The 3D FEMs were constructed based on those reported by Kasai et al. [14], Kayumi et al. [15] and consisted of a mandible, natural teeth with the periodontal ligament (PDL), and titanium implant(s) with superstructures in the left premolar and molar regions.

The surface of the mandible was generated using measurements of a commercially available model (QS7, SOMSO) of the dentate mandible with a 3D laser scanner (LPX-250, Roland DG). Appropriate thickness of the cortical bone was determined according to the anatomical findings [16, 17] and given with computer-aided design software (Rhinoceros, AppliCraft). The mass/volume and shape of the mandible were assumed to be 2 and B, respectively, according to the classification of Lekholm and Zarb [18]. The dimensions of the natural teeth and PDL were based on previous literature [19, 20]. The PDL had approximately the same surface area as the anatomical value, with uniform thickness of 0.25 mm at all sites [20]. The diameter and the length of the implants measured 3.75 and 10 mm, respectively. All materials were assumed to be linear and isotropic except for the PDL, which had biphasic properties as previously described [21,22,23,24]. The properties of other materials were based on previous studies [25,26,27,28,29] (Table 1).

The occlusal surfaces of the implants and the teeth were simplified and flattened in agreement with Monson’s sphere (10 cm diameter) and included two condyle points and the incisal point. Six models with varying numbers of teeth and implants were constructed. Model Im67 contained two implants placed in the left molar region (Fig. 1a). Model Im6 contained one implant placed in the left molar region (Fig. 1b). Model Im4567 contained four implants placed in the left premolar and molar regions (Fig. 1c). Model Im456 contained three implants placed in the left premolar and molar regions (Fig. 1d). Model MT67 contained no implant placed in the molar defect (Fig. 1e). Model MT7 represented natural dentition without the second molar as a control (Fig. 1f).