A Dual Quaternion-Based Method for Estimating PTV Margins in Intensity-Modulated Radiotherapy

Abstract

While a general pose or spatial displacement in 3D space is often decomposed into a translation and a rotation, they are not independent of each other, as the composition of translation and rotation does not commute. ICRU 83 notes that most planning systems allow margins along the three Cartesian dimensions only, although inter- and intra- fraction rotations are routinely imaged and reported. Efforts to construct PTV margins that account for systematic errors from rotations typically sample imaged anatomic rotations and translations in a population, and then use Monte Carlo or other random sampling techniques to construct a space of displacements for a structure by treating the rotations and translations as independent. Significant errors can occur for off-axis rotations as the translational shift of the reference frame origin depends on the axis and magnitude of rotation. Because most rigid rotations in the body occur about the spine, resulting in roll, pitch and yaw, a process to produce a PTV for a distant target that accounts for these shifts is needed. Given two spatial positions of a tumor, the relative spatial displacement between them can be reduced to a screw displacement, which is a combined rotation about and translation along the same axis, called screw axis. Everything about the screw displacement, i.e., the location and orientation of the screw axis as well as the angle of rotation about and distance of translation along the screw axis, can be represented holistically by an 8-dimensional hypercomplex number called dual quaternion. In this way, a set of spatial displacements is represented by a set of points in the space of dual quaternions (S) and the problem of finding the “workspace”, i.e., the enveloping space of all attainable spatial displacements from the sampled 6D population data can be converted into that of fitting the closest fit volume in S that contains all dual-quaternion points. Given the shape and size of a tumor, the resulting PTV that accounts for kinematic errors can then be obtained as swept volume of the kinematic workspace. For a given 3D model of a tumor as well as a set of dual quaternions that represent errors in both rotation and translation, an algorithm has been developed to compute the convex hull of the given displacements and then the swept volume generated by the tumor model. The swept volume represents the expanded PTV that accounts for position and orientation errors. Dual quaternion provides a compact presentation for translational and rotational errors that not only circumvents the non-commutative issue in combining rotation and translation in different order but also are independent of the coordinate systems. In addition, this representation captures naturally the effect of off-axis rotation on the swept volume.