Abstract
This paper deals with numerical behavior of a recently presented column generation approach for optimization of so called step-and-shoot radiotherapy treatment plans. The approach and variants of it have been reported to be efficient in practice, finding near-optimal solutions by generating only a low number of columns.
The impact of different restrictions on the columns in a column generation method is studied, and numerical results are given for quadratic programs corresponding to three patient cases. In particular, it is noted that with a bound on the two-norm of the columns, the method is equivalent to the conjugate-gradient method. Further, the above-mentioned column generation approach for radiotherapy is obtained by employing a restriction based on the infinity-norm and non-negativity.
The column generation method has weak convergence properties if restricted to generating feasible step-and-shoot plans, with a “tailing-off” effect for the objective values. However, the numerical results demonstrate that, like the conjugate-gradient method, a rapid decrease of the objective value is obtained in the first few iterations. For the three patient cases, the restriction on the columns to generate feasible step-and-shoot plans has small effect on the numerical efficiency.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahnesjö, A., Hårdemark, B., Isacsson, U., & Montelius, A. (2006). The IMRT information process—mastering the degrees of freedom in external beam therapy. Physics in Medicine and Biology, 51(13), R381–R402.
Alber, M., Meedt, G., Nüsslin, F., & Reemtsen, R. (2002). On the degeneracy of the IMRT optimization problem. Medical Physics, 29(11), 2584–2589.
Boland, N., Hamacher, H. W., & Lenzen, F. (2004). Minimizing beam-on time in cancer radiation treatment using multileaf collimators. Networks, 43(4), 226–240.
Carlsson, F. (2008). Combining segment generation with direct step-and-shoot optimization in intensity-modulated radiation therapy. Medical Physics, 35(9), 3828–3838.
Carlsson, F., & Forsgren, A. (2006). Iterative regularization in intensity-modulated radiation therapy optimization. Medical Physics, 33(1), 225–234.
Chvetsov, A. V., Calvetti, D., Sohn, J. W., & Kinsella, T. J. (2005). Regularization of inverse planning for intensity-modulated radiotherapy. Medical Physics, 32(2), 501–514.
Engl, H., Hanke, M., & Neubauer, A. (2000). Regularization of inverse problems. Dordrecht: Kluwer Academic.
Galvin, J. (1999). The multileaf collimator: a complete guide. In Proc. AAPM annual meeting.
Gersem, W. D., Claus, F., Wagter, C. D., Duyse, B. V., & Neve, W. D. (2001). Leaf position optimization for step-and-shoot IMRT. International Journal of Radiation Oncology, Biology, Physics, 51(5), 1371–1388.
Gondzio, J., & Sarkissian, R. (1996). Column generation with a primal-dual method (Technical Report, Logilab Technical Report 96.6). Department of Management Studies, University of Geneva, Switzerland.
Gustafsson, A., Lind, B. K., & Brahme, A. (1994). A generalized pencil beam algorithm for optimization of radiation therapy. Medical Physics, 21(3), 343–356.
Karush, W. (1939). Minima of functions of several variables with inequalities as side constraints. Master’s thesis, Department of Mathematics, University of Chicago.
Kuhn, H. W., & Tucker, A. W. (1951). Nonlinear programming. In J. Neyman (Ed.), Proceedings of the second Berkeley symposium on mathematical statistics and probability (pp. 481–492). Berkeley: University of California Press.
Lübbecke, M. E., & Desrosiers, J. (2005). Selected topics in column generation. Operations Research, 53(6), 1007–1023.
Men, C., Romeijn, H. E., Taskin, Z. C., & Dempsey, J. F. (2007). An exact approach to direct aperture optimization in IMRT treatment planning. Physics in Medicine and Biology, 52(24), 7333–7352.
Romeijn, H. E., Ahuja, R. K., Dempsey, J. F., & Kumar, A. (2005). A column generation approach to radiation therapy treatment planning using aperture modulation. SIAM Journal on Optimization, 15(3), 838–862.
Saad, Y. (2003). Iterative methods for sparse linear systems. Philadelphia: Society for Industrial and Applied Mathematics.
Shepard, D. M., Earl, M. A., Li, X. A., Naqvi, S., & Yu, C. (2002). Direct aperture optimization: A turnkey solution for step-and-shoot IMRT. Medical Physics, 29(6), 1007–1018.
Tervo, J., Kolmonen, P., Lyyra-Laitinen, T., Pintér, J., & Lahtinen, T. (2003). An optimization-based approach to the multiple static delivery technique in radiation therapy. Annals of Operations Research, 119, 205–227.
Webb, S. (2003). The physical basis of IMRT and inverse planning. The British Journal of Radiology, 76(910), 678–689.
Acknowledgements
We thank Göran Sporre and Henrik Rehbinder for their careful reading of the manuscript and for a number of helpful comments. We also thank the anonymous referees for many constructive comments that significantly improved the presentation. The research was supported by the Swedish Research Council (VR).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Carlsson, F., Forsgren, A. On column generation approaches for approximate solutions of quadratic programs in intensity-modulated radiation therapy. Ann Oper Res 223, 471–481 (2014). https://doi.org/10.1007/s10479-013-1360-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-013-1360-1