Numerical simulation of the seeded batch cooling crystallization process of active pharmaceutical ingredients was performed. Crystal size distribution was observed by focused-beam reflectance measurements and concentration by simultaneous Fourier transform infrared spectroscopy. First, kinetic parameters were estimated. Nucleation rate coefficient and order were estimated from the linear regression equation between cooling rate and metastable zone width. The remaining parameters were estimated from the evaluation function. Then, the numerical simulation was performed by utilizing the method of moments. Finally, observed data were compared with simulated data. The simulated values of mean mass size corresponded reasonably with the observed ones.An ethanolic solution of acetaminophen (CH 3 CONHC 6 H 4 OH, AAP) and an aqueous solution of L-arginine (C 6 H 14 N 4 O 2 , Arg) were selected as target substances. Physical properties of these substances are listed in Tab. 1.

The kinetic parameters of stochastic primary nucleation were estimated for the batch-cooling crystallization of L-arginine. It is difficult for process analytical tools to detect the first nucleus. In this study, the latent period for the total number of crystals to be increased to a predetermined threshold was repeatedly measured with focused-beam reflectance measurements. Consequently, the latent periods were different in each measurement due to the stochastic behavior of both primary and secondary nucleation. Therefore, at first, the distribution of the latent periods was estimated by a Monte Carlo simulation for some combinations of the kinetic parameters of primary nucleation. In the simulation, stochastic integrals of the population and mass balance equations were solved. Then, the parameters of the distribution of latent periods were estimated and correlated with the kinetic parameters of primary nucleation. The resulting correlation was represented by a mapping. Finally, the parameters of the actual distribution were input into the inverse mapping, and the kinetic parameters were estimated as the outputs. The estimated kinetic parameters were validated using statistical techniques, which implied that the observed distribution function of the latent periods for the thresholds used in the estimation coincided reasonably with the simulated one based on the estimated parameters.

Chord length distribution (CLD) can be determined by an in‐line measuring system with focused‐beam reflectance measurement, but it can differ from crystal size distribution (CSD). However, expected values of CLD can be calculated from CSD by statistical methods and vice versa. In this study, a correlation equation between crystal size and aspect ratio during cooling crystallization was obtained and a mapping matrix was calculated based on the correlation equation. Then, the suspension obtained in cooling crystallization was sampled and CSDs were measured by microscopy at the same time that CLDs were measured with FBRM. As a result of error evaluation, transformation of CLD into CSD reduced the errors between CLDs and CSDs except in the early stage of crystallization.

In manufacturing process, it is necessary to measure change in CSD (Crystal Size Distribution) with time accurately because CSD is one of the most important indices that evaluate quality of products. FBRM (Focused Beam Reflectance Measurement) can measure CLD (Chord Length Distribution) in line, but CLD is different from CSD because of principle of FBRM. However, if CSD is determined beforehand, CLD can be calculated from the CSD with statistical method. First, when crystal shape is defined from the characteristic crystal size, the matrix of each crystal shape which transforms CSD into CLD in a uniform manner is calculated with Monte Carlo analysis. Characteristic crystal size is added to the variables defining chord length in order to avoid complex integrals and apply the change in crystal shape with characteristic crystal size to the transforming matrix. Secondly, CSD and CLD are actually measured in suspension of acetaminophen in ethanol and suspension of Larginine in water to demonstrate the validity of 2 matrices. Lastly, these matrices are multiplied by some simple CSD models to test the properties of these matrices and demonstrate the utility of this transformation.

Numerical simulations of seeded batch cooling crystallization were performed to investigate the e ect of stochastic nucleation on crystal quality parameters. In a typical deterministic mathematical model for crystallization, which contains population and mass balance equations, primary and secondary nucleation are regarded as Poisson processes in order to derive a stochastic model. In this study, these stochastic model equations were repeatedly solved to achieve a stochastic simulation, and statistical analysis of the results revealed di erences in product quality when the simulations were run under certain conditions. In particular, the statistics, such as the mean and the coe cient of variation, of the product crystal size distribution were found to uctuate as a result of stochastic primary and secondary nucleation. Further, stochastic primary nucleation was also found to be the source of the di erences between the statistics obtained using the deterministic and stochastic simulations when the seed-loading ratio was very low. This di erence was attributed to the growth of crystals when the total number of crystals was less than one, as well as the accompanying secondary nucleation, in the standard deterministic simulation. Thus, a novel deterministic model in which secondary nucleation does not occur until the total number of crystals reaches one was used, and the results and statistics were found to agree with those obtained using the stochastic numerical simulation. In addition, a stochastic model ignoring stochastic secondary nucleation omitted to predict the signi cant statistical uctuations under certain conditions. Finally, the statistical uctuations were predicted for several crystallizer scales, and crystallizer scale-up was found to reduce the uctuations caused by stochastic primary and secondary nucleation.

Partial seeding, in which the nuclei originating from seed crystals grow to yield the product crystals, was simulated and optimized for controlling the batch cooling crystallization process of L-arginine. The product quality was evaluated by the coefficient of variation (CV) of the crystal size distribution. First, the optimum seed amount for partial seeding was estimated by simulation. Then, the simulated values of the optimum seed amount and the resulting local minimum CV were correlated with the seed crystal size and the cooling rate. The correlation can be utilized for estimating the seed amount in the case that the seed crystals are added in a slurry. Finally, these simulated values were compared to the measured ones. Consequently, the optimum seed amount was suggested to be reasonably predicted.

No abstract

The dispersion of batch time, i.e., the time for finalizing batch crystallization satisfying batch end conditions, in internally seeded cooling crystallization with direct nucleation control (DNC) was estimated by computer simulation. The batch time is considered to disperse at such crystallization due to stochastic nucleation. In this study, first, a population balance equation was digitized for numerical calculation, and the simulation was developed in MATLAB. Then, repetitive simulations of internally seeded cooling crystallization considering stochastic nucleation with DNC were performed. Finally, the batch time of each simulation was arranged. As a result, it was found that there is little batch time dispersion in crystallization controlled by DNC and without adding seed.