Multiple local minima in IMRT optimization based on dose-volume criteria.

Multiple local minima traps are known to exist in dose-volume and dose-response objective functions. Nevertheless, their presence and consequences are not considered impediments in finding satisfactory solutions in routine optimization of IMRT plans using gradient methods. However, there is often a concern that a significantly superior solution may exist unbeknownst to the planner and that the optimization process may not be able to reach it. We have investigated the soundness of the assumption that the presence of multiple minima traps can be ignored. To find local minima, we start the optimization process a large number of times with random initial intensities. We investigated whether the occurrence of local minima depends upon the choice of the objective function parameters and the number of variables and whether their existence is an impediment in finding a satisfactory solution. To learn about the behavior of multiple minima, we first used a symmetric cubic phantom containing a cubic target and an organ-at-risk surrounding it to optimize the beam weights of two pairs of parallel-opposed beams using a gradient technique. The phantom studies also served to test our software. Objective function parameters were chosen to ensure that multiple minima would exist. Data for 500 plans, optimized with random initial beam weights, were analyzed. The search process did succeed in finding the local minima and showed that the number of minima depends on the parameters of the objective functions. It was also found that the consequences of local minima depended on the number of beams. We further searched for the multiple minima in intensity-modulated treatment plans for a head-and-neck case and a lung case. In addition to the treatment plan scores and the dose-volume histograms, we examined the dose distributions and intensity patterns. We did not find any evidence that multiple local minima affect the outcome of optimization using gradient techniques in any clinically significant way. Our study supports the notion that multiple minima should not be an impediment to finding a good solution when gradient-based optimization techniques are employed. Changing the parameters for the objective function had no observable effect on our findings.