An analytical expression for R50% dependent on PTV surface area and volume: A cranial SRS comparison

The intermediate dose spill for a stereotactic radiosurgery (SRS) plan can be quantified with the metric R50%, defined as the 50% isodose cloud volume (V_IDC50% ) divided by the volume of the planning target volume (PTV). By coupling sound physical principles with the basic definition of R50%, we derive an analytical expression for R50% for a spherical PTV. Our analytical expression depends on three quantities: the surface area of PTV (SA_PTV ), the volume of PTV (V_PTV ), and the distance of dose drop-off to 50% (Δr). The value of ∆r was obtained from a simple set of cranial phantom plan calculations. We generate values from our analytical expression for R50% (R50%_Analytic ) and compare the values to clinical R50% values (R50%_Clinical ) extracted from a previously published SRS data set that spans the V_PTV range from 0.15 to 50.1 cm³ . R50%_Analytic is smaller than R50%_Clinical in all cases by an average of 15% ± 7%, and the general trend of R50%_Clinical vs V_PTV is reflected in the same trend of R50%_Analytic . This comparison suggests that R50%_Analytic could represent a theoretical lower limit for the clinical SRS data; further investigation is required to confirm this. R50%_Analytic could provide useful guidance for what might be achievable in SRS planning.