Abstract
Radiation therapy is one of the commonly used cancer therapies. The radiation treatment poses a tuning problem: it needs to be effective enough to destroy the tumor, but it should maintain the functionality of the organs close to the tumor. Towards this goal the design of a radiation treatment has to be customized for each patient. Part of this design are intensity matrices that define the radiation dosage in a discretization of the beam head. To minimize the treatment time of a patient the beam-on time and the setup time need to be minimized. For a given row of the intensity matrix, the minimum beam-on time is equivalent to the minimum number of binary vectors with the consecutive “1”s property that sum to this row, and the minimum setup time is equivalent to the minimum number of distinct vectors in a set of binary vectors with the consecutive “1”s property that sum to this row. We give a simple linear time algorithm to compute the minimum beam-on time. We prove that the minimum setup time problem is APX-hard and give approximation algorithms for it using a duality property. For the general case, we give a \(\frac {24}{13}\) approximation algorithm. For unimodal rows, we give a \(\frac 97\) approximation algorithm. We also consider other variants for which better approximation ratios exist.
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References
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 44, 36–41 (2005)
Albers, S., Arora, S., Khanna, S.: Page replacement for general caching problems. In: Proc. 10th ACM-SIAM Symp. on Discrete Algorithms, pp. 31–40 (1999)
Bar-Noy, A., Bar-Yehuda, R., Freund, A., Naor, J., Schieber, B.: A unified approach to approximating resource allocation and scheduling. Journal of the ACM 48, 1069–1090 (2001)
Bernstein, S.: Theory of Probability. Moscow (1927)
Boyer, A.L.: Use of MLC for intensity modulation. Medical Physics 21, 1007 (1994)
Bortfeld, T.R., Kahler, D.L., Waldron, T.J., Boyer, A.L.: X-ray field compensation with multileaf collimators. Int. Journal of radiation Oncology, Biology, Physics 28, 723–730 (1994)
Boland, N.H., Hamacher, H.W., Lenzen, F.: Minimizing beam-on time in cancer radiation therapy using multileaf collimators. Networks 43, 226–240 (2004)
Chen, D.Z., Hu, X.S., Luan, S., Wang, C., Wu, X.: Mountain reduction, block matching, and applications in intensity modulation radiation therapy. In: Proc. 21st ACM Symp. on Computational Geometry, pp. 35–44 (2005)
Dai, J., Zhu, Y.: Minimizing the number of segments in a delivery sequence for intensity modulated radiation therapy with multileaf collimator. Medical Physics 28, 2113–2120 (2001)
Gabow, H.N., Bentley, J.L., Tarjan, R.E.: Scaling and related techniques for geometry problems. In: Proc. 16th ACM Symp. on Theory of Computing, pp. 135–143 (1984)
Hurkens, C.A.J., Schrijver, A.: On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems. SIAM Journal on Discrete Mathematics 2, 68–72 (1989)
Kamath, S., Sahni, S., Palta, J., Ranka, S.: Algorithms for optimal sequencing of dynamic multileaf collimators. Physics in Medicine and Biology 49, 33–54 (2004)
Kamath, S., Sahni, S., Palta, J., Ranka, S., Li, J.: Optimal leaf sequencing with elimination of tongue-and-groove underdosage. Physics in Medicine and Biology 49, N7–N19 (2004)
Petrank, E.: The hardness of approximation: gap location. Computational Complexity 4, 133–157 (1994)
Vuillemin, J.: A Unifying Look at Data Structures. Comm. ACM 23, 229–239 (1980)
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Bansal, N., Coppersmith, D., Schieber, B. (2006). Minimizing Setup and Beam-On Times in Radiation Therapy. In: Díaz, J., Jansen, K., Rolim, J.D.P., Zwick, U. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2006 2006. Lecture Notes in Computer Science, vol 4110. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11830924_5
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DOI: https://doi.org/10.1007/11830924_5
Publisher Name: Springer, Berlin, Heidelberg
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