Advertisement

Annals of Operations Research

, Volume 119, Issue 1–4, pp 165–181 | Cite as

Integer Programming Applied to Intensity-Modulated Radiation Therapy Treatment Planning

  • Eva K. LeeEmail author
  • Tim Fox
  • Ian Crocker
Article

Abstract

In intensity-modulated radiation therapy (IMRT) not only is the shape of the beam controlled, but combinations of open and closed multileaf collimators modulate the intensity as well. In this paper, we offer a mixed integer programming approach which allows optimization over beamlet fluence weights as well as beam and couch angles. Computational strategies, including a constraint and column generator, a specialized set-based branching scheme, a geometric heuristic procedure, and the use of disjunctive cuts, are described. Our algorithmic design thus far has been motivated by clinical cases. Numerical tests on real patient cases reveal that good treatment plans are returned within 30 minutes. The MIP plans consistently provide superior tumor coverage and conformity, as well as dose homogeneity within the tumor region while maintaining a low irradiation to important critical and normal tissues.

intensity-modulated radiation therapy external beam radiotherapy optimization mixed integer programming treatment planning 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. Ahnesj, Collapsed cone convolution of radiant energy for photon dose calculation in heterogeneous media, Medical Physics 16 (1989) 577–592.Google Scholar
  2. [2]
    G. Bahr, J. Kereiakes, H. Horwitz, R. Finney, J. Galvin and K. Goode, The method of linear programming applied to radiation treatment planning, Radiology 91 (1968) 686–693.Google Scholar
  3. [3]
    E. Balas, Disjunctive programming: cutting planes from logical conditions, in: Nonlinear Programming, Vol. 2, eds. O.L. Mangasarian et al. (Academic Press, New York, 1975) pp. 279–312.Google Scholar
  4. [4]
    E. Balas, Disjunctive programming, Annals of Discrete Mathematics 5 (1979) 3–51.Google Scholar
  5. [5]
    E. Balas, S. Ceria and G. Cornuéjols, A lift-and-project cutting plane algorithm for mixed 0/1 programs, Mathematical Programming 58 (1993) 295–324.Google Scholar
  6. [6]
    E. Balas, S. Ceria and G. Cornuéjols, Mixed 0–1 programming by lift-and-project in a branch-and-cut framework, Management Science 42 (1996) 1229–1246.Google Scholar
  7. [7]
    E. Balas, A modified lift-and-project procedure, Mathematical Programming 79 (1997) 19–32.Google Scholar
  8. [8]
    J. Barbiere, M.F. Chan, J. Mechalakos, D. Cann, K. Schupak and C. Burman, A parameter optimization algorithm for intensity-modulated radiotherapy prostate treatment planning, J. Appl. Clin. Medical Physics 3(3) (2002) 227–234.Google Scholar
  9. [9]
    R.E. Bixby, W. Cook, A. Cox and E.K. Lee, Computational experience with parallel mixed integer programming in a distributed environment, Annals of Operations Research 90 (1995) 19–43.Google Scholar
  10. [10]
    T. Bortfeld, K. Jokivarsi, M. Goitein, J. Kung and S.B. Jiang, Effects of intra-fraction motion on IMRT dose delivery: statistical analysis and simulation, Physics in Medicine and Biology 47(13) (2002) 2203–2220.Google Scholar
  11. [11]
    T. Bortfeld, U. Oelfke and S. Nill, What is the optimum leaf width of a multileaf collimator? Medical Physics 27(11) (2000) 2494–2502.Google Scholar
  12. [12]
    ?. Bourland and ?. Chaney, A finite-size pencil beam model for photon dose calculations in three dimensions, Medical Physics 19 (1992) 1401–1412.Google Scholar
  13. [13]
    A.L. Boyer and E.G. Mok, A photon dose distribution model employing convolution calculations, Medical Physics 12 (1985) 169–177.Google Scholar
  14. [14]
    S. Ceria and G. Pataki, Solving integer and disjunctive programs by lift and project, in: Integer Programming and Combinatorial Optimization, Proceedings of the 6th International IPCO Conference, eds. R.E. Bixby et al., Lecture Notes in Computer Science, Vol. 1412 (Springer, Berlin, 1998) pp. 271–283.Google Scholar
  15. [15]
    S.M. Crooks and L. Xing, Linear algebraic methods applied to intensity modulated radiation therapy, Physics in Medicine and Biology 46(10) (2001) 2587–2606.Google Scholar
  16. [16]
    D. Gale, The Theory of Linear Economic Models (McGraw-Hill, New York, 1960).Google Scholar
  17. [17]
    R.E. Gomory, Solving linear programs in integers, in: Combinatorial Analysis, eds. R.E. Bellman and M. Hall, Jr. (American Mathematical Society, Providence, RI, 1960), pp. 211–216.Google Scholar
  18. [18]
    R.E. Gomory, An algorithm for the mixed integer problem, RM-2597, The Rand Corporation, 1960.Google Scholar
  19. [19]
    T. Fox, A dose calculation engine for an intensity-modulated treatment planning system, Working Paper (2002).Google Scholar
  20. [20]
    M. Hartmann, L. Bogner, M. Fippel, J. Scherer and S. Scherer, IMCO(++) – a Monte Carlo based IMRT system, Medical Physics 12(2) (2002) 97–108.Google Scholar
  21. [21]
    M. Hilbig, R. Hanne, P. Kneschaurek, F. Zimmermann and A. Schweikard, Design of an inverse planning system for radiotherapy using linear optimization, Medical Physics 12(2) (2002) 89–96.Google Scholar
  22. [22]
    P.W. Hoban, D.C. Murray and W.H. Round, Photon beam convolution using polyenergetic energy deposition kernels, Physics in Medicine and Biology 39 (1994) 669–685.Google Scholar
  23. [23]
    M. Langer, R. Brown, M. Urie, J. Leong, M. Stracher and J. Shapiro, Large scale optimization of beam weights under dose-volume restrictions, International Journal of Radiation Oncology and Biological Physics 18 (1990) 887–893.Google Scholar
  24. [24]
    E.K. Lee, Computational experience of a general purpose mixed 0/1 integer programming solver (MIPSOL), Technical Report, Georgia Institute of Technology (1997).Google Scholar
  25. [25]
    E.K. Lee, Generating cutting planes for mixed integer programming problems in a parallel computing environment, Technical Report, Georgia Institute of Technology, to appear in INFORMS Journal on Computing (2000).Google Scholar
  26. [26]
    E.K. Lee, T. Fox and I. Crocker, Effects of beam configuration and tumor representation on dosimetry and plan quality, Medical Physics, in review (2002).Google Scholar
  27. [27]
    E.K. Lee, T. Fox and I. Crocker, Sensitivity analysis of clinical objectives to plan qualities in intensity modulated radiation therapy treatment planning optimization, Medical Physics, in review (2002).Google Scholar
  28. [28]
    T.R. Mackie, J.W. Scrimger and J.J. Battista, A convolution method of calculating dose for 15 MV X rays, Medical Physics 12 (1985) 188–196.Google Scholar
  29. [29]
    T.R. Mackie, A.F. Bielajew, D.W.O. Rogers and J.J. Battista, Generation of photon energy deposition kernels using the EGS Monte Carlo code, Physics in Medicine and Biology 33 (1988) 1–20.Google Scholar
  30. [30]
    P.E. Metcalfe, P.W. Hoban, D.C. Murray and W.H. Round, Beam hardening of 10 MV radiotherapy X-rays: analysis using a convolution/superposition method, Physics in Medicine and Biology 35 (1990) 1533–1549.Google Scholar
  31. [31]
    O.L. Mangasarian, Nonlinear Programming (McGraw-Hill, New York, 1959).Google Scholar
  32. [32]
    R. Mohan, C. Chui and L. Lidofsky, Energy and angular distributions of photons from medical linear accelerators, Medical Physics 12 (1985) 592–597.Google Scholar
  33. [33]
    R. Mohan, C. Chui and L. Lidofsky, Differential pencil beam dose computation model for photons, Medical Physics 13 (1986) 64–73.Google Scholar
  34. [34]
    N. Papanikolaou, T.R. Mackie, C. Meger-Wells, M. Gehring and P. Reckwerdt, Investigation of the convolution method for polyenergetic spectra, Medical Physics 20 (1993) 1327–1336.Google Scholar
  35. [35]
    A.B. Pugachev, A.L. Boyer and L. Xing, Beam orientation optimization in intensity-modulated radiation treatment planning, Medical Physics 27(6) (2000) 1238–1245.Google Scholar
  36. [36]
    O.Z. Ostapiak, J. Van Dyk and Y. Zhu, A model for 3D photon dose calculations based on finite size pencil beams generated using FFT convolutions, AAPM Annual Meeting Program, Medical Physics 22 (1995) 976.Google Scholar
  37. [37]
    R.J. Schulz and A.R. Kagan, On the role of intensity-modulated radiation therapy in radiation oncology, Medical Physics 29(7) (2002) 1473–1482.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  1. 1.Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Radiation OncologyEmory University School of MedicineAtlantaUSA
  3. 3.Radiation OncologyEmory University School of MedicineAtlantaUSA

Personalised recommendations