Probabilistic modeling of an injectable aqueous crystalline suspension using influence networks

Probabilistic modeling using influence networks is an efficient, intuitive, and easy to communicate strategy in the development of complex pharmaceutical products. This study was aimed to use a risk-based approach to explore the complex interactions between product and process design parameters affecting size and shape of the particles in injectable aqueous crystalline suspensions (ACS). Based on a risk assessment, a design of experiments (DOE) was applied to evaluate the most important parameters, i.e., four critical material attributes and two critical process parameters. A model hydrophobic drug (carbamazepine) was milled and homogenized in a multistep process (dispersion and milling steps). The final formulations were characterized with automated at-line image analysis of thousands of individual particles. The particle size and shape distributions were summarized with descriptive parameters, and the relationship of these parameters and the DOE was modeled using influence networks (INs). This approach was compared and contrasted with a classical modeling approach based on multivariate linear regression (MVLR). INs had a superior visual interpretation capability of the complex and multivariate ACS system making the risk-based decision making more accessible. The probability and causality were included in the IN, i.e., the relationships between size and shape. Moreover, IN allowed to incorporate prior knowledge in a systematic way by implementing a 'black and white list'. An IN based model was created with the following model performance: a mean absolute percentage error of 1.7% and 1.1% for the size and 6.2% and 5.0% for the shape, respectively for dispersed and milled ACS. Parameters with the highest and lowest probability to control the critical quality attributes of ACS could be identified. Consequently, the parameter settings giving the optimum particle size and shape could be predicted using a Monte Carlo simulation to calculate the probability of success including the uncertainty of the model. The cubic MVLR model for the size of milled ACS was comparable to the IN in terms of the mean absolute percentage error, i.e., 1.1%. However, IN was more efficient in visualizing the complex and multivariate data set, including all the critical quality attributes and formulation/process parameters of the ACS at the same time. Moreover, the prior knowledge used in probabilistic modeling of IN could be systematically documented.