## 2 Materials and Methods  
The study consisted of three separate parts, each involving a group of orthodontic patients from whom CBCT data and 3D facial scans were obtained before and after treatment. All participants provided informed consent before the study, were in good general health, and the research was conducted according to the guidelines of the American Dental Association (ADA) and the tenets of the Declaration of Helsinki. Data confidentiality and anonymity were strictly maintained. The study was approved by the Ethics Committee (EK-973 IGA 1.12/11). Our methodology focused on comparing three-dimensional images of the skull or its components, supplemented by 3D facial scans to enhance the robustness of our findings. The proposed methodology includes:  
1. taking cranial measurements on 3D cranial models,
2. collecting discrete feature points on 3D cranial models,
3. using surface face models and discrete feature points collected on 3D face models. Our study focused on adult orthodontic patients, both male and female. A 3D treatment plan was prepared for each participant, with surgical intervention as a core component of the therapy. Inclusion criteria included planned therapy, surgical intervention, comparison with a diagnostic standard, and desired outcomes (PICO). The study aimed to contribute to a methodology that can enhance quality of life and facial aesthetic appearance. The initial set of patients included 45 individuals, selected based on age, general health condition, orthodontic defects, and surgery types. This set was later reduced to 30 individuals from whom 3D scans of both soft and hard facial tissues were obtained. Analyses of these scans included: (i) surgically altered cranial metrics, (ii) hard tissue landmark assessment, and (iii) soft tissue surface and landmark matching. No facial soft tissue scans were available for 15 patients in the first group, which was closely related to upper and lower jaw therapy, leading to their exclusion from this part of the study. **The surgically altered cranial metrics** in the first part of the study consisted of 45 individuals who had undergone corrective surgery of the upper and lower jaw. Two medical imaging data sets were available for each patient, obtained by cone beam computed tomography (CBCT). The sample consisted of 19 women and 26 men. The mean age at the start of treatment was 22.21 years (females: 22.31, males: 22.13). The mean age after surgery was 23.01 years. The mean time between data sets was 0.85 years (approximately 10 months), ranging from 4.5 months to 2 years and 11 months. According to a Mann-Whitney test, no statistically significant differences in age or time interval were found between males and females at the 5% significance level. For each image dataset, a set of 14 linear measurements were taken using the functionalities available in the Anatomage Invivo software (Figure 1). Eight of these measurements were bilateral (comprising four pairs), while the remaining six were unilateral. The differences between the preoperative (\(PRE\)) and postoperative (\(POST\)) data were analyzed using univariate statistical tests, both unpaired and paired. In addition, identical tests were performed separately for males and females. Furthermore, the signed relative preoperative to postoperative differences \(PPD\) were calculated using the formula:  
\[PPD=(POST-PRE)/mean(POST,PRE)\times 100 \tag{1}\]  
These differences were evaluated using \(t\)-tests and Mann-Whitney tests concerning biological sex, the time interval between image datasets, and patient age at baseline. Finally, the overall differences were analyzed by multivariate analysis of variance with biological sex and time interval as factors. For bilateral linear distances, side differences were assessed using both unpaired and paired tests. Within-group variations according to biological sex and preoperative and postoperative status were also examined.