## 2 Materials and Methods  
### CAD Design  
A maxillary arch model considering teeth from the second molar to the second molar, the periodontal ligament (PDL) of each tooth, rectangular attachments from the second molar to the canine on both sides and dedicated orthodontic aligners, was obtained using SpaceClaim Computer-aided Design (CAD) software (SpaceClaim Corporation; Canonsburg, PA, USA) [Figure 1]. The teeth were created by one of the authors (GR), starting from a full arch STL file derived from a real patient CBCT. The PDL was designed using the SpaceClaim offset and Boolean intersection functions, on the basis of root shapes. The PDL average thickness was 0.25 mm, according to the existing literature [20]. Attachments were built, based on ClinCheck(r) software (Align Technology, Inc., San Jose, CA, USA) auxiliaries, with a vertical rectangular shape, 3 mm height, 2 mm width and 1 mm thickness [7]. The shape, size and position of composite attachments were determined by simulating rectangular attachments designed for distalization on a real case with ClinCheck(r) software. Aligners were obtained by applying the SpaceClaim software offset function on all tooth crowns and attachments and then manually refined to remove redundant surfaces and increase the accuracy of the aligners' contours. In this way, the shape of the obtained virtual model corresponded to that of a real aligner. After repeated measurements with a Micro-CT Scan (SkyScan 1172: Bruker-microCT; Kontich, Belgium) of brand new Invisalign(r) aligners (Align Technology, Inc., San Jose, CA, USA), the virtual aligner thickness was set at 0.5 mm. The adopted FE software was ANSYS, (ANSYS 18.2, Inc.; Canonsburg, PA, USA), in which the model was imported after the CAD process. After an initial phase of alignment and leveling of the teeth, the model was modified in order to reproduce a clinical condition in which the second molar was already distalized by 2.5 mm [9] and the first molar was already distalized by 1.2 mm, in a 50% sequential distalization protocol configuration. This is a sequentialization pattern where the upper second molar started to distalize; when it reached 50% of its movement, the upper first molar started to distalize. When the second molar completed distalization and the first molar completed 50% of distalization, the upper second premolar started to distalize; when the first molar completed distalization and the second premolar reached 50% of its movement, the upper first premolar started to distalize; and so on until the upper distalization of the arch was completed. From canine to canine, the arch will distalize en masse, according to the protocol. Therefore, the simulated movement was represented by 0.2 mm distalization of the upper second premolar and upper first molar, reproducing a real clinical setting [21]. According to the existing literature, every CAD element, with the exception of the PDL, was considered isotropic and homogeneous [22]. The teeth and their attachments were considered as a single body with a Young's Modulus of 19,600 MPa and Poisson's Ratio of 0.3 [15]. The aligner was considered as an isotropic and homogeneous material with a Young's Modulus of 528 MPa and Poisson's Ratio of 0.36 [Table 1]. The skeletal anchorage requested for the distalization movement was provided through the simulation of a buccal interradicular TAD located in the space between the first and second upper molar. Furthermore, in order to avoid proclination of the upper incisors, the effect of a Class I elastic was analyzed in two possible configurations: (1) stretched from the TAD to the upper canine and (2) from the TAD to the first upper premolar. The following parameters were considered for the mesh [6]:  
* Mesh size: 0.09 mm;  
Figure 1: CAD Design.