# American Institute of Mathematical Sciences

August & September  2019, 12(4&5): 1527-1534. doi: 10.3934/dcdss.2019105

## A novel approach to improve the accuracy of the box dimension calculations: Applications to trabecular bone quality

 1 University Centre of Defence at the Spanish Air Force Academy, MDE-UPCT, 30720 Santiago de la Ribera, Murcia, Spain 2 University of Murcia, Department of Dermatology, Stomatology, Radiology and Physical Medine, Morales Messeguer General Universitary Hospital, Avda. Marqus de los Velez, 30008 Murcia, Spain

* Corresponding author: Yolanda Guerrero-Sánchez

Received  August 2017 Revised  January 2018 Published  November 2018

Fund Project: The first author has been partially supported by Grants No. 19219/PI/14 from Fundación Séneca of Región de Murcia and No. MTM2015-64373-P from Spanish Ministry of Economy and Competitiveness.

Fractal dimension and specifically, box-counting dimension, is the main tool applied in many fields such as odontology to detect fractal patterns applied to the study of bone quality. However, the effective computation of such invariant has not been carried out accurately in literature. In this paper, we propose a novel approach to properly calculate the fractal dimension of a plane subset and illustrate it by analysing the box dimension of a trabecular bone through a computed tomography scan.

Citation: M. Fernández-Martínez, Yolanda Guerrero-Sánchez, Pía López-Jornet. A novel approach to improve the accuracy of the box dimension calculations: Applications to trabecular bone quality. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1527-1534. doi: 10.3934/dcdss.2019105
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##### References:
First two levels, $\Gamma_1$ and $\Gamma_2$, of a (quite natural) fractal structure on the Sierpiński gasket. The elements in each covering are depicted in pale blue
First two levels of the natural fractal structure on $[0, 1]\times [0, 1]$ as a Euclidean subset
The two images above show the first two levels, $\Delta_1$ and $\Delta_2$, of a fractal structure induced by $\mathbf{\Gamma }$ on the image set of a Brownian motion, where $\mathbf{\Gamma }$ is the natural fractal structure on $[0, 1]$
The two images above depict the construction of the first two levels of the Hilbert's plane-filling curve
CT scan from a periodontitis patient (left) and detailed view of a trabecular bone from that scan where it can be identified the fractal nature of periodontal tissues
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