Cure Model

Among patients who have finished cancer treatment, how many remain free of recurrence even after several years? Could the risk of recurrence be high in the first few years after treatment and then drop sharply? Do the long-term non-recurrence proportion and the time to recurrence differ by treatment type or stage?

The method that assumes some units may belong to a group that never experiences the event of interest in the long run, and estimates both that proportion and the event times of the group that can experience the event, is the cure model. Here 'cure' is a term of the statistical model; it does not mean a confirmed diagnosis that an individual patient has been medically cured.

Standard survival analysis uses observed events and censoring to look at survival probabilities or hazards over time. A cure model, by contrast, separately models the possibility that a group exists in which the event does not occur for a long time. But when follow-up is short, it is hard to tell people in whom recurrence simply has not been observed yet from the long-term non-recurrence group.

Cure models are used with time-to-event data such as recurrence after cancer treatment, equipment that goes a long time without failing, and credit delinquency. In this article we use recurrence after cancer treatment as an example to look at the basic principle, the main models, interpreting the long-term non-recurrence proportion and time to recurrence, checking fit, and how cure models differ from other survival analyses.

A cure model thinks of all subjects as split into a latent group that does not experience the event for a long time and a latent group that can still experience it. In real data you cannot directly observe who belongs to which group; the model estimates the overall proportion and the distribution of event times.

For cancer recurrence, the event is 'recurrence after treatment', and patients whose follow-up ends without recurrence are recorded as censored. Not every censored patient is in the long-term non-recurrence group.

Even if two treatments have similar recurrence curves early in follow-up, the proportion remaining free of recurrence after enough time has passed may differ. Conversely, the long-term non-recurrence proportions may be similar while the timing of recurrence among patients who do recur differs.

Separating the two lets you examine the questions "Who is likely to belong to the long-term non-recurrence group?" and "How quickly does recurrence happen in the group that can recur?" one at a time.

In a mixture cure model, the overall recurrence-free survival probability can be expressed as S(t) = π + (1 − π)Sᵤ(t). Here π is the proportion in the long-term non-recurrence group, and Sᵤ(t) is the recurrence-free survival probability in the group that can recur.

If the survival probability of the group that can recur approaches 0 once enough time has passed, the overall curve approaches the level π. Just because a curve looks flat, you cannot conclude that its value is the exact long-term non-recurrence proportion.

Suppose 360 hypothetical patients who finished cancer treatment were followed for up to 8 years. Stage at diagnosis, treatment type, age and date of recurrence are recorded. All the data and results below are hypothetical examples for learning.

You also need to define in advance how deaths other than recurrence are handled in the analysis. Death can be a competing event that prevents recurrence from being observed, so whether simple censoring is appropriate is judged by the purpose of the study.

P002 and P003 are both censored, but they differ in that they were observed through 96 months and 43 months respectively. Lumping them into a single 'no recurrence' category without reflecting this information loses the time information the analysis needs.

If a hypothetical mixture cure model estimates π = 0.38, it is interpreted as about 38% of all patients belonging, under the model, to the long-term non-recurrence group. This is not the same as the simple proportion of patients who did not actually recur during the observation period.

A mixture cure model also includes a model for the time to recurrence in the group that can recur. Distributions such as Weibull or lognormal can be used, and the shape of the late tail and the estimates change depending on the distribution chosen.

Below is a hypothetical summary of two groups to explain how to report results. It is not the result of fitting a model to real data.

In this example, Treatment B has a higher long-term non-recurrence proportion and a longer median time to recurrence in the group that can recur. But you cannot claim a treatment effect or a significant difference from figures alone without examining sampling error, confidence intervals and confounders.

A survival curve may drop early and then look flat later. But you cannot conclude that follow-up was sufficient from a leveled-off shape alone, so also check the time of the last event, the size of the risk set and the amount of loss to follow-up.

Is the follow-up period long enough?

To estimate the long-term non-recurrence proportion stably, you need to be able to observe the late period in which the hazard nearly disappears. With short follow-up, it is hard to separate slow recurrence from long-term non-recurrence.

How did censoring occur?

If dropping out of follow-up is related to the likelihood of recurrence, the assumption of non-informative censoring may not hold. Examine the amount of and reasons for loss to follow-up.

How do you handle people who died without recurrence?

Death before recurrence can be a competing event. Check whether the estimation target of a model that treats death as simple censoring matches the actual clinical question.

Which cure model did you use?

Mixture and non-mixture cure models express the survival function differently. A sensitivity analysis of the distributional assumption for the group that can recur is also useful.

This does not mean that either standard survival models or cure models are always better. First judge whether there is a substantive reason to assume a long-term non-recurrence group and whether the data support it.

A paper should report the number of subjects, the observation period, the numbers of recurrences and censored cases, the type of cure model used, the time distribution for the group that can recur, the explanatory variables included and the uncertainty of the estimates. When comparing groups, describe the non-recurrence proportion part and the time-to-recurrence part separately so they are not confused.

If the confidence interval of the cure fraction and the time-to-recurrence coefficients were not actually estimated, do not arbitrarily add p-values or claims of statistical significance.

When analyzing recurrence after cancer treatment, simply comparing mean time to recurrence or the recurrence rate at a particular time may not adequately describe the characteristics of patients who go a long time without recurrence. A cure model helps you analyze separately the proportion of all subjects in the long-term non-recurrence group and the time to recurrence in the group that can recur. Its distinctive feature is therefore that the same treatment effect can be examined from two angles.

However, if follow-up is not long enough, or death before recurrence and censoring are not handled appropriately, the long-term non-recurrence proportion can be estimated unstably. And a statistical cure fraction does not mean an individual patient is cured. When interpreting the results, look at the model used, the distributional assumptions, the sufficiency of late follow-up data and the confidence intervals together.