## 2 Methods  
This study included 341 Vietnamese participants (skeletal Class I, 123; Class II, 124; Class III, 94), including 126 male and 169 female individuals, aged between 17 and 25 years (mean age, 22 years), who were recruited from the Department of Orthodontics of Hanoi Medical University (Hanoi, Vietnam). Participants with a full set of permanent dentition were included in this study. Participants who had previously undergone orthodontic or surgical treatments, had a BMI exceeding 22.9 and had symptoms or a history of upper respiratory infections, adenoid hypertrophy, tonsillitis, adenoidectomy, or tonsillectomy were excluded from the study. This study focused on investigating the natural progression and potential associations between vertical skeletal patterns and the upper airway space in a more controlled manner. This approach was implemented to reduce confounding factors that may affect outcomes and to facilitate a clearer understanding of the association between skeletal morphology and airway dynamics among the study participants. This study was reviewed and approved by the Institutional Review Board of the authors' institution (HMU IRB675). CBCT volume scans of all the patients were obtained with maximum intercuspation to minimize variability in the mandibular and soft tissue measurements [9] using the Sirona DS Plus [Sirona Dental Systems, Bernsheim, Germany]. The imaging protocol included a 12-inch field of view to encompass the entire craniofacial anatomy. The axial slice thickness was set at 0.3 mm, and the voxels were isotropic. The scanning time ranged from 6 to 20 s, with resolutions varying between 0.25 mm and 0.6 mm and voxel sizes between 3.5 \(\times\) 3.5 pixels and 1024 \(\times\) 1024 pixels. The information was transmitted directly to the researchers' computer and saved in Digital Imaging and Communications in Medicine format. To account for the influence of head posture on the airway volume, the craniocortical inclinations of all the patients were examined to confirm that they fell within the standard range [10]. A 3D coordinate system and image were constructed using the In Vivo Dental software 6.0 (Anatomage, San Jose, CA, USA), a medical image analysis tool. This software allows the precise construction of a 3D representation, enabling researchers to accurately analyze anatomical structures. The system provides a standardized framework for measurements and comparisons across subjects. With advanced features for image processing and analysis, In Vivo Dental 6.0 facilitates detailed visualization and manipulation of dental and skeletal structures derived from imaging data, such as CBCT data. This approach offers valuable insights into the relationships among skeletal patterns, dental alignment, and upper airway anatomy, which are essential for enhancing orthodontic diagnosis and treatment planning. To minimize errors and standardize the measurements, the 3D image was reoriented using the nasion point as the origin of the 3D coordinate system (Figure 1). In contrast to other studies that employed 2D cephalometric analysis, our study used a 3D analysis system derived from CBCT volumetric images. This involved establishing a set of landmarks, reference lines, and reference planes to conduct comprehensive 3D assessments. [PERSON] et al. conducted a study indicating that deep learning algorithms demonstrated exceptional accuracy in automatically identifying 3D cephalometric landmarks [1]. The landmark identifications and physical measurements were performed by the same orthodontist (V.T.T.T). For the 3D cephalometric analysis, we identified 16 hard tissue landmarks and performed seven anteroposterior and five vertical measurements. These are defined in Table 1 and illustrated in Figure 2.  
Figure 1: The right portion, the left located in the most laterosuperior points of the external auditory meatus, and the right orbitals defined the standard Frankfort horizontal plane. A frontal plane was constructed through origin points.