(200d) 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.