Pharmacodynamic Modelling of Discrete Outcomes

Andrew Hooker, Mats Karlsson, Elodie Plan,
Sebastian Ueckert

For many diseases the main outcome is of discrete nature: stage category, symptom severity, number of events, or occurrence of events. In pharmacometrics we distinguish (non-)ordered categorical data, count data, and (repeated) time-to-event data. Models handling this type of data are based on probabilities and, even if they have been around for ~20 years in pharmacometrics, are still not widely used and subject to important innovations. In this project we aim to study and develop new methodologies for discrete data, in order to better describe disease progression, characterize exposure-response with a higher power, as well as simulate clinical trials in a more realistic manner.

In terms of applications, sleep stages have been analysed using Markov models in patients with insomnia. Pain scores rated on a Likert scale by neuropathic patients have also been modelled by including features for under-dispersion and serial correlation to count models. Daily numbers of seizures have been used in the investigation of over-dispersion and Markov patterns in count data. Simultaneous characterization of drug effect on severity and time to acid reflux events has been made possible by a repeated time-to-categorical event model.

Methodology-wise, parametric time-to-event models have been compared to semi-parametric Cox proportional hazard models and an approach to simulate large scale unbiased repeated time-to-event data has been developed. Methods to handle within-subject variability in count models were studied since these data are often collected on a regular basis in clinical trials, resulting in the time and lenght of potential occasions to not be predefined and a dynamic implementation of inter-occasion variability as well as stochastic differential equations were proposed. 

The performance of estimation methods available for discrete models was also explored. We have studied how Laplace behaves in situations with non-even distributions of ordered categories as well as for different Poisson-type models. In another study, the accuracy of parameter estimation with SAEM and importance sampling was compared to the one of Laplace in repeated time-to-event models where the frequency of individuals with events was low. We have also conducted a study investigating all methods available in NONMEM 7 for all types of discrete models.

We also introduced to the field the item response theory (IRT) approach, allowing to connect subscores of a composite scale to a continuous latent disease variable. Such models have now been applied in the group to Alzheimer’s (with ADAS-cog), Schizophrenia (with PANSS), multiple sclerosis (with EDSS), and we are presently working with Parkinson’s (with UPDRS) and COPD (with EXACT).

Currently, we are interested in the implementation of hidden Markov models (HMM) in NONMEM, enabling to characterize non-observed discrete stages. Through simulations, we demonstrate how time and drug effects can be investigated on the transition probability from a disease state to another and how correlation can be explored in bivariate models. An ongoing application for HMMs concerns the exploration of antidrug antibodies.