Evolution of cancer survival analysis
Article Outline
Survival analysis is a useful tool for evaluating survival differences between groups, the effect of prognostic factors on survival, and the efficacy of cancer treatments. It is important to note that no survival analysis is done without an assumption (survival model). Even the Kaplan-Meier curve assumes that censored subjects will follow the same survival pattern after withdrawal as non-censored subjects. Depending on the model used, survival analysis may yield different results, and hence may have various impacts on the outcome of future patients. Oncologists, therefore, should have a sufficient knowledge of survival model.
In 1972, Cox [1] introduced the proportional hazards model and thereby achieved a major breakthrough in cancer survival analysis. This model assumes that the ratio of death rate (hazard) in one group to the death rate in the other group is constant at any point and does not change with time. As long as this assumption holds, the hazard ratio (HR) provides a valid measure of survival difference between groups. Furthermore, the extension of this model to regression analysis allows us to quantitatively evaluate the efficacy of treatment and the prognostic effect of clinical, pathological and biological variables in terms of the HR. As a component of the proportional hazards model, the HR has been widely used in randomized controlled trials (RCTs), as has the log-rank statistic (LR), which is often used to test the hypothesis that there is no survival difference between groups [2].
Although the Cox model has prevailed in cancer survival analysis, its validity has rarely been confirmed by the life-long observation of patients. In his original report, Cox used data with a maximum follow-up of only 23 weeks. The data were from an RCT of antileukemic chemotherapy conducted in 1959–1960 [3], where the effect of 6-mercaptopurine (6-MP) on the length of steroid-induced remission was compared with the effect of a placebo. Most of the patients were children with acute lymphoblastic leukemia (ALL). The resulting HR of 6-MP to placebo was about 0.2 (LR: P < 0·0001). According to the definition of HR, this figure implies that, if this model holds, about 80% (1-HR) of the relapses that would occur with the placebo can be prevented by 6-MP. Clearly, this implication does not match the fact that before 1960, chemotherapy failed to prevent death from relapse [4]; only after the discovery of combination chemotherapy was relapse-free cure achieved in a substantial number of ALL children [5]. Thus it is clear that, despite a marked reduction in HR, 6-MP did not prevent relapse, but merely delayed it.
If 6-MP is only a palliative treatment, perhaps patients and clinicians can better understand its true benefit if the results are expressed in terms of an increase in the length of remission rather than as a reduction in HR, assisting patients in making their therapeutic decisions. Specifically, physicians might explain to their patients that 6-MP postponed the time to relapse (failure time) by a factor of four (95% CI: 2.0–7.8), or from a median of 6 weeks to a median of 25 weeks, but it did not increase the likelihood of cure. These values were obtained by assuming that 6-MP increases failure time by a constant factor and are illustrated in the Fig. 1. This assumption was mentioned by Cox as the accelerated life model [6] and is also called the accelerated failure time model. Cox, however, chose the proportional hazards model to analyze the 6-MP data.
Figure 1
Prolonged remission of acute leukemia with 6-MP chemotherapy The smooth curves are estimated using the lognormal accelerated failure time model. The distance between the two curves are nearly equal on a semilogarithmic scale. Arrow indicates a gain in median relapse-free time.
Given the dramatic improvements brought about by new cancer treatments, perhaps the time has come to modify the statistical methods we use to measure outcome. When assessing palliative treatments, the accelerated failure time model might help to distinguish between two drugs that affect only survival time. On the other hand, when treatment offers the possibility of cure, perhaps a model that incorporates the cure rate will provide a better measure of the long-term benefit since cured patients gain much greater benefit including decades of life and freedom from recurrence. In either case, as shown above, the HR provides a less than ideal measure of survival benefit.
Unfortunately, in order to know the failure time of every patient in a RCT, we must wait until all have died of their cancer, died from an unrelated cause, or lived so long without recurrence that a cure can be assumed. Though such an approach is feasible with rapidly proliferating cancers such as childhood leukemia, most adult cancers would demand a maximum follow-up of several decades. To address this problem, Boag pioneered a parametric cure model that assumes a fraction of patients are permanently cured and the time to death from a given cancer follows a lognormal distribution [7]. This model provides an estimate of both the percentage of patients cured by treatment and the median survival time of uncured patients – i.e., the median time from treatment to death from their tumor. To obtain accurate estimates, the Boag model demands a larger data set and longer follow-up than is needed for more conventional statistical methods.
Despite their computational advantages, conventional methods such as the Cox model also suffer from limitations. For example, as noted above, the assumption of proportional hazards may be violated if treatment merely prolongs time to death. A number of simulation studies have shown that when this occurs, and when follow-up is limited, the LR or HR may overestimate the benefit derived from treatment [8], [9], [10], [11]. Such violations of proportionality may not be detected by conventional tests. It is possible that Cox’s analysis of the results from 6-MP chemotherapy might be just such a case.
It is important to note that Cox regression analysis, which is an extension of the proportional hazards model, suffers the same limitations as the HR. Thus the Cox regression measures the effect of covariates (prognostic factors such as tumor size, nodal status, or treatment) on HR and not their effect on cure rate, which is clinically more important than HR and easier for both patients and their doctors to understand.
To measure the association of covariates with the cure rate (or likelihood of cure), the Boag regression model was extended to allow multivariate analysis [12]. This model also measures the impact of covariates on the mean and standard deviation of log failure time. Thus we can now determine whether the association of a covariate with an unfavorable outcome results from a decreased cure rate, an accelerated failure time, or both. This provides oncologists with a more accurate prognosis and allows them to offer their patients greater insight into the potential benefit from therapy.
Using this model, we analyzed follow-up data on the re-excision of pelvic relapse from rectal cancer after abdominoperineal excision, a condition which was once considered incurable by any modality of treatment. The results showed that late relapse is the only factor contributing to cure of the disease after re-excision. This was confirmed by further follow-up: in six of the seven patients who survived more than 10-years after re-excision, the first relapse had occurred later than three years after the initial surgery, whereas in 24 of the 29 patients who died of the second relapse, the first relapse had occurred in less than three years after the initial abdominoperineal excision [13].
In addition to predicting the outcome from surgery, the lognormal model also allows us to determine whether an effective treatment achieves its goal via a curative or palliative effect – i.e., whether the treatment enhances the likelihood of cure or merely delays the time to death. Thus far, two trials have shown that postoperative adjuvant chemotherapy is associated with a significant increase in cure rate: one trial examined patients with Stage 2 breast cancer [14] while the other examined patients with gastric cancer who had subserosal invasion [15]. These results suggest that some regimens can cure a certain subset of patients by eradicating residual malignancies – i.e., patients who otherwise would have died of the disease. Further follow-up is needed to confirm these results.
Even when a cancer is cured by a treatment, older patients, who tend to die earlier as well as more often from other causes, generally derive less survival benefit from the treatment. Hence, the cure rate is not always the most relevant measure of survival benefit. A more useful measure may be the life expectancy (mean survival time or mean time to event) [15], [16], [17], which allows for deaths from other causes. The mean survival time is estimated as the area under the overall survival curve, which in turn is estimated using the competing risk model [18]. This model assumes that at any point in time, the probability of being free from any death (overall survival curve) is equal to the probability of being free from disease-related death times the probability of being free from unrelated death. The first term on the right side of the equation is derived from the three parameters of the Boag Model (the disease-specific survival curve) while the second term is approximated by the survival curve for the age- and sex-matched general population. If the model fits well, both curves can be extrapolated beyond the actual known follow-up of the patients in a trial.
This combination of the Boag model and the competing risk model was tested in 3597 gastric cancer patients who were operated on between 1950 and 1969 with 97% followed to their deaths. The results showed that the mean survival is predictable with reasonable accuracy. For 27 groups each with 500 or more patients, the predicted mean survival at 5 postoperative years ranged from 95% to 117% of the actual observation; in 22 of the 27 groups, the prediction error was less than 10% [17]. When the mean survival time is calculated for each individual patient in this fashion, it allows us to better determine whether the risk of treatment is justified by the potential benefit (personalized medicine).
In conclusion, with the development of various new models we have gained deeper insight into the biological behavior of cancer, the effect of cancer therapy, and the long-term outcome of the host than can be obtained with more traditional statistical models. Nevertheless the ideal survival model has yet to be developed. Even the evidence gained from RCTs may be disputed unless the underlying model is validated by long-term clinical observations.
References
- . Regression model and life-table. J Roy Statist Soc. 1972;B34:187–220
- Design and analysis of randomized clinical trials requiring prolonged observation of each patient: part II. Analysis and examples. Br J Cancer. 1977;35:1–39
- The effect of 6-mercaptopurine on the duration of steroid-induced remission in acute leukemia: a model for evaluation of other potentially useful therapy. Blood. 1963;21:699–716
- . Old man river: the flow of pediatric oncology. Hematol Oncol Clin North Am. 2001;15:599–607
- . Joe Burchanal and the birth of combination chemotherapy. Br J Haematol. 2006;133:493–503
- . Analysis of survival data. 1st ed.. London: Chapman and Hall; 1984;
- . Maximum likelihood estimates of the proportion of patients cured by cancer therapy. J Roy Statist Soc. 1949;11B:15–53
- . Comparison of parametric and non-parametric survival methods using simulated clinical trials. Stat Med. 1997;16:1629–1643
- . Cure model analysis in cancer: an application to data from the children’s cancer group. Stat Med. 2002;21:293–312
- . Parametric models for accelerated and long-term survival: a comment on proportional hazards. Stat Med. 2002;21:3279–3289
- Simulation of randomized controlled trials for cancer therapy using the Boag model and the competing risk model. Tenri Medical Bulletin. 2008;11:22–40
- . A stable, multivariate extension of the log-normal survival model. Comput Biomed Res. 1994;27:148–155
- . Long-term cure in surgery for extrarectal pelvic recurrence of rectal cancer. Dis Colon Rectum. 2007;50:1588–67
- . Parametric survival analysis of adjuvant therapy for stage II breast cancer. Cancer. 1994;74:2483–2490
- . Evaluation of intensive adjuvant chemotherapy in gastric cancer using life expectancy compared with logrank test as a measure of survival benefit. Ann Surg Oncol. 2007;14:348–354
- . Refined measurement of outcome for adjuvant breast carcinoma therapy. Cancer. 2003;97:1139–1146
- . Parametric mean survival analysis in gastric cancer patients. Med Decis Making. 2004;24:131–141
- . Survival distributions: reliability applications in the biomedical sciences. New York: Johns Wiley & Sons; 1975;
PII: S0960-7404(10)00028-9
doi:10.1016/j.suronc.2010.03.002
© 2010 Published by Elsevier Inc.
