The use of mixed-integer programming for inverse treatment planning with pre-defined field segments

Abstract

Purpose: To investigate the optimization aspect of a new inverse planning approach devised by our institution. This method uses pre-defined segments instead of relying on a translation from a derived intensity pattern to segments. This work compares a mixed-integer programming with a previously used back projection algorithm and uses beamlet-based simulated annealing as a benchmark. Materials and methods: Complex intensity patterns generated by traditional inverse treatment plans are often difficult to deliver. The technique devised in our institution uses a set of simple rules to define a pool of delivery segments. The problem then becomes one of optimizing segment weights to satisfy prescribed dose constraints. In this report, we investigate the use of the mixed-integer programming(MIP) for the weight optimization. In the MIP approach, real variables describing segment weights were combined with binary variables, used to enumerate volume voxels in targets and critical structures. These binary variables provided a mechanism of imposing multiple dose-volume constraints. Limits on segment weights were also included in the optimization. The MIP method was compared to previously used optimization with the Cimmino back-projection algorithm (CBP). These algorithms represent two different optimization approaches. The MIP will not yield a solution to the optimization problem if the constraints are not strictly satisfied, however it will yield the optimal solution if the problem is feasible (i.e., solvable). In contrast, the CBP attempts to converge on a solution, which satisfies a given set of constraints according to the pre-set proximity parameter but it does not provide information about the problem feasibility and it does not seek the best solution. These two methods were compared to the NOMOS CORVUS inverse planning with a MLC beamlet-based simulated annealing as a benchmark. We analyzed five complex cases of patients treated for oropharyngeal cancer. Two of the cases involved three dose levels to treat the gross tumor to 66 Gy, subclinical PTV at high risk to 60 Gy and bilateral lymph nodes to 54 Gy. The other cases involved only 66 and 54 Gy dose levels. Critical normal structures included brain stem (<54 Gy), spinal cord (<45 Gy), mandible (<70 Gy) and parotid glands (50% of at least one gland below 30 Gy). Nine equally spaced gantry angles were used to define the pool of segments. The search strategies for the mixed-integer solution involved either minimization of the dose to critical structures and/or maximization of the target doses or minimization of the number of non-compliant voxels. Results: The mixed-integer programming, the Cimmino back-projection and the CORVUS benchmark solutions produced similar dose-volume histograms. Both in-house optimization methods met their constraints for critical structures sparing. The mixed-integer solutions held slight advantage over the Cimmino algorithm in limiting the high and low dose tails in the target coverage. The CORVUS solutions showed advantages of optimizing over an unrestricted intensity map. Generally, they provided better coverage at the 54 Gy level and better sparing of salivary glands. The segmental methods held advantage in minimizing the number of segments needed to deliver an acceptable solution. On average, 50 segments were needed to deliver these complex treatments. In selected cases, the optimizer allowed the restriction on differences in segment weights to limit the infiltration of high dose regions into the healthy tissue. Conclusion: The mixed-integer programming was successfully used for inverse treatment planning for IMRT dose delivery with pre-defined segments for complex head and neck cases. If the initial dose constraints could not be satisfied, the optimizer sought the best solution for a given pool of available segments through the minimization of the number of voxels inside the under- or overdosed volumes. The mixed-integer programming could work in tandem with the Cimmino method to allow for fast identification of the desired solution.