  In radiotherapy, the dose-volume histogram (DVH) curve is an important means
of evaluating the clinical feasibility of tumor control and side effects in
normal organs against actual treatment. Fractionation, distributing the amounts
of irradiation, is used to enhance the treatment effectiveness of tumor control
and mitigation of normal tissue damage. Therefore, dose and volume receive
time-varying effects per fractional treatment event. However, the difficulty of
DVH superimposition of different situations prevents evaluation of the total
DVH despite different shapes and receiving dose distributions of organs in each
fraction. However, an actual evaluation is determined traditionally by the
initial treatment plan because of summation difficulty. Mathematically, this
difficulty can be regarded as a kind of optimal transport of DVH. For this
study, we introduced DVH transportation on the curvilinear orthogonal space
with respect to arbitrary time ($T$), time-varying dose ($D$), and time-varying
volume ($V$), which was designated as the TDV space embedded in the Riemannian
manifold.Transportation in the TDV space should satisfy the following: (a) the
metrics between dose and volume must be equivalent for any fractions and (b)
the cumulative characteristic of DVH must hold irrespective of the lapse of
time. With consideration of the Ricci-flat condition for the $D$-direction and
$V$-direction, we obtained the probability density distribution, which is
described by Poisson's equation with radial diffusion process toward $T$. This
geometrical requirement and transportation equation rigorously provided the
feasible total DVH.
