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The effect of microcomputed tomography scanning and reconstruction voxel size on the accuracy of stereological measurements in human cancellous bone
Bone 35 (2004) 1375 – 1382 www.elsevier.com/locate/bone
The effect of microcomputed tomography scanning and reconstruction voxel size on the accuracy of stereological measurements in human cancellous bone Do-Gyoon Kima,*, Gregory T. Christophersona, X. Neil Donga, David P. Fyhrieb, Yener N. Yenia b
a Bone and Joint Center, Henry Ford Hospital, Detroit, MI, 48202, USA Orthopaedic Research Laboratories, UC Davis, School of Medicine, Sacramento, CA 95817, USA
Received 16 March 2004; revised 14 September 2004; accepted 20 September 2004
Abstract Stereological parameters have been used as an approximation for the architecture of trabecular bone. Structural indices such as bone volume fraction (BV/TV), trabecular number (Tb.N), trabecular thickness (Tb.Th), trabecular separation (Tb.Sp), bone surface-to-volume ratio (BS/BV), degree of anisotropy (MIL1/MIL3), and connectivity density (Euler/Vol) have been widely studied to investigate pathological conditions in bone. Due to its high resolution and nondestructiveness, microcomputed tomography (micro-CT) has been utilized to take precise three-dimensional (3D) images of trabecular microstructures. However, spatial limitations for applying micro-CT-based analyses to large specimens, such as whole vertebral bodies, require using larger scanning and reconstruction voxel sizes. In this study, combinations of three different scanning and reconstruction voxel size were used to represent best possible voxel size (21 Am; best in our scanner for the specimen size used) relative to other voxel sizes used in this study, commonly used intermediate voxel sizes (50 Am), and those applicable to scans of whole human vertebral bodies (110 Am) in order to examine the effect of scanning and reconstruction voxel size on stereological measures for human cancellous bone. The error in stereological parameters calculated using combinations of large voxel sizes compared to the gold standard (best possible case) ranged from 0.1% to 102%. The signed magnitude of the error in other cases relative to the gold standard was a function of either scanning or reconstruction voxel size or both (r 2 = 0.55–0.95). For most of the structural indices, the results from analysis of images with larger voxel sizes were correlated with those from the gold standard (r 2 = 0.55–0.99) except for Tb.N at 110/110 Am, MIL1/MIL3 at larger than 110 Am reconstruction voxel size, and Euler/Vol at any combination of voxel sizes. Overall, it was observed that resampling a high resolution image at lower resolutions (corresponding to increasing reconstruction voxel size in this study) had different effects on the calculated parameters than scanning at the same low resolution (corresponding to increasing scanning voxel size in this study). Our results show that investigations of image resolution should include actual scans at the resolution of interest rather than simply coarsening of high-resolution images as is customarily done. D 2004 Elsevier Inc. All rights reserved. Keywords: Stereology; Trabecular bone; Micro-CT; Resolution; Calibration
Introduction Trabecular architecture has been considered as an important determinant of osteoporosis and other pathological conditions in bone. The microstructure of the * Corresponding author. Bone and Joint Center, Henry Ford Hospital, 2799 West Grand Boulevard, Detroit, MI, 48202. Fax: +1 313 916 8064. E-mail address: [email protected] (D.-G. Kim). 8756-3282/$ - see front matter D 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.bone.2004.09.007
trabecular network is often measured using measures such as bone volume fraction (BV/TV), trabecular number (Tb.N), trabecular thickness (Tb.Th), separation (Tb.Sp), and surface-to-volume ratio of bone (BS/BV). These parameters have been examined for different anatomical sites and physiological conditions such as aging [4,24,27], bone disease [5,11,20], and recently, remodeling caused by chemical treatments [3,18,26]. A related parameter, the degree of anisotropy of trabecular
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bone, was observed to highly correlate with increase in fracture risk of the hip [2]. Trabecular connectivity based on topological measurements has also been useful for predicting changes in mechanical properties with bone loss [14,28]. Microcomputed tomography (micro-CT) was introduced to explore the three-dimensional (3D) architecture of bone [8]. Due to its high resolution (as small as a few micrometers), micro-CT can obtain precise 3D images at the microlevel of trabecular structure. For in vitro microCT studies, different voxel sizes have been used, ranging from 8 to 120 Am [10,16,28,30,31]. Although high resolution is achievable using micro-CT, scanning large specimens such as a whole vertebral body may require use of a spatial resolution corresponding to a voxel size greater than 100 Am [6,13]. Because 100 Am is in the order of typical trabecular thickness, partial volume effects will cause errors when computing the morphological parameters for trabecular bone [15]. Ding and Hvid [4] found that the magnitudes of stereological parameters were strongly dependent on voxel size for voxel size larger than 100 Am. Ruegsegger et al. [25] indicated that the morphological values could be corrected to a specific resolution up to 200-Am voxel size based on the monotonic dependence of morphological parameters on image resolution. However, the effect of scanning and reconstruction voxel size has not been considered separately in previous studies [12,17,30]. The separate and combined effects of these voxel sizes on morphological variables need to be evaluated. Scanning voxel size is a measure of the quality of the raw data images and determines the best level of detail that can be resolved in the image. The raw data can be appropriately reconstructed at any reconstruction voxel size that is larger than (or equal to) the scanning voxel size. Reconstruction voxel size is the actual voxel size chosen for the 3D image reconstruction. It is ordinarily assumed that a larger reconstruction voxel size will decrease the accuracy of the image. Nevertheless, mainly to avoid large computational costs, images may have to be coarsened using greater reconstruction voxel sizes than scanning voxel size in some applications such as microCT-based, large-scale finite element models [10,16,29] (note that choosing a reconstruction voxel size smaller than the scanning voxel size cannot improve the quality of the reconstructed image because the raw data do not support the smaller reconstructed voxels). The goal of this study was to evaluate the effect of scanning and reconstruction voxel size on the micro-CTbased 3D stereological analyses of human trabecular tissue. To this end, six different combinations of scanning/reconstruction voxel size were considered on the basis of three different voxel sizes that represent high-resolution scans (best possible voxel size in our scanner for the specimen size used), commonly used intermediate voxel sizes and those applicable to large specimens, such as a whole vertebral
body. The variations of morphological parameters were quantified at different combinations of voxel sizes. The errors and relationships of each voxel size case relative to the best possible voxel size case were estimated.
Methods Eight cylindrical cancellous bone specimens (10-mm length, 8-mm diameter) were cored in the inferosuperior direction from L2–L4 vertebrae of a 63-year-old male and the metaphyseal tibia of a 52-year-old male. Each specimen was scanned with micro-CT at a scanning voxel size of 21, 50, and 110 Am voxel size. Images were reconstructed based on Feldkamp’s convolutionback projection algorithm [7] using reconstruction voxel sizes that were the same or larger than the scanning voxel size (21, 50, and 110 Am). This procedure resulted in scanning/reconstruction voxel size combinations of 21/ 21, 21/50, 21/110, 50/50, 50/110, and 110/110 Am for each specimen. Twenty-one micrometers is about the best possible scanning voxel size that can be achieved in our scanner for specimens of this size, and 50 Am is a commonly employed intermediate value. The choice of 110 Am was based on our experience with micro-CT scanning of whole human vertebral bodies and represents a bbest-caseQ scenario for the future in vivo scanning of the spine. The 21/21 case was used as the gold standard for determining the level of inaccuracy in the coarser scanning/reconstruction combinations. After reconstruction, bone and nonbone voxels were segmented using a heuristic segmentation algorithm developed specifically for bone tissue with highly nonhomogeneous CT density distributions that have large overlap in density values between bone and bone marrow [32]. A special purpose code for stereology was utilized to compute the morphological parameters directly from microCT images that had been thresholded following reconstruction. The volume of interest was determined as the largest sphere that could be fit inside the cylindrical specimen volume. Seven morphological parameters were examined: Bone volume fraction (BV/TV), trabecular number (Tb.N), trabecular thickness (Tb.Th), trabecular separation (Tb.Sp), bone surface-to-volume ratio (BS/BV), degree of anisotropy (MIL1/MIL3), and connectivity density (Euler/Vol). All computations were based on formulation of the plate model for trabecular bone structure [8,9,23]. BV/TV (mm3/mm3) was given by P P that was obtained dividing the number of bone voxels by the total number of voxels in the volume of interest. Tb.N (mm1) was expressed by P L, the number of intersections between bone and nonbone voxels per total length of test lines. Tb.Th, Tb.Sp, and BS/BV are calculated based on P P and P L: Tb:Th ðmmÞ ¼ P P =P L
ð1Þ
D.-G. Kim et al. / Bone 35 (2004) 1375–1382
Tb:Sp ðmmÞ ¼ ð1 P P Þ=P L
ð2Þ
BS=BV mm2 =mm3 ¼ 2P L =P P
ð3Þ
Mean intercept length (MIL) represents the mean distance between intersections of bone and nonbone components for each test line at a different angle of measurement. Therefore, the degree of anisotropy was found as the ratio between the maximum (MIL1) and minimum (MIL3) mean intercept length, which were measured at the major (1) and minor (3) axis of a trabecular network [2]. It can be denoted in terms of P L as MIL1=MIL3 ¼ P L3 =P L1
ð4Þ
where P L3 and P L1 are the trabecular number at the minor and major axis, respectively. Connectivity of the trabecular network was obtained based on topological measurements depending on Euler-Poincare number [22] independent from the plate model assumption. If there is only one connected bone phase and only one connected bone marrow phase, connectivity density can be defined as Connectivity density mm3 ¼ ð1 #EulerÞ=Volc #Euler=Vol
ð5Þ
where (1 Euler number) represents the maximum number of branches that can be removed without breaking the structure into separate parts. Frequently, the connectivity is also defined as the number of bone marrow loops in the trabecular network [14]. To effectively quantify the number of connections, the Euler number was multiplied by negative one and normalized by the volume of interest. This allowed for reasonable comparison between specimens with different volume. The present stereology code assumes the trabeculae at the boundary of the volume of interest as connected to the adjacent trabecular network so that the so-called bedgeQ effects [22] related to connectivity computation were corrected. Two-way repeated measures ANOVA was used for analyzing the effect of scanning/reconstruction combinations with each specimen as the subject and scanning/reconstruction (21/21, 21/50. . .) as repeated factors using Sigma Stat (SPSS Inc., Chicago). The error (D) in a parameter of concern from a large voxel size case was calculated as the signed difference in this parameter between the large voxel size case
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and the 21/21 case assuming the 21/21 case error-free. Multivariable linear regression was performed to examine the relationship between the error (D) and scanning/reconstruction voxel size (21, 50, and 110 Am). Dummy variables [1] were used in multivariable regressions in order to account for within-specimen variability. Another multivariable linear regression was performed to examine the relationship between scanning/reconstruction voxel size and the standard deviation (SD) of the error in a parameter of concern. If either scanning or reconstruction voxel size was significant only, a simple linear regression was performed. The relationships between parameters calculated from the 21/21 Am images and those from other combinations of scanning/reconstruction voxel size were examined using regression analysis (Microsoft Excel).
Results All stereological parameters varied with different combinations of scanning and reconstruction voxel sizes (Table 1). A two-way repeated-measures ANOVA suggested that scanning voxel size is the major factor causing the difference of values in BV/TV and in Euler/Vol ( P b 0.001 for both parameters), whereas reconstruction voxel size did not have a significant effect ( P = 0.72 and P = 0.30, respectively). For Tb.N and Tb.Sp, the differences were significantly related with reconstruction voxel size ( P b 0.001) but not with scanning voxel size ( P = 0.22 and 0.97, respectively). The differences in Tb.Th and in BS/BV were significantly related with both the scanning and reconstruction voxel size ( P b 0.001). However, MIL1/MIL3 did not depend on either scanning or reconstruction voxel size ( P N 0.20). Most regressions between scanning/reconstruction voxel size and the error (D) (Table 2) agreed with the results of RMANOVA. Scanning voxel size but not reconstruction voxel size was a major source of the errors in BV/TV and Euler/Vol. The error in Tb.N and Tb.Sp significantly correlated with increasing reconstruction voxel sizes but not with scanning voxel size. In Tb.Th and BS/BV, the error was significantly related to both scanning and reconstruction voxel sizes. While the differences in MIL1/MIL3 were not significant between voxel size combinations, the error in MIL1/MIL3 was correlated with reconstruction voxel size.
Table 1 Average values (standard deviation) of stereological parameters for the scanning/reconstruction voxel size groups Voxel size
BV/TV
Tb.N (mm1)
Tb.Th (mm)
Tb.Sp (mm)
BS/BV (mm2/mm3)
MIL1/MIL3
Euler/Vol (mm3)
21/21 21/50 50/50 21/110 50/110 110/110
0.174 0.176 0.209 0.174 0.203 0.276
1.39 1.31 1.23 1.13 1.10 1.06
0.12 0.13 0.17 0.15 0.18 0.25
0.62 0.65 0.66 0.77 0.76 0.75
16.68 15.77 12.31 14.05 11.62 8.18
1.33 1.32 1.30 1.20 1.21 1.22
6.38 5.38 3.21 4.75 3.23 1.42
(0.047) (0.052) (0.053) (0.060) (0.069) (0.103)
(0.24) (0.21) (0.19) (0.22) (0.20) (0.25)
(0.025) (0.031) (0.034) (0.039) (0.046) (0.046)
(0.14) (0.16) (0.15) (0.23) (0.23) (0.35)
(3.40) (3.70) (2.43) (3.55) (2.88) (1.58)
(0.19) (0.18) (0.18) (0.11) (0.11) (0.12)
(2.69) (2.26) (1.31) (2.65) (1.80) (0.60)
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Table 2 Multivariable linear regression models of the error (D) and the scatter (SD) of error for stereological measurements with scanning (V S) and reconstruction (V R) voxel size as dependent variables Parameter
Y
BV/TV
D
Tb.N (mm1)
SD D
Tb.Th (mm)
SD D
Tb.Sp (mm)
SD D
BS/BV (mm2/mm3)
MIL1/MIL3
Euler/Vol (mm3)
SD D
SD D SD D SD
r2
X 0.001 VS (Am) 0.024 + 0.028 D1 + 0.029 D2 + 0.009 D3 + 0.009 D4 0.032 D5 0.008 D6 0.024 D7 0.001 VS (Am) + 0.0002 VR (Am) 0.013 0.003 VR (Am) + 0.051 + 0.084 D1 + 0.139 D2 + 0.000 D3 0.062 D4 0.153 D5 + 0.010 D6 0.010 D7 0.002 VS (Am) + 0.012 0.001 VS (Am) + 0.0003 VR (Am) 0.029 + 0.015 D1 + 0.009 D2 + 0.014 D3 + 0.007 D4 0.016 D5 0.010 D6 0.012 D7 0.00022 VR (Am) 0.003 0.002 VR (Am) 0.040 0.082 D1 0.108 D2 0.028 D3 0.022 D4 + 0.044 D5 0.011 D6 + 0.165 D7 0.002 VS (Am) + 0.001 VR (Am) 0.064 0.070 VS (Am) 0.025 VR (Am) + 1.468 + 0.028 D1 + 0.662 D2 + 0.616 D3 + 0.092 D4 + 0.549 D5 + 0.071 D6 1.061 D7 0.018 VS (Am) + 0.092 0.001 VR (Am) + 0.046 + 0.149 D1 + 0.022 D2 + 0.114 D3 0.147 D4 0.162 D5 0.074 D6 + 0.108 D7 0.002 VR (Am) 0.034 0.047 VS (Am) 0.167 + 0.303 D1 + 1.280 D2 1.512 D3 4.699 D4 + 0.922 D5 + 2.200 D6 + 0.852 D7 NS
P VS
VR
0.75
b0.001
NS (0.78)
0.99 0.72
b0.001 NS (0.06)
b0.01 b0.001
0.84 0.95
b0.02 b0.001
NS (0.08) b0.001
0.87 0.57
b0.007 NS (0.83)
NS (0.16) b0.001
0.97 0.89
b0.01 b0.001
b0.022 b0.001
0.84 0.55
b0.011 NS (0.84)
NS (0.29) b0.015
0.74 0.76
NS (0.16) b0.001
b0.03 NS (0.09)
NS
NS (0.55)
NS (0.19)
Seven dummy variables (D1 to D7) were included in the regression equations of the error to account for the variability within eight specimens used. (NS: not significant).
The scatter (SD) of DBV/TV and DTb.Sp within a scanning/reconstruction group increased with both scanning and reconstruction voxel sizes (Table 2). In the cases of DTb.N and DBS/BV, the scatter significantly increased with increasing scanning voxel sizes but not with reconstruction voxel sizes. In contrast, the scatter of DTb.Th increased with reconstruction voxel sizes only. The scatter in anisotropy (DMIL1/MIL3) increased with reconstruction but not with scanning voxel sizes. No correlation between the scatter and voxel sizes was observed for D(Euler/Vol). All the 21/21 results (taken as the gold standard) of BV/TV, Tb.N, Tb.Th, Tb.Sp, and BS/BV were positively correlated with the 21/50 (r 2 = 0.94–0.99; P b 0.001), 50/50 (r 2 = 0.77–0.93; P b 0.03), 21/110 (r 2 = 0.85– 0.96; P b 0.002), and 50/110 (r 2 = 0.75–0.92; P b 0.006) results (Table 3, Figs. 1–5). BV/TV, Tb.Th, Tb.Sp, and BS/BV results from the 21/21 case were correlated with those from the 110/110 case (r 2 = 0.63– 0.91; P b 0.02) as well. However, Tb.N from the 21/21 case was not correlated with the 110/110 measurements ( P N 0.13). For the case of MIL1/MIL3, 21/21 results were related with the 21/50 (r 2 = 0.89; P b 0.001), 50/ 50 (r 2 = 0.60; P b 0.03), and 110/110 (r 2 = 0.55; P b 0.04) but not with the 21/110 and 50/110 cases (Fig. 6). Euler/Vol from the 21/21 case was not correlated with any combination of voxel sizes (Fig. 7) but the 110/110 case (r 2 = 0.56; P b 0.04).
Discussion Both scanning and reconstruction voxel sizes had significant effects on micro-CT-based 3D stereological measurements of human cancellous bone microstructure (Table 1). As expected, the largest differences were observed between the 21/21 case and the 110/110 case for all parameters. For most parameters, the error (D) relative to the 21/21 case and the scatter (SD) of error in each group of voxel size combination were correlated with either scanning or reconstruction voxel size or both (Table 2). Increases in these errors gradually deteriorated the strength of relationships between the 21/21 case calculations and those from other voxel size cases (Table 3, Figs. 1–7). BV/TV, Tb.Th, and Tb.Sp increased with voxel size (up to 58.9%, 102.9%, and 22.1%, respectively), whereas Tb.N, Euler/Vol, and BS/BV decreased with voxel size (up to 23.5%, 77.7%, and 50.9%, respectively) (Table 1). MIL1/MIL3 did not considerably change with voxel size (less than 10%). Since all parameters based on the plate model are derived from two independent parameters (Eqs. (1)–(4)), the magnitude of each parameter inherently varied depending on the variations of P P and P L. For instance, Tb.Th (PP/PL; Eq. (1)) had the largest difference at 110/110 Am relative to the 21/21 case (102.9%) due to P P and P L reaching their largest and lowest values, respectively. Although not independent measurements, each parameter has a physical meaning and therefore can be
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Table 3 Regressions between the 21/21 case and other scanning/reconstruction voxel size cases for each parameter Y 21/21 BV/TV
Tb.N (mm1)
Tb.Th (mm)
Tb.Sp (mm)
BS/BV (mm2/mm3)
MIL1/MIL3
Euler /Vol (mm3)
Eqn r2 P slope P int Eqn r2 P reg Eqn r2 P slope P int Eqn r2 P reg Eqn r2 P slope P int Eqn r2 P reg Eqn r2 P slope P int Eqn r2 P reg Eqn r2 P slope P int Eqn r2 P reg Eqn r2 P slope P int Eqn r2 P reg Eqn r2 P slope P int Eqn r2 P reg
21/50
50/50
21/110
50/110
110/110
=0.884X + 0.018 0.97 b0.001 N0.16 Y =0.981X 0.96 b0.001 =1.111X 0.065 0.94 b0.001 N0.65 Y =1.062X 0.93 b0.001 =0.808X + 0.017 0.99 b0.001 b0.02
=0.752X + 0.043 0.92 b0.001 b0.05
=0.6X + 0.052 0.78 b0.005 N0.1 Y =0.834X 0.65 b0.012 =1.071X + 0.214 0.75 b0.006 N0.45 Y =1.261X 0.73 b0.005 =0.519X + 0.03 0.92 b0.001 b0.05
=0.384X + 0.068 0.71 b0.009 N0.06 Y =0.603X 0.45 b0.05 NS
=0.992X + 0.015 0.89 b0.001 N0.93 Y =1.003X 0.89 b0.001 NS
=0.777X + 0.012 0.77 b0.005 N0.76 Y =0.829X 0.76 b0.004 =1.137X 0.016 0.81 b0.003 N0.9 Y =1.124X 0.81 b0.002 =0.701X + 0.006 0.93 b0.001 N0.6 Y =0.737X 0.93 b0.001 =0.857X + 0.051 0.84 b0.002 N0.64 Y =0.93X 0.83 b0.002 =1.349X + 0.077 0.93 b0.001 N0.97 Y =1.355X 0.93 b0.001 =0.812X + 0.275 0.60 b0.03 N0.45 Y =1.021X 0.56 b0.025 NS
N0.5
N0.35
=0.91X + 0.024 0.96 b0.001 N0.65 Y =0.945X 0.96 b0.001 =0.915X + 2.256 0.99 b0.001 b0.005
=1.035X + 0.214 0.85 b0.002 N0.3 Y =1.218X 0.83 b0.002 =0.619X + 0.031 0.94 b0.001 b0.03
=0.585X + 0.169 0.90 b0.001 b0.04
NS
=0.56X + 0.189 0.82 b0.002 N0.06 Y =0.789X 0.68 b0.009 =1.125X + 3.608 0.91 b0.001 N0.085 Y =1.419X 0.84 b0.001 NS
N0.05
N0.3
NS
NS
N0.5
N0.4
=0.939X + 3.491 0.96 b0.001 b0.025
N0.13
=0.519X 0.007 0.92 b0.001 N0.6 Y =0.494X 0.92 b0.001 =0.324X + 0.372 0.63 b0.02 b0.005
=2.046X 0.056 0.91 b0.001 N0.98 Y =2.039X 0.91 b0.001 =1.148X 0.069 0.55 b0.04 N0.89 Y =1.091X 0.55 b0.03 =3.362X + 1.605 0.56 b0.035 N0.4 Y =4.341X 0.50 b0.04
If the slope of the regression was significant but the intercept was not, another linear model was applied to the data forcing the fit through the origin. For the latter, P values associated with the regression are reported.
considered a special model of P P and P L. Because the stereological parameters derived from these two were not regressed together in the same analysis, their interdependence was not a problem. The narrow range (up to 10%) of variation in MIL1/MIL3 indicated that P L changed at similar rates in the major and minor axis of trabecular microstructure. Consistent with the present results, Kothari et al. [15] found a strong resolution dependency of the traditional
histomorphological parameters when they compared 40-Am images with images degraded (100- to 1000-Am resolutions) from the 40-Am image. They observed thickening of the vertical trabeculae and loss of horizontal trabeculae in degraded images. Consistent with these observations, loss of trabeculae or merging of adjacent branches of trabeculae due to thickening (Fig. 8) could decrease the trabecular number, whereas the bone volume fraction would continue to increase with increasing voxel size within the same volume of
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Fig. 1. Estimation of BV/TV calculated from 21/21 Am images by BV/TV calculated from other combinations of scanning/reconstruction voxel size. All relationships are significant (Table 3).
Fig. 3. Estimation of Tb.Th calculated from 21/21 Am images by Tb.Th calculated from other combinations of scanning/reconstruction voxel size. All relationships are significant (Table 3).
interest. Other morphological parameters would change accordingly (Eqs. (1)–(4)) except the topologically measured Euler/Vol (Eq. (5)). Odgaard and Gundersen [22] pointed out that reconstruction of 3D images has a potential noise problem that might cause assignment of a wrong phase membership of bone and nonbone voxels. If a set of bone marrow voxels is surrounded by bone voxels in threedimensional space, it is disconnected from other marrow. This results in violation of the basic assumption of one connected bone marrow in Eq. (5). Separation of phases might occur due to loss or merging of trabeculae as the scanning and reconstruction voxel size increase. If loss or merging of the trabecular network happens in an unpredictable way, the values of connectivity would not be confidently calculated with changes of voxel sizes using the current computational concept. These unpredictable changes in connectivity might be one reason that no significant relationship existed in connectivity density between the values of the 21/21 case and those of other cases.
Significant correlations between the 21/21 Am and 110/110 Am measurements of anisotropy and connectivity density were exceptional to the general trend between voxel size and other stereological measurements (Table 3, Figs. 6, 7). Compared with other morphological parameters, the magnitudes of MIL1/MIL3 and Euler/Vol in each specimen changed more randomly with each voxel size case (Figs. 6 and 7). This randomness immediately deteriorated the relationships between the 21/21 case and others. On the other hand, when voxel size increased, the magnitudes of MIL1/ MIL3 and Euler/Vol gradually converged to a range of values, and simultaneously, the scattering of magnitudes decreased (Table 1). As a result, a correlation with the best possible case could be recovered in the 110/110 case for MIL1/MIL3 and Euler/Vol. Nonetheless, the nonsystematic change in correlations between voxel size combinations indicates that these two parameters should be interpreted cautiously for poorresolution images.
Fig. 2. Estimation of Tb.N calculated from 21/21 Am images by Tb.N calculated from other combinations of scanning/reconstruction voxel size. All relationships except for the 110/110 Am case are significant (Table 3).
Fig. 4. Estimation of Tb.Sp calculated from 21/21 Am images by Tb.Sp calculated from other combinations of scanning/reconstruction voxel size. All relationships are significant (Table 3).
D.-G. Kim et al. / Bone 35 (2004) 1375–1382
Fig. 5. Estimation of BS/BV calculated from 21/21 Am images by BS/BV calculated from other combinations of scanning/reconstruction voxel size. All relationships are significant (Table 3).
The present study made use of the plate model to estimate the stereological parameters, which have been widely used [2,9,23,24]. However, a number of studies indicated that the rod-type trabeculae should not be ignored in order to describe the 3D bone structure more accurately [3,19,21,22]. Ding and Hvid [4] observed a drastic structural change in trabeculae from plate to rod type for old age (N79 years). They indicated that trabecular thickness indirectly computed by the 2D plate model formula alone was underestimated (32%) when compared to the 3D model including both plates and rods. Laib and Ruegsegger [17] were able to directly measure the trabecular number using ridge number density (RND) modified by the distance transformation. Because the values of stereological parameters computed by these studies were strongly related with those based on the plate model (r 2 = 0.87 to 0.95), the plate-model assumption can be considered an acceptable limitation for the purpose of the current study.
Fig. 6. Estimation of MIL1/MIL3 calculated from 21/21 Am images by MIL1 / MIL3 calculated from other combinations of scanning/reconstruction voxel size. The relationships are significant except for the 21/110 and 50/110 Am cases (Table 3).
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Fig. 7. Estimation of connectivity (Euler/Vol) calculated from 21/21 Am images by Euler/Vol calculated from other combinations of scanning/ reconstruction voxel size. Significance was detected for the 110/110 Am case only (Table 3).
The present study has strength in that it provided a systematic understanding of the separate effects of scanning and reconstruction voxel sizes. Our use of multiple specimens provided insight into the variability of error within specimens (Table 2). Although it could be deduced from the dummy-variable equations, the regressions of the standard deviation of error against voxel size more explicitly indicated that the increased scatter of error could be explained as a function of voxel size. Our results show that the level of variation of morphological parameters with voxel size can be more than 100% for the range of voxel sizes examined. This finding reinforces that caution should be exercised while evaluating pathological conditions based on morphological parameters that are determined at different resolutions. Our results also show that reconstructing high-resolution data (small scanning voxel size) at a larger voxel size can have a smaller effect than decreasing scanning resolution (Table 3). Thus, studies of image resolution should include actual scans at the resolution of interest rather than simply coarsening high-resolution reconstructed images as is customarily done.
Fig. 8. A typical comparison of matched micro-CT slice at selected combination of voxel sizes. Loss or merging of adjacent branches of trabeculae resulted from thickening.
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The effect of microcomputed tomography scanning and reconstruction voxel size on the accuracy of stereological measurements in human cancellous bone Do-Gyoon Kima,*, Gregory T. Christophersona, X. Neil Donga, David P. Fyhrieb, Yener N. Yenia b
a Bone and Joint Center, Henry Ford Hospital, Detroit, MI, 48202, USA Orthopaedic Research Laboratories, UC Davis, School of Medicine, Sacramento, CA 95817, USA
Received 16 March 2004; revised 14 September 2004; accepted 20 September 2004
Abstract Stereological parameters have been used as an approximation for the architecture of trabecular bone. Structural indices such as bone volume fraction (BV/TV), trabecular number (Tb.N), trabecular thickness (Tb.Th), trabecular separation (Tb.Sp), bone surface-to-volume ratio (BS/BV), degree of anisotropy (MIL1/MIL3), and connectivity density (Euler/Vol) have been widely studied to investigate pathological conditions in bone. Due to its high resolution and nondestructiveness, microcomputed tomography (micro-CT) has been utilized to take precise three-dimensional (3D) images of trabecular microstructures. However, spatial limitations for applying micro-CT-based analyses to large specimens, such as whole vertebral bodies, require using larger scanning and reconstruction voxel sizes. In this study, combinations of three different scanning and reconstruction voxel size were used to represent best possible voxel size (21 Am; best in our scanner for the specimen size used) relative to other voxel sizes used in this study, commonly used intermediate voxel sizes (50 Am), and those applicable to scans of whole human vertebral bodies (110 Am) in order to examine the effect of scanning and reconstruction voxel size on stereological measures for human cancellous bone. The error in stereological parameters calculated using combinations of large voxel sizes compared to the gold standard (best possible case) ranged from 0.1% to 102%. The signed magnitude of the error in other cases relative to the gold standard was a function of either scanning or reconstruction voxel size or both (r 2 = 0.55–0.95). For most of the structural indices, the results from analysis of images with larger voxel sizes were correlated with those from the gold standard (r 2 = 0.55–0.99) except for Tb.N at 110/110 Am, MIL1/MIL3 at larger than 110 Am reconstruction voxel size, and Euler/Vol at any combination of voxel sizes. Overall, it was observed that resampling a high resolution image at lower resolutions (corresponding to increasing reconstruction voxel size in this study) had different effects on the calculated parameters than scanning at the same low resolution (corresponding to increasing scanning voxel size in this study). Our results show that investigations of image resolution should include actual scans at the resolution of interest rather than simply coarsening of high-resolution images as is customarily done. D 2004 Elsevier Inc. All rights reserved. Keywords: Stereology; Trabecular bone; Micro-CT; Resolution; Calibration
Introduction Trabecular architecture has been considered as an important determinant of osteoporosis and other pathological conditions in bone. The microstructure of the * Corresponding author. Bone and Joint Center, Henry Ford Hospital, 2799 West Grand Boulevard, Detroit, MI, 48202. Fax: +1 313 916 8064. E-mail address: [email protected] (D.-G. Kim). 8756-3282/$ - see front matter D 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.bone.2004.09.007
trabecular network is often measured using measures such as bone volume fraction (BV/TV), trabecular number (Tb.N), trabecular thickness (Tb.Th), separation (Tb.Sp), and surface-to-volume ratio of bone (BS/BV). These parameters have been examined for different anatomical sites and physiological conditions such as aging [4,24,27], bone disease [5,11,20], and recently, remodeling caused by chemical treatments [3,18,26]. A related parameter, the degree of anisotropy of trabecular
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bone, was observed to highly correlate with increase in fracture risk of the hip [2]. Trabecular connectivity based on topological measurements has also been useful for predicting changes in mechanical properties with bone loss [14,28]. Microcomputed tomography (micro-CT) was introduced to explore the three-dimensional (3D) architecture of bone [8]. Due to its high resolution (as small as a few micrometers), micro-CT can obtain precise 3D images at the microlevel of trabecular structure. For in vitro microCT studies, different voxel sizes have been used, ranging from 8 to 120 Am [10,16,28,30,31]. Although high resolution is achievable using micro-CT, scanning large specimens such as a whole vertebral body may require use of a spatial resolution corresponding to a voxel size greater than 100 Am [6,13]. Because 100 Am is in the order of typical trabecular thickness, partial volume effects will cause errors when computing the morphological parameters for trabecular bone [15]. Ding and Hvid [4] found that the magnitudes of stereological parameters were strongly dependent on voxel size for voxel size larger than 100 Am. Ruegsegger et al. [25] indicated that the morphological values could be corrected to a specific resolution up to 200-Am voxel size based on the monotonic dependence of morphological parameters on image resolution. However, the effect of scanning and reconstruction voxel size has not been considered separately in previous studies [12,17,30]. The separate and combined effects of these voxel sizes on morphological variables need to be evaluated. Scanning voxel size is a measure of the quality of the raw data images and determines the best level of detail that can be resolved in the image. The raw data can be appropriately reconstructed at any reconstruction voxel size that is larger than (or equal to) the scanning voxel size. Reconstruction voxel size is the actual voxel size chosen for the 3D image reconstruction. It is ordinarily assumed that a larger reconstruction voxel size will decrease the accuracy of the image. Nevertheless, mainly to avoid large computational costs, images may have to be coarsened using greater reconstruction voxel sizes than scanning voxel size in some applications such as microCT-based, large-scale finite element models [10,16,29] (note that choosing a reconstruction voxel size smaller than the scanning voxel size cannot improve the quality of the reconstructed image because the raw data do not support the smaller reconstructed voxels). The goal of this study was to evaluate the effect of scanning and reconstruction voxel size on the micro-CTbased 3D stereological analyses of human trabecular tissue. To this end, six different combinations of scanning/reconstruction voxel size were considered on the basis of three different voxel sizes that represent high-resolution scans (best possible voxel size in our scanner for the specimen size used), commonly used intermediate voxel sizes and those applicable to large specimens, such as a whole vertebral
body. The variations of morphological parameters were quantified at different combinations of voxel sizes. The errors and relationships of each voxel size case relative to the best possible voxel size case were estimated.
Methods Eight cylindrical cancellous bone specimens (10-mm length, 8-mm diameter) were cored in the inferosuperior direction from L2–L4 vertebrae of a 63-year-old male and the metaphyseal tibia of a 52-year-old male. Each specimen was scanned with micro-CT at a scanning voxel size of 21, 50, and 110 Am voxel size. Images were reconstructed based on Feldkamp’s convolutionback projection algorithm [7] using reconstruction voxel sizes that were the same or larger than the scanning voxel size (21, 50, and 110 Am). This procedure resulted in scanning/reconstruction voxel size combinations of 21/ 21, 21/50, 21/110, 50/50, 50/110, and 110/110 Am for each specimen. Twenty-one micrometers is about the best possible scanning voxel size that can be achieved in our scanner for specimens of this size, and 50 Am is a commonly employed intermediate value. The choice of 110 Am was based on our experience with micro-CT scanning of whole human vertebral bodies and represents a bbest-caseQ scenario for the future in vivo scanning of the spine. The 21/21 case was used as the gold standard for determining the level of inaccuracy in the coarser scanning/reconstruction combinations. After reconstruction, bone and nonbone voxels were segmented using a heuristic segmentation algorithm developed specifically for bone tissue with highly nonhomogeneous CT density distributions that have large overlap in density values between bone and bone marrow [32]. A special purpose code for stereology was utilized to compute the morphological parameters directly from microCT images that had been thresholded following reconstruction. The volume of interest was determined as the largest sphere that could be fit inside the cylindrical specimen volume. Seven morphological parameters were examined: Bone volume fraction (BV/TV), trabecular number (Tb.N), trabecular thickness (Tb.Th), trabecular separation (Tb.Sp), bone surface-to-volume ratio (BS/BV), degree of anisotropy (MIL1/MIL3), and connectivity density (Euler/Vol). All computations were based on formulation of the plate model for trabecular bone structure [8,9,23]. BV/TV (mm3/mm3) was given by P P that was obtained dividing the number of bone voxels by the total number of voxels in the volume of interest. Tb.N (mm1) was expressed by P L, the number of intersections between bone and nonbone voxels per total length of test lines. Tb.Th, Tb.Sp, and BS/BV are calculated based on P P and P L: Tb:Th ðmmÞ ¼ P P =P L
ð1Þ
D.-G. Kim et al. / Bone 35 (2004) 1375–1382
Tb:Sp ðmmÞ ¼ ð1 P P Þ=P L
ð2Þ
BS=BV mm2 =mm3 ¼ 2P L =P P
ð3Þ
Mean intercept length (MIL) represents the mean distance between intersections of bone and nonbone components for each test line at a different angle of measurement. Therefore, the degree of anisotropy was found as the ratio between the maximum (MIL1) and minimum (MIL3) mean intercept length, which were measured at the major (1) and minor (3) axis of a trabecular network [2]. It can be denoted in terms of P L as MIL1=MIL3 ¼ P L3 =P L1
ð4Þ
where P L3 and P L1 are the trabecular number at the minor and major axis, respectively. Connectivity of the trabecular network was obtained based on topological measurements depending on Euler-Poincare number [22] independent from the plate model assumption. If there is only one connected bone phase and only one connected bone marrow phase, connectivity density can be defined as Connectivity density mm3 ¼ ð1 #EulerÞ=Volc #Euler=Vol
ð5Þ
where (1 Euler number) represents the maximum number of branches that can be removed without breaking the structure into separate parts. Frequently, the connectivity is also defined as the number of bone marrow loops in the trabecular network [14]. To effectively quantify the number of connections, the Euler number was multiplied by negative one and normalized by the volume of interest. This allowed for reasonable comparison between specimens with different volume. The present stereology code assumes the trabeculae at the boundary of the volume of interest as connected to the adjacent trabecular network so that the so-called bedgeQ effects [22] related to connectivity computation were corrected. Two-way repeated measures ANOVA was used for analyzing the effect of scanning/reconstruction combinations with each specimen as the subject and scanning/reconstruction (21/21, 21/50. . .) as repeated factors using Sigma Stat (SPSS Inc., Chicago). The error (D) in a parameter of concern from a large voxel size case was calculated as the signed difference in this parameter between the large voxel size case
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and the 21/21 case assuming the 21/21 case error-free. Multivariable linear regression was performed to examine the relationship between the error (D) and scanning/reconstruction voxel size (21, 50, and 110 Am). Dummy variables [1] were used in multivariable regressions in order to account for within-specimen variability. Another multivariable linear regression was performed to examine the relationship between scanning/reconstruction voxel size and the standard deviation (SD) of the error in a parameter of concern. If either scanning or reconstruction voxel size was significant only, a simple linear regression was performed. The relationships between parameters calculated from the 21/21 Am images and those from other combinations of scanning/reconstruction voxel size were examined using regression analysis (Microsoft Excel).
Results All stereological parameters varied with different combinations of scanning and reconstruction voxel sizes (Table 1). A two-way repeated-measures ANOVA suggested that scanning voxel size is the major factor causing the difference of values in BV/TV and in Euler/Vol ( P b 0.001 for both parameters), whereas reconstruction voxel size did not have a significant effect ( P = 0.72 and P = 0.30, respectively). For Tb.N and Tb.Sp, the differences were significantly related with reconstruction voxel size ( P b 0.001) but not with scanning voxel size ( P = 0.22 and 0.97, respectively). The differences in Tb.Th and in BS/BV were significantly related with both the scanning and reconstruction voxel size ( P b 0.001). However, MIL1/MIL3 did not depend on either scanning or reconstruction voxel size ( P N 0.20). Most regressions between scanning/reconstruction voxel size and the error (D) (Table 2) agreed with the results of RMANOVA. Scanning voxel size but not reconstruction voxel size was a major source of the errors in BV/TV and Euler/Vol. The error in Tb.N and Tb.Sp significantly correlated with increasing reconstruction voxel sizes but not with scanning voxel size. In Tb.Th and BS/BV, the error was significantly related to both scanning and reconstruction voxel sizes. While the differences in MIL1/MIL3 were not significant between voxel size combinations, the error in MIL1/MIL3 was correlated with reconstruction voxel size.
Table 1 Average values (standard deviation) of stereological parameters for the scanning/reconstruction voxel size groups Voxel size
BV/TV
Tb.N (mm1)
Tb.Th (mm)
Tb.Sp (mm)
BS/BV (mm2/mm3)
MIL1/MIL3
Euler/Vol (mm3)
21/21 21/50 50/50 21/110 50/110 110/110
0.174 0.176 0.209 0.174 0.203 0.276
1.39 1.31 1.23 1.13 1.10 1.06
0.12 0.13 0.17 0.15 0.18 0.25
0.62 0.65 0.66 0.77 0.76 0.75
16.68 15.77 12.31 14.05 11.62 8.18
1.33 1.32 1.30 1.20 1.21 1.22
6.38 5.38 3.21 4.75 3.23 1.42
(0.047) (0.052) (0.053) (0.060) (0.069) (0.103)
(0.24) (0.21) (0.19) (0.22) (0.20) (0.25)
(0.025) (0.031) (0.034) (0.039) (0.046) (0.046)
(0.14) (0.16) (0.15) (0.23) (0.23) (0.35)
(3.40) (3.70) (2.43) (3.55) (2.88) (1.58)
(0.19) (0.18) (0.18) (0.11) (0.11) (0.12)
(2.69) (2.26) (1.31) (2.65) (1.80) (0.60)
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Table 2 Multivariable linear regression models of the error (D) and the scatter (SD) of error for stereological measurements with scanning (V S) and reconstruction (V R) voxel size as dependent variables Parameter
Y
BV/TV
D
Tb.N (mm1)
SD D
Tb.Th (mm)
SD D
Tb.Sp (mm)
SD D
BS/BV (mm2/mm3)
MIL1/MIL3
Euler/Vol (mm3)
SD D
SD D SD D SD
r2
X 0.001 VS (Am) 0.024 + 0.028 D1 + 0.029 D2 + 0.009 D3 + 0.009 D4 0.032 D5 0.008 D6 0.024 D7 0.001 VS (Am) + 0.0002 VR (Am) 0.013 0.003 VR (Am) + 0.051 + 0.084 D1 + 0.139 D2 + 0.000 D3 0.062 D4 0.153 D5 + 0.010 D6 0.010 D7 0.002 VS (Am) + 0.012 0.001 VS (Am) + 0.0003 VR (Am) 0.029 + 0.015 D1 + 0.009 D2 + 0.014 D3 + 0.007 D4 0.016 D5 0.010 D6 0.012 D7 0.00022 VR (Am) 0.003 0.002 VR (Am) 0.040 0.082 D1 0.108 D2 0.028 D3 0.022 D4 + 0.044 D5 0.011 D6 + 0.165 D7 0.002 VS (Am) + 0.001 VR (Am) 0.064 0.070 VS (Am) 0.025 VR (Am) + 1.468 + 0.028 D1 + 0.662 D2 + 0.616 D3 + 0.092 D4 + 0.549 D5 + 0.071 D6 1.061 D7 0.018 VS (Am) + 0.092 0.001 VR (Am) + 0.046 + 0.149 D1 + 0.022 D2 + 0.114 D3 0.147 D4 0.162 D5 0.074 D6 + 0.108 D7 0.002 VR (Am) 0.034 0.047 VS (Am) 0.167 + 0.303 D1 + 1.280 D2 1.512 D3 4.699 D4 + 0.922 D5 + 2.200 D6 + 0.852 D7 NS
P VS
VR
0.75
b0.001
NS (0.78)
0.99 0.72
b0.001 NS (0.06)
b0.01 b0.001
0.84 0.95
b0.02 b0.001
NS (0.08) b0.001
0.87 0.57
b0.007 NS (0.83)
NS (0.16) b0.001
0.97 0.89
b0.01 b0.001
b0.022 b0.001
0.84 0.55
b0.011 NS (0.84)
NS (0.29) b0.015
0.74 0.76
NS (0.16) b0.001
b0.03 NS (0.09)
NS
NS (0.55)
NS (0.19)
Seven dummy variables (D1 to D7) were included in the regression equations of the error to account for the variability within eight specimens used. (NS: not significant).
The scatter (SD) of DBV/TV and DTb.Sp within a scanning/reconstruction group increased with both scanning and reconstruction voxel sizes (Table 2). In the cases of DTb.N and DBS/BV, the scatter significantly increased with increasing scanning voxel sizes but not with reconstruction voxel sizes. In contrast, the scatter of DTb.Th increased with reconstruction voxel sizes only. The scatter in anisotropy (DMIL1/MIL3) increased with reconstruction but not with scanning voxel sizes. No correlation between the scatter and voxel sizes was observed for D(Euler/Vol). All the 21/21 results (taken as the gold standard) of BV/TV, Tb.N, Tb.Th, Tb.Sp, and BS/BV were positively correlated with the 21/50 (r 2 = 0.94–0.99; P b 0.001), 50/50 (r 2 = 0.77–0.93; P b 0.03), 21/110 (r 2 = 0.85– 0.96; P b 0.002), and 50/110 (r 2 = 0.75–0.92; P b 0.006) results (Table 3, Figs. 1–5). BV/TV, Tb.Th, Tb.Sp, and BS/BV results from the 21/21 case were correlated with those from the 110/110 case (r 2 = 0.63– 0.91; P b 0.02) as well. However, Tb.N from the 21/21 case was not correlated with the 110/110 measurements ( P N 0.13). For the case of MIL1/MIL3, 21/21 results were related with the 21/50 (r 2 = 0.89; P b 0.001), 50/ 50 (r 2 = 0.60; P b 0.03), and 110/110 (r 2 = 0.55; P b 0.04) but not with the 21/110 and 50/110 cases (Fig. 6). Euler/Vol from the 21/21 case was not correlated with any combination of voxel sizes (Fig. 7) but the 110/110 case (r 2 = 0.56; P b 0.04).
Discussion Both scanning and reconstruction voxel sizes had significant effects on micro-CT-based 3D stereological measurements of human cancellous bone microstructure (Table 1). As expected, the largest differences were observed between the 21/21 case and the 110/110 case for all parameters. For most parameters, the error (D) relative to the 21/21 case and the scatter (SD) of error in each group of voxel size combination were correlated with either scanning or reconstruction voxel size or both (Table 2). Increases in these errors gradually deteriorated the strength of relationships between the 21/21 case calculations and those from other voxel size cases (Table 3, Figs. 1–7). BV/TV, Tb.Th, and Tb.Sp increased with voxel size (up to 58.9%, 102.9%, and 22.1%, respectively), whereas Tb.N, Euler/Vol, and BS/BV decreased with voxel size (up to 23.5%, 77.7%, and 50.9%, respectively) (Table 1). MIL1/MIL3 did not considerably change with voxel size (less than 10%). Since all parameters based on the plate model are derived from two independent parameters (Eqs. (1)–(4)), the magnitude of each parameter inherently varied depending on the variations of P P and P L. For instance, Tb.Th (PP/PL; Eq. (1)) had the largest difference at 110/110 Am relative to the 21/21 case (102.9%) due to P P and P L reaching their largest and lowest values, respectively. Although not independent measurements, each parameter has a physical meaning and therefore can be
D.-G. Kim et al. / Bone 35 (2004) 1375–1382
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Table 3 Regressions between the 21/21 case and other scanning/reconstruction voxel size cases for each parameter Y 21/21 BV/TV
Tb.N (mm1)
Tb.Th (mm)
Tb.Sp (mm)
BS/BV (mm2/mm3)
MIL1/MIL3
Euler /Vol (mm3)
Eqn r2 P slope P int Eqn r2 P reg Eqn r2 P slope P int Eqn r2 P reg Eqn r2 P slope P int Eqn r2 P reg Eqn r2 P slope P int Eqn r2 P reg Eqn r2 P slope P int Eqn r2 P reg Eqn r2 P slope P int Eqn r2 P reg Eqn r2 P slope P int Eqn r2 P reg
21/50
50/50
21/110
50/110
110/110
=0.884X + 0.018 0.97 b0.001 N0.16 Y =0.981X 0.96 b0.001 =1.111X 0.065 0.94 b0.001 N0.65 Y =1.062X 0.93 b0.001 =0.808X + 0.017 0.99 b0.001 b0.02
=0.752X + 0.043 0.92 b0.001 b0.05
=0.6X + 0.052 0.78 b0.005 N0.1 Y =0.834X 0.65 b0.012 =1.071X + 0.214 0.75 b0.006 N0.45 Y =1.261X 0.73 b0.005 =0.519X + 0.03 0.92 b0.001 b0.05
=0.384X + 0.068 0.71 b0.009 N0.06 Y =0.603X 0.45 b0.05 NS
=0.992X + 0.015 0.89 b0.001 N0.93 Y =1.003X 0.89 b0.001 NS
=0.777X + 0.012 0.77 b0.005 N0.76 Y =0.829X 0.76 b0.004 =1.137X 0.016 0.81 b0.003 N0.9 Y =1.124X 0.81 b0.002 =0.701X + 0.006 0.93 b0.001 N0.6 Y =0.737X 0.93 b0.001 =0.857X + 0.051 0.84 b0.002 N0.64 Y =0.93X 0.83 b0.002 =1.349X + 0.077 0.93 b0.001 N0.97 Y =1.355X 0.93 b0.001 =0.812X + 0.275 0.60 b0.03 N0.45 Y =1.021X 0.56 b0.025 NS
N0.5
N0.35
=0.91X + 0.024 0.96 b0.001 N0.65 Y =0.945X 0.96 b0.001 =0.915X + 2.256 0.99 b0.001 b0.005
=1.035X + 0.214 0.85 b0.002 N0.3 Y =1.218X 0.83 b0.002 =0.619X + 0.031 0.94 b0.001 b0.03
=0.585X + 0.169 0.90 b0.001 b0.04
NS
=0.56X + 0.189 0.82 b0.002 N0.06 Y =0.789X 0.68 b0.009 =1.125X + 3.608 0.91 b0.001 N0.085 Y =1.419X 0.84 b0.001 NS
N0.05
N0.3
NS
NS
N0.5
N0.4
=0.939X + 3.491 0.96 b0.001 b0.025
N0.13
=0.519X 0.007 0.92 b0.001 N0.6 Y =0.494X 0.92 b0.001 =0.324X + 0.372 0.63 b0.02 b0.005
=2.046X 0.056 0.91 b0.001 N0.98 Y =2.039X 0.91 b0.001 =1.148X 0.069 0.55 b0.04 N0.89 Y =1.091X 0.55 b0.03 =3.362X + 1.605 0.56 b0.035 N0.4 Y =4.341X 0.50 b0.04
If the slope of the regression was significant but the intercept was not, another linear model was applied to the data forcing the fit through the origin. For the latter, P values associated with the regression are reported.
considered a special model of P P and P L. Because the stereological parameters derived from these two were not regressed together in the same analysis, their interdependence was not a problem. The narrow range (up to 10%) of variation in MIL1/MIL3 indicated that P L changed at similar rates in the major and minor axis of trabecular microstructure. Consistent with the present results, Kothari et al. [15] found a strong resolution dependency of the traditional
histomorphological parameters when they compared 40-Am images with images degraded (100- to 1000-Am resolutions) from the 40-Am image. They observed thickening of the vertical trabeculae and loss of horizontal trabeculae in degraded images. Consistent with these observations, loss of trabeculae or merging of adjacent branches of trabeculae due to thickening (Fig. 8) could decrease the trabecular number, whereas the bone volume fraction would continue to increase with increasing voxel size within the same volume of
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Fig. 1. Estimation of BV/TV calculated from 21/21 Am images by BV/TV calculated from other combinations of scanning/reconstruction voxel size. All relationships are significant (Table 3).
Fig. 3. Estimation of Tb.Th calculated from 21/21 Am images by Tb.Th calculated from other combinations of scanning/reconstruction voxel size. All relationships are significant (Table 3).
interest. Other morphological parameters would change accordingly (Eqs. (1)–(4)) except the topologically measured Euler/Vol (Eq. (5)). Odgaard and Gundersen [22] pointed out that reconstruction of 3D images has a potential noise problem that might cause assignment of a wrong phase membership of bone and nonbone voxels. If a set of bone marrow voxels is surrounded by bone voxels in threedimensional space, it is disconnected from other marrow. This results in violation of the basic assumption of one connected bone marrow in Eq. (5). Separation of phases might occur due to loss or merging of trabeculae as the scanning and reconstruction voxel size increase. If loss or merging of the trabecular network happens in an unpredictable way, the values of connectivity would not be confidently calculated with changes of voxel sizes using the current computational concept. These unpredictable changes in connectivity might be one reason that no significant relationship existed in connectivity density between the values of the 21/21 case and those of other cases.
Significant correlations between the 21/21 Am and 110/110 Am measurements of anisotropy and connectivity density were exceptional to the general trend between voxel size and other stereological measurements (Table 3, Figs. 6, 7). Compared with other morphological parameters, the magnitudes of MIL1/MIL3 and Euler/Vol in each specimen changed more randomly with each voxel size case (Figs. 6 and 7). This randomness immediately deteriorated the relationships between the 21/21 case and others. On the other hand, when voxel size increased, the magnitudes of MIL1/ MIL3 and Euler/Vol gradually converged to a range of values, and simultaneously, the scattering of magnitudes decreased (Table 1). As a result, a correlation with the best possible case could be recovered in the 110/110 case for MIL1/MIL3 and Euler/Vol. Nonetheless, the nonsystematic change in correlations between voxel size combinations indicates that these two parameters should be interpreted cautiously for poorresolution images.
Fig. 2. Estimation of Tb.N calculated from 21/21 Am images by Tb.N calculated from other combinations of scanning/reconstruction voxel size. All relationships except for the 110/110 Am case are significant (Table 3).
Fig. 4. Estimation of Tb.Sp calculated from 21/21 Am images by Tb.Sp calculated from other combinations of scanning/reconstruction voxel size. All relationships are significant (Table 3).
D.-G. Kim et al. / Bone 35 (2004) 1375–1382
Fig. 5. Estimation of BS/BV calculated from 21/21 Am images by BS/BV calculated from other combinations of scanning/reconstruction voxel size. All relationships are significant (Table 3).
The present study made use of the plate model to estimate the stereological parameters, which have been widely used [2,9,23,24]. However, a number of studies indicated that the rod-type trabeculae should not be ignored in order to describe the 3D bone structure more accurately [3,19,21,22]. Ding and Hvid [4] observed a drastic structural change in trabeculae from plate to rod type for old age (N79 years). They indicated that trabecular thickness indirectly computed by the 2D plate model formula alone was underestimated (32%) when compared to the 3D model including both plates and rods. Laib and Ruegsegger [17] were able to directly measure the trabecular number using ridge number density (RND) modified by the distance transformation. Because the values of stereological parameters computed by these studies were strongly related with those based on the plate model (r 2 = 0.87 to 0.95), the plate-model assumption can be considered an acceptable limitation for the purpose of the current study.
Fig. 6. Estimation of MIL1/MIL3 calculated from 21/21 Am images by MIL1 / MIL3 calculated from other combinations of scanning/reconstruction voxel size. The relationships are significant except for the 21/110 and 50/110 Am cases (Table 3).
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Fig. 7. Estimation of connectivity (Euler/Vol) calculated from 21/21 Am images by Euler/Vol calculated from other combinations of scanning/ reconstruction voxel size. Significance was detected for the 110/110 Am case only (Table 3).
The present study has strength in that it provided a systematic understanding of the separate effects of scanning and reconstruction voxel sizes. Our use of multiple specimens provided insight into the variability of error within specimens (Table 2). Although it could be deduced from the dummy-variable equations, the regressions of the standard deviation of error against voxel size more explicitly indicated that the increased scatter of error could be explained as a function of voxel size. Our results show that the level of variation of morphological parameters with voxel size can be more than 100% for the range of voxel sizes examined. This finding reinforces that caution should be exercised while evaluating pathological conditions based on morphological parameters that are determined at different resolutions. Our results also show that reconstructing high-resolution data (small scanning voxel size) at a larger voxel size can have a smaller effect than decreasing scanning resolution (Table 3). Thus, studies of image resolution should include actual scans at the resolution of interest rather than simply coarsening high-resolution reconstructed images as is customarily done.
Fig. 8. A typical comparison of matched micro-CT slice at selected combination of voxel sizes. Loss or merging of adjacent branches of trabeculae resulted from thickening.
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