(595a) In-Silico Design of Experiments As a Tool for Nonlinear Sensitivity Analysis of Detailed Models

Mathematical modelling
provides a systematic way to study quantitatively the behaviour of complex
systems composed of numerous components and some of which might be described at
different levels of detail. Engineers and scientists who postulate such  models
invariably chose to include more rather than less detail is their description
of the physical, chemical or/and biological system of interest. The inclusion
of more and more details in such models increases the number of parameters that
need be estimated from experimental data. The estimation of an increased number
of parameters increases the number of experimental data needed. However, even
if this requirement is met, the estimation of parameters associated with
insignificant model details is extremely difficult because of their minimal
impact on the model predictions being matched to the experimental data.  This
difficulty cannot be easily removed because we do not know a priori
which model components are insignificant in the region of present interest.
Even in the case where the values of the model parameters are known, it is of
interest to investigate the sensitivity of the model predictions on the values
of the input variables of the process. Exhaustive simulations, especially of
very complex models, do not provide an effective answer to the problem because
of the computational cost.  Knowing which parameters and to what extent
have a significant effect on the model output allows for their accurate
estimation through tailor-made experiments (Kiparissides et al 2011).
Parameters with negligible sensitivity, quantified by a Sensitivity Index (SI),
can be fixed at their literature or their initially assumed value as they do
not affect the model predictions

Existing model sensitivity methods
are commonly classified in three broad categories: screening, local and global
methods (Saltelli, 2000). For the general case of non-linear ODE models,
commonly met in engineering applications, global methods and variance based
global methods in particular have the edge over their counterparts as discussed
previously (Chan et al 1997, Saltelli et al 2000, Kiparissides et al 2009). Briefly, the ability to estimate higher-level indices, which quantify the
effect on the output of parameter-parameter interactions, gives global methods
the edge over their local counterparts. Alas, global methods usually require
the estimation of a large, often prohibitive, number of model evaluations.

Here we propose to
use, in a different context, a methodology developed more than fifty years ago.
It is called ?Design of Experiments? and has become a classic experimental tool
in understanding how different operating conditions affect the performance of a
process (see for example Box and Draper (1987)). Herein, we will apply this technique
on the mathematical model of the process instead of on an experimental set-up, hence
the use of the term in silico. Similar to its original use in extracting
the maximum amount of information from the process with the minimum number of
properly designed experiments, we will use it here so that we can obtain the maximum
information on the sensitivity of the model to its parameters with the minimum number
of simulations of the full model. This parallel use of the design of
experiments concepts and tools has also been used to understand the operability
characteristics of continuous chemical processes by Li and Georgakis (2010),
where it was called design of selective calculations. 

We utilise a DAE
model that describes in substantial detail hybridoma cell growth and proliferation
in batch cultures (Bang and Jarford 2000). We show that the use of DoE for the
generation of a response surface model (RSM) approximation of the full model can
lead to the same quality of sensitivity information as the established variance
based GSA methods at a significantly lower computational cost. We show that the
explicit information in the RSM model provides the needed sensitivity
information with respect to the parameters of the detailed model. Besides their
relative absolute magnitude, the RSM model parameters provide additional
information that previous methods did not avail. The results of the proposed
approach are in full agreement with those of the Sobol' method, one of the most
widely used previously, and this is done at a fraction of the computational
cost.

The proposed
approach offers directional and interactional information about the full model at
a minimal computational cost. Moreover, we show that by estimating an approximate
RSM model, minimal information is lost in terms of parameter significance and sensitivity.
This is highlighted by comparing the results of the Sobol' method when applied
to the full model to the results of the Sobol' method when applied to the estimated
RSM model.

References:

1. Box G.E.P., Draper N.R. (1987) Empirical Model Building and Response Surfaces. Wiley Press.
2. Chan K., Saltelli A., Tarantola S. (1997) Sensitivity Analysis of the model output: variance-based methods make the difference. Proceedings of the 1997 Winter Simulation Conference.
3. Jang J.D., Barford J.P. (2000) An unstructured kinetic model of macromolecular metabolism in batch and fed-batch cultures of hybridoma cells producing monoclonal antibodies. Biochem. Eng. Journal, 4, 153-168.
4. Kiparissides A., Kucherenko S., Mantalaris A., Pisitikopoulos E.N. (2009) Global Sensitivity Analysis Challenges in biological systems modeling. Ind. Eng. Chem. Res., 48 (15), pp 7168?7180.
5. Kiparissides A., Koutinas M., Kontoravdi C., Mantalaris A., Pistikopoulos E.N. (2011) ?Closing the loop? in biological systems modeling ? From the in silico to the in vitro. Automatica, doi:10.1016/j.automatica.2011.01.013.
6. Li, L., Georgakis, C. (2010), ?On the Calculation of Operability Sets of Nonlinear High-Dimensional Processes?  Ind. & Eng. Chem. Res. 49, (17) pp.8035-8047
7. Saltelli A., Chan K., Scott E.M. (2000) Sensitivity Analysis, Wiley Press.