To analyze the biomechanics of complex systems, such as a tooth, the application of complex research methods is required . Finite element analysis has previously been used to successfully simulate clinically observed orthodontic tooth movement , as well as endodontic treatment protocols under static, dynamic, and multiple loading scenarios . It is ideal for solving complex design problems. In this study, a finite element biomedical model was established for analyzing the mechanical stress distribution and transient displacement of the mandibular first molar upon application of a load on the occlusal surface using different abrasion models.
In this study, on the basis of the equivalent stress and instantaneous displacement in the control group, we found that stress from occlusal loading was concentrated in the central fossa, and that the maximal displacement contour was in the mesial crown near the cheek, extending down to the tooth neck. The stress on the first molar was concentrated at the crown and at the furcation of the mesial and distal roots. However, with the simulation of increased abrasion, the displacement contour gradually became V-shaped. This might account for a significant increase in the prevalence of wedge-shaped defects in patients with severe tooth wear .
In the wear models with tilted abrasion, the stress was concentrated in the mesial root, especially when the tilt angle was increased from 5° to 15°. This concentration resulted from the vertical loading force, which would have been distributed differently with inclined wear of the tooth. This force can be decomposed into vertical and horizontal components of force, and with an increase in tilt angle, the magnitude of the horizontal component of force gradually increases, which in turn increases the load on the mesial root and makes it a stress concentration area. Thus, the occurrence of VRFs becomes more likely. This was also confirmed by the instantaneous displacement diagram, where the maximal contour was situated obliquely to the mesial and gingival side from the cut surface and formed an acute angle with the mesial root. With an increase of inclined wear, the angle between the displacement contour line and the mesial root decreases. In the 15° inclined model, the displacement contours coincided with the long axis of the proximal root, showing that the mesial root was prone to vertical fractures. Thus, we propose that we could reduce the risk of vertical fractures caused by inclination via occlusal adjustment.
However, when the tilt angle was increased to 20°, the stress on the mesial root was reduced, and it was concentrated on the crown; additionally, the displacement contour became circular, and the maximum displacement appeared in the crown. Therefore, when the crown is worn out severely and the cusps become tilted, they become a stress concentration area that is prone to fracture. Appropriate jaw tuning could reduce the risk of such a cusp fracture.
According to previous work, there are several causes for vertical mesial root fracture of the first molar . First, loading on a mandibular first molar that has some anatomical defects, like long vertical grooves on the distal side of the mesial root, whose oval canals bend to the distal side, would cause uneven stress on the tooth. A previous study  found that the diameter of a root that has vertically fractured was typically smaller than that of a normal root. Secondly, when the upper and lower first molars occlude, cusp inclination could disperse the loading force without causing severe damage to the tooth. However, when severe wear causes cusp disappearance, more force will be transferred to the root. Thirdly, when wear on the mandibular first molar crown is more severe on the distal side, a condition that might be related to the frequency, time, and intensity of loading, the likelihood of eruption of the distal side is higher than that on the mesial side due to compensation. Therefore, the crown is prone to tilt to the mesial side. Based on our models, when the distal tipping angle of the abrasion plane of the mandibular first molar increases, the angle between the instantaneous displacement contours and the tooth long axis becomes smaller. The mesial root suffers from the highest stress concentration, thereby increasing the possibility of longitudinal root fracture.
Our study does have some limitations to understanding the etiopathology of VRF. Various diets cause tooth abrasion in diverse ways , and we have modeled just two types of abrasion. In addition, the etiology of tooth wear is commonly multifactorial, and commonly a combination of erosion, attrition, abrasion, and abfraction . Future modelling studies must be undertaken with a wider variety of tooth wear scenarios, as well as clinical studies aimed at preventing premature abrasion of the mandibular first molar.
In conclusion, this study model provided a theoretical basis for clinical grinding and jaw adjustment. In our models, the vertical fracture of the mesial root of the mandibular first molar was largely due to excessive tooth wear that results from occlusal tilt to the distal side and increased stress concentration on the mesial root. Therefore, we propose that VFRs could be prevented by adjusting the jaw.