Towards unlocking the full potential of Multileaf Collimators Paul - - PowerPoint PPT Presentation
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,
Guillaume Blin1 Paul Morel1 Romeo Rizzi2 St´ ephane Vialette1
1Universit´
e Paris-Est, LIGM - UMR CNRS 8049, France.
2Department of Computer Science - University of Verona, Italy.
January, 27 2014
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Radiotherapy Algorithmics and Radiotherapy Multi-Leaf Collimators (MLCs) Algorithmic results for dual MLCs and rotating MLCs
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Radiotherapy
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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.
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Radiotherapy
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Radiotherapy
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Radiotherapy
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Radiotherapy
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Radiotherapy
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Radiotherapy
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Algorithmics and Radiotherapy
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Algorithmics and Radiotherapy
Fluence map ⇔ 2D intensity matrix MLC Configurations ⇔ Matrix decomposition Time (Intensity) ⇔ Weight
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Algorithmics and Radiotherapy
2D intensity matrix:
Positive integer matrix
Matrix representing an MLC configuration:
Binary matrix Consecutive ones property (C1P)
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Algorithmics and Radiotherapy
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]
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Multi-Leaf Collimators
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Multi-Leaf Collimators
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Multi-Leaf Collimators
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Multi-Leaf Collimators
Interest ? 1 1 1 . . . 1 1 . . . 1 1 1 1 . . . 1 1 . . . 1 1 1 1 . . . . . . . . . . . . . . . . . . ... . . . 1 1 . . . 1
=
1
↑
1
↑
1 . . .
↑ ← + ← + ←
. . .
+
1
↑
1
↑
1 . . .
↑ ← + ← + ←
. . .
+
1
↑
1
↑
1 . . .
↑
. . . . . . . . . . . . . . . ... . . .
← + ← + ←
. . .
+
+
+ → + → +
. . .
→ ↑
1
↑
1
↑
. . . 1
+ → + → +
. . .
→ ↑
1
↑
1
↑
. . . 1
+ → + → +
. . .
→
. . . . . . . . . . . . . . . ... . . .
↑
1
↑
1
↑
. . . 1 2 configurations instead of a linear number!
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Multi-Leaf Collimators
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Multi-Leaf Collimators
Interest ? 1 4 2 5 1 3 3 2 1 3 2 5 6 4 6 = 1 1 1 1 1 1 1 1
H
+ 1 1 1 1 1 1 1 1 1
H
+ 1 1 1 1 1 1 1 1 1 1 1 1
H
+ 1 1 1 1 1 1 1 1 1 1 1
H
+ 1 1 1 1 1 1
V
+ 1 1 1 1
V
6 configurations instead of 8. Note: H: horizontal configuration, V: vertical configuration.
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Multi-Leaf Collimators
What happens to the problems: Setup time minimization ? Total beam-on time minimization ?
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Algorithmic results for dual MLCs and rotating MLCs
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Algorithmic results for dual MLCs and rotating MLCs Result for the dual MLCs
Theorem The Dual-MLC Decomposition problem is NP-Hard when minimizing the total setup time.
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Algorithmic results for dual MLCs and rotating MLCs Results for the 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.
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Algorithmic results for dual MLCs and rotating MLCs Results for the rotating MLCs
Integer Linear Program minimizing the total beam-on time for MOD: minimize H + V subject to ∀1 ≤ k ≤ m,
Hk
ij ≤ H
(1) ∀1 ≤ k ≤ m,
V k
ij ≤ V
(2) ∀k, k′ ∈ {1, . . . m}2,
Hk
ij +
V k′
i′j′ = M[k][k′]
(3) ∀i, j, k, Hk
ij ≥ 0, V k ij ≥ 0
1 ≤ i ≤ j ≤ m, 1 ≤ i′ ≤ j′ ≤ m, H ≥ 0, V ≥ 0.
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Algorithmic results for dual MLCs and rotating MLCs Results for the rotating MLCs
Relax of the integrality constraint ⇒ Fractional Linear Program. Rounding of the fractional solution ⇒ Integral solution not too far from
Principle of the algorithm:
1
Compute in polynomial time an optimal fractional solution for horizontal configurations.
2
Provide an integral rounding for horizontal configurations.
3
Compute the corresponding vertical configurations in linear time.
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Algorithmic results for dual MLCs and rotating MLCs Results for the rotating MLCs
Let us consider the kth row of the horizontal configurations:
6.6 3.9 10.5 10.5 11.7 15.6 11.7 10.5 7.8 3.9 7.8
Mk
[6, 6] 3.9 [8, 9] 1.2 [5, 7] [4, 7] [1, 1] 2.7 [3, 8] [11, 11] [1, 3] [1, 9] [3, 11]
Original set of intervals Transform the original set into a set where intervals are either nested or independent.
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Algorithmic results for dual MLCs and rotating MLCs Results for the rotating MLCs
6.6 3.9 10.5 10.5 11.7 15.6 11.7 10.5 7.8 3.9 7.8
Mk Mk
[6, 6] [9, 9] [5, 7] [4, 7] [1, 1] [4, 7] [8, 8] [11, 11] [3, 3] [1, 2] [1, 2] [3, 9] 6.6 [10, 11]
Transformed set without crossing Transform the new set into a set of nested intervals.
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Algorithmic results for dual MLCs and rotating MLCs Results for the rotating MLCs
Transformations used to build a set of nested intervals.
[1, 2] w1 [1, 2] w2 [1, 2] w1 + w2
Case: Copy of an interval
[2, 3] w2 [1, 5] w1 [4, 5] w3 [2, 3] w2 [4, 5] w3 [1, 3] w1 [4, 5] w1
Case 3 intervals: a)
[2, 3] w2 [1, 5] w1 [3, 4] w3 [3, 4] w3 − w2 [2, 4] w2 [1, 5] w1
Case 3 intervals: b) w2 < w3
If w3 < w2, the process is similar to the case b).
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Algorithmic results for dual MLCs and rotating MLCs Results for the rotating MLCs
Set of independent subsets of nested intervals (stack):
6.6 3.9 10.5 10.5 11.7 15.6 11.7 10.5 7.8 3.9 7.8
Mk
[6, 6] [5, 7] [1, 1] [8, 8] [11, 11] [1, 2] [3, 7] 10.5 [8, 9] [10, 11]
Final set composed of independent towers Round each stack separately.
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Algorithmic results for dual MLCs and rotating MLCs Results for the rotating MLCs
11 11 12 15 12
Mk
[6, 6] [5, 7] [3, 7]
Rounded stack Finally:
1
Subtract the horizontal integral solution from the intensity matrix.
2
Compute the corresponding vertical configurations in linear time.
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Conclusion
Radiotherapy is an interesting application field for algorithmics. Minimizing the total setup time is NP-Hard for Dual MLCs. The MOD problem is NP-Hard when minimizing the beam-on time or the total setup time. Approximation algorithm to minimize the beam-on time for rotating MLCs.
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Acknowledgments
Members of the Radiotherapy Group of Institut Bergoni´ e (Bordeaux, France): Especially:
Guy Kantor Christina Zacharatou
Members of the Dpt. of Radiation Oncology (University of Iowa, USA): Especially:
Xiaodong Wu Dongxu Wang Ryan Flynn
Work partially supported by ANR project BIRDS JCJC SIMI 2-2010
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Acknowledgments
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Bibliography
Ahuja, R. and Hamacher, H. (2005). A network flow algorithm to minimize beam-on time for unconstrained multileaf collimator problems in cancer radiation therapy. Networks, 45(1):36–41. Baatar, D., Hamacher, H., Ehrgott, M., and Woeginger, G. (2005). Decomposition of integer matrices and multileaf collimator sequencing. Discrete Applied Mathematics, 152(1-3):6–34. Webb, S. (2010). Does the option to rotate the Elekta Beam Modulator MLC during VMAT IMRT delivery confer advantage? - a study of ’parked gaps’. Physics in Medicine and Biology, 55(11):N303–19. Yao, J. (1997). US Patent 5,591,983: Multiple layer multileaf collimator.
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