Estimation of nonlinear differential equation model for glucose-insulin dynamics in type I diabetic patients using generalized smoothing

TL;DR

This paper develops a novel method for estimating nonlinear differential equation models of glucose-insulin dynamics in type 1 diabetic patients, using generalized smoothing with optimized penalty weights and knot selection, enabling accurate parameter estimation from limited noisy data.

Contribution

It introduces an optimized generalized smoothing approach with covariance penalties for estimating nonlinear ODE models from sparse data, specifically applied to glucose-insulin dynamics in T1DM patients.

Findings

Accurate estimation of model parameters from limited noisy data.

Improved prediction errors using optimized smoothing.

Physiologically meaningful parameter estimates obtained.

Abstract

In this work we develop an ordinary differential equations (ODE) model of physiological regulation of glycemia in type 1 diabetes mellitus (T1DM) patients in response to meals and intravenous insulin infusion. Unlike for the majority of existing mathematical models of glucose-insulin dynamics, parameters in our model are estimable from a relatively small number of noisy observations of plasma glucose and insulin concentrations. For estimation, we adopt the generalized smoothing estimation of nonlinear dynamic systems of Ramsay et al. [J. R. Stat. Soc. Ser. B Stat. Methodol. 69 (2007) 741-796]. In this framework, the ODE solution is approximated with a penalized spline, where the ODE model is incorporated in the penalty. We propose to optimize the generalized smoothing by using penalty weights that minimize the covariance penalties criterion (Efron [J. Amer. Statist. Assoc. 99 (2004) 619-642]). The covariance penalties criterion provides an estimate of the prediction error for nonlinear estimation rules resulting from nonlinear and/or nonhomogeneous ODE models, such as our model of glucose-insulin dynamics. We also propose to select the optimal number and location of knots for B-spline bases used to represent the ODE solution. The results of the small simulation study demonstrate advantages of optimized generalized smoothing in terms of smaller estimation errors for ODE parameters and smaller prediction errors for solutions of differential equations. Using the proposed approach to analyze the glucose and insulin concentration data in T1DM patients, we obtained good approximation of global glucose-insulin dynamics and physiologically meaningful parameter estimates.