Open Access
June 2014 Estimation of nonlinear differential equation model for glucose–insulin dynamics in type I diabetic patients using generalized smoothing
Inna Chervoneva, Boris Freydin, Brian Hipszer, Tatiyana V. Apanasovich, Jeffrey I. Joseph
Ann. Appl. Stat. 8(2): 886-904 (June 2014). DOI: 10.1214/13-AOAS706


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.


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Inna Chervoneva. Boris Freydin. Brian Hipszer. Tatiyana V. Apanasovich. Jeffrey I. Joseph. "Estimation of nonlinear differential equation model for glucose–insulin dynamics in type I diabetic patients using generalized smoothing." Ann. Appl. Stat. 8 (2) 886 - 904, June 2014.


Published: June 2014
First available in Project Euclid: 1 July 2014

zbMATH: 06333780
MathSciNet: MR3262538
Digital Object Identifier: 10.1214/13-AOAS706

Keywords: covariance penalties , Generalized profiling , parameter cascading , penalized smoothing , prediction error , profiled penalty estimation

Rights: Copyright © 2014 Institute of Mathematical Statistics

Vol.8 • No. 2 • June 2014
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