Quality by design: optimization of a freeze-drying cycle via design space in case of heterogeneous drying behavior and influence of the freezing protocol

Abstract

Purpose: This paper shows how to optimize the primary drying phase, for both product quality and drying time, of a parenteral formulation via design space.

Methods: A non-steady state model, parameterized with experimentally determined heat and mass transfer coefficients, is used to define the design space when the heat transfer coefficient varies with the position of the vial in the array. The calculations recognize both equipment and product constraints, and also take into account model parameter uncertainty.

Results: Examples are given of cycles designed for the same formulation, but varying the freezing conditions and the freeze-dryer scale. These are then compared in terms of drying time. Furthermore, the impact of inter-vial variability on design space, and therefore on the optimized cycle, is addressed. With this regard, a simplified method is presented for the cycle design, which reduces the experimental effort required for the system qualification.

Conclusions: The use of mathematical modeling is demonstrated to be very effective not only for cycle development, but also for solving problem of process transfer. This study showed that inter-vial variability remains significant when vials are loaded on plastic trays, and how inter-vial variability can be taken into account during process design.

Keywords::

• Freeze-drying
• design space
• primary drying
• optimization
• PAT
• annealing

Appendix A

Modeling and determination of the heat transfer coefficient

The overall heat transfer coefficient (K_v) is:

A1

Where K′_v,1 is the heat transfer coefficient between the tray surface and the vial bottom, K′_v,2 is the heat transfer coefficient between the shelf and the tray surface, and K_s is the heat transfer coefficient between the heating fluid and the shelf. The parameter K′_v,1 is a nonlinear function of the operating pressure:

A2

Where C₁, C₂ and C₃ are characteristics of the vial and of the equipment used.^[21] The shelf-tray and tray-vial heat transfer coefficients can be described by the same law, therefore an effective heat transfer coefficient (K′_v) is introduced to take into account both contributions:

A3

In order to describe the pressure dependence of K′_v, the values of C₁, C₂ and C₃ are determined by the regression of experimental values for K_v vs. P_DC.

Experimental determination of K_v

Various methods were proposed in literature to estimate the parameter K_v. These techniques are based on the measurement of the sublimation flux (J_w), as it can be correlated to the value of K_v by the following equation:

A4

To estimate K_v, therefore, the following variables have to be monitored during the drying: the fluid temperature, the product temperature at the vial bottom (T_B), and the sublimation flux (J_w). In this study, J_w is measured by gravimetric procedure. The fluid temperature varies along the shelf, as the heat is transferred from the heat transfer fluid to the product. However, in our equipment the difference between inlet and outlet fluid temperature, under full-load conditions, is of the same order of magnitude of temperature sensor uncertainty, i.e. less than 1 K. Therefore, in order to calculate K_v, the fluid temperature gradient along the shelf does not have to be taken into account explicitly, but can be included in the parameter uncertainty.

In order to calculate the value of K_v vs. P_DC, the measurement of K_v has to be repeated at different values of chamber pressure. Parameters C₁, C₂ and C₃ in Equation (A2) are determined by the nonlinear regression of experimental values of K_v vs. P_DC.

Appendix B

Modeling and determination of the mass transfer coefficient

The resistance to vapor flow (R_p) depends on the thickness of the dried layer (L_dried). According to Pikal^[26] the value of R_p vs. L_dried can be expressed by the following empirical equation:

B1

Where R_p,0, P₁ and P₂ are determined by fitting of experimental data. The product resistance to mass transfer is only marginally affected by the processing conditions used during the drying, except for amorphous products. These products, in fact, may undergo a modification of their structure. For example, if the product temperature is slightly higher than the glass transition value, small openings appears in the dried solid matrix. Because of this structure, the product resistance to mass transfer is lower than the value observed when the dried product does not undergo modifications of its structure.^[27–29]

Experimental determination of R_p

The product resistance to vapor flow can be correlated to the vapor flow rate by the following equation:

B2

Where, the resistance to mass transfer can be expressed (according to the International System of Units) in Pa m²s·kg⁻¹, and hence in m·s⁻¹. To estimate R_p, three variables have to be monitored: the sublimation flux, the chamber pressure, and the product temperature at the interface of sublimation (T_i). As this temperature cannot be measured directly, the PRT technique is here used.^[20] The value of p_w,i in Equation (B2) is, instead, calculated using the relationship proposed by Goff and Gratch.^[30] The sublimation flux can be still measured by the gravimetric procedure. However, if the value of the heat transfer coefficient is known, the sublimation flux during the primary drying can also be calculated using Equation (A4), and monitoring the fluid and product temperature.