## 4 Discussion
Regarding stress distribution in the tooth, the present study shows similar mean stress values and stress distribution patterns for cementum- and dentine-reinforced teeth in mesial load, palatal inclination and rotation. In addition, the maturation of the root, with either dentine or cementum, was shown to be beneficial for all scenarios, where the immature tooth and its periodontal ligament showed higher mean stress values. The disadvantageous stress distribution of immature teeth compared to mature teeth has been observed in FEA studies [14; 32] and poorer fracture resistance [33] and prognosis [22] have been shown in clinical and animal studies as well [34], and therefore, it is necessary to limit the orthodontic force to prevent root resorption. The finite element model can simulate tooth-periodantal ligament complex under orthodontic loads. It allows the construction of standardised models to simulate the biomechanical performance of teeth and supporting tissues. Remodelling of the periodontal ligament for FEA analyses has been studies extensively, and the large inconsistency in results can be explained by the modelling assumptions (i.e., linear or non-linear, elastic or viscoelastic) given to the models [35]. The periodontal ligament is known to be a non-linear and time-dependent material, as proven by several experimental studies [34; 35; 36; 37]. However, finite element model studies simulating orthodontic loads often incorporate homogeneous, linear elastic, isotropic and continuous periodontal ligament properties [17]. Non-linear simulations of the periodontal ligament provide more accurate and reliable calculation for the stresses and strains over several tooth movements than the ones provided by linear simulations [37]. Similarly, viscoelastic time-dependent properties of the ligament are the reason of strain energy dissipation, without which excessive load would cause tooth fracture [38; 39]. In this study, we assumed non-linear periodontal ligament properties [17] to capture the elastic nature and deformation capacity of the periodontal ligament. The inclusion of the time-dependent viscoelastic properties was discarded in this study because of the comparative nature of the work. When comparing the same scenario under the three different models we expected the same qualitative results before the relaxation than after the relaxation. Thus, it must be considered that 'time' was not a variable included in our analysis, and in consequence this study shows the stress distribution in a specific moment and not the fatigue caused by the orthodontic treatment. The results of this research must be validated in clinical studies, since no computational study can represent the complex response of the teeth and surrounding tissues under orthodontic loads. To the best of our knowledge, the mechanical properties of the newly formed tissues after regenerative endodontics (periodontal ligament and cementum-like tissue) have not been studied so far. The histological evidence of human teeth treated with regenerative endodontics, limited to case reports and mainly focused on the analysis of the intracanal tissues [6,8,40], do not report differences regarding the shape, thickness nor distribution of the collagen fibres of the newly formed periodontal ligament (the ligament formed between the newly formed hard tissue and the alveolar bone) and the pre-existing one (the periodontal ligament between the immature root and the alveolar bone). Similarly, the tissue responsible for root maturation is histologically similar to the cellular cementing [6,8,40], tissue present in the apical third of mature teeth, or osteocementum. Considering this data, in the present study the shape and mechanical properties given to the periodontal ligament and cementum of the cementum-reinforced tooth were assumed equal to that of a physiologically developed tooth.