Multinomial logistic functions in Markov-chain models for modeling sleep architecture: external validation and covariate analysis
Roberto Bizzotto (1), Stefano Zamuner (2), Giuseppe De Nicolao (3), Roberto Gomeni (2), Andrew C Hooker (4), Mats O Karlsson (4)
(1) Dept. of Information Engineering, University of Padova, Italy; (2) Clinical Pharmacology/Modeling&Simulation, GlaxoSmithKline, Verona, Italy; (3) Dept. of Computer Engineering and Systems Science, University of Pavia, Italy (4) Dept. of Pharmaceutical Biosciences, Uppsala University, Sweden
Objectives: A mixed-effect Markov-chain model based on piecewise linear multinomial logistic functions has been recently proposed [1] to characterize the time course of transition probabilities between sleep stages in insomniac patients treated with placebo. The aims of this work were to further develop the model, explore the covariate effects, and perform the external validation of model structure and parameters estimates.
Methods: Polysomnography data were obtained from the first night after placebo administration to patients affected by primary insomnia and belonging to two different placebo-controlled, parallel studies (study A and study B with NA=116 and NB=81). Time courses of sleep stages (awake stage, stage 1, stage 2, slow-wave sleep and REM sleep) were assumed to be realizations of a Markov-chain process and modeled using multinomial logit functions, from which transition probabilities can be easily back-calculated [1]. In this work, a thorough investigation of the structure and predictors of the logit functions has been performed, including: a) the choice of ratios entering the logits, b) the estimation of significant correlation terms, c) the implementation of a longer memory for the Markov-chains, d) a different description of transitions during and after initial sleeplessness, e) a reduced parameterization of the logits time dependence. Model building was based on dataset A and guided by model adequacy criteria (log likelihood ratio test and Akaike Information Criteria) and visual predictive checks (presented in [3]). External validation of the final model was based on dataset B and relied on the evaluation of objective function value (OFV), empirical Bayes estimates (EBEs) distributions, and posterior predictive checks (PPCs). Finally, stepwise covariate analysis within NONMEM [2] was performed on dataset A.
Results: The changes introduced in the model before including covariates led to an increase of data likelihood, without significantly affecting runtimes and model size. Although sleep maintenance parameters resulted still slightly biased, the final PPC showed a definite improvement in the distributions of aggregated sleep parameters. When using dataset B, PPC provided very similar outcomes; moreover, OFV and EBEs distributions did not change substantially when parameters estimated from dataset A were plugged in. Finally, age, gender and BMI were found as statistically significant covariates affecting many transition probabilities in different night time intervals; however, their inclusion did not improve substantially PPC performance and provided limited reduction in inter-individual variability.
Conclusions: Previously proposed mixed-effect Markov-chain models for describing sleep architecture of insomniac patients treated with placebo [1,4] were further improved. External validation has shown that the developed framework provides not only an adequate sleep model but also reliable parameter estimates for a general population with similar sleep characteristics and study conditions. Some influential covariates have been detected whose clinical relevance deserves further exploration in a wider population of insomniac subjects.
References:
[1] Bizzotto R, Zamuner S, De Nicolao G, Karlsson MO, Gomeni R. Multinomial logistic estimation of Markov-chain models for modeling sleep architecture in primary insomnia patients. J Pharmacokinet Pharmacodyn, 2010(online) DOI 10.1007/s10928-009-9148-2
[2] Jonsson EN, Karlsson MO. Automated covariate model building within NONMEM. Pharm Res Sep 1998;15(9):1463-1468
[3] Mezzalana E, Bizzotto R, Sparacino G, Zamuner S. Multinomial logistic functions in Markov-chain models for modeling sleep architecture: internal validation based on VPCs. PAGE 2010;19
[4] Karlsson M O et al. A pharmacodynamic Markov mixed-effect model for the effect of temazepam on sleep. Clin Pharmacol Ther 2000;68(2):175-88
