Bayesian Analysis of Growth Curves Using Mixed Models Defined by Stochastic Differential Equations

Abstract

Growth curve data consist of repeated measurements of a continuous growth process over time in a population of individuals. These data are classically analyzed by nonlinear mixed models. However, the standard growth functions used in this context prescribe monotone increasing growth and can fail to model unexpected changes in growth rates. We propose to model these variations using stochastic differential equations (SDEs) that are deduced from the standard deterministic growth function by adding random variations to the growth dynamics. A Bayesian inference of the parameters of these SDE mixed models is developed. In the case when the SDE has an explicit solution, we describe an easily implemented Gibbs algorithm. When the conditional distribution of the diffusion process has no explicit form, we propose to approximate it using the Euler-Maruyama scheme. Finally, we suggest validating the SDE approach via criteria based on the predictive posterior distribution. We illustrate the efficiency of our method using the Gompertz function to model data on chicken growth, the modeling being improved by the SDE approach.

Bayesian Inference for Stochastic Differential Equation Mixed Effects Models of a Tumour Xenography Study

“…For instance, Donnet et al . () found that a stochastic differential equation (SDE) version of the Gompertz growth model is superior to its non‐linear deterministic mixed model counterpart, for prediction of the body weight of growing chickens. Donnet and Samson () reported similar findings from pharmacokinetic experiments and Whitaker et al .…”

Gaussian Process Modeling of Protein Turnover

“…A similar approach has employed a mixed-effects modeling for growth curves resulting from the Gompertz model. 42 Using each point of peptide data separately rather than averaging the peptide data will allow us to estimate the variability due to the heavy-isotope labeling of peptides. In addition, we will extend the model to describe the turnover of proteins that are not in a steady-state condition.…”

Stochastic modeling of systems mapping in pharmacogenomics

“…The approaches for estimating the curve parameters that specify a group of SDEs using observed data at discrete time points have been proposed [30]. Donnet et al [31] developed a Bayesian method to estimate SDE-based mixed model parameters using PK/PD data, typical of high sparsity and subject-dependent measure schedule. A review of the estimation of SDEs in PK/PD models is given by Donnet and Samson [21].…”