Simultaneous optimization of mixed photon energy beams in volumetric modulated arc therapy

Despite the availability of multiple energy photon beams on clinical linear accelerators, volumetric modulated arc therapy (VMAT) optimization is currently limited to a single photon beam. The purpose of this work was to present a proof-of-principle study on an algorithm for simultaneous optimization of mixed photon beams for VMAT (MP - VMAT), utilizing an additional photon energy as an additional degree of freedom.

The MP - VMAT optimization algorithm is presented as a two-step heuristic approach. First, a convex linear programming problem is solved for simultaneous optimization of nonuniform dual energy intensity maps (DEIMs) for an angular resolution of 36 equi-spaced beam segments. Subsequently, for a given gantry speed schedule, the second step aims to best replicate each DEIM by dispersing MP - VMAT apertures along with their corresponding intensities over their respective beam segment. This constitutes a nonlinear problem, which is linearized using McCormick relaxation. The final large-scale mixed integer linear programming (MILP) dispersion model ensures a contiguous and smooth transition of multileaf collimators (MLCs) from one beam segment to the next. To demonstrate the proof-of-principle, we first compared the quality of dose volume histograms (DVHs) of MP - VMAT to the ones calculated from 36 DEIMs following the step 1 of MP - VMAT model. Additionally, the MLCs motion violations were evaluated for the complete 360° gantry rotation for gantry speeds ranging from 1 to 6° per second. The quality of MP - VMAT plans were also compared to conventional single energy VMAT plans via DVH, homogeneity index (HI), and conformity number (CN) for two prostate cases.

The MP - VMAT model resulted in a successful convergence of DVHs relative to the ones from DEIMs with HI and CN of 0.05 and 0.9, respectively, for 1 and 2° per second gantry speed schedules. In replicating the DEIMs, the MILP dispersion model was able to achieve optimality for almost all segments at 1° per second and for majority of segments at 2° per second. Although, DVHs quality was slightly inferior for 3° per second gantry speed, the target conformity of 0.9 and heterogeneity of 0.08 were achievable even for the suboptimal solutions. No violations of the MLC constraints were observed throughout the complete 360 degree arc rotation for any gantry speed schedule, thereby confirming MILP dispersion model. For the two prostate cases, the results showed MP - VMAT's ability to achieve substantial dose reduction in rectum and bladder while yielding similar target coverage compared to single energy VMAT. Bladder volume was mostly spared in low-to-intermediate dose region. Rectal volume sparing (3 % to 12 %) was observed in the intermediate (from 25 to 50 Gy) dose region.

We demonstrate the first formalism of a large-scale simultaneous optimization of mixed photon energy beams for VMAT. Dosimetric comparison of MP - VMAT to single energy VMAT demonstrated potential advantages of using mixed photon energy beams for prostate plans, thus providing an impetus for further testing on a large clinical cohort.