Modeling a Reactive Crystallization with a Reduced Population Balance Equation for Dissolution and Growth to Control Critical Quality Attributes
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Modeling has always been a powerful weapon in a chemical engineer’s arsenal. Use of modeling should be considered a key piece in the application of Quality by Design (QbD) in drug substance and drug product development. In this study, reaction, dissolution and growth kinetics are determined for a reactive crystallization of an intermediate-grade drug substance (DS) currently in development at Cephalon, Inc. Initial data showed that decreasing particle size of the starting material (in the DS formation step) increased the levels of entrapped starting material in the DS product. Smaller particles of starting material dissolve faster enhancing the rate of product formation. This leads to product coating the remaining starting material, thus increasing the level of entrapped starting material. The level of starting material in the DS is a critical quality attribute (CQA). Also, decreasing the particle size of the product led to lower levels of starting material entrapment. Breakage exposes the interior starting material to additional reagent. DS particle size is a potential CQA. The goal of this work is to model and hence predict entrapment levels and particle size as a function of dissolution, reaction, and crystallization kinetics. In the solution phase, a Mannich reaction occurs which is modeled as two second order reversible reactions in series. The model equations to describe this reactive crystallization consist of a solution phase mass balance for the reaction kinetics coupled with two population balance equations (PBEs). The PBEs describe the particle size densities for the nearly insoluble starting material and the nearly insoluble DS product during dissolution and precipitation. Due to the low solubility of the starting material and the DS product, we assume the solution phase concentration is always saturated with respect to these two materials. Therefore, a reduced fast dissolution and high growth model was developed to describe this process. The model in this study is a new model for reactive crystallizations because of the reduced nature and the ability to describe multiple crystallization phenomena and two second order reversible reactions in series with minimal parameters. Using the modeling software package Octave, the model parameters (reaction, nucleation and nuclei-disappearance rate constants) were estimated by fitting model predictions to experimental measurements including particle number (via Mettler’s Lasentec FBRM probe), reactor temperature, and percent entrapped starting material in DS. This type of modeling approach is a macroscopic modeling approach for QbD and can be superior to a factorial or central-composite design approach because it can offer physical insights. Such insights include knowledge of the reaction and crystallization rate constants and the ability to extrapolate beyond previously studied experimental conditions.