Bayesian approach to investigate a two-state mixed model of COPD exacerbations

Abstract

Chronic obstructive pulmonary disease (COPD) is a chronic obstructive disease of the airways. An exacerbation of COPD is defined as shortness of breath, cough, and sputum production. New therapies for COPD exacerbations are examined in clinical trials frequently based on the number of exacerbations that implies long-term study due to the high variability in occurrence and duration of the events. In this work, we expanded the two-state model developed by Cook et al. where the patient transits from an asymptomatic (state 1) to a symptomatic state (state 2) and vice versa, through investigating different semi-Markov models in a Bayesian context using data from actual clinical trials. Of the four models tested, the log-logistic model was shown to adequately characterize the duration and number of COPD exacerbations. The patient disease stage was found a significant covariate with an effect of accelerating the transition from asymptomatic to symptomatic state. In addition, the best dropout model (log-logistic) was incorporated in the final two-state model to describe the dropout mechanism. Simulation based diagnostics such as posterior predictive check (PPC) and visual predictive check (VPC) were used to assess the behaviour of the model. The final model was applied in three clinical trial data to investigate its ability to detect the drug effect: the drug effect was captured in all three datasets and in both directions (from state 1 to state 2 and vice versa). A practical design investigation was also carried out and showed the limits of reducing the number of subjects and study length on the drug effect identification. Finally, clinical trial simulation confirmed that the model can potentially be used to predict medium term (6-12 months) clinical trial outcome using the first 3 months data, but at the expense of showing a non-significant drug effect.

Trial registration: ClinicalTrials.gov NCT01218126 NCT01017952 NCT01009463.

Keywords: Bayesian; COPD; Exacerbations; Negative Binomial; Two-state model.

Conflict of interest statement

AL was a post-doc researcher of GlaxoSmithKline at the time of conducting this work. She now works at Certara Strategic Consulting. SY and MB are employees and shareholders of GlaxoSmithKline.

Figures

Fig. 1

Schematic of the different analysis steps reporting which dataset and which tools/diagnostics were used

Fig. 2

PPC on total number of observations, and number of observations in each state (top—red vertical line is the observed value) and VPCs on number of exacerbations (bottom left—solid bars are observed values, error bars are 95% CI obtained from model simulation), observations in state 1 (bottom middle) and observations in state 2 (bottom right) of the final integrated model (two-state model and the dropout mechanism with the inclusion of the disease stage covariate implemented with the log-logistic model) (Color figure online)

Fig. 3

Transition rate ratios (placebo/active) using the log-logistic model in Data 1 (on the top), Data 2 (in the middle) and Data 3 (on the bottom)

Fig. 4

Transition probability using the log-logistic model for different disease stage level in Data 1 (a), in Data 2 (b) and in Data 3 (c) [red lines—placebo arm; blue lines—treatment arm] (Color figure online)

Fig. 4

Transition probability using the log-logistic model for different disease stage level in Data 1 (a), in Data 2 (b) and in Data 3 (c) [red lines—placebo arm; blue lines—treatment arm] (Color figure online)

Fig. 5

Transition rates ratio using log-logistic model for Data 3 using respectively 150, 100 and 50 subjects per arm (from left to right)

Fig. 6

Transition rates ratio using log-logistic model for Data 3 using study length equal to 168, 120 and 90 days (from left to right), respectively and keeping 150 subjects per arm

Fig. 7

Transition rates ratio (on top) and number of exacerbations (on the bottom—solid bars are observed values, error bars are 95% CI obtained from model simulation) using log-logistic model for Data 1 extrapolating to 6 months (middle) and 1 year (right) from estimates of 3-months data (left)