A Mathematical Framework for Determining the Effect of Rotational Errors on Single-Isocenter Multi-target Stereotactic Radiosurgery.

Introduction Single-isocenter multi-target (SIMT) radiosurgery enhances efficiency by treating multiple lesions with a single plan. However, this technique is highly susceptible to patient positioning errors, particularly rotational deviations. Even minor rotations can lead to significant reductions in target dose coverage, with the degree of degradation being influenced by target size, the magnitude of rotation, and the target's distance from the treatment isocenter. While previous studies have investigated these effects through simulations, retrospective analyses, or statistical models, a robust mathematical framework for predicting these interactions is lacking. Purpose This study aims to develop and apply mathematical models to quantify the impact of rotational errors on target coverage in SIMT radiosurgery. The primary objective is to elucidate the interplay between target size, distance from the isocenter, and the magnitude of rotational error, with a view to informing individualized planning target volume (PTV) margin selection for optimized target coverage and minimized normal tissue irradiation. Methods We developed mathematical models to simulate the dosimetric consequences of rotational errors under the assumption of a dose distribution encompassing the target perfectly, in the absence of setup errors. The target and radiation fields were modeled as three-dimensional spheres, and equations were derived to calculate the volume of intersection between the rotated target and the prescribed radiation field. The percentage of target volume maintained within the prescription dose sphere was then calculated. Graphical representations of Gross Tumor Volume (GTV) coverage as a function of rotational error (ranging from 0° to 3°), target diameter (0.5 cm to 3.0 cm), PTV margins (0 mm, 0.5 mm, 1.0 mm), and distance from the isocenter (1.5 cm to 7.5 cm) were generated. The methodology was applied to clinical cases to demonstrate its utility in deriving customized margins. Results The mathematical analysis demonstrated that small targets located further from the isocenter are most vulnerable to coverage degradation due to rotational errors, especially with minimal or no PTV margins. For a 0.5 cm diameter target, even a 0.5° rotation resulted in coverage below 95% at 1.5 cm from the isocenter without a margin, dropping significantly at greater distances. Conversely, larger targets maintained adequate coverage even at 2.5° rotation without a margin. Introduction of a 1.0 mm uniform margin generally ensured over 95% GTV coverage for 1° rotations across all evaluated target sizes and distances from the isocenter. Applying this framework to a clinical case, differential margins derived from our model resulted in a decreased treated volume and reduced brain V12Gy compared to a uniform 1 mm margin plan. Conclusions The findings underscore the impact of rotation based on target size and location and the critical importance of individualized PTV margin selection. By leveraging this analysis, clinicians can determine optimal, customized margins based on specific target characteristics and acceptable rotation thresholds, thereby ensuring robust target coverage while minimizing unnecessary irradiation of healthy tissue and enhancing the precision of SIMT treatments.