(M1330-03-16) From Desired Dissolution Profile To Particle Size Distribution: Solving The Inverse Problem

Purpose: Dissolution testing is a common practice in the pharmaceutical industry, measuring the release of a drug from solid dosage forms. The rate of drug release is influenced by particle size distribution (PSD) of the drug. Typically, the relationship between PSD and drug release rate is determined through empirical experimentation, covering only a limited portion of the design space. To broaden the design space, researchers developed computational models, utilizing PSD as a design parameter to predict the desired dissolution curve. However, these models primarily address the forward problem, calculating dissolution profiles from known monomodal PSD only, limiting the range of achievable dissolution profiles. In real cases, any dissolution profile can be attained through appropriate combination of multiple particle size fractions, resulting in complex multimodal PSDs. To accurately mirror the real case, we developed a computational model which predicts the dissolution profile arising from an arbitrary multimodal PSD. The model not only estimates dissolution profiles from given PSDs (the forward problem) but also determines the necessary particle mixtures required to achieve a desired dissolution profile (the inverse problem, Figure 1, i). Our model thus introduces a shift from monomodal to multimodal PSD modeling, reducing the reliance on dissolution-testing experiments and thus broadening the design space.

Methods: 1. Experimental Investigation: To validate the model, we choose KCl as our reference substance due to its straightforward dissolution. Three different PSD’s of KCl microparticles were prepared: namely monomodal, bimodal and trimodal ones (Figure 2, ii). To measure the effect of such PSD’s on dissolution profile, PSD evolution and KCl dissolution were measured in real time in the sample funnel of the static light scattering (SLS) instrument (Figure 2, iii). The sample funnel served as a dissolution tank in which a conductometer was inserted to measure the KCl concentration decrease over time. Since KCl is a fast-soluble salt, its dissolution was slowed down in an isopropanol-water mixture (IPA-H2O).
2. Computational Methodology: The mathematical model was developed in Matlab by coupling the derivative form of the Nernst-Brunner equation with population balance equation (PBE) (Figure 2, iv.). The inverse problem is posed as an optimization problem, where the objective was to minimize the distance between the desired dissolution profile and the calculated one, obtained by the solution of equations with respect to mass fractions, which make up the monomodal bimodal and trimodal PSD’s.

Results: To demonstrate the model's predictive abilities, a direct comparison between experimental and simulated data is shown in Figure 2. First, we compared the evolution of PSD's (Figure 2) for monomodal (i), bimodal (ii) and trimodal case (iii), a good agreement being obtained between two sets of data. Notably, the experimental PSD evolution for monomodal case moved faster than the simulated one. To explain this phenomenon, we calculated the total number of particles from known concentration to understand if the number of particles is increasing or decreasing during the dissolution. The calculation showed that the number of particles for monomodal case is increasing, indicating possible breakage (Figure 2, iv). Moving on, the comparison between experimental and simulated dissolution curves for all three cases is shown in Figure 2, v, where an excellent agreement was achieved, demonstrating that the time course of dissolution is heavily controlled by the initial PSD. Furthermore, the model generated an attainable range of various dissolution profiles for the bimodal and trimodal case respectively (Figure 2, iv). These parametric curves are very useful in generic drug product development, where it is required to fine-tune a dissolution profile by adjusting the PSD. The limiting cases in Figure 2, iv indicate that if a desired dissolution profile lies outside of these boundaries, a more dramatic adjustment of PSD is required. We then automated the model in such a way that we can obtain an exact particle fraction mixture from a desired dissolution profile. To do so, we considered the bimodal and trimodal case. Since their particle composition is known from experimental investigation, we tested the algorithm's ability to converge to this solution, i.e. predict their particle size fractions from their dissolution profiles. The shape of their error functions is shown in Figure 3 (i, ii). The experimental versus predicted particle composition values and their residual errors for both bimodal and trimodal cases are depicted in Figure 3, iii. These results illustrate that the optimization algorithm accurately identified particle mixtures necessary to attain a desired dissolution profile, effectively solving the inverse problem.

Conclusion: We developed and validated a PBE model for the dissolution of arbitrary multimodal particle population. We posed the inverse problem, by employing an optimization algorithm which finds the precise PSD from a desired dissolution profile. The ability of the algorithm to find accurate solutions was demonstrated through bimodal and trimodal PSD cases. This systematic design of PSD broadens the design space during drug development and thus reduces the number of time-consuming experiments.

References: 1. Bhaskarwar, A.N., General population balance model of dissolution of polydisperse particles. 1989. 35(4): p. 658-661.
2. Hänchen, M., et al., Validation of a population balance model for olivine dissolution. Chemical Engineering Science, 2007. 62(22): p. 6412-6422.
3. Zhao, W., et al., PBE model and experimental validation for reactive dissolution of particle agglomerates. Computers & Chemical Engineering, 2018. 108: p. 240-249.

Acknowledgements: The authors would like to thank the Operating Program Jan Amos Comenský (OP-JAK) for financial support, co-funded by MŠMT, and the European Union. S.Đ is grateful for the financial support of the Martina Roeselová Memorial Fellowship granted by the IOCB Tech Foundation. The support of Zentiva is gratefully acknowledged.


Figure 1: i) The concept of the inverse problem, ii) Experimental and modeled PSD’s of monomodal, bimodal and trimodal cases, iii) Experimental methodology, iv) Computational methodology



Figure 2: i) Monomodal PSD evolution, ii) Bimodal PSD evolution, iii)Trimodal PSD evolution, iv) Attainable range of dissolution profiles by combining different particle fractions, v) Comparison between experimental and simulated dissolution curves, vi) Calculated number of particles during dissolution


Figure 3: i) Shape of error function for bimodal case, ii) Shape of error function for trimodal case, iii) Comparison of exact and calculated solution of the inverse problem for bimodal and trimodal case