## 2 Materials and Methods
### Sample Selection  
Patients attending the private orthodontic clinic of one of the authors were enrolled in the study if they fulfilled specific eligibility criteria. Inclusion criteria were: age over 20 years, presence of dentoalveolar compression of the upper arch with unilateral, bilateral, or single-tooth crossbite; absence of missing teeth (excluding wisdom teeth) in the upper arch; complete permanent dentition, with the exception of third molars; no scheduled dental extraction; need for orthodontic expansion and orthodontic treatment on both arches; and willingness to be treated using clear aligners. Clinical and radiographic records were evaluated to determine the need for orthodontic expansion. Exclusion criteria were: being in the growth phase; reported previous orthodontic treatment; presence of craniofacial syndrome, systemic disease, periodontal disease, or active joint disorder before or during treatment; and need for treatment requiring therapeutic dental extraction or orthognathic surgery. Patients who fulfilled the eligibility criteria were then scheduled for treatment using differential anchorage as the biomechanics of choice. Patient were divided into three groups according to their crossbite: bilateral crossbite, with at least two teeth (upper premolars and/or upper molars) in crossbite in both the right and left hemiarches (Group A); unilateral crossbite, with at least two teeth (upper premolars and/or upper molars) in crossbite in one hemiarch (Group B); or single-tooth crossbite of the upper first or second premolar or upper first molar (Group C). All patients were treated using the Invisalign(r) clear aligner system (Align Technology, San Jose, CA, USA), fabricated with Ex30(tm) and SmartTrack(tm) material. Treatment planning was completed using the ClinCheck(r) virtual model, which is a virtual treatment plan design tool developed by Align Technology, San Jose, CA, USA). Anchorage was determined by the type of posterior crossbite. In the unilateral group, anchorage was provided by the opposite dental arch. The bilateral group used the upper second molars and anterior teeth as anchorage. In the single-tooth group, adjacent teeth served as anchorage. To minimize anchorage loss, sequential movements were planned. Patients were instructed to use the aligners 22 h/ day for 10 days. Aligners change was scheduled every 10 days. Control appointments were fixed at 6-week intervals. Compliance was also verbally confirmed at each appointment. Stereolithography (stl) files of digital models of all patients were obtained to perform the measurements using Orthoanalyzer(tm 1.7 analytical software (3 Shape, Copenhagen, Denmark). Three digital models were developed for each patient to show: the initial malocclusion, pretreatment (T1), the outcome predicted by the planning software (T2), and the post-treatment outcome before initial (T3). Pretreatment stl files were obtained from the initial intraoral scans of the patients (T1). Final-stage stl files were exported from the ClinCheck software and labeled as \"predicted outcome\" (T2). Post-treatment files were also obtained from the intraoral scans taken after the first set of aligners and labeled as \"achieved outcome\" (T3) since they represented the actual outcome. All stl files were deidentified. The following measurements were recorded in mm at each time point: intercanine width, first and second interpremolar width (mm), and intermolar width (mm). The effectiveness of expansion at the canine, premolar, and molar level was evaluated as the difference in widths between T3 and T1, calculating the percentage of the initial width achieved by treatment (T3 \(-\) T1%). The predictability of expansion at the canine, premolar, and molar level was assessed as the difference in measurements between T2 and T3, calculating the percentage of the final width not achieved by the treatment (T2 \(-\) T3%) and the percentage of relative predictability, i.e., the observed expansion relative to the predicted expansion (T3 \(-\) T1 \(\times\) 100/ \(\mathrm{T2}-\mathrm{T1}\)).