  We present a method of directly optimizing on deviations in clinical goal
values in radiation therapy treatment planning. Using a new mathematical
framework in which metrics derived from the dose-volume histogram are regarded
as functionals of an auxiliary random variable, we are able to obtain
volume-at-dose and dose-at-volume as infinitely differentiable functions of the
dose distribution with easily evaluable function values and gradients.
Motivated by the connection to risk measures in finance, which is formalized in
this framework, we also derive closed-form formulas for mean-tail-dose and
demonstrate its capability of reducing extreme dose values in tail
distributions. Numerical experiments performed on a prostate and a
head-and-neck patient case show that the direct optimization of dose-volume
histogram metrics produced marginally better results than or outperformed
conventional planning objectives in terms of clinical goal fulfilment, control
of low- and high-dose tails of target distributions and general plan quality
defined by a pre-specified evaluation measure. The proposed framework
eliminates the disconnect between optimization functions and evaluation metrics
and may thus reduce the need for repetitive user interaction associated with
conventional treatment planning. The method also has the potential of enhancing
plan optimization in other settings such as multicriteria optimization and
automated treatment planning.
