## 2 Materials and Methods
### Therapeutic and Measurement Protocol  
For each patient, pre-treatment (T1) and post-treatment (T2) digital dental casts were taken using an intraoral scanner iTero(r) Orthodontic ver. 5.2.1.290 (Align Technology Inc., Santa Clara, CA, USA). Final position of the corresponding ClinCheck representation was also collected to establish the predictability of the final virtual model (T2 ClinCheck) with respect to the movements observed in the post-treatment models. On the initial digital casts, the upper permanent right and left first molars' rotations were assessed using the angular measurement of Henry [10] (Figure 1). All the upper permanent first molars with a Henry's angle more than 11 degrees were analyzed in the study, making a sample of 53 teeth. Patients received a non-extraction treatment protocol consisting of Invisalign System clear aligners. The ClinCheck for each patient was planned with the same standardized disto-rotation protocol, according to [PERSON]'s line [3]. Using ClinCheck software, a mesial-out buccal rotation of upper first molars was programmed with an amount of 2 degrees for each aligner. During the rotational correction, an arch expansion was always required. No distalization movements were included in the treatment protocol. All patients were instructed to wear their aligners full-time, excluding meals and tooth brushing. The patient changed the aligners every 7 days and every 4 stages the clinician checked for good aligner fitting and the position of the attachments. The mean number of aligners was 32 for the maxillary arch. If new scans were necessary to improve the fit of the aligners due to dental exfoliation and eruption, the prescription form was settled to continue the treatment until reaching the same final position as the first approved ClinCheck. Optimized expansion support attachments and optimized retention attachments were automatically placed by the software on the posterior teeth. The average time between the initial and final scans was 8.6 months. At the delivery appointment, the patients gained awareness that they were part of a research study, and honest compliance reporting was difficult. Patient's compliance was noted in the clinical diary. A single investigator conducted a face-to-face interview with each patient and their parents to assess his/her cooperation. Compliance was appraised on a 3-point Likert-type scale (poor, moderate, good) [11]. Poor compliance was reported when the patient wore the aligners less than 16 h/day; moderate from 16-20 h/day; good when the patient wore the aligner full time as suggested by the clinicians. All the maxillary digital dental casts were exported in.stl format and for each of the 53 upper permanent first molars included in the study the following measurements were carried out at T1, T2, and T2 ClinCheck using Viewbox 4 (dHAL software, Kifissia, Greece):  
* Henry's angle (HA): the angle formed between the mid-palatal raphe (identified by an anterior and a posterior point along the mid-palatal raphe) and the line passing through the mesiobuccal (MB) and distobuccal (DB) cusps of the maxillary first molar [8] (Figure 1);
* Mesiobuccal Expansion (MBE): the perpendicular distance between the mesiobuccal cusp of the first right or/and left permanent molar and midpalatal raphe (Figure 2a);
* Distobuccal Expansion (DBE): the perpendicular distance between the distobuccal cusp of the first right or/and left permanent molar and midpalatal raphe (Figure 2a);
* Mesiobuccal Sagittal (MBS): the distance between the midpoint of the first palatal rugae and the projection of the mesiobuccal cusp of the first right or/and left permanent molar on the midpalatal raphe (Figure 2b);
* Distobuccal Sagittal (DBS): the distance between the midpoint of the first palatal rugae and the projection of the distobuccal cusp of the first right or/and left permanent molar on the midpalatal raphe (Figure 2b).