Affinely adjustable robust optimization for radiation therapy under evolving data uncertainty via semi-definite programming

Abstract

Static robust optimization has played an important role in radiotherapy, where the decisions aim to safeguard against all possible realizations of uncertainty. However, it may lead to overly conservative decisions or too expensive treatment plans, such as delivering significantly more dose than necessary. Motivated by the success of adjustable robust optimization in reducing highly conservative decision-making of static robust optimization in applications, in this paper, we present an affinely adjustable robust optimization (AARO) model for hypoxia-based radiation treatment planning in the face of evolving data uncertainty. We establish an exact semi-definite program reformulation of the model under a so-called affine decision rule and evaluate our model and approach on a liver cancer case as a proof-of-concept. Our AARO model incorporates uncertainties both in dose influence matrix and re-oxygenation data as well as inexactness of the revealed (re-oxygenation) data. Our numerical experiments demonstrate that the adjustable model successfully handles uncertainty in both re-oxygenation and the dose matrix. They also show that, by utilizing information halfway through the treatment plan, the adjustable solutions of the AARO model outperform a static method while maintaining a similar total dose.

Keywords:

• Affinely adjustable robust optimization
• semi-definite programming
• radiation therapy optimization
• evolving uncertainty

Acknowledgments

The authors are grateful to the referees for their comments and suggestions on the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

We used a dataset provided by [6] for our study on a liver cancer case. The data generated during the current study are available from the fourth author on reasonable request.

Notes

1 This paper considered a multi-stage robust optimization model, incorporating intermediate information in treatment planning with accrued additional cost. This multi-stage model results in an equivalent linear optimization problem.

2 A better estimate of the size of the uncertainty set r for our problem could be found by applying max-pooling over the 3-dimensional neighbourhood surrounding each voxel in the treatment volume, which can be interpreted as describing the maximal possible change in dosage throughout the treatment due to patient movement (under certain assumptions). We leave the evaluation of this estimate and other such choices for future work.

Additional information

Funding

Research was partially supported by a grant from the Australian Research Council.