A parallel algorithm for generating Pareto-optimal radiosurgery treatment plans

Using inverse planning tools to create radiosurgery treatment plans is an iterative process, where clinical trade-offs are explored by changing the relative importance given to different objectives and rerunning the optimizer until a desirable plan is found. Simultaneously generating many plans corresponding to different objective weights, while the patient is awaiting treatment, would allow the planner to navigate clinical trade-offs interactively, without optimizing a new plan between each update.

We seek to optimize hundreds of Gamma Knife radiosurgery treatment plans, corresponding to different weightings of objectives, fast enough to allow interactive Pareto navigation of clinical trade-offs to be incorporated into the clinical workflow.

We apply the alternating direction method of multipliers (ADMM) to the linear-program formulation of the optimization problem used in the clinical Lightning optimizer. We implement both a CPU and a GPU version of ADMM in Matlab and compare them to Matlab's built-in, single-threaded dual-simplex solver. The ADMM implementation is adapted to the optimization procedure used in the clinical software, with a bespoke algorithm for maximizing the overlap between low-dose points for different objective weights. The method is evaluated on a test dataset consisting of 20 cases from three different indications, with between one and nine targets and total target volumes ranging from 0.66 to 52 cm³.

The total optimization time to create 81 plans corresponding to different objective weightings varied from 63 to 520 s on CPU and from 1.8 to 40 s GPU, for the different test cases. As a reference, optimizing 81 plans using simplex took 100-51000 s, corresponding to ADMM speedups of 1.6-97 and 54-1500 times for the CPU and GPU, respectively. Increasing the number of plans to 441, corresponding to all combinations of slider values between 0.0 and 1.0 in steps of 0.05 in the clinical software, the total ADMM optimization time on GPU was between 3.0 and 110 s for the different test cases. Plan quality was evaluated by rerunning the ADMM optimization 20 times, each with a different random seed, for each test case and for nine objective weightings per case. The resulting relative differences in clinical metrics ( $\mathrm{mean}\pm \mathrm{SD}$ ) were 0.0 $\pm$ 0.2%, 0.0 $\pm$ 1.6%, 0.1 $\pm$ 0.8%, and 0.1 $\pm$ 3.0%, for coverage, selectivity, gradient index and beam-on time, respectively, compared to mean values for the corresponding reference simplex results. The standard deviations in these metrics closely mimicked those obtained when rerunning the simplex solver, verifying the validity of the method.

We show how ADMM can be adapted for radiosurgery plan optimization, allowing hundreds of high-quality Gamma Knife treatment plans to be created in under two minutes on a single GPU, also for very large cases. The presented method would allow streamlined multicriteria optimization on the day of treatment, with interruption-free navigation of clinical trade-offs.