Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content

Advertisement

Springer Nature Link
Log in

Strong valid inequalities for fluence map optimization problem under dose-volume restrictions

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

Fluence map optimization problems are commonly solved in intensity modulated radiation therapy (IMRT) planning. We show that, when subject to dose-volume restrictions, these problems are NP-hard and that the linear programming relaxation of their natural mixed integer programming formulation can be arbitrarily weak. We then derive strong valid inequalities for fluence map optimization problems under dose-volume restrictions using disjunctive programming theory and show that strengthening mixed integer programming formulations with these valid inequalities has significant computational benefits.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203–228.

    Article  Google Scholar 

  • Balas, E. (1979). Disjunctive programming. Annals of Discrete Mathematics, 5, 3–51.

    Article  Google Scholar 

  • Bortfeld, T., Stein, J., & Preiser, K. (1997). Clinically relevant intensity modulation optimization using physical criteria. In D. Leavitt & G. Starkschall (Eds.), XIIth international conference on the use of computers in radiation therapy, XIIth ICCR (pp. 1–4). Madison: Medical Physics Publishing.

    Google Scholar 

  • Censor, Y., Ben-Israel, A., Xiao, Y., & Galvin, J. (2007). On linear infeasibility arising in intensity-modulated radiation therapy inverse planning. Linear Algebra and Its Applications, 428(5–6), 1406–1420.

    Google Scholar 

  • Deasy, J. (1997). Multiple local minima in radiotherapy optimization problems with dose-volume constraints. Medical Physics, 24(7), 1157–1161.

    Article  Google Scholar 

  • Ferris, M. C., Meyer, R. R., & D’Souza, W. (2006). Radiation treatment planning: mixed integer programming formulations and approaches. In Handbook on modelling for discrete optimization (pp. 317–340). Berlin: Springer.

    Google Scholar 

  • Horst, R., & Tuy, H. (1996). Global optimization: deterministic approaches (third edn.). Berlin: Springer.

    Google Scholar 

  • Jeroslow, R. G. (1977). Cutting plane theory: disjunctive methods. Annals of Discrete Mathematics, 1, 293–330.

    Article  Google Scholar 

  • Langer, M., & Leong, J. (1987). Optimization of beam weights under dose-volume restrictions. International Journal of Radiation Oncology Biology Physics, 13(8), 1255–1260.

    Article  Google Scholar 

  • Larsen, N., Mausser, H., & Uryasev, S. (2002). Algorithms for optimization of value-at-risk. In P. Pardalos & V. K. Tsitsiringos (Eds.), Financial engineering, E-commerce and supply chain (pp. 129–157). Norwell: Kluwer Academic.

    Google Scholar 

  • Lee, E. K., Fox, T., & Crocker, I. (2003). Integer programming applied to intensity-modulated radiation therapy treatment planning. Annals of Operations Research, 119(14), 165–181.

    Article  Google Scholar 

  • Mageras, G. S., & Mohan, R. (1993). Application of fast simulated annealing to optimization of conformal radiation treatments. Medical Physics, 20(3), 639–647.

    Article  Google Scholar 

  • Nemhauser, G. L., & Wolsey, L. A. (1999). Integer and combinatorial optimization. New York: Wiley-Interscience.

    Google Scholar 

  • Parker, R. G., & Rardin, R. L. (1988). Discrete optimization. Boston: Academic Press.

    Google Scholar 

  • Preciado-Walters, F. (2003). Optimal external radiation therapy planning for cancer: a mixed integer approach. PhD thesis, Purdue University.

  • Preciado-Walters, F., Rardin, R., Langer, M., & Thai, V. (2004). A coupled column generation, mixed integer approach to optimal planning of intensity modulated radiation therapy for cancer. Mathematical Programming, 101(2), 319–338.

    Article  Google Scholar 

  • Preciado-Walters, F., Langer, M. P., Rardin, R. L., & Thai, V. (2006). Column generation for IMRT cancer therapy optimization with implementable segments. Annals of Operations Research, 148(1), 65–79.

    Article  Google Scholar 

  • Romeijn, H. E., Ahuja, R. K., Dempsey, J. F., Kumar, A., & Li, J. G. (2003). A novel linear programming approach to fluence map optimization for intensity modulated radiation therapy treatment planning. Physics in Medicine and Biology, 48, 3521–3542.

    Article  Google Scholar 

  • Sherali, H. D., & Sen, S. (1985). On generating cutting planes from combinatorial disjunctions. Operations Research, 33(4), 928–933.

    Article  Google Scholar 

  • Spirou, S. V., & Chuia, C. (1998). A gradient inverse planning algorithm with dose-volume constraints. Medical Physics, 25(3), 321–333.

    Article  Google Scholar 

  • Webb, S. (1989). Optimization of conformal radiotherapy dose distributions by simulated annealing. Physics in Medicine and Biology, 34, 1349–1370.

    Article  Google Scholar 

  • Wu, Q., & Mohan, R. (2000). Algorithms and functionality of an intensity modulated radiotherapy optimization system. Medical Physics, 27(4), 701–711.

    Article  Google Scholar 

  • Wu, Q., & Mohan, R. (2002). Multiple local minima in IMRT optimization based on dose-volume criteria. Medical Physics, 29(7), 1524–1527.

    Article  Google Scholar 

  • Wu, X., Zhu, Y., Jianrong, D., & Wang, Z. (2000). Selection and determination of beam weights based on genetic algorithms for conformal radiotherapy treatment planning. Physics in Medicine and Biology, 45, 2547–2558.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IBM Global Business Services, Washington, DC, USA

    Ali T. Tuncel

  2. Smeal College of Business, Penn State University, University Park, PA, USA

    Felisa Preciado

  3. Department of Industrial Engineering, University of Arkansas, Fayetteville, AR, USA

    Ronald L. Rardin

  4. School of Medicine, Indiana University, Indianapolis, IN, USA

    Mark Langer

  5. Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA

    Jean-Philippe P. Richard

Authors
  1. Ali T. Tuncel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Felisa Preciado
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ronald L. Rardin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Mark Langer
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Jean-Philippe P. Richard
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ali T. Tuncel.

Additional information

This work was supported in part by NCI STTR grant number 1 R01 CA12345-01.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tuncel, A.T., Preciado, F., Rardin, R.L. et al. Strong valid inequalities for fluence map optimization problem under dose-volume restrictions. Ann Oper Res 196, 819–840 (2012). https://doi.org/10.1007/s10479-010-0759-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-010-0759-1

Keywords

Search

Navigation