(83h) Statistical Modeling of Image-Based Crystal Observation

Solution crystallization produces materials with specific crystal shape and size distributions. Both the crystal size and shape play an important role in deciding the processing rate of material in various downstream operations, for instance filtration, washing, and drying. Crystals of different shapes have different dissolution rates and henceforth bioavailability, the properties that are important in engineering crystals of active pharmaceutical ingredients. Thus, the significant impact of crystal size and shape demands for its tighter control. Therefore, the measurement of crystal shape distributions becomes important.

Using advanced mathematical techniques, such as multidimensional population balance equations, models for crystallization processes can take into account the complex behavior of crystal shape evolution as a result of crystal growth [1,2]. In order to experimentally validate model predictions, the inherent multidimensionality of the modeling framework must be taken into account when a measurement system is selected. Due to the rich information contained in crystal photographs, the observation of crystallization processes using imaging techniques has drawn the attention of researchers in chemical engineering in recent years. The images acquired give a good qualitative visual impression of the crystal morphologies produced. Clearly, the quantitative evaluation of these images with regards to crystal morphology demands for well developed mathematical tools. Besides image processing tools, the quantitative analysis requires the linking between the geometrical crystal model and the information which can be extracted from an image. This contribution seeks to provide such a link.

The three-dimensional convex crystal is characterized by an n-dimensional geometrical state vector, where n is the number of faces exposed on the crystal surface. The crystal is projected on a two-dimensional surface from which the boundary curve is extracted. Subsequently, the boundary curve is characterized by a set of m scalar measures, for instance several shape factors or Fourier Descriptors. The shape of the boundary curve depends highly on the crystal orientation. That is, there is no one-to-one mapping from the m-dimensional vector describing the boundary curve and the n-dimensional geometrical state vector of the crystal model. However, the usage of an orientation model, for instance assuming randomly or preferentially oriented crystals, allows the design of state estimators for the geometrical state vector.

[1] Borchert, C.; Nere, N.; Ramkrishna, D.; Voigt, A.; Sundmacher, K., ?On the Prediction of Crystal Shape Distributions in a Steady State Continuous Crystallizer?, Chem. Eng. Sci. 64 (2008), 686-696.

[2] Borchert, C.; Voigt, A.; Sundmacher, K.; Nere, N.; Ramkrishna, D., ?Evolution of Crystal Shape Distributions and Morphology Classification?, Proceedings of the 17th International Symposium on Industrial Crystallization, 14-17 September 2008, Maastricht, Netherlands.