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[align=left][align=left][align=left][align=left][align=left][align=left][align=left][align=left][align=left][align=left][align=left][align=left][align=left][align=left][align=left][align=left][align=left][align=left][align=left][align=left][align=left]5. Conclusions and FutureResearch Directions We have demonstrated that the NP metaheuristic providesan effective framework for obtaining highquality solutions to the combined BASand DO problems in both 3DCRT and IMRT. Relative to good quality beam-anglesets constructed via expert clinical judgment and other approaches, the beamsets generated via NP with HTC showed significant relative reduction (up to32%) in the radiation delivered to noncancerous OAR near the tumors. UtilizingHTC provides clinicians an efficient tool to deal with parallel computationthat is inherent in RTP. Furthermore, treatment couch angles could also beconsidered to introduce an extra degree of treatment flexibility. Thus, inaddition to providing a method for automating beam-angle and couch-angleselection, the NP framework yields higher-quality beam sets that significantly reduceradiation overdose to critical organs.We have described indetail an NP framework for the combined BAS and DO problems. If we start from theviewpoint of the mixed-integer programming formulation of the problem, thefeasible solutions that NP provides are upper bounds for the problem. The NPframework is very flexible in terms of allowing alternative lower-boundingapproaches. A hybrid NP approach can combine both an upper bound and a lowerbound within the framework. It has been demonstrated in several applications,such as large-scale job-shop scheduling, transportation, and logisticsplanning, that NP combined with a good lower-bounding scheme makes thealgorithm more powerful and efficient in terms of obtaining optimal solutions.The current mostsuccessful lower-bounding approach applied to IMRT planning is columngeneration (Preciado-Walters et al. 2004, Romeijn et al. 2005).In contrast withthe commonly used inverse-planning approach, the column-generation approachgenerates treatment plans explicitly in terms of apertures and their associatedintensities. This model thus integrates the second-stage problem of solving theDO problem (i.e., obtain intensity maps for each angle) with the third-stageproblem of decomposing the intensity maps into a set of deliverable aperturesand intensities. Because this approach focuses on apertures instead ofbeamlets, each column in the algorithm corresponds to a single aperture. Aftersolving the problem, a set of apertures to deliver radiation and correspondingintensities is obtained. Another advantage of using column generation based onapertures is that the number of apertures used for delivering radiation can beeasily controlled. Thus, the important factor of treatment delivery time couldbe addressed. This paper illustrates the efficiency of using NP to obtain agood upper bound. A hybrid algorithm with column generation may provide a goodlower bound and a way to incorporate intensity-map segmentation. This wouldprovide a unified approach to solve all three stages of the IMRT planningproblem efficiently.AcknowledgmentsThis research wassupported in part by the National Science Foundation under Grants DMI-0400294and DMI-0355567, and by the Air Force of Scientific Research under Grants FA9550-04-1-0179and FA9550-07-1-0390. [/align][/align][/align][/align][/align][/align][/align][/align][/align][/align][/align][/align][/align][/align][/align][/align][/align][/align][/align][/align][/align]