A feasible dose-volume estimation of radiotherapy treatment with optimal transport using a concept for transportation of Ricci-flat time-varying dose-volume

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.