Combining Mathematical Modeling and Thermal Infrared Data in the Freezing of Pharmaceutical Liquid Formulations

Abstract

Infrared-based (IR) thermal imaging data was combined here with mathematical modeling to describe the freezing process of a pharmaceutical formulation being lyophilized using two different loading configurations; (i) vials in direct contact with the shelf and (ii) vials suspended over it. In all the experiments, the nucleation event was trigged at a specific time instant using the vacuum induced surface freezing (VISF) method. The IR thermal data was given as input to three different mathematical models for freezing and used to estimate the resulting cake’s pore size (d p ) distribution. The resulting d p values were then compared to experimental data obtained through SEM images coupled with an image segmentation tool. The supersaturation model showed the best agreement between the estimated d p and experimental values, while minor discrepancies were shown by the other two models. Nonetheless, the outcomes of these last two models, given as inputs to a mathematical model for the primary drying phase, resulted in satisfactory predictions of the product temperature at the moving front, the product resistance to vapor flow, and the primary drying end point. It follows that the combination of the IR thermocamera and freezing modeling is a promising tool for the in-line monitoring and optimization of a freeze-drying cycle.

Thermodynamics Explains How Solution Composition Affects the Kinetics of Stochastic Ice Nucleation

“…Despite a long history of research on the freezing of aqueous solutions, the nature of some underlying phenomena remains elusive. This is especially true for ice nucleation, which denotes the formation of the first ice crystal from a clear solution. − Its kinetics is of great relevance to multiple fields, from cloud microphysics, − to cryobiology, , to pharmaceutical manufacturing. , The formation of a nucleus is a stochastic event, , so that solutions of identical composition that are stored under identical conditions nucleate at different, randomly distributed times. , Such behavior is highly undesirable in pharmaceutical manufacturing because of its strict quality control regulations for the freezing of biopharmaceutical drug products. , The majority of biopharmaceuticals, including most commercially available vaccines against COVID-19, is formulated and distributed in a frozen or freeze-dried state in vials of milliliter scale. − Still, mechanistic descriptions of ice nucleation in models for the freezing of these products have become available only recently. ,, …”

“…With reference to the schematic in Figure 3a, we note that the distance of the solution's state to the equilibrium can be expressed either (i) as difference of the activities, i.e., Δa w = a w 0 − a w eq (T), as typically used in the atmospheric sciences; 20,23,29 or (ii) as difference of temperatures, i.e., ΔT = T eq − T, which is also called degree of supercooling and is preferred in pharmaceutical applications due to its experimental accessibility. 12,15,17 The expression for the chemical potential can be rewritten in terms of these quantities when taking the appropriate simplifications:…”

“…The last expression follows when considering that fugacity is the product of a reference fugacity and the activity of water. With reference to the schematic in Figure a, we note that the distance of the solution’s state to the equilibrium can be expressed either (i) as difference of the activities, i.e., Δ a w = a w 0 – a w eq ( T ), as typically used in the atmospheric sciences; ,, or (ii) as difference of temperatures, i.e., Δ T = T eq – T , which is also called degree of supercooling and is preferred in pharmaceutical applications due to its experimental accessibility. ,, The expression for the chemical potential can be rewritten in terms of these quantities when taking the appropriate simplifications: normalΔ μ = R T ln ( a w a w eq ) ≈ R T a normalw eq ( a w − a w eq ) = α ( T ) normalΔ a normalΔ μ = R T ln ( a w a w eq ) ≈ normalΔ H T eq ( T eq − T ) = β <...…”

“… 12 , 13 Such behavior is highly undesirable in pharmaceutical manufacturing because of its strict quality control regulations for the freezing of biopharmaceutical drug products. 14 , 15 The majority of biopharmaceuticals, including most commercially available vaccines against COVID-19, is formulated and distributed in a frozen or freeze-dried state in vials of milliliter scale. 16 − 18 Still, mechanistic descriptions of ice nucleation in models for the freezing of these products have become available only recently.…”

“…With reference to the schematic in Figure 3 a, we note that the distance of the solution’s state to the equilibrium can be expressed either (i) as difference of the activities, i.e., Δ a w = a w 0 – a w eq ( T ), as typically used in the atmospheric sciences; 20 , 23 , 29 or (ii) as difference of temperatures, i.e., Δ T = T eq – T , which is also called degree of supercooling and is preferred in pharmaceutical applications due to its experimental accessibility. 12 , 15 , 17 The expression for the chemical potential can be rewritten in terms of these quantities when taking the appropriate simplifications: As can be seen, interpreting Δμ as a function solely of Δ a requires (i) linearization of the logarithm and (ii) neglecting the temperature-dependency of the prefactor α. In contrast, to arrive at the expression for Δ T , no linearization is required; however, the Δ c p term in the Schröder–van Laar equation is neglected.…”