The Annals of Applied Statistics

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, and Jeffrey I. Joseph

Full-text: Open access


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.

Article information

Ann. Appl. Stat., Volume 8, Number 2 (2014), 886-904.

First available in Project Euclid: 1 July 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


Chervoneva, Inna; Freydin, Boris; Hipszer, Brian; Apanasovich, Tatiyana V.; Joseph, Jeffrey I. Estimation of nonlinear differential equation model for glucose–insulin dynamics in type I diabetic patients using generalized smoothing. Ann. Appl. Stat. 8 (2014), no. 2, 886--904. doi:10.1214/13-AOAS706.

Export citation


  • Bergman, R. Y., Ider, Z., Bowden, C. and Cobelli, C. (1979). Quantitative estimation of insulin sensitivity. Am. J. Physiol. 236 E667–E676.
  • Brunel, N. J.-B. (2008). Parameter estimation of ODE’s via nonparametric estimators. Electron. J. Stat. 2 1242–1267.
  • Cao, J. and Ramsay, J. O. (2009). Generalized profiling estimation for global and adaptive penalized spline smoothing. Comput. Statist. Data Anal. 53 2550–2562.
  • Chen, J. and Wu, H. (2008). Efficient local estimation for time-varying coefficients in deterministic dynamic models with applications to HIV-1 dynamics. J. Amer. Statist. Assoc. 103 369–384.
  • Claeskens, G., Krivobokova, T. and Opsomer, J. D. (2009). Asymptotic properties of penalized spline estimators. Biometrika 96 529–544.
  • Cobelli, C., Bier, D. M. and Ferrannini, E. (1990). Modeling glucose metabolism in man: Theory and practice. Horm. Metab. Res. Suppl. 24 1–10.
  • Craven, P. and Wahba, G. (1979). Smoothing noisy data with spline functions. Estimating the correct degree of smoothing by the method of generalized cross-validation. Numer. Math. 31 377–403.
  • De Gaetano, A. and Arino, O. (2000). Mathematical modelling of the intravenous glucose tolerance test. J. Math. Biol. 40 136–168.
  • Donnet, S. and Samson, A. (2007). Estimation of parameters in incomplete data models defined by dynamical systems. J. Statist. Plann. Inference 137 2815–2831.
  • Efron, B. (2004). The estimation of prediction error: Covariance penalties and cross-validation. J. Amer. Statist. Assoc. 99 619–642.
  • Fischer, U., Salzsieder, E., Jutzi, E., Albrecht, G. and Freyse, E. J. (1984). Modeling the glucose–insulin system as a basis for the artificial beta cell. Biomed. Biochim. Acta 43 597–605.
  • Gu, C. (2002). Smoothing Spline ANOVA Models. Springer, New York.
  • Heckman, N. E. and Ramsay, J. O. (2000). Penalized regression with model-based penalties. Canad. J. Statist. 28 241–258.
  • Hipszer, B., Joseph, J. and Kam, M. (2005). Pharmacokinetics of intravenous insulin delivery in humans with type 1 diabetes. Diabetes Technol. Ther. 7 83–93.
  • Hu, Y. (1993). An algorithm for data reduction using splines with free knots. IMA J. Numer. Anal. 13 365–381.
  • Huang, Y., Liu, D. and Wu, H. (2006). Hierarchical Bayesian methods for estimation of parameters in a longitudinal HIV dynamic system. Biometrics 62 413–423.
  • Jupp, D. L. B. (1978). Approximation to data by splines with free knots. SIAM J. Numer. Anal. 15 328–343.
  • Lehmann, E. D. Interactive educational simulators in diabetes care. Med. Inform. (Lond.) 22 47–76.
  • Lehmann, E. D. and Deutsch, T. A. (1992). A physiological model of glucose–insulin interaction in T1DM mellitus. J. Biomed. Eng. 14 235–242.
  • Li, Z., Osborne, M. R. and Prvan, T. (2005). Parameter estimation of ordinary differential equations. IMA J. Numer. Anal. 25 264–285.
  • Lindstrom, M. J. (1999). Penalized estimation of free-knot splines. J. Comput. Graph. Statist. 8 333–352.
  • Monks, J. (1990). The construction of diabetic diets using linear programming. M.Sc. thesis, Univ. Salford.
  • Mukhopadhyay, A., De Gaetano, A. and Arino, O. (2004). Modeling the intra-venous glucose tolerance test: A global study for a single-distributed-delay model. Discrete Contin. Dyn. Syst. Ser. B 4 407–417.
  • Ramsay, J. O., Hooker, G., Campbell, D. and Cao, J. (2007). Parameter estimation for differential equations: A generalized smoothing approach. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 741–796.
  • Ruppert, D. (2002). Selecting the number of knots for penalized splines. J. Comput. Graph. Statist. 11 735–757.
  • Scheiner, G. and Boyer, B. A. (2005). Characteristics of basal insulin requirements by age and gender in type-1 diabetes patients using insulin pump therapy. Diabetes Res. Clin. Pract. 69 14–21.
  • Sherwin, R., Kramer, K., Tobin, J., Insel, P., Liljenquist, J., Berman, M. and Andres, R. (1974). A model of the kinetics of insulin in man. J. Clin. Invest. 53 1481–1492.
  • Shimoda, S., Nishida, K., Sakakida, M., Konno, Y., Ichinose, K., Uehara, M., Nowak, T. and Shichiri, M. (1997). Closed-loop subcutaneous insulin infusion algorithm with a short-acting insulin analog for long-term clinical application of a wearable artificial endocrine pancreas. Front. Med. Biol. Eng. 8 197–211.
  • Sorensen, J. T. (1985). A physiological model of glucose metabolism in man and its use to design and assess improved insulin therapies for diabetes. Ph.D. thesis, Massachusetts Institute of Technology.
  • Spiriti, S., Eubank, R., Smith, P. W. and Young, D. (2013). Knot selection for least-squares and penalized splines. J. Stat. Comput. Simul. 83 1020–1036.
  • Steil, G., Rebrin, K., Mastrototaro, J., Bernaba, B. and Saad, M. (2003). Determination of plasama glucose during rapid glucose excursions with a subcutaneous glucose sensor. Diabetes Tech. Ther. 5 27–31.
  • Stein, C. M. (1981). Estimation of the mean of a multivariate normal distribution. Ann. Statist. 9 1135–1151.
  • Thorsteinsson, B. (1990). Kinetic models for insulin disappearance from plasma in man. Dan. Med. Bull. 37 143–153.
  • Trajanoski, Z., Wach, P., Kotanko, P., Ott, A. and Skraba, F. (1993). Pharmacokinetic model for the absorption of subcutaneously injected soluble insulin and monomeric insulin analogues. Biomed. Tech. (Berl.) 38 224–231.
  • Varah, J. M. (1982). A spline least squares method for numerical parameter estimation in differential equations. SIAM J. Sci. Statist. Comput. 3 28–46.
  • Wilinska, M. E., Chassin, L. J., Schaller, H. C., Schaupp, L., Pieber, T. R. and Hovorka, R. (2005). Insulin kinetics in type-I diabetes: Continuous and bolus delivery of rapid acting insulin. IEEE Trans. Biomed. Eng. 52 3–12.
  • Worthington, D. R. (1997). Minimal model of food absorption in the gut. Med. Informatics 22 35–45.
  • Wu, H., Zhu, H., Miao, H. and Perelson, A. S. (2008). Parameter identifiability and estimation of HIV/AIDS dynamic models. Bull. Math. Biol. 70 785–799.
  • Xue, H., Miao, H. and Wu, H. (2010). Sieve estimation of constant and time-varying coefficients in nonlinear ordinary differential equation models by considering both numerical error and measurement error. Ann. Statist. 38 2351–2387.
  • Ye, J. (1998). On measuring and correcting the effects of data mining and model selection. J. Amer. Statist. Assoc. 93 120–131.