On coupled Boltzmann transport equation related to radiation therapy

We consider a system of Boltzmann transport equations which models the charged particle evolution in media. The system is related to the dose calculation in radiation therapy. Although only one species of particles, say photons is invasing these particles mobilize other type of particles (electrons and positrons). Hence in realistic modelling of particle transport one needs a coupled system of three Boltzmann transport equations. The solution of this system must satisfy the inflow boundary condition. We show existence and uniqueness result of the solution applying generalized Lax–Milgram Theorem. In addition, we verify that (in the case of external therapy) under certain assumptions the “incoming flux to dose operator” D1 is compact. Also the adjoint is analyzed. Finally we consider the inverse planning problem as an optimal control problem. Its solution can be used as an initial solution of the actual inverse planning problem.