Modeling the Secondary Drying Stage of Freeze Drying: Development and Validation of an Excel-Based Model

Journal of Pharmaceutical Sciences xxx (2016) 1-13

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Pharmaceutical Biotechnology

Modeling the Secondary Drying Stage of Freeze Drying: Development and Validation of an Excel-based Model Ekneet K. Sahni 1, 2, Michael J. Pikal 2, * 1 2

Global Manufacturing Science and Technology, Pfizer Inc., McPherson, Kansas 67460 Department of Pharmaceutical Sciences, School of Pharmacy, University of Connecticut, Storrs, Connecticut 06269

a r t i c l e i n f o

a b s t r a c t

Article history: Received 1 August 2016 Revised 23 October 2016 Accepted 25 October 2016

Although several mathematical models of primary drying have been developed over the years, with significant impact on the efficiency of process design, models of secondary drying have been confined to highly complex models. The simple-to-use Excel-based model developed here is, in essence, a series of steady state calculations of heat and mass transfer in the 2 halves of the dry layer where drying time is divided into a large number of time steps, where in each time step steady state conditions prevail. Water desorption isotherm and mass transfer coefficient data are required. We use the Excel “Solver” to estimate the parameters that define the mass transfer coefficient by minimizing the deviations in water content between calculation and a calibration drying experiment. This tool allows the user to input the parameters specific to the product, process, container, and equipment. Temporal variations in average moisture contents and product temperatures are outputs and are compared with experiment. We observe good agreement between experiments and calculations, generally well within experimental error, for sucrose at various concentrations, temperatures, and ice nucleation temperatures. We conclude that this model can serve as an important process development tool for process design and manufacturing problem-solving. © 2016 American Pharmacists Association®. Published by Elsevier Inc. All rights reserved.

Keywords: secondary drying Excel-based model mass transfer coefficient lyophilization process freeze drying heat and mass transfer

Introduction Freeze drying is increasingly being employed to manufacture the final dosage form for a variety of injectable products, including therapeutic proteins and vaccines. Although the freeze-drying process is complex, with many interacting variables, the basic physics is relatively well understood, which means that modeling the process can be a very useful endeavor, particularly for the drying stages. Modeling freeze drying has a long history1-7 with some of the approaches being very rigorous approaches based on solving coupled differential equations governing heat and mass transfer, both in primary drying and in secondary drying.4-6 However, very simple pseudo steady state models for primary drying have also proved very useful.8 Primary drying, or the ice sublimation stage, is normally the longest phase of the process, and it is during primary drying that improper heat input can either result in exceeding the collapse temperature and suffering loss of product elegance or lead to unnecessarily long processes. Thus, primary

Conflicts of interest: The authors have no conflict of interest. * Correspondence to: Michael J. Pikal (Telephone: 1-860-486-3202). E-mail address: [email protected] (M.J. Pikal).

drying is usually the target for process optimization, and simple steady state models can be very useful in such exercises. Examples of applications include development of a design space for primary drying9,10 as well as estimation of the impact of variability in the process on product temperature history and product quality.8 These applications use an Excel-based version of the original algorithm that is often termed the “LyoCalculator.”8 Here, the software (Excel) is generally available, the input is quick, and the product temperature history and drying times are evaluated essentially instantly with high accuracy based on comparisons with experimental data8 or the more rigorous nonesteady state differential equationebased models.5 Experimental or estimated values of the vial heat transfer coefficient, specific to the vials used, and values of the mass transfer coefficient for vapor flow through the dry layer, specific to the product, are needed input parameters. Secondary drying is the stage of the process that begins in a local region of the product once ice sublimes from that region, meaning some secondary drying occurs during the ice sublimation or primary drying stage. For example, with sucrose, the water content at the top of the cake is only about half the water content at the bottom where the ice was last present. In spite of this partial secondary drying during primary drying, the usual terminology defines the start of the secondary drying stage for a given vial as the

http://dx.doi.org/10.1016/j.xphs.2016.10.024 0022-3549/© 2016 American Pharmacists Association®. Published by Elsevier Inc. All rights reserved.

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time when all ice has been sublimed from that vial. However, for a batch, this definition is somewhat ambiguous because not all vials finish primary drying at the same time, and therefore enter the “secondary drying stage” at the same time. In most discussions, and in the context of this research, we define the start of secondary drying as the average start of secondary drying for the vials in the batch as a whole, meaning the end of primary drying for the average vial in the batch. The nonesteady state differential equationebased models consider both primary drying and secondary drying, including the secondary drying that occurs while ice is still in the vial.5 This rigor is compromised, however, by the fact that not all vials finish primary drying at the same time, so comparison with experiment is not entirely straightforward. In this research, we seek to develop a highly simplified model for secondary drying, which runs on Excel as does the LyoCalculator that is much easier to use than the rigorous models but still gives useful prediction for the impact of freezing variations and shelf temperature variations on the time course of residual water. The Excel-based model that we develop makes a number of simplifying approximations that are described later in this document. Two approximations are most critical. First, we consider the dry layer to be a composite of 2 homogenous regions, top and bottom. This is a crude approximation to the actual situation of variable water content from top to bottom at the end of primary drying. Secondly, the model is not based on solving coupled heat and mass transfer differential equations, but is in essence a series of steady state calculations of heat and mass transfer in the 2 halves of the dry layer where the time of drying is divided into a large number of time steps, Dt, where in each time step steady state conditions prevail. The assumptions are justified by the good agreement between calculations and experiment as well as the essentially exact agreement between the Excel-based model and the more rigorous differential equationebased model (Passage II).5 Applications would include optimizing the secondary drying process, as well as investigating the impact of formulation and freezing process on secondary drying. Such calculations would be particularly useful in cases where the optimal residual water content was “intermediate” between “dry” and “wet.” However, as with the more rigorous approach (Passage II),5 several key input parameters need to be estimated or evaluated experimentally. The 2 most important are the secondary drying mass transfer coefficient and the water desorption isotherm. The water desorption isotherm may be evaluated as a function of temperature and water activity by employing a “moisture balance,” and the mass transfer coefficient for secondary drying needs a detailed study of residual moisture during secondary drying, usually using a “sample thief” and traditional residual moisture assay.

Table 1 Solute Concentration, and Ice Nucleation and Secondary Drying Temperatures for Freeze-drying Experiments Sucrose Concentration (w/v)

Controlled Ice Nucleation Temperatures ( C)

Secondary Drying Temperatures ( C)

5% 5% 5% 5% 5% 10% 15%


5 C
5 C
5 C
7 C
10 C
5 C
5 C

25 C 40 C 50 C 40 C 40 C 40 C 40 C

times during secondary drying for residual water assay by Karl Fischer analysis. Aqueous solution of the solute was prepared and filtered through a 0.22-mm membrane filter. A total of 160 vials were filled with appropriate fill volume (5 mL for 5% w/w sucrose; 3 mL for 10% w/w and 15% w/w sucrose) and loaded onto the lowermost temperature-controlled shelf of the freeze dryer. The height of the shelf was adjusted to facilitate easy removal of vials during primary and secondary drying stages using the sample thief. Product temperature was measured using 30-gauge copperconstantan (type T) thermocouples (Omega Engineering, Inc., Stamford, CT) with a resolution of ±0.1 C. The thermocouples were calibrated using ice-water slush and those within ±0.5 C were used in the experiments. Thermocouples were placed in the bottom center of vials in the center of the vial array. The arrangement of thermocouple vials is shown in Figure 1. A Pirani/capacitance manometer comparative pressure measurement was employed to determine the end of primary drying for essentially all vials. Freeze drying was performed without collapse.

Materials and Methods Crystalline sucrose was obtained from Sigma-Aldrich Company (St. Louis, MO). Vials used for freeze drying were 20-mL tubing vials from West Pharmaceutical Company (Lionville, NJ) with 20-mm finish Daikyo Flurotec® stoppers (West Pharmaceutical Company) designed for freeze drying. Freeze Drying Freeze-drying experiments were performed in a laboratoryscale freeze dryer (Lyostar II, SP Scientific, Stone Ridge, NY), using sucrose at different concentrations (5% w/w, 10% w/w, and 15% w/ w) at several controlled ice nucleation temperatures (
5 C,
7 C, and
10 C) and secondary drying temperatures (25 C, 40 C, and 50 C) with conditions as shown in Table 1. A chamber door with a sampling thief was used to periodically remove samples at different

Figure 1. Representation of the vial map for the freeze-drying experiments. The outermost rows of vials (dark circles) represent edge vials and “X” represents thermocouple vials.

E.K. Sahni, M.J. Pikal / Journal of Pharmaceutical Sciences xxx (2016) 1-13

A reduced pressure ice fog technique11 was employed to carry out the controlled ice nucleation. The purpose of performing controlled ice nucleation was to reduce variability in drying rate and thereby reduce the variability in the residual moisture contents when sampling. When the vials reached the desired nucleation temperature (
5 C,
7 C, or
10 C), the chamber pressure was reduced on the order of 50 Torr. Subsequently, the valve separating chamber and condenser was closed and cold nitrogen gas was passed into the chamber in order to nucleate ice at that temperature. Next, the shelf temperature was lowered to
40 C at a ramp rate of 1 C/min to complete the freezing process. This was followed by primary and secondary drying stages at 60 mT chamber pressure as described below in the protocol for freeze drying. Experimental Procedure Protocol for Freeze Drying Freezing Step. Cool from room temperature to 5 C (1 C/min), equilibrate for 30 min; cool to
5 C (1 C/min), equilibrate for 30 min. When using controlled nucleation, this step was followed by the ice-fog procedure before finally cooling to the terminal freezing temperature of
40 C at 1 C/min. After a hold time of 120 min, primary drying was started. For controlled ice nucleation at
5 C,
7 C, and
10 C, the ramp to
40 C was begun after a delay period of 20 min. Primary Drying. The shelf temperature was raised to
25 C for sucrose concentration of 5% and 10%. For the higher sucrose concentration of 15%,
28 C was used as the shelf temperature during primary drying to minimize the possibility of collapse during primary drying. The chamber pressure was set to 60 mTorr for all conditions. Secondary Drying. The shelf temperature was raised to 25 C, 40 C, or 50 C in the separate lyophilization runs at a ramp rate of 0.2 C/ min for 5% w/w sucrose. For higher sucrose concentrations of 10% and 15%, the ramp rate was reduced to 0.1 C/min to minimize the possibility of exceeding the glass transition temperature at the bottom of the vial during early secondary drying. Samples were extracted after the average thermocouple vial showed the sharp increase in temperature indicating primary drying is over in that vial. This means, of course, that some vials were not yet quite out of primary drying and therefore it is possible that some vials removed for moisture analysis still contained ice. In order to minimize variation in moisture content, only the vials from the center of the vial array were removed. We note that, due to radiation effects, the drying of edge vials will be much quicker, leading to a significant difference in the residual water contents between edge and center vials at any given time during the drying process. First, at any given time during secondary drying, the average edge vial will have been in secondary drying longer than the average center vial. In addition, the higher vial heat transfer coefficient will provide a higher rate of heat transfer and therefore slightly higher product temperature for an average edge vial. These differences are larger earlier in secondary drying. Even within the center vial class, inter-vial variations in residual water are large early in the process. Thus, only center vials were studied, and more vials were extracted at a given time early in secondary drying. Four or 5 vials were extracted at regular intervals before the shelf temperature was increased toward the final secondary drying temperature. Later, when the Pirani gauge versus capacitance manometer gauge comparison indicated essentially all vials were finished with primary drying, the number of samples withdrawn was reduced to 3 vials. Thus, water content as a function of time is generated for the secondary drying cycle being investigated. These data are then used to

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evaluate the mass transfer coefficient for secondary drying by one of several procedures which do also require the water desorption isotherm as a function of water activity and temperature. Karl Fischer Residual Moisture Analysis Residual moisture contents of the sample vials were determined using Karl Fischer titration. A Metrohm (KF756) Karl Fischer Coulometer (Riverview, FL) with Hydranal Coulomat titration solvent from Sigma Chemical Company was used. To the sample vial, 10 mL of dried methanol was added, stirred briefly, and left to equilibrate for about 10 min. Next, 0.5 mL of that solution was injected into the titration cell. A blank measurement was made which was then subtracted from the sample reading. Thus, knowing the amount of water content and the weight of the sample, the % residual (w/w) water content in the sample was determined. Specific Surface Area Measurement The specific surface area measurement of the dried sample was measured using a Brunauer-Emmett-Tellerespecific surface area analyzer Flowsorb II (Micromeritics Instrument Corporation, Norcross, GA). Samples were degassed for at least 4 h at 40 C by passing a stream of a gas mixture of helium and krypton through the sample. The composition of the mixture was 0.10 mole% krypton in helium. The instrument was calibrated using 100% krypton under room conditions. After the sample was degassed, single point analysis of adsorption of krypton was carried out by placing the sample holder in liquid nitrogen to reduce the sample temperature to liquid nitrogen (~77 K) and flowing the He-Kr gas mixture over the sample. For desorption, the sample holder was immersed in a water bath at room temperature, and the Kr released was measured. Reproducibility in specific surface area measurement was better than 0.04 m2/g. The Secondary Drying Excel-based Model In the previous treatment of secondary drying, denoted “Passage,” the rate of change in water content of the solute phase with time is assumed proportional to the difference between the water content at time “t” and the water content that would represent equilibrium with the surrounding water activity, aw, at time “t” and temperature T, denoted C*(aw,T),


vCw;s ¼
kg Cw;s
C*ðaw ; TÞ vt

(1)

where kg is a “rate constant” assumed to exhibit Arrhenius temperature dependence. C* is the water content of the system if the system were at equilibrium at temperature T with water activity aw and must be determined by water sorption-desorption experiments, ideally using a moisture balance. The parameterized functional form of C* for sucrose was previously determined5 and will be used in the present work for all calculations. In the Passage procedure, the desorption rate constant, kg, was calculated using data obtained from a freeze-drying experiment, as described in that report5 and summarized below. At each sampling time, values of the partial pressure of water vapor, Pw, were calculated from the dew point measurement, and the vapor pressure of supercooled water, P0w , was calculated from the product temperature and the known relationship between temperature and vapor pressure. These values were then used to calculate C* from the Equation 2 below. C* is a function of water activity at selected

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E.K. Sahni, M.J. Pikal / Journal of Pharmaceutical Sciences xxx (2016) 1-13

temperatures and can be determined experimentally using the dynamic vapor sorption balance.

  Ef 1 Pw $ 0 C* ¼ Af 0 $exp T Pw

!n

AL , þ

Pw

!

P0w

1 þ BL ,

    Ef 2 Ef 3 ; BL ¼ BL0 $exp AL ¼ AL0 $exp T T

Pw P0w

! (2)

where Af0, Efi (i ¼ 1, 2, 3), n, AL0, and BL0 are constants evaluated by a fit of the equation to experimental data.5 Pw is the partial pressure of water, and P0w is the vapor pressure of pure (liquid) water at the sample temperature. At each time, the values of Cw,s (t), C*, and dCw,s/dt (obtained by numerical differentiation) were calculated. The desorption parameter kg was then calculated from the experimental values of the time derivative of water concentration and the difference between instantaneous water concentration and that which would represent equilibrium, C*, as indicated by Equation 1. As expected, kg did exhibit Arrhenius temperature dependence according to



 E ln kg ¼ ln kg0
T

(3)

where E is related to the activation energy (activation energy for water diffusion through the solid divided by the gas constant, R) and kg0 is the pre-exponential factor, presumably directly proportional to specific surface area.12 A couple of experiments were performed which serve as the “calibration runs” to get the average of E and kg0. This same Passage procedure was also used in the present work to verify that the Excel-based model could reproduce the calculations in the more rigorous and complex Passage software, using all the same input parameters as used in the Passage calculation.5 However, when this procedure was used in Excel to predict the water content versus time curve for secondary drying cycles other than the one originally used to determine kg, it was found that agreement with experiment was frequently not quantitative. The major issue appeared to be that Equation 1 does not describe the time course of water content particularly well. For example, when C* << Cw,s, Equation 1 predicts linearity in log(Cw,s) versus time, t, but experimentally, this is a very poor approximation from both studies performed as part of the present research and previously published studies.5 Rather, as a good first approximation, empirically we find that log(Cw,s) is linear in the square root of time. Thus, except when directly comparing the Excel-based calculations with Passage, in the present Excel-based work we use


vCw;s pffiffi ¼
kg Cw;s
C*ðaw ; TÞ v t

secondary drying cycles. While using the average of E and kg0 determined from the calibration runs does serve this need, a far simpler procedure that provides much better results is to use “Solver” function within Excel, which uses a generalized reduced gradient nonlinear algorithm, to evaluate E and kg0. The procedure is to first perform 1 or 2 “calibration” freeze-drying runs, and then to run Excel to calculate Cw versus time with initial estimates of E and kg0, and use regression analysis, with Solver within Excel, to find the “best fit” values of E and kg0. That is, determine the value that minimizes the square of the deviations between calculated and experimental values of the water content. This is the procedure used for all calculations in this report except in the direct comparison between Passage and Excel calculations. To initiate the Excel calculations, all material properties, such as heat capacity of the product and vial, thermal conductivity of the dry layer, heat transfer coefficients, the parameters E and kg0 that define the temperature dependence of the mass transfer coefficient, kg, cake thickness, concentration of solute, as well as time dependence of the shelf temperature are input much as in Passage. However, the procedure for executing the calculations does not involve solution of differential equations, as the Passage procedure uses, but rather is a process whereby initial values of product temperature and water content are set as input parameters and after small time increments, the temperature and water content are recalculated using the mass transfer equation (Eq. 1), Equations 3 and 4, and the heat transfer equations governing the dependence of the product temperature on time. Changes in water concentration are calculated in the top half of the cake and the bottom half using Equations 2 and 4. This process continues throughout the secondary drying process. The net result is a calculation with tabulation of calculation results and plots, of the average water concentration in the top and bottom halves of the dry later as a function of time. Additional details of the calculation are provided in Appendix I (Supplementary Table 1). Copies of the Excel program are available from the authors upon request.

Results and Discussion Comparison of Theoretical Results With Passage Time-based Versus Square Root of Time-based Determination of kg Figure 2 illustrates a plot of versus time which compares Excel-calculated results (calculated) with Passage-calculated values (Passage II) and with experimental data where calculations

(4)

As will be demonstrated later, Equation 4 provides a quantitative description of the experimental data. While using the Passage procedure for evaluation of kg from the derivative data, as described above by Equation 4, does always result in acceptable agreement with experiment, there is considerable scatter in the experimental kg values, largely due to uncertainty in the water contents and the corresponding scatter in the derivative data. Thus, we often find poor reproducibility in the values of E and kg0 between different secondary drying experiments. In an actual application, one would evaluate E and kg0 from 1 (or 2) experiment(s) and then use these data, with possible adjustment for specific surface area differences (kg0 should be directly proportional to specific surface area) for different

Figure 2. Comparison of Excel-based calculated results with Passage calculations and experimental studies with time (dCw/dt)-based kg data. Procedure for evaluation of kg and all input information provided was same as in the Passage.5

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Figure 3. Comparisons for 5% and 10% sucrose using the numerical differentiation method of analysis: The figure illustrate versus time obtained with input from square root of time-based (dCw/d√t) kg analysis for 2 individual runs, R1 and R2, for (a) 5% sucrose concentration,
10 C ice nucleation temperature and (b) 10% sucrose concentration,
5 C ice nucleation temperature.

involved time-based kg data (dCw/dt) for both the Excel and Passage calculations. This uses the time-based kg evaluation from the original literature data,5 but an improved procedure for derivative evaluation (dCw/dt). The raw data for water concentration versus square root of time were smoothed using MathLab and numerically differentiated to obtain dCw/d(√t). The mass transfer coefficient, kg, was then calculated for secondary drying. Here, we also use thermal conductivity data and heat capacity data for sucrose, and include the heat capacity of the glass vial as the vial plus contents are heated during the shelf temperature increase. The Passage calculations used input data for heat capacity and thermal conductivity for skim milk, which are smaller than the values for sucrose, and ignored the heat capacity of the glass vial. However, what is most important for the calculations is the ratio of the total heat capacity and the thermal conductivity. These ratios are nearly identical for both the Excel calculations here and the Passage calculations, and when we used identical input data as used in the Passage calculations, we find good agreement with the data and the Passage calculations. But it is necessary to use much smaller time increments within the Excel calculation when the heat capacity is small, which makes the Excel calculation cumbersome. Agreements of water content versus time between Excel calculations and both the Passage calculations5 and the experimental data are excellent. The problem though with this approach is that if we run a temperature program much different than the one used for the kg “calibration,” we do not always get acceptable agreement

with the data, and this problem prompted us to go to a square root of time basis for time in order to determine kg. Comparisons of experimental data versus calculations are shown in Figures 3 and 4 for kg evaluation using the numerical differentiation and Solver approaches, respectively, both using square root of time-based kinetics. For both cases, comparisons are made between the theoretical calculations and the experiments conducted for sucrose at different processing conditions. Figures 3a and 4a compare the different approaches for kg evaluation (i.e., numerical differentiation vs. Solver, respectively) for 5% w/w sucrose and
10 C ice nucleation temperature, and Figures 3b and 4b compare 10% w/w sucrose performed at
5 C ice nucleation temperature. For each plot, 2 replicate runs for each condition are compared. As observed from Figures 3 and 4, the agreement between experiment and calculation is not as good with kg evaluation using the numerical differentiation approach as the agreement using Solver. With Solver-based calculations, almost exact agreement was seen between the replicate runs. This is presumably due to the more consistent (and accurate) values of the pre-exponential factor and activation energy with the Solver approach. Therefore, because the Solver approach gives superior results and is easier to use, this approach is used throughout the rest of this work. Table 2 displays the individual activation energies and preexponential factors computed using Solver for different parametric conditions. Although not perfect, agreement between replicate runs and different conditions is relatively good. E and kg0

Figure 4. Comparisons for 5% and 10% sucrose using the Solver method of analysis: The figure displays as a function of time for (a) 5% sucrose concentration (R1: E ¼
2181, kg0 ¼ 4.11; R2: E ¼
1858, kg0 ¼ 2.94) and (b) 10% sucrose concentration (R1: E ¼
2417, kg0 ¼ 4.73; R2: E ¼
2347, kg0 ¼ 4.33) The plot shows 2 different runs for each condition with individual E and kg0 values evaluated by Solver from kinetics based upon square root of time (dCw/d√t). The symbol represents the experimental points and the lines represent the calculated results.

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Table 2 Solver Evaluation of Activation Energy Parameter, E, and Pre-Exponential Factor, kg0, for Individual Freeze-Drying Runs Parameters

Conditions

No. of Runs

E (K)

ln(kg0) (s
1)

Ice nucleation temperatures


5 C

R1 R2 R1 R2 R1 R2 R1 R2 R1 R2 R1 R2 R1 R2


2361
2977
2312
2855
2181
1858
2067
3257
2715
3297
2417
2347
1815
1996

4.80 6.88 4.36 6.41 4.11 2.94 3.61 8.72 6.06 8.42 4.73 4.33 2.30 3.00


7 C
10 C Secondary drying temperatures

25 C 50 C

Sucrose concentrations

10% 15%

Means slope ¼
2461 K and mean kg0 ¼ 155.7 s
1 (mean activation energy ¼
2461 K).

have a coupled effect in their impact of mass transfer coefficient. That is, an increase in E is coupled with an increase in kg0 with the mass transfer coefficient changing typically only slightly. We note that, while in principle the “E” constants (related to activation energy) could depend on sucrose concentration, one would expect this dependency to be relatively small. The kg0 values should, in principle, be directly proportional to specific surface area.

Matching the Timescales of Drying When comparing calculations with experiment, the time interval between the end of primary drying and the time at which the shelf temperature is ramped up from the primary drying setting to the secondary drying setting is important. This interval determines the time when the average vial is undergoing secondary drying at the primary drying shelf temperature setting, which is needed to set the proper shelf temperature versus time profile for the calculation. The major problem is that this number is not accurately known. We can obtain an approximation for the average primary drying time of the vials containing thermocouples. Here, the end of drying time for a given vial is determined by the onset of the sharp increase in measured temperature that nominally corresponds to removal of all ice from that vial. However, there is always some error in this number, and the average primary drying time for the thermocouple vials may not be exactly representative of the average for the batch as a whole. We use this average number as a starting point, but when we compare the calculated water

contents with the experimental water contents, there is often clear evidence of a “time shift.” That is, the shape of the experimental curve is well reproduced by the calculations, but the entire curve is displaced by a few hours. Thus, our procedure is to adjust the time interval during which secondary drying is being conducted at the shelf temperature setting used for primary drying. This is equivalent to using this interval as an “adjustable parameter” to be used in comparing experiment and calculation. An additional complication is that due to “radiation effects,” the product temperature typically approaches a plateau temperature above the shelf temperature set point for secondary drying, meaning the effective shelf temperature is somewhat higher than the actual shelf temperature. Thus, we begin the simulations with a shelf temperature setting equivalent to this plateau level, which is about
20 C for our runs where the actual shelf temperature is
25 C. We then use a ramp time to the secondary drying setting(s) to match the experimental settings. Figure 5 shows an example of estimating average end point of primary drying for 5% w/w sucrose at ice nucleation temperature of
5 C. The sharp spikes seen in the product temperature-time profile in Figure 5a are a result of sample extraction during secondary drying, resulting in temperature excursions. The sampling procedure was pretty quick. As soon as the samples were taken (not more than 2 min), the external vacuum was released to allow the regular experiment running. Thus, any collapse should be minor and confined to a small region of the cake and therefore without a significant effect on specific surface area. The thermocouple response (sharp increase in temperature) shows an average end point of primary drying time to be about 44.5 h (solid arrow), and the shelf temperature ramp started at 52.7 h as shown by the solid line, a difference of 8.2 h. The plot in Figure 5b shows acceptable agreement between experimental and calculated water contents, meaning that 8.2 h in secondary drying at a shelf temperature of
20 C is approximately correct. Of course, the time course of product temperatures for vials not containing thermocouples is unknown, and we assume, as a first approximation, it is the same as for the thermocouple vials. This was the case for the data in Figure 5. So no adjustment in average primary drying time was required as thermocouple behavior was approximately correct. Thermocouple response does not always provide an accurate estimate of the average end point of primary drying as is the case for the data shown in Figure 6 for 5% w/w sucrose at ice nucleation temperature of
10 C. The average end point of primary drying in this case was calculated (from average thermocouple “break”) to be about 35.1 h (solid arrow in Fig. 6a), and the time marking the

Figure 5. Temperature history during secondary drying: (a) depicts the sharp increase in product temperatures indicating end of primary drying, which averages about 44.5 h on the timescale, (b) shows the comparison of < Cw > versus time for calculated and experimental results.

E.K. Sahni, M.J. Pikal / Journal of Pharmaceutical Sciences xxx (2016) 1-13

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Figure 6. The figure shows adjustment to average end point of primary drying for 5% w/w sucrose concentration,
10 C ice nucleation temperature where (a) depicts the average end point observed from the thermocouple behavior and was computed to be 35.1 h but was adjusted to 45.1 h, (b) shows the comparison of < Cw > versus time for calculated and experimental results before and after adjustment of primary drying times. Lines represent the calculated results: solid (before adjustment) and dotted (after adjustment); the square symbols represent experimental data points: solid (before adjustment) and open (after adjustment). The thermocouple data represent only center vials.

increase in shelf temperature is 50.6 h (solid line), 15.5 h later than the estimated end of primary drying (in the average vial). Thus, with this estimate of primary drying time, the average vial was in secondary drying at a shelf temperature of
20 C for 15.5 h. As can be observed from Figure 6b, there appears to be a significant time shift in the water content versus time curve between calculated and experiment, here about 10 h. The difference in the initial shape of the curve exists because it is required to run the Solver again when time base is shifted, which in this case gives a much larger value of E, so that the drying rate at low temperature is extremely slow. This is turn gives a different shape of the Cw versus time curve. The values of E and kg0 we get after the time shift is likely more accurate and give a value of E much closer to those values determined from data where no time shift is needed. Thus, the actual end of primary drying seems to be about 10 h longer, or 45.1 h which is shown in Figure 6a. In other words, 45.1 h was the “adjusted” average end point of primary drying time, and because the shelf temperature was ramped up at the 50.6 h mark, the average vial was in secondary drying at a shelf temperature of
20 C for 5.5 h. Figure 6b shows the comparison of calculated with experimental water content versus time for both the “incorrect” 35.1 h average primary drying time and the “corrected” 45.1 h primary drying time. Obviously, the agreement between experiment and calculation is far superior with this “corrected” primary drying time. This correction procedure was followed whenever a significant time shift was noted.

Comparison of Experimental and Calculated Product Temperatures Accurate calculated product temperatures are required for accurate water contents. While late in secondary drying, the shelf and product temperatures are nearly the same, significant differences do exist early in secondary drying and in the shelf temperature ramp phase. Calculations are made using sequence of small time steps where steady state is assumed, and the accuracy of the calculations during these periods is also sensitive to the values of heat capacity and thermal conductivity of the product and vial. A comparison of calculated and experimental temperatures is given in Figure 7. Note that the agreement is excellent, meaning both the simple steady state procedures and input heat capacity and thermal conductivity data were sufficiently accurate. Evaluation of “Universal” Values of the Constants in kg, E, and kg0 After the individual values for E and kg0 have been obtained using Solver (as shown in Table 2), an average of E and kg0 was computed to be
2461 K and 155.7 s
1, respectively, which were then treated as the universal parameters, for use in any drying scenario for sucrose, provided a suitable adjustment is made for differences in specific surface area. Note that the value of E is the activation energy divided by the gas constant. The drying rate should be roughly directly proportional to the specific surface area of the solid, other factors being constant.12 Thus, the kg0 data were

Figure 7. The figure shows the comparison of experimental and calculated product temperatures as a function of time in secondary drying for (a) 5% sucrose concentration,
5 C ice nucleation temperature and (b) 5% sucrose concentration,
10 C ice nucleation temperature. Experimental product temperatures are the solid triangles, and calculated product temperatures are open circles. Solid lines represent the shelf temperatures for the theoretical calculation.

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E.K. Sahni, M.J. Pikal / Journal of Pharmaceutical Sciences xxx (2016) 1-13

Table 3 Average Specific Surface Areas at Different Freeze-Drying Conditions for Different Parameters With All Controlled Ice Nucleation Except the Effect of 25 C and 50 C Parameters

Conditions

Average Specific Surface Area (m2/g)

Ice nucleation temperatures


5 C
7 C
10 C 25 C 50 C 10% 15%

0.61 0.65 0.72 0.64 0.66 0.56 0.45

Secondary drying temperatures Sucrose concentrations

± ± ± ± ± ± ±

0.05 0.05 0.04 0.04 0.02 0.03 0.09

Comparison of the Fits From the Universal Parameters to the Individual Runs

adjusted for specific surface area differences (as shown in Table 3 for specific freeze-drying run). The specific surface areas were nearly all the same, so this adjustment was of little significance in this particular data set. We also note that the fill volume (height of dry layer) was different for the 10% and 15% w/w sucrose runs (3 mL as compared to 5 mL for 5% w/w sucrose). However, this difference should not be significant as long as specific surface area adjustments are made as summarized by Equation 5.12

Specific kg0 ¼ SSA*


 avg kg0 avgðSSAÞ

Equation 5 assumes that the specific kg0 for a particular experiment is directly proportional to its specific surface area. Thus, the value of kg0 specific to any experiment may be computed using the specific surface area of that run as well as the average values of kg0 and specific surface areas for a number of comparator runs on the same formulation.

(5)

We next used these universal values, that is, E ¼
2461 K and kg0 ¼ 155.7 s
1as input parameters for all conditions listed in Table 1. The effect of secondary drying temperature (25 C, 40 C, and 50 C) was investigated for 5% w/w sucrose solutions (at a constant controlled ice nucleation temperature). Next, the effect of ice nucleation temperature (
5 C,
7 C, and
10 C) on the drying behavior of 5% w/w sucrose solutions was investigated at a constant secondary drying temperature. Lastly, the secondary drying behavior of sucrose was evaluated at different concentrations (5%, 10%, and 15% w/w) at a constant ice nucleation temperature (
5 C) and a constant secondary drying temperature (40 C). Figure 8 shows the semi-log plots of < Cw > versus time for different parametric conditions using the universal E and kg0. The solid lines indicate the theoretical calculations and the data points show the

Figure 8. Comparison of calculations with experiment using “universal” values for E and kg0. Log plots for versus time showing comparison of theoretical (lines) and experimental results (symbols) for different drying runs: (1) ice nucleation temperatures: (a)
5 C, (b)
7 C, and (c)
10 C; (2) secondary drying temperatures: (a) 25 C, (b) 40 C, and (c) 50 C; and (3) sucrose concentrations: (a) 5%, (b) 10%, and (c) 15%.

E.K. Sahni, M.J. Pikal / Journal of Pharmaceutical Sciences xxx (2016) 1-13

9

Figure 9. Example applications calculated for 5% sucrose. Ramp rate: 0.2 C/min, Pc: 60 mT: (a) Prediction of the impact of different Tsdtime sequences for different secondary drying temperatures. The different curves represent different terminal shelf temperatures in secondary drying after the ramp from the primary drying shelf temperature, with the 25 C and 40 C representing a combination of 25 C (hold of 3 h) and then 40 C. (b) Depicts for top and bottom halves for secondary drying temperatures of 20 C and 60 C as a function of time. Dotted lines represent the water contents for 20 C, and solid lines represent the water contents for 60 C.

experimental values. We note that the calculated curves for runs 1 and 2 for each given condition are the same, which means the timing for increase in shelf temperature was correctly assigned and the calculated curves should be the same, as observed in Figure 8. All the plots generally show 2 slopes, one for the period where the shelf temperature is at the primary drying setting and early ramp into secondary drying and a second slope when the system is well into secondary drying. The agreement between experiment and calculation with the universal constants is essentially exact with the exception of 15% w/w sucrose concentration in Figure 8 (3c) where the calculations underpredicted the moisture content starting the secondary drying phase. This apparent shortcoming may indicate that the kg for sucrose (or other materials) does depend somewhat on solute concentration, beyond the possible difference in specific surface area. Applications of the Theoretical Model There are 2 classes of applications for the secondary drying model. First, the calculations can be used to study differences due to formulation or freezing behavior. For a material other than sucrose, different freezing conditions may lead to much larger differences in specific surface area than found for the sucrose systems studied here. The impact of such differences in procedure on moisture content history could be predicted with useful accuracy with the secondary drying model. Moreover, it is likely that different formulations may show very different secondary drying behavior, even modest differences in materials such as comparing 2 disaccharides like sucrose and trehalose. Certainly, larger differences may be expected when comparing a series of disaccharide:protein formulations with very different disaccharide:protein ratios. The second class of applications involve comparing very different shelf temperature versus time profiles for a given product to assess the practical outcome in residual moisture at the end of the process to facilitate optimizing the process. One example of this application is to determine what shelf temperature versus time profile is needed to obtain reproducible intermediate levels of residual moisture, which could have stability benefits.13 Figure 9 illustrates 2 application examples. Figure 9a shows the difference in the secondary drying profile by different shelf temperature versus time sequences for varying secondary drying temperatures (
20 C, 0 C, 20 C, 40 C, and 60 C) for 5% sucrose. This Excel model has the capability of setting 2 different secondary drying temperatures; therefore, Figure 9a also includes a plot where 2 different secondary drying temperatures were used (25 C for 3 h

followed by ramp to a final 40 C). For all variations in the secondary drying temperatures, the calculation begin with the primary drying settings (of
20 C for the shelf) for 0.5 h followed by ramp into the different secondary drying temperatures. Because of being at primary drying settings initially and then with the shelf temperature not approaching 20 C until after about 4 h, no difference is seen in the semi-log plots for < Cw > versus time in Figure 9a for first 4-5 h. After 5 h, differences certainly can be noticed in < Cw > as the final shelf temperature changes, with < Cw > decreasing in the order expected, 60 C > 40 C > 25 C (3 h) and 40 C > 20 C > 0 C >
20 C. Figure 9b shows semi-log plots of < Cw > versus time for the top and bottom halves of the dry layer for 2 secondary drying temperatures of 20 C and 60 C. The average water concentration is more in the bottom half of the dry layer which is shown by the solid lines as compared to the top half represented by the dashed lines for both 20 C and 60 C. The average water concentration for the cake will be roughly in the middle of the water concentrations represented by the top and bottom halves. Also as evident in Figure 9b, the cake becomes more homogeneous with respect to moisture content when the shelf temperature is set to a high value of 60 C as opposed to a low shelf temperature of 20 C in secondary drying, where the heterogeneity is significant even after more than 10 h in secondary drying. The model was also used to study the difference in the < Cw > versus time profiles for edge and center vials for 5% w/w sucrose

Figure 10. Application of the model to studies of the difference in the versus time curves for edge and center vials. 5% Sucrose, ramp at 0.2 C/min to 40 C, Pc ¼ 60 mT. Solid and dashed lines represent the effect of center and edge vials, respectively.

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E.K. Sahni, M.J. Pikal / Journal of Pharmaceutical Sciences xxx (2016) 1-13

Table 4 Input Parameters for the Excel-Based Model Symbol

Description

Pc KC KD Av 104Kv(top) kg0

Chamber pressure (Torr) Pressure-independent part of vial heat transfer coefficient, Kv Constant term in denominator of Kv multiplying P Outer cross-sectional area of vial (cm2) Top heat flow only (top radiation) The pre-exponential factor in the Arrhenius expression for the secondary drying mass transfer coefficient, kg (s
1) The slope of ln(kg) versus 1/T, which is the negative of the activation energy divided by R (gas constant) The amount of “nonequilibrium” water at the top of the cake at the start of secondary drying, estimated The number of vials in the batch The fill volume per vial (mL) The concentration of the fill solution (%, w/w) The resistance of pumping noncondensables, evaluated by experiment The mass of the glass, g, for 1 vial The ramp rate of the shelf temperature used in proceeding to secondary drying The final temperature desired for stage 1 in secondary drying The final temperature desired for stage 2 in secondary drying The heat of vaporization of water from the solid (cal/g) The area of the duct between chamber and condenser (m2), used to evaluate Jinerts 14 Thermal conductivity of the solid; for sucrose is 6.4  10
4 cal s
1 cm
1 K
1

S Top noneq w N Vfill conc, wt% Rp',inerts Mass vial, g Ramp rate Ts final 1 Ts final 2 DHv, cal/g Aduct ksolid

(not studied experimentally), with a shelf temperature ramp at 0.2 C/min to 40 C and a chamber pressure of 60 mT (Fig. 10). In order to plot the < Cw > versus time for the edge vials, 2 factors were considered. The vial heat transfer coefficient for the center vial (Kv,bot) is 0.000382 cal/(K$s$cm2) which was increased for the edge vial by about 30% (consistent with previous experiments) to account for the “edge vial effect” and was therefore adjusted to 0.000497 cal/(K$s$cm2). Secondly, edge vials complete primary drying earlier than center vials (on average), and hence the edge vial primary drying time was decreased by 30%, meaning edge vials enter secondary drying sooner than center vials. During the first 13.3 h of secondary drying for edge vials, center vials are still in primary drying. Thus, as illustrated in Figure 10, there is no change in water content for center vials for the first 13.3 h, simply because there is no secondary drying in the center vials during that time. Once primary drying is over for center vials, they then start secondary drying, meaning nonzero values of kg. The net result, shown in Figure 10, is that edge vials have lower water content throughout the drying process, although near the 35 h mark, the moisture content is near zero in both classes of vials.

Impact of Variation in Input Parameters In order to test the effect of other factors, a variation in the input parameters was performed to draw a generalized conclusion on the impact of variation on average water content. Chamber pressure only impacts heat transfer, and secondary drying is essentially totally mass transfer limited by transport processes in the solid state through kg. Therefore, changes to parameters contributing to the heat transfer coefficients, that is, KC and KD (Table 4), did not make any noticeable difference as the demand for heat during secondary drying is so small such that any change in heat transfer does not significantly impact the product temperature and therefore impact on drying rate is not significant. Solute concentration would only impact the drying behavior through variation in specific surface area, which should be minimal. Shelf temperature though

has a significant impact on the drying rate in secondary drying. Increasing the shelf temperature and the ramp rate considerably decreases the calculated < Cw > moisture content versus time with significant changes only in the second half of the curve (suggesting changes in secondary drying phase). Conclusions and Significance An Excel-based secondary drying model is developed which is simple and easy to use and whose use can impart insight into the impact of changes in process and formulation on water content during secondary drying. Uses include investigation of changes in process on water content as well as impact of formulation changes and investigation of differences in residual water between edge vials and center vials. The calculated results are generally in good agreement with experimental values, suggesting that this model can serve as an important process development tool for process design and manufacturing problem-solving. There is a clear need for additional studies, particularly those involving different formulations. Future studies should focus on extending this approach and comprehensively validating the model for different materials, including both small molecules and proteins, involving representative amorphous as well as semicrystalline formulations, different concentrations, as well as different ice nucleation temperatures. We expect qualitatively similar behavior to that found for sucrose, but quantitatively the results may be quite different for other systems. Subjects that may be of particular interest are formulations with high protein content and high fill depth. Acknowledgments Financial support from Pfizer is gratefully acknowledged. The authors also acknowledge the discussions with Serguei Tchessalov and Bakul Bhatnagar. References 1. Millman MJ, Liapis AI, Marchello JM. An analysis of the lyophilization process using a sorption-sublimation model and various operational policies. AICHE J. 1985;3:1594-1604. 2. Tang MM, Liapis AI, Marchello JM. A multi-dimensional model describing the lyophilization of a pharmaceutical product in a vial. In: Mujumdar AS, ed. Proc. Fifth Int. Drying Symposium. Vol. 1. NY: Hemisphere Publishing Corporation; 1986:57-64. 3. Mascarenhas WJ, Akay HU, Pikal MJ. A computational model for finite element analysis of the freeze-drying process. Comput Methods Appl Mech Eng. 1997;148:105-124. 4. Sheehan P, Liapis AI. Modeling of the primary and secondary drying stages of the freeze drying of pharmaceutical products in vials: numerical results obtained from the solution of a dynamic and spatially multi-dimensional lyophilization model for different operational policies. Biotechnol Bioeng. 1998;60(6):712-728. 5. Pikal M, Mascarenhas W, Akay H, et al. The non-steady state modeling of freeze drying: in-process product temperature and moisture content mapping and pharmaceutical product quality applications. Pharm Dev Technol. 2005;10(1): 17-32. 6. Velardi SA, Barresi AA. Development of simplified models for the freeze-drying process and investigation of the optimal operating conditions. Chem Eng Res Des. 2008;86:9-22. 7. Fissore D, Pisano R, Barresi AA. Monitoring of the secondary drying in freezedrying of pharmaceuticals. J Pharm Sci. 2010;100(2):732-742. 8. Pikal MJ. Use of laboratory data in freeze drying process design: heat and mass transfer coefficients and the computer simulation of freeze drying. J Parenter Sci Technol. 1985;39(3):115-139. 9. Nail S, Searles J. Elements of quality by design in development and scale up of freeze dried parenterals. Biopharm Int. 2008;21(1):44-52. 10. Koganti VR, Shalaev EY, Berry MR, 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):854-861. 11. Patel SM, Bhugra C, Pikal MJ. Reduced pressure ice fog technique for controlled ice nucleation during freeze-drying. AAPS PharmSciTech. 2009;10(4):14061411.

E.K. Sahni, M.J. Pikal / Journal of Pharmaceutical Sciences xxx (2016) 1-13 12. Pikal MJ, Shah S, Roy ML, Putman R. The secondary drying stage of freeze drying: drying kinetics as a function of temperature and chamber pressure. Int J Pharm. 1990;60(3):203-217. 13. Chang L, Pikal M. Mechanisms of protein stabilization in the solid state. J Pharm Sci. 2009;98(9):2886-2908. 14. Lin Y, Shi Z, Wildfong P. Thermal conductivity measurements for small molecule organic solid materials using modulated differential scanning calorimetry

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(MDSC) and data corrections for sample porosity. J Pharm Biomed Anal. 2010;51(4):979-984. 15. Pikal MJ, Roy ML, Shah S. Mass and heat transfer in vial freeze-drying of pharmaceuticals: role of the vial. J Pharm Sci. 1984;73(9):1224-1237. 16. Chang L, Milton N, Rigsbee D, et al. Using modulated DSC to investigate the origin of multiple thermal transitions in frozen 10% sucrose solutions. Thermochim Acta. 2006;444(2):141-147.

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Appendix I. Details of the Excel Calculations of the Kinetics of Secondary Drying

0.84, independent of vapor composition, pressure, or nature of the product.15 Thermal resistance of a significant dry layer thickness is then included in similar fashion to Equation A1, to give the top radiation equivalent of Equation A1,

Initial Value Input: Step n ¼ 0 Values given in the following discussion are those for sucrose as used in the experiments reported here. In this version of the program, 2 “final” shelf temperatures and 1 ramp rate for shelf temperature increase are allowed. These are provided as input parameters. The program allows a starting shelf temperature, which is the shelf temperature at the end of primary drying for the average vial. The program then allows a ramp to another shelf temperature (i.e., the first “final” shelf temperature) and also, if desired, another ramp to the second and last “final” temperature. The dry cake is divided into 2 sections, top half and bottom half, and the calculations proceed assuming each half is homogenous, which of course is an approximation. The calculations begin by input of initial values (step, n ¼ 0) for the effective shelf temperature (
20 C), the product temperature at the bottom of the vial (
35 C, based on experimental temperatures), and the mole fraction of water vapor in the chamber, Xw (1.0). The temperature of the top half of the cake is higher and is estimated to be higher than the bottom temperature by half the difference between the shelf and bottom temperatures.5 Next, the effective vial heat transfer coefficient for heat transfer (Supplementary Table 1) through the bottom of the vial (Kvbe) is calculated as a function of the mole fraction of water, Xw, the vial heat transfer coefficient for heat flow to the bottom of the vial, Kv(bot), and the heat transfer coefficient for heat flow through the dry layer from the bottom to the center of the bottom half of the cake, kcake/d, by:

Kvbe ¼

Kv ðbotÞ$½Xw þ ð1
Xw Þ=1:3 h i 1 þ Kkv ðbotÞ =d

(A1)

cake

The factor of 1.3 arises because the vial heat transfer coefficient depends on vapor composition, which changes during secondary drying, and this heat transfer coefficient with pure nitrogen in the chamber is roughly 30% less than with pure water vapor for the usual chamber pressures used.15 Also, note that we approximate heat flow to the dry layer as the steady state heat flow from the bottom of the vial to the center of the bottom half of the cake. A separate calculation evaluates heat transfer from the top shelf to the dry layer. Here, we use the empirical observation15 that heat transfer to the top of the product during primary drying can be calculated by using an “effective” emissivity for top radiation of

Kve ðtopÞ ¼

0:84 1 þ k0:84=d

(A2)

cake

Next, the initial vapor pressures of supercooled water, Pw*, for both top and bottom halves of the cake are calculated from the initial temperatures using the relationship between vapor pressure and temperature for water.5 The initial partial pressure of water (supercooled) at the bottom of the cake is assumed equal to the partial pressure when the last trace of ice was still present, meaning this quantity is calculated as the vapor pressure of ice at the initial bottom product temperature. The initial partial pressure of water at the top of the cake is given by the product of the mole fraction of water initially, Xw ¼ 1, and the chamber pressure. The initial concentration of water in the solid, C0(Cw bot, I), in g H2O/g sucrose is taken as 0.227, which corresponds to the experimental value of 18.5% water,16 and the initial concentration of water vapor in the top half of the cake is calculated from the cake temperature and the partial pressure of water vapor in the chamber using the experimentally derived empirical desorption isotherm (Passage II). To this calculated equilibrium value is added an estimate of the nonequilibrium water content in the top half of the cake (0.034 g/g). The assumption here is that perfect equilibrium between the water vapor and the water in the solid is not maintained during primary drying, and the small amount of water added reflects (roughly) this plausible assumption that is consistent with previous work (Passage II). The next series of calculations to establish initial behavior focus on calculating the changes in water content in the top and bottom halves of the cake. We start by evaluating the value of the secondary drying mass transfer coefficient, kg, using input values of the term reflecting the activation energy term, E, and the preexponential term, kg0. Then, we calculate the rate of change in water content with square root of time, dCw/d√t, from the relationship given in Equation 3 using the initial values of both the concentration of water in the solid, Cw,s, and the concentration that would represent equilibrium between solid and vapor, C*(aw, T), for both top and bottom halves of the dry layer. For both top and bottom halves, the activity of water, aw, is calculated as the ratio of the partial pressure of water to the vapor pressure of pure water at the same temperature for the n ¼ 0 step and all following steps. This step completes calculation of the initial quantities of interest.

Supplementary Table 1 Derived Parameters for the Excel-Based Model Symbol 4

10 Kv(total) 104Kv(bot) Kvbe Kve(top) aw(top, I) Ttop, I DTp top-bot, I C0(Cw bot, I) Pw*(top, I) Cw (top, I) Pw*(bot, I) Mass solid Jinerts Cp, vialþdry lmax 104kcake/d

Description Total vial heat transfer coefficient including bottom and top heat transfer, cal s
1 cm
2 K Bottom vial heat transfer coefficient only The effective vial heat transfer coefficient for heat transfer through the bottom of the vial, cal s
1 cm
2 K The effective vial heat transfer coefficient for heat transfer through the top of the vial, cal s
1 cm
2 K The activity of water in the chamber at the start of secondary drying The temperature of the dry layer at the top of the cake at the start of secondary drying, estimated The difference in temperature between the dry layer at the top of the cake and the bottom, at the start of secondary drying, estimated The concentration of water, Cw, g/g, at the bottom of the cake at the start of secondary drying The vapor pressure (Torr) of pure water (liquid) at the temperature of the top of the cake at the start of secondary drying, calculated The concentration of water at the top of the cake, Cw, at the start of secondary drying. The vapor pressure of pure liquid water at the temperature of the bottom of the cake at the start of secondary drying, calculated The total mass of solid in the batch, g The molar flux (mol/m2s) of inerts in this run, calculated from Rp’, inerts, and Pc5 The total heat capacity of all the glass and solid content in the batch (cal/K) The total thickness of the dry layer, cm The effective thermal conductivity of the dry layer divided by the distance over which heat flows to the center of the section, ¼ of lmax

E.K. Sahni, M.J. Pikal / Journal of Pharmaceutical Sciences xxx (2016) 1-13

First Time Increment: Step n ¼ 1 The next step is to increment the time by an arbitrary time interval, Dt, which we choose as 10 min. This time interval is suitable because the changes that occur do not occur rapidly and with this large time interval, the number of increments needed to include the entire secondary drying process is quite manageable. For step n ¼ 1, and all following steps, the partial pressure of water vapor at the bottom of the cake is estimated as the product of the chamber pressure and the mole fraction of water in the chamber. This approximation assumes the composition of vapor in the cake becomes independent of depth in the cake after 10 mins, which is only a rough approximation, but has little impact on the calculation accuracy. The partial pressure of water in the top section of the cake is taken as the product of the chamber pressure and mole fraction of water vapor in the chamber throughout the secondary drying calculations. Using the initial values for the drying rates for bottom and top halves of the cake, dCw/d√t, the change in Cw after the first time interval, D√t, is calculated for both bottom and top half. This change in Cw (DCw) is then added to the previous value of Cw, which in step n ¼ 1 is the initial value, to give the updated value of Cw for both top and bottom halves. The heat removed during this time step (heat of vaporization) for both halves is then calculated from the corresponding values of DCw assuming a heat of vaporization of 598 cal/g. The heat flows into the product, at top and bottom, are then calculated from the heat transfer coefficients, Equations A1 and A2, and the time interval, Dt. The net input of heat is then calculated from the sum of the heat flow and the heat of vaporization. The temperature change in both bottom and top is then evaluated from the net heat input into that half divided by the heat capacity of half the vial and product. The product temperatures are then “updated” by adding these temperature changes to the values for the previous time, in this case to the initial values. The partial pressure of water is assumed uniform throughout the cake at step

13

n ¼ 1 and thereafter and is therefore calculated as the product of the chamber pressure and the mole fraction of water in the chamber. This concludes the calculations for step n ¼ 1. Steps n > 1 Step 3 and all following steps build on the procedure described for steps 0 and 1. That is, the property of interest, P, for step n, Pn, is evaluated from the value in the previous step plus the change calculated for step n. That is, Pn ¼ Pn-1 þDPn. This procedure continues until the entire time span of secondary drying has been covered. The program can be used to evaluate the impact of changes in shelf temperature profile and chamber pressure changes given reliable input values of the constants that define the rate constant for secondary drying, kg, E, and kg0. Also, given a set of experimental values for water content versus time for a given shelf temperature program, values of E and kg0 may be evaluated by starting with estimates of E and kg0, running the program, and calculating the mean square of the deviations between calculated and measured values of Cw (Solver method). Optimum values of E and kg0 are then easily determined by using the Excel Solver to find the values of E and kg0 that minimize the mean square of the deviations. As noted earlier, this procedure is far superior to that used in Passage (Passage II) because it provides more reproducible values of E and kg0 and is much less labor-intensive in its use. A number of simplifying assumptions are used, including assuming steady state transport behavior and breaking the dry layer into 2 homogenous sections. The heat transfer assumptions are judged of sufficient accuracy by the close correspondence of the product temperature profiles measured experimentally and those calculated. The close agreement of experimental and calculated water contents as a function of time reflects acceptable accuracy of both the assumptions made in evaluating heat transfer and product temperature as well as the simple approximate treatment of the kinetics of drying.