Optimal solution for a cancer radiotherapy problem with a maximal damage constraint on normal tissues

In Bertuzzi et al. [1] we addressed the problem of finding the optimal radiotherapy fractionation scheme, representing the response to radiation of tumour and normal tissues by the LQ model including exponential repopulation and sublethal damage due to incomplete repair. We formulated the nonlinear programming problem of maximizing the overall tumour damage, while keeping the damages to the late and early responding normal tissues within a given admissible level. In the present paper we show the results for a simpler optimization problem, containing only the constraint related to the late normal tissue that reduces to an equality constraint under suitable assumptions. In fact, it has been shown in [1] that, suitably choosing the maximal damage values, the problem here considered is equivalent to the more general one, in that their extremals, and then their optimal solutions, coincide. The optimum is searched over a single week of treatment and its possible structures are identified. We characterize the optimal solution in terms of model parameters. Apart from limit values of the parameters, we prove that the optimal solution is unique and never consisting of five equal fractions per week. This is interesting in comparison to the uniform fractionation schemes commonly used in radiotherapy.