Data-driven model and model paradigm to predict 1D and 2D particle size distribution from measured chord-length distribution

Determining particle‐size distributions from chord length measurements for different particle morphologies

“…The model approach used here has been introduced in previous work and is summarized below. The model architecture consists of three sequential steps to compute PSDs from CLDs, that is, CLD synthesis, regression models for low‐order PSD moments, and a two‐layer network defined by a generating function.…”

“…The model architecture consists of three sequential steps to compute PSDs from CLDs, that is, CLD synthesis, regression models for low‐order PSD moments, and a two‐layer network defined by a generating function. The standard model architecture is shown in Figure and a detailed description of all model variations can be found in previous work . The inputs for this model are the measured CLD ( Q = { q 1 , …, q M }) and the corresponding solids concentration ( c ).…”

“…In pharmaceutical crystallization, focused beam reflectance measurement (FBRM) has become a standard tool for in situ particle monitoring due to its ease of process implementation and ability to operate at high solids concentrations. However, a major limitation of this technique is that its measurement signal, that is, the chord length distribution (CLD), differs fundamentally from particle‐size distribution (PSD) data due to the FBRM measurement principle, which is explained in detail elsewhere . Thus, the measured CLD is not only a function of particle number and size, but is generally also affected by particle morphology and various process parameters, for example, probe positioning, particle concentration, and optical effects of the continuous and dispersed phase.…”

“…Thus, a single ab initio model alone cannot capture all physical phenomena and reflection properties of liquid and solid phases affecting a CLD measurement and therefore will fail to predict the PSD of the dispersed phase accurately. Generally, the measured PSD depends on the type of characterization, which is why the selection of a reference technique is also important when aiming at extracting PSD from CLD data. PSDs in the pharmaceutical industry are generated as quality control under laboratory conditions.…”

“…Although laser diffraction measurements suffer also from limitations, particularly for nonspherical particles as described in the literature, PSD data from laser diffraction will be considered as the reference data due to its importance in the quality release process during pharmaceutical production. Previously, a data‐driven modeling approach has been introduced which allowed the quantitative determination of one‐dimensional and two‐dimensional PSDs from measured CLDs, suggesting an effective solution to the above‐mentioned limitations of the FBRM . The model architecture is based on three sequential steps: First, CLD data are compressed into a small set of CLD descriptors (low‐order moments). Second, the CLD descriptors are mapped into a small number of PSD moments using a regression model. Finally, the PSD moments are expanded into a full PSD using a two‐layer network model. …”