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Measurement of intraspecimen variations in vertebral cancellous bone architecture
Bone Vol. 25, No. 2 August 1999:245–250
Measurement of Intraspecimen Variations in Vertebral Cancellous Bone Architecture M. KOTHARI,1 T. M. KEAVENY,2,3 J. C. LIN,1 D. C. NEWITT,1 and S. MAJUMDAR1 1
Magnetic Resonance Science Center, Department of Radiology, University of California, San Francisco, California, USA Orthopedic Biomechanics Laboratory, Department of Mechanical Engineering, University of California, Berkeley, California, USA 3 Department of Orthopedic Surgery, University of California, San Francisco, California, USA 2
and the number of horizontal and vertical trabeculae.11 Measures that quantify this network geometry could therefore play an important role not only in characterizing degenerative changes, but also in the evaluation of therapeutic measures for combating osteoporosis. The architecture of vertebral cancellous bone has been described as a three-dimensional (3D) lattice composed of vertical, nearly parallel supporting plates and columns interconnected by thinner horizontal struts.8 It has been postulated that with osteoporosis, the vertical plates become increasingly rodlike, and the number of horizontal trabeculae decrease. Common two- and three-dimensional measures of bone architecture such as mean trabecular thickness (Tb.Th.), mean trabecular spacing (Tb.Sp.), mean trabecular number (Tb.N.), and mean intercept length do not differentiate between the length and thickness of trabeculae and hence cannot provide specific information regarding the geometry of individual trabecula.5,9,10 Techniques for the estimation of structure using two-dimensional images are confounded by the fact that all stereologic measures are restricted to the image plane and the resulting measures depend on the choice of the sectioning plane.9 Measurements cannot be made out of the image plane, and hence it is difficult to differentiate between rods and plates. A three-dimensional approach based on volumetric images would allow for analysis of different two-dimensional sectioning planes as well as out-of-plane measurements, and would therefore address the limitations associated with the two-dimensional approach. Wakamatsu and Sissons used two-dimensional images of cancellous bone from the iliac crest to obtain thickness distributions of individual trabecula, and showed that the coefficient of variation was between 45% and 51%.18 A study by Aaron also used two-dimensional images from the iliac crest and determined the coefficient of variation to be between 36% and 52%.1 Both studies noted that the thickness distribution was right skewed. Hildebrand and Ru¨eggseger were the first to develop a threedimensional technique that could estimate the distribution of trabecular thickness within a cancellous bone specimen using a model-independent approach.5 They used a concept of a “volume-based thickness,” in which the diameter of the largest circle (or sphere in three dimensions) in any cross section represented its thickness, and reported trabecular thickness distributions in one sample from the femur and another from the vertebral body. This approach, however, did not attempt to determine the thickness variations among those trabeculae oriented along a particular direction. Such information may be important from a biomechanical standpoint, since it is well established that the
A three-dimensional technique was developed for the quantification of the number and cross-sectional geometry of individual trabeculae oriented along a given direction. As an example application, the number of vertical and horizontal trabeculae and their respective cross-sectional geometry were determined for a set of six vertebral cancellous bone specimens (L3–L4 female vertebral bodies; age range 39 – 63 years). Three-dimensional optical images at a spatial resolution of 20 m were obtained using an automated serial milling technique. The thickness distributions were generally right skewed. The mean true thickness for both the vertically and horizontally oriented trabeculae showed a strong relationship with volume fraction (vertical: r2 ⴝ 0.86; p < 0.05; horizontal: r2 ⴝ 0.80; p < 0.05), and mean trabecular thickness (Tb.Th.) (vertical: r2 ⴝ 0.81; p < 0.05; horizontal: r2 ⴝ 0.72; p < 0.05). The horizontal trabeculae were greater in number and were thinner than the vertical trabeculae. The coefficient of variation of the intraspecimen vertical trabecular thicknesses ranged from 25% to 42%, and showed a weak, albeit insignificant, positive correlation with volume fraction (r2 ⴝ 0.46). The findings demonstrated substantial intraspecimen variations exist in trabecular thickness that are not related to volume fraction. Further studies are recommended to determine the potential role of such intraspecimen variations in architecture on biomechanical properties. (Bone 25:245–250; 1999) © 1999 by Elsevier Science Inc. All rights reserved. Key Words: Bone; Cancellous; Morphometry; Osteoporosis; Anisotropy. Introduction With aging, there is a change in the density,4 architecture,3,14 and tissue properties16 of cancellous bone, all of which degrade mechanical properties. Architecture has so far been quantified by a variety of mean parameters that ignore the effect of intraspecimen variations in trabecular geometry. Finite element models, however, have suggested that the intraspecimen variations in trabecular thickness may strongly affect biomechanical properties.17,19 It has also been hypothesized that the efficacy of therapeutic measures in combating vertebral fractures may depend on existing bone architecture as defined by the thickness Address for correspondence and reprints: Manish Kothari, Ph.D., Synarc Inc., 455 Market St. Suite 1850, San Francisco, CA 94105. © 1999 by Elsevier Science Inc. All rights reserved.
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morphological characteristics of vertical vs. horizontal trabeculae are different for human spine cancellous bone.8 The primary aim of this study was to develop and describe a technique that determines measures of the thickness of individual trabecula oriented along a particular direction, and does so without any repeated measurements of the same trabecula. To this end, we: (a) describe a technique for three-dimensional analysis of cancellous bone geometry that quantifies trabecular cross-sectional geometry as a function of direction; (b) use the technique to obtain estimates of intraspecimen trabecular thickness variations; and (c) compare the mean trabecular thickness determined using traditional morphometry with the estimates obtained using the new technique. Algorithm An algorithm was developed to evaluate the geometry of individual trabeculae oriented along a specific direction. The software was written using commercial software (Interactive Display Language; Research Systems Inc., Boulder, CO) on an Ultra SPARC Workstation (Sun Microsystems, Mountainview, CA). It was validated using simulated images that incorporated realistic variability (Figure 1A), and then applied to six human vertebral specimens. The algorithm consisted of five sequential steps. Geometrical Description The geometry of individual trabecula is complex: the thickness varies over the length of the trabecula and the trabecula itself maybe curved. For this study, the geometry is represented by a characteristic length L and two cross-sectional dimensions T1 and T2 (L ⬎ T1 ⬎ T2), where T2 is defined as the thickness of the individual trabecula (Figure 2). A rodlike geometry would have L ⬎⬎ T1 ⬵ T2, whereas a platelike structure would have L ⬵ T1 ⬎⬎ T2. The length of the trabecula is defined as the longest straight line that can pass through a bone segment along the direction of interest. Choice of Search Direction A set of lines is passed through the volumetric image (a simulated image is shown in Figure 1A with thick vertical and thin horizontal trabeculae). The direction of the lines, i.e., the search direction, can be arbitrarily chosen. Vertebral bone consists primarily of long, thick vertical trabeculae and small thin horizontal trabeculae.8,12 The bone has been shown to be transversely isotropic, with its preferred orientation along the inferiorsuperior direction.15 Therefore, the choice of search direction in the vertebral cancellous bone would be taken along the inferiorsuperior direction if the focus were the long, thick vertical trabeculae. Alternatively, the grid of lines could be rotated at different angles within the horizontal plane to obtain estimates of trabecular thickness for trabeculae oriented within the horizontal plane. Finally, the choice of search directions in this study was chosen based on a priori knowledge of vertebral cancellous bone structure. For cancellous bone from regions such as the femur, a fabric tensor-based approach could be first used to determine the principal orientations, and the set of grid lines could then be passed along those directions (although this was not attempted in this study). Estimation of Length L The set of lines is then passed through the volumetric image (shown passing in the vertical direction in Figure 1B). Whenever a bone intercept is encountered, the length of that segment is
Figure 1. (A) A test image with thick rods aligned close to the vertical, and thin horizontal struts. (B) A set of grid lines is passed along a given direction through the 3D image. This set of lines is shown passing in the vertical direction. The white regions are the cross sections of the vertical trabeculae seen in A. (C) The actual length of the trabeculae is estimated; (D) cross-sectional measures are made at five points along the length of the trabeculae; (E) the cross-sectional measures are analogous to inscribing and circumscribing a circle in the cross section; (F) with a fine grid spacing, a trabeculae may be measured a number of times (shown twice). A proximity check is done to ensure that a given trabeculae is not counted more than once. If it is, the results corresponding to the shorter lengths are discarded.
termed as the initial length estimate. By using the mid-point of the segment, and rotating the grid line within a range of angles of the line direction, one can obtain a length estimate at orientations that are slightly oblique to the preferred orientation (Figure 1C). A range of ⫾ 8 degrees was chosen here based on the variation that was seen in the preferred direction of vertebral bone.6 The direction corresponding to the maximum length estimate for the various orientations was deemed to be the true orientation of that particular trabecula. Estimation of Cross-Sectional Dimensions Estimates of the cross-section dimensions T1 and T2 (T1 ⬎ T2) are made in directions perpendicular to the true trabecular orientation as determined in the previous step (Figure 1D). They are measured at five equispaced points along the line representing the length of the trabecula (Figure 1D). The cross-sectional dimension T1 is defined as the diameter of the circle circumscribing the cross section (Figure 1E). It is measured by determining the maximum distance between any two points on the cross section. Thickness T2 is defined as the diameter of the largest circle that can be inscribed within the cross section
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true measure of trabecular dimension. The other sets corresponding to the same trabecula are removed and are not considered in the evaluation (the proximity check algorithm is described using pseudocode in Appendix I). Example Application
Figure 2. The geometry of rods and plates can be represented as length measures, L, and two cross-sectional measures T1 and T2 (L ⬎ T1 ⬎ T2), where T2 represents the thickness. Transitions from a platelike structure to a rodlike structure would result in a corresponding decrease in T1. Note that the technique does not require the trabeculae to be ideal rods or plates.
(Figure 1E), similar to the approach suggested by Hildebrand and Ru¨egsegger.5 It is determined by applying a circular structural element of different diameters, and eroding the image (Interactive Display Language; Research Systems Inc.). The thickness T2 is the smallest diameter for which the erosion process deletes the cross section. The algorithm to approximate the diameter of the inscribed and circumscribed circles was validated by manually inscribing and circumscribing circles on the cross section of interest. The cross-sectional dimensions T1 and T2 were taken to be the median of the five measured values. Use of a median value ensured that occasional bone callus or connection point for other trabeculae did not skew the results. In this way, each trabecula can thus be characterized three-dimensionally using one length measure (L) and two cross-sectional dimension parameters (T1 and T2) (Figure 2). As stated earlier, T2 corresponds to the trabecula thickness. The set of lines will intersect trabeculae oriented parallel as well as perpendicular to the grid lines. These intercepts will therefore be comprised of thickness measures of trabeculae oriented perpendicularly to the line direction, as well as length measures of trabeculae oriented along the line direction. By definition, an oriented trabecula is one for which L ⬎ T1 ⬎ T2. If T1 ⬎ L, then the trabecula is not considered to be oriented along the search direction and the dimensions are not stored. The abovementioned steps are carried out for all bone intercepts encountered by the grid lines. For each oriented intercept, the true orientation and location of the intercept is stored, in addition to its length L and cross-sectional measures T1 and T2. At this stage, no attempt is made to determine a unique measure for an individual trabecula. Depending on the grid spacing, some of these measures may correspond to the same trabecula.
The software was first validated on simulated volumetric images generated using a two-step procedure (typical image shown in Figure 1A). First of all, a volumetric image was created with thick, vertically oriented trabeculae (cross-sectional dimension range: 140 –500 m) and thinner trabeculae trabeculae (crosssectional dimensions range: 80 – 400 m) oriented within the horizontal plane. The trabeculae were then perturbed so that their alignments were no longer exactly vertical, or exactly within the horizontal plane. The validation process indicated that: (a) the cross-sectional dimensions of each trabeculae could be predicted correctly; (b) the proximity check ensured that the trabeculae were not being counted twice; and (c) any trabecula whose cross section was limited by the edge of the specimen was not included in the measurement. A serial milling technique was used to create high-resolution 3D images of six 12-mm3 specimens of vertebral cancellous bone (L3–L4 vertebral bodies, six female cadavers; age range 39 – 63 years; specimens cut from different regions within the body). The donors had no history of metastatic bone disease or any other condition that might have influenced the vertebral bone structure. In this technique, the defatted cancellous bone cubes were first stained black in silver nitrate solution, and then exposed to ultraviolet light. The samples were embedded in white pigmented methyl methacrylate (MMA), and a computer numerically controlled (CNC) milling machine (SpectreLite 0200; Light Machines Corp., Manchester, NH) was programmed to serially mill off 20 m from the surface of each specimen. The resulting surface was imaged using a low-magnification image acquisition system (camera: CCD model XC-75, Sony, Japan; low-magnification lens: Pan-Cinor 1:3.8 f ⫽ 17, Som-Berthoit, France), and the entire process is repeated until the entire
Choice of Grid Spacing and Proximity Checking The grid spacing should be kept as fine as possible to ensure that no trabeculae are overlooked. If the grid spacing is chosen to be less than typical trabecular cross-sectional dimensions, it is probable that two or more grid lines may pass through the same trabeculae (Figure 1F). If a trabecula is measured more than once, the set of dimensions corresponding to the greatest length is considered as the
Figure 3. A 6-mm cube of vertebral cancellous bone (L4 vertebral body; 59 year-old female). Large vertical and small horizontal trabeculae can be seen in the vertebral body.
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Figure 4. Thickness distributions for vertical (left) and horizontal (right) trabeculae (L4 vertebral body; 63-year-old female; volume fraction: 0.04). b) Thickness distributions for vertical (left) and horizontal (right) trabeculae (L4 vertebral body; 49-year-old female; volume fraction: 0.15).
specimen has been milled and imaged. Each two-dimensional (2D) image was acquired at resolution of 20 ⫻ 20 ⫻ 20 m.2 The 2D images were stacked and combined into a 3D array using commercial software (Interactive Display Language; Research Systems Inc.) on an Ultra SPARC workstation (Sun Microsystems). Preprocessing of the image was limited to bone marrow segmentation using a dual-reference thresholding algorithm.7 The images were grey scale inverted so that the bone appears white on a black background (Figure 3). The technique was applied to a 5-mm cube extracted from the center of each image. A grid of lines with a line spacing of 20 m was passed along the inferior-superior direction to determine the geometry of vertically oriented trabeculae. For the horizontally oriented trabeculae, the grid of lines was passed along the horizontal plane, and rotated at increments of 15 degrees. Coupled with the ⫾8-degree search for the orientation of each trabecula, the 10-degree increments ensured that no trabeculae were overlooked. If the cross section of any trabecula lay along the edge of the cube, it was not included in results, as the thickness would then be underpredicted. Trabeculae whose length (as defined earlier) were truncated by the end of the cube were included in the study. An in-house code was developed to estimate 3D measures of bone architecture (Interactive Display Language; Research Systems Inc.). The code was validated using a similar program,6,13 and was run on an Ultra SPARC workstation (Sun Microsystems). Traditional morphometric measures of bone structure such as cancellous bone volume fraction (BV/TV), Tb.N., Tb.Th., and Tb.Sp. were computed using the volumetric images based on 3D extensions of the parallel plate model.
Results Thickness distributions for individual trabecula were obtained for each bone specimen (Figure 4, Tables 1 and 2). The thickness distributions revealed that specimens with higher volume fractions had a number of platelike trabeculae, i.e., where T1 ⬵ T2, while specimens with lower volume fraction showed mainly rodlike trabeculae, i.e., where T1 ⬎ T2 (Figure 4a, b). Although the true orientation of each bone segment was assessed within a cone of ⫾8 degrees, all vertically oriented trabeculae lay within ⫾4 degrees of the vertical direction. For the five specimens measured, removal of trabeculae that abut the edge resulted in one to four and two to nine trabeculae not being included in the calculation of vertical and horizontal trabeculae thickness respectively. Table 1. Intraspecimen variation in vertical trabeculae for six human female vertebral cancellous bone specimens Vertical Specimen 1 2 3 3 5 6
Age (years)
Volume fraction
Number
Mean T1 (SD) (mm)
Mean T2 (SD) (mm)
63 59 61 58 46 39
0.04 0.07 0.10 0.15 0.18 0.20
5 9 11 13 9 12
0.38 (0.16) 0.52 (0.22) 0.74 (0.30) 0.92 (0.34) 0.88 (0.30) 0.90 (0.52)
0.22 (0.06) 0.20 (0.06) 0.22 (0.10) 0.24 (0.08) 0.32 (0.16) 0.30 (0.14)
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Table 2. Intraspecimen variation in horizontal trabeculae for six human female vertebral cancellous bone specimens Horizontal Specimen 1 2 3 4 5 6
Age (years)
Volume fraction
63 59 61 55 46 39
0.04 0.07 0.10 0.15 0.18 0.20
Number
Mean T1 (SD) (mm)
Mean T2 (SD) (mm)
25 26 43 34 32 37
0.26 (0.14) 0.38 (0.20) 0.56 (0.28) 0.54 (0.24) 0.52 (0.40) 0.58 (0.44)
0.14 (0.04) 0.16 (0.06) 0.16 (0.04) 0.16 (0.06) 0.18 (0.08) 0.20 (0.06)
The mean true thickness t (i.e., mean T2 for each specimen) for both the vertically and horizontally oriented trabeculae showed a strong positive correlation with the volume fraction (vertical: r2 ⫽ 0.86, p ⬍ 0.05; horizontal: r2 ⫽ 0.80, p ⬍ 0.05). The horizontal trabeculae were greater in number and were thinner than the vertical trabeculae (Tables 1 and 2). The coefficient of variation of the intraspecimen vertical trabecular thicknesses ranged from 25% to 42%. This coefficient of variation shows a weak, albeit insignificant, positive correlation with volume fraction (r2 ⫽ 0.46, p ⬎ 0.05). The mean true trabecular thickness t along the preferred orientation showed a strong positive correlation with Tb.Th. (vertical: r2 ⫽ 0.81, p ⬍ 0.05; horizontal: r2 ⫽ 0.72, p ⬍ 0.05). The computed Tb.Th., however, was systematically lower than the mean true vertical trabecular thickness, with the error varying between 33% and 48% (mean 39%, SD 5%). Discussion A technique was introduced to quantify trabecular thickness as a function of direction. This thickness computation method was not based on the parallel plate model. Repeat measurements of a given trabecula were accounted for by incorporating a proximity checking mechanism. The technique was first validated using artificial images with known geometry; accurate estimates of the cross-sectional dimensions could be determined for all trabeculae oriented along a given direction. It was then applied to a set of vertebral cancellous bone specimens, quantifying intratrabecular variations in thickness for both horizontal and vertical trabeculae. The relationship between mean true thickness t and volume fraction was consistent with trends seen in other studies.8 The coefficient of variations of 25% to 42% seen in this study were somewhat smaller, but consistent with the results obtained by 2D studies, which found the coefficient of variation of thickness to lie between 36% and 52% for cancellous bone from the iliac crest.1,18 The thickness distributions displayed a similar skewed distribution as obtained by both the 2D studies.1,18 The results suggest that the volume fraction may be the predominant factor controlling the mean structural characteristics of vertebral cancellous bone.8 However, there was a large intraspecimen variability in trabecular thickness that showed only a weak relationship with volume fraction. Both 2D and 3D finite element models suggest that a intraspecimen variability of this order may reduce the elastic modulus by 20% to 40%.17,19 Mean cross-sectional thickness determined here for both the vertical and horizontal trabeculae was greater than that seen in the only other study using a 3D approach. That study, however, presented results for only one specimen, and it is possible that this difference may just be due to the specimen. Further specimens need to be evaluated using both techniques before comparisons can be drawn. It must be emphasized that the mean true trabecular thickness
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t computed here was different than the 3D estimates of traditional trabecular thickness Tb.Th. made using the parallel plate model.10 The definition of Tb.Th. (and a measure such as directional intercept length) does not differentiate between length and thickness, and is statistically equivalent to the mean of all intercept length measures. Given the larger number of thinner horizontal trabeculae as compared with the number of thicker vertically oriented trabeculae, it is not surprising that Tb.Th. falls between the mean true thickness computed for the horizontal and vertical trabeculae. The mean true trabecular thickness t, on the other hand, was based on true thickness estimates of the oriented trabeculae. The strong relationship found between t and Tb.Th. in the specimens studied here suggests that Tb.Th. could possibly serve as a surrogate measure for true thickness in regression analyses trying to elucidate the impact of bone structure on mechanical properties. The skewed nature of the distributions, however, reflects the fact that mean parameters may not adequately describe the variation in trabecular thickness, and other models that account for the skewness have to be considered. The architecture of vertebral cancellous bone has been described as a 3D lattice composed of vertical nearly parallel supporting plates and columns connecting thin horizontal struts.8 It has been postulated that, with osteoporosis, the vertical plates become increasingly rodlike, and the number of horizontal trabeculae decrease. This postulate could be validated by determining whether there is indeed a decrease in cross-sectional dimension T1 over time; as T1 becomes smaller and closer to thickness T2, the bone is becoming increasingly rodlike. Furthermore, the very definition of what is a plate or a rod in context of cancellous bone is unclear. It therefore may be better to focus on the changes in thickness itself. The focus of this study was to introduce a technique for the evaluation of thickness distributions of the trabeculae and not on the length measures. For this study, the definition of the length of an oriented trabecula was chosen to be the longest straight line along the direction of interest (Figure 5). Different measures of length have been proposed, although no definition has yet been accepted as the norm.9 It is likely that edge effects and highly curved trabeculae may cause the length to be underestimated. We have tried to minimize the affect of this inherent variability in cancellous architecture by adopting a 3D approach, incorporating some of the variability in trabecular orientation, and using a fine grid spacing combined with a proximity check that maximizes the determined length of a trabecula. Adopting a 3D skeletonizing approach to length measures may result in a more accurate assessment of the trabecular length, although this remains to be shown. It is also recognized that trabecular geometry is highly irregular, with the thickness varying from site to site along a trabecula. The measures computed for each trabecula are therefore characteristic measures of that trabecula. The results presented here serve to exemplify the usefulness of the technique. The vertebral specimens used in the study were all representative of cancellous bone from peri- or postmenopausal women. The relationship between true trabecular thickness and Tb.Th., as well as the overall geometry of trabeculae, may show different trends if bone from different ages and regions of the vertebra were sampled, and if a larger sample size were to be used. The model presented here opens the question as to how to evaluate the geometry of individual trabecula, as well as the question of adequate mechanistic definitions of length and thickness. Appendix The pseudocode given below describes the logic used to remove repeat measurements of the same trabeculae. Steps I–IV in the
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References
Figure 5. Magnified image of vertebral bone showing vertical rodlike and platelike structures. The estimated length measures are superimposed on the image. The algorithm was capable of estimating the length of the vertical struts.
algorithm described earlier have found N oriented bone segments. Some of these will be repeat measurements of the same trabeculae. for j ⫽ 1,N use dimensions and starting location to mark out where the trabecula lies for k ⫽ 2,N use dimensions and starting location to mark out where the trabecula lies does trabecula k lies within the region of trabecula j If yes and Lj ⬍ Lk, then delete set j If yes and Lj ⬎ Lk, then delete set k Update k and iterate Update j and iterate
Acknowledgments: The authors thank Brian Canfield, Sean Haddock, and Tony Chen for their help with the serial milling of the bone specimens. Support for this work came from National Institutes of Health grants R01-AG-13612 and AR-43784, and National Science Foundation grant BES-9625030, as well as from the University of California Academic Senate.
1. Aaron, J., Makins, N., and Sagreiya, K. The microanatomy of bone loss in normal aging men and women. Clin Orthopaed Relat Res 215:260 –271; 1987. 2. Beck, J. D., Canfield, B. L., Haddock, S. M., Chen, T. J. H., Kothari, M., and Keaveny, T. M. Three-dimensional imaging of cancellous bone using the computer numerically controlled milling technique. Bone 21:281–287; 1997. 3. Goldstein, S. A., Goulet, R., and McCubbrey, D. Measurement and significance of three-dimensional architecture to the mechanical integrity of cancellous bone. Calcif Tissue Int 53(Suppl. 1):127–133; 1993. 4. Hedlund, L. and Gallagher, J. The effect of age and menopause on bone mineral density of the proximal femur. J Bone Miner Res 4:639 – 642; 1989. 5. Hildebrand, T. and Ru¨egsegger, P. A new method for the model-independent assessment of thickness in three-dimensional images. J Microscopy 185:67–75; 1995. 6. Kothari, M., Keaveny, T., Lin, J., Newitt, D., Genant, H., and Majumdar, S. Impact of spatial resolution on the prediction of trabecular architecture parameters. Bone 22:437– 442; 1997. 7. Majumdar, S., Newitt, D., Jergas, M., et al. Evaluation of technical factors affecting the quantitation of cancellous bone structure using magnetic resonance imaging. Bone 17:417– 430; 1995. 8. Mosekilde, L. Age related changes in vertebral bone architecture-assessed by a new method. Bone 9:247–250; 1988. 9. Odgaard, A. Three-dimensional methods for quantification of cancellous bone architecture. Bone 20:315–328; 1997. 10. Parfitt, A., Drezner, M., Glorieux, F., et al. Bone histomorphometry: standardization of nomenclature, symbols, and units. J Bone Miner Res 2:595– 610; 1987. 11. Silva, M. and Gibson, L. Modeling the mechanical behavior of vertebral cancellous bone: effects of age related change in microstructure. Bone 21:191– 199; 1997. 12. Silva, M. J. and Gibson, L. J. Loss of trabeculae reduces strength more than trabecular thinning in a finite element model of vertebral cancellous bone. Transactions of the 43rd Meeting of the Orthopaedic Research Society. San Francisco, CA 21:63; 1996. 13. Simmons, C. and Hipp, J. A. Three dimensional mean intercept length ellipsoid software, T3M. Orthopedics Biomechanics Laboratory, Harvard Medical School, Boston, MA. 14. Snyder, B. D., Piazza, S., Edwards, W. T., Hayes, W. C., and Schaffler, W. C. Role of trabecular morphology in the etiology Of age-related vertebral Fractures. Calcified Tissue Int 53(Suppl.):14 –22; 1993. 15. Turner, C. H., Cowin, S. C., Rho, J. Y., Ashman, R. B., and Rice, J. C. The fabric dependence of the orthotropic elastic constants of cancellous bone. J Biomech 23:549 –561; 1990. 16. Very, J., Gilbert, R., Guilhot, B., Debout, M., and Alexandre, C. Effect of aging on the amide group of bone matrix, measured by FTIR spectroscopy, in adult subjects deceased as a result of violent death. Calcif Tiss Int 60:271–275; 1997. 17. Wachtel, E. and Keaveny, T. M. Intra-specimen variation in trabecular thickness decrease trabecular modulus. Eds. Transactions of the 43rd meeting of the Orthopaedic Research Society, San Francisco, CA; 21:61; 1996. 18. Wakamatsu, E. and Sissons, H. A. The cancellous bone of the iliac crest. Calcif Tiss Res 4:147–161; 1969. 19. Yeh, O. C. and Keaveny, T. M. Biomechanical effects of intraspecimen variations in trabecular architecture: a three-dimensional finite element study. Bone. 25:223–228; 1999.
Date Received: June 16, 1998 Date Revised: April 28, 1999 Date Accepted: April 28, 1999
Measurement of Intraspecimen Variations in Vertebral Cancellous Bone Architecture M. KOTHARI,1 T. M. KEAVENY,2,3 J. C. LIN,1 D. C. NEWITT,1 and S. MAJUMDAR1 1
Magnetic Resonance Science Center, Department of Radiology, University of California, San Francisco, California, USA Orthopedic Biomechanics Laboratory, Department of Mechanical Engineering, University of California, Berkeley, California, USA 3 Department of Orthopedic Surgery, University of California, San Francisco, California, USA 2
and the number of horizontal and vertical trabeculae.11 Measures that quantify this network geometry could therefore play an important role not only in characterizing degenerative changes, but also in the evaluation of therapeutic measures for combating osteoporosis. The architecture of vertebral cancellous bone has been described as a three-dimensional (3D) lattice composed of vertical, nearly parallel supporting plates and columns interconnected by thinner horizontal struts.8 It has been postulated that with osteoporosis, the vertical plates become increasingly rodlike, and the number of horizontal trabeculae decrease. Common two- and three-dimensional measures of bone architecture such as mean trabecular thickness (Tb.Th.), mean trabecular spacing (Tb.Sp.), mean trabecular number (Tb.N.), and mean intercept length do not differentiate between the length and thickness of trabeculae and hence cannot provide specific information regarding the geometry of individual trabecula.5,9,10 Techniques for the estimation of structure using two-dimensional images are confounded by the fact that all stereologic measures are restricted to the image plane and the resulting measures depend on the choice of the sectioning plane.9 Measurements cannot be made out of the image plane, and hence it is difficult to differentiate between rods and plates. A three-dimensional approach based on volumetric images would allow for analysis of different two-dimensional sectioning planes as well as out-of-plane measurements, and would therefore address the limitations associated with the two-dimensional approach. Wakamatsu and Sissons used two-dimensional images of cancellous bone from the iliac crest to obtain thickness distributions of individual trabecula, and showed that the coefficient of variation was between 45% and 51%.18 A study by Aaron also used two-dimensional images from the iliac crest and determined the coefficient of variation to be between 36% and 52%.1 Both studies noted that the thickness distribution was right skewed. Hildebrand and Ru¨eggseger were the first to develop a threedimensional technique that could estimate the distribution of trabecular thickness within a cancellous bone specimen using a model-independent approach.5 They used a concept of a “volume-based thickness,” in which the diameter of the largest circle (or sphere in three dimensions) in any cross section represented its thickness, and reported trabecular thickness distributions in one sample from the femur and another from the vertebral body. This approach, however, did not attempt to determine the thickness variations among those trabeculae oriented along a particular direction. Such information may be important from a biomechanical standpoint, since it is well established that the
A three-dimensional technique was developed for the quantification of the number and cross-sectional geometry of individual trabeculae oriented along a given direction. As an example application, the number of vertical and horizontal trabeculae and their respective cross-sectional geometry were determined for a set of six vertebral cancellous bone specimens (L3–L4 female vertebral bodies; age range 39 – 63 years). Three-dimensional optical images at a spatial resolution of 20 m were obtained using an automated serial milling technique. The thickness distributions were generally right skewed. The mean true thickness for both the vertically and horizontally oriented trabeculae showed a strong relationship with volume fraction (vertical: r2 ⴝ 0.86; p < 0.05; horizontal: r2 ⴝ 0.80; p < 0.05), and mean trabecular thickness (Tb.Th.) (vertical: r2 ⴝ 0.81; p < 0.05; horizontal: r2 ⴝ 0.72; p < 0.05). The horizontal trabeculae were greater in number and were thinner than the vertical trabeculae. The coefficient of variation of the intraspecimen vertical trabecular thicknesses ranged from 25% to 42%, and showed a weak, albeit insignificant, positive correlation with volume fraction (r2 ⴝ 0.46). The findings demonstrated substantial intraspecimen variations exist in trabecular thickness that are not related to volume fraction. Further studies are recommended to determine the potential role of such intraspecimen variations in architecture on biomechanical properties. (Bone 25:245–250; 1999) © 1999 by Elsevier Science Inc. All rights reserved. Key Words: Bone; Cancellous; Morphometry; Osteoporosis; Anisotropy. Introduction With aging, there is a change in the density,4 architecture,3,14 and tissue properties16 of cancellous bone, all of which degrade mechanical properties. Architecture has so far been quantified by a variety of mean parameters that ignore the effect of intraspecimen variations in trabecular geometry. Finite element models, however, have suggested that the intraspecimen variations in trabecular thickness may strongly affect biomechanical properties.17,19 It has also been hypothesized that the efficacy of therapeutic measures in combating vertebral fractures may depend on existing bone architecture as defined by the thickness Address for correspondence and reprints: Manish Kothari, Ph.D., Synarc Inc., 455 Market St. Suite 1850, San Francisco, CA 94105. © 1999 by Elsevier Science Inc. All rights reserved.
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morphological characteristics of vertical vs. horizontal trabeculae are different for human spine cancellous bone.8 The primary aim of this study was to develop and describe a technique that determines measures of the thickness of individual trabecula oriented along a particular direction, and does so without any repeated measurements of the same trabecula. To this end, we: (a) describe a technique for three-dimensional analysis of cancellous bone geometry that quantifies trabecular cross-sectional geometry as a function of direction; (b) use the technique to obtain estimates of intraspecimen trabecular thickness variations; and (c) compare the mean trabecular thickness determined using traditional morphometry with the estimates obtained using the new technique. Algorithm An algorithm was developed to evaluate the geometry of individual trabeculae oriented along a specific direction. The software was written using commercial software (Interactive Display Language; Research Systems Inc., Boulder, CO) on an Ultra SPARC Workstation (Sun Microsystems, Mountainview, CA). It was validated using simulated images that incorporated realistic variability (Figure 1A), and then applied to six human vertebral specimens. The algorithm consisted of five sequential steps. Geometrical Description The geometry of individual trabecula is complex: the thickness varies over the length of the trabecula and the trabecula itself maybe curved. For this study, the geometry is represented by a characteristic length L and two cross-sectional dimensions T1 and T2 (L ⬎ T1 ⬎ T2), where T2 is defined as the thickness of the individual trabecula (Figure 2). A rodlike geometry would have L ⬎⬎ T1 ⬵ T2, whereas a platelike structure would have L ⬵ T1 ⬎⬎ T2. The length of the trabecula is defined as the longest straight line that can pass through a bone segment along the direction of interest. Choice of Search Direction A set of lines is passed through the volumetric image (a simulated image is shown in Figure 1A with thick vertical and thin horizontal trabeculae). The direction of the lines, i.e., the search direction, can be arbitrarily chosen. Vertebral bone consists primarily of long, thick vertical trabeculae and small thin horizontal trabeculae.8,12 The bone has been shown to be transversely isotropic, with its preferred orientation along the inferiorsuperior direction.15 Therefore, the choice of search direction in the vertebral cancellous bone would be taken along the inferiorsuperior direction if the focus were the long, thick vertical trabeculae. Alternatively, the grid of lines could be rotated at different angles within the horizontal plane to obtain estimates of trabecular thickness for trabeculae oriented within the horizontal plane. Finally, the choice of search directions in this study was chosen based on a priori knowledge of vertebral cancellous bone structure. For cancellous bone from regions such as the femur, a fabric tensor-based approach could be first used to determine the principal orientations, and the set of grid lines could then be passed along those directions (although this was not attempted in this study). Estimation of Length L The set of lines is then passed through the volumetric image (shown passing in the vertical direction in Figure 1B). Whenever a bone intercept is encountered, the length of that segment is
Figure 1. (A) A test image with thick rods aligned close to the vertical, and thin horizontal struts. (B) A set of grid lines is passed along a given direction through the 3D image. This set of lines is shown passing in the vertical direction. The white regions are the cross sections of the vertical trabeculae seen in A. (C) The actual length of the trabeculae is estimated; (D) cross-sectional measures are made at five points along the length of the trabeculae; (E) the cross-sectional measures are analogous to inscribing and circumscribing a circle in the cross section; (F) with a fine grid spacing, a trabeculae may be measured a number of times (shown twice). A proximity check is done to ensure that a given trabeculae is not counted more than once. If it is, the results corresponding to the shorter lengths are discarded.
termed as the initial length estimate. By using the mid-point of the segment, and rotating the grid line within a range of angles of the line direction, one can obtain a length estimate at orientations that are slightly oblique to the preferred orientation (Figure 1C). A range of ⫾ 8 degrees was chosen here based on the variation that was seen in the preferred direction of vertebral bone.6 The direction corresponding to the maximum length estimate for the various orientations was deemed to be the true orientation of that particular trabecula. Estimation of Cross-Sectional Dimensions Estimates of the cross-section dimensions T1 and T2 (T1 ⬎ T2) are made in directions perpendicular to the true trabecular orientation as determined in the previous step (Figure 1D). They are measured at five equispaced points along the line representing the length of the trabecula (Figure 1D). The cross-sectional dimension T1 is defined as the diameter of the circle circumscribing the cross section (Figure 1E). It is measured by determining the maximum distance between any two points on the cross section. Thickness T2 is defined as the diameter of the largest circle that can be inscribed within the cross section
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true measure of trabecular dimension. The other sets corresponding to the same trabecula are removed and are not considered in the evaluation (the proximity check algorithm is described using pseudocode in Appendix I). Example Application
Figure 2. The geometry of rods and plates can be represented as length measures, L, and two cross-sectional measures T1 and T2 (L ⬎ T1 ⬎ T2), where T2 represents the thickness. Transitions from a platelike structure to a rodlike structure would result in a corresponding decrease in T1. Note that the technique does not require the trabeculae to be ideal rods or plates.
(Figure 1E), similar to the approach suggested by Hildebrand and Ru¨egsegger.5 It is determined by applying a circular structural element of different diameters, and eroding the image (Interactive Display Language; Research Systems Inc.). The thickness T2 is the smallest diameter for which the erosion process deletes the cross section. The algorithm to approximate the diameter of the inscribed and circumscribed circles was validated by manually inscribing and circumscribing circles on the cross section of interest. The cross-sectional dimensions T1 and T2 were taken to be the median of the five measured values. Use of a median value ensured that occasional bone callus or connection point for other trabeculae did not skew the results. In this way, each trabecula can thus be characterized three-dimensionally using one length measure (L) and two cross-sectional dimension parameters (T1 and T2) (Figure 2). As stated earlier, T2 corresponds to the trabecula thickness. The set of lines will intersect trabeculae oriented parallel as well as perpendicular to the grid lines. These intercepts will therefore be comprised of thickness measures of trabeculae oriented perpendicularly to the line direction, as well as length measures of trabeculae oriented along the line direction. By definition, an oriented trabecula is one for which L ⬎ T1 ⬎ T2. If T1 ⬎ L, then the trabecula is not considered to be oriented along the search direction and the dimensions are not stored. The abovementioned steps are carried out for all bone intercepts encountered by the grid lines. For each oriented intercept, the true orientation and location of the intercept is stored, in addition to its length L and cross-sectional measures T1 and T2. At this stage, no attempt is made to determine a unique measure for an individual trabecula. Depending on the grid spacing, some of these measures may correspond to the same trabecula.
The software was first validated on simulated volumetric images generated using a two-step procedure (typical image shown in Figure 1A). First of all, a volumetric image was created with thick, vertically oriented trabeculae (cross-sectional dimension range: 140 –500 m) and thinner trabeculae trabeculae (crosssectional dimensions range: 80 – 400 m) oriented within the horizontal plane. The trabeculae were then perturbed so that their alignments were no longer exactly vertical, or exactly within the horizontal plane. The validation process indicated that: (a) the cross-sectional dimensions of each trabeculae could be predicted correctly; (b) the proximity check ensured that the trabeculae were not being counted twice; and (c) any trabecula whose cross section was limited by the edge of the specimen was not included in the measurement. A serial milling technique was used to create high-resolution 3D images of six 12-mm3 specimens of vertebral cancellous bone (L3–L4 vertebral bodies, six female cadavers; age range 39 – 63 years; specimens cut from different regions within the body). The donors had no history of metastatic bone disease or any other condition that might have influenced the vertebral bone structure. In this technique, the defatted cancellous bone cubes were first stained black in silver nitrate solution, and then exposed to ultraviolet light. The samples were embedded in white pigmented methyl methacrylate (MMA), and a computer numerically controlled (CNC) milling machine (SpectreLite 0200; Light Machines Corp., Manchester, NH) was programmed to serially mill off 20 m from the surface of each specimen. The resulting surface was imaged using a low-magnification image acquisition system (camera: CCD model XC-75, Sony, Japan; low-magnification lens: Pan-Cinor 1:3.8 f ⫽ 17, Som-Berthoit, France), and the entire process is repeated until the entire
Choice of Grid Spacing and Proximity Checking The grid spacing should be kept as fine as possible to ensure that no trabeculae are overlooked. If the grid spacing is chosen to be less than typical trabecular cross-sectional dimensions, it is probable that two or more grid lines may pass through the same trabeculae (Figure 1F). If a trabecula is measured more than once, the set of dimensions corresponding to the greatest length is considered as the
Figure 3. A 6-mm cube of vertebral cancellous bone (L4 vertebral body; 59 year-old female). Large vertical and small horizontal trabeculae can be seen in the vertebral body.
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Figure 4. Thickness distributions for vertical (left) and horizontal (right) trabeculae (L4 vertebral body; 63-year-old female; volume fraction: 0.04). b) Thickness distributions for vertical (left) and horizontal (right) trabeculae (L4 vertebral body; 49-year-old female; volume fraction: 0.15).
specimen has been milled and imaged. Each two-dimensional (2D) image was acquired at resolution of 20 ⫻ 20 ⫻ 20 m.2 The 2D images were stacked and combined into a 3D array using commercial software (Interactive Display Language; Research Systems Inc.) on an Ultra SPARC workstation (Sun Microsystems). Preprocessing of the image was limited to bone marrow segmentation using a dual-reference thresholding algorithm.7 The images were grey scale inverted so that the bone appears white on a black background (Figure 3). The technique was applied to a 5-mm cube extracted from the center of each image. A grid of lines with a line spacing of 20 m was passed along the inferior-superior direction to determine the geometry of vertically oriented trabeculae. For the horizontally oriented trabeculae, the grid of lines was passed along the horizontal plane, and rotated at increments of 15 degrees. Coupled with the ⫾8-degree search for the orientation of each trabecula, the 10-degree increments ensured that no trabeculae were overlooked. If the cross section of any trabecula lay along the edge of the cube, it was not included in results, as the thickness would then be underpredicted. Trabeculae whose length (as defined earlier) were truncated by the end of the cube were included in the study. An in-house code was developed to estimate 3D measures of bone architecture (Interactive Display Language; Research Systems Inc.). The code was validated using a similar program,6,13 and was run on an Ultra SPARC workstation (Sun Microsystems). Traditional morphometric measures of bone structure such as cancellous bone volume fraction (BV/TV), Tb.N., Tb.Th., and Tb.Sp. were computed using the volumetric images based on 3D extensions of the parallel plate model.
Results Thickness distributions for individual trabecula were obtained for each bone specimen (Figure 4, Tables 1 and 2). The thickness distributions revealed that specimens with higher volume fractions had a number of platelike trabeculae, i.e., where T1 ⬵ T2, while specimens with lower volume fraction showed mainly rodlike trabeculae, i.e., where T1 ⬎ T2 (Figure 4a, b). Although the true orientation of each bone segment was assessed within a cone of ⫾8 degrees, all vertically oriented trabeculae lay within ⫾4 degrees of the vertical direction. For the five specimens measured, removal of trabeculae that abut the edge resulted in one to four and two to nine trabeculae not being included in the calculation of vertical and horizontal trabeculae thickness respectively. Table 1. Intraspecimen variation in vertical trabeculae for six human female vertebral cancellous bone specimens Vertical Specimen 1 2 3 3 5 6
Age (years)
Volume fraction
Number
Mean T1 (SD) (mm)
Mean T2 (SD) (mm)
63 59 61 58 46 39
0.04 0.07 0.10 0.15 0.18 0.20
5 9 11 13 9 12
0.38 (0.16) 0.52 (0.22) 0.74 (0.30) 0.92 (0.34) 0.88 (0.30) 0.90 (0.52)
0.22 (0.06) 0.20 (0.06) 0.22 (0.10) 0.24 (0.08) 0.32 (0.16) 0.30 (0.14)
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Table 2. Intraspecimen variation in horizontal trabeculae for six human female vertebral cancellous bone specimens Horizontal Specimen 1 2 3 4 5 6
Age (years)
Volume fraction
63 59 61 55 46 39
0.04 0.07 0.10 0.15 0.18 0.20
Number
Mean T1 (SD) (mm)
Mean T2 (SD) (mm)
25 26 43 34 32 37
0.26 (0.14) 0.38 (0.20) 0.56 (0.28) 0.54 (0.24) 0.52 (0.40) 0.58 (0.44)
0.14 (0.04) 0.16 (0.06) 0.16 (0.04) 0.16 (0.06) 0.18 (0.08) 0.20 (0.06)
The mean true thickness t (i.e., mean T2 for each specimen) for both the vertically and horizontally oriented trabeculae showed a strong positive correlation with the volume fraction (vertical: r2 ⫽ 0.86, p ⬍ 0.05; horizontal: r2 ⫽ 0.80, p ⬍ 0.05). The horizontal trabeculae were greater in number and were thinner than the vertical trabeculae (Tables 1 and 2). The coefficient of variation of the intraspecimen vertical trabecular thicknesses ranged from 25% to 42%. This coefficient of variation shows a weak, albeit insignificant, positive correlation with volume fraction (r2 ⫽ 0.46, p ⬎ 0.05). The mean true trabecular thickness t along the preferred orientation showed a strong positive correlation with Tb.Th. (vertical: r2 ⫽ 0.81, p ⬍ 0.05; horizontal: r2 ⫽ 0.72, p ⬍ 0.05). The computed Tb.Th., however, was systematically lower than the mean true vertical trabecular thickness, with the error varying between 33% and 48% (mean 39%, SD 5%). Discussion A technique was introduced to quantify trabecular thickness as a function of direction. This thickness computation method was not based on the parallel plate model. Repeat measurements of a given trabecula were accounted for by incorporating a proximity checking mechanism. The technique was first validated using artificial images with known geometry; accurate estimates of the cross-sectional dimensions could be determined for all trabeculae oriented along a given direction. It was then applied to a set of vertebral cancellous bone specimens, quantifying intratrabecular variations in thickness for both horizontal and vertical trabeculae. The relationship between mean true thickness t and volume fraction was consistent with trends seen in other studies.8 The coefficient of variations of 25% to 42% seen in this study were somewhat smaller, but consistent with the results obtained by 2D studies, which found the coefficient of variation of thickness to lie between 36% and 52% for cancellous bone from the iliac crest.1,18 The thickness distributions displayed a similar skewed distribution as obtained by both the 2D studies.1,18 The results suggest that the volume fraction may be the predominant factor controlling the mean structural characteristics of vertebral cancellous bone.8 However, there was a large intraspecimen variability in trabecular thickness that showed only a weak relationship with volume fraction. Both 2D and 3D finite element models suggest that a intraspecimen variability of this order may reduce the elastic modulus by 20% to 40%.17,19 Mean cross-sectional thickness determined here for both the vertical and horizontal trabeculae was greater than that seen in the only other study using a 3D approach. That study, however, presented results for only one specimen, and it is possible that this difference may just be due to the specimen. Further specimens need to be evaluated using both techniques before comparisons can be drawn. It must be emphasized that the mean true trabecular thickness
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t computed here was different than the 3D estimates of traditional trabecular thickness Tb.Th. made using the parallel plate model.10 The definition of Tb.Th. (and a measure such as directional intercept length) does not differentiate between length and thickness, and is statistically equivalent to the mean of all intercept length measures. Given the larger number of thinner horizontal trabeculae as compared with the number of thicker vertically oriented trabeculae, it is not surprising that Tb.Th. falls between the mean true thickness computed for the horizontal and vertical trabeculae. The mean true trabecular thickness t, on the other hand, was based on true thickness estimates of the oriented trabeculae. The strong relationship found between t and Tb.Th. in the specimens studied here suggests that Tb.Th. could possibly serve as a surrogate measure for true thickness in regression analyses trying to elucidate the impact of bone structure on mechanical properties. The skewed nature of the distributions, however, reflects the fact that mean parameters may not adequately describe the variation in trabecular thickness, and other models that account for the skewness have to be considered. The architecture of vertebral cancellous bone has been described as a 3D lattice composed of vertical nearly parallel supporting plates and columns connecting thin horizontal struts.8 It has been postulated that, with osteoporosis, the vertical plates become increasingly rodlike, and the number of horizontal trabeculae decrease. This postulate could be validated by determining whether there is indeed a decrease in cross-sectional dimension T1 over time; as T1 becomes smaller and closer to thickness T2, the bone is becoming increasingly rodlike. Furthermore, the very definition of what is a plate or a rod in context of cancellous bone is unclear. It therefore may be better to focus on the changes in thickness itself. The focus of this study was to introduce a technique for the evaluation of thickness distributions of the trabeculae and not on the length measures. For this study, the definition of the length of an oriented trabecula was chosen to be the longest straight line along the direction of interest (Figure 5). Different measures of length have been proposed, although no definition has yet been accepted as the norm.9 It is likely that edge effects and highly curved trabeculae may cause the length to be underestimated. We have tried to minimize the affect of this inherent variability in cancellous architecture by adopting a 3D approach, incorporating some of the variability in trabecular orientation, and using a fine grid spacing combined with a proximity check that maximizes the determined length of a trabecula. Adopting a 3D skeletonizing approach to length measures may result in a more accurate assessment of the trabecular length, although this remains to be shown. It is also recognized that trabecular geometry is highly irregular, with the thickness varying from site to site along a trabecula. The measures computed for each trabecula are therefore characteristic measures of that trabecula. The results presented here serve to exemplify the usefulness of the technique. The vertebral specimens used in the study were all representative of cancellous bone from peri- or postmenopausal women. The relationship between true trabecular thickness and Tb.Th., as well as the overall geometry of trabeculae, may show different trends if bone from different ages and regions of the vertebra were sampled, and if a larger sample size were to be used. The model presented here opens the question as to how to evaluate the geometry of individual trabecula, as well as the question of adequate mechanistic definitions of length and thickness. Appendix The pseudocode given below describes the logic used to remove repeat measurements of the same trabeculae. Steps I–IV in the
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References
Figure 5. Magnified image of vertebral bone showing vertical rodlike and platelike structures. The estimated length measures are superimposed on the image. The algorithm was capable of estimating the length of the vertical struts.
algorithm described earlier have found N oriented bone segments. Some of these will be repeat measurements of the same trabeculae. for j ⫽ 1,N use dimensions and starting location to mark out where the trabecula lies for k ⫽ 2,N use dimensions and starting location to mark out where the trabecula lies does trabecula k lies within the region of trabecula j If yes and Lj ⬍ Lk, then delete set j If yes and Lj ⬎ Lk, then delete set k Update k and iterate Update j and iterate
Acknowledgments: The authors thank Brian Canfield, Sean Haddock, and Tony Chen for their help with the serial milling of the bone specimens. Support for this work came from National Institutes of Health grants R01-AG-13612 and AR-43784, and National Science Foundation grant BES-9625030, as well as from the University of California Academic Senate.
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Date Received: June 16, 1998 Date Revised: April 28, 1999 Date Accepted: April 28, 1999