Dose/volume parameters are widely used to estimate the probability of normal tissue injury (1–4). However, these parameters are only part of the information contained in the dose volume histogram (DVH) curve. For instance, dose delivered to 1 cc (D_{1 cc}) is only a discrete point on DVH. Some normal tissue complication probability (NTCP) models utilize all information of the DVH curve by compressing the entire curve into a single factor (5–7). However, most DVH reduction models are based on estimated complication probability under uniform irradiation (5), which could not be used for non-uniform dose distributions directly. To solve this problem, the equivalent uniform dose (EUD) is often introduced (8–10).

EUD is a simple power-law assumption that may not correspond to the weight changes of all doses. If the doses vary over a wide range, this uncertainty may increase and lead to a decrease in predictive ability—the fitting of power exponent (a or 1/n) should be more accurate in a small dose range than in a large dose range. Since both volume and dose are factors of EUD formula, the proportions of volume bins in different dose ranges should theoretically influence the volume-effect parameter. To increase the weight of critical dose, the power exponent might be larger in the case that the proportion of inessential low-dose volume becomes larger. In order to identify these potential defects and try to improve them, this study explored the impact of dose distribution on the EUD-based prediction model of radiation-induced brain injury (BI) for nasopharyngeal carcinoma (NPC) patients, taking into account the large enough low-dose volume and typical symmetrical objects including structure, field setup and location of injury.

Modern radiotherapy techniques such as IMRT, volumetric arc radiotherapy (VMAT), and stereotactic body radiotherapy (SBRT) are able to spare normal tissue and reduce side effects (5), resulting in rather non-uniform dose distributions for normal tissue (14). To make sure the traditional NTCP models based on uniform dose still work, effective volume reduction scheme and effective dose reduction scheme have been proposed (15–17). Together, the EUD equation and the Lyman assumptions are often referred to as the Lyman–Kutcher–Burman (LKB) model which has been widely used (5). Other EUD-related models include Logit-EUD model, Schultheiss model (18), Poisson-EUD model, Källman model (19), and Parallel model (20). Regardless of which model is selected to fit clinical data, the intrinsic difference of dichotomous data is the basis of prediction accuracy. In the study Ospina, the six NTCP models above showed almost identical predictive ability for late bladder complications with similar ROC (21). EUD is essentially a redistribution function of dichotomous data. To improve its predictive power, EUD must be able to improve the AUC.

EUD-based optimization of radiation plan has been reported in some studies (22–24). However, there is no guarantee that the parameters are generic and stable in different radiotherapy techniques (such as IMRT and SBRT). As mentioned in the introduction, the volume-effect parameter might be influenced by the dose distribution. In this study, radiation-induced BI model for NPC patients was selected as the object due to its following advantages: 1) fewer setup errors and organ movements; 2) clear boundaries for delineation; 3) clear imaging findings; 4) clear dose–injury relationship with almost no effects of infection or other factors; 5) symmetrical structure and dose distribution for symmetrical segmentation; and 6) relatively concentrated high-dose region and relatively independent injury in temporal pole for asymmetric segmentation. In this study, average-DVH curves of left half-brain, right half-brain and brain overlapped almost exactly, and the differences were negligible, indicating that half-brain could be viewed as an independent structure with a dose distribution similar to that of brain. In addition, V _D with different D could be viewed as reference volumes with different dose distributions.

The purpose of half-brain delineation was to study whether the a value and mAUC were influenced by volume size. In this study, both the optimal a values and their changes were similar in brain group and half-brain group, and also mAUC, indicating that volume size may not be a significant factor. Instead, the dose distribution may be an important factor. The results showed that the optimal a value of EUD_{V D} was correlated with D strongly, and it gradually decreased to 1 with the increase of D, indicating that the optimal a value is variable if the dose distribution is uncertain. The variation of volume-effect parameter in different volumes may be due to the variation of dose distribution rather than to the variation of volume size itself.

An important role of EUD is to convert the doses of entire organ to suit the findings of Emami (25). However, modern radiotherapy techniques rarely involve one third, one half or entire organ being irradiated uniformly. The EUD conversion of entire organ doses may reduce the predictive efficacy, especially in a large organ with only a small volume exposed to high dose, since the low-dose volume may affect the role of crucial doses and increase the uncertainty of a value. The study of Heemsbergen showed that the local dose–effects were most pronounced in intermediate-high dose regions for late gastrointestinal toxicity (26). If the EUD was limited to a defined high-dose region, the a value would be more stable and the AUC should be larger. However, if the volume of critical dose was removed excessively, the AUC would decrease. In this study, the mAUC of EUD_{V D} reached the maximum value when D was around 50–55 Gy, indicating that 50–55 Gy might be the critical low dose for BI. On the other hand, regardless of what a value was selected, AUC of EUD_{V55 Gy} was always larger than that of traditional EUD, further supporting the removal of low-dose volume.

It looked like the brain tissue went from being a series organ to being a parallel organ when a value decreased from 21.5/22 to 1, which might not fit the original definition of volume-effect parameter. However, volume-effect parameter of series/parallel organ is essentially an indirect estimation for complication which depends on the location and extent of direct biological damage. In this study, volume-effect parameter of whole volume seemed to only play a role in reducing the interference of low-dose volume. If a novel radiotherapy technique leads to more/less low-dose volume, a larger/smaller a value may be needed. Therefore, a parallel/series organ should not be defined directly by the volume-effect parameter.

In this study, whole-volume EUD was not superior to all simple dose/volume points, while EUD_{V55 Gy} could improve the predictive ability, although the difference may not be significant (with AUCs within the 95% CI of the others). However, an overwhelming predictor is almost impossible considering that the continuous variables are highly correlated (for example, D_{1 cc} is never significantly superior to D_{1.01 cc}), so the significant trend of AUC may also be important. Given a larger AUC and a more stable a value, EUD_{V D} with an optimal D should be a better choice which is less influenced by dose distribution. From the results, in order to fit the low-dose volume together, the best weight relationship of critical doses lost, which may be the reason why whole-volume EUD did not show an advantage.

Mean dose may be the case in which EUD_{V D} has the best predictive ability, but the critical dose needs to be determined first. In this study, the optimal reference volume of EUD model for BI may be V_{50 Gy}–V_{55 Gy} (V_{46 Gy}–V_{53 Gy} for equivalent dose in 2 Gy per fraction). When the reference volume ranged from V_{45 Gy} to V_{65 Gy}, the AUC of mean dose was not inferior to traditional EUD, and when the reference volume ranged from V_{50 Gy} to V_{60 Gy}, the AUC of mean dose was better than traditional EUD and dose/volume points, indicating that even if there is some error in critical dose, the predictive ability is still advantageous in a certain dose range. In fact, it is easier and more reliable to find an optimal mean dose than to fit an unstable parameter. On the other hand, since the volume covered by 55 Gy is related to BI and contributes to the predictive ability, the sub-high dose volumes of brain should be fully delineated (not just temporal lobe) and enrolled in optimization plan to better protect brain tissue. EUD_{V55 Gy} may be a suitable dose constraint index for brain.

It should be clarified that the fundamental purpose of different reference volumes is not to study a sub-structure EUD against traditional EUD, but to enlarge the potential errors (EUD with a large low-dose volume is not better than simple dose/volume points; volume-effect parameter is variable in different dose distribution) in a typical organ and try to improve them. In addition, the higher incidence of BI may be related to advanced T-stage and follow-up bias (symptomatic patients were more likely to complete follow-up). There are several limitations in this study. Firstly, the optimal a value was selected by simple calculation with limited accuracy, although the optimal EUD_{V D} does not need an “accurate” a value other than 1. Secondly, the optimal a value of 1 is just an assumption that requires more evidence. Thirdly, the conclusions may not be applicable to other organs and more researches are needed. Finally, the division of reference volume is crude, and further studies and more accurate control groups are needed.

Volume-effect parameter of EUD is variable and related to dose distribution. Before referencing a EUD-based model, the similarity of dose distribution should be confirmed. EUD with large low-dose volume may not be better than simple dose/volume points. EUD of critical-dose volume could improve the predictive ability and has an invariant volume-effect parameter. No matter what a value is selected, critical-dose-volume EUD may always be better than whole-volume EUD. Mean dose may be the case in which critical-dose-volume EUD has the best predictive ability.

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

The studies involving human participants were reviewed and approved by the Ethics committee of Second Affiliated Hospital of Guangxi Medical University. The patients/participants provided their written informed consent to participate in this study.

W-QL and R-SW participated in the study concept and design. H-JZ, H-YY, X-DL, XO, Q-LZ, D-JL, Y-TX, and Q-FL participated in the acquisition of data. Q-HD and JL participated in the analysis and interpretation of data. Q-HD participated in the statistical analysis. Q-HD and Y-XG participated in the drafting of the manuscript. All authors contributed to the article and approved the submitted version.

This work was supported by the Science Foundation of Guangxi Zhuang Autonomous Region Health and Family Planning Commission [award Z20181010] and the Science Foundation of Second Affiliated Hospital of Guangxi Medical University [award EFYKY2020008].

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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