Estimation of Tamoxifen s Efficacy for Preventing the Formation and Growth of Breast Tumors

Transcription

1 Estimation of Tamoxifen s Efficacy for Preventing the Formation and Growth of Breast Tumors Michael D. Radmacher, Richard Simon Background: Several randomized clinical trials have tested the hypothesis that tamoxifen is effective in preventing breast cancer. The largest such trial, the National Surgical Adjuvant Breast and Bowel Project s Breast Cancer Prevention Trial (BCPT), reported a 49% reduction in risk of invasive breast cancer for the tamoxifen group. However, it is unclear whether the effect of tamoxifen in this trial was mainly due to prevention of newly forming tumors or to treatment of occult disease. Methods: We used various tumor growth models (i.e., exponential and Gompertzian [growth limited by tumor size]) and a computer simulation to approximate the percentage of detected tumors that were initiated after study entry. Maximum likelihood techniques were then used to estimate separately the efficacy of tamoxifen in treating occult disease and in preventing the formation and growth of new tumors. Results: Under the assumptions of most of the growth models, the trial was sufficiently long for substantial numbers of new tumors to form, grow, and be detected during the trial. With the Gompertzian model and all available incidence data from the BCPT, it was estimated that 60% (95% confidence interval [CI] = 40% 80%) fewer new tumors were detected in the tamoxifen group than in the placebo group. Likewise, 35% (95% CI = 6% 63%) fewer occult tumors were detected in the tamoxifen group. With this model, the estimated incidence rate of invasive breast cancer among women in the placebo group of the BCPT was 7.7 (95% CI = ) per 1000 women per year. Similar results were obtained with three exponential tumor growth models. Conclusions: These results support the concept that tamoxifen reduced cancer incidence in the BCPT through both treatment of occult disease and prevention of new tumor formation and growth. However, data from prevention trials may never be sufficient to completely distinguish prevention of new tumor formation from treatment of occult disease. [J Natl Cancer Inst 2000;92:48 53] The National Surgical Adjuvant Breast and Bowel Project (NSABP) recently reported the results of a randomized, placebo-controlled clinical trial to evaluate the efficacy of tamoxifen in preventing breast cancer in women considered to be at high risk for the disease (1). Tamoxifen was chosen as the agent for evaluation in the Breast Cancer Prevention Trial (BCPT) in part because of its proven effectiveness in reducing tumor recurrence when used as postoperative adjuvant therapy as well as because of the finding that patients treated with tamoxifen had a statistically significant decrease in incidence of contralateral breast cancer (2 5). Participants of the BCPT received either placebo or a 20-mg dose of tamoxifen daily; the median patient follow-up was 4.5 years. In the study, the cumulative incidence of invasive breast cancer per 1000 women was reduced by 49% in the tamoxifen group. [The term invasive breast cancer in this report conforms with that used by the NSABP.] Consistent with tamoxifen s proposed mode of action as an estrogen antagonist (6), the reduction in incidence was more pronounced when only estrogen receptor-positive (ER + ) tumors were considered: Tamoxifen administration led to a 69% decrease in the detection of ER + tumors. 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 tumors i.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 shortlived, whereas prevention is usually thought of as a long-term or permanent effect. Indeed, in another tamoxifen prevention trial with a longer median followup (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. METHODS The National Surgical Adjuvant Breast and Bowel Project s Breast Cancer Prevention Trial 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 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 ( mdradmac@helix.nih.gov). See Notes following References. 48 REPORTS Journal of the National Cancer Institute, Vol. 92, No. 1, January 5, 2000

2 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 followup clinical examinations at 3 and 6 months and every 6 months thereafter; they also had annual followup mammograms. All positive or suspicious pathology or mammogram reports were submitted to NSABP headquarters for medical review. A total of 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, N t, is related to the initial size N 0 and growth rate b by the formula: N t = N 0 e bt. Exponential models are characterized by unbounded growth (i.e., there is no upper limit) and a constant doubling time (t d ) that is related to the growth rate b by the equation t d 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, t 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 t 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 t d (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. Although exponential growth serves as an appropriate model for early stages of tumor growth, it is generally the case that the doubling time of a tumor begins to increase as the tumor grows larger (15). Gompertzian growth accounts for this size-limited growth, with tumor size a function of the time t, an initial size N 0, limiting size N, and growth rate b. The parameters are related by the following equation: N t = N 0 exp ln N N 0 1 e bt. 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 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. 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 X i denote the number of tumors detected in the placebo group during year i of the study. Let Y i denote the analogous number for the tamoxifentreated group. We assume that X i and Y i 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 1 r, for i = Xi = nxi n Xi, fori 1, 1000 where n Xi 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 Tr represent the efficacy of tamoxifen in the treatment of occult tumors. Likewise, let Pr 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 Tr (for occult tumors) and Pr (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 n Yi represent the number of women in the placebo group in year i of the study, Y i is Poisson-distributed with parameter Yi (i.e., the incidence rate of invasive breast cancer for the tamoxifen group in year i), where, Fig. 1. Variation in the time of growth from a single malignant cell to the threshold size of mammographic detection, with the use of exponential (Exp) and Gompertzian models. Box plots show the 2.5 th (left whisker), 25 th (left end of rectangle), 50 th (middle line), 75 th (right end of rectangle), and 97.5 th (right whisker) percentiles of the time required to grow from 1 to cells (approximate number in a tumor of diameter 5 mm) for each of the tumor growth models. Journal of the National Cancer Institute, Vol. 92, No. 1, January 5, 2000 REPORTS 49

3 Table 1. Tumor incidence data from the Breast Cancer Prevention Trial (BCPT)* Placebo group Tamoxifen group Year Person-years at risk ER + tumors Total tumors Person-years at risk ER + tumors Total tumors *Incidence data for the BCPT including person-years at risk, number of estrogen receptor-positive (ER + ) tumors detected, and total number of tumors detected are shown for each year of the study. Data were provided by the National Surgical Adjuvant Breast and Bowel Project. 1 r 1000 i 1 Tr + 1 i 1 Pr ), for i = 1 Yi = nyi n Yi 1000 i 1 Tr + 1 i 1 Pr, for i 1. 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 Tr ), 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 Pr ). The likelihood for the number of tumors detected in the placebo group (X i ) and the tamoxifen-treated group (Y i ) is then 6 = i = 1 e X i Xi Xi e Yi Yi Yi. X i! Y i! The likelihood is a function of, Tr, Pr,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, Tr, and Pr for each tumor growth model. Maximum likelihood estimates (MLEs) of, Tr, and Pr 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 Tr and Pr 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 ) status the 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. RESULTS Contribution of New Tumors to Incidence Rates During Six Years of Study 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 ˆi 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%. Efficacy of Tamoxifen in Treating Occult Tumors Versus Preventing New Tumors 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 Tr and Pr was also observed among the different tumor growth models for the ER + analysis, with ˆTr slightly larger than ˆPr for each case. The 95% CIs for Tr and Pr are positive and exclude zero for every growth model except Pr from the expo- Table 2. Estimated contribution of occult and new tumor detection to the Breast Cancer Prevention Trial (BCPT) among women not treated with tamoxifen* Growth model ˆ1 ˆ2 ˆ3 ˆ4 ˆ5 ˆ6 % new tumors in BCPT Gomperzian Exponential t d 0.7 mo t d 2.1 mo t d 4.0 mo *Estimates from simulations of the proportion, i, of occult tumors (i.e., tumors that existed before but were not detected at study entry) of the total detected in the placebo group in year i of study are shown for each of four tumor growth models; also shown is the predicted percentage of tumor incidence in the placebo group that was due to detection of new tumors. Computed with the use of the total tumor incidence numbers from the placebo group of the BCPT. t d mean doubling time. 50 REPORTS Journal of the National Cancer Institute, Vol. 92, No. 1, January 5, 2000

4 Table 3. Maximum likelihood estimates (MLEs) and 95% confidence intervals (CIs) of combined rate of mammographically and clinically detected tumors among women in the placebo group ( ), efficacy of tamoxifen in treating occult tumors ( Tr ), and efficacy of tamoxifen in preventing formation and growth of new tumors ( Pr )* Growth model nential model with t 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. Results from the BCPT indicated a slight increase in the incidence of ER tumors in the tamoxifen group compared with the placebo group (1). Although the difference was not statistically significant, it is possible that tamoxifen caused the conversion of some ER + tumors to ER status. Limiting the analysis to ER + case subjects ignores this potential negative consequence of tamoxifen use, and the resulting estimates of Tr and Pr may be artificially inflated. Because of this possibility, an analysis was performed that included all tumor incidence data from the BCPT (Table 3 complete analysis). The results of the complete analysis differed from the ER + analysis in several ways: The estimates of were higher, the estimates of Tr were lower, and there was more variation in the estimates of Pr among the growth models. However, similar to the ER + analysis, the 95% CIs for Tr and Pr fell entirely above zero for all cases except for Pr from the exponential (t d 4.0 months) model. DISCUSSION MLE (95% CI) ˆ, incidence case subjects/y per 1000 women ˆ Tr ˆPr ER + analysis Gomperzian 5.7 ( ) 0.71 ( ) 0.67 ( ) Exponential t d 0.7 mo 5.3 ( ) 0.80 ( ) 0.63 ( ) t d 2.1 mo 5.8 ( ) 0.71 ( ) 0.63 ( ) t d 4.0 mo 6.1 ( ) 0.70 ( ) 0.64 ( 0.40 to 1.00) Complete analysis Gomperzian 7.7 ( ) 0.35 ( ) 0.60 ( ) Exponential t d 0.7 mo 7.2 ( ) 0.50 ( ) 0.49 ( ) t d 2.1 mo 7.8 ( ) 0.32 ( ) 0.82 ( ) t d 4.0 mo 8.2 ( ) 0.37 ( ) 1.00 *MLEs and 95% CIs are shown for, Tr, and Pr for an analysis that only considers estrogen receptorpositive (ER + ) tumor incidence data from the Breast Cancer Prevention Trial (BCPT) (ER + analysis) and for an analysis that considers all tumor incidence data from the BCPT (complete analysis). t d mean doubling time. We performed this post-study statistical analysis of the BCPT incidence data to distinguish between tamoxifen s effectiveness in the treatment of occult disease and prevention of new tumor formation and growth. The first step was to determine whether the study was long enough for sufficient numbers of tumors to form, grow, and be detected entirely within the span of the trial. If not, then most of the reduction in incidence for the BCPT was probably due to treatment of occult disease. Three of the four tumor growth models used in our analysis (the Gompertzian model and the exponential models with t d 0.7 and 2.1 months) indicated that a substantial number of the tumors detected during the trial were newly formed. For these growth models, it was estimated that at least 34% of the tumors detected in the placebo group during the BCPT formed after study entry. It is thus possible that, under the assumptions of these three growth models, at least some of the observed reduction in incidence was due to prevention of new tumor formation and growth. In contrast, using the fourth growth model (exponential, t d 4.0 months), we estimated that 81% of the tumors detected in the placebo group during the BCPT were occult at study entry; under the assumptions of this model, nearly all of the reduction in incidence during the trial is explained by treatment of occult disease. 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 separately in 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, ˆTr ranged from 0.32 to 0.50, indicating a 32% 50% reduction in the incidence of occult disease in the tamoxifen group. Similarly, ˆPr 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 ˆPr for the exponential model with t 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 Tr and Pr. 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 Journal of the National Cancer Institute, Vol. 92, No. 1, January 5, 2000 REPORTS 51

5 transplant as few as one malignant cell by the method used. It was shown that the tumor-doubling times between 10 and 10 6 cells were relatively constant and, above a size of 10 6 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 10 6 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 poststudy 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 Tr and Pr 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. APPENDIX Reduction in First-Year Detection Due to Decreased Mammographic Detection 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 cm 3 for clinical detection ( and 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 10 8 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 cells) (10) at the start of the study were further considered. Tumors that reached the threshold of clinical detection (10 9 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 ˆi 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. Confidence Intervals CIs for, Tr, and Pr were computed on the basis of the asymptotic properties of MLEs 52 REPORTS Journal of the National Cancer Institute, Vol. 92, No. 1, January 5, 2000

6 Appendix table 1. Estimated reduction factors ( ˆr ) for incidence of breast tumors in the first year of the Breast Cancer Prevention Trial Growth model Gomperzian 0.51 Exponential* t d 0.7 mo 0.23 t d 2.1 mo 0.55 t d 4.0 mo 0.73 * t d mean doubling time. (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 ˆPr was located at the upper constraint of 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 ˆ, the variance is Var = E Var + Var E. We utilized computer simulations in making the above computations; 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. ˆr REFERENCES (1) Fisher B, Costantino JP, Wickerham DL, Redmond CK, Kavanah M, Cronin WM, et al. Tamoxifen for prevention of breast cancer: report of the National Surgical Adjuvant Breast and Bowel Project P-1 study. J Natl Cancer Inst 1998;90: (2) Controlled trial of tamoxifen as single adjuvant agent in management of early breast cancer. Analysis at six years by Nolvadex Adjuvant Trial Organisation. Lancet 1985;1: (3) Fisher B, Redmond C, Brown A, Fisher ER, Wolmark N, Bowman D, et al. Adjuvant chemotherapy with and without tamoxifen in the treatment of primary breast cancer: 5-year results from the National Surgical Adjuvant Breast and Bowel Project Trial. J Clin Oncol 1986;4: (4) Fisher B, Costantino J, Redmond C, Poisson R, Bowman D, Couture J, et al. A randomized clinical trial evaluating tamoxifen in the treatment of patients with node-negative breast cancer who have estrogen-receptorpositive tumors. N Engl J Med 1989;320: (5) Rutqvist LE, Cedermark B, Glas U, Mattsson A, Skoog L, Somell A, et al. Contralateral primary tumors in breast cancer patients in a randomized trial of adjuvant tamoxifen therapy. J Natl Cancer Inst 1991;83: (6) Wakeling AE, Valcaccia B, Newboult E, Green LR. Non-steroidal antioestrogens receptor binding and biological response in rat uterus, rat mammary carcinoma and human breast cancer cells. J Steroid Biochem 1984; 20: (7) Bruzzi P. Tamoxifen for the prevention of breast cancer. Important questions remain unanswered, and existing trials should continue. BMJ 1988;316: (8) Pritchard KI. Is tamoxifen effective in prevention of breast cancer? Lancet 1998;352: (9) Powles T, Eeles R, Ashley S, Easton D, Chang J, Dowsett M, et al. Interim analysis of the incidence of breast cancer in the Royal Marsden Hospital tamoxifen randomised chemoprevention trial. Lancet 1998;352: (10) Gallager HS, Martin JE. An orientation to the concept of minimal breast cancer. Cancer 1971;28: (11) Heuser L, Spratt JS, Polk HC Jr. Growth rates of primary breast cancers. Cancer 1979;43: (12) Brown BW, Atkinson EN, Bartoszynski R, Thompson JR, Montague ED. Estimation of human tumor growth rate from distribution of tumor size at detection. J Natl Cancer Inst 1984;72:31 8. (13) Brown BW, Atkinson EN, Thompson JR, Montague ED. Lack of concordance of growth rates of primary and recurrent breast cancer. J Natl Cancer Inst 1987;78: (14) Norton L. A Gompertzian model of human breast cancer growth. Cancer Res 1988;48: (15) Shackney SE, McCormack GW, Cuchural GJ Jr. Growth rate patterns of solid tumors and their relation to responsiveness to therapy: an analytical review. Ann Intern Med 1978;89: (16) Steel GG. Growth kinetics of tumours. Oxford (U.K.): Oxford University Press; p (17) Moolgavkar SH, Day NE, Stevens RG. Twostage model for carcinogenesis: epidemiology of breast cancer in females. J Natl Cancer Inst 1980;65: NOTES We thank the National Surgical Adjuvant Breast and Bowel Project, and especially Dr. Joseph Costantino, for providing data from the Tamoxifen Breast Cancer Prevention Trial. Manuscript received April 24, 1999; revised October 7, 1999; accepted October 21, Journal of the National Cancer Institute, Vol. 92, No. 1, January 5, 2000 REPORTS 53