A statistical model for analyzing the rotational error of single isocenter for multiple targets technique

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Abstract

Purpose

To develop a statistical model that incorporates the treatment uncertainty from the rotational error of the single isocenter for multiple targets technique, and calculates the extra PTV (planning target volume) margin required to compensate for this error.

Methods

The random vector for modeling the setup (S) error in the three-dimensional (3D) patient coordinate system was assumed to follow a 3D normal distribution with a zero mean, and standard deviations of σx, σy, σz. It was further assumed that the rotation of clinical target volume (CTV) about the isocenter happens randomly and follows a three-dimensional (3D) independent normal distribution with a zero mean and a uniform standard deviation of σδ. This rotation leads to a rotational random error (R), which also has a 3D independent normal distribution with a zero mean and a uniform standard deviation of σR equal to the product of Symbol and Symbol, the distance between the isocenter and CTV. Both (S and R) random vectors were summed, normalized, and transformed to the spherical coordinates to derive the Chi distribution with three degrees of freedom for the radial coordinate of S+R. PTV margin was determined using the critical value of this distribution for a 0.05 significance level so that 95% of the time the treatment target would be covered by the prescription dose. The additional PTV margin required to compensate for the rotational error was calculated as a function of σR and Symbol.

Results

The effect of the rotational error is more pronounced for treatments that require high accuracy/precision like stereotactic radiosurgery (SRS) or stereotactic body radiotherapy (SBRT). With a uniform 2-mm PTV margin (or σx = σy = σz = 0.715 mm), a σR = 0.328 mm will decrease the CTV coverage probability from 95.0% to 90.9%, or an additional 0.2-mm PTV margin is needed to prevent this loss of coverage. If we choose 0.2 mm as the threshold, any σR > 0.328 mm will lead to an extra PTV margin that cannot be ignored, and the maximal σδ that can be ignored is 0.45° (or 0.0079rad) for Symbol = 50 mm or 0.23° (or 0.004rad) for Symbol = 100 mm.

Conclusions

The rotational error cannot be ignored for high-accuracy/-precision treatments like SRS/SBRT, particularly when the distance between the isocenter and target is large.

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