The minimal continuous-time Markov pharmacometric model
Emilie Schindler, Mats O. Karlsson
Department of Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden.
Objectives: To present the minimal continuous-time Markov model (mCTMM) as an alternative to discrete-time Markov model (DTMM) or standard CTMM for analyzing ordered categorical data with Markov properties.
Methods: In the mCTMM, the probability of each score is defined by an ordinary differential equation and the transition rate between two consecutive states is assumed to be independent on the state. The steady-state probabilities are described by a proportional odds (PO) model. The Markov property is accounted for by a single parameter, the mean equilibration time (MET). Covariate effects can be implemented on parameters related to steady-state probabilities or on the MET. The performance of the mCTMM was evaluated and compared to the PO model, which ignores Markov features, and to published models with Markov properties using three examples: the 4-state fatigue data and hand-foot syndrome (HFS) data in cancer patients, initially described by DTMMs [1], and the 11-state Likert pain score data in diabetic patients previously analysed with a count model including Markovian transition probability inflation [2]. The PO models and mCTMM reproduced as closely as possible the random effect structure and covariate effects on the score probabilities as in the published models.
Results: In all three examples, mCTMM adequately predicted the average number of transitions per patient and the maximum achieved score per patient, which were not well described by PO models. As expected, the mCTMM could not describe the data as well as more flexible DTMM but required fewer estimated parameters (7 vs 20 for fatigue and 7 vs 19 for HFS). The mCTMM better fitted Likert data than the count model while being more parsimonious (18 vs 23 parameters). The estimated covariate effects in the mCTMM were consistent with previously published DTMM and count models.
Conclusions: The mCTMM allows the exploration of potential covariate effects on ordered categorical data while accounting for Markov features, in cases where DTMM is not applicable (non-uniform time intervals between observations) and/or CTMM not conveniently implemented (large number of states).
References:
[1] Hansson, E.K., et al., PKPD Modeling of Predictors for Adverse Effects and Overall Survival in Sunitinib-Treated Patients With GIST. CPT Pharmacometrics Syst Pharmacol, 2013. 2: p. e85.
[2] Plan, E.L., et al., Likert pain score modeling: a Markov integer model and an autoregressive continuous model. Clin Pharmacol Ther, 2012. 91(5): p. 820-8.
Acknowledgements: This work was supported by the Swedish Cancer Society.