Innovative Applications of O.R.A robust-CVaR optimization approach with application to breast cancer therapy
Introduction
The value-at-risk (VaR) of a loss distribution at the confidence level is the smallest loss such that the probability of exceeding such a loss is at most . The VaR concept is useful to measure quantities associated with the tail of a loss distribution. It has found application in areas such as finance (Duffie and Pan, 1997, Linsmeier and Pearson, 2000, El Ghaoui et al., 2003) and healthcare (Bortfeld, 1999, Wu and Mohan, 2000, Erkut et al., 2008), but solving optimization problems with VaR metrics are challenging since these problems are non-convex (Artzner, Delbaen, Eber, & Heath, 1999). Conditional value-at-risk (CVaR) is an alternative metric that measures the average of the tail loss values (Rockafellar & Uryasev, 2000). One of the attractive features of CVaR is that it is convex (Rockafellar & Uryasev, 2002). However, uncertainty in the underlying loss distribution results in uncertainty in the CVaR measurements.
Many researchers have considered robust optimization in a CVaR framework, particularly in financial engineering. Robust optimization is a technique that can be used to manage uncertainties in the data of an optimization problem. In traditional robust optimization, an uncertain hard constraint is replaced by its robust counterpart, to which the solution remains feasible under any realization of the uncertain data within some specified set (Ben-Tal and Nemirovski, 1999, Ben-Tal and Nemirovski, 2000, Bertsimas and Sim, 2004). In the literature of robust CVaR methods, one general approach aims to optimize the worst-case with respect to the choice of the underlying distribution of the stochastic parameter that characterizes the loss distribution (Jabbour et al., 2008, Huang et al., 2008, Huang et al., 2010, Zhu and Fukushima, 2009). Quaranta and Zaffaroni (2008) considered optimizing a CVaR objective, subject to standard linear constraints with uncertain coefficients, whereas Natarajan, Pachamanova, and Sim (2009) considered CVaR with uncertain parameters in the objective function and formulated worst-case CVaR models for different types of uncertainty sets. Our model is similar in mathematical structure to the mixture and discrete distribution models of Zhu and Fukushima (2009), who consider a likelihood distribution on the sample points and derive a min–max approach to minimize the worst-case scenario. However, we consider a different interpretation of the uncertainty and optimize the expected tail loss as opposed to minimizing the worst-case.
In this paper, we consider a system that can be in one of many states at any point in time. The number of states and the loss distribution corresponding to each state are known, but the fraction of time that the system spends in each state is unknown. For example, system states could correspond to states of the economy (e.g., recession, growth) and a different distribution of stock market returns would exist in each state. Our approach models the uncertain fraction of time the system spends in each state and aims to optimize or constrain the accumulated mean tail loss (CVaR) over some period of time. Our robust-CVaR model generalizes some of the existing CVaR models in the literature and is less conservative than worst-case approaches.
The development of our robust-CVaR model is motivated by an application in cancer therapy. Intensity-modulated radiation therapy (IMRT) is a cancer treatment method that aims to deliver sufficient radiation dose to a tumor while sparing healthy organs. An IMRT treatment generates a dose distribution – a distribution of dose to points inside the body. In the literature of IMRT treatment planning, the CVaR concept has been used to formulate constraints on the tails of a dose distribution (Romeijn, Ahuja, Dempsey, Kumar, & Li, 2003; Romeijn, Ahuja, Dempsey, & Kumar, 2006). For example, treatments generally aim to minimize underdose (i.e., lower tail of the dose distribution) to the tumor or overdose to healthy organs. Separately, robust optimization has been used to deal with setup uncertainty and day-to-day positioning errors (Chu, Zinchenko, Henderson, & Sharpe, 2005), dose calculation errors (Ólafsson & Wright, 2006), and motion during the treatment session (Chan et al., 2006, Bortfeld et al., 2008). However, there is no literature that combines robust optimization and CVaR in IMRT. We will apply the robust-CVaR method developed in this paper to breast cancer IMRT, where CVaR constraints are used to limit tail dose to the organs and robust optimization is used to mitigate the effects of breathing motion uncertainty, which causes the chest to move unpredictably during treatment.
Our specific contributions in this paper are as follows:
- 1.
We develop a robust-CVaR framework that models the tail of a distribution, which changes over time depending on the state of some system. Our framework is tractable through a duality-based reformulation, and generalizes the stochastic and worst-case CVaR models in the optimization literature.
- 2.
We develop the first optimization model in IMRT treatment planning that embeds robust optimization within a CVaR framework. This model generalizes the existing clinical treatment planning methods for breast cancer.
Section snippets
Clinical motivation
Breast cancer is the most frequently diagnosed type of cancer and the leading cause of cancer death in females worldwide (Jemal et al., 2011). Patients receiving radiation therapy for left-sided breast cancer demonstrate increased cardiac morbidity (Giordano et al., 2005, Harris et al., 2006, Hooning et al., 2007), which is highly dependent on the volume of heart exposed to radiation during treatment (Marks et al., 2005, Lind et al., 2003). This fact motivates the development of specialized
A robust-CVaR model
First, we briefly review the standard (stochastic) CVaR model, closely following the conceptual framework of Rockafellar and Uryasev (2000). Then, we extend this model to formulate CVaR in systems with multiple loss distributions corresponding to different states.
Current clinical treatment methods
Treatment planning for breast cancer is most commonly implemented using a simple radiation beam geometry in which two opposed tangential beams (Fig. 2) are used to irradiate the whole breast volume (Kestin et al., 2000, Landau et al., 2001, Vicini et al., 2002). The concave shape of the chest wall results in unavoidable dose to parts of the heart. The tangential opposed beam setup with IMRT is currently used in clinical practice (Purdie, Dinniwell, Letourneau, Hill, & Sharpe, 2011).
To create a
Results
In this section, we demonstrate the application of (w-CVaR), (CVaR-a) formulations using clinical data. We divided the patient’s breathing cycle into five phases. The nominal PMF had 50% of the probability at the exhale phase, and the remaining 50% was evenly distributed across the remaining four phases. This PMF is structurally representative of regular, exhale-weighted breathing. We define the uncertainty set to be the set of all PMFs that in each phase have a probability within of
Potential impact of the robust-CVaR approach in IMRT
Here, we expand on the clinical implications of the results seen in Section 5. Recall that we set out to develop an optimization-based IMRT treatment planning method for left-sided breast cancer that combines the advantages of the average and ABC methods. Our robust-CVaR treatments can be delivered during free breathing for any patient (like the average method) and produces dose distributions that push the frontier forward and towards the ABC dose distribution. While the average method often
Conclusions
In this paper, we develop two robust-CVaR definitions for formulating the tail of a loss distribution under uncertainty. We use a novel interpretation of a system with a loss distribution that is state dependent and the time spent in each state is uncertain. The cumulative loss depends on the uncertain fraction of time the system spends in each state. Our robust-CVaR framework generalizes existing models in the literature and is less conservative than worst-case methods.
We demonstrate an
Acknowledgments
Computational infrastructure used in this paper was provided by the High Performance Computing Virtual Laboratory (HPCVL). This research was supported in part by the Government of Ontario, the Canadian Breast Cancer Foundation, and the Canadian Breast Cancer Research Alliance.
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