## 1. Introduction

In several fields of science and technology, the objects under investigation can be represented as porous media. The solid and void phases of such media determine their physical properties. For example, trabecular bone may be considered as a porous medium with low solid fraction (0.05–0.25). Note that the trabecular microarchitecture of the bone solid phase is very important in terms of bone mechanical strength. Several imaging techniques are available to obtain three-dimensional (3D) reconstruction of porous media (Kinney *et al*., 1995; Majumdar & Genant, 1995; Ruegsegger *et al*., 1996; Peyrin *et al*., 1998). Image analysis techniques are used in order to obtain morphological (Cruz-Orive, 1979; Levitz & Tchoubar, 1992; Guilak, 1994) and topological (DeHoff *et al*., 1972; MacDonald *et al*., 1986; Odgaard & Gundersen, 1993) characterizations. As a general rule, these techniques are applied in a global way and give some mean information. These usual global techniques are interesting but do not capture all the complex microarchitecture properties of disordered porous media.

The morphological techniques are generally based on a physical model (Whitehouse, 1974a; Parfitt *et al*., 1983; Kragstrup, 1985). Consequently, if the structure differs from this assumed model, the morphological parameters will be biased to an unknown extent (Hildebrand & Ruegsegger, 1997). The mean size evaluation (Cruz-Orive, 1979; Levitz & Tchoubar, 1992) takes into account only a size averaging of various shapes of elements. For example the mean trabecular size evaluation (mean intercept length) does not take into account the real complex shape of the trabeculae. The requirement of independent trabecular size estimation free of any assumption was reported by Kinney *et al*. (1995) and by Hildebrand & Ruegsegger (1997). Concerning topological characterization, a technique based on 2D skeletonization has been developed (Garrahan *et al*., 1986; Compston *et al*., 1987, 1989). The results are expressed as a set of ratios between nodes, termini and loops of the 2D skeleton graph. Nevertheless, these 2D topological parameters must be cautiously interpreted because on a theoretical ground, their topological signification is not valid for the complete 3D object (DeHoff, 1983; Gundersen *et al*., 1993). Some works have been reported concerning the estimation of the connectivity from serial sections (DeHoff *et al*., 1972; MacDonald *et al*., 1986; Gundersen *et al*., 1993). However, these approaches are relatively complex and involve more efforts than the direct 3D evaluation. In fact, the connectivity can be directly evaluated in 3D from the Euler–Poincaré characteristic (Odgaard & Gundersen, 1993; Vogel, 1997). Nevertheless, a global connectivity index does not capture all the topological properties as the numbers of connection nodes and their relative positions.

3D skeleton graph analysis is a powerful technique to describe architecture of porous media. For example, this approach has been used to characterize granular porous media in the case of sedimentary rock (Lin & Cohen, 1982). Each vertex of this graph represented a grain and each branch represented a contact between grains. Several parameters such as size distributions, aspect ratio distributions or coordination number distributions were calculated from the skeleton graph. In the same way, 3D skeleton graph analysis can be applied to trabecular bone microarchitecture. The analysis of the 3D skeleton graph of trabecular bone microarchitecture allows us to obtain real 3D topological parameters and to extract each trabecula of the bone network. This extraction permits a morphological characterization of each trabecula independently of any model.

The purpose of this study was to develop a 3D skeleton graph analysis of trabecular bone microarchitecture, in order to obtain a local characterization of the trabecular bone network. Bone microarchitecture was imaged by magnetic resonance imaging (MRI) (Section 2.1). A 3D skeletonization technique is reported in Section 2.2. Section 2.3 presents a new 3D skeleton graph analysis taking into consideration several artefacts due to the digitization process. We have introduced the notion of pixels clustering and redefined the notion of pixels neighbourhood in order to eliminate topological artefacts. Taking into consideration these two concepts, we have used a graph point classification to count and localize the vertices and branches of the skeleton graph. Our approach was entirely based on the topology preservation between initial structure and its skeleton graph. A topological equation based on the three Betti numbers and Euler–Poincaré characteristic is used to check topology conservation (Section 2.4). The numbers of vertices and branches are expressed by unit volume. Each trabecula is extracted from its associated skeleton branch (Section 2.5). Local morphological parameters such as length and volume of each trabecula are evaluated (Section 2.6), as well as global morphological parameters (Section 2.7). In the last part, we used our new graph analysis technique to analyse 12 trabecular bone samples (six osteoporosis and six osteoarthritis) (Section 3).