Designing optically tracked instruments for image-guided surgery

Most image-guided surgery (IGS) systems track the positions of surgical instruments in the physical space occupied by the patient. This task is commonly performed using an optical tracking system that determines the positions of fiducial markers such as infrared-emitting diodes or retroreflective spheres that are attached to the instrument. Instrument tracking error is an important component of the overall IGS system error. This paper is concerned with the effect of fiducial marker configuration (number and spatial distribution) on tip position tracking error. Statistically expected tip position tracking error is calculated by applying results from the point-based registration error theory developed by Fitzpatrick et al. Tracking error depends not only on the error in localizing the fiducials, which is the error value generally provided by manufacturers of optical tracking systems, but also on the number and spatial distribution of the tracking fiducials and the position of the instrument tip relative to the fiducials. The theory is extended in two ways. First, a formula is derived for the special case in which the fiducials and the tip are collinear. Second, the theory is extended for the case in which there is a composition of transformations, as is the situation for tracking an instrument relative to a coordinate reference frame (i.e., a set of fiducials attached to the patient). The derivation reveals that the previous theory may be applied independently to the two transformations; the resulting independent components of tracking error add in quadrature to give the overall tracking error. The theoretical results are verified with numerical simulations and experimental measurements. The results in this paper may be useful for the design of optically tracked instruments for image-guided surgery; this is illustrated with several examples.