SEARCH

SEARCH BY CITATION

Keywords:

  • breast carcinoma;
  • adjuvant therapy;
  • log normal model;
  • likelihood of cure

Abstract

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION
  6. REFERENCES

BACKGROUND

Traditional nonparametric statistical methods do not provide a quantitative measure of the lifetime benefit from adjuvant therapy. This deficiency makes it difficult to determine the long-term difference in impact between the two treatment arms of a clinical trial.

METHODS

To assess the impact of breast carcinoma recurrence, parametric survival models were derived from two randomized, controlled clinical trials of adjuvant therapy for Stage II breast carcinoma. To assess time to death from causes other than breast carcinoma, actuarial models derived from 1980 Census data were used. These two models were then combined to estimate the mean time to event (MTE) as a function of patient age, with the event being either recurrence or death from other causes. The MTE was then used to measure the differential benefit between two arms of a clinical trial.

RESULTS

In the first trial, differences in MTE between treatment groups varied from 2.7 years for 35-year-old patients to 1.4 years for 75-year-old patients. For this trial, the mechanism of survival benefit was an increase in time to recurrence. In the second trial, differences in MTE varied from 7.6 to 1.6 years over the same age ranges. For this trial, the mechanism of survival benefit was an increase in the likelihood of cure, i.e., an increase in the asymptote of the curve that represents proportion of patients without relapse.

CONCLUSIONS

When applied to data from controlled clinical trials, MTE offers a quantitative measure of long-term outcome from adjuvant therapy. The greatest benefit is achieved when therapy that increases the likelihood of cure is provided to young patients. Cancer 2003;97:1139–46. © 2003 American Cancer Society.

DOI 10.1002/cncr.11171

Breast carcinoma is the most common malignancy among American women. Because of the enormous personal and social impact of this disease, it is imperative that we select the most effective method to measure the benefit of treatment. Oncologists have conducted a number of clinical trials over the past two decades, many of which have confirmed the benefit of adjuvant therapy for patients with breast carcinoma. These trials have produced specific conclusions, often in the format: “For patients with stage II breast cancer, regimen A provided a 5-year survival advantage of X percent over regimen B.” Therefore, clinicians have often presumed that all women with Stage II breast carcinoma should be treated with Regimen A.

Within the past few years, however, such information has been viewed with increasing circumspection. When the difference in survival curves at only one fixed point in time is considered, the survival advantage that is propagated over the remaining lifetimes of those at risk is not acknowledged.1, 2 In addition, survival curves measure outcome for treatment groups, but they do not provide specific information regarding the time course of individual patients.1, 2 Therefore, we began to focus on an essential question that extends beyond the 5- or 10-year survival: How is treatment likely to affect the overall quality and duration of life for individual patients?

To address this question, we must develop quantitative, objective measures of outcome from treatment. To provide insight into the clinical outcomes of individual patients, these measures must include mitigating factors. Among such factors, age is especially important, because progressive mortality from other causes may overshadow an otherwise significant survival benefit for older patients.

In seeking the optimum measure of benefit from adjuvant therapy, we undertook a study focused on the three events that determine the clinical course of breast carcinoma: recurrence, death from breast carcinoma, and death from other causes. In treating patients, the goal is to delay recurrence and death by using palliative therapy or to remove the risk of recurrence altogether by using curative therapy.

To emphasize quality of life, we define the event as either recurrence or death from other causes, presuming that the side effects of recurrence and salvage therapy substantially diminish the value of the patient's remaining years. To measure the overall impact of a specific therapy, we compute the mean time to event (MTE) for all of the patients in a treatment group. Finally, to measure the difference in impact between two therapies in the context of a clinical trial, we estimate the difference in MTE between the two treatment arms.

Researchers can estimate MTE by performing numerical integration on a crossproduct of two survival curves.3, 4 The first curve represents cumulative survival from all causes other than breast carcinoma. This curve is generated by adapting data from population surveys, such as the U.S. Census Bureau report. The second curve represents cumulative survival related to breast carcinoma, with the survival end point being either recurrence or death from breast carcinoma. Unfortunately, the generation of such a curve poses a more serious computational challenge.

Researchers can estimate breast carcinoma survival (or recurrence-free survival) by using actuarial data from clinical trials. However, for most trials of current adjuvant regimens, the duration of follow-up is limited, usually to less than 10 years. Breast carcinoma is characterized by late recurrence, sometimes more than 20 years after diagnosis.5 Therefore, to obtain an accurate estimate of MTE, follow-up must be available for the entire time that any patient remains at risk from breast carcinoma. As a result, data from recent clinical trials cannot be used in its “raw” format to compute an accurate estimate of MTE. Instead, to obtain an accurate but expedient estimate of MTE from contemporary clinical trials, parametric survival methods must be used. These methods allow us to project outcome beyond available follow-up (see Fig. 1).

thumbnail image

Figure 1. Irregular lines represent Kaplan–Meier estimates of the proportion of breast carcinoma patients who have not had disease recurrence (recurrence-free breast carcinoma survival). Smooth curves with symbols represent the log normal model fit to these data. (A) The clinical trial by Bonadonna et al.20 used to derive the log-normal model. The two survival curves approach the same asymptote, suggesting that the two treatment groups differ only in their times to recurrence, but share the same asymptote, or cured fraction. (B) The clinical trial by Buzzoni et al.22 used to derive the log-normal model. The two survival curves approach different asymptotes, suggesting that the two treatment groups have different cured fractions.

Download figure to PowerPoint

Parametric survival analysis also serves an additional purpose by providing insight into the mechanism by which therapy alters outcome. Specifically, parametric methods allow a distinction to be made between palliative and curative therapies, i.e., therapies that only increase time to recurrence or death versus those that increase the likelihood of cure.6 These insights are especially significant for younger patients, for whom curative therapy can provide a many-fold greater benefit than palliation.4

We have developed an event-based survival model that contains two components. We derived the first component, which measures the impact of death from causes other than breast carcinoma, by using actuarial data from the U.S. Census Bureau 1980 report. We derived the second component, which measures the impact of either recurrence or death from breast carcinoma, by using a parametric survival model to extrapolate outcome beyond available follow-up. The parameters of this model were based on the data from selected clinical trials.

Collectively, these components provide objective, quantitative, and inclusive measures of long-term clinical course (i.e., the MTE for each treatment group and the difference in MTE between groups). Because death from other causes is highly dependent on age, MTE is expressed as a function of patient age. Using currently available data, MTE can be determined for a variety of adjuvant regimens studied in clinical trials. As an added advantage, this method helps to distinguish between curative and palliative therapy.

MATERIALS AND METHODS

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION
  6. REFERENCES

Survival from Other Causes

For causes other than breast carcinoma, survival was estimated by using the U.S. Census data from 1980. Adjustments were made to remove the small proportion of deaths from breast carcinoma in this population. The resulting data express the likelihood of survival for each year of life from birth through age 110 years.4

Recurrence-Free Survival from Breast Carcinoma

The log normal model was used to estimate breast carcinomasurvival beyond available follow-up. This model was developed originally by Boag7 in 1949 as a general method for estimating the likelihood of cure for various malignancies. The model incorporates the likelihood of cure (i.e., the asymptote of the curve that represents survival from the cancer under study) as one of three basic survival parameters. The two other basic parameters represent the mean and standard deviation of log survival time among uncured patients. The log normal function is used to represent the distribution of time to death among uncured patients. The resulting method provides a good approximation to survival data from a variety of malignancies.7–12 The value of the log normal function is especially well documented with respect to breast carcinoma.13–17 It is important to note that the model works equally well when the end point to survival is either recurrence13, 17 or death from breast carcinoma.14–16

To refine Boag's original model, the three basic survival parameters are expressed as functions of prognostic covariates, such as treatment group.8, 18 Further refinements allow other distribution functions, such as the log logistic or Weibull, to be substituted for the log normal.6, 19

Assumptions

The parametric survival model is based on two fundamental assumptions. First, the distribution of time to event (recurrence or death from cancer) follows a specific distribution, such as the log-normal, during the period of observed follow-up. Second,this distribution is unaffected by the termination of follow-up. That is, when follow-up terminates, time to event continues to follow the same distribution observed before termination.

Mechanism of therapy

To apply parametric analysis to the results of clinical trials, the mechanism by which therapy produces a difference in survival between the two treatment arms must be determined. The two alternative mechanisms are an increase in the likelihood of a cure and an increase in time to event among the uncured patients. First, tests are performed to determine the distribution function that most closely approximates time to event in the trial under study. For breast carcinoma, best results are usually achieved with the log-normal function, as was true in the current study.13–16 Next, a comparison is made between the two arms of the clinical trial. This comparison produces a P value, using the log likelihood method, to determine whether the two arms differ in the likelihood of a cure, median time to event, or both.6 A negative result with respect to the likelihood of a cure does not suggest that both treatment arms result in a zero likelihood of cure. Rather, the result suggests that the two arms produce the same likelihood of cure. In this context, any survival benefit of one arm over the other must result from a difference in the median time to event.

Measuring stability

To assure reproducible results, we tested the stability of the parametric model. These tests were performed with the same clinical data sets used to construct the model. Our goal was to measure the impact of the specific distribution function selected to approximate time to event and the impact of duration of follow-up. First, we selected the log normal function because it provided the best approximation to observed data. This function was used to estimate differences in MTE between the two treatment arms of each clinical trial included in the study. We repeated this process, using the log logistic and the Weibull functions instead of the log normal, and compared results with the estimated differences in MTE obtained by log normal analysis. In the next phase of testing, using the log normal model, we reduced the duration of follow-up by 1-year intervals and compared the resulting estimates with those produced by the full duration of follow-up. With this methodology, we determined the sensitivity of our method to changes in the distribution function selected and to the duration of available follow-up. The longer the duration of follow-up, the shorter the extrapolation of the model beyond available follow-up. Therefore, this methodology tests the sensitivity of the model to the degree of extrapolation.

Data from Two Clinical Trials

In a previous study, the log normal model was applied to data from five controlled, randomized clinical trials, with recurrence as the end point to survival.17 For each trial, the parametric model was used to determine whether the two treatment arms differed with respect to the likelihood of cure, median time to recurrence, or by both mechanisms. In the trial by Bonadonna et al.,20 results after 20 years of follow-up showed that the 2 treatment groups suffered the same proportion of breast carcinoma recurrences. This suggests that there was no difference in the likelihood of cure between the groups. Conversely, there was a difference between groups in time to recurrence, suggesting that therapy imparted a palliative benefit.

By applying the log normal model to the trial by Bonadonna et al.20 and by using recurrence as the end point to survival, it was possible to document a palliative effect with less than 10 years of follow-up.17, 21 The recurrence-free survival curves merged as they approached the same asymptote, or the likelihood of cure (Fig. 1A). This is the expected finding when there is no difference in the likelihood of cure between the two treatment arms, as was confirmed quantitatively using the parametric model.

The five clinical trials studied by parametric analysis also include one by Buzzoni et al.22 Results showed that the two treatment groups differed with respect to the likelihood of cure.17 The recurrence-free survival curves diverged toward distinct asymptotes. This is the expected finding when there is a difference in the likelihood of cure between the two treatment arms. This difference in the likelihood of cure was also confirmed quantitatively using the parametric model. Extensive testing of the parametric model has shown that the incidence of false positives (with respect to the likelihood of cure or median survival time) is within the alpha limit selected (i.e., P = 0.05).6

The two clinical trials described above, i.e., Bonadonna et al.20 and Buzzoni et al.,22 were used in this study as templates. Specifically, a log normal model was derived from the survival data of each trial by using recurrence as the end point to survival. However, in practice, the methodology described can be applied to the arms of any similarly structured clinical trial. Furthermore, the selected end point can be either recurrence or death from breast carcinoma.

The model derived from the Bonadonna et al. trial allows us to evaluate the impact of palliative therapy because parametric testing showed that the two arms differed only with respect to the median time to event. Conversely, the model derived from the Buzzoni et al. trial allows us to evaluate the impact of curative therapy because parametric testing showed that the two arms differed only with respect to the likelihood of cure. The two resulting models were used to generate estimates of recurrence-free survival from breast carcinoma. One model demonstrated that the impact of therapy alters only the median time to event, whereas the other demonstrated that the impact of therapy alters only the likelihood of cure.

Estimates of MTE

Our goal was to estimate the MTE in each treatment arm of a clinical trial, with the event being either breast carcinoma recurrence or death from other causes. To estimate MTE, we combined the effect of these two events into an estimate of recurrence-free/other-cause survival. Recurrence-free/other-cause survival represents the proportion of patients who, at any given time after treatment, have not suffered either a recurrence or death from causes other than breast carcinoma.

This survival curve is generated by taking the product of the two survival curves described, i.e., one that represents recurrence-free survival from breast carcinoma (derived from a parametric survival model) and one that represents survival from other causes (derived from the U.S. Census data). To estimates MTE, we performed numerical integration (using 1-year intervals) of the curve representing recurrence-free/other-cause survival. This integration was extended until age 110 years, by which time the curve had converged to zero survival. Details of this method are provided by Gamel and Vogel.4

RESULTS

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION
  6. REFERENCES

The basic findings of this study are shown in Table 1. For both treatment arms of both clinical trials, values of MTE diminished with increasing patient age. Within each trial, differences in MTE between the two treatment groups also decreased with increasing patient age. For most of the age intervals, however, the estimated values of MTE for the two treatment groups were different from one another (P < 0.05), suggesting that treatment had a statistically significant differential effect on MTE.

Table 1. Mean Time to Event (MTE) with Standard Error (SE) Estimated Using a Log Normal Modela
Age at diagnosis (yrs)TreatmentDifference in MTEP value
Group 1Group 2
MTESEMTESE
  • a

    Bonadonna trial: Group 1, adjuvant therapy; Group 2, no therapy. Buzzoni trial: Group 1, adjuvant therapy regimen A; Group 2, adjuvant therapy regimen B.

Trial by Bonadonna et al.20      
 3514.61.417.31.32.70.16
 4512.31.114.71.02.50.10
 5510.00.812.20.82.20.05
 657.80.69.70.61.80.03
 755.80.47.10.41.30.01
Trial by Buzzoni et al.22      
 3521.42.214.02.37.40.02
 4517.51.811.81.85.70.03
 5513.81.39.71.44.10.03
 6510.30.97.70.92.70.04
 757.30.65.80.61.50.06

It is important to assure that these results are not highly sensitive to either of two important factors: 1) the particular distribution function used to represent time to recurrence or 2) the duration of available follow-up. To achieve such assurance, analysis was repeated after substituting first the log logistic and then the Weibull for the log normal function.19 The values shown in Table 1 were consistent among these three functions. That is, for both clinical trials, results with the log logistic and Weibull distributions were within 5% of those found with the log normal distribution, when measured by differences in MTE between treatment groups. By the same measure, the log normal model deviated less than 10% when maximum follow-up was sequentially reduced by 1-year intervals, from 19 to 7 years in the first trial and from 10 to 5 years in the second trial. These findings provide assurance that the findings shown in Table 1 are not highly sensitive to the specific distribution used to approximate time to event or to the duration of follow-up available.

When follow-up was reduced beyond these limits, the results became inconsistent. Inconsistency is unavoidable with parametric survival methods because it is computationally difficult to approximate the distribution of recurrence times, unless available data extend beyond the modal value of the distribution. To derive stable parameter estimates, follow-up must include a number of events (recurrences) beyond the modal failure time.6, 8, 18, 19

Figure 2 shows recurrence-free/other-cause survival in the Bonadonna et al. (Fig. 2A) and Buzzoni et al. (Fig. 2B) trials. The curves in each figure correspond to the two treatment groups. To understand these figures, keep in mind that recurrence-free/other-cause survival is the product of recurrence-free survival (estimated by parametric analysis) and survival from other causes (estimated by census data). Within each trial, the two recurrence-free/other-cause curves represent the two treatment arms. These two curves first separate and then reunite (or close) to form an “envelope,” which represents the differences over time in survival between treatment groups.

thumbnail image

Figure 2. The curves represent recurrence-free, all-cause survival (i.e., the proportion of patients surviving both recurrence and death from causes other than breast carcinoma). The curves with symbols represent 35-year-old patients in the two treatment arms, whereas the symbols without lines represent 55-year-old patients. (A) The clinical trial by Bonadonna et al.20 (B) The clinical trial by Buzzoni et al.22

Download figure to PowerPoint

The initial separation of the curves reflects the differential impact of therapy on the two treatment groups. Closure of the curves to form an envelope occurs either when recurrence-free survival from breast carcinoma approaches the same asymptote in the two treatment groups or when other-cause survival approaches zero with progressive age, driving recurrence-free/other-cause survival to zero in both groups.

Closure occurs relatively quickly in the first clinical trial, even for young patients (Fig. 2A). Rapid closure occurs because treatment has had no impact on the likelihood of cure. As a result, recurrence-free survival approaches the same asymptote in both groups (Fig. 1A). Among middle-aged and older patients, rapid closure occurs for an additional reason: The impact of death from other causes is manifest within a few years of treatment.

In the second trial, however, the envelope of young patients stays open for several decades (Fig. 2B). This persistent difference between treatment groups represents a combination of two effects. First, recurrence-free survival approaches different asymptotes in the two treatment groups because they have a different likelihood of cure (Fig. 1B). Second, the impact of death from other causes is delayed for 25–30 years because the patients are young. Therefore, to produce a broad and enduring envelope, treatment must alter the likelihood of cure and must be directed toward young patients. This envelope, in turn, determines the difference in MTE between treatment arms. Therefore, the optimum benefit from treatment, as measured by MTE, is achieved when curative therapy is given to young patients. This conclusion, although well established by previous research, can now be confirmed with the specific data presented in Table 1.

DISCUSSION

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION
  6. REFERENCES

It is well known among clinical oncologists that adjuvant therapy is most effective in younger patients because they have the greatest life expectancy. Even with younger patients, however, we must consider the mechanism of therapy. In the case of palliative therapy, which prolongs life without enhancing the likelihood of cure, the advantage from therapy may be only marginal. Consider, for example, a regimen that provides a “statistically significant” survival benefit, but parametric analysis reveals that this benefit produces only a 1-year increase in the MTE, with the event being recurrence or death from other causes. Would a young woman consider this advantage to be worth the side effects of adjuvant therapy? In contrast, parametric analysis of clinical trials has shown that one therapy can offer a greater likelihood of cure than the other.17 This benefit offers young patients the possibility of many extra years of life, diminishing substantially the cost/benefit ratio by which side effects are measured.4

Adjuvant therapy provides optimum benefit when treatment that increases the likelihood of cure is given to young patients. This conclusion does not demand sophisticated statistical methodology, such as parametric modeling. Nevertheless, at the bedside of women with breast carcinoma, general terminology may prove insufficient. Patients may want to know whether a specific regimen is curative or palliative. Furthermore, the question arises: How young is “young”? That is, at what age does the diminished life expectancy of older patients shift the cost/benefit ratio against treatment?

Parametric methods have proved useful in addressing these questions. Previous studies have established a number of important points about parametric methodology: 1) The log normal model provides a good approximation for the time to recurrence or death from breast carcinoma;13–16 2) this model can be adapted to compare the treatment arms of a clinical trial;6, 17 3) when provided data comparable to data used in clinical trials, a distinction can be made between palliative and curative therapy;6 4) when other functions are substituted for the log normal, or when the duration of follow-up is diminished, there is no consistent trend for the method to find a treatment benefit when none exists;6 5) the parametric model can be combined with all-cause survival data from actuarial tables, and the combined model can estimate the long-term benefit of therapy;3, 4 and 6) by using mathematically generated data, the combined model provides quantitative measurements that confirm the advantage of curative versus palliative therapy, as well as the advantage of treating younger versus older patients.4

The major limitation of parametric analysis is related to the duration of follow-up. In some datasets with limited follow-up, the parametric method may not detect a significant difference in survival between treatment arms, even though more conventional methods (e.g., Cox model or log rank test) are able to establish such a difference.6 This limitation may be remedied by an increase in the duration of follow-up. Even with limited data, however, there is no consistent tendency for parametric analysis to find a significant difference when none exists. Therefore, a positive finding carries the same level of reliability as is found with the more conventional methods.6

We have shown, for the first time, a modification of the parametric survival model beyond those outlined above. This new modification incorporates the raw survival data from a clinical trial (as opposed to mathematically generated data) and produces a specific estimate of long-term survival in the treatment arms of that trial. That is, for a woman of a given age at the time of treatment, the model estimates the MTE in each arm, as well as the difference in MTE between the two arms. This method also generates estimates of standard error for all values.

We used recurrence as the survival end point, rather than death from breast carcinoma. Because recurrence generally occurs months or years before death, this expedient substantially reduces the time needed to detect a difference between treatment groups. Furthermore, the mathematical methods described can be adapted easily to incorporate tumor-related death as the end point. In addition, the models can be expanded to include both end points, providing an estimate of the mean survival interval from recurrence to death.

In addition to the impact of therapy on the quantity of life, we may also consider the subjective factors that affect the quality of life, such as the side effects of treatment or the suffering that follows recurrence. To achieve this goal, Gelber et al.23 estimated the impact of treatment on survival time. They then modified this estimate with subjective factors to produce “Q-TWIST,” an index that combines both quantity and quality of life into one number. It is important to note, however, that MTE, as derived herein, could easily be adapted to generate an analogous index.

Besides the estimates of MTE, the extrapolative models also provide supplemental information that patients may find useful in deciding between treatment options. When treatment options have been compared in a clinical trial, the parametric survival model can be used to distinguish between a survival advantage due to palliation and an advantage due to an increase in the likelihood of cure.6, 17

If the treatment under consideration offers only an increase in survival time, as was noted in the clinical trial by Bonadonna et al.18 used to estimate MTE, a physician might say to the patient: “The alternative treatment does not increase your likelihood of being cured, but it does increase your expected survival time. We estimate that, for a woman of your age, recurrence-free survival will be increased by an average of X months.” If the treatment under consideration offers an increase in the likelihood of cure, as was noted in the clinical trial by Buzzoni et al.19 used to estimate MTE, a physician might say to the patient: “The alternative treatment increases your likelihood of cure by Y percent. For a woman of your age, we estimate that this will yield an increase in recurrence-free survival of Z months.”

We propose an extrapolative method that refines our interpretation of clinical trials. These insights are particularly applicable to diseases such as breast carcinoma, which can recur many years after the initial diagnosis. This knowledge greatly facilitates the decision-making process for patients. By coupling estimates of MTE with measures of the likelihood of cure, a patient may be empowered with the information required to make the most important personal decision.

REFERENCES

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION
  6. REFERENCES