Radiotherapy and Oncology, 25 (1992) 313-315 © 1992 Elsevier Science Publishers B.V. All rights reserved. 0167-8140/92/$05.00

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R A D I O N 01096

Short Communication

Dose-time factors in head and neck data Jeremy M. G. Taylor and H. Rodney Withers Department of Radiation Oncology and JCCC, UCLA Medical Center, Los Angeles, CA, USA (Received 27 April 1992, revision received 18 August 1992, accepted 24 August 1992)

Key words: Tumor control; Power calculation; Non-parametric regression; Squamous cell carcinoma

to analyze the institutional control rate data using a method which does not require any extrapolation from the average dose to the isoeffect dose, and thus is not dependent on the steepness of the dose-response relationship.

Summary

Sample size necessary to detect a dose--response relationship

This paper discusses two points regarding the interpretation of dosetime effects on tumor control in head and neck data. It is shown that the sample size in many clinical series will be too small to be able to statistically detect a dose--response relationship. The results from a non-parametric regression technique applied to control rate data from a large number of institutions suggest an influence of both dose and time on the control rate and qualitatively agree with a previous analysis of these data.

As an illustration, we consider a simple clinical series of 50, 100, 150, 200, or 250 patients. All patients are assumed to be stage 2 head and neck cancer and receive 2 Gy fractions but with some variation in the overall treatment times and total doses. The total doses ranged from 58 Gy to 72 Gy (mean = 64.6 Gy, S.D. = 3.1 Gy). The overall treatment times ranged from 45 days to 69 days (mean= 51.6 days, S.D. = 5.5 days). The total dose and overall treatment time are correlated, Pearson's p = 0.55. This type of variation and association is typical of that seen in scattergram clinical head and neck data. The correlation coefficient of 0.55 between time and dose is the average of the coefficients from the 26 scattergrams with sample size at least 25 as described previously [4]. The range of the correlation coefficients in these scattergrams was 0-0.9. It is interesting that this average association between dose and time is not as high as hypothesized by Bentzen and Thames who suggest that dose and time are highly entangled and carry very similar information. In 4 of the 26 series, the correlation is greater than 0.7. In these, dose and time are highly entangled and probably carry similar information. Three different statistical models are assumed for the probability of-2 year local control (P). They are:

In a recent article [ 1 ] Bentzen and Thames discussed the time effect in fractionated radiotherapy for head and neck cancer. In particular, they looked critically at the method of analysis used in a previous study [4] to infer an effect of overall treatment time from the data on control rates at different institutions. Because of the lack of specific data such as stage, dose per fraction, overall time, and total dose for individual patients, the authors of the previous paper extrapolated from the institutional dose with the observed control rate to the dose to give a 50% control rate. These estimated isoeffect doses were then plotted against the treatment times and shown [4] to follow a "dog-leg" shape and to be in good agreement with the results from numerous scattergrams. The interpretation of this "dogleg" shape is that there is no effect of treatment time on the control rate up to about 28 days, and that beyond 28 days, extension of the treatment time is detrimental to tumor control and that roughly 0.6 Gy/day is needed to balance the effect of accelerated tumor growth. Bentzen and Thames criticize the method used to construct the "dog-leg" curve because the estimated isoeffect doses depend strongly on the assumed steepness of the dose-response curves. Critical to their criticism is the assumption that, in a series of patients, there is little or no effect of total dose on tumor control probability and they point out that for many clinical series, dose is not a statisticaUy significant factor. In particular, they suggest and provide some evidence that Withers et al, [4] have used too steep a dose-response relationship in extrapolating from the average dose actually administered to the estimated isoeffect dose. The purpose of this note is firstly to investigate the sample size necessary to statistically detect a dose-response relationship in a typical clinical series, and secondly

(A) log(P/(1- P))= - 1.693 + 0.082 D o s e - 0.049 Time. (B) log(P/(1 - P)) = - 3.661 + 0.140 Dose - 0.084 Time. (C) log(P/(1 - P)) = - 8.454 + 0.280 Dose - 0.168 Time. The coefficients in all three equations are chosen so that (i) 64 Gy in 50 days gives a 75% control rate, and (ii) 0.6 Gy is needed to balance 1 day's extension of treatment time [4]. Equation (A) corresponds to a shallow dose-response relationship (effective D O~ 18 Gy, see ref. [4]) and in which 64 Gy in 57 days gives a 68% control rate. Equation (B) corresponds to an intermediate dose-response relationship (effective Do--- 10 Gy) in which 64 Gy in 57 days gives a 63~o control rate. Equation (C) corresponds to a steep dose-response relationship (effective Do g 5 Gy) in which 64 Gy in 57 days gives a 47% control rate. With the above specifications it is possible [3] to calculate the approximate power of the 2-sided test of the hypothesis at the 5 % level that there is zero dose effect. These approximate powers are given in Fig. 1. The power is the probability of being able to detect a significant effect of dose. Except for the largest sample sizes and

Address for correspondence: J. M. G. Taylor, Depa_rlment of Radiation Oncology and JCCC, U C L A Medical Center, Los Angeles, CA 900241714, USA.

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Fig. 1. Statistical power to detect a significant effect of dose in a typical clinical series. The DO value indicates the assumed steepness of the true doseresponse relationship, Do = 5 Gy is the steepest and Do = 18 Gy is the shallowest (see text for details). the steepest assumed dose-response relationship, the probabilities in the figure are all low. Bentzen and Thames indicate that frequently dose is found to be not significant suggesting that the effect of dose on local control is very small. As the number of patients in most clinical series is less than 100, Fig. 1 suggests that even when there is an effect of dose it will usually be found to be not statistically significant. A lack of statistical significance should not necessarily be interpreted as no dose-response relationship. Rather than just indicating whether dose is statistically significant, a more informative summary would be to give an estimate together with a confidence interval of the steepness of the dose-response. Non-parametric estimate of isoeffect lines in institutional data

A different approach to analysis of the institutional control rate data is through non-parametric regression. Imagine a 3-dimensional surface (a "hillside") in which the two planar directions are total dose and overall treatment time, and the height of the "hill" is the control rate. Non-parametric regression consists of estimating the smooth "hillside" which best approximates the control rates at the specific institutions. Once the "hillside" is estimated, isoeffect contours can be easily obtained for any probability of local control. For example, the TCDs0 could be obtained for any specific overall treatment time, and then the locus of TCDso estimates for different treatment times could be plotted. This non-parametric regression analysis was performed using the ACE algorithm [2]. This algorithm fits a model of the form 0 (control rate) = #l(dose) + #2(time) + #3(stage). The functions 0, #1, #2, and #3 are estimated from the data, and required to be smooth and monotonic. In addition, institutions with

Fig. 3. Isoeffect tumor control probability contours for T 3 tumors estimated from institutional control rate data using a non-parametric regression technique. larger numbers of patients were given greater influence in the analysis by using a weighting term equal to the sample size. Figures 2, 3, and 4 show selected contours for 3 stages. The data used to obtained these lines are the 59 observations listed in Table I of ref. 4. The main point to notice from the graphs, which should be interpreted qualitatively, not quantitatively, is that they have a roughly similar shape to the dog-leg curve given in ref. 4. There is no effect of time up to about 40 days, and a strong effect at times larger than 40 days. There is also an apparent dose-response relationship. An increase in dose of 10 Gy increases the control rate by somewhere between 3 ~ and 20% depending upon the initial control rate selected. Because of the nature of the institutional control rate data, it is important that the reader does not overinterpret the exact position and shape of the curves in Figs. 2-4. Our purpose in presenting the curves is to show that the conclusions reached in ref. 4 concerning the influence of time on control rate are qualitatively replicated when a method of analysis is used which does not use the extrapolation technique and does not rely on a value of D o. Bentzen and Thames are skeptical about the way clinical data have been interpreted to adduce the influence of proliferation. As shown in Figs. 2-4, the interpretations are qualitatively similar even when no assumptions are made regarding the steepness of tumor control probability curves. In our view, a much greater source of concern, even in the absence of any extrapolations, is the usefulness and quality of the retrospective data. Even for one radiation oncologist in one institution, there will be patient and logistic selection biases. Furthermore, it is questionable whether it is valid to compare data between institutions. The above concerns, as well as those of Bentzen and Thames, do not invalidate the use of retrospective analysis to define important questions, but do underline the need for large randomized trials and standardized data collection procedures to obtain objective data for definitive analysis and interpretation.

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Fig. 4. Isoeffect tumor control probability contours for T2/T3 mixed tumors estimated from institutional control rate data using a non-parametric regression technique.

315 References 1 Bentzen, S. M., and Thames, H . D . Clinical evidence after tumor clonogen regeneration: interpretations of data. Radiother. Oncol. 22: 161-166, 1991. 2 Breiman, L. and Friedman, J.H. Estimating optimal transformations for multiple regression and correlation. J. Am. Star. Assoc. 80: 580-619, 1985.

3 Hsieh, F.Y. Sample size tables for logistic regression. Stat. Med. 8: 795802, 1989. 4 Withers, H. R., Taylor, J. M. G. and Maciejewski, B. The hazard of accelerated tumor clonogen repopulation during radiotherapy. Acta Oncol. 27: 131-146, 1988.

Dose-time factors in head and neck data.

This paper discusses two points regarding the interpretation of dose-time effects on tumor control in head and neck data. It is shown that the sample ...
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