The dimensions of Micro-arch brackets (Tomy, Tokyo, Japan) were simulated as attachments to the teeth. The interface between the teeth and the bracket was completely joined, to omit the intervention of the composite bonding material. The distance from the incisal edge of the maxillary central incisor to the bracket slot was 4.5 mm (perpendicular to the occlusal plane), 11 mm to the labial CEJ, and 11.8 mm to the labial alveolar crest. The main archwire was modeled according to the dimensions of a 0.017 × 0.025-inch (in) stainless steel archwire, and it was assumed that there was no play between the brackets and the archwire. At the interfacial nodes between the archwire and the brackets, translational degrees of freedom along the axial direction of the archwire were not constrained, to simulate free sliding of the archwire.15 (link) The retraction hook was modeled using a rigid (0.036-in) stainless steel wire, in order to reduce the deflection when retraction force was applied. The midpoints of incisal edges, buccal cusp tips, and root apices were used as landmarks for the assessment of displacement, and the occlusal plane was defined by connecting the midpoint of the central incisal edge and the mesiobuccal cusp of the first molar (
The teeth, alveolar bone, brackets, periodontal ligament, and archwire were all constructed using fine tetrahedron solid elements, and were all assumed to be isoparametric and homogeneous linear elastic bodies. Altogether, the model was constructed with 53,665 nodes and 272,118 elements. Due to the large number of elements, teeth and bone were approximated as uniform structures, without differentiation into enamel/dentin or cortical/trabecular bone respectively.16 (link) Each tooth contacted the adjacent tooth at the contact point as an individual element. At the interface between the archwire and the brackets, transitional degrees of freedom were not constrained and the friction in the interface was ignored.15 (link) Other flexural directions of the archwire were coupled at the connected nodes at the junction, to eliminate possible bracket-wire play. The Young's modulus and Poisson's ratio of the elements were obtained from previous studies (