# Modeling of freezing step during vial freeze-drying of pharmaceuticalsЧinfluence of nucleation temperature on primary drying rate.

код для вставкиСкачатьASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING Asia-Pac. J. Chem. Eng. 2011; 6: 288–293 Published online 19 January 2010 in Wiley Online Library (wileyonlinelibrary.com) DOI:10.1002/apj.424 Short Communication Modeling of freezing step during vial freeze-drying of pharmaceuticals – influence of nucleation temperature on primary drying rate Kyuya Nakagawa,1 Aurelie Hottot,2 Severine Vessot2 and Julien Andrieu2 * 1 Department of Mechanical and System Engineering, University of Hyogo, 2167 Shosha, Himeji, Hyogo 671 2280, Japan Laboratoire Automatique etGénie des Procédés, LAGEP.UMR Q 5007 CNRS UCBL ESCPE Bât. 308 G, 43, Bd du 11 Novembre 1918, 69622 Villeurbanne, Cédex, France 2 Received 28 September 2009; Accepted 24 November 2009 ABSTRACT: To optimize the primary drying step of freeze-drying processes, it is very important to control morphological parameters of the frozen matrix. A mathematical model that simulates temperature profiles during freezing process of standard pharmaceutical formulations was set up and the ice crystal mean sizes were semi-empirically estimated from the simulated thermal profiles. Then, water vapor mass transfer kinetics during sublimation step was estimated from ice phase morphological parameters. All these numerical data were compared with experimental data and a quite good agreement was observed, which confirmed the adequacy of the present model calculations. It was confirmed that, for a given formulation, the mass transfer parameters during freeze-drying were strongly dependent on morphological textural parameters, and consequently, on the nucleation temperatures that fix the ice phase morphology. The influence of freezing rates was also predicted from the simulations, proving that an increase of cooling rates led to slower primary drying rates. 2010 Curtin University of Technology and John Wiley & Sons, Ltd. KEYWORDS: freezing; freeze-drying; ice crystal size; nucleation temperature INTRODUCTION Ice nucleation process is well known to be a spontaneous and stochastic phenomenon. Due to the heterogeneous and stochastic nature of nucleation phenomena, morphologies of frozen products vary from one vial to another all through their location on the plate of the sublimation chamber. These structural heterogeneities mainly result in large distributions of primary drying rates during freeze-drying.[1 – 2] Especially, for small scale frozen system, it is considered that the nucleation temperature is a key factor for the optimization of the principal quality factors of the freeze-dried matrix. As a matter of fact, the undercooling degree of the solution determines the number of nuclei and, consequently, greatly influences the ice crystals morphology in the frozen sample.[3 – 4] In previous studies, we have investigated an ultrasound system that allows the ice crystal nucleation *Correspondence to: Julien Andrieu, Laboratoire Automatique etGénie des Procédés. LAGEP.UMR Q 5007 CNRS UCBL ESCPE Bât. 308 G, 43, Bd du 11 Novembre 1918, 69622 Villeurbanne, Cédex, France. E-mail: andrieu@lagep.cpe.fr 2010 Curtin University of Technology and John Wiley & Sons, Ltd. Curtin University is a trademark of Curtin University of Technology control during the freeze-drying process of model formulations in commercial glass vials.[5] As results, we have observed that the ice crystal sizes were closely dependent on nucleation temperatures, and that the water vapor permeabilities of the freeze-dried cake during the sublimation step increased with the nucleation temperatures. In this article, a mathematical model of the freezing step was proposed. It allowed the simulations of the temperature profiles of standard pharmaceutical formulations by using a commercial finite element code taking account of actual vial geometry and of the freezedrying operating conditions. From these temperatures profiles, a semi-empirical model was set up to estimate the mean ice crystal sizes and, consequently, the water vapor-dried layer permeabilities for an optimization of the whole freeze-drying cycle. EXPERIMENT Ten percentage (w/w) mannitol solutions prepared from distilled water and mannitol powder (Fluka Chemie AG) were used as model pharmaceutical formulations. A heat exchanger combined with ultrasound transducer Asia-Pacific Journal of Chemical Engineering MODELING OF FREEZING STEP DURING VIAL FREEZE-DRYING (SODEVA, France) allowed the control of the cooling and of the nucleation processes. The nucleation of the sample at the selected temperature was realized by ultrasound wave propagation during 1 s through the system cooled at −1 ◦ C/min (−0.72 ◦ C/min at vial bottom) and tubing glass vials (Verretubex, France) of 3 ml (vial diameter d = 12 mm) were only used in this work. Sample solution (0.75 ml) was filled in each vial (solution height h = 8.0 mm). Frozen samples were sublimated (primary drying period) with a laboratory pilot freeze-dryer (USIFROID SMH45, France) under the following standard sublimation conditions for therapeutic proteins: shelf temperature at −40 ◦ C, and chamber total pressure at 10 Pa. moderately. The freezing modeling was based again on the heat conduction equation recalled below, by adding two source terms, namely the heat generation due to the ice nucleation latent heat, Qn , and the heat generation due to the ice crystallization latent heat, Qc . ρCp−apparent (2) As well, it was assumed that the rate of ice nucleation was proportional to the supercooling degree Tf − T ∗ [K], so that the positive source term Qn was expressed as follows[6] : Qn = Hf ki (Tf − T ∗ ) (3) where T ∗ represents the temperature in the supercooled liquid, and Tf , the freezing front temperature. This term became zero as soon as the supercooled water has been totally crystallized. Here, ki (kg/m3 s K) represents the nucleation rate constant estimated from freezing front velocity v (m/s). Besides, the positive source term, Qc , corresponding to the latent heat generated by the ice crystallization was expressed by the classical equation: MODEL CALCULATION To simulate these freezing temperature profiles, our modeling was divided into two periods, namely the cooling step and then the freezing step as explained below. The 2D axi-symmetric model taking into account the real geometry of the vial was written under FEMLAB 3.0 (COMSOL). Firstly, cooling step was simulated to obtain the initial condition for the temperature profile throughout the vial bulk for the following freezing step. Thus, the cooling step from t = 0 up to the nucleation time, noted tn , was simulated with the classical conduction heat equation, namely: ∂T = ∇(k ∇T ) (1) ρCp−liquid ∂t Qc = Hf ∂ (ρXice ) ∂t (4) As illustrated in Fig.1, during the ice crystal growth period, the sample could be divided in three zones corresponding to three different physical states, namely, a solid zone, a mushy zone and a liquid zone. The mushy zone is a suspension of ice in the undercooled solution (ice fraction, Xice ) which was defined by the temperature difference (T ) between the freezing front Then, during the freezing step, the nucleation started at the fixed time tn corresponding to nucleation temperature, Tn , and subsequently, ice crystal growth progressed Sample height [m] ∂T = ∇(k ∇T ) + Qn + Qc ∂t Ambient temperature 280 K Heat transfer by air contact h1 = 6 W/m2K Liquid zone Cp-liquid Thermocouple Mushy zone Cp-mushy Freezing front Cp-ice Ice zone Tf Tf + ∆T Controlled temperature at -0.72 K/min Heat transfer via vial glass h2 = 11 W/m2K 8 mm 4 mm Temperature [K] 12 mm f = 0.2 mm Figure 1. Model of the physical formulation states through the sample. 2010 Curtin University of Technology and John Wiley & Sons, Ltd. Asia-Pac. J. Chem. Eng. 2011; 6: 288–293 DOI: 10.1002/apj 289 NAKAGAWA et al. Asia-Pacific Journal of Chemical Engineering RESULTS AND DISCUSSIONS and the liquid zone. ThisT parameter was a fitting parameter, the value of which was identified from experimental data. Moreover, we supposed that the ice crystal growth rate was entirely controlled by the heat transfer and that the ice fraction, Xice , varied linearly with the temperature in the domain of phase change. Thus, this term could be introduced in the accumulation term be using an apparent heat capacity value, Cp−apparent , defined as follows.[7] Cp−apparent C : T > Tf + T p−liquid Cp−liquid + Cp−ice f + H T Xice : 2 = Tf ≤ T ≤ Tf + T Cp−ice : T < Tf Estimation of mean ice crystal sizes As a result of model calculation, we could obtain an history of temperature for each freezing procedure as a function of nucleation temperature Tn . Quantitative estimation of ice crystal mean size, L∗ , was achieved by using a relationship between the freezing front rate, R, and the temperature gradient in frozen zone, G. For each position, the time corresponding to the freezing front passing across was determined at the equilibrium freezing temperature (Tf ). The R and G values were estimated from temperature profiles at different locations from the vial bottom. Based on literature reviews and due to our ice phase morphology type, we adopted the following relationship type to interpret our ice morphology data[8 – 10] : (5) The variations of apparent thermal conductivity and of bulk density values as a function of temperature were represented by a linear law all along the domain between the liquid and the solid zone. The numerical values that were used in this freezing model calculation are listed in Table 1. L∗ = aR −0.5 G −0.5 Freezing model calculation Ice crystal size estimation Form factor estimation A kliquid (W/m K) kice (W/m K) Cp−liquid (J/kg K) Cp−ice (J/kg K) Tf (K) Hf (J/kg) Xice (−) T (K) a (mK0.5 s−0.5 ) ε/τ (−) 0.6 2.5 3852 1967 270.3 333 500 0.9 0.98 12 0.225 B 140 Tn = -2 °C Tn = -6 °C Tn = -10 °C Tn = -14 °C 120 100 140 100 L* [mm] 60 80 60 40 40 20 20 0 0.002 0.004 0.006 Position in Vial (height) [m] Experiment Calculation 120 80 0 (6) where a = 12 (mK0.5 s−0.5 ) is an empirical constant that was identified by experimental data (cf. Table 1). Then, we could obtain the different ice crystal values corresponding to each R and G values set. During freezing step, these values distributed all along the heat transfer direction and, moreover, they depended on nucleation temperature values. Thus, one can observe on the plots of Fig. 2(a) that the ice crystal sizes generally increase as a function of local vial height position, but that this size becomes smaller at the upper layer of the vial. Then, mean ice crystal sizes values were calculated by averaging local ice crystal sizes throughout the whole sample bulk; the calculated mean values, noted L∗ , are plotted as a function of nucleation temperatures and Table 1. Numerical values used in the calculation. L* [mm] 290 0.008 0 -14 -12 -10 -8 -6 -4 -2 0 Tn [°C] Figure 2. Ice crystal mean sizes (Frozen 10% mannitol solution), (A) distribution in vial vertical direction; (B) nucleation temperature dependency. 2010 Curtin University of Technology and John Wiley & Sons, Ltd. Asia-Pac. J. Chem. Eng. 2011; 6: 288–293 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering MODELING OF FREEZING STEP DURING VIAL FREEZE-DRYING compared with the experimental values obtained from image analysis in Fig. 2(b). Although some deviations were observed around Tn = −2 ◦ C, most of other data are in quite good agreement. This agreement was also observed with Bovin Serum Albumin formulations.[11] 1.5 Experiment Model Permeability during freeze-drying Dried zone permeability, noted K , could be estimated from mean ice crystals diameters values by assuming that the dried cake texture is represented by a bundle of capillary tubes (diameter d p ). In our case, we could estimate the value of Knudsen number around Kn = 4 and, consequently, from the molecular diffusion theory in Knudsen regime, the dried layer permeability, K , by the following equations: Kmodel = ε Dk τ (7) where represents the total flow contribution factor, and the Knudsen diffusivity Dk is expressed by: 1 8Rg Ts Dk = · dp (8) 3 π Mw and the total flow contribution factor, , is equal to: = 1 1 + dp λ (9) where λ represents the mean free path, given by: kb Ts λ= √ 2π dm2 P (10) For this estimation, we assumed that the ice crystal sizes corresponded to the pore diameter (d p ) in the Eqn (8). In Eqn (7) the form factor ε/τ = 0.225 is an empirical constant that was identified from experimental sublimation rate data (cf. Table 1). For determining the sublimation rates, the freeze-drying runs were stopped after 3–4 h of sublimation that corresponded to 30–40 wt% of ice sublimated and, then, the weight loss was measured. Otherwise, the experimental water vapor permeability values were estimated by the following equation Kexp = edry Rg Ts ·m Ps − Pc Mw (11) which assumes that the water vapor pressure decreases linearly through the dried layer thickness, noted edry , and where Ps and Pc represents, respectively, the water vapor pressure at the sublimation front and in the sublimation chamber. Then, these experimental 2010 Curtin University of Technology and John Wiley & Sons, Ltd. K [10-3 m2/s] 1.0 0.5 0 -14 -12 -10 -8 -6 Tn [°C] -4 -2 0 Figure 3. Dried layer permeability as a function of nucleation temperature. Kexp values are plotted as a function of nucleation temperature in Fig. 3. Thus, we observed that the mean Kmodel values plot represented pretty well the tendency of variation of the corresponding experimental values during the primary drying step. Moreover, as previously discussed, frozen samples presented always ice crystal sizes distributions depending on their thermal history, namely their cooling rate and their nucleation temperature. Then, sublimation kinetics at each sublimation time can be estimated by freezing conditions by using Eqs (7)–(9) and the ice crystal mean values in Fig. 2(b). For these calculations, the value of the form factor ε/τ was still identified by fitting several values of Kmodel and Kexp as mentioned before Thus, we have calculated sublimation rates at certain vial heights by using Eqn (11) with Kmodel value deduced from ice crystal size distributions (cf. Fig. 2(a)) for two nucleation temperatures, namely −2 ◦ C and at around −6 ◦ C, and the results are plotted in Fig. 4. It was observed, firstly, that the experimental sublimation rates decrease slowly with the progress of sublimation and, secondly, that a large discrepancy between Kexp and Kmodel values existed at the beginning and during the first half of sublimation period with, nevertheless, a better agreement in the second half of sublimation period. This discrepancy could be due to a crust effect at the top of freeze-dried layer, the effect resulting from the cryoconcentration of the solute (mannitol) at the top of frozen sample. Consequently, it was quantitatively confirmed that nucleation temperatures are key parameters that determined the ice morphology and, consequently, the sublimation times during freeze-drying processes. However, Asia-Pac. J. Chem. Eng. 2011; 6: 288–293 DOI: 10.1002/apj 291 NAKAGAWA et al. Asia-Pacific Journal of Chemical Engineering 0.4 1.0 Exp Tn = -2 °C Model Tn = -2 °C Exp 0.3 Tn = -6 °C 0.8 Cooling Rate 0.2 K/min Model Tn = -6 °C K [10-3 m2/s] Drying Rate [10-3kg/m2s] 292 0.2 0.6 0.72 K/min 5.0 K/min 0.4 0.1 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Amount Dried [kg-sublimed water/kg-initial water] 1.0 Figure 4. Drying rate kinetics during primary drying step. cooling rate is the most experimentally accessible freezing operating parameter that controls the ice crystals morphology. Thus, it is also worthwhile to know the direct influence of this cooling rate on ice crystal sizes and freeze-dried layer permeability. Figure 5 shows the estimated dried layer permeability values simulated for different cooling rates for mannitol samples by using the same values for the empirical constant reported above. These simulations clearly showed that an increase of cooling rates led to smaller permeability values of the dried layer. Moreover, it is noteworthy that this tendency becomes much more significant by increasing the nucleation temperature because the permeability dependency on the nucleation temperatures becomes much higher by decreasing the cooling rate. For example, when samples were prepared at cooling rate equal to −5 K/min, which corresponded to a fast cooling rate, the variation of nucleation temperature did not result in too large variations of permeability. The physical explanation of this behavior certainly relies on the complex relationships that exist between cooling rates, internal thermal gradients, crystals growth and nucleation rates that took place all along the freezing step in the undercooled solution inside the glass vials of small size. CONCLUSIONS A physical modeling of standard pharmaceutical formulations freezing processes was proposed. Ice crystals mean sizes were estimated from the obtained temperature profiles and they were found in fine agreement with experimental values. We also estimated the ice crystal sizes distribution in the axial position of the vial along the heat flux direction. Based on molecular diffusion 2010 Curtin University of Technology and John Wiley & Sons, Ltd. 0 -14 -12 -10 -8 -6 Tn [°C] -4 -2 0 Figure 5. Simulation of influence of cooling rates on dried layer permeability. theory, dried layer permeability values were estimated from simulated ice crystal sizes and we still observed a satisfactory agreement between experimentally determined mean permeability values and calculated ones. Sublimation kinetics was also evaluated from estimated permeability values. A discrepancy between experimental sublimation rates and simulated ones was observed at the beginning of sublimation period possibly due to a crust effect induced by the solute cryoconcentration at the top of frozen sample. Moreover, permeability dependency on cooling rate was also predicted, that is to say, when the samples were frozen at faster cooling rates, the variation of nucleation temperature does not result in too large distribution of freeze-dried cake permeability values. Thus, it was experimentally and theoretically confirmed that the mass transfer parameters during freeze-drying are strongly dependent on morphological parameters of the ice crystals phase and, consequently, on the nucleation temperatures. NOMENCLATURE a Empirical parameter in Eqn (6) Cp Heat capacity (J/kg K) Dk Knudsen molecular diffusion coefficient (m2 /s) dm Diameter of water molecule (m) dp Pore diameter (m) d p Ice crystal mean diameter (m) edry Dried layer thickness (m) G Temperature gradient in frozen zone (K/m) Hf Latent heat of crystallization (J/kg) Asia-Pac. J. Chem. Eng. 2011; 6: 288–293 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering h K k kB ki Kn L∗ Mw m P Pc Ps Qc Qn R Rg s T t T∗ T◦ Tf Tn Ts Xice ν MODELING OF FREEZING STEP DURING VIAL FREEZE-DRYING Convective heat transfer coefficient (W/m2 K) Dried layer permeability (m2 /s) Thermal conductivity (W/m K) Boltzmann constant (J/K) Nucleation rate constant (kg/m3 s K) Knudsen number (−) Ice crystal mean size (m) Molecular weight of water (kg/kmol) Water vapor sublimation rate (kg/m2 s) Total pressure (Pa) Chamber total pressure (Pa) Sublimation front total pressure (Pa) Latent heat of crystallization (W) Latent heat of nucleation (W) Freezing front rate (m/s) Perfect gas constant (J/kmol K) Thickness of undercooled zone (m) Temperature (K) Time (s) Temperature in supercooled liquid (K) Homogeneous undercooled temperature (K) Freezing front temperature (K) Nucleation temperature (◦ C) Sublimation front temperature (K) Ice fraction Freezing front velocity (m/s) Greek letters ε Porosity of dried layer (−) λ Water molecules mean free path (m) ρ Density (kg/m3 ) τ Tortuosity factor (−) Total flow contribution (−) REFERENCES [1] M. 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