Model-Based Framework for the Analysis of Failure Consequences in

Aug 29, 2012 - Here a monodimensional model is used as it requires a limited number of parameters, which are the heat transfer coefficient and the pro...
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Model-Based Framework for the Analysis of Failure Consequences in a Freeze-Drying Process Davide Fissore, Roberto Pisano,* and Antonello A. Barresi Dipartimento di Scienza Applicata e Tecnologia, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 - Torino, Italy ABSTRACT: This paper shows how mathematical modeling can be used to assess the effect of process failures on product quality during the freeze-drying of pharmaceutical and biotech products. In particular, the model is used to calculate the product temperature response after an unexpected variation in the temperature of the heat transfer fluid and/or in the chamber pressure. Here a monodimensional model is used as it requires a limited number of parameters, which are the heat transfer coefficient and the product resistance to vapor flow. In this paper, the problem of determination of model parameters is addressed, with emphasis on the nonuniformity of the batch and on the uncertainty of the method of measurement used. The heat transfer coefficient was determined by gravimetric method, while the product resistance to vapor flow was measured by a weighing device placed directly in the chamber. Results obtained by the above devices agreed with the values estimated by the pressure rise test technique. This paper also shows how mathematical modeling can be used to calculate the probability that product temperature is maintained below its limit value, as well as to estimate the new duration for the primary drying.

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INTRODUCTION The market is growing rapidly for biological and biopharmaceutical products1 such as growth factors, recombinant vaccine components, and monoclonal antibodies. These molecules are usually very heat sensitive, therefore they can be subjected to degradation and deterioration during a manufacturing process, mainly during drying. It is clear that traditional drying techniques cannot be used, but alternative processes such as freeze-drying are required to stabilize the product and obtain long-term preservation.2 A freeze-drying process comprises three different phases: 1. During freezing, the material is cooled below its triple point. 2. During primary drying the pressure is lowered below the triple point to cause ice sublimation, while heat is continuously supplied as the sublimation process is endothermic. 3. During secondary drying the amount of residual water, which is bound to the partially dried product, is reduced to the target level by using high vacuum and moderate temperatures. At the end of the process a porous cake is obtained, which can easily be rehydrated.3−7 This study focuses on the primary drying phase since it is the most risky phase of the entire process. To preserve product quality, three variables have to be monitored and controlled during freeze-drying: 1. The temperature of the product, which has to be maintained below a limit value in order to avoid melting, shrinkage, or collapse phenomenon. This limit value is the eutectic temperature when the solute crystallizes or the glass transition temperature when the solute does not crystallize. 2. The amount of ice in the product, which indicates the end of the primary drying and, thus, the time at which © 2012 American Chemical Society

the secondary drying can be started without product damage due to the collapse phenomenon. 3. The amount of residual moisture in the product at the end of the secondary drying phase. To obtain a product with acceptable quality, the freezedrying cycle is often designed by means of an experimental campaign based on a trial and error approach, which is carried out in small-scale equipment. However, this approach does not often lead to the optimal cycle and the effort required is very large. In the past decade, various authors proposed the use of automatic control to manage processing conditions during primary drying. Examples of these systems are the SMART Freeze-Dryer,8,9 the LyoDriver,10 or more sophisticated control logics as the model predictive control.11,12 All these control systems aim to carry out the primary drying at a desired value for the product temperature. For this purpose, the controller compares the product temperature measured during the drying with the desired set value. This error is then used to calculate an appropriate value for the temperature of the heat transfer fluid. In this way, primary drying is carried out at its maximum rate and the product constraint is satisfied. Whichever control algorithm is used, a regular estimation is needed for the temperature and the position of the moving front, as well as for the heat and the mass transfer coefficient. For this purpose, model-based tools such as the pressure rise test technique13−16 or software sensors17−21 can be used. Nowadays, various manufacturers offer freeze-dryers equipped with an automatic control system for the primary drying. Nevertheless, the use of these tools is still limited to small units for cycle development. On the contrary, in manufacturing equipment the regulatory requirements make troublesome the Received: Revised: Accepted: Published: 12386

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geometry of the interface of sublimation. Both multidimensional and monodimensional models were proposed. However, various authors showed that monodimensional models are adequate to describe the dynamics of the system as radial gradients for the product temperature are usually small, even when radiation from the environment is considered.26−29 Furthermore, monodimensional models involve few parameters, which can be measured experimentally and require a small computational time. In this study, the monodimensional model proposed by Velardi and Barresi28 is used to design a tool for the analysis of the effect of process failures.

use of the above tools. In industrial practice, freeze-drying is carried out using a predefined cycle. The above approach is very simple, but the quality of the product cannot be guaranteed when processing conditions deviate unexpectedly from the set-point value. Furthermore, as any monitoring tool is available to track the dynamics of the system, it is not possible to know the effect of these events on the quality of the product, apart from waiting for the end of the cycle to analyze its characteristics. In addition, even if the quality of the product is not compromised by the above process deviations (e.g., thanks to an appropriate safety margin introduced during the development of the cycle22), the time required by the primary drying phase can significantly change and, therefore, its duration has to be properly modified to avoid starting secondary drying before all the ice is sublimated. In this paper, mathematical modeling is used to calculate the variation in product temperature after an unexpected change on the operating conditions with respect to the original cycle. Furthermore, mathematical modeling is used to calculate the design space, which is used here not only for cycle development,23−25 but also for analyzing the effect of process disturbances on the product quality. This paper also discusses how model parameters can be determined, with particular emphasis on simple and reliable methods, which can be employed both in small equipment and industrial plants. For example, in order to measure the resistance of the dried cake to vapor flow, a weighing device was used. Moreover, it is shown how mathematical modeling can be used when the drying behavior is not uniform. Finally, a simple procedure is proposed in order to account for parameter uncertainty in the above analysis. The result of this calculation is the probability that the product temperature is maintained below its limit value, and this value is compared with an acceptance threshold. This provides a true risk-based framework to analyze the effects of process failure on product quality. If the probability that product temperature exceeds the limit value is higher than the threshold value, the batch can be discarded without the necessity to wait for the end of the cycle and afterward test the quality of the product. On the contrary, if the above probability is lower than the threshold value, the cycle is not stopped but the duration for the primary drying has to be modified. In this study, an example is given for the calculation of the drying time by means of mathematical modeling after an unexpected process disturbance. The structure of the paper is the following: first model equations are introduced, with the approach used to take into account model parameters uncertainty as well as the nonuniformity of the batch; then, the experimental methods used to determine the parameters of the model are discussed, and examples of results are given for a representative case study. Finally, the proposed approach is validated by experiments.

−1 ⎛1 Lfrozen ⎞ 1 + ⎜ ⎟ (Tfluid − Ti ) = ΔHs (pw , i − pw , c ) k frozen ⎠ Rp ⎝ Kv

(1)

dLfrozen 1 1 (p − pw , c ) =− dt ρfrozen − ρdried R p w , i

(2)

Equation 1 is the energy balance at the interface of sublimation, stating that the heat flux from the heat transfer fluid to the product is used for ice sublimation with no accumulation, while eq 2 is the mass balance for the frozen product. In addition to some physical properties, there are two parameters to be specified in eqs 1 and 2: 1. The overall heat transfer coefficient Kv, which should be regarded as an effective heat transfer coefficient. This coefficient also accounts for the heat transfer from vial walls and from radiation. 2. The overall resistance to vapor flow Rp, which is used to express the sublimation flux as a function of the driving force (pw,i − pw,c), differently from Pikal26 who used various parameters to express the resistance to vapor flow due to the dried layer, the stopper, and the drying chamber. The vapor pressure pw,i is a function of the ice temperature at the moving front,30 while the partial pressure of water inside the drying chamber pw,c can reasonably be approximated to the total gas pressure Pc. As it will be discussed in the following sections, an effective value of Kv and Rp can be obtained by experiments. Parameter Uncertainty. When using a mathematical model to simulate the dynamics of a process, it is important to understand the effect of parameter uncertainty on model predictions. In this study, the values of Kv and Rp are assumed to be distributed around their mean values according to a density function f Kv,Rp. Giordano et al.24 proposed to describe the above density function as a Gaussian function characterized by the mean values of the two model parameters (K̅ v and R̅ p) and by their standard deviations (σKv and σRp):

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THEORETICAL CALCULATIONS Model Equations. Mathematical modeling can be used to predict the evolution of the product temperature and of the residual amount of ice as a function of the operating conditions, namely the temperature of the heat transfer fluid and the pressure in the drying chamber. Various models have been proposed in the literature to describe the dynamics of a freezedrying process. These models are based on mass and energy balances, which were written for both the frozen layer and the dried layer. The above models differ in the transport phenomena considered and in the assumption about the

fK , R = v

p

1 2πσK vσR p

2⎤ ⎡⎛ 2 K − K̅ ⎞ ⎛ R p − R̅ p ⎞ ⎥ ⎟⎟ −1/2⎢⎜ vσ v ⎟ + ⎜⎜ σ ⎢⎝ K v ⎠ ⎝ R p ⎠ ⎥ ⎦ ⎣ e

(3)

31

Pisano et al. confirmed that the values of Kv for a batch of vials are normally distributed. The values of Rp for a batch of vials can also be described by a normal distribution. In fact, the vials of a batch usually show different temperatures of nucleation during freezing, as this phenomenon is stochastic. Nonuniformity of the Batch. As it has been recently discussed by various authors,31,32 the evolution of the product temperature can be different for vials placed in different 12387

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positions on the shelf, e.g. because only some vials of the batch receive heat by side-wall radiation or because of temperature and pressure gradients over the shelf. As Kv is an effective heat transfer coefficient, which accounts for the various mechanisms involved in the heat transfer between the container and the environment, it is clear that Kv varies with the position of the vial on the shelf. Therefore, an appropriate value of Kv has to be used to simulate the product dynamics for a vial placed in a specific position on the shelf. In this study, the effect of pressure gradients over the shelves is assumed to be negligible. However, it must be said that pressure gradients can be relevant in very large apparatus in case of high sublimation rates and small clearance between the shelves.32,33 In this case, the above algorithm can take it into account during the calculation, provided that the distribution of pressure over the shelves is known. In fact, the value of pw,c appearing in eqs 1 and 2 can be expressed as a function of the vial position on the shelf and of the sublimation flux. The temperature distribution for the heat transfer fluid depends on the equipment load and on the desired value for the temperature of the heat transfer fluid. In well-designed equipment the above distribution is minimized, but a few degrees of temperature difference between the edges of the shelf can exist. Here, we assumed that the temperature of the heat transfer fluid is uniform over the shelf, but, as discussed above for pressure distribution, it would be very easy to take into account temperature gradients over the shelf in the calculations if this value is available. However, if temperature gradients are limited, they do not have to be taken into account in the calculation of Kv as their effect can be included in its uncertainty. Use of Mathematical Simulation for the Analysis of Failure Consequences. In this section specific details are given on how mathematical modeling can be used effectively for assessing the consequences of process deviations on product quality. In particular, here the design space is used not only to assist freeze-drying practitioners during the process development phase, but also to investigate the impact of process disturbances on the product dynamics during a production cycle. Furthermore, this paper shows how mathematical modeling can be used to obtain a detailed picture of process deviation consequences in case of heterogeneous drying behavior, as well as to take into account model parameter uncertainty in the analysis. The design space is a multidimensional diagram, which identifies the values of processing conditions that preserve the product quality. Fissore at al.25 showed that the above diagram varies as the sublimation goes on. Therefore, a combination of Tfluid and Pc, which belongs to the design space at the beginning of the drying, might exit from the design space as the sublimation of ice proceeds. Here, the method proposed by Fissore et al.25 is used to calculate the design space for a sucrose-based formulation. In fact, this method is the only one that can calculate how the design space changes as the drying proceeds. Once the design space has been calculated for a specific formulation, the following multistep procedure can be used to analyze the effect of a process disturbance on the product quality.

2. The design space is calculated for each group of vials identified at point 1 and at various values of Ldried/L. 3. The value of Ldried is determined for the various groups of vials when the undesired variations in the operating conditions occurred. For this purpose, the mathematical model of the process can be used for calculating the product history until the time of interest. 4. In each diagram, the design space curve calculated for the value of Ldried/L determined at point 3 is identified. 5. The point representing the new operating conditions can be placed in each diagram, thus evidencing whether or not the product is inside the design space. The above analysis can be used not only to check if a specific combination of (Tfluid, Pc) can maintain the temperature of the product below its limit value, but also to understand if, how, and when, processing conditions have to be modified to meet product constraints. As stated above, the mathematical model at the basis of design space calculation can also be used to obtain a more detailed picture of process failure consequences. For example, mathematical modeling can calculate the probability that product temperature is below its limit value for a given combination of Tfluid and Pc. Details are given for the sequence of the calculations by Giordano et al.24 In case of an undesired and significant variation in the process parameters, the above procedure can establish whether the product temperature remains below its limit value within a target probability value. Then, the new duration of the primary drying can also be calculated. Instead, if the product temperature is higher than its limit value, the cycle can be stopped.

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EXPERIMENTAL METHODS Experimental Determination of Kv. Theoretical correlations were proposed in the past to calculate Kv (see, among the others, ref 34) as a function of the geometrical characteristics of the vial and of the operating conditions. The application of the above correlations can be troublesome, because it is difficult to precisely modeling the heat transfer from the shelf to the vial in the contact area, the heat conduction in the gas layer between the shelf and the vial bottom in case of a strong curvature of the vial bottom, and the heat flux due to radiation from chamber walls and from the upper shelf. However, theoretical calculations show that Kv depends on the characteristics of both the vial and the equipment, as well as on chamber pressure. K v = C1 +

C2Pc 1 + C3Pc

(4)

Reliable values for the coefficients C1, C2, and C3 can be obtained by regression of experimental values for Kv vs Pc. As Kv depends on chamber pressure, at least three experiments (carried out at different values of Pc) are required in order to solve the above parameter regression. Various techniques were proposed in order to measure the heat transfer coefficient, e.g. tunable diode laser absorption spectroscopy (TDLAS)35−38 and the pressure rise test technique.16 These methods determine an average value of Kv for the batch as a whole. To determine the distribution of Kv on the shelf and hence the variance to be used by the model, in this work the heat transfer coefficient is measured by gravimetric method.

1. Vials are classified into groups by their position on the shelf. 12388

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Experimental Determination of Rp. Given the freezing protocol, the type of product, and the characteristics of the equipment, the resistance to vapor flow is assumed to be a function of the thickness of the dried layer. R p = R p,0 +

PL 1 dried 1 + P2Ldried

(5)

The parameters Rp,0, P1, and P2 have to be determined by means of experiments, looking for the best-fit between the measured values of Rp vs Ldried and the reference curve (eq 5). The value of Rp can be determined by means of the pressure rise test technique, as it also provides estimations of Rp and Ldried. The test has to be repeated during the primary drying in order to determine the dependence of R p on L dried . Unfortunately, the estimation of the model parameters and of the state of the product obtained using this method is inaccurate in the second part of the drying, even if the use of DPE+ algorithm16 significantly improves the robustness of the method. As an alternative, it is possible to calculate Rp from the measurement of the solvent flux obtained using the TDLAS sensor: pw , i − pw , c Rp = Jw (6) The main drawbacks of the TDLAS sensor are the cost and the difficulty of calibration, as fluid flow modeling is required to provide acceptable velocity determinations.39 Once the evolution of Rp has been determined, the values of Rp vs Ldried can be retrieved if the values of Ldried vs time are known. For this purpose, the evolution of Ldried can be calculated by integrating the following differential equation: dLdried 1 = J dt ρfrozen − ρdried w

(7)

In this study, a weighing device was used to determine the sublimation flux and, then, the resistance to vapor flow using eq 6. Different types of balances were proposed in the past to weigh in-line one or more vials in the drying chamber, with the goal to monitor primary drying and to detect the completion of ice sublimation.40−42 However, the mass measurement supplied by the above balances is not representative of the batch as a whole, because the weighed vials are exposed to heating conditions different from the rest of the batch. Furthermore, the above balances often use vials with a specific geometry, which does not often correspond to the geometry of the other vials of the batch.43−46 In this work, the balance designed by Carullo and Vallan47 was used to determine the value of Rp vs Ldried. This balance weighs up to 15 vials and is equipped with a miniaturized radio-controlled thermometer, which can also measure the temperature in one or more of the monitored vials without altering the mass measurement because of the force transmitted by the thermocouple wires (see Figure 1). The weighed vials are almost always in contact with the shelf and they are lifted just during the measurement. The balance is connected to a PC by means of a serial interface; the resolution is about 10 mg and the total uncertainty is about 100 mg from 233 to 313 K. The number of vials and their size can be easily varied by using an appropriate loading cell. If the monitored vials and the balance case are properly shielded to minimize radiation effects, the heat input for the monitored vials is similar to the heat input for the other vials of the batch.48,49

Figure 1. Weighing device (with a detail of the moving plate) used in this study.

Nevertheless, it must be noted that there is no hexagonal packing array when using the above balance and the vial arrangement certainly impacts the heat transfer. However, experiments carried out by the authors showed that the value of Kv observed for the vials loaded on the balance is comparable with the value observed for vials placed at the edge of the shelf.48 Furthermore, although the heat transfer coefficient can be different for the vials monitored by the balance and the vials of the batch, there is no evidence of variations in the product resistance to vapor flow. In fact, in this work the balance is used to estimates the values of Rp vs Ldried. The following issues have to be noted: 1. The sublimation flux is measured for vials placed in the drying chamber that have been frozen with all the other vials of the batch. The monitored vials undergo the same freezing conditions as the rest of the batch and, therefore, the problems related to the devices proposed by Pikal et al.40 and by Bruttini41 are solved. 2. The device is simple and can easily be placed in manufacturing equipment without requiring any modification, thus solving the problems related to the use of the TDLAS sensor.50 12389

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3. It is no longer required to carry out any pressure rise test and, thus, the measurement of Rp does not affect the dynamics of the vials of the batch. Moreover, a much more reliable estimation of this parameter is obtained, as it is no longer determined using a best-fit algorithm.16 4. The dynamics of the product in the weighed vials can be different from that of the other vials of the batch as monitored vials are exposed to different heat transfer conditions. However, this does not affect the curve Rp vs Ldried as the temperature of the product in the weighed vials is measured. 5. Finally, the device designed by Vallan allows the measurement of the weight loss in a (large) group of vials: this is a relevant issue as it is well-known that the insertion of a thermocouple in a vial can affect the ice nucleation in the freezing step and, thus, the value of Rp, which is a function of ice crystals size and distribution. Thus, if we weigh only one vial, e.g. as done by the devices described in refs 43−46, and we insert the thermocouple in that vial, the value of Rp so measured may be not representative of the Rp for the batch. In this study, instead, we have used the former configuration in which we monitor the mass of 15 vials, and we insert the thermocouple in only one vial. In this manner, the contribution of the vial equipped with a thermocouple to the average value of Jw detected by the balance is limited, i.e. less than about 7%. Furthermore, although the insertion of the temperature probe can alter the structure of the cake for the monitored vial, the temperature of the product is not significantly modified and, thus, it can be considered representative of the entire batch of vials. Case Study. The case study discussed here is the freezedrying of a 5% (w/w) sucrose (Riedel de Haën) solution processed in tubing vials ISO 8362-1 2R. These vials have an internal diameter of 14 ± 0.25 mm, with a wall thickness of 1 ± 0.08 mm, and a bottom thickness of 0.7 mm. The maximum gap between the bottom and the shelf is 0.4 mm.51 Each vial was filled with 1.5 mL of solution. For the determination of the overall heat transfer coefficient Kv, gravimetric tests were carried out using the above vials but filled with 3 mL of ultrapure water obtained by a Millipore water system (Milli-Q RG, Millipore, Billerica, MA). All the experiments were carried out in a prototype freezedryer (LyoBeta25 by Telstar, Spain) with a chamber volume of 0.2 m3 and equipped with thermocouples (T type, Tersid S.p.A., Milano, Italy), capacitance (MKS Type 626A Baratron) and thermal conductivity (Pirani PSG-101-S) gauges, and with LyoMonitor software.49 The pressure in the chamber was regulated by controlled leakage of an inert gas. To check the quality of the product, and in particular the presence of collapse phenomenon, freeze-dried samples were fixed on aluminum circular stubs and sputtered with chrome. The morphology of metalized samples was then examined using a scanning electron microscope (FEI, Quanta Inspect 200, Eindhoven, The Netherlands) at 15 kV and under high vacuum.

Figure 2. Effect of vial position on the value of the heat transfer coefficient. (a) Contour plot of Kv at Pc = 10 Pa (only half of the batch is shown). (b) Vial groups as classified by the position on the shelf: (●) A; (gray dot) B; (slant-hashed dot) C; (horizontal-hashed dot) D, and (○) E. (c) Distribution of Kv for E vials. The solid line is the Gaussian distribution calculated according to the mean value and standard deviation shown in Table 1.

emphasized by refs 52 and 53, vials at the edge of the batch have a higher value of Kv with respect to central ones. Edge vials, in fact, receive an additional heat flow due to side radiation, as well as conduction through the metal band when it is used to load the vials. However, the effect of these additional contributions is strongly reduced beyond the first row of vials and, de facto, is limited to only vials placed just next to the additional heat sources.31 In this study, vials were classified into various groups by their position on the shelf and, hence, the heat transfer mechanisms involved. In the configuration used in this study, vials are arranged in clusters of hexagonal arrays, surrounded by a metal band and loaded directly on the heating shelf. Therefore, vials were classified into five groups as shown in Figure 2b. Vials in the corner were not considered in the analysis as their contribution was not statistically relevant. Graph c displays the distribution of Kv for central vials. It can be observed that the above distribution is well described by a Gaussian distribution. Therefore, eq 3 can effectively be used to describe the density function f Kv,Rp. Figure 3 compares the value of Kv vs Pc for the various groups of vials. As stated above, the pressure dependence of Kv can be described by a nonlinear function (see eq 4), where the model parameters C1, C2, and C3 are obtained by a nonlinear regression of experimental values for Kv vs Pc. The values of C1, C2, and C3 are summarized in Table 1 for the vial used. Table 1 also shows the variance of C1 as observed experimentally for the various groups of vials. This variance is the sum of two contributions: the uncertainty of the method of measurement and the intervial variability. Figure 3 shows a good agreement between the values of Kv vs Pc as measured by gravimetric

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RESULTS AND DISCUSSION Estimation of Model Parameters. Figure 2a shows the value of the heat transfer coefficient as measured by gravimetric method in a batch of vials at Pc = 10 Pa. The value of Kv significantly varies with the position of the vial on the shelf. As 12390

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Figure 4. Resistance to mass transfer measured by Lyobalance (solid line) and estimated by the pressure rise test technique (symbol), as well as the value calculated by using eq 5 with the parameters of Table 1 (dashed line).

Figure 3. Heat transfer coefficient as a function of chamber pressure for the four groups of vials identified in Figure 2b.

Table 1. Parameters Required to Calculate the Value of Kv vs Pc for the Various Groups of Vials [C2 = 1.4 J s−1 m−2 K−1 Pa−1, C3 = 0.04 Pa−1], and of Rp vs. Ldried for a 5% (w/w) Solution of Sucrose heat transfer coefficient (Kv) group of vials B C D E Rp,0 P1 P2

C1, W m−2 K−1 21.9 13.6 9.7 7.8 resistance to mass transfer (Rp) 2.1 × 104 1.4 × 108 1.1× 103

σC1, W m−2 K−1 6.3 1.5 0.5 0.1 m s−1 s−1 m−1

method and as calculated by using eq 4 and the parameters of Table 1. The value of the resistance to vapor flow vs the thickness of the dried layer as measured by Lyobalance is displayed in Figure 4 for a sucrose-based formulation. Rp sharply increased at the beginning of the drying and, then, continued to increase but much more slowly. This trend is typical of porous materials with a nonuniform structure that are characterized by the presence of a compact layer at the top of the cake.25 Scanning electron microscope images confirmed the presence of a compact layer at the top surface for the formulation used, see Figure 5. The values of Rp vs Ldried were calculated using eqs 6 and 7, where the mean value of Jw was retrieved from the weight loss measured by the balance, while the product temperature was measured by a thin thermocouple. The calculation of Rp (by eq 6) requires the knowledge of the ice vapor pressure, which is a function of the temperature of the product at the moving

Figure 5. Cake structure of a 5% (w/w) sucrose sample: enlargement (a) of the top surface and (b) of the internal structure of the solid matrix.

interface. This temperature cannot directly be measured, but can easily be retrieved from the value measured close to the bottom of the vial. Moreover, since the dependence of pw,i on temperature is exponential, a small error on the value of Ti has a significant impact on the final value of Rp, therefore the uncertainty of this parameter is dictated by that of the temperature sensor. In particular, a variation of 0.5 K in temperature reading implies a maximum variation of 10% on the final value of Rp. In this work, we have assumed that all the 12391

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uncertainty on Rp is contained in the parameter P1 of eq 5 as it influences both the final value and the shape of the curve Rp vs Ldried. Therefore, the calculation of the design space in presence of parameter uncertainty is carried out by considering an uncertainty of 10% on the value of P1. Figure 4 also compares the value of the resistance to vapor flow through the dried layer as measured by Lyobalance and as estimated by the pressure rise test technique coupled with DPE+ algorithm. A fairly good agreement can be observed between the two methods both in terms of final value and shape of the curve. However, it must be remarked that the value of Rp as measured by Lyobalance refers to the 15 vials loaded on the balance tray, while the pressure rise test technique refers to the batch as a whole. Nevertheless, the value of sublimation flux as measured by the Lyobalance was higher than the average value as estimated by the pressure rise test technique (data not shown). This difference was due to the fact that vials loaded on the balance tray, with respect to rest of the batch, are exposed to slightly different heat transfer conditions and therefore they have higher values of Kv and show higher product temperatures. It is clear that in order to calculate the resistance to mass transfer by using eq 6, care must be paid in coupling the measured value of Jw with an appropriate value of product temperature. For example, if we use the value of Jw supplied by the Lyobalance to calculate Rp, we have to use the value of product temperature as measured by a thermocouple placed within one of the vials loaded on the tray balance. Finally, the nonlinear dependence of Rp vs Ldried was described by eq 5, where the model parameters Rp,0, P1 and P2 were obtained by a nonlinear regression of experimental values. An example of results is shown in Table 1, where the experimental values obtained by the balance were used to solve the above regression problem. Validation of the Mathematical Model of the Process. The adequacy of the mathematical model to describe the process dynamics was tested on two process variables: the temperature of the product and the duration of the primary drying phase. The evolution of product temperature at the bottom of the vial predicted by the mathematical model was compared with the experimental value supplied by thermocouples and by the pressure rise test technique coupled with DPE+ algorithm. As already emphasized in the previous section, vials located in different positions on the heating shelf can show a different evolution of product temperature because of a different value of Kv vs Pc. Therefore, the comparison between model predictions and thermocouples readings was extended to both edge and central vials, confirming for both of them the adequacy of the model to describe their dynamics. On the other hand, the completion of the sublimation phase was determined experimentally by monitoring the composition of the gas inside the drying chamber, e.g. by comparison of pressure readings obtained by Pirani and Baratron sensors. In particular, in this work the primary drying end-point was associated with the end of the decreasing part of the Pirani− Baratron pressure ratio curve. An example of results is given in Figure 6, where a 5% (w/w) aqueous solution of sucrose was freeze-dried using a two-step cycle for the primary drying: (step 1) from t = 0 to 2.5 h, Pc = 5 Pa, and Tfluid = 258 K; (step 2) from t = 2.5 h to the end of the primary drying, Pc = 5 Pa, and Tfluid = 253 K. Graphs a and b show a good agreement between the experimental measurements of product temperature (as

Figure 6. Example of freeze-drying cycle for a 5% (w/w) sucrose solution. The evolution of (top graphs) Ldried/L and of (bottom graphs) product temperature are given for (a) edge and (b) central vials. In particular, model predictions of product temperature (solid line) are compared with the values measured by thermocouples (dotted line) and estimated by PRT (symbol). The model predictions calculated considering parameter uncertainty are also displayed (dashed line). The horizontal line indicates the limit temperature.

measured by thermocouples) and model predictions, at least within parameter uncertainty (see the dashed lines displayed in a and b) and until ice was still present in the monitored vials. The temperature of the product of both central and edge vials was always below its limit value; this behavior was observed for that fraction of B vials that was characterized by the highest value of Kv and Rp, i.e. K̅ v + σKv, R̅ p + σRp. The pressure ratio between Pirani and Baratron sensor was used here for detecting the completion of ice sublimation (data not shown). This signal was almost constant during the primary drying phase and started to decrease only when most of the vials had completed the drying, which was after 34 h of drying. This result was in good agreement with model predictions of the frozen layer thickness, provided that the drying time estimated by pressure ratio is compared with the value predicted by the model for central vials, and in particular to that fraction of E vials having the lowest value of Kv (i.e., K̅ v − σKv) and the highest value of Rp (i.e., R̅ p + σRp). As these vials are the last to complete ice sublimation, pressure ratio curve approaches the unity only when ice sublimation is completed in all the vials of the batch; that is, only when even the last fraction of central vials (that have the lowest value of Kv because of the parameter variability) have finished sublimating. Analysis of a Freeze-Drying Cycle via Mathematical Modeling. The design space can also be used for the analysis of process failure consequences, that is, to evaluate if 12392

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unexpected variations in Tfluid and Pc can impair the quality of the product that, in this study, corresponds to evaluate if the temperature of the product overcomes the collapse value. However, the same analysis can be used independently of the type of quality criterion considered. In the previous sections, we have observed that the batch of vials was not uniform, thus to obtain a complete vision of the consequences of the above disturbances on the quality of the entire batch, we have to carry out the failure analysis for each group of vials identified in Figure 2b. In this way, the failure analysis can also be employed in case of heterogeneous drying behavior. This approach represents a significant improvement with respect to what proposed by Fissore et al.25 where the batch was assumed to be uniform. In fact, it can happen that the investigated process disturbances damage only certain groups of vials, e.g. those placed at the edge of the shelf as they are more sensitive to product overheating, while the product temperature of the other groups is kept below the limit value. For the sake of clarity, in the following analysis we will refer to only B and E vials, as these two groups are characterized by the highest and the lowest values of Kv and, therefore, are the most and the less sensitive vials to product overheating. Let us now consider an example, that is, the investigated formulation is freeze-dried by using the same two-step recipe for the primary drying phase that was previously used for validating the mathematical model of the process. This cycle was designed to guarantee the quality of the product for the batch as a whole. As a first step, we used the design space to check if the above cycle can effectively maintain the product temperature below its limit value for the entire batch, and then to study if this requirement is still respected after a process disturbance. To calculate the design space, the following parameters have to be set: the range of operating conditions of interest, the model parameters, and product and equipment constraints. In this work, the minimum value of Tfluid and Pc was imposed by the characteristics of the dryer used, i.e. Tfluid,min = 223 K and Pc,min = 2.5 Pa, while the maximum value was set to Tfluid,max = 283 K and Pc,max = 20 Pa. The maximum value for the temperature of the product was set a few degrees higher than the glass transition value observed for the formulation used, that is, 240 K. Finally, the values of Kv vs Pc, and of Rp vs Ldried were calculated by a nonlinear equation (see eqs 4 and 6) whose coefficients are shown in Table 1. Figure 7 displays the design space of the investigated formulation when it is processed in B and E vials (identified in Figure 2b). The points no. 1 and no. 2 indicate the set-point values for the two-step recipe investigated. Instead, the point no. 3 indicates how Tfluid and Pc were intentionally modified to simulate a process disturbance on chamber pressure, which was indeed a pressure increase of 6 Pa (the final pressure is 11 Pa). The various curves delimit the set of (Tfluid, Pc) that, for a fixed value of Ldried, meets the product quality requirement (filled area under the curve), that is, the temperature of the product is maintained below Tmax until the completion of ice sublimation. As it can be expected, the design space of central vials was larger than that of edge vials, as they were characterized by a lower value of Kv (see Figure 3). Therefore, if the objective is to check if the two-step cycle described above guarantees the product quality for all the vials of the batch, we have to refer to the design space of edge vials, as these vials are the most sensitive to product overheating.

Figure 7. Design space of a 5% (w/w) sucrose solution processed in B and E vials. The various curves refer to various values of Ldried/L: (solid line, dark gray area) 1%, (dashed line, gray area) 12%, and (dotted line, light gray area) 99%. The symbol (○) corresponds to (1) Pc = 5 Pa and Tfluid = 258 K, (2) Pc = 5 Pa and Tfluid = 253 K, and (3) Pc = 11 Pa and Tfluid = 253 K.

To carry out the analysis, we have to determine which is the value of Ldried/L (for the selected group of vials) when processing conditions are modified (from step no. 1 to no. 2). With this regard, as the duration of the two steps is known, the value of Ldried/L at the end of each step (and for each group of vials) was calculated by using the mathematical model of the process (data not shown) as already described by Fissore at al.25 According to the cycle investigated here, the drying was carried out at constant chamber pressure (= 5 Pa), while the temperature of the fluid was 258 K for the first 2.5 h and then was reduced to 253 K. At this point, the mathematical model of the process was used to calculate which was the value of Ldried/ L when the modification of Tfluid was introduced, which resulted 12% for B vials. Therefore, to be sure that the constraint on the maximum product temperature was respected, the two points representing the processing conditions for the investigated recipe have to belong to the design space calculated at Ldried/L = 12% for the first step and at Ldried/L = 99% for the second one, respectively. As shown in Figure 7, the two combinations of (Tfluid, Pc) are within the area under the corresponding design space curves of both central and edge vials, therefore the product quality was guaranteed for all the vials of the batch. Figure 6 shows an example of the results that can be obtained carrying out a freeze-drying cycle by using the above recipe. The drying time, as measured by Pirani−Baratron pressure ratio, resulted to be about 34 h that corresponds to the duration of the sublimation phase of central vials. Graphs a and b show a comparison between the product temperatures estimated by the pressure rise test technique (coupled with DPE+ algorithm) 12393

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above, it is also possible to simulate the dynamics of the product for the various vial groups, considering the uncertainty/variability on model parameters, as well as the real evolution of Tfluid and Pc. This analysis gives the probability that the product temperature is below its limit value and the cumulative distribution of the drying time for each group of vials. An example of results is shown in Figure 8 (dashed curves), where it can be noted that the temperature of the product always remained below the limit value of 240 K only for central vials, while a significant fraction of edge vials were processed at a temperature above this value. As already stated in the previous section, the drying time is defined by central vials, thus it would have to be at least 32 h to guarantee that ice sublimation was completed for the entire batch. Furthermore, the drying time observed above was shorter than the value observed for the previous test (i.e., 34 h). This time reduction was due to the fact that the second part of the drying was carried out at a higher value of Pc. Figure 9 shows an example of experimental cycle carried out using the operating conditions identified in Figure 7, but

and measured through thermocouples. It must be remarked that the pressure rise test technique refers to the entire batch of vials,49 therefore it estimated an average value of the system state that includes also the contribution of vials placed at the edge of the batch. The above results confirmed that the cycle designed in Figure 7 can effectively satisfy product quality requirements. In fact, the temperature of the product of both central and edge vials was below its limit value throughout the sublimation stage. The mathematical model was used also for calculating the probabilistic distribution of the product temperature and of the drying time for specific values of uncertainty/variability of Kv and Rp. An example of results is given in Figure 8 (solid lines).

Figure 8. Comparison between the probabilistic distribution of the maximum product temperature and of the drying time for B and E vials as calculated for the two-step cycle identified in Figure 7 (solid line), and for the same recipe but in presence of a process deviation (dashed line). The area evidenced in gray indicates when the temperature of the product is higher than its limit value.

Figure 9. Example of model validation in case of freeze-drying of a sucrose solution (5% w/w), and in presence of a step disturbance on chamber pressure. Comparison of TB predicted by the model (solid line) and measured by thermocouples (dashed line) for B and E vials. The evolution of the frozen layer thickness as predicted by the model is also shown. The horizontal line indicates the maximum allowable product temperature.

The temperature of the product (of both B and E vials) remained always below the limit value of 240 K, while the drying time would have to be at least 34 h to guarantee that all the vials have completed the primary drying. These results are in a fairly good agreement with the above experimental observations. In conclusion, experiments and mathematical simulations confirm that, as already emphasized by the previous analysis carried out by using the design space, the investigated recipe can effectively guarantee the quality of the product for all the vials of the batch. Let us now consider what happens when a pressure disturbance is introduced during the sublimation phase. If the variation of chamber pressure occurs at the beginning of primary drying, e.g. when Ldried/L is smaller than 12%, the product remains inside the design space after the variation (see Figure 7) and, hence, its quality is preserved. Nevertheless, if the original chamber pressure is not restored before the completion of the sublimation, vials at the edge of the shelf exit the design space. On the contrary, central vials are always inside the design space, even if the pressure remains at 11 Pa. The above analysis was carried out considering the average properties of each group, thus intervial variability for Kv was not considered in the calculations. However, as already done

introducing a pressure disturbance when Ldried/L for edge vials was almost 12% that, according to mathematical model simulation, happened after about 3 h of drying. The new combination of Tfluid and Pc (point no. 3 in Figure 7) maintained the maximum value for the product temperature (as inferred by thermocouples) lower than its limit value only for central vials, see Figure 9 (bottom graph). Edge-vials, instead, exited the design space at about Ldried/L = 12%, see Figure 9 (top graph). This evidence is in good agreement with the results of the previous analysis. A fairly good agreement between model predictions and experimental observations for the product temperature was observed also in this test; however, the drying time estimated by the mathematical model for E vials (as the time at which Lfrozen = 0) was 29 h, therefore it is slightly shorter than the experimental observation. In fact, Pirani−Baratron pressure ratio detected the end of the primary drying after 32 h of drying. However, it 12394

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must be noted that the above pressure ratio signal starts to decrease when most of the vials have completed the ice sublimation. Here, model predictions refer to the average properties of central vials; therefore they do not take into account intervial variability. Therefore, after 29 h of drying some vials were still sublimating and impeded the pressure ratio signal to decrease. The nonuniformity of central vials is, instead, considered in the results shown in Figure 8, where it can be observed a good agreement between the drying times predicted by the model and estimated by the pressure ratio. The visual inspections of freeze-dried samples confirmed that edge vials actually overcame the limit temperature as they were partially collapsed, while neither micro- nor macro-collapse of the structure was observed in central vials.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +39-(0)11-0904679. Fax: +39-(0)11-0904699. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS A valuable contribution of Dr. Alberto Vallan (Politecnico di Torino, Italy) for the setup of the weighing device is gratefully acknowledged.

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CONCLUSIONS In this paper it has been shown how mathematical simulation can rightly predict the behavior of the system after an unexpected change of Pc, both in terms of total drying time and product temperature response. The above method can also be employed in case of heterogeneous drying behavior. It has been proven that even a small variation in Pc can bring the product above its collapse temperature and might impair the quality of the final product, at least for a part of the batch. Therefore, the mathematical simulation can be used for establishing in-line, as the computation time requested by mathematical simulation is very short (few minutes), if an unexpected variation in processing conditions is acceptable and eventually which vials of the batch might be damaged. This information is important to establish if the cycle has to be rejected or a significant fraction of vials can still satisfy the product requirements. The above analysis can be used for both laboratory units and manufacturing equipment, provided that an appropriate value is used for the heat transfer coefficient of the container. Furthermore, it must be noted that the analysis of process failure can be carried out in-line as the computational effort is small and few minutes are required for the calculations by a standard PC. Intelligent methods (e.g., fuzzy systems or neural networks) can also be used to carry out the above analysis. In this way, the computational effort can be further diminished. However, the huge amount of work required to develop the black-box model limits its application. In fact, the black-box model should correlate the operating conditions to the product temperature, and this relation depends on the residual amount of ice, the heat transfer coefficient for the vial used, and the product resistance to vapor flow. The strong nonlinear dependence of these variables makes the development of the above model a really challenging task, because a large amount of data can be required in the training phase. A similar analysis can also be carried out to support the cycle development phase, e.g. to define appropriate margins of safety, which can guarantee the robustness of the cycle. In fact, the operator in charge of selecting a proper combination of Tfluid and Pc for the primary drying should consider, besides the sublimation flux and the consequent drying time, also the robustness of the cycle, that is, remaining far enough from the curves identifying the design space in the perspective of potential disturbances on the operating conditions. The same approach can also be used to evaluate the robustness of a cycle, which has been already developed in a laboratory scale dryer but has to be transferred to different freeze-drying apparatus.

NOTATION C1 = parameter expressing the dependence of Kv from chamber pressure, J m−2 s−1 K−1 C2 = parameter expressing the dependence of Kv from chamber pressure, J m−2 s−1 K−1 Pa−1 C3 = parameter expressing the dependence of Kv from chamber pressure, Pa−1 f Kv,Rp = density function describing the distribution of the values of Kv and Rp ΔHs = heat of sublimation, J kg−1 Jw = sublimation flux of the solvent, kg s−1 m−2 Kv = overall heat transfer coefficient between the shelf and the product, J m−2 s−1 K−1 K̅ v = mean value of the overall heat transfer coefficient between the shelf and the vial, J m−2 s−1 K−1 kfrozen = thermal conductivity of the frozen layer, J K−1 s−1 m−1 L = total product thickness, m Ldried = thickness of the dried layer, m Lfrozen = thickness of the frozen layer, m P1 = parameter used to calculate the resistance to vapor flow, s−1 P2 = parameter used to calculate the resistance to vapor flow, m−1 Pc = total pressure in the drying chamber, Pa pw,c = solvent partial pressure in the drying chamber, Pa pw,i = solvent partial pressure at the interface of sublimation, Pa Rp = resistance to vapor flow, m s−1 Rp,0 = parameter used to calculate the resistance to vapor flow, m s−1 R̅ p = mean value of the resistance to vapor flow, m s−1 TB = temperature of the frozen product at the bottom, K Tfluid = temperature of the heating fluid, K Ti = temperature of the product at the interface of sublimation, K Tmax = maximum temperature of the product, K t = time, s td = duration of the primary drying, h

Greeks

ρfrozen = density of the frozen product, kg m−3 ρdried = apparent density of the dried product, kg m−3 σ = standard deviation

Subscripts

max = maximum value min = minimum value Abbreviations

DPE = dynamic parameters estimation TDLAS = tunable diode laser absorption spectroscopy 12395

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dx.doi.org/10.1021/ie300505n | Ind. Eng. Chem. Res. 2012, 51, 12386−12397