Generalized Geometric Approaches for Leaf Sequencing Problems in Radiation Therapy

Chen, Danny Z.; Hu, Xiaobo Sharon; Luan, Shuang; Naqvi, S; Wang, Chao; Yu, C

doi:10.1142/s0218195906001999

Cited by 16 publications

(33 citation statements)

References 28 publications

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“…(A constant window function W (·) is defined on an interval I such that W (x) is a fixed value h > 0 for any x ∈ I and is 0 otherwise.) Since the shape rectangularization problems are NP-hard [7,8] (in fact, APX-hard [1]), approximation algorithms were given, and some special cases were solved optimally [1,7]. Note that constant window functions are a more restricted form of unimodal functions.…”

Section: Motivations and Related Workmentioning

confidence: 99%

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Representing a Functional Curve by Curves with Fewer Peaks

Chen

Wang

2011

Discrete Comput Geom

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In this paper, we study the problems of (approximately) representing a functional curve in 2-D by a set of curves with fewer peaks. Representing a function (or its curve) by certain classes of structurally simpler functions (or their curves) is a basic mathematical problem. Problems of this kind also find applications in applied areas such as intensity-modulated radiation therapy (IMRT). Let f be an input piecewise linear functional curve of size n. We consider several variations of the problems.(1) Uphill-downhill pair representation (UDPR): Find two nonnegative piecewise linear curves, one nondecreasing (uphill) and one nonincreasing (downhill), such that their sum exactly or approximately represents f. (2) Unimodal representation (UR): Find a set of unimodal (single-peak) curves such that their sum exactly or approximately represents f. (3) Fewer-peak representation (FPR): Find a piecewise linear curve with at most k peaks that exactly or approximately represents f. Furthermore, for each problem, we consider two versions. For the UDPR problem, we study its feasibility version: Given > 0, determine whether there is a feasible UDPR solution for f with an approximation error ; its min-version: Compute the minimum approximation error * such that there is a feasible UDPR solution for f with error * . For the UR problem, we study its min-k version: Given > 0, find a feasible solution Discrete Comput Geom (2011) 46:334-360 335 with the minimum number k * of unimodal curves for f with an error ; its min-version: given k > 0, compute the minimum error * such that there is a feasible solution with at most k unimodal curves for f with error * . For the FPR problem, we study its min-k version: Given > 0, find one feasible curve with the minimum number k * of peaks for f with an error ; its min-version: given k ≥ 0, compute the minimum error * such that there is a feasible curve with at most k peaks for f with error * . Little work has been done previously on solving these functional curve representation problems. We solve all the problems (except the UR min-version) in optimal O(n) time, and the UR min-version in O(n + m log m) time, where m < n is the number of peaks of f. Our algorithms are based on new geometric observations and interesting techniques.

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Section: Motivations and Related Workmentioning

confidence: 99%

“…It is also interesting to note that while the unimodal representation problems are nearly linear time solvable, in contrast, the shape rectangularization problems, which can be viewed as a more restricted case of the unimodal representation problems, are NP-hard [1,7,8].…”

Section: Our Contributionsmentioning

confidence: 99%

Representing a Functional Curve by Curves with Fewer Peaks

Chen

Wang

2011

Discrete Comput Geom

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“…They also discuss further improvements of the formulation through cuts and bounds. Chen et al (2004aChen et al ( ,b, 2005Chen et al ( , 2006 consider the decomposition cardinality problem with interleaf motion, width, and tongue-and-groove constraints. The first two groups of constraints are considered by a geometric argumentation.…”

Section: Theorem 16mentioning

confidence: 99%

Mathematical optimization in intensity modulated radiation therapy

Ehrgott

Güler

Hamacher

et al. 2008

4OR

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The design of an intensity modulated radiotherapy treatment includes the selection of beam angles (geometry problem), the computation of an intensity map for each selected beam angle (intensity problem), and finding a sequence of configurations of a multileaf collimator to deliver the treatment (realization problem). Until the end of the last century research on radiotherapy treatment design has been published almost exclusively in the medical physics literature. However, since then, the attention of researchers in mathematical optimization has been drawn to the area and important progress has been made. In this paper we survey the use of optimization models, methods, and theories in intensity modulated radiotherapy treatment design.H.W. Hamacher and Ç.

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“…Baatar et al [9] showed that NP-hardness holds for input IMs even with a single row, using a reduction from the 3-partition problem [20]. Chen et al [16], independently, gave an NP-hardness proof of this problem based on a reduction from the 0-1 knapsack problem [20]. The GSR problem, as a generalization of the SR problem, is clearly NP-hard.…”

Section: Previous and Related Workmentioning

confidence: 99%

Shape Rectangularization Problems in Intensity-Modulated Radiation Therapy

et al. 2009

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In this paper, we present a theoretical study of several shape approximation problems, called shape rectangularization (SR), which arise in intensity-modulated Partial preliminary versions of this work appeared in Proc. of the 9th Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX) and in Proc. of the 17th International Symposium on Algorithms and Computation (ISAAC). 422Algorithmica (2011) 60: 421-450 radiation therapy (IMRT). Given a piecewise linear function f such that f (x) ≥ 0 for any x ∈ R, the SR problems seek an optimal set of constant window functions to approximate f under a certain error criterion, such that the sum of the resulting constant window functions equals (or well approximates) f . A constant window function W (·) is defined on an interval I such that W (x) is a fixed value h > 0 for any x ∈ I and is 0 otherwise. A constant window function can be viewed as a rectangle (or a block) geometrically, or as a vector with the consecutive a's property combinatorially. The SR problems find applications in setup time and beam-on time minimization and dose simplification of the IMRT treatment planning process. We show that the SR problems are APX-Hard, and thus we aim to develop theoretically efficient and provably good quality approximation SR algorithms. Our main contribution is to present algorithms for a key SR problem that achieve approximation ratios better than 2. For the general case, we give a 24 13 -approximation algorithm. For unimodal input curves, we give a 9 7 -approximation algorithm. We also consider other variants for which better approximation ratios are possible. We show that an important SR case that has been studied in medical literature can be formulated as a k-MST (k-minimum-spanning-tree) problem on a certain geometric graph G; based on a set of geometric observations and a non-trivial dynamic programming scheme, we are able to compute an optimal k-MST in G efficiently.

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Generalized Geometric Approaches for Leaf Sequencing Problems in Radiation Therapy

Cited by 16 publications

References 28 publications

Representing a Functional Curve by Curves with Fewer Peaks

Representing a Functional Curve by Curves with Fewer Peaks

Mathematical optimization in intensity modulated radiation therapy

Shape Rectangularization Problems in Intensity-Modulated Radiation Therapy

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