Error prediction for probes guided by means of fixtures

Probe guides are surgical fixtures that are rigidly attached to bone anchors in order to place a probe at a target with high accuracy (RMS error < 1 mm). Applications include needle biopsy, the placement of electrodes for deep-brain stimulation (DBS), spine surgery, and cochlear implant surgery. Targeting is based on pre-operative images, but targeting errors can arise from three sources: (1) anchor localization error, (2) guide fabrication error, and (3) external forces and torques. A well-established theory exists for the statistical prediction of target registration error (TRE) when targeting is accomplished by means of tracked probes, but no such TRE theory is available for fixtured probe guides. This paper provides that theory and shows that all three error sources can be accommodated in a remarkably simple extension of existing theory. Both the guide and the bone with attached anchors are modeled as objects with rigid sections and elastic sections, the latter of which are described by stiffness matrices. By relating minimization of elastic energy for guide attachment to minimization of fiducial registration error for point registration, it is shown that the expression for targeting error for the guide is identical to that for weighted rigid point registration if the weighting matrices are properly derived from stiffness matrices and the covariance matrices for fiducial localization are augmented with offsets in the anchor positions. An example of the application of the theory are provided for ear surgery.