A Mathematical Approach to Optimizing the Radiation dose Distribution in Heterogeneous Tumours

This paper offers a general mathematical approach to dose distribution optimization which allows tumours with different degrees of complexity to be considered. Two different biological criteria - A) keeping the control probability of the different parts of the tumour (local tumour control probability) uniform throughout the tumour and B) minimizing the mean dose delivered to the tumour are studied. For both criteria we impose the requirement that the whole tumour control probability be kept on a certain desired level. It is proved that the adoption of the first criterion requires a dose distribution logarithmic with the cell density and proportional to the inverse of the cell radiosensitivity while the adoption of the second criterion necessitates a homogeneous dose distribution when the cell radiosensitivity is constant. The corresponding formula for the dose distribution in case of heterogeneous cell radiosensitivity is also given. The two criteria are compared in terms of local tumour control probability and mean dose delivered to the tumour. It is concluded that maintaining constant local tumour control probability (criterion A) may be of greater clinical importance then minimizing the mean dose (criterion B).