Mixture Cure Model

Mixture Cure Model for Survival Extrapolation in HTA

A mixture cure model fits trial survival data as a blend of two latent groups of patients, one cured of the disease and one not, and estimates both the size of the cured group and the survival curve of the rest. In health technology assessment (HTA) it extrapolates overall survival over a lifetime when a treatment, such as an immunotherapy, appears to leave a share of patients with no excess risk of dying from their cancer. NICE Decision Support Unit Technical Support Document 21 (TSD 21) calls the mixture form the most common type of cure model. The model's two parts can be read separately, so an analysis can show whether a new drug cures more patients or extends survival in those it does not cure. The wider family, including the non-mixture form, is described on the cure model page, and the cured proportion has its own cure fraction page.

Incidence and latency, the two parts of the model

Othus and colleagues put the model in words as the probability of being cured plus the probability of not being cured multiplied by the probability of being alive at time $t$ if not cured. Cai and colleagues, who credit Boag and, separately, Berkson and Gage with first introducing the model, call the model for the probability of not being cured the incidence and the survival of the uncured the latency. In HTA the model is usually fitted in a relative survival framework, which TSD 21 writes by multiplying the mixture by expected survival in the general population.

$$S(t) = S^{*}(t)\,\bigl[\pi + (1-\pi)\,S_u(t)\bigr], \qquad \log\frac{\pi}{1-\pi} = b_0 + b_1 z$$

where $S(t)$ is all-cause survival at time $t$ since randomisation, $S^{*}(t)$ is expected survival in a general population matched on age, sex and calendar year, $\pi$ is the cure fraction, $S_u(t)$ is the survival function of the uncured, $z$ is a covariate such as treatment arm, and $b_0$ and $b_1$ are regression coefficients. The bracket is relative survival, which levels off at $\pi$ once no uncured patients remain.

The second equation is the logit link. Othus and colleagues note that the probability of cure is typically modelled with logistic regression, and Cai and colleagues list complementary log-log and probit links as alternatives. For the latency, Felizzi and colleagues and Latimer and Rutherford describe standard parametric survival models such as the Weibull or log normal.

Cure here is statistical. Felizzi and colleagues stress that no patient can be identified as cured, and Latimer and Rutherford add that each patient has a probability of cure, so the fraction belongs to the population. As a weighted sum of two distributions, the model is a special case of a mixture model in which one component has no disease-related deaths.

Why the mixture produces a falling excess hazard

The hazard form shows how the model fits a flattening curve. Felizzi and colleagues express the overall hazard as background mortality plus an excess term.

$$h(t) = h^{*}(t) + \frac{(1-\pi)\,f_u(t)}{\pi + (1-\pi)\,S_u(t)}$$

where $h(t)$ is the all-cause hazard, $h^{*}(t)$ is the general population hazard, $f_u(t)$ is the density of survival time among the uncured and the other symbols are as above. The second term is the excess hazard, the extra risk of death from the disease.

As uncured patients die, survivors are increasingly cured patients, so the excess hazard falls even when the uncured hazard is constant and the curve settles onto background mortality. Felizzi and colleagues point out that no extra assumption about the long-term hazard is needed.

Fitting the model when cure status is never observed

Felizzi and colleagues note that maximum likelihood estimation is generally used. Their log-likelihood adds, over patients, the log hazard at each observed death and the log survival at each patient's last follow-up time.

$$\log L = \sum_{i=1}^{N} \delta_i \log h(t_i) + \sum_{i=1}^{N} \log S(t_i)$$

where $L$ is the likelihood, $N$ is the number of patients, $t_i$ is the death or censoring time of patient $i$, and $\delta_i$ equals 1 if patient $i$ died and 0 if the patient was censored.

Censored patients carry most of the information about cure, because a censored patient could belong to either group. Cai and colleagues treat cure status as missing data and fit the model with the expectation-maximisation (EM) algorithm. In each E-step a patient who died gets a weight of 1 for being uncured, and a censored patient gets the conditional probability of remaining uncured given survival to $t_i$. The complement is the probability of cure among survivors.

$$P(\text{cured} \mid T > t) = \frac{\pi}{\pi + (1-\pi)\,S_u(t)}$$

where $T$ is the time of death from the disease and the other symbols are as above. Background mortality affects both groups equally, so $S^{*}(t)$ cancels and the same expression holds in the relative survival framework.

Choosing the distribution for the uncured

The latency distribution decides much of the answer. Latimer and Rutherford explain that a log-logistic latency can represent a falling hazard and long-term survivors among the uncured, so the estimated cure fraction may be low, whereas a Weibull latency with a rising hazard cannot, so the fraction may be high, even when combined curves look alike. They conclude that differing fractions need not be a concern and that the effort belongs in choosing a distribution suited to the uncured.

Fit statistics give limited help, and TSD 21 warns that the Akaike information criterion (AIC) and Bayesian information criterion weight the start of follow-up, where events are dense, and give very little weight to the tail, so the uncured distribution should be chosen on biological plausibility. In Felizzi and colleagues' simulated melanoma data, lognormal and generalised gamma latencies had similar AIC values but cure fractions 2.7 percentage points apart. Latimer and Rutherford add that if hazards are still rising during follow-up, a mixture model will estimate a 0% cure fraction and adds nothing.

Covariates, treatment effects and an informed cure fraction

The separate parts let a treatment act in two ways. With a logistic incidence and Weibull latency, Othus and colleagues explain, the model returns an odds ratio for being cured and a hazard ratio for survival of the uncured, so a therapy can be shown to raise the cured share without changing survival in those not cured.

The cure fraction need not come from the trial. Felizzi and colleagues contrast an uninformed approach, in which $\pi$ is estimated with the other parameters, with an informed approach, in which $\pi$ is set equal to an external estimate from expert opinion or long-term real-world data, for example at an interim analysis when follow-up is too short for cure to be identifiable. They recommend sensitivity analysis around the fraction and stress that any value chosen in the informed approach needs careful justification.

What the model changes in a cost-effectiveness analysis

Each cured patient adds close to a full remaining lifetime, so the cure fraction moves results strongly. Othus and colleagues analysed a trial of ipilimumab against glycoprotein 100 (gp100) in advanced melanoma. The mixture cure model attributed the survival difference to cure, with 21% cured on ipilimumab and 6% on gp100, mean overall survival of 10 and 9 months among the uncured, and 26 years among the cured on both arms.

NICE's manual (PMG36, section 4.6.25) lists cure among the more flexible approaches to extrapolation and asks for the clinical plausibility of the underlying hazard functions to be routinely assessed and for uncertainty in the extrapolated portion to be explored. The effect of cure assumptions on QALYs and appraisal decisions is discussed under cure rate model.

Worked example with an exponential latency

All numbers in this example are illustrative. A mixture cure model in the relative survival framework has $\pi = 0.30$ and an exponential latency in which the uncured have an excess hazard of 0.4 a year, so $S_u(t) = e^{-0.4t}$ and relative survival is $R(t) = 0.30 + 0.70\,e^{-0.4t}$. Background mortality is a constant 0.01 a year, so $S^{*}(t) = e^{-0.01t}$, where a real model would use life tables.

1. Relative survival. At two years $e^{-0.8} = 0.4493$, so $R(2) = 0.30 + 0.70 \times 0.4493 = 0.6145$. At five years $R(5) = 0.30 + 0.70 \times 0.1353 = 0.3947$, and at ten years $R(10) = 0.30 + 0.70 \times 0.0183 = 0.3128$.

2. Probability of cure among survivors. Dividing the cure fraction by relative survival gives 0.30 / 0.6145 = 0.488 at two years, 0.30 / 0.3947 = 0.760 at five years and 0.30 / 0.3128 = 0.959 at ten years.

3. Excess hazard. The excess hazard equals the uncured hazard times the uncured share of survivors. At two years that is 0.4 × 0.512 = 0.205. At five years it is 0.4 × 0.240 = 0.096, although the uncured hazard never changes.

4. Likelihood contributions. A patient who dies at two years contributes $S^{*}(2) \times R(2) \times h(2) = 0.9802 \times 0.6145 \times 0.2147 = 0.129$, where $h(2) = 0.01 + 0.2047 = 0.2147$. A patient censored at five years contributes $S^{*}(5) \times R(5) = 0.9512 \times 0.3947 = 0.3755$, and in an EM fit carries a weight of 0.760 towards the cured group.

5. A second model matched at five years. A model with $\pi = 0.20$ and an uncured hazard of 0.2826 a year gives $R(5) = 0.20 + 0.80 \times e^{-1.413} = 0.20 + 0.80 \times 0.2434 = 0.3947$, the same as the first. At ten years it gives $R(10) = 0.20 + 0.80 \times 0.0593 = 0.2474$.

6. Interpretation. The two models agree at five years but differ by about 6 percentage points of all-cause survival at ten years and 8 at twenty, so five years of data could not separate two models that a lifetime economic model would treat very differently.

How the mixture form relates to neighbouring models

A non-mixture cure model bounds the cumulative excess hazard instead of splitting the population. Othus and colleagues note that its survival covariates describe a short-term effect, not survival of the uncured, which makes the mixture form easier to read when the question is whether a drug cures.

Limitations and common misreadings

Most problems arise when a short trial's plateau is taken at face value. Trials submitted for HTA are often immature, so the assumptions behind the plateau carry much of the weight.

• Cure at baseline. Latimer and Rutherford note that, strictly, the model assumes cured patients are cured at baseline, and Felizzi and colleagues add that this may not suit diseases where cure follows a long period of stable disease.
• Short follow-up. Othus and colleagues note that the model assumes enough follow-up to identify patients censored at the end as cured. Refitting mixture cure models to six leukaemia trials with longer follow-up, they found mean overall survival estimates higher from the first, shorter data cut in four trials.
• Weights read as diagnoses. The probability of cure among survivors is a population quantity under the fitted model and says nothing about whether a particular patient is free of disease.
• A lognormal latency as a precaution. Latimer and Rutherford found lognormal mixture models among the worst performers when a cure existed, and warn that fitting any cure model where there is no cure is likely to overestimate long-term survival.

Sources

• Cai C, Zou Y, Peng Y, Zhang J. smcure: an R-package for estimating semiparametric mixture cure models. Computer Methods and Programs in Biomedicine. 2012;108(3):1255-1260. https://doi.org/10.1016/j.cmpb.2012.08.013
• 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
• Othus M, Barlogie B, LeBlanc ML, Crowley JJ. Cure models as a useful statistical tool for analyzing survival. Clinical Cancer Research. 2012;18(14):3731-3736. https://doi.org/10.1158/1078-0432.CCR-11-2859
• Othus M, Bansal A, Koepl L, Wagner S, Ramsey S. Accounting for cured patients in cost-effectiveness analysis. Value in Health. 2017;20(4):705-709. https://doi.org/10.1016/j.jval.2016.04.011
• Othus M, Bansal A, Erba H, Ramsey S. Bias in mean survival from fitting cure models with limited follow-up. Value in Health. 2020;23(8):1034-1039. https://doi.org/10.1016/j.jval.2020.02.015
• 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). Sections 3.6.2 and 3.6.3. https://doi.org/10.15131/shef.data.33849118