A variational approach to the problem of optimizing the radiation dose distribution in tumours

This paper presents a precise mathematical formulation of a biological criterion by which the radiation dose distribution in tumours homogeneous or heterogeneous in cell density and radiosensitivity can be optimized. The criterion is formulated as search for a dose distribution that would minimize the mean dose delivered to the tumour under the constraint that the tumour control probability reaches a given desired value. Using a method from the calculus of variations it has been proven that a homogeneous dose distribution is the solution in case of tumours homogeneous in radiosensitivity independent of their cell spatial density status. Thus the usual requirement for homogeneous dose distribution in case of homogeneous tumours is proven if the leading clinical criterion is the described one. The formula for the dose distribution in case of tumours heterogeneous in cell radiosensitivity is given too.