Robust stochastic optimization of needle configurations for robotic HDR prostate brachytherapy.

Background: Ideally, inverse planning for HDR brachytherapy (BT) should include the pose of the needles which define the trajectory of the source. This would be particularly interesting when considering the additional freedom and accuracy in needle pose which robotic needle placement enables. However, needle insertion typically leads to tissue deformation, resulting in uncertainty regarding the actual pose of the needles with respect to the tissue.

Purpose: To efficiently address uncertainty during inverse planning for HDR BT in order to robustly optimize the pose of the needles before insertion, that is, to facilitate path planning for robotic needle placement.

Methods: We use a form of stochastic linear programming to model the inverse treatment planning problem. To account for uncertainty, we consider random tissue displacements at the needle tip to simulate tissue deformation. Conventionally for stochastic linear programming, each simulated deformation is reflected by an addition to the linear programming problem which increases problem size and computational complexity substantially and leads to impractical runtime. We propose two efficient approaches for stochastic linear programming. First, we consider averaging dose coefficients to reduce the problem size. Second, we study weighting of the slack variables of an adjusted linear problem to approximate the full stochastic linear program. We compare different approaches to optimize the needle configurations and evaluate their robustness with respect to different amounts of tissue deformation.

Results: Our results illustrate that stochastic planning can improve the robustness of the treatment with respect to deformation. The proposed approaches approximating stochastic linear programming better conform to the tissue deformation compared to conventional linear programming. They show good correlation with the plans computed after deformation while reducing the runtime by two orders of magnitude compared to the complete stochastic linear program. Robust optimization of needle configurations takes on average 59.42 s. Skew needle configurations lead to mean coverage improvements compared to parallel needles from 0.39 to 2.94 percentage points, when 8 mm tissue deformation is considered. Considering tissue deformations from 4  to 10 mm during planning with weighted stochastic optimization and skew needles generally results in improved mean coverage from 1.77 to 4.21 percentage points.

Conclusions: We show that efficient stochastic optimization allows selecting needle configurations which are more robust with respect to potentially negative effects of target deformation and displacement on the achievable prescription dose coverage. The approach facilitates robust path planning for robotic needle placement.