(470d) Optimal Design of Experiments for Building Fundamental Models of Pharmaceutical Production Processes

Authors: 
Shahmohammadi, A., Queens University
McAuley, K. B., Queen's University
In many chemical and pharmacological systems, fundamental models play a significant role in design and optimization of industrial processes. When building a fundamental model, knowledge about chemical, physical and biological laws is often used to develop a model or propose rival models that can describe the underlying process behaviour. Usually, these models contain adjustable parameters that need to be estimated using available experimental data. Often available experimental data are not sufficiently informative for model building and validation, so modeler developers must perform additional experiments to obtain data to achieve improved parameter estimates and model predictions. Experimental runs may be expensive and time-consuming, especially if the new operating conditions lie outside of the routine operating range of the system. Therefore, experiments should be designed carefully to save time and effort while obtaining sufficient additional information.

Different experimental design strategies can be applied to obtain improved parameter estimates and model predictions Factorial and fractional-factorial experimental designs are commonly used because they are easy to implement and interpret. However, these designs are somewhat inflexible because they do not account for prior experimental data, nor for any structural information contained in the fundamental model equations. To overcome these difficulties, sequential model-based optimal design techniques can be used to select appropriate experimental settings that will lead to accurate parameter estimates and model predictions

Objective functions for sequential design may be A-optimal (aimed at reducing variance of parameter estimates), D-optimal (aimed at reducing the size of the joint confidence region for the parameters) or V-optimal (aimed at reducing uncertainty in model predictions at key operating conditions). An important computational step when performing traditional A-,D- and V-optimal design calculations is inversion of the Fisher information matrix (FIM). Unfortunately, in practical sequential design situations for fundamental models with numerous parameters, the FIM may be non-invertible. The main objective of the current study is to investigate the effectiveness of the several approaches for addressing the problem of noninvertible FIM when designing new sequential model-based experiments for pharmaceutical production processes. Proposed approaches for dealing with this problem include: i) a “leave out” approach where problematic parameters are identified and left out of the design problem (LO approach), ii) a pseudo-inverse approach where the Moore Penrose pseudo inverse is used (PI approach) and iii) an approach where modelers select new experiments based on their prior knowledge without using a formal experimental design technique (MS approach). For the MS approach, the new experimental settings are chosen randomly to be either the lower or the upper bounds for the corresponding decision variables.

This study focuses on sequential A-optimal design because the A-optimality criterion is relatively easy for modelers to interpret and is the most readily usable design criterion for situations where the FIM is noninvertible. Two case studies are considered. The first case study is a simple linear example model with 7 parameters. In this example, old experimental results are available that result in perfectly correlated parameters, which makes the FIM noninvertible. Additional sequential experiments are designed to obtain parameter estimates with the lowest total variance. The second case study involves a Michael addition reaction used by a pharmaceutical company. Old experimental data are available at settings that result in a noninvertible FIM. New A-optimal sequential experiments are designed to select the recipes for one or two additional batch experiments that will lead to the best possible parameter estimates. The investigations are carried out using Monte Carlo simulations. Simulated datasets with random noise are generated for both case studies. The effectiveness of the LO, PI and MS approaches are compared using an A-optimal criterion.

The results for the two case studies show that the LO approach is better on average for achieving parameter estimates and model prediction that are closer to their true values. In addition, the results illustrate that, even with a badly conditioned or non-invertible FIM, it is advantageous to design sequential experiments using the LO approach rather than arbitrarily choosing new experimental settings at the bounds of the operating range. Designing more than one new experiment improves the quality of the parameter estimates. Also, if the number of sequential experiments is large enough, the FIM can become invertible so that the LO and PI approaches give the same results.