Background. A property of the integral dose-volume histogram (DVH) space is analytically investigated in this work. A curve called an α-iso-NTCP (normal tissue complication probability) envelope is constructed by connecting points belonging to step-like integral DVHs, each corresponding to homogeneous partial organ irradiation of a relative volume vk to dose Dk such that the resulting NTCP has, in all cases, a particular value α. The two subspaces into which the envelope divides the DVH space are analytically explored in terms of the equivalent uniform doses (EUDs) corresponding to the different DVHs. It is theoretically proven that any DVH, other than the step-like DVH, passing through a point (Dk, vk) from the α-iso-NTCP envelope, will result in an NTCP > α.
Conclusions. Thus, it is proven that any DVH that at least partially lies above the envelope results in an NTCP > α For some of the DVHs lying under the envelope, e.g. those that are tangential to the envelope, it is also true that the resulting NTCP > α. However, it was numerically demonstrated elsewhere that there exist DVHs lying entirely in the lower subspace that result in an NTCP < α. Therefore, one can conclude that since there is a chance that a DVH lying under the α-iso-NTCP envelope will result in NTCP < α, it would be preferable in the treatment optimization process to seek solutions for DVHs lying entirely under an iso-NTCP envelope and avoid solutions that have DVHs above an iso-NTCP envelope.
