Conversion of Chord Length Distribution to Particle Size Distribution in Needle Systems
Information regarding the time-evolving Particle Size Distribution (PSD) of the product is of crucial importance in any crystallization processes. The real-time measurement of the PSD along with the use of population balance equation provides valuable information about the nucleation and growth kinetics. By modeling the complex crystallization kinetics, the final PSD can then be accurately predicted and the process controlled to achieve the desired one. The final characteristics of the PSD are critical since it directly affects the ease of solids handling in downstream operations. It also affects the product quality such as the dissolution rate of an Active Pharmaceutical Ingredient (API). Although the available on-line analytical tools, such as Focus Beam Reflectance Measurement (FBRM), provide real time measurement capability, additional steps are required to convert the raw Chord Length Distribution (CLD) data into useful information such as PSD. This presentation addresses a new methodology developed to convert Chord Length Distribution (CLD) obtained from the FBRM measurement into PSD. We are also comparing the results to two other existing methods.
In this presentation, the forward CLD-to-PSD modeling method is proposed to estimate PSD from the measured CLD with FBRM. It is different form the exiting inverse modeling methods[1], [2] in that it does not attempt to invert the ill-conditioned matrix that characterized the PSD-to-CLD information. It assumes a probability density function of the particles with 2 or 3 unknown parameters. It also assumes a constant aspect ratio as all other methods have done. It estimates the 2 or 3 parameters probability density function by minimizing error between the measured CLD and the calculated one. Furthermore, the forward CLD-to-PSD modeling method is capable of estimating PSDs of crystals with two different crystal shapes. This is particularly important in cases where the crystallization system has undergone secondary nucleation, polymorphic transformation, or another diastereomer has appeared in the slurry.
The advantage of the forward CLD-to-PSD modeling method is that it does not invert the conversion matrix that is ill-conditioned in nature. As a result, the solution will not have oscillation or sometime result in negative number of particle size. In order to quantify the conversion accuracy, the proposed method was compared with two inverse modeling methods, the regularization method1 and the Project Over Convex Sets (POCS) method2, in eight different simulated cases and with real FBRM measurements from one crystallization experiments. The simulated cases were necessary to evaluate the relative merits of all three approaches. The experimental cases, even though they are affected by measurement errors, provide a cross-examination of the approaches discussed.
In all eight simulated test cases, we focused on a needle crystal system and we considered different realistic examples of particle size distributions (PSD). Knowing the shape of the crystals, one can calculate the CLD of a single particle or of an assumed PSD. The calculated CLD is compared to the measured CLD and the probability density function parameters are set to minimize the difference between the two CLDs. In the experimental case, the CLDs measured by the FBRM instrument in a crystallization experiment were used. It should be noted that the estimated PSDs from all methods are compared with the PSDs obtained via image analysis. The PSD differences were due to three causes, CLD measurement errors, conversion errors, and PSD measurement errors.
The forward CLD-to-PSD modeling method can easily be applied on-line for process modeling and control purposes. It requires minimal computation time of less than 30 seconds. With the implementation of the forward CLD-to-PSD model on-line, the PSD can then be controlled on-line within specification provided the real-time measurements of PSD and solute concentration are made available.
[1] Hukkanen, E. J. and R. D. Braatz (2003). "Measurement of particle size distribution in suspension polymerization using in situ laser backscattering." Sensors and Actuators B 96: 451-459.
[2] Worlitschek, J., T. Hocker, et al. (2005). "Restoration of PSD from Chord Length Disribution Data using the Method of Projections onto Convex Sets." Particle & Particle Systems Characterization 22(2): 81-98.