Margins for translational and rotational uncertainties: a probability-based approach.

PURPOSE To define margins for systematic rotations and translations, based on known statistical distributions of these deviations. METHODS AND MATERIALS The confidence interval-based expansion method for translations, known as the "rolling ball algorithm," was extended to include rotations. This new method, which we call the Rotational and Translational Confidence Limit (RTCL) method, is exact for a point with arbitrary rotations and translations or for a finite shape with rotations only. The method was compared with two existing expansion methods: a rolling ball algorithm without rotations, and a convolution (blurring) method which included rotations. On the basis of these methods, planning target volumes (PTVs, expanded clinical target volumes [CTVs]) were constructed for a number of shapes (a sphere, a sphere with an extension, and three prostate cases), and evaluated in several ways by means of a Monte Carlo method. The accuracy of each method was measured by determining the probability of finding the CTV completely inside the PTV (P(CTVinPTV)), using parameters that yield a 90% probability for a sphere-shaped CTV without rotations. Furthermore, with the expansion parameters adjusted to give an equal P(CTVinPTV) for all methods, PTV volumes were compared. RESULTS With the expansion algorithm parameters chosen to yield P(CTVinPTV) = 90% for a sphere, an average P(CTVinPTV) of 84%, 57%, and 46% was obtained for the other shapes, using the RTCL method, coverage probability, and rolling ball, respectively. With the parameters adjusted to yield an equal P(CTVinPTV) for all methods, the PTV volume was on average 8% larger for the coverage probability method and 15% larger for the rolling ball algorithm compared to the RTCL method. CONCLUSION The RTCL method provides an accurate way to include the effects of systematic rotations in the margin. Compared to other algorithms, the method is less sensitive to the shape of the CTV, and, given a fixed probability of finding the CTV inside the PTV, a smaller PTV volume can be obtained.