Relativistic Optimization Force Concept for gEUD Biological Optimization and Novel a-value Selection Viewpoint
Authors: Anetai Yusuke , Takegawa Hideki , Koike Yuhei , Nakamura Satoaki , Tanigawa Noboru 
Published: February 2021

Abstract
Generalized equivalent uniform dose (gEUD) optimization is a biological optimization method used for intensity modulated radiation therapy (IMRT). Although parametric analyses have been widely reported, the use of parameter a-value in the optimization method remains elusive. This study aims to clarify the mathematical characteristics of the gEUD and to provide effective a-value selection. The gEUD is typically obtained using a differential dose volume histogram (DVH). This can be rewritten using a cumulative DVH (cDVH) and applied to variational analysis. The equivalence between the gEUD and the dose is then obtained; a low or high a-value corresponds to a wide or narrow dose range of optimization, respectively. Next, we focused on the gEUD curve behavior against a-value shifts and it retained its curve characteristics despite optimization. Using differential geometry, this curve shift can be considered as a geodesic deviation between pre- and post-optimization by a relativistic optimization force. The total action enacted by the force includes the curvature of the gEUD curve. This idea provides a novel viewpoint that the curvature of the gEUD curve is influenced by the optimization effect. The curvature stationary point of the gEUD curve (the vertex point, a = a_k) is expected to be a special point that leads to effective a-value selection. Eleven head and neck patient cases were used to verify the curvature effect. We used the Photon Optimizer (PO) of Eclipse for optimization and focused the upper gEUD to simplify the dose constraint for the organ at risk (OAR) that requires balancing of the overlapped planning target volume(PTV). Static seven-field IMRT was used for optimization, changing the a-value of the affected side of the parotid and retaining PTV D95% = 70Gy at the different a-value optimization. Finally, cDVH shift (ΔDVH), gEUD shift (ΔgEUD), their average values, and a_k were evaluated. The a = a_k optimization showed an intermediate effect of lower and higher a-values on ΔDVH, ΔgEUD, and their averages. “Lower” (a=0.5/1.0/2.0/3.0), “middle” (a=4.0/5.0/6.0/8.0/10/a_k), and “higher” (a=12/15/20/40) were defined using a=a_k as a base point. Lower a-value optimization was effective for the low-dose region and weakly affected the whole range of cDVH weight. In contrast, higher a-value optimization addressed the high-dose region and strongly affected the high-dose range of the cDVH weight as theoretically predicted. In addition, the middle range of the a-value optimization induced a decrease in the clinically important middle-to-high dose range, which retained the high dose of the PTV. Interestingly, the average ΔDVH and ΔgEUD corresponded exponentially to the curvature and the gradient of the gEUD curve. Using our relativistic optimization force concept, gEUD optimization is represented as a gEUD curve shift, highlighting that the curvature of the gEUD curve is the essence of gEUD optimization. The curvature stationary point (a = a_k), namely the vertex point of the gEUD curve, played an intermediate role in the low-to-high a-value condition. We can effectively select a lower/middle/higher a-value from a base point of a = a_k under clinically complex optimization situation.