Affiliation of authors: Biometric Research Branch, Division of Cancer Treatment and Diagnosis, National Cancer Institute, Bethesda, MD.
Correspondence to: Michael D. Radmacher, Ph.D., National Institutes of Health, 7550 Wisconsin Ave., MS 9015, Bethesda, MD 20892-9015 (e-mail: mdradmac{at}helix.nih.gov).
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ABSTRACT |
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INTRODUCTION |
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Although the BCPT resulted in a large reduction in the incidence of breast cancer, inferring from these results that tamoxifen prevented breast cancer is controversial (7,8). It is unclear whether the added benefit of tamoxifen in the BCPT was due to preventing the formation and growth of new tumors or to treating undetected, subclinical disease that existed at the time of study entry. This distinction is an important one: If the reduction in incidence was due exclusively to treatment of subclinical disease, then tamoxifen only benefited BCPT participants who had occult tumorsi.e., tumors that were not detected by an initial screening mammogram. If this were the case, then the observed reduction in breast cancer incidence due to tamoxifen would be short-lived, whereas prevention is usually thought of as a long-term or permanent effect. Indeed, in another tamoxifen prevention trial with a longer median follow-up (9), no statistically significant reduction in breast cancer incidence was observed. We have therefore developed a post-study analysis to investigate tamoxifen's efficacy in both preventing the formation and growth of new tumors and treating subclinical disease during the BCPT; we discuss our findings in this report.
It is quite likely that, regardless of stringent screening, a substantial number of BCPT participants had occult breast tumors at their points of entry into the study: The limit of tumor detection by mammography is at a diameter of approximately 5 mm (10), and the time necessary for the growth of a breast tumor from a single malignant cell at formation to a mammographically detectable size is on the order of years (11). Utilizing previously published growth models for breast tumors (12-14), we estimated the relative proportions of occult and new (i.e., formed after study entry) tumors that were detected in each year of the study. We used these estimates to judge whether the BCPT was sufficiently long to evaluate effects on new tumors and also to provide separate estimates of the efficacy of tamoxifen for preventing the formation and growth of new breast cancers and for treating occult disease.
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METHODS |
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A detailed description of the NSABP BCPT has been given elsewhere (1). Briefly, women who met specified eligibility criteria indicative of a high-risk breast cancer profile were accrued into the study during the period from 1992 to 1997 from 131 participating clinical centers throughout the United States and Canada. Each woman had a screening mammogram no more than 6 months before her entry into the study that showed no evidence of breast cancer, as well as a breast examination at entry that demonstrated no clinical evidence of cancer. The women were randomly assigned in a double-blind fashion to receive either placebo or 20 mg of tamoxifen daily for 5 years. Participants were monitored for development of disease. They had follow-up clinical examinations at 3 and 6 months and every 6 months thereafter; they also had annual follow-up mammograms. All positive or suspicious pathology or mammogram reports were submitted to NSABP headquarters for medical review. A total of 13 175 women were randomly assigned during the trial that had some follow-up information; the median follow-up was 54.6 months. We received data on the BCPT participants from the NSABP for statistical analysis.
Tumor Growth Models
We focused on two of the most widely used tumor growth models: exponential and Gompertzian growth. For the exponential growth model, the number of cells in a tumor at time t, Nt, is related to the initial size N0 and growth rate b by the formula:
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Exponential models are characterized by unbounded growth (i.e., there is no upper limit) and
a constant doubling time (td) that is related to the growth rate b by
the equation td = ln(2)/b. Assuming that doubling times for
individual tumors are gamma distributed (a distribution chosen for its ability to represent a wide
range of possibilities and its mathematical convenience), Brown et al. (13)
estimated the mean doubling time of a primary breast tumor, d
to be 2.1 months but cautioned that this estimate was not precisely determined and could vary by
a factor of 3. We initially used this model with the same three values of
d (0.7, 2.1, and 6.3 months) but determined that a mean doubling time of 6.3 months led
to longer subclinical growth intervals than were consistent with the BCPT tumor incidence data
(data not shown). We therefore retained the two smaller means of td (0.7 and
2.1 months) and a new mean value, 4.0 months, that was chosen to be marginally consistent with
the BCPT data. Fig. 1
compares the amount of time necessary for growth
of a single malignant cell to a mammographically detectable size for the three exponential models
assuming three different mean doubling times. This figure illustrates the variability of tumor
growth rates, given different underlying assumptions of the model.
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In addition to the three exponential models discussed above, we estimated time to
mammographic detection using a Gompertzian model proposed by Norton (14), in which the growth rates for individual breast tumors are log-normally distributed
with a mean ln(b) of -2.9 and standard deviation of ln(b) of 0.71, and N = 3.1 x 1012. The variation in time to
detection among tumors conforming to this model of growth is displayed alongside those of the
exponential models in Fig. 1
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Likelihood Model
We developed a likelihood model and used it to make a comparison between tamoxifen's efficacy in preventing the formation and growth of new tumors and in treating occult disease. For a specified tumor growth model and the given BCPT incidence data, our likelihood equation is a function of three parameters: measures of the efficacy of tamoxifen in both the prevention of new tumor formation and growth and the treatment of occult disease and an incidence rate of invasive breast cancer per 1000 women. The likelihood equation is maximized with respect to these three parameters, resulting in the best model fit for the given incidence data.
Let Xi denote the number of tumors detected in the placebo group during
year i of the study. Let Yi denote the analogous number for the
tamoxifen-treated group. We assume that Xi and Yi have
Poisson distributions, as is conventional for the analysis of rare events. Let be the combined
rate of mammographic and clinical detection per 1000 untreated women; we assume that
is
constant throughout the study. However, in year 1, the incidence rate per 1000 women is (1
- r)
, where the factor (1 - r) accounts for a reduced level of
mammographic detection in year 1 (see "Appendix" section). We
estimated r by computer simulations for each of the tumor growth models considered (see "Appendix" section). The incidence rate for the placebo group in year i is thus
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where nXi is the number of subjects at risk in the placebo group during year i.
For the tamoxifen group, we must account for the effects of the drug on invasive breast
cancer in determining the rates of tumor detection. Tamoxifen has a potential treatment effect on
occult tumors and a preventive effect on new tumors. Let ßTr represent
the efficacy of tamoxifen in the treatment of occult tumors. Likewise, let ßPr represent the efficacy of tamoxifen in preventing the formation and growth of new tumors.
The efficacy parameters are continuous with an upper limit of 1.0. A value of 1.0 indicates that
100% of tumors are effectively eliminated or prevented from forming, a value of 0.0
indicates that tamoxifen has no effect, and a negative value indicates that tamoxifen leads to an
increase in tumor incidence. Estimation of ßTr (for occult tumors) and
ßPr (for new tumors) requires estimates of the proportions of occult
tumors and new tumors out of the total detected in each year of study. Let i be the proportion of occult tumors out of the total detected in year i. Hence, (1
-
i) is the proportion of new tumors out of the total detected in
year i. Values of
i were estimated for each of the tumor
growth models by simulation (see "Appendix" section). Letting nYi represent the number of women in the placebo group in year i of the study, Yi is Poisson-distributed with parameter
Yi (i.e., the incidence rate of invasive breast cancer for the tamoxifen group
in year i), where,
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For example, the incidence rate per 1000 women in the tamoxifen group during the second
year of the study is equal to the incidence rate (per 1000 women) for occult tumors in the second
year, ·
2 (where
2 is the proportion of occult
tumors detected in the second year), times the probability that such a tumor is not effectively
treated by tamoxifen, (1 - ßTr), plus the incidence rate (per 1000
women) for new tumors in the second year,
· (1 -
2), times
the probability that tamoxifen is not effective in preventing the formation and growth of such
tumors, (1 - ßPr).
The likelihood for the number of tumors detected in the placebo group (Xi) and the tamoxifen-treated group (Xi) is then
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The likelihood is a function of , ßTr, ßPr, r, and the six values of
(one for each year). For a given growth model, r and
i are estimated separately from simulations (see
"Appendix" section). Thus, the likelihood becomes a function of
, ßTr, and ßPr for each tumor growth model. Maximum
likelihood estimates (MLEs) of
, ßTr, and ßPr were computed with the use of a FORTRAN program that incorporates the International
Mathematical and Statistical Library (IMSL) routine bconf for function minimization in
multiple dimensions (Visual Numerics, Inc., Houston, TX, 1994). Approximation of 95%
confidence intervals (CIs) for MLEs is discussed in the "Appendix" section.
Tamoxifen was considered to have statistically significant treatment and prevention effects if the
respective two-sided, approximate 95% CIs for ßTr and ßPr did not contain zero.
We performed two maximum likelihood analyses with the BCPT incidence data. In the first analysis, we only considered incidence data involving ER+ tumors, since tamoxifen's effectiveness in treatment seems to be confined to this subtype of breast tumor. However, it is possible that treatment with tamoxifen created an artificial environment in which breast tumors underwent selective pressure to convert to an ER-negative (ER-) statusthe ER+ analysis ignores this potential effect of tamoxifen. Hence, we performed a second maximum likelihood analysis that included incidence data on all breast tumors from the BCPT, regardless of their ER status.
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RESULTS |
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Tumor incidence data for the NSABP BCPT are displayed in Table
1. It is impossible to discern from these data how
many tumors detected in a given year were actually occult (i.e.,
existed at the start of the trial but were smaller than the threshold
of mammographic detection). For our likelihood model, however, it was
necessary to estimate the proportion,
i , of
occult tumors among the total detected in year i of the study
for untreated women. This was accomplished for each tumor growth model
by a computer simulation discussed in the "Appendix" section;
estimates are shown in Table 2.
The values of
i and the placebo
group data from Table 1
were used to estimate the percentage of tumors
detected in untreated women over the course of the study that formed
after participant entry (Table 2
). The estimated percentages of tumors
initiated after study entry range between 19% and 70%.
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The BCPT data and likelihood model described in the "Methods"
section were used to estimate the treatment and preventive efficacy of
tamoxifen as well as the rate of tumor detection. Since tamoxifen's
treatment efficacy seems to be limited to hormone-dependent tumors, we
first computed MLEs of the three parameters using only ER+
tumor data from Table 1; the estimates and 95% CIs are displayed
in
Table 3.
For the four growth models, estimates of
the combined rate of mammographic and clinical detection (
) among
women in the placebo group ranged between 5.3 and 6.1 ER+
invasive breast tumors per 1000 women. Good relative consistency for
the estimates of ßTr and
ßPr was also observed among the different tumor
growth models for the ER+ analysis, with
Tr slightly larger
than
Pr for each
case. The 95% CIs for ßTr and
ßPr are positive and exclude zero for every
growth model except ßPr from the exponential
model with
d = 4.0 months,
suggesting that tamoxifen had a statistically significant positive
effect on both the treatment of occult ER+ tumors and the
prevention of ER+ tumor formation and growth under the
assumptions of these three growth models.
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DISCUSSION |
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Next, we performed maximum likelihood analyses to estimate the efficacy of tamoxifen in the
treatment of occult disease and prevention of new tumor formation and growth. Two analyses
were performed separatelyin the complete group and in the ER+ group. We
focus our discussion on the results of the complete analysis because of the aforementioned
possibility that some ER+ tumors converted to ER- status in the
presence of tamoxifen. Moreover, the estimates of tamoxifen's efficacy for the complete
analysis tended to be more conservative (i.e., smaller) than those for the ER+
analysis. For the complete analysis, Tr ranged from 0.32 to 0.50, indicating a 32%-50% reduction in the incidence of
occult disease in the tamoxifen group. Similarly,
Pr ranged from 0.49 to 1.0, indicating a 49%-100% reduction in the
incidence of new tumors in the tamoxifen group. Most of these estimates were statistically greater
than zero, indicating that tamoxifen had a significant effect in both treating occult disease and
preventing the formation and growth of new tumors. The only exception was
Pr for the exponential model with
d = 4.0 months; a 95% CI could not be computed for this case
because the estimate occurred at a bound.
The results discussed above depend on estimates of the growth rate of subclinical breast cancer. For this purpose, we used models and growth rate estimates from two different sources and studied sensitivity to a range of values. However, the ability of any model to describe the growth characteristics of subclinical breast cancer is problematic, since, by definition, tumors are not observable at this stage. Because of the unavailability of data on subclinical growth rates of breast cancer, we used data from the clinically observable stage. Parameter values for the Gompertzian model were obtained by fitting the model to clinical data on detected, untreated breast tumors (14), while the exponential parameters were estimated from data on detected breast tumor recurrences (12,13). It is possible that subclinical growth rates are not accurately represented by these observable growth rates. If tumor growth is actually slower in subclinical than in clinical stages, our analysis underestimates the number of women with occult tumors in the BCPT, leading to inaccurate estimation of ßTr and ßPr.
However, some evidence from tumor transplantation models in mice supports the view that tumors are rapidly growing, even in their very earliest stages (16). In these experiments, suspensions of individual melanoma cells were inoculated into groups of mice. It was possible to transplant as few as one malignant cell by the method used. It was shown that the tumor-doubling times between 10 and 106 cells were relatively constant and, above a size of 106 cells, began to decrease slightly, indicating that the fastest growth in the transplanted tumors occurred early and supporting the view that subclinical growth is not slower than clinical growth. Doubling times for tumors composed of between one and 10 cells were slightly longer than those for tumors composed of between 10 and 106 cells, but they resulted in only a small increase in the estimated time for a tumor to reach a detectable size.
Still, it is possible that the prevention effect that we have measured in this post-study analysis does not truly represent an ability of tamoxifen to block cells from becoming malignant but rather represents an ability of tamoxifen to inhibit newly formed malignancies from growing to a detectable size. Even if this is the case, the estimates of ßTr and ßPr from our analysis, along with results from a trial (B-14) evaluating tamoxifen as a postsurgical adjuvant in the treatment of patients with ER+ tumors (4), support the concept that the effectiveness of tamoxifen in treating invasive breast cancer is a function of the initial tumor size being treated. First, from the BCPT, we predicted that tamoxifen reduced the incidence of occult disease by between 32% and 50%, whereas the B-14 trial resulted in only a 26% reduction in treatment failure; these measures are consistent with the view that occult, undiagnosed tumors may be smaller, less disseminated, and more effectively treated than residual tumor in a patient after detection and surgery. (We caution, however, that treatment failure in trial B-14 was defined as recurrence of breast cancer, a second primary cancer, or non-cancer-related death and, therefore, may not be directly comparable to the reduction of tumor incidence in the BCPT.) Furthermore, in our complete analysis, the predicted efficacy of tamoxifen in the treatment and/or prevention of newly forming tumors tended to be greater than for occult disease for a given growth model, indicating that treatment of tumors in their earliest stage of development and growth is even more effective than treatment of larger, though still undetectable, tumors.
Our analysis supports the view that the BCPT was long enough for a reduction in breast cancer incidence due to new tumors (as opposed to occult disease) to be measured, if one existed. Furthermore, the likelihood model that we developed suggests that tamoxifen did indeed reduce the incidence of breast cancer by both treating occult disease and preventing the formation and growth of new tumors. However, our analysis is limited by a lack of direct measurements on the growth function of occult breast tumors and by a lack of long-term follow-up data from the BCPT. In addition, many breast cancer risk factors (e.g., is the patient a carrier of a BRCA mutation?) and risks of tamoxifen use (e.g., increased risk of endometrial cancer) must be considered before recommending the use of tamoxifen to individual women for the prevention of breast cancer.
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APPENDIX |
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After a screening mammogram at a participant's entry into the
study (which excluded women with detectable breast cancer from the
study), the protocol of the BCPT required a follow-up mammogram
annually. Hence, it is possible that many patients did not receive
their first follow-up mammograms until the start of the second year.
Also, the date of diagnosis implies the date of biopsy diagnosis, which
may not have occurred until after the start of the second year for
women who had suspicious mammograms late in the first year. Mammography
is a more sensitive method of detection than a clinical examination.
The threshold size of mammographic detection is at a diameter of about
5 mm (10), whereas it is at a size of 1 cm3 for
clinical detection (6.5 x 107 and 1 x 109
cells, respectively). Thus, the number of tumors detected in the first
year of the BCPT was likely reduced because of a paucity of first-year
mammograms. Indeed, a chi-squared goodness-of-fit test of the BCPT
placebo group data rejected the hypothesis that the combined rate of
mammographic and clinical detection per 1000 untreated women ()
was constant over all 6 years of the study (P = .01).
Furthermore, reducing the incidence in the first year by a factor
r allowed for a good fit of the BCPT data to a model with
constant
(data not shown).
Simulation to Measure i and r
The proportion of occult tumors detected out of the total number
in year i of the BCPT (i) and the
fractional amount that detection was reduced in the first year of
study (r) were estimated from simulations executed in
FORTRAN. r ranges from 0.0 to 1.0; a value of 0.0 indicates
that no reduction occurred in the first year compared with later years,
whereas a value of 1.0 indicates that no tumors were detected in the
first year.
i also ranges from 0.0 to 1.0; a
value of 0.6 indicates that 60% of the tumors detected in the placebo
group in year i were occult (and the other 40% formed after
study entry).
Tumor formation was simulated as a Poisson process in a population of 108 high-risk women. The assumption that tumor formation is a Poisson process means that a tumor is as likely to arise at any one time in the interval of interest as at any other time (i.e., the rate of formation is constant). A Poisson model was chosen because it was suggested by the pattern of tumor occurrence for the BCPT; a more complicated multistage model of breast carcinogenesis has been shown to fit epidemiologic data well (17), but the Poisson model is justified for the high-risk participants of the prevention trial by viewing the women as being in the final stage of tumor progression before formation of a malignant tumor.
After tumor formation, growth was simulated according to the dynamics of a specific tumor growth model. The growth rate for individual tumors was randomly generated according to the distribution of growth rates and/or doubling times for the specific model. At a distant point in simulated time (25 years), a study was begun; this ensured that the distribution of the size of a simulated tumor in women who had a tumor at the start of the study was accurately represented. Only individuals without tumors or with tumors below the threshold of mammographic detection (assumed to be of a diameter of 0.5 cm or roughly 6.5 x 107 cells) (10) at the start of the study were further considered.
Tumors that reached the threshold of clinical detection (109 cells) within the
first year of the simulated study were detected in that year. At the beginning of the second year, a
mammogram was simulated. Women with tumors larger than the threshold for mammographic
detection, but that were not previously detected, were counted in the tumor incidence data for
year 2. Any other tumors that reached the threshold of clinical detection during the second year
were also counted in the incidence data for year 2. This mode of computing incidence data was
continued for 6 simulated years. The program tabulated whether or not detected tumors were
initiated prior to the start of the simulated study, making it possible to directly compute i for every year of study (see Table
2
). We estimated r by comparing the number of tumors detected
per 1000 women in year 1 with the average of the number detected per 1000 women in each of
years 2 through 6. The estimates of r derived from these simulations are shown in
Appendix table 1.
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CIs for , ßTr, and
ßPr were computed on the basis of the asymptotic
properties of MLEs (i.e., Gaussian distributions were used in the
construction of CIs). We verified the asymptotic approximation by
comparing the asymptotic distributions of parameters to empirical ones
from Monte Carlo simulations; all cases produced valid results except
where
Pr was
located at the upper constraint of 1.00.
Since the BCPT participants actually represent a small sample of the entire population, the
proportion of detected tumors in a given year that were occult varied about the mean value of
i. This variability was accounted for in the variance computations of the
MLEs: For any MLE
, the variance is
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We utilized computer simulations in making the above computations; 10 000 replicates were performed in which values were randomly assigned to represent the true proportion of occult tumors detected in a given year, and MLEs and asymptotic variances were computed for each replicate with the use of the simulated proportions and BCPT data. We estimated the first term on the right hand side of the above equation by taking the mean of the asymptotic variances over all of the replicates for a single parameter. We estimated the second term by calculating the variability in the MLEs of a given parameter over all replicates.
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NOTES |
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REFERENCES |
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Manuscript received April 24, 1999; revised October 7, 1999; accepted October 21, 1999.
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