American Institute of Mathematical Sciences

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, doi: 10.3934/dcdss.2019105
References:
 [1] A. Estrugo-Devesa, J. Segura-Egea, L. García-Vicente, M. Schemel-Suárez, A. Blanco-Carrión, E. Jané-Salas and J. López-López, Correlation between mandibular bone density and skeletal bone density in a Catalonian postmenopausal popultion, Oral Surgery, Oral Medicine, Oral Pathology and Oral Radiology, 125 (2018), 495-502. [2] M. Fernández-Martínez, A survey on fractal dimension for fractal structures, Applied Mathematics and Nonlinear Sciences, 2 (2016), 437-472. [3] M. Fernández-Martínez and M. A. Sánchez-Granero, A new fractal dimension for curves based on fractal structures, Topology and its Applications, 203 (2016), 108-124. doi: 10.1016/j.topol.2015.12.080. [4] M. Fernández-Martínez and M. A. Sánchez-Granero, Calculating the fractal dimension in higher dimensional spaces, preprint. [5] M. Fernández-Martínez and M. A. Sánchez-Granero, Fractal dimension for fractal structures, Topology and Its Applications, 163 (2014), 93-111. doi: 10.1016/j.topol.2013.10.010. [6] M. Fernández-Martínez and M. A. Sánchez-Granero, How to calculate the Hausdorff dimension using fractal structures, Applied Mathematics and Computation, 264 (2015), 116-131. doi: 10.1016/j.amc.2015.04.059. [7] E. Jagelavičienė and R. Kubilius, The relationship between general osteoporosis of the organism and periodontal diseases, Medicina (Kaunas), 42 (2006), 613-618. [8] A. Jordão Camargo, E. Saito Arita, M. C. Cortéz de Fernández and P. C. Aranha Watanabe, Comparison of Two Radiological Methods for Evaluation of Bone Density in Postmenopausal Women, International Journal of Morphology, 33 (2015), 732-736. [9] A. N. Law, A.-M. Bollen and S.-K. Chen, Detecting osteoporosis using dental radiographs: A comparison of four methods, The Journal of American Dental Association, 127 (1996), 1734-1742. [10] P. L. Lin, P. W. Huang, P. Y. Huang and H. C. Hsu, Alveolar bone-loss area localization in periodontitis radiographs based on threshold segmentation with a hybrid feature fused of intensity and the H-value of fractional Brownian motion model, Computer Methods and Programs in Biomedicine, 121 (2015), 117-126. [11] F. Martínez-López, M. A. Cabrerizo-Vílchez and R. Hidalgo-Álvarez, An improved method to estimate the fractal dimension of physical fractals based on the Hausdorff definition, Physica A: Statistical Mechanics and its Applications, 298 (2001), 387-399. [12] K. R. Phipps, B. K. S. Chan, T. E. Madden, N. C. Geurs, M. S. Reddy, C. E. Lewis and E. S. Orwoll, Longitudinal Study of Bone Density and Periodontal Disease in Men, Journal of Dental Research, 86 (2007), 1110-1114. [13] E. Sener, S. Cinarcik and B. Guniz Baksi, Use of fractal analysis for the discrimination of trabecular changes between individuals with healthy gingiva or moderate periodontitis, Journal of Periodontology, 86 (2015), 1364-1369. [14] M. Tezal, J. Wactawski-Wende, S. G. Grossi, A. W. Ho, R. Dunford and R. J. Genco, The Relationship Between Bone Mineral Density and Periodontitis in Postmenopausal Women, Journal of Periodontology, 71 (2000), 1492-1498. [15] B. Tolga Suer, Z. Yaman and B. Buyuksarac, Correlation of Fractal Dimension Values with Implant Insertion Torque and Resonance Frequency Values at Iimplant Recipient Sites, The International Journal of Oral & Maxillofacial Implants, 31 (2016), 55-62. [16] S. X. Updike and H. Nowzari, Fractal analysis of dental radiographs to detect periodontitis-induced trabecular changes, Journal of Periodontal Research, 43 (2008), 658-664. [17] A. Yoshihara, Y. Seida, N. Hanada and H. Miyazaki, A longitudinal study of the relationship between periodontal disease and bone mineral density in community-dwelling older adults, Journal of Clinical Periodontology, 31 (2004), 680-684. [18] A. Yoshihara, Y. Seida, N. Hanada, K. Nakashima and H. Miyazaki, The relationship between bone mineral density and the number of remaining teeth in community-dwelling older adults, Journal of Oral Rehabilitation, 32 (2005), 735-740.

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References:
 [1] A. Estrugo-Devesa, J. Segura-Egea, L. García-Vicente, M. Schemel-Suárez, A. Blanco-Carrión, E. Jané-Salas and J. López-López, Correlation between mandibular bone density and skeletal bone density in a Catalonian postmenopausal popultion, Oral Surgery, Oral Medicine, Oral Pathology and Oral Radiology, 125 (2018), 495-502. [2] M. Fernández-Martínez, A survey on fractal dimension for fractal structures, Applied Mathematics and Nonlinear Sciences, 2 (2016), 437-472. [3] M. Fernández-Martínez and M. A. Sánchez-Granero, A new fractal dimension for curves based on fractal structures, Topology and its Applications, 203 (2016), 108-124. doi: 10.1016/j.topol.2015.12.080. [4] M. Fernández-Martínez and M. A. Sánchez-Granero, Calculating the fractal dimension in higher dimensional spaces, preprint. [5] M. Fernández-Martínez and M. A. Sánchez-Granero, Fractal dimension for fractal structures, Topology and Its Applications, 163 (2014), 93-111. doi: 10.1016/j.topol.2013.10.010. [6] M. Fernández-Martínez and M. A. Sánchez-Granero, How to calculate the Hausdorff dimension using fractal structures, Applied Mathematics and Computation, 264 (2015), 116-131. doi: 10.1016/j.amc.2015.04.059. [7] E. Jagelavičienė and R. Kubilius, The relationship between general osteoporosis of the organism and periodontal diseases, Medicina (Kaunas), 42 (2006), 613-618. [8] A. Jordão Camargo, E. Saito Arita, M. C. Cortéz de Fernández and P. C. Aranha Watanabe, Comparison of Two Radiological Methods for Evaluation of Bone Density in Postmenopausal Women, International Journal of Morphology, 33 (2015), 732-736. [9] A. N. Law, A.-M. Bollen and S.-K. Chen, Detecting osteoporosis using dental radiographs: A comparison of four methods, The Journal of American Dental Association, 127 (1996), 1734-1742. [10] P. L. Lin, P. W. Huang, P. Y. Huang and H. C. Hsu, Alveolar bone-loss area localization in periodontitis radiographs based on threshold segmentation with a hybrid feature fused of intensity and the H-value of fractional Brownian motion model, Computer Methods and Programs in Biomedicine, 121 (2015), 117-126. [11] F. Martínez-López, M. A. Cabrerizo-Vílchez and R. Hidalgo-Álvarez, An improved method to estimate the fractal dimension of physical fractals based on the Hausdorff definition, Physica A: Statistical Mechanics and its Applications, 298 (2001), 387-399. [12] K. R. Phipps, B. K. S. Chan, T. E. Madden, N. C. Geurs, M. S. Reddy, C. E. Lewis and E. S. Orwoll, Longitudinal Study of Bone Density and Periodontal Disease in Men, Journal of Dental Research, 86 (2007), 1110-1114. [13] E. Sener, S. Cinarcik and B. Guniz Baksi, Use of fractal analysis for the discrimination of trabecular changes between individuals with healthy gingiva or moderate periodontitis, Journal of Periodontology, 86 (2015), 1364-1369. [14] M. Tezal, J. Wactawski-Wende, S. G. Grossi, A. W. Ho, R. Dunford and R. J. Genco, The Relationship Between Bone Mineral Density and Periodontitis in Postmenopausal Women, Journal of Periodontology, 71 (2000), 1492-1498. [15] B. Tolga Suer, Z. Yaman and B. Buyuksarac, Correlation of Fractal Dimension Values with Implant Insertion Torque and Resonance Frequency Values at Iimplant Recipient Sites, The International Journal of Oral & Maxillofacial Implants, 31 (2016), 55-62. [16] S. X. Updike and H. Nowzari, Fractal analysis of dental radiographs to detect periodontitis-induced trabecular changes, Journal of Periodontal Research, 43 (2008), 658-664. [17] A. Yoshihara, Y. Seida, N. Hanada and H. Miyazaki, A longitudinal study of the relationship between periodontal disease and bone mineral density in community-dwelling older adults, Journal of Clinical Periodontology, 31 (2004), 680-684. [18] A. Yoshihara, Y. Seida, N. Hanada, K. Nakashima and H. Miyazaki, The relationship between bone mineral density and the number of remaining teeth in community-dwelling older adults, Journal of Oral Rehabilitation, 32 (2005), 735-740.
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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