Transformation of CSD When Crystal Shape Changes with Crystal Size into CLD from FBRM by Using Monte Carlo Analysis

Abstract

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.

Determination Methods for Crystal Nucleation Kinetics in Solutions

“…Usually, for simplicity, the crystal morphology or size is described by only one variable. Furthermore, the poor conversion from the chord length distribution (CLD) and total counts/s measured by FBRM to the crystal size distribution (CSD) and crystal number density restricted the wider use of FBRM and the PBE model method. − …”

Parameter Estimation of the Stochastic Primary Nucleation Kinetics by Stochastic Integrals Using Focused-Beam Reflectance Measurements

“…However, the CLD obtained with FBRM is different from CSD, and hence, the CLD needs to be converted into the CSD [22,23]. In our previous work [18,24], the method for the conversion was investigated for the Arg crystals and is represented as follows:…”

“…Each element of S corresponds with the probability that a crystal with a certain size will be measured as a certain chord length [18]. In this study, S was computed with a Monte Carlo analysis in MATLAB (version R2019a made by The MathWorks, Inc., Natick, MA, USA) [25,26], in which crystals with a predetermined shape were rotated and translated randomly [24,27]. Settings of the Monte Carlo analysis and the instruments for FBRM are listed in Table 2.…”

Computing Crystal Size Distribution by Focused‐Beam Reflectance Measurement when Aspect Ratio Varies

“…In previous work [29], the algorithm of the computation of the mapping matrix which can afford to take into account the change in crystal shape was described. In this study, the suspension of L-arginine (Arg) in water was used.…”

“…Each element of the shape transformation matrix S 1) corresponds with the ratio of the probability event of each chord length range in the sample space of the domains of five independent variables and can be computed with Monte Carlo analysis [29]. For example, (S) ij corresponds with the probability that the chord length defined by five independent variables, the values of which are uniformly random within the domain of the j-th bin in CSD, is present within the range of the i-th bin in CLD.…”