Historical Use of Markov Model and Posterior Predictive Checks in Pharmacometrics

Abstract

We would like to congratulate the authors for their excellent “Tutorial on pharmacometric Markov models” published in a recent issue of CPT-PSP [1]. Their publication emphasizes the increasing use of such models in the field of pharmacometrics, is very complete, and presents in one single paper the theoretical framework of discret-time Markov model (DTMM), continuous-time Markov model, and Hidden Markov model.

However, by restricting their Pubmed search to “Markov pharmacometric” the authors missed two important seminal papers in this field, both authored by Girard, Sheiner, Kastrissios and Blashke, related to the analysis of dosing regimen compliance (or adherence) data and population pharmacokinetic (pop-PK) modeling [2, 3]. The first paper [2] addressed via DTMM and pop-PK simulations the question of the loss of information (bias and precision) in pop-PK analysis when using partial information on patients' dose intakes before concentration measurements versus using the full dosing history as provided by an electronic device. The paper concluded that the use of a limited number of dose records (chosen based on an a priori estimate of the half-life of the drug) would be sufficient to get unbiased and precise PK parameter estimates. Interestingly, the sequence of dose intakes was simulated using a DTMM that was calibrated using real data from electronically monitored patients, which to our best knowledge is the first time a Markov model was used in the field of pharmacometrics.

For p(n), a Markov model was postulated and logits were derived with proper constraints for n = 0, 1, or > 1. The full log likelihood was derived, and parameter estimation was performed with the Laplacian method in NONMEM. The covariates, time of the day (morning, mid-day, evening), weekend days, and age were found to be significant. Figure 4 of that paper visualizes observed dosing patterns (corresponding to the number of times the pill bottle was opened by the patients) [3] (reproduced here as Figure 1). Interestingly, it is quite similar to panels (a) and (b) of Figure 1 of the tutorial paper that shows a visualization to explore the Markovian features of a categorical response, although the latter paper goes one step further by showing a correlation plot of current versus previous response [1].

The original work [3] was published in a statistical journal and was written based on a statistical background rather than a pharmacometric one. However, it is an important reminder that our work [2, 3] was supervised and guided by Prof Lewis B. Sheiner, a giant in the field of pharmacometrics, even before it became a newly coined discipline in 1982 [4]. Coming back to our original published work [3], it is also worth noting that this paper was the first time where pharmacometricians used the ‘posterior predictive check’ concept for model checking and qualification, a concept borrowed from Bayesian statisticians Gelman et al. [5].

All of this does not diminish in any way the merits and the benefits for the pharmacometric community of the current tutorial on Markov models in pharmacometrics [1], which will be of great help for a new generation of pharmacometricians, and for this we would like to thank Qing Xi Ooi, Elodie Plan, and Martin Bergstrand for their valuable contribution.

Editor's Note: The authors submitted this letter in memory of Prof. Lewis B. Sheiner.

The authors declare no conflicts of interest.