The paper studies the radiation dose distribution when a tumor is subject to radiotherapy at a prescribed dose, such that the patient’s organs at radiation risk are protected as best as possible. Therefore, external beam radiation therapy (EBRT) utilizing dose volume histograms (DVH) is optimized. The cumulative DVH is a graph with the absorbed dose on the \(x\)-axis and the relative amount of a planning target volume (PTV) volume receiving the absorbed close \(\leq\) dose \(d\) on the \(y\)-axis. DVHs are used as inputs for inverse planning. The authors study the equivalent uniform dose (EUD), a radiobiological index defined as the absorbed dose that, homogeneously given to the target volume, would yield the same biological effect as the dose distribution described by the DVH. EUD is described as a real functional on the set of DVH normalized or transformed versions. A Lévy distance is defined on the set of DVHs and bounds on the variation of the EUD functionals can be found which can be translated into DVH constraints when the reference distribution is from a uniform radiation at the prescribed dose. The sensitivity of functionals of DVHs such as the EUD is studied when a small change in the absorbed dose distribution takes place.
The paper provides a theoretical base of the absorbed dose distributions and DVHs, Lévy distance and DVHs, PTV and the references dose distribution for Lévy distances, and functionals of the DVHs and the EUD. The authors show how the method can be applied and how a minimum of input information from a dose distribution is enough to determine whether or not it complies with a given EUD tolerance. This allows the assessment of computed plans for different anatomical data under different uncertainty conditions and under missing information.