  In recent years, volumetric modulated arc therapy (VMAT) has been becoming a
more and more important radiation technique widely used in clinical application
for cancer treatment. One of the key problems in VMAT is treatment plan
optimization, which is complicated due to the constraints imposed by the
involved equipments. In this paper, we consider a model with four major
constraints: the bound on the beam intensity, an upper bound on the rate of the
change of the beam intensity, the moving speed of leaves of the multi-leaf
collimator (MLC) and its directional-convexity. We solve the model by a
two-stage algorithm: performing minimization with respect to the shapes of the
aperture and the beam intensities alternatively. Specifically, the shapes of
the aperture are obtained by a greedy algorithm whose performance is enhanced
by random sampling in the leaf pairs with a decremental rate. The beam
intensity is optimized using a gradient projection method with non-monotonic
line search. We further improve the proposed algorithm by an incremental random
importance sampling of the voxels to reduce the computational cost of the
energy functional. Numerical simulations on two clinical cancer date sets
demonstrate that our method is highly competitive to the state-of-the-art
algorithms in terms of both computational time and quality of treatment
planning.
