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Estimation of nonlinear differential equation model for glucose-insulin dynamics in type I diabetic patients using generalized smoothing. (English) Zbl 1454.62319

Summary: 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 J. O. Ramsay et al. [“Parameter estimation for differential equations: a generalized smoothing approach”, J. R. Stat. Soc. Ser. B, Stat. Methodol. 69, No. 5, 741–796 (2007; doi:10.1111/j.1467-9868.2007.00610.x)]. 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 [B. Efron, J. Am. Stat. Assoc. 99, No. 467, 619–632 (2004; Zbl 1117.62324)]. 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.

MSC:

62P10 Applications of statistics to biology and medical sciences; meta analysis

Citations:

Zbl 1117.62324

Software:

gss
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References:

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