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 [1] addressed this issue by developing a parametric cure model that allowed him to estimate the cure rate (c) and the mean of log failure times (m) among uncured patients (failure time means time to death from cancer under study). Twenty-three years later, Cox [2] published the proportional hazards model, which uses the hazard ratio to measure the survival difference between groups. Since then non-parametric or semi-parametric methods based on his model (e.g., log-rank test and Cox regression) have remained the mainstay of cancer survival analysis. Hereafter these methods will be referred to as “standard survival analysis”. The purpose of our paper is to compare the usefulness of the Boag and Cox models, whose primary parameters are the cure rate and the hazard ratio, respectively, and also to confirm the validity of the Boag model using the data from breast cancer and other neoplastic diseases.

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 [28]. Although the hazard ratio in the Cox model is closely related to the cure rate in the Boag model [29], HR is less comprehensible to non-statisticians than c, which more accurately depicts the long-term survival benefit of a given treatment.

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 Figure 3. In the right lower panel, the two hazard curves representing the control (red) and the test treatment (blue), respectively, separate at the beginning, but then come closer and cross. If follow-up is truncated earlier than this crossing (for example, at 2 years), the hazard curve of the life-prolonging treatment remains lower than that of the control throughout the study. Thus the HR of treatment versus control would be overstated, while violation of the proportional hazards assumption may be missed. For these reasons, standard analysis might mislead clinicians and patients into selecting a less effective regimen. Though pharmaceutical industries may benefit from the consumption of such drugs, the welfare of patients and their families will suffer; if they were correctly informed, they might have refused to have a regimen that merely delays death. When Heyland et al. [34] interviewed 278 elderly patients who were at high risk of dying in the next 6 months. Only 12% preferred life-prolonging care.

On the other hand, the parametric cure models also suffer limitations. They are less sensitive to difference in failure time during the early period [31], although they gain power with longer follow-up. Furthermore, under certain conditions, they may provide a poor fit to the observed data. To avoid this problem, some statisticians recommend alternative parametric models.

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 (Figure 1). Such a unimodal hazard curve is seen only in the lognormal or log-logistic model, but not in the Weibul or gamma model. If death occurs very early, the cause of death should be checked carefully. If the cause is ambiguous, such cases should be classified as censored at the time of death. Otherwise, the maximum likelihood estimation may fail to converge, or the result may be biased toward a shorter estimate of m.

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.