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Decomposition of matrices and static multileaf collimators: a survey

In: Optimization in Medicine

Author

Listed:
  • Matthias Ehrgott

    (The University of Auckland)

  • Horst W. Hamacher

    (Technische Universität Kaiserslautern)

  • Marc Nußbaum

    (Technische Universität Kaiserslautern)

Abstract

Summary Multileaf Collimators (MLC) consist of (currently 20-100) pairs of movable metal leaves which are used to block radiation in Intensity Modulated Radiation Therapy (IMRT). The leaves modulate a uniform source of radiation to achieve given intensity profiles. The modulation process is modeled by the decomposition of a given non-negative integer matrix into a non-negative linear combination of matrices with the (strict) consecutive ones property. In this paper we review some results and algorithms which can be used to minimize the time a patient is exposed to radiation (corresponding to the sum of coefficients in the linear combination), the set-up time (corresponding to the number of matrices used in the linear combination), and other objectives which contribute to an improved radiation therapy.

Suggested Citation

  • Matthias Ehrgott & Horst W. Hamacher & Marc Nußbaum, 2008. "Decomposition of matrices and static multileaf collimators: a survey," Springer Optimization and Its Applications, in: Carlos J. S. Alves & Panos M. Pardalos & Luis Nunes Vicente (ed.), Optimization in Medicine, pages 25-46, Springer.
    • Handle: RePEc:spr:spochp:978-0-387-73299-2_2
    DOI: 10.1007/978-0-387-73299-2_2
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    Citations

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    Cited by:

    1. Z. Caner Taşkın & J. Cole Smith & H. Edwin Romeijn & James F. Dempsey, 2010. "Optimal Multileaf Collimator Leaf Sequencing in IMRT Treatment Planning," Operations Research, INFORMS, vol. 58(3), pages 674-690, June.
    2. Andreas T. Ernst & Vicky H. Mak & Luke R. Mason, 2009. "An Exact Method for the Minimum Cardinality Problem in the Treatment Planning of Intensity-Modulated Radiotherapy," INFORMS Journal on Computing, INFORMS, vol. 21(4), pages 562-574, November.
    3. Luke Mason & Vicky Mak-Hau & Andreas Ernst, 2015. "A parallel optimisation approach for the realisation problem in intensity modulated radiotherapy treatment planning," Computational Optimization and Applications, Springer, vol. 60(2), pages 441-477, March.

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