## Introduction

The Frankfort horizontal (FH) is a plane that intersects both porions and the left orbitale. However, other combinations of points have also been used to define this plane in 3-dimensional cephalometry. These variations are based on the hypothesis that they do not affect the cephalometric analysis. We investigated the validity of this hypothesis.

## Methods

The material included cone-beam computed tomography data sets of 82 adult subjects with Class I molar relationship. A third-party method of cone-beam computed tomography-based 3-dimensional cephalometry was performed using 7 setups of the FH plane. Six lateral cephalometric hard tissue measurements relative to the FH plane were carried out for each setup. Measurement differences were calculated for each pair of setups of the FH plane. The number of occurrences of differences greater than the limits of agreement was counted for each of the 6 measurements.

## Results

Only 3 of 21 pairs of setups had no occurrences for the 6 measurements. No measurement had no occurrences for the 21 pairs of setups. Setups based on left or right porion and both orbitales had the greatest number of occurrences for the 6 measurements.

## Conclusions

This investigation showed that significant and undesirable measurement differences can be produced by varying the definition of the FH plane.

## Highlights

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Test method shows that different setups of the FH plane affect the cephalometric analysis.

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Replacing the left orbitale by the right one in the FH plane affects the results.

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Replacing the right orbitale by the left one in the FH plane affects the results.

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Replacing the left porion by the right one in the FH plane affects the results.

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Replacing the right porion by the left one in the FH plane affects the results.

The Frankfort horizontal (FH) is an imaginary cranial reference plane used in orienting the skull for standardizing and unifying the cephalometric and craniometric measurements. Previously called the “German horizontal,” it was renamed in the Frankfort Craniometric Agreement established in the 13th General Congress of the German Anthropological Society held in Frankfort in August 1882. According to the agreement, the FH plane is defined by 2 lines, one on either side of the skull, connecting the uppermost point on the margin of the external acoustic meatus (porion) with the lowermost point on the orbital margin (orbitale). Since 2 lines are coplanar solely when they are either parallel or intersecting, this definition of the FH plane is not formally consistent from a purely mathematical point of view. After the inconsistency was identified, the definition was modified to the mathematically correct definition of a plane as uniquely determined by 3 noncollinear points. In this definition, the FH plane is determined by both porions and the left orbitale.

Despite the agreement, other combinations of craniometric points have often been used to set up the FH plane by dental researchers. For example, just 1 porion and both orbitales were used in the studies of Bayome et al and Swennen et al. The left porion was used in the first study, and the right porion was used in the second study. Compiling other related studies, we found that 6 points have been used for setting up the FH plane: 4 points representing anatomic structures of the skull (anatomic points) and 2 points secondarily obtained from these points (constructed points). The left porion (PoL), right porion (PoR), left orbitale (OrL), and right orbitale (OrR) points are the 4 anatomic points. The midpoint between both the porions (PoC) and the midpoint between the orbitales (OrC) are the 2 constructed points. As a result, there are 7 acceptable possibilities for connecting these points to set up the FH plane: (1) PoL, PoR, and OrL ; (2) PoL, PoR, and OrR ; (3) PoL, PoR, and OrC ; (4) PoL, OrL, and OrR ; (5) PoR, OrL, and OrR ; (6) PoC, OrL, and OrR ; and (7) PoL, PoR, OrL, and OrR (by least-squares plane fitting). Considering the large number of studies based on different setups of the FH plane, we supposed that these variations do not impact the cephalometric measurements for subjects with craniofacial symmetry. To check this hypothesis is the purpose of this study.

Cephalometric measurement values are used by surgeons and orthodontists for diagnosis, treatment planning, and treatment evaluation of orthodontic and orthognathic surgery patients. The greater the accuracy of these values, the lower the probability of disparate or erroneous clinical outcomes. For this reason, validating this hypothesis is of paramount importance. To the best of our knowledge, no study has been published to confirm specifically this hypothesis. A literature review showed 3 related studies. Oh et al investigated the influence of different horizontal reference planes for evaluating occlusal cant. They used distances based on teeth. Yoon et al analyzed the influence of different reference systems for evaluating facial asymmetry. They used horizontal and vertical deviations of landmarks. Lin et al evaluated the influence of different horizontal reference planes in analyzing patients with facial symmetry or asymmetry. They also used distances based on teeth. Our study is based on cephalometric hard tissue measurements originally performed on lateral cephalometric radiography and frequently used by surgeons.

Our goal in this article was to evaluate whether variations in the FH plane impact cephalometric analysis.

## Material and methods

This study was conducted in accordance with the ethical guidelines for biomedical research involving human subjects from the National Health Council of Brazil. It was approved by the Human Research Ethics Committee of the School of Medical Sciences at the University of Campinas (protocol number: 27917314.0.0000.5404).

Out of a total of 135, 82 cone-beam computed tomography (CBCT) data sets acquired from living subjects (27 men, 55 women) were selected from the Division of Oral Radiology’s archive at the Piracicaba Dental School at the University of Campinas in Brazil. The eligibility criteria were based not only on the subjects’ features, but also on the quality of the CBCT data sets. The inclusion criteria were the following: subjects with Class I molar relationship, subjects aged 18 to 32 years, CBCT data sets with a large field of view (diameter, 16 cm; height, 13-22 cm), and CBCT data sets with a voxel size of 0.4 mm ^{3 }or less. The exclusion criteria were subjects with craniofacial asymmetry or deformity, subjects who had surgery of the facial bones, and CBCT data sets with severe noise.

A custom software toolkit for visualizing and interacting with the CBCT data sets was developed by the first author using Python (version 2.7.7; Python Software Foundation; Beaverton, Ore) programming environment and NumPy (version 1.7.1; NumFOCUS; Austin, Tex), PyQt (version 4.11; Riverbank Computing; Wimborne, Dorset, United Kingdom), and VTK (version 6.1.0; Kitware; Clifton Park, NY) extension packages. The marching cubes algorithm was applied on each CBCT data set to build a high-resolution 3-dimensional (3D) geometric model representing the hard tissue surface of the skull, producing a reconstructed skull.

The method of CBCT-based 3D cephalometry described by Swennen et al was adopted for this study. It provided a suitable procedure to accurately evaluate the hypothesis. The method has 4 stages: (1) setting up the 3D cephalometric reference system (semiautomatic); (2) identifying the 3D cephalometric hard tissue landmarks (manual); (3) setting up the 3D cephalometric planes (automatic); and (4) performing the 3D cephalometric hard tissue measurements (automatic). These stages were sequentially performed for each of the 82 reconstructed skulls. The technical terms adopted by Swennen et al were followed in our study.

In stage 1, 5 tasks were carried out to set up a 3D cephalometric reference system based on the anterior cranial base, called the sella-nasion plane; this is required to standardize the 3D cephalometric hard tissue measurements. The first task was to position the reconstructed skull to the FH and median planes ( Fig 1 ). In the anterior view, the skull is symmetrically placed facing forward, with the horizontal line in the screen crossing the OrL and OrR points, and the vertical line crossing the point where the frontonasal and internasal sutures intersect (nasion). In right lateral view, the skull is positioned so that the horizontal line intersects the PoR and OrR points. Horizontal and vertical lines are mutually orthogonal and represent the FH and median planes, respectively. Orthogonal projection is required in this task. The second task was to generate the simulated lateral and posteroanterior cephalometric radiographs ( Fig 2 ). An image-order volume-rendering algorithm is used on the CBCT data set to build both radiographs. The third task was to identify nasion (N) and sella (S) 3D cephalometric hard tissue landmarks ( Fig 3 ). The S landmark is situated in the center of the hypophyseal fossa. The fourth task was to set up the sella-nasion 3D cephalometric plane, which intersects the S and N landmarks and is orthogonal to lateral radiography ( Fig 2 ). The fifth task was to set up the horizontal, vertical, and median 3D cephalometric reference planes ( Fig 2 ). The horizontal plane is generated by rotating the sella-nasion plane 6° clockwise around the S landmark on the lateral radiography. The vertical plane intersects the S landmark and is orthogonal to the lateral radiography and to the horizontal plane. The median plane intersects the S landmark and is orthogonal to both planes.

In stage 2, a total of 12 3D cephalometric hard tissue landmarks were identified on the reconstructed skull: 4 landmarks located in the median plane (unpaired landmarks) and 8 landmarks on either side of this plane (paired landmarks). The 4 unpaired landmarks were A-point (A), the most posterior point on the intermaxillary suture; pogonion (Pog), the most anterior point on the mandibular symphysis; menton (Me), the lowermost point on the mandibular symphysis; and gnathion (Gn), the most anteroinferior point on the mandibular symphysis. The 8 paired landmarks were PoL and PoR; OrL and OrR; left gonion (GoL) and right gonion (GoR), the points where the line bisecting the angle formed by extending the posterior ramus border and the inferior body border intersect the gonial angle of each mandibular ramus in the lateral view; and left upper incisor (UIL) and right upper incisors (U1R), the most mesial points on the cusp of each maxillary central incisor. These landmarks are shown in the Figure 3 . The simulated cephalometric radiographs help the observer in the 3D positioning of the landmarks.

In stage 3, the facial, y-axis, mandibular, and FH 3D cephalometric planes were set up on the reconstructed skull ( Fig 4 ). The facial plane intersects the N and Pog 3D cephalometric hard tissue landmarks and is orthogonal to the median 3D cephalometric reference plane. The y-axis plane intersects the S and Gn landmarks and is orthogonal to the median plane. The mandibular plane intersects the GoL, GoR, and Me landmarks. As already mentioned, there are 7 possible definitions for the FH plane. Therefore, the FH 3D cephalometric plane can be FH1, the plane that intersects the PoL, PoR, and OrL landmarks; FH2, the plane that intersects the PoL, PoR, and OrR landmarks; FH3, the plane that intersects the PoL, PoR, and OrC landmarks; FH4, the plane that intersects the PoL, OrL, and OrR landmarks; FH5, the plane that intersects the PoR, OrL, and OrR landmarks; FH6, the plane that intersects the PoC, OrL, and OrR landmarks; or FH7, the least-squares best-fit plane to the PoL, PoR, OrL, and OrR landmarks (ie, the plane with the lowest square sum of the shortest distances from each landmark to the plane itself).

In stage 4, a total of 6 3D cephalometric hard tissue measurements related to the FH plane were performed on the reconstructed skull: 3 linear projective measurements and 3 angular projective measurements. These measurements were denoted projective because they were performed on the median 3D cephalometric reference plane using the orthogonal projections of the A, N, Pog, UIL, and UIR 3D cephalometric landmarks onto this plane and the lines where the facial, y-axis, mandibular, and FH 3D cephalometric planes intersect the median plane. The 3 linear projective measurements were A-point to nasion perpendicular (A-Np), the distance from A-point to the line that intersects N and is orthogonal to the FH line (nasion perpendicular); pogonion to nasion perpendicular (Pog-Np), the distance from Pog to nasion perpendicular; and upper incisor to A-point vertical (UI-Av), the distance from the midpoint between the UIL and UIR points to the line that intersects A-point and is orthogonal to the FH line. The negative sign was used to indicate that the point was more posteriorly positioned than the reference line. The 3 angular projective measurements were facial angle (FA), the inferoposterior angle between the FH and the facial lines; y-axis angle (YAA), the inferoanterior angle between the FH and the y-axis lines; and mandibular plane angle (MPA), the inferoanterior angle between the FH and themandibular lines. The lines and points used in these measurements are shown in Figure 5 . The linear and angular measurements are expressed in millimeters and degrees, respectively. The 6 measurements were performed 7 times, one for each setup of the FH plane. They were chosen because they were originally performed on lateral cephalometric radiography and are usually performed by surgeons.

For estimating the measurement variability introduced by an observer, stages 2 to 4 were performed twice by the same observer (R.M.G.S.). The second observation was done almost 2 years after the first observation. In the second observation, the 3D cephalometric hard tissue measurements (stage 4) were performed only for the FH6 3D cephalometric plane, which is the set up used by Swennen et al.

## Statistical analysis

Intraobserver variability is described by 2 values: the mean (bias) and the standard deviation of the measurement differences between 2 observations. Based on the Bland-Altman analysis, these values were used to calculate the limits of agreement, with 95% confidence intervals around the limits, for each 3D cephalometric hard tissue measurement. The numbers of occurrences in which the measurement differences between 2 setups of the FH plane were greater than the limits of agreement were counted. This was done for all pairwise comparisons of setups of the FH plane, for a total of 21 pairwise comparisons. The R (version 3.1.0; R Foundation for Statistical Computing; Vienna, Austria) statistical computing environment was used for this statistical evaluation.

## Results

A total of 3936 (82 subjects × 6 measurements × 7 setups + 82 subjects × 6 measurements × 1 setup) 3D cephalometric hard tissue measurements were performed for this study: 1968 linear projective measurements and 1968 angular projective measurements. Intraobserver variability of the 3D cephalometric hard tissue measurements is shown in Table I . The numbers and percentages of occurrences in which the measurement differences between pairwise comparisons of setups of the FH plane were greater than the limits of agreement are presented in Table II . The FH3-FH6, FH3-FH7, and FH6-FH7 pairwise comparisons were the only ones with no occurrences for all 3D cephalometric hard tissue measurements. No 3D cephalometric hard tissue measurement had no occurrences for all pairwise comparisons. The FH4-FH5 pairwise comparison had the greatest number of occurrences for all the 3D cephalometric hard tissue measurements.