Towards unlocking the full potential of Multileaf Collimators Paul Morel 1 Romeo Rizzi 2 Guillaume Blin 1 ephane Vialette 1 St´ 1 Universit´ e Paris-Est, LIGM - UMR CNRS 8049, France. 2 Department of Computer Science - University of Verona, Italy. January, 27 2014 G. Blin, P.Morel, R. Rizzi, S. Vialette Unlocking the full potential of Multileaf Collimators January, 27 2014 1 / 34
Outline: Radiotherapy Algorithmics and Radiotherapy Multi-Leaf Collimators (MLCs) Algorithmic results for dual MLCs and rotating MLCs G. Blin, P.Morel, R. Rizzi, S. Vialette Unlocking the full potential of Multileaf Collimators January, 27 2014 2 / 34
Radiotherapy Radiotherapy G. Blin, P.Morel, R. Rizzi, S. Vialette Unlocking the full potential of Multileaf Collimators January, 27 2014 3 / 34
Radiotherapy Radiotherapy Radiation-Therapy Cancer treatment relying on radiations aiming at killing cancerous cells. Radiotherapy (Step and Shoot) External X-ray(photon) cone beam rotating around a patient, stopping at specific angles to deliver a prescribed treatment. G. Blin, P.Morel, R. Rizzi, S. Vialette Unlocking the full potential of Multileaf Collimators January, 27 2014 4 / 34
Radiotherapy Radiotherapy: Gantry G. Blin, P.Morel, R. Rizzi, S. Vialette Unlocking the full potential of Multileaf Collimators January, 27 2014 5 / 34
Radiotherapy Radiotherapy: Multi-Leaf Collimator (MLC) G. Blin, P.Morel, R. Rizzi, S. Vialette Unlocking the full potential of Multileaf Collimators January, 27 2014 6 / 34
Radiotherapy Radiotherapy: Treatment Planning G. Blin, P.Morel, R. Rizzi, S. Vialette Unlocking the full potential of Multileaf Collimators January, 27 2014 7 / 34
Radiotherapy Radiotherapy: Treatment Planning G. Blin, P.Morel, R. Rizzi, S. Vialette Unlocking the full potential of Multileaf Collimators January, 27 2014 8 / 34
Radiotherapy Radiotherapy: Treatment Planning G. Blin, P.Morel, R. Rizzi, S. Vialette Unlocking the full potential of Multileaf Collimators January, 27 2014 9 / 34
Radiotherapy Radiotherapy: Treatment Planning G. Blin, P.Morel, R. Rizzi, S. Vialette Unlocking the full potential of Multileaf Collimators January, 27 2014 10 / 34
Algorithmics and Radiotherapy Algorithmics and Radiotherapy G. Blin, P.Morel, R. Rizzi, S. Vialette Unlocking the full potential of Multileaf Collimators January, 27 2014 11 / 34
Algorithmics and Radiotherapy Algorithmics and Radiotherapy Fluence map ⇔ 2D intensity matrix MLC Configurations ⇔ Matrix decomposition Time (Intensity) ⇔ Weight G. Blin, P.Morel, R. Rizzi, S. Vialette Unlocking the full potential of Multileaf Collimators January, 27 2014 12 / 34
Algorithmics and Radiotherapy Algorithmics and Radiotherapy: Matrices properties 2D intensity matrix: Positive integer matrix Matrix representing an MLC configuration: Binary matrix Consecutive ones property (C1P) G. Blin, P.Morel, R. Rizzi, S. Vialette Unlocking the full potential of Multileaf Collimators January, 27 2014 13 / 34
Algorithmics and Radiotherapy Algorithmics and Radiotherapy: Problems Total beam-on time minimization: total irradiation time. Solvable in linear time.[Ahuja and Hamacher, 2005] Setup time minimization: time shaping the apertures. Strongly NP-Hard, even for 1 row matrices.[Baatar et al., 2005] G. Blin, P.Morel, R. Rizzi, S. Vialette Unlocking the full potential of Multileaf Collimators January, 27 2014 14 / 34
Multi-Leaf Collimators Multi-Leaf Collimators G. Blin, P.Morel, R. Rizzi, S. Vialette Unlocking the full potential of Multileaf Collimators January, 27 2014 15 / 34
Multi-Leaf Collimators Multi-Leaf Collimators: Conventional MLC G. Blin, P.Morel, R. Rizzi, S. Vialette Unlocking the full potential of Multileaf Collimators January, 27 2014 16 / 34
Multi-Leaf Collimators Multi-Leaf Collimators: Dual MLC G. Blin, P.Morel, R. Rizzi, S. Vialette Unlocking the full potential of Multileaf Collimators January, 27 2014 17 / 34
Multi-Leaf Collimators Multi-Leaf Collimators: Dual MLC Interest ? 1 0 1 0 1 0 1 1 1 ↑ ↑ ↑ . . . . . . 0 1 0 1 0 1 . . . ← + ← + ← . . . + 1 0 1 0 1 0 1 1 1 ↑ ↑ ↑ . . . . . . 0 1 0 1 0 1 + + + . . . ← ← ← . . . = 1 0 1 0 1 0 1 1 1 ↑ ↑ ↑ . . . . . . . . . . . . . . . . . . ... ... . . . . . . . . . . . . . . . . . . . . . . . . 0 1 0 1 0 1 . . . ← + ← + ← . . . + + → + → + . . . → 1 1 1 ↑ ↑ ↑ . . . + → + → + . . . → 1 1 1 ↑ ↑ ↑ . . . + + + + → → . . . → . . . . . . ... . . . . . . . . . . . . 1 1 1 ↑ ↑ ↑ . . . 2 configurations instead of a linear number! G. Blin, P.Morel, R. Rizzi, S. Vialette Unlocking the full potential of Multileaf Collimators January, 27 2014 18 / 34
Multi-Leaf Collimators Multi-Leaf Collimators: Rotating MLC G. Blin, P.Morel, R. Rizzi, S. Vialette Unlocking the full potential of Multileaf Collimators January, 27 2014 19 / 34
Multi-Leaf Collimators Multi-Leaf Collimators: Rotating MLC Interest ? 1 4 2 5 0 0 0 1 0 0 0 1 0 1 1 1 1 3 3 2 0 0 1 1 0 1 1 1 1 1 1 0 = + + + 1 3 2 5 0 0 1 1 0 0 1 1 0 1 1 1 6 4 6 0 1 1 1 0 1 1 1 0 1 1 1 0 H H H 1 1 1 1 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 + + 1 1 1 1 0 1 0 1 0 0 0 0 1 1 1 0 1 0 1 0 1 0 1 0 H V V 6 configurations instead of 8. Note: H: horizontal configuration, V: vertical configuration. G. Blin, P.Morel, R. Rizzi, S. Vialette Unlocking the full potential of Multileaf Collimators January, 27 2014 20 / 34
Multi-Leaf Collimators Multi-Leaf Collimators: Dual and Rotating MLCs What happens to the problems: Setup time minimization ? Total beam-on time minimization ? G. Blin, P.Morel, R. Rizzi, S. Vialette Unlocking the full potential of Multileaf Collimators January, 27 2014 21 / 34
Algorithmic results for dual MLCs and rotating MLCs Algorithmic results for dual MLCs and rotating MLCs G. Blin, P.Morel, R. Rizzi, S. Vialette Unlocking the full potential of Multileaf Collimators January, 27 2014 22 / 34
Algorithmic results for dual MLCs and rotating MLCs Result for the dual MLCs Algorithmic result for dual MLCs Theorem The Dual-MLC Decomposition problem is NP-Hard when minimizing the total setup time. G. Blin, P.Morel, R. Rizzi, S. Vialette Unlocking the full potential of Multileaf Collimators January, 27 2014 23 / 34
Algorithmic results for dual MLCs and rotating MLCs Results for the rotating MLCs Algorithmic results for rotating MLCs Matrix Orthogonal Decomposition (MOD) problem: Decompose a fluence map using horizontal and vertical MLC configurations. Theorem The MOD problem is NP-Hard when minimizing either the total setup time or the total beam-on time. G. Blin, P.Morel, R. Rizzi, S. Vialette Unlocking the full potential of Multileaf Collimators January, 27 2014 24 / 34
Algorithmic results for dual MLCs and rotating MLCs Results for the rotating MLCs Algorithmic results for rotating MLCs: Approximation algorithm for the beam-on time minimization Integer Linear Program minimizing the total beam-on time for MOD: minimize H + V � H k subject to ∀ 1 ≤ k ≤ m , ij ≤ H (1) i ≤ j � V k ∀ 1 ≤ k ≤ m , ij ≤ V (2) i ≤ j ∀ k , k ′ ∈ { 1 , . . . m } 2 , � � V k ′ H k i ′ j ′ = M [ k ][ k ′ ] ij + (3) i ≤ k ′ ≤ j i ′ ≤ k ≤ j ′ H k ij ≥ 0 , V k ∀ i , j , k , ij ≥ 0 1 ≤ i ≤ j ≤ m , 1 ≤ i ′ ≤ j ′ ≤ m , H ≥ 0 , V ≥ 0 . G. Blin, P.Morel, R. Rizzi, S. Vialette Unlocking the full potential of Multileaf Collimators January, 27 2014 25 / 34
Algorithmic results for dual MLCs and rotating MLCs Results for the rotating MLCs Algorithmic result for rotating MLCs: Approximation algorithm for the beam-on time minimization Relax of the integrality constraint ⇒ Fractional Linear Program. Rounding of the fractional solution ⇒ Integral solution not too far from optimal. Principle of the algorithm: Compute in polynomial time an optimal fractional solution for horizontal 1 configurations. Provide an integral rounding for horizontal configurations. 2 Compute the corresponding vertical configurations in linear time. 3 G. Blin, P.Morel, R. Rizzi, S. Vialette Unlocking the full potential of Multileaf Collimators January, 27 2014 26 / 34
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