# Optimization of ultrafiltration/diafiltration processes for partially bound impurities.

**ABSTRACT** Ultrafiltration and diafiltration processes are used extensively for removal of a variety of small impurities from biological products. There has, however, been no experimental or theoretical analysis of the effects of impurity- product binding on the rate of impurity removal during these processes. Model calculations were performed to account for the effects of equilibrium binding between a small impurity and a large (retained) product on impurity clearance. Experiments were performed using D-tryptophan and bovine serum albumin as a model system. The results clearly demonstrate that binding interactions can dramatically reduce the rate of small impurity removal, leading to large increases in the required number of diavolumes. The optimal product concentration for performing the diafiltration shifts to lower product concentrations in the presence of strong binding interactions. Approximate analytical expressions for the impurity removal were developed which can provide a guide for the design and optimization of industrial ultrafiltration/diafiltration processes.

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Page 1

Optimization of Ultrafiltration/Diafiltration

Processes for Partially Bound Impurities

Jiahui Shao, Andrew L. Zydney

Department of Chemical Engineering, The Pennsylvania State University,

University Park, Pennsylvania 16802; telephone: 814-863-7113;

fax: 814-865-7846; e-mail: zydney@engr.psu.edu

Received 23 April 2003; accepted 2 July 2003

Published online 7 July 2004 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/bit.20113

Abstract: Ultrafiltration and diafiltration processes are used

extensively for removal of a variety of small impurities from

biological products. There has, however, been no experi-

mental or theoretical analysis of the effects of impurity—

productbindingontherateofimpurityremovalduringthese

processes. Model calculations were performed to account

for the effects of equilibrium binding between a small im-

purity and a large (retained) product on impurity clearance.

Experiments were performed using D-tryptophan and bo-

vine serum albumin as a model system. The results clearly

demonstrate that binding interactions can dramatically

reduce the rate of small impurity removal, leading to large

increasesintherequirednumberofdiavolumes.Theoptimal

product concentration for performing the diafiltration

shifts to lower product concentrations in the presence of

strong binding interactions. Approximate analytical expres-

sions for the impurity removal were developed which can

provide a guide for the design and optimization of in-

dustrial ultrafiltration/diafiltration processes. B 2004 Wiley

Periodicals, Inc.

Keywords: impurity removal; ultrafiltration; diafiltration;

clearance; binding interactions; buffer exchange

INTRODUCTION

Ultrafiltration (UF) is used throughout the downstream sep-

aration process for the purification of recombinant proteins,

blood components, natural protein products, and industrial

enzymes (Madsen, 2001). Specific applications include pro-

tein concentration (i.e., volume reduction), desalting, and

buffer exchange, all of which are used to condition the

product prior to, or immediately after, other separation pro-

cesses or as part of the final product formulation (van Reis

and Zydney, 1999). In addition, ultrafiltration is used ex-

tensively for the clarification of antibiotics, amino acids,

and other small biological molecules.

Buffer exchange and de-salting are typically performed

using a diafiltration (DF) mode in which the small im-

purities and buffer components are effectively washed

away from the product by the continuous (or discontinuous)

addition of new buffer with the desired composition and

purity (Kurnick et al., 1995). The most common approach

is to perform the diafiltration using a constant retentate

volume, in which case the impurity concentration in the

product solution can be evaluated from a simple mass

balance as:

Ci

Cio

¼ exp½?NDSi?ð1Þ

where Cio is the initial impurity concentration in the

product solution, Ciis the impurity concentration remaining

in the retentate at any time, and NDis the number of dia-

volumes, which is equal to the total collected filtrate vol-

ume divided by the constant retentate volume during the

diafiltration process. Siis the solute sieving coefficient,

which is defined as the ratio of the solute concentration in

the filtrate solution to that in the retentate. A diafiltration

process with ND= 10 will provide more than a 104reduction

in the concentration of a completely unretained component.

In many applications of diafiltration, the overall filtra-

tion process involves a combination of ultrafiltration and

diafiltration to achieve both the desired volume reduction

(i.e., the desired final protein concentration) and the re-

quired impurity or salt removal. The total process time

under these conditions will depend upon the point at which

the diafiltration is performed (Ng et al., 1976). If the

diafiltration is performed before the protein is concentrated,

a very large diafiltration volume will be needed although

the filtrate flux during the diafiltration process will be quite

high. The amount of diafiltration buffer decreases as the

retentate volume is reduced, but the resulting increase in

product concentration causes a reduction in the filtrate flux.

This effect is usually described using a simple stagnant film

model (Zeman and Zydney, 1996):

Jv¼ kmln

Cw

Cp

??

ð2Þ

where Jvis the filtrate flux (volumetric flow rate per unit

membrane area), kmis the product mass transfer coefficient

in the bulk solution, Cwis the product concentration at the

membrane surface, and Cpis the product concentration in

the bulk solution. Equation (2) is strictly valid for a system

in which the membrane is fully retentive to the product of

B 2004 Wiley Periodicals, Inc.

Correspondence to: Andrew Zydney

Page 2

interest. At high transmembrane pressures, the product

concentration at the membrane reaches a critical value, at

which point the flux becomes essentially independent of the

transmembrane pressure. This critical concentration may be

related to the protein solubility or it may arise from osmotic

pressure effects (Zeman and Zydney, 1996).

Ng et al. (1976) have used the stagnant film model

to evaluate the bulk protein concentration that will min-

imize the total time for the ultrafiltration/diafiltration

process as:

C?

p¼Cw

e

ð3Þ

where e = 2.718. If the diafiltration is performed at smaller

bulk protein concentrations, the process time increases

because of the greater volume of diafiltration buffer needed

to obtain the desired impurity removal. If the diafiltration is

performed at higher bulk protein concentrations, the

process time increases due to the reduction in the filtrate

flux with increasing Cpas given by Eq. (2).

The analysis leading to Eqs. (1) and (3) implicitly as-

sumes that the transmission of the impurity is unaffected by

the concentration of the retained product. This assumption

is only valid if the impurity has no interactions with the

product. However, many proteins are able to bind a wide

range of small molecules (Denizli and Piskin, 2001; Falsey

et al., 2001), amino acids (Bowen and Nigmatullin, 2002;

Romero and Zydney, 2001), and ionic species (Pidcock and

Moore, 2001; Ueda et al., 2003; Yamauchi et al., 2001).

Under these conditions, the rate of small molecule removal

will be determined by the amount of free (unbound) im-

purity since the bound solute will be completely retained by

the membrane.

The objective of this study was to examine the behavior

of a UF/DF process for the removal of an impurity that has

a reversible binding interaction with the retained product.

Calculations were performed using a Michaelis–Menten

binding expression, with the flux evaluated using the clas-

sical stagnant film model. The model was confirmed ex-

perimentally using data for the removal of D-tryptophan

from bovine serum albumin. Equations were also devel-

oped for the optimal diafiltration conditions to minimize

the total process time accounting for the reversible binding

of the impurity.

THEORETICAL DEVELOPMENT

We assume that the impurity and product undergo a revers-

ible association with the equilibrium constant defined as:

K ¼

Ci;bound

Ci;freeCp;free

ð4Þ

where Ci,bound and Ci,free are the bound and free con-

centrations of the impurity and Cp,freeis the free (unbound)

concentration of the protein product. Equation (4) can be

combined with simple mass balance expressions for the

total amount of product and impurity to give the following

equation for the concentration of free impurity:

Ci;freeþnKCpCi;free

1 þ KCi;free

¼ Ci

ð5Þ

where n is the number of binding sites on the protein, and Ci

and Cpare the total (bound + free) concentrations of the

impurity and the product, respectively.

Ultrafiltration

The impurity mass balance during an ultrafiltration pro-

cess is:

d

dtðVCiÞ ¼ ?JvASiCi; free

ð6Þ

where V is the volume of the retentate solution, Jvis the

filtrate flux through the membrane, and A is the exposed

membrane area. Note that in writing Eq. (6) we have

assumed that the protein product, and thus the bound

impurity, is fully retained by the membrane. Equations (5)

and (6) can be integrated numerically to evaluate Ciafter a

given volume reduction. Relatively simple analytical so-

lutions can be developed under two important limiting

conditions. First, if the membrane is completely permeable

to the unbound impurity, i.e., if Si= 1, it is straightforward

to show that Ci,freemust remain constant throughout the

ultrafiltration, thus:

Ci

Cio

¼1 þ KCi;freeþ nKCp

1 þ KCi;freeþ nKCpo

ð7Þ

The protein concentration at the end of the UF step is

directly related to the volume concentration ratio (X) as:

Cp

Cpo

¼Vinitial

Vfinal

¼ X

ð8Þ

The total impurity concentration increases during the

ultrafiltration due to the increase in bound concentration

associated with the greater product concentration. A second

limiting case occurs when KCi,freeis much less than one,

i.e., for a weakly bound impurity or where the concen-

tration of the protein is much larger than that of the

impurity. Under these conditions Eq. (6) can be integrated

to give:

Ci

Cio

¼

1 þ nKCp

1 þ nKCpo

??

Cpoð1 þ nKCpÞ

Cpð1 þ nKCpoÞ

??Si?1

ð9Þ

Equation (9) reduces to the classical expression for the

concentration of a partially retained solute when the term

nKCp= 0, with the protein concentration again given in

terms of the volume reduction ratio by Eq. (8).

SHAO AND ZYDNEY: ULTRAFILTRATION/DIAFILTRATION PROCESSES FOR PARTIALLY BOUND IMPURITIES287

Page 3

Diafiltration

The impurity mass balance during a diafiltration process is

also given by Eq. (6) assuming that the concentration of the

small impurity (or ion) is zero in the diafiltration buffer.

The change in solute concentration during a constant

volume diafiltration can be evaluated by direct integration

under conditions where the protein product is fully retained

by the membrane to give:

? ?

lnCi;free;oð1 þ KCi;freeÞ

Ci;freeð1 þ KCi;free;oÞ

ND¼

l

Si

ln

Ci;free;o

Ci;free

??

þ

nKCp

Si

?

??

?

?

KðCi;free;o? Ci;freeÞ

ð1 þ KCi;free;oÞð1 þ KCi;freeÞ

??

where Ci,free,o is the concentration of free impurity at

the start of the diafiltration and ND is the number of

diavolumes defined in terms of the retentate volume during

the diafiltration step (and not the initial retentate volume).

Equation (10) can be placed in much simpler form under

conditions where KCi,freeV 1:

?

ND¼

1 þ nKCp

Si

?

ln

Cio

Ci

??

ð11Þ

where Ciois the total impurity concentration at the start of

the diafiltration. Note that Ciowill not be equal to the

impurity concentration in the initial feed unless the dia-

filtration is performed immediately since the amount of

bound impurity increases during the UF step even if Si= 1.

Equation (11) predicts an exponential decline in the

impurity concentration with increasing number of diavol-

umes, similar to Eq. (1), but the rate of decline is reduced by

a factor of 1 + nKCpwhere Cpis the protein concentration

during the diafiltration process. Thus, more diavolumes are

needed to reduce the impurity concentration by a given

factor in the presence of impurity binding, and the extent of

this increase depends upon the degree of protein concen-

tration used prior to the diafiltration.

The final impurity concentration at the end of a UF—

DF—UF process can be evaluated analytically under con-

ditions where KCi,freeV 1 and Si= 1 by combining Eqs. (9)

and (11) to give:

Ci;final

Cio

¼1 þ nKCpoXtotal

1 þ nKCpo

exp ?

ND

1 þ nKCpoX1

??

ð12Þ

where Xtotalis the total volume concentration factor at the

end of the full UF—DF—UF process and X1is the volume

concentration factor for the first UF step.

UF/DF Optimization

The time required for a given UF/DF process depends upon

the product concentration at which the diafiltration is

performed (the time for the ultrafiltration depends only on

the degree of product concentration as long as the mass

transfer coefficient and Cwvalues are unaffected by the

change in buffer conditions). The diafiltration time can be

evaluated using the simple stagnant film model (Eq. 2) as:

tD¼

NDVoCpo

kmACpln

Cw

Cp

??

ð13Þ

where Voand Cpoare the volume and product concentra-

tion in the initial feed solution (prior to the UF) and Cpis

the product concentration during the diafiltration. The re-

quired number of diavolumes for a given overall impurity

removal must be evaluated by considering the concen-

tration changes that occur during the initial ultrafiltra-

tion, the diafiltration, and then the final ultrafiltration since

the impurity concentration changes in all three stages of

the process.

Under conditions where KCi,freeV 1 it is possible to

evaluate the required number of diavolumes analytically as:

?

Ci;feed

Ci;final

ND¼

1 þ nKCp

Si

?

?

?

ln

þ ðSi? 1Þln

Cp;feed

Cp;final

??

? Siln

1 þ nKCp;feed

1 þ nKCp;final

????

where Cp is the product concentration at which the

diafiltration is performed and Ci,finaland Cp,finalare the

impurity and product concentrations in the final solution

obtained at the end of the UF—DF—UF process. The value

of the product concentration that minimizes the process

time (C*

p) is evaluated by differentiating Eq. (13) using

Eq. (14) to evaluate NDto give:

ln

Cw

C?

p

!

¼ 1 þ nKC?

p

ð15Þ

Equation (14) must be solved iteratively to evaluate C*

for given values of Cw and nK. In the absence of any

binding, Eq. (15) reduces to Eq. (3). The optimal con-

centration decreases as the binding constant increases; thus,

the diafiltration should be performed at a lower protein con-

centration when the impurity is strongly bound to the prod-

uct. Note that at large values of nKC*

time will be predicted to occur at a bulk protein concen-

tration that is smaller than the initial feed concentration.

Under these conditions it could be beneficial to dilute the

feed before performing the diafiltration, although it should

be noted that Eq. (14) couldn’t be used to determine the

optimal extent of dilution since this equation does not

account for the increase in ultrafiltration time caused by

any initial dilution.

p

pthe minimum process

MATERIALS AND METHODS

Experiments were performed using D-tryptophan as the

small impurity and bovine serum albumin (BSA) as the

protein product. D-tryptophan (Sigma Chemical, St. Louis,

(10)

(14)

288BIOTECHNOLOGY AND BIOENGINEERING, VOL. 87, NO. 3, AUGUST 5, 2004

Page 4

MO) was dissolved in 10 mM borate buffer prepared from

sodium tetraborate (Sigma) and deionized distilled water

(resistivity > 18 Mohm-cm) obtained from a Barnstead

water purification system (Dubuque, IA). All buffer

solutions were prefiltered through a 0.2 Am microfiltration

membrane to remove particulates prior to use.

Bovine serum albumin (Sigma A-8022) was slowly

added to the tryptophan solution, with the resulting solution

gently mixed to ensure equilibrium binding. The pH of the

final solution was measured using a 420APlus pH meter

(Thermo Orion, Beverly, MA) as pH 8.5.

D-tryptophan concentrations were evaluated spectro-

photometrically using a UVmini-1240 spectrophotometer

(Shimadzu, Kyoto, Japan). The D-tryptophan concentra-

tion in the presence of BSA was determined by reducing

the pH by adding 13 AL of a 1M HCl/0.2M KCl solu-

tion to a 1 mL sample. D-tryptophan binding to the posi-

tively charged BSA at this low pH was negligible. The

resulting solution was filtered through a 30,000 molecular

weight cut-off membrane to remove the BSA, with the

concentration of D-tryptophan in the collected filtrate eval-

uated spectrophotometrically.

All filtration experiments were performed with Biomax

polyethersulfone membranes (Millpore Corp., Bedford,

MA) having a nominal molecular weight cut-off of 30,000

g/mol. The membranes were thoroughly flushed with 100 L/

m2offiltereddeionizedwatertoremoveanyglycerin,which

was used as a wetting and storage agent. The membrane

hydraulic permeability was evaluated before and after each

filtration experiment to determine if there was any signi-

ficant fouling.

Diafiltration experiments were performed in an Amicon

stirred cell (Millipore Corp.) with volume of 16.8 mL. The

stirred cell was connected to a solution reservoir containing

borate buffer (without any tryptophan or BSA). Filtrate

samples were collected for subsequent determination of the

D-tryptophan concentration. At the end of the diafiltration,

the stirred cell was opened and a sample taken to evaluate

the tryptophan concentration in the final retentate solution.

RESULTS AND DISCUSSION

Figure 1 shows the effect of BSA on D-tryptophan removal

during a diafiltration process. Data are shown for three

separate experiments with BSA concentrations of 0, 15, and

40 g/L. The open symbols represent the D-tryptophan con-

centrations in the retentate solution calculated from the

measured filtrate concentrations using an overall material

balance. These results were in good agreement with the

retentate concentration measured directly in the stirred cell

at the end of the diafiltration (shown by the filled symbols

in Fig. 1), confirming the overall mass balance closure in

this system. The data in the absence of any BSA are in

excellent agreement with Eq. (1) using Si= 1 for the small

D-tryptophan molecule. The solid curves for the diafiltra-

tions performed in the presence of BSA are model cal-

culations given by Eq. (10), with the best-fit values for the

number of binding sites (n = 3) and the equilibrium binding

constant (K = 370M?1) determined by minimizing the

sum of the squared residualsbetweenthe model calculations

and experimental data. These values are in very good

agreement with independent results for the binding of D-

tryptophan to BSA using a two-site binding model which

gaven1=0.83andn2=2,andK1=510M?1andK2=300M?1

(Romero and Zydney, 2001). The model is in excellent

agreement with the data at both BSA concentrations,

properly capturing the reduction in clearance caused by the

binding interactions. For example, the number of diavol-

umes needed to obtain a 100-fold reduction in Ci,total

increased from ND= 4.6 in the absence of BSA to ND= 7.6

for the solution with Cp= 40 g/L.

The dashed curves in Figure 1 represent the results de-

veloped from the approximate solution [Eq. (11)] using the

same values of n and K as for the full model. The approx-

imate solution is also in good agreement with the data,

even though KCi,free,owas as large as 0.3 for the run with

Cp= 40 g/L. The largest discrepancy between the full and

approximate models occurs right at the start of the dia-

filtration, where the D-tryptophan concentration is highest,

with the two solutions converging at large numbers of dia-

volumes since the D-tryptophan concentration becomes very

small towards the end of the diafiltration. Thus, Eq. (11)

will provide an excellent estimate for the required number

of diavolumes for impurities that must be reduced to very

low concentrations since the condition KCi,freeV 1 will be

valid over the majority of the diafiltration process.

The variation in the total tryptophan concentration in the

retentate solution during a simulated UF—DF—UF is

Figure 1.

function of the number of diavolumes. Open symbols represent calculated

values of the retentate concentration determined from measured filtrate

concentrations using an overall mass balance. Filled symbols represent

direct measurements from the retentate at the end of the diafiltration. Solid

curves are full model given by Eq. (10); dashed curves are approximation

given by Eq. (11).

Normalizedimpurityconcentrationintheretentatesolutionasa

SHAO AND ZYDNEY: ULTRAFILTRATION/DIAFILTRATION PROCESSES FOR PARTIALLY BOUND IMPURITIES289

Page 5

examined in Figure 2. Results are shown for a feed solution

(V/A = 0.16 m) with initial concentrations of 5 g/L BSA

and 0.184 g/L D-tryptophan (corresponding to 0.9 mM D-

tryptophan). The initial ultrafiltration provides a fourfold

reduction in process volume (yielding a BSA concentration

of 20 g/L), the diafiltration is performed for 10 diavolumes,

and the final ultrafiltration provides an additional 2.5-fold

volume reduction (yielding a final protein concentration of

50 g/L). Simulations are shown for several values of the

equilibrium binding constant, with n = 3 and Si= 1 for all

calculations. The solid curves are the full model calcu-

lations, while the dashed curves represent the approximate

solution developed using Eq. (7) for the ultrafiltration steps

and Eq. (11) during the diafiltration. The results with K = 0

show no change in impurity concentration during the

ultrafiltration followed by an exponential decay during the

diafiltration. The behavior in the presence of product-

impurity binding is very different. The total impurity con-

centration increases during the ultrafiltration process due

to the increase in impurity binding as the protein (product)

concentration increases. This effect is significantly more

pronounced for the higher binding constant, with Ci,total=

1.55 mM at the end of the first UF for K = 37,000M?1

compared to only Ci,total= 1.06 mM when K = 370M?1. The

larger value of the binding constant also significantly

reduces the rate of impurity removal during the diafiltra-

tion. The net result is that at the end of the UF—DF—UF

process, there is actually a small increase in the impurity

concentration for the simulations with K = 37,000M?1(from

0.9–1.13 mM) compared to the more than 20,000-fold

reduction in the impurity concentration that occurs in the

absence of any binding interactions. Even the weak binding

interactions between D-tryptophan and BSA (corre-

sponding to the simulations with K = 370M?1) result in

only a 1,200-fold reduction in the impurity concentration

over this UF—DF—UF process. This value is in good

agreement with that calculated from the analytical approx-

imation given by Eq. (12) which yields Ci/Cio= 0.00094

for the same overall process, corresponding to 1,100-fold

reduction in the impurity concentration. The approximate

solution (dashed curves) is in excellent agreement with the

full model when the binding is relatively weak, but begins to

significantly underpredict the rate of impurity removal for

the larger values of K.

Figure 3 shows the calculated values of the scaled pro-

cess time (top panel) and the required number of diavol-

umes (bottom panel) for a UF—DF—UF process involving

a 10-fold volume concentration and a 10,000-fold reduction

in the concentration of the impurity. The results are plotted

as a function of the protein concentration at which the

diafiltration is actually performed. The scaled process time

is simply equal to the product of the actual time, the bulk

mass transfer coefficient, and the membrane area divided by

the initial feed volume [see Eq. (13)]. Thus, a scaled process

time of 20 corresponds to an actual time of 4 h for an initial

volume of 1000 L, a membrane area of 100 m2, and a bulk

mass transfer coefficient of 50 L m?2h?1. Simulations were

Figure 2.

of the scaled cumulative filtrate volume for a UF—DF—UF process in-

volvingafourfoldvolumeconcentration,a10-diavolumediafiltration,anda

2.5-fold final volume concentration.

Normalizedimpurityconcentrationintheretentateasafunction

Figure 3.

process time (top panel) for a UF—DF— process designed to achieve a

10-fold volume concentration and a 10,000-fold impurity removal. Results

are plotted as a function of the protein concentration during the diafiltra-

tion step.

Required number of diavolumes (bottom panel) and total

290BIOTECHNOLOGY AND BIOENGINEERING, VOL. 87, NO. 3, AUGUST 5, 2004