## 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 **û _{1}**,

**û**,

_{2}**û**and eigenvalues , , . The latter are frequently summarized using the degree of anisotropy

_{3}*DA*(= / ), as well as isotropy index

*I*(= / ) and elongation index

*E*(= 1 − (/ )) (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

where *n* is the number of points examined and *L*_{i} 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

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.