Cure models in survival analysis: mixture and non-mixture

A cure model is a survival model that allows for patients who will never have the event. Instead of assuming that everyone eventually does, it assumes that a fraction of patients, the cured fraction, are long-term survivors, so that the survival curve levels off at a plateau above zero. “Cured” is meant in a statistical sense: these patients are taken to be at no further risk of the event or, for death from any cause, at the same risk as the general population. Cure models are used where lasting remission or cure is plausible, as after treatment of some cancers (Othus et al., 2012). They are also used to extrapolate survival beyond the end of follow-up, which is the use that most needs justifying.

Most standard parametric families, the Weibull, the log-normal and the log-logistic among them, have survival that falls to zero in the end. A Gompertz model with a falling hazard is the exception and levels off above zero. A cure model builds the plateau in from the start and estimates where it lies.

Two kinds of cure model

Mixture cure models. The population is a mixture of cured and uncured patients. With $π$ the cured fraction and $S_{u} (t)$ the survival of those not cured, $S (t) = π + (1 - π) S_{u} (t)$ (see Farewell, 1982). Covariates can act on the cured fraction, usually through a logistic regression, and on the survival of the uncured, through a hazard model.
Non-mixture cure models. The survival of the whole population is $S (t) = \exp {- θ F (t)}$ , where $F (t)$ is a proper distribution function, so that the cured fraction is $\exp (- θ)$ (Chen et al., 1999). If covariates act only on $θ$ , the hazards of two groups are proportional.

The two kinds can fit the same data about equally well and still differ when extrapolated. Lambert et al. (2007) compare them in population-based cancer data and investigate the importance of modelling the ancillary parameters of the chosen distribution.

A worked example

Suppose 30% of patients are cured and the others have a Weibull survival, $S_{u} (t) = \exp {- (t / 3)^{1.2}}$ with t in years. The survival of the population is then 0.84 at one year, 0.68 at two, 0.56 at three and 0.41 at five, 0.31 at ten and 0.30 for ever after. The hazard of the population is about 0.21 per year at one year, 0.12 at five and 0.017 at ten, and then goes to nearly zero, because by then almost everyone left is cured. (This example sets aside deaths from other causes, which a section below returns to.)

Now suppose that only the first five years are observed and a Weibull model is fitted to them. The Weibull that fits those five years best by least squares (scale 5.3 years, shape 0.99, so close to an exponential) stays within about 0.02 of this curve at every time, and has a survival of 0.39 at five years, 0.15 at ten, 0.02 at twenty and 0.004 at thirty. Other Weibulls that fit as closely give 0.02 to 0.03 at twenty years. Restricted mean survival over the first five years is 3.24 years for the cure model and 3.23 for the Weibull. Over 40 years it is 14.0 years for the cure model and 5.3 for the Weibull, 2.6 times as much. With a few hundred patients the first five years cannot reliably tell the two apart, and the extrapolations differ by more than a factor of two. Once a constant mortality of 2% a year, standing in for other causes of death, is applied to both curves, the ratio over 40 years falls from 2.6 to 2.1. That gap is one reason cure models are argued over in health technology assessments.

Can the data show that patients are cured?

Only with long follow-up. The cured fraction is estimated mainly from the plateau in the Kaplan–Meier curve, so it is determined by the data only if follow-up runs well past the time when most events happen and many patients are still at risk at the end. A plateau that rests on a handful of patients and ends in censoring is weak evidence, and the cured fraction is then sensitive to the distribution assumed for the uncured, which the data cannot check beyond the end of follow-up. Even a long, well-supported plateau shows that the hazard became very small during follow-up, not that it is zero for ever or will not rise again. Cure models are reviewed by Amico and Van Keilegom (2018).

Cure and general-population mortality

A cured patient still dies of other causes. A model for all-cause survival that plateaus at 0.30 for forty years says that cured patients never die, which is not plausible. Cure models for extrapolation are therefore often fitted on the excess-hazard scale. In population-based cancer studies, cure is said to occur when the mortality of the patients returns to the level expected in the general population (Lambert et al., 2007), and the survival of the cured then follows the general population's, matched for age and sex. This is the relative survival set-up, and it is how the hazard of an extrapolation is kept above the population's.

What reviewers ask for

A reason to expect cure. Clinical and external evidence that some patients are cured: long-term registry data on similar patients, a plateau in other studies, the biology of the disease. A good fit to the trial alone is not enough.
Follow-up that supports the plateau. How long it is, how many patients are at risk at its end, and how the cured fraction changes if the model is fitted to less follow-up.
The comparison with models without cure. The extrapolations side by side, with their fit statistics, and a scenario in which no one is cured.
General-population mortality. That cured patients are given at least the mortality of the general population, matched for age and sex.
Guidance. The NICE Decision Support Unit's technical support document on flexible methods for survival analysis covers cure models among its approaches (TSD 21, 2020).

In software

In R, the flexsurvcure package fits mixture and non-mixture cure models, with a bhazard argument for the excess-hazard scale, and the smcure package a semiparametric mixture cure model on the all-cause scale. In Stata, strsmix and strsnmix fit mixture and non-mixture models (Lambert, 2007), and stpm2 fits cure models within the flexible parametric framework (Andersson and Lambert, 2012).

Extrapolating a survival curve?

Whether to use a cure model, with which distribution and on which scale, is one of the choices that decides an extrapolation, and the one a reviewer is most likely to question. We can help you justify it for a NICE, TLV or other submission: see our HTA survival modelling service, or talk to us about your data.

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