The generalized equivalent uniform dose function as a basis for intensity-modulated treatment planning

The efficiency of intensity-modulated radiation therapy (IMRT) treatment planning depends critically on the presence or absence of multiple local minima in the feasible search space. We analyse the convexity of the generalized equivalent uniform dose equation (Niemierko A 1999 Med. Phys. 26 1100) when used either in the objective function or in the constraints. The practical importance of this analysis is that convex objective functions minimized over convex feasibility spaces do not have multiple local minima, likewise for concave objective functions maximized over convex feasibility spaces. Both of these situations are referred to as 'convex problems' and computationally efficient local search methods can be used for their solution. We also show that the Poisson-based tumour control probability objective function is strictly concave (if one neglects inter-patient heterogeneity), and hence it implies a single local minimum if maximized over a convex feasibility space. Even when including inter-patient heterogeneity, multiple local minima, although theoretically possible, are expected to be of minimal concern. The generalized equivalent uniform dose function (EUDa) is proved to be convex or concave depending on its only parameter a: when a is equal to or greater than 1, minimizing EUDa, on a convex feasibility space leads to a single minimum; when a is less than 1, maximizing EUDa, on a convex feasibility space leads to a single minimum. We also study a recently proposed practical, yet difficult, IMRT treatment planning formulation: unconstrained optimization of the objective function proposed by Wu et al (2002 Int. J. Radiat. Oncol. Biol. Phys. 52 224-35), which is expressed in terms of the EUDa for the target and normal tissues. This formulation may theoretically lead to multiple local minima. We propose a procedure for improving resulting solutions based on the convexity properties of the underlying objective function terms.