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Ha, Kim, Choi, and Baek: Characterization of facial asymmetry phenotypes in adult patients with skeletal Class III malocclusion using three-dimensional computed tomography and cluster analysis
Original Article

Korean J Orthod 2022;52(2):85-101.

Published online: 25 March 2022

DOI: https://doi.org/10.4041/kjod.2022.52.2.85

Characterization of facial asymmetry phenotypes in adult patients with skeletal Class III malocclusion using three-dimensional computed tomography and cluster analysis

Sang-Woon Haa, Su-Jung Kimb, Jin-Young Choic, Seung-Hak Baekd,

aDepartment of Orthodontics, School of Dentistry, Seoul National University, Seoul, Korea

bDepartment of Orthodontics, Kyung Hee University School of Dentistry, Seoul, Korea

cDepartment of Oral and Maxillofacial Surgery, School of Dentistry, Seoul National University, Seoul, Korea

dDepartment of Orthodontics, School of Dentistry and Dental Research Institute, Seoul National University, Seoul, Korea

Corresponding author: Seung-Hak Baek. Professor, Department of Orthodontics, School of Dentistry and Dental Research Institute, Seoul National University, 101, Daehak-ro, Jongno-gu, Seoul 03080, Korea., Tel +82-2-2072-3952 e-mail drwhite@unitel.co.kr

How to cite this article: Ha SW, Kim SJ, Choi JY, Baek SH. Characterization of facial asymmetry phenotypes in adult patients with skeletal Class III malocclusion using three-dimensional computed tomography and cluster analysis. Korean J Orthod 2022;52:85-101.

Received 17 June 2021       Revised 22 July 2021       Accepted 1 September 2021

© 2022 The Korean Association of Orthodontists.

(open-access):

This is an open-access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Objective

To classify facial asymmetry (FA) phenotypes in adult patients with skeletal Class III (C-III) malocclusion.

Methods

A total of 120 C-III patients who underwent orthognathic surgery (OGS) and whose three-dimensional computed tomography images were taken one month prior to OGS were evaluated. Thirty hard tissue landmarks were identified. After measurement of 22 variables, including cant (°, mm), shift (mm), and yaw (°) of the maxilla, maxillary dentition (Max-dent), mandibular dentition, mandible, and mandibular border (Man-border) and differences in the frontal ramus angle (FRA, °) and ramus height (RH, mm), K-means cluster analysis was conducted using three variables (cant in the Max-dent [mm] and shift [mm] and yaw [°] in the Man-border). Statistical analyses were conducted to characterize the differences in the FA variables among the clusters.

Results

The FA phenotypes were classified into five types 1) non-asymmetry type (35.8%); 2) maxillary-cant type (14.2%; severe cant of the Max-dent, mild shift of the Man-border); 3) mandibular-shift and yaw type (16.7%; moderate shift and yaw of the Man-border, mild RH-difference); 4) complex type (9.2%; severe cant of the Max-dent, moderate cant, severe shift, and severe yaw of the Man-border, moderate differences in FRA and RH); and 5) maxillary reverse-cant type (24.2%; reverse-cant of the Max-dent). Strategic decompensation by pre-surgical orthodontic treatment and considerations for OGS planning were proposed according to the FA phenotypes.

Conclusions

This FA phenotype classification may be an effective tool for differential diagnosis and surgical planning for Class III patients with FA.

Keywords: Facial asymmetry, Class III malocclusion, Cluster analysis

INTRODUCTION

The etiology of facial asymmetry (FA) includes genetic or congenital malformations, acquired or developmental deformities, environmental factors including habits or trauma, and functional deviations.1 Piao et al.2 reported that the prevalence of FA in Korean orthodontic patients was the highest in skeletal Class III malocclusion, followed by skeletal Class I and Class II malocclusions (16.6%, 10.1%, 6.9%, p < 0.001), despite data from a single dental hospital.
FA has been diagnosed using facial photographs, posteroanterior (PA) cephalograms, or submentovertex projections. Although facial photographs can provide an intuitive impression of FA, it is difficult to accurately measure the amount of asymmetry. PA cephalograms and/or submentovertex projections are useful for diagnosis and orthognathic surgical planning of oral and maxillofacial deformities. However, these modalities have limitations due to magnification, projection errors, and two-dimensional (2D) assessments of the three-dimensional (3D) structures.3 Therefore, the use of 3D computed tomography (3D-CT) or cone-beam CT has become popular for the accurate measurement of craniofacial anatomic structures.4-8
The major limitations in previous studies on FA can be summarized as follows: 1) Most studies investigated subjects with skeletal Class I, II, and III malocclusion, causing a problem in sample purity;9-12 2) PA cephalograms or facial photographs have inevitable errors in the 2D analysis of a 3D object;9,11 3) Although some previous study used 3D-CT analysis, the number of FA patients was insufficient to draw a robust statistical significance;10 4) Some previous studies did not provide statistical evidence for the classification of FA;13-15 5) Although FA was classified in a 3D manner, most previous studies have focused only on cant and shift. Furthermore, they did not pay attention to the yaw as well as the differences in the frontal ramus angle (FRA) and ramus height (RH);9-13,15,16 6) It is important to analyze the asymmetry in the maxilla, maxillary dentition, mandibular dentition, mandible, and mandibular border separately; and 7) Some previous studies failed to consider the clinical significance in relation to pre-operative orthodontic treatment and orthognathic surgery.12,16
For clinically significant and statistically valid classification of FA, it is important to conduct studies using 3D-CT images with a large number of patients and full consideration of cant, shift, and yaw. Therefore, the purpose of this study was to classify and characterize the FA phenotypes in Korean adult patients with skeletal Class III malocclusion who had undergone orthognathic surgery using 3D-CT and cluster analysis.

MATERIALS AND METHODS

The initial samples were Korean adult patients who had undergone pre-operative orthodontic treatment and orthognathic surgery at Seoul National University Dental Hospital (SNUDH) in Seoul, Republic of Korea between 2015 and 2020. The inclusion criteria were 1) patients with completed facial growth (over the age of 18 years); 2) patients who were diagnosed with skeletal Class III malocclusion; and 3) patients whose 3D-CT images were taken at least one month prior to orthognathic surgery. The 3D-CTs taken before orthognathic surgery were used to analyze the skeletal problems and set up precise surgical planning and to minimize cost and radiation exposure issues from sequential 3D-CT taken from the initial visit to the post-operative stage. The exclusion criteria were 1) patients whose posterior teeth were missing or abnormally shaped; 2) patients who had a degenerative joint disease, tumor, or trauma history in the temporomandibular joints; and 3) patients who had hemifacial microsomia or other craniofacial anomaly syndromes.
As a result, 120 Korean adult patients who had undergone pre-operative orthodontic treatment and orthognathic surgery for correction of skeletal Class III malocclusion were recruited as the final sample (72 males and 48 females; mean age at the time of 3D-CT taking, 22.9 ± 4.4 years). This study was reviewed and approved by the Institutional Review Board Committee of SNUDH (ERI20029).
3D-CTs (Sensation 10; Siemens, München, Germany; axial slice thickness, 1.0 mm) were taken with centric relation and lips in repose. After each data set was imported into the ON3D program (3DONS, INC., Seoul, Korea), 3D-CT images were re-orientated using the horizontal, coronal, and mid-sagittal planes (Figures 1 and 2, Table 1). The definitions of landmarks and reference planes used in the present study were adopted from the methodology of Hong et al.17 The N point was registered as the origin (0, 0, 0) of the Cartesian coordinate system.
The definitions of 30 hard tissue landmarks, six lines, and seven planes are enumerated in Figures 1 and 2 and Table 1. These landmarks were identified on each 3D-CT image by a single operator (SWH) with ON3D software.
The definitions of 22 measurement variables are enumerated in Figure 3 and Table 2. In the present study, a novel 3D measurement method was developed to express the cant, shift, and yaw of the maxilla, maxillary dentition, mandibular dentition, mandible, and mandibular border. When asymmetry occurred in the same direction of the Me deviation, the sign of the measurement variables was designated as positive (+); otherwise, the sign was designated as negative (–).
Twelve randomly selected CT images were re-digitized and re-measured after two weeks by the same operator (SWH). Since there was no significant difference in the values of the measurement variables between the first and second measurements in the Wilcoxon signed rank test (p > 0.05), the first set of measurements was used for further analysis.
K-means cluster analysis was conducted to classify the FA phenotypes using the three representative variables (cant in the maxillary dentition [molar height difference, mm] and shift [Me deviation, mm] and yaw [°] in the mandibular border), which provide significant clinical information for diagnosis and surgical planning. The reasons were as follows: 1) The cant of the maxillary dentition was used in clustering because it is one of the main targets of orthognathic surgery; and 2) Since the degree of asymmetry worsens from top to bottom, the shift and yaw of the mandibular border were used for clustering. In addition, although we measured both the angle and distance of the cant in the maxillary dentition, only the cant distance (molar height difference) was utilized for clustering to prevent overfitting and to provide clinical information for easy application in surgical planning.
According to the total within-cluster sum of squares, the appropriate number of clusters was 4 to 6. After analyzing the results of clustering by the number of clusters with a 3D scatter plot, the final number of clusters was determined to be five.
Among the five clusters, the cluster with the least amounts of shift, cant, and yaw was designated as the non-asymmetry type (Table 3). The degree of asymmetry in each measurement variable was classified into normal, mild, moderate, and severe based on the means and standard deviations (SD) of the non-asymmetry type (Table 3). The cut-points were 1 SD to 2 SD or –1 SD to –2 SD for mild degree, 2 SD to 3 SD or –2 SD to –3 SD for moderate degree, and > 3 SD or < –3 SD for severe degree. The one-way analysis of variance test and multiple comparisons with Tukey’s honestly significant difference test were conducted to characterize the differences in the FA variables among the five clusters.
All statistical analyses were conducted using Language R, version 3.6.3 (The R Foundation for Statistical Computing, Vienna, Austria). p-value less than 0.05 was considered statistically significant.

RESULTS

Morphological characteristics of five FA phenotypes (Table 4, Figure 4)

The FA phenotypes in Class III patients were classified into five types according to their distinct morphological characteristics: non-asymmetry type (n = 43, 35.8%), maxillary-cant (Max-Cant) type (n = 17, 14.2%), mandibular-shift and yaw (Man-Shift-Yaw) type (n = 20, 16.7%), complex type (n = 11, 9.2%), and maxillary reverse-cant (Max-Rev-Cant) type (n = 29, 24.2%). In the non-asymmetry type, no significant asymmetry was observed.
The Max-Cant type demonstrated severe cant in the maxillary dentition (MxD-Cant, 2.9°, 2.7 mm) and mild cant in the mandibular dentition (MdD-Cant, 2.4°, 2.0 mm). A mild shift was observed in the mandible and mandibular border (MdS-Shift, 3.1 mm; MdB-Shift, 4.1 mm).
The Man-Shift-Yaw type exhibited moderate shift and mild yaw in the mandibular dentition (MdD-Shift, 4.7 mm; MdD-Yaw, 3.7°), moderate shift and yaw in the mandible (MdS-Shift, 5.4 mm; MdS-Yaw, 4.1°) and the mandibular border (MdB-Shift, 6.5 mm; MdB-Yaw, 4.5°), and mild RH-difference (3.4 mm). In addition, a mild cant was observed in the mandibular dentition (MdD-Cant, 2.2°, 1.8 mm).
The complex type demonstrated severe cant in the maxillary dentition (MxD-Cant, 3.4°, 3.2 mm); severe shift and yaw in the mandibular dentition (MdD-Shift, 8.7 mm; MdD-Yaw, 7.0°), mandible (MdD-Shift, 10.3 mm; MdD-Yaw, 7.1°), and mandibular border (MdB-Shift, 13.7 mm; MdB-Yaw, 8.5°); and moderate FRA-difference (7.3°) and RH-difference (8.4 mm). Mild cant in the maxilla (MxS-Cant, 2.1°, 2.1 mm), severe cant in the mandibular dentition (MdD-Cant, 5.4°, 4.7 mm), and moderate cant in the mandible (MdS-Cant, 3.8°, 5.7 mm) and mandibular border (MdB-Cant, 3.9°, 6.4 mm) were also observed.
The Max-Rev-Cant type revealed that the cant in the maxillary dentition (MxD-Cant, −0.9°, −0.8 mm) was expressed into the opposite direction of the Me deviation. However, there was no significant asymmetry in the mandibular border (MdB-Cant, 0.2°, 0.3 mm; MdB-Shift, 2.2 mm; MdB-Yaw, 1.4°).

Comparison of cant, yaw, and shift among the FA phenotypes (Table 4, Figure 4)

Cant was most significant in all parts in the complex type (MxS-Cant, 2.1°, 2.1 mm; MxD-Cant, 3.4°, 3.2 mm; MdD-Cant, 5.4°, 4.7 mm; MdS-Cant, 3.8°, 5.7 mm; MdB-Cant, 3.9°, 6.4 mm); followed by the maxillary and mandibular dentition in the Max-Cant type (MxD-Cant, 2.9°, 2.7 mm; MdD-Cant, 2.4°, 2.0 mm); and the mandibular dentition in the Man-Shift-Yaw type (MdD-Cant 2.2°, 1.8 mm) (all p < 0.001). However, in the Max-Rev-Cant type, reverse cant was observed at the maxilla and maxillary dentition (−0.8°, −0.8 mm and −0.9°, −0.8 mm; all p < 0.001).
Shift was most significant in the mandibular dentition, mandible, and mandibular border of the complex type (MdD-Shift, 8.7 mm; MdS-Shift, 10.3 mm; MdB-Shift, 13.7 mm); followed by the Man-Shift-Yaw type (MdD-Shift, 4.7 mm; MdS-Shift, 5.4 mm; MdB-Shift, 6.5 mm); and the mandible and mandibular border of the Max-Cant type (MdS-Shift, 3.1 mm; MdB-Shift, 4.1 mm) (all p < 0.001).
Yaw was most significant in the mandibular dentition, mandible, and mandibular border of the complex type, followed by the Man-Shift-Yaw type (MdD-Yaw, 7.0°, 3.7°; MdS-Yaw, 7.1°, 4.1°; MdB-Yaw, 8.5°, 4.5°; all p < 0.001).

DISCUSSION

Classification of the FA phenotypes (Tables 4 and 5, Figure 5)

Approximately two-thirds (64.2%) of Class III patients who underwent orthognathic surgery had a significant FA. They were divided into four major FA clusters according to the existence of the shift and yaw in the mandibular border, the cant in the maxillary dentition, and a combination of these asymmetries (Man-Shift-Yaw type, Max-Cant type, Max-Rev-Cant type, and complex type; Figure 6). This finding was somewhat different from the classification of previous cluster analysis studies9,10,18 and a non-cluster analysis study (Table 5).15

Characteristics of each FA phenotype (Tables 4 and 5, Figure 5)

The Max-Cant type was characterized by severe cant and molar height differences in the maxillary dentition, and mild cant and molar height differences in the mandibular dentition. These values indicated the remaining amounts of transverse occlusal plane (OP) cant and molar height difference after vertical decompensation by pre-operative orthodontic treatment. In addition, a mild shift of the mandibular border (Me deviation) into the deviated side, which might be induced by the cant of the maxilla. Since there was no significant difference in RH, the FA of the Max-Cant type in this study might not be caused by unilateral condylar hyperplasia.
The Man-Shift-Yaw type was characterized by moderate shift and yaw in the mandibular border and mild RH-difference. When the shift of the mandible occurs in the deviated side, it usually exhibits different molar inclinations (transverse buccolingual compensation). When the yaw of the mandible occurs in the deviated side, it results in an asymmetric molar relationship, distorted dental arch form, and discrepancy between the dental and basal arches (horizontal compensation). Moreover, the Man-Shift-Yaw type exhibited a mild cant and molar height difference in the mandibular. Since there was no significant cant in the maxillary dentition, the cant and molar height difference in the mandibular dentition might occur due to the shift and yaw of the mandible. Joondeph19 suggested that a functional shift at an early age might lead to remodeling of the condyle and glenoid fossa, resulting in asymmetric growth of the mandible.
Since the complex type was characterized by a combination of the Max-Cant type and the Man-Shift-Yaw type, it showed more complicated features of vertical, transverse, and horizontal dentoalveolar compensation. Since patients with hemifacial microsomia and other craniofacial anomalies were excluded in this study, the asymmetric growth of the mandible and compensation of the maxilla might be the causes of this FA type rather than a problem of the cranial base, at least in Class III patients.
In the Max-Rev-Cant type, since there was no significant asymmetry in the mandibular border, the reverse cant in the maxillary dentition may conceal the shift of the maxilla. Tay20 suggested that unilateral mastication on the Me deviation side was related to this FA type.

Strategic decompensation according to the FA phenotypes (Figure 6)

Cant correction is required for the Max-Cant type, complex type, and Max-Rev-Cant type. Maxillary and mandibular molar height discrepancies less than 3 mm and 1.5 mm, respectively, can be treated by unilateral intrusion/extrusion of these teeth using miniscrews (known as a temporary anchorage device [TAD]) during pre-operative orthodontic treatment.21,22 Then, the remaining molar height discrepancy and OP cant after pre-operative orthodontic treatment should be corrected by orthognathic surgery.
In the Man-Shift-Yaw type and the complex type, both transverse and horizontal decompensations are required to correct the shift and yaw in the mandibular dentition simultaneously. In terms of transverse decompensation of the molar inclinations, we can upright the posterior teeth on the basal arch and expand the inter-premolar width up to 4 mm and the inter-molar width up to 2 mm.23 Then, we have to decide whether we should intentionally create the posterior crossbite on the deviated side and increase the amount of maxillary and mandibular dental midline-off for maximizing the surgical correction of the shift.
During horizontal decompensation, it is important to coordinate the dental and basal arch forms and correct the asymmetric molar relationship. Since the amount of distalization of the mandibular posterior teeth is considered to be 3 mm,24,25 we can apply unilateral distalization of the mandibular posterior teeth in the deviated side up to 3 mm with TAD to maximize the surgical correction of the yaw.

Strategic surgical planning according to the FA phenotypes (Figure 6)

For correction of the remaining OP cant and molar height difference of the maxillary dentition after pre-operative orthodontic treatment in the Max-Cant type and the complex type, differential impaction of the maxilla can be performed. However, interference by the inferior nasal concha and nasal septum during impaction of the maxilla can result in deviation of the nasal septum and asymmetry of the nasal tip.26 To prevent these adverse effects, trimming of the nasal septum and inferior turbinectomy can be considered.26
If the difference in the amount of mandibular setback between the deviated and non-deviated sides is large in the Man-Shift-Yaw type and the complex type, there is a high probability of bony interferences between the proximal and distal segments of the mandible. This can cause displacement and torque of the condyle, which is one of the main causes of post-surgical relapse. To prevent these problems, the following should be considered: (1) meticulous removal of the bony interference between the proximal and distal segments, (2) a distal cutting technique and/or posterior bending osteotomy of the distal segment, (3) minimization of the vertical step interference between the proximal and distal segments, and (4) 3D simulation to correct yaw and FRA asymmetry.27-29
When the patient has a large difference in the soft tissue thickness or the degree of bony protuberance of the mandibular border between the deviated and non-deviated sides, angle reduction, centering genioplasty, and border contouring procedures can be considered in conjunction with orthognathic surgery or as the secondary operation after orthognathic surgery.
This FA phenotype classification might be an effective tool for the differential diagnosis and for proper surgical planning of Class III patients with FA. Although this study provided meaningful results and compared the results with previous studies (Table 5), the morphology of the mandible was not analyzed in this study. In the future, it would be necessary to investigate the morphological abnormalities, such as hemimandibular elongation and hemimandibular hyperplasia.

CONCLUSION

• In the present study, the classification and percentage distribution of the FA phenotypes obtained from K-means cluster analysis were as follows: Non-asymmetry type (35.8%); Max-Cant type (14.2%), which showed severe cant of the maxillary dentition, mild shift of the mandibular border; Man-Shift-Yaw type (16.7%), which presented moderate shift and yaw of the mandibular border, mild RH-difference; Complex type (9.2%) with severe cant of the maxillary dentition, moderate cant, severe shift, and severe yaw of the mandibular border, moderate differences in FRA and RH; and Max-Rev-Cant type (24.2%), which showed reverse-cant of the maxillary dentition.
• Important measurement variables for differential diagnosis and a primary guideline of pre-operative orthodontic treatment and orthognathic surgery planning according to FA classification in Class III patients were also presented.

ACKNOWLEDGEMENTS

This research was supported by a grant of the Korea Health Technology R&D Project through the Korea Health Industry Development Institute, funded by the Ministry of Health & Welfare, Republic of Korea (grant number: HI18C1638).
This article is partly from the PhD thesis of S.W.H.

Notes

CONFLICTS OF INTEREST

No potential conflict of interest relevant to this article was reported.

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crossref

Figure 1
The landmarks used in this study. A, Nasion (N), frontozygomatic point, (FZP, R and L), porion (Po, R and L), orbitale (Or, R and L), mesiobuccal (MB) cusp tip of the mandibular first molar (L6CP, R and L), lower dental midline point (L1MP), inferior gonion (Inf-Go, R and L), and menton (Me). B, A point (A), jugal process point (J, R and L), mandibular foramen (MdF, R and L), MB cusp tip of the maxillary first molar (U6CP, R and L), B point (B), and upper dental midline point (U1MP). C and D, Condylion (Cd-S, R and L), lateral pole of condyle (Cd-L, R and L), medial pole of condyle (Cd-M, R and L), and condylar center (Cd-C, R and L).
R, right; L, left.
kjod-52-2-85-f1.tif
Figure 2
The planes and lines used in this study. A, Horizontal plane, coronal plane, mid-sagittal (MidS) plane, frontozygomatic point (FZP) line, right Frankfort horizontal (RFH) line, and ramal line (R and L). B, Condylar sagittal plane (R and L) and condylar axis line (R and L). C, Condylar coronal plane (R and L).
R, right; L, left.
See Figure 1 for definitions of the other landmarks.
kjod-52-2-85-f2.tif
Figure 3
The variables used in this study. Maxillary skeletal (MxS)-Cant (°), MxS-Cant (mm), MxS-Shift (mm), MxS-Yaw (°), maxillary dental (MxD)-Cant (°), MxD-Cant (mm), MxD-Shift (mm), MxD-Yaw (°), mandibular dental (MdD)-Cant (°), MdD-Cant (mm), MdD-Shift (mm), MdD-Yaw (°), mandibular skeletal (MdS)-Cant (°), MdS-Cant (mm), MdS-Shift (mm), MdS-Yaw (°), mandibular border (MdB)-Cant (°), MdB-Cant (mm), MdB-Shift (mm), MdB-Yaw (°), frontal ramus angle (FRA, °) (R and L, difference), Ramus height (RH, mm) (R and L, difference).
MidS, mid-sagittal; R, right; L, left.
See Figure 1 for definitions of the other landmarks.
kjod-52-2-85-f3.tif
Figure 4
Examples of the non-asymmetry type and 4 facial asymmetry phenotypes. 1) Non-asymmetry type (35.8%); 2) maxillary-cant (Max-Cant) type (14.2%; severe cant of the maxillary dentition, mild shift of the mandibular border); 3) mandibular-shift and yaw (Man-Shift-Yaw) type (16.7%; moderate shift and yaw of the mandibular border, mild ramus height difference); 4) complex type (9.2%; severe cant of the maxillary dentition, moderate cant, severe shift, and severe yaw of the mandibular border, moderate differences in frontal ramus angle and ramus height); and 5) maxillary reverse-cant (Max-Rev-Cant) type (24.2%; reverse-cant of the maxillary dentition).
kjod-52-2-85-f4.tif
Figure 5
Flow chart that explains the characteristics of the facial asymmetry phenotypes in skeletal Class III patients who underwent orthognathic surgery.
Max-Cant, maxillary-cant; Man-Shift-Yaw, mandibular-shift and yaw; Max-Rev-Cant, maxillary reverse-cant; FRA, frontal ramus angle; RH, ramus height.
kjod-52-2-85-f5.tif
Figure 6
Strategic decompensation and surgical planning according to the facial asymmetry phenotypes.
Max-Cant, maxillary-cant; Man-Shift-Yaw, mandibular-shift and yaw; Max-Rev-Cant, maxillary reverse-cant; TAD, temporary anchorage device; IPW, inter-premolar width; IMW, inter-molar width; 3D, three-dimensional.
kjod-52-2-85-f6.tif
Table 1
Definition of the landmarks, lines, and planes used in this study
Landmarks, lines and planes Abbreviation Definition
Cranial landmarks
Nasion N The middle point of nasofrontal suture
Right frontozygomatic suture RFZP The intersection of the right frontozygomatic suture and the inner rim of the orbit
Left frontozygomatic suture LFZP The intersection of the left frontozygomatic suture and the inner rim of the orbit
Right porion RPo The most superior point of the right external auditory meatus
Left porion LPo The most superior point of the left external auditory meatus
Maxillary skeletal landmarks
Right orbitale ROr The most inferior point of the right orbital contour
Left orbitale LOr The most inferior point of the left orbital contour
Right jugal process point RJ The intersection point between the lateral contour of the alveolar process and the lower contour of the zygomatic buttress of the right maxilla
Left jugal process point LJ The intersection point of the lateral contour of the alveolar process and the lower contour of the zygomatic buttress of the left maxilla
A point A The deepest point between the anterior nasal spine and the upper incisal alveolus in the mid-sagittal plane
Maxillary dental landmarks
Maxillary central incisor U1MP Upper dental midline point
Cusp of the maxillary right first molar RU6CP The tip of the mesiobuccal cusp of the maxillary right first molar crown
Cusp of the maxillary left first molar LU6CP The tip of the mesiobuccal cusp of the maxillary left first molar crown
Mandibular skeletal landmarks
Right mandibular foramen RMdF The most Inferior point of the right mandibular foramen
Left mandibular foramen LMdF The most inferior point of the left mandibular foramen
B point B The deepest point between pogonion and the lower incisal alveolus in the mid-sagittal plane
Menton Me The most inferior point in the middle of the mandibular chin in the coronal plane
Right condylion RCd-S The most superior point of the right condyle in the condylar sagittal plane within the range of glenoid fossa
Left condylion LCd-S The most superior point of the left condyle in the condylar sagittal plane within the range of glenoid fossa
Lateral pole of the right condyle RCd-L The most lateral point of the right condylar head
Lateral pole of the left condyle LCd-L The most lateral point of the left condylar head
Medial pole of the right condyle RCd-M The most medial point of the right condylar head
Medial pole of the left condyle LCd-M The most medial point of the left condylar head
Right condylar center RCd-C A mid-point between the RCd-L and RCd-M
Leftt condylar center LCd-C A mid-point between the LCd-L and LCd-M
Right inferior gonion RInf-Go The most inferior point of the inferior border of the lower half of the right ramus
Left inferior gonion LInf-Go The most inferior point of the inferior border of the lower half of the left ramus
Mandibular dental landmarks
Mandibular central incisor L1MP Lower dental midline point
Cusp of mandibular right first molar RL6CP The mesiobuccal cusp tip of the mandibular right first molar
Cusp of mandibular left first molar LL6CP The mesiobuccal cusp tip of the mandibular left first molar
Lines
Frontozygomatic point line FZP line A line passing through the RFZP and LFZP
Right Frankfort horizontal line RFH line A line passing through the RPo and ROr
Right ramal line RRamal line A line passing through the RInf Go and RCd-S
Left ramal line LRamal line A line passing through the LInf Go and LCd-S
Right condylar axis line RCondylar axis line A line passing through the RCd-L and RCd-M
Left condylar axis line LCondylar axis line A line passing through the LCd-L and LCd-M
Planes
Horizontal plane Horizontal plane A plane parallel to both the RFH line and the FZP line passing N
Coronal plane Coronal plane A plane perpendicular to both the horizontal plane and the FZP line passing through N
Mid-sagittal plane MidS plane A plane perpendicular to both the horizontal plane and the coronal plane passing through N
Right condylar coronal plane RCondylar coronal plane A plane including the RCondylar axis line and perpendicular to the horizontal plane
Left condylar coronal plane LCondylar coronal plane A plane including the LCondylar axis line and perpendicular to the horizontal plane
Right condylar sagittal plane RCondylar sagittal plane A plane including the RCd-C point and perpendicular to the horizontal plane and the RCondylar coronal plane
Left condylar sagittal plane LCondylar sagittal plane A plane including the LCd-C point and perpendicular to the horizontal plane and the LCondylar coronal plane

Adapted from the article of Hong et al. (Korean J Orthod 2020;50:293-303).17

Table 2
Definition of the variables used in this study
Variable Abbreviation Definition
Cant Angulation Maxillary skeletal Cant (°) MxS-Cant (°) The angle formed between the RJ–LJ line and the FZP line projected to the Coronal plane
Maxillary dental Cant (°) MxD-Cant (°) The angle formed between the RU6CP–LU6CP line and the FZP line projected to the Coronal plane
Mandibular dental Cant (°) MdD-Cant (°) The angle formed between the RL6CP–LL6CP line and the FZP line projected to the Coronal plane
Mandibular skeletal Cant (°) MdS-Cant (°) The angle formed between the RMdF–LMdF line and the FZP line projected to the Coronal plane
Mandibular border Cant (°) MdB-Cant (°) The angle formed between the RInf-Go–LInf-Go line and the FZP line projected to the Coronal plane
Distance Maxillary skeletal Cant (mm) MxS-Cant (mm) The difference in the coordinates on the z-axis of RJ and LJ
Maxillary dental Cant (mm) MxD-Cant (mm) The difference in the coordinates on the z-axis of RU6CP and LU6CP
Mandibular dental Cant (mm) MdD-Cant (mm) The difference in the coordinates on the z-axis of RL6CP and LL6CP
Mandibular skeletal Cant (mm) MdS-Cant (mm) The difference in the coordinates on the z-axis of RMdF and LMdF
Mandibular border Cant (mm) MdB-Cant (mm) The difference in the coordinates on the z-axis of RInf-Go and LInf-Go
Shift Maxillary skeletal Shift (mm) MxS-Shift (mm) The x-axis coordinate of A point
Maxillary dental Shift (mm) MxD-Shift (mm) The x-axis coordinate of U1MP
Mandibular dental Shift (mm) MdD-Shift (mm) The x-axis coordinate of L1MP
Mandibular skeletal Shift (mm) MdS-Shift (mm) The x-axis coordinate of B point
Mandibular border Shift (mm) MdB-Shift (mm) The x-axis coordinate of Me
Yaw Maxillary skeletal Yaw (°) MxS-Yaw (°) The angle between the bisector of the RJ–A–LJ angle and the MidS plane projected to the Horizontal plane
Maxillary dental Yaw (°) MxD-Yaw (°) The angle between the bisector of the RU6CP–U1MP–LU6CP angle and the MidS plane projected to the Horizontal plane
Mandibular dental Yaw (°) MdD-Yaw (°) The angle between the bisector of the RL6CP–L1MP–LL6CP angle and the MidS plane projected to the Horizontal plane
Mandibular skeletal Yaw (°) MdS-Yaw (°) The angle between the bisector of the RMdF–B–LMdF angle and the MidS plane projected to the Horizontal plane
Mandibular border Yaw (°) MdB-Yaw (°) The angle between the bisector of the RInf-Go–Me–LInf-Go angle and the MidS plane projected to the Horizontal plane
Ramus Frontal ramus angle difference between the right and left sides (°) FRA-difference (°) The difference between the RFRA (the angle formed between RRamal line and the FZP line projected to the Coronal plane) and the LFRA (the angle formed between the LRamal line and the FZP line projected to the Coronal plane)
Ramus height difference between the right and left sides (mm) RH-difference (mm) The difference between the RRH (the linear distance from RInf-Go to RCd-S) and the LRH (the linear distance from LInf-Go to LCd-S)

See Table 1 for definitions of each landmark, line, and plane.

Table 3
Severity of the measurement variables
Variable Severe
(< −3 SD)		 Moderate
(≥ −3 SD and < −2 SD)		 Mild
(≥ −2 SD and < −1 SD)		 Normal
(≥ −1 SD and ≤ 1 SD)		 Mild
(> 1 SD and ≤ 2 SD)		 Moderate
(> 2 SD and ≤ 3 SD)		 Severe
(> 3 SD)
Cant MxS-Cant (°) < −3.30 ≥ −3.30 and < −2.06 ≥ −2.06 and < −0.82 ≥ −0.82 and ≤ 1.65 > 1.65 and ≤ 2.89 > 2.89 and ≤ 4.13 > 4.13
MxD-Cant (°) < −0.69 ≥ −0.69 and < −0.18 ≥ −0.18 and < 0.32 ≥ 0.32 and ≤ 1.33 > 1.33 and ≤ 1.84 > 1.84 and ≤ 2.34 > 2.34
MdD-Cant (°) < −2.07 ≥ −2.07 and < −1.11 ≥ −1.11 and < −0.16 ≥ −0.16 and ≤ 1.75 > 1.75 and ≤ 2.71 > 2.71 and ≤ 3.67 > 3.67
MdS-Cant (°) < −2.72 ≥ −2.72 and < −1.61 ≥ −1.61 and < −0.50 ≥ −0.50 and ≤ 1.72 > 1.72 and ≤ 2.82 > 2.82 and ≤ 3.93 > 3.93
MdB-Cant (°) < −3.92 ≥ −3.92 and < −2.44 ≥ −2.44 and < −0.96 ≥ −0.96 and ≤ 2.00 > 2.00 and ≤ 3.48 > 3.48 and ≤ 4.96 > 4.96
MxS-Cant (mm) < −3.50 ≥ −3.50 and < −2.18 ≥ −2.18 and < −0.87 ≥ −0.87 and ≤ 1.76 > 1.76 and ≤ 3.08 > 3.08 and ≤ 4.40 > 4.40
MxD-Cant (mm) < −0.60 ≥ −0.60 and < −0.15 ≥ −0.15 and < 0.30 ≥ 0.30 and ≤ 1.20 > 1.20 and ≤ 1.65 > 1.65 and ≤ 2.10 > 2.10
MdD-Cant (mm) < −1.72 ≥ −1.72 and < −0.92 ≥ −0.92 and < −0.13 ≥ −0.13 and ≤ 1.46 > 1.46 and ≤ 2.26 > 2.26 and ≤ 3.05 > 3.05
MdS-Cant (mm) < −4.03 ≥ −4.03 and < −2.39 ≥ −2.39 and < −0.74 ≥ −0.74 and ≤ 2.54 > 2.54 and ≤ 4.19 > 4.19 and ≤ 5.83 > 5.83
MdB-Cant (mm) < −6.36 ≥ −6.36 and < −3.96 ≥ −3.96 and < −1.55 ≥ −1.55 and ≤ 3.25 > 3.25 and ≤ 5.65 > 5.65 and ≤ 8.05 > 8.05
Shift MxS-Shift (mm) < −3.14 ≥ −3.14 and < −2.01 ≥ −2.01 and < −0.87 ≥ −0.87 and ≤ 1.39 > 1.39 and ≤ 2.52 > 2.52 and ≤ 3.66 > 3.66
MxD-Shift (mm) < −4.07 ≥ −4.07 and < −2.56 ≥ −2.56 and < −1.05 ≥ −1.05 and ≤ 1.96 > 1.96 and ≤ 3.46 > 3.46 and ≤ 4.97 > 4.97
MdD-Shift (mm) < −3.01 ≥ −3.01 and < −1.50 ≥ −1.50 and < 0.02 ≥ 0.02 and ≤ 3.06 > 3.06 and ≤ 4.58 > 4.58 and ≤ 6.10 > 6.10
MdS-Shift (mm) < −3.08 ≥ −3.08 and < −1.53 ≥ −1.53 and < 0.03 ≥ 0.03 and ≤ 3.13 > 3.13 and ≤ 4.69 > 4.69 and ≤ 6.24 > 6.24
MdB-Shift (mm) < −2.21 ≥ −2.21 and < −0.74 ≥ −0.74 and < 0.73 ≥ 0.73 and ≤ 3.67 > 3.67 and ≤ 5.14 > 5.14 and ≤ 6.61 > 6.61
Yaw MxS-Yaw (°) < −4.84 ≥ −4.84 and < −3.27 ≥ −3.27 and < −1.71 ≥ −1.71 and ≤ 1.43 > 1.43 and ≤ 3.00 > 3.00 and ≤ 4.56 > 4.56
MxD-Yaw (°) < −6.16 ≥ −6.16 and < −4.29 ≥ −4.29 and < −2.42 ≥ −2.42 and ≤ 1.32 > 1.32 and ≤ 3.19 > 3.19 and ≤ 5.06 > 5.06
MdD-Yaw (°) < −5.39 ≥ −5.39 and < −3.46 ≥ −3.46 and < −1.52 ≥ −1.52 and ≤ 2.34 > 2.34 and ≤ 4.27 > 4.27 and ≤ 6.21 > 6.21
MdS-Yaw (°) < −4.17 ≥ −4.17 and < −2.66 ≥ −2.66 and < −1.15 ≥ −1.15 and ≤ 1.87 > 1.87 and ≤ 3.39 > 3.39 and ≤ 4.90 > 4.90
MdB-Yaw (°) < −3.76 ≥ −3.76 and < −2.33 ≥ −2.33 and < −0.90 ≥ −0.90 and ≤ 1.97 > 1.97 and ≤ 3.40 > 3.40 and ≤ 4.83 > 4.83
Ramus FRA-difference (°) < −5.65 ≥ −5.65 and < −3.21 ≥ −3.21 and < −0.77 ≥ −0.77 and ≤ 4.10 > 4.10 and ≤ 6.53 > 6.53 and ≤ 8.97 > 8.97
RH-difference (mm) < −7.25 ≥ −7.25 and < −4.62 ≥ −4.62 and < −1.98 ≥ −1.98 and ≤ 3.29 > 3.29 and ≤ 5.92 > 5.92 and ≤ 8.56 > 8.56

The mean and standard deviation (SD) from the non-asymmetry group were used for deciding the severity of each variable.

See Table 2 for definitions of each measurement variable.

Table 4
Comparison of measurement variables among the facial asymmetry types
Parameter Non-asymmetry type
(n = 43, 35.8%, 1)		 Max-Cant type
(n = 17, 14.2%, 2)		 Man-Shift-Yaw type
(n = 20, 16.7%, 3)		 Complex type
(n = 11, 9.2%, 4)		 Max-Rev-Cant type (n = 29, 24.2%, 5) p-value Multiple comparison
Mean SD Mean SD Mean SD Mean SD Mean SD
Cant Angulation MxS-Cant (°) 0.41 1.24 1.15 1.11 1.04 2.24 2.05† 1.54 −0.80 1.17 < 0.001*** 5 < (1, 3, 2) < (3, 2, 4)
MxD-Cant (°) 0.83 0.50 2.87c 0.96 1.03 0.85 3.40c 1.67 −0.87c 0.61 < 0.001*** 5 < (1, 3) < (2, 4)
MdD-Cant (°) 0.80 0.96 2.35† 1.05 2.15† 1.56 5.44c 1.46 0.01 1.13 < 0.001*** (5, 1) < (3, 2) < 4
MdS-Cant (°) 0.61 1.11 1.05 0.91 1.30 1.58 3.81b 1.46 −0.30 1.74 < 0.001*** (5, 1) < (1, 2, 3) < 4
MdB-Cant (°) 0.52 1.48 1.17 0.98 1.36 1.60 3.94b 1.80 0.17 1.73 < 0.001*** (5, 1, 2, 3) < 4
p-value 0.363 < 0.001*** 0.207 < 0.001*** < 0.011*
Multiple
comparison		 (MdS, MxS, MdB) < (MdD, MxD) (MxS, MxD, MdS, MdB) < (MdS, MdB, MdD) (MxD, MxS, MdS, MdD) < (MxS, MdS, MdD, MdB)
Distance MxS-Cant (mm) 0.45 1.32 1.26 1.22 1.07 2.22 2.13† 1.51 −0.84 1.22 < 0.001*** 5 < (1, 3, 2) < (3, 2, 4)
MxD-Cant (mm) 0.75 0.45 2.69§ 0.91 0.94 0.79 3.20§ 1.56 −0.81§ 0.58 < 0.001*** 5 < (1, 3) < (2, 4)
MdD-Cant (mm) 0.67 0.79 2.01† 0.89 1.82† 1.34 4.65§ 1.35 0.02 0.93 < 0.001*** (5, 1) < (3, 2) < 4
MdS-Cant (mm) 0.90 1.64 1.63 1.39 2.00 2.36 5.74‡ 2.12 −0.42 2.54 < 0.001*** (5, 1) < (1, 2, 3) < 4
MdB-Cant (mm) 0.85 2.40 1.92 1.67 2.17 2.57 6.39‡ 2.95 0.27 2.83 < 0.001*** (5, 1, 2, 3) < 4
p-value 0.664 < 0.028* 0.204 < 0.001*** < 0.096
Multiple
comparison		 (MxS, MdS, MdB, MdD) < (MdS, MdB, MdD, MxD) (MxS, MxD) < (MxD, MdD) < (MdD, MdS, MdB)
Shift MxS-Shift (mm) 0.26 1.13 −0.23 1.86 0.69 1.30 0.88 1.17 0.48 0.79 0.122
MxD-Shift (mm) 0.45 1.51 0.91 2.10 1.37 1.28 1.92 1.52 0.49 1.45 0.029*
MdD-Shift (mm) 1.54 1.52 2.53 2.17 4.68‡ 1.58 8.65§ 1.74 1.83 1.68 < 0.001*** (1, 5, 2) < 3 < 4
MdS-Shift (mm) 1.58 1.55 3.14† 2.13 5.39‡ 1.47 10.25§ 2.11 1.82 1.49 < 0.001*** (1, 5) < (5, 2) < 3 < 4
MdB-Shift (mm) 2.20 1.47 4.05† 2.07 6.51‡ 1.58 13.73§ 2.79 2.19 1.49 < 0.001*** (5, 1) < 2 < 3 < 4
p-value < 0.001*** < 0.001*** < 0.001*** < 0.001*** < 0.001***
Multiple
comparison		 (MxS, MxD) < (MdD, MdS, MdB) (MxS, MxD)
< (MxD, MdD)
< (MdD, MdS, MdB)		 (MxS, MxD)
< (MdD, MdS)
< (MdS, MdB)		 (MxS, MxD)
< (MdD, MdS)
< MdB		 (MxD, MxS, MdS, MdD) < (MxS, MdS, MdD, MdB)
Yaw MxS-Yaw (°) −0.14 1.57 −0.03 1.81 0.64 1.49 1.19 1.37 −0.24 1.35 0.042*
MxD-Yaw (°) −0.55 1.87 −0.12 1.60 0.18 2.00 −0.14 1.20 −0.32 1.92 0.677
MdD-Yaw (°) 0.41 1.93 1.42 1.74 3.72† 1.63 6.97§ 2.29 1.03 1.85 < 0.001*** (1, 5, 2) < 3 < 4
MdS-Yaw (°) 0.36 1.51 1.62 1.47 4.05‡ 1.40 7.13§ 2.29 1.08 1.09 < 0.001*** (1, 5) < (5, 2) < 3 < 4
MdB-Yaw (°) 0.53 1.43 1.73 1.29 4.52‡ 1.42 8.54§ 2.37 1.36 1.32 < 0.001*** (5, 1, 2) < 3 < 4
p-value 0.017* < 0.001*** < 0.001*** < 0.001*** < 0.001***
Multiple
comparison		 (MxD, MxS, MdS, MdD) < (MxS, MdS, MdD, MdB) (MxD, MxS, MdD)
< (MdD, MdS, MdB)		 (MxD, MxS)
< (MdD, MdS, MdB)		 (MxD, MxS)
< (MdD, MdS, MdB)		 (MxD, MxS)
< (MdD, MdS, MdB)
Ramus FRA-difference
(°)		 1.66 2.44 2.25 2.54 2.74 2.01 7.33‡ 2.54 1.35 2.27 < 0.001*** (5, 1, 2, 3) < 4
RH-difference
(mm)		 0.65 2.63 2.08 2.26 3.42† 2.46 8.41‡ 3.06 0.24 2.87 < 0.001*** (5, 1, 2) < (2, 3) < 4

The K-mean cluster analysis was conducted using three representative variables, which provide significant clinical information: MxD-Cant (mm), MdB-Shift (mm), and MdB-Yaw (°).

The one-way analysis of variance test with Tukey’s honestly significant difference post hoc comparison was conducted.

*p < 0.05; ***p < 0.001.

†Mild; ‡Moderate; §Severe.

See Table 2 for definitions of each measurement variable.

Table 5
Comparison of facial asymmetry classifications between this and previous studies
Phenotype This study Hwang et al.9
Hwang et al.18 		Baek et al.10 Chen et al.15
Samples
Ethnicity Korean Korean Korean Taiwanese
Subjects Patients who underwent orthognathic surgery Patients who were diagnosed with facial asymmetry Patients who wanted correction of facial asymmetry Patients who wanted correction of facial asymmetry
Number 120 100 43 70
Study design
Angle classification Skeletal Class III Skeletal Class I, II, and III Skeletal Class I, II, and III Skeletal Class III
Exclusion criteria

  • Posterior teeth were missing or abnormally shaped

  • Degenerative joint disease, tumor, or trauma history in the temporomandibular joints

  • Hemifacial microsomia and other craniofacial syndromes

  • Not mentioned

  • Cleft lip and/or palate, plagiocephaly, hemifacial microsomia

  • Congenital muscular torticollis

  • Degenerative temporomandibular joint disease

  • Mandibular tumors

  • History of trauma to the jaw

  • Congenital anomalies

  • History of maxillofacial trauma
Reference planes Horizontal plane Frankfort horizontal plane Frankfort horizontal plane Mid-sagittal plane
Coronal plane Mid-sagittal reference line Coronal plane Horizontal plane
Mid-sagittal plane Mid-sagittal plane Coronal plane
Origin (0, 0, 0) N point None Not mentioned Not mentioned
Variables for Max-dental Cant (mm) Menton deviation (°) Upper midline deviation (mm) The direction and magnitude oftransverse ramus discrepancy relative to menton deviation
classification Man-border Shift (mm) Apical base midline discrepancy (mm) Maxilla canting of U3 and U6 (mm)
Man-border Yaw (°) Vertical difference of the right and left Ag (mm) Arch form discrepancy (mm)
Horizontal difference of the right and left Ag (mm) Gonion to the mid-sagittal plane (mm)
Maxillary base canting (°) Ramus height (mm)
Maxillary alveolar canting (°) Frontal ramus inclination (°)
Bulkiness difference of the mandibular inferior border Menton deviation (mm)
Lip line canting (°)
Inclusion of yaw Yes No No No
Analyzed data 3D-CT PA cephalogram and frontal photograph 3D-CT CBCT
Statistical method Cluster analysis Cluster analysis Cluster analysis No cluster
Types Non-asymmetry type (35.8%) Group E (Normal, 28%) Not mentioned Not mentioned

  • All variables were within normal limits
Max-Cant type (14.2%) Not mentioned Group 2 (Universal lateral condylar hyperplasia asymmetry, 39%) Not mentioned

  • Significant difference between the left and right ramus height

  • Menton deviation to the short side
Man-Shift-Yaw type (16.7%) Group C (Menton type, 21%) Group 1 (Mandibular body asymmetry, 44%) Group 2 (27%)

  • No ramus height difference between the deviated and non-deviated sides

  • Deviation of menton and lower apical base midline to the same side

  • Shift or lateralization of the mandibular body

  • Menton and ramus deviation to the same side

  • Discrepancy in the ramus width was larger than the menton shift

  • Asymmetry resulted from a bodily side shift of the maxillomandibular complex without significant cant and yaw
Complex type (9.2%) Group A (Ramus–Menton type, 7%) Group 4 (C-shaped asymmetry, 5%) Group 1 (47%)

  • Significant canting of the maxillary basal and alveolar bone

  • Deviation of menton and mandibular apical base midline to the side of the shorter ramus

  • Difference of the right and left ramus height

  • Severe maxillary canting

  • Significant ramus height differences

  • Significant menton deviation to the short side

  • Large shift of menton

  • Synchronous but smaller ramus deviation

  • The maxillomandibular complex had cant and yaw to the menton deviation side

  • Maxillary and dental asymmetry was obvious in the transverse and vertical dimensions

  • Cant of occlusal plane was apparent
Max-Rev-Cant type (24.2%) Group B (Ramus–Angle type, 16%) Group 3 (Atypical asymmetry, 12%) Group 3 (26%)

  • Same ramus height difference between the deviated and non-deviated sides as group A

  • Menton was deviated in the opposite direction to shorter ramus

  • Reverse maxillary canting

  • Deviation of the menton to the short side

  • Prominence of the angle/gonion on the larger side

  • Menton and ramus deviated in opposite directions, which seemed secondary to a yaw rotation
Not matched Group D (Bulkiness type, 28%)

  • Similar to group A

  • Small difference in magnitude

Max-Cant, maxillary-cant; Man-Shift-Yaw, mandibular-shift and yaw; Max-Rev-Cant, maxillary reverse-cant; Ag, antegonion; U3, distance between Frankfort horizontal plane and midpoint of bracket slot of upper canine; U6, distance between Frankfort horizontal plane and midpoint of bracket slot of upper first molar; 3D-CT, three-dimensional computed tomography; PA, posteroanterior; CBCT, cone beam computed tomography.

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