On coupled Boltzmann transport equation related to radiation therapy. (English) Zbl 1115.92035

Summary: 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 types 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 results of the solutions applying a generalized Lax-Milgram theorem. In addition, we verify that (in the case of external therapy) under certain assumptions the “incoming flux to dose operator” \(D_{1}\) is compact. Also, the adjoint \(D^{*}_{1}\) 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.


92C50 Medical applications (general)
82C70 Transport processes in time-dependent statistical mechanics
49N90 Applications of optimal control and differential games
46N60 Applications of functional analysis in biology and other sciences
47N60 Applications of operator theory in chemistry and life sciences


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