307592 The Role of the Confidence Intervals in Parameter Estimation and Model Refinement for Dynamical Systems
Parameter estimation in dynamical systems has been used since the 1960s, mainly for the estimation of chemical reaction rate parameters. Recently there is a considerable interest in estimation of kinetic parameters in biological models. The computation of such parameters represents specific challenges (see, for example, Leppävuori et al, 2011) as the models often contain a large number of unknown kinetic parameters that cannot be measured. Moreover, the number of parameters that can be reliably estimated, based on available experimental data, is often smaller than the total number of parameters. To reduce the number of equations and parameters, the model is usually simplified (using, for example the pseudo steady state assumption, Ji and Luo, 2000) and some of the parameter values are a-priory are set. The presently used parameter estimation techniques (for a recent review, see Michalic et al., 2009) may converge to a local minimum. However, even if they do converge to a global minimum, the predicted values may differ significantly from the experimental data because of incorrectness of the assumptions that were associated with the model derivation and the assignment of fixed values to some of the parameters. Insufficient amount and/or low precision of available experimental data may also render the determination of statistically significant parameter values impossible.
Because of the difficulties associated with the estimation of the parameter values of the dynamic models, it is important to check the validity of the models and the parameter values using statistical metrics, in addition to ensuring convergence to the global minimum of the maximum likelihood function. Following this premise we checked the use of confidence intervals for validating the significance of the various terms in the proposed dynamic models and the accuracy of the parameter values. Our hypothesis was that superfluous terms in the model may render the corresponding parameter values not significantly different from zero. On the other hand, convergence to a local minimum, insufficient amount and/or low precision of the experimental data will yield results where all (or the great majority) of the parameters are not significantly different from zero. Six test problems (from the book of Floudas et al., 1999, models containing between two to five parameters) with known optimal parameter values were used to verify this hypothesis. The method was used to validate the results for a larger scale problem (Merchuk et al., 2013), where the model contains 12 unknown parameter values.
The algorithm developed for maximum likelihood determination of the parameter values uses the so called, "sequential approach" for parameter estimation. In an outer loop the weighted squared error between the experimental data set and the corresponding model predictions is minimized (using either the simplex-search method or the Levenberg-Marquardt algorithm (Seber and Wild, 2003). In the inner loop, a non-stiff (or stiff) integration routine is used to determine the state-variable values at time intervals where experimental data are available. Upon identifying optimal parameter values, the linearized parameter covariance matrix is used to calculate the confidence intervals (Seber and Wild, 2003).
The results of this study show that in cases where sufficient amount of experimental data are available, the confidence intervals can help to eliminate superfluous terms and parameters from the model and ensure statistical significance of the model and its parameter values. They also help to diagnose cases where more data is essential to obtain statistically significant results.
References
1. Floudas, C. A., Pardalos, P. M. et al., Handbook of Test Problems in Local and Global Optimization, Kluwer, Dordrecht, The Netherlands, 1999
2. Ji, F. and L. Luo, A hyper cycle theory of proliferation of viruses and resistance to the viruses of transgenic plant, Journal of Theoretical Biology, 2000, 204(3), 453-465.
3. Leppävuori, J. T.; Domach, M.M.; Biegler, L.T., Parameter Estimation in Batch Bioreactor Simulation Using Metabolic Models: Sequential Solution with Direct Sensitivities, Ind. Eng. Chem. Res. 2011, 50, 12080-12091
4. Merchuk, J. C.; Miron, A. S.; Asurmendi, S. and Shacham, M., Modeling TMV Proliferation in Protoplasts, submitted for publication, 2013.
5. Michalik, C.; Chachuat, B.; Marquardt, W., Incremental Global Parameter Estimation in Dynamical Systems, Ind. Eng. Chem. Res. 2009, 48, 5489–5497.
6. Seber, G.A.F, and Wild, C.J. Nonlinear Regression, Wiley-Interscience, Hoboken, NJ, 2003.
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