Difference in LET‐based biological doses between IMPT optimization techniques: Robust and PTV‐based optimizations

Abstract Purpose While a large amount of experimental data suggest that the proton relative biological effectiveness (RBE) varies with both physical and biological parameters, current commercial treatment planning systems (TPS) use the constant RBE instead of variable RBE models, neglecting the dependence of RBE on the linear energy transfer (LET). To conduct as accurate a clinical evaluation as possible in this circumstance, it is desirable that the dosimetric parameters derived by TPS (DRBE=1.1) are close to the “true” values derived with the variable RBE models (DvRBE). As such, in this study, the closeness of DRBE=1.1 to DvRBE was compared between planning target volume (PTV)‐based and robust plans. Methods Intensity‐modulated proton therapy (IMPT) treatment plans for two Radiation Therapy Oncology Group (RTOG) phantom cases and four nasopharyngeal cases were created using the PTV‐based and robust optimizations, under the assumption of a constant RBE of 1.1. First, the physical dose and dose‐averaged LET (LETd) distributions were obtained using the analytical calculation method, based on the pencil beam algorithm. Next, DvRBE was calculated using three different RBE models. The deviation of DvRBE from DRBE=1.1 was evaluated with D 99 and D max, which have been used as the evaluation indices for clinical target volume (CTV) and organs at risk (OARs), respectively. The influence of the distance between the OAR and CTV on the results was also investigated. As a measure of distance, the closest distance and the overlapped volume histogram were used for the RTOG phantom and nasopharyngeal cases, respectively. Results As for the OAR, the deviations of DmaxvRBE from DmaxRBE=1.1 were always smaller in robust plans than in PTV‐based plans in all RBE models. The deviation would tend to increase as the OAR was located closer to the CTV in both optimization techniques. As for the CTV, the deviations of D99vRBE from D99RBE=1.1 were comparable between the two optimization techniques, regardless of the distance between the CTV and the OAR. Conclusion Robust optimization was found to be more favorable than PTV‐based optimization in that the results presented by TPS were closer to the “true” values and that the clinical evaluation based on TPS was more reliable.

Conclusion: Robust optimization was found to be more favorable than PTV-based optimization in that the results presented by TPS were closer to the "true" values and that the clinical evaluation based on TPS was more reliable. To take full advantage of IMPT, it is necessary to incorporate the biological effects of protons in the treatment planning process. In current clinical practice, a proton beam is delivered assuming a constant relative biological effectiveness (RBE) of 1.1. On the contrary, extensive preclinical evidence shows that the RBE varies across treatment fields. Particularly, it depends on linear energy transfer (LET), tissue-specific parameters (α and β), dose per fraction, and other factors. 7 Various phenomenological RBE models considering LET have been proposed, [8][9][10][11] and these are herein referred to as "variable RBE models." Some researchers use the variable RBEweighted dose in both the calculation and optimization of IMPT, 12 while others use both the physical dose and the biological surrogate (which is defined as the sum of LET Â physical dose and physical dose, yielding values similar to the variable RBE-weighted dose) 13 simultaneously in the optimization, to increase LET in tumors. 14 Furthermore, the biological surrogate is used to avoid the occurrence of high LET areas in critical organs. 15 However, as far as the authors' knowledge holds, no commercial TPS has so far been able to provide any option of utilizing LET during the optimization process, or to compute dose distributions weighted by a variable RBE.
To conduct as accurate a clinical evaluation as possible in this circumstance, it is desirable that the deviation of the TPS biological dose that is calculated using a constant RBE from the biological dose computed with a variable RBE, which is herein referred to as a "true" biological dose, is as small as possible. In this research, the authors have looked into the practical perspective of focusing on the PTV-based and robust plans created using commercial TPS. Biological dose distributions are computed using a variable RBE model, and their deviations from those computed with the constant RBE are evaluated for both the clinical target volume (CTV) and OARs. The influence of the distance between the OARs and CTV on the deviation size is also investigated.

2.A | Treatment planning
PTV-based and robust plans were made using VQA (Hitachi Ltd., Tokyo, Japan) for a Radiation Therapy Oncology Group (RTOG) benchmark phantom 2,8,19 and four nasopharyngeal tumor cases ( Fig. 1). In both phantom and patient plans, the PTV was generated by isotropically expanding the CTV by 3 mm. 20 In the RTOG phantom, different diameters of the OAR (15 and 12 mm) were used to examine the influence of the distance between the OAR and the CTV. The OAR was surrounded by the horseshoe-shaped PTV with inner and outer radii of 18 and 40 mm, respectively. For the nasopharyngeal cases, the brainstem and spinal cord were regarded as the OARs. The beam angles are also illustrated in Fig. 1.  In both PTV-based and robust plans, the prescription dose, D pres , was administered to the CTV, such that the D 99 of the CTV > D pres (RBE = 1.1). In PTV-based plans, an additional constraint was applied to the PTV, such that the D 95 of the PTV > D pres (RBE = 1.1). In the robust plans, the minimum D 98 of the CTV among the nine dose distributions, D 98,worst , was made higher than the 95% of the D pres (RBE = 1.1). 21 Spot-to-spot intervals were set to 5 mm for both plans.
In the RTOG phantom plan, D pres was set to 200 cGy (RBE = 1.1) and the dose constraint was applied to the OAR such that the maximum dose, D max , of the OAR < 140 cGy (RBE = 1.1). The dose was administered in one fraction. In patient plans, in accordance with the institutional protocol, D pres was set to 7140 cGy (RBE = 1.1) and the dose constraints to the spinal cord and the brainstem were set as follows: D max of the brainstem < 5400 cGy (RBE = 1.1) and D max of the spinal cord < 4600 cGy (RBE = 1.1). The dose was administered in 34 fractions.

2.B | Dose and dose-averaged LET calculations
The variable RBE and the biological dose were obtained by calculating the physical dose and dose-averaged LET (LET d ) distributions using analytical methods based on the pencil beam algorithm. 22,23 In this algorithm, the physical dose was calculated by convolving the fluence with the dose kernel. The dose kernel was represented by a triple Gaussian to include protons that underwent not only multiple Coulomb scattering but also large-angle scattering due to nuclear reactions. 22 For the LET d calculation, three-dimensional LET d distribution of an infinitesimal proton beam in water is defined as the LET kernel. Then, LET d of some point is derived by taking the dose average of all the LET kernels that contribute to that point. 23 Different LET kernels were created for primary Gaussian dose kernel and second, third Gaussian kernels, respectively. Each LET kernel was assumed to vary only in the depth direction and was constant in the lateral direction. A 2-mm calculation grid was used in both the dose and LET d calculations.

2.C | Biological dose calculation considering LET d
The RBE was calculated voxel by voxel using the phenomenological RBE model proposed by McNamara et al. 11 : where d i , L d;i , and α=β ð Þ x;i represent the physical dose per fraction, the dose-averaged LET, and the α=β ð Þ x parameter at the i th voxel, respectively. Nasopharyngeal tumor cases were also evaluated using the RBE models proposed by Wilkens et al. 8 and Wedenberg et al. 10 These results are shown in the discussion. For the RTOG phantom case, α=β ð Þ x parameters for the CTV and OAR were set to 10 and 3 Gy, respectively, while in the nasopharyngeal case, they were set to the values shown in Table 1. 24,25 The biological dose was  and D max were used as the evaluation indices for the CTV and OARs, respectively: where the deviations were normalized by the prescribed dose, D pres .
In addition, for the nasopharyngeal case, the authors verified whether the biological dose distribution evaluated with the variable RBE satisfied their institutional criteria of the OARs: D vRBE max of the spinal cord < 5000 cGy (RBE) and D vRBE max of the brainstem < 6000 cGy (RBE).

2.D.2 | Order of CTV-to-OAR distance
To investigate whether the distance between the OAR and CTV affected the magnitudes of ΔD 99 and ΔD max , ΔD 99 and ΔD max were compared between plans with different CTV-to-OAR distances. A definition of the order of CTV-to-OAR distances is described in the subsequent text.
First, for the RTOG phantom, the closest distance was used as the measure of the distance. Therefore, the OAR was closer to the CTV at an OAR radius of 15 mm. For the nasopharyngeal case, it is not unique to define CTV-to-OAR distances because the shapes of the targets and the OARs were more complicated than the RTOG phantom. Consequently, the authors used the overlapped volume histogram (OVH), which is generally exploited to characterize the three-dimensional spatial relationship between the CTV and the OAR for DVH prediction, 26,27 to define the order of the CTV-to-OAR distance.
The OVH indicated the overlapped volume fraction between the OAR and the tumor when the tumor was expanded at different distances. More specifically, the kth element of the OVH for the OAR O, OVH O;k , was calculated using the formula where O j j is the volume of the OAR, d p; CTV ð Þis the distance from the position p to the boundary of the CTV, and δ is the finite distance interval, set herein to 3 mm. The numerator represents the subset of the OAR whose distance from the CTV boundary is less than kδ: In this study, OVH was evaluated at k = 3 and greater value of OVH O;3 was regarded as the geometry with closer CTV-to-OAR distance.  where the OAR tended to be spared using lateral penumbra.    the CTV-to-OAR distance decreases, as was observed in the cases of the RTOG phantom.

| DISCUSSION
In this study, the deviations of the biological doses indicated by the TPS from those computed with the variable RBE model were compared between PTV-based and robust plans, which were created using a commercial TPS. For the CTV, the difference in the magnitude of ΔD 99 between the optimization techniques was negligible in both the RTOG phantom and nasopharyngeal cases, regardless of the values of the parameter α=β ð Þ x . For the OARs, the ΔD max for the robust plans was much smaller than that for the PTV-based plans (see Table 2 and Fig. 3), which indicates that the biological dose derived by the TPS is closer to the "true" biological dose, and thus is As for the dosimetric parameters for the CTV calculated with the variable RBE, the results of α=β   13,15 Using these techniques, it should become possible to further reduce the deviation not only for the OARs close to the tumor but also for the OARs distant from the tumor.
As described above, robust optimization has an advantage that it is capable of handling not only the physical uncertainty against setup and range errors but also (though not intentionally) the uncertainty against the biological dose. Though we have shown that the maximum biological dose in OAR is smaller in robust plans, the OAR volume receiving a low dose is larger than PTV-based plans, as shown in Fig. 2. This is because the robust plans often use the lateral penumbra to avoid the OAR instead of distal fall-off. Therefore, it should be decided which optimization to be used in clinics within these trade-offs.
Finally, as the scope of this study focused on the comparison between the RTOG phantoms and nasopharyngeal cases with only the brainstem and the spinal cord examined as the OARs, the authors believe that the findings herein could be established in more general settings in a future study that involves the use of different treatment sites and OARs.

| CONCLUSION
Under the circumstance that the current commercial TPS indicates only the biological dose evaluated with a constant RBE, it is of practical importance that such biological doses derived by the TPS should be as close to the "true" biological dose (the biological dose calculated with variable RBE) as possible. The result of the comparison between the PTV-based and robust plans of the RTOG phantom and nasopharyngeal tumor cases indicated that the deviations of D vRBE max from D RBE¼1:1 max of OARs tend to be smaller in robust plans as compared to PTV-based plans. In addition, the deviation becomes larger as the OAR is located closer to the CTV. Similar tendencies were observed in three different RBE models. Therefore, robust optimization was found to be more favorable than PTV-based optimization in that the results presented by the TPS were closer to the "true" values, and thus clinical evaluation based on these results will be more reliable when employing robust optimization.

CONFLI CT OF INTERESTS
We disclose Shusuke Hirayama and Takaaki Fujii are paid from Hitachi, Ltd., Tokyo, Japan.