A two-component nonlinear mixed effects model for longitudinal data, with application to gastric emptying studies

Abstract

Gastric emptying studies are of great interest in human and veterinary medical research to evaluate effects of medications or diets for promoting gastrointestinal motility and to examine unintended side-effects of new or existing medications, diets, or procedures. Summarizing gastric emptying data is important to allow easier comparison between treatments or groups of subjects and comparisons of results among studies. The standard method for assessing gastric emptying is by using scintigraphy and summarizing the nonlinear emptying of the radioisotope. A popular model for fitting gastric emptying data is the power exponential model. This model can only describes a globally decreasing pattern and thus has the limitation of poorly describing localized intragastric events that can occur during emptying. Hence, we develop a new model for gastric emptying studies to improve population and individual inferences using a mixture of nonlinear mixed effects models. One mixture component is based on a power exponential model which captures globally decreasing patterns. The other is based on a locally extended power exponential model which captures both local bumping and rapid decay. We refer to this mixture model as a two-component nonlinear mixed effects model. The parameters in our model have clear graphical interpretations that provide a more accurate representation and summary of the curves of gastric emptying pattern. Two methods are developed to fit our proposed model: one is the mixture of an Expectation Maximization algorithm and a global two-stage method and the other is the mixture of an Expectation Maximization algorithm and the Monte Carlo Expectation Maximization algorithm. We compare our methods using simulation, showing that the two approaches are comparable to one another. For estimating the variance and covariance matrix, the second approach appears approximately more efficient and is also numerically more stable in some cases. Our new model and approaches are applicable for assessing gastric emptying in human and veterinary medical research and in many other biomedical fields such as pharmacokinetics, toxicokinetics, and physiological research. An example of gastric emptying data from equine medicine is used to demonstrate the advantage of our approaches.

Figure 1

The estimated individual curves obtained from fitting the two-component nonlinear mixed effects model using 14 horses that received the test meal without atropine: the solid line curve of power exponential random effects model using GTS (black), the dashed-dotted line curves of mixture model using EMGTS (gray), and the dashed line curves of mixture model using EMMCEM (black).

Figure 2

The estimated population curve (solid thick line) and individual curves (thin lines) of the two-component nonlinear mixed effects model (two-component NLME) which is obtained from EMMCEM using 14 horses that received the test meal without atropine.

Figure 3

The estimated individual curves obtained from fitting the two-component nonlinear mixed effects model using seven horses that received the test meal with atropine: the solid line curve of power exponential random effects model using GTS (black), the dashed-dotted line curves of mixture model using EMGTS (gray), and the dashed line curves of mixture model using EMMCEM (black).

Figure 4

The estimated population curve (solid thick line) and individual curves (thin lines) of the two-component nonlinear mixed effects model (two-component NLME) which is obtained from EMMCEM using seven horses that received the test meal with atropine.