Cure Model

Cure Model for Long-Term Survival Extrapolation

A cure model lets a fitted survival curve level off, because after some point the disease stops adding to the risk of death and the people still alive die only at the rate expected in the general population. In health technology assessment (HTA) it is used to extrapolate trial survival over a lifetime when a treatment appears to cure some patients, as with some immunotherapies and CAR T-cell therapies. The choice matters because these trials end with many patients still alive, so whether the model allows cure, and in what form, decides much of the lifetime QALY gain. Mixture and non-mixture models, both described in the NICE Decision Support Unit's Technical Support Document 21 (TSD 21), differ in who is cured and when. The share of patients cured has its own cure fraction page, and the effect of the cure assumption on appraisal results is discussed under cure rate model.

Why standard survival models struggle when some patients are cured

Latimer and Rutherford's HTA tutorial explains why. Exponential, Weibull, Gompertz and Gamma models cannot represent any turning point in the hazard function, and log-logistic, log normal and Generalised Gamma models can represent only one. In a trial of a serious disease the hazard of death may rise at first, fall as the patients with the worst prognosis die, drop towards general population levels in those who are cured, and later rise again with age, a pattern with two turning points.

In TSD 21 a cure model describes a population in which some people never have the event, so the hazard reaches zero and survival plateaus above zero. All-cause survival must eventually reach zero, so TSD 21 states that the event cannot be all-cause mortality, and cure models are fitted to cause-specific survival or in an excess mortality, or relative survival, framework.

Statistical cure in the relative survival framework

Latimer and Rutherford fit cure models in the relative survival framework, in which each patient's all-cause hazard is split into the expected general population hazard and an excess hazard due to the disease. They prefer it because it is intuitive, needs no cause-of-death data and avoids choosing when to start adding general population mortality to an all-cause curve.

$$h(t) = h^{*}(t) + \lambda(t), \qquad R(t) = \frac{S(t)}{S^{*}(t)}$$

where $h(t)$ is the all-cause hazard at time $t$ since the start of follow-up, $h^{*}(t)$ is the expected hazard in the matched general population, $\lambda(t)$ is the excess hazard, $S(t)$ is all-cause survival, $S^{*}(t)$ is expected survival in the general population and $R(t)$ is relative survival. Cure occurs when $\lambda(t)$ approaches zero, at which point $R(t)$ plateaus and background mortality from life tables governs the rest of the curve. All-cause survival is recovered by multiplying the fitted relative survival by $S^{*}(t)$.

Mixture cure models with latent cured and uncured groups

A mixture cure model treats the population as two groups, those cured of their disease and those who are not. TSD 21 writes it in the relative survival setting as follows.

$$S(t) = S^{*}(t)\,\bigl[\pi + (1-\pi)\,S_u(t)\bigr]$$

where $S(t)$ is all-cause survival, $S^{*}(t)$ is expected survival in the general population, $\pi$ is the proportion cured and $S_u(t)$ is the survival function of the uncured; the bracket is relative survival. Without the $S^{*}(t)$ term the same form is the cause-specific mixture model, and TSD 21 warns that its survival functions are then not real-world probabilities until other-cause mortality is taken into account.

Latimer and Rutherford note that, strictly, a mixture model assumes the cured patients are cured at baseline, and that it can estimate a cure fraction of 0%. The distribution chosen for the uncured drives the estimated fraction, so the same data can give low or high fractions under different distributions.

Non-mixture cure models with a bounded excess hazard

A non-mixture cure model does not split the population into cured and uncured groups, although the cure fraction and the survival of the uncured can still be derived from it. TSD 21 describes it as a function that defines an asymptote for the survival function, the value approached as time tends to infinity.

$$R(t) = \pi^{F_z(t)}, \qquad \Lambda(t) = -\ln(\pi)\,F_z(t), \qquad \lambda(t) = -\ln(\pi)\,f_z(t)$$

where $R(t)$ is relative survival, $\pi$ is the cure fraction, $F_z(t)$ is a cumulative distribution function with density $f_z(t)$, $\Lambda(t)$ is the cumulative excess hazard and $\lambda(t)$ is the excess hazard. Because $F_z(t)$ rises to 1, the cumulative excess hazard is bounded at $-\ln(\pi)$, relative survival falls towards $\pi$, and the excess hazard has the shape of the density.

TSD 21 notes that $F_z(t)$ usually takes a simple parametric form such as the Weibull or log normal, and that it can instead be built from restricted cubic splines as an extension of flexible parametric models. In that version the analyst fixes the time at which cure is assumed through the placement of the final boundary knot, after which the all-cause hazard is set by other-cause mortality. Latimer and Rutherford report that mixture and non-mixture models with the same standard parametric form typically give similar estimates, and that flexible parametric non-mixture models had not been used in HTA although they suit it when cure is plausible but follow-up is limited.

Fitting a cure model to trial data

Cure models are estimated from time-to-event data, usually by maximum likelihood. Boag's 1949 paper on maximum likelihood estimates of the proportion of patients cured by cancer therapy is an early example, and Felizzi and colleagues note that the method is generally used for mixture cure models. In the relative survival framework each participant is assigned a background mortality rate from life tables by calendar year, age and sex, so Latimer and Rutherford recommend patient-level data; data reconstructed from a published Kaplan-Meier curve lack these details.

Models can be fitted to each arm separately, as is common in HTA, or with treatment as a covariate acting on the cure proportion and the survival of the uncured. If disease-related deaths are associated with age, Latimer and Rutherford expect models that include age to do better, with regression standardisation giving the marginal survival curve. If cured patients keep a raised risk of death, a standardised mortality ratio can be applied to the life-table rates in either family.

Choosing a cure model for extrapolation in HTA

The NICE technology appraisal manual (PMG36, section 4.6.25) lists cure among the more flexible approaches to extrapolation and asks for the clinical plausibility of the hazard functions to be routinely assessed and for uncertainty in the extrapolated part to be explored. TSD 21 rests any cure model on three assumptions, that the data can support a reliable cure fraction, that cure really occurs by some time point, and that the distribution for the uncured fits. It warns that the Akaike information criterion (AIC) and Bayesian information criterion (BIC) weight the start of follow-up, where events are dense, and give very little weight to the tail, so the distribution for the uncured should be chosen on biological plausibility.

Latimer and Rutherford note that if hazards are still increasing during follow-up, a mixture model will estimate a 0% cure fraction and adds nothing, whereas a flexible parametric non-mixture model is forced to turn the hazard down at the boundary knot, so it is the only logical cure option when cure can be confidently predicted from external data or strong clinical opinion. Predicted hazards begin to fall sharply well before the knot; with a 5-year boundary knot they fell steeply at around 2 years. The authors therefore advise relatively late knots and a range of knots as sensitivity analysis.

Statistical fit alone cannot settle the choice. In Latimer and Rutherford's colon cancer case study, set out on the cure fraction page, mixture and non-mixture models with similar AIC values gave very different cure fractions and 10-year survival from the same 24 months of data. When cure is uncertain the authors recommend fitting both cure and non-cure models, treating the assumption as a structural uncertainty explored in scenario analysis.

Worked example comparing mixture and non-mixture curves

All numbers here are illustrative. Both models use a cure fraction of $\pi = 0.40$ in the relative survival framework. The mixture model gives the uncured an exponential survival curve with excess hazard 0.5 a year, and the non-mixture model uses the same exponential distribution for $F_z(t)$. Background mortality is a constant hazard of 0.03 a year, so $S^{*}(t) = e^{-0.03t}$; a real model would use life tables.

1. Write the two relative survival curves. The mixture model gives $R_m(t) = 0.40 + 0.60\,e^{-0.5t}$ and the non-mixture model gives $R_n(t) = 0.40^{\,1 - e^{-0.5t}}$. At five years $e^{-2.5} = 0.0821$, so $R_m(5) = 0.40 + 0.60 \times 0.0821 = 0.449$ and $R_n(5) = 0.40^{0.918} = 0.431$.

2. Convert to all-cause survival. Multiplying by $S^{*}(t)$ gives the curves an economic model uses.

3. Compare the early excess hazards. At time zero the mixture model's excess hazard is 0.60 × 0.5 = 0.30, while the non-mixture model's is 0.916 × 0.5 = 0.458, since $-\ln(0.40) = 0.916$. The non-mixture model loses more patients early, and its 20-year restricted mean survival is 6.91 years against 7.15 for the mixture model.

4. Interpretation. The same cure fraction does not fix the curve, because the two structures spread the excess deaths differently over the first years. Cured patients still die of other causes, so reading the fraction of 0.40 as the share alive overstates survival at 10 years (about 0.30) and 20 years (0.22) by a third or more.

How a cure model relates to nearby approaches

Some neighbouring methods give a similar plateau without being cure models. Piecewise models switch to general population mortality at a chosen landmark time, but Latimer and Rutherford note that they do not use the data to estimate the cure and may create discontinuities in the hazard. What counts as cure also changes with the endpoint, overall survival or progression-free survival.

Limitations and common misreadings

Most errors come from treating the plateau as evidence when it is an assumption. TSD 21 calls cure models a tool to force a plateau, and warns that cure may look reasonable within follow-up only because samples are small and mortality is low but not zero.

• Fitting a cure model where there is no cure. Latimer and Rutherford state that this is likely to overestimate long-term survival, and that fitting log normal mixture models just in case is inadvisable because they performed among the worst when there was a cure.
• Comparing cure fractions across models. Fractions depend on the framework and distribution and should be expected to differ, so the authors recommend judging models on predicted survival over time.
• Equating statistical and clinical cure. The fraction describes the end of excess mortality in a population; it does not show that any patient is free of disease or say anything about quality of life.

Sources

• Boag JW. Maximum likelihood estimates of the proportion of patients cured by cancer therapy. Journal of the Royal Statistical Society Series B (Methodological). 1949;11(1):15-44. https://doi.org/10.1111/j.2517-6161.1949.tb00020.x
• Felizzi F, Paracha N, Pöhlmann J, Ray J. Mixture cure models in oncology: a tutorial and practical guidance. PharmacoEconomics - Open. 2021;5(2):143-155. https://doi.org/10.1007/s41669-021-00260-z
• Latimer NR, Rutherford MJ. Mixture and non-mixture cure models for health technology assessment: what you need to know. PharmacoEconomics. 2024;42(10):1073-1090. https://doi.org/10.1007/s40273-024-01406-7
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). 2022, last updated 13 October 2026. Section 4.6.25. https://www.nice.org.uk/process/pmg36/chapter/economic-evaluation-2
• Rutherford MJ, Lambert PC, Sweeting MJ, Pennington B, Crowther MJ, Abrams KR, Latimer NR. NICE DSU Technical Support Document 21: Flexible methods for survival analysis. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2020 (updated March 2022). Section 3.6. https://doi.org/10.15131/shef.data.33849118