Fractal organization of trabecular bone images on calcaneus radiographs

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Date of Original Version



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 α = 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 μm, then above 900 μm, where linear regression produced high mean correlation coefficients (r ≥ 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. Copyright © 1994 ASBMR

Publication Title

Journal of Bone and Mineral Research