### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Material and methods
- 3. Results
- 4. Discussion and conclusion
- Acknowledgements
- References

This paper introduces a new three-dimensional analysis of complex disordered porous media. Skeleton graph analysis is described and applied to trabecular bone images obtained by high resolution magnetic resonance imaging. This technique was developed bearing in mind topological considerations. The correspondence between vertices and branches of the skeleton graph and trabeculae is used in order to get local information on trabecular bone microarchitecture. In addition to real topological parameters, local structural information about trabeculae, such as length and volume distributions, are obtained. This method is applied to two sets of samples: six osteoporosis and six osteoarthritis bone samples. We demonstrate that skeleton graph analysis is a powerful technique to describe trabecular bone microarchitecture.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Material and methods
- 3. Results
- 4. Discussion and conclusion
- Acknowledgements
- References

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).

### 4. Discussion and conclusion

- Top of page
- Abstract
- 1. Introduction
- 2. Material and methods
- 3. Results
- 4. Discussion and conclusion
- Acknowledgements
- References

In this study, we have developed a new technique to analyse 3D digitized skeleton graph of porous media. This technique is validated from a topological criterion with a relative error of 0.42%. In the particular case of trabecular bone microarchitecture, the skeleton graph gives a template of the trabecular bone network. Our 3D skeleton graph analysis, which localizes each vertex and branch of the skeleton graph, allows us to extract each trabecula of the bone network as related to a branch.

A 3D skeletonization technique is used, based on topological invariant, and implemented with a sequential thinning algorithm. The topology is tested from β_{0}, β_{2} and N_{3} numbers, and is rigorously preserved between the initial 3D image and its 3D digitized skeleton graph. This validation is performed from a topological analysis of the initial image and the skeleton graph.

We then developed a 3D skeleton graph analysis to count and to isolate all the vertices and branches of the skeleton graph. This critical step is disturbed by topological artefacts (Thovert *et al*., 1993) due to both the digitization process and a loss of symmetry between solid and void phases in the case of a cubic grid. Hence, we introduce the concept of pixels clustering involving detection of artificial loops between neighbouring pixels. As a result, the digitized skeleton graph is analysed as a thin network, where each point of the network was either a single point (one pixel) or a multiple point (cluster of several pixels). Furthermore, we have introduced an order relationship between six-, 12- and eight-connectivity in order to define the degree of proximity between two skeleton points. The graph points are classified according to the number of their neighbours and the neighbourhood configuration of these neighbours. The numbers of vertices and branches are directly deduced from this classification. A topological criterion is used in order to validate the counting of vertices and branches.

A global analysis applied to our 12 femoral head samples permits us to distinguish the OP samples from the OA samples by global morphological parameters (Table 2). This distinction, indicating a loss of solid in OP samples, was confirmed by local analysis of the volume of trabeculae (Fig. 7). On the contrary, the length of the trabeculae (Fig. 6) indicates no difference about the relative position of the vertices, interpreted as connecting site of the trabeculae. The topology of the two groups evaluated from the number of loops and the number of coordination (mean number of branches linked to a same vertex) is the same. These results show that the two groups of samples have equivalent trabecular network, and only the thickness of the trabeculae is lower in OP samples than in OA samples. This interpretation must be cautious: firstly due to the limited size of the two groups of samples, and secondly because the osteoarthritis alterations at the head–neck boundary have not been previously studied (Ciarelli *et al*., 2000), the preferential site of osteoarthritis changes being the subchondral area of the femoral head.

Our new technique consisting of the use of the 3D skeleton graph allows local information on each trabecula independently of a size or shape model (Kinney *et al*., 1995; Hildebrand & Ruegsegger, 1997). The localization of each trabecula of the bone network is interesting firstly to get a local morphological analysis but allows several new interesting analyses. Such an approach could be used either to perform a 3D simulation of the bone remodeling process (Reeve *et al*., 1993; Thomsen *et al*., 1994, 1996; Smith *et al*., 1997), or a 3D simulation of architectural changes due to bone diseases or bone treatments (Steiniche *et al*., 1989; Boyce *et al*., 1995b), or in order to provide an evaluation of bone mechanical strength (Silva & Gibson, 1997; Jiang *et al*., 1998; Ladd *et al*., 1998; Mullender *et al*., 1998).

Finally, such a 3D skeleton graph analysis is not limited to trabecular bone study. It can be extended to the case of natural disordered porous media (Levitz, 1998).