LyoPRONTO: an Open-Source Lyophilization Process Optimization Tool
- 285 Downloads
This work presents a new user-friendly lyophilization simulation and process optimization tool, freely available under the name LyoPRONTO. This tool comprises freezing and primary drying calculators, a design-space generator, and a primary drying optimizer. The freezing calculator performs 0D lumped capacitance modeling to predict the product temperature variation with time which shows reasonably good agreement with experimental measurements. The primary drying calculator performs 1D heat and mass transfer analysis in a vial and predicts the drying time with an average deviation of 3% from experiments. The calculator is also extended to generate a design space over a range of chamber pressures and shelf temperatures to predict the most optimal setpoints for operation. This optimal setpoint varies with time due to the continuously varying product resistance and is taken into account by the optimizer which provides varying chamber pressure and shelf temperature profiles as a function of time to minimize the primary drying time and thereby, the operational cost. The optimization results in 62% faster primary drying for 5% mannitol and 50% faster primary drying for 5% sucrose solutions when compared with typical cycle conditions. This optimization paves the way for the design of the next generation of lyophilizers which when coupled with accurate sensor networks and control systems can result in self-driving freeze dryers.
KeywordsFreeze-drying Lyophilization Heat and mass transfer Quality by design (QBD) Process optimization Freezing model
Lyophilization refers to the energy and time intensive process of solvent (typically water) removal used to improve the long-term storage stability of perishable materials (1,2). A typical pharmaceutical lyophilization cycle is composed of three stages namely, freezing, primary drying, and secondary drying. An optimal lyophilization cycle is one that achieves the highest drug quality for the least cost (3). Of the three stages, primary drying is usually the longest part of the cycle (4) and its optimization shortens the cycle time resulting in a higher throughput for a given lyophilizer and thereby, lower manufacturing cost (3). On the other hand, a non-optimal cycle not only takes longer and costs more than essential but may also compromise drug stability (3).
Nakagawa et al. (5) and Hottot et al. (6) developed a two-dimensional finite-element model based on the models proposed by Qin et al. (7) and Lunardini (8) for freezing analysis in lyophilization vials. In the present work, we propose a simplified mathematical model of the freezing step using the lumped capacitance method of heat transfer analysis (9). We compare the simulated results with experimental measurements. The simplified freezing calculator is one of the modes of operation of the Lyophilization Process Optimization Tool (LyoPRONTO) (10), written using Python.
The current practice in lyophilization is to control the chamber pressure and shelf temperature at constant setpoints based on rules of thumb using open-loop control. The setpoint values are chosen conservatively which leads to an energy efficiency of less than 5% for the cycle. Typically, the chamber pressure is maintained at a setpoint between 50 and 200 mTorr (3). The shelf temperature is maintained at a constant value that ensures that the product temperature does not exceed the target temperature by a predetermined tolerance of 1–2°C at any point during the primary drying process. The target product temperature is taken to be 2–5°C below the critical value beyond which the product appearance is unacceptable (3). The entire freeze-drying process could take days or even weeks to finish when these input conditions are not optimized (2,17).
Although operating at any condition within the yellow region in Fig. 2 is acceptable, the most optimal operating point is at the intersection of the equipment capability curve and the product temperature isotherm corresponding to the maximum allowable product temperature on the design space (12). This point, indicated by the blue triangle, provides the constant setpoints of chamber pressure and shelf temperature for the primary drying for maximum cycle efficiency. However, at every instant during the drying, this point changes due to the variation of product resistance with cake length. The design space is constructed using the largest value of product resistance which occurs towards the end of the cycle when the cake length is maximum. This ensures that the product temperature limit is not exceeded at any point in the cycle. However, varying the chamber pressure and shelf temperature in real time during the drying, instead of maintaining them at constant setpoints, allows for the most optimal operation throughout the drying process and not just at the end, leading to higher energy efficiency.
In this work, we present the capabilities of LyoPRONTO which can be used as a simple lyophilization calculator for the freezing and primary drying steps, as a primary drying design-space generator, and as a process optimizer. The 0D freezing model in the “freezing calculator” mode predicts the time variation of the product temperature during the freezing and the total freezing time. The 1D quasi-steady heat and mass transfer analysis through a vial in the “primary drying calculator” mode predicts the time variation of product temperature and cake length, as well as the total primary drying time when the chamber pressure and shelf temperature setpoints are specified along with the heat transfer and product parameters. In addition, the heat transfer parameters can be determined based on the experimental drying time, and product resistance can be calculated based on the experimental product temperature profile. Moreover, the input chamber pressure and shelf temperature setpoints specified need not stay constant throughout during drying but can be specified as a function of time, and the shelf temperature ramping from freezing or annealing to primary drying is accounted for. The “design-space generator” mode extends this to a range of specified chamber pressure and shelf temperature setpoints. The “optimizer” mode determines the most optimal chamber pressure and/or shelf temperature in real time within their bounding values for given product, load, and equipment characteristics. This method, when coupled with accurate measurements of instantaneous sublimation rates using a network of pressure and/or temperature sensors, can achieve closed-loop control of the process resulting in self-driving freeze dryers.
The remainder of this paper is organized as follows. The next section presents the modeling methodology used in LyoPRONTO. The following section describes the experimental setup used for comparing typical freeze-drying cycles with modeling results. “BENCHMARKING OF THE MODEL” presents the studies used to compare LyoPRONTO results with published modeling and experimental results, as well as our own experimental measurements. Finally, the “PROCESS OPTIMIZATION” outlines the need for variable input parameter lyophilization cycles, and the reduction in primary drying time for 5% mannitol and 5% sucrose formulations when optimized variable chamber pressures and shelf temperatures are used instead of typical cycle setpoints.
LyoPRONTO performs a 0D calculation of the transient heat conduction during the freezing process. The model assumptions are (i) lumped capacitance heat transfer, (ii) constant product temperature during ice crystallization, and (iii) heat is transferred to the product only from the shelf. Lumped capacitance method refers to the assumption of low resistance to conduction within the product when compared with the heat transfer between the product and its surroundings and thereby, of a spatially uniform temperature within the product. The temperature gradient within the product is small when the Biot number, Bi ≪ 1, and large for Bi ≫ 1 (9). Biot number is defined as Bi = hLc/k where, h is the heat transfer coefficient, Lc is the characteristic length taken as the maximum of the fill height of the product or diameter of the vial, and k is the thermal conductivity of the product. In general, the error associated with the lumped capacitance method is very small for Bi < 0.1 (9). During freezing of the product in a vial, the average Bi during cooling (based on water) is ~ 0.6, and during crystallization and solid cooling is ~ 0.1. Since these values are slightly greater than the cut-off value, the lumped capacitance method leads to a higher error, but the agreement is taken to be satisfactory owing to the stochastic nature of the nucleation process which leads to a significant error even when the time-intensive finite-element method is used. This can be seen in the work of Nakagawa et al. (5), where a reduction in the specified nucleation temperature from − 2.5 to − 4.5°C leads to a 20% reduction in the ice crystallization time, and a further reduction in the specified nucleation temperature from − 4.5 to − 10°C leads to a 34% reduction in the ice crystallization time.
where, Hf is the latent heat of fusion of ice. The temperature is assumed to be constant and equal to Tf during the crystallization. It is significant to note that this method neglects the heat transfer between the ambient air and the product and only considers heat transfer between the shelf, the vial, and the product. Solid cooling occurs once the crystallization is complete and is modeled using Eq. 1 with material properties corresponding to the solid phase of the product (ice).
A limitation of this model is that experimentally measured nucleation and freezing temperatures must be provided as inputs to the model. Ice nucleation is spontaneous and random, and the nucleation temperature depends on the solution properties, process parameters, environmental factors, container characteristics, and the presence of particulate matter (18). One method of overcoming the large variability in these stochastic parameters is to use a range of typical temperatures at which nucleation and freezing occur based on the literature and previously obtained experimental data. The model provides a range of expected output properties and the process parameters can be modified in order to ensure that the outputs lie in the desired range of values. Controlled nucleation technology (18,19) can also be used to eliminate the uncertainty in the nucleation process and there are multiple methods of achieving this. Seeding the product with inorganic compounds or bacteria, pre-treating the vials to increase the surface roughness, ultrasound nucleation, and vacuum-induced freezing are known methods of controlling nucleation. The ice-fog technique is another popular approach to control nucleation by introducing cold nitrogen gas at pressures close to atmospheric conditions into the chamber to freeze the moisture content. The ice crystals thus formed induce nucleation at the surface of the solution. Using controlled nucleation, it is possible to accurately predict the nucleation temperature and mitigate the requirement of experimental calibration for the model.
Primary Drying Calculator
LyoPRONTO performs heat and mass transfer modeling to determine primary drying time as well as the sublimation flux and maximum product temperature as functions of time during primary drying. The model assumptions are (i) 1D quasi steady state heat and mass transfer, (ii) center vial is representative of the entire batch, and (iii) convective heat transfer can be neglected. The lyophilization calculator and design-space generator are based on previously published 1D models (4,20, 21, 22, 23). We ignore the non-uniformity in heat transfer due to edge effect and consider the center vial to be representative of the entire batch. The heat transfer can thus be approximated as one-dimensional due to symmetry about the central axis of the vial.
where, Kc, Kp , and KD are fitting parameters that depend on the vial and the lyophilizer.
LyoPRONTO also has modes where Kv and Rp can be determined if they are unknown. Instead of additional time-consuming experiments to determine the heat transfer characteristics, Kv is determined by iterating through a range of values to match the calculated drying time with a known experimental drying time for the same input conditions. Moreover, if this can be determined at three or more chamber pressures, a curve fit through the points provides the three coefficients, Kc, Kp , and KD for the given vial and lyophilizer combination. If the product temperature profile is known from experiments, this is used to calculate Rp as a function of cake length, which in turn is curve fit to determine the coefficients, R0, A1 , and A2.
where, a and b can be determined using experimental choked flow tests (29), minimum controllable pressure tests (30), or using computational fluid dynamics (CFD) modeling of choked flow through a lyophilizer (23,31,32). The critical product temperature isotherm is determined by fixing the vial bottom temperature at the maximum allowable value (Tpr, max), which is typically taken to be 2 to 3°C below the critical temperature. In this case, the chamber pressure is fixed, but the shelf temperature is not. The sublimation front temperature, and thereby the sublimation rate, at every time step are determined based on the fixed Tbot. It is worth noting that LyoPRONTO accounts for the shelf temperature ramping from the freezing to the primary drying stages based on the ramping rate and freezing shelf temperature provided.
We perform experiments at constant chamber pressure and shelf temperature setpoints for comparison with the modeling results in order to evaluate the performance of LyoPRONTO. We use the laboratory-scale freeze-dryer REVO by Millrock Technology, Kingston, NY for our experiments. One of the four loadable shelves of the lyophilizer is loaded fully with 398 Schott 6R vials filled with 2 mL of 5% mannitol solution each. For the freezing step, we ramp the shelf temperature from room temperature to − 20°C at a rate of 1°C/min and hold for 2 h. Next, we ramp the shelf temperature to − 5°C and hold for 1 h. The primary drying is carried out at a shelf temperature of − 5°C with the chamber pressure in the 100 to 1500 mTorr range. The experiment at 300 mTorr is repeated for the same conditions using pure water as the product in order to evaluate the performance of the lumped capacitance freezing model.
Product Resistance Parameters According to Eq. 5 for Products Modeled Here
× 104 m/s
cm2 h Torr/g
× 107 s−1
cm h Torr/g
× 102 m−1
5% mannitol (11)
5% povidone (11)
0.5% lysozyme at Tsh = − 25 ° C (34)
0.5% lysozyme at Tsh = 25 ° C (34)
2% lysozyme at Tsh = − 25 ° C (34)
2% lysozyme at Tsh = 25 ° C (34)
0.5% BSA at Tsh = − 25 ° C (34)
0.5% BSA at Tsh = 25 ° C (34)
2% BSA at Tsh = − 25 ° C (34)
2% BSA at Tsh = 25 ° C (34)
0.5% IgG at Tsh = − 25 ° C (34)
0.5% IgG at Tsh = 25 ° C (34)
2% IgG at Tsh = − 25 ° C (34)
2% IgG at Tsh = 25 ° C (34)
5% sucrose (22)
BENCHMARKING OF THE MODEL
Comparison with Experimental Measurements
The crystallization and solid cooling steps can be clearly distinguished in the product temperature profiles. The simulated crystallization time is about twice the experimental value for both cases due to the assumption of lumped capacitance. The product temperature profiles during solid cooling obtained using LyoPRONTO and thermocouples display similar trends and converge to the shelf temperature by the end of the solid cooling process. The fluctuation in the measured product temperature profile near the shelf temperature setpoint value towards the end of solid cooling is expected to be due to the variation in the shelf surface temperature itself as shown by the measured shelf temperature profile. While setting up a freezing process during a freeze-drying cycle, it is required to specify the time for which the shelf temperature must be maintained at the value corresponding to the freezing step. Beyond this time, the setpoints are changed to the annealing or primary drying step values. If a range of expected Tf and Tn values are known based on previous experiments, the freezing model can predict a range of times required for the completion of the freezing process. Since the model overpredicts the crystallization time when compared with the experiments, it is safe to use these values to predict the total time required for the freezing step. Since the uncontrolled nucleation process is stochastic, the agreement between the model and experiments can be taken to be satisfactory when designing a freezing cycle where a factor of safety can be incorporated. This model has the added advantage of being user friendly, computationally less expensive than a multi-dimensional finite element model, and fast.
Schott 6R Vial Heat Transfer Coefficient in REVO Based on Experimental Drying Time at Different Chamber Pressures
Chamber pressure (mTorr)
Experimental drying time (h)
Simulated drying time (h)
10−4 cal/s K cm2
Heat Transfer Parameters According to Eq. 3 for Different Vial/Tray and Lyophilizer Combinations
Heat transfer coefficients
Kc = 11.51 W/m2 K = 2.75 × 10−4cal/s K cm2
Kp = 0.28 W/m2 K Pa = 8.93 × 10−4cal/s K cm2 Torr
KD = 3.45 × 10−3Pa−1 = 0.46 Torr−1
5800 W (11)
Highly modified commercial laboratory-scale lyophilizer
Kc = 11.04 W/m2 K = 2.64 × 10−4cal/s K cm2
Kp = 1.03 W/m2 K Pa = 33.2 × 10−4cal/s K cm2 Torr
KD = 27.3 × 10−3Pa−1 = 3.64 Torr−1
5816 W (11)
Kc = 8.49 W/m2 K = 2.03 × 10−4cal/s K cm2
Kp = 1.03 W/m2 K Pa = 33.2 × 10−4cal/s K cm2 Torr
KD = 29.8 × 10−3Pa−1 = 3.97 Torr−1
Kc = 6.36 W/m2 K = 1.52 × 10−4cal/s K cm2
Kp = 1.03 W/m2 K Pa = 33.2 × 10−4cal/s K cm2 Torr
KD = 52.3 × 10−3Pa−1 = 6.97 Torr−1
Warped stainless steel tray (11)
KTc = 2.51 W/m2 K = 0.6 × 10−4cal/s K cm2
KTp = 2.04 W/m2 K Pa = 65.9 × 10−4cal/s K cm2 Torr
KTD = 0.2 Pa−1 = 27 Torr−1
Wheaton Science 2 mL type-1 tubing vials (34)
SP Scientific LyoStar II
For Pch = 57 mTorr and Tsh = − 25 ° C
Using thermocouple measurements:
Kv = 15.9 W/m2 K = 3.8 × 10−4cal/s K cm2
Using Pirani gauge measurements:
Kv = 11.3 W/m2 K = 2.7 × 10−4cal/s K cm2
For Pch = 57 mTorr and Tsh = 25 ° C
Using thermocouple measurements:
Kv = 8.37 W/m2 K = 2.0 × 10−4cal/s K cm2
Using Pirani gauge measurements:
Kv = 7.95 W/m2 K = 1.9 × 10−4cal/s K cm2
Comparison with Published Data
Effect of the Iterative Method
The popular iterative method among a majority of the existing lyophilization calculators in the pharmaceutical community is to divide the product length equally into a fixed number of divisions, and perform calculations from a product length of zero to the maximum. In general, the number of divisions used is 5 or 10, and in some rare cases goes up to 100. On the other hand, LyoPRONTO uses the time stepping method to proceed through the primary drying process. The time starts at 0 and proceeds till the entire length of the product is dried in steps which can be specified as an input.
For the faster cycle shown in Fig. 13b, the drying time is less than 6 h and assuming a time step of 5 h leads to a 16% underestimation of the drying time. However, the solutions are converged for the 1 h time step case as well as the ten product length divisions case. This is because ten divisions produce time steps of approximately 0.6 h. Thus, it is essential to verify whether time step convergence has been achieved for each new cycle condition that is simulated. For the freeze drying of products with low critical temperatures using manufacturing scale lyophilizers, the cycle length may be very large and using 10 or sometimes even 100 divisions of the product length might be insufficient for a converged solution. LyoPRONTO allows for the specification of the time step as an input and the default value used is 0.05 h (3 min). The added advantage of using time stepping instead of product length stepping is the ability to specify the chamber pressure and shelf temperature as functions of time in order to evaluate their effect on the primary drying characteristics.
Design-Space Variation During a Cycle
When the drying begins, and the cake length is 0, the most optimal chamber pressure at the maximum shelf temperature of 120°C is 480 mTorr. This value falls to 120 mTorr when half the drying is complete, and the cake length is equal to half the initial product length. Towards the end of drying, the inputs are purely limited by the maximum product temperature and the most optimal values are the minimum chamber pressure of 50 mTorr beyond which leakage may occur in the chamber, and a shelf temperature of 110°C. This shows that the optimal primary drying parameters varies significantly and continuously throughout the process and that it is imperative to optimize them in real time in order to improve the process time and energy efficiency.
Cycle Optimization with Variable Inputs
The constraints imposed for the optimized cycle are the maximum product temperature of − 5°C and equipment capability as described in the previous subsection. The maximum shelf temperature and minimum chamber pressure are limited to 120°C and 50 mTorr, respectively. The primary drying time for the typical cycle is 5.11 h. As seen in Fig. 14, these conditions are not the most optimal values and the sublimation flux could be improved significantly by maintaining the cycle parameters at their optimal values. This can also be seen in the maximum product temperature in Fig. 15a which is 12 to 30°C below the limiting value throughout the cycle. Figure 15b shows the results for a variable pressure cycle when the shelf temperature is maintained at 30°C. The drying time reduces by 41% to 2.99 h. Since only one shelf is loaded, the cycle is never limited by the equipment capability on account of low overall sublimation rates. The limiting factor, therefore, is the product temperature and the optimization is carried out by maintaining Tbot at the limiting product temperature of − 5°C, Tsh at 30°C, and determining the chamber pressure to obtain the highest sublimation rate possible under the given conditions.
Figure 15c shows that optimizing the shelf temperature at a constant chamber pressure of 150 mTorr leads to a further improvement in drying time which drops to 2.11 h. The cycle parameters are limited by the maximum shelf temperature of 120°C till 40% of the product is dried. This is because at 150 mTorr chamber pressure and 120°C shelf temperature, the highest sublimation flux possible does not produce a maximum product temperature which is greater than the limiting value. Tpr, max increases continuously till it reaches the limiting value, and beyond this point, the cycle parameters become product temperature limited. The shelf temperature decreases continuously to ensure the − 5°C limit for the product is not exceeded.
Figure 15dshows the most optimal primary drying process possible for the given conditions and constraints. Both the chamber pressure and shelf temperature are variable. The figure shows that for most of the cycle, the maximum shelf temperature and maximum product temperature are both limiting factors. In other words, the most optimal chamber pressure is the one that produces the maximum sublimation flux when the shelf temperature and product bottom temperature are fixed at their maximum limiting values. The driving force for sublimation is the pressure difference between the sublimation front and the chamber. As the drying progresses, the product resistance increases, and a greater pressure difference is required to sustain the maximum sublimation flux. Consequently, the Pch, opt reduces continuously till the minimum chamber pressure limit is reached when 83% of the product is dried. For the rest of the cycle, this minimum Pch is the limiting factor. The total drying time is 1.96 h and the optimization leads to 62% reduction in the primary drying time when compared with the typical cycle.
In all these simulations, we do not consider the effects of heat transfer non-uniformity encountered in the lyophilization chamber. We assume that the vial heat transfer coefficient and drying time are the same for all the vials loaded in the chamber and equal to those of the center vials on account of their longer drying time. A simple way for incorporating this in the simulations is to divide the total number of vials into center and edge vials and use different vial heat transfer coefficients for each set. With more accurate process monitoring using sensor networks, the measurements could be coupled with the code to determine the most optimal conditions that not only minimize the total drying time within the constraints, but also minimize the non-uniformity in the chamber. The maximum shelf temperature that can be used without product lift-off or vial blow-out must be characterized experimentally and this must be imposed as the limiting value. Analysis of the residual moisture content and specific surface area tests to evaluate cake shrinkage in addition to the experimental demonstration of variable process parameter optimization are to be included in future work.
The Lyophilization Process Optimization Tool or LyoPRONTO is an open-source application that can be used to model various aspects of the lyophilization process for a given cycle and to design more efficient cycles. The existing freezing models require 2D or 3D finite element analysis of the product in the vial which is time consuming and computationally expensive. LyoPRONTO presents a novel method of applying a 0D lumped capacitance heat transfer model to predict the crystallization time and product temperature profile during the freezing process. The overprediction of the ice crystallization time when compared with experiments can be attributed to the stochastic nature of the nucleation process and the assumption of a 0D model. The results are qualitatively accurate and can quantitatively be used to design freezing cycles provided a factor of safety is included to account for the stochasticity. The 1D quasi-steady heat and mass transfer analysis of primary drying is based on existing models but using time steps instead of steps of product length. The modeling results deviate on average by 3% from experimental measurements. The tool is also capable of determining the vial heat transfer characteristics and product resistance parameters based on the experimental drying time and product temperature profile respectively which reduces the number of overall experiments required for the complete characterization of a lyophilization cycle.
LyoPRONTO can generate the primary drying design space for a given product, vial, load, and lyophilizer combination, with the most optimal setpoint of operation being the point of intersection of the equipment capability curve and the maximum allowable product temperature isotherm. Since this optimal point varies at each instant of time during a cycle, the most efficient cycle is one with variable time-dependent chamber pressure and shelf temperature profiles set at their optimal values at all times. The optimizer tool in LyoPRONTO is designed to determine such a variable profile for chamber pressure and/or shelf temperature within the constraints of the equipment, product, and practical limitations. The time interval for the variation can also be changed as an input parameter. Optimal cycles with 3 min variations in chamber pressure and shelf temperature result in a 62% reduction in the primary drying time for a 5% mannitol cycle in a laboratory scale when compared with a typical single setpoint cycle. This reduction is 50% for 5% sucrose solution and the optimization results in a significant primary drying time reduction under both partial and full load conditions.
This work was supported by NSF Partnerships for Innovation-Research Partnerships (PFI-RP) Grant Number 1827717. The authors would like to thank Dr. Steven Nail and Dr. Gregory Sacha for useful discussions and acknowledge the use of the Cake Resistance Library in the Excel-based Lyocycle Design and Transfer Template developed by Dr. Serguei Tchessalov at Pfizer Inc.
- 1.Pikal MJ. Freeze-drying of proteins. Part I: process design. BioPharm. 1990;3(8):18–27.Google Scholar
- 2.Gatlin LA, Nail SL. Freeze drying: a practical overview. Bioprocess Technology. 1993;18:317–7.Google Scholar
- 4.Koganti VR, Shalaev EY, Berry MR, Osterberg T, Youssef M, Hiebert DN, et al. Investigation of design space for freeze-drying: use of modeling for primary drying segment of a freeze-drying cycle. AAPS PharmSciTech. 2011;12(3):854–61.Google Scholar
- 8.Lunardini VJ. Heat transfer in cold climates: Van Nostrand Reinhold Company; 1981.Google Scholar
- 9.Bergman TL, Incropera FP, DeWitt DP, Lavine AS. Fundamentals of heat and mass transfer: Wiley; 2011.Google Scholar
- 10.G. Shivkumar, P. S. Kazarin and A. A. Alexeenko. Lyopronto lyophilization process optimization tool. Purdue University [Online]. 2019. Available from lyopronto.org.Google Scholar
- 11.Pikal MJ. Use of laboratory data in freeze drying process design: heat and mass transfer coefficients and the computer simulation of freeze drying. PDA J Pharm Sci Technol. 1985;39(3):115–39.Google Scholar
- 12.Nail SL, Searles JA. Elements of quality by design in development and scale-up of freeze-dried parenterals. Biopharm Int. 2008;21(1).Google Scholar
- 15.Sundaram J, Hsu CC, Shay Y-HM, Sane SU. Design space development for lyophilization using DOE and process modeling. BioPharm International. 2010;23(9).Google Scholar
- 16.Searles J. Observation and implications of sonic water vapor flow during freeze-drying. Am Pharm Rev. 2004;7:58–69.Google Scholar
- 17.Franks F. Freeze drying: from empericism to predictability. Cryo-Letters. 1990;11:93–110.Google Scholar
- 18.H. Gieseler. Controlled nucleation in freeze-drying. European Pharmaceutical Review, No. 5, 2012.Google Scholar
- 22.Tchessalov S, Dassu D, Latshaw D, Nulu S. An industry perspective on the application of modeling to lyophilization process scale up and transfer. Am Pharm Rev. 2017;20.Google Scholar
- 23.Zhu T, Moussa EM, Witting M, Zhou D, Sinha K, Hirth M, et al. Predictive models of lyophilization process for development, scale-up/tech transfer and manufacturing. Eur J Pharm Biopharm. 2018;128:363–78.Google Scholar
- 32.Shivkumar G, Kshirsagar V, Zhu T, Sebastiao IB, Nail SL, Sacha GA, et al. Freeze-dryer equipment capability limit: comparison of computational modeling with experiments at laboratory scale: Journal of Pharmaceutical Sciences; 2019.Google Scholar
- 34.Lewis LM, Johnson RE, Oldroyd ME, Ahmed SS, Joseph L, Saracovan I, et al. Characterizing the freeze-drying behavior of model protein formulations. AAPS PharmSciTech. 2010;11(4):1580–90.Google Scholar
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.