Markovianity of Trabecular networks

Clinically established methods to diagnose osteoporosis, a systemic disease with multiple implications to the human’s well being are bound to the bone mineral density. Since it is well known, that the bone structure changes under osteoporosis, this thesis investigates, whether structural changes represent information independent of bone mineral density, and whether this structural information, obtained by innovative mathematical techniques improves first the ability to distinguish between osteoporotic and healthy structures, second the measurement of a risk to belong to the osteoporotic group, and third the prediction of a biomechanical measure, the failure load, compared to existing standard structural measures. This in vitro study utilised mainly Micro-Computer Tomography (μ-CT) and for some proof of concept High-Resolution Computer Tomography (HRCT) to obtain digital datasets of trabecular structures. Having the aim to preserve and exploit the complexity inherent to trabecular structures, the following methods of the theory of Markov processes were investigated: Markov random fields, Markov point processes, a Markov graph based method of conditional entropy as well as Hidden Markov Models. These techniques allowed both to focus on the investigation of simpler, non-complex substructures and at the same time keep structural information of the whole structure by investigating these substructures in parallel. The study confirmed the independency of structural information from bone mineral density. Moreover, both a measure of clustering based on Markov point processes and the Markov graph based conditional entropy improved the discrimination of osteoporotic and healthy trabecular networks compared to standard structural variables. Furthermore, the explanation of failure load was improved beyond bone mineral density by the application of entropy. Additionally, the standardised odds ratios were higher again for the Markov graph based conditional entropy than for the standard variables and the bone mineral density. Summarising, by building a bridge between natural patterns of human cancellous bone to mathematical and stochastic algorithms capable of modeling complex structures this in vitro study proposed a successful way to catch the specifics of these natural patterns and to improve the current structural analysis methods.

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