Optimal Treatment Planning in Radiotherapy Based on Boltzmann Transport Equations

  • Richard C. BarnardEmail author
  • Martin Frank
  • Michael Herty
Part of the International Series of Numerical Mathematics book series (ISNM, volume 165)


We look at the optimization of radiotherapy treatment planning. By using a deterministic model of dose deposition in tissue derived from the Boltzmann transport equations, we can improve on the accuracy of existing models near tissue inhomogeneities while also making use of adjoint calculus for developing necessary conditions for optimality. We describe the relevant model and consider the planning problem in an optimal control framework. Two versions of the problem are discussed, optimality conditions are derived, and numerical methods are described. Numerical examples are presented.


Kinetic equation Optimal control Radiotherapy 

Mathematics Subject Classification (2010)

35Q20 49J20 


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Richard C. Barnard
    • 1
    Email author
  • Martin Frank
    • 2
  • Michael Herty
    • 3
  1. 1.Institute for Mathematics and Scientific ComputingUniversity of GrazGrazAustria
  2. 2.MATHCCES, Department of MathematicsRWTH Aachen UniversityAachenGermany
  3. 3.IGPM, Department of MathematicsRWTH Aachen UniversityAachenGermany

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