Modeling of expanded-bed protein adsorption by taking into account the axial particle size distribution

Biochemical Engineering Journal 16 (2003) 265–272

Modeling of expanded-bed protein adsorption by taking into account the axial particle size distribution Xiao-Dong Tong, Bo Xue, Yan Sun∗ Department of Biochemical Engineering, School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, China Received 4 December 2002; accepted after revision 3 February 2003

Abstract A mathematical model was established to describe protein adsorption behavior in expanded-bed chromatography by taking into account the axial size classification of adsorbent within the bed. This model has been used to simulate the expanded-bed adsorption (EBA) of bovine serum albumin (BSA) to Streamline DEAE under various operating conditions, such as different feed protein concentrations, liquid velocities and liquid viscosities. The parameters involved in the model are determined by the independent experiments or calculated from correlation to ensure a reliable comparison between the experimental and simulated results. Using these independently determined parameters, the model prediction agreed reasonably well to the experimental data. Moreover, the model was found more precise than that without considering the axial particle size distribution. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Expanded-bed chromatography; Streamline DEAE; Bovine serum albumin; Modeling; Particle size distribution; Simulation

1. Introduction Expanded-bed is a low back-mixing liquid fluidized bed achieved by the purpose-design of solid matrix with a defined size and/or density distribution. As an innovative chromatography mode, the EBA technology offers the advantages of combining the unit operations of particulate removal, product concentration and product capture into one single step, benefiting in the reduction of production cost, operation time and the number of operation steps in biological production [1,2]. Currently, expanded-bed adsorption (EBA) has been widely studied for application in the various aspects of downstream bioprocesses, including protein recovery [3], cell disruption [4], cell sort [5], nucleic acid purification [6] and flow-ELISA analysis [7]. To widely extend the EBA technology in biological industries, better understanding of the expanded-bed behavior is necessary to maximize its efficient applications [8]. Recently, Chase and coworkers [9,10] have reported the control of adsorption process within an expanded-bed system by on-line monitoring method. On-line monitoring of the target product at different bed heights within the bed provided the possibility of tighter control of the separation and greater ∗ Corresponding author. Tel.: +86-22-2740-2048; fax: +86-22-2740-6590. E-mail address: [email protected] (Y. Sun).

process efficiency. In addition to the technical effort, process modeling can also lead to better understanding of the adsorption/desorption performance within an expanded-bed. In 1960, Fan et al. [11] developed a mathematical model for phenol adsorption to granular activated carbon in liquid fluidized bed by taking in account the flow hydrodynamics, mass transfer and diffusion resistance within the particles. In a recent publication, Wright and Glasser [12] introduced the pore diffusion–adsorption mechanism into Veeraraghavan and Fan’s model to make it useful in expanded-bed chromatography for protein adsorption. These existing models have generally assumed that the adsorbents are mono-size and mono-density particles, and the voidage of bed is constant through the column. However, it has been found that adsorbent in expanded-bed system have size, density and concentration distributions along the column [13–15]. Therefore, the above assumptions may result in the decrease of accuracy in simulations. In a previous work, we measured the particle size and density distributions of STREAMLINETM gel (Amersham Biosciences, Uppsala, Sweden) in expanded-bed systems [15]. We found that the density distribution of the solid-phase was negligible, while the size classification in the axial direction was significant. Hence, an empirical correlation for the mean particle size as a function of bed height was obtained for the commercial solid-phase [15]. Thus, in this work, we have made efforts to incorporate the correlation into the

1369-703X/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S1369-703X(03)00058-5

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Nomenclature Bo c

Bodenstein number aqueous protein concentration in equilibrium (mg/ml) c0 initial bulk-phase protein concentration (mg/ml) cf protein concentration at the particle surface (mg/ml) ci protein concentration at the particle pore (mg/ml) DAB BSA diffusivity in free solution (m2 /s) DL liquid-phase dispersion coefficient (m2 /s) Dm molecular diffusion coefficient of BSA (m2 /s) Dp BSA pore diffusivity in adsorbents (m2 /s) DS solid-phase dispersion coefficient (m2 /s) d¯ mean particle diameter at different axial position (m) d0 mean particle diameter at unclassified state (m) h bed height at different axial position (m) H expanded-bed height (m) H0 settled bed height (m) kf,e liquid film mass transfer coefficient (m/s) Kd dissociation constant (mg/ml) q adsorbed protein density in equilibrium (mg/ml) q¯ average solid-phase protein concentration (mg/ml) qi particle adsorbed protein concentration (mg/ml) qm adsorption capacity (mg/ml) r coordinate in radial direction (m) R mean particle radius (m) ¯ Rep particle Reynolds number (=ρp dU/µε) RSSE relative sum of the squares of the error (%) Sc Schmidt number (=µ/ρDm ) t time (min) U superficial velocity (m/s) z axial position in the column (m) Greek ε ε0 εp µ ρ ρp σθ2

letters liquid void fraction in the expanded-bed liquid void fraction in the packed-bed effective porosity of agarose gel for BSA liquid viscosity (Pa s) liquid density (kg/m3 ) solid density (kg/m3 ) variance

ous operating conditions. The prediction from the improved model was also compared with that from the model without considering the size classification. The effectiveness of the improved model has been validated by the better agreement between the model simulations and experimental data. 2. Materials and methods 2.1. Materials Bovine serum albumin (BSA) was purchased from Sigma. Streamline DEAE (Amersham Biosciences, Uppsala, Sweden) was an agarose-based anion exchanger, which was composed of a crystalline quartz core and covered by a 6% cross-linked agarose. The size distribution analyzed with a Mastersizer 2000 unit (Malvern Instruments Ltd., UK) was found in the range of 80–500 ␮m, with a volume-weighted mean diameter of 210 ␮m. The average density measured by a pycnometer (Anhui Fengyang Instruments Ltd., Anhui, China) was 1135 kg/m3 at 20 ◦ C. Glycerol was obtained from Tianjin Letai Chemical Company (Tianjin, China). All other chemicals were of analytical grade from local sources. 2.2. Expanded-bed operation A standard Streamline 25 column (25 mm i.d., 100 cm height) designed by Amersham Biosciences (Uppsala, Sweden) and ÄKTA explorer 100 with Unicorn 3.21 control were connected for the expanded-bed adsorption. The hydraulic adapter was positioned 0.5 cm above the bed surface. Proper column vertical alignment was confirmed in all experiments. Liquid dispersion behavior in the expanded-bed was determined by residence time distribution (RTD) experiments at 25 ◦ C using the step-input technique [16]. Phosphate buffer (0.02 mol/l) and 20% (v/v) glycerol solution in the buffer were used as the liquid-phases for expanded-bed operations. In the RTD experiments, 0.25% (v/v) acetone solution was injected as the tracer solution at the bottom inlet of the column. Individual experiments were performed for the complete experimental rig in the presence or absence of adsorbent in order to identify the contribution of the volume of fittings and the 0.5 cm zone above the bed surface. Moment analyses of the RTD data brought out the mean residence time and the variance for the expanded-bed system. Considering the expanded-bed as a close vessel, the Bodenstein number, Bo, can be calculated from the following formula [17]: σθ2 =

mathematical model for expanded-bed protein adsorption described by Wright and Glasser [12]. Predicted with unadjustable parameters, the model simulations were compared with the experimental results of protein breakthrough in the expanded-bed of Streamline DEAE investigated under vari-

2 2 − (1 − e−Bo ) Bo Bo2

(1)

The Bo number is expressed by the ratio of convective to dispersion mass transport: Bo =

UH εDL

(2)

X.-D. Tong et al. / Biochemical Engineering Journal 16 (2003) 265–272

Thus, the liquid-phase dispersion coefficient DL (m2 /s) can be calculated from Eq. (2). 2.3. Protein adsorption experiments The adsorption equilibrium and kinetics of BSA to Streamline DEAE were characterized by batch adsorption method [18]. The aqueous phase was 0.02 mol/l phosphate buffer (pH 7.0) or the buffer containing 20 or 40% (v/v) glycerol. In adsorption equilibrium experiments, 0.1 ml of the drained adsorbent was added to 10 ml of BSA solution of different concentrations. Adsorption experiments were conducted at 25 ◦ C for 8 h in a shaking incubator. At the end of adsorption, the solid-phase was separated, and the supernatant was analyzed spectrophotometrically at 280 nm for protein concentration. The adsorbed mass of protein was calculated by mass balance. In kinetic experiments, 0.5 ml of the drained gel was mixed to 50 ml of BSA solution. The adsorption was carried out in the shaking incubator at 25 ◦ C. Every few minutes, about 2 ml of the liquid-phase was aspirated using a suction tube to determine protein concentration, and the sample was returned to the vessel immediately. This procedure took less than 20 s. By this procedure, the time course of BSA concentration decrease was determined. The pore diffusion model combining the external film mass transfer resistance [19] was used for fitting the experimental data to determine the pore diffusivity.

267

can be represented by the Langmuir equation: qm c q= Kd + c

(3)

(2) The rate of protein adsorption to the adsorbent is controlled by pore diffusion and external liquid-film mass transfer resistances. (3) The hydrodynamic behavior of the solid and liquidphases can be described by the axial dispersion model [17]. (4) The local size distribution in a definite expanded-bed height is neglected. Instead, the volume-weighted mean size is used to express the local particle size. The axial distribution of the mean particle size can be expressed by the empirical correlation of Tong and Sun [15]: d¯ h = 1.21 − 0.46 (4) d0 H The axial dispersion equation for the liquid-phase and its initial and boundary conditions are written as ∂2 c U ∂c 3kf,e (1 − ε)(c − cf ) ∂c = DL 2 − − ∂t ε ∂z εR ∂z IC :

t = 0,

c(z, 0) = 0

BC1 :

z = 0,

c = c0 +

BC2 :

z = H,

∂c =0 ∂z

(5) (5a)

DL ε ∂c U ∂z

(5b) (5c)

2.4. Protein breakthrough experiments

In addition, the solid-phase dispersion equation and its initial and boundary conditions are given by

BSA adsorption in the expanded-bed of Streamline DEAE was investigated by frontal analysis. All experiments were carried out at 25 ◦ C with a settled bed height of 10.0±0.5 cm. Before applying BSA solution, the bed was allowed to expand stably at least 30 min with a proper liquid-phase. Prior to adsorption experiment, RTD in the expanded-bed was measured to predict axial dispersion coefficient. In all the breakthrough experiments, the liquid flow rate was kept constant, and the bed height decrease due to protein adsorption was found less than 0.5 cm.

(1 − ε) IC :

∂2 q¯ 3kf,e (1 − ε)(c − cf ) ∂q¯ = DS 2 + ∂t R ∂z

t = 0,

The breakthrough model taking into account the particle size distribution along the column was introduced to predict BSA breakthrough performance in the expanded-bed of Streamline DEAE. The governing equations of the present model were derived based on the work of Wright and Glasser [12]. Basic assumptions for the model are as follows: (1) Protein concentration in the adsorbent pore is in local equilibrium with its concentration adsorbed on the inner surface of the pore wall. This adsorption equilibrium

q¯ (z, 0) = 0

(6a)

BC1 :

z = 0,

∂q¯ =0 ∂z

(6b)

BC2 :

z = H,

∂q¯ =0 ∂z

(6c)

The solid-phase dispersion coefficient DS can be calculated from the empirical correlation [20]: DS = 0.04U 1.8

3. Model development

(6)

(7)

The liquid film mass transfer coefficient for expanded-bed adsorption kf,e can be calculated as a function of the bed void fraction [11]: DAB 1/2 (8) {2 + [1.5(1 − ε)Rep Sc1/3 ]} kf,e = d¯ Based on assumption (2), the pore diffusion model [20] is used to analyze the kinetics of protein adsorption to Streamline DEAE. The intra-particle continuity equation and its initial and boundary conditions (IC and BC) are as follows:
2  ∂ci ∂ ci 2∂ci ∂qi + + = εp Dp (9) εp ∂t ∂t r∂r ∂r 2

268

IC :

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t = 0,

c = c0 ,

ci = 0,

qi = 0

(9a)

BC1 :

r = 0,

∂ci =0 ∂r

(9b)

BC2 :

r = R,

R ∂q¯ ∂ci = ∂r 3εp Dp ∂t

(9c)

The liquid- and solid-phase concentration as a function of time and axial position is described by Eqs. (5) and (6). These equations were solved by the finite element method with 20 evenly spaced finite elements along the column length. In this way, the liquid-phase and average solid-phase protein concentration at each collocation point in every finite element can be determined in a short time interval. The model was solved by the orthogonal collocation method with five evenly orthogonal collocation points along the adsorbent radius. Further increase in the number of finite elements or orthogonal collocation points gave little influence on the simulation results. This indicates that the number of finite element and collocation points can give sufficient calculation accuracy.

4. Results and discussion 4.1. Determination of model parameters Before predicting the breakthrough curves of BSA in expanded-bed of Streamline DEAE, the parameters in the breakthrough model should be determined by independent experiments or calculated from correlation. It is to be noted that the model simulation validity is closely related to the sensitivity of model parameters. Wright and Glasser [12] investigated the effects of some parameters on breakthrough behaviors by perturbing each of the parameters while holding the rest of the conditions constant. Within the range of increasing or decreasing two-fold the experimental values, the superficial liquid velocity and particle size had the most significant effect on the breakthrough behaviors, while the effect of liquid and solid axial dispersion, film mass transfer and pore diffusion coefficient had a secondary contribution. Compared to this observation, Veeraraghavan and Fan found that the shape of breakthrough curves became to be significantly influenced by the magnitude of the film mass transfer and diffusion coefficient when their experimental values increased or decreased by a factor of 3 [11]. Thus, it is crucial to carefully determine the model parameters to ensure a reliable comparison between the experimental and simulated results. Some of the model parameters have been measured in triplicates to test the reliability of the experimental results, and the average values were used in the simulations. In this work, three major operation parameters, that is, feed BSA concentration, liquid velocity and liquid viscosity, were varied to validate the present model.

Fig. 1. Effect of flow velocity on ( ) the Bodenstein number and ( ) the axial mixing coefficient in buffer.

4.1.1. Axial dispersion behavior in expanded-bed The liquid-phase axial dispersion behavior in the expanded-bed of Streamline DEAE as a function of liquid flow rate was investigated and estimated by the step-input technique (Fig. 1). From Fig. 1, it can be seen that increasing liquid velocity led to an increase of Bo from 17 to 48. Chang and Chase [21] have stated that expanded-bed system behaves similarly to a packed-bed at Pe > 20 (the expression for Pe is the same as Bo as defined by Eq. (1)). So, the present expanded-bed is considered to be high efficient for protein adsorption. The liquid-phase dispersion coefficients DL could be calculated from Eq. (2) with the Bo value. Fig. 1 indicates that the DL value in the expanded-bed of Streamline DEAE increased with increasing liquid velocity. The DL values are between 2.2 × 10−6 and 1.1 × 10−5 m2 /s at liquid velocities of 60–470 cm/h, which are comparable with those reported in literature for the expanded-bed of Streamline gel [21–23]. Together with the experimental conditions for expanded-bed protein adsorption, the corresponding DL values for model simulations are summarized in Table 1. 4.1.2. Adsorption equilibrium The adsorption isotherms of BSA to Streamline DEAE are shown in Fig. 2. As can be seen, the experimental data are well fitted by the Langmuir equation (Eq. (3)). Moreover, the addition of glycerol to the buffer has little effect on the static adsorption of BSA to Streamline DEAE. Similar results were observed by Chang and Chase [21] for lysozyme adsorption to Streamline SP. The adsorption capacity and dissociation constant were determined by least square fitting the equilibrium data to the isotherm with qm = 75.3 mg/ml and Kd = 0.06 mg/ml. These data are used in the following model simulations.

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Table 1 Partial parameters for the BSA breakthrough model Run

Buffer

A B C D

c0 (mg/ml)

F

20% glycerol

U (cm/h)

DL (10−6 m2 /s)

RSSEb (%)

RSSEc (%)

0.5

2.0

190

7.1

96.6

94.5

1.0

2.0 2.5 3.0

190 310 400

7.1 10.2 9.8

95.6 95.4 97.0

79.0 91.9 97.2

2.0

2.0

190

7.1

92.5

85.9

1.0

2.0

125

5.7

84.4

76.4

93.6

87.5

0.02 mol/l PBS

E

H/H0 a

Average H0 = 10.0 ± 0.5 cm. b Predicted with the model by taking in account the axial particle size distribution. c Predicted with the model without considering the axial particle size distribution. a

4.1.3. Adsorption kinetics A series of experiments were undertaken to determine the effect of protein concentration or liquid viscosity on the uptake rate of BSA to Streamline DEAE. Fig. 3 displays the uptake curves of BSA to Streamline DEAE in phosphate buffer at different initial protein concentrations. The solid lines in the figure are calculated from the pore diffusion model [19]. As indicated in Fig. 3, the theoretical solutions at the three different protein concentrations are in good agreement with the experimental data. The pore diffusion coefficients obtained at the three initial protein concentrations are nearly the same, so its average value can be regarded as (6.5 ± 0.1) × 10−11 m2 /s. It indicates that the pore diffusivity of BSA in Streamline DEAE is independent of protein concentration. Fig. 4 shows the effects of liquid viscosity on the uptake curves of BSA to Streamline DEAE. Increasing liquid-phase viscosity resulted in the decrease of the pore diffusivity of BSA. For example, when the addition of glycerol increased from 20 to 40% (v/v), the pore diffusivities of BSA remarkably decreased from 2.5 × 10−11 to 0.7 × 10−11 m2 /s. A

similar phenomenon has been found with the effect of solution viscosity on lysozyme adsorption to Streamline SP and S-HyperD by Wright et al. [24]. They explained the observation as the decrease of film mass transfer coefficients at high liquid viscosity because they incorporated the film mass transfer into the pore diffusivity. For our model, however, we have separately considered the external film mass transfer in the pore diffusion model. Therefore, we think that the viscosity increase due to the addition of glycerol is the main reason why diffusivity is inversely proportional to liquid viscosity [25]. In addition, an increasing discrepancy between the simulated and experimental data was observed with increasing the liquid-phase viscosity due to the addition of glycerol. It is probably because of the agglomeration of particles in viscous solution, which may also interfere protein diffusion into the beads.

Fig. 2. Adsorption isotherms of BSA to Streamline DEAE in ( ) phosphate buffer, ( ) 20% (v/v) glycerol solution, and ( ) 40% (v/v) glycerol solution. Solid lines are calculated from the Langmuir equation.

Fig. 3. Kinetics of BSA adsorption to Streamline DEAE in buffer at the feed BSA concentrations of ( ) 0.5, ( ) 1.0, and ( ) 2.0 mg/ml. Solid lines are calculated from pore diffusion model [19].

4.2. Simulation of breakthrough curves To predict the breakthrough curves of BSA in expanded-bed of Streamline DEAE, the parameters in the breakthrough

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Glasser [12]. The RSSE is calculated as follows: m

RSSE =

1  |(ei )2 − (ei − si )2 | × 100% m e2i

(12)

i=1

Fig. 4. Kinetics of BSA adsorption to Streamline DEAE at c0 = 1.0 mg/ml in ( ) phosphate buffer, ( ) 20% (v/v) glycerol solution, and ( ) 40% (v/v) glycerol solution. Solid lines are calculated from pore diffusion model [19].

model were determined by the above independent experiments or calculated from correlation. In each finite element space, the value of kf,e was calculated from Eq. (8) using ¯ Using the indethe local volume-weighted mean size (d). pendently determined model parameters, the breakthrough curves can be predicted and compared with the experimental data. The simulation results under different experimental conditions are plotted in Figs. 5–7. For a quantitative assessment of the agreement between experimental and simulated breakthrough curves, a relative sum of the squares of the errors (RSSE) was estimated, as described by Wright and

where ei and si represent the experimental and simulated data, respectively, and m is the total amount of the data points. The RSSE was used to express the simulation validity as a percentage where 100% means a perfect agreement. The values of the RSSE at different operating conditions were thus obtained, and are listed in Table 1. It can be seen from the table and figures that in most cases the numerical solutions with the independently determined model parameters are well fitted to the experimental data at less than 90% breakthrough. In most cases, discrepancies between experimental data and simulated results at the later stage of the breakthrough curves are observed. It may be due to the following reasons. (1) The dimer formation of segmental BSA molecules in the column resulted in additional adsorption capacity beyond that estimated by the Langmuir equation, as observed in fixed-bed chromatography by Skidmore et al. [26]. (2) Bruce and Chase [13] have reported lower axial dispersion and higher void fraction at the top region of the column than those at the bottom. Therefore, the assumptions of uniform particle concentration distribution and even liquid-phase dispersion coefficient along the column used in the present model may also lead to the discrepancies. Because no enough experimental data of these parameters are available at present, they cannot yet be considered in the present model. As for the larger discrepancy between experimental data and simulated results at the later stage of the breakthrough curves when liquid velocity is

Fig. 5. Simulated and experimental BSA breakthrough curves in the expanded-bed of Streamline DEAE at feed BSA concentrations of ( ) 0.5, ( ) 1.0, and ( ) 2.0 mg/ml. The bed was expanded twice at 15.5 ml/min. The solid and dashed lines are predicted using the model with and without considering axial particle size distribution, respectively.

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271

Fig. 6. Simulated and experimental BSA breakthrough curves in the expanded-bed of Streamline DEAE at liquid velocities of ( ) 15.5, ( ) 25.5, and ( ) 33.0 ml/min. Feed BSA concentration was 1.0 mg/ml. The solid and dashed lines are predicted using the model with and without considering axial particle size distribution, respectively.

25.5 ml/min in Fig. 6, it might be due to the imprecision of experimental data. It should be noted that in Fig. 7 when the liquid-phase was changed from buffer to 20% (v/v) glycerol solution, the liquid flow velocity was reduced from 15.5 to 9.7 ml/min to reach the same expanded-bed height. The reduced liquid velocity due to the increase of liquid-phase viscosity resulted

in a longer breakthrough time. The simulation showed an evident discrepancy with the experimental data at the beginning of the breakthrough stage. Except for the reasons discussed above for the discrepancies observed in the later stage, this deviation may be additionally due to the group migration of agglomerated particles in the viscous solution. This may result in breakthrough of BSA in some regions

Fig. 7. Simulated and experimental BSA breakthrough curves in the expanded-bed of Streamline DEAE in ( ) buffer and ( ) 20% (v/v) glycerol solution. Settled bed height was 10.0 ± 0.5 cm, and feed BSA concentration was 1.0 mg/ml. The bed was expanded twice at ( ) 15.5 and ( ) 9.7 ml/min, respectively. The solid and dashed lines are predicted using the model with and without considering axial particle size distribution, respectively.

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of the bed more rapidly than in others as a result of flow irregularities in the expanded-bed [21]. To further confirm the effectiveness of the improved model, the breakthrough profiles were also predicted by the model without considering the particle size distribution (dashed lines in Figs. 5–7), and the RSSE values thus obtained are also listed in Table 1. Without considering the particle size distribution, the RSSE values became smaller in most cases (Table 1). It is worth noting that increasing liquid velocity resulted in the increase of the simulation agreement with this model (see RSSE values in runs B–D in the last column in Table 1), and at a liquid velocity of 400 cm/h, the two models gave similar satisfactory simulation agreement (run D in Table 1). The reason is unclear. Anyhow, by comparison of the two models, more accurate predictions by the model taking into account the axial size distribution has been obtained because the average RSSE values increased from 87.5 to 93.6% (Table 1). Therefore, it can be concluded that the model by taking into account the particle size distribution is a better approximate to protein adsorption in the expanded-bed system.

5. Conclusions In this article, a mathematical model taking into account the axial particle size distribution of adsorbent is developed to describe protein breakthrough behavior in expanded-bed chromatography. This model has been tested against experimental results under various operating conditions, and compared with the model without considering the particle size distribution. Analysis of the RSSE values has proved better description of the model for the breakthrough profiles. To further improve the model accuracy, more information on the axial solid-phase particle concentration distribution and liquid-phase dispersion at different regions should be accumulated and taken into account in the model.

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