# On the Symmetry of the Bone Structure Density over the Nasopalatine Foramen via Accurate Fractal Dimension Analysis

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## Abstract

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## 1. Introduction

- Body: it is most of the bone, pyramidal, is part of orbit, nasal cavity, infratemporal fossa, and the middle third of the face. It presents, in its anterior region, both the anterior nasal spine and the nasal notch.
- Frontal apophysis, which is articulated with nasal, frontal, ethmoid, and lacrimal bones.
- Zygomatic apophysis, that is articulated medially with the maxillary process of zygomatic bone.
- Palatine process, extending medially by forming the greatest part of hard palate, articulating in the middle line with the contralateral maxilla one, and later with palatal bone, and
- Alveolar process, which supports the upper teeth. The convex region that covers canine by vestibular is the canine eminence. There is a concavity mesial to this, i.e., the incisive fossa. Also, the canine fossa is a concavity which is distal to the canine one. The most posterior region of the alveolar process is the tuberosity of the maxilla [2].

- Anterior zone, which covers from the intermaxillary suture to the canine eminence.
- Middle zone: canine eminence and zygomatic-alveolar or infratemporal crest, and
- Posterior zone: distal to zygomatic-alveolar crest.

## 2. Materials and Methods

^{®}equipment, Planmeca ProMax 3D Max (Planmeca Oy, Helsinki, Finland) calibrated according to technical considerations. X-rays were obtained with the patient in the same position (prone position). The beam emission parameters were kV $=\phantom{\rule{3.33333pt}{0ex}}96$, mA $=8$, exposure time of 12 s ($11.94$ s) with an image size of $501\times 501\times 466$ voxels (each voxel being equivalent to 200 $\mathsf{\mu}$m). The evaluation software used was the Romexis 2.5.1

^{®}program (Planmeca Oy, Helsinki, Finland), which allowed observing the image in a multiple window where the axial, coronal and sagittal planes can be visualized in $0.2\phantom{\rule{0.166667em}{0ex}}$mm intervals, in addition to a 3D vision. As indicated above, the sample was divided into three groups. We proceed to select a specific ROI that was obtained in the axial plane at the height of the nasal spine, visualising the nasopalatine foramen and the canine mamelons on both sides, see Figure 1.

^{®}code was written to perform all the statistical analyses.

## 3. Results and Discussion

#### 3.1. Description of the Sample

- Age: it was found a mean age equal to $53.67$ years with a standard deviation of $8.20$ years.
- DCV: a mean of $7.54$ and a standard deviation equal to $1.53$ were found.
- DCP: with a mean of $3.99$ and a standard deviation equal to $1.65$.
- DVD: a mean of $12.95$ and a standard deviation of $1.75$ were obtained.
- DVI: with a mean of $12.98$ and a standard deviation equal to $1.34$.
- Area: a mean of $5.63$ and a standard deviation equal to $2.18$ were found.
- W: a mean equal to $2.96$ and a standard deviation of $0.71$ were found.
- H: with a mean equal to $2.23$ and a standard deviation of $0.59$.
- Mean: a mean equal to $158.66$ and a standard deviation of $120.83$ was obtained.
- DIM: a mean fractal dimension of $1.69$ and a standard deviation equal to $0.09$ were found.

#### 3.2. Sample Description by Sex

#### 3.2.1. Female Population

- Age: a mean age of $54.62$ years with a standard deviation equal to $8.99$ years was found.
- DCV: a mean equal to $7.29$ and a standard deviation of $1.63$ were obtained.
- DCP: with a mean equal to $4.05$ and a standard deviation equal to $1.69$.
- DVD: a mean of $12.82$ and a standard deviation equal to $1.62$ were found.
- DVI: it was found a mean equal to $12.38$ with a standard deviation of $1.62$.
- Area: with a mean equal to $5.33$ and a standard deviation equal to $1.98$.
- W: a mean equal to $3.03$ and a standard deviation of $0.72$ were found.
- H: a mean of $2.26$ and a standard deviation equal to $0.44$ were found.
- Mean: a mean of $142.65$ and a standard deviation of $107.52$ were obtained.
- DIM: a mean fractal dimension of $1.68$ and a standard deviation equal to $0.13$ were found.

#### 3.2.2. Male Population

- Age: it was found a mean age equal to $53.03$ years with a standard deviation of $7.05$ years.
- DCV: a mean of $8.75$ and a standard deviation equal to $1.35$ were found.
- DCP: with a mean of $4.13$ and a standard deviation equal to $1.26$.
- DVD: a mean of $13.71$ and a standard deviation of $1.96$ were obtained.
- DVI: with a mean equal to $13.13$ and a standard deviation of $1.70$.
- Area: a mean of $5.18$ and a standard deviation equal to $2.60$ were found.
- W: a mean equal to $2.64$ and a standard deviation of $0.73$ were found.
- H: with a mean of $2.01$ and a standard deviation equal to $0.66$.
- Mean: a mean equal to $160.16$ and a standard deviation of $98.83$ were obtained.
- DIM: a mean fractal dimension equal to $1.69$ and a standard deviation of $0.10$ were found.

#### 3.3. Some Comparisons by Sex

- Age: no significative differences were found. In fact, a p-value equal to $0.12$ was obtained by the Mann–Whitney test.
- DCV: in this case, significative differences were found by a p-value of 0.01* (* means that significative differences were found at a confidence level of 95%.) in the Mann–Whitney test.
- DCP: no significative differences were found by a Mann–Whitney p-value equal to $0.25$.
- DVD: a p-value of $0.25$ was provided by the Mann–Whitney test. As such, no significative differences were found.
- DVI: the Mann–Whitney test provided a p-value equal to $0.11$. Thus, no significative differences were found.
- Area: no differences were observed. In fact, the Mann–Whitney test provided a p-value equal to $0.13$.
- W: a p-value of $0.71$ was found in the Mann–Whitney test. As such, no significative differences were found.
- H: There were no significative differences. In fact, the Mann–Whitney test provided a p-value of $0.26$.
- Mean: the Mann–Whitney test threw a p-value equal to $0.45$, so no significative differences were found.
- DIM: no significative differences were found. In fact, a p-value of $0.33$ was provided by the Mann–Whitney test.

#### 3.4. A First Step towards Symmetry

#### 3.5. Fractal Dimension Analysis

#### 3.6. Fractal Dimension Analysis for Group One Patients

#### 3.7. Fractal Dimension Analysis for Patients in Group Two

#### 3.8. Analysis of Fractal Dimension by Groups

#### Fractal Dimension Comparison between Groups One and Two

#### 3.9. A Note on the Multiple Comparisons Problem

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Mathematical Foundations on Fractal Dimension Calculations

**Definition**

**A1.**

**Definition**

**A2.**

**Γ**is a fractal structure provided that the two following statements hold:

- (i)
- for each $A\in {\Gamma}_{n+1}$, there exists $B\in {\Gamma}_{n}$ such that $A\subseteq B$.
- (ii)
- $B=\cup \{A\in {\Gamma}_{n+1}:A\subseteq B\}$ for all $B\in {\Gamma}_{n}$.

**Figure A1.**First two levels of the natural fractal structure on $[0,1]\times [0,1]$. Notice that the first level consists of four squares with sides equal to $\frac{1}{2}$, ${\Gamma}_{2}$ contains ${4}^{2}$ squares with sides equal to $\frac{1}{{2}^{2}}$, and in general, level n consists of ${4}^{n}$ squares with sides equal to $\frac{1}{{2}^{n}}$.

**Theorem**

**A1**

**.**Let

**Δ**be the natural fractal structure on $[0,1]\times [0,1]$ and assume that $[0,1]$ is endowed with the fractal structure

**Γ**with levels given by ${\Gamma}_{n}=\{[{\textstyle \frac{k}{{2}^{2n}}},{\textstyle \frac{1+k}{{2}^{2n}}}]:k=0,1,\dots ,{2}^{2n}-1\}$. In addition, let F be a subset of $[0,1]\times [0,1]$ and $\alpha :[0,1]\to [0,1]\times [0,1]$ a function with $\mathbf{\Delta}=\alpha (\mathbf{\Gamma})$. The (lower/upper) box dimension of F can be calculated by the following expressions:

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**Figure 2.**Graphical representation of an actual cone beam computed tomography (CBCT) from a patient that took part in our study (left) and its corresponding binary images.

**Figure 3.**The blue line in each plot at the left illustrates the empirical distribution of the fractal dimension values for all the binary images analyzed from each group in our patient sample. Further, the discontinuous line marks the mean fractal dimension of the binary images from all the scanners analyzed. Notice also that the orange line (resp., the green line) represents the empirical distribution of the fractal dimensions of the left side (resp., the right side) binary images from each group of patients. On the other hand, each graph at the right depicts the empirical distribution of the differences (in absolute value) between each left side binary image and its corresponding right side one from each group of patients. The discontinuous line represents the mean of such differences.

**Figure 4.**Empirical distributions of the fractal dimensions of the binary images from both groups one (blue line) and two (picture at the first row) and empirical distributions of the fractal dimensions of their lateral binary images (second row). The discontinuous straight lines mark the mean fractal dimension of each group and each kind of binary images.

**Table 1.**Sample description by attributes. Recall that DCV denotes distance anterior wall nasopalatine hole to anterior nasal spine, DCP means distance back wall foramen (NF) to border palate bone, DVD is distance right side wall NF to right canine mamelon, DVI refers to distance left lateral wall NF to left canine mamelon, Area is the area of the NF, and W, H, mean, and standard deviation are provided by the software.

Whole Sample (n = 130) | Female Group (n = 68) | Male Group (n = 62) | ||||
---|---|---|---|---|---|---|

Attribute | Mean | Std. Dev. | Mean | Std. Dev. | Mean | Std. Dev. |

Age | 53.67 | 8.20 | 54.62 | 8.99 | 53.03 | 7.05 |

DCV | 7.54 | 1.53 | 7.29 | 1.63 | 8.75 | 1.35 |

DCP | 3.99 | 1.65 | 4.05 | 1.69 | 4.13 | 1.26 |

DVD | 12.95 | 1.75 | 12.82 | 1.62 | 13.71 | 1.96 |

DVI | 12.98 | 1.34 | 12.38 | 1.62 | 13.13 | 1.70 |

Area | 5.63 | 2.18 | 5.33 | 1.98 | 5.18 | 2.60 |

W | 2.96 | 0.71 | 3.03 | 0.72 | 2.64 | 0.73 |

H | 2.23 | 0.59 | 2.26 | 0.44 | 2.01 | 0.66 |

Mean | 158.66 | 120.83 | 142.65 | 107.52 | 160.16 | 98.83 |

DIM | 1.69 | 0.09 | 1.68 | 0.13 | 1.69 | 0.10 |

Dim. of Binary Images | Dim. of Left Images | Dim. of Right Images | |||||
---|---|---|---|---|---|---|---|

n | Mean | Std. Dev. | Mean | Std. Dev. | Mean | Std. Dev. | |

Whole sample | 130 | 1.70 | 0.09 | 1.68 | 0.08 | 1.72 | 0.08 |

Group One | 65 | 1.67 | 0.06 | 1.66 | 0.04 | 1.68 | 0.04 |

Group Two | 65 | 1.70 | 0.06 | 1.68 | 0.04 | 1.70 | 0.04 |

**Table 3.**Analysis of the differences among the fractal dimensions of each left binary image and its corresponding right side one for each group.

Differences (in abs.) among Left and Right DIMs | |||
---|---|---|---|

Mean | Std. Dev. | Mann–Whitney (p-Value) | |

Whole sample | 0.09 | 0.07 | 0.21 |

Group One | 0.04 | 0.03 | 0.19 |

Group Two | 0.05 | 0.04 | 0.38 |

**Table 4.**Comparison of empirical distributions of (lateral) fractal dimensions from each group. They are provided the p-values from the Mann–Whitney tests involving both groups. No significative differences were found at a confidence level of $95\%$.

Group One vs. Two Comparison | |
---|---|

CBCT images | M-W p-Value |

Whole images | 0.28 |

Left images | 0.29 |

Right images | 0.19 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Bornstein, M.M.; Fernández-Martínez, M.; Guirao, J.L.G.; Gómez-García, F.J.; Guerrero-Sánchez, Y.; López-Jornet, P.
On the Symmetry of the Bone Structure Density over the Nasopalatine Foramen via Accurate Fractal Dimension Analysis. *Symmetry* **2019**, *11*, 202.
https://doi.org/10.3390/sym11020202

**AMA Style**

Bornstein MM, Fernández-Martínez M, Guirao JLG, Gómez-García FJ, Guerrero-Sánchez Y, López-Jornet P.
On the Symmetry of the Bone Structure Density over the Nasopalatine Foramen via Accurate Fractal Dimension Analysis. *Symmetry*. 2019; 11(2):202.
https://doi.org/10.3390/sym11020202

**Chicago/Turabian Style**

Bornstein, Michael M., Manuel Fernández-Martínez, Juan L. G. Guirao, Francisco J. Gómez-García, Yolanda Guerrero-Sánchez, and Pía López-Jornet.
2019. "On the Symmetry of the Bone Structure Density over the Nasopalatine Foramen via Accurate Fractal Dimension Analysis" *Symmetry* 11, no. 2: 202.
https://doi.org/10.3390/sym11020202