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Spatiotemporal fractionation schemes, that is, treatments deliveringdifferent dose distributions in different fractions, may lower treatment sideeffects without compromising tumor control. This is achieved byhypofractionating parts of the tumor while delivering approximately uniformlyfractionated doses to the healthy tissue. Optimization of such treatments isbased on biologically effective dose (BED), which leads to computationallychallenging nonconvex optimization problems. Current optimization methods yieldonly locally optimal plans, and it has been unclear whether these are close tothe global optimum. We present an optimization model to compute rigorous boundson the normal tissue BED reduction achievable by such plans.  The approach is demonstrated on liver tumors, where the primary goal is toreduce mean liver BED without compromising other treatment objectives. First auniformly fractionated reference plan is computed using convex optimization.Then a nonconvex quadratically constrained quadratic programming model issolved to local optimality to compute a spatiotemporally fractionated plan thatminimizes mean liver BED subject to the constraints that the plan is no worsethan the reference plan with respect to all other planning goals. Finally, wederive a convex relaxation of the second model in the form of a semidefiniteprogramming problem, which provides a lower bound on the lowest achievable meanliver BED.  The method is presented on 5 cases with distinct geometries. The computedspatiotemporal plans achieve 12-35% mean liver BED reduction over the referenceplans, which corresponds to 79-97% of the gap between the reference mean liverBEDs and our lower bounds. This indicates that spatiotemporal treatments canachieve substantial reduction in normal tissue BED, and that local optimizationprovides plans that are close to realizing the maximum potential benefit.

Optimization of spatiotemporally fractionated radiotherapy treatments with bounds on the achievable benefit