MULTIPLE LOCAL MINIMA IN RADIOTHERAPY OPTIMIZATION PROBLEMS WITH DOSE-VOLUME CONSTRAINTS

The cause of multiple local minima in beam weight optimization problem s subject to dose-volume constraints is analyzed. Three objective func tions were considered: (a) maximization of turner control probability (TCP), (b) maximization of the minimum target dose, and (c) minimizati on of the mean-squared-deviation of the target dose from the prescript ion dose. It is shown that: (a) TCP models generally result in strongl y quasiconvex objective functions; (b) maximization of the minimum tar get dose results in a strongly quasiconvex objective function; and (c) minimizing the root-mean-square dose deviation results in a convex ob jective function. Dose-volume constraints are considered such that, fo r each region at risk (RAR), the volume of tissue whose dose exceeds a certain tolerance dose (D-Tol) is kept equal to or below a given frac tional level (U-Tol). If all RARs lack a ''volume effect'' (i.e., U-To l = 0 for all RARs) then there is a single local minimum. But if volum e effects are present, then the feasible space is possibly nonconvex a nd therefore possibly leads to multiple local minima. These conclusion s hold for all three objective functions. Hence, possible local minima come not from the nonlinear nature of the objective functions conside red, but from the ''either this volume or that volume but not both'' n ature of the volume effect. These observations imply that optimization algorithms for dose-volume constraint types of problems should have e ffective strategies for dealing with multiple local minima. (C) 1997 A merican Association of Physicists in Medicine.