Statistical Approximation of the Inverse Problem in Multivariate Population Balance Modeling

Abstract

This paper deals with the estimation of model parameters and their uncertainties encountered in granulation modeling. The outcome of a multivariate, detailed population balance model of a high shear granulation process is locally approximated in the parameter space by first and second order response surfaces, allowing a fast computation of the model response. The response surfaces are used in three different objective functionssmoment matching, expected least-squares, and expected weighted least-squaressin order to estimate ranges for the rate constants for particle coalescence, particle compaction, particle breakage, and reaction, which appear as free parameters in the granulation model. First, second-order response surfaces for the population balance model are constructed and used as approximations of the model in the objective functions for the numerical solution of the inverse problem. Second, the choice of objective function is investigated. It is found that the uncertainties of the model predictions differ for the three objective functions only slightly. The estimates for the intervals of the model parameters either overlap or are very close. However, the moment matching objective function is recommended because the number of estimated parameters and experimental data sets can be chosen independently. IntroductionGranulation is an important and widely used process in many industries. In order to improve the process, e.g. produce a product which is cheaper and/or closer to the required specification, the process needs to be better understood, for instance, through additional experiments and/or modeling of the process. Detailed modeling of the granulation processes, such as by population balances, captures the complex structure of the granules and the various transformations in a granulator such as nucleation, coalescence, and breakage, to name but a few, thereby aiming to bridge between the micro-and macroscales. The rates with which these processes occur are normally not known and have to be estimated using experimental results.This so-called "inverse problem" is encountered in virtually all modeling applications. Ramkrishna and Mahoney 1 emphasize the importance of the inverse problem in population balance modeling. Population balance models are not only used in granulation, 2-7 but also in other fields such as precipitation and crystallization, 8-10 combustion, 11-14 polymerization, 15,16 and modeling of biological systems. 17 In terms of parameter estimation, rates, such as the growth rate in a precipitation 18 or the aggregation rate in a granulation process, 19 are very often the center of interest. This parameter identification can be performed in different ways. For instance, Vikhansky and Kraft 20 developed a stochastic algorithm that can be used to calculate the sensitivities of the parameters of a population balance for coagulation only processes. On the basis of this method, they solve the inverse problem for a liquid-liquid extractor. 21 Following a technique developed by Sheen et al. 22 for par...

The inverse problem in granulation modeling—Two different statistical approaches

“…The surrogate model is a second order response surface, an approach that has already been used in previous work. 38 …”

“…Braumann et al 36 presented an approach using a surrogate model to solve the inverse problem for a multivariate population balance model for granulation. Given that experimental findings possess uncertainties, the methodology was developed further, 37,38 to estimate the unknown parameters along with their uncertainties, building on findings by Sheen et al 39 These uncertainties then carry on through the model response, enabling further investigation such as model discrimination. The existing methodology is somewhat restrictive in the sense that the scientist concerned with the problem must have a good initial guess for the values of the unknown parameters.…”

Model-based analysis of high shear wet granulation from batch to continuous processes in pharmaceutical production – A critical review

“…However, an alternative way of retrieving the 7 kernels based on experimental data is to solve the inverse problem [51,52]. Braumann and Kraft studied the 8 inverse problem occurring in a multidimensional population balance model describing granulation employing 9 linear response surfaces [39] and second order response surfaces [53]. There are different collision frequency 10 functions for kernels available in the literature based on theoretical, empirical and experimental calculations 11 and observations (Table 6).…”

“…Therefore, experiments have to be carried out in 12 order to identify and measure the unknown model parameters, e.g. aggregation rate constants [51][52][53]73]. …”

Estudo do problema inverso em balanço populacional aplicado a degradação de polímeros.

“…Para trabalhos futuros recomenda-se utilizar uma abordagem mais abrangente que admita uma cinética de segunda ordem na fragmentação, como a proposta por BRAUMANN; MAN; KRAFT (2010), o que permitiria estudar sistemas mais complexos, não apenas com quebra de partículas, e sim com a inclusão de outros fenômenos.…”