Assessment of local tumor control using censored tumor response data

036(r3016/83/030383w4s03.00/0 Copyright 5 1983 Pcrgamon Press Ltd.

Inf. 3. Radiation Oncology Biol. Phys.. Vol. 9. pp. 383-386 Printed in the U.S.A. All rights resctvod.

0 Oncology Intelligence ASSESSMENT OF LOCAL TUMOR CONTROL USING CENSORED TUMOR RESPONSE DATA ALEXANDER M. WALKER, M.D., DR.P.H.’ HERMAN D. SUIT, M.D., D.PHIL.’

AND

‘Division of Biostatistics and Epidemiology, Sidney Farber Cancer Institute, and Department of Epidemiology, Harvard School of Public Health, Boston, MA 02115; 2Department of Radiation Medicine, Massachusetts

General

Hospital,

Boston, MA 02114

When animals die in the course of tumor control assays, the loss of information need not be absolute. Actuarial survival curves may be used to estimate local tumor control fractions, and these unbiased estimates may be weighted and used to estimate the parameters of dose/cure functions. The current standard procedure of discarding from analysis animals who do not complete a pre-assigned observation period is biased, and should not be used. Censored data, Kaplan-Meier

survival estimator, Tumor control.

INTRODUCTION The purpose of this communication is to provide a simple method for employing censored survival information in dose-response assays. The essential components of this method are already part of the statistical literature, but the proposed application is new. Consider as an example an experiment to estimate a TCDSO with a planned observation period of I20 days. A mouse may die before the end of the observation period, say at 80 days, because of the development of metastatic tumor or intercurrent disease. If the tumor has persisted or recurred in the irradiated area prior to death then the mouse is scored as a local failure. If at the time of death there is no evident tumor at the irradiated site then the tumor status which the mouse would have had at the end of planned observation period (120 days) is unknown: the datum is “censored” beyond the time of death. The animal can be counted as a tumor control for 80 days. The amount of information lost when an animal’s experience is censored is not known, because the incidence of tumor recurrence is almost always an unknown function of elapsed time in the observation period, and the dependence of tumor recurrence on time is in general not known for different radiation doses, among different kinds of tumors, or under different experimental conditions for which one might want to compare TCDSO’s. Uncertainty about the distribution of local failure times in any given tumor-radiation protocol contravenes the assumptions which underlie most strategies for dealing with censored data. Counting a censored animal as

half an animal at risk’ ignores the nonuniform distribution of failure times. A proportional hazards analysis’ is twice ruled out by the differences in failure-time distributions: first because the hazards of local failure in different treatment groups do not stand in a fixed ratio to one another over time; second because the proportion of tumors controlled at the end of the fixed observation time, is, in fact, the measurement of interest, and proposed methods for calculating survival curves from proportional hazards coefficients presuppose a known failure time distribution.*.’ “ Reduced sample” estimates, obtained by removing from analysis all animals that die prior to the end of follow-up, even if they have failed, are unbiased if death is unrelated to tumor recurrence, but they discard information and have the disconcerting property that estimated survivals may sometimes increase with time.4 Two common crude procedures for dealing with censoring-removing censored animals from the analysis altogether, or counting them as if they had continued tumor-free to the end of follow-up-are both biased. The former method, proposed some years ago,X has become a common practice. Tumor control fractions at fixed times can be calculated for each radiation dose as actuarial, or product-limit survival estimates4 without placing undue constraints on the distribution of failure times. The question remains of how to employ a series of these fractions as the realization of a dependent variable in an appropriately weighted regression on radiation dose. Kaplan and Meier4 suggested comparing the variance of the product-limit esti-

Reprint requests to: Dr. Walker, Division of Biostatistics and Epidemiology, Sidney Farber Cancer Institute, 44 Binney Street, Boston, MA 02115.

Supported by Grants CA 133 I 1 and CA 065 16, awarded the National Cancer Institute, DHEW. Accepted for publication 12 November 1982. 383

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mate of the survival fraction to a binomial variance to derive an “effective sample size . . which in the absence of losses would give the same variance” as that of the product-limit estimator. Taken together with the estimated survival fraction, this effective sample size can then be used to construct “effective failure” and “effective survival” counts (these would not in general be integers), which could be entered into the likelihood function of a logistic regression. Unfortunately the effective sample size estimate is indeterminate when the surviving proportion is 1 or 0, an unacceptable limitation for the extremes of survival which characterize dose-response assays. However, this defect can be remedied by assigning reasonable effective sample sizes for these two extreme situations. Throughout the sections that follow the term “recurrence” should be taken to mean a local recurrence or local failure of tumor control. at the irradiated site.

METHODS

AND MATERIALS

Calculating tumor control fraction and effective sample Size in the presence of censoring. The proportion of animals surviving an observation period may be viewed as the product of the proportions surviving a sequence of time intervals which, added together, make up the observation period. That is, if an observation period t is broken up into K intervals, then the relation between the proportion surviving to t, S(t), and the proportion surviving each interval i, s(i) is

S(t)

= fJ s(i) 1-l

If we let K become very large, so that the intervals become so small that at most a single event (censoring or recurrence) occurs in each interval, then the values s(i) take on a dichotomous character, depending on whether a tumor has recurred during the interval. Let n(i) be the number of animals under observation (alive and recurrence-free) at the beginning of the ilh interval. Then, if a tumor recurs in the ifh interval

s(i) =

n(i) - 1 n(i)

and if no tumor recurs s(i) = 1 S(t) is unaffected by intervals in which there is no recurrence. Therefore, even with an “infinite” number of intervals, the product can be evaluated by considering only those infinitesimal time periods in which a tumor does recur. If there are r recurrences up to and including

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1983. Volume 9. Number

3

time t, then

S(t) =

r n(j) n j-1

-

1

n(j)

where the n(j) are the numbers of animals under observation (alive and recurrence-free) just prior to each local tumor recurrence. Censoring of animals has been taken into account implicitly in the n(j)‘s. Successive n(j)‘s are decremented not only by recurrences, but also by intervening censored observations. S(t) is the Kaplan-Meier4 product-limit estimator of local recurrence free survival probability through time t. The variance of S(t) is

V[S(t)]

= S’(t)

2 [n’(j) ,= I

- n(j)]

~’

If there were no censored animals then S(t) would equal the simple proportion surviving. If there were m animals in the experiment to begin with, then the variance of the uncensored S(t), as an estimate of the true local survival probability would be the binomial variance:

v[s(t)l

=

S(f)[J - S(t)1 m

Kaplan and Meier4 suggested that in the presence of censoring, an effective sample size could be calculated by considering the size of an uncensored experiment, which would yield an estimate of the survival probability to the same degree of accuracy. This effective sample size is obtained by setting the above two expressions equal to one another and solving for m: 1 - s(t)

m=

S(t)

2

[n”(j) -

n(j>l-'

,=I

Two extreme cases need to be specified in order to define m for all possible values. When S(t) is zero or one, the approximations on which the variance estimates above are based are no longer valid. When S(t) = 0, it is appropriate to set m equal to the number of recurrences, since the zero estimate of tumor control probability depends only on the recurrences and not on the distribution of censored values prior to the last recurrence. When S(t) = 1, m can be set to equal the number of survivors through time t, since the estimate of S(t) again depends in no way on the censored observations. Given m(t) (the effective sample size through time t), the effective number of failures, e(t), in a group at time t can be calculated from the Kaplan-Meier tumor control fraction, S(t), as e(t) = [I - S(t)]m(t)

Assessment of local tumor control 0 A. M.

Appendix tions.

The e’s and m’s from several groups observed at different dose levels can be entered into commonly used analytic procedures just as if they represented integral, uncensored values. Use of m in regression procedures Point and interval estimation in logistic regression are accomplished by procedures analogous to those used when there is no censoring. Let a be the log(TCD50) and b be the logistic regression slope. Then the fitted tumor control fraction for log-dose d is s(d) = {exp[b(d The likelihood

function

- a)]\/{1 + exp[b(d

- a)]

is

f. = Wrn

“( 1 - s)e

the product being taken over all dose levels observed. Values of a and b which maximize I!. are taken as estimates of the log(TCD50) and slope parameters. The inverse of the matrix of second derivatives of the loglikelihood function with respect to a and 6, when multiplied by 1, provides an estimate of the variancecovariance matrix, whose asymptotic validity derives from the fact that the various dose groups are weighted as a function of their information content in a way that renders groups with and without censoring directly comparable. The proposed methodology is not limited to logistic regression: in fact, it can be adapted to any dose-response estimating procedure which uses a maximum likelihood criterion to define parameter estimates. For example, Porter6 has presented methods for estimating the parameters of a tumor control function, taking as his starting point a careful mathematical interpretation of the single clonogen theory of tumor recurrence. In order to employ data from groups with censoring, one substitutes m(t) and e(t), as defined above, for n and r throughout Porter’s

385

WALKER AND H. D. SUIT

A to obtain

the appropriate

estimating

equa-

Example Table 1 shows the results of a hypothetical tumor control assay in which six groups of 10 mice each are exposed to radiation doses ranging from 50 to 100 Gy and censoring occurs in every dose group. The proposed, unbiased estimated local tumor control fractions are the Kaplan-Meier product-limit estimates of survival up to the close of the observation period; the effective group sizes have been calculated using the methods previously described in the section on Calculating Tumor Control Fraction. The TCDSO, calculated using the logistic model described in the section on Use of m in Regression Procedures is 79.1 Gy, with 95% confidence limits of 73.0 Gy and 85.8 Gy. Tumor control fractions and sample sizes obtained by eliminating censored animals altogether are also given. Every tumor control fraction is less than or equal to its unbiased counterpart. The corresponding TCDSO is 80.5 Gy (95% confidence interval 73.9 to 87.6 Gy); this is higher than the unbiased value because the artifactual lowering of tumor control fractions makes it appear that a higher dose is required to achieve 50% control. Applicability and limitations The method of calculating the effective sample size presented here was chosen because it is relatively well established in the statistical literature. Rothman,’ for example, used it in calculating, for survival curves, approximate confidence intervals which Anderson and Bernstein’ have shown to have excellent statistical properties. The effective sample size depends entirely on the relative sequence of censoring events and tumor control failures. In effect, animals censored prior to the occurrence of any failure are eliminated from the study; animals censored after the last failure are counted as if they had survived the entire observation period. Interme-

Table I. Results of a tumor control assay Proposed Radiation dose (CY) 50 60 70 80 90 100

Tumor control pattern* [CFFFFFFFFF] [ FFCFFFFFFIX [FFFFCFFFIXX [ FFFFFCIXXXX [CFFC]XXXXXX [CFC]XXXXXXX

Tumor control fraction+ ,000 .I 14 ,240 500 .778 ,889

procedure

Standard

procedure3

Effective sample size

Tumor control fraction

Sample size

9.0 8.8 8.6 10.0 9.0 9.0

,000 .I 1 I ,222 ,444 ,750 ,875

9.0 9.0 9.0 9.0 8.0 8.0

*Read from left to right for the temporal sequence of events. [ = Initiation of observation period; ] = Close of observation period; C = Censored, mouse dies free of tumor before 1; F = Failure, tumor occurs before 1; X = Survival, mouse is tumor-free through 1. tKaplan-Meier product-limit estimate of the “actuarial survival” to the close of the observation period. SObtained by excluding censored animals from all calculations.

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diate censoring patterns result in partial degrees of inclusion in the effective sample size. The suggested method for calculating m is free from prior assumptions regarding the underlying, expected distribution of failure times, resulting in a flexibility of useful applications. Assumption-free statistical procedures, however, are generally inferior to procedures which incorporate valid assumptions: a loss of efficiency is the price usually paid for a robust statistical technique. Several other procedures for calculating m can in fact be contemplated; however, the final estimate of TCDSO turns out to be relatively insensitive to different choices of m, so that almost any set of plausible values yields substantially the same estimate. The importance of the proposed method then does not derive from the algorithm for calculating effective sample size. The most important aspect of the proposed methodology is its use of an unbiased estimate of local tumor control probability. The Kaplan-Meier product-limit estimate of survival (the “actuarial survival”) provides a valid estimate of the local tumor control probability, even

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with small test groups, so long as the censoring is independent of local tumor recurrence probability within each of the dose groups. By contrast, the generally applied technique of simply removing censored animals from the analysis leads to a consistent negative bias in local tumor control probability estimates and a resulting positive bias in estimates of the TCDSO. The validity of the Kaplan-Meier estimate of local tumor control probability is unaffected when censoring is purely a function of radiation dose or any factor which, at any given dose, is unrelated to local tumor control. Nonetheless, when censoring is very common (with for example half of the animals dying before local failure or the end of the observation period), then small, undocumentable correlations between local tumor recurrence risk and censoring can introduce major errors into the estimation of local tumor control probability. Experiments in which even a quarter of the animals are censored probably should not be accepted for analysis by this or any other technique because of the potential for large, uncontrollable errors in the final estimates.

REFERENCES Anderson, J.R., Bernstein, L.: Approximate confidence intervals for probabilities of survival and quantiles in life1982. table analysis. Biometrics 38: 407416, Cox, D.R.: Regression models and life tables (with discussion). J.R. Statist. Sot. B 34: 187-220, 1972. Kalbfleisch, J.D., Prentice, R.: Marginal likelihoods based on Cox’s regression and life model. Biometrika 60: 267-278, 1973. Kaplan, E.L., Meier, P.: Nonparametric observation from incomplete observation. J. Am. Statist. Assoc. 53: 457-48 I, 1958.

Littel, A.S.: Estimation of the T-year survival rate from follow-up studies over a limited period of time. Hum. Biol. 24: 87-l 16, 1952. Porter, E.H.: The statistics of dose/cure relationships for irradiated tumors. Br. J. Radial. 53: 210-217, 1980. Rothman, K.J.: Estimation of confidence limits for the cumulative probability of survival in life table analysis. J. Chron. Dis. 31: 557-560, 1978. 8. Suit, H.D., Shalek, R.J., Wette, R.: Radiation response of C3H mammary carcinoma. In Cellular Radiation Biology. Baltimore, Williams and Wilkins. 1965, pp. 5 14-530.