[Paper Review] Robust and probabilistic optimization of dose schedules in radiotherapy

Hamidreza Badri, Yoichi Watanabe+1 more|arXiv (Cornell University)|Jan 1, 2015

Advanced Radiotherapy Techniques48 references4 citations

TL;DR

This paper proposes robust and probabilistic optimization frameworks to improve radiotherapy dose scheduling by accounting for inter-patient variability in tumor and normal tissue radiosensitivity (α, β). Using the linear-quadratic model and transformation to two dimensions, it shows that uncertainty shifts optimal schedules from corners of the feasible region to its boundary, requiring branch-and-bound for global solution. Key results demonstrate that uncertainty leads to reduced total dose or dose-squared when fraction numbers differ from nominal schedules.

ABSTRACT

We consider the effects of parameter uncertainty on the optimal radiation schedule in the context of the linear-quadratic model. Our interest arises from the observation that if inter-patient variations in OAR and tumor sensitivities to radiation or sparing factor of the OAR are not accounted for during radiation scheduling, the performance of the therapy may be strongly degraded or the OAR may receive a substantially larger dose than the maximum threshold. This paper proposes two radiation scheduling concepts to incorporate inter-patient variability into the scheduling optimization problem. The first approach is a robust formulation that formulates the problem as a conservative model that optimizes the worst case dose scheduling that may occur. The second method is a probabilistic approach, where the model parameters are given by a set of random variables. This formulation insures that our constraints are satisfied with a given probability, and that our objective function achieves a desired level with a stated probability. We used a transformation to reduce the resulting optimization problem to two dimensions. We showed that the optimal solution lies on the boundary of the feasible region and we used a branch and bound algorithm to find the global optimal solution. We observed that if the number of fractions in the optimal conventional schedule is the same as the robust and stochastic solutions, it is preferable to administer equal or smaller total dose. In addition if there exist more (fewer) treatment sessions in the probabilistic or robust solution compared to the conventional schedule, a reduction in total dose squared (total dose) will be expected. Finally, we performed numerical experiments in the setting of head-and-neck tumors to reveal the effect of parameter uncertainty on optimal schedules and to evaluate the sensitivity of the model to the choice of key model parameters.

Motivation & Objective

• To address the degradation in radiotherapy outcomes caused by unaccounted inter-patient variability in tumor and normal tissue radiosensitivity.
• To develop optimization models that ensure OAR dose constraints are met with high probability despite uncertainty in α and β parameters.
• To compare robust and probabilistic scheduling approaches with conventional equal-fractionation schedules in terms of total dose and fractionation.
• To evaluate the sensitivity of optimal schedules to key parameters like tumor growth rate and normal tissue radiobiological parameters.
• To provide a computational framework using branch-and-bound to globally optimize dose schedules under uncertainty.

Proposed method

• Proposes a robust optimization formulation that minimizes the worst-case tumor control probability under interval uncertainty in α and β.
• Develops a probabilistic model where α and β are random variables, ensuring constraints are satisfied with pre-specified probability (pi) and objective achieves desired level with probability (pz).
• Applies a transformation from [37] to reduce the two-dimensional optimization problem to a line segment search over feasible dose-fraction combinations.
• Uses a branch-and-bound algorithm to globally optimize the problem, with convergence certified by ϵ-suboptimality and interval length reduction.
• Implements a partitioning strategy that dynamically refines intervals and prunes inactive nodes based on upper and lower bounds of the objective function.
• Employs a bisection-based node selection and refinement process to efficiently explore the feasible region while maintaining computational tractability.

Experimental results

Research questions

• RQ1 How does inter-patient variability in α and β affect the optimality of conventional radiotherapy fractionation schedules?
• RQ2 What is the impact of uncertainty on the location of the optimal dose schedule—does it remain at a corner of the feasible region or shift to the boundary?
• RQ3 How do robust and probabilistic optimization approaches compare to conventional equal-dose scheduling in terms of total dose and number of fractions?
• RQ4 What is the effect of uncertainty on the required total dose when the number of fractions remains unchanged?
• RQ5 How sensitive are optimal schedules to variations in tumor growth rate and normal tissue radiobiological parameters?

Key findings

• When α and β are deterministic, the optimal schedule lies at a corner of the feasible region; under uncertainty, the optimal solution shifts to the boundary of the feasible region.
• If the number of fractions in the robust or stochastic solution matches the conventional schedule, the total dose must be reduced or kept equal to maintain safety.
• When the robust or stochastic solution uses fewer fractions than the conventional schedule, total dose is reduced (dose-squared effect), while more fractions lead to reduced total dose.
• In case 1 (normal β), the robust solution reduces tumor BED from 12.10 to 9.93 Gy, and stochastic from 12.10 to 4.36 Gy, with increased fractionation.
• In case 2 (very low β), the robust solution maintains similar tumor BED (5.53 vs 5.69 Gy) but reduces total dose and increases fractionation.
• Numerical experiments on head-and-neck tumors show that uncertainty significantly alters optimal schedules, and treatment outcome is sensitive to tumor growth rate γ, especially in long treatment times.