Bone density is not the unique factor conditioning bone strength. Trabecular bone microarchitecture also plays an important role. We have developed a fractal evaluation of trabecular bone microarchitecture on calcaneus radiographs. Fractal models may provide a single numeric evaluation (the fractal dimension) of such complex structures. Our evaluation results from an analysis of images with a varying range of gray levels, without binarization of the image. It is based on the fractional brownian motion model, or more precisely on the analysis of its increment, the fractional gaussian noise (FGN). The use of this model may be considered validated if two conditions are fulfilled: the gaussian repartition and the self-similarity of our data. The gaussian repartition of intermediate lines of these images was tested on a sample of 32,800 lines from 82 images. Following a chi-square goodness-of-fit test, it was checked in 86% of these lines for alpha = 0.01. The self-similarity was tested on 20 images by two estimators, the variance method of Pentland and the spectrum method of Fourier. Self-similarity is defined by lined-up points in a log-log plot of the FGN spectrum or of the variance as a function of the lag. We found two self-similarity areas between scales of analysis ranging from 105 to 420 microns, then above 900 microns, where linear regression produced high mean correlation coefficients (r > or = 0.97). Following this validation, we studied the reproducibility of this new technique. Intra- and interobserver reproducibility, influence of transferring the region of interest, and long-term reproducibility were assessed and given CV of 0.61 +/- 0.15, 0.68 +/- 0.47, 0.53 +/- 0.16, and 2.07 +/- 0.84%, respectively. These data have allowed us to validate the use of this fractal model by checking the fractal organization of our radiographic images analyzed by the model. The good reproducibility of successive x-rays in the same subject allows us to undertake population studies and to envisage longitudinal series.

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The purpose of this communication is to compare the quality of synthesis and analysis methods for fractional Brownian motion (fBm). Three conditions are necessary and sufficient for a process to be fBm. This allows to check the correspondance between a given synthesis method and the fBm model. Two synthesis methods among 6 tested give good results. Reference signals obtained by these 2 methods are used to evaluate 8 analysis methods. Three estimators have a low bias and a variance close to the CramerRao lower bound.

A fractal analysis based on the Fractional Brownian Motion is proposed to characterize the architecture of the trabecular bone network. This analysis is applied on digital image lines of X-ray views. The study of self-similarity on images of calcaneus radiographs allows the determination of two length scales where the fractal hypothesis is verified. The first one is ranging from about 100 to 550 µm. This area is used to characterize the bone structure at the trabecular thickness scale. The second one starts about above 900 µm, it can be related to the trabecular spacing. A statistical approach allow to choose the fractal area given a level of significance.

Non-invasive and in vivo assessment of bone architectural changes at high resolution is of considerable interest in osteoporosis. In this note, the use of an x-ray acquisition system in the evaluation of the architectural quality of trabecular bone by radiographic texture analysis is optimized to achieve good long-term reproducibility. First, radiographic and digitization processes are modelled and defined. Procedures to make radiographs and their digital images are fixed. Then, measurements of the modulation transfer function (MTF) of the entire acquisition chain were completed. These measurements provide an MTF in excess of 30% at a spatial frequency of 2.5 lp/mm. Also, results of a fractal texture analysis made on digital images of calcaneus radiographs show a mean coefficient of variation of 2.07%. These data show that good long-term reproducibility can make the x-ray acquisition system efficient for patient follow-up, or evaluation of treatment regimes for osteoporosis. Finally, it is shown that fractal texture parameters are statistically different in an osteoporotic population and in a control group. Therefore, this system should also be of medical interest.

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