Toward semi-automatic biologically effective dose treatment plan optimisation for Gamma Knife radiosurgery.

Objective. Dose-rate effects in Gamma Knife radiosurgery treatments can lead to varying biologically effective dose (BED) levels for the same physical dose. The non-convex BED model depends on the delivery sequence and creates a non-trivial treatment planning problem. We investigate the feasibility of employing inverse planning methods to generate treatment plans exhibiting desirable BED characteristics using the per iso-centre beam-on times and delivery sequence. Approach. We implement two dedicated optimisation algorithms. One approach relies on mixed-integer linear programming (MILP) using a purposely developed convex underestimator for the BED to mitigate local minima issues at the cost of computational complexity. The second approach (local optimisation) is faster and potentially usable in a clinical setting but more prone to local minima issues. It sequentially executes the beam-on time (quasi-Newton method) and sequence optimisation (local search algorithm). We investigate the trade-off between time to convergence and solution quality by evaluating the resulting treatment plans' objective function values and clinical parameters. We also study the treatment time dependence of the initial and optimised plans using BED 95 (BED delivered to 95% of the target volume) values. Main results. When optimising the beam-on times and delivery sequence, the local optimisation approach converges several orders of magnitude faster than the MILP approach (minutes versus hours-days) while typically reaching within 1.2% (0.02-2.08%) of the final objective function value. The quality parameters of the resulting treatment plans show no meaningful difference between the local and MILP optimisation approaches. The presented optimisation approaches remove the treatment time dependence observed in the original treatment plans, and the chosen objectives successfully promote more conformal treatments. Significance. We demonstrate the feasibility of using an inverse planning approach within a reasonable time frame to ensure BED-based objectives are achieved across varying treatment times and highlight the prospect of further improvements in treatment plan quality.