  In a treatment plan optimization problem for radiotherapy, a clinically
acceptable plan is usually generated by an optimization process with weighting
factors or reference doses adjusted for organs. Recent discoveries indicate
that adjusting parameters associated with each voxel may lead to better plan
quality. However, it is still unclear regarding the mathematical reasons behind
it. To answer questions related to this problem, we establish in this work a
new mathematical framework equipped with two theorems. The new framework
clarifies the different consequences of adjusting organ-dependent and
voxel-dependent parameters for the treatment plan optimization of radiation
therapy, as well as the different effects of adjusting weighting factors versus
reference doses in the optimization process. The main discoveries are
threefold: 1) While in the organ-based model the selection of the objective
function has an impact on the quality of the optimized plans, this is no longer
an issue for the voxel-based model since the entire Pareto surface could be
generated regardless the specific form of the objective function as long as it
satisfies certain mathematical conditions; 2) A larger Pareto surface is
explored by adjusting voxel-dependent parameters than by adjusting
organ-dependent parameters, possibly allowing for the generation of plans with
better trade-offs among different clinical objectives; 3) Adjusting voxel
weighting factors is preferred to adjusting the voxel reference doses since the
Pareto optimality can be maintained.
