Optimal weekly scheduling in fractionated radiotherapy: effect of an upper bound on the dose fraction size (Q2355786)

The paper deals with the optimization of cancer radiotherapy schedules to maximize the effectiveness of destroying cancerous cells and minimize the damage to normal cells. These two objectives are usually conflicting requirements. Mathematically, such a situation can be viewed as a multi-objective optimization problem. More specifically, the dose fractions are optimized for one fraction per day, five fractions per week. The overall treatment time is fixed. The LQ (linear-quadratic) model is used as a starting point. It means that the objective function is nonlinear. As the constraints that are introduced to limit the radiation damages to surrounding normal cells and that limit the daily fraction size result in nonlinear terms we are faced with a highly nonlinear and non-convex mathematical programming problem. The paper defines the Lagrangian function of the problem and works out the Kuhn-Tucker necessary conditions of optimality. The resulting large scale set of equations and inequalities is not directly solvable. The authors adopt a three-stage procedure that successively reduces the domain of feasible solutions. In this way they obtain several local optima. The solutions are expressed in terms of the tumor and normal tissue parameters as well as the upper bound on dose fractions. Finally, the best (in terms of the objective value) solution is selected by direct comparison. The paper includes a detailed analytical development of the entire procedure. Some numerical examples are also presented for optimal treatment schedules.