Multinomial logistic functions in Markov-chain models for modelling sleep architecture after placebo administration
Bizzotto R (1), Zamuner S (2), De Nicolao G (3), Cobelli C (1), Gomeni R (2)
(1) Dept. of Information Engineering, University of Padova, Padova, Italy; (2) Clinical Pharmacology/Modeling&Simulation, GlaxoSmithKline, Verona, Italy; (3) Dept. of Computer Engineering and Systems Science, University of Pavia, Pavia, Italy.
Objectives: The aim of this work was to generalize the previously proposed [1] mixed-effect Markov-chain model based on piecewise linear binary logistic functions through the implementation of multinomial logistic functions, in order to characterize the time course of transition probabilities between sleep stages in insomniac patients.
Methods: Polysomnography data were obtained from the first night of a placebo-controlled treatment of insomniac patients. Assuming that the time course of sleep stages (awake stage, stage 1, stage 2, slow-wave sleep and REM sleep) obeys to a Markov-chain model, a population approach was implemented with NONMEM VI. In particular, the relationship between time and individual transition probabilities between sleep stages was modeled through piecewise linear multinomial logistic functions. For example, assuming that ST1, ST2 and ST3 are three sleep stages, two multinomial logit functions can be defined as:
g1T = log ( Pr(ST2T|ST3T-1) / Pr(ST1T|ST3T-1) ),
g2T = log ( Pr(ST3T|ST3T-1) / Pr(ST1T|ST3T-1) ).
Using these equations and recalling that the sum of the three probabilities conditional on ST3T-1 must be equal to one, the transition probabilities Pr(ST1T|ST3T-1), Pr(ST2T|ST3T-1), Pr(ST3T|ST3T-1) can be easily derived from the logits. The choice of the multinomial model was motivated by the following reasons: (1) to assure that the sum of all probabilities of transitions starting from a certain stage is equal to one; (2) to reduce the number of sub-models to be identified: from 20 sub-models using the binary-logit approach to 5 sub-models in the new approach (one for each sleep stage); (3) to estimate probabilities of all transitions at any time avoiding the need for a preliminary analysis aimed to identify zero-probability transitions. Performance was evaluated through visual inspection of model fitting on post-hocs and through posterior predictive check as suggested by Gelman et al. [2].
Results: The identification of the five sub-models produced a good adherence of mean post-hocs to the observed transition frequencies. Parameters were generally well estimated in terms of CV, shrinkage and distribution of empirical Bayes estimates around the typical values. The posterior predictive check showed good adherence of most of the simulated distributions of sleep macro parameters to the observed parameter values. A slight overestimation of the transition probabilities from the slow-wave sleep stage to other stages was found. This outcome may be explained by the small number of occurrences of these transitions, although further work is needed to investigate the reason for this finding.
Conclusions: This work confirms the adequacy of mixed-effect Markov-chain models for describing sleep architecture of insomniac patients treated with placebo. Moreover, the use of multinomial logit functions in place of binary ones yields physiologically constrained parameters, reduces the influence of exploratory data analysis, and requires the identification of fewer sub-models.
References:
[1] 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
[2] Gelman A et al. Bayesian data analysis. Chapman & Hall 1995, London, UK
