On linear infeasibility arising in intensity-modulated radiation therapy inverse planning

Yair Censor, Adi Ben-Israel, Ying Xiao, James M. Galvin

Research output: Contribution to journalArticlepeer-review


Intensity-modulated radiation therapy (IMRT) gives rise to systems of linear inequalities, representing the effects of radiation on the irradiated body. These systems are often infeasible, in which case one settles for an approximate solution, such as an {α, β}-relaxation, meaning that no more than α percent of the inequalities are violated by no more than β percent. For real-world IMRT problems, there is a feasible {α, β}-relaxation for sufficiently large α, β > 0, however large values of these parameters may be unacceptable medically. The {α, β}-relaxation problem is combinatorial, and for given values of the parameters can be solved exactly by Mixed Integer Programming (MIP), but this may be impractical because of problem size, and the need for repeated solutions as the treatment progresses. As a practical alternative to the MIP approach we present a heuristic non-combinatorial method for finding an approximate relaxation. The method solves a Linear Program for each pair of values of the parameters {α, β} and progresses through successively increasing values until an acceptable solution is found, or is determined non-existent. The method is fast and reliable, since it consists of solving a sequence of linear progrms.

Original languageEnglish
Pages (from-to)1406-1420
Number of pages15
JournalLinear Algebra and Its Applications
Issue number5-6
StatePublished - 1 Mar 2008

Bibliographical note

Funding Information:
We thank two anonymous reviewers for their constructive reports which helped us to improve the presentation. We thank Arik F. Hatwell for his skillful programming and computational work. We thank Thomas Bortfeld for many fruitful discussions and Wei Chen for his comments on an earlier draft. The work of Y. Censor was supported by Grant No. 2003275 of the United States-Israel Binational Science Foundation (BSF), by a National Institutes of Health (NIH) Grant No. HL70472 and by Grant No. 522/04 of the Israel Science Foundation (ISF) at the Center for Computational Mathematics and Scientific Computation (CCMSC) in the University of Haifa. Preliminary results of this study were presented at “The Interdisciplinary Experts’ Workshop on Intensity-Modulated Radiation Therapy (IMRT), Medical Imaging, and Optimization Theory”, June 6–11, 2004, that took place at the University of Haifa, Haifa, Israel. We thank Eva Lee, Ron Rardin and Mark Langer for their useful comments at that workshop.

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Geometry and Topology
  • Numerical Analysis
  • Algebra and Number Theory


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