Absence of multiple local minima effects in intensity modulated optimization with dose volume constraints

Abstract

This paper reports on the analysis of intensity modulated radiation treatment optimization problems in the presence of non-convex feasible parameter spaces caused by the specification of dose-volume constraints for the organs-at-risk (OARs). The main aim was to determine whether the presence of those non-convex spaces affects the optimization of clinical cases in any significant way. This was done in two phases: (1) Using a carefully designed two-dimensional mathematical phantom that exhibits two controllable minima and with randomly initialized beamlet weights, we developed a methodology for exploring the nature of the convergence characteristics of quadratic cost function optimizations (deterministic or stochastic). The methodology is based on observing the statistical behaviour of the residual cost at the end of optimizations in which the stopping criterion is progressively more demanding and carrying out those optimizations to very small error changes per iteration. (2) Seven clinical cases were then analysed with dose-volume constraints that are stronger than originally used in the clinic. The clinical cases are two prostate cases differently posed, a meningioma case, two head-and-neck cases, a spleen case and a spine case. Of the 14 different sets of optimizations (with and without the specification of maximum doses allowed for the OARs), 12 fail to show any effect due to the existence of non-convex feasible spaces. The remaining two sets of optimizations show evidence of multiple minima in the solutions, but those minima are very close to each other in cost and the resulting treatment plans are practically identical, as measured by the quality of the dose-volume histograms (DVHs). We discuss the differences between fluence maps resulting from those similar treatment plans. We provide a possible reason for the observed results and conclude that, although the study is necessarily limited, the annealing characteristics of a simulated annealing method may not be justified in clinical optimization in the presence of dose-volume constraints. The results of optimizations by the Newton gradient (NG) method with a quadratic cost function are reported in detail. An adaptive simulated annealing method, optimizing the same function, and the dynamically penalized likelihood method, optimizing a log likelihood function, have also been used in the study. The results of the latter two methods have only been discussed briefly, as they yielded the same conclusions as the NG method.

Minimax optimization for handling range and setup uncertainties in proton therapy

“…This bound constrained program is convex whenever the objective function f is (since d is linear in x) and can be solved by gradient based methods. Using dosevolume optimization functions leads to a nonconvex problem, but in practice, there seems to be little difficulty with local minima [10].…”

Inverse 4D conformal planning for lung SBRT using particle swarm optimization

“…The main impact of non-convexity is multiple local minima in the solution space. Some studies suggest that, due to degeneracy, the non-convexity in radiotherapy planning optimization does not cause a significant issue [32]; however, we believe that the way an objective function is framed defines how significant the roles of non-convexity and degeneracy are [33]. One way to manage non-convexity is by limiting computation of dose-volume objectives to intervals in between consecutive convex optimization steps [34].…”

Mathematical optimization in intensity modulated radiation therapy

“…The author demonstrates that for models with dose-volume constraints there is the possibility of multiple local minima due to the concavity of the feasible set formed by DVCs. Furthermore, , Rowbottom and Webb (2002), Llacer et al (2003), Jeraj et al (2003), Wu et al (2003a) verify the existence of local minima by performing case studies. They also show that most of the local minima are very close to a global minimum.…”