(34c) A Constrained Version of the Dynamic Response Surface Methodology for Challenging Time-Resolved Pharmaceutical Reaction Data

The newly proposed Design of Dynamic Experiments (DoDE)¹ and the Dynamic Response Surface Methodology (DRSM)² are two generalizations of Design of Experiments (DoE) and Response Surface Methodology (RSM). DoDE and DRSM methodologies have the advantages of allowing time-varying inputs, accommodating time-resolved output measurements, and modeling their interrelationship. Here, we apply the DRSM-2 algorithm³ to some challenging example cases and further introduce knowledge driven constraints on the expected shape of the predictions. This constrained DRSM-2 algorithm consistently provides a better representation of the time evolution of species concentrations during reactions. We test the updated approach against a complex simulated pharmaceutical reaction network consisting of eight reactions and involving ten species. We also develop accurate DRSM models for two experimental data sets from the two collaborating companies.

The initial DRSM-1 approach² leads to oscillatory behavior in the modeling of some concentration profiles, especially for experiments with non-constant sampling intervals or for intermediate species whose final concentrations are comparable to the level of noise. To address these issues, the DRSM-2 approach uses an exponential transformation of time as the independent variable which results in more parsimonious models. This eliminates the occurrence of any oscillatory behavior on the prediction when none is expected, and it provides much more accurate models for the case of non-equidistant sampling strategies. The constrained improvements introduced here include: fixing the initial concentrations when they are known, prohibiting the model from temporarily predicting negative concentration for species that start at a zero value, and enforcing that all concentration predictions must be non-negative. We conclude with a demonstration of the effectiveness of the constrained DRSM-2 algorithm in cases that a substantial fraction of the concentration data are missing, but still leading to an accurate DRSM model.

References

1. Georgakis C. Design of Dynamic Experiments: A Data-Driven Methodology for the Optimization of Time-Varying Processes. Industrial & Engineering Chemistry Research. 2013;52(35):12369-12382.
2. 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.
3. Wang ZY, Georgakis C. New Dynamic Response Surface Methodology for Modeling Nonlinear Processes over Semi-infinite Time Horizons. Industrial & Engineering Chemistry Research. 2017;56(38):10770-10782.