Fractal organization of trabecular bone images on calcaneus radiographs. 1994

C L Benhamou, and E Lespessailles, and G Jacquet, and R Harba, and R Jennane, and T Loussot, and D Tourliere, and W Ohley
Pôle d'activité Rhumatologie, CHR Orléans La Source, France.

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.

UI MeSH Term Description Entries
D008962 Models, Theoretical Theoretical representations that simulate the behavior or activity of systems, processes, or phenomena. They include the use of mathematical equations, computers, and other electronic equipment. Experimental Model,Experimental Models,Mathematical Model,Model, Experimental,Models (Theoretical),Models, Experimental,Models, Theoretic,Theoretical Study,Mathematical Models,Model (Theoretical),Model, Mathematical,Model, Theoretical,Models, Mathematical,Studies, Theoretical,Study, Theoretical,Theoretical Model,Theoretical Models,Theoretical Studies
D011859 Radiography Examination of any part of the body for diagnostic purposes by means of X-RAYS or GAMMA RAYS, recording the image on a sensitized surface (such as photographic film). Radiology, Diagnostic X-Ray,Roentgenography,X-Ray, Diagnostic,Diagnostic X-Ray,Diagnostic X-Ray Radiology,X-Ray Radiology, Diagnostic,Diagnostic X Ray,Diagnostic X Ray Radiology,Diagnostic X-Rays,Radiology, Diagnostic X Ray,X Ray Radiology, Diagnostic,X Ray, Diagnostic,X-Rays, Diagnostic
D002111 Calcaneus The largest of the TARSAL BONES which is situated at the lower and back part of the FOOT, forming the HEEL. Heel Bone,Bone, Heel
D006801 Humans Members of the species Homo sapiens. Homo sapiens,Man (Taxonomy),Human,Man, Modern,Modern Man
D015203 Reproducibility of Results The statistical reproducibility of measurements (often in a clinical context), including the testing of instrumentation or techniques to obtain reproducible results. The concept includes reproducibility of physiological measurements, which may be used to develop rules to assess probability or prognosis, or response to a stimulus; reproducibility of occurrence of a condition; and reproducibility of experimental results. Reliability and Validity,Reliability of Result,Reproducibility Of Result,Reproducibility of Finding,Validity of Result,Validity of Results,Face Validity,Reliability (Epidemiology),Reliability of Results,Reproducibility of Findings,Test-Retest Reliability,Validity (Epidemiology),Finding Reproducibilities,Finding Reproducibility,Of Result, Reproducibility,Of Results, Reproducibility,Reliabilities, Test-Retest,Reliability, Test-Retest,Result Reliabilities,Result Reliability,Result Validities,Result Validity,Result, Reproducibility Of,Results, Reproducibility Of,Test Retest Reliability,Validity and Reliability,Validity, Face
D015588 Observer Variation The failure by the observer to measure or identify a phenomenon accurately, which results in an error. Sources for this may be due to the observer's missing an abnormality, or to faulty technique resulting in incorrect test measurement, or to misinterpretation of the data. Two varieties are inter-observer variation (the amount observers vary from one another when reporting on the same material) and intra-observer variation (the amount one observer varies between observations when reporting more than once on the same material). Bias, Observer,Interobserver Variation,Intraobserver Variation,Observer Bias,Inter-Observer Variability,Inter-Observer Variation,Interobserver Variability,Intra-Observer Variability,Intra-Observer Variation,Intraobserver Variability,Inter Observer Variability,Inter Observer Variation,Inter-Observer Variabilities,Inter-Observer Variations,Interobserver Variabilities,Interobserver Variations,Intra Observer Variability,Intra Observer Variation,Intra-Observer Variabilities,Intra-Observer Variations,Intraobserver Variabilities,Intraobserver Variations,Observer Variations,Variabilities, Inter-Observer,Variabilities, Interobserver,Variabilities, Intra-Observer,Variabilities, Intraobserver,Variability, Inter-Observer,Variability, Interobserver,Variability, Intra-Observer,Variability, Intraobserver,Variation, Inter-Observer,Variation, Interobserver,Variation, Intra-Observer,Variation, Intraobserver,Variation, Observer,Variations, Inter-Observer,Variations, Interobserver,Variations, Intra-Observer,Variations, Intraobserver,Variations, Observer
D017709 Fractals Patterns (real or mathematical) which look similar at different scales, for example the network of airways in the lung which shows similar branching patterns at progressively higher magnifications. Natural fractals are self-similar across a finite range of scales while mathematical fractals are the same across an infinite range. Many natural, including biological, structures are fractal (or fractal-like). Fractals are related to "chaos" (see NONLINEAR DYNAMICS) in that chaotic processes can produce fractal structures in nature, and appropriate representations of chaotic processes usually reveal self-similarity over time. Fractal

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