Continuous time Markov modelling of relapse sojourns for relapse-remitting multiple sclerosis patients
Gordon Graham, Francois, Mercier, Mick Looby Amy Racine
Objectives: To develop an exposure-response model for the sojourns in either a relapse or remitting disease state in relapse-remitting multiple sclerosis (RRMS) patients using data from two phase III longitudinal studies. Secondly, to explore the effect of demographic and disease status covariates to describe the transition rates from a relapse state to a remitting state and vice versa.
Methods: A total of 2552 MS patients were included in the analysis from two phase III studies: study A was of one year duration (1272 patients) and study B was of two years duration (1280 patients). Data were available from two dose levels of FTY720 in both studies. Study A had an active comparator arm and study B was placebo controlled. A population pharmacokinetic model was developed to describe the steady state concentrations of FTY720 phosphate (FTY720-P), investigating the effect of demographic variables on the estimated steady state concentrations. A continuous-time two-state Markov model was developed to describe the times at which relapses began and ended since the start of FTY720 treatment (Jones et al, 2006). Proc NLMIXED in SAS v9.1 was used to perform the analysis. The mean sojourn in a state was calculated as 1/lambda_lm, where lambda_lm is the model estimated transition rate from state l to state m. The aggregate relapse rate (ARR) was estimated by 1/(lambda_12+lambda_21).
Results: The transition rate from a relapse state to a remitting state was found to be independent of FTY720-P concentration and patient demographic and disease status covariates. The transition rate from a remitting state to a relapse state was modeled as an inhibitory Emax model of FTY720-P concentration. The baseline was found to be a function of the number of gadolinium enhanced lesions (T1B), the disability score (EDSS) and the number of relapses in the two years prior to the study start, such that patients with the most active or severe disease had a higher ARR. The Emax parameter (maximum reduction in transition rate) was dependent on T1B and EDSS. The sojourn in a remitting state was estimated to be at least twice as long for the FTY720 treated patients than for the active control or placebo assigned patients.
Conclusions: The exposure-response model demonstrated the efficacy of FTY720 to increase the sojourn in a remitting state compared to the active control or placebo, and that the transition rate was dependent on the disease activity and severity. A natural modelling extension would be to develop a model linking the number of lesions, the relapse rate and the disability score to further improve the understanding of the disease progression and the beneficial effect of FTY720.
Jones, R.H., Xu, S. and Grunwald, G.K. Continuous time Markov models for binary longitudinal data. Biometrical Journal, 48: 411-419 (2006).