## 4 Discussion
\[\alpha=2\cdot arctg\bigg{(}\frac{AnsRight-AnsLeft/2}{Ans-N}\bigg{)}\]  
Imagining a further right-angled triangle whose base is again represented by the unilateral transverse expansion obtained (AnsRight-AnsLeft/2) and the height by the length of the bispinal plane (Ans-Pns), it is possible to calculate the expansion angle \(\delta\) with the following formula:  
\[\delta=2\cdot arctg\bigg{(}\frac{AnsRight-AnsLeft/2}{Ans-Pns}\bigg{)}\]  
These theoretical-experimental formulas can be transposed clinically by taking only an endoral periapical radiograph in the incision area with the expander screw turns just completed (before the ossification process is achieved), which allows calculation of the transverse expansion obtained by measuring the distance between the right and left anterior nasal spines (AnsRight-AnsLeft), as well as the opening of the suture in the anterior area (Figure 7a). Instead, the N-Ans and Ans-Pns distances can be obtained from latero-lateral projection teleradiography of the skull performed at time T0; since this examination is routinely required prior to undertaking any orthodontic orthopedic treatment and since these distances are constant, there is no need to request additional post-treatment radiographic examinations from the patient with the only purpose of assessing that the molars have decompensated (Figure 7b,c). In addition, using an intraoral scan and any measurement software, it is possible to draw the MVC and MPC points of the maxillary right and left first molar (taken as reference points considering that during treatment the anatomy of the cusp is not modified), join the ipsilateral points by forming two straight lines, and then very easily measure the \(\beta\) angle subtended by them. By drawing the MPC and DBC points and the straight lines connecting them on either side, it is also possible to measure the \(\gamma\) angle. Such measurements can be made on files both before and after treatment; it is also possible to superimpose both STL (Standard Triangle Language) files to obtain a better perception of the displacements that have occurred (Figure 8). These considerations are a significative advantage for patient care given an important clinical implication, as they make it possible for all clinicians to be able to assess whether the treatment performed has been favorable by additionally requiring only a periapical intraoral radiograph of the incisors without the need for post-expansion CBCT scans. Using simple radiographic measurements with the hemimaxillary reference system allows clinicians to assess treatment outcomes while respecting the ALARA (As Low As Reasonably Achievable) radiation protection and justification principles, thereby reducing radiation exposure. This method is particularly beneficial for pediatric patients, where minimizing radiographic exposure is a critical concern. Improved predictability of molar position changes enables orthodontists to better manage treatment plans, ensuring optimal alignment and reducing the risk of undesired outcomes. As such, the method can streamline clinical practice while enhancing patient safety and outcomes. Figure 8: Intraoral scan of the upper arch of a clinical case with expander on miniscrews in anterior (superiorly) and occlusal (inferiorly) views showing upper permanent molar displacements at T0 (**sa**—displayed in green) and at T1 (**sb**—displayed in red); overlapping scan at both times (**sc**). Figure 7: Radiographic examinations on which measurements necessary to calculate orthopedic expansion angles can be conducted. Periapical radiograph at upper incisors level (**7a**), cone beam computed tomography (**7b**), latero-lateral projection teleradiography (**7c**). In this study, all devices that allow rapid palatal expansion with bands on deciduous second molars or anterior paramedian miniscrews were considered. A greater expansion angle was expected in the former than in the latter where the opening of the medial palatine suture generally occurs more parallel.