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

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.

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 σ_x , σ_y , σ_z . 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 σ_δ . 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 σ_R equal to the product of σδπ180 and dI⇔T, 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 σ_R and dI⇔T.

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 σ_x = σ_y = σ_z = 0.715 mm), a σ_R = 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 σ_R > 0.328 mm will lead to an extra PTV margin that cannot be ignored, and the maximal σ_δ that can be ignored is 0.45° (or 0.0079^rad ) for dI⇔T = 50 mm or 0.23° (or 0.004^rad ) for dI⇔T = 100 mm.

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.