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Figure from:A Column Generation Approach to Radiation Therapy Treatment Planning Using Aperture Modulation

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Number of (used) apertures needed to satisfy each criterion.

Table 3 Number of (used) apertures needed to satisfy each criterion.

Related Figures (15)

Fic. 1.1. Irradiating a head-and-neck cancer patient using beams from different directions.
Fic. 1.1. Irradiating a head-and-neck cancer patient using beams from different directions.
Fic. 1.2. A multileaf collimator (MLC) system.
Fic. 1.2. A multileaf collimator (MLC) system.
It immediately follows that the pricing problem for beam ¢ € B in this case decom- poses by beamlet row. In particular, for beamlet row r in beam @ we need to solve 3.3.1. Allowing interdigitation. The most widely used commercial MLC sys- tem is a system that allows interdigitation. In the simple approach discussed above the pricing problems decompose by beamlet. However, if we take into account that an aperture is formed by pairs of left and right leaves for each beamlet row, this de- composition is not valid anymore. We can reformulate the pricing problem for beam £ by letting c,(r) and c2(r) denote the index of the last beamlet that is blocked by the left leaf and the first beamlet that is blocked by the right leaf in row r of beam , respectively. When interdigitation is allowed, the pricing problem becomes
It immediately follows that the pricing problem for beam ¢ € B in this case decom- poses by beamlet row. In particular, for beamlet row r in beam @ we need to solve 3.3.1. Allowing interdigitation. The most widely used commercial MLC sys- tem is a system that allows interdigitation. In the simple approach discussed above the pricing problems decompose by beamlet. However, if we take into account that an aperture is formed by pairs of left and right leaves for each beamlet row, this de- composition is not valid anymore. We can reformulate the pricing problem for beam £ by letting c,(r) and c2(r) denote the index of the last beamlet that is blocked by the left leaf and the first beamlet that is blocked by the right leaf in row r of beam , respectively. When interdigitation is allowed, the pricing problem becomes
Fic. 3.2. Network for the pricing problem in case interdigitation is not allowed.
Fic. 3.2. Network for the pricing problem in case interdigitation is not allowed.
4.3. The pricing problem. The KKT-conditions for optimality of (AM2) are
4.3. The pricing problem. The KKT-conditions for optimality of (AM2) are
Note that it is easy to see that there will exist an optimal solution to (AM2) for which constraints (4.1) and (4.2) are binding. we let G? denote a convex and nonincreasing penalty function associated with the lower a-CVaR. constraint for structure s. Similarly, if SAt C S x (0,1) denotes the set of upper CVaR constraints, we let G denote a convex and nondecreasing penalty function associated with the upper a-CVaR constraint for structure s. This leads to the following aperture modulation formulation of the FMO problem:
Note that it is easy to see that there will exist an optimal solution to (AM2) for which constraints (4.1) and (4.2) are binding. we let G? denote a convex and nonincreasing penalty function associated with the lower a-CVaR. constraint for structure s. Similarly, if SAt C S x (0,1) denotes the set of upper CVaR constraints, we let G denote a convex and nondecreasing penalty function associated with the upper a-CVaR constraint for structure s. This leads to the following aperture modulation formulation of the FMO problem:
Values of the coefficients of the voxzel-based penalty functions. Values of the coefficients for the CVaR constraints.
Values of the coefficients of the voxzel-based penalty functions. Values of the coefficients for the CVaR constraints.
Regression results of the relationship between generated and used apertures.
Regression results of the relationship between generated and used apertures.
Fic. 5.1. (a) Objective function value as a function of number of apertures generated and used, and comparison with the optimal value to the beamlet FMO problem. (b) Number of apertures used as a function of number of apertures generated.
Fic. 5.1. (a) Objective function value as a function of number of apertures generated and used, and comparison with the optimal value to the beamlet FMO problem. (b) Number of apertures used as a function of number of apertures generated.
Fic. 5.2. Coverage of (a) PTV1 and (b) PTV2 as a function of number of apertures used.
Fic. 5.2. Coverage of (a) PTV1 and (b) PTV2 as a function of number of apertures used.
Fic. 5.3. Sparing of saliva glands according to (a) DVH criterion (relative volume > 30 Gy) and (b) mean dose as a function of number of apertures used.
Fic. 5.3. Sparing of saliva glands according to (a) DVH criterion (relative volume > 30 Gy) and (b) mean dose as a function of number of apertures used.
Fic. 5.4. Sparing of (a) spinal cord (volume > 45 Gy) and brainstem (volume > 54 Gy) and (b) unspecified tissue as a function of number of apertures used.
Fic. 5.4. Sparing of (a) spinal cord (volume > 45 Gy) and brainstem (volume > 54 Gy) and (b) unspecified tissue as a function of number of apertures used.