Propagation of Error and Uncertainty in a Computer-Assisted Orthopedic Surgical System

Computer-assisted orthopedic surgical systems (CAOSSs) require accurate estimation of the relative position/orientation of points and surface on the bone to reproduce the preoperative resection plan accurately, such as the osteotomy lines for resection. This position/orientation is determined by a series of measurements ranging from bone registration, system calibration, and real-time tracking. Thus, the error and uncertainty of position/orientation in each measurement step would propagate to and accumulate the uncertainty of the final stage for resection. In this article, an analytical method is proposed to calculate the propagation of errors and uncertainties in CAOSS. Furthermore, a method to illustrate the uncertainty of resection is introduced. An example using the proposed analytical method, based on a projection guidance system, is presented to illustrate the effect of tracking uncertainty of resection. The outcome of resection uncertainty is demonstrated as a hyperboloid with a 98% confidence region. This is applied together with the tumor in the computer display to check the possibility of the resection to intrude the tumor and failing the surgery, and to determine if the preoperative resection plan should be adjusted with a larger safety margin. The methodology of the propagation of error and uncertainty presented in this article can facilitate surgeons to make better decisions about surgery by creating and monitoring the resection plan with the consideration of the accuracy of the system. Furthermore, the theoretical principle of uncertainty propagation can be applied to other systems that utilize the
$4\times 4$
homogeneous transformation matrices.