## 2 Materials and Methods  
The finite element model employed in this study included the maxilla and the mandible with associated teeth and periodontal ligaments and a mandibular advancement device (Figure 1). The model was developed according to the dental cone beam computed tomography (CBCT) (Voxel size 75 micron, FOV 11 \(\times\) 13 cm. [PERSON], Cefla, Italy) images of a 29-year-old female patient. The patient is in good health and does not suffer from OSA or mandibular disorders. Before proceeding with the CT scan, the patient was informed of the purpose of the study and gave her consent to the procedure. Her data were used anonymously in the simulations. The volumetric image of the anatomy was reconstructed in Mimics (v.12.11, Materialise NV, Leuven, Belgium) using an appropriate threshold based on the Hounsfield unit. The periodontal ligaments (PDLs) were reconstructed through the 3D modelling software Rhinoceros (v.5.0 by McNeel & Associates, Seattle, WA) since soft tissues are not identifiable in CT scans. Although the average thickness of periodontal ligament is 0.15 to 0.38 mm and it has an hourglass shape (it is thinnest at the middle third of the root [39]), PDLs were modelled by offsetting each tooth root surface of 0.3 mm to fill the space between each tooth and the alveolar socket [38; 40; 41]. The devices chosen are among the most used in OSA therapy. The authors have selected to consider two different types of mechanics: Lateral and anterior propulsion. In Herbst'sTM case, two lateral telescopic arms protrude the mandible. SomnodentTM also has a lateral mechanism, but the propulsion is no longer given by arms but by two screws that push the wings of the lower splint forward as they move forward. OrthoapneaTM is an entirely different device that has an anterior inverse connecting rod that is activated through the central screw that produces the protrusion of the mandible. In the simulations, SomnodentTM was reproduced without vertical elastics that are used to improve the outcome of the therapy. To simulate the effect of different MAD models, three MADs were created through a RE approach starting from the physical prototype due to the personalised nature. An optical laser scanner (Konica Minolta Range 7) was used for digitising a model of MAD to be used as a reference for the design of the two other devices. All the triangular mesh models were further processed and reconstructed to obtain the non-uniform rational basis spline (NURBS) models ready to be imported into Ansys v.19 R1 (Canonsburg, PA, USA) for the final arrangement of the model in terms of material properties, meshing, boundary conditions, and loading conditions. The material properties of teeth and MAD as presented in Table 1 were assumed to be linear elastic, homogeneous, and isotropic. At the same time, a hyperelastic law was used to simulate large non-linear strains associated with the non-linear nature of PDLs [42]. A 2 nd-order Ogden model was used to define the strain energy function [43; 44; 45]. The parameter values listed in Table 2 were obtained through the fitting of uniaxial experimental data by [PERSON] et al. [46]. Perfect bonding was assumed between the bodies since MADs are anchored on teeth to force the mandible to a protruded position, and PDLs strongly bind the tooth root to the supporting alveolar bone. Thus, a continuous mesh (shared topology) has been created between MAD and teeth and teeth and PDLs with common nodes at the boundary between the parts. The contact between the bone and the PDL was set as bonded meaning that no sliding or separation is allowed. The two splints of the MAD were coupled in a configuration that will enable mutual sliding without separation. The bony components were assumed as rigid bodies since they have more rigid behaviour than PDLs. Moreover, they were constrained in all directions to prevent rigid body motion. The authors decided to not differentiate bone in its cortical and cancellous components since the main objective of the study is to compare the results in the three devices.