257668 Co-Current Parameter Estimation and Model Refinement in 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 challenge (as pointed out, for example by Leppävuori et al, 2011) as the models often contain a large number of unknown kinetic parameters that cannot be measured and 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 parameters are assigned a constant value a-priory. 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 constant, fixed values to some of the parameters.
To alleviate these difficulties we have developed an interactive tool that incorporates the human investigator in the parameter estimation and model refinement loop. This tool is basically a program which 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. In the inner loop an integration routine is used to determine the state variable values at time intervals where experimental data are available. The user can select the parameters that remain fixed and the ones that can be changed by the optimization algorithm. After an optimization run he obtains the list of the resultant parameter values, the final sum of squares of errors and plot of predicted versus experimental values of the state variables. The differences between the shapes of the predicted and experimental curves of the state variables can be used as the basis for refining the model and/or assigning a better initial estimate to some of the parameters.
The proposed interactive parameter estimation program has been implemented in POLYMATH 7.0 (POLYMATH is a product of Polymath Software, http://www.polymath-software.com ). The Levenberg-Marquardt algorithm is used for minimization of the objective function and several non-stiff and stiff integration algorithms are available for integrating the model equations. The user may select more relaxed error tolerances during the early stages of the model refinement and the parameter search process and he can tighten the tolerances when getting closer to the optimal solution.
In the extended abstract and the presentation the proposed method will be demonstrated by identifying the parameters of a model representing the dynamics of the TMV (Tobacco Mosaic Virus) replication inside a protoplast. The importance of the visual feedback for comparing experimental and predicted curves for model refinement and parameter identification will be emphasized.
1. 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.
2. 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
3. Michalik, C.; Chachuat, B.; Marquardt, W., Incremental Global Parameter Estimation in Dynamical Systems, Ind. Eng. Chem. Res. 2009, 48, 5489–5497.
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