An analytical formulation underpinning the precipitous drop and asymptotic values observed in measured virtual or dynamic filter transmission factors as they fall with segmentation widths of the photon beam modulating collimator

When a shielding jaw or multiple leaf collimator (MLC) traverses across an X-ray field, it modulates the field photon intensity, thereby inducing dose distribution patterns in cross-beam profiles usually termed dynamic (DW) or virtual wedges (VW), or virtual filter for intensity modulated radiation treatment (IMRT). Measured virtual filtration or transmission factors have, unlike those of conventional metals, exhibited rapid drops with modulated field sizes and at times, shown a discontinuity in the transmission when the stepping segmental width is altered amid modulation. We formulated a generalized analytical virtual transmission factor VTF_Δ for a modulated photon beam irrespective of the accelerator (Varian, Siemens, Philips-Elekta) schemes of modulation. WTF_Δ equals a time integral of TR_p(t) plus GR_s(t), and respectively, are the filter modulation primary fraction and the scatter gradient. The modulation duration t or equivalently, is the accumulating counts of partial monitor units (MU). When tested against actual segmental treatment tables (SST Varian), the stated virtual transmission equation reproduced acceptably the measured (DW) factors and predicted their downward trends in the 6 MV photon beams with 60° and 45° DW filters. Further, it is shown that upon taking the primary fraction as the integrated MU-step function δ, and taking the number (n) of the modulating collimator steps to infinity n→∞, one has the analytical virtual transmission approach the asymptotic value limit VTF _Δ(n→∞)=½. This corollary theorem is shown supported by (Varian DW) virtual transmission factors as they taper to 0.48 and 0.51 experimental values