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.