Model Identification Using Stochastic Differential Equation Grey-Box Models in Diabetes

BackgroundThe acceptance of virtual preclinical testing of control algorithms is growing and thus also the need for robust and reliable models. Models based on ordinary differential equations (ODEs) can rarely be validated with standard statistical tools. Stochastic differential equations (SDEs) offer the possibility of building models that can be validated statistically and that are capable of predicting not only a realistic trajectory, but also the uncertainty of the prediction. In an SDE, the prediction error is split into two noise terms. This separation ensures that the errors are uncorrelated and provides the possibility to pinpoint model deficiencies.MethodsAn identifiable model of the glucoregulatory system in a type 1 diabetes mellitus (T1DM) patient is used as the basis for development of a stochastic-differential-equation-based grey-box model (SDE-GB). The parameters are estimated on clinical data from four T1DM patients. The optimal SDE-GB is determined from likelihood-ratio tests. Finally, parameter tracking is used to track the variation in the “time to peak of meal response” parameter.ResultsWe found that the transformation of the ODE model into an SDE-GB resulted in a significant improvement in the prediction and uncorrelated errors. Tracking of the “peak time of meal absorption” parameter showed that the absorption rate varied according to meal type.ConclusionsThis study shows the potential of using SDE-GBs in diabetes modeling. Improved model predictions were obtained due to the separation of the prediction error. SDE-GBs offer a solid framework for using statistical tools for model validation and model development.     

Grey-box Modelling of Pharmacokinetic /Pharmacodynamic Systems

Grey-box pharmacokinetic/pharmacodynamic (PK/PD) modelling is presented as a promising way of modelling PK/PD systems. The concept behind grey-box modelling is based on combining physiological knowledge along with information from data in the estimation of model parameters. Grey-box modelling consists of using stochastic differential equations (SDEs) where the stochastic term in the differential equations represents unknown or incorrectly modelled dynamics of the system. The methodology behind the grey-box PK/PD modelling framework for systematic model improvement is illustrated using simulated data and furthermore applied to Bergmans minimal model of glucose kinetics using clinical data from an intravenous glucose tolerance test (IVGTT). The grey-box estimates of the stochastic system noise parameters indicate that the glucose minimal model is too simple and should preferably be revised in order to describe the complicated in vivo system of insulin and glucose following an IVGTT.