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282525 A Multidimensional Population Balance Model for Growth and Dissolution Identified From a Designed Temperature-Cycling Experiment

A large proportion of pharmaceutical crystallizations produce rod-like crystals, which can cause problems in process operations downstream from the crystallizer. Many recent efforts have been directed towards the construction of multidimensional population balance models suitable for the design of process operations to optimize crystal shape [1-11]. These models require the estimation of growth kinetics along multiple axes, with most methods based on sampling the slurry during crystallization, or on employing imaging technology that only applies for low crystal number densities or is not easy to implement in an industrial environment [12-21]. While it is not possible to construct an arbitrary two-dimensional crystal size distribution (2D CSD) from measurements of the chord length distribution (CLD) obtained by Focused Beam Reflectance Measurement (FBRM, which is a probe for in-situ laser backscattering measurements of particles), application to a specific solute-solvent system is simpler due to limited range of 2D CSDs that can occur for the allowed variation in seed mass and temperature profiles.

Previously,
we have reported the application of the commonly available FBRM for the *in-situ* simultaneous measurement of
crystal shape and size for rod-like organic
crystals during a temperature cycling experiment, which resulted
in substantial continual increases in aspect ratio, mean crystal length, and
mean crystal width with cycle number. This
presentation will describe the construction of a multidimensional population
balance model for growth and dissolution rates along the two axes. The
experimental observations motivated the
incorporation of size-dependent dissolution into the multidimensional population
balance model

where *f* is
the 2D population density, *t* is time,
and the growth and dissolution rates in the length and width directions are 2D
generalizations of the 1D equations [22], i.e.,

and the *d _{ij}* and

*γ*define size dependencies. The size dependency broadens the crystal size distribution, which result in more crystals becoming dissolved in each cycle, as required to explain the experimentally observed systematic increase in both the average length and width of crystals with cycle number. A parameter estimation code (the numerical algorithms and software are similar to that used in [23] and [24]) employed a successive quadratic programming algorithm to minimize the sum-of-squared-deviations between the model predictions and experiments,

where *θ*
is the vector of model parameters and *α*
weighs the confidence in each of the mean length and width measurements
relative to the solute concentration measurements and takes into account the
differing units and number of experimental data points. A Pareto-optimality
plot of squared-error values was constructed with each point being associated
with a value of *α*, which ensured
that the value of *α* of 0.00008 reasonably
weighed the sets of solute concentration and mean length/width data.

Compared to previous multidimensional population balance models (PBMs), this PBM includes the effects of dissolution rates that are estimated from experimental data. Although the PBM's modeling of the size dependency of dissolution is rather simplistic, the model fits the experimental data with limited number of parameters with a relatively small level of uncertainty in the parameters that was quantified by first-order multivariate statistical analysis. In addition, the model simulation at optimal kinetic parameters was able to predict the crystal size and shape distribution in a different temperature-cycling experiment reasonably well. These results suggest that growth and dissolution rates estimated in a single temperature-cycling experiment may be suitable for designing a temperature-cycling protocol for optimizing the mean length, mean width, mean aspect ratio, or some weighted combination of the three characteristics of the product crystals. These results also suggest that an automated protocol may be developed that can assess the potential usefulness of temperature cycling for removing or reducing undesirable crystal size and shape characteristics produced by more standard crystallization operations.

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