A Minimal Continuous-Time Markov Pharmacometric Model

Emilie Schindler,Mats O Karlsson

doi:10.1208/s12248-017-0109-1

Abstract

In this work, an alternative model to discrete-time Markov model (DTMM) or standard continuous-time Markov model (CTMM) for analyzing ordered categorical data with Markov properties is presented: the minimal CTMM (mCTMM). Through a CTMM reparameterization and under the assumption that the transition rate between two consecutive states is independent on the state, the Markov property is expressed through a single parameter, the mean equilibration time, and the steady-state probabilities are described by a proportional odds (PO) model. The mCTMM performance was evaluated and compared to the PO model (ignoring Markov features) and to published Markov models using three real data examples: the four-state fatigue and hand-foot syndrome data in cancer patients initially described by DTMM and the 11-state Likert pain score data in diabetic patients previously analyzed with a count model including Markovian transition probability inflation. The mCTMM better described the data than the PO model, and adequately predicted the average number of transitions per patient and the maximum achieved scores in all examples. As expected, mCTMM could not describe the data as well as more flexible DTMM but required fewer estimated parameters. The mCTMM better fitted Likert data than the count model. The mCTMM enables to explore the effect of potential predictive factors such as drug exposure and covariates, on ordered categorical data, while accounting for Markov features, in cases where DTMM and/or standard CTMM is not applicable or conveniently implemented, e.g., non-uniform time intervals between observations or large number of categories.

Highlights

In clinical trials and practice, efficacy and safety evaluation often involves endpoints that are of categorical nature
We present a version of continuous-time Markov model (CTMM), the minimal CTMM, as an alternative to discrete-time Markov model (DTMM) or standard CTMM
An improvement in OFV was observed in all three examples when accounting for Markov elements; i.e., DTMM or count model and minimal CTMM (mCTMM) always performed better than proportional odds (PO) models

Summary

Introduction

In clinical trials and practice, efficacy and safety evaluation often involves endpoints that are of categorical nature. In more complex cases, when the intensity of a variable that is not directly or quantifiable is of interest, ordered categorical variables are collected, for example, pain intensity, sedation grade, or adverse effect severity (e.g., none, mild, moderate, severe, or life threatening). Depending on the frequency of collection, the outcome (state) of two consecutive assessments may not be independent but intercorrelated beyond what is predicted by taking standard predictors (dose, time, individual patient characteristics) into account. A way to handle such correlation is to make an outcome conditional on the observed state at the previous assessment in the individual (but not on the whole history). Such models are referred to as (first-order) Markov models

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Conclusion