Fundamental properties of the delivery of volumetric modulated arc therapy (VMAT) to static patient anatomy

Purpose: The primary goal of this article is to formulate volumetric modulated arc therapy (VMAT) delivery problem and study interdependence between several parameters (beam dose rate, gantry angular speed, and MLC leaf speed) in the delivery of VMAT treatment plan. The secondary aim is to provide delivery solution and prove optimality (minimal beam on time) of the solution. An additional goal of this study is to investigate alternative delivery approaches to VMAT (like constant beam dose rate and constant gantry angular speed delivery). Method: The problem of the VMAT delivery is formulated as a control problem with machine constraints. The relationships between parameters of arc therapy delivery are derived under the constraint of treatment plan invariance and limitations on delivery parameters. The nonuniqueness of arc therapy delivery solutions is revealed from these relations. The most efficient delivery of arc therapy is then formulated as optimal control problem and solved by geometrical methods. A computer program is developed to find numerical solutions for deliveries of specific VMAT plan. Results: Explicit examples of VMAT plan deliveries are computed and illustrated with graphical representations of the variability of delivery parameters. Comparison of delivery parameters with that of Varian's delivery are shown and discussed. Alternative delivery strategies such as constant gantry angular speed delivery and constant beam dose rate delivery are formulated and solutions are provided. The treatment times for all the delivery solutions are provided. Conclusion: The investigations derive and prove time optimal VMAT deliveries. The relationships between delivery parameters are determined. The optimal alternative delivery strategies are discussed.