## 2 Materials and Methods
### Three-Dimensional Evaluation  
Afterwards, the digital models were exported to a specialized software (Geomagic Control X, 3D Systems, Rock Hill, SC, USA- version 2018.1.1) to assess dento-alveolar and  
Figure 2: Maximillary digital model of a patient with FPCB with the representation of Raphe plane, reference plane, and the linear transverse and sagittal distances measured on Crossbite Side (CB) and Non-Crossbite Side (NCB) sides. A: the most distal point of the incisive papilla; B: the most posterior visible point on the median palatal raphe. RP: Raphe plane; rP: reference plane; C TW CB: Canine Width on the Crossbite Side; C TW NCB: Canine Transverse Width on the Non-Crossbite Side; M TW NCB: Molar Transverse Width on the Non-Crossbite Side; C DEP CB: Canine Depth on the Crossbite Side; C DEP NCB: Canine Depth on the Non-Crossbite Side; M DEP CB: Molar Depth on the Crossbite Side; M DEP NCB: Molar Depth on the Non-Crossbite Side, UM: upper midline.  
palatal vault asymmetry. This was achieved by superimposing the mirrored models for each patient and performing a 3D analysis of deviations between the two halves of the maxillary arch. The deviations found in the FPCB group were compared with those of the controls.  
The workflow for the superimposition--which is a modified version of the method described by [PERSON] et al., 2018 [PERSON] et al. (2018)--was carried out by the same operator ([PERSON]), with 20 years of orthodontic experience, for each subject in this study. The process involved the following five key steps:  
1. Orientation: The occlusal plane was set identifying the same three reference points used in OrthoAnalyzer(tm). Then, the digital models were aligned so that the occlusal view was parallel to the occlusal plane (Figure 3a).
2. Segmentation: To define the surface of the model to be analysed, a gingival line was drawn along the most apical regions of the buccal detongigival junction of all maxillary teeth. The posterior limit was delineated from the most posterior point of the right and the left first molars (Figure 3b).
3. Mirroring: A Raphe plane was set perpendicular to the occlusal plane and the maxillary model was transversely flipped so that the right-side points of the median raphe appeared on the left and vice versa (Figure 3c,d).
4. Superimposition: The \"best-fit alignment\" function in the Geomagic Control X was used to superimpose the original maxillary model with the mirrored one. This fine-matching technique uses thousands of reference points and is based on an iterative closest-point algorithm. The precision of the superimposition was set to at least 0.2 mm, and the number of polygons for surface representation was fixed at the maximum of 100.000 (Figure 3e).
5. Three-Dimensional Analysis: Distances between corresponding points of the original maxillary model and the mirrored version were calculated to generate colour-coded maps (Figure 4). A tolerance range of \(\pm\)0.25 mm with a height of \(\pm\)1.5 mm was established. In the colour-coded maps, areas within the tolerance range (indicating matching between the two mirrored 3D casts) are shown in green. Yellow-to-red ranges indicate that the mirrored models are more expanded than the original casts, whereas light blue-to-dark blue ranges indicate that the mirrored model is narrower than the original (Figure 4).  
Figure 3: _Cont._  
Figure 4: Maximly model 3D analysis. RGB-coloured scale rod (millimetres) is shown on the right: the scale denotes both positive (red) and negative (blue) height differences. Green signifies the tolerance range.  
Figure 3: **(a–e)**: (**a**) Orientation based on the occlusal plan; (**b**) Segmentation of the digital model; (**c**) Raphe plane set perpendicular to occlusal plane; (**d**) Mirrored maxillary model; (**e**) Best-fit alignment of mirrored models.