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As cancer therapy has progressed dramatically, its goal has shifted toward cure of the disease (curative therapy) rather than prolongation of time to death (life-prolonging therapy). Consequently, the proportion of cured patients (c) has become an important measure of the long-term survival benefit derived from therapy. In 1949, Boag addressed this issue by developing the parametric log-normal cure model, which provides estimates of c and m where m is the mean of log times to death from cancer among uncured patients. Unfortunately, traditional methods based on the proportional hazards model like the Cox regression and log-rank tests cannot provide an estimate of either c or m. Rather, these methods estimate only the differences in hazard between two or more groups. In order to evaluate the long-term validity and usefulness of the parametric cure model compared with the proportional hazards model, we reappraised randomized controlled trials and simulation studies of breast cancer and other malignancies. The results reveal that: 1) the traditional methods fail to distinguish between curative and life-prolonging therapies; 2) in certain clinical settings, these methods may favor life-prolonging treatment over curative treatment, giving clinicians a false estimate of the best regimen; 3) although the Boag model is less sensitive to differences in failure time when follow-up is limited, it gains power as more failures occur. In conclusion, unless the disease is always fatal, the primary measure of survival benefit should be c rather than m or hazard ratio. Thus, the Boag lognormal cure model provides more accurate and more useful insight into the long-term benefit of cancer treatment than the traditional alternatives.

In recent decades, as more cancer victims have enjoyed long-term, relapse-free survival, cure has become a reality for both patients and clinicians. Thus, the primary goal of cancer therapy has shifted toward cure of the disease rather than prolongation of time to death. To achieve a cure, selecting the best regimen is vital. This is especially true with children, for whom curative treatment can yield many years of healthy life, while prolongation of life offers only a limited benefit before relapse takes the child’s life. Furthermore, cured patients are saved from cancer-associated sufferings, which could be more unbearable to patients than death itself. Hence, the proportion of cured patients (cure rate) has become an important measure of long-term survival benefit.

As early as 1949, Boag [

In 1972, Cox [

It is generally assumed that, provided the proportional hazards model holds, the proportion of patients cured (i.e., saved from the event) can be estimated by 1-HR [3-5], which in turn is calculated from standard survival analysis. For example, consider a trial that compared adjuvant chemotherapy with or without trastuzumab and found a HR of 0.67 in HER2-positive breast cancer patients [

However, according to Peto et al. [

Moreover, the relationship between the cure rate and HR became even more questionable when we read the Cox original paper [

When we re-analyzed the data using the Cox model, the HR of 6-MP versus placebo was 0.22 (95% CI: 0.10 to 0.49) [^{ }If the relationship between the cure rate and HR were valid, this finding would indicate the following: 78% (1-HR = 100% - 22%) of the relapse that would have occurred in the placebo group were prevented by 6-MP.^{ }On the contrary, further follow-up of 6-MP-treated children revealed that almost all died from relapse [12,13].

Although the Cox model is still commonly used in cancer survival analysis, Cox [

The Boag log normal model incorporates the cure rate as one of its parameters [

In 1977, Farewell [

The multivariate model was then extended to allow the analysis of grouped data where information on the cause of death for individual patients may not be available [17,18] (http://survillance.cancer.gov/cansurv/).

When the nultivariate Boag model was applied to the 6 MP data, we found that the chemotherapy failed to cure the disease (Wald P = 0.99), but rather prolonged time to relapse 3.8 times longer (95% CI: 2.07 to 7.15) than in the placebo group [

These results show that at least two types of anticancer treatments are available: a curative one that increases the

fraction of patients cured of the disease and a life-prolonging one that merely delays tumor-related death. To illustrate the differential effects of these two regimens on survival and hazard curves, these curves were simulated by increasing one of the two parameters (either c or m) of the Boag model (

The red curves show the base-line values of the control group (Group 1), while the blue curves represent the effects of therapies (Group 2). The middle and right panels show the effects of curative and life-prolonging regimens, respectively.

It is readily seen that increase in parameter m alone

from 2.5 to 3.0 results in crossing of the hazard curves (right lower panel), so the proportional hazards assumption does not hold. The left panels show the survival and hazard curves satisfying the proportional hazards assumption. Note that these curves are very similar to the corresponding ones derived from the Boag model (middle panels) in which the parameter c alone is increased from 0.3 to 0.4 while m is kept unchanged.

To further illustrate the difference between these models, two examples of randomized controlled trials will be shown in which both models are applied.

Using the three parametric versions of the Boag cure model (lognormal, log logistic and Weibul) plus the log-rank statistic, Gamel et al. [

The results showed that in three of the five trials there were statistically significant survival differences between the treatment and control groups. However, a curative effect was found only in the trial with doxorubicin plus CMF, whereas in the other two positive trials (with CMF regimens), the treatment merely prolonged the time to relapse. The stepwise log likelihood ratio test and chi-square statistics showed that the lognormal distribution provided the better fit to the pooled data than the log-logistic or Weibul versions of the Boag cure model.

From 1989 to 1993, the Dutch Gastric Cancer Group [

Such negative results did not agree with the clinical experience of Japanese surgeons, so we applied the Boag model to the same data. Our findings showed a significant difference in cure rate (11.5%; 95% CI: 3.1 to 20.0)

between the two groups [

Boag is the first scientist who attached special importance to cure in the analysis of cancer-related survival. The multivariate extension of his model estimates the impact of treatment and other prognostic variables on the likelihood of cure, thus providing both patients and clinicians the information they need to make vital decisions [

In contrast, the parameter m of the Boag model plays a subordinate role unless all patients being studied are incurable. If only time to death is studied to assess the effect of treatment, this could be likened to counting only the coins in a cash transaction while leaving the bills uncounted.

Clinical trials and simulation studies have shown that standard survival analysis suffer a number of failings; 1) they cannot distinguish between curative and life-prolonging treatments [21,30,31]; 2) they are more sensitive to an increase in failure time than to an increase in cure rate, especially when the follow-up is limited; 3) as a result, they may favor a death-delaying treatment over one that is curative; 4) they tend to loose power with increasing follow-up [30,32,33].

These limitations are graphically shown in

On the other hand, the parametric cure models also suffer limitations. They are less sensitive to difference in failure time during the early period [

However, the cause of the poor fit may not be use of the wrong model (misspecification) but misclassification of events. For example, patients who actually died from therapeutic complications in the early postoperative period may be misclassified as dead from cancer (failure). It must be kept in mind, however, that early failure is very rare, since most clinical trials in the adjuvant setting require that participants are in remission or have undergone potentially curative surgery (i.e., these are the eligibility criteria for candidates to be enrolled in the trial). So it is unlikely that failure occurs shortly after its cause has been eliminated.

Even if the trial is conducted in the non-adjuvant setting, imminently fatal cases should have been excluded from most trials. Consequently, the actual hazard curve should begin at zero, rise to a peak and then gradually decline to zero (

Another criticism against the parametric cure models is that they rely on “extrapolation of a survival curve outside the available data” [5,35]. It is important to note, however, that predicting events beyond observed data has long served many branches of sciences. An excellent example is meteorology, where predicting the course of hurricanes has enjoyed great success. In addition, few models have been reported to provide a better fit to observed survival data than the lognormal or loglogistic models [1,20,35-39]. Nevertheless, we must continue our effort to find a better model using large, accurate, lifelong follow-up data sets.

In conclusion, unless the disease is always fatal, the primary measure of survival benefit should be the proportion of patients cured rather than hazard ratio or median time to failure. Thus, the Boag lognormal cure model provides more accurate and more useful insight into the long-term benefit of cancer treatment than the standard non-parametric alternatives.