Variable Dose-Constraints Method Based on Multiplicative Dynamical Systems for High-Precision Intensity-Modulated Radiation Therapy Planning

An optimization framework that effectively balances dose–volume constraints and treatment objectives is required in intensity-modulated radiation therapy (IMRT) planning. In our previous work, we proposed a dynamical systems-based approach in which dose constraints, along with beam coefficients, are treated as state variables and dynamically evolve within a continuous-time system. This method improved the accuracy of the solution by dynamically adjusting the dose constraints, but it had a significant drawback. Specifically, because it is as an iterative process derived from discretization of a linear differential equation system using the additive Euler method, a lower-bound clipping procedure is required to prevent the state variables for both beam coefficients and dose constraints from taking negative values. This issue could prevent constrained optimization from functioning properly and undermine the feasibility of the treatment plan. To address this problem, we propose two types of multiplicative continuous-time dynamical system that inherently preserve the nonnegativity of the state variables. We theoretically prove that the initial value problem for these systems converges to a solution that satisfies the constraints of consistent IMRT planning. Furthermore, to ensure computational practicality, we derive discretized iterative schemes from the continuous-time systems and confirm that their iterations maintain nonnegativity. This framework eliminates the need for artificial clipping procedures and leads to the multiplicative variable dose-constraints method, which dynamically adjusts dose constraints during the optimization process. Finally, numerical experiments are conducted to support and illustrate the theoretical results, showing how the proposed method achieves high-precision IMRT planning while ensuring physically meaningful solutions.