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Robust Radiotherapy Planning with Spatially Based Uncertainty Sets

Noam Goldberg, Mark P. Langer, Shimrit Shtern

TL;DR

This work tackles robust radiotherapy planning under voxel-level biological uncertainty by introducing a spatially bound uncertainty set for radiosensitivity that couples personalized imaging data with spatial interdependencies. The authors derive a compact, provably equivalent reformulation that remains polynomial in size and develop a scalable row-generation algorithm to solve large-scale robust FMO problems; they further extend the framework to handle dose-volume constraints through a parametric LP relaxation and a predictive DV-constraint approach. Empirical results on brain and liver data show the spatially robust plans maintain performance in nominal settings while delivering superior worst-case adjusted dose and dose homogeneity under uncertainty, outperforming nominal and box-uncertainty models. The method demonstrates practical potential for personalized, biomarker-driven RTP and provides a foundation for future work on temporal and adaptive robust planning.

Abstract

Radiotherapy treatment planning is a challenging large-scale optimization problem plagued by uncertainty. Following the robust optimization methodology, we propose a novel, spatially based uncertainty set for robust modeling of radiotherapy planning, producing solutions that are immune to unexpected changes in biological conditions. Our proposed uncertainty set realistically captures biological radiosensitivity patterns that are observed using recent advances in imaging, while its parameters can be personalized for individual patients. We exploit the structure of this set to devise a compact reformulation of the robust model. We develop a row-generation scheme to solve real, large-scale instances of the robust model. This method is then extended to a relaxation-based scheme for enforcing challenging, yet clinically important, dose-volume cardinality constraints. The computational performance of our algorithms, as well as the quality and robustness of the computed treatment plans, are demonstrated on simulated and real imaging data. Based on accepted performance measures, such as minimal target dose and homogeneity, these examples demonstrate that the spatially robust model achieves almost the same performance as the nominal model in the nominal scenario, and otherwise, the spatial model outperforms both the nominal and the box-uncertainty models.

Robust Radiotherapy Planning with Spatially Based Uncertainty Sets

TL;DR

This work tackles robust radiotherapy planning under voxel-level biological uncertainty by introducing a spatially bound uncertainty set for radiosensitivity that couples personalized imaging data with spatial interdependencies. The authors derive a compact, provably equivalent reformulation that remains polynomial in size and develop a scalable row-generation algorithm to solve large-scale robust FMO problems; they further extend the framework to handle dose-volume constraints through a parametric LP relaxation and a predictive DV-constraint approach. Empirical results on brain and liver data show the spatially robust plans maintain performance in nominal settings while delivering superior worst-case adjusted dose and dose homogeneity under uncertainty, outperforming nominal and box-uncertainty models. The method demonstrates practical potential for personalized, biomarker-driven RTP and provides a foundation for future work on temporal and adaptive robust planning.

Abstract

Radiotherapy treatment planning is a challenging large-scale optimization problem plagued by uncertainty. Following the robust optimization methodology, we propose a novel, spatially based uncertainty set for robust modeling of radiotherapy planning, producing solutions that are immune to unexpected changes in biological conditions. Our proposed uncertainty set realistically captures biological radiosensitivity patterns that are observed using recent advances in imaging, while its parameters can be personalized for individual patients. We exploit the structure of this set to devise a compact reformulation of the robust model. We develop a row-generation scheme to solve real, large-scale instances of the robust model. This method is then extended to a relaxation-based scheme for enforcing challenging, yet clinically important, dose-volume cardinality constraints. The computational performance of our algorithms, as well as the quality and robustness of the computed treatment plans, are demonstrated on simulated and real imaging data. Based on accepted performance measures, such as minimal target dose and homogeneity, these examples demonstrate that the spatially robust model achieves almost the same performance as the nominal model in the nominal scenario, and otherwise, the spatial model outperforms both the nominal and the box-uncertainty models.
Paper Structure (28 sections, 7 theorems, 29 equations, 17 figures, 4 tables, 5 algorithms)

This paper contains 28 sections, 7 theorems, 29 equations, 17 figures, 4 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Modeling Robust Radiotherapy Planning
  3. Model Introduction - Nominal Model
  4. Robust Optimization Model with Spatially Bound Uncertainty Set
  5. The Spatially Bound Uncertainty Set.
  6. Compactly Reformulating The Robust Problem
  7. Solving Large-scale RO
  8. Dose-Volume Constraints
  9. Experiments
  10. Computational Performance
  11. Robust Model Evaluation
  12. Uncertainty Set Construction
  13. Robust Formulation Experiments
  14. Dose-Volume Experiments
  15. Conclusions and Future Work
  16. ...and 13 more sections

Key Result

Lemma 2

Suppose that Assumption ass:gamma_struct holds. Then, $\gamma_{\cdot}=\Gamma(d(\cdot)):T\times T\rightarrow [0,1]$ is a metric.

Figures (17)

  • Figure 1: Relationship between radiosensitivity difference and pairwise PTV voxel distance in the first-visit data of Patient 1 from the TCIA brain dataset Clark2013.
  • Figure 2: An illustration of the one-dimensional projection result of Proposition \ref{['prop:one_dim_proj']}. Assuming only three voxels, $v, u,$ and $w$, the bounds on the radiosenstivity of voxel $v$ are determined by tightening its original interval ${[{\underaccent{\bar{}}{\phi}}^0_v,\bar{\phi}^0_v]}$ by the intersection with $[\underaccent{\bar{}}{\phi}^0_u - \gamma_{uv},\bar{\phi}^0_u + \gamma_{uv}]$ and $[\underaccent{\bar{}}{\phi}^0_w-\gamma_{wv},\bar{\phi}^0_w+\gamma_{wv}]$, imposed by voxels $u$ and $w$, respectively.
  • Figure 3: An illustration of two different two-dimensional projections of $\mathcal{U}_{\text{SB}}$ for voxels $v$ and $u$. The shaded square depicts the $\delta$-bounds on the deviation from the measured $\widehat{\phi}$. The gray $45^0$ band depicts the $\gamma_{vu}$ bound on the radiosensitivity difference between the voxels. The green polygons are the projected uncertainty sets. The red points indicate the vertices that potentially maximize \ref{['eq:reform_second_const']}.
  • Figure 4: Radiosensitivity difference vs. voxel distance for PTV voxel pairs of Patient 4 in the TCIA archive Brain dataset.
  • Figure 5: Nominal biologically-adjusted performance of solutions optimal to model \ref{['prob:SR_robust']} for $\delta=0.14$ and different values of $\gamma$, compared to the nominal solution.
  • ...and 12 more figures

Theorems & Definitions (13)

  • Lemma 2: Corazza1999
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Theorem 6
  • proof
  • Lemma 7
  • Proposition 8
  • proof
  • ...and 3 more