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12 May 2004 The degree of order determined by local conditional entropy: a new technique to assess osteoporotic changes of trabecular networks
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Abstract
Trabecular networks of cancellous bone show complex and stochastic characteristics, which have to be modelled in an adequate way to determine pathological changes of the network induced by osteoporosis. The analysis of complexity may be handled by the use of the Markov Theory, which is based on local interactions of a set of elements and thus can be applied to trabecular networks. The conditional entropy which is investigated as a potential measure of complexity, estimates the order of a structure and thus provides a means for classification of healthy versus osteoporotic bone structures. Since the conditional entropy is based on transition probabilities, stochastic characteristics are modelled, too. From a set of 29 female human vertebra T12, classified into two groups of 18 non-osteoporotic and 11 osteoporotic vertebrae axial biopsies were excised from the centre of the vertebral body. A digital model of the trabecular network was extracted with a Micro-CT device (FanBeam Microscope, Stratec, Pforzheim, Germany). Transition probabilities between neighboured voxels were coded as a set of 18 symbols describing the local dimension of a voxel and its relationship to its neighbours within a certain distance. A tree graph of the symbolic transitions coded the transition probabilities and founded a basis for the calculation of the local conditional entropy as a measure of order. The estimated local entropy for a distance at and above 10 voxels showed significantly higher values for the non-osteoporotic subjects than for the osteoporotic ones. This difference indicates, that non-osteoporotic trabecular networks show a higher degree of disorder compared to the osteoporotic ones.
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Wolfram H. W. Timm, Claus Christian Gluer, and Gerald Sommer "The degree of order determined by local conditional entropy: a new technique to assess osteoporotic changes of trabecular networks", Proc. SPIE 5370, Medical Imaging 2004: Image Processing, (12 May 2004); https://doi.org/10.1117/12.535218
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