Studies have shown that AP leads to resorption of the periapical bone, resulting in a reduction of the biomechanical resistance of the affected tooth and making it more susceptible to vertical root fractures [26,27,28]. However, the effects of different degrees of periapical bone defects on the biomechanical state of teeth have not been elucidated. In our current FEA study, periapical spherical defects of different diameters were modeled to simulate different periapical bone defects in teeth with AP and to analyze their effects on the biomechanical state of affected teeth. Our results showed that (1) the presence of periapical bone defects led to increased stress concentration and tooth displacement, and (2) this effect became more pronounced as the size of the bone defect increased. Therefore, the null hypotheses were rejected.

FEA model construction is the basis of FEA, and good models simulate real situations better and provide more accurate data [29]. In numerous previous FEA studies, mandibular second premolars were selected for analysis because of their relatively simple anatomical structure [30,31,32]. Therefore, we chose this tooth as the reconstructed object for our AP models. AP can be categorized into different types depending on the size of the bone defects [10,11,12]. In general, AP with periapical bone defects > 10 mm in diameter are clinically referred to as periapical cysts, and those < 10 mm are referred to as periapical granulomas [33]. For modeling rigor, we applied two modeling approaches in our pre-experiments (Approach 1: Set periapical bone defects as a separate contact body. Approach 2: Not set periapical bone defects as a separate contact body). We analyzed the von Mises stress distribution and tooth displacement distribution of two modeling approaches. The results showed that the two modeling approaches produce almost identical mechanical effects (Supplementary Figs. 1, 2).

The validity of the model and the accuracy of the analysis are closely related. This study conducted a tentative analysis after the initial construction of the model and confirmed that the stress distribution and tooth displacement distribution cloud map of the normal model and the periapical bone defect model showed similar trends to those in the previous literature [7, 17, 18].They constructed a normal tooth model and a periapical bone defect tooth model (anterior and premolar) and performed the corresponding finite element analyses. Von Mises stress analysis showed that the periapical bone defect portion of the periapical bone defect tooth model showed a significant decrease in stress. Tooth displacement analysis demonstrate a large increase in Tooth displacement at the periapical bone defect region of tooth. This characteristic biomechanical change is consistent with our modeling analysis.

Notably, our analysis reveals that stresses were concentrated in the coronal cervix of the tooth and would increase with the generation of periapical bone defects. Experimental and clinical case studies have also demonstrated that the coronal cervix of the tooth is prone to fracture and that the incidence of tooth fracture is higher in teeth with periapical inflammation [17, 34]. Therefore, this characteristic clinical feature is consistent with our finite element results. At the same time, in order to improve result accuracy, a sensitivity analysis was conducted to refine the mesh until stress values were convergent. In summary, the FEA model established in this study is reasonable, qualified and validated for subsequent studies. Meanwhile, the digital image correlation (DIC) and model construction analysis in vitro also are useful tool for validating FEA models [35, 36].

In the overall analysis, the maximum von Mises stresses in the periapical bone defect models were lower compared to that in the normal model (Fig. 3A, B), which seems to be inconsistent with the consensus that teeth with AP are prone to root fracture [17]. Considering that only the maximum von Mises stress was analyzed in the overall analysis of the models, this may have neglected the change in stresses in the internal parts of the tooth. Therefore, we further investigated the von Mises stress changes in six parts of the tooth in the coronal (occlusal surface, middle, and cervix) and root (cervix, middle, and apical) sections. First, we found that the stresses in the tooth were concentrated in the coronal cervix (Fig. 4A, B), where the von Mises stresses were the highest, which is consistent with previous study findings [17]. This phenomenon may be attributable to the oblique nature of the masticatory force [37]. Maintaining intact coronal and radicular tooth structure as well as cervical tissue to generate a ferrule effect is thought to be critical for optimizing the biomechanical behavior of a restored tooth [38, 39]. Therefore, we should aim to preserve as much dental tissue as possible to strengthen the resistance of the coronal cervix during the later restoration of the affected tooth.

Second, our results clearly showed that as the size of the bone defects increased, the maximum von Mises stress in the coronal cervix of the tooth increased considerably, while the maximum von Mises stress in the middle and apical part of the tooth root decreased considerably (Fig. 4A, B). These results indicate that although periapical bone defects have no noticeable effect on the overall maximum von Mises stress of the tooth, they lead to stress concentration in the coronal region of the tooth, while the stress at the root decreases. This interesting change we may be able to explain in terms of the mechanical integrity of the tooth-alveolar bone [40]. During mastication, the tooth needs to rely on the surrounding bone to disperse the chewing force. However, due to the thinning or defect of bone around the root, the tooth is unable to disperse the pressure well, which results in stress concentration in the coronal region. Meanwhile, the stress reduction at the root may be due to the loss of contact between the root and the bone tissue due to the defect of preapical bone, which naturally leads to a decrease in root stress. What’s more, this change in stress can lead to a polarization of stresses within the tooth and finally have an extremely destructive effect on the residual dental hard tissue [41, 42].

In the overall analysis of the models in this study, the maximum tooth displacement was much higher in the periapical bone defect models than in the normal group and increased with increasing periapical bone defect size (Fig. 3C, D). This could be the result of the tooth loss of its restriction by the periapical alveolar bone. And increased tooth displacement due to alveolar bone destruction is considered a risk factor for the preservation and restoration of the affected tooth [43] The maximum tooth displacement of each part of the tooth showed the same results in the subsequent internal analysis of the tooth (Fig. 5A, B). Under normal circumstances, the displacement of natural teeth during functional loading ranges from 0.02 to 0.2 mm [44]. At a periapical bone defect diameter of 1 mm, the maximum tooth displacement is already 0.216 mm, and the tooth displacement continues to rise as the bone defect increases in size, and such unreasonable tooth displacement are unacceptable because they can lead to periodontal tissue dysfunction and destruction, and even tooth loosening or loss. Notably, the increase in maximum tooth displacement was most significant in the apical portion of the tooth compared to that in other parts. Therefore, this phenomenon should be considered in the preservation and restoration of teeth with periapical bone defects, especially when establishing an occlusal relationship with the contralateral tooth. As the displacement between an affected tooth with periapical bone defects and a normal tooth differs under the same masticatory load, this may prevent a uniform distribution of the load. Therefore, considering the physiological mobility of the adjacent tooth, it is recommended that when periapical bone defects are present in a tooth, occlusion of the affected tooth should be appropriately lowered [45, 46].

Meanwhile, for the treatment of AP with periapical bone defects, it is suggested that the periapical bone defects are filled with bone to increase the support of the affected tooth, considering its increased displacement. However, it is unclear whether the biomechanical state of the model will change after the filling of the apical region with the bone defect filling material. Further analysis is required to clarify this aspect.

In this study, we use static linear analysis because of its simple computational procedure, fast solution speed and easy understanding and verification of the results. However, masticatory cycle in the oral cavity is a dynamic process, so in the subsequent study, we need to use dynamic analysis to more realistically reproduce the clinical conditions and obtain more accurate data. In addition, only FEA was performed in this experiment and no in vitro experiments were conducted in conjunction with the desired corresponding realistic model. In the next experimental design, we should consider constructing 3D printed models to verify the validity of the FEA model.

This study focused on the biomechanical response of teeth with different sizes of periapical bone defects; However, aspects such as subsequent corresponding root canal therapy treatment, full crown restoration, and the biomechanical situations in the corresponding states were not considered.