(259b) Optimal Time Sampling Strategy in Pharmaceutical Reactions for the Estimation of Accurate Drsm Models

Through the available robotic experimental devices, one is able to perform multiple experiments and sample the reaction mixture at multiple time instants. Such time resolved data have been successfully modeled through the recently introduced Dynamic Response Surface Methodology (DRSM). The development of the DRSM method has gone through phases, from the initial approach where the model parameters were estimated with linear stepwise regression¹, to the more complete and accurate method incorporating an exponential transformation of time², constrained regression³ and non-linear parameter estimation of one of the critical parameters.

This presentation addresses the optimal time instants to take compositional samples that lead to the most accurate model. We first analyze two industrial practices. The most traditional approach is to take samples equidistantly in time. A second approach is to double the sampling interval as time increases, since most of the compositional changes happen at initial times.

We propose a novel sampling strategy, which is equidistant in an exponentially transformed time, denoted by θ. The proposed approach follows two steps. First, one estimates the time constant, t_c, characterizing how fast or slow the evolution of the chemical transformations will be represented by the DRSM model. This can be estimated from prior process knowledge, from historical data, or through a single new experiment. Second, depending on the number of measurements one is willing to take, the proposed equidistant in θ strategy is translated into the sampling times in through the relationship.

We will show that this strategy is the most desirable one with respect to the following interrelated aspects: (1) it makes the non-diagonal elements of the Fisher information matrix as small as possible and reduces the cross-correlation between model parameters; (2) it maximizes the determinant of the Fisher information matrix; and (3) it reduces the prediction interval of the estimated responses. Another approach for determining the sampling times is to define and solve a set of non-linear equations requiring that the non-diagonal entries of the Fisher information matrix are zero. Nevertheless, solving such a problem can be time-consuming when the number of samples is large and offers only small incremental improvements in the accuracy of the model. Because all versions of the DRSM modeling approach use shifted Legendre polynomials, we will also demonstrate their advantage over the simple power series expansions, in t or θ, in providing superior correlation properties and higher accuracy models.

Reference:

1. Klebanov N, Georgakis C. Dynamic Response Surface Models: A Data-Driven Approach for the Analysis of Time-Varying Process Outputs. Ind Eng Chem Res. 2016;55(14):4022-4034.
2. Wang Z, Georgakis C. New Dynamic Response Surface Methodology for Modeling Nonlinear Processes over Semi-infinite Time Horizons. Ind Eng Chem Res. 2017;56(38):10770-10782.
3. Dong Y, Georgakis C, Mustakis J, et al. A Constrained Version of the Dynamic Response Surface Methodology. Ind Eng Chem Res. 2019;(submitted).