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1 doi: /j.ijrobp Int. J. Radiation Oncology Biol. Phys., Vol. 82, No. 1, pp. e17 e24, 2012 Copyright Ó 2012 Elsevier Inc. Printed in the USA. All rights reserved /$ - see front matter CLINICAL INVESTIGATION Genitourinary Cancer DOSE-FRACTIONATION SENSITIVITY OF PROSTATE CANCER DEDUCED FROM RADIOTHERAPY OUTCOMES OF 5,969 PATIENTS IN SEVEN INTERNATIONAL INSTITUTIONAL DATASETS: a/b = 1.4 ( ) GY RAYMOND MIRALBELL, M.D.,* y STEPHEN A. ROBERTS, PH.D., z EDUARDO ZUBIZARRETA, M.D., x AND JOLYON H. HENDRY, PH.D. k *University Hospital, Geneva, Switzerland; y Institut Oncologic Teknon, Barcelona, Spain, z Health Sciences Methodology, Manchester Academic Health Sciences Centre, University of Manchester, Manchester, United Kingdom; x International Atomic Energy Agency, Vienna, Austria; and k Adlington, Macclesfield, United Kingdom Purpose: There are reports of a high sensitivity of prostate cancer to radiotherapy dose fractionation, and this has prompted several trials of hypofractionation schedules. It remains unclear whether hypofractionation will provide a significant therapeutic benefit in the treatment of prostate cancer, and whether there are different fractionation sensitivities for different stages of disease. In order to address this, multiple primary datasets have been collected for analysis. Methods and Materials: Seven datasets were assembled from institutions worldwide. A total of 5969 patients were treated using external beams with or without androgen deprivation (AD). Standard fractionation ( Gy per fraction) was used for 40% of the patients, and hypofractionation ( Gy per fraction) for the remainder. The overall treatment time ranged from 1 to 8 weeks. Low-risk patients comprised 23% of the total, intermediate-risk 44%, and high-risk 33%. Direct analysis of the primary data for tumor control at 5 years was undertaken, using the Phoenix criterion of biochemical relapse free survival, in order to calculate values in the linear-quadratic equation of k (natural log of the effective target cell number), a (dose-response slope using very low doses per fraction), and the ratio a/b that characterizes dose-fractionation sensitivity. Results: There was no significant difference between the a/b value for the three risk groups, and the value of a/b for the pooled data was 1.4 (95% CI = ) Gy. Androgen deprivation improved the bned outcome index by about 5% for all risk groups, but did not affect the a/b value. Conclusions: The overall a/b value was consistently low, unaffected by AD deprivation, and lower than the appropriate values for late normal-tissue morbidity. Hence the fractionation sensitivity differential (tumor/normal tissue) favors the use of hypofractionated radiotherapy schedules for all risk groups, which is also very beneficial logistically in limited-resource settings. Ó 2012 Elsevier Inc. Prostate cancer, Radiotherapy, Alpha/beta value, Radiobiology, Fractionation sensitivity. INTRODUCTION Although a dose of radiotherapy (RT) >70 Gy for prostate cancer has shown improved biochemical control rates in randomized trials a higher risk of late toxicity has frequently been observed for patients treated in the high-dose arms of these studies (1 5). Several strategies have been proposed to limit the radiation-induced toxicity by optimizing treatment conformation by using either intensity-modulated radiotherapy (IMRT) or high-dose rate brachytherapy (HDR-BT) techniques (6 13). Hypofractionation has been proposed as an additional strategy to optimize the therapeutic ratio by taking advantage of the Reprint requests to: Raymond Miralbell. M.D., Radiation Oncology Department, University Hospital of Geneva, 1211 Geneva 14, Switzerland. Tel: ; Fax: ; Raymond.Miralbell@hcuge.ch Conflict of interest: none. Acknowledgments The authors are very grateful to the following investigators for providing their primary data for this analysis: e17 potential high sensitivity of prostate cancer to RT dose fractionation compared with the late-responding tissues nearby (rectum, bladder, and urethra) (14, 15).This has prompted several trials of hypofractionation schedules. Based on the assumption that the fractionation sensitivity of prostate cancer cells is characterized by a low a/b ratio in the range of 0.8 to 2.2 Gy, delivering treatment fractions higher than the standard 2-Gy/fraction may be radiobiologically and therapeutically effective. Furthermore, delivering a higher dose in a reduced number of fractions may be most convenient for patients and logistically advantageous for busy RT departments. Such a treatment strategy may also P. Kupelian, F. Leborgne, J. Logue, H. Lukka, R. Miralbell, and E. Yeoh. Thanks are also expressed to Professors Jack Fowler and Howard Thames for very helpful comments on the paper. SAR is supported by the NIHR National Institute for Health Research Manchester Biomedical Research Centre. Received July 22, 2010, and in revised form Oct 20, Accepted for publication Oct 22, 2010.

2 e18 I. J. Radiation Oncology d Biology d Physics Volume 82, Number 1, 2012 succeed to reduce the cost of RT for prostate cancer in payper-fraction reimbursement systems. Nevertheless, moderately hypofractionated RT (i.e., fractions of Gy) to treat prostate cancer is not a novel treatment approach. Indeed, it has been standard in the United Kingdom, Canada, and Australia for many years with results very similar to standard fractionated treatments with 1.8 to 2 Gy fractions (16 19). It still remains unclear, however, whether hypofractionation will provide a significant therapeutic benefit in the treatment of prostate cancer. Also, whether there are different fractionation sensitivities for different disease risk groups, with some of them treated with androgen deprivation (AD) in addition to RT. Thus, to better assess these questions we decided to undertake a retrospective study on nearly 6,000 prostate cancer patients from seven international institutional primary datasets treated with curative external RT stratified by risk groups and AD status. A direct analysis of 5-year biochemical relapse free survival (brfs) data with the linear-quadratic (LQ) model was implemented to estimate the dose fractionation sensitivity for this group of patients. METHODS AND MATERIALS Clinical material and data collection Seven datasets were assembled from seven institutions worldwide: Australia, Canada, Switzerland, the United Kingdom, Uruguay (one dataset each), and the United States (two datasets). All institutions but one (data taken from a freshly published report by Madsen et al. at the time of the study) (20) were requested to provide recently updated information concerning the patients outcome (5-year brfs according to the Phoenix criterion) (21), which was the primary and single endpoint in this study. Results were stratified by risk groups and AD status. Risk grouping was undertaken according to the National Comprehensive Cancer Network (NCCN) guidelines risk group classification (22). Substratification of androgen deprivation modalities (duration, neoadjuvant vs. concomitant vs. adjuvant) was not investigated. Nevertheless, as acknowledged by the investigators who shared their clinical material for the purpose of the study, almost all patients on AD were treated in a neoadjuvant/concomitant setting and for a period as short as 6 months. The total dose of radiotherapy, the dose per fraction, the number of fractions, and the overall treatment time in weeks were recorded in the database along with the treatment technique used by each institution and the treatment target (local vs. loco-regional). Statistical analysis Standard LQ models for tumor control at 5 years of the form: P ¼ exp exp k ad aðb=aþd 2 =N ; (1) were fitted. Here, P is the tumor control probability (brfs); D, the total dose; N, the number of fractions; k can be interpreted as the natural logarithm of an effective target cell number, a as a dose response slope for small fractions and b/a the ratio that characterizes dose-fractionation sensitivity. The parameters k, a, and b/a were allowed to vary between risk groups and patients treated with and without hormone therapy. Preliminary analyses indicated that a full model where each of the parameters was allowed to take different values for each combination of risk group and hormone treatment was overspecified given the data available. Therefore we considered all possible models in which each of the parameters was allowed to vary between risk groups and/or AD status and selected the optimal models according to the Akaike information criterion (AIC). The AIC is a measure of the goodness of fit that adjusts for the added complexity of models with more parameters. Two models had identical AIC values, and so both are presented. Fitting was performed by direct maximisation of the beta-binomial likelihood using custom-written code in Stata version 10 (23). The beta-binomial approach (24) models the overdispersion of the data explicitly in a full likelihood approach, rather than the more usual quasi-likelihood method, and was preferred here as it allows direct estimation of the 95% confidence intervals (CI) using a profile-likelihood method, rather than relying on Wald approximations which were considered likely to be invalid in this dataset. The quasi-binomial model was fitted as a sensitivity analysis. Statistical inference used likelihood ratio tests between models. The beta-binomial model, like the quasi-binomial model, makes explicit assumptions about the nature and form of the overdispersion. In particular, both approaches assume that the variation in response is between individual patient groups rather than between centers. Therefore we undertook a sensitivity analysis by considering formal random effect models of the form: P ¼ exp exp k ad aðb=aþd 2 =N þ u i ; (2) Table 1. Radiotherapy characteristics, by first author Author Dose/fraction Total dose No. fractions No. fractions/wk OTT (wk) Pelvic nodes RT Technique Kupelian 2 Gy 78 Gy No 3d-CRT 2.5 Gy 70 Gy No IMRT-BAT Leborgne 2 Gy 76 Gy No 3d-CRT Logue Gy 50 Gy No 3d-CRT Lukka 2 Gy 66 Gy No 2d-CRT 2.62 Gy 52.4 Gy No 2d-CRT Madsen et al (20) 6.7 Gy 33.5 Gy No SRT-IGRT Miralbell Gy Gy No/yes 3d-CRT 4 Gy 56 Gy No SRT Yeoh 2 Gy 64 Gy No 2d-RT 2.75 Gy 55 Gy No 2d-RT Abbreviations: BAT = transabdominal ultrasound system; IGRT = image guided radiotherapy; OTT = overall treatment time; SRT = stereotactic radiotherapy; 2D-RT = two-dimensional radiotherapy treatment planning; 3D-CRT = three-dimensional conformal radiotherapy.

3 Dose fractionation sensitivity and prostate cancer d R. MIRALBELL et al. e19 where the u i are normally distributed random effects between patient group or center depending on the interpretation of the index i. This model was fitted in a BayesianMarkov chain Monte Carlo (MCMC) framework using WinBUGS (25) with noninformative priors and 9,000,000 iterations following a 1,000,000 iteration burn-in period using three independent chains. A final sensitivity analysis considered excluding one outlying center. A priori we did not expect an influence of overall treatment time. Data were pooled by scheduled treatment time as individual treatment durations were not available so it would be expected that any estimate of treatment time effects may be biased because of the loss of information, and the strong correlations between time and the other dose parameters would be expected to make any estimation of such effects difficult. Nevertheless we did consider time effects as a sensitivity analysis, fitting a model of the form: P ¼ exp exp k ad aðb=aþd 2 =N þ aðg=aþ½t T k Š þ ; (3) Here we explicitly parameterized the time effect as a ratio to a (26) and expressed it as Gy/d. The repopulation kicks in at time T k and the subscript on the ½T T k Š þ term indicates that only positive values were considered. RESULTS A total of 5969 patients were treated with external beams in an overall treatment time ranging from 1 to 8 weeks. Treatment characteristics are shown in Table 1. Patient characteristics (risk groups) and treatment stratification (fractionation and AD status) are summarized in Table 2. Standard fractionation ( Gy per fraction) was used for 2,410 patients (40%) and hypofractionation ( Gy per fraction) for the remainder. Low-risk patients comprised 23% of the total (1,405 patients), intermediate-risk, 44% (2,616 patients); and high-risk, 33% (1,928 patients). Neoadjuvant and/or concomitant androgen deprivation was given to 2,203 patients (37%). The 5-year brfs for each institution, stratified by risk group, fractionation, and AD status, are presented in Table 3. The median follow-up was <5 years for the hypofractionated cohorts of patients reported by Madsen et al. and by the Swiss group (41 and 52 months, respectively) but was >5 years for all remaining patients in the study. Two LQ models provided equally optimal fits to the data as assessed by the AIC. In both models, hormones only affect the a component of the model, whereas risk group affects both k and a; the two models differ in that risk group is or is not included in the b/a component. The parameter estimates from the two models are shown in Table 4. Figure 1 shows the raw and fitted values for the simpler model (B). A significant improvement of brfs with dose was observed for all risk groups independently of their AD status. As expected, there was a significant difference among the three risk groups with a decrease in control as the risk group increased from low to intermediate to high. In addition, AD + RT improved significantly brfs in all risk groups by about 5% (Fig. 1) compared with those patients treated with RT alone (p = 0.013). The value of a/b for the pooled data was 1.4 Gy (95% CI = ). If we allowed a/b to vary between risk groups, the values for low-, intermediate-, and high-risk groups were 0.6 Gy (95% CI = Gy), 1.7 Gy (95% CI = ), and 1.6 Gy (95% CI = ), respectively. Although these values would not be considered to differ significantly (p = 0.13) allowing them to take different values would be considered to be an equally good fit to the data according to the AIC. There is no evidence that a/b differs between patients receiving or not receiving AD (p = 0.23). The value for low-risk patients was not well specified, and all these values were within the low range generally reported for prostate cancer, and are consistently low for all subgroups of patients according to their risk and AD status. Table 2. Patient distribution according to dose/fraction and centers stratified by risk group and androgen deprivation (AD) status Without AD With AD Author Dose/fraction Low risk Intermediate risk High risk Low risk Intermediate risk High risk Total Kupelian* 2 Gy Gy Leborgne y 2 Gy Logue z Gy ,082 Lukka x 2 Gy Gy Madsen et al (20)* 6.7 Gy Miralbell k 1.8/2 Gy Gy Yeoh { 2 Gy Gy Total ,969 * United States. y Uruguay. z United Kingdom. x Canada. { Australia. k Switzerland.

4 e20 I. J. Radiation Oncology d Biology d Physics Volume 82, Number 1, 2012 Table 3. Five-year biochemical relapse free survival probability stratified by risk groups and androgen deprivation (AD) status Without AD With AD Author Dose/fraction Low Risk Intermediate risk High risk Low risk Intermediate risk High risk Kupelian* 2 Gy Gy Leborgne y 2 Gy Logue z Gy Lukka x 2 Gy Gy Madsen et al (20)* 6.7 Gy 0.93 Miralbell k 1.8/2 Gy Gy Yeoh { 2 Gy Gy * United States. y Uruguay. z United Kingdom. x Canada. { Australia k Switzerland. It would be expected that there might be appreciable interinstitutional and inter-group variability in outcomes due to differences in technique, supportive care and environment between the centers in this study. Additionally the dosefractionation groups represent different historical, demographic or clinical groups and might be expected to differ in outcomes beyond the sampling (binomial) variability. The primary analysis used a beta-binomial model to represent this heterogeneity. In Table 5, we show the results of refitting the simpler model with a variety of assumptions and models of the heterogeneity. A standard quasi-binomial approach, where the variance of the outcome measure is increased by a multiplicative factor gives a somewhat lower a/b than the beta-binomial model with an explicit b-distribution for the outcome variable. The two models with a normally distributed variance (in k) give very similar results, and slightly larger a/b estimates, suggesting that in this dataset center effects are dominated by differences in patient groups. The magnitude of this variability is modest, but statistically significant, with the standard deviation estimates of 0.3 being small compared with the value of k and equivalent to a 10-Gy (e.g., five fractions of 2 Gy) dose difference. Table 4. Parameter estimates with 95% CI from the two optimal models as selected by the Akaike information criterion (AIC) Model A Model B Estimate (95% CI) p Estimate (95% CI) p k High 3.0 ( ) < ( ) <0.001 Intermediate 4.5 ( ) 4.5 ( ) Low 5.4 ( ) 5.3 ( ) H a (Gy 1 ) High ( ) ( ) Intermediate ( ) ( ) Low ( ) ( ) H ( ) ( ) b/a (Gy 1 ) High 1.69 (0.66 N*) ( ) 0.13 Intermediate 0.59 ( ) Low 0.64 ( ) H a/b (Gy) High 0.6 (0 1.5) ( ) 0.13 Intermediate 1.7 ( ) Low 1.6 ( ) H * Limit undefined.

5 Dose fractionation sensitivity and prostate cancer d R. MIRALBELL et al. e21 Low risk Intermediate risk High risk 5 Year brfs EQD2 (2Gy Fx) EQD2 (2Gy Fx) EQD2 (2Gy Fx) Fig. 1. Outcomes for each patient/treatment group along with fitted values from Model B. Error bars represent 95% CI on the binomial proportions in each group. Solid lines and filled symbols represent AD-treated patients; and broken lines and open symbols represent non AD-treated patients. Data are normalized to 2-Gy fractions using the fitted a/b values. Overall the various approaches give a/b estimates that differ by a factor of 2, which is within the confidence intervals of the primary model. Although a priori we did not expect there to be a time factor, we considered a model including the presence of a time factor starting at time T k (kick-off time) after a lag period. Surprisingly this model gave a significantly better fit to the data (p = 0.005) with zero lag (95% CI = 0 26d) and a time factor (g/a) of 0.51 Gy/d (95% CI = Gy/ d) (Table 5). This corrresponds to 0.21 Gy/d when using 2-Gy fractions. Including this time factor increases the a/b ratio to 4.0 ( ). However this result was strongly dependent on one of the datasets (Miralbell 4-Gy per fraction, 6.5 weeks overall time) and after excluding this dataset the time factor was not significant. DISCUSSION Several experimental and clinical studies have suggested a relatively low a/b ratio for prostate cancer control ranging between 0.8 and 2.2 Gy, lower than the corresponding a/b ratio for late-responding tissues (i.e., 3 5 Gy) (14, 15). Such Table 5. Sensitivity analyses. Fitted parameters with 95%CI from a number of alternative representations of the between-group variability and the inclusion of a time fac r (g/a). Models are described in the methods section. s represents the standard deviation of a normally distributed error where explicitly available Beta-binomial group heterogenetity (Model B) Quasi-binomial group heterogeneity MCMC group heterogeneity MCMC center heterogeneity Beta-binomial adding time factor k High 2.8 ( ) 2.5 ( ) 2.7 ( ) 0.0 ( ) 3.7 ( ) Interm. 4.5 ( ) 5.7 ( ) 4.2 ( ) 1.9 ( ) 5.4 ( ) Low 5.3 ( ) 6.0 ( ) 5.6 ( ) 2.3 ( ) 6.4 ( ) a Gy 1 High ( ) ( ) ( ) ( ) ( ) Interm ( ) ( ) ( ) ( ) ( ) Low ( ) ( ) ( ) ( ) ( ) H ( ) ( ) ( ) ( ) ( ) b/a Gy 1 Pooled 0.70 ( ) 0.96 ( ) 0.58 ( ) 0.55 ( ) 0.25 ( ) a/b Gy Pooled 1.4 ( ) 1.0 ( ) 1.8 ( ) 2.1 ( ) 4.0 ( ) g/a Gy per 0.51 ( ) day s 0.32 ( ) 0.34 ( )

6 e22 I. J. Radiation Oncology d Biology d Physics Volume 82, Number 1, 2012 a low a/b value, related to a relatively long doubling time of prostate cancer cells (27, 28) and to a very effective repair capacity of sublethal radiation damage at low dose per fraction, supports hypofractionation as an optimal RT option for localized prostate cancer. Several authors, however, have challenged the assumption of a low a/b value for prostate cancer control (29 32). A study on 3,571 patients failed to obtain reliable a/b results, and wide and insignificant 95% CI for a/b ratios (i.e., 1.1, N) were estimated for patients treated with external beam RT and different fractionation schedules when early PSA recurrence was the endpoint (33). Such uncertainty may be explained by the large majority of patients (3,144 or 91%) in that study being treated with standard fractions of 1.8 to 2.1 Gy with only a small proportion of patients treated with >2.1-Gy fractions. The authors acknowledged that outcome data from external beam RT studies using substantially higher doses per fraction might be needed to show increased precision in estimating reliable a/b values for prostate cancer. A subsequent analysis of 5093 patients showed an a/ b value of 1.55 Gy, but still fairly wide 95% CI ( Gy) (34). The majority of patients in our study (60%) were treated with moderately hypofractionated protocols (i.e., Gy/fraction) with only a small group of 40 patients treated with extreme hypofractionation (i.e., 6.7 Gy/fraction). The much higher proportion of hypofractionationtreated patients in the present dataset allowed a more consistent and reliable low a/b value to be calculated with narrow 95% CI ( Gy). Several radiobiological considerations need to be addressed when using the LQ model to estimate the fractionation sensitivity of prostate cancer cells. Factors conditioning the response (brfs) to changes in dose fractionation may be tumor-related (e.g., number of clonogens, cell proliferation rate, intrinsic radiosensitivity, oxygenation status, and repair rate of the irradiated tumor cells) as well as treatment-related factors (total dose and treatment time). Prolonging the overall treatment time (OTT) may have a negative influence on the outcome after RT as shown in tumor sites such as the head and neck, lung, cervix, and bladder (35). This is probably a consequence of accelerated tumor cell repopulation of clonogenic cells in these faster growing neoplasms, as opposed to prostate cancer that has been considered by many to be a very slow-growing cancer with negligible tumor clonogenic cell repopulation during the first 8 to 9 weeks of treatment (36, 37). One would expect that if repopulation exists in prostate cancer, this would probably happen mostly in high-risk tumors with consequently higher a/b values. However, recent studies have suggested a clinically significant repopulation effect exclusively for low-risk prostate cancer patients with an onset time of accelerated repopulation of days and an effective clonogen doubling time of 12 days (38, 39). Thus, for low-risk patients, an OTT >6 weeks might influence the outcome especially in those patients treated with suboptimally low total doses. Also, a recent retrospective analysis of a nine-institution database with 4,839 prostate cancer patients showed that dose and overall time above a selected cut point of 52 days (i.e., 7 weeks) were significant determinants of outcome of radiotherapy in low- and intermediate-risk patients, but only in those treated to $70 Gy (40). Most hypofractionated treatment protocols in the present dataset were delivered in <6 weeks, thus with almost no opportunity for clonogens to start their presumed accelerated repopulation. Therefore, the low a/b value estimated for low-risk patients in our study also may have been consistent with negligible cell repopulation. Nevertheless, the sensitivity analysis in our study showed that adding a lagged time factor to the model gave a significantly better fit to the data with an increase of the a/b value to 4 Gy (95% CI, ). However this result was dependent on one outlying dataset among the 11 datasets used for this analysis. Including a time factor in the model with the present number of centers and treatments is almost certainly overfitting the data, and it is likely that this is a spurious result. Certainly the lack of a lag phase is inconsistent with the biological and clinical data discussed above, and if we impose a reasonable lag period of 6 weeks, no repopulation can be detected in this total dataset. The present data is based on prescribed treatment, and thus overall treatment time is strongly correlated with dose and fractionation; individual patient data, with the inevitable variability in treatment time due to gaps in therapy may be better able to separate out the effects of fractionation and treatment time, albeit that such data are subject to other biases. It has been suggested that high-risk patients may be more likely to present with subclinical metastatic disease which might overshadow the OTT modulation effect for these patients (38). Furthermore, Gao et al., have suggested that locally advanced tumors may require a longer time (69 days) to improve blood/nutrient supply and trigger accelerated repopulation while on treatment (39). For this group of patients protracting the treatment up to 10 weeks or more may have a negligible effect on outcome. Again, either the presence of micrometastases and/or a long onset time to accelerated repopulation for intermediate- or high-risk patients may be consistent with the relatively low a/b values estimated in our study for these groups of patients. The question of how proposed AD mediated tumor reoxygenation in a neoadjuvant setting before RT influences cell repopulation especially in advanced stage tumors is a matter of speculation. Indeed, the presumed cell proliferation trigger from reoxygenation because of the enhanced tumor-cell killing effect may be prevented by an enhanced recruitment of tumor cells into the non-proliferative phase of the cell cycle ( G 0 ) as a consequence of the same AD therapy. Also, this argues against the higher estimated a/b value of 7.1 Gy (95% CI = ) as reported by Williams et al. (41). Other radiobiological parameters to be concerned about when estimating fractionation sensitivity of prostate cancer are the intrinsic radiosensitivity and the number of tumor clonogens. Both parameters are directly related in clinical

7 Dose fractionation sensitivity and prostate cancer d R. MIRALBELL et al. e23 responses and are strong determinants of tumor control probability. Repair during protracted irradiation is also a factor that can affect the derivation of a/b values. A slow in vivo repair half time of prostate tumor cells of 2 h or longer has been proposed (15, 42). Although the influence of slow repair in estimating the a/b value for prostate cancer may be less contributory than is proliferation/repopulation, incomplete repair may become an important factor for treatments with multiple fractions per day, in such cases artefactually increasing the a/b values for prostate cancer if one does not take it in account (32, 43). On the other hand, a slow repair in late responding normal tissues may be favored by the longer treatment intervals between fractions used in many hypofractionated treatment protocols (44). The sensitivity analyses using alternative methods for modelling the variability between patient groups and centers give a/b estimates that differ by a factor of (at most) 2. Although this is of a magnitude similar to the 95% CI for the primary fit, and does not cast doubt on the conclusion that a/b is small for these tumors, the variability does suggest that accurate estimates will require careful modelling of the heterogeneity, including data with multiple schedules from each center and sufficient data (large numbers of centers) to allow exploration of the distributions and assumptions. The Bayesian MCMC framework is potentially expandable to allow such investigation, although it is computationally intensive and requires large datasets. CONCLUSION In summary, the present analysis showed no significant evidence of a different value of a/b for different stages of disease. 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