Time-dependent Closed Loop Control of the Radiative Transfer Equations with Applications in Radiotherapy

Patient motion during treatment is one of the future challenges in the field of external beam radiotherapy. In this paper, we address this problem by considering a time–dependent Boltzmann transport model for dose calculation and by deriving closed-loop control laws for the treatment planning problem. We formulate an optimal control problem for the desired dose using boundary and distributed control and derive optimality conditions. For the construction of the closed-loop control laws we use an inexact variant of model predictive control called instantaneous control. We compare numerical results obtained with instantaneous control to those obtained by optimal open-loop control, and present numerical simulations in one and two spatial dimensions.

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