Simulation study for a finite helical axis analysis of tooth movement

The description of motion using a finite helical axis (FHA) is independent of the chosen coordinate system because the values of the helical axis parameters do not change with coordinate transformations. However, an intuitive understanding of tooth movement expressed by the FHA can be slightly difficult for orthodontists who have never used it. The purpose of this study was to clarify the basic behavior of the FHA to increase our understanding of the FHA during tooth movement. Parameters of the FHA were calculated in two different simulations of canine retraction. In simulation 1, as the tipping angle of the canine was increased from 5° to 30°, the direction vector of the FHA ν approached the x-axis (tipping axis), thus increasing the rotation angle about the FHA θ. These results demonstrated that the direction vector of the FHA can indicate the axis of rotation that is most affected in three-dimensional transformation. In simulation 2, bodily tooth movement of the canine was increased from one to five mm, and the shortest distance from the origin to the FHA d increased in a linear manner. In this case, the percent increase in bodily tooth movement could be determined by the d value. This study showed that the use of FHA makes it possible to determine the torques applied to the tooth and the true bodily tooth movement during orthodontic treatment, and this may lead to a better understanding of how to move teeth.