Relativistic Optimization Force Concept for gEUD Biological Optimization and Novel a-value Selection Viewpoint


 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.


Introduction
In radiotherapy, intensity modulation radiotherapy (IMRT) planning, which includes the static-field 50 technique and volumetric modulated arc therapy (VMAT), is calculated using the inverse optimization 51 process. 1 Dose-volume histogram (DVH)-based optimization is widely used for the optimization 52 process. 2-8 DVH-based optimization systematically addresses the three-dimensional (3D) dose 53 distribution using the parameters of dose and volume with a priority weight ratio for optimization 54 (hereinafter called the optimization weight), whereas treatment regions in the planning target volume 55 (PTV) often occur near organs at risk (OARs), and this complicated balance often causes difficulties for 56 the treatment plan. 3 Therefore, the optimization process is usually dependent on the experience of the 57 planner and treatment clinical goals that reflect the facility characteristics of radiotherapy. 9 58 Maximal and minimal dose-volume (DV) constraints are typically used to perform clinical inverse 59 treatment planning, 4,5 which is a standard criterion to evaluate whether the treatment plans are clinically 60 acceptable. 6,8 Moreover, DV optimization efficiently and effectively addresses the complicated relationship 61 of between the tumor target and OARs. Although the organ evaluation criteria Pareto surface is naturally 62 included by the DVH Pareto surface in the optimization, 9,10 the volume control method is not always 63 directly linked with the details of the dose distribution of interest. This is because it lacks the spatial 64 information between the dose-volume and dose distribution. Biological optimization is an alternative 65 solution to control the optimization; 11 this is originally derived from the idea of introducing the radiation 66 response in the tissue into the optimization process 12 . The generalized equivalent uniform dose (gEUD) is 67 one of the criteria for a biological dose, assuming a uniform dose proposed by Niemierko,13,14 which 68 provides the same biological effect to the tissue as an inhomogeneous dose distribution; the 69 tissue-specific difference is represented by the -value. However, this biological optimization allows a 70 wide range of DV domain control; therefore, dose constraints for optimization become easier to implement. 71 Multiple DV constraints can be replaced with a single EUD-based cost function 15 . Meanwhile, biological 72 optimization provides a wide range of dose-volume constraints, and it allows the voxels to deliver an 73 inhomogeneous dose distribution to the same organ. 16,17 For the proper use of biological optimization, it is 74 essential to evaluate whether the optimized result and final dose distribution are clinically appropriate 15 . 75 The Photon Optimizer (PO) In the Eclipse planning system (Varian Medical Systems, Palo Alto, 76 CA, USA) is the optimization engine for IMRT and VMAT. This system provides gEUD optimization, which 77 is primarily controlled by the -value and its optimization weight. The role of the gEUD during the 78 optimization process has been reported by Fogliata et.,al. 18 They reported different -value effects for 79 the OAR in the optimization process for a virtual phantom to investigate the -value and the distance 80 between the OAR and target. The results indicate that the PO showed a trade-off between the tumor 81 target and OAR dose, resulting in a decrease in the target high dose with an increase in the -value; 82 however, the effectiveness of the variable -value and optimization weight of the OAR against the target 83 have not been clarified under clinically complex optimization conditions. Originally, the -value 84 represented a tissue-specific parameter, whereas the -value behaves as the DV cost function for particular domains of the DVH in the optimization process. An appropriate use of the gEUD enables simple and effective optimization, whereas elusive characteristics that depend on an arbitrary DVH shape 87 remain in the empirical usage. 19 The aim of this study is to clarify the effect of different -values in the 88 optimization process among existing multiple-body optimization objectives by using the originally 89 formularized concept of relativistic force in optimization. This provides a novel and effective use of the 90 -value in the optimization process under clinically complex situations such as when OARs are adjacent 91 to the tumor target requiring harmful high doses for the normal tissue because of PTV marginal regions. 92 93

94
Mathematical characteristics of gEUD 95 The DVH curve can be represented by the following two types: the cumulative DVH (cDVH) and the 96 differential DVH (dDVH). The latter is used for gEUD calculations as follows: 97 where ( is the domain volume that received a dose of ( , and * is the total volume domain of the 98 organ. The gEUD is a typical representation of the weighted Minkowski distance. In the integral 99 expression for the derivative analysis form with the cDVH ( ), Eq. (1) is naturally expanded as 100 dose constraints for the target and OARs, resulting in an optimized cDVH. Now, we consider the change 117 in the cDVH during the optimization process, which results from the pre-optimized cDVH curve receiving 118 an external force and changing its form into an appropriate optimized cDVH curve. During the cDVH curve 119 change with proper time in the optimization, the corresponding gEUD curve ( ) changes its form as 120 well retaining the relation of cDVH and gEUD which is provided by Eq. (2). Let the external force that 121 addresses the gEUD curve change be the optimization force. Selecting an appropriate coordinate system, 122 this gEUD curve deviation caused by the optimization force is represented by the following geodesic 123 deviation system with an external force 21 : 124 denotes an optimization force that represents an input factor as a priority weight of optimization, which is 128 an arbitrary defined balancing factor between the targets and OARs. During the optimization, the 129 optimization priority weight value is a variable parameter; however, this parameter is typically fixed. The 130 absolute derivative operator / has the following expression: 131 We define the orthogonal , coordinate system to evaluate a non-linear , system as follows: the 132 -coordinate represents geodesic lines obtained from the gEUD curve ( ) , and the orthogonal 133 -coordinate can be defined when is selected during every change in the proper time. Therefore, we 134 can obtain , coordinates from the , coordinate transformation and its relation of inverse function is 135 as follows: 136 The metric tensor \] from ,^ is: 137 Therefore, the non-zero affine connection coefficients are: where (b is the Ricci curvature tensor and vanishes in the coordinates of \] , and the Gauss curvature 143 = 0 (Supplementary Information Appendix Eq. (B2)). Therefore, , and , coordinate surfaces 144 can transforme each other without distortion by planar deployment. This characteristic of zero Gaussian 145 curvature is called a developable surface in differential geometry. Here, we focus the total action ∫ I , 146 which is the term used for representing the effects of optimized conditions on the change in the gEUD. 147 Integrating both sides of Eq. (9) and arranging the terms, we obtain the following equation: 148 where I is a constant vector against . Eq. (10) indicates that the action of the optimization force is 149 weakened by the reaction of the change in the gEUD. To maximize the action of the optimization force, i.e., 150 to maximize I , the following equations are required, 151 As for \ , depicted in Fig. 1(b), by using the relations of Eq. (6), the former differential equation of Eq. (11) 152 becomes 153 where represents the curvature of the gEUD curve in an , coordinate system (Supplementary 154 Information Eqs. (B3) and (B4)), 155 Then, integrating both sides of Eq. (12) with , the norm of \ becomes 156 Introducing time mean curvature ( ) against to Eq. (12), and considering that the magnitude of the ( The term of ( / ) converges to a specific value (a function of ) in a finite optimization time o\T . Let 160 the converged value of the integration term be ( ) and its average ( ), then the final form is obtained as a pure function of : This relation exhibits the exponential curvature effect for the shifting gEUD. Let the newly defined be 163 the total optimization effect, 164 ( where ? @ of curvature of the gEUD curve is 0, i.e., κ/ = 0. The vertex point is numerically obtained by the 185 below Eq. (18) using the first, second, and third numerical derivative calculation of the gEUD curve with 186 second-order accuracy, which is followed by solving the equation using the Newton method because the 187 Eclipse PO is limited to receive only the first decimal place of the -value; therefore, numerical accuracy 188 Let and be the cDVH and gEUD deviations from the control case, respectively.
To numerically verify these evaluation from the theory, we used the following numerical summation for = IMRT with gantry angles of 50°, 70°, 150°, 180°, 210°, 290°, and 310° were used, and the optimization 204 weights for the regions are shown in Table 1. Case #00 was selected as a representative case to verify 205 the optimization weight ratio between H_Parotid and the PTVs. The remaining ten cases were calculated 206 with a fixed optimization weight ratio condition (PTVs vs OARs = 200:60). We changed the -value of the control case (zero optimization weight for H_Parotid). To normalize the optimization effect by optimization 209 force, we obtained the cDVH and the shifted gEUD according to -value changed optimization after the 210 D95 normalization of PTV70opt defined in Fig. 2

241
First, we verified the optimization weight condition in the representative case with a detailed -value (Fig.  242 3). The intended dose-volume naturally decreased with an increasing optimization weight of H_Parotid. 243 The weak optimization weight ratio did not address the higher dose-volume and -value but instead 244 exacerbated it. In contrast, a strong optimization weight ratio can achieve preferable conditions for 245 H_Parotid; however, this sacrificed the higher dose range of PTV and adversely influenced the D95 246 scaling. Therefore, in this study, we selected an optimization weight ratio between H_Parotid and PTVs as 247 60:200 for balancing. 248 Next, we verified the -value changed effect for the optimization under clinically multi-body 249 optimized objects. The average ± SD of the DV clinical criteria for the eleven head and neck cases are 250 shown in Table 2. This table indicated that the gEUD optimization for the H_Parotid achieved the selective 251 dose-range optimization with the change in -value retaining the PTV condition of the high doses. The 252 detailed analyses for the cDVH are as follows. The representative patient case (Case #00) is shown in Fig.  253 4, exhibiting dose distribution, the DVH, and the gEUD. In the extremely low -value case ( = 0.5), the 254 optimization force weakly addressed the whole range of dose-volume and was not effective for the higher dose region. Lower -value optimization ( = 1, 2, 3) was primarily effective for the lower-to-middle dose 256 region (<30Gy) and achieved a mean dose decrease, whereas the gEUD shift was small. Higher -value 257 optimization ( = 12, 15, 20, 40) focused on the higher dose region (>60Gy) and the gEUD shift was large, optimization. However, the rate was not increased by more than 7.0%; that is, not more than 266 approximately 80-100 MU (Fig. 6(c) deformed compared with the control case, resulted in drifts of the curvature of the gEUD curve (Fig. 7(c)). 272 The length of the drifts did not exceed ±1~2 orders for each patient case excluding higher -value 273 conditions, namely = 20 and 40 (Fig. 8) The vertex point of the gEUD curve also exhibited a drift. This is equivalent to the curvature stationary 278 point ( / = 0), which was not scattered in the range (Fig. 7(d)).  0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 V54Gy [cc] 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 We applied the relativistic optimization force concept to the gEUD optimization and theoretically 339 ! = ! # to the tumor target under clinical multi-body optimization objects. The concept is focused on the fact that 341 the gEUD of an optimization object retains its curve characteristics despite the optimization process. Such 342 a group of curves can be treated as geodesics in differential geometry. The optimization effect is regarded 343 as the deviation of the geodesics, deductively leading to the evaluation of the curvature of the gEUD 344 curve as shown in Eqs. (4)-(17). The original optimization effect in this theory contains the proper time 345 effect evaluated as the differential equation of Eq. (12). There is a possibility that the curvature of the 346 gEUD provides a more effective method for optimization against local minima problems by applying 347 dynamic -value during optimization, which is beyond the scope of this study. The exponential relations 348 of the curvature or the gradient of the gEUD curve to the average values or were shown 349 in this study, and these were in accordance with the theory, although there was a translation against the 350 -value (Figs. 7(a) and 7(b)).
grounds the deviation at the same ≃ , represented by \ , 351 the dominant term is ≪ 1 for negative exponential due to ( G / G ) < 0. grounds the 352 deviation of the same , the exponential effect is from the transformation of \ → , (Supplementary 353 Information Eq. (B7)). The minus scale term is required for the exact match. 354 In this study, we used a moderate optimization weight ratio between the H_Parotid and PTVs to 355 verify the relativistic optimization force. Parametric analysis for a change in the -value has been widely 356 evaluated 9,22-24 as a promising and effective tool to improve the OAR dose. The 0mm-cropped case, 357 reported by Fogliata et. al., 18 is in good agreement with the results of and in this study. 358 They concluded that the gEUD relationship to the -value depends on the DVH shape, and there is no 359 correlation between the gEUD and the mean dose. However, the gEUD optimization has been understood 360 in an empirical manner, which is not essential comprehension. From our results, gEUD optimization was 361 represented as the curvature of the gEUD curve and its change. The middle range of -values was very 362 effective, despite different level of PTV and other OAR optimization forces. In particular, = s indicated 363 the intermediate optimization effect of lower and higher -value effects (Figs. 5(a)-5(c), 6(a) and 6(b)). 364 The representative dose distribution, with optimization for a changing -value, explicitly depicted clinical 365 applications ( Fig. 4(a)). Focusing on the head and neck cases, a middle-to-high range of dose decrease 366 was required to decrease the mean dose of the parotid gland. In other words, lower and middle -value 367 gEUD optimization was effective for this case. If a high dose region is a clinical matter such as the region 368 adjacent to spinal cord, a higher -value gEUD optimization is appropriate. Note that a higher -value 369 gEUD optimization required a trade-off of the erosion of the PTV dose coverage (Fig. 4(b) and 4(c)). In 370 addition, gEUD optimization was only a range effect for the input dose; thus, it did not necessarily 371 decrease the intended dose. However, this is useful and is permitted if the OAR is heavily prioritized over 372 the PTV, such as aiming for a dose decrease of spinal cord to be hollowed out. Extremely low -value 373 gEUD optimization also encountered problems that required sequentially lower dose-range optimization, 374 leading to the prevention of beam paths and increasing the tangential effect. However, this characteristic 375 is very useful for limiting the lower dose of lung regions. The particular use of the -value should depend 376 on the clinical situation. gEUD curve = s (Fig. 9). We can determine the appropriate -value using this, whereas the 379 information for the gEUD curve of the control case is typically not visible. Additionally, gEUD curves 380 deform; therefore, curvature drifts are observed during and post optimization, despite the fixed -value 381 condition, as shown in Fig. 8(c). However, the curvature stationary point is not scattered among 382 optimizations that do not have extreme changes in the -value as shown in Fig. 8(d) and Fig. 9. The 383 outlier cases in Fig. 8 were mostly for = 20 and excluded = 40. The optimization for the extremely 384 high -values affected the lower dose to increase in addition to erosion of the higher dose regions, 385 causing of the vertex of the gEUD curve to shift to a lower -value. Nevertheless, it is sufficient to grasp 386 the current state of the gEUD curve and its vertex point to consider a strategy for better optimization. We 387 propose the following process to maximize the gEUD optimization effect: (1) creating a temporal plan, 388 which is a PTV-focused optimization with no dose constraint for the OARs; (2) obtaining = s for the 389 gEUD curves of the OARs from the cDVH of the temporal plan; and (3) applying an appropriate -value 390 to the optimization, considering = s for lower than/ higher than/ both sides of the dose we intend to 391 intensively decrease. This usage of -value based optimization would help the planner to avoid the 392 optimization labor without highly-diversified dose-volume setting and promotes an effective equalization of 393 optimization cuisines without requiring the planner to have empirical knowledge. 394 395 Figure 9. Schematic concept of strategical optimization for a changing -value. The vertex point of the gEUD 396 curve becomes a base point against a "lower" dose and "higher" dose.

399
This study clarified further mathematical characteristics of the gEUD and the novelty that the gEUD 400 optimization essentially corresponds to the curvature of the gEUD curve, which is derived from originally 401 defined relativistic optimization force concept. We defined the vertex point of the gEUD curve as a base 402 point and this leads to effective a-value selection against lower, higher, and both sides of the dose 403 distribution. Thereby, the mathematical characteristics of the gEUD facilitate its effective use for 404 optimization in a clinically complex situation without empirical manners. 405 406