(302c) Adventures with and Insights Into the Bayesian Design of Experiments In Complex Polymerizations

For many polymerizations, experimental data may be available either from industrial or exploratory laboratory work. In addition, mathematical models usually do exist, albeit often with unreliable and/or highly correlated parameters, and sometimes even unverified mechanistic bases. In complex polymerization systems, these problems are even more prominent. Hence, ideas from the (statistical) design of experiments applied as early as possible can be very beneficial for the clarification of polymerization kinetics. Classical experimental design methods (for example, (fractional) factorial designs) have been used extensively and are useful in optimizing a wide variety of systems. However, these designs usually ignore the prior knowledge available and are often limited when the system digresses from linear behavior. Using more efficient experimental designs, which can accommodate both prior information and nonlinearities, could (hopefully) lead to optimal performance in fewer trials. 

Bayesian design is a powerful and largely unstudied (in the polymerization area) experimental design methodology, which can accommodate practical limitations encountered in classical (fractional) factorial designs. In addition, Bayesian design allows the use of a nonlinear (fully mechanistic) model along with experimental information (hence, it is essentially an optimal model-based design of experiments). The approach can shed light on the most uncertain parts of our process understanding, identify the least reliable (less well known) parameters (e.g., uncertain values of kinetic rate constants), and further guide sensitivity analysis studies focusing on key uncertain parameters in one’s model. 

The Bayesian design approach is an experimental design technique which has many advantages over standard experimental designs. The approach incorporates prior knowledge about the process (or product) into the design to suggest a set of future experiments in an optimal, sequential and iterative fashion. Since for many complex polymerizations prior information is available, either in the form of experimental data or mathematical models, use of a Bayesian design methodology could be highly beneficial. Hence, exploiting this technique could hopefully lead to optimal performance in fewer trials, thus saving time and money. Advantages of the Bayesian design approach will be illustrated via case studies drawn from different complex polymerization systems. However, since this technique is perfectly general, and exploiting the duality between many systems in science and engineering, it can potentially be applied to other chemical engineering processes. 

Why Bayesian design? As mentioned above, standard experimental design methods (for example, (fractional) factorial designs) have been employed extensively and are useful in optimizing a wide variety of systems. However, these designs usually suffer from several limitations and cannot handle certain situations, as listed in Table 1. In addition, these approaches do not take direct advantage of the considerable prior knowledge that is available about the reaction system to design experiments. As prior information is already available within existing data, it is logical that it should be used in order to contribute to the optimality of the designed experiments, and hence to improved models and performance of the process in question. Using more efficient experimental designs, like the family of Bayesian designs, can accommodate these restrictions and could lead to optimal performance in fewer trials. 

Table 1. Typical Limitations in Standard Designs

The Bayesian experimental design is based on Bayes’ theorem which is well established, especially among statisticians. Recent applications more relevant to chemical and process engineering include heat transfer in packed beds [1], emulsion terpolymerization kinetics [2], drug and cell transport kinetics [3], particle size distribution studies in suspension polymerization [4], pharmaceutical kinetics [5], and analyses of catalytic systems [6]. The common characteristic of all these Bayesian applications is that they are concerned with, admittedly very important, (multi)parameter estimation questions (of kinetic rate constants), but not with the issue of the design of experiments (with the exception of [2], [4], and [6]). In other words, looking at the last 15 years or so, whereby many advances have taken place in polymerization, Bayesian design of experiments has not been exploited extensively nor frequently in complex polymerization systems, which could benefit tremendously from its important traits. 

What could one gain from the Bayesian design? In our presentation, principles and capabilities/benefits of the Bayesian design approach will be illustrated and its superiority to the currently practised (standard) design of experiments will be presented via case studies drawn from representative complex polymerization processes. These include examples from nitroxide-mediated controlled radical polymerization (both bimolecular and unimolecular), emulsion copolymerization of acrylonitrile/butadiene (nitrile or NBR) and styrene/butadiene (SBR) rubber, and crosslinking copolymerizations (of styrene and divinyl benzene under controlled radical polymerization conditions). All these case studies address important, yet practical, issues in not only the study of polymerization kinetics but also, in general, in process engineering and improvement. A preview of these important process issues and, accordingly, topics that can be handled efficiently by the Bayesian design approach are listed in Table 2.

Table 2. Overview of Gains in Bayesian Design Approach

In addition, the analysis is amenable to a series of statistical diagnostic tests that one can carry out in parallel. These diagnostic tests serve to quantify the relative importance of the parameters (intimately related to the significance of the estimated factor effects) and their interactions, as well as the quality of prior knowledge (in other words, the adequacy of the model or the expert’s opinions used to generate the prior information, as the case might be). 

The case studies will clearly show that what is novel in the Bayesian approach is the simplicity and the natural way with which it follows the logic of sequential model building paradigm, taking full advantage of the researcher’s expertise and information (knowledge about the process or product) prior to the design, and invoking enhanced information content measures (the Fisher Information matrix is maximized, which corresponds to minimizing the variances and reducing the 95% joint confidence regions, hence improving the precision of the parameter estimates). 

1. Duran, M. A.; White, B. S.  Chem. Eng.  Sci. 1995, 50, 495.
2. Dube, M. A.; Penlidis, A.; Reilly, P. M.  J. Polym. Sci. Part A-Polym. Chem. 1996, 34, 811.
3. Murphy, E. F.; Gilmour, S. G.; Crabbe, M. J. C. Febs Letters 2004, 556, 193.
4. Vivaldo-Lima, E.; Penlidis, A.; Wood, P. E.; Hamielec, A. E. J. Appl. Polym. Sci. 2006, 102, 5577.
5. Hermanto, M. W.; Kee, N. C.; Tan, R. B. H.; Chiu, M. S.; Braatz, R. D. AIChE J. 2008, 54, 3248.
6. Hsu, S. H.; Stamatis, S. D.; Caruthers, J. M.; Delgass, W. N.; Venkatasubramanian, V.; Blau, G. E.; Lasinski, M.; Orcun, S. Ind. Eng. Chem. Res. 2009, 48, 4768.