Nonlinear dependency of tooth movement on force system directions

Moment-to-force ratios (M:F) define the type of tooth movement. Typically, the relationship between M:F and tooth movement has been analyzed in a single plane. Hence, limited information is available to evaluate a load system elicited by an appliance in 3D. Here, to increment 3-D tooth movement theory, we test the hypothesis that the mathematical relationships between M:F and tooth movement are distinct depending on force system directions. A finite element model of a first maxillary premolar, scaled to average tooth dimensions, was constructed based on a CBCT scan. We conducted finite element analysis (FEA) of the M:F and tooth movement relationships, represented by the projected axis of rotation (C.Rot) in each plane, for 510 different Loads. We confirmed that an hyperbolic equation relates the Distance (C.Res-C.Rot) and M:F; however, the constant of proportionality ("k") varied with non-linearly the force direction. With a force applied parallel to the tooth long axis, "k" was 12 times higher than with a force parallel to the mesio-distal direction and 7 times higher than with a force parallel to the bucco-lingual direction. The M:F has differential influence on tooth movement depending on load directions, and it is an incomplete parameter to describe the quality of an orthodontic load system if not associated with force and moment directions. Moreover, incremental differences in M:F in each plane have different incremental effects on C.Rot position.