Mathematical cancer therapy planning in deep regional hyperthermia*

This paper surveys the mathematics required for a typically challenging problem from computational medicine: cancer therapy planning in deep regional hyperthermia. In the course of many years of close cooperation with clinics, the medical problem has given rise to many subtle mathematical problems, some of which were unsolved when the project started. Efficiency of numerical algorithms, i.e., computational speed and monitored reliability, plays a decisive role in the medical treatment. Off-the-shelf software had turned out to be insufficient to meet the requirements of medicine. Instead, new mathematical theory as well as new numerical algorithms had to be developed. In order to make our algorithms useful in the clinical environment, new visualization software, i.e., a ‘virtual lab’, including three-dimensional geometry processing of individual virtual patients, had to be designed and implemented. Moreover, before the problems could be attacked by numerical algorithms, careful mathematical modelling had to be done. Finally, parameter identification and constrained optimization for the PDEs had to be newly analysed and realized over the individual patient's geometry. Our new techniques had an impact on the specificity of the treatment of individual patients and on the construction of an improved hyperthermia applicator.