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.
Appendix A
Modeling and determination of the heat transfer coefficient
The overall heat transfer coefficient (Kv) is:
A1Where 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 Ks 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:
A2Where C1, C2 and C3 are characteristics of the vial and of the equipment used.[Citation21] 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:
A3In 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 Kv vs. PDC.
Experimental determination of Kv
Various methods were proposed in literature to estimate the parameter Kv. These techniques are based on the measurement of the sublimation flux (Jw), as it can be correlated to the value of Kv by the following equation:
A4To estimate Kv, therefore, the following variables have to be monitored during the drying: the fluid temperature, the product temperature at the vial bottom (TB), and the sublimation flux (Jw). In this study, Jw 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 Kv, 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 Kv vs. PDC, the measurement of Kv 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 Kv vs. PDC.
Appendix B
Modeling and determination of the mass transfer coefficient
The resistance to vapor flow (Rp) depends on the thickness of the dried layer (Ldried). According to Pikal[Citation26] the value of Rp vs. Ldried can be expressed by the following empirical equation:
B1Where Rp,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.[Citation27–29]
Experimental determination of Rp
The product resistance to vapor flow can be correlated to the vapor flow rate by the following equation:
B2Where, 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 Rp, three variables have to be monitored: the sublimation flux, the chamber pressure, and the product temperature at the interface of sublimation (Ti). As this temperature cannot be measured directly, the PRT technique is here used.[Citation20] The value of pw,i in Equation (B2) is, instead, calculated using the relationship proposed by Goff and Gratch.[Citation30] 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.