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Mathematical Medicine and Biology Advance Access originally published online on May 25, 2008
Mathematical Medicine and Biology 2008 25(2):141-155; doi:10.1093/imammb/dqn011
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Maximum likelihood estimation of a time-inhomogeneous stochastic differential model of glucose dynamics

Umberto Picchini{dagger}

Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark and Biomathematics Laboratory, IASI-CNR, Largo A. Gemelli 8, 00168 Rome, Italy

Susanne Ditlevsen{ddagger}

Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark

Andrea De Gaetano§

Biomathematics Laboratory, IASI-CNR, Largo A. Gemelli 8, 00168 Rome, Italy

{dagger} Email: umberto{at}math.ku.dk

{ddagger} Email: susanne{at}math.ku.dk

§ Email: andrea.degaetano{at}gmx.net

Received on November 16, 2007. Revised on March 5, 2008. Accepted on April 1, 2008.

Stochastic differential equations (SDEs) are assuming an important role in the definition of dynamical models allowing for explanation of internal variability (stochastic noise). SDE models are well established in many fields, such as investment finance, population dynamics, polymer dynamics, hydrology and neuronal models. The metabolism of glucose and insulin has not yet received much attention from SDE modellers, except from a few recent contributions, because of methodological and implementation difficulties in estimating SDE parameters. Here, we propose a new SDE model for the dynamics of glycemia during a euglycemic hyperinsulinemic clamp experiment, introducing system noise in tissue glucose uptake and apply for its estimation a closed-form Hermite expansion of the transition densities of the solution process. The present work estimates the new model parameters using a computationally efficient approximate maximum likelihood approach. By comparison with other currently used methods, the estimation process is very fast, obviating the need to use clusters or expensive mainframes to obtain the quick answers needed for everyday iterative modelling. Furthermore, it can introduce the demonstrably essential concept of system noise in this branch of physiological modelling.

Keywords: stochastic differential equations; dynamical models; non-autonomous differential equations; system noise; parameter estimation; closed-form transition density expansion; Hermite expansion; insulin; euglycemic hyperinsulinemic clamp


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