Mean Arc Distance (MAD): a quantity to compare trajectory 4 pi sampling in single target cranial stereotactic radiotherapy

Purpose. C-arm linac-based radiotherapy has seen a recent interest in 4 pi methods of delivery using simultaneous rotations of couch and gantry to reduce doses to organs-at-risk (OARs) and increase dose compactness. While many methods use heuristics to generate trajectories that avoid OARs, combined with arbitrary trajectory restrictions to prevent oversampling, a quantity has not yet been developed to succinctly compare sampling of the 4 pi space for candidate trajectories as a surrogate for dosimetric compactness. Methods. Evenly spaced sampling points were distributed across a 4 pi sphere centred on isocentre. A metric, mean arc distance (MAD), was defined that quantifies the average arc distance between all sampling points and their nearest field in a radiotherapy trajectory. The relationship between isodose volume and MAD was examined in 2,047 plans: 900 unique trajectories of fixed port DCA plans, 900 unique trajectories of contiguous field DCA plans, 192 VMAT plans (eight volumes in four locations, each with six trajectories) in matRad with 5 VMAT plans repeated for validation in a clinical planning system, and 10 clinical VMAT cases replanned with five trajectories in a clinical treatment planning system. Results. All isodose volumes greater than 10% of the prescription dose decreased with decreasing MAD in all comparisons. In the range of 10% to 100% of the prescription dose, the rate of isodose volume decrease was exponential as a function of MAD in all comparisons. Reduction of absolute isodose volume is seen with increased 4 pi sampling, with larger target volumes exhibiting larger absolute reductions. Very low isodoses (0% to 10% of prescription) increased with decreasing MAD. Conclusions. MAD is a 4 pi sampling quantity useful in quantifying the decrease of isodose volume, relevant for sparing normal tissues. By quantifying this feature, candidate dynamic trajectories can be efficiently compared for 4 pi sampling. This quantity is explored here for single target cranial radiotherapy but may have applications to other radiotherapy treatment sites.