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The vector d represents the doses to each voxel of the patient (a) The vector d represents the doses to each voxel of the patient Optimization of Nonuniformly Fractionated Radiotherapy Treatments Melissa R. Gaddy 1 , Sercan Yildiz 2 , Jan Unkelbach 3 , Dávid Papp 1 1. NC State, 2. UNC Chapel Hill, 3. University Hospital Zurich The SDP Relaxation After introducing auxiliary variables to rewrite the model as a QCQP, the quadratic constraints involving x t are linearized introducing the matrix variables X t = x t x t T . Relaxing these nonconvex constraints yields an SDP. • Solving this SDP relaxation gives a lower bound on the mean BED to the liver. • This bound is better than the one obtained by simply relaxing the nonconvex constraints in the above model. The Nonuniform Optimization Model Our primary measure of plan quality is the mean BED to the healthy liver. In order to prioritize this clinical objective, we formulate a constrained model to minimize the mean liver BED, F 1 (b), subject to the constraint that the solution must be at least as good as the uniform reference plan with respect to all other clinical objectives. This model is nonconvex, thus we can only find locally optimal solutions. Results • Nonuniformly fractionated plans achieved a 12-35% mean liver BED reduction over the optimal uniform reference plans. • Nonuniformly fractionated plans close 79-97% of the gap between the mean liver BED in the uniform reference plans and the lower bounds on the lowest achievable mean liver BED. Fig 1. An example of a nonuniform, 5-fraction treatment plan for a large liver tumor. Each panel is a dose distribution (Gy) that is delivered during a single fraction. The high single-fraction doses to different subregions of the tumor allow for a substantial reduction of total physical dose compared to the uniformly-fractionated reference plan. Radiotherapy Treatment Optimization The objective function in the treatment plan optimization is the sum where each term F i is a piecewise quadratic function that penalizes the BED above or below prescription values for the voxel set V i of a given structure: This material was based upon work partially supported by the National Science Foundation under Grant DMS- 1127914 to the Statistical and Applied Mathematical Sciences Institute. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Fig 2. (a) Dose distribution (Gy) for the uniformly fractionated reference plan. (b) Equivalent uniform dose that is isoeffective in delivering the same BED as the locally-optimal nonuniformly fractionated treatment plan. (c) The difference between (a) and (b), which shows that the nonuniformly fractionated plans maintain dose in the tumor while reducing dose in the healthy liver tissue outside the tumor and the beam entrance regions. What is Fractionation? • In uniform fractionation, the same treatment (beamlet weights x) is delivered on each treatment day. • In nonuniform fractionation, the vector of beamlet weights on each day is allowed to be different. The vectors x t and d t denote the beamlet weights and doses delivered in fraction t (t=1,…,N). The Objective Function x: vector of beamlet weights, determines the treatment d: vector of doses absorbed in each voxel of the patient D: dose-influence matrix that relates the beamlet weights to absorbed dose, Dx=d (voxel) The BED Model • Fractionated schemes are compared via the BED model. • BED (biologically effective dose) for a nonuniformly fractionated treatment for a single patient voxel: Image credit: Reemsten and Alber. “Continuous Optimization of Beamlet Intensities for Intensity Modulated Photon and Proton Radiotherapy.” Handbook of Optimization in Medicine, Pardalos and Romeijn (eds.), Springer. 2009. min x,X,p,q,r r 1 s.t. X v 2V i p 2 iv F i (b ⇤ ) 8i 2 I + X v 2V i q 2 iv F i (b ⇤ ) 8i 2 I - r i F i (b ⇤ ) 8i 2 I m ,i 6=1 p iv ≥ N * 1 x T x X , " - b hi iv N e T v D 2 D T e v 2 C v #+ p iv ≥ 0 8v 2 V i , 8i 2 I + q iv ≥-N * 1 x T x X , " - b lo iv N e T v D 2 D T e v 2 C v #+ q iv ≥ 0 8v 2 V i , 8i 2 I - r i ≥ N |V i | X v 2V i * 1 x T x X , " - m hi i N e T v D 2 D T e v 2 C v #+ r i ≥ 0 8i 2 I m 1 x T x X < 0, x ≥ 0, X ≥ 0. Table 1. A comparison of mean liver BED values and lower bounds from the SDP relaxation for five clinical liver cases. (a) (b) (c) Mean liver BED (Gy) in uniform reference plan Mean liver BED (Gy) in nonuniform plan Mean liver BED reduction Mean liver BED lower bound (Gy) Gap closed by SDP bound 1 84.54 75.87 12.75% 73.38 79.52% 2 26.14 19.47 34.26% 18.58 88.23% 3 59.54 50.24 18.51% 48.03 80.80% 4 47.51 38.65 22.92% 37.65 89.86% 5 88.67 77.38 14.59% 77.02 96.91% Objectives • To evaluate the benefit of nonuniform fractionation over conventional uniform fractionation in radiotherapy treatments. • To determine the proximity of locally optimal nonuniformly fractionated plans to the globally optimal solution.
