Field Splitting Problems in Intensity-Modulated Radiation Therapy

  • Danny Z. Chen
  • Chao Wang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4288)


Intensity-modulated radiation therapy (IMRT) is a modern cancer treatment technique that delivers prescribed radiation dose distributions, called intensity maps (IMs), to target tumors via the help of a device called the multileaf collimator (MLC). Due to the maximum leaf spread constraint of the MLCs, IMs whose widths exceed a given threshold cannot be delivered as a whole, and thus must be split into multiple subfields. Field splitting problems in IMRTnormally aim to minimize the total beam-on time (i.e., the total time when a patient is exposed to actual radiation during the delivery) of the resulting subfields. In this paper, we present efficient polynomial time algorithms for two general field splitting problems with guaranteed output optimality. Our algorithms are based on interesting observations and analysis, as well as new techniques and modelings. We formulate the first field splitting problem as a special integer linear programming (ILP) problem that can be solved optimally by linear programming due to its geometry; from an optimal integer solution for the ILP, we compute an optimal field splitting by solving a set of shortest path problems on graphs. We tackle the second field splitting problem by using a novel path-sweeping technique on IMs.


Direct Acyclic Graph Integer Linear Programming Short Path Problem Multileaf Collimator Short Path 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ahuja, R.K., Hamacher, H.W.: A Network Flow Algorithm to Minimize Beam-on Time for Unconstrained Multileaf Collimator Problems in Cancer Radiation Therapy. Networks 45, 36–41 (2005)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Boland, N., Hamacher, H.W., Lenzen, F.: Minimizing Beam-on Time in Cancer Radiation Treatment Using Multileaf Collimators. Networks 43(4), 226–240 (2004)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bortfeld, T.R., Boyer, A.L., Schlegel, W., Kahler, D.L., Waldron, T.L.: Realization and Verification of Three-Dimensional Conformal Radiotherapy with Modulated Fields. Int. J. Radiat. Oncol. Biol. Phys. 30, 899–908 (1994)Google Scholar
  4. 4.
    Chen, D.Z., Hu, X.S., Luan, S., Naqvi, S.A., Wang, C., Yu, C.: Generalized Geometric Approaches for Leaf Sequencing Problems in Radiation Therapy. Int. Journal of Computational Geometry and Applications 16(2-3), 175–204 (2006)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chen, D.Z., Hu, X.S., Luan, S., Wang, C., Wu, X.: Geometric Algorithms for Static Leaf Sequencing Problems in Radiation Therapy. In: Proc. of 19th ACM Symposium on Computational Geometry, pp. 88–97 (2003)Google Scholar
  6. 6.
    Chen, D.Z., Hu, X.S., Luan, S., Wang, C., Wu, X.: Mountain Reduction, Block Matching, and Applications in Intensity-Modulated Radiation Therapy. In: Proc. of 21th ACM Symposium on Computational Geometry, pp. 35–44 (2005)Google Scholar
  7. 7.
    Engel, K.: A New Algorithm for Optimal Multileaf Collimator Field Segmentation. Discrete Applied Mathematics 152(1-3), 35–51 (2005)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hong, L., Kaled, A., Chui, C., Losasso, T., Hunt, M., Spirou, S., Yang, J., Amols, H., Ling, C., Fuks, Z., Leibel, S.: IMRT of Large Fields: Whole-Abdomen Irradiation. Int. J. Radiat. Oncol. Biol. Phys. 54, 278–289 (2002)CrossRefGoogle Scholar
  9. 9.
    Kamath, S., Sahni, S., Li, J., Palta, J., Ranka, S.: A Generalized Field Splitting Algorithm for Optimal IMRT Delivery Efficiency. In: The 47th Annual Meeting and Technical Exhibition of the American Association of Physicists in Medicine (AAPM) (2005), Also Med. Phys. 32(6), 1890 (2005) Google Scholar
  10. 10.
    Kamath, S., Sahni, S., Palta, J., Ranka, S.: Algorithms for Optimal Sequencing of Dynamic Multileaf Collimators. Phys. Med. Biol. 49(1), 33–54 (2004)CrossRefGoogle Scholar
  11. 11.
    Kamath, S., Sahni, S., Ranka, S., Li, J., Palta, J.: Optimal Field Splitting for Large Intensity-Modulated Fields. Med. Phys. 31(12), 3314–3323 (2004)CrossRefGoogle Scholar
  12. 12.
    Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. John Wiley, Chichester (1988)MATHGoogle Scholar
  13. 13.
    Papadimitriou, C.H., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall, New Jersey (1982)MATHGoogle Scholar
  14. 14.
    Siochi, R.A.C.: Minimizing Static Intensity Modulation Delivery Time Using an Intensity Solid Paradigm. Int J. Radiation Oncology Biol. Phys. 43(3), 671–680 (1999)CrossRefGoogle Scholar
  15. 15.
    Webb, S.: The Physics of Three-Dimensional Radiation Therapy. Institute of Physics Publishing, Bristol (1993)CrossRefGoogle Scholar
  16. 16.
    Webb, S.: The Physics of Conformal Radiotherapy — Advances in Technology. Institute of Physics Publishing, Bristol (1997)CrossRefGoogle Scholar
  17. 17.
    Wu, Q., Arnfield, M., Tong, S., Wu, Y., Mohan, R.: Dynamic Splitting of Large Intensity-Modulated Fields. Phys. Med. Biol. 45, 1731–1740 (2000)CrossRefGoogle Scholar
  18. 18.
    Wu, X.: Efficient Algorithms for Intensity Map Splitting Problems in Radiation Therapy. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 504–513. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  19. 19.
    Xia, P., Verhey, L.J.: MLC Leaf Sequencing Algorithm for Intensity Modulated Beams with Multiple Static Segments. Med. Phys. 25, 1424–1434 (1998)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Danny Z. Chen
    • 1
  • Chao Wang
    • 1
  1. 1.Department of Computer Science and EngineeringUniversity of Notre DameNotre DameUSA

Personalised recommendations