# Computer-Aided Framework for the Design of Freeze-Drying Cycles: Optimization of the Operating Conditions of the Primary Drying Stage

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^{†}

## Abstract

**:**

## 1. Introduction

_{fluid}) and the pressure in the drying chamber (P

_{c}). This is particularly true in the primary drying stage, which is the most risky part of the process because of the lower value of the limit temperature (due to the higher water content). In this framework mathematical modeling can support freeze-drying practitioners, avoiding the extended experimental investigation based on trial-and-error approaches. For example, mathematical modeling (coupled with few experiments) can be used to calculate the design space of the process, i.e., the “the multidimensional combination of input variables and process parameters that have been demonstrated to provide assurance of quality” as defined in Ref. [15]. Using mathematical modeling product quality can be obtained by design (and it is no longer tested at the end of the manufacturing process), following the guidelines of the Guidance for Industry PAT issued by the US-FDA in 2004 and encouraging the pharmaceutical industry to take advantage of the various Process Analytical Technologies (PAT) that can be used to monitor a manufacturing process, thus moving from a quality-by-testing to a quality-by-design approach.

_{fluid}and P

_{c}on product temperature in the primary drying stage a one-dimensional model can be used [16,17,18,19,20,21]. The heat flux to the product (J

_{q}) is proportional to the temperature difference between the heating fluid and the product at the bottom of the container (T

_{B}):

_{w}) is proportional to the difference between the vapor pressure at the interface of sublimation (p

_{w,i}) and the water vapor partial pressure in the drying chamber (p

_{w,c}):

_{w,i}is a function of product temperature at the interface of sublimation (T

_{i}), and p

_{w,c}can be considered equal to P

_{c}as the composition of the gas in the chamber is about 100% water vapor.

_{v}, the overall heat transfer coefficient between the heating fluid and the product in the container, and R

_{p}, the resistance of the dried product to vapor flux. Various experimental methods were developed in the past to evaluate K

_{v}and R

_{p}, e.g., the test of pressure rise: the drying chamber is isolated from the condenser for a short time interval (usually ranging from 5–10 to 30 s) closing a valve in the duct connecting the chamber to the condenser and the pressure variation in the chamber, due to water vapor accumulation, is measured. Then, model parameters (K

_{v}and R

_{p}), as well as other variables like product temperature and the residual amount of ice, are estimated looking for the best fit between the values of chamber pressure measured and those calculated using a mathematical model of the process [22,23,24,25,26,27]. As an alternative to this method, it is possible to use Tunable Diode Laser Absorption Spectroscopy (TDLAS) to estimate K

_{v}and R

_{p}, beside product temperature and the residual amount of ice [28,29,30]. Moreover, a soft-sensor based on the measurement of product temperature, on a mathematical model of the process and on the Extended Kalman Filter algorithm was also proposed to get the values of K

_{v}and R

_{p}[31,32,33,34,35,36,37].

_{dried}):

_{frozen}and ρ

_{dried}are, respectively, the density of the frozen product and the effective density of the dried cake. As already described in Velardi and Barresi [21] the energy balance at the interface of sublimation gives:

_{s}is the heat of sublimation. Substituting Equations (1) and (2) into Equation (4), we obtain an expression that relates the temperature of the product at the interface of sublimation to that close to the bottom of the container. This last temperature (T

_{B}) can then be calculated from the energy balance for the frozen product, assuming steady-state conditions:

_{frozen}is the thickness of the frozen product, whose thermal conductivity is k

_{frozen}. By solving Equations (3)–(5) it is possible to calculate the evolution of the product and, thus, to evaluate if the selected values of T

_{fluid}and P

_{c}belong to the design space (or not), and also the drying time.

_{p}. The paper is thus organized as follows: at first the method used to describe the dependence of R

_{p}on the operating conditions is described. Then, details of the experimental investigations required to identify model parameters are given, as well as the application of this method to the calculation of the design space. Finally, results are presented and discussed, proving the effectiveness of the proposed method.

## 2. Materials and Methods

#### 2.1. Modeling of the Effect of the Operating Conditions on Dried Cake Resistance

_{e}is the effective Knudsen diffusivity and c

_{w,i}and c

_{w,c}are, respectively, the water vapor concentrations at the interface of sublimation and in the drying chamber. Using the ideal gas law, Equation (6) can be written as:

_{w}is the water molecular weight, R is the ideal gas constant and T is the “mean” temperature of the dried product. With this respect it has to be pointed out that T usually ranges in a narrow interval, e.g., from −35 to −20 °C, i.e., from 238 to 253 K: this means that the percentage effect of a variation of T on J

_{w}is small. As T can be hardly known, it can be replaced by T

_{i}without significantly affecting the accuracy of the results. Taking into account Equation (2) it comes that:

_{k}) by means of the following equation:

_{e}) and of the dried cake temperature (assumed equal to T

_{i}in these calculations, as previously discussed) [43]:

^{0.5}. Replacing Equation (10) into Equation (9) we get:

_{e}/τ is not a function of the axial position. In this case, Equation (8) becomes:

_{p}on L

_{dried}. In various cases (e.g., with sucrose-based formulations), the structure of the cake is not uniform, and the following equation was demonstrated to be effective to fit the experimental values of dried cake resistance vs. cake thickness in a wide range of formulations and operating conditions [30]:

_{0}and a

_{1}(and, thus A and B) can change with product temperature.

#### 2.2. Experimental Investigation

_{p}is significantly affected by the operating conditions. Differential Scanning Calorimetry (DSC type Q200, TA Instruments, New Castle, DE, USA) was used to detect the glass transition temperature: the samples were frozen at −60 °C and heated up to room temperature at 10 °C·min

^{−1}(the analysis was carried out in inert atmosphere). A cryo-microscope (type BX51, Olympus Europa, Hamburg, Germany; temperature controller: PE95-T95, Linkam, Scientific Instruments, Tadworth, Surrey, UK) was used to detect the collapse temperature of the dried cake.

^{3}and four shelves (the total area available is 0.5 m

^{2}), with an external condenser (the maximum ice capacity is 40 kg). Controlled leakage of inert gas was used to regulate chamber pressure. The system was equipped with T type thermocouples (Tersid, Milano, Italy) and a capacitance gauge (Baratron type 626A, MKS Instruments, Andover, MA, USA).

_{v}and R

_{p}as a function of the operating conditions [27]. In order to determine the parameters a

_{0}and a

_{1}(and, thus, A and B), the values of R

_{p}vs. L

_{dried}were fitted with Equation (16), thus obtaining the three parameters R

_{p}

_{0}, A and B. Then, a

_{0}and a

_{1}were calculated using Equation (15).

#### 2.3. Calculation of the Design Space

_{fluid}and P

_{c}) to calculate the temperature of the product at the interface of sublimation: if this temperature remains below the limit value, than the couple of values of T

_{fluid}and P

_{c}belongs to the design space. When performing these calculations care has to be paid to the variation of the parameters A and B describing the effect of product temperature on the resistance of the dried cake to the water vapor flux. The detailed procedure is summarized in the following:

- Selection of the minimum and maximum values of the fluid temperature (T
_{fluid,min}and T_{fluid,max}) and of the pressure in the chamber (P_{c,min}and P_{c,max}) to be considered in the design space. - Calculation of the arrays of values of T
_{fluid}and P_{c}:$${T}_{fluid}\left(k\right)={T}_{fluid,min}+\left(k-1\right)\Delta {T}_{fluid}\hspace{1em}\text{with}k=\left[1,{n}_{T}\right]$$$${P}_{c}\left(j\right)={P}_{c,\mathrm{min}}+\left(j-1\right)\Delta {P}_{c}\hspace{1em}\text{with}k=\left[1,{n}_{P}\right]$$_{fluid}and ∆P_{c}are, respectively, the sampling intervals for T_{fluid}and P_{c}in the design space, and n_{T}and n_{P}are the length of the two arrays:$${n}_{T}=\frac{{T}_{fluid,max}-{T}_{fluid,min}}{\Delta {T}_{fluid}}+1,\hspace{1em}{n}_{P}=\frac{{P}_{c,max}-{P}_{c,min}}{\Delta {P}_{c}}+1$$A grid of n_{T}x n_{P}points with coordinates (T_{fluid,k}, P_{c,j}) is thus defined. - Given a couple of values of T
_{fluid}and P_{c}, the evolution of the product is simulated, as previously described, thus calculating the maximum value of its temperature. If this value is lower than the limit temperature, then the selected values of T_{fluid}and P_{c}belongs to the design space.

## 3. Results and Discussion

**Figure 1.**DSC thermogram for the 5% w/w sucrose formulation used in this study (the glass transition temperature is evidenced).

**Figure 2.**Images obtained at the cryo-microscope, showing the onset of the cake collapse at −33.4 °C.

**Figure 3.**Comparison between the values of heating fluid temperature (upper graphs), product temperature at the bottom of the container (middle graphs) and dried cake resistance (lower graphs) in three different cycles for a 5% w/w sucrose solution.

_{dried}, as shown in the bottom graphs of Figure 3, thus indicating a non-uniform cake structure in the axial direction, but also that the temperature of the product strongly influences the values of R

_{p}, as dried cake resistance strongly decreases when the glass transition temperature is trespassed. This is due to micro-collapse phenomena, resulting in the formation of larger holes in the dried cake, as shown in Figure 4. This behavior was different to that shown in Overcashier et al. [44] where holes in the dried structure were observed. From this analysis, it is not however clear which is the maximum temperature reached by product during the drying process. We can hypothesize that the formation of these holes was due to the fact that the product temperature was very close to, or slightly higher than, the collapse temperature, thus promoting a more marked mobility of the freeze-concentrate.

**Figure 4.**Cake structures at 1 mm from the top obtained by Scanning Electron Microscope in the run I (

**A**); run II (

**B**); and in the run III (

**C**).

**Figure 5.**Values of r

_{e}/τ as a function of L

_{dried}calculated for type-I (■), type-II (●), and type-III (▲) runs.

_{e}/τ as a function of L

_{dried}calculated for the three types of runs are shown in Figure 5, evidencing that Equation (13) is suitable for describing the “structure” of the drying cake at different temperatures, providing that the values of the parameters a

_{0}and a

_{1}are modified. It appears that the structure of the cake, i.e., the values of r

_{e}/τ, are different depending on the temperature of the product, and two sets of values of a

_{0}and a

_{1}(and, thus A and B) can be calculated depending on the fact that product temperature is higher or lower than the glass transition temperature.

**Figure 6.**Results obtained in a run where the values T

_{fluid}and P

_{c}are modified during the primary drying (as shown in the top graph). Values of product temperature measured using a thermocouple and of the dried cake resistance estimated using the pressure rise test are shown in the middle and in the bottom graph respectively.

_{fluid}and P

_{c}are shown in the upper graph, as well as product temperature (where a sharp decrease appears in the measured values when T

_{fluid}and P

_{c}are decreased) and the dried cake resistance in the middle and bottom graphs respectively. The higher pressure and temperature gave smaller resistance to vapor flow (bottom graph), because the product temperature was higher than the glass transition value in the first part of the drying and, thus, it promoted the formation of larger pores within the upper part of the lyophilized cake (see Figure 6, middle graph).

_{p}were close to those obtained in a type-III run, where product temperature trespasses the glass transition value, and it appeared to be almost unaffected by the dried cake thickness. In the second part of the drying R

_{p}increased with a trend similar to a type-I run, where product temperature remains below the glass transition value and, thus, micro-collapse did not occur and R

_{p}increased to higher values as dried cake thickness increased.

_{e}/τ as a function of L

_{dried}, as shown in Figure 7: in the first part of the drying the values of r

_{e}/τ are similar to those obtained in a type-III run and when product temperature decreases, the values of r

_{e}/τ are close to those obtained in a type-I run, thus proving the adequacy of the proposed model to account for variations of R

_{p}vs. L

_{dried}as a function of product temperature.

_{p}. Therefore, in this study we hypothesized that the variation in R

_{p}occurred only if the product temperature is higher than the glass transition value, independently of how much higher this difference is.

**Figure 7.**Values of r

_{e}/τ as a function of L

_{dried}calculated for the run shown in Figure 6 (symbols). The values obtained in type-I and type-III runs are shown for comparison (lines).

_{p}can have important effects on the design space of the primary drying stage. Figure 8 shows the maximum value of the temperature of the heating fluid as a function of chamber pressure. Dashed line corresponds to the values calculated assuming that the limit value is the glass transition temperature and, thus, the parameters corresponding to a type-I run are used to calculate R

_{p}. This approach can be too precautionary, as higher values of product temperature (below the collapse value) are allowed. Therefore, for each value of P

_{c}product dynamics has been simulated for higher values of T

_{fluid}. Taking into account results shown in Figure 4, the resistance of the dried cake to vapor flux was described by two sets of values, one in case micro-collapse does not occur (when the temperature is below the glass transition temperature), and one in case of micro-collapse (when the temperature is above the glass transition temperature, and below the collapse temperature). Therefore, according to the algorithm described in Section 2.3, product dynamics has been simulated, thus resulting in the determination of the new maximum values of T

_{fluid}, represented in Figure 8 by the solid line.

**Figure 8.**Design space of the primary drying stage calculated in case product temperature remains below the glass transition value (dashed line) and below the collapse value (solid line) for the 5% w/w sucrose solution investigated in this study.

**Figure 9.**Values of the sublimation flux (colored lines) as a function of the operating conditions, in case the product has to be maintained below the glass transition temperature, or in case micro-collapse is allowed. Black line corresponds to the maximum allowed fluid temperature.

_{w}reaches a steady-state value, remaining almost constant until the end of the process. In order to optimize the process, if the product has to remain below the glass transition temperature, a suitable value of T

_{fluid}appears to be −10 °C and that of P

_{c}appears to be 5 Pa; the sublimation flux obtained in this case is slightly higher than 0.4 kg h

^{−1}m

^{−2}. In case micro-collapse is allowed, it is possible to use T

_{fluid}= −5 °C and P

_{c}= 20 Pa, thus obtaining a sublimation flux higher than 1.0 kg·h

^{−1}m

^{−2}, that significantly reduced the duration of the primary drying stage.

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Fissore, D.; Pisano, R.
Computer-Aided Framework for the Design of Freeze-Drying Cycles: Optimization of the Operating Conditions of the Primary Drying Stage. *Processes* **2015**, *3*, 406-421.
https://doi.org/10.3390/pr3020406

**AMA Style**

Fissore D, Pisano R.
Computer-Aided Framework for the Design of Freeze-Drying Cycles: Optimization of the Operating Conditions of the Primary Drying Stage. *Processes*. 2015; 3(2):406-421.
https://doi.org/10.3390/pr3020406

**Chicago/Turabian Style**

Fissore, Davide, and Roberto Pisano.
2015. "Computer-Aided Framework for the Design of Freeze-Drying Cycles: Optimization of the Operating Conditions of the Primary Drying Stage" *Processes* 3, no. 2: 406-421.
https://doi.org/10.3390/pr3020406