(202e) 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.

References

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.