## 2 Materials and Methods  
This study followed a randomized prospective longitudinal design and enrolled subjects who were seeking orthodontic treatment and who had never been treated before, presenting at the Department of Orthodontics (University of Milan, Milan, Italy). As a routine procedure, a signed informed consent for releasing diagnostic records for scientific purposes was obtained from the parents of the patients prior to the beginning of treatment. The protocol was reviewed and approved by the Ethical Committee of University of Milan (updated protocol code #51/21) and all the procedures adhered to the World Medical Organization Declaration of Helsinki. The following inclusion and exclusion criteria were applied to select the population study: (1) no systemic disease or syndromic features, (2) early mixed dentition with incisors and first permanent molars erupted and non-mobile maxillary primary second molars, (3), no pathologic periodontal status or tooth decay on both deciduous and permanent teeth, (4) skeletal Class I and II, (5) maxillary transverse deficiency with or without unilateral/bilateral posterior cross-bite. Sample size was calculated, considering as the primary outcome the changes in the transverse diameter at the pulp chambers, on the data of a preliminary selected pilot sample of 5 patients per group [16]. To retrieve \(\beta=0.80\) with \(\alpha\) set at 0.05, a sample of at least 13 subjects per group was necessary. The initial sample involved 44 consecutive patients (28 females and 16 males) enrolled in 2019. An electronic computer program was used for the allocation sequence generation. Each new patient presenting, and adherent to the inclusion and exclusion criteria, was allocated to the tooth-borne Hyrax expander group or the tissue-borne Haas expander group (using www.random.org; accessed on 1 March 2020). Then, the allocation information (randomization results) was concealed in opaque and sealed envelopes by the statistician. The treating clinicians were blinded from the randomization procedure but not during the treatment period (registration number \"RCT_4_19_2.3\"). Figure 1 shows the consort diagram showing the flow of patients through the trial. Patients who did not correctly follow the protocol of activation, who did not come back for control appointments, or whose skeletal and dental structures were difficult to visualize on the CBCT due to the low quality of scans, were dropped from the study sample. The final sample involved 34 patients (20 females and 14 males): Hyrax group (16 patients) included 6 females and 10 males (mean age 8 \(\pm\) 3 years; range 7.3-9.0 years), Haas group (18 patients) included 14 females and 4 males (mean age 8 \(\pm\) 2 years; range 7.1-9.0 years). Palatal expanders were composed of two bands bonded on second deciduous molars and with palatal extensions bonded onto deciduous canines. The expander's screw was initially activated and turned twice (0.45 mm initial activation) during the first appointment. Afterwards, it was turned once per each of the following day (0.225 mm activation/day). Maximilary expansion was performed until dental overcorrection, defined as when the lingual cusps of the upper first molars occluded onto the buccal cusps of the lower first molars, was achieved. The screw was then locked, and the expander was kept on the teeth as a passive retainer. During this period, none of the patients underwent any further orthodontic treatment. The mean activation and retention times were 3 \(\pm\) 1 months (range 1-5 months) and 10 \(\pm\) 2 months (range 6-12 months), respectively. CBCTs and dental casts were taken before (T0) and after treatment (T1). Ultra low dose 3D imaging protocol and adjusted field of view (FOV) were used to respect ALARA principle (Planmeca ProMax 3D Mid; Planmeca, Helsinki, Finland). The entire treatment lasted a mean time of 13 \(\pm\) 3 months (range of 9-13 months). CBCT scans were traced, and data was subsequently recollected by the same expert orthodontist (C.V.).