Approximated matrix decomposition for IMRT planning with multileaf collimators. (English) Zbl 1231.90399
Summary: A standard problem in intensity modulated radiation therapy is the representation of a given intensity matrix, i.e. a matrix of nonnegative integers, as a nonnegative linear combination of special 0-1-matrices, called segments. These segments can be practically realized by multileaf collimators. One important aim is the minimization of the sum of the coefficients of the linear combination, i.e. the delivery time. In this article, we study the question how much the delivery time can be reduced if some small deviation from the given intensity matrix is allowed. We characterize the optimal solutions for one-row matrices and show that the approximation can be carried out in an iterative way. The structural characterization yields a fast algorithm that minimizes the delivery time and then also the deviation. Moreover, algorithms for the general case together with numerical results are presented.
MSC:
| 90C90 | Applications of mathematical programming |
| 92C50 | Medical applications (general) |
| 15B48 | Positive matrices and their generalizations; cones of matrices |
| 90B50 | Management decision making, including multiple objectives |
References:
| [1] | Ahuja RK, Hamacher HW (2005) A network flow algorithm to minimize beam-on time for unconstrained multileaf collimator problems in cancer radiation therapy. Networks 45(1): 36–41 · Zbl 1061.92033 · doi:10.1002/net.20047 |
| [2] | Baatar D, Hamacher HW, Ehrgott M, Woeginger GJ (2005) Decomposition of integer matrices and multileaf collimator sequencing. Discrete Appl Math 152(1–3): 6–34 · Zbl 1084.15014 · doi:10.1016/j.dam.2005.04.008 |
| [3] | Boland N, Hamacher HW, Lenzen F (2004) Minimizing beam-on time in cancer radiation treatment using multileaf collimators. Networks 43(4): 226–240 · Zbl 1044.92030 · doi:10.1002/net.20007 |
| [4] | Engel K (2005) A new algorithm for optimal multileaf collimator field segmentation. Discrete Appl Math 152(1-3): 35–51 · Zbl 1076.92026 · doi:10.1016/j.dam.2004.10.007 |
| [5] | Engel K, Tabbert E (2005) Fast simultaneous angle, wedge, and beam intensity optimization in inverse radiotherapy planning. Optim Eng 6(4): 393–419 · Zbl 1168.92312 · doi:10.1007/s11081-005-2065-3 |
| [6] | Kalinowski T (2006) Realization of intensity modulated radiation fields using multileaf collimators. In Ahlswede R et al (eds) Information transfer and combinatorics. LNCS, vol 4123. Springer, Berlin, pp 1010–1055 · Zbl 1158.92314 |
| [7] | Kalinowski T (2008) Multileaf collimator shape matrix decomposition. In: Lim GJ(eds) Optimization in Medicine and Biology. Auerbach Publishers Inc., Pennsauken, pp 253–286 |
| [8] | Kalinowski T (2009) The algorithmic complexity of the minimization of the number of segments in multileaf collimator field segmentation. Discrete Appl Math (in press) · Zbl 1190.68030 |
| [9] | Kamath S, Sartaj S, Palta J, Ranka S, Li J (2004) Optimal leaf sequencing with elimination of tongue-and-groove underdosage. Phys Med Biol 49: N7–N19 · doi:10.1088/0031-9155/49/3/N01 |
| [10] | Lim J, Ferris MC, Wright SJ, Shepard DM, Earl MA (2007) An optimization framework for conformal radiation treatment planning. INFORMS J Comput 19(3): 366–380 · Zbl 1241.90199 · doi:10.1287/ijoc.1060.0179 |
| [11] | Xia P, Verhey L (1998) Multileaf collimator leaf-sequencing algorithm for intensity modulated beams with multiple static segments. Med Phys 25: 1424–1434 · doi:10.1118/1.598315 |
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