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Incorporation of delivery times in stereotactic radiosurgery treatment optimization

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Abstract

Although the use of mathematical optimization techniques can greatly improve the quality of treatment plans in various radiation therapy treatment settings, one complication is the potentially clinically unrealistic nature of optimized treatments. The difficulty arises from two factors: (1) machine limitations that govern the minimum amount of radiation delivery time, and (2) long treatment times due to the complexity of optimized treatments. In the first scenario, if a particular configuration of the radiation delivery device is used, then it typically must deliver radiation for a minimum length of time. Incorporation of such requirements in a mathematical model generally requires additional constraints and binary variables, increasing the difficulty of the optimization. In the second scenario, mathematically optimized treatments commonly assign (small amounts of) radiation to be delivered from many configurations, drastically increasing the time needed to deliver the treatment (beam-on time). We examine these two issues within the penalty-based sector duration optimization model for Leksell Gamma Knife\(^{\textregistered }\) Perfexion\(^{\mathrm{TM}}\) (Elekta, Stockholm, Sweden) and the combined sector duration and isocentre optimization model to reduce beam-on time and to ensure that machine limitations regarding delivery times are met.

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Notes

  1. The general case of the inequality is \(\frac{1}{\sqrt{m}}\Vert A\Vert _{1}\le \sqrt{mn}\Vert A\Vert _\mathrm{max}\), where A is an \(m\times n\) matrix. In this case, \(A = t_I\), a matrix of size \(3 \times 8\) (\(m=3\) collimator sizes and \(n=8\) sectors for isocentre I), yielding \(\sqrt{m} \times \sqrt{mn} = m\sqrt{n} = 3 \sqrt{8} = 6 \sqrt{2}\).

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Acknowledgements

Funding was provided by Ontario Research Fund (Grant No. RE-04-026).

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Authors and Affiliations

  1. Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, ON, M5S 3G8, Canada

    Hamid R. Ghaffari & Dionne M. Aleman

  2. Radiation Medicine Program, Princess Margaret Cancer Centre, 150 College Street, Toronto, ON, M5S 3E2, Canada

    David A. Jaffray

  3. Odette Cancer Centre, Sunnybrook Health Sciences Centre, Toronto, ON, M4N 3M5, Canada

    Mark Ruschin

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  1. Hamid R. Ghaffari
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  2. Dionne M. Aleman
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  3. David A. Jaffray
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  4. Mark Ruschin
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Correspondence to Dionne M. Aleman.

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Ghaffari, H.R., Aleman, D.M., Jaffray, D.A. et al. Incorporation of delivery times in stereotactic radiosurgery treatment optimization. J Glob Optim 69, 103–115 (2017). https://doi.org/10.1007/s10898-017-0527-8

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