Optimal stochastic correction strategies for rigid-body target motion

Purpose: To derive optimal correction strategies for setup errors, including the uncertainty in their measurement, and to analyze their impact on treatment margins. Methods and Materials: New concepts like image-guided radiotherapy aim to provide an increasing amount of targeting information during treatment. Future treatment devices incorporating imaging capabilities will facilitate frequent correction of treatment setup errors. It is, therefore, possible to design new correction protocols that reduce not only systematic but also random setup errors. A novel, very general approach to developing optimal correction strategies in the presence of measurement uncertainties is derived from linear systems theory. In the simplest approach, the state variable of the system, which represents the patient, is the spatial displacement of the center-of-mass of the clinical target volume with respect to the planning CT. This displacement is the sum of a systematic and a random component. Uncertainties in the measured value of the state variable due to the measurement process, image processing technique, or organ deformation are naturally incorporated into a linear system. The true value of the displacement can be estimated from the noisy measurements with a stochastic filter (Kalman filter). These estimates provide an optimal control law for the system and therefore optimal values for the setup corrections. In the case of unknown systematic and random error variances, an adaptive version of the filter was implemented. The statistical properties of the filter were investigated by performing simulations of the state space model and assessed for individual patients and a large patient population subject to different action criteria. Results: Over a patient population, the corrections by the Kalman filter estimates are always advantageous compared with the corrections by the measured values themselves. For a small percentage of individual patients however, the Kalman corrections worsen the results. For large measurement error, the residual standard deviation of the random setup errors can be reduced by approximately 28% for over 90% of the patients. The uncertainty in the measured value impairs the ability to completely account for uncertainties. Conclusions: The Kalman estimates provide an effective means to perform daily setup corrections in the presence of measurement errors. The linear system approach is very versatile and can be extended to more general state variables. (C) 2003 Elsevier Science Inc.