Navigation-based multi-criteria optimization has been introduced to radiotherapy planning in order to allow the interactive exploration of trade-offs between conflicting clinical goals. However, this has been mainly applied to fluence map optimization. The subsequent leaf sequencing step may cause dose discrepancy, leading to human iteration loops in the treatment planning process that multi-criteria methods were meant to avoid. To circumvent this issue, this paper investigates the application of direct aperture optimization methods in the context of multi-criteria optimization. We develop a solution method to directly obtain a collection of apertures that can adequately span the entire Pareto surface. To that end, we extend the column generation method for direct aperture optimization to a multi-criteria setting in which apertures that can improve the entire Pareto surface are sequentially identified and added to the treatment plan. Our proposed solution method can be embedded in a navigation-based multi-criteria optimization framework, in which the treatment planner explores the trade-off between treatment objectives directly in the space of deliverable apertures. Our solution method is demonstrated for a paraspinal case where the trade-off between target coverage and spinal-cord sparing is studied. The computational results validate that our proposed method obtains a balanced approximation of the Pareto surface over a wide range of clinically relevant plans.

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