Application of fractal dimension to trabecular bone and its implications with pathologies related to aging

Abstract

In the first work, published in the Journal of Intelligent & Fuzzy Systems, we developed the novel idea of computing the box-counting dimension by analyzing the cut points with the boxes that crosslink the subsets of the plane of a covering curve modeled by a one-dimensional function. We emphasize that this allows the computation of the box-counting dimension in a much more precise way than up to the present in which the counting was done in a more rudimentary way. As an application of our theoretical mathematical results in this work, we explore the fractal nature of a patient with periodontitis whose trabecular structure can vary with respect to healthy subjects. We selected this patient specifically because his CBCT was of very high quality and resolution. We perform an analysis of computation times between the three fractal invariants. In the second paper, published in the journal Discrete and Continuous Dynamical Systems S, we provide all the mathematical tools that we apply to carry out the clinical study on the samples collected. More specifically, this paper contained the basic concepts of the box-counting dimension and described the concept of fractal structure (on which our theoretical results are based). In particular, we define the natural fractal structure in the Euclidean plane. Since we were interested in the calculation of the box-counting dimension for flat CBCT images, all our theoretical development is in the context of the subsets of the plane. We present a section where our computational results are discussed. In fact, we explain how the calculations with respect to the box-counting dimension of the CBCT images have been carried out through three different approaches, namely the standard box-counting algorithm and two novel approaches based on our theorems. In the third work, published in Symmetry, the objective was to describe all the anatomical considerations that surround the nasopalalatin foramen, relating them to the study of bone density, through the fractal dimension, in that same area. We consecutively selected a sample of 100 patients, all with CBCT performed for treatment needs. We chose a specific window (ROI), which coincides with an axial cut at the level of the nasopalatine foramen exit. We analyze different anthropometric measures together with the analysis of the fractal dimension. This was an observational and transversal clinical study, in which we selected a sample of 100 patients consecutively from the dental clinic of the University of Murcia (Spain). The study was approved by the Bioethics Committee of the University of Murcia. All individuals gave their informed consent in writing before participating. The inclusion criteria were applied: patients in both systemic and dental health conditions, not pregnant, that did not contain artifacts. Of these 100 initial patients, we had 77 who met all the criteria described above (5 were discarded because they were treated with bisphosphonates and 18 from the beginning). The images that presented artifacts or were not considered with sufficient quality to be able to apply the mathematical computation algorithm of the fractal dimension were discarded. We proceeded to select a specific ROI that was obtained in the axial plane, visualizing the nasopalatine foramen and the canine mamelons on both sides. Thanks to the third article we can conclude that the fractal dimension is a mathematical invariant that behaves symmetrically for binary images of the scanner of each subject of our study sample for the contour of the nasopalalatin foramen. We also concluded that there were no significant differences between all the anthropometric measurements used neither in the subjects themselves nor in the different groups. Therefore, a symmetry pattern is observed at all levels.