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Background: Prostate cancer (PCa) is known to be suitable for hypofractionated radiotherapy due to the very low α/β ratio (about 1.5–3 Gy). However, several randomized controlled trials have not shown the superiority of hypofrac‑ tionated radiotherapy over conventionally fractionated radiotherapy. Besides, in vivo and in vitro experimental results show that the linear‑ quadratic (LQ) model may not be appropriate for hypofractionated radiotherapy, and we guess it may be due to the influence of fractionation schedules on the α/β ratio. Therefore, this study attempted to estimate the α/β ratio in different fractionation schedules and evaluate the applicability of the LQ model in hypofractionated radiotherapy. Methods: The maximum likelihood principle in mathematical statistics was used to fit the parameters: α and β values in the tumor control probability ( TCP) formula derived from the LQ model. In addition, the fitting results were substi‑ tuted into the original TCP formula to calculate 5‑ year biochemical relapse‑free survival for further verification. Results: Information necessary for fitting could be extracted from a total of 23,281 PCa patients. A total of 16,442 PCa patients were grouped according to fractionation schedules. We found that, for patients who received conventionally fractionated radiotherapy, moderately hypofractionated radiotherapy, and stereotactic body radiotherapy, the aver‑ age α/β ratios were 1.78 Gy (95% CI 1.59–1.98), 3.46 Gy (95% CI 3.27–3.65), and 4.24 Gy (95% CI 4.10–4.39), respectively. Hence, the calculated α/β ratios for PCa tended to become higher when the dose per fraction increased. Among all PCa patients, 14,641 could be grouped according to the risks of PCa in patients receiving radiotherapy with different fractionation schedules. The results showed that as the risk increased, the k (natural logarithm of an effective target cell number) and α values decreased, indicating that the number of effective target cells decreased and the radiore ‑ sistance increased. Conclusions: The LQ model appeared to be inappropriate for high doses per fraction owing to α/β ratios tending to become higher when the dose per fraction increased. Therefore, to convert the conventionally fractionated radiation *Correspondence: firstname.lastname@example.org; email@example.com‑ cu.ac.jp Xian‑Shu Gao and Yuta Shibamoto contributed equally to this paper Department of Radiation Oncology, Peking University First Hospital, Peking University, Beijing, People’s Republic of China Department of Radiology, Nagoya City University Graduate School of Medical Sciences, Nagoya 467‑8601, Japan Full list of author information is available at the end of the article © The Author(s) 2022. 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The Creative Commons Public Domain Dedication waiver (http:// creat iveco mmons. org/ publi cdoma in/ zero/1. 0/) applies to the data made available in this article, unless otherwise stated in a credit line to the data. Cui et al. Radiation Oncology (2022) 17:54 Page 2 of 11 doses to equivalent high doses per fraction using the standard LQ model, a higher α/β ratio should be used for calculation. Keywords: LQ model, α/β ratio, Hypofractionated radiotherapy, High dose per fraction, Prostate cancer Background using an established mathematical calculation method to With the development of high-precision radiotherapy, estimate the variation in the α/β ratio of PCa according to fractionation schedules to treat various tumors are the daily fractional dose and to verify the applicability of changing . Hypofractionated radiotherapy is being the LQ model in hypofractionated radiotherapy. increasingly employed in clinics as stereotactic body radiotherapy (SBRT) and moderately hypofractionated Methods intensity-modulated radiotherapy (IMRT), which have Clinical data collection become valuable therapeutic approaches for a variety of We searched for relevant articles in PubMed with key tumors owing to the improved dose distribution. In addi- words of “radiotherapy” or “radiation therapy” and “pros- tion, for prostate cancer (PCa), hypofractionated IMRT tate cancer” or “prostatic carcinoma”. The inclusion cri - and SBRT seem to have radiobiological advantages based teria were as follows: (1) patients with PCa undergoing on the linear-quadratic (LQ) model estimation. conventionally fractionated radiotherapy, moderately Since definitive hypofractionated radiotherapy is a hypofractionated radiotherapy, or SBRT, with or with- relatively novel treatment, optimal dose fractionation out androgen deprivation therapy (ADT) and (2) 5-year schedules often need to be inferred from mathemati- biochemical relapse-free survival (5y-bRFS), number cal calculation, and an LQ model-based formula is fre- of patients, total dose, and fraction number or dose per quently used to convert the conventionally fractionated fraction available from the articles. Articles that lacked radiation doses to high doses per fraction by clinicians the necessary fitting data or that used other fractiona - due to its convenience and simplicity . Recently, how- tion schedules, such as hyperfractionated radiotherapy, ever, several investigators demonstrated that the stand- were excluded. Conventionally fractionated radiotherapy ard LQ model may not be applicable to hypofractionated was defined as that using 1.8–2.1 Gy per fraction. Moder - radiotherapy especially in SBRT [2–6], although other ately hypofractionated radiotherapy was defined as that researchers insist that the LQ model can be used to esti- using 2.19–3.5 Gy in our study. SBRT was defined as that mate the antitumor effects of hypofractionated radio - using 6.5 Gy per fraction or greater according to National therapy [7, 8]. Comprehensive Cancer Network (NCCN) guidelines. The α/β ratio is a key factor in the LQ model . Basi - Five-year bRFS according to the ASTRO or Phoenix defi - cally, the α/β ratio of a tumor is obtained from an in vitro nition was evaluated. The ASTRO definition of biochem - dose-survival curve of tumor cells [2, 6], but this method ical relapse is three consecutive rises in prostate-specific cannot be applied to human tumors in patients. The α/β antigen (PSA) from the nadir . The Phoenix definition ratio can also be obtained from in vivo tumor or normal of biochemical relapse is a rise of PSA over 2 ng/mL from tissue responses to different fractionation schedules, and the nadir . Risk stratification of PCa was mostly made following this in vivo method, a mathematical method according to the NCCN guidelines risk group classifica - was elaborated to estimate the α/β ratio from clinical tion and in part D’Amico’s classification. data employing various fractionation schedules [9, 10]. Using the method, the α/β ratios for various tumors have Estimation of the α/β ratios been reported, and PCa was found to have a low α/β ratio Statistical analyses were carried out exactly following [9, 10], which was lower than the α/β ratio for normal tis- the method of Miralbell and coworkers [15, 16]. Briefly, sue late reactions. Accordingly, moderately hypofraction- standard LQ models for tumor control at 5 years of the ated IMRT and SBRT are being increasingly used in the form: treatment of PCa. However, since the reliability of the LQ P = exp −exp k − αD − βD /N model in SBRT was questioned in recent studies [4–6], it (1) may be necessary to re-evaluate the validity of converting or conventionally fractionated doses to hypofractionated doses with the LQ model. Previous studies only tried to P = exp(−exp(k − αD − βDd)) (2) demonstrate that PCa has a low α/β ratio [9–12], and var- iability of the α/β ratios with the dose per fraction has not were fitted to the obtained data. In the formula, P is been investigated. Therefore, we carried out an analysis interpreted as the tumor control probability (5y-bRFS); C ui et al. Radiation Oncology (2022) 17:54 Page 3 of 11 D is the total dose; N is the number of fractions during method was checked whether there was statistical differ - the whole radiotherapy; d is the dose per fraction; and k ence between the calculated P and the P from the original represents the natural logarithm of an effective target cell study. The statistical software SPSS 22.0 was used. number. α represents unrepairable lethal damage caused by a one-track action and β represents repairable sub- Results lethal damage caused by a two-track action in the DNA There were 45 articles (23,281 PCa patients) incorpo - damage repair kinetics . So, the α/β ratio can be con- rated in this study published during 2003 to February sidered as the balance between the two forms of damage. 2021. Detailed information on the 45 articles is shown Fitting was performed by maximum-likelihood meth- in Table 1. Among them, 5 articles reported on more ods and parameter estimates were obtained using the than 1000 patients [20–24], and 3 reported on 500–1000 custom-written code in Stata version 12.0 [15, 16, 18]. patients [25–27]. Among all 45 studies, a total of 38 arti- The custom code is shown below: cles (16,442 PCa patients) could be grouped according to fractionation schedules of radiotherapy and complete information could be extracted based on the inclusion capture program drop myprog and exclusion criteria; 15 were enrolled in the conven- program myprog tionally fractionated radiotherapy group, 24 were enrolled args lnf k d c rab in the moderately hypofractionated radiotherapy group, quietly replace `lnf’ = $ML_y1*$ML_y2*ln(exp(- and 8 were enrolled in the SBRT group. Nine articles exp(`k’-`d’-`c’/`rab’))) + $ML_y1*(1-$ML_y2)*ln(1- were duplicated since the studies investigated both con- exp(-exp(`k’-`d’-`c’/`rab’))) ventional fractionation and moderate hypofractionation. end Detailed data from each article are shown in Tables 2, 3 constraint 1 D = C and 4 according to the three different regimens of radio - ml model lf myprog (k:All P =) (d:D, nocons) (c:C, therapy. To explore the relationship between α/β ratios nocons) (rab:), constraint(1) and risks of PCa, we also divided each group into three ml search subgroups by the risks in patients receiving radiotherapy ml maximize with different fractionation schedules. Of all the 45 stud - ml graph ies, 21 studies (14,641 PCa patients) could be grouped and the characteristics are shown in Additional file 1: The parameter estimation was directed by Professor Tables S1 to S3. Geng and Dr. Yin of the School of Mathematical Sci- Estimated α/β ratios are shown in Table 5. Among all ences, Peking University. 16,442 PCa patients, 7793 patients received convention- In order to reduce the errors of fitting parameters, we ally fractionated radiotherapy, and the average α/β ratio performed jackknife method. We removed one group was 1.78 Gy (95% confidence intervals (CI): 1.59–1.98, every time from the 22 groups of data (taking conven- P < 0.001). There were 6822 patients in the moderately tionally fractionated radiotherapy group as an example); hypofractionated radiotherapy group. The α/β ratio we could get 22 groups of sample data, which is C21 22in was 3.46 Gy (95% CI 3.27–3.65, P < 0.001). In the SBRT mathematics. Using these 22 groups of data, the aver- group of 1827 patients, the α/β ratio was 4.24 Gy (95% age and standard error of the sample can be obtained, CI 4.10–4.39, P < 0.001). We go a step further and calcu- and then the population mean and the confidence inter - late the BCa intervals of each groups:1.78 Gy (Bca 95% val of fitting parameters can be obtained. Moreover, we confidence intervals (CI): 1.62–1.97), 3.46 Gy (Bca95% CI go a step further and calculate the Bias-Corrected and 3.30–3.66), 4.24 Gy (BCa95% CI 4.14–4.34). Hence, the accelerated(BCa) intervals according to the Jung et al.’s calculated α/β ratios for PCa tended to become higher methods. For the parameters of the LQ model, sig- when the dose per fraction increased. However, the k and nificant figures were rounded to the 2nd decimal place. α values were not affected by fractionation schedules. In order to verify the accuracy of the results, we sub- The k value was calculated as 5.35 (95% CI 4.61–6.08, stituted the fitting results: k, α and α/β values and P < 0.001), 1.15 (95% CI 0.21–2.09, P = 0.017), and 1.67 known parameters, i.e., total dose (D), single dose (d) (95% CI − 4.80–8.15, P < 0.61), respectively, in patients and total fraction number (N), into the original TCP for- receiving conventionally fractionated radiotherapy, mod- mula (1) or (2) and got a calculated P (5y-bRFS). Using erately hypofractionated radiotherapy and SBRT. The α goodness of fit test by chi-square test, the formula was −1 k (O −T ) 2 i i value was 0.043 Gy (95% CI 0.029–0.056, P < 0.001), X = , where T means theoretical fre- i=1 i −1 0.026 Gy (95% CI 0.016–0.036, P < 0.001), and quency (original P) and O means observed frequency −1 0.042 Gy (95% CI − 0.27–0.36, P < 0.79), respectively. (calculated P). Then, we put calculated X into the criti- cal value table and obtained corresponding P value. This Cui et al. Radiation Oncology (2022) 17:54 Page 4 of 11 Table 1 Detailed information on 45 cited articles Only 21 of 45 studies (14,641 PCa patients) could be grouped by the risks of PCa. For different risk sub - Study First author Year Journal Volume Pages groups, the results were shown in Table 6. At the same 1 Aizer 2009 Radiother Oncol 93 185–191 fractionation schedules, there were no practical clini- 2 Alicikus 2011 Cancer 117 1429–1437 cal significance or significant differences in α/β values 3 Cahlon 2008 IJROBP 71 330–337 among the three risk groups. For example, the α/β ratios 4 Eade 2007 IJROBP 68 682–689 were 1.10 Gy (1.04–1.15, P < 0.001), 1.34 Gy (1.25–1.44, 5 Kuban 2008 IJROBP 70 67–74 P < 0.001), and 0.39 Gy (0.33–0.45, P < 0.001) in the three 6 Lukka 2005 J Clin Oncol 23 6132–6138 risk groups, respectively, in the moderately hypofraction- 7 Valdagni 2005 Radiother Oncol 75 74–82 ated radiotherapy group. We also calculated Bca intervals 8 Zelefsky 2008 IJROBP 71 1028–1033 and the results were showen in Table 6. In the conven- 9 Miralbell 2012 IJROBP 82 e17–24 tionally fractionated radiotherapy group, although the 10 Arcangeli 2012 IJROBP 84 1172–1178 α/β ratios were significantly different among the three 11 Catton 2017 J Clin Oncol 35 1884–1890 risk groups, there was no practical clinical significance. 12 Dearnaley 2016 The Lancet Oncology 17 1047–1060 The calculated k value was 7.68 (95% CI 6.15–9.22, 13 Incrocci 2016 The Lancet Oncology 17 1061–1069 P < 0.001), 6.62 (95% CI 5.85–7.38, P < 0.001), and 4.93 14 Kim 2014 Radiat Oncol J 32 187–197 (95% CI 4.00–5.87, P < 0.001), respectively, in the low-, 15 Kupelian 2005 IJROBP 63 1463–1468 intermediate-, and high-risk groups in the moderately 16 Leborgne 2009 IJROBP 74 1441–1446 hypofractionated radiotherapy group and the α value was 17 Leborgne 2012 IJROBP 82 1200–1207 −1 −1 0.047 Gy (95% CI 0.026–0.067, P < 0.001), 0.044 Gy 18 Pollack 2013 J Clin Oncol 31 3860–3868 −1 (95% CI 0.032–0.057, P < 0.001) and 0.011 Gy (95% 19 Yeoh 2006 IJROBP 66 1072–1083 CI 0.0002–0.022, P = 0.046), respectively. According to 20 Cheung 2016 IJROBP 96 S33 the results, we found that the k and α values tended to 21 Di Muzio 2016 Clin Oncol (R Coll 28 490–500 decrease when the risks of PCa increased. In the con- Radiol) ventionally fractionated radiotherapy group, the same 22 Faria 2011 Radiother Oncol 101 486–489 23 Fonteyne 2012 IJROBP 84 e483–490 conclusion could be drawn. In the SBRT groups, the α/β 24 Hashimoto 2017 Int J Clin Oncol NA NA ratios were − 10.7 Gy (95% CI − 12.6– − 8.7, P < 0.001), 25 Kuban 2010 IJROBP 78 S58–59 25.6 Gy (95% CI 21.6–29.6, P < 0.001), and 2.94 Gy (95% 26 Kupelian 2007 IJROBP 68 1424–1430 CI − 14.4–20.2, P = 0.74) in the low-, intermediate-, and 27 Lieng 2017 Radiother Oncol 122 93–98 high-risk groups, respectively. Since the α/β ratio in the 28 Livsey 2003 IJROBP 57 1254–1259 low-risk patients was negative, we imposed non-neg- 29 Mai 2010 IJROBP 78 S59 ativity restrictions; thereafter, the α/β ratio in the low- 30 Patel 2013 IJROBP 86 534–539 risk group was 0.032 Gy (95% CI − 0.40–0.47, P = 0.89). 31 Pervez 2017 Am J Clin Oncol 40 200–206 The conclusion which came out from the conventionally 32 Shimizu 2017 Anticancer Res 37 5829–5835 fractionated radiotherapy group and moderately hypo- 33 Thomson 2012 Prostate Cancer NA NA fractionated radiotherapy group could not be drawn in 34 Viani 2016 Rep Pract Oncol 21 162–167 the SBRT group due to the limited number of articles Radiother involved. 35 Bolzicco 2013 BMC Urol 13 NA The preliminary results of verification of fitting results 36 Fuller 2014 Front Oncol 4 NA were shown in Table 7. The X were all < 1 in all three risk 37 Kang 2011 Tumori 97 43–48 groups and the P values were all > 0.995 that meant there 38 Katz 2013 Radiat Oncol 8 NA was no statistical difference between the calculated TCP 39 King 2013 Radiother Oncol 109 217–221 and the TCP from the original study. In other words, our 40 Lee 2014 Medicine (Baltimore) 93 e290 fitting was accurate. 41 Loblaw 2013 Radiother Oncol 107 153–158 In summary, for PCa patients receiving conventionally 42 Mantz 2014 Front Oncol 4 NA fractionated radiotherapy, moderately hypofractionated 43 Meier 2016 IJROBP 96 S33–34 radiotherapy, and SBRT, the mean α/β ratios were 1.78, 44 Tsang 2021 Radiother Oncol 158 184–190 3.46, and 4.24 Gy, respectively. Meanwhile, as the risks of 45 Chin,S 2020 IJROBP 107 288–296 PCa increased, the k and α values decreased. IJROBP Int J Radiat Oncol Biol Phys, NA not available Journal abbreviations follow the PubMed style C ui et al. Radiation Oncology (2022) 17:54 Page 5 of 11 Table 2 Conventionally fractionated radiotherapy group Study number Author Number of 5y-bRFS Total dose (D) Fractions (N)/ D /N (C) Definition patients single doseof bRFS 1 Aizer 352 0.748 75.6 42 136.08 P 2 Arcangeli 85 0.79 80 40 160 P 3 Catton 598 0.85 78 39 156 P 4 Dearnaley 1065 0.883 74 37 148 P 5 Eade 43 0.7 69 2.1 144.9 P Eade 552 0.81 72.5 2.1 152.25 P Eade 568 0.83 77.5 2.1 162.75 P Eade 367 0.89 81 2.1 170.1 P 6 Incrocci 397 0.771 78 39 156 P 7 Kim 56 0.641 70.2 39 126.36 P 8 Kuban 150 0.78 70 35 140 P Kuban 151 0.85 78 39 156 P 9 Kupelian 310 0.78 78 39 156 A 10 Leborgne 160 0.887 78 39 156 P 11 Lukka 470 0.4705 66 33 132 A 12 Pollack 152 0.852 76 38 152 P 13 Valdagni 161 0.7 74 37 148 A 14 Yeoh 109 0.555 64 32 128 A 15 Zelefsky 358 0.61 70.2 39 126.36 P Zelefsky 471 0.74 75.6 42 136.08 P Zelefsky 741 0.85 81 45 145.8 P Zelefsky 477 0.82 86.4 48 155.52 P P Phoenix, A ASTRO Discussion may be no repopulation during radiotherapy , and The α/β ratio proposed in the early 1970’s derives from the α/β ratio increase is not due to tumor cells repopu- the LQ models [28, 29]. Factors that can influence α and/ lation. Also the time factors should not be considered in or β independently increase or decrease the α/β ratio. late-responding tissues [33, 34]. u Th s, we did not take The major influencing factors are internal factors from the time factor into consideration when converting doses cells themselves and external factors from physical or using the standard LQ model. chemical effects [17, 30]. The internal factors include Recent randomized trials demonstrated that hypofrac- cell cycle regulation, cell repopulation, and DNA dam- tionated radiotherapy was not superior to conventional age repair after irradiation. The external physical fac - radiotherapy in PCa. In the Radiation Therapy Oncol - tors include temperature (hyperthermia), oxygenation ogy Group (RTOG) 0415 , Hypofractionated Irradia- (hypoxia), characteristics of radioactive rays-like linear tion for Prostate Cancer trial (HYPRO) , and the Fox energy transfer, and the dose rate. The external chemical Chase trial (ClinicalTrials.gov identifier: NCT00062309) factors are some anticancer drugs such as cisplatin, EGFR , biological effective doses (BEDs) in hypofraction - inhibitors, and PARP1 inhibitors. Thus, there are multi - ated vs. conventionally fractionated radiotherapy groups ple factors that affect the α/β ratio and modify the radio - were calculated as 186.7 vs. 162.4 Gy, 211.0 vs 182.0 Gy, sensitivity of tumors. and 196.6 vs 177.3 Gy, respectively, using an α/β ratio of Our study showed that the α/β ratio tended to become 1.5 Gy. All BEDs in the hypofractionated groups were higher when the dose per fraction increased. The α/β 19–29 Gy higher than BEDs of the conventionally frac- ratios may increase also dynamically during treatment, tionated groups. Nevertheless, the higher BEDs did not from approximately 4 Gy for ‘short’ fractionation sched- lead to satisfactory improvements in the outcome. This ules to about 1.5 Gy for long schedules, which probably may be attributable to the inaccurate conversion of radia- reflects the process of accelerated repopulation in nor - tion doses using the LQ model. In our study, the α/β mal acute skin reactions [31, 32]. For late-responding tis- ratio tended to become higher when the dose per frac- sues and slow-growing tumors like PCa, however, there tion increased. When the doses in the three trials were Cui et al. Radiation Oncology (2022) 17:54 Page 6 of 11 Table 3 Moderately hypofractionated radiotherapy group Study number Author Number of 5y-bRFS Total dose (D) Fractions (N) D /N (C) Definition patientsof bRFS 1 Arcangeli 83 0.85 62 20 192.2 P 2 Catton 608 0.85 60 20 180 P 3 Cheung 230 0.837 67.5 25 182.25 P 4 Chin,S 112 0.68 52.5 20 137.8125 P 5 Dearnaley 1077 0.859 57 19 171 P Dearnaley 1074 0.906 60 20 180 P 6 Di Muzio 80 0.911 74.2 28 196.63 P Di Muzio 78 0.946 71.4 28 182.07 P Di Muzio 53 0.962 74.2 28 196.63 P 7 Faria 82 0.954 66 22 198 P 8 Fonteyne 113 0.94 56 16 196 P 9 Hashimoto 195 0.924 66 22 198 P 10 Lieng 96 0.81 60 20 180 P Lieng 27 0.88 66 22 198 P 11 Incrocci 407 0.805 64.6 19 219.64 P 12 Kim 30 0.929 70 28 175 P 13 Kuban 102 0.96 72 30 172.8 A 14 Kupelian 100 0.88 70 28 175 P 15 Kupelian 770 0.83 70 28 175 P 16 Leborgne 114 0.894 61.2 20 187.272 P 17 Mai 596 0.927 76.65 35 167.8635 P 18 Patel 129 0.97 66 22 198 P 19 Pervez 60 0.9167 67.5 25 182.25 P 20 Pollack 151 0.81 70.2 26 189.54 P 21 Shimizu 73 0.77 74.8 34 164.56 P Shimizu 21 0.92 74.8 34 164.56 P Shimizu 44 0.95 72.6 33 159.72 P 22 Thomson 30 0.5 57 19 171 P Thomson 30 0.58 60 20 180 P 23 Viani 149 0.946 69 23 207 P 24 Yeoh 108 0.574 55 20 151.25 A P Phoenix, A ASTRO Table 4 SBRT group Study number Author Number of 5y-bRFS Total dose (D) Fractions (N) D /N (C) Definition patientsof bRFS 1 Bolzicco 100 0.944 35 5 245 P 2 Kang 44 0.936 34 4 289 P 3 King 1100 0.93 36.25 5 262.8 P King 385 0.925 35 5 245 P King 589 0.907 36.25 5 262.8 P King 126 0.958 39 5 304.2 P 4 Lee 45 0.897 36 5 259.2 P 5 Loblaw 84 0.98 35 5 245 P 6 Mantz 102 1 40 5 320 P 7 Meier 309 0.971 40 5 320 P 8 Tsang 43 0.92 36.25 5 262.8 P P Phoenix C ui et al. Radiation Oncology (2022) 17:54 Page 7 of 11 Table 5 Parameters estimated with 95% CIs in different regimens of radiotherapy −1 Kα (Gy ) α/β (Gy) Estimate 95% CI P Estimate 95% CI P Estimate 95% CI BCa95% CI P Conventional fractionation 5.35 4.61–6.08 < 0.001 0.043 0.029–0.056 < 0.001 1.78 1.59–1.98 < 0.001 1.62–1.97 Moderate hypofractionation 1.15 0.21–2.09 0.017 0.026 0.016–0.036 < 0.001 3.46 3.27–3.65 < 0.001 3.30–3.66 SBRT 1.67 − 4.80–8.15 0.61 0.042 − 0.27–0.36 0.79 4.24 4.10–4.39 < 0.001 4.14–4.34 BCa 95% CI Bias-Corrected and accelerated intervals. We go a step further and calculate BCa intervals on the base of jackknife methods converted with the LQ model using the α/β ratios that we the probability of intracellular cell repair, and higher α/β estimated (1.78 Gy for conventionally fractionation and ratios were shown with a linear survival curve [39, 42]. 3.46 Gy for moderate hypofractionation), BEDs of the An in vivo study involving a murine tumor model dem- hypofractionated and conventionally fractionated groups onstrated that an equivalent single high dose converted were 120.6 and 148.4 Gy, 128.1 and 165.6 Gy, and 125 and from fractionated radiotherapy was lower than the actual 161.4 Gy in the RTOG0415, HYPRO, and Fox Chase trial, dose. However, when a higher α/β ratio was used, the respectively. The BEDs in the hypofractionated group discrepancy became smaller . Our data agreed with were significantly lower than in the conventionally frac - their results. At different fractional doses, the α/β ratio tionated group. u Th s, the non-superiority of the hypof - tended to be higher when the dose per fraction increased ractionated group could be in part explained by these (1.78 Gy for conventional fractionation, 3.46 Gy for mod- BEDs calculated based on our results. erate hypofractionation, and 4.24 Gy for SBRT). Espe- Several studies investigated the appropriateness of the cially in the SBRT groups, the high α/β ratio was marked. LQ model at high doses per fraction. Previous in vitro We also found that α and k values decreased with risk and in vivo studies demonstrated that the LQ model elevation in the conventional fractionation and moder- overestimated the efficacy of tumor cell killing with ate hypofractionation groups. These results were similar a high dose per fraction [3, 37, 38]. Thus, several mod - to those in the previous study . A decrease in the α els were proposed modifying the standard LQ model to values with escalation of the risk group can be attributed reasonably convert conventionally fractionated doses to to higher radio-resistance of tumor cells in higher risk equivalent single or hypofractionated doses. The lethal- patients. k represents the natural logarithm of an effec - potentially-lethal (LPL) model considered DNA lesion tive target cell number, and a decrease in k values means repair and could explain very effectively the shoulder that the effective target cell number is reduced with esca - on survival curves ; The modified LQ (MLQ) model lation of the risk group. made a better fit to the iso-effect data than the LQ A recent publication was based on the dose distribu- model in a single high dose . The “universal survival tion delivered to patients and provided another method curve”(USC) model proposed by Park et al. combined for fitting parameter . This method used the linear- two classical radiobiological models: the multitarget quadratic Poisson TCP model with dose distribution like model and the standard LQ model that provide superior GTV (prostate gland), mpMRI-GTV, D and dose vol- approximation of survival curves in the high-dose range. ume histograms (DVH) to estimate α/β ratio and obtain , and generalized LQ (gLQ) model encompasses the solution space, initial parameter values, and optimal solu- full dose range of possible dose delivery patterns and spe- tion by optimizer. Our research approach was different cial radiotherapy schemes. . Characteristics of these from this method and used maximum likelihood princi- models have already been described . Wang et al.  ple in mathematical statistics to fit the α/β ratio accord - demonstrated that the problems in the LQ model derived ing to the Miralbell model with tumor control probability from the amount of sublethal damage were reduced (5y-bRFS), total dose, and number of fractions or dose owing to conversion to lethal damage at a single high per fraction; this method was much easier than the meth- dose; if sublethal damage is converted to lethal dam- ods mentioned above. age, then the α/β ratio is elevated with a single high dose The α/β ratio in the low-risk patients of the SBRT according to the definition of the α/β ratio. Other stud - group was in the negative range. A study using exter- ies also revealed that cell death at high doses exceeded nal beam radiation therapy alone also had negative α/β Cui et al. Radiation Oncology (2022) 17:54 Page 8 of 11 Table 6 Parameters estimated with 95% CIs in different risks under different regimens of radiotherapy −1 Fractionation regimen Risk group Kα (Gy ) α/β (Gy) Estimate 95% CI P Estimate 95% CI P Estimate 95% CI BCa 95% CI P Conventional fractionation Low risk 10.2 7.34–13.1 < 0.001 0.081 0.012–0.15 0.022 1.80 1.46–2.14 < 0.001 1.56–2.11 Intermediate risk 8.20 6.85–9.56 < 0.001 0.073 0.041–0.10 < 0.001 2.29 2.12–2.47 < 0.001 2.15–2.44 High risk 4.31 2.80–5.83 < 0.001 0.023 − 0.0053–0.051 0.11 0.98 0.91–1.05 < 0.001 0.94–1.04 Moderate hypofractionation Low risk 7.68 6.15–9.22 < 0.001 0.047 0.026–0.067 < 0.001 1.10 1.04–1.15 < 0.001 1.04–1.14 Intermediate risk 6.62 5.85–7.38 < 0.001 0.044 0.032–0.057 < 0.001 1.34 1.25–1.44 < 0.001 1.26–1.43 High risk 4.93 4.00–5.87 < 0.001 0.011 0.0002–0.022 0.046 0.39 0.33–0.45 < 0.001 .34–0.45 SBRT Low risk − 4.81 − 16.3–6.64 0.41 − 0.14 − 0.66–0.39 0.61 − 10.7 − 12.6– − 8.7 < 0.001 Intermediate Risk 10.7 3.16–18.2 0.005 0.27 − 0.019–0.56 0.067 25.6 21.6–29.6 < 0.001 High risk 16.0 − 42.7–74.8 0.59 0.14 − 0.89–1.17 0.79 2.94 − 14.4–20.2 0.74 BCa 95% CI Bias-Corrected and accelerated intervals. We go a step further and calculate BCa intervals on the base of jackknife methods The BCa intervals could not be drawn in the SBRT group due to the limited number of articles involved C ui et al. Radiation Oncology (2022) 17:54 Page 9 of 11 Table 7 Preliminary results of verification of fitting results −1 2 Fractionation regimen kα (Gy ) α/β ratio (Gy) X P value (goodness of fit test) Conventional fractionation 5.35 0.043 1.78 0.10 > 0.995 Moderate hypofractionation 1.15 0.026 3.46 0.51 > 0.995 SBRT 1.67 0.042 4.24 0.01 > 0.995 ratios . Repeated measures of PSA at 6 institutions In conclusion, our study using mathematical statis- were analyzed and data from 3 institutions including tics with 5y-bRFS data in PCa patients demonstrated RTOG showed negative α/β ratios. In the Peter Mac- that the α/β ratio was dependent on the fractionation Callum Cancer Center, the α/β ratio was − 2.05 Gy (95% schedule. In SBRT, the estimated α/β ratio was > 4 Gy. CI −∞ − +∞ ). Another study found that the α/β ratio Therefore, to convert conventionally fractionated radia - of arteriovenous malformation obliteration after radio- tion doses to an equivalent single high dose, it may be surgery was markedly negative (α/β = − 49.3 ± 5.3) . necessary to use either a modified formula or a higher However, neither study explained why the α/β ratio was α/β ratio with the standard LQ model. negative. These results as well as ours suggest a limitation of this calculation method in that it could possibly yield Abbreviations unrealistic α/β ratios, especially when the patient number PCa: Prostate cancer; LQ: Linear‑ quadratic; TCP: Tumor control probability; is small. Although, we imposed non-negativity restric- SBRT: Stereotactic body radiotherapy; IMRT: Intensity‑modulated radiotherapy; ADT: Androgen deprivation therapy; 5y‑bRFS: 5‑ Year Biochemical Relapse‑Free tions on the α/β ratio of the low-risk group (− 10.7 Gy to Survival; NCCN: National Comprehensive Cancer Network; BCa: Bias‑ corrected 0.032 Gy), it is merely a mathematical statistics method and accelerated; PSA: Prostate specific antigen; CI: Confidence intervals; RTOG: and may affect real result. Radiation Therapy Oncology Group; BEDs: Biological effective doses; HYPRO: Hypofractionated Irradiation for Prostate Cancer Trial; LPL: Lethal‑potentially‑ There are several limitations in our study. Since we lethal; MLQ: Modified LQ; USC: Universal survival curve; gLQ: Generalized LQ; divided the whole group into three subgroups according DVH: Dose volume histograms. to the fractionation schedule, the dose ranges per fraction were relatively narrow in each fractionation group. This Supplementary Information may increase the variability of the estimated α/β ratios, The online version contains supplementary material available at https:// doi. but we tried to solve this problem by including as many org/ 10. 1186/ s13014‑ 022‑ 02010‑9. patients as possible. Subtle variations in patient evalua- tion including the definition of PSA failure and treatment Additional file 1. Supplementary Table 1. Conventionally fractionated radiotherapy in different risk groups. Supplementary Table 2. Moderately including the dose prescription method among respec- hypofractionated radiotherapy in different risk groups. Supplementary tive studies would also contribute to variability in the Table 3. SBRT in different risk groups. estimated α/β ratios; this problem is common to all stud- ies of this kind, and is considered to be ameliorated by Acknowledgements including a large number of patients. An analysis of over The authors with to thank Prof. Zhi Geng, Dr. Yunjian Yin, and Dr. Han Wang 14,000 patients showed that the derived α/β ratios were (School of Mathematical Sciences, Peking University) for their valuable advices on the statistical methods. not different between studies using the ASTRO definition and those using the Phoenix definition . The patient Authors’ contributions number in our study was larger than in any other studies Corresponding author: X‑SG and YS contributed equally to this paper. All authors read and approved the final manuscript. investigating the α/β ratio for PCa. Also, the influence of ADT was not considered, as was the case with other pre- Funding vious studies, since a previous study indicated the mini- This research was funded in part by the Japanese Ministry of Education, Culture, Sports, Science and Technology. mal influence of ADT . Another limitation is that our analysis was based on the prescription dose and the treat- Availability of data and materials ment outcome of the incorporated studies. Thus, 3D dose Not applicable. description or the DVHs were not incorporated into the analysis, because, for fitting parameters, we only needed Declarations prescription dose and treatment outcome according to Ethics approval and consent to participate TCP formula. Furthermore, the incorporated studies all Not applicable. used prescription dose which was widely used in clinical practice rather than dose distribution. Cui et al. Radiation Oncology (2022) 17:54 Page 10 of 11 Consent for publication brachytherapy (3145 multicentre patients) combined with, or in contrast All authors have agreements on publication. to, external‑beam radiotherapy. Radiother Oncol. 2014;111(1):114–9. 17. Barendsen GW, Van Bree C, Franken NA. 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Radiation Oncology – Springer Journals
Published: Mar 18, 2022
Keywords: LQ model; α/β ratio; Hypofractionated radiotherapy; High dose per fraction; Prostate cancer
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