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Most pharmaceutical manufacturing processes include a series of crystallization processes where their product quality is often associated with the crystal final form (such as crystal habit, shape and size distribution). Recently, there is a rapid growth of interest in polymorphism (when a substance has multiple crystal forms) motivated by patent, operability, profitability, regulatory, and scientific considerations [1-3]. The different polymorphs can have orders-of-magnitude differences in properties such as solubility, chemical reactivity, and dissolution rate, which can have an adverse effect on downstream operability and performance of the crystal product [2]. As a result, controlling polymorphism to ensure consistent production of the desired polymorph is very critical in those industries, including in drug manufacturing where safety is of paramount importance.

This presentation considers the crystallization of L-Glutamic acid which has two polymorphic forms (the metastable alpha-form and the stable beta-form). A mathematical model based on population balance equations [4] is developed along with the fundamental equations describing the nucleation and growth kinetics for both polymorphs. Compared to past models for L-glutamic acid crystallization [5,6], this appears to be the first to estimate activation energies for the nucleation and growth processes, and the first to quantify the uncertainties in the kinetic parameters. Bayesian inference is used to determine the kinetic parameters from experimental data which includes on-line temperature measurement, in-situ concentration measurement by Attenuated Total Reflection Fourier Transform Infrared (ATR-FTIR) spectroscopy, and in-situ Chord Length Distribution (CLD) measurement by Focused Beam Reflectance Measurement (FBRM). While weighted least-squares methods are adequate for many problems, Bayesian inference is able to include prior knowledge in the statistical analysis which can produce models with higher predictive capability. Bayesian estimation produces a posterior distribution for the estimated parameters, which can be incorporated into robust control strategies for crystallization process [7].

Numerical simulation for polymorphic crystallizations enables the investigation of the effects of various operating conditions and can be used for optimal design and control [6,8,9]. Solving population balance equations is particularly challenging when the PDEs are hyperbolic with sharp gradients or discontinuities in the distribution [10]. Standard first-order methods require a very small grid size in order to reduce the numerical diffusion (i.e., smearing), whereas standard higher order methods introduce numerical dispersion (i.e., spurious oscillations), which usually results in a crystal size distribution with negative values. Efficient and sufficiently accurate computational methods for simulating the population balance equations are required to ensure the behaviour of the numerical solution is determined by the assumed physical principles and not by the chosen numerical method. This presentation considers a class of numerical algorithms known as weighted essentially non-oscillatory (WENO) methods [11-14] which were developed for especially accurate simulation of shock waves and provide much higher order accuracy than the previously considered methods for solving PBMs. These WENO methods are compared to the high resolution (HR) finite volume method [15] and a second-order finite difference (FD2) method, for the crystallization of L-glutamic acid polymorphs under conditions in which the distribution has sharp gradients.

References

[1] Fujiwara M, Nagy ZK, Chew JW, Braatz RD. First-principles and direct design approaches for the control of pharmaceutical crystallization. J Process Control 2005; 15:493-504.

[2] Blagden N, Davey R. Polymorphs take shape. Chem. Brit. 1999; 35:44-47.

[3] Brittain HG. The impact of polymorphism on drug development: A regulatory viewpoint. Amer. Pharm. Rev. 2000; 3:67-70.

[4] Randolph AD, Larson MA. Theory of particulate processes: analysis and techniques of continuous crystallization. San Diego: Academic Press. 1988.

[5] Ono T, Kramer HJM, Ter Horst JH, Jansens PJ. Process modeling of the polymorphic transformation of L-glutamic acid. Crystal Growth & Design 2004; 4(6):1161-1167.

[6] Scholl J, Bonalumi D, Vicum L, Mazzotti M. In situ monitoring and modeling of the solvent-mediated polymorphic transformation of L-Glutamic acid. Crystal Growth & Design 2006; 6(4):881-891.

[7] Nagy ZK, Braatz RD. Worst-case and distributional robustness analysis of finite-time control trajectories for nonlinear distributed parameter systems. IEEE Trans. Control Systems Technology 2003; 11(5):694-704.

[8] Hermanto MW, Chiu MS, Woo XY, Braatz RD. Robust optimal control of polymorphic transformation in batch crystallization. AIChE J. 2007; 53(10):2643-2650.

[9] Roelands CP, Jiang SF, Kitamura M, ter Horst JH, Kramer HJ, Jansens PJ. Antisolvent crystallization of the polymorphs of L-histidine as a function of supersaturation ratio and of solvent composition. Crystal Growth & Design 2006; 6(4):955-963.

[10] Sotowa KI, Naito K, Kano M, Hasebe S, Hashimoto I. Application of the method of characteristics to crystallizer simulation and comparison with finite difference for controller performance evaluation. J. Process Control 2000; 10:203-208.

[11] Liu XD, Osher S, Chan T. Weighted essentially non-oscillatory schemes. J. Comput. Phys. 1994; 115:200-212.

[12] Jiang GS, Shu CW. Efficient implementation of weighted ENO schemes. J. Comput. Phys. 1996; 126:202-228.

[13] Henrick AK, Aslam TD, Powers JM. Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points. J. Comput. Phys. 2005; 207:542-567.

[14] Serna S, Marquina A. Power ENO methods: a fifth-order accurate weighted power ENO method. J. Comput. Phys. 2004; 194:632-658.

[15] Gunawan R, Fusman I, Braatz RD. High resolution algorithms for multidimensional population balance equations. AIChE J. 2004; 50(11):2738-2748.

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