Proton therapy treatment planning via the alternating direction method of multipliers: a tutorial review

Inverse treatment planning is central to intensity-modulated proton therapy, where high-quality dose distributions must be achieved under machine, biological, delivery-efficiency, and robustness constraints. The alternating direction method of multipliers (ADMM) is a flexible operator-splitting framework that is well suited to such structured optimization problems because it separates smooth dose-fidelity terms from nonsmooth constraints or regularizers. This tutorial review summarizes representative ADMM formulations for proton therapy treatment planning, including minimum monitor unit optimization, linear energy transfer optimization, energy-layer sparsity, robust optimization, and spot sparsity for proton arc therapy. We also place these formulations in the broader context of radiation therapy optimization, compare them with alternative methods, report illustrative computation times from five matRad examples, and discuss practical limitations, including parameter sensitivity, nonconvex convergence issues, and barriers to clinical translation.