As “the theater begins with the cloakroom” (as attributed to a famous Russian art director Konstantin Stanislavsky), any freeze-drying recipe begins with the loading temperature. Freeze-dryers are usually loaded at either room temperature (usually 20 to 25°C) or 5°C when liquid stability is a potential concern. On rare occasions, a product is loaded onto shelves cooled below 0°C (e.g., to -50°C); this could be a case when a fast cooling rate is required if there are major concerns with liquid stability or for scheduling reasons in a manufacturing environment to reduce cycle time by eliminating a long cooling step. In such cases, water from the atmosphere could condense on the shelves to form ice, which is highly undesirable.

After loading a freeze-dryer, vials are typically equilibrated for at least 30 min before cooling to minimize vial-to-vial temperature variations. In selecting the shelf cooling rate, one should remember that the shelf temperature/time program is not the same as the product temperature/time response, as illustrated in Fig. 1a. A faster cooling rate results in a larger difference between shelf and product temperature (Fig. 1b). In this example the larger dryer (Lyomax 42 with 42 sq.m shelf area) shows a more significant difference between inlet temperature and product temperature even when it was only partially loaded (2 shelves out of 15) as opposed to a fully loaded pilot dryer (6 sq.m). As discussed below, an approach to a cooling program depends on formulation type, i.e., if the formulation is completely amorphous or partially crystalline.

As illustrated in Fig. 1, product behavior during cooling can be described by two factors, i.e., ice nucleation temperature and freezing time. Ice nucleation can be measured directly by detecting an abrupt increase in the product temperature due to an exothermic process of ice formation. An exact definition of freezing time would require monitoring water-to-ice conversion in real-time, which would be a difficult task in practice. As a conceptual (and approximate) definition, freezing time can be defined as the period between ice nucleation and the time when the product temperature trend resumes following the shelf temperature trend. At this point, most “freezable” (in a kinetic sense) water is converted to ice, although water-to-ice transformation could probably continue below this point, depending on formulation composition and specifics of heat/mass transfer in a particular container. Operationally, the freezing time is commonly defined as the hold time at the final freezing temperature, sometimes including freezing ramp duration.

Lower supercooling (i.e., higher ice nucleation temperature) is considered beneficial, as it is expected to result in larger ice crystals with lower surface area [2, 3]. Frozen mass with larger and inter-connected ice crystals would have a lower resistance to water vapor transfer and a lower risk for protein destabilization because of a lower ice/solution interface and, therefore, a lower fraction of protein molecules exposure to the interface. A temperature equilibration step prior to the initiation of cooling is recommended [1] to reduce the vial-to-vial difference in ice nucleation. There is no definite agreement on the relationships between cooling rate and ice nucleation temperature. While it has been suggested that slow cooling causes larger supercooling [4], and it also reported that the cooling rate (range 0.6–40 K/min) did have some impact on the homogeneous ice nucleation temperature in solutions of LiCl with a concentration above 6.8 mol % [5], there are also reports to the contrary. Indeed, no impact of the cooling rate on the homogeneous ice nucleation temperature was observed for LiCl solutions at 5 mol % [5].

Furthermore, it was reported that an increase in the cooling rate from 0.1 to 1000 K/min resulted in only a 2°​C difference in the heterogeneous ice nucleation temperature and 4°​C in homogeneous ice nucleation [6]. No difference in the ice nucleation temperature was observed with cooling rates of 0.5 to 3.2°​C/min [7], 0.07-​7°​/min [8], and 0.05 to 1°​C/min [2] Also, nucleation rate coefficients (nucleation events per unit time per unit area) of ice on kaolinite did not demonstrate any significant difference in cooling rates between 0.8 and 10 K/min [9].

As noted in [1], “it is not practical to manipulate the supercooling by changing the cooling rate in a freeze dryer because the cooling rates are usually limited to less than 2°C/min, and the degree of supercooling is unlikely to change within such a small range.” It remains a true statement based on almost two decades since the publication of that paper.

One approach to eliminating the differences in freezing due to varying degrees of supercooling across a shelf of vials is by employing controlled ice nucleation. The product temperature is reduced below the equilibrium freezing point (or melting temperature). After a brief equilibration, ice nucleation is initiated by a variety of approaches: (i) pressurization and depressurization of the drying chamber [10]; (ii) introduction of an ice-fog [3]; (iii) reduction of chamber pressure [11]; and (iv) utilization of ultrasound [12, 13]. In addition, vacuum-induced surface freezing has also been explored [11, 14].

The use of higher ice nucleation temperatures results in the formation of larger ice crystals, which lead to larger pores post-ice sublimation and a lower cake resistance during drying. Consequently, primary drying duration can be reduced along with decreased inter-vial heterogeneity in drying rates [10, 15].

Controlled ice nucleation (CIN) could be beneficial for amorphous formulations as it is associated with a lower dry layer resistance and less variability between vials [10, 15]. However, controlled ice nucleation may need longer/higher temperature secondary drying because of the lower specific surface area of the freeze-dried solids [16]. Solute crystallization may not always benefit from controlled ice nucleation, as observed in [17] using model water-NaCl mixtures. Controlled ice nucleation enabled control of the physical form of mannitol during freezing and facilitated the formation of anhydrous mannitol instead of the less desirable mannitol hemihydrate [18]. Implementing controlled ice nucleation in manufacturing is currently limited by the availability of freeze-dryers equipped with ice nucleation capabilities. Evaluating this technology at a laboratory scale is advisable as an alternative to annealing. Different methods of CIN were utilized to compare dried product attributes when nucleated at the same temperature [19–22]. Gitter et al.[19] conducted ice nucleation was achieved at -5°C using ice fog or depressurization methods in systems containing excipients (sucrose, trehalose) or their mixtures (mannitol and sucrose at a 4:1 ratio) and in monoclonal antibodies containing formulation (in sucrose, or trehalose, or mannitol-sucrose and 10 mM histidine buffer and 0.02% w/v polysorbate 80). The authors observed comparable product quality attributes covering a specific surface area, water content, phase behavior, turbidity post reconstitution, sub-visible particle counts, and % monomer / % High molecular weight (HMW) species (for the mAb containing formulation only). Vollrath et al. extended the earlier work to focus on ice fog methods only, namely, using the VERISEQ® nucleation system from IMA Life / Linder, the FreezeBooster® method (Millrock Technology) and the method described by Geidobler and coworkers [15]. Differences exist in the three methods in the temperature at which the ice is generated and the induction pressure, which could lead to the nucleation seeds of different ice morphologies (dendritic, hexagonal, and dispersed spherulitic) and, thereby, may later impact the ice fog texture. Ice nucleation was conducted at either -3°C or -10°C in monoclonal antibody formulations (10 mg/mL) in either sucrose or trehalose (disaccharide at 7.5% w/v). The authors observed a correlation between specific surface area (SSA) and ice nucleation temperature for the FreezeBooster® and Geidobler et al. methods for the sucrose-based formulations. No differences were observed in the SSAs of the trehalose-based formulations nucleated at -3(C or -10(C), suggesting that the disaccharide formulation can impact SSA, leading to differences in SSA between sucrose vs. trehalose-based formulations. The authors reported that the benefit of utilizing a higher ice nucleation temperature from the perspective of solid-state attributes could outweigh the higher risk of less complete nucleation. Luoma and coworkers investigated the effect of depressurization, partial vacuum, and ice fog techniques on product attributes of a monoclonal antibody formulation (10 and 100 mg/mL mAb in sucrose and histidine buffer-based formulations) and also an enzyme formulation (2.5 mg/mL in 500 mM arginine phosphate and polysorbate 80) post drying and during storage in vials of different sizes. Comparable product attributes were observed across the multiple mAb and enzyme formulations and vial sizes using different ice nucleation methods. The enzyme formulation cakes produced using the partial vacuum method were the exception to the general comparable behavior.

Freezing time is important because the material might be partially frozen at this stage, with the freeze-concentrated fraction containing protein in a liquid state. Protein molecules are susceptible to various stresses during this period, including, e.g., freeze-concentration, pH and ionic strength changes, ice/freeze concentrate interface, local mechanical stresses, and protein/cryoprotector phase separation. The freezing time may be reduced by using higher cooling rates, therefore reducing the risk of protein destabilization due to freezing stresses, but one needs to pay attention to a potential increase in heterogeneity of product properties.

Considering that a relatively fast cooling rate would probably have a minimal impact on supercooling and a potential benefit for reducing cycle time, a faster shelf cooling rate can be a logical recommendation for amorphous formulations. In the author's experience, an increase in cooling rate from 0.3 to 1°C/min resulted in a notable improvement of cake appearance for one of the small molecule drugs due to a reduction in phase separation. However, this advice may not apply to formulations with a crystalline bulking agent, as discussed below. The effect of fast cooling has been extensively studied in mannitol solutions, where vial breakage was reported post fast cooling and warming and attributed to mannitol + ice crystallization [23–25]. In addition, vial breakage during freezing might be a risk even for amorphous formulations if a faster cooling rate is used (unpublished data).

As for commercial cycles, cooling rates of 0.5°C/min or lower are recommended to ensure product temperature homogeneity across freeze-dryer shelves [26].

Regarding final freezing temperature and hold time, Tang and Pikal[1] recommended targeting a final freezing product temperature of -40°C in most cases. This considers that if the Tg’ (for completely amorphous products) or temperature of the secondary crystallization (amorphous/crystalline formulations) is usually higher than − 38°. Due to the radiation effect, shelf temperature should be lower (-45°C or lower) to ensure that product temperature in edge vials remains below Tg’. Tang and Pikal [1] recommended one-hour hold time at the final freezing temperature for products with fill depth <  = 1 cm, and two hours for a 1–2 cm fill depth. Fill heights above 2 cm are undesirable, but certain practical applications may require higher fill depth; in such cases, the freezing hold time should be extended beyond two hours. To assess freezing time, a simplified freezing model can be used [38]. The freezing model is based on the assumption of 0D transient heat conduction with lumped capacitance heat transfer. The lumped capacitance approach assumes that the internal resistance to thermal conduction within the product is much lower than the heat transfer between the product and its environment, resulting in a uniform temperature profile throughout the product. The error associated with lumped capacitance method can be characterized by the Biot number (Bi = hL/k, where h – heat transfer coefficient, L – characteristic length, which can be fill height of the product or vial diameter,k – product thermal conductivity). For low Biot numbers (typically Bi < 0.1), the temperature gradient in the product is small [39], and inaccuracy linked to lumped capacitance method is low. In addition, it is assumed that the product temperature is constant during crystallization, and heat is supplied to the product from the shelf only. The lumped capacitance model agrees well with the experimental data [38]. The model requires several input parameters: fill volume, heat transfer coefficient, nucleation, and freezing temperatures. The stochastic nature of the nucleation can introduce uncertainty to the model predictions. The input parameters are to be obtained from the experimental data, or a typical value should be assigned. Use of controlled ice nucleation would reduce the parametric variability [10, 15].

To demonstrate potential applications of the freezing model, freezing times are calculated for different experimental protocols. Two shelf ramp rates and different vials and fill volumes are considered. The heat transfer coefficient, which was determined experimentally for the SCHOTT 6R vial, is assumed to be similar to other vials used in the simulations. The results are presented in Fig. 2. Input parameters in calculations are presented in Table A1 (Appendix).

Figure 2 demonstrates that 1 h hold time would be sufficient for fill volumes of less than 1.5 mL (2R vial) and 5.5 mL (20R vial) at 1C/min ramp rate (fill depth 1 cm for both 2R and 20 R- table A1) which is in alignment with Tang and Pikal [1].

Note that freezing model data for high-fill volume products need to be used carefully since the model does not account for the time of ice propagation. Additional soak time (about 30 min) can be added to calculated values shown in Fig. 2.

The temperature of the fully frozen product is assumed to be -44.8° C rather than -45° C. Indeed, the product temperature is slowly approaching the final shelf temperature (-45° C) during the solid cooling stage in most cases. The final temperature change between -44.8° C and -45° C takes a significant amount of time and is not practical to be accounted for in real applications.

While reducing primary drying time would require running the process at a higher temperature, there is an upper limit for a product temperature from a product perspective. The upper-temperature limit is defined by critical product temperature. This section discusses the identification of a target product temperature for two main types of formulations.

In most cases, three main set points define primary drying: chamber pressure, shelf temperature, and primary drying time. The transition from freezing to the primary drying step is usually straightforward, with a vacuum applied at the end of the freezing segment, followed by heating the shelf to the pre-determined set-point, usually at the ramp rate of 0.5 or 1 °C/min. The heating power of commercial freeze-dryers can limit the ramp rate due to the high mass of shelves and product vials. Also, a fast heating rate may compromise the ability of the condenser to maintain the temperature below -40°C, and pressure control could be lost due to the high sublimation rate at the initial step. Chamber pressure during primary drying is almost universally maintained between 50 and 200 mTorr, with 100 to 150 mTorr representing a typical range for high Tg’ products. Such pressure range provides a compromise between the need to accelerate ice sublimation, which usually requires a lower pressure, and improves heat transfer from shelf to product, which benefits from higher pressure. Per Tang and Pikal [1], “Pc should be well below the ice vapor pressure at the target product temperature to allow a high sublimation rate. The sublimation rate is proportional to pressure difference between the vapor pressure of ice and the partial pressure of water in the chamber (Pi), this difference being the driving force for ice sublimation. Pi is essentially the same as chamber pressure during primary drying. However, very low chamber pressure may cause problems, such as contamination of product with volatile stopper components or pump oil, and also produce larger heterogeneity in heat transfer, thereby giving larger product temperature heterogeneity between vials”. Note that a distinction should be made between P_i (partial pressure of water in the chamber) and P_ice (pressure over sublimation surface). In calculations of sublimation rate, pressure difference P_ice-P_c is used.

Also, note that a minimum sustainable pressure value depends on the equipment, sublimation rate, and chamber load, as discussed later in the section.

As a starting point, chamber pressure for primary drying can be selected using the following equation [1]:

where P_c is chamber pressure (Torr) and T_p is product temperature (°C).

A common approach for the shelf temperature program during primary drying is maintaining the same temperature during the entire primary drying. However, in some (rare) cases, variable shelf temperature during primary drying can be beneficial to accelerate drying while keeping the product temperature below the critical value. Such an approach can be easily executed in laboratory-scale experiments but requires caution in a manufacturing environment to minimize errors in programming multiple primary drying steps.

Two parameters, chamber pressure and shelf temperature, define product temperature and duration of the primary drying for a given product and freeze-dryer. The product temperature at the sublimation interface is always lower than the shelf temperature during the sublimation stage, and the difference can sometimes reach more than 40°C.

As an efficient way to select the initial combination of chamber pressure and shelf temperature for a particular product and freeze-dryer, we recommend using a heat/mass transfer quasi-steady-state mathematical model as the first step. The basis for the model can be found in [54], with additional details (e.g., vials heat transfer coefficients) provided in [1]. More complicated (non-steady-state) mathematical models are also available [55]. Any mathematical heat/mass transfer model, which describes product temperature and water loss during primary drying, has two main inputs, i.e., the amount of heat supplied to the product from the environment and the heat extracted from the product by ice sublimation. For quasi-steady-state models, these inputs are provided by the vial heat transfer coefficient and the resistance of the product dry layer to water vapor flow. The authors’ experience is that the more straightforward quasi-steady-state approach has demonstrated appropriate accuracy in most cases studied, and this paper focuses on these models.

STca Lyo is the Excel-based steady-state primary drying model [56] in which heat and mass transfer over the sublimation surface is balanced using Solver. It uses vial/shelf heat transfer coefficients as a function of pressure (for both center and edge vial), the resistance of cake as a function of dry layer height, and minimum controllable pressure as a function of sublimation rate as input parameters for the calculation of product temperature profile and drying time for a particular dryer (listed in the database provided in this paper). It allows the estimation of conditions when the freeze-dryer cannot control pressure set point at a full load.

VMCa Lyo is a Python™ based steady-state primary drying model based on the same assumptions as those in the STca Lyo model. The model was designed to execute multiple scenarios that enable the creation of a design space for input parameters. This model uses “while loops” as its solver to increase or decrease the product temperature until it satisfies the conditions.

LyoPRONTO [38] is a lyophilization simulation and process optimization tool. The tool includes several instruments for predicting different stages of the lyophilization process: freezing and primary drying. It is also capable of design space generation, cycle optimization, product resistance, and heat transfer coefficient determination. The freezing calculator is based on a 0D lumped capacitance model, which can successfully predict the freezing process time and its steps. The primary drying calculator is based on a vial's 1-D quasi-steady-state heat-and-mass transfer. It is assumed that the central vial is representative of the entire batch, and convective heat transfer is neglected. The primary drying optimizer determines the optimal chamber pressure and shelf temperature at each time step to minimize the total drying time under the existing constraints.

Resistance of the dry product layer to the flow of water vapor depends primarily on the composition of the formulation and the dry layer thickness. The resistance can depend on the product history during freezing/annealing and primary drying, in particular, if the product experience change in the phase state (e.g., amorphous-to-crystalline transformation) or morphology (e.g., as a result of microcollapse) (Fig. 3). Dry layer resistance can be measured by MTM [57], PDM [58], TDLAS [59], micro X-ray tomography [60], and calculated from a product temperature profile measured by a thermocouple [61]. In most cases, resistance increases as primary drying proceeds because of the increase in the dry layer thickness. A classical paper describes the measurement of cake resistances for different types of products [62], while examples of both common R_p patterns and unusual cases are provided below in Figs. 3, 4 and 5. Representative R_p patterns for typical amorphous formulations (model systems with different sucrose concentrations) and formulations with a crystalline bulking agent (mannitol) are shown in Fig. 3. These R_p patterns are used to assemble the primary drying database, as described later in the paper. For the sucrose formulations, the R_p increases at the beginning of the primary drying and tends to level off as the thickness of the dry layer increases due to likely microscopic collapse or difference in cake structure toward the bottom of the vial. The graph also shows two significant trends: higher resistance for formulations with higher solid content and a significant decrease in the resistance in cases when primary drying is performed to allow microcollapse. For crystalline formulations, the R_p tends to be higher than the amorphous of the same solid content, showing a near-linear increase with the dry layer thickness. It should be mentioned that the impact of the process on pore structure and cake resistance should be further investigated in a separate study correlating the pore structure of dry solids (for example, by micro CT) and resistance to mass flow.

The R_p for these formulations is described by an empirical equation [62]:

R_p = a + b*h/(1 + c*h), where h is a dry layer in cm, and a, b, and c are adjustable constants.

There are also cases when the R_p behavior deviates from the “standard” patterns. An example of such unusual behavior is presented in Fig. 4. for a model lactose system, a significantly higher resistance (i.e., a very low sublimation rate) was determined at the beginning of the primary drying [63]. Visual recording during primary drying showed a gradual progression of the sublimation front in a vial with 10% sucrose, while no sublimation was observed in the vial with lactose for the first 2 h of the primary drying (Fig. 4). This observation is consistent with the R_p measurements for 10% lactose performed using MTM. The very low sublimation rate for the lactose solution was attributed to a " skin " formation on the top of the frozen product. This would lead to a higher resistance to water vapor transfer, especially at the beginning of freeze-drying. However, it is still unclear why the skin would form in the lactose solution but not in the sucrose system.

According to M.Pikal [54], a heat transfer coefficient is defined as the ratio of the area-normalized heat flow to the temperature difference between the heat source and heat sink. For the vials directly sitting on the shelf of the dryer (most common case), he defined the heat transfer coefficient, K_v, as.

Kv=dqdt1Aout1(TSh.S-Tbottom), where A_out is the cross-sectional area of the vial calculated from the vial outer diameter, T_Sh.S is the temperature of the shelf surface, and T_bottom is the temperature of the product at the bottom center of the vial.

As described by [54], the heat transfer coefficient (K_v) is a function of pressure, heat radiation from the shelves (bottom and top), and direct contact between the vial and shelf. These three contributing factors are vial specific (depending, e.g., on the vial’s dimensions, wall thickness, and bottom concavity) and need to be measured in the experiment. Figure 5 (left panel) shows K_v = f(P) for W5816, K5816, and 5303 vials.

Measuring shelf surface temperature is difficult, especially for fully loaded dryers. As an alternative, the inlet temperature of heat transfer fluid is used to calculate K_v (Fig. 5, right panel). Note that K_v, calculated from the temperature difference between inlet temperature and product temperature, is not only pressure or vial position-dependent (edge vials) but also varies from one freeze-dryer to another due to differences in shelf heat transfer coefficient. One can also notice a trend, with the heat transfer coefficient K_v typically decreasing with an increase in the size of vials, especially at high pressures.

According to [65] atypical radiation heat transfer is responsible for higher sublimation rates for vials located at the front and side of an array. This so-called “edge effect” could be as high as 50% [66] The edge effect (factor 1.5 used in simulations) is also an input parameter to the model.

Two critical characteristics of freeze-dryers are needed for the successful scale-up and transfer of lyophilization processes: minimum controllable pressure (or maximum sublimation rate at a given pressure) and maximum sublimation rate (typically due to condenser overload) for a particular dryer. The common understanding is that the dryer's performance could be limited by 1) the onset of choked flow, which limits the ability to control pressure, or by condenser capacity, where condenser temperature increases excessively. Due to geometric differences and limitations in the design laboratory, pilot and commercial dryers could respond differently to sublimation rates [67]. A total sublimation rate profile during simulation is calculated as a sublimation rate profile from the individual vial multiplied by the number of vials. Then, it is compared to the pressure the freeze-dryer could maintain at a particular sublimation rate (minimum controllable pressure, P_min). Minimum controllable pressure could be obtained by either a sublimation test or by using computational fluid dynamics modeling (CFD) [68, 69]. An example of P_min as a function of sublimation rate for different freeze-dryers is shown in Fig. 6.

Note that normalized sublimation rate values, plotted in Fig. 6, showed up to 1.5 kg/hr/m² while actual maximum sublimation rates values tested for these dryers were 0.8 kg/hr (Lyostar II), 5.45 kg/hr for Lyomax 6 and 50 kg/hr for Lyomax 42. In sublimation tests used during the characterization of these dryers, condenser temperature never exceeded -40°C, which we consider a critical point beyond which water vapor may not be able to condense fully, potentially resulting in a vacuum pump failure. Therefore, the true value of the maximum sublimation rate for these dryers would be when the condenser temperature is above -40°C. For most laboratory-scale dryers, the condenser temperature remains below -70C even when the flow is fully choked due to normally high refrigeration capacity. Therefore, the relationship between minimum controllable pressure and sublimation rate is typically linear (red line in Fig. 6). More details on this subject can be found in a recent paper [32]_.

The normalized sublimation rate of 1.5 kg/hr/m2 could appear to be very high and nonrealistic. It corresponds to shelf temperature values above 40°C and pressures above 200 mT cycle conditions for the products with moderate cake resistance values (15 Torr*hr*cm² at 1 cm for 10% solids, calculated as an example). Product temperatures, in this case, could exceed -10°C and stay just below the formulation's melting point during primary drying. While cycle conditions discussed above were not common in the past, in recent days and, especially in the future, one should expect more aggressive cycles to be implemented using bulking agents and freeze-drying well above Tg’. Most modern freeze-dryers can support high sublimation rates (up to 1.5 kg/hr/m²).

With R_p and K_v established and knowing the limitation of the freeze-dryer, primary drying conditions can be established using a computer model. Several primary drying-based models are listed in Table I. To compare the models, product temperature, sublimation rate, and primary drying time are calculated for multiple combinations of formulation/vial/fill volume/shelf T/chamber pressure; the complete dataset is provided in Supporting Information, Appendix (Tables A2-A4)1. Product temperature, primary drying time, and sublimation rate are similar between all three models As an example, product temperature at the bottom of the vial and primary drying time of the same product (5% mannitol in a 2-ml vial with a fill volume of 1.35 ml) and identical shelf temperature and chamber pressure (15 °C and 0.225 Torr, respectively) outputs for three models are provided in Table I. All three models provide comparable results. VMCa and LyoPRONTO product temperature traces for this example are also similar at every timepoint, with a maximum difference below 0.6 °C in two instances, within a range of 0.4 to 0.5 °C in 12 instances and below 0.3 °C for the rest of the data set (data not shown). The differences in the product temperature profiles between the VMCa and STCa data are negligible. The LyoPRONTO model shows a minor difference in the sublimation rate form the other 2 Lyocalculators. This can be explained because it uses a different type of solver, i.e., an optimizer package in Scipy, while the two other models use Excel-based and “while loops” solvers.

For any computer model to be valid, it is essential to confirm that computational results agree with the experiment. There are several publications in which different computer models of primary drying were compared with experiments. Specifically, a non-steady state PASSAGE model, which solves the heat and mass transfer equations in frozen and dried regions of a single vial by finite element formulation in 2-D axisymmetric space, has been verified for several freeze-dried products in glass vials [70]. Quasi-steady state approach, which commonly employs Excel-based calculations, has also been shown to accurately predict product temperature and primary drying time for several products [56].

In this study, the STCa Lyo model is compared with the experiment for four products (Table II), and the results are presented in Fig. 7 and Table III. The product temperature predicted by the model shows an excellent agreement with the experimental data, with the difference not exceeding 0.5° C. For the primary drying time, the most significant error (about 32%) was found in the prediction of drying time for high solids formulation (Table III). For the rest of the cases, the error in the prediction of drying time was less than 13%. Note that a comparison was made between the calculated and end point values estimated by the offset of Pirani, which is the most conservative approach to time prediction.

Overall, the STCa model, and by extension, two other models from Table I, are expected to provide an accurate estimate of the product temperature and a reasonable (within 30%, except for high solids case) prediction of the primary drying time.

All cycles were performed above Tg’ of formulations but below collapse (measured by freeze-drying microcopy) temperature.

Data of Table III show that the primary drying model (STCa) well predicts product temperature at the middle point of drying (approximately 60% from the beginning of the ramp) for four types of products. At the same time, the difference between the predicted value and actual measured product temperature is about 2°C in the beginning of PD, when the shelf temperature reaches the set point. Note that predicted drying time is always shorter than drying time measured either by product temperature or Pirani (in both cases using offset values). However, differences do generally not exceed 15% except for high solids content amorphous formulation.

To build the database, the following variables are included (Table IV):

These parameters are used to calculate product temperature, primary drying time, and sublimation rates for the center and edge vials for 46,962 product/primary drying set-points combinations and presented as a database. Needless to say, it would be impossible to cover all combinations of input parameters. Nevertheless, the range of parameters selected here should cover most practical cases. A user could bracket a particular input parameter if a particular condition is not covered in the database. In some cases, two or more brackets could be required. Several examples of using the database are given later in the paper.