## Introduction

In this article, we aimed to establish an ideal definition for the craniofacial midsagittal plane (MSP) by first finding an optimal “plane of best fit” and then deriving a simple approximation for clinical use that is highly accurate.

## Methods

For 60 adolescent patients, 3-dimensional coordinates of 8 central landmarks and 6 pairs of lateral landmarks were collected. Across all patients, the coplanarity of the central landmarks was compared with that of the midpoints of the lateral landmarks. The MSP of best fit was then found by minimizing the mean square distance of the 8 central landmarks to a plane. Across all patients, each possible 3-point plane was compared with the MSP of best fit with respect to both orientation and proximity.

## Results

The central landmarks were more coplanar and thus more accurate than the midpoints of the lateral pairs. The plane defined by nasion, basion, and incisive foramen was the closest to the MSP of best fit in both orientation and proximity.

## Conclusions

The nasion-basion-incisive foramen plane should be used for skull orientation and 3-dimensional cephalometric analyses because it approximates the MSP of best fit with high accuracy, avoids the use of horizontal reference planes, avoids influence from upper and midface asymmetry, uses easily identifiable relevant landmarks, and is simple to define.

## Highlights

- •
Central landmarks of the skull agree on a midsagittal plane (MSP) better than lateral landmarks.

- •
“MSP of best fit” is a highly accurate plane that best fits multiple central craniofacial landmarks.

- •
A plane passing through Na, Ba, and IF is an optimal approximation of the “MSP of best fit.”

- •
The N-Ba-IF MSP definition is recommended for skull orientation and 3D cephalometric analyses.

The midsagittal plane (MSP) is the only major plane of symmetry in the craniofacial complex, and it sets the foundation for skull orientation and 3-dimensional (3D) cephalometric analyses. Skull orientation is an important first step when carrying out 3D diagnoses for the following reasons. First, it dramatically affects our ability to visually assess the skull for asymmetries. Second, if a cone-beam computed tomography (CBCT) image is being reformatted into a 2-dimensional radiograph, angular deviations foreshorten and skew the proportions of the resulting image. Finally, multiple 3D landmarks are orientation-dependent. For instance, it is not uncommon for a landmark to be defined as “the most lateral, anterior, inferior, or posterior point” on a given structure. Aside from being used for skull orientation, the MSP serves as a reference plane for both 3D cephalometric analyses and for the superimposition of radiographs from different time points.

Authors primarily use the MSP as a reference plane for skull orientation. However, a true craniofacial plane of symmetry exists only as a theoretical construct, since no human is perfectly symmetric. Thus, there is wide variation among MSP definitions in the literature. Various definitions rely on other horizontal reference planes (eg, Frankfort horizontal), midpoints of lateral landmarks (eg, anterior clinoid processes, foramina spinosum, orbitale, porion, and so on), central landmarks (eg, sella, nasion, crista galli, anterior nasal spine, and so on), or a combination of these. It stands to reason, however, that an ideal definition of the MSP would incorporate as many relevant landmarks as possible. The purposes of this study were to (1) establish this ideal definition, to be called the MSP of best fit, and (2) derive a simple and accurate protocol for approximating it.

## Material and methods

Approval from the Institutional Review Board at the University of Detroit Mercy School of Dentistry was obtained. Volumetric data from CBCT radiographs acquired with the Next Generation scanner (i-CAT, Hatfield, Pa) were collected from the Department of Orthodontics at the University of Detroit Mercy School of Dentistry database. A total of 60 pretreatment radiographs were selected from adolescent patients who had no prior orthodontic treatment and no obvious craniofacial anomalies. Ten patients from each of the following groups (designations recommended by the Food and Drug Administration and the National Institutes of Health ) were randomly selected: 10 black boys (average age, 13.3 years), 10 black girls (average age, 13.2 years), 10 Hispanic boys (average age, 12.0 years), 10 Hispanic girls (average age, 12.4 years), 10 white boys (average age 12.8 years), and 10 white girls (average age, 12.1 years). The average age of all 60 patients was 12.6 years.

The images were obtained in DICOM format and imported into Dolphin software (version 11.7 premium; Dolphin Imaging, Chatsworth, Calif). Each DICOM image was imported into the 3D imaging tool. The volumetric image was then adjusted for segmentation and opacity to allow for adequate visualization of the anterior nasal spine and minimize scatter. The radiographs were visually oriented from the front, right, and top views using the orientation tool.

Twenty craniofacial landmarks were selected for each patient with the Dolphin digitize/measurement tool. The landmarks were categorized as either central or lateral. Central landmarks were expected to sit on the MSP ( Fig 1 ), whereas lateral landmarks were expected to have mirror symmetry across the MSP. Although some of the 3D landmarks used in this study share names with 2-dimensional landmarks (eg, nasion, sella, basion, anterior nasal spine, posterior nasal spine, orbitale), these landmarks are fundamentally different. Although traditional cephalometric 2-dimensional landmarks display adequate reliability when used in 3D analyses, each 3D landmark has been explicitly defined for added accuracy.

Eight central landmarks ( Fig 1 ) were defined 3-dimensionally as follows.

- 1.
Nasion (N): the most anterior aspect of the frontonasal suture from a sagittal view and centered mediolaterally from the axial and coronal views.

- 2.
Crista galli (CG): the most posterior and inferior point of the perpendicular plate of the ethmoid bone where it joins the cribriform plate from a sagittal view and centered mediolaterally on the junction of the perpendicular plate of the ethmoid bone and the cribriform plate from the axial and coronal views.

- 3.
Sella (S): the center of the space in sella turcica from a sagittal view, centered mediolaterally on the base of sella turcica from the axial and coronal views.

- 4.
Basion (Ba): the middorsal point of the anterior margin of the foramen magnum on the basilar part of the occipital bone, located at the most posterior and inferior point of the basilar part of the occipital bone from a sagittal view and centered middorsally from the axial and coronal views.

- 5.
Vomer (V): the most posterior and inferior aspect of the sulcus vomeris from a sagittal view and centered mediolaterally from the axial and coronal views.

- 6.
Posterior nasal spine (PNS): the most posterior point of the posterior nasal spine from the sagittal and axial views. If bilateral processes are visible, their midpoint is selected.

- 7.
Incisive foramen (IF): the anteroposterior and mediolateral center of the incisive foramen as it exits the maxilla viewed from the sagittal and axial views, respectively.

- 8.
Anterior nasal spine (ANS): the most anterior point of the anterior nasal spine from the sagittal and axial views. If bilateral processes are visible, their midpoint is selected.

Six pairs of lateral landmarks were defined as follows.

- 1.
Zygomaticofrontal suture (ZFS): the center of the area of the axial slice of the zygomaticofrontal suture.

- 2.
Anterior clinoid process (ACP): the most posterior point of the anterior clinoid process when viewed sagittally and axially.

- 3.
Porion (Po): the superior aspect of the external auditory meatus when viewed sagittally, positioned mediolaterally where the superior epithelium tapers to its thinnest point.

- 4.
Foramen spinosum (FSp): the axial center of the area of the foramen spinosum at its most superior point as it joins the cranial fossa.

- 5.
Orbitale (Or): the most inferior point of the orbital sphere when oriented to the Frankfort horizontal.

- 6.
Lateral foramen magnum (LFM): the most lateral point of the foramen magnum when viewed axially, vertically centered on the most convex point of the foramen when viewed coronally.

The central landmarks were chosen for their ease of identification and biologic relevance. From a developmental standpoint, the cranial base sets the foundation for craniofacial development. This cartilaginous core is represented well by Ba, V, CG, and S. N, ANS, IF, and PNS represent the anterior and inferior portion of the craniofacial complex. The lateral landmarks were also chosen because of their ease of identification and to represent various regions of the skull. ZFS and Or represent the orbit, FSp and ACP represent the middle cranial fossa, Po represents the lateral extent of the skull, and LFM represents the posterior aspect of the posterior extent of the area of interest.

Landmarks of the mandible were not used since the mandible does not rigidly articulate with the rest of the skull. Because it is free to move, its development and position at the time of exposure may be subject to functional and environmental influence. Additionally, landmarks posterior to foramen magnum (eg, inion, opisthion) were not used since they are not visually relevant.

The general protocol for landmark selection included 3 steps: (1) locating the desired landmark in the sagittal view anteroposteriorly and superoinferiorly, (2) refining the landmark position mediolaterally from the coronal and axial views, and (3) selecting the landmark from the sagittal view. The 20 landmarks were selected for each of the 60 patients by the first author. The x, y, and z coordinates of all 1200 points were exported to Excel (Microsoft, Redmond, Wash), converted to Comma Separated Value format (*.csv), and then imported into the statistical computing software R (available at www.r-project.org ).

For each of the 60 adolescents, 3D coordinates of 8 central landmarks and 6 pairs of lateral landmarks were collected. The 6 midpoints of the lateral landmarks were then calculated as well. To establish an ideal definition of the MSP, the degree to which the 8 central points agree on 1 plane, compared with the 6 midpoints, was considered. To measure the coplanarity of a set of points, the following mathematical notions were used.

Let *p *_{1 }*, … , p *_{n }be n points in 3D space. Let *P *be a plane. Let *d *_{i }be the (perpendicular) distance from the point p _{i }to the plane P. The mean absolute distance or mean absolute error (MAE) of the points with respect to the plane P is the mean of the distances. The mean square distance or mean square error (MSE) of the points with respect to the plane P is the mean of the squared distances.

Although MAE may be a more intuitive notion, MSE is more sensitive to outliers and has better mathematical properties. The plane of best fit for the points p _{1 },…, p _{n }is the plane P that minimizes the MSE. It is a unique plane (assuming the points are not colinear) and may be calculated using linear algebra. The plane of best fit and the resulting MAE and MSE were then computed for both the 8 central landmarks and the 6 midpoints across all patients.

Once the higher degree of central landmark coplanarity was established, the MSP of best fit using all 8 central landmarks was constructed for each patient. The 3 central landmarks whose plane most accurately approximated this MSP of best fit was then determined as follows. For each of the 56 combinations of 3 central landmarks, the plane through those 3 landmarks was compared with the MSP of best fit across all 60 patients with respect to both orientation and proximity.

For orientation, the angle between the 3-point plane and the MSP of best fit was measured. The angle between 2 planes is defined as the angle between the normal vectors (perpendicular directions) to the planes, chosen to be between 0° and 90°. For proximity, the MSE and MAE between each 3-point plane and the remaining 5 central landmarks were measured.

In addition, 3 landmarks (N, Ba, IF) across 5 randomly selected patients were analyzed for interoperator and intraoperator reliability. To test interoperator reliability, copies of written instructions were given to 5 operators; these instructions outlined the relevant features in the Dolphin Imaging software and the protocol for landmark identification and selection. No additional guidance was given to the operators during landmark selection. For each landmark, axis, and patient (eg, N along the mediolateral x-axis for 1 patient), the standard deviation between all 5 operators was calculated. The means (across the 5 patients) of these standard deviations were also calculated. To test intraoperator reliability, the same landmarks were identified by the first author at 2 time points 3 days apart. Similarly, the mean absolute deviations of each landmark and axis were calculated.

## Results

For the central landmarks, for each patient, the MSE and MAE for the 8 central points with respect to their plane of best fit were computed. Across all patients, the mean MSE was 0.17 mm ^{2 }, and the mean MAE was 0.32 mm. For the lateral landmarks, for each patient, the MSE and MAE for the 6 midpoints with respect to their plane of best fit were also computed. Across all patients, the mean MSE was 0.22 mm ^{2 }, and the mean MAE was 0.35 mm.

The mean, median, and maximum angles (across all patients) for each of the 56 triples of central landmarks were calculated ( Table I ). The plane defined by N-Ba-IF showed the lowest mean (0.52°), median (0.46°), and maximum (1.51°) angles across all patients relative to the 8-point MSP of best fit. Additionally, in the 5 planes with the lowest mean angle, N appeared 3 times, Ba appeared 4 times, and IF appeared 3 times. By contrast, Ba-PNS-ANS had the largest mean angle of 18.67°.

L1 | L2 | L3 | Mean angle (°) | Median angle (°) | Maximum angle (°) |
---|---|---|---|---|---|

N | Ba | IF | 0.52 | 0.46 | 1.51 |

CG | Ba | IF | 0.6 | 0.51 | 2.02 |

N | Ba | ANS | 0.61 | 0.5 | 1.57 |

N | S | IF | 0.69 | 0.54 | 1.58 |

CG | Ba | ANS | 0.71 | 0.61 | 2.51 |

N | V | IF | 0.72 | 0.59 | 1.93 |

N | S | ANS | 0.81 | 0.69 | 2.17 |

N | PNS | IF | 0.82 | 0.64 | 2.49 |

N | V | ANS | 0.84 | 0.88 | 1.94 |

N | PNS | ANS | 0.86 | 0.87 | 2.44 |

N | S | PNS | 0.91 | 0.78 | 2.59 |

CG | PNS | ANS | 0.91 | 0.86 | 2.36 |

S | Ba | ANS | 0.92 | 0.81 | 2.7 |

S | Ba | IF | 0.92 | 0.8 | 2.69 |

CG | PNS | IF | 0.95 | 0.81 | 2.77 |

CG | V | IF | 1.02 | 0.82 | 3.85 |

CG | V | ANS | 1.07 | 0.96 | 3.93 |

N | CG | IF | 1.1 | 0.89 | 4.58 |

CG | Ba | PNS | 1.13 | 1 | 2.94 |

N | CG | ANS | 1.2 | 0.96 | 4.61 |

S | Ba | PNS | 1.21 | 1.1 | 2.74 |

S | PNS | ANS | 1.24 | 1.03 | 4.24 |

CG | S | PNS | 1.28 | 1.09 | 3.39 |

N | Ba | PNS | 1.29 | 1.13 | 3.77 |

CG | S | IF | 1.35 | 1.08 | 4.6 |

N | S | V | 1.53 | 1.49 | 4.08 |

S | PNS | IF | 1.59 | 1.26 | 6.73 |

N | CG | PNS | 1.66 | 1.31 | 6.37 |

CG | S | ANS | 1.72 | 1.41 | 5.51 |

N | S | Ba | 1.76 | 1.4 | 5.11 |

CG | S | V | 1.8 | 1.73 | 4.31 |

S | Ba | V | 1.84 | 1.75 | 4.84 |

N | V | PNS | 1.96 | 1.9 | 6.22 |

Ba | V | IF | 2.02 | 1.49 | 6.63 |

CG | S | Ba | 2.03 | 1.8 | 6.08 |

S | IF | ANS | 2.06 | 1.48 | 7.95 |

CG | IF | ANS | 2.12 | 1.5 | 8.07 |

Vo | IF | ANS | 2.31 | 1.69 | 8.29 |

S | V | ANS | 2.31 | 2.1 | 5.37 |

Ba | V | PNS | 2.45 | 2.15 | 6.24 |

Vo | PNS | ANS | 2.46 | 1.95 | 7.52 |

Ba | IF | ANS | 2.7 | 2.02 | 9.38 |

PNS | IF | ANS | 2.8 | 2.28 | 10.87 |

N | CG | V | 2.83 | 2.28 | 12.56 |

CG | Vo | PNS | 2.9 | 2.22 | 32.33 |

N | IF | ANS | 2.96 | 2.17 | 13.61 |

Ba | V | ANS | 3.11 | 2.45 | 13.84 |

Vo | PNS | IF | 3.19 | 2.69 | 15.06 |

CG | Ba | V | 3.21 | 2.75 | 15.25 |

S | Vo | IF | 4.16 | 2.76 | 80.46 |

N | CG | Ba | 4.75 | 3.08 | 23.96 |

N | Ba | V | 6.88 | 2.96 | 67.87 |

Ba | PNS | IF | 14.09 | 8.82 | 86.04 |

N | CG | S | 17.87 | 10.86 | 79.27 |

S | V | PNS | 18.44 | 12.55 | 79.33 |

Ba | PNS | ANS | 18.67 | 11.4 | 82.75 |

For proximity, the MSE and MAE between each 3-point plane and the remaining 5 central landmarks were measured. The mean, median, and maximum MSE and MAE (across all patients) for each of the 56 triples of central landmarks were calculated ( Table II ). The plane defined by N-Ba-IF showed the lowest mean MSE (0.31°), median MSE (0.24°), mean MAE (0.34°), and mean MSE (0.30°), as well as the second lowest maximum MSE (1.61°) and the third lowest maximum MAE (0.81°). S-N-IF had the lowest maximum MSE (1.51°), and CG-Ba-IF had the lowest maximum MAE (0.78°).

L1 | L2 | L3 | Mean MSE (mm) | Median MSE (mm) | Maximum MSE (mm) | Mean MAE (mm) | Median MAE (mm) | Maximum MAE (mm) |
---|---|---|---|---|---|---|---|---|

N | Ba | IF | 0.31 | 0.24 | 1.54 | 0.34 | 0.3 | 0.81 |

CG | Ba | IF | 0.33 | 0.24 | 1.61 | 0.34 | 0.33 | 0.78 |

N | Ba | ANS | 0.34 | 0.27 | 1.55 | 0.36 | 0.33 | 0.81 |

CG | Ba | ANS | 0.35 | 0.25 | 1.62 | 0.36 | 0.33 | 0.85 |

N | S | IF | 0.42 | 0.28 | 1.51 | 0.39 | 0.34 | 0.81 |

N | V | IF | 0.43 | 0.34 | 1.73 | 0.4 | 0.37 | 0.87 |

N | V | ANS | 0.47 | 0.34 | 1.7 | 0.41 | 0.38 | 0.87 |

S | Ba | ANS | 0.47 | 0.32 | 1.83 | 0.42 | 0.38 | 0.8 |

N | S | PNS | 0.49 | 0.36 | 2.45 | 0.42 | 0.38 | 0.83 |

S | Ba | IF | 0.5 | 0.32 | 2 | 0.42 | 0.39 | 0.85 |

CG | PNS | ANS | 0.5 | 0.37 | 2.32 | 0.43 | 0.37 | 1.06 |

N | PNS | ANS | 0.55 | 0.39 | 3.82 | 0.44 | 0.39 | 1.33 |

N | S | ANS | 0.56 | 0.35 | 2.04 | 0.45 | 0.39 | 1.02 |

CG | V | ANS | 0.56 | 0.41 | 2.39 | 0.45 | 0.43 | 1.05 |

N | PNS | IF | 0.56 | 0.36 | 3.48 | 0.44 | 0.39 | 1.27 |

CG | PNS | IF | 0.58 | 0.39 | 2.87 | 0.46 | 0.41 | 1.22 |

CG | V | IF | 0.59 | 0.39 | 3.31 | 0.45 | 0.43 | 1.2 |

CG | Ba | PNS | 0.67 | 0.46 | 2.65 | 0.49 | 0.45 | 1.14 |

S | PNS | ANS | 0.68 | 0.42 | 3.9 | 0.48 | 0.43 | 1.28 |

N | Ba | PNS | 0.71 | 0.47 | 2.71 | 0.5 | 0.45 | 1.24 |

CG | S | PNS | 0.9 | 0.46 | 4.76 | 0.57 | 0.48 | 1.61 |

S | Ba | PNS | 0.91 | 0.6 | 3.5 | 0.57 | 0.52 | 1.35 |

CG | S | IF | 0.91 | 0.53 | 6.79 | 0.57 | 0.49 | 1.82 |

N | S | V | 1.01 | 0.78 | 3.47 | 0.61 | 0.57 | 1.24 |

N | CG | IF | 1.13 | 0.53 | 10.02 | 0.59 | 0.45 | 2.16 |

S | PNS | IF | 1.14 | 0.59 | 10.26 | 0.61 | 0.52 | 2.23 |

N | CG | ANS | 1.3 | 0.58 | 11.89 | 0.64 | 0.49 | 2.4 |

N | CG | PNS | 1.31 | 0.63 | 10.62 | 0.67 | 0.56 | 2.39 |

N | V | PNS | 1.38 | 0.89 | 7.66 | 0.72 | 0.65 | 2.04 |

CG | S | ANS | 1.38 | 0.67 | 9.42 | 0.7 | 0.52 | 2.17 |

CG | S | V | 1.45 | 0.92 | 7.73 | 0.73 | 0.64 | 1.98 |

S | V | ANS | 1.49 | 1.14 | 5.53 | 0.74 | 0.72 | 1.55 |

N | S | Ba | 1.57 | 0.88 | 8.65 | 0.71 | 0.62 | 2.08 |

S | IF | ANS | 1.6 | 0.74 | 12.77 | 0.74 | 0.58 | 2.56 |

Ba | V | IF | 1.82 | 0.77 | 12.17 | 0.74 | 0.57 | 2.32 |

V | IF | ANS | 1.83 | 0.9 | 13.03 | 0.79 | 0.68 | 2.56 |

V | PNS | ANS | 1.87 | 1.06 | 11.24 | 0.81 | 0.72 | 2.32 |

S | Ba | V | 2.01 | 1.04 | 13.76 | 0.84 | 0.76 | 2.73 |

Ba | V | PNS | 2.14 | 1.16 | 11.35 | 0.88 | 0.75 | 2.3 |

CG | S | Ba | 2.35 | 1.16 | 15.32 | 0.87 | 0.76 | 2.69 |

CG | IF | ANS | 2.95 | 0.98 | 22.23 | 0.97 | 0.71 | 3.54 |

Ba | V | ANS | 2.97 | 1.25 | 28.62 | 0.96 | 0.76 | 3.76 |

N | CG | V | 3.08 | 1.51 | 24.5 | 0.99 | 0.83 | 3.37 |

Ba | IF | ANS | 3.16 | 1.34 | 20.7 | 1 | 0.82 | 3.13 |

PNS | IF | ANS | 3.45 | 1.55 | 36.6 | 1.04 | 0.79 | 4.43 |

V | PNS | IF | 4.09 | 1.7 | 51.48 | 1.1 | 0.87 | 4.95 |

CG | Ba | V | 4.66 | 1.77 | 36.73 | 1.15 | 0.86 | 4.09 |

CG | V | PNS | 5.93 | 1.34 | 203.92 | 1.14 | 0.85 | 11.13 |

N | CG | Ba | 10.66 | 2.71 | 130.93 | 1.62 | 1.07 | 7.99 |

N | IF | ANS | 10.77 | 3.26 | 121.15 | 1.76 | 1.34 | 8.39 |

S | V | IF | 13.51 | 1.83 | 660.51 | 1.23 | 0.92 | 18.33 |

N | Ba | V | 24.21 | 2.2 | 570.64 | 2.09 | 1.03 | 16.62 |

Ba | PNS | IF | 93.42 | 20.85 | 925.94 | 4.58 | 3.04 | 20.95 |

Ba | PNS | ANS | 127.19 | 26.22 | 979.71 | 5.6 | 3.49 | 21.87 |

S | V | PNS | 173.21 | 50.79 | 1270.45 | 7.5 | 5.41 | 26.98 |

N | CG | S | 173.49 | 40.54 | 1012.68 | 6.99 | 4.82 | 24.01 |