Principal direction of inertia for 3D trajectories from patient-specific TMJ movement

Accurate simulation and evaluation of mandibular movement is fundamental for the analysis of functional changes and effects of the mandible and maxilla before and after surgical treatments. We applied principal axes of inertia to the three-dimensional (3D) trajectories generated by patient-specific simulations of TMJ movements for the functional evaluations of mandible movement. Three-dimensional movements of the mandible and the maxilla were tracked based on a patient-specific splint and an optical tracking system. The dental occlusion recorded on the sprint provided synchronization for initial movement in the tracking and the simulation phases. The translation and rotation recorded during movement tracking was applied sequentially to the mandibular model in relation to a fixed maxilla model. The sequential 3D positions of selected landmarks on the mandible were calculated based on the reference coordinate system. The landmarks selected for analysis were bilateral condyles and pogonion points. The moment of inertia tensor was calculated with respect to the 3D trajectory points. Using the unit vectors along the principal axes derived from the tensor matrix, α, β and γ rotations around z-, y- and x-axes were determined to represent the principal directions as principal rotations respectively. The γ direction showed the higher standard deviation, variation of directions, than other directions at all the landmarks. The mandible movement has larger kinematic redundancy in the γ direction than α and β during mouth opening and closing. Principal directions of inertia would be applied to analyzing the changes in angular motion of trajectories introduced by mandibular shape changes from surgical treatments and also to the analysis of the influence of skeletal deformities on mandibular movement asymmetry.