(164d) Modeling and Optimization of Agglomeration in Suspension through a Coupled Population Balance Framework

Since its introduction by Randolph and Larson¹ (1971) and the solution methods described by Ramkrishna² (2000), the population balance model (PBM) has been widely used and accepted as model formulation method for simulation and prediction of the size distribution and other properties of particulate systems. PBMs allow for systems that include any or all of the following mechanisms: nucleation, growth, breakage and agglomeration. Following the initial work by Smoluchowski³ (1917) on the rate of aggregation for spherical particles, there have many contributions for systems that exhibit agglomeration including dispersion (bubble) coalescence^4,5, granulation^6,7 and particle aggregation during crystallization^8,9,10. The commonality in the limitations of many of the previous studies were physically irrelevant and/or empirically based agglomeration kernels, difficulty in assessing the influence of process conditions (e.g. hydrodynamics, particulate physical properties), solution method efficiency for optimization and control applications, and loss of information of constituent particles. These limitations present obstacles for the simulation, optimization, and control of the increasingly popular agglomeration technique of spherical crystallization.

Peña and Nagy¹¹ studied and showed the benefits of spherical crystallization as a process intensification technique whereby both internal (primary crystals) and external (agglomerates) properties can be controlled experimentally through a decoupled continuous spherical crystallization (CSC) approach; providing the means by which both biopharmaceutical (bioavailability, dissolution) and manufacturing (flowability, filtration, drying) properties can be simultaneously adapted to meet desired quality specifications. This technique opens the door for combined experimental and modeling approaches for the optimization and control of both the primary crystal and agglomerate properties in spherical crystallization processes. However, many of the PBMs currently in literature would fail to accomplish this because of the aforementioned limitations and loss of information.

To overcome the issues of loss information a coupled PBM formulation is required. A coupled PBM formulation could simultaneously track the evolution in the primary crystals and the evolution of the agglomerates. The relationship between primary crystal properties and their effect on final agglomerate properties would thereby be more evident and more efficient than traditional approaches. To the best of our knowledge, the only previous work that presented this approach is that of Ochsenbein et al.¹². In their study, Ochsenbein et al.¹² focused on the agglomeration of needle-like crystals in suspension. Through a coupled PBM framework Ochsenbein et al.¹² were able to develop a population balance equation (PBE) to describe the evolution of the primary crystals by a two-dimensional growth rate to represent the needle like structure of the crystal. They then used another PBE to describe the evolution of the agglomerates as a function of the primary crystals. For the agglomeration kernel, they derived their own modified kernel that include both characteristic lengths of the primary crystals participating in the agglomeration. The new PBM formulation also allowed for the development of new parameters that add value to the simulations due to their experimental relevance. However, the work of Ochsenbein et al.¹² neglected nucleation something common to previously developed agglomeration models. The coupled population balance model framework will be extended herein.

The contribution of this work is the extension of the coupled PBM framework for application in the simulation and optimization of a spherical crystallization system. A coupled PBM framework has been developed for a semi-batch, reverse addition, anti-solvent crystallization system with agglomeration. The system includes nucleation and growth the primary crystals and subsequent agglomeration. The purpose of the work is to exploit the advantages presented by a coupled PBM framework; for example, the ability to optimize for specific primary and agglomerate sizes. This presents an opportunity to find optimal operating conditions that meet both bioavailability and manufacturability demands. It also allows for the ability to develop first principles based parameters for agglomeration efficiency and porosity. Additionally, through the retention of the information of the primary particles the interplay between the effects of operating conditions on the properties of the primary crystals versus the agglomerates will be clear.

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