Quantification and visualization of anisotropy in trabecular bone

Authors

  • R. A. Ketcham,

    Corresponding author
    1. Department of Geological Sciences, The University of Texas at Austin, Austin, TX 78712, U.S.A.
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  • T. M. Ryan

    1. Department of Geological Sciences, The University of Texas at Austin, Austin, TX 78712, U.S.A.
    2. Division of Fossil Primates, Duke University Primate Center, Durham, NC 27705, U.S.A.
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    • 1

      Present address: Department of Anthropology, Pennsylvania State University, University Park, PA 16702, U.S.A.


Dr Richard A. Ketcham. E-mail: ketcham@mail.utexas.edu

Summary

A number of methods for measuring anisotropy in trabecular bone using high-resolution X-ray computed tomography exist, which give different answers but have not been compared in detail. In this study, we examine the mean-intercept length (MIL), star volume distribution (SVD) and star length distribution (SLD) methods, their algorithmic implementation for three-dimensional (3D) data, and how their results relate to each other. A uniform ordered sampling scheme for determining which orientations to sample during analysis enhances the reproducibility of anisotropy and principal component direction determinations, with no evident introduction of biasing. This scheme also facilitates the creation of a 3D rose diagram that can be used to gain additional insights from the data. The directed secant algorithm that is frequently used for traversing pixel and voxel grids for these calculations is prone to bias unless a previously unreported normalization is used. This normalization ameliorates the bias present when using cubic voxels, and also permits calculations on data sets in which the slice spacing is not equal to the pixel spacing. Overall, the three methods for quantification of anisotropy give broadly similar results, but there are systematic divergences that can be traced to their differences in data and processing, and which may impact on their relative utility in estimating mechanical properties. Although discussed in the context of computed tomography of trabecular bone, the methods described here may be applied to any 3D data set from which fabric information is desired.

Introduction

Trabecular bone plays a crucial role in determining bone strength, and when its structure is anisotropic its load-bearing properties vary with direction as well (Turner et al., 1990; Odgaard et al., 1997). As a result, there is considerable interest in quantifying fabric anisotropy in trabecular bone (Ulrich et al., 1999). Current methods for doing so are primarily geared toward determining a second-rank fabric tensor, from which can be extracted estimates of principal component directions and magnitudes represented by the eigenvectors û1, û2, û3 and eigenvalues inline image, inline image, inline image. The latter are frequently summarized using the degree of anisotropy DA (= inline image/ inline image), as well as isotropy index I (= inline image/ inline image) and elongation index E (= 1 − (inline image/ inline image)) (Benn, 1994).

There are a number of methods for calculating a fabric tensor, and the results are to some degree method-dependent. The earliest and still most widely employed is the mean intercept length (MIL) (Whitehouse, 1974; Harrigan & Mann, 1984; Cowin, 1986), which uses the mean distance between material intersections (bone–marrow interfaces) along linear traverses over a range of orientations. Because MIL traverses cross both materials, the result is a combined measure that incorporates features of each, and subsequent methods were developed that concentrate on a single material at a time. The star volume distribution (SVD) (Cruz-Orive et al., 1992; Karlsson & Cruz-Orive, 1993) and star length distribution (SLD) (Odgaard et al., 1997; Smit et al., 1998) are determined by placing a series of points within the material of interest, and measuring the lengths of the lines emanating from them in various directions until they encounter a boundary. For the SVD, these lines are considered infinitesimal cones, with their vertex at the origin and subtending a solid angle as they approach the material interface, whereas for the SLD they are left as lines. In practical terms, this means that the star volume component in a particular direction ω is estimated as

image(1)

where n is the number of points examined and Li is the length of the line with orientation ω passing through each point that stays entirely within the material of interest. The corresponding star length component is

image(2)

Because the components are normalized to calculate anisotropy, the main practical difference between these measures is that the SVD uses cubed lengths, which will tend to amplify differences between major and minor components, increasing the inferred anisotropy. These methods are described more fully elsewhere (Odgaard et al., 1997; Odgaard, 1997), although comparisons were limited to principal axis orientation from one type of bone from one species, and algorithmic details were not supplied.

All of these analyses were originally developed for measuring two-dimensional (2D) sections, and if orthogonal sections (i.e. in three mutually perpendicular planes) are measured then sufficient information is available to estimate 3D structure. However, with the increasing capability and availability of volumetric imaging techniques such as high-resolution X-ray computed tomography (HRXCT) and magnetic resonance micro-imaging, routine acquisition of 3D data sets is becoming more commonplace (Müller et al., 1996, 1998; Kapadia et al., 1998; Borah et al., 2001; Fajardo & Müller, 2001b; MacLatchy & Müller, 2002; Ryan & Ketcham, 2002a,b). Such data allow these analyses to be done directly without the structural form assumptions sometimes required to extrapolate from 2D to 3D (Hildebrand et al., 1999). A number of 3D implementations have been mentioned in the literature (Hildebrand & Rüegsegger, 1997; Simmons & Hipp, 1997; Kothari et al., 1998) but these calculations have not been discussed in detail. In one case, source code is available (Simmons & Hipp, 1997).

In this study, the SVD, SLD and MIL methods are analysed with respect to both each other and to various methodological factors, including sampling strategy, number of orientations and points sampled, vector sampling method, and data processing. We also propose some algorithmic improvements that serve to make calculations more efficient and reproducible while diminishing bias. Finally, we introduce a 3D rose diagram that opens up new possibilities for data visualization and analysis.

Materials and methods

Data

HRXCT data were used for this study. Figure 1 provides a schematic overview of the form of computed tomography (CT) data as applicable to the calculations discussed here. Each CT image, also termed a slice, represents a finite thickness of material (Fig. 1a), and thus the pixels in CT images are termed voxels (‘volume elements’). By acquiring a contiguous series of CT slices, data describing a complete volume are obtained (Fig. 1b). From this ‘data brick’ a volume of interest (VOI) may be extracted for analysis, such as a sphere (Fig. 1c). Alternatively, a number of stereological approaches employ circular orthogonal sections (Fig. 1d), although in the case of CT this approach discards a lot of the available data. A frequent feature of CT data is that the in-plane resolution is superior to the between-slice resolution, resulting in voxels having a non-unit aspect ratio (Fig. 1e), and in such cases sampling and analysis strategies must be adjusted to compensate (e.g. Fig. 1f).

Figure 1.

Schematic illustration of CT data. Each cube represents a single data voxel. A CT slice image (a) represents a finite thickness of material, and by acquisition of a contiguous series of slices (b) a volume may be described. From these data subvolumes are typically extracted for fabric analysis, optimally a sphere (c) or with more loss of information orthogonal circular sections (d). In many cases voxel thickness will not equal in-plane size (e), requiring adaptation of sampling (f) and voxel traversal methods, as discussed in this study.

The data for this study were obtained at the High-resolution X-ray CT Facility at the University of Texas at Austin (Ketcham & Carlson, 2001). A microfocal X-ray source and image intensifier detector with a 512 × 512 CCD video camera was used to acquire 36-µm resolution CT data on trabeculae within the femoral heads of 2–11 specimens of seven species of strepsirrhine primates, for a total of 48 analyses (Ryan, 2000; Ryan & Ketcham, 2002b). The specimens were dry bones from adult individuals displaying no pathology, and the analysis was non-destructive. Complete details of the scanning and specimens for this study are provided by Ryan & Ketcham (2002b).

To enable comparison of data across specimens and taxa, a common VOI within the femoral head was defined by Ryan & Ketcham (2002b) as the cube centred at the intersection of the superoinferior, mediolateral and anteroposterior planes bisecting the femoral head, and its edge length was 50% of the shortest of the three measured dimensions – usually the mediolateral. In this study, the VOI was changed to the sphere enclosed within the cube (Figs 1 and 2), owing to the biasing effects of a non-spherical VOI as discussed below. Each VOI was segmented for bone using an iterative thresholding algorithm (Ridler & Calvard, 1978; Trussell, 1979) as described in detail by Ryan & Ketcham (2002b). It should be noted that these volumes are fairly small, both physically and in number of voxels; this is an inevitable outgrowth of using these techniques to study small animals. In all cases the diameter of the VOI is more than 10 times the mean trabecular thickness, and BV/TV (bone volume/total volume) is roughly 50%.

Figure 2.

CT image of the femoral head of specimen PP-k. The specimen is 13.4 mm across, and slice thickness is 0.036 mm. The square shows the cross-section of VOI used in an earlier study, and the inscribed circle is the cross-section of VOI used in this study.

SVD, SLD, MIL estimation

All fabric tensors were calculated for this study based on 3D analyses of the VOIs using a voxel-traversing directed secant algorithm, as implemented by the software Quant3D (Ryan & Ketcham, 2002b), written in the IDL programming language (Research Systems, Inc., Boulder, CO, U.S.A.). Although MIL analysis is usually performed using 2D data (e.g. Fig. 1d), the similarity of measurements necessary for the three methods made 3D implementation straightforward and more efficient. For osteology, the traditional process for determining the MIL fabric tensor from orthogonal sections consists of assuming that the distribution of mean intercept lengths forms an ellipsoid in 3D, the parameters for which can be estimated from the three cross-sections (Harrigan & Mann, 1984; Cowin, 1986). With volumetric data, there are two potential approaches to obtaining a fabric tensor. First, an ellipsoid can be fitted to the data in three dimensions, providing an exactly analogous result to the 2D analysis (Hildebrand et al., 1999; Simmons & Hipp, 1997; Ulrich et al., 1999). The resulting material tensor, termed M, is converted to a fabric tensor H using the relation H = M−½ (the square root of the matrix inverse) (Harrigan & Mann, 1984; Cowin, 1986). Second, the data can be processed in the same way as is done for SVD and SLD determinations, using a moment of inertia calculation to define an orientation matrix T (Fisher et al., 1987; Launeau & Robin, 1996) from which a fabric tensor F is derived by normalizing the eigenvalues to sum to 1. This method does not make the prior assumption that the data are distributed on an ellipsoid. These methods, hereafter termed MIL-H and MIL-F, are compared below.

Traversal method

Intercept length and star component measurements are typically performed using an algorithm that traverses the pixel or voxel grid (Fig. 1) detecting and measuring the distance between material intersections. In most cases these algorithms are a version of a digital differential analyser (DDA), a frequently used line-drawing algorithm for pixel displays in computer graphics (Pokorny & Gerald, 1989) that is easily extensible to 3D; this is also known as a directed secant algorithm. In 2D, the DDA works by calculating for a particular angle φ a pair of finite displacements:

Δx = a cos φ, Δy = a sin φ,(3)

where a is a scaling constant determining the size of the step. The pixels sampled for a traverse are then calculated by starting at an origin point and sequentially adding the displacements and rounding to the nearest pixel. When used for computer graphics, the value for a is chosen so that the traverse always progresses at least one pixel in the fastest-changing direction; for example, if φ = 30°, a = 1/cos(30°), or 1.155, making Δx = 1 pixel and Δy = 0.577 pixels. This standardization is hereafter termed a ‘graphics’ sampling rate.

The result of this scheme is a proper-looking line on a computer graphics display, but it is not appropriate for traversing a grid to count material intersections because it varies the traversal rate. For example, for a given number of pixels a 45° line travels further than a 0° line by a factor of √2 (1.41). In the context of the MIL algorithm this means that intersections are tested for up to 1.41 times less frequently per unit distance. Although for most traverses this will not matter, in cases that are near tangential to a trabecular surface this will tend to underestimate intersections at near-45° angles compared with near-orthogonal angles, artificially increasing the relative MIL in the former.

Because most fabric analysis source code is private or proprietary, it is not known how prevalent the use of a graphics sampling rate is among software packages. Inspection of the source code for the 3D implementation used by Simmons & Hipp (1997) reveals that it does use graphics-type sampling, which because of 3D effects produces an even more severe biasing (up to 1.73 : 1).

A more accurate measurement is achieved by adjusting the DDA to provide as constant a sampling rate as possible across all angles, although the discrete cubic shape of pixels and voxels prevents the creation of a perfectly continuous solution. In 2D the simplest method is to set a equal to 1 pixel width at all times. On a graphics display this creates the appearance of an uneven line of varying thickness as a result of the additional pixels, and we therefore refer to this technique as ‘dense’ sampling. Note that this will also produce a certain amount of inconsequential over-sampling (pixels tested twice) at non-orthogonal angles.

The principle of dense sampling can also be applied to 3D voxel grids, and extended to allow more accurately for grids in which the slice spacing is not equal to the pixel spacing (e.g. Fig. 1e,f). Given an azimuth φ and elevation θ, and a voxel aspect ratio AR (pixel spacing/slice spacing), the 3D extension of Eq. (3) is:

Δx = sin θ cos φ, Δy = sin θ sin φ, Δz = AR cos θ.(4)

In this case the constant a has been omitted because once again it is set to 1 pixel length, assuming AR is less than 1 (if AR > 1, then all displacements above should be multiplied by AR−1).

We tested the correctness of this approach and in particular Eq. (4) in two ways. First, we analysed synthetic data of spheres with various voxel aspect ratios and verified that the results showed no anisotropy or preferred orientations. Second, we ran 20 replicate analyses of two data sets (PP-c and PP-k) with both graphics and dense sampling, both at the original data resolution and also after combining adjacent slice data (along the superior–inferior axis) to create voxels with aspect ratios of 2 : 1 and 4 : 1. The data for the third data set (GS-d) were not used because the data dimensions are too small to allow further reduction without inducing significant boundary effects.

Volume of interest shape

Various VOI shapes have been used, including spheres (MacLatchy & Müller, 2002), cubes (Hildebrand et al., 1999; Ryan & Ketcham, 2002b) and irregular polyhedra conformed to the shape of the region being examined (Fajardo & Müller, 2001a). We investigated the use of cubes and spheres, and found that in some cases these gave different answers, which tended to be in the form of higher component values in orientations towards the edges or corners of the cubic VOI. Because differently shaped VOIs by definition sample different material, it is difficult to gauge whether these effects are an artefact or reflect true specimen features. However, it is evident that the uneven sampling inherent in a cubic VOI makes features orientated near 45° to one or more axes more likely to be included and measured than near-orthogonal features, particularly with small specimens and/or when the VOI is small compared with the fabric elements. Thus, a spherical VOI is preferable when possible.

Point sampling

The traversal origin points for MIL or star analyses can be chosen randomly or by defining a uniform gridded sampling. Although a uniform grid is usually used for MIL, it is not optimal for SVD and SLD because of the requirement that all measurements be taken within bone material. A certain proportion of points must thus be discarded, on average 1 − BV/TV; aliasing can also occur if the grid spacing is near some multiple of the average material spacing. Thus, in this study, only randomly placed points are used, and for SVD and SLD only points within bone are selected. The number of points is not a directly comparable parameter between MIL and the star-based analyses, as the line projected through a point will tend to produce multiple intersection length measurements but only a single star component measurement.

Orientation sampling

Random orientation sampling requires a statistically homogeneous distribution of points on a unit hemisphere, which is achieved by sampling the azimuth φ evenly over the range [0, 2π] and the cosine of elevation θ over [0, 1] (Odgaard et al., 1997). Although there is no unique definition of a uniform regular (non-random) distribution of points on a hemisphere or sphere, a visualization of a reasonable implementation is shown in Fig. 3. The sphere is divided into triangular facets that are defined by recursively subdividing the faces of an octahedron and then projecting them outward to the sphere surface. The analysis directions correspond to the vertices, which are symmetric about the equator, so each direction is represented twice on the sphere surface, once in each hemisphere. The angular sampling thus defined will hereafter be termed uniform sampling (not to be confused with uniform random or isotropic random sampling). The algorithm that defines this structure generates unique directions in increments of 2(2n+1) + 1, where n is a positive integer. The available values closest to those employed in previous work are 513 and 2049, and in this study these numbers of orientations are used for both uniform and random sampling. To impart a degree of randomness to the uniform distribution of directions, and to minimize biasing from the cubic form of the voxels, a random rotation is applied to the vector set for each analysis.

Figure 3.

Unit sphere illustrating uniform orientation distribution used in this study. Line intersections are used as orientations for directional analysis, and triangles (facets) are used to aid in construction of 3D rose diagrams.

Uniform regular sampling on a unit sphere has also been approached using standard map projections (e.g. Mattfeldt et al., 1990). Karlsson & Cruz-Orive (1993, appendix B) provide an equi-areal partition of a hemisphere based on appropriate selection of latitudinal and longitudinal divisions. The method presented here may be considered an improvement in certain possibly influential respects: the distance from each point to its nearest neighbours on the sphere is more consistent, and there is no change in the character of the distribution between the equator and poles.

Two approaches are used to study the reproducibility of the various anisotropy measures and the relationships among them. First, to gauge precision, 20 replicate analyses were made for all of the measures using various parameters (sampling method, number of points and orientations, and uniform vs. random orientations) on three specimens (Table 1). Second, to compare methods the entire 48-specimen data set was analysed using parameters judged acceptable based on analysis of replicate analyses (8000 points and 513 uniformly distributed directions for SVD and SLD, and 1000 points and 513 uniformly distributed directions for MIL).

Table 1.  Specimen and scan data for this study
SpecimenSpeciesMuseum no.BV/TVMean trabecular thickness (mm)*Voxel edge length (µm)VOI edge length (mm)VOI voxel dimensionsTotal no. of voxels
  • *

    Mean trabecular thickness estimated as the mean minimum measured length in all directions though a random point placed within bone.

GS-dGalago senegalensisUSNM 3979690.4760.1135.91.43640 × 40 × 40 33 552
PP-cPerodicticus pottoUSNM 1842310.4380.1636.03.56499 × 99 × 99508 371
PP-kPerodicticus pottoUSNM 4817380.5320.1836.03.56499 × 99 × 99508 371

3D rose diagrams

Although a second-rank tensor provides an excellent summary of a continuum material, in many cases it may be an oversimplification of a framework architecture such as trabecular bone. A second-rank tensor defines three orthogonal axes, whereas the alignment of skeletal elements may be organized in a non-orthogonal fashion. A simple example is a capital letter ‘X’, or a grid-work made up thereof, in which the two major material directions are at 60° to each other. The intuitive misfit between measured data and tensor summaries has been noted previously (Odgaard, 1997; Smit et al., 1998), with the suggested remedy of using higher-rank tensors in some cases (e.g. Kanatani, 1984).

One added benefit of the uniform sampling method described above is that it makes it straightforward to define a structure that can be viewed using standard computer graphics tools. By displacing each point along its respective orientation by the relative magnitude of its corresponding measurement (MIL or star component), and maintaining the grouping of points and edges into facets, a surface can be defined that represents a 3D rose diagram (Fig. 4). All vertex positions are normalized so that the maximum measurement plots at the same displacement from the origin as the end of the coordinate axes, and the rest are scaled accordingly. Also plotted on these figures are the calculated eigenvectors, scaled according to their eigenvalues, again so that the maximum length matches the coordinate axis length. Because of their nature these figures are best viewed interactively, and are exported in VRML format, making them potentially viewable with a variety of software, including browsers. Colours are used to improve clarity; in these examples a rainbow pattern is used, with red corresponding to the highest value(s) measured in a given analysis, and violet to the lowest.

Figure 4.

Examples of 3D rose diagrams depicting trabecular bone measurements. Distance from origin and colour (violet = minimum, red = maximum) indicate relative component value within a single analysis. Red axes show principal component directions and relative magnitudes. (a,b) SVD data for specimen PP-k, with (b) viewing the diagram directly down the axis of the secondary eigenvector. (c) SVD data for specimen PP-c. (d) SLD data for specimen PP-k. (e,f) MIL data for PP-k using dense and graphics sampling, respectively.

Results

3D rose diagrams

Several example rose diagrams are shown in Fig. 4, demonstrating the general level of detail available with each method. The SVD is the most starkly illustrative, often clearly revealing the form of multiple fabric elements. In general, planar fabric components, such as predominant orientations of trabecular plates, appear as discs in the diagrams, whereas alignments of linear elements such as struts appear as rod-shaped features. Figure 4(a,b) show two views of the SVD data for specimen PP-k, revealing that the fabric is dominated by two planar components at a low (∼10°) angle to each other. The SVD diagram for PP-c (Fig. 4c) shows a similar distribution, with the principal planes more preferentially elongated along a superomedial axis, and an additional semiplanar feature that cross-cuts the principal planes. The diagrams showing the SLD and MIL data (Fig. 4d,e) are in general less feature-rich, but still show some interesting details. Figure 4(f) illustrates the consequences of using graphics sampling instead of dense sampling for MIL calculation: the distribution has become less ellipsoidal, with measurements at 45° to two or three axes artificially inflated compared with Fig. 4(e).

The relationship of the underlying fabric elements to the directions provided by principal component analysis is instructive. In many cases the orientations do not necessarily correspond to one's intuition; for example, the primary eigenvector often does not coincide with the direction having the highest SVD values. Figure 4(a) provides a particularly striking example, in which neither of the two local maxima is associated with a principal component orientation. Even in the MIL analysis (Fig. 4e) the primary eigenvector does not correspond to the clear locus of highest values. As a corollary, also evident from the diagrams is that often neither distribution has a true ellipsoidal shape, each featuring slight but persistent departures covering significant solid angles. The seeming inconsistency between inferences based on the diagrams and principal components lies in the fact that the latter are mutually constraining and represent global averages, whereas the former highlight localized features.

Anisotropy

Table 2 shows the results of a series of sets of 20 replicate MIL analyses for the three specimens listed in Table 1. In general, it is apparent that increasing the number of measurements reduces the uncertainty in the anisotropy results. When fabric tensor H is used, uncertainties tend to be small. There is some indication that increasing the number of traverse origin points is more effective than increasing the number of traverse orientations in reducing uncertainty, and there is little difference between using a uniform or random distribution of orientations. The orientation matrix-based tensor F suffers from significantly worse precision when a random distribution of orientations is used, but results obtained using a uniform distribution are more comparable. This stems from the fact that the ellipsoid method assumes a reasonable underlying form for the data, effectively interpolating orientations that are under-represented in a random sampling, whereas orientation matrix results will be skewed by whatever range of directions is over-sampled as a result of randomization. When this randomness is removed by using a uniform orientation distribution, reproducibility improves markedly.

Table 2.  Anisotropy results from MIL analyses
SpecimenRun parametersH (ellipsoid model)F (orientation matrix)
DAIEDAIE
  1. Run parameter codes signify number of orientations (513 or 2049), followed by orientation distribution type (r = random, u = uniform), followed by number of points measured for each orientation (traverse origins). DA = degree of anisotropy. I = isotropy index. E = elongation index. Column headings H and F signify the method used to calculate the fabric tensor. All results shown are means with one standard deviation in parentheses.

GS-d513r5001.82 (03)0.55 (01)0.16 (01)1.63 (10)0.62 (04)0.13 (06)
 513r10001.83 (02)0.55 (01)0.16 (01)1.67 (13)0.60 (05)0.14 (07)
 2049r5001.84 (03)0.54 (01)0.16 (01)1.65 (07)0.61 (03)0.11 (03)
 2049r10001.83 (02)0.55 (01)0.16 (01)1.64 (04)0.61 (01)0.12 (03)
 513u5001.82 (04)0.55 (01)0.15 (02)1.64 (05)0.61 (02)0.12 (02)
 2049u10001.83 (02)0.55 (01)0.16 (01)1.64 (03)0.61 (01)0.12 (02)
PP-c513r5001.46 (02)0.69 (01)0.12 (02)1.42 (11)0.71 (05)0.10 (04)
 513r10001.45 (02)0.69 (01)0.12 (02)1.38 (07)0.73 (04)0.11 (04)
 2049r5001.46 (02)0.68 (01)0.13 (01)1.37 (05)0.73 (03)0.11 (03)
 2049r10001.46 (01)0.69 (01)0.12 (01)1.38 (05)0.73 (03)0.11 (03)
 513u5001.46 (01)0.68 (01)0.12 (01)1.36 (03)0.74 (02)0.09 (01)
 2049u10001.46 (01)0.69 (01)0.11 (01)1.35 (03)0.74 (02)0.09 (01)
PP-k513r5001.64 (02)0.61 (01)0.07 (02)1.52 (10)0.66 (04)0.09 (05)
 513r10001.63 (01)0.61 (01)0.06 (01)1.55 (10)0.65 (04)0.08 (05)
 2049r5001.64 (02)0.61 (01)0.06 (01)1.52 (05)0.66 (02)0.07 (02)
 2049r10001.63 (02)0.61 (01)0.06 (01)1.51 (05)0.66 (02)0.06 (02)
 513u5001.64 (02)0.61 (01)0.06 (01)1.50 (04)0.66 (02)0.04 (02)
 2049u10001.63 (01)0.61 (01)0.06 (01)1.49 (02)0.67 (01)0.04 (01)

Values of DA, I and E vary systematically between results based on MIL-F and MIL-H. Extending this comparison to all of the specimens in the data set, it is evident that the relationship between the eigenvalues provided by the two can be largely described by a simple power law (Fig. 5a):

Figure 5.

Relationship of eigenvalue determinations derived from different methods, for 48 primate specimens. (a) MIL-H determinations vs. MIL-F converted to MIL-H using Eq. (5). (b) SVD determinations vs. SLD converted to SVD using Eq. (6). (c) SVD determination vs. MIL-H converted to SVD using Eq. (7).

image(5)

This relation is only approximate, as the residuals show some structure. However, the mean residuals for the three eigenvalues are only 0.001, −0.003 and 0.002, respectively, which are close to the magnitude of their uncertainties.

The replicate analyses of SVD and SLD (Table 3) show similar trends to the MIL-F results. In particular, the effect of a uniform orientation distribution is clear, as the standard deviations from analyses based on 513 orientations and 500 points are sharply lower everywhere when this one parameter is changed. However, errors for the random and uniform cases converge to similar values once sufficiently large numbers of points and orientations are used; for the random case a large number of orientations seems necessary, probably reflecting a component of variance dominated by overly sparse and uneven angular sampling. For DA values, the SVD shows the least precision, but it also provides the greatest range. As a percentage of the available range of values (σ/(DA − 1)), the uncertainties for SVD and SLD are very similar, starting at 15–20% for the worse cases and improving to 6–7% with the most advantageous parameters. These results in turn are somewhat similar to those for the MIL-F method, although direct comparisons are inappropriate because of the different character of the measurements.

Table 3.  Anisotropy results from SVD and SLD analyses
SpecimenRun parametersSVD SLD
DAIEDAIE
GS-d513r10004.82 (58)0.21 (03)0.33 (09)1.56 (11)0.65 (04)0.16 (07)
 513r20004.71 (44)0.21 (02)0.35 (06)1.52 (08)0.66 (04)0.16 (07)
 2049r10004.57 (34)0.22 (02)0.28 (06)1.48 (05)0.68 (02)0.10 (04)
 2049r20004.40 (26)0.23 (01)0.27 (05)1.45 (04)0.69 (02)0.10 (03)
 513u10004.45 (42)0.23 (02)0.29 (05)1.47 (04)0.68 (02)0.11 (01)
 513u80004.56 (26)0.22 (01)0.29 (03)1.48 (03)0.68 (01)0.11 (01)
 2049u20004.48 (24)0.22 (01)0.30 (03)1.47 (03)0.68 (01)0.11 (01)
PP-c513r10002.60 (30)0.39 (04)0.29 (05)1.44 (12)0.70 (05)0.13 (05)
 513r20002.48 (23)0.41 (04)0.28 (06)1.38 (09)0.73 (05)0.12 (06)
 2049r10002.58 (14)0.39 (02)0.28 (03)1.40 (05)0.71 (02)0.11 (02)
 2049r20002.53 (12)0.40 (02)0.28 (03)1.39 (05)0.72 (03)0.11 (03)
 513u10002.54 (17)0.39 (02)0.29 (04)1.37 (04)0.72 (02)0.11 (02)
 513u80002.54 (13)0.40 (02)0.28 (03)1.38 (03)0.73 (02)0.11 (01)
 2049u20002.58 (14)0.39 (02)0.29 (03)1.39 (03)0.72 (01)0.11 (01)
PP-k513r10002.74 (20)0.37 (03)0.10 (06)1.45 (09)0.69 (04)0.09 (04)
 513r20002.82 (20)0.36 (02)0.13 (06)1.50 (10)0.67 (05)0.11 (04)
 2049r10002.70 (11)0.37 (02)0.10 (04)1.44 (05)0.70 (02)0.07 (03)
 2049r20002.63 (14)0.38 (02)0.07 (04)1.40 (05)0.71 (03)0.06 (03)
 513u10002.70 (16)0.37 (02)0.08 (03)1.41 (04)0.71 (02)0.04 (01)
 513u80002.70 (12)0.37 (02)0.07 (03)1.41 (03)0.71 (02)0.04 (01)
 2049u20002.72 (11)0.37 (01)0.06 (03)1.41 (03)0.71 (01)0.04 (01)

There is a very close relationship between the eigenvalues and anisotropy indices for SVD and SLD. Figure 5(b) relates the eigenvalues between the two methods; in this case, the power-law function is predictable based on the conversion of length to volume:

image(6)

The correlation is not as tight, probably because of the lower precision of the eigenvalue determinations. Mean residuals for the three eigenvalues are −0.001, −0.007 and 0.008, respectively, smaller than the typical standard deviation inferred from replicate analyses.

Anisotropy estimates can also be converted between the star-based methods and MIL using an empirical power-law relationship; for example (Fig. 5c):

image(7)

The correlation is worse owing to the large number of discrepancies between the methods. The dumbbell-shaped distribution of points around the 1 : 1 line suggests that increasing anisotropy indicated by either method leads to increasing disagreement between them.

Principal directions

Results from replicate analyses show that estimates of principal directions are influenced both by the analytical method and by the character of the underlying distributions. Figure 6(a) shows the results of MIL analysis of specimen PP-k, using a relatively low number of traverses and randomly distributed orientations. Principal directions for MIL-F are considerably more scattered than for MIL-H, particularly the primary and secondary eigenvectors, which are distributed almost randomly around a great circle normal to the relatively more constrained tertiary eigenvector. This pattern can be traced directly to the ‘square-disc’ distribution of the data seen in Fig. 4(a) and also reflected in the low elongation index E, which leads to sudden fluctuations with relatively minor changes as a result of randomization. Results based on the ellipsoid method are more resistant to this effect, although even here the primary directions are distributed along a roughly 20° arc of the great circle. By acquiring many more measurements and using a uniform orientation distribution (Fig. 6b), the MIL-F results become much better organized, although they still vary over a 40° arc; there is also a modest improvement in the ellipsoid method results (15° arc). When a specimen with greater spread between its three eigenvalues is analysed, such as GS-d (Fig. 6c,d), the same patterns are observed but to a considerably lesser degree.

Figure 6.

Equal-area stereographic projections of replicate trabecular fabric principal axis estimates (û1, û2, û3) derived from MIL analyses of two specimens using various algorithm parameters. The centre point of the diagrams represents the inferior direction. Latitude and longitude lines are at 10° increments. Parameter codes are from Table 2. Although for sample PP-k some scatter in the MIL-F method using random orientation sampling (a) is removed by using a larger number of uniform orientations (b), primary and secondary component estimates still show spread along a great circle, owing to the underlying distribution (Fig. 4a). A similar but less severe pattern is observed with sample GS-d (c,d).

The reproducibility of principal directions in the SVD and SLD methods (Fig. 7) is similar to that of MIL-F. When a random distribution of orientations is used, results are very scattered (Fig. 7a), whereas a uniform distribution removes this scatter without any shift that would imply a relative bias. The pattern for specimen PP-k (Fig. 7b) is similar to that obtained with MIL analysis, with a tightly constrained third direction and the first and second distributed around a great circle. Interestingly, however, the results are somewhat different among all four methods. The location of û3 shifts about 5° between the MIL and star-based methods, and about 2° between the SVD and SLD. A much larger shift is observed among the primary and secondary directions, with the mean SVD û1 being closest to the inferior–superior axis and the SLD, MIL-F and MIL-H means progressively rotating toward the lateral–medial axis by 15°, 27° and 32°, respectively. A similar shift of approximately 3° for û3 and 8° for û1 between MIL-H and SVD is observed for specimen GS-d (Figs 7c and 6d). With specimen PP-c, there is an approximately 5° separation between SVD and SLD in all three axes (Fig. 7d), and the MIL results (data not shown) closely match the SVD.

Figure 7.

Equal-area stereographic projections of replicate trabecular fabric principal axis estimates (û1, û2, û3) derived from SVD and SLD analyses illustrating reproducibility for three specimens. The centre point of the diagrams represents the inferior direction. Parameter codes are from Table 3. As with MIL-F analyses (Fig. 6), some scatter for PP-k estimates (a) is removed by utilizing a uniform orientation distribution (b), although some spread around the great circle persists and primary direction estimates have shifted closer to the superior–inferior axis. There is less spread among replicate analyses of GS-d (c) and PP-c (d).

The degree to which dispersion of primary and secondary directions on a great circle occurs can be linked to the shape indices: the effect is greatest when the elongation index E is very low (PP-k), and can also occur at somewhat higher E values when the isotropy index I is sufficiently low (GS-d). Presumably at very high E values it is the secondary and tertiary directions that will be distributed around a great circle, although that circumstance does not occur for any of the specimens in this study.

Traversal sampling and voxel aspect ratio

Table 4 shows the results of varying the DDA algorithm sampling rate and increasing the voxel aspect ratio for each tensor method of calculating DA and E. For the unreduced data most results are comparable, with the exception of the MIL determinations for sample PP-c, for which graphics sampling gives a significantly higher DA. Although E values are within error of each other, graphics sampling invariably gives equal or slightly higher results. In the transition from 1 : 1 to 2 : 1 aspect ratio data, the advantage of dense sampling becomes apparent. In most cases, the anisotropy measures show no appreciable shift owing to the loss in resolution when dense sampling is used, but graphics sampling results in systematic increases in DA and more strongly in E. At a 4 : 1 aspect ratio the slice spacing (0.144 mm) is of similar magnitude to the mean trabecular thickness (Table 1), resulting in more significant divergences regardless of sampling. In some cases values re-approach 1 : 1 results, but we consider this a random effect.

Table 4.  Anisotropy results from test of voxel anisotropy and data traversal method
SpecimenTestAspect ratioDAE
DenseGraphicsDenseGraphics
  1. Aspect ratio = voxel z dimension (slice spacing) : voxel xy dimension (pixel spacing). All results shown are means with one standard deviation in parentheses.

PP-cMIL-H1 : 11.45 (01)1.54 (01)0.12 (01)0.15 (01)
  2 : 11.38 (02)1.63 (01)0.09 (01)0.23 (01)
  4 : 11.30 (02)1.56 (03)0.05 (01)0.21 (01)
 MIL-F1 : 11.36 (02)1.43 (03)0.09 (01)0.12 (01)
  2 : 11.38 (02)1.63 (01)0.09 (01)0.23 (01)
  4 : 11.22 (02)1.41 (04)0.03 (01)0.16 (02)
 SVD1 : 12.55 (11)2.52 (12)0.28 (02)0.30 (03)
  2 : 12.57 (14)2.76 (17)0.31 (03)0.37 (02)
  4 : 12.22 (10)2.56 (16)0.23 (04)0.36 (03)
 SLD1 : 11.38 (02)1.38 (03)0.11 (01)0.12 (01)
  2 : 11.36 (03)1.41 (04)0.11 (01)0.15 (01)
  4 : 11.24 (02)1.32 (03)0.05 (01)0.11 (02)
PP-kMIL-H1 : 11.63 (01)1.67 (01)0.06 (01)0.08 (01)
  2 : 11.62 (02)1.92 (02)0.06 (01)0.22 (01)
  4 : 11.56 (02)1.89 (03)0.03 (01)0.21 (01)
 MIL-F1 : 11.49 (03)1.53 (04)0.04 (02)0.06 (02)
  2 : 11.47 (03)1.70 (05)0.03 (01)0.17 (02)
  4 : 11.44 (02)1.65 (05)0.03 (01)0.15 (02)
 SVD1 : 12.71 (12)2.67 (10)0.06 (02)0.07 (03)
  2 : 12.79 (12)3.03 (14)0.06 (03)0.16 (03)
  4 : 12.73 (09)3.11 (13)0.06 (03)0.17 (03)
 SLD1 : 11.41 (03)1.42 (02)0.04 (01)0.04 (01)
  2 : 11.41 (03)1.48 (03)0.04 (01)0.08 (01)
  4 : 11.37 (02)1.45 (03)0.03 (01)0.06 (02)

For principal directions, Fig. 8 shows results that are fairly typical. Each stereo plot illustrates the results for the principal eigenvector direction û1 for MIL-H in sample PP-k for each voxel aspect ratio. The dense sampling case (Fig. 8a) shows that the initial dispersal of results around a great circle is shifted slightly from 1 : 1 to 2 : 1, and at 4 : 1 there is greater dispersal around the great circle. The great circle described by û1 and û2 shifts by 3° and 7° in the 2 : 1 and 4 : 1 cases, as calculated using vector averages (Mardia & Jupp, 2000). When graphics sampling is used, the 2 : 1 and 4 : 1 aspect ratio cases are displaced strongly toward the vertical (undersampled) axis, an effect also observed by Kothari et al. (1998). The great circle defined by û1 and û2 is somewhat more affected, shifting by 5° and 9° in the 2 : 1 and 4 : 1 cases, respectively.

Figure 8.

Equal-area stereographic projections of replicate trabecular fabric primary axis estimates (û1) for sample PP-k when using dense (a) and graphics (b) voxel traversal sampling methods, showing the effect of increasing voxel aspect ratio. The centre point of the diagrams represents the inferior direction, along which scan images were combined to create elongated voxels. Corresponding degree of anisotropy estimates are provided in Table 4.

The SVD and SLD results for specimen PP-k behave in essentially the same manner, with increased dispersion around the great circle as discussed previously. For specimen PP-c, results were overall less divergent, owing to the lack of a great circle distribution of û1; however, the graphics sampling again resulted in a larger shift toward vertical and a reduction in dispersion as a result of concentration near the vertical axis at aspect ratios higher than 1 : 1.

Discussion

The results presented here corroborate previous investigations in which MIL, SVD and SLD analyses gave results that were qualitatively similar (Odgaard et al., 1997; Smit et al., 1998). However, it is evident that there also are systematic departures that can be traced to their differences in data and processing. The MIL-F bridges the gap between the MIL-H and SLD methods, as it utilizes the data of the former and processing of the latter. The moment of inertia calculation leads to somewhat lower reproducibility in MIL determinations compared with the ellipse method, owing to the rather rigid structure imposed by the latter on the assumed pattern of the data. The ellipse method also leads to a wider range of anisotropy values with lower uncertainties, thereby ostensibly increasing the sensitivity of the method. By contrast, it is clear from the 3D rose plots that the data are not perfectly ellipsoidal, and it may be the case that the ellipse model slightly skews results because of this misfit.

The MIL and SLD methods are based on very similar data. For a given set of traverses, the SLD essentially discards half of the measurements (those not within the material of interest), and has different biasing: whereas in the MIL all line segments are equally likely to be sampled, SLD segment sampling is proportional to segment length. It is thus interesting that the SLD gives statistically indistinguishable DA results for PP-c and PP-k, whereas MIL-F results are much more divergent. This could reflect the fact that the difference quantified by the MIL analysis in this case lies predominantly in the distribution of non-calcified tissue, and thus may be of lesser relevance for inferring mechanical properties.

The power-law scaling among eigenvalues reflected in Eqs (5–7) may also be of relevance in estimating the mechanical characteristics of bone from anisotropy measurements. The MIL DA has been shown to be an effective parameter for improving prediction of mechanical properties (Ulrich et al., 1999), and the non-linear differences between methods observed here would be expected to impact such estimates. We can speculate that both confining measurements to a single material and utilizing the moment of inertia rather than ellipsoid dimensions may improve the correlation with mechanical properties, but further study is required.

Based on the comparison between analyses utilizing uniform and random angular sampling, there is no evidence in either the orientation or the relative magnitudes of the principal components that a bias is introduced by utilizing the uniform scheme implemented here. Because of its advantages in improving reproducibility and allowing visualization, we can without reservation recommend it as a preferable method for conducting anisotropy measurements.

Although the methods in this paper have been developed in the context of HRXCT scanning of trabecular bone, they have much wider application in terms of both instrumentation (e.g. confocal microscopy, magnetic resonance imagery) and subject (e.g. rock foliations, particle preferred orientations).

Supplementary material

Interactive versions of the 3D rose diagrams are given in VRML 2.0 format files, available at: http://www.black wellpublishing.com.products/journals/suppmat/JMI/JMI1277/JMI1277sm.htm

A number of free applications and browser plug-ins for viewing VRML files can be accessed through http://www.web3d.org/vrml/browpi.htm.

Acknowledgements

This research was supported by the National Science Foundation (awards EAR-0113480 and EAR-0004082 to RAK and BCS-9908847 to T.M.R.) and by the Leakey Foundation (to T.M.R.). Parts of this work were inspired by conversations with and preliminary studies by M. Maga, A. Gordon and M. Zylstra. Comments and suggestions from T. Mattfeldt and an anonymous reviewer improved this manuscript.

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