## INTRODUCTION

Trabecular bone microarchitecture plays a main role in osteoporosis and prediction of fractures.^{(1,2)} In recent years, important advances have been made concerning three-dimensional (3D) imaging techniques. Automated serial sectioning,^{(3)} quantitative computerized tomography,^{(4)} microcomputed tomographic scanning,^{(5,6)} X-ray tomographic microscopy,^{(7,8)} or magnetic resonance imaging ^{(9–11)} are all recent techniques designed to obtain high-resolution 3D reconstruction of trabecular bone micro-architecture. Several image-analysis tools can characterize these 3D images and give some morphological and topological information about trabecular bone. Concerning morphological evaluation, the most current indicators are porosity, trabecular thickness, trabecular spacing,^{(12)} and anisotropy.^{(13)} Concerning topological evaluation, the most current indicator is the connectivity number.^{(14)} Another technique evaluates the fractal dimension of bone-marrow surface,^{(15,16)} giving some information on the degree of complexity or disorder of the bone microarchitecture. Three-dimensional trabecular microarchitecture characteristics are a potent source of information concerning bone strength. Several authors have reported high correlation between 3D microarchitecture characteristics and bone strength.^{(11,17–19)} Finite element analysis was recently used to estimate mechanical properties of bone from its 3D reconstruction.^{(20–23)}

All of these 3D techniques are well accepted. However, they actually remain in a research stage and are not used in clinical routine. Another recent trend characterizes trabecular bone microarchitecture on a two-dimensional (2D) radiographic image. This technique based on plain radiograph analysis can currently be performed in routine evaluation. It is very convenient for the patient and would be suitable for large populations. The 2D radiographic projection image appears as a gray-level texture. Several techniques for analyzing such a texture have been reported and can be classified according to the type of the 2D indicators measured. Structural measurements^{(24–26)} characterize the distribution and shape of the radiographic patterns appearing on the texture after thresholding. Fractal analysis^{(16,27–29)} expresses the roughness of the texture and characterizes the self-similarity of its gray-level variations over different scales.

Some interesting correlations have been reported between fractal analysis on 2D projection texture and microarchitecture parameters measured by histomorphometry^{(30,31)}, or between texture analysis and bone strength.^{(32–34)} Furthermore, clinical studies have shown a high level of statistically significant difference between normal subjects and osteoporotic patients with fractures. Caligiuri et al.^{(29)} calculated fractal dimension on lateral lumbar spine radiographs of 43 osteoporotic patients, with and without spine fractures, by a method based on surface area measurement. They used receiver operating characteristic (ROC) analysis to show that fractal dimension was a better discriminator than lumbar spine bone mineral density (BMD) to distinguish spine fracture cases (statistical level *P* < 0.01). Khosrovi et al.^{(35)} calculated fractal dimension on radius radiographs by a method based on the Fourier transform. They found that fractal dimension was statistically different (*P* < 0.002) between a group of 10 osteoporotic cases (bone density below normal and/or vertebral fractures) and a group of 10 controls. Benhamou et al.^{(36)} performed a preliminary study in osteoporosis to validate a fractal analysis based on the fractional Brownian motion.^{(37)} They found that fractal dimension calculated on calcaneus radiographs statistically distinguished (*P* < 0.0001) the osteoporotic group (*n* = 17) from the control group (*n* = 12). Pothuaud et al.^{(38)} have shown that fractal dimension calculated on calcaneus radiographs was a better discriminator than BMD (*P* < 0.0001) to distinguish osteoporotic patients with vertebral crush fractures (*n* = 39) from an age-matched control group (*n* = 39). Furthermore, this discrimination remained with high statistical level (*P* = 0.006) from osteoporotic and control subgroups with overlapping BMD values. These clinical studies have confirmed the growing interest in the 2D fractal analysis of bone radiographic texture. Nevertheless, such an image offers only 2D projection information of complex trabecular bone structure. The correspondence between gray-level variations on 2D projection texture and 3D trabecular bone microarchitecture remains unclear.

The purpose of this study was to understand how fractal dimension on 2D trabecular bone projection image could be related to 3D structural properties such as porosity or connectivity. In this aim, we used numerical algorithms to simulate trabecular bone microarchitecture changes. Such a numerical approach has already been used to find a correlation between 3D architectural parameters and bone strength,^{(17,19)} and in a recent work^{(39)} to characterize anisotropy relationships between plain radiographic patterns and 3D trabecular microarchitecture.

We have recently developed^{(40)} a new technique based on 3D skeletonization and 3D skeleton graph analysis. This technique permits detection and extraction of each trabecula of the bone network, and thereby permits modification of the bone microarchitecture by trabeculae thinning or thickening, as well as by trabeculae removal. These two alteration processes were applied to 3D bone images obtained by magnetic resonance imaging (MRI). Porosity and/or connectivity were progressively modified to obtain several models of bone structure. All of these 3D models were analyzed, and porosity and connectivity were evaluated, after which they were projected onto a 2D gray-level texture, and fractal dimension was calculated.