This study aimed to establish mathematical equations to describe the dental and basal arch forms in skeletal Class II malocclusion and to investigate correlations between these forms to facilitate clinical diagnosis and treatment design.
Cone-beam computed tomography images of 60 patients (30 each with skeletal Class II malocclusion and normal occlusion) were subjected to 3-dimensional volume rendering, image reconstruction, and measurement. Using a computer program written in Java on the Eclipse platform (Eclipse Foundation, Ottawa, Canada), a beta function was used to establish mathematical models of dental and basal bone arch forms, and model-fitting was performed.
A mathematical model was developed to describe the dental and basal arch forms in skeletal Class II malocclusion. The maxillary and mandibular dental arch lengths were significantly longer in skeletal Class II malocclusion than in normal occlusion, whereas the width of the maxillary molar segment was smaller. The maxillary molar and mandibular intercranial segments in the basal arch were significantly longer in skeletal Class II malocclusion than in normal occlusion, whereas the mandibular intercranial segment width was smaller.
The dental arch and basal bone arch forms in the maxilla and mandible follow individual curves, described by a beta function. In skeletal Class II malocclusion, the dental and basal arches are discrepant in horizontal and anteroposterior dimensions, which should be considered during orthodontic treatment planning to improve arch matching.
Dental and basal bone arch forms were compared in skeletal Class II malocclusion.
Compared with normal occlusion, arch lengths are greater in skeletal Class Ⅱ malocclusion.
Basal bone arch width is smaller in the maxilla and larger in the mandible in Class Ⅱ malocclusion.
Length differences between dental and basal arches are larger in Class II malocclusion patients.
Clinicians should use individualized archwires to improve arch matching.
The relationship between dental and basal forms have significant effects on orthodontic diagnosis and treatment planning. If teeth are moved excessively, over the limits of their supporting basal bone, periodontal complications and relapse may occur. However, previous studies that investigated basal form dimensions using different definitions have yielded conflicting results. Moreover, previous studies that investigated the relationship of the basal bone and dental arches did not investigate both jaws concurrently; thus, this relationship between the maxillary and mandibular arches.
In 1925, Lundström proposed the term “apical base” to refer to the junction of the alveolar and basal bones of the maxilla and mandible near the apices of the teeth. In early studies, , based on casts and lateral cephalometric radiographs, investigators described the lowest point of the anterior alveolar border in the maxilla and mandible as the basal bone. Later, different points of the alveolar bone were used in various studies to determine the basal bone. The Bonwill-Hawley arch form, which is based on the principle of individualized arches, was the first selected for mathematical modeling. , Later, various mathematical functions were applied to dental arch curve fitting. Braun et al considered that the beta function provided the closest fit to the curve of the human dental arch, with correlation coefficients of curve fit exceeding 0.98 for all types of arch forms. Mina et al also proved that the beta function provided a higher degree of fit for dental or basal arch forms in normal occlusion. In our study, we used a novel least squares method to describe the basal bone arch on cone-beam computed tomography (CBCT) images and evaluate the transverse dimensions of skeletal Class malocclusion, without interference by the variable overlying soft tissues.
Several studies have focused on characterizing the dental arch or basal bone arch forms of patients with Class Ⅱ malocclusion. , Patients with Class II Division I malocclusion are considered to have a narrower maxillary dental arch than those with Class Ⅰ occlusion. However, the mandibular characteristics of the dental and basal arch forms were not consistent across studies. Ball et al and Gupta et al stated that there was no significant difference in the dental and basal arch forms between subjects with Class II Division 1 malocclusion and those with Class I ideal occlusion. Sayin et al and Uysal et al argued that the mandibular intercanine width was greater in Class II Division 1 arches than in normal occlusions.
In contrast, Al-Hilal determined that the mandibular dental and basal arch widths of patients with Class II Division 1 malocclusion were smaller than those of patients with Class I occlusion. In general, most studies only focused on evaluating the transverse dimensions of arch forms or describing the dental and basal forms of the maxilla or the mandible. Seldom previous studies paid attention to the incoordination between maxillary and mandibular arch forms, particularly at the basal bone arch level.
Thus, the objectives of this study were as follows: (1) to establish mathematical equations to describe the dental and basal arch forms in skeletal Class II malocclusion by computer programming and (2) to compare the matching between the dental and basal bone arch forms, which could facilitate clinical diagnosis, thereby revealing disharmonious relationships and ensuring the formulation of individualized treatment plans.
Material and methods
This study was approved by the ethics committee of the Tianjin Stomatological Hospital, Tianjin, China. CBCT records of patients who visited the Tianjin Stomatological Hospital between September 2014 and January 2017 were studied. The need to obtain informed patient consent was waived as all data were anonymized. The study population comprised 60 subjects aged between 18 and 28 years (male: 18-26 years, female: 18-28 years), including 30 skeletal Class II (GII group; 11 men and 19 women) and 30 skeletal Class I (GN group; 13 men and 17 women) malocclusions. There was no sex-specific difference between the 2 groups ( P = 0.792). The demographic information is shown in Table I .
|Variables||GⅡ group||GN group|
|Mean age, y, mean ± SD||21.47 ± 2.75||20.83 ± 2.54|
|Female||21.78 ± 2.91||20.23 ± 2.51|
|Male||20.90 ± 2.35||21.61 ± 2.37|
The inclusion criteria were as follows: (1) full permanent dentition (excluding third molars); (2) minor length discrepancy for each arch (crowding ≤3 mm, spacing ≤3 mm); (3) for GⅡ group, bilateral Class Ⅱ canine and molar relationships and protrusive incisors with ANB ≥5°; and (4) for the GN group, bilateral Class Ⅰ canine and molar relationships and a normal overbite and overjet with 0° ≤ ANB ≤ 4°.
Exclusion criteria were as follows: (1) dentition with decayed or cracked teeth; (2) dentition with a prosthetic crown or restoration-altered tooth size and clinical crown form; (3) unilateral or bilateral crossbite; (4) a history of orthodontic treatment; and (5) a history of trauma or oral and maxillofacial surgery.
All CBCT images were acquired using a KaVo 3D eXam device (Imaging Sciences International, Hatfield, Pa) (120 kVp; 3-7 mA; 0.40-mm voxel size; scan time, 26.9 seconds; and field of view of 242 × 193 mm). CBCT data were exported in digital imaging, and imaging information was imported into the Invivo software (version 5.4; Anatomage, San Jose, Calif) for 3-dimensional (3D) volume rendering, image reconstruction, and measurement and recording of coordinate values. Then, the 3D coordinates were transformed into 2-dimensional coordinates by performing mathematical operations of the work plane. We improved the calculation process and completed software development with regard to data processing and acquirement. Operational programs were written in Java on the Eclipse development platform (Eclipse Foundation, Ottawa, Canada).
For dental measurements, the fa point is defined as the midpoint of the facial axis of the clinical crown. The fa points from the right second molar to the left second molar in the maxilla and mandible were recorded (n = 28). Four quadrants of teeth (from the upper right ) were marked in order as A, B, C, and D. The specific landmarks of teeth in the 4 quadrants were displayed in Figure 1 .
The 3D intracranial reference planes were oriented according to the following 4 reference planes. The occlusion plane (O-plane) was an optimization plane calculated using the Artificial Bee Colony (ABC) algorithm by analyzing 28 markers of maxillary and mandibular teeth using the programming software. The algorithm is used for solving multimodal optimization problems and simplifies the calculation. The maxillary basal bone (A-plane) was parallel to the O-plane, which passed through the A point. The mandibular basal planes (B-plane) was parallel to the O-plane, which passed through the B point. The midsagittal plane was defined by the Nasion, Anterior nasal spine, and Basion points. The reference planes are shown in Figure 2 .
The anatomic apical point (Ap) of all 28 teeth were recorded. For premolars and molars, the position of the Ap was calculated with the following formula: ([X 0 , Y 0 , Z 0 ] = [X 1 + X 2 +…+ X n ]/n + [Y 1 + Y 2 +…+Y n ]/n + [Z 1 + Z 2 +…+ Z n ]/n [n ≥ 2]). The T-plane was parallel to the A-plane and passed through the Ap point. The H-plane was perpendicular to the A-plane and parallel to the midsagittal plane. The T- and H-plane intersected at a line referred to as the line T. The landmarks of the basal bone (ba point) were intersections of line T and the anterior border of the basal bone. The basal bone points in the mandible were defined similarly by referring to the B-plane. All basal bone makers are displayed in Figure 3 .
The 2-dimensional coordinate values were programmed by the projection of the 3D coordinate values on the occlusion plane (standard work plane). Projections of fa and ba were marked as fa′ and ba′. The origin (0, 0) of the coordinate system, and 2 axes (x, y) were constructed. For the convenience of calculations, the transverse axis (x-axis) was set as the line connecting the projection points of the maxillary second molars (right: Afa-7′, coordinate [−W 7 /2, 0]; left: Bfa-7′, coordinate [W 7 /2, 0]); the midpoint of the connecting line was recorded as the origin O point (0, 0) ( Fig 4 ). The vertical axis (y-axis) was perpendicular to the x-axis and passed through the origin. The projection of the connecting point of the maxillary central incisors was recorded as point O’ (0, D 7 ). The conversion of the projection points is shown in Table II .
|Dental arch landmarks in 2D coordinate (fa′ point)|
|Right upper quadrant||Afa-11′, Afa-12′, Afa-13′, Afa-14′, Afa-15′, Afa-16′, Afa-17′|
|Left upper quadrant||Bfa-21′, Bfa-22′, Bfa-23′, Bfa-24′, Bfa-25′, Bfa-26′, Bfa-27′|
|Right lower quadrant||Cfa-41′, Cfa-42′, Cfa-43′, Cfa-44′, Cfa-45′, Cfa-46′, Cfa-47′|
|Left lower quadrant||Dfa-31′, Dfa-32′, Dfa-33′, Dfa-34′, Dfa-35′, Dfa-36′, Dfa-37′|
|Basal bone arch landmarks in 2D coordinate (ba′ point)|
|Right upper quadrant||Aba-11′, Aba-12′, Aba-13′, Aba-14′, Aba-15′, Aba-16′, Aba-17′|
|Left upper quadrant||Bba-21′, Bba-22′, Bba-23′, Bba-24′, Bba-25′, Bba-26′, Bba-27′|
|Right lower quadrant||Cba-41′, Cba-42a′, Cba-43a′, Cba-44a′, Cba-45a′, Cba-46a′, Cba-47a′|
|Left lower quadrant||Dba-31′, Dba-32′, Dba-33′, Dba-34′, Dba-35′, Dba-36′, Dba-37′|
The relationships between dental arch width and depth can be described by the beta function, which is simplified as follows: Y = D [1 − (2x/W) 2 ] e . Compared with other mathematical functions, the beta function represents the dental arch form the closest. D is the vertical distance from the point adjacent to the left and right central incisors to the second permanent molar line. W represents the horizontal distance between the second permanent molars on both sides. The constant value e is affected by all teeth or basal bone positions, especially by the relative position of the canine, which determines the curve shape of the dental or basal arch. To obtain the best-fitting curve for each group, we applied the ABC algorithm to acquire the optimal coordinates of each tooth position in the dental and basal arches. The e value was acquired by finding the optimal curve that could fit multiple positions using our program based on the algorithm.
A fitting analysis of all markers was carried out between the beta function and arch form curve with SPSS software (version 17.0; SPSS Inc, Chicago, Ill).
The width and length differences between the basal bone and dental arch in the maxilla and mandible were recorded ( Table III ). Intercanine and intermolar lengths and widths in the GⅡ and GN groups were compared using 2 independent sample t tests. Correlation statistics were used to assess the correlation between the fa and ba projection point distances for both groups.
|AB-Wfb-3||The difference in width between the basal bone and dental arch projections at the maxillary canines.|
|AB-Wfb-7||The difference in width between the basal bone and dental arch projections at the maxillar y second molars.|
|CD-Wfb-3||The difference in width between the basal bone and dental arch projections at the mandibular canine.|
|CD-Wfb-7||The difference in width between the basal bone and dental arch projections at the mandibular second molars.|
|AB-Lfb-3||The difference in length between the basal bone and dental arch projections at the maxillary canine.|
|AB-Lfb-7||The difference in length between the basal bone and dental arch projections at the maxillar y second molars.|
|CD-Lfb-3||The difference in length between the basal bone and dental arch projections at the mandibular canine.|
|CD-Lfb-7||The difference in length between the basal bone and dental arch projections at the mandibular second molars.|
|W-fa-3||The difference in width between the maxillary and mandibular canine projections in the dental arch.|
|W-fa-7||The difference in width between the maxillary and mandibular second molars projections in the dental arch.|
|W-ba-3||The difference in width between the maxillary and mandibular canine projections in the basal bone arch.|
|W-ba-7||The difference in width between the maxillary and mandibular second molars projections in the basal bone arch.|
|L-fa-3||The difference in length between the maxillary and mandibular canine projections in the dental arch.|
|L-fa-7||The difference in length between the maxillary and mandibular second molars projections in the dental arch.|
|L-ba-3||The difference in length between the maxillary and mandibular canine projections in the basal bone arch.|
|L-ba-7||The difference in length between the maxillary and mandibular second molars projections in the basal bone arch.|
The distances between fa′ and ba′ points for each tooth in skeletal Class II malocclusion are shown in Table IV . These values were positive in both the maxilla and the mandible. The curvatures of the projection points for the dental arch and basal form are compared in Figure 5 ; the fa′ points were located buccally relative to the corresponding ba′ points. The curve fitting analysis is provided in Supplementary Tables I and II .
|Projection points||Relative value||Correlation coefficient (r)||Regression equation|
|A 7 fa′-ba′||1.75||0.747||y = 0.513x + 0.725|
|A 6 fa′-ba′||1.54||0.905||y = 0.829x + 1.339|
|A 5 fa′-ba′||2.09||0.829||y = 0.818x + 4.538|
|A 4 fa′-ba′||2.93||0.704||y = 0.822x + 1.078|
|A 3 fa′-ba′||7.19||0.917||y = 0.624x + 3.828|
|A 2 fa′-ba′||9.56||0.773||y = 0.479x + 9.037|
|A 1 fa′-ba′||11.59||0.870||y = 0.562x + 6.051|
|B 1 fa′-ba′||11.62||0.742||y = 0.574x + 5.567|
|B 2 fa′-ba′||9.93||0.842||y = 0.642x + 3.008|
|B 3 fa′-ba′||7.33||0.916||y = 0.564x + 5.499|
|B 4 fa′-ba′||4.30||0.897||y = 0.589x + 4.962|
|B 5 fa′-ba′||2.35||0.760||y = 0.485x + 7.186|
|B 6 fa′-ba′||2.71||0.837||y = 0.425x + 6.537|
|B 7 fa′-ba′||3.76||0.746||y = 0.719x + 4.395|
|C 7 fa′-ba′||7.62||0.697||y = 0.324x + 9.808|
|C 6 fa′-ba′||4.43||0.730||y = 0.247x + 12.11|
|C 5 fa′-ba′||1.40||0.881||y = 0.555x + 8.091|
|C 4 fa′-ba′||3.00||0.906||y = 0.624x + 6.237|
|C 3 fa′-ba′||4.92||0.726||y = 0.618x + 6.850|
|C 2 fa′-ba′||6.04||0.882||y = 0.755x + 2.579|
|C 1 fa′-ba′||6.41||0.847||y = 0.703x + 4.617|
|D 1 fa′-ba′||6.41||0.817||y = 0.771x + 2.092|
|D 2 fa′-ba′||5.96||0.914||y = 0.746x + 2.983|
|D 3 fa′-ba′||5.02||0.763||y = 0.599x + 7.320|
|D 4 fa′-ba′||2.95||0.786||y = 0.619x + 6.461|
|D 5 fa′-ba′||1.69||0.785||y = 0.431x + 10.356|
|D 6 fa′-ba′||4.58||0.759||y = 0.273x + 10.766|
|D 7 fa′-ba′||7.44||0.883||y = 0.502x + 8.938|