A Multistate Model Incorporating Relative Survival Extrapolation and Mixed Time Scales for Health Technology Assessment

Abstract

Background: Multistate models have been widely applied in health technology assessment. However, extrapolating survival in a multistate model setting presents challenges in terms of precision and bias. In this article, we develop an individual-level continuous-time multistate model that integrates relative survival extrapolation and mixed time scales.

Methods: We illustrate our proposed model using an illness-death model. We model the transition rates using flexible parametric models. We update the hesim package and the microsimulation package in R to simulate event times from models with mixed time scales. This feature allows us to incorporate relative survival extrapolation in a multistate setting. We compare several multistate settings with different parametric models (standard vs. flexible parametric models), and survival frameworks (all-cause vs. relative survival framework) using a previous clinical trial as an illustrative example.

Results: Our proposed approach allows relative survival extrapolation to be carried out in a multistate model. In the example case study, the results agreed better with the observed data than did the commonly applied approach using standard parametric models within an all-cause survival framework.

Conclusions: We introduce a multistate model that uses flexible parametric models and integrates relative survival extrapolation with mixed time scales. It provides an alternative to combine short-term trial data with long-term external data within a multistate model context in health technology assessment.

Fig. 1

An irreversible illness–death model with three transitions: $h_1(\text{t})$, progression free → progression; $h_2(\text{t})$, progression free → death, and $h_3(t-u)$, progression → death, with a semi-Markov approach

Fig. 2

An irreversible illness–death model incorporating a relative survival framework, with five transitions: $h_1(t)$, progression free → progression; ${\lambda }_2(t)$, progression free → excess death; $h_2^{\ast }(t+a,t+c)$, progression free → expected death; ${\lambda }_3(t-u)$, progression → excess death; and $h_3^{\ast }(t+a,t+c)$, progression → expected death, with a semi-Markov approach

Fig. 3

Observed and extrapolated all-cause hazard functions by treatment arm for transitions. a Transition 1: progression free $\to$ progression, b Transition 2: progression free $\to$ death, and c transition 3: progression $\to$ death. The observed hazard functions were with maximum 4 years of follow-up; the parametric hazard functions were extrapolated by flexible parametric models (FPMs) within an all-cause survival framework (ASF) or a relative survival framework (RSF), the Gompertz model, and the generalized gamma model. Vertical dashed lines indicate time points at 4, 8, and 15 years. FC, fludarabine and cyclophosphamide; RFC, rituximab, fludarabine and cyclophosphamide

Fig. 4

Observed and extrapolated survival functions for patients of the rituximab, fludarabine and cyclophosphamide (RFC) arm (left) and the fludarabine and cyclophosphamide (FC) arm (right) in the CLL-8 trial. Observed overall survival (OS), estimated with the Kaplan–Meier estimator, is shown by Hallek et al. (observed 4-year OS in black) and Fischer et al. (observed 8-year OS in green). Extrapolated OS for 50 years is shown by the following models: models using standard parametric models (SPMs) within an all-cause survival framework (ASF) and using flexible parametric models (FPMs) within an ASF or a relative survival framework (RSF). The dotted lines indicate 4, 8, and 15 years

Fig. 5

Cost-effectiveness plane (left) and cost-effectiveness acceptability curve (right). The dashed line for the right panel indicates £30,000. ASF all-cause survival framework, CET cost-effectiveness threshold, FPM flexible parametric model, QALY quality-adjusted life-years, RFC rituximab, fludarabine and cyclophosphamide, RSF relative survival framework, SPM standard parametric model