New methods for complex models defined by a large number of ODEs. Application to a Glucose/Insulin model
Celia Barthelemy and Marc Lavielle
INRIA Saclay and University Paris-Sud
Objectives: Modelers are increasingly faced with complex physiological models represented by a large number of ordinary differential equations (ODEs). Widely used modeling algorithms need to evaluate the structural model a large number of times, and for instance SAEM, MCMC and Monte-Carlo algorithms can be extremely time-consuming when the model is defined by a large system of ODEs. There is therefore a need for new efficient numerical tools to help the modeler deal with such complex models. We propose an extension of these algorithms that limits the total number of times the ODEs need to be solved.
Methods: This new approach consists in first evaluating the structural model on a well-defined grid of parameters. Then, the structural model is approximated by interpolating these isolated values. The number of points of the grid defines the quality of the approximation. “Exact methods” are obtained when this number tends to infinity.
The proposed method has been evaluated using simulations. Performance was assessed by computing the root mean square error (RMSE) and the computing time required for several tasks: estimation of the population and individual parameters, evaluation of the Fisher information matrix, evaluation of the log-likelihood, and creation of VPCs.
Results: We illustrate the method on simulated datasets from the glucose/insulin model proposed by Alvehag [1]. This model is composed of 29 EDOs, and 5 parameters are estimated. For a grid of 11^5 points, the elapsed time for each task is approximately divided by 7. For example, the times for the original and extended SAEM algorithms are respectively 3 minutes and 42 seconds, and the increase in the RMSE does not exceed 5%.
Conclusions: Encouraging results have been obtained with models defined by a large system of ODEs and a relatively small number of unknown parameters. In particular, the population parameters are estimated with little bias whereas the estimation is significantly faster compared to standard SAEM. This method makes feasible the use of more and more realistic physiological models.
Attempts can now be made to extend such approaches to models defined with a larger number of parameters. Application to spatial models defined by Partial Differential Equations (PDEs) could also be considered.
References:
[1] Karin Alvehag. Glucose regulation, a mathematical model (2006).
[2] Donnet and Samson. Estimation of parameters in missing data models defined by differential equations. ESAIM: Probability and Statistics (2005)
Acknowledgments: This work was supported by the DDMoRe project (www.ddmore.eu).
