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Theoretical analysis of the ultrafiltration behavior of highly concentrated protein solutions
Journal of Membrane Science 494 (2015) 216–223
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Journal of Membrane Science journal homepage: www.elsevier.com/locate/memsci
Theoretical analysis of the ultrafiltration behavior of highly concentrated protein solutions Elaheh Binabaji a, Junfen Ma b, Suma Rao b, Andrew L. Zydney a,n a b
Department of Chemical Engineering, The Pennsylvania State University, University Park, PA 16802, United States Amgen Inc., One Amgen Center Drive, Thousand Oaks, CA 91320, United States
art ic l e i nf o
a b s t r a c t
Article history: Received 28 May 2015 Received in revised form 23 July 2015 Accepted 30 July 2015 Available online 4 August 2015
Ultrafiltration is currently used for the concentration and formulation of monoclonal antibody solutions with target protein concentrations of up to 200 g/L or higher. The filtrate flux and maximum achievable antibody concentration in these systems is strongly influenced by the intermolecular interactions and non-ideal behavior in these highly concentrated protein solutions. The objective of this work was to develop a theoretical framework for analyzing the ultrafiltration of highly concentrated protein solutions accounting for the complex thermodynamic and hydrodynamic behavior in these systems. A modified polarization model was developed to describe the bulk mass transfer characteristics. In addition, the model accounts for the back-filtration phenomenon that occurs at very high protein concentrations due to the large pressure drop through the module associated with the high viscosity of the antibody solutions. Model parameters were evaluated from independent data for the protein osmotic pressure, osmotic virial coefficients, and viscosity. Model calculations demonstrate the importance of back-filtration, with numerical results in good agreement with experimental data for both the filtrate flux and maximum achievable antibody concentration obtained in a Pellicon 3 tangential flow filtration module. These results provide important insights into the key factors controlling the ultrafiltration behavior of highly concentrated protein solutions as well as a framework for the design and optimization of these ultrafiltration processes. & 2015 Elsevier B.V. All rights reserved.
Keywords: Ultrafiltration Monoclonal antibody Concentration polarization Protein–protein interactions
1. Introduction Ultrafiltration is the most commonly used method for concentration and final formulation of recombinant therapeutic proteins, most of which are delivered by injection [1–3]. Ultrafiltration of highly active hormones (e.g., insulin), cytokines (e.g., interferon), and clotting factors (e.g., Factor VIII) is relatively straightforward since these proteins are delivered at low to moderate concentrations [2]. In contrast, monoclonal antibodies need to be formulated in highly concentrated solutions (up to and exceeding 200 g/L) to achieve the desired dosage in the limited volumes that can be delivered by subcutaneous injection, creating significant challenges for ultrafiltration [1]. Most theoretical descriptions of the filtrate flux in ultrafiltration systems are developed using a “stagnant film” model based on solution of the steady-state one-dimensional diffusion equation: n
Corresponding author. Fax: þ1 814 865 7846. E-mail address: [email protected] (A.L. Zydney).
http://dx.doi.org/10.1016/j.memsci.2015.07.068 0376-7388/& 2015 Elsevier B.V. All rights reserved.
⎛ dC ⎞ Ns = Jv C − D ⎜ ⎟ ⎝ dy ⎠
(1)
where Ns is the protein flux, Jv is the filtrate flux (volumetric filtrate flow rate per unit membrane area), and C is the local protein concentration at position y measured from the membrane surface into the bulk solution. Eq. (1) can be integrated across the concentration boundary layer (with thickness δ) assuming that the protein diffusion coefficient (D) is constant to give the classical concentration polarization model for a fully retentive membrane (i.e., Ns ¼0) [4]:
⎡C ⎤ Jv = k m ln ⎢ w ⎥ ⎣ Cb ⎦
(2)
where km ¼D/δ is the protein mass transfer coefficient, and Cw and Cb are the protein concentrations at the membrane surface and in the bulk solution, respectively. A variety of expressions are available for the mass transfer coefficient in different modules [5], with these correlations developed from experimental data (obtained from both mass and heat transfer experiments) and from solution of the appropriate convection–diffusion equation in a particular
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flow system (e.g., laminar flow over a flat plate). Although, Eq. (2) often provides an accurate description of the filtrate flux, this simple polarization model cannot be used with highly concentrated solutions due to the assumptions of a constant diffusion coefficient and constant viscosity, both of which neglect the significant protein–protein interactions that occur within the concentration polarization boundary layer. A number of different approaches have been proposed to address the limitations of the classical stagnant film model. For example, Aimar and Field [6] accounted for the effects of the high viscosity of the concentrated solution near the membrane surface by multiplying the mass transfer coefficient in Eq. (2) by the correction factor (nb/nw)0.27 where ηb and ηw are the viscosities evaluated using the bulk and wall concentrations, respectively. Gekas and Hallstrom [7] used a similar approach but with a correction factor based on the Schmidt η number, ( Scb / Scw )0.11 where Sc = ρD , to account for the dependence of both the solution viscosity (η) and diffusion coefficient (D) on the protein concentration (with ρ the solution density). Kozinski and Lightfoot [8] argued that the product of the diffusion coefficient and viscosity was constant based on the Stokes–Einstein equation, and that this was the key parameter governing mass transfer in a rotating disk ultrafiltration module. Zydney [9] took a different approach by incorporating the concentration dependence of the diffusion coefficient directly in Eq. (1), both through the protein mobility and the use of the gradient in the chemical potential (instead of the gradient in the protein concentration) for the driving force for diffusion, although no attempt was made to account for the concentration dependence of the viscosity. The objective of this work was to develop a more accurate model for the filtrate flux in the ultrafiltration of highly concentrated protein solutions that properly accounts for: (1) the effects of intermolecular interactions on the thermodynamic driving force for diffusion, (2) the concentration dependence of the solution viscosity, and (3) the large parasitic pressure losses due to flow through the tangential flow filtration (TFF) module which can, under some circumstances, lead to back-filtration near the device exit. The key thermodynamic (virial coefficients) and hydrodynamic (viscosity) properties were evaluated from independent experimental measurements using a highly purified monoclonal antibody. The model predictions are compared with experimental data for the ultrafiltration of the same antibody in a linearly scalable screened TFF cassette used extensively in bioprocessing applications for final product formulation [2]. The model is in good agreement with the experimental results, providing important additional insights into the key factors controlling the filtrate flux during ultrafiltration of highly concentrated protein solutions.
2. Theoretical development 2.1. Modified polarization model As discussed by Zydney [9], the diffusive solute flux in highly concentrated solutions is proportional to the gradient in the chemical potential (m) instead of the gradient in the solute concentration, so that Eq. (1) becomes:
Ns = Jv C −
DC ⎛ dμ ⎞ ⎜ ⎟ RT ⎝ dy ⎠
(3)
where R is the ideal gas constant and T is the absolute temperature. The gradient in the chemical potential can be rewritten in terms of the protein osmotic pressure (Π) as:
217
⎛ dμ ⎞ Mp ⎛ dΠ ⎞ ⎛ dC ⎞ ⎟⎜ ⎜ ⎟ ⎜ ⎟= C ⎝ dC ⎠ ⎝ dy ⎠ ⎝ dy ⎠
(4)
where Mp is the protein molecular weight. The osmotic pressure (Π) is conveniently expressed using a virial expansion as [5]: 1 ⎧ ⎫ 2 ⎤2 ⎪ ⎡ ⎛ ZC ⎞ ⎪ 2⎥ ⎢ ⎨ ⎟ + ms Π = RT 2 ⎜ − 2ms ⎬ + RT B1C + B2 C2 + B3 C3 ⎥ ⎪ ⎢⎣ ⎝ 2Mp ⎠ ⎪ ⎦ ⎩ ⎭
(
) (5)
where the first term is the Donnan contribution with Z the net protein charge, ms is the molar salt concentration, and B1, B2, and B3 are the osmotic virial coefficients. Eq. (5) has been truncated after the third osmotic virial coefficient (B3), which is sufficient to describe the behavior of concentrated monoclonal antibody solutions up to concentrations of at least 250 g/L [10]. Note that the analysis presented by Zydney [9] only considered the term involving the second virial coefficient without the Donnan contribution. The diffusion coefficient in Eq. (3) is also a function of the protein concentration due to the dependence of the protein mobility on the local solution viscosity:
⎛η ⎞ D = D0 ⎜ 0 ⎟ ⎝ n⎠
(6)
where D0 and η0 are the diffusivity and viscosity in the limit of an infinitely dilute solution. Note that Eq. (6) has been used previously by Kozinski and Lightfoot [7] (among others) for describing diffusion in concentrated protein solutions; this form is also consistent with Einstein's analysis of Brownian diffusion in terms of the particle mobility. Eqs. (4) to (6) can be substituted into Eq. (3), with the resulting equation integrated over the concentration boundary layer thickness (δ) to give the following expression for the filtrate flux:
Jv =
D0 δ
∫C
Cw
b
⎛ η0 ⎞ ⎛ Mp ⎞ ⎛ dΠ ⎞ dC ⎟ ⎟⎜ ⎜ ⎟⎜ ⎝ η ⎠ ⎝ RT ⎠ ⎝ dC ⎠ C
(7)
where D0/δ is related to the mass transfer coefficient and the derivative of the osmotic pressure is evaluated from Eq. (5). Eq. (7) reduces to the classical stagnant film model (Eq. (2)) when the solution viscosity is constant (η ¼ η0) and the osmotic pressure is a linear function of concentration, i.e., under conditions where there are no intermolecular interactions. Eq. (7) can be integrated using an appropriate relationship for the viscosity as a function of the protein concentration; this is discussed in more detail in the results. The boundary layer thickness in Eq. (7) is determined by the module geometry (e.g., channel height, spacer, module length, etc.) as well as the device hydrodynamics (e.g., feed flow rate). In addition, the variation in solution viscosity alters the growth of the concentration polarization boundary layer thickness as discussed by Aimar and Field [6]. Previous experimental and theoretical studies have shown that the boundary layer thickness typically depends on Sc1/3 where Sc is the Schmidt number [5]. Eq. (7) was thus rewritten as:
⎛ η ⎞1/3 Jv = k 0 ⎜ b ⎟ ⎝ η0 ⎠
∫C
Cw
b
⎛ η0 ⎞ ⎛ Mp ⎞ ⎛ dΠ ⎞ dC ⎟ ⎟⎜ ⎜ ⎟⎜ ⎝ η ⎠ ⎝ RT ⎠ ⎝ dC ⎠ C
(8)
where k0 ¼ D0/δ is the mass transfer coefficient that would exist in the absence of any non-idealities, i.e., with constant viscosity and with dΠ/dC ¼constant. There is considerable debate in the literature over the factors that determine the wall concentration in the stagnant film model. Some investigators have evaluated the wall concentration based on the solubility (or “gel”) concentration for the particular protein, while others have assumed that the wall concentration is
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determined by the osmotic pressure, in which case Cw can be evaluated by simultaneous solution of the concentration polarization model (Eq. (2) or (8)) and the equation describing the volumetric flux across the membrane [11]:
Jv = L p (ΔP TM − ΔΠ )
(9)
where Lp is the membrane hydraulic permeability, ΔPTM is the local transmembrane pressure difference, and ΔΠ is the local osmotic pressure difference across the membrane. Eq. (9) is valid for a fully retentive membrane, i.e., under conditions where the osmotic reflection coefficient is equal to one. In this case, the protein concentration in the filtrate is zero and the osmotic pressure difference is simply equal to the osmotic pressure evaluated using the protein concentration at the membrane surface, Cw. The osmotic pressure model was used to evaluate the wall concentration in this work as discussed in more detail in the results. Eq. (8) describes the local filtrate flux in a tangential flow filtration (TFF) device. The length-average filtrate flux (or the volumetric filtrate flow rate) can then be evaluated by integrating Eq. (8) along the length of the module. 2.2. Pressure effects In dilute solutions, the pressure drop due to flow through the channel is relatively small, in which case the transmembrane pressure drop (ΔPTM) is often taken as a constant. However, the large viscosity of the highly concentrated antibody solutions leads to a gradient in the local transmembrane pressure:
12aQηb d ΔP TM = − dz wh3
(10)
where Q is the local volumetric flow rate, z is the position along the length of membrane, w is the channel width, and h is the channel height. Note that the viscosity in Eq. (10) should be evaluated at an appropriate mean value to account for the variation in protein concentration (and thus viscosity) across the concentration polarization boundary layer. If the boundary layer is sufficiently thin, ηb can be approximated using the bulk protein concentration. The coefficient a is a correction factor that accounts for the additional parasitic pressure losses associated with the screen in the TFF cassette; a¼ 1 in an open channel. Previous work by Subramani et al. [12] in a spacer filled submerged channel gave a ¼1.5–5 depending upon the channel geometry and flow rate. Eq. (10) is only valid for laminar flow; the maximum Reynolds number in our experiments was less than 500 (and well below 100 at the
higher bulk protein concentrations). The effect of the pressure gradient on the ultrafiltration is shown schematically in Fig. 1. The y-axis shows the dimensionless transmembrane and the bulk osmotic pressure, both scaled by the mean transmembrane pressure in the module. The osmotic pressure is taken as a constant throughout the channel using the value determined at the inlet protein concentration; this neglects the small variation in bulk concentration as one moves through module. This effect is relatively small since typical TFF modules used in bioprocessing operate at low conversion. At low feed concentrations (left panel), the transmembrane pressure will be relatively uniform throughout the module with a value that is significantly greater than the osmotic pressure evaluated at the bulk protein concentration. There is thus a net (positive) driving force for filtration along the entire length of the module. The situation is very different at high feed concentrations due to the large increase in both the osmotic pressure and the solution viscosity under these conditions. The large parasitic pressure loss due to flow through the module causes a large variation in the transmembrane pressure over the length of the channel, with ΔPTM dropping below the bulk osmotic pressure over the final portion of the module (right panel). This leads to a reverse or backfiltration near the module exit as given by Eq. (9) with ΔPTM o Πb. The implications of this back-filtration on the overall ultrafiltration behavior are discussed subsequently.
3. Materials and methods 3.1. Protein solution A highly purified humanized monoclonal antibody was provided by Amgen with a molecular weight of 142 kDa and isoelectric point of 8.16. The antibody was stored at 80 °C and slowly thawed prior to use. The antibody was placed in the desired buffer by diafiltration at low filtrate flux through a fully retentive Ultracel™ composite regenerated cellulose membrane with 30 kDa nominal molecular weight cut-off (Millipore Corp., Bedford, MA). The resulting protein solution was kept at 4 °C for up to a week; solutions used for longer periods of time were kept at 30 °C. There was no visual evidence of any protein precipitation in any of the protein solutions, either during storage or after the ultrafiltration experiment. More details regarding preparation of the protein solutions are provided elsewhere [10].
Fig. 1. Schematic diagram showing the variation in the transmembrane pressure (solid lines) and the osmotic pressure (dashed lines) at both low (left panel) and high (right panel) feed concentrations.
E. Binabaji et al. / Journal of Membrane Science 494 (2015) 216–223
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3.2. Ultrafiltration Ultrafiltration experiments were performed using Pellicon 3 tangential flow filtration (TFF) cassettes with C screen channel provided by EMD Millipore (Bedford, MA). The cassettes housed 30 kDa Ultracel composite regenerated cellulose membranes with a total membrane area of 88 cm2. The channel dimensions were measured to be approximately L¼ 20 cm and w¼ 2.2 cm, yielding a membrane area of 88 cm2 since the channel is defined by two flat sheet membranes. The channel height was estimated as h¼ 0.024 cm based on manufacturer's data for the feed channel hold-up volume (1.5 mL) assuming that the spacer occupies 30% of the channel. Cassettes were installed in the cassette holder using the procedures provided by the manufacturer. The modules were then flushed with at least 30 L/m2 of the specific buffer to be used in the ultrafiltration experiment. Modules were cleaned by recirculating a 0.3 N NaOH solution for 30–60 min and stored in 0.1 N NaOH between experiments. The feed was driven through the module using a positive displacement pump (Masterflex, Gelsenkirchen, Germany), with the inlet (feed) and exit (retentate) pressures evaluated using analog pressure gauges (Ashcroft, Stratford, CT). The permeate line was simply kept at atmospheric pressure. The membrane hydraulic permeability was evaluated from data for the filtrate flux (using DI water) as a function of the transmembrane pressure. The feed reservoir was then filled with the antibody solution. The transmembrane pressure was set by adjusting a pinch valve on the retentate exit line, with the permeate flow rate evaluated by timed collection of permeate samples. Experiments were performed in a batch concentration mode, with the retentate recycled back to the feed reservoir while the permeate was removed. The filtrate flux was measured as a function of time, corresponding to different bulk protein concentrations, as the feed volume was reduced. Permeate and feed samples were taken periodically to evaluate the bulk protein concentration and sieving coefficient using a NanoDrop 2000c spectrophotometer (Thermo Scientific, Waltham, MA) with the absorbance measured at 280 nm.
4. Results and discussion 4.1. Filtrate flux Fig. 2 shows experimental data for the filtrate flux as a function of the bulk protein concentration during ultrafiltration of the monoclonal antibody in a 5 mM acetate buffer with 20 mM NaCl at pH 5. Data are shown for two repeat experiments using different Pellicon 3 modules, each obtained as part of a single experimental run with the protein concentration increasing as a function of time as permeate was removed. The feed flow rate was fixed at Q¼45 mL/min (7.5 10 7 m3/s), which corresponds to a feed flux of Q/A¼ 85 mm/s (310 L/m2/h). The mean transmembrane pressure P feed + Pretentate
( − Pfiltrate ) was kept constant at 15 psi (105 kPa) by 2 continually adjusting the clamp on the retentate exit. This pressure was sufficiently high to insure that the flux was in the pressuredependent regime. At the highest protein concentrations, the retentate clamp was fully opened so that Pretentate E Pfiltrate, which caused a small increase in the mean transmembrane pressure due to the very high solution viscosity. Antibody concentrations in the initial filtrate samples were undetectable (corresponding to concentrations less than 0.01 g/L). Low levels of antibody were detected at the highest feed concentrations, corresponding to a sieving coefficient around 0.006 (99.4% retention). Overall mass balance closure was good, with the final mass of antibody (accounting for the volume in the feed
Fig. 2. Experimental data for the filtrate flux as a function of the bulk monoclonal antibody concentration during ultrafiltration through the Pellicon 3 module. Data were obtained using a 5 mM acetate buffer with 20 mM added NaCl at pH 5 with a feed flow rate of 45 mL/min. Dashed line is best fit given by the classical polarization model (Eq. (2)). Dotted curve is the prediction using the correction factor proposed by Aimar and Field [6]. Solid curve is the prediction using the modified polarization model (Eq. (19)) described subsequently. (For interpretation of the references to color in the text, the reader is referred to the web version of this article.)
reservoir, tubing, and module) being more than 90% of the initial mass – most of the “loss” was in the feed samples taken during the course of the experiment. The membrane permeability evaluated at the end of the experiment (after gently rinsing the module with buffer) was 13% smaller than the initial permeability, suggesting a small amount of irreversible fouling. The permeability could be restored to within a few percent of the initial value after cleaning with 0.3 N NaOH. The filtrate flux during the ultrafiltration decreased with increasing bulk protein concentration as expected, with the data looking nearly linear when plotted as a function of the logarithm of the bulk protein concentration (dashed blue line in Fig. 2). The maximum achievable protein concentration for this experiment was approximately 225 g/L, which was attained after 1 h of ultrafiltration; continued circulation of the feed through the module for an additional 3 h gave no measurable filtrate flux and thus no change in the feed concentration. The dotted black curve is the prediction using the classical concentration polarization model (Eq. (2)) with the correction factor proposed by Aimar and Field [6] using the experimentally determined viscosity for this monoclonal antibody. This model predicts a clear downward curvature in sharp contrast to the experimental data. The solid curve in Fig. 2 is the filtrate flux given by the modified concentration polarization model developed subsequently. Although the logarithmic dependence on Cb given by the simple stagnant film model appears to be in fairly good agreement with the data, there are two major problems with this analysis. First, the calculated value of the wall concentration determined using the simple stagnant film model (dashed blue line showing linear regression of the data using Eq. (2)) is only Cw ¼ 225 g/L. Binabaji et al. [10] evaluated the osmotic pressure for the same antibody in identical buffer conditions at this protein concentration as ΔΠ ¼44 kPa. The filtrate flux evaluated from Eq. (9) using this value of ΔΠ and a transmembrane pressure of 105 kPa is Jv ¼42 mm/s (based on the “fouled” membrane permeability of Lp ¼0.69 mm/s/kPa), which is several times larger than the measured filtrate flux even at the very start of the experiment (i.e., at a bulk protein concentration of o30 g/L). A much higher wall concentration, or a much lower membrane permeability, would be needed to give the measured values of the filtrate flux seen in Fig. 2 using the simple stagnant film model. There is no evidence
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E. Binabaji et al. / Journal of Membrane Science 494 (2015) 216–223
Fig. 3. Calculated values of the slope, dJv/d(lnCb) as a function of the bulk protein concentration from the data in Fig. 2. Dashed curve is developed using Eq. (8) which ignores the effect of back-filtration. Solid curve is given by Eq. (19).
to support either of these. In addition to the problem with Cw, the filtrate flux data in Fig. 2 show a small but significant curvature on the semi-log plot. This can be seen more easily by evaluating the slope:
dJv dJ = Cb v d (ln Cb ) dCb
(11)
using finite difference representations for the derivative accounting for the non-uniform spacing of the bulk concentration data. Fig. 3 shows the calculated values of the slope as a function of the bulk protein concentration based on the filtrate flux data for the repeat experiments in Fig. 2. The slope is highly variable at the lowest Cb but then decreases with increasing bulk protein concentration between about 60 and 170 g/L. The slope then appears to go through a minimum around Cb ¼ 170 g/L before increasing slightly at very high protein concentrations. This is in sharp contrast to the constant value of the slope (equal to the mass transfer coefficient) given by the classical stagnant film model (Eq. (2)) and the continuously increasing value of the slope given by the correction proposed by Aimar and Field [6] (curve not shown). The dashed curve in Fig. 3 is developed from the modified polarization model (Eq. (8)), with the viscosity of the antibody solution given by the Mooney equation [13] as:
⎛ ⎞ bC η = exp ⎜ ⎟ η0 ⎝ 1 − C /Cmax ⎠
(12)
Eq. (12) was also used by Ross and Minton [14] to describe the viscosity of concentrated hemoglobin solutions. Binabaji et al. [15] evaluated the parameters b ¼0.0108 L/g and Cmax ¼800 g/L for the same antibody and buffer conditions examined in this work. Eq. (8) was integrated numerically in Mathematica with B2 ¼4.4 10 4 m3 mol/kg2 and B3 ¼ 11 10 7 m6 mol/kg3 as determined from osmotic pressure measurements [10] with k0 ¼1.6 mm/s and Cw ¼330 g/L (the evaluation of k0 and Cw is discussed in more detail subsequently). The slope was then evaluated by numerical differentiation of an interpolation function that was fit to the integral using Mathematica. The slope given by this modified polarization model (dashed curve in Fig. 3) is in very good agreement with the experimental data for Cb o170 g/L, properly capturing the reduction in slope with increasing bulk protein concentration. However, the model is unable to explain the increase in the slope at very high protein concentrations – the predicted slope given by Eq. (8) decays continuously to zero as Cb increases.
Fig. 4. Experimental data for the module pressure drop as a function of the bulk protein concentration. Solid curve is the model calculation obtained using Eq. (13) with the viscosity given by Eq. (12).
4.2. Back-filtration As discussed previously, the ultrafiltration behavior at high protein concentrations is complicated by the large pressure drop due to flow of the highly viscous protein solution through the TFF module. Fig. 4 shows experimental data for the module pressure drop, ΔPmodule ¼Pfeed Pretentate, as a function of the bulk protein concentration for the repeat experiments from Fig. 2. The pressure drop is less than 50 kPa for dilute solutions but increases to more than 400 kPa (58 psi) at a bulk protein concentration of 210 g/L. The solid curve in Fig. 4 shows the calculated values of the pressure drop given by integration of Eq. (10):
ΔP =
12aQηb (L + 2L o ) wh3
(13)
where the pressure drop across the inlet and outlet headers of the module was modeled as an additional effective length of Lo ¼0.02 m. In order to simplify the analysis, Eq. (13) was developed assuming that the retentate flow rate and viscosity are both independent of axial position. Note that a filtrate flux of 10 mm/s (corresponding to a bulk protein concentration of 40 g/L) gives a volumetric filtrate flow rate of 5.3 mL/min, which is less 12% of the feed flow rate; the retentate flow rate varies by less than 5% across the length of the module for Cb 4100 g/L. In addition, the small reduction in Q will be at least partially compensated for by the corresponding increase in the bulk viscosity due to the corresponding increase in the bulk protein concentration. The best fit value of the channel correction factor (a ¼2.4) was determined by minimizing the sum of the squared residuals between the model and data. This value of a is in good agreement with results presented previously by Subramani et al. [12]. The model calculations are in excellent agreement with the data at high antibody concentrations, but slightly under-predict the data at low Cb. This could be due to an offset in the pressure gauges or to an additional pressure loss in the inlet/outlet headers. This small discrepancy had no effect on the calculations of the filtrate flux since the module pressure drop was only important at high antibody concentrations. The increase in the module pressure drop leads to a reduction in the local transmembrane pressure at the module exit since the mean transmembrane pressure drop was maintained constant at ΔPTM ¼105 kPa. The value of the bulk protein concentration at the onset of reverse filtration, i.e., when ΔPTM at the module exit first drops below the osmotic pressure can be evaluated as:
E. Binabaji et al. / Journal of Membrane Science 494 (2015) 216–223
ΔPTM, exit = ΔP¯TM −
6aQηbL wh3
= Πb
(14)
where ηb and Πb are the viscosity and osmotic pressure evaluated at C ¼Cb using Eqs. (12) and (5), respectively, and the exit transmembrane pressure has been evaluated from Eq. (10). Eq. (14) can be solved iteratively giving Cb ¼172 g/L at a feed flow rate of 45 mL/min. This value is very good agreement with the bulk antibody concentration at which the experimental data for the slope goes through a minimum in Fig. 3, i.e., the inability of Eq. (8) to accurately predict the slope at high bulk protein concentrations appears to be directly due to the onset of back-filtration in the TFF cassette. 4.3. Modified polarization model A modified concentration polarization model was developed by accounting for the effect of back-filtration on the total ultrafiltration flux at high bulk protein concentrations, i.e., under conditions where the local transmembrane pressure becomes less than the osmotic pressure. The local transmembrane pressure within the module is evaluated using Eqs. (10) and (14) as:
ΔP TM = ΔP¯TM +
⎞ 12aQηb ⎛ L ⎜ − z⎟ 3 ⎝2 ⎠ wh
(15)
The location within the module at which back-filtration first occurs, denoted as the “osmotic pinch point” z*, can then be determined from Eq. (15) by setting the local transmembrane pressure equal to Πb giving:
z* =
wh3 ⎡ ¯ L ⎣ ΔPTM − Πb⎤⎦ + 12aQηb 2
(16)
where Eq. (16) is only valid under conditions where back-filtration occurs. The osmotic pinch point moves inward (to smaller z) as Πb increases, going from z*/L¼1 at Cb ¼172 g/L to z*/L¼ 0.77 at Cb ¼200 g/L (for an inlet feed flow rate of 45 mL/min). The calculation of the reverse ultrafiltration flux arising from back-filtration is quite complicated. The protein concentration at the membrane surface in the presence of reverse filtration will be a complex function of the module geometry and fluid flow rates due to “reverse concentration polarization” that will occur when the flow is from the filtrate into the retentate. In addition, it will take time for the highly concentrated boundary layer that accumulates in the region of positive filtration to fully disperse as the direction of filtration is reversed. A rigorous description of this behavior would require a detailed numerical analysis of the coupled flow and transport within the module, particularly focused on the transition region as one moves from positive to reverse filtration. Instead, the reverse flux was simply evaluated as:
Jback = L p, eff ⎡⎣ ΔP TM − ΔΠb ⎤⎦
221
membrane, using Eq. (8) when the local transmembrane pressure is greater than the osmotic pressure and Eq. (17) when ΔPTM o Πb. In order to simplify the analysis, the bulk protein concentration (Cb), the wall concentration (Cw), the bulk osmotic pressure (Πb), and the effective permeability in the presence of back-filtration (Lp,eff) were all assumed to be constant (independent of position z). The local transmembrane pressure in Eq. (17) was evaluated from Eq. (15) giving:
⎡ η ⎤1/3 ⎡ Jv = k 0 ⎢ b ⎥ ⎢ ⎣ η0 ⎦ ⎣
∫C
Cw
b
f
⎛ LP, eff ⎞ ⎛ 6Qη aL ⎞ ⎛ z* ⎞2 dC ⎤ * b ⎟⎜ ⎟⎜ ⎟ ⎥ z /L − ⎜ C ⎦ ⎝ η0 ⎠ ⎝ wh3 ⎠ ⎝ L ⎠
(
)
(19)
where z* is the position at which ΔPTM ¼ Πb as given by Eq. (16). The negative sign in front of the second term in Eq. (19) is due to the reverse filtration, which reduces the value of the average filtrate flux. The intermolecular interaction function:
⎛ Mp ⎞ ⎛ η0 ⎞ ⎛ dΠ ⎞ f=⎜ ⎟ ⎟⎜ ⎟⎜ ⎝ RT ⎠ ⎝ η ⎠ ⎝ dC ⎠
(20)
describes the effects of protein–protein interactions on bulk mass transfer in the region of positive filtration (with f ¼1 for an ideal solution) while the second term on the right hand-side of Eq. (19) describes the contribution due to back-filtration. Note that when z*(given by Eq. (16)) is greater than L, the filtrate flux is simply given by the first term in Eq. (19) with z*/L¼ 1. The calculated values of the intermolecular interaction function were evaluated using the previously determined expression for the osmotic pressure (Eq. (5)) and viscosity (Eq. (12)) for the monoclonal antibody solution with the results for the pH 5 acetate buffer with 20 mM NaCl shown in Fig. 5. The horizontal dashed line shows f ¼1 as would be the case in the absence of any intermolecular interactions (i.e., as assumed in the classical stagnant film model). The intermolecular interaction function for the monoclonal antibody is equal to one at C ¼0 and then initially increases with increasing bulk protein concentration due to the higher order terms in the expression for the protein osmotic pressure. The intermolecular interaction function attains its maximum value of f ¼4.4 at Cb ¼55 g/L and then decreases at higher protein concentrations due to the rapid increase in the solution viscosity, approaching a value of zero at large Cb. In order to evaluate the integral in the first term in Eq. (19), it is first necessary to determine the protein concentration at the membrane surface, Cw. An average value for the wall concentration was determined based on the assumption that the filtrate flux is limited by the osmotic pressure of the highly concentrated protein solution in the immediate vicinity of the membrane surface (i.e., at C ¼Cw). However, the virial expansion developed by Binabaji et al.
(17)
where ΔPTM is the local transmembrane pressure difference and Lp,eff is the effective membrane permeability accounting for the effects of concentration polarization as discussed by Opong and Zydney [16]:
L p, eff =
1+
LP Lp Cb dΠ k0 dC C b
( )
(18)
Note that Eq. (18) assumes that the reverse concentration polarization is described by the mass transfer coefficient k0, which neglects the complex effects of intermolecular interactions on bulk mass transfer during back-filtration. The length average filtrate flux in the module can then be evaluated by integration of Eqs. (8) and (17) over the length of the
Fig. 5. Intermolecular interaction function in Eq. (20) as a function of the antibody concentration. Dashed line is f ¼1 as in the classical stagnant film model.
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[10] goes through a maximum at Cw ¼330 g/L due to the negative value of the third virial coefficient; extrapolation to higher concentrations would require values of the higher order virial coefficients. Thus, all model calculations were performed using a constant value of Cw ¼330 g/L in the region of positive ultrafiltration; limited calculations performed over a range of Cw values (from 300 to 500 g/L) gave essentially identical values of the filtrate flux. The best fit value of k0 was then determined by minimizing the sum of the squared residuals between the model (Eq. (19)) and the experimental data under conditions where there was no back-filtration, i.e., for filtrate flux values at Cb o 170 g/L. This gave k0 ¼1.6 mm/s based on the experimental data in Fig. 2. The solid curves in Figs. 2 and 3 represent the calculated values of the filtrate flux and the slope given by Eq. (19). The model calculations for the filtrate flux are in excellent agreement with the experimental data over the full range of antibody concentrations. Note that the maximum achievable antibody concentration predicted by the model is much smaller than the wall concentration (Cw ¼330 g/L) since the condition of zero net flux occurs when the positive filtrate flux over the first part of the channel exactly balances the reverse (back-filtration) flux in the region near the channel outlet. The model does slightly under-estimate the maximum achievable concentration (Cmax ¼206 g/L while the data gave 225 g/L); this is discussed in more detail subsequently. The solid curve in Fig. 3 is identical to the dashed curve for Cb o172 g/L, i.e. under conditions where there is no back-filtration. However, the complete model, which accounts for the effects of back-filtration, predicts a transition in the slope at Cb ¼172 g/L, in good qualitative agreement with the experimental results. The large increase in the slope at this value of Cb, which corresponds to a rapid reduction in the net filtrate flux, is due to the loss of filtrate caused by the back-filtration. The discontinuity in the slope at 172 g/L is not seen in the experimental data, at least in part because the slope was evaluated by numerical differentiation of data for the filtrate flux obtained at discrete protein concentrations (typically differing by 20 g/L at relatively high antibody concentrations). Fig. 6 shows a comparison between the model calculations and the experimental data for the filtrate flux during ultrafiltration of the monoclonal antibody at feed flow rates of 25, 45, and 65 mL/ min (feed flux of 47, 85, and 120 mm/s). The solid curves are the model calculations given by Eq. (19), with the k0 values at 25 and 65 ml/min determined by scaling as Q1/2 based on available correlations for mass transfer in a screened module [5]. The wall
Fig. 6. Experimental data for the filtrate flux during ultrafiltration of the monoclonal antibody in the Pellicon 3 module at different feed flow rates. Data were obtained using a 5 mM acetate buffer with 20 mM added NaCl at pH 5. Solid curves are model calculations developed using the modified polarization model accounting for back-filtration as given by Eq. (19).
Fig. 7. Maximum achievable antibody concentration as a function of feed flow rate. Data were obtained using a 5 mM acetate buffer with 20 mM added NaCl at pH 5. Solid curve is the modified polarization model accounting for back-filtration given by Eq. (19).
concentration was fixed at 330 g/L for all three feed flow rates since the data were all obtained at the same mean transmembrane pressure. The model calculations are in very good agreement with the experimental data at all 3 flow rates, properly capturing the increase in filtrate flux (for most values of Cb) and the reduction in the maximum achievable antibody concentration with increasing feed flow rate. The effect of the feed flux on the maximum achievable concentration is examined more explicitly in Fig. 7. The values of Cmax were determined by solving for the bulk protein concentration at which the net filtrate flux given by Eq. (19) is equal to zero. In the limit of very low feed flow rates, the pressure drop through the module becomes very small, eliminating back-filtration so that the filtrate flux goes to zero as Cb-Cw. As the feed flow rate increases, the pressure drop through the module increases, leading to an increase in back-filtration and a corresponding reduction in Cmax as seen in both the model and experimental results. The model calculations slightly under-predict the maximum achievable wall concentrations at all three feed flow rates, which is likely due to the over-estimation in the extent of back-filtration as described by Eqs. (17) and (18). This discrepancy likely reflects the approximations involved in estimating the rate of back-filtration, including the effects of “reverse polarization” on the concentration profile. In addition, the mean transmembrane pressure in the module tended to increase at very high antibody concentrations due to the very large pressure drop due to flow through the module (with the pinch valve on the retentate exit line fully opened). This effect was not included in the model calculations. Note that the model calculations suggest that the maximum achievable protein concentration for this antibody under these buffer conditions in the Pellicon 3 module could be increased to 270 g/L by decreasing the feed flux to below 11 mm/s (corresponding to a feed flow rate less than 6 mL/min). Further confirmation of the model results is presented in Fig. 8, which shows the calculated values of the slope as a function of the bulk antibody concentration for the three feed flow rates. In this case, the model curves were evaluated by numerical differentiation of the calculated values of the filtrate flux using centered difference representations with ΔCb ¼10 g/L; this numerically “smooths” the model calculations so that they are consistent with the numerical evaluation of the slopes from the filtrate flux data. The minimum in the slope shifts to lower bulk protein concentrations as the feed flow rate increases due to the increase in the pressure drop associated with the flow through the module. The increase in slope at high Cb is due to the large reduction in the filtrate flux associated with the back-filtration. The model
E. Binabaji et al. / Journal of Membrane Science 494 (2015) 216–223
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concentration due to the increase in back-filtration arising from the larger pressure loss associated with flow through the channel. Based on the experimental results and model calculations, it should be possible to obtain antibody formulations with higher protein concentration by decreasing the feed flow rate. High feed flow rates could be used over most of the ultrafiltration to provide high filtrate flux, with the feed flow rate reduced in the final stage of the ultrafiltration to minimize the extent of back-filtration by reducing the parasitic pressure losses. The modified polarization model developed in this work provides new insights into the ultrafiltration behavior of these highly concentrated protein solutions as well as an appropriate framework for the design and analysis of ultrafiltration processes/systems for the formulation of monoclonal antibody products. Fig. 8. Calculated values of the slope, dJv/d(lnCb) as a function of the bulk protein concentration from the data in Fig. 6. Solid curves are calculations based on the modified polarization model accounting for back-filtration as given by Eq. (19).
calculations are in good qualitative agreement with the experimental results, with the deviations at very high bulk protein concentrations likely due to the approximations made in the evaluation of the filtrate flux due to back-filtration.
Acknowledgment The authors would like to acknowledge Amgen for their financial support and donation of the monoclonal antibody and EMD Millipore Corp. for their donation of the Pellicon 3 modules.
References 5. Conclusions The need to formulate monoclonal antibody products at very high protein concentrations (above 200 g/L) has created major challenges for the design and application of ultrafiltration processes for final formulation of monoclonal antibodies. A new mathematical model was developed specifically to describe the filtrate flux behavior during ultrafiltration of these highly concentrated monoclonal antibody solutions. The model includes two basic phenomena: (1) the filtrate flux over most of the channel is evaluated using a modified form of the stagnant film model in which the driving force is expressed in terms of the gradient in the chemical potential (evaluated using data for the osmotic virial coefficients) with the protein mobility accounting for the concentration dependence of the viscosity, and (2) the large parasitic pressure losses due to flow through the screened channel causes a negative filtrate flux, or back-filtration, near the exit of the module since the transmembrane pressure becomes less than the osmotic pressure at high bulk protein concentrations. Model calculations were performed for a highly purified monoclonal antibody (provided by Amgen) based on independent measurements for the osmotic pressure and virial coefficients [10] and the solution viscosity [15], both obtained at protein concentrations up to at least 250 g/L. The model was in good agreement with experimental data obtained in a linearly-scalable Pellicon 3 tangential flow filtration module commonly used in downstream processing of monoclonal antibodies. The model properly captures the observed variation of the filtrate flux with bulk protein concentration, including the complex dependence of the slope in a plot of flux versus the logarithm of Cb. In addition, the model provides reasonably accurate predictions for the maximum achievable protein concentration; increasing the feed flow rate leads to a reduction in the maximum achievable protein
[1] S.J. Shire, Z. Shahrokh, J. Liu, Challenges in the development of high protein concentration formulations, J. Pharm. Sci. 93 (2004) 1390–1402. [2] R. van Reis, A. Zydney, Bioprocess membrane technology, J. Membr. Sci. 297 (2007) 16–50. [3] R.J. Harris, S.J. Shire, C. Winter, Commercial manufacturing scale formulation and analytical characterization of therapeutic recombinant antibodies, Drug Dev. Res. 61 (2004) 137–154. [4] W. Blatt, A. Dravid, A. Michaels, L. Nelsen, Solute polarization and cake formation in membrane ultrafiltration: causes, consequences, and control techniques, in: J.E. Flinn (Ed.), Membrane Science and Technology, Plenum Press, New York, 1970, pp. 47–97. [5] L.J. Zeman, A.L. Zydney, Microfiltration and Ultrafiltration: Principles and Applications, Marcel Dekker, New York, 1996. [6] P. Aimar, R. Field, Limiting flux in membrane separations: a model based on the viscosity dependency of the mass transfer coefficient, Chem. Eng. Sci. 47 (1992) 579–586. [7] V. Gekas, B. Hallström, Mass transfer in the membrane concentration polarization layer under turbulent cross flow: I. Critical literature review and adaptation of existing Sherwood correlations to membrane operations, J. Membr. Sci. 30 (1987) 153–170. [8] A.A. Kozinski, E.N. Lightfoot, Ultrafiltration of proteins in stagnation flow, AIChE J. 17 (1971) 81–85. [9] A.L. Zydney, Concentration effects on membrane sieving: development of a stagnant film model incorporating the effects of solute–solute interactions, J. Membr. Sci. 68 (1992) 183–190. [10] E. Binabaji, S. Rao, A.L. Zydney, The osmotic pressure of highly concentrated monoclonal antibody solutions: effect of solution conditions, Biotechnol. Bioeng. 111 (2014) 529–536. [11] O. Kedem, A. Katchalsky, Thermodynamic analysis of the permeability of biological membranes to non-electrolytes, Biochim. Biophys. Acta 27 (1958) 229–246. [12] A. Subramani, S. Kim, E.M.V. Hoek, Pressure, flow, and concentration profiles in open and spacer-filled membrane channels, J. Membr. Sci. 277 (2006) 7–17. [13] M. Mooney, The viscosity of a concentrated suspension of spherical particles, J. Colloid Sci. 6 (1951) 162–170. [14] P.D. Ross, A.P. Minton, Hard quasispherical model for the viscosity of hemoglobin solutions, Biochem. Biophys. Res. Commun. 76 (1977) 971–976. [15] E. Binabaji, J. Ma, A.L. Zydney, Intermolecular interactions and the viscosity of highly concentrated monoclonal antibody solutions, Pharm. Res. 32 (2015) 3102–3109. [16] S. Opong, A.L. Zydney, Effect of membrane structure and solute concentration on the osmotic reflection coefficient, J. Membr. Sci. 72 (1992) 277–292.
Contents lists available at ScienceDirect
Journal of Membrane Science journal homepage: www.elsevier.com/locate/memsci
Theoretical analysis of the ultrafiltration behavior of highly concentrated protein solutions Elaheh Binabaji a, Junfen Ma b, Suma Rao b, Andrew L. Zydney a,n a b
Department of Chemical Engineering, The Pennsylvania State University, University Park, PA 16802, United States Amgen Inc., One Amgen Center Drive, Thousand Oaks, CA 91320, United States
art ic l e i nf o
a b s t r a c t
Article history: Received 28 May 2015 Received in revised form 23 July 2015 Accepted 30 July 2015 Available online 4 August 2015
Ultrafiltration is currently used for the concentration and formulation of monoclonal antibody solutions with target protein concentrations of up to 200 g/L or higher. The filtrate flux and maximum achievable antibody concentration in these systems is strongly influenced by the intermolecular interactions and non-ideal behavior in these highly concentrated protein solutions. The objective of this work was to develop a theoretical framework for analyzing the ultrafiltration of highly concentrated protein solutions accounting for the complex thermodynamic and hydrodynamic behavior in these systems. A modified polarization model was developed to describe the bulk mass transfer characteristics. In addition, the model accounts for the back-filtration phenomenon that occurs at very high protein concentrations due to the large pressure drop through the module associated with the high viscosity of the antibody solutions. Model parameters were evaluated from independent data for the protein osmotic pressure, osmotic virial coefficients, and viscosity. Model calculations demonstrate the importance of back-filtration, with numerical results in good agreement with experimental data for both the filtrate flux and maximum achievable antibody concentration obtained in a Pellicon 3 tangential flow filtration module. These results provide important insights into the key factors controlling the ultrafiltration behavior of highly concentrated protein solutions as well as a framework for the design and optimization of these ultrafiltration processes. & 2015 Elsevier B.V. All rights reserved.
Keywords: Ultrafiltration Monoclonal antibody Concentration polarization Protein–protein interactions
1. Introduction Ultrafiltration is the most commonly used method for concentration and final formulation of recombinant therapeutic proteins, most of which are delivered by injection [1–3]. Ultrafiltration of highly active hormones (e.g., insulin), cytokines (e.g., interferon), and clotting factors (e.g., Factor VIII) is relatively straightforward since these proteins are delivered at low to moderate concentrations [2]. In contrast, monoclonal antibodies need to be formulated in highly concentrated solutions (up to and exceeding 200 g/L) to achieve the desired dosage in the limited volumes that can be delivered by subcutaneous injection, creating significant challenges for ultrafiltration [1]. Most theoretical descriptions of the filtrate flux in ultrafiltration systems are developed using a “stagnant film” model based on solution of the steady-state one-dimensional diffusion equation: n
Corresponding author. Fax: þ1 814 865 7846. E-mail address: [email protected] (A.L. Zydney).
http://dx.doi.org/10.1016/j.memsci.2015.07.068 0376-7388/& 2015 Elsevier B.V. All rights reserved.
⎛ dC ⎞ Ns = Jv C − D ⎜ ⎟ ⎝ dy ⎠
(1)
where Ns is the protein flux, Jv is the filtrate flux (volumetric filtrate flow rate per unit membrane area), and C is the local protein concentration at position y measured from the membrane surface into the bulk solution. Eq. (1) can be integrated across the concentration boundary layer (with thickness δ) assuming that the protein diffusion coefficient (D) is constant to give the classical concentration polarization model for a fully retentive membrane (i.e., Ns ¼0) [4]:
⎡C ⎤ Jv = k m ln ⎢ w ⎥ ⎣ Cb ⎦
(2)
where km ¼D/δ is the protein mass transfer coefficient, and Cw and Cb are the protein concentrations at the membrane surface and in the bulk solution, respectively. A variety of expressions are available for the mass transfer coefficient in different modules [5], with these correlations developed from experimental data (obtained from both mass and heat transfer experiments) and from solution of the appropriate convection–diffusion equation in a particular
E. Binabaji et al. / Journal of Membrane Science 494 (2015) 216–223
flow system (e.g., laminar flow over a flat plate). Although, Eq. (2) often provides an accurate description of the filtrate flux, this simple polarization model cannot be used with highly concentrated solutions due to the assumptions of a constant diffusion coefficient and constant viscosity, both of which neglect the significant protein–protein interactions that occur within the concentration polarization boundary layer. A number of different approaches have been proposed to address the limitations of the classical stagnant film model. For example, Aimar and Field [6] accounted for the effects of the high viscosity of the concentrated solution near the membrane surface by multiplying the mass transfer coefficient in Eq. (2) by the correction factor (nb/nw)0.27 where ηb and ηw are the viscosities evaluated using the bulk and wall concentrations, respectively. Gekas and Hallstrom [7] used a similar approach but with a correction factor based on the Schmidt η number, ( Scb / Scw )0.11 where Sc = ρD , to account for the dependence of both the solution viscosity (η) and diffusion coefficient (D) on the protein concentration (with ρ the solution density). Kozinski and Lightfoot [8] argued that the product of the diffusion coefficient and viscosity was constant based on the Stokes–Einstein equation, and that this was the key parameter governing mass transfer in a rotating disk ultrafiltration module. Zydney [9] took a different approach by incorporating the concentration dependence of the diffusion coefficient directly in Eq. (1), both through the protein mobility and the use of the gradient in the chemical potential (instead of the gradient in the protein concentration) for the driving force for diffusion, although no attempt was made to account for the concentration dependence of the viscosity. The objective of this work was to develop a more accurate model for the filtrate flux in the ultrafiltration of highly concentrated protein solutions that properly accounts for: (1) the effects of intermolecular interactions on the thermodynamic driving force for diffusion, (2) the concentration dependence of the solution viscosity, and (3) the large parasitic pressure losses due to flow through the tangential flow filtration (TFF) module which can, under some circumstances, lead to back-filtration near the device exit. The key thermodynamic (virial coefficients) and hydrodynamic (viscosity) properties were evaluated from independent experimental measurements using a highly purified monoclonal antibody. The model predictions are compared with experimental data for the ultrafiltration of the same antibody in a linearly scalable screened TFF cassette used extensively in bioprocessing applications for final product formulation [2]. The model is in good agreement with the experimental results, providing important additional insights into the key factors controlling the filtrate flux during ultrafiltration of highly concentrated protein solutions.
2. Theoretical development 2.1. Modified polarization model As discussed by Zydney [9], the diffusive solute flux in highly concentrated solutions is proportional to the gradient in the chemical potential (m) instead of the gradient in the solute concentration, so that Eq. (1) becomes:
Ns = Jv C −
DC ⎛ dμ ⎞ ⎜ ⎟ RT ⎝ dy ⎠
(3)
where R is the ideal gas constant and T is the absolute temperature. The gradient in the chemical potential can be rewritten in terms of the protein osmotic pressure (Π) as:
217
⎛ dμ ⎞ Mp ⎛ dΠ ⎞ ⎛ dC ⎞ ⎟⎜ ⎜ ⎟ ⎜ ⎟= C ⎝ dC ⎠ ⎝ dy ⎠ ⎝ dy ⎠
(4)
where Mp is the protein molecular weight. The osmotic pressure (Π) is conveniently expressed using a virial expansion as [5]: 1 ⎧ ⎫ 2 ⎤2 ⎪ ⎡ ⎛ ZC ⎞ ⎪ 2⎥ ⎢ ⎨ ⎟ + ms Π = RT 2 ⎜ − 2ms ⎬ + RT B1C + B2 C2 + B3 C3 ⎥ ⎪ ⎢⎣ ⎝ 2Mp ⎠ ⎪ ⎦ ⎩ ⎭
(
) (5)
where the first term is the Donnan contribution with Z the net protein charge, ms is the molar salt concentration, and B1, B2, and B3 are the osmotic virial coefficients. Eq. (5) has been truncated after the third osmotic virial coefficient (B3), which is sufficient to describe the behavior of concentrated monoclonal antibody solutions up to concentrations of at least 250 g/L [10]. Note that the analysis presented by Zydney [9] only considered the term involving the second virial coefficient without the Donnan contribution. The diffusion coefficient in Eq. (3) is also a function of the protein concentration due to the dependence of the protein mobility on the local solution viscosity:
⎛η ⎞ D = D0 ⎜ 0 ⎟ ⎝ n⎠
(6)
where D0 and η0 are the diffusivity and viscosity in the limit of an infinitely dilute solution. Note that Eq. (6) has been used previously by Kozinski and Lightfoot [7] (among others) for describing diffusion in concentrated protein solutions; this form is also consistent with Einstein's analysis of Brownian diffusion in terms of the particle mobility. Eqs. (4) to (6) can be substituted into Eq. (3), with the resulting equation integrated over the concentration boundary layer thickness (δ) to give the following expression for the filtrate flux:
Jv =
D0 δ
∫C
Cw
b
⎛ η0 ⎞ ⎛ Mp ⎞ ⎛ dΠ ⎞ dC ⎟ ⎟⎜ ⎜ ⎟⎜ ⎝ η ⎠ ⎝ RT ⎠ ⎝ dC ⎠ C
(7)
where D0/δ is related to the mass transfer coefficient and the derivative of the osmotic pressure is evaluated from Eq. (5). Eq. (7) reduces to the classical stagnant film model (Eq. (2)) when the solution viscosity is constant (η ¼ η0) and the osmotic pressure is a linear function of concentration, i.e., under conditions where there are no intermolecular interactions. Eq. (7) can be integrated using an appropriate relationship for the viscosity as a function of the protein concentration; this is discussed in more detail in the results. The boundary layer thickness in Eq. (7) is determined by the module geometry (e.g., channel height, spacer, module length, etc.) as well as the device hydrodynamics (e.g., feed flow rate). In addition, the variation in solution viscosity alters the growth of the concentration polarization boundary layer thickness as discussed by Aimar and Field [6]. Previous experimental and theoretical studies have shown that the boundary layer thickness typically depends on Sc1/3 where Sc is the Schmidt number [5]. Eq. (7) was thus rewritten as:
⎛ η ⎞1/3 Jv = k 0 ⎜ b ⎟ ⎝ η0 ⎠
∫C
Cw
b
⎛ η0 ⎞ ⎛ Mp ⎞ ⎛ dΠ ⎞ dC ⎟ ⎟⎜ ⎜ ⎟⎜ ⎝ η ⎠ ⎝ RT ⎠ ⎝ dC ⎠ C
(8)
where k0 ¼ D0/δ is the mass transfer coefficient that would exist in the absence of any non-idealities, i.e., with constant viscosity and with dΠ/dC ¼constant. There is considerable debate in the literature over the factors that determine the wall concentration in the stagnant film model. Some investigators have evaluated the wall concentration based on the solubility (or “gel”) concentration for the particular protein, while others have assumed that the wall concentration is
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determined by the osmotic pressure, in which case Cw can be evaluated by simultaneous solution of the concentration polarization model (Eq. (2) or (8)) and the equation describing the volumetric flux across the membrane [11]:
Jv = L p (ΔP TM − ΔΠ )
(9)
where Lp is the membrane hydraulic permeability, ΔPTM is the local transmembrane pressure difference, and ΔΠ is the local osmotic pressure difference across the membrane. Eq. (9) is valid for a fully retentive membrane, i.e., under conditions where the osmotic reflection coefficient is equal to one. In this case, the protein concentration in the filtrate is zero and the osmotic pressure difference is simply equal to the osmotic pressure evaluated using the protein concentration at the membrane surface, Cw. The osmotic pressure model was used to evaluate the wall concentration in this work as discussed in more detail in the results. Eq. (8) describes the local filtrate flux in a tangential flow filtration (TFF) device. The length-average filtrate flux (or the volumetric filtrate flow rate) can then be evaluated by integrating Eq. (8) along the length of the module. 2.2. Pressure effects In dilute solutions, the pressure drop due to flow through the channel is relatively small, in which case the transmembrane pressure drop (ΔPTM) is often taken as a constant. However, the large viscosity of the highly concentrated antibody solutions leads to a gradient in the local transmembrane pressure:
12aQηb d ΔP TM = − dz wh3
(10)
where Q is the local volumetric flow rate, z is the position along the length of membrane, w is the channel width, and h is the channel height. Note that the viscosity in Eq. (10) should be evaluated at an appropriate mean value to account for the variation in protein concentration (and thus viscosity) across the concentration polarization boundary layer. If the boundary layer is sufficiently thin, ηb can be approximated using the bulk protein concentration. The coefficient a is a correction factor that accounts for the additional parasitic pressure losses associated with the screen in the TFF cassette; a¼ 1 in an open channel. Previous work by Subramani et al. [12] in a spacer filled submerged channel gave a ¼1.5–5 depending upon the channel geometry and flow rate. Eq. (10) is only valid for laminar flow; the maximum Reynolds number in our experiments was less than 500 (and well below 100 at the
higher bulk protein concentrations). The effect of the pressure gradient on the ultrafiltration is shown schematically in Fig. 1. The y-axis shows the dimensionless transmembrane and the bulk osmotic pressure, both scaled by the mean transmembrane pressure in the module. The osmotic pressure is taken as a constant throughout the channel using the value determined at the inlet protein concentration; this neglects the small variation in bulk concentration as one moves through module. This effect is relatively small since typical TFF modules used in bioprocessing operate at low conversion. At low feed concentrations (left panel), the transmembrane pressure will be relatively uniform throughout the module with a value that is significantly greater than the osmotic pressure evaluated at the bulk protein concentration. There is thus a net (positive) driving force for filtration along the entire length of the module. The situation is very different at high feed concentrations due to the large increase in both the osmotic pressure and the solution viscosity under these conditions. The large parasitic pressure loss due to flow through the module causes a large variation in the transmembrane pressure over the length of the channel, with ΔPTM dropping below the bulk osmotic pressure over the final portion of the module (right panel). This leads to a reverse or backfiltration near the module exit as given by Eq. (9) with ΔPTM o Πb. The implications of this back-filtration on the overall ultrafiltration behavior are discussed subsequently.
3. Materials and methods 3.1. Protein solution A highly purified humanized monoclonal antibody was provided by Amgen with a molecular weight of 142 kDa and isoelectric point of 8.16. The antibody was stored at 80 °C and slowly thawed prior to use. The antibody was placed in the desired buffer by diafiltration at low filtrate flux through a fully retentive Ultracel™ composite regenerated cellulose membrane with 30 kDa nominal molecular weight cut-off (Millipore Corp., Bedford, MA). The resulting protein solution was kept at 4 °C for up to a week; solutions used for longer periods of time were kept at 30 °C. There was no visual evidence of any protein precipitation in any of the protein solutions, either during storage or after the ultrafiltration experiment. More details regarding preparation of the protein solutions are provided elsewhere [10].
Fig. 1. Schematic diagram showing the variation in the transmembrane pressure (solid lines) and the osmotic pressure (dashed lines) at both low (left panel) and high (right panel) feed concentrations.
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3.2. Ultrafiltration Ultrafiltration experiments were performed using Pellicon 3 tangential flow filtration (TFF) cassettes with C screen channel provided by EMD Millipore (Bedford, MA). The cassettes housed 30 kDa Ultracel composite regenerated cellulose membranes with a total membrane area of 88 cm2. The channel dimensions were measured to be approximately L¼ 20 cm and w¼ 2.2 cm, yielding a membrane area of 88 cm2 since the channel is defined by two flat sheet membranes. The channel height was estimated as h¼ 0.024 cm based on manufacturer's data for the feed channel hold-up volume (1.5 mL) assuming that the spacer occupies 30% of the channel. Cassettes were installed in the cassette holder using the procedures provided by the manufacturer. The modules were then flushed with at least 30 L/m2 of the specific buffer to be used in the ultrafiltration experiment. Modules were cleaned by recirculating a 0.3 N NaOH solution for 30–60 min and stored in 0.1 N NaOH between experiments. The feed was driven through the module using a positive displacement pump (Masterflex, Gelsenkirchen, Germany), with the inlet (feed) and exit (retentate) pressures evaluated using analog pressure gauges (Ashcroft, Stratford, CT). The permeate line was simply kept at atmospheric pressure. The membrane hydraulic permeability was evaluated from data for the filtrate flux (using DI water) as a function of the transmembrane pressure. The feed reservoir was then filled with the antibody solution. The transmembrane pressure was set by adjusting a pinch valve on the retentate exit line, with the permeate flow rate evaluated by timed collection of permeate samples. Experiments were performed in a batch concentration mode, with the retentate recycled back to the feed reservoir while the permeate was removed. The filtrate flux was measured as a function of time, corresponding to different bulk protein concentrations, as the feed volume was reduced. Permeate and feed samples were taken periodically to evaluate the bulk protein concentration and sieving coefficient using a NanoDrop 2000c spectrophotometer (Thermo Scientific, Waltham, MA) with the absorbance measured at 280 nm.
4. Results and discussion 4.1. Filtrate flux Fig. 2 shows experimental data for the filtrate flux as a function of the bulk protein concentration during ultrafiltration of the monoclonal antibody in a 5 mM acetate buffer with 20 mM NaCl at pH 5. Data are shown for two repeat experiments using different Pellicon 3 modules, each obtained as part of a single experimental run with the protein concentration increasing as a function of time as permeate was removed. The feed flow rate was fixed at Q¼45 mL/min (7.5 10 7 m3/s), which corresponds to a feed flux of Q/A¼ 85 mm/s (310 L/m2/h). The mean transmembrane pressure P feed + Pretentate
( − Pfiltrate ) was kept constant at 15 psi (105 kPa) by 2 continually adjusting the clamp on the retentate exit. This pressure was sufficiently high to insure that the flux was in the pressuredependent regime. At the highest protein concentrations, the retentate clamp was fully opened so that Pretentate E Pfiltrate, which caused a small increase in the mean transmembrane pressure due to the very high solution viscosity. Antibody concentrations in the initial filtrate samples were undetectable (corresponding to concentrations less than 0.01 g/L). Low levels of antibody were detected at the highest feed concentrations, corresponding to a sieving coefficient around 0.006 (99.4% retention). Overall mass balance closure was good, with the final mass of antibody (accounting for the volume in the feed
Fig. 2. Experimental data for the filtrate flux as a function of the bulk monoclonal antibody concentration during ultrafiltration through the Pellicon 3 module. Data were obtained using a 5 mM acetate buffer with 20 mM added NaCl at pH 5 with a feed flow rate of 45 mL/min. Dashed line is best fit given by the classical polarization model (Eq. (2)). Dotted curve is the prediction using the correction factor proposed by Aimar and Field [6]. Solid curve is the prediction using the modified polarization model (Eq. (19)) described subsequently. (For interpretation of the references to color in the text, the reader is referred to the web version of this article.)
reservoir, tubing, and module) being more than 90% of the initial mass – most of the “loss” was in the feed samples taken during the course of the experiment. The membrane permeability evaluated at the end of the experiment (after gently rinsing the module with buffer) was 13% smaller than the initial permeability, suggesting a small amount of irreversible fouling. The permeability could be restored to within a few percent of the initial value after cleaning with 0.3 N NaOH. The filtrate flux during the ultrafiltration decreased with increasing bulk protein concentration as expected, with the data looking nearly linear when plotted as a function of the logarithm of the bulk protein concentration (dashed blue line in Fig. 2). The maximum achievable protein concentration for this experiment was approximately 225 g/L, which was attained after 1 h of ultrafiltration; continued circulation of the feed through the module for an additional 3 h gave no measurable filtrate flux and thus no change in the feed concentration. The dotted black curve is the prediction using the classical concentration polarization model (Eq. (2)) with the correction factor proposed by Aimar and Field [6] using the experimentally determined viscosity for this monoclonal antibody. This model predicts a clear downward curvature in sharp contrast to the experimental data. The solid curve in Fig. 2 is the filtrate flux given by the modified concentration polarization model developed subsequently. Although the logarithmic dependence on Cb given by the simple stagnant film model appears to be in fairly good agreement with the data, there are two major problems with this analysis. First, the calculated value of the wall concentration determined using the simple stagnant film model (dashed blue line showing linear regression of the data using Eq. (2)) is only Cw ¼ 225 g/L. Binabaji et al. [10] evaluated the osmotic pressure for the same antibody in identical buffer conditions at this protein concentration as ΔΠ ¼44 kPa. The filtrate flux evaluated from Eq. (9) using this value of ΔΠ and a transmembrane pressure of 105 kPa is Jv ¼42 mm/s (based on the “fouled” membrane permeability of Lp ¼0.69 mm/s/kPa), which is several times larger than the measured filtrate flux even at the very start of the experiment (i.e., at a bulk protein concentration of o30 g/L). A much higher wall concentration, or a much lower membrane permeability, would be needed to give the measured values of the filtrate flux seen in Fig. 2 using the simple stagnant film model. There is no evidence
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Fig. 3. Calculated values of the slope, dJv/d(lnCb) as a function of the bulk protein concentration from the data in Fig. 2. Dashed curve is developed using Eq. (8) which ignores the effect of back-filtration. Solid curve is given by Eq. (19).
to support either of these. In addition to the problem with Cw, the filtrate flux data in Fig. 2 show a small but significant curvature on the semi-log plot. This can be seen more easily by evaluating the slope:
dJv dJ = Cb v d (ln Cb ) dCb
(11)
using finite difference representations for the derivative accounting for the non-uniform spacing of the bulk concentration data. Fig. 3 shows the calculated values of the slope as a function of the bulk protein concentration based on the filtrate flux data for the repeat experiments in Fig. 2. The slope is highly variable at the lowest Cb but then decreases with increasing bulk protein concentration between about 60 and 170 g/L. The slope then appears to go through a minimum around Cb ¼ 170 g/L before increasing slightly at very high protein concentrations. This is in sharp contrast to the constant value of the slope (equal to the mass transfer coefficient) given by the classical stagnant film model (Eq. (2)) and the continuously increasing value of the slope given by the correction proposed by Aimar and Field [6] (curve not shown). The dashed curve in Fig. 3 is developed from the modified polarization model (Eq. (8)), with the viscosity of the antibody solution given by the Mooney equation [13] as:
⎛ ⎞ bC η = exp ⎜ ⎟ η0 ⎝ 1 − C /Cmax ⎠
(12)
Eq. (12) was also used by Ross and Minton [14] to describe the viscosity of concentrated hemoglobin solutions. Binabaji et al. [15] evaluated the parameters b ¼0.0108 L/g and Cmax ¼800 g/L for the same antibody and buffer conditions examined in this work. Eq. (8) was integrated numerically in Mathematica with B2 ¼4.4 10 4 m3 mol/kg2 and B3 ¼ 11 10 7 m6 mol/kg3 as determined from osmotic pressure measurements [10] with k0 ¼1.6 mm/s and Cw ¼330 g/L (the evaluation of k0 and Cw is discussed in more detail subsequently). The slope was then evaluated by numerical differentiation of an interpolation function that was fit to the integral using Mathematica. The slope given by this modified polarization model (dashed curve in Fig. 3) is in very good agreement with the experimental data for Cb o170 g/L, properly capturing the reduction in slope with increasing bulk protein concentration. However, the model is unable to explain the increase in the slope at very high protein concentrations – the predicted slope given by Eq. (8) decays continuously to zero as Cb increases.
Fig. 4. Experimental data for the module pressure drop as a function of the bulk protein concentration. Solid curve is the model calculation obtained using Eq. (13) with the viscosity given by Eq. (12).
4.2. Back-filtration As discussed previously, the ultrafiltration behavior at high protein concentrations is complicated by the large pressure drop due to flow of the highly viscous protein solution through the TFF module. Fig. 4 shows experimental data for the module pressure drop, ΔPmodule ¼Pfeed Pretentate, as a function of the bulk protein concentration for the repeat experiments from Fig. 2. The pressure drop is less than 50 kPa for dilute solutions but increases to more than 400 kPa (58 psi) at a bulk protein concentration of 210 g/L. The solid curve in Fig. 4 shows the calculated values of the pressure drop given by integration of Eq. (10):
ΔP =
12aQηb (L + 2L o ) wh3
(13)
where the pressure drop across the inlet and outlet headers of the module was modeled as an additional effective length of Lo ¼0.02 m. In order to simplify the analysis, Eq. (13) was developed assuming that the retentate flow rate and viscosity are both independent of axial position. Note that a filtrate flux of 10 mm/s (corresponding to a bulk protein concentration of 40 g/L) gives a volumetric filtrate flow rate of 5.3 mL/min, which is less 12% of the feed flow rate; the retentate flow rate varies by less than 5% across the length of the module for Cb 4100 g/L. In addition, the small reduction in Q will be at least partially compensated for by the corresponding increase in the bulk viscosity due to the corresponding increase in the bulk protein concentration. The best fit value of the channel correction factor (a ¼2.4) was determined by minimizing the sum of the squared residuals between the model and data. This value of a is in good agreement with results presented previously by Subramani et al. [12]. The model calculations are in excellent agreement with the data at high antibody concentrations, but slightly under-predict the data at low Cb. This could be due to an offset in the pressure gauges or to an additional pressure loss in the inlet/outlet headers. This small discrepancy had no effect on the calculations of the filtrate flux since the module pressure drop was only important at high antibody concentrations. The increase in the module pressure drop leads to a reduction in the local transmembrane pressure at the module exit since the mean transmembrane pressure drop was maintained constant at ΔPTM ¼105 kPa. The value of the bulk protein concentration at the onset of reverse filtration, i.e., when ΔPTM at the module exit first drops below the osmotic pressure can be evaluated as:
E. Binabaji et al. / Journal of Membrane Science 494 (2015) 216–223
ΔPTM, exit = ΔP¯TM −
6aQηbL wh3
= Πb
(14)
where ηb and Πb are the viscosity and osmotic pressure evaluated at C ¼Cb using Eqs. (12) and (5), respectively, and the exit transmembrane pressure has been evaluated from Eq. (10). Eq. (14) can be solved iteratively giving Cb ¼172 g/L at a feed flow rate of 45 mL/min. This value is very good agreement with the bulk antibody concentration at which the experimental data for the slope goes through a minimum in Fig. 3, i.e., the inability of Eq. (8) to accurately predict the slope at high bulk protein concentrations appears to be directly due to the onset of back-filtration in the TFF cassette. 4.3. Modified polarization model A modified concentration polarization model was developed by accounting for the effect of back-filtration on the total ultrafiltration flux at high bulk protein concentrations, i.e., under conditions where the local transmembrane pressure becomes less than the osmotic pressure. The local transmembrane pressure within the module is evaluated using Eqs. (10) and (14) as:
ΔP TM = ΔP¯TM +
⎞ 12aQηb ⎛ L ⎜ − z⎟ 3 ⎝2 ⎠ wh
(15)
The location within the module at which back-filtration first occurs, denoted as the “osmotic pinch point” z*, can then be determined from Eq. (15) by setting the local transmembrane pressure equal to Πb giving:
z* =
wh3 ⎡ ¯ L ⎣ ΔPTM − Πb⎤⎦ + 12aQηb 2
(16)
where Eq. (16) is only valid under conditions where back-filtration occurs. The osmotic pinch point moves inward (to smaller z) as Πb increases, going from z*/L¼1 at Cb ¼172 g/L to z*/L¼ 0.77 at Cb ¼200 g/L (for an inlet feed flow rate of 45 mL/min). The calculation of the reverse ultrafiltration flux arising from back-filtration is quite complicated. The protein concentration at the membrane surface in the presence of reverse filtration will be a complex function of the module geometry and fluid flow rates due to “reverse concentration polarization” that will occur when the flow is from the filtrate into the retentate. In addition, it will take time for the highly concentrated boundary layer that accumulates in the region of positive filtration to fully disperse as the direction of filtration is reversed. A rigorous description of this behavior would require a detailed numerical analysis of the coupled flow and transport within the module, particularly focused on the transition region as one moves from positive to reverse filtration. Instead, the reverse flux was simply evaluated as:
Jback = L p, eff ⎡⎣ ΔP TM − ΔΠb ⎤⎦
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membrane, using Eq. (8) when the local transmembrane pressure is greater than the osmotic pressure and Eq. (17) when ΔPTM o Πb. In order to simplify the analysis, the bulk protein concentration (Cb), the wall concentration (Cw), the bulk osmotic pressure (Πb), and the effective permeability in the presence of back-filtration (Lp,eff) were all assumed to be constant (independent of position z). The local transmembrane pressure in Eq. (17) was evaluated from Eq. (15) giving:
⎡ η ⎤1/3 ⎡ Jv = k 0 ⎢ b ⎥ ⎢ ⎣ η0 ⎦ ⎣
∫C
Cw
b
f
⎛ LP, eff ⎞ ⎛ 6Qη aL ⎞ ⎛ z* ⎞2 dC ⎤ * b ⎟⎜ ⎟⎜ ⎟ ⎥ z /L − ⎜ C ⎦ ⎝ η0 ⎠ ⎝ wh3 ⎠ ⎝ L ⎠
(
)
(19)
where z* is the position at which ΔPTM ¼ Πb as given by Eq. (16). The negative sign in front of the second term in Eq. (19) is due to the reverse filtration, which reduces the value of the average filtrate flux. The intermolecular interaction function:
⎛ Mp ⎞ ⎛ η0 ⎞ ⎛ dΠ ⎞ f=⎜ ⎟ ⎟⎜ ⎟⎜ ⎝ RT ⎠ ⎝ η ⎠ ⎝ dC ⎠
(20)
describes the effects of protein–protein interactions on bulk mass transfer in the region of positive filtration (with f ¼1 for an ideal solution) while the second term on the right hand-side of Eq. (19) describes the contribution due to back-filtration. Note that when z*(given by Eq. (16)) is greater than L, the filtrate flux is simply given by the first term in Eq. (19) with z*/L¼ 1. The calculated values of the intermolecular interaction function were evaluated using the previously determined expression for the osmotic pressure (Eq. (5)) and viscosity (Eq. (12)) for the monoclonal antibody solution with the results for the pH 5 acetate buffer with 20 mM NaCl shown in Fig. 5. The horizontal dashed line shows f ¼1 as would be the case in the absence of any intermolecular interactions (i.e., as assumed in the classical stagnant film model). The intermolecular interaction function for the monoclonal antibody is equal to one at C ¼0 and then initially increases with increasing bulk protein concentration due to the higher order terms in the expression for the protein osmotic pressure. The intermolecular interaction function attains its maximum value of f ¼4.4 at Cb ¼55 g/L and then decreases at higher protein concentrations due to the rapid increase in the solution viscosity, approaching a value of zero at large Cb. In order to evaluate the integral in the first term in Eq. (19), it is first necessary to determine the protein concentration at the membrane surface, Cw. An average value for the wall concentration was determined based on the assumption that the filtrate flux is limited by the osmotic pressure of the highly concentrated protein solution in the immediate vicinity of the membrane surface (i.e., at C ¼Cw). However, the virial expansion developed by Binabaji et al.
(17)
where ΔPTM is the local transmembrane pressure difference and Lp,eff is the effective membrane permeability accounting for the effects of concentration polarization as discussed by Opong and Zydney [16]:
L p, eff =
1+
LP Lp Cb dΠ k0 dC C b
( )
(18)
Note that Eq. (18) assumes that the reverse concentration polarization is described by the mass transfer coefficient k0, which neglects the complex effects of intermolecular interactions on bulk mass transfer during back-filtration. The length average filtrate flux in the module can then be evaluated by integration of Eqs. (8) and (17) over the length of the
Fig. 5. Intermolecular interaction function in Eq. (20) as a function of the antibody concentration. Dashed line is f ¼1 as in the classical stagnant film model.
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[10] goes through a maximum at Cw ¼330 g/L due to the negative value of the third virial coefficient; extrapolation to higher concentrations would require values of the higher order virial coefficients. Thus, all model calculations were performed using a constant value of Cw ¼330 g/L in the region of positive ultrafiltration; limited calculations performed over a range of Cw values (from 300 to 500 g/L) gave essentially identical values of the filtrate flux. The best fit value of k0 was then determined by minimizing the sum of the squared residuals between the model (Eq. (19)) and the experimental data under conditions where there was no back-filtration, i.e., for filtrate flux values at Cb o 170 g/L. This gave k0 ¼1.6 mm/s based on the experimental data in Fig. 2. The solid curves in Figs. 2 and 3 represent the calculated values of the filtrate flux and the slope given by Eq. (19). The model calculations for the filtrate flux are in excellent agreement with the experimental data over the full range of antibody concentrations. Note that the maximum achievable antibody concentration predicted by the model is much smaller than the wall concentration (Cw ¼330 g/L) since the condition of zero net flux occurs when the positive filtrate flux over the first part of the channel exactly balances the reverse (back-filtration) flux in the region near the channel outlet. The model does slightly under-estimate the maximum achievable concentration (Cmax ¼206 g/L while the data gave 225 g/L); this is discussed in more detail subsequently. The solid curve in Fig. 3 is identical to the dashed curve for Cb o172 g/L, i.e. under conditions where there is no back-filtration. However, the complete model, which accounts for the effects of back-filtration, predicts a transition in the slope at Cb ¼172 g/L, in good qualitative agreement with the experimental results. The large increase in the slope at this value of Cb, which corresponds to a rapid reduction in the net filtrate flux, is due to the loss of filtrate caused by the back-filtration. The discontinuity in the slope at 172 g/L is not seen in the experimental data, at least in part because the slope was evaluated by numerical differentiation of data for the filtrate flux obtained at discrete protein concentrations (typically differing by 20 g/L at relatively high antibody concentrations). Fig. 6 shows a comparison between the model calculations and the experimental data for the filtrate flux during ultrafiltration of the monoclonal antibody at feed flow rates of 25, 45, and 65 mL/ min (feed flux of 47, 85, and 120 mm/s). The solid curves are the model calculations given by Eq. (19), with the k0 values at 25 and 65 ml/min determined by scaling as Q1/2 based on available correlations for mass transfer in a screened module [5]. The wall
Fig. 6. Experimental data for the filtrate flux during ultrafiltration of the monoclonal antibody in the Pellicon 3 module at different feed flow rates. Data were obtained using a 5 mM acetate buffer with 20 mM added NaCl at pH 5. Solid curves are model calculations developed using the modified polarization model accounting for back-filtration as given by Eq. (19).
Fig. 7. Maximum achievable antibody concentration as a function of feed flow rate. Data were obtained using a 5 mM acetate buffer with 20 mM added NaCl at pH 5. Solid curve is the modified polarization model accounting for back-filtration given by Eq. (19).
concentration was fixed at 330 g/L for all three feed flow rates since the data were all obtained at the same mean transmembrane pressure. The model calculations are in very good agreement with the experimental data at all 3 flow rates, properly capturing the increase in filtrate flux (for most values of Cb) and the reduction in the maximum achievable antibody concentration with increasing feed flow rate. The effect of the feed flux on the maximum achievable concentration is examined more explicitly in Fig. 7. The values of Cmax were determined by solving for the bulk protein concentration at which the net filtrate flux given by Eq. (19) is equal to zero. In the limit of very low feed flow rates, the pressure drop through the module becomes very small, eliminating back-filtration so that the filtrate flux goes to zero as Cb-Cw. As the feed flow rate increases, the pressure drop through the module increases, leading to an increase in back-filtration and a corresponding reduction in Cmax as seen in both the model and experimental results. The model calculations slightly under-predict the maximum achievable wall concentrations at all three feed flow rates, which is likely due to the over-estimation in the extent of back-filtration as described by Eqs. (17) and (18). This discrepancy likely reflects the approximations involved in estimating the rate of back-filtration, including the effects of “reverse polarization” on the concentration profile. In addition, the mean transmembrane pressure in the module tended to increase at very high antibody concentrations due to the very large pressure drop due to flow through the module (with the pinch valve on the retentate exit line fully opened). This effect was not included in the model calculations. Note that the model calculations suggest that the maximum achievable protein concentration for this antibody under these buffer conditions in the Pellicon 3 module could be increased to 270 g/L by decreasing the feed flux to below 11 mm/s (corresponding to a feed flow rate less than 6 mL/min). Further confirmation of the model results is presented in Fig. 8, which shows the calculated values of the slope as a function of the bulk antibody concentration for the three feed flow rates. In this case, the model curves were evaluated by numerical differentiation of the calculated values of the filtrate flux using centered difference representations with ΔCb ¼10 g/L; this numerically “smooths” the model calculations so that they are consistent with the numerical evaluation of the slopes from the filtrate flux data. The minimum in the slope shifts to lower bulk protein concentrations as the feed flow rate increases due to the increase in the pressure drop associated with the flow through the module. The increase in slope at high Cb is due to the large reduction in the filtrate flux associated with the back-filtration. The model
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concentration due to the increase in back-filtration arising from the larger pressure loss associated with flow through the channel. Based on the experimental results and model calculations, it should be possible to obtain antibody formulations with higher protein concentration by decreasing the feed flow rate. High feed flow rates could be used over most of the ultrafiltration to provide high filtrate flux, with the feed flow rate reduced in the final stage of the ultrafiltration to minimize the extent of back-filtration by reducing the parasitic pressure losses. The modified polarization model developed in this work provides new insights into the ultrafiltration behavior of these highly concentrated protein solutions as well as an appropriate framework for the design and analysis of ultrafiltration processes/systems for the formulation of monoclonal antibody products. Fig. 8. Calculated values of the slope, dJv/d(lnCb) as a function of the bulk protein concentration from the data in Fig. 6. Solid curves are calculations based on the modified polarization model accounting for back-filtration as given by Eq. (19).
calculations are in good qualitative agreement with the experimental results, with the deviations at very high bulk protein concentrations likely due to the approximations made in the evaluation of the filtrate flux due to back-filtration.
Acknowledgment The authors would like to acknowledge Amgen for their financial support and donation of the monoclonal antibody and EMD Millipore Corp. for their donation of the Pellicon 3 modules.
References 5. Conclusions The need to formulate monoclonal antibody products at very high protein concentrations (above 200 g/L) has created major challenges for the design and application of ultrafiltration processes for final formulation of monoclonal antibodies. A new mathematical model was developed specifically to describe the filtrate flux behavior during ultrafiltration of these highly concentrated monoclonal antibody solutions. The model includes two basic phenomena: (1) the filtrate flux over most of the channel is evaluated using a modified form of the stagnant film model in which the driving force is expressed in terms of the gradient in the chemical potential (evaluated using data for the osmotic virial coefficients) with the protein mobility accounting for the concentration dependence of the viscosity, and (2) the large parasitic pressure losses due to flow through the screened channel causes a negative filtrate flux, or back-filtration, near the exit of the module since the transmembrane pressure becomes less than the osmotic pressure at high bulk protein concentrations. Model calculations were performed for a highly purified monoclonal antibody (provided by Amgen) based on independent measurements for the osmotic pressure and virial coefficients [10] and the solution viscosity [15], both obtained at protein concentrations up to at least 250 g/L. The model was in good agreement with experimental data obtained in a linearly-scalable Pellicon 3 tangential flow filtration module commonly used in downstream processing of monoclonal antibodies. The model properly captures the observed variation of the filtrate flux with bulk protein concentration, including the complex dependence of the slope in a plot of flux versus the logarithm of Cb. In addition, the model provides reasonably accurate predictions for the maximum achievable protein concentration; increasing the feed flow rate leads to a reduction in the maximum achievable protein
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