Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Coupled Path Planning, Region Optimization, and Applications in Intensity-modulated Radiation Therapy


In this paper, we consider an optimization problem in discrete geometry, called coupled path planning (CPP). Given a finite rectangular grid and a non-negative function f defined on the horizontal axis of the grid, we seek two non-crossing monotone paths in the grid, such that the vertical difference between the two paths approximates f in the best possible way. This problem arises in intensity-modulated radiation therapy (IMRT), where f represents an ideal radiation dose distribution and the two coupled paths represent the motion trajectories (or control sequence) of two opposing metal leaves of a delivery device for controlling the area exposed to the radiation source. By finding an optimal control sequence, the CPP problem aims to deliver precisely a prescribed radiation dose, while minimizing the side-effects on the surrounding normal tissue. We present efficient algorithms for different versions of the CPP problems. Our results are based on several new ideas and geometric observations, and substantially improve the solutions based on standard techniques. Implementation results show that our CPP algorithms run fast and produce better quality clinical treatment plans than the previous methods.

This is a preview of subscription content, log in to check access.


  1. 1.

    Aggarwal, A., Park, J.: Notes on searching in multidimensional monotone arrays. In: Proc. 29th Annual IEEE Symp. on Foundations of Computer Science, pp. 497–512 (1988)

  2. 2.

    Aggarwal, A., Klawe, M.M., Moran, S., Shor, P., Wilber, R.: Geometric applications of a matrix-searching algorithm. Algorithmica 2, 195–208 (1987)

  3. 3.

    Apostolico, A., Atallah, M.J., Larmore, L., McFaddin, H.S.: Efficient parallel algorithms for string editing and related problems. SIAM J. Comput. 19, 968–988 (1990)

  4. 4.

    Bortfeld, T.R., Kahler, D.L., Waldron, T.J., Boyer, A.L.: X-ray field compensation with multileaf collimators. Int. J. Radiat. Oncol. Biol. Phys. 28, 723–730 (1994)

  5. 5.

    Cameron, C.: Sweeping-window arc therapy: an implementation of rotational IMRT with automatic beam-weight calculation. Phys. Med. Biol. 50(18), 4317–4336 (2005)

  6. 6.

    Cao, D., Afghan, M., Ye, J., Wong, T., Shepard, D.: Generalized inverse planning tool for arc-based IMRT delivery. In: The 50th Annual Meeting of the American Association of Physicists in Medicine (AAPM) (2008)

  7. 7.

    Evans, P.M., Hansen, V.N., Swindell, W.: The optimum intensities for multiple static collimator field compensation. Med. Phys. 24(7), 1147–1156 (1997)

  8. 8.

    Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction. Springer, New York (1985)

  9. 9.

    Shepard, D., Cao, D., Afghan, M.: Initial experience with the delivery of volumetric modulated arc therapy. In: The 50th Annual Meeting of the American Association of Physicists in Medicine (AAPM) (2008)

  10. 10.

    Spirou, S.V., Chui, C.S.: Generation of arbitrary intensity profiles by dynamic jaws or multileaf collimators. Med. Phys. 21, 1031–1041 (1994)

  11. 11.

    Tang, G., Earl, M., Luan, S., Naqvi, S., Yu, C.: Converting multiple-arc intensity-modulated arc therapy into a single arc for efficient delivery. In: Proc. The 49th Annual Meeting of American Society of Therapeutic Radiology and Oncology (ASTRO) (2007)

  12. 12.

    Tang, G., Earl, M., Luan, S., Wang, C., Cao, D., Yu, C., Naqvi, S.: Stochastic vs. deterministic kernel based superposition approaches for dose calculation of intensity-modulated arcs. Phys. Med. Biol. 53, 4733–4746 (2008)

  13. 13.

    Tang, G., Earl, M., Luan, S., Wang, C., Chen, D., Yu, C.: Is dose variation crucial for single-arc radiation therapy delivery. In: Proc. The 50th Annual Meeting of American Association of Physicists in Medicine (AAPM) (2008)

  14. 14.

    Wang, C., Luan, S., Tang, G., Chen, D.Z., Earl, M., Yu, C.X.: Arc-modulated radiation therapy (AMRT): a single-arc form of intensity-modulated arc therapy. Phys. Med. Biol. 53, 6291–6303 (2008)

  15. 15.

    Webb, S.: The Physics of Three-Dimensional Radiation Therapy. Institute of Physics Publishing, Bristol (1993)

  16. 16.

    Webb, S.: The Physics of Conformal Radiotherapy—Advances in Technology. Institute of Physics Publishing, Bristol (1997)

  17. 17.

    Yu, C.X., Luan, S., Wang, C., Chen, D.Z., Earl, M.: Single-arc dose painting: an efficient method of precision radiation therapy. Provisional patent, University of Maryland (2006)

Download references

Author information

Correspondence to Chao Wang.

Additional information

This research was supported in part by the National Science Foundation under Grants CCF-0916606, CCF-0515203, and CBET-0755054, and the National Institutes of Health under Grants R01-EB004640-01A2 and R01CA117997.

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

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Chen, D.Z., Luan, S. & Wang, C. Coupled Path Planning, Region Optimization, and Applications in Intensity-modulated Radiation Therapy. Algorithmica 60, 152–174 (2011).

Download citation


  • Intensity-modulated radiation therapy (IMRT)
  • Intensity-modulated arc therapy (IMAT)
  • Single-arc IMRT
  • Arc-modulated radiation therapy (AMRT)
  • Coupled-path planning
  • Dynamic leaf sequencing