In the context of evaluating overall survival, which is the focus of this tutorial, all cure models estimate the cure fraction (the proportion of patients who will not die from their disease), and survival for uncured patients. If cure models are used to analyse different endpoints, definitions change. For instance, if used to analyse progression-free survival, cure models estimate the proportion of patients who will not die or experience disease progression, and progression-free survival for uncured patients. Irrespective of the endpoint being analysed, MCMs and NMCs estimate cure and survival in different ways, and can be applied using different frameworks, with implications for the definition of the cure fraction. In this Section we describe frameworks for fitting cure models, key characteristics of MCMs and NMCs, model specification and the use of lifetables and SMRs.

3.1 Frameworks for Cure Models

It is important to determine what is meant when a cure model is used, and this depends on the framework in which the model is fitted. In the context of healthcare interventions that prevent people from dying from the disease the treatment is for, it is logical to consider cure as occurring when the all-cause hazard function for death, \(_\left(t\right),\) converges with the general population hazard function, \(_^(t)\). In this context, it is typical to fit cure models in an ‘excess mortality’ framework, also known as a ‘relative survival’ framework; where we model the difference between the hazard function observed in the trial and the general population hazard (i.e. the excess hazard that is associated with the disease of interest, \(_\left(t\right)\)). That is, we partition the total all-cause hazard as follows:

$$_\left(t\right)=_^\left(t\right)+_\left(t\right).$$

We can transform from the hazard to the survival scale, and rearrange the above equation to show that relative survival [\(_\left(t\right)\)] is the ratio of the all-cause survival \(_\left(t\right)\) and the expected survival in the background population \(_^(t)\):

$$_\left(t\right)=\frac_\left(t\right)}_^\left(t\right)}.$$

Using this framework, cure models are fitted to the relative survival function, which will plateau if and when the all-cause hazard approaches general population mortality rates, and the excess hazard function approaches 0. All-cause survival estimates are derived by multiplying the estimated relative survival function by the expected survival function for the background population. Taking this approach means that we directly use general population mortality rates when fitting models to the trial data, and allow these rates to govern the long-term survival function. If we expect that the disease will have a lasting effect on survival, even in cured patients, SMRs can be applied to general population mortality rates (see Sect. 3.5), and the relative survival function will plateau if and when the all-cause hazard approaches general population mortality rates with the SMR applied.

It is possible to instead fit cure models using different frameworks. A ‘disease-specific’ framework could be used, where models are fitted to disease-specific survival functions (using cause of death information), or an ‘all-cause’ framework could be used, where models are fitted to all-cause survival functions. Under either approach, estimates from these models would need to be combined with general population mortality rates in the long term. For models fit to disease-specific survival functions, this could be done using data on other-cause deaths for the trial period, and then from lifetables beyond the trial period. For models fit to all-cause survival functions it is more complex – in particular, decisions have to be made about when to build in background mortality rates: if this is done from time 0, this would double count early events; if it is done at a later timepoint questions would be asked about how the timepoint was chosen.

Because the relative survival approach is intuitive, does not need information on cause of death and avoids the need to make assumptions around when to begin incorporating general population mortality rates, we focus the rest of this tutorial on cure models fitted in a relative survival framework. This has implications for the definition of the cure fraction: in a relative survival framework, because we directly model the relative survival function, the cure fraction corresponds to the proportion of patients alive in a world where patients can only die from the disease of interest. Therefore, while the cure fraction does represent the proportion estimated not to die from their disease, it will always be higher than the all-cause survival function at the cure timepoint (the point at which excess disease-related hazards fall to zero), because some people will have died of other causes before this timepoint.

3.2 Mixture Cure Models (MCMs)

MCMs assume that there are two groups of individuals – those who are cured of their disease and those who are not [22,23,24]. When fitted in a relative survival framework, general population mortality rates are incorporated directly into the model and the model uses these, combined with the parametric distribution chosen to represent the uncured patients, to estimate the cure fraction. General population mortality rates are taken from relevant lifetables, with rates from the appropriate calendar year used, and these are further stratified by characteristics such as age and sex, so that each trial participant can be assigned an expected background mortality rate. MCMs can be fitted using standard software packages, such as strsmix in Stata [25], and flexsurv and cuRe in R [26, 27].

3.2.1 When Does Cure Occur?

Strictly speaking, MCMs assume that at the study baseline there is a group of patients who experience no excess mortality compared with the general population – that is, ‘cured’ patients are ‘cured’ at baseline [25]. This makes interpretation awkward, but it is also useful to consider MCMs with respect to the timepoint at which no uncured patients remain – representing the timepoint after which all remaining patients are assumed to be cured. MCMs place no constraint on this timepoint – it could occur early in the trial if the difference between hazards observed in the trial and hazards in the general population disappears quickly, or it could occur much later or not at all: sometimes MCMs will predict a 0% cure fraction. It is also important to note that the assignment of cure in an MCM is probabilistic rather than deterministic – individuals are not segregated into cured and uncured groups, they are assigned a probability of being cured and the cure fraction is estimated at the population level.

3.2.2 What is Assumed for Uncured Patients?

MCMs can be fitted using a range of ‘standard’ parametric models – for example, Weibull, log normal, log-logistic, etc., to represent survival for the uncured group of patients. Hence, it is important to consider which models are appropriate for uncured patients – for example, is it likely that the hazard function in uncured patients will be monotonically increasing or decreasing, or will have turning points? Formulations of MCMs using flexible parametric models have been developed, but are seldom used in practice [28].

3.2.3 What About the Cure Fraction?

The choice of parametric distribution used within an MCM to represent uncured patients can have an important impact on the cure fraction estimated by the model. For example, if a log-logistic MCM is used, the survival distribution for uncured patients is likely to have a decreasing hazard in the long term, and the estimated cure fraction may be low because the distribution used for the uncured group is able to represent a reducing hazard and long-term survivors. In contrast, if a Weibull MCM is used, the survival distribution for uncured patients may have an increasing hazard, and the estimated cure fraction may be high, because the distribution used for the uncured group is unable to represent long-term survivors. The two models may in fact result in similar survival curves for the cured and uncured populations combined, but these survival curves would be based on very different assumptions about survival in uncured patients, and would therefore be associated with very different cure fractions.

It should not necessarily be a concern if different MCMs give very different estimates of cure fractions: this is a function of the parametric distribution chosen for uncured patients. Instead, the focus should be on selecting appropriate distributions for the uncured patient group.

3.3 Non-Mixture Cure Models (NMCs)

The key difference between MCMs and NMCs is that NMCs do not split the population into cured and uncured groups directly, although the cure fraction and the survival of the uncured can still be estimated from these models [24, 25, 29, 30]. NMCs can be fitted using standard parametric or flexible parametric distributions. As for MCMs, when fitted in a relative survival framework, general population mortality rates are incorporated directly into the model and should be taken from relevant lifetable sources, using appropriate calendar years and stratifying for key characteristics such as age and sex. NMCs can be fitted using standard software packages such as strsnmix and stpm2 in Stata [25, 31], and flexsurv, cuRe and rstpm2 in R [26, 27].

3.3.1 When Does Cure Occur?

Unlike MCMs, NMCs do not assume that there is a group of patients who are ‘cured’ at baseline. The timepoint at which cure occurs depends on when the modelled hazards converge with those observed in the general population. When fitted using standard parametric models, there is no constraint on when this convergence will occur and typically the estimates of MCMs and NMCs using the same parametric form will be similar to one another. When NMCs are applied using flexible parametric models, the analyst can specify the point at which hazards meet background population levels – these have also been referred to as latent cure models.[30]

3.3.2 What is Assumed for Uncured Patients?

NMCs do not split patients into cured and uncured groups, so there is not a survival model specific to uncured patients. The parametric distribution used within the NMC must be sufficiently flexible to model the survival experience of the cohort as it approaches the cure fraction.

Fitting NMCs using flexible parametric models allows for a complex hazard function to be captured prior to the cure timepoint, and provides the analyst with an additional tool to control when the cure timepoint will occur, dictated by the placement of a ‘boundary knot’ [31]. This may be useful if external data or clinical expert opinion allows for the cure timepoint to be estimated with some level of confidence. Care must be taken with this – specifying a boundary knot at (say) 5 years may mean that hazards begin to decrease rapidly much earlier, allowing gradual convergence at 5 years. To protect against an unrealistically early sharp decrease in the hazards, it may be necessary to set the boundary knot further into the future. Research has been undertaken to investigate the estimation of cure timepoints [32, 33], and this may help inform boundary knots used within flexible parametric NMCs.

3.3.3 What About the Cure Fraction?

The cure fractions associated with NMCs fitted with different parametric distributions are likely to differ. As for MCMs, this is to be expected. The focus should be on selecting a distribution that is likely to be adequate for representing the hazard function prior to the cure timepoint.

3.4 Model Specifications

MCMs and NMCs can be fitted to each treatment group independently, or with treatment group as a covariate. In HTA, it is common to fit independent survival models to each treatment arm, due to concerns around assuming that the treatment effect (in the form of a hazard ratio for proportional hazards models, or a time ratio for accelerated failure time models) is constant over time [2, 5]. In a MCM and NMC setting, the considerations are similar. If dependent models are used, it is possible to allow a treatment effect on the cure proportion parameter, and also on the parameters that define the survival function for the uncured [24, 25]. When choosing between dependent and independent cure models, it is important to consider the validity of assumptions around the treatment effect enforced by using dependent models, to inspect modelled estimates of hazards and survival compared with the observed data, and to consider the validity of long-term extrapolations. Independently fitted models do not enforce assumptions around the treatment effect, but the treatment effect implied by the models should still be assessed, as recommended by TSD 21 [12].

It is rare to include baseline covariates in survival models used for HTA. However, this may be more relevant for cure models because of their use of lifetable data, stratified by age and sex. Over time, if older patients are more likely to die from their cancer, the relative age mix in the remaining trial population will change, and the conditional hazard function at later timepoints will be based on younger people than would have been the case if general population mortality were the only cause of death. Therefore, if disease-related deaths are likely to be associated with age (or sex), cure models that include these as baseline variables should be superior to models that exclude them. If covariates are included in the model, it will often be necessary to obtain the marginal all-cause survival function for the cohort as a whole, especially in a HTA setting where typically marginal rather than conditional survival functions are required. This can be achieved through regression standardisation and re-incorporation of general population mortality rates. Accounting for changes in age and sex distributions over time is relatively rare in survival models used for HTA – however, this can be important and more details on standardisation are available from Lambert et al. [34].

3.5 Standardised Mortality Ratios (SMRs)

SMRs relate to whether cancer survivors are at a higher risk of death than the age- and sex-matched general population, separate from their disease-related mortality risk, perhaps due to co-morbidities or lasting ill-effects of intensive treatment. Various studies have reported SMRs relevant for different groups of patients with cancer, ranging from values of 6 or higher [35, 36] to values close to 1 [37].

When cure models have been used in NICE appraisals, it has been common for SMRs to be applied. Recent appraisals in ovarian cancer and gastric cancer tested SMRs ranging between 1.4 and 1.8, though these appeared to be based primarily on assumption and clinical opinion, rather than data [38,39,40,41].

SMRs can be particularly important when the cure fraction is large. It is important to assess hazards observed during trial periods to see whether they approach background levels, but if the cure timepoint has not been reached during the study, this will not provide information on whether applying an SMR > 1 would be appropriate. It may be helpful to analyse relevant registry datasets that contain long-term survivors with the same disease to investigate whether those patients exhibit mortality rates that are similar to, or higher than, background population levels.

When SMRs are applied, they should be applied to the general population mortality rates in the lifetables being used, ensuring that the adjusted rates are incorporated directly into the cure models fitted. This is done in the same way, irrespective of whether MCMs or NMCs are being used.

3.6 Lifetables

Cure models require the use of general population lifetables, and therefore the source of lifetable data must be chosen. Lifetables are available for different countries (and sometimes regions) and are usually split by sex, calendar year and age. In the context of fitting a cure model to data from an international clinical trial, it could be argued that it is most appropriate to use different lifetables for each patient according to their country (or region) of residence, their age and sex and the year that they entered the trial. Alternatively, it could be argued that it is more relevant to use lifetables for the country the analysis is designed for (still stratified for age, sex and calendar year).

We suggest that this choice should be dictated by the specified purpose of the analysis – is the aim purely to project survival for trial participants, or is it to project survival for a different population (i.e. the population that a decision is being made for)? In practice, as shown by TSD 21, the use of ‘incorrect’ lifetables is unlikely to have a large impact on survival predictions [12], although this might not be the case if the cure fraction is large and if alternative lifetables have large differences in life expectancy.

When cure models are used, they should ideally be fitted to patient-level data. This allows for the distribution of age, sex and calendar year to be accounted for in the model, as well as how the age and sex mix in the surviving population changes over time. If patient-level data are not available, published survival curves can be digitised to reconstruct the data using commonly used methods [42], but the reconstructed dataset will not include information on age, sex and calendar year of recruitment for each patient. Assumptions are then required to assign values for these variables for each patient (likely based on published means and distributions), which make resulting model predictions prone to additional error, though these are likely to be small.