  In this article we provide a method to generate the trade-off between
delivery time and fluence map matching quality for volumetric modulated arc
therapy (VMAT). At the heart of our method lies a mathematical programming
model that, for a given duration of delivery, optimizes leaf trajectories and
dose rates such that the desired fluence map is reproduced as well as possible.
This model was presented for the single map case in a companion paper (Fast
approximate delivery of fluence maps: the single map case). The resulting
large-scale, non-convex optimization problem was solved using a heuristic
approach. The single-map approach cannot directly be applied to the full arc
case due to the large increase in model size, the issue of allocating delivery
times to each of the arc segments, and the fact that the ending leaf positions
for one map will be the starting leaf positions for the next map. In this
article the method proposed in \cite{dm1} is extended to solve the full map
treatment planning problem. We test our method using a prostate case and a head
and neck case, and present the resulting trade-off curves. Analysis of the leaf
trajectories reveal that short time plans have larger leaf openings in general
than longer delivery time plans. Our method allows one to explore the continuum
of possibilities between coarse, large segment plans characteristic of direct
aperture approaches and narrow field plans produced by sliding window
approaches. Exposing this trade off will allow for an informed choice between
plan quality and solution time. Further research is required to speed up the
optimization process to make this method clinically implementable.
