Instantaneous closed loop control of the radiative transfer equations with applications in radiotherapy

In this paper we consider a time‐dependent Boltzmann equation for particle transport in biological tissues, which can be used for radiation dose calculation. We assume that the dose should be delivered to a target volume which moves over time. We formulate this as an optimal control problem and derive closed‐loop control laws for the dose delivery problem using boundary and distributed control. Optimality conditions are derived. 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. This work could be applied to image‐guided radiation therapy, where patient motion during treatment is one of the future challenges.

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