Optimal control model for radiation therapy inverse planning applying the Boltzmann transport equation

We consider an inverse problem related to external radiation therapy treatment planning. The dose calculation (the forward problem) is based on the Boltzmann Transport Equation (BTE) which models exactly the transport of charged particles in tissue. The inverse planning (the inverse problem) is formulated as an optimal boundary control problem. The optimal control variable is the incoming (external) flux and the output is the dose distribution in patient domain. Both physical and biological cost functions are defined but here we concentrate on the physically based optimization. The discretization is done by finite element approximations for which the BTE is expressed in its variational form. In the optimization process of the discrete model we apply so called parametrization (which may essentially diminish the decision variables in optimization) of the matrix equations. Parametrization is based on certain linear algebraic decompositions of matrices. One simulation is presented which show the functionality of the developed methods.