Signature
T_bar = pi * E_B + (1 - pi) * E_U
| Inputs | Definition | Unit |
|---|---|---|
pi | Proportion of patients who will not die from their disease | proportion from 0 to 1 |
E_B | Area under the background survival curve S_star(t); with a constant background hazard mu it is 1/mu | years |
E_U | Area under S_star(t) times S_u(t); with constant background and excess hazards mu and lambda it is 1/(mu plus lambda) | years |
T_bar | Mean overall survival of the whole cohort | years |
|---|
Function
Cure rate model survival and lifetime mean survival with a cure fraction
Maps a cure fraction, the expected survival of the matched general population and the survival of uncured patients to all-cause survival over time and to the lifetime mean survival and QALYs that drive an appraisal. Cured patients face background mortality only; uncured patients also carry an excess hazard from the disease. The mixture form weights the two groups directly, the non-mixture form bounds the excess cumulative hazard so that relative survival falls to the cure fraction, and mean survival is the cure-fraction weighted average of the two group means. Reused, not repeated here: the all-cause hazard as background plus excess hazard HE-FM-BTH-005, the continuous discount rate log(1 plus d) HE-FM-CONT-003, the per-cycle probability from a hazard HE-FM-TP-001 and the restricted mean from a Kaplan-Meier curve HE-FM-ADMC-003. Notation follows the Cure Rate Model article.
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Implementations
Excel
Cure-fraction mean survival from named cells
With CureFrac, MeanBg and MeanUncured named, the formula returns mean overall survival, held in MeanSurv. For the constant-hazard check, BgHazard and ExcessHazard hold mu and lambda.
=CureFrac*MeanBg+(1-CureFrac)*MeanUncured
Assumptions
Cured and uncured patients share the same background mortality
Both groups face the same general population mortality, and uncured patients carry the excess hazard in addition, so E_U is below E_B.
Lifetime areas under the cure model survival curves
E_B and E_U are areas over a horizon long enough for the cohort to have died. Areas cut at an earlier horizon give a restricted mean, as in HE-FM-ADMC-003, and understate T_bar while cured patients are still alive.
Worked examples
Mean survival with a cure fraction of 0.10
With E_B of 25 years (background hazard 0.04 a year) and E_U of 2 years (total hazard 0.50 a year for the uncured), a cure fraction of 0.10 gives 2.5 plus 1.8, or 4.3 years, as in the article's table.
pi = 0.10; E_B = 25; E_U = 2; T_bar = 4.3
Mean survival with a cure fraction of 0.25
A cure fraction of 0.25 gives 6.25 plus 1.5, or 7.75 years. Against the comparator with 0.10 the gain is 3.45 years, which is 0.15 times 23, all from the extra cure, as in the article.
pi = 0.25; E_B = 25; E_U = 2; T_bar = 7.75
Mean survival with a cure fraction of 0.40
A cure fraction of 0.40 gives 10 plus 1.2, or 11.2 years, the last row of the article's table.
pi = 0.40; E_B = 25; E_U = 2; T_bar = 11.2
Common errors
Uncured mean survival from the excess hazard alone
E_U must include background mortality. With an excess hazard of 0.46 a year, one over 0.46 gives about 2.17 years instead of one over 0.50, or 2 years, and T_bar at pi = 0.25 becomes about 7.88 years instead of 7.75.
Cure-fraction mean survival over a horizon that ends too soon
A 40-year horizon leaves cured patients alive at its end. With the article's inputs the restricted mean at pi = 0.25 is about 6.49 years against a lifetime mean of 7.75, so about 1.26 years of the benefit of cure is cut off (computed here for illustration).
Sources
Mean survival of a population with a cure fraction as a weighted average
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. Section on cure models in economic evaluation, equation 1, S(t) = S_B(t)[p plus (1 minus p)S_E(t)]: mean survival of the population is the weighted average of the mean survival of the cured and not cured subpopulations, the mean for the cured is the integral of S_B(t) and for the not cured the integral of S_B(t)S_E(t).
All-cause survival from relative and expected survival in cure models
Latimer NR, Rutherford MJ. Mixture and non-mixture cure models for health technology assessment. PharmacoEconomics. 2024;42(10):1073-1090. Section on frameworks for cure models: all-cause survival estimates are derived by multiplying the estimated relative survival function by the expected survival function for the background population.
Canonical Identity
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