Significance of mandibular molar replacement with a dental implant: a theoretical study with nonlinear finite element analysis

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 [21‐24]. The properties of other materials were based on previous studies [25‐29] (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).

Nonlinear characteristics according to the load displacement curve of teeth [21‐24] and cartilage [30] were given to the springs for the opposing teeth and TMJs, respectively (Fig. 2). The nonlinear elasticity of the springs on the teeth and implants simulated displaceability of opposing natural teeth at compression and separation of the occlusal surface from opposing teeth at tension.

The PDLs of natural teeth and the springs corresponding to opposing teeth demonstrated a two-stage displaceability corresponding to the measurement of the load displacement curve of real teeth [21, 22] (Fig. 3).

Kumagai et al. [31] investigated occlusal force distribution in human using the Dental Prescale System. They described that the distribution of the occlusal force was greatest at molar region followed by the premolar and anterior region. Our theoretical model was based on Kasai [14] and Kayumi’s [15] reports. The theoretical model with natural teeth and no defect in Kayumi’s report and the occlusal force distribution in this model under load 400 N described later are shown in Figs. 4 and 5, respectively. This distribution was similar to the measurement in human body by Kumagai et al. [31].

The boundary conditions are shown in Fig. 1a–f. To simplify the FEM, TMJs and maxillary teeth were replaced with appropriate springs (Fig. 6). The springs for the maxillary teeth, except for the anterior teeth, were directed perpendicular to the occlusal plane, which was defined using Monson’s sphere. The apical end of each spring was restrained in all directions. The other end of the spring was attached to the node corresponding to the occlusal central pit on a mandibular tooth, which allowed displacement perpendicular to the occlusal plane. The springs for TMJs linked an external restricted node to the top of the mandibular condyle [14, 29].

The loading conditions simulated intercuspal clenching. With the assumption that occlusal force was generated by the contractile force of four bilateral masticatory muscles, masseter, temporalis, and medial and lateral pterygoids, loading points and directions of the loads were generated based on previous literature [19, 25] (Fig. 6). The summation of the reaction force at the occlusal surfaces of the teeth and superstructures was used as a standard for the load amount. For example, the load condition that resulted in a total reaction force of 400 N in the preliminary FEM with natural dentition was designated as “load 400 N.”

Analysis was performed according to the report by Kayumi et al. [15]. In linear finite element analysis (FEA), all teeth maintain perfect contact with antagonists with no stress on occlusal surfaces before loading. However, there must be some occlusal force on the occlusal surface when a mandible is in the intercuspal position. Since the displaceabilities of osseointegrated implants, natural teeth with PDLs, and TMJs are quite different from one another, the results of linear FEA under some occlusal loads should not correspond to real biomechanical conditions.

To verify the similarity of the FE models to the real stomatognathic system, “initializing” of models, i.e., modification of the condition of occlusal contact assuming the intercuspal position, was performed before analysis. This procedure was performed by modifying occlusal contacts on the implants so that the distribution of occlusal force was symmetrical under load 400 N [15]. The occlusal contacts were modified by altering the load displacement curves of the springs on the implants. The load displacement curve was shifted so that the spring provided little resistance to compressive forces until the gap closed, i.e., occlusal surface came in contact with antagonists (Fig. 7). The size of the gap was determined by trial and error, such that the occlusal force, namely, the reaction force of the springs on the occlusal surface, was distributed with approximate symmetry [14, 15] in models Im6, Im67, Im456, and Im4567. This size was determined so that the amount of reaction force on the most posterior teeth on both sides became as equal as possible. After initializing was completed, symmetric distribution of the reaction force on the teeth and the superstructures was confirmed in each model. Thereafter, the FEA was performed under the load conditions of loads 100, 200, and 800 N using the software package MSC.Marc2010 (MSC Software). The distribution of the reaction forces on the occlusal surface and the mandibular condyle, which were regarded as the occlusal force and the load on the TMJ, respectively, was investigated.

FEA is useful for mechanical simulations of a living body and has been used in implant dentistry research under careful consideration of the analysis conditions [32, 33]. Although some reports have demonstrated that bone density varies according to bone type and location, the material properties of the mandible were homogenous and isotropic in this study. However, the effect of this difference was considered to be negligible under the confirmation of the displacement of teeth and implants because of its far larger elastic modulus and far smaller strain than those of soft tissues, such as TMJs and PDL. Since the purpose of the present study was to examine the distribution of occlusal forces on the occlusal surface, occlusal forces should be mainly affected by the displaceability of TMJs and teeth, not by that of osseointegrated implants.

The FEMs in this study were based on those reported by Kasai et al. [14] and Kayumi et al. [15]. The displaceability of TMJs was regarded as that of the cartilage [30] because it has a far smaller elastic modulus than that of the TMJ disc [34, 35]. Therefore, the elastic modulus of the springs corresponding to TMJs was determined based on the thicknesses of the TMJ disc [36] and articular cartilage [35], the stress-strain curve of the intervertebral discs [30], and the displacement of the condyle [37, 38] in intercuspal clenching by indirect measurement. Although the material properties of the human body vary on an individual basis, the models used in this study were therefore considered to be appropriate to investigate the distribution of occlusal forces on the teeth, implants, and TMJs.

Based on previous literature [31, 39], occlusal loading of 200 N was considered to correspond with hard clenching. However, a previous study indicated that the maximum biting force (400 N) was better for occlusal adjustment with intercuspal clenching. Therefore, this study was performed with the assumption that the maximum functional force was 400 N. Occlusal loading of 100 N was considered to correspond with light clenching while loading of 200 N was considered to correspond with middle clenching. Calculations were also performed under a load of 800 N, which was assumed to be the maximum nonfunctional occlusal force, such as that exerted in nocturnal bruxism. Because of the difficulty of controlling nocturnal bruxism, this value was considered to be sufficient to include as the condition under maximum force [40]. However, Hattori et al. [41] described that neuromuscular regulatory systems control maximum clenching strength under various occlusal conditions. Therefore, the large force used in this study may not occur clinically in the SDA except in the case of nocturnal bruxism.

Occlusal adjustment is usually performed to obtain symmetrical occlusal force distribution in natural dentition. However, occlusal force distribution among natural teeth and implants depends on occlusal force because of the difference of displaceability between a natural tooth and an implant [14, 15]. Therefore, we evaluated the result of the analysis from viewpoints of symmetry of occlusal force distribution and existence of harmful load on implants, residual teeth, and TMJs.

In this study, occlusal force distribution under load 400 N was adjusted to maintain symmetry. In models Im67 and Im4567, occlusal load on premolars did not show large asymmetry even in lower and higher loading conditions. Thus, restoration with implants of the same number of missing teeth was recommended for unilateral defect(s) of the mandibular premolar and molar regions. In models Im6 and Im456, occlusal force distribution was altered more than in models Im67 and Im4567. However, occlusal force distribution in models Im6 and Im456 was similar to that in model MT7. Since most patients missing only second molars do not always receive replacements, as described above [12], restoring mandibular distally extended edentulism with a second molar defect might be acceptable, similar to receiving no prosthetic treatment for a second molar defect in natural dentition. However, a much larger occlusal force on the superstructure of the first molar was observed compared with the first molar on the natural dentition side of model Im6, the first molar on the natural dentition side of model Im456, and the first molar on the defect side of model MT7 under load 800 N. These finding might indicate a risk of damage to implants and surrounding bone tissue.

In this study, force distribution on the left TMJ (defect side) was similar to that on the right TMJ (natural dentition side) under not only load 100 N but also other loading conditions. Hattori et al. [41] did not find evidence that the SDA causes overloading of the joints or the teeth, which suggests that neuromuscular regulatory systems control maximum clenching strength under various occlusal conditions. Reissmann et al. [42] also demonstrated no significant difference between SDA patients and removable dental prosthesis patients. Therefore, apprehension concerning overloading of TMJs in an SDA with implants is considered to be negligible.

It should be noted that our results were obtained under conditions of vertical loading by bilaterally balanced muscle activity with tight intercuspation in the correct mandibular position because the horizontal displacement of the premolars and molars was restrained. The actual distribution of occlusal forces may differ due to individual differences in the material properties of the soft tissue. Additionally, the lateral load, which may occur in lateral movement of the mandible during mastication, was not considered. Furthermore, this study investigated only the biomechanical aspect of the SDA using implants. The function and QoL of real patients with SDA should be investigated in future studies. Moreover, it is unclear whether the occlusal force of each tooth and implant observed in this study is harmful in human. Further investigation should include analysis of bone strain.