5% buffered formalin until µCT scanning. The scans were acquired for the central 6 mm of the specimen using a µCT 20 scanner (Scanco Medical, Bassersdorf, Switzerland) as described previously.11 In brief, the resolution was set to 26 µm (isotropic), similar to a previous study on human trabecular bone,3 with medium scan mode and at an integration time of 100 ms. The total scan time per sample was 4.1 hours. Maximum compressive strength (MCS) of the contralateral radius was calculated from the first local maximum of the stress-strain curve using a three-point bending test. Bone mineral density was evaluated using dual X-ray absortiometry.
2.2. Trabecular bone structure analysis Within a defined volume of interest (VOI: diameter 6mm, length 6mm) we determined the following 3D-structural parameters, using the following settings (σ = 0.8; Support 1.0; Threshold 22% of maximal gray value) and the software provided by the manufacturer: 1) bone volume fraction (BV/TV) in %; 2) trabecular number (Tb.N) in 1/mm; 3) trabecular thickness (Tb.Th) in µm; 4) trabecular separation (Tb.Sp) in µm. These structural parameters were calculated as previously described by Majumdar et al.12 In addition, we apply the SIM to µ-CT images in order to assess the trabecular bone structure (for a detailed description of the SIM see4, 6, 7 ). Briefly, consider a 3D tomographic image G(x, y, z), where G is
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Figure 2. Left: Color-coded α-image obtained using a 3D image embedding. Color varies from blue to yellow as increasing the α value. Right: Idem for 4D image embedding. Note that plate-like structures are now represented by small α-values while lines posses higher α-values.
the grey-level value at position (x, y, z). Pixel information can be encompassed in a four dimensional vector p = [x, y, z, G(x, y, z)]. Thus, G(x, y, z) can be mapped onto a 4D point distribution which contains spatial and grey-level information. The SIM characterizes the structural information of images via the estimation of local scaling properties of the above mentioned point distribution. The scaling index αi at pi is the local logarithmic derivative of the density of points with respect to a characteristic scaling parameter. For µ-CT images, the trabecular structure can easily be segmented by thresholding the grey-level distribution in such a way that grey-level values below the threshold are set to zero. Thus, the selected trabecular bone net still possess a grey-level distribution. In contrast to previous applications of the SIM to MR images,4, 6, 7, 9, 10 we apply the SIM only to bone pixels. For 4D image embedding a 4D point distribution is generated by considering the vectors p = [x, y, z, G(x, y, z)], where (x, y, z) is the position of a bone pixel and G(x, y, z) its grey-level value. For 3D image embedding, the first step is to set all grey-level values for bone pixels to one. Thus, a 3D point distribution is automatically generated by the vectors p = (x, y, z), where (x, y, z) is the position of a bone pixel. This property of µ-CT images allows us to compare the two different SIM approaches for assessing the trabecular bone structure. In order to allow for comparison and as an attempt to account for partial volume effects at the trabecular bone borders, the grey-level distribution of trabecular bone pixels is linearly scaled to satisfy σ(G) R before applying the 4D image embedding modality of the SIM, where σ(G) is the standard deviation of G(x, y, z) and R is the scaling range parameter of the SIM.7 In both cases, the local structural information given by the scaling indices is compiled in the so-called P (α) probability distribution. Figure 1 shows P (α) frequency distributions for two bone specimens which have quite different biomechanical properties. When using a 3D image embedding, stronger bones lead to P (α) spectra which are shifted rightwards in comparison to the ones obtained for weaker bones (Fig. 1 left panel). The reason for this behavior is that stronger bones contain a large amount of plate-like structural elements which lead to α values in the range α ∈ [2, 2.5]. On the other hand, weaker bones are composed mainly by rod-like structural elements leading to α values in the range α ∈ [1, 2]. The right panel of Fig. 1 shows results for the same specimens obtained using a 4D image embedding. In this case, stronger bones differentiate from weaker ones since they display a stronger signal at small α values and significantly narrower P (α) spectra. In contrast to the results for 3D image embedding, 4D embedding leads to higher α values for rod-like structural elements. These differences can be clearly observed in Fig. 2 which shows α-images (images generated using the scaling index values) for the two SIM modalities. The high resolution of µ-CT images allows us to visualize structural elements that compose the trabecular
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Figure 3. Upper left: Original µ-CT image of a human distal radius specimen. Upper right: Color-coded image of the scaling indices obtained using a 4D image embedding. Lower left: Filtered image showing the backbone of the trabecular net. Lower right: Filtered image showing plates (blue region) and rods (yellow region).
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Figure 4. Upper left: Color-coded image of the scaling indices obtained using a 3D image embedding. Upper right: Filtered image where dense areas (plates) have been selected. Lower left: Filtered image showing where mainly rod-like structures are observed. Lower right: Filtered image where dense areas (yellow) and approximately 1D regions (blue) are observed
network like rod-like and plate-like structures (see Fig. 3 upper left). A structural filter of the image can be realized by considering different regions of the P (α) frequency distribution. The upper right panel of Fig. 3 shows the color-coded α-tomographic image for the 4D image embedding. The lower panels of Fig. 3 show different filtered structures where the structure backbone, plates and rods can be clearly visualized. Similar results are obtained using the 3D embedding representation. Figure 4 (upper left) shows the color-coded α-tomographic image and other filtered structures. It should be noted that dense plate-like structures can be nicely filtered using this SIM modality (see upper right panel of Fig. 4). We perform a correlation analysis based on a structural reconstruction of the image using the P (α) spectrum. Given a P (α) spectrum, a binary image can be obtained by separating the pixels having α values below a given threshold from the pixels having α values above it. The resulting binary image is then analyzed using a nearest
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Figure 5. Upper left: Result of the clustering analysis for a weak bone (MCS = 473 N) using the SIM for a 3D image embedding. Upper right: Result of the clustering analysis for a dense bone (MCS = 2442.2 N) using the SIM for a 3D image embedding. Lower left: Idem upper left using the SIM for a 4D image embedding. Lower right: Idem upper right using the SIM for a 4D image embedding.
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neighbor clustering algorithm in order to identify the most prominent clusters. By increasing the threshold value, these clusters gradually grow and new clusters appear as well. We repeat this procedure until the first cluster that spans the image along the z direction is created. The assumption is that the spanning cluster and other large clusters which develop during this procedure contribute most to the biomechanical properties of the bone. Figure 5 shows images obtained by means of this analysis for the two image embedding modalities and for bones possessing different biomechanical properties. Prominent clusters are shown in different colors. Clusters in the lower panel appear blurred in order to indicate that the cluster growing processes for 4D image embedding occurs from the center of thick trabeculae outwards (see Fig. 2 left panel) By means of this analysis several quantities which provide a measure of bone quality can be extracted. First of all, we consider the set of threshold values {αc } which satisfy the cluster spanning condition and the corresponding set of amplitudes {P (αc )}. In addition, we also calculate some properties of the clusters, namely mass, connectivity, moments of inertia, etc, and typical quantities characterizing the P (α) spectrum. Only quantities leading to significant correlation coefficients are reported in this paper.
3. RESULTS In this study, all 3D standard morphometric parameters have non-significant Spearman’s correlation coefficient (R). On the other hand, the correlation coefficient for MCS versus BMD is R = 0.83. For 3D image embedding, the structure parameters which lead to better correlations with MCS were αc and the mean connectivity of relevant clusters C. The correlation coefficient for MCS versus αc is R = 0.70 (p < 0.001) and for MCS versus C is R = 0.62 (p < 0.001). Correlations coefficients improve between 4 − 9% (R(M CS vs αc ) = 0.73 and R(M CS vs C) = 0.675) when normalizing the MCS values with the projected coronal bone area obtained using DXA. It should be noted that µ-CT imaging only depict the trabecular bone network and completely ignores both cortical bone and bone size. The improved results obtained after normalization are then consistent since bone area roughly quantifies the contribution of bone size to bone strength. It should be remarked that BMD obtained using DXA already accounts for the contribution of the cortical bone which is particularly important when MCS values are obtained using a tree-point bending test. We also performed a multiregression analysis first combining the structure measures and then structure measures with BMD. A combination of the αc and the cluster connectivity C leads to a slightly higher correlation coefficient R = 0.74. However, a combination of structure measures and BMD leads to significantly higher correlation coefficient (R = 0.89). Table 1. Summary of correlation coefficients R for MCS versus the different bone quality measures above defined (∗ correlation coefficient obtained using normalized MCS values). In all cases, p < 0.001
R
MCS
BMD
0.83
αc
0.73∗
C
0.675∗
F13D
0.74∗
Fc3D
0.89
σ(P (α))
0.81
Fc4D
0.89
For 4D image embedding, the structure measure αc lead to a significant but poor correlation coefficient R = 0.58. Other measures extracted out of the prominent clusters lead to non-significant correlations. In this case, typical parameters characterizing the P (α) spectrum lead to better results. We considered the set of amplitude values at the maximum of the spectrum {P (αm )} and the set of standard deviations {σ(P (α))}. In this case, normalization using the bone area does not lead to significantly better correlation coefficients. The
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Figure 6. Scatter plots of MCS versus the bone quality measures described in the text. Red lines indicate the quality of the linear regressions. Upper left: M CS versus BMD. Upper right: The maximum compressive strength normalized √ 2by the2 3D quantity F = αc + C bone area M CS n versus αc for 3D image embedding. Middle left: M CS n versus the combined 1 p for 3D image embedding. Middle right: M CS versus the combined quantity Fc3D = (F13D )2 + BM D2 for 3D image embedding. p Lower left: M CS versus σ(P (α)) for 4D image embedding. Lower right: M CS versus the combined quantity Fc4D = (σ(P (α)))2 + BM D2
for 4D image embedding.
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correlation coefficient for MCS versus σ(P (α)) is R = 0.81 (p < 0.001) and for MCS versus P (αm ) is R = 0.60 (p < 0.001). A combination of these structure measures using a multiregression model does not lead to better results. However, when combining σ(P (α)) and BMD a higher correlation coefficient is obtained (R = 0.89) (see caption of Fig. 6).
4. CONCLUSIONS The structure characterization of µ-CT images using the different modalities of the SIM allows for a structural decomposition of the trabecular bone network where various structural elements can easily be spotted (see Figs. 3 and 4). Here, we applied for the first time this methodology to µ-CT images for assessing the biomechanical strength of trabecular bone in vitro. Preliminary results obtained using a clustering analysis based on a structural reconstruction applying the SIM are superior to the ones obtained using the 3D standard morphometric parameters. However, correlation coefficients obtained using our method are slightly lower than the one obtained using BMD. This is in contrast to results reported in previous studies.6, 8, 10 However, it should be noted that BMD was calculated using DXA: a technique which also accounts for the mineral content of the cortical bone. It is a well known fact that properties of the cortical bone play an important role in the prediction of the biomechanical bone strength. In addition, µ-CT images and the mechanical testing were not performed at the same skeletal site. This asymmetry and the small sample size may lead to lower correlation values. A higher correlation coefficient was obtained using a multiregression analysis combining BMD and structure parameters. This result indicates that structure characterization and BMD measurements are complementary techniques and provide a great deal of information on the biomechanical properties of the bone.
5. ACKNOWLEDGMENTS This work is partially supported by the Deutsche Forschungsgemeinschaft (DFG) under grant MU/2288/2-2.
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