Thank you for your comments on the calculation of tooth movement. I agree that tooth displacement should be described in 3-dimensional space, and a helical axis is a way to express it. The helical axis method describes a rigid body’s motion by using the following parameters: a vector, P, pointing at the axis and showing the location of the axis, a unit vector showing the orientation of the axis, the amount of translation along the axis, and the rotation about the axis. The location of the axis has a wide range. When a tooth translates, the location of the axis is undefined and might go to infinity. When a tooth has controlled tipping, the axis is located close to the root. Theoretically, these parameters can be calculated by using displacements of at least 3 noncollinear anatomic markers on the tooth. However, the calculation requires precise measurements of the displacements. To increase the accuracy, displacements of more points can be used. The more points measured, the more accurate the calculation.

We evaluated the technology and found it inappropriate for describing the tooth displacement for the following reasons.

• 1.

  The helical axis method is not suitable for describing small tooth displacements because the orientation of the helical axis and the location vector, P, is very sensitive to errors when the positions of the anatomic markers are measured. As you mentioned, artifacts of the computed tomography images can distort the tooth; this makes measurement errors inevitable. With the difficulty of finding the same point before and after the displacement, the calculated results will not be reliable.

• 2.

  Tooth displacement is normally used to evaluate treatment effects. Helical axis parameters are not intuitive to clinicians and do not correlate to treatment strategies—ie, the applied force systems. Thus, the method does not serve the purpose.

Furthermore, the expression is not suitable for a longitudinal comparison of tooth movement because the translations and rotations are about different axes with different orientations and locations. A tooth moves in response to an applied force system. Prediction of tooth displacement requires understanding of the applied force systems and quantification of the resulting 3-dimensional tooth displacement. These technologies are also necessary to study the effects of various biologic factors, such as periodontal tissues, genotypes, and bone modeling and remodeling, on tooth movement. Thus, tooth movement should be quantified and expressed in the coordinate system of the applied force systems. The coordinate systems we used can serve the purpose.

I agree that 3-dimensional tooth displacement needs to be properly described. I am interested in the results of your study.