We propose and study a unified model for handling dose constraints (physical dose, equivalent uniform dose (EUD), etc) and radiation source constraints in a single mathematical framework based on the split feasibility problem. The model does not impose on the constraints an exogenous objective (merit) function. The optimization algorithm minimizes a weighted proximity function that measures the sum of the squares of the distances to the constraint sets. This guarantees convergence to a feasible solution point if the split feasibility problem is consistent (i.e., has a solution), or, otherwise, convergence to a solution that minimally violates the physical dose constraints and EUD constraints. We present computational results that demonstrate the validity of the model and the power of the proposed algorithmic scheme.

Treatment delivery with active beam scanning in proton radiation therapy introduces the problem of interplay effects when pencil beam motion occurs on a similar time scale as intra-fractional tumor motion. In situations where fractionation may not provide enough repetition to blur the effects of interplay, repeated delivery or 'repainting' of each field several times within a fraction has been suggested. The purpose of this work was to investigate the effectiveness of different repainting strategies in proton beam scanning. To assess the dosimetric impact of interplay effects, we performed a series of simulations considering the following parameters: tumor motion amplitude, breathing period, asymmetry in the motion trajectory for the target and time required to change the beam energy for the delivery system. Several repainting strategies were compared in terms of potential vulnerability to a dose delivery error. Breathing motion perpendicular to the beam direction (representing superior-inferior type tumor motion in patients) was considered and modeled as an asymmetric sine function with a peak-to-peak amplitude of between 10 and 30 mm. The results show that motion effects cause a narrowing of the high-dose profile and widening of the penumbra. The 90% isodose area was reduced significantly when considering a large motion amplitude of 3 cm. The broadening of the penumbra appears to depend only on the amplitude of tumor motion (assuming harmonic motion). The delivered dose exhibits a shift of 10-15% of the tumor amplitude (or 1-5 mm) in the caudal direction due to breathing asymmetry observed for both sin(4)(x) and sin(6)(x) motion. Of the five repainting techniques studied, so-called 'breath sampling' turned out to be most effective in reducing dose errors with a minimal increase in treatment time. In this method, each energy level is repainted at several evenly spaced times within one breathing period. To keep dose delivery errors below 5% while minimizing treatment time, it is recommended that breath sampling repainting be employed using 5-10 paintings per field for an assumed tumor volume of 8.5 x 8.5 x 10 cm(3). For smaller tumor volumes more repaintings will be required, while for larger volumes five repaintings should be sufficient to achieve the required dose accuracy.

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