🔬 materials science

Direct Modeling of Pore Size Evolution and Microcollapse in Lyophilization by Population Balance

This paper presents a computationally efficient population balance model that directly simulates pore size evolution and microcollapse during pharmaceutical freeze-drying, successfully predicting mass transfer resistance and product temperature across various conditions using a single material-specific rate constant.

Original authors: Isaac Stonewall Wheeler (Purdue University Davidson School of Chemical Engineering), Vivek Narsimhan (Purdue University Davidson School of Chemical Engineering), Alina A. Alexeenko (Purdue University

Published 2026-10-02

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CC BY 4.0

Original authors: Isaac Stonewall Wheeler (Purdue University Davidson School of Chemical Engineering), Vivek Narsimhan (Purdue University Davidson School of Chemical Engineering), Alina A. Alexeenko (Purdue University Davidson School of Chemical Engineering, Purdue University School of Aeronautics and Astronautics, Purdue University Birck Nanotechnology Center), Davide Fissore (Politecnico di Torino Dipartimento di Scienza Applicata e Tecnologia)

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of medicine, many life-saving drugs cannot survive the heat of traditional drying. To preserve them, manufacturers use a process called freeze-drying, or lyophilization. The drug is frozen solid, then placed in a vacuum where the ice turns directly into vapor and disappears, leaving behind a dry, porous cake that can be reconstituted later. This method is essential for vaccines, antibodies, and other delicate biologics, but it is also incredibly slow and expensive. The bottleneck is the primary drying phase, where ice sublimates away. The speed of this process depends on how easily water vapor can escape through the tiny tunnels of the dried cake. If the tunnels are narrow or blocked, the process drags on for days, driving up costs and limiting how much medicine can be made.

For decades, scientists have used a standard mathematical model to predict how fast this drying happens. This model treats the resistance to vapor flow as a simple, fixed curve that changes only as the layer of dried material gets thicker. It works well enough for many situations, but it struggles when the drug formulation is amorphous—a glass-like state rather than a crystal. In these cases, as the product warms up during drying, the delicate glassy structure can soften and collapse slightly. This phenomenon, known as "microcollapse," changes the size of the pores inside the cake, altering how easily vapor can escape. The old models could not predict this change; they required researchers to run separate, time-consuming experiments for every new temperature setting to see how the resistance shifted.

A team of researchers at Purdue University and the Politecnico di Torino has developed a new way to model this process that looks directly at the changing shape of the pores. Instead of guessing how resistance changes, they built a model that tracks the population of pores inside the cake as they evolve. They imagined the pores not as static holes, but as a living population that grows and merges. As the ice sublimates, new pores are born. At the same time, if the temperature gets too high, the glassy walls between neighboring pores can soften and merge, creating larger, fewer tunnels. The researchers used a mathematical approach called a population balance to track these births and mergers in real-time. By counting how many pores exist and how their sizes change, the model can calculate the resistance to vapor flow without needing to measure it experimentally for every single condition.

The team tested this new approach against real-world data from previous studies and their own experiments. They looked at two types of materials: crystalline dextran, which does not collapse, and amorphous sucrose and protein mixtures, which do. For the crystalline material, the model worked perfectly by simply tracking the size of the pores created by the ice crystals, confirming that the underlying math was sound. For the amorphous materials, where microcollapse is a major factor, the model proved its true value. In one set of experiments involving a protein formulation, the researchers fitted the model to just two extreme drying conditions—one very cold and one warmer. Once calibrated with those two cases, the model successfully predicted the drying behavior for three other intermediate conditions that it had never seen before. The old, standard model could only match the data if it was fitted separately for each condition, making it useless for predicting new scenarios.

The researchers also explored whether the rate at which these pores merge is a fixed property of the material itself. They tested different concentrations of sucrose and different freezing methods to see if the "microcollapse rate constant" remained the same. Their results suggest that for a given material, this rate is indeed a consistent physical property, independent of how much of the drug is in the solution or how it was frozen. This finding is significant because it implies that engineers could determine this rate once for a specific drug and then use it to design drying cycles for any temperature range, rather than running endless trial-and-error tests.

While the new model is not yet a perfect replacement for the old one, it offers a much clearer window into what is happening inside the drying vial. It connects the physical reality of collapsing glass structures to the mathematical prediction of drying speed. The authors caution that the model still relies on some estimated parameters and that the exact relationship between pore size and flow resistance in these complex, rarefied gas environments needs further study. However, the work demonstrates that by treating the pore structure as a dynamic system that changes over time, it is possible to reduce the number of experiments needed to design efficient freeze-drying cycles. This could eventually lead to faster production times for critical medicines, ensuring that life-saving drugs reach patients more quickly and at a lower cost.

Technical Summary: Direct Modeling of Pore Size Evolution and Microcollapse in Lyophilization by Population Balance

Problem Statement
In the primary drying phase of pharmaceutical lyophilization (freeze drying), the mass transfer resistance ( $R_p$ ) of the dried cake is a critical parameter governing process duration and product temperature. For amorphous formulations, $R_p$ is known to decrease as the product temperature approaches the glass transition temperature ( $T_g'$ ), a phenomenon termed "microcollapse." Existing modeling approaches, such as the standard Pikal model, treat $R_p$ as an empirical, weakly nonlinear function of the dried layer height ( $h_d$ ) with parameters ( $a_0, a_1, a_2$ ) that must be fitted for every specific set of process conditions. This necessitates extensive experimental campaigns to characterize resistance across different shelf temperatures and chamber pressures. Furthermore, these empirical models lack a direct physical link to the evolving porous structure, making it difficult to predict behavior under untested conditions or to utilize post-hoc pore size measurements for model calibration.

Methodology
The authors propose a new vial-scale model that replaces the empirical resistance expression with a mechanistic description based on the evolution of the pore size distribution (PSD). The approach integrates the standard Pikal heat and mass transfer framework with a Population Balance Model (PBM).

Population Balance Formulation: The model treats the void space in the dried layer as a collection of discrete pores. Microcollapse is modeled as an irreversible aggregation process where two pores coalesce into one larger pore, governed by a rate constant $k(T)$ . New pores are continuously generated at the sublimation front via ice crystal sublimation.
Moment Method: To maintain computational efficiency comparable to standard models, the authors apply the method of moments to the PBM master equation. The model tracks the zeroth moment (total pore count, $\mu_0$ ), the first moment (total pore volume, $\mu_1$ ), and the second moment ( $\mu_2$ ).
Mass Transfer Resistance Relation: The mass transfer resistance $R_p$ is derived as a function of these moments. Assuming Knudsen diffusion (the dominant regime in freeze drying), $R_p$ is related to the moments via an expression involving the tortuosity and a fractional moment ( $\mu_{3/2}$ ), which is estimated using an interpolative closure (Lagrange interpolation of log-moments).
Temperature Dependence: The aggregation rate constant $k(T)$ is modeled using either a sigmoid function or a MYEGA-like equation to capture the rapid acceleration of microcollapse as the product temperature approaches the critical temperature.
Experimental Validation: The model was validated against four datasets:
- Two literature datasets: An amorphous formulation (BSA/sucrose) with TDLAS mass flow data and a crystalline formulation (dextran) with SEM-measured pore size distributions.
- Two new experimental datasets: Sucrose and dextran solutions lyophilized under varying shelf temperatures and concentrations, with product temperatures and drying times recorded.

Key Contributions

Mechanistic Link: The work establishes a direct mathematical link between the physical evolution of pore structures (specifically the moments of the PSD) and the macroscopic mass transfer resistance, moving beyond purely empirical correlations.
Predictive Capability: By modeling the underlying physics of pore aggregation, the model can predict $R_p$ and product temperatures for intermediate process conditions using parameters fitted only to extreme cases (e.g., low and high shelf temperatures), whereas the standard Pikal model requires re-fitting for every condition.
Utilization of Structural Data: The model allows for the direct incorporation of measured pore size distributions (e.g., from SEM) to constrain empirical parameters ( $N$ and $S$ , representing pore density and size scaling), reducing the reliance on curve-fitting to process data alone.
Computational Efficiency: Despite the added physical complexity, the moment-based approach keeps the computational burden comparable to the standard literature models, making it suitable for process design and optimization.

Results

Amorphous Formulations: When fitted to two extreme drying conditions for an amorphous formulation, the PBM successfully predicted product temperatures and drying times for three intermediate conditions with accuracy comparable to case-by-case Pikal fits. This demonstrates the model's ability to capture the temperature-dependent evolution of resistance without re-parameterization.
Crystalline Formulations: For crystalline dextran (where microcollapse is negligible, $k \approx 0$ ), the model was fitted to SEM-measured pore size distributions. Using these measured moments to fix $N$ and $S$ , the model accurately predicted $R_p$ across four different experimental cycles using only two global fitting parameters ( $C$ and $R_s$ ). This confirmed that variations in resistance among different crystalline batches could be explained by differences in pore size distributions.
Multi-Experiment Fitting: A simultaneous fit of six sucrose experiments (varying concentration and freezing protocols) yielded a single set of microcollapse rate constants ( $k(T)$ ) and tortuosity parameters ( $C$ ). While the model showed some deviation in predicting drying times for specific high-concentration cases, it successfully captured the general trends in resistance evolution.
Flow Regime: Analysis suggests that flow in the dried cake operates primarily in the transitional regime between Knudsen and viscous flow, though the Knudsen assumption remains a reasonable approximation for the derived resistance expressions.

Significance and Claims
The paper claims that this population balance approach offers a more rigorous foundation for lyophilization process modeling. By describing the physical mechanism of microcollapse (pore aggregation) rather than just its empirical consequence, the model reduces the number of experiments required to design optimal freeze-drying cycles for amorphous formulations. The authors suggest that this framework could facilitate the design of cycles that deliberately trigger microcollapse to shorten cycle times, provided product quality attributes are validated. Additionally, the ability to integrate direct pore structure measurements (e.g., from SEM) into the model offers a pathway to reduce the sensitivity of model parameters to specific process data, potentially making the model more robust and transferable across different formulations and equipment. The work concludes that the microcollapse rate constant may be a material parameter independent of the precise porous geometry, a hypothesis that warrants further investigation.

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