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Shape Rectangularization Problems in Intensity-Modulated Radiation Therapy

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Abstract

In this paper, we present a theoretical study of several shape approximation problems, called shape rectangularization (SR), which arise in intensity-modulated 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 \(\frac{24}{13}\)-approximation algorithm. For unimodal input curves, we give a \(\frac{9}{7}\)-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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Authors and Affiliations

  1. IBM T.J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY, 10598, USA

    Nikhil Bansal & Baruch Schieber

  2. Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN, 46556, USA

    Danny Z. Chen, Xiaobo S. Hu & Chao Wang

  3. IDA Center for Communications Research, 805 Bunn Drive, Princeton, NJ, 08540, USA

    Don Coppersmith

  4. Department of Computer Science, University of New Mexico, Albuquerque, NM, 87131-0001, USA

    Shuang Luan

  5. Mathematics Department, Saint Mary’s College, Notre Dame, IN, 46556, USA

    Ewa Misiołek

Authors
  1. Nikhil Bansal
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  2. Danny Z. Chen
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  3. Don Coppersmith
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  4. Xiaobo S. Hu
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  5. Shuang Luan
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  6. Ewa Misiołek
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  7. Baruch Schieber
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  8. Chao Wang
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Corresponding author

Correspondence to Chao Wang.

Additional information

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).

The research of D.Z. Chen was supported in part by the Faculty Research Program of the University of Notre Dame and the National Science Foundation under Grants CCR-9988468, CCF-0515203 and CCF-0916606.

The work of D. Coppersmith was done while at IBM T.J. Watson Research Center.

The research of S. Luan was supported in part by a faculty start-up fund from the Department of Computer Science, University of New Mexico.

The research of C. Wang was supported in part by two Fellowships in 2004–2006 from the Center for Applied Mathematics of the University of Notre Dame.

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Bansal, N., Chen, D.Z., Coppersmith, D. et al. Shape Rectangularization Problems in Intensity-Modulated Radiation Therapy. Algorithmica 60, 421–450 (2011). https://doi.org/10.1007/s00453-009-9354-8

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