(448a) A High-Performance Numerical Library for Solving Population Balance Equations for Crystallization Processes

Due to the widespread implementation of Process Analytical Technology, the process-product design in the pharmaceutical industry has gradually shifted from the factorial experimental design to model-based design (MBD) for robust and rapid process development. The benefits of MBD include accelerating the development cycle, enhancing process reliability, efficient risk management, and determining recipes for robust optimal control. Moreover, MBD provides a common language for engineers to communicate the design concept while dealing with complex systems such as crystallization processes. Crystallization plays a vital role in the pharmaceutical industry since its practical design can reduce the burden of downstream processes. However, developing a reliable model for MBD is always a challenging task. Although the population balance equations (PBEs) had been developed to describe the evolution of particle properties in crystallization, the complexity of model structure and the uncertainty and sloppiness of parameters often cause severe issues for implementing MBD^1,2. The most common approach to resolving this issue is to utilize a proper workflow for model discrimination, parameter identifiability, and reducing model uncertainty^3,4. However, this systematic approach usually requires many iterations of solving PBEs. Furthermore, the computational expense becomes significant when the model structure becomes complicated, or a generalized numerical solution is required to encompass different scenarios.

The central idea of this work is to reduce the computational expense of solving PBEs by developing a C++ dynamic shared library known as OpenPBM with application potential for cooling, antisolvent, and combined cooling-antisolvent crystallization processes. The numerical algorithms within the library for solving PBEs stem from the high-resolution finite volume method (HRFVM)⁵, which can deal with various mechanisms like primary and secondary nucleation, growth, dissolution, agglomeration, and breakage. An optimal mesh adaption strategy⁶ is applied to streamline and relocate the grid point in the simulation to retain the maximum resolution by avoiding numerical diffusion. In order to further reduce the overhead of each iteration and ensure the positivity of the solution, an adaptive time step scheme is used by considering the Courant–Friedrichs–Lewy conditions that consider both growth rate and change of number density due to the agglomeration or breakage. Further, a parallel computing structure is used to improve the hardware's central processing unit scheduling.

This work divides the discussions of the library's numerical efficiency and accuracy into several parts. First, the computational performance between variational mesh adaption schemes is examined through case studies of crystallization processes by changing the monitor functions that evaluate the arduousness of numerical approximation on the physical domain or the numerical algorithms for the iterations. Second, through different case studies, the numerical accuracy of the library is compared with other numerical techniques, such as the standard method of moments and the quadrature method of moments. Last, the library's run time and relative errors under different scenarios are compared with the HRFVM with linear grids to demonstrate the benefits of using this library for model-based design in crystallization processes.

Acknowledgement:

Funding from Takeda Pharmaceuticals International Co. is gratefully acknowledged.

References:

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