Basic behavior of the finite helical axis in a simple tooth movement simulation

A finite helical axis (FHA) analysis can provide precise three-dimensional information on orthodontic tooth movement compared to an analysis based on a rectangular coordinate system. The FHA has already been applied in an analysis of orthodontic tooth movement. Interestingly, the position of the FHA changes dramatically in different stages of treatment; however, no previous report has provided detailed information of its basic behavior for clinicians. The purpose of this study was to clarify the basic behavior of the FHA in simple tooth movement, which could promote a better understanding of the 3-D orientation of and rotation about the FHA during tooth movement. Parameters of the FHA were calculated from a simulation of canine retraction. As the tipping angle of the canine was increased from 5° to 30°, the orientation vector of the FHA approached the most affected axis of rotation (on a rectangular coordinate system) in a non-linear manner. The angle of rotation about the FHA also increased in a non-linear manner. This non-linear problem was solved analytically. The basic behavior of the orientation vector of the FHA and the non-linear characteristics of the FHA parameters clarified in this study should be important for the future analysis of actual tooth movement based on the FHA. However, we must be aware that the non-linearity of the FHA itself may affect the analysis of the mechanical properties of the periodontal tissue.