, Volume 16, Issue 2, pp 173–194 | Cite as

CP and IP approaches to cancer radiotherapy delivery optimization

  • Davaatseren Baatar
  • Natashia Boland
  • Sebastian Brand
  • Peter J. Stuckey


We consider the problem of decomposing an integer matrix into a positively weighted sum of binary matrices that have the consecutive-ones property. This problem is well-known and of practical relevance. It has an important application in cancer radiation therapy treatment planning: the sequencing of multileaf collimators to deliver a given radiation intensity matrix, representing (a component of) the treatment plan. Two criteria characterise the efficacy of a decomposition: the beam-on time (the length of time the radiation source is switched on during the treatment), and the cardinality (the number of machine set-ups required to deliver the planned treatment). Minimising the former is known to be easy. However finding a decomposition of minimal cardinality is NP-hard. Progress so far has largely been restricted to heuristic algorithms, mostly using linear programming, integer programming and combinatorial enumerative methods as the solving approaches. We present a novel model, with corresponding constraint programming and integer programming formulations. We compare these computationally with previous formulations, and we show that constraint programming performs very well by comparison.


Modelling Symmetry-breaking Search Integer programming Intensity-modulated radiation therapy Multileaf collimator leaf sequencing 


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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Davaatseren Baatar
    • 1
  • Natashia Boland
    • 2
  • Sebastian Brand
    • 3
  • Peter J. Stuckey
    • 3
  1. 1.Department of MathematicsUniversity of MelbourneMelbourneAustralia
  2. 2.School of Mathematical and Physical SciencesUniversity of NewcastleNewcastleAustralia
  3. 3.NICTA Victoria Research Lab, Department of Computer Science and Software EngineeringUniversity of MelbourneMelbourneAustralia

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