  Spatiotemporal fractionation schemes, that is, treatments delivering
different dose distributions in different fractions, may lower treatment side
effects without compromising tumor control. This is achieved by
hypofractionating parts of the tumor while delivering approximately uniformly
fractionated doses to the healthy tissue. Optimization of such treatments is
based on biologically effective dose (BED), which leads to computationally
challenging nonconvex optimization problems. Current optimization methods yield
only locally optimal plans, and it has been unclear whether these are close to
the global optimum. We present an optimization model to compute rigorous bounds
on the normal tissue BED reduction achievable by such plans.
  The approach is demonstrated on liver tumors, where the primary goal is to
reduce mean liver BED without compromising other treatment objectives. First a
uniformly fractionated reference plan is computed using convex optimization.
Then a nonconvex quadratically constrained quadratic programming model is
solved to local optimality to compute a spatiotemporally fractionated plan that
minimizes mean liver BED subject to the constraints that the plan is no worse
than the reference plan with respect to all other planning goals. Finally, we
derive a convex relaxation of the second model in the form of a semidefinite
programming problem, which provides a lower bound on the lowest achievable mean
liver BED.
  The method is presented on 5 cases with distinct geometries. The computed
spatiotemporal plans achieve 12-35% mean liver BED reduction over the reference
plans, which corresponds to 79-97% of the gap between the reference mean liver
BEDs and our lower bounds. This indicates that spatiotemporal treatments can
achieve substantial reduction in normal tissue BED, and that local optimization
provides plans that are close to realizing the maximum potential benefit.
