(218d) A Simple Method for Incorporating Prior Parameter Information during Model-Based D-Optimal Design of Experiments in Pharmaceutical Production

Mathematical models are important for process design, control, and optimization for chemical, biological and pharmaceutical systems.¹ These fundamental models often contain many unknown parameters, such as kinetic rate constants and activation energies, that require estimation from experimental data. Typically, only limited data are available for parameter estimation, making it difficult to estimate all the unknown parameters reliably.² When modelers wish to obtain more-accurate estimates for model parameters, they may choose to perform new experiments to obtain additional information. Performing new experiments can be expensive and time-consuming, so researchers want to carefully select operating conditions so that as much information as possible is obtained from the new experimental runs. Model-based design of experiments (MBDoE) is an efficient way to plan experiments that lead to accurate parameter estimates and improved reliability of the model.³

For nonlinear models with many unknown parameters, computation of MBDoE objective functions is a challenge due to a singular Fisher information matrix (FIM).^4,5,6 Several methods have been developed to overcome the problem of a singular FIM, such as parameter-subset selection⁴ and Bayesian approaches.⁵ Unfortunately, closed-form objective functions for Bayesian MBDoE are difficult to obtain.⁷ Therefore, sampling-based approaches, such as Markov chain Monte Carlo (MCMC) methods, are typically used for Bayesian MBDoE.^8,9 However, these methods tend to be complex to understand and require excessive computation for large models, such as many chemical and pharmaceutical process models.⁸

In this study, we propose a simplified linearization-based Bayesian D-optimal objective function for MBDoE in multi-response nonlinear models. The proposed objective function will be useful for modelers who want to design new experiments to obtain better parameter estimates for nonlinear pharmaceutical process models with many unknown parameters. We also developed a new method to test the effectiveness of experimental design methods for nonlinear models in a D-optimal sense. This new method relies on the computation of the volumes of minimum volume ellipsoids (MVEs) that are centered at the true parameter values. The proposed MVE methodology will be helpful to future researchers who want to compare the effectiveness of different experimental design methods aimed at achieving accurate parameter estimates and model predictions used for mapping of permissible operating regions.

A pharmaceutical case study is used to illustrate and test the proposed Bayesian D-optimal objective function for designing new experiments. This case study involves a nonlinear dynamic model of a batch reactor with Michaelis-Menten kinetics, which is used for the production of a pharmaceutical product.¹⁰ Initially, only four replicate runs are available to estimate the 14 unknown parameters (kinetic rate constants and activation energies). The proposed simple Bayesian D-optimal objective function is used to design two new experiments in a variety of ways: two experiments one-at-a-time and two experiments at-the-same-time. We confirm that, as new D-optimal experiments are included in parameter estimation, the MVE of the resulting parameter estimates gets smaller. We also use a similar Bayesian A-optimal objective function, developed in a previous study⁵, to design new A-optimal experiments. We show that estimates from D-optimal experiments have a much smaller MVE than corresponding estimates from A-optimal experiments or randomly-selected corner-point experiments. As a result, we confirm that the proposed Bayesian D-optimal objective function is effective in a D-optimal sense. We anticipate that the proposed Bayesian D-optimal methodology will be useful to a wide variety of modelers who develop pharmaceutical models with a large number of parameters, because of its computational simplicity relative to MCMC-based methods. When planning new experiments, future modelers will be able to readily account for prior beliefs about plausible parameter values, the structure of their model equations, and old data that may be available.

References

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