  In the treatment plan optimization for intensity modulated radiation therapy
(IMRT), dose-deposition coefficient (DDC) matrix is often pre-computed to
parameterize the dose contribution to each voxel in the volume of interest from
each beamlet of unit intensity. However, due to the limitation of computer
memory and the requirement on computational efficiency, in practice matrix
elements of small values are usually truncated, which inevitably compromises
the quality of the resulting plan. A fixed-point iteration scheme has been
applied in IMRT optimization to solve this problem, which has been reported to
be effective and efficient based on the observations of the numerical
experiments. In this paper, we aim to point out the mathematics behind this
scheme and to answer the following three questions: 1) whether the fixed-point
iteration algorithm converges or not? 2) when it converges, whether the fixed
point solution is same as the original solution obtained with the complete DDC
matrix? 3) if not the same, whether the fixed point solution is more accurate
than the naive solution of the truncated problem obtained without the
fixed-point iteration? To answer these questions, we first performed
mathematical analysis and deductions using a simplified fluence map
optimization (FMO) model. Then we conducted numerical experiments on a
head-and-neck patient case using both the simplified and the original FMO
model. Both our mathematical analysis and numerical experiments demonstrate
that with proper DDC matrix truncation, the fixed-point iteration can converge.
Even though the converged solution is not the one that we obtain with the
complete DDC matrix, the fixed-point iteration scheme could significantly
improve the plan accuracy compared with the solution to the truncated problem
obtained without the fixed-point iteration.
