Mountain reduction, block matching, and applications in intensity-modulated radiation therapy
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
The intensity level reduction in radiation therapy
Proceedings of the 2005 ACM symposium on Applied computing
Decomposition of integer matrices and multileaf collimator sequencing
Discrete Applied Mathematics
A new algorithm for optimal multileaf collimator field segmentation
Discrete Applied Mathematics
Discrete Applied Mathematics
Approximation algorithms for minimizing segments in radiation therapy
Information Processing Letters
Efficient intensity map splitting algorithms for intensity-modulated radiation therapy
Information Processing Letters
Optimal Field Splitting, with Applications in Intensity-Modulated Radiation Therapy
FAW '08 Proceedings of the 2nd annual international workshop on Frontiers in Algorithmics
ESA '08 Proceedings of the 16th annual European symposium on Algorithms
Computers and Operations Research
Discrete approximations to real-valued leaf sequencing problems in radiation therapy
Discrete Applied Mathematics
Optimal matrix-segmentation by rectangles
Discrete Applied Mathematics
Decomposition of integer matrices and multileaf collimator sequencing
Discrete Applied Mathematics
A new algorithm for optimal multileaf collimator field segmentation
Discrete Applied Mathematics
Discrete Applied Mathematics
Optimal Multileaf Collimator Leaf Sequencing in IMRT Treatment Planning
Operations Research
Algorithmics in intensity-modulated radiation therapy
Algorithms and theory of computation handbook
A note on improving the performance of approximation algorithms for radiation therapy
Information Processing Letters
A closest vector problem arising in radiation therapy planning
Journal of Combinatorial Optimization
A shortest path-based approach to the multileaf collimator sequencing problem
Discrete Applied Mathematics
Efficient algorithms for intensity map splitting problems in radiation therapy
COCOON'05 Proceedings of the 11th annual international conference on Computing and Combinatorics
The matrix orthogonal decomposition problem in intensity-modulated radiation therapy
COCOON'06 Proceedings of the 12th annual international conference on Computing and Combinatorics
Minimizing setup and beam-on times in radiation therapy
APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
Realization of intensity modulated radiation fields using multileaf collimators
General Theory of Information Transfer and Combinatorics
Minimum decomposition into convex binary matrices
Discrete Applied Mathematics
Field splitting problems in intensity-modulated radiation therapy
ISAAC'06 Proceedings of the 17th international conference on Algorithms and Computation
Hybrid methods for the multileaf collimator sequencing problem
CPAIOR'10 Proceedings of the 7th international conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems
Iterative variable aggregation and disaggregation in IP: An application
Operations Research Letters
How to decompose a binary matrix into three hv-convex polyominoes
DGCI'13 Proceedings of the 17th IAPR international conference on Discrete Geometry for Computer Imagery
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In this article the modulation of intensity matrices arising in cancer radiation therapy using multileaf collimators (MLC) is investigated. It is shown that the problem is equivalent to decomposing a given integer matrix into a positive linear combination of (0, 1) matrices. These matrices, called shape matrices, must have the strict consecutive-1-property, together with another property derived from the technological restrictions of the MLC equipment. Various decompositions can be evaluated by their beam-on time (time during which radiation is applied to the patient) or the treatment time (beam-on time plus time for setups). We focus on the former, and develop a nonlinear mixed-integer programming formulation of the problem. This formulation can be decomposed to yield a column generation formulation: a linear program with a large number of variables that can be priced by solving a subproblem. We then develop a network model in which paths in the network correspond to feasible shape matrices. As a consequence, we deduce that the column generation subproblem can be solved as a shortest path problem. Furthermore, we are able to develop two alternative models of the problem as side-constrained network flow formulations, and so obtain our main theoretical result that the problem is solvable in polynomial time. Finally, a numerical comparison of our exact solutions with those of well-known heuristic methods shows that the beam-on time can be reduced by a considerable margin. © 2004 Wiley Periodicals, Inc.