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Modeling of transient flow through a viscoelastic preparative chromatography packing

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Modeling of Transient Flow Through a Viscoelastic Preparative ChromatographyPacking
Dariusch Hekmat, Michael Kuhn, Verena Meinhardt, and Dirk Weuster-Botz
Inst. of Biochemical Engineering, Technische Universit
€
at M
€
unchen, 85748 Garching, Germany
DOI 10.1002/btpr.1768 Published online June 25, 2013 in Wiley Online Library (wileyonlinelibrary.com)
The common method for puriﬁcation of macromolecular bioproducts is preparative packed-bed chromatography using polymer-based, compressible, viscoelastic resins. Becauseof a downstream processing bottleneck, the chromatography equipment is often operated at its hydrodynamic limit. In this case, the resins may exhibit a complex behavior which resultsin compression–relaxation hystereses. Up to now, no modeling approach of transient ﬂowthrough a chromatography packing has been made considering the viscoelasticity of the res-ins. The aim of the present work was to develop a novel model and compare model calcula-tions with experimental data of two agarose-based resins. Fluid ﬂow and bed permeabilitywere modeled by Darcy’s law and the Kozeny–Carman equation, respectively. Fluid ﬂowwas coupled to solid matrix stress via an axial force balance and a continuity equation of adeformable packing. Viscoelasticity was considered according to a Kelvin–Voigt material.The coupled equations were solved with a ﬁnite difference scheme using a deformable mesh.The model boundary conditions were preset transient pressure drop functions which resem-ble simulated load/elution/equilibration cycles. Calculations using a homogeneous model(assuming constant variables along the column height) gave a fair agreement with experi-mental data with regard to predicted ﬂow rate, bed height, and compression–relaxation hys-teresis for symmetric as well as asymmetric pressure drop functions. Calculations using aninhomogeneous model gave proﬁles of the bed porosity as a function of the bed height. Inaddition, the inﬂuence of medium wall support and intraparticle porosity was illustrated.The inhomogeneous model provides insights that so far are not easily experimentally acces-sible.
V
C
2013 American Institute of Chemical Engineers
Biotechnol. Prog
., 29:958–967,2013
Keywords: preparative chromatography, viscoelastic resins, modeling, transient ﬂow
Introduction
The most common method for large-scale puriﬁcation of macromolecular pharmaceutical bioproducts is preparativechromatography.
1
Since conventional packed-bed chromatog-raphy often represents a downstream processing bottleneckbecause of limited volumetric ﬂow rates, other approacheslike membrane chromatography have been under investiga-tion for more than 20 years.
2,3
Membrane chromatographycan be a viable alternative, especially for ﬂow-through appli-cations as a polishing step as ﬂow limitations are reducedcompared to packed-bed chromatography,
4,5
However, whenoverall advantages and disadvantages are compared, theinclination of the pharmaceutical industry to apply mem-brane chromatography is rather limited. Therefore, themajority of preparative chromatography is still performedusing conventional packed beds. Because of the mentionedbottleneck, equipment and chromatography media have to beutilized in an optimized manner. Hence, the media have tobe used for a maximum number of cycles while maintainingthe column performance at a high level during this timeperiod.
6
However, it is well known that the operation of thecolumns at high ﬂow rates near the hydrodynamic designlimit may lead to column integrity breaches.
7
These integritybreaches typically comprise medium wall detachment, devel-opment of cracks and ﬂow channels near the column walland/or inside the bed, partial subsidence of the top of thepacking, and even bed collapse.
8
Up to now, the detailedcauses of these phenomena are largely unknown. However,some evidence was reported by Ladisch and Tsao as early as1978 that the compressibility behavior of the commonlyused polymer-based resins may be the srcin of the prob-lem.
9
In fact, the majority of the polymer-based resins is rep-resented by compressible, viscoelastic, porous media andexhibit a complex, dynamic compression behavior that hasto be considered properly. Therefore, a necessity exists for mathematical models, which describe the resins as a visco-elastic material. J
€
onsson and J
€
onsson reported the occurrenceof compression–relaxation hysteresis behavior for the ﬁrsttime. In their work, steady-state ﬂow through a chromatogra-phy column was modeled considering both compressibilityand permeability of the media.
10
Dynamic modeling yieldedthe time dependency of the extra-particle porosity as a func-tion of bed height at different mechanical and/or hydraulicloads.
11
However, these calculations were not validated
Correspondence concerning this article should be addressed toD. Hekmat at hekmat@lrz.tum.de.
958
V
C
2013 American Institute of Chemical Engineers
experimentally.
€
Ostergren et al. developed a two-dimensional model for steady-state ﬂow through a chroma-tography column assuming pure elastic deformation of themedia.
12,13
A one-dimensional model based on pure elastic-ity of the media to describe mechanical deformation of com-pressible chromatography beds was published by Keener et al.
14
Further publications deal with modeling of packingand scale-up of chromatography columns again assumingpure elastic deformation of the media.
15–18
A comparison of mechanical compression and ﬂow packing was given.The mathematical models of all above mentioned publica-tions are based on the theory of Biot where consolidation of a porous material with pure elastic properties wasdescribed.
19
As to the knowledge of the authors, no modelof ﬂow through a chromatography column exists so far thattakes the viscoelastic properties of the media into considera-tion. Therefore, the aim of the present work was to developtwo novel mathematical models with different levels of com-plexity of transient incompressible ﬂow through a packing of compressible, viscoelastic, porous media. The ﬁrst modelconsiders the packing as homogeneous where bed variableslike stress, strain, and bed porosity are not a function of bedheight. Simulation results shall be compared with experimen-tal data. The second model considers the packing to be inho-mogeneous where bed variables like stress, strain, and bedporosity are in fact a function of bed height. However, in thelatter case, no sufﬁcient experimental data are up to nowavailable in order to validate the model. On the other hand,the inhomogeneous model may give insight into which of the process parameters are signiﬁcant. It is expected that themodels to be developed are able to predict the mentionedcomplex, hydrodynamic behavior, which leads to compres-sion–relaxation hystereses. The focus of the study lies solelyon the investigation of the hydrodynamics of the chromatog-raphy column. Issues regarding the separation process itself or the inﬂuence of other physical or chemical processes werenot taken into consideration.
Materials and Methods
Column, stationary phases, and experimental setup
The column was a BPG 140 glass column with an internaldiameter of 140 mm (GE Healthcare, Uppsala, Sweden). Themaximum allowed pressure drop of the column was 6 bar.Two different agarose-based media from GE HealthcareEurope, Munich, Germany, were chosen for the experiments:(1) CM Sepharose 6 FF
V
R
(resin SEP). This resin is a weakcation exchange resin made of 6% cross-linked agarose. Themean particle diameter is 90
l
m. (2) Capto Q ImpRes
V
R
(resin CAP). This is a strong anion exchange resin made of highly cross-linked agarose (the degree of cross-linking isnot disclosed by the manufacturer). The mean particle diam-eter is 40
l
m. Particle size distributions were determined bypipetting approximately 5
l
L of diluted particle suspensionto a hemocytometer (Jessen Z
€
ahlkammer, depth: 0.4 mm,markings: 1/16 mm
2
). The dilution was chosen so that noparticles overlapped. Photomicrographs were taken with anoptical microscope (type: Axioplan, supplier: Carl Zeiss,G
€
ottingen, Germany) and evaluated with the open-sourcesoftware Image J (version: 1.46, available online: http:// rsbweb.nih.gov/ij/ ). The experimental setup was designed toperform automated long-term experiments using LabVIEW.The schematics are given in Figure 1.The computer-controlled precision pump was a gear pump(type Micropump GB-P25.JVS.A, Idex Health & Science,Oak Harbor, WA, USA), which was magnetically coupled toits drive. The driving magnet was a rare earth type (NdFeB).The ﬂow rate was measured by an inductive ﬂow meter (typePromag 50 H, Endress
1
Hauser, Weil am Rhein, Germany). Adigital pressure gauge (type 692.30111151, Huba Control,Walddorfh
€
aslach, Germany) was used to measure the pressuredrop of the column. The height of the chromatography bedwas measured using an intelligent, high-resolution CCD-linesensor (type IZS 1024, Asentics, Siegen, Germany). Theexperimental setup was connected to the microcomputer via aUSB-6008 data acquisition device (National Instruments Ger-many, Munich, Germany). The pump was controlled by thePC according to one of two alternative control schemes: (1)pump speed controlled in order to achieve a given ﬂow rateproﬁle, and (2) pump speed controlled in order to achieve agiven pressure drop proﬁle.All experiments were performed at a constant temperatureof 20
C by using a cooling thermostat. The mobile phasewas de-ionized water (
l
5
1 mPa s). The column waspacked to a height approximately equal to the column diam-eter using ﬂow packing procedures at different constant pre-determined pressure drops. Packing experiments were
Figure 1. Schematics of the experimental setup.
The height of the chromatography bed was measured using a high-resolution CCD-line sensor.
Biotechnol. Prog.,
2013, Vol. 29, No. 4
959
performed with varying packing rates. The column pressuredrop as a function of the superﬁcial velocity for resin SEPwas measured by applying a controlled linear increase of theﬂow rate until a preset threshold gauge pressure of 3.9 bar was reached. For resin CAP, a controlled linear increase of pressure drop up to 6 bar was preset and the resulting ﬂowrate was measured. In order to examine the compression– relaxation behavior, linear pressure drop gradients wereapplied with varying slopes for compression and relaxation(the transition rates of the simulated separation cycle).Between each compression and relaxation event, enoughtime was provided in order to achieve a steady-state of thechromatography bed.
Modeling
For a simple account of viscoelasticity of materials, twomodels are established. These are the Maxwell model andthe Kelvin–Voigt model.
20
The Maxwell model consists of apurely elastic spring and a purely viscous damper connectedin series. This model describes a time-delayed stressresponse
r
(
t
) to an applied strain
k
(
t
). The Kelvin–Voigtmodel consists of a purely elastic spring and a purely vis-cous damper connected in parallel. This model describes atime-delayed strain response
k
(
t
) to an applied stress
r
(
t
)and is applicable to characterize the behavior of a packing of viscoelastic chromatography media during hydrodynamicloading. Previously, Singh et al. successfully applied theKelvin–Voigt model to describe moisture transport in porousfood, which may be considered similar to a packing of viscoelastic particles.
21,22
The governing equation of the Kel-vin–Voigt model is
20
r
ð
t
Þ
5
E
k
ð
t
Þ
1
g
d
k
ð
t
Þ
dt
(1)
The parameter Young’s modulus
E
characterizes the elas-tic spring, whereas the damping constant
g
characterizes thedamper. Figure 2 shows the strain response to an appliedrectangular stress step according to the Kelvin–Voigt model.When the stress is raised instantaneously, the strainmomentarily stays at zero (1). During the period of constantstress, the strain converges to its maximum value (2). At thetime when the stress is abruptly reduced to zero, the strainmomentarily stays constant (3). During the following timeperiod of zero stress, the strain responds with a time delayuntil it ﬁnally reaches zero as well (4). If strain
k
(
t
) is plot-ted against stress
r
(
t
), a hysteresis curve results. The visco-elastic parameters Young’s modulus
E
and damping constant
g
for both media were determined from relaxation experi-ments. Properties from single particle measurements cannotbe used as the bed behavior as a whole is of interest here(including particle-particle interactions). Therefore, theviscoelastic parameters were derived from measurementsusing the chromatography packing. For this purpose, a givenpressure drop resulting in a known bed stress was applied tothe packing. After reaching equilibrium, the pressure dropwas abruptly reduced to zero and the compressed bedrelaxed with a certain time delay. The measured relaxationcurves were used to evaluate the model parameters. The ana-lytical solution of Eq. 1 for stress relaxation is
k
5
r
0
E
e
2
E
g
t
(2)
The initially applied pressure drop is equal to the initialbed stress
r
0
if wall effects are neglected. From the currentbed height
h
and the known bed height of the gravity settledbed
h
0
, the bed compression
k
could be calculated from
k
5
h
0
2
hh
0
(3)
With the experimental values of
r
0
,
k
, and time
t
, themodel parameters
E
and
g
were calculated from a curve-ﬁtby applying the common method of nonlinear least squares.For reasons of simplicity, the model to be developed wasone-dimensional. Laminar ﬂow through porous media iscommonly described by Darcy’s law
23
u
52
1
l
k
@
p
@
z
(4)
which relates the superﬁcial ﬂuid velocity
u
to a pressuregradient
@
p
/
@
z
along the axial direction
z
.
l
is the dynamicviscosity of the ﬂuid and
k
is the bed permeability. TheKozeny–Carman equation is a model for the permeability
k
in Darcy’s law.
23
It relates
k
to the bed porosity
e
and themean particle diameter
d
p
k
5
d
2p
180
e
3
ð
1
2
e
Þ
2
(5)
The constant 180 is derived from a model approach whichreduces the complex array of ﬂuid path lines in a poroussample with a given height to simple ﬂow through an arrayof parallel pipes where the path length is assumed to be 2.5times the same sample height.
23
The behavior of the compressible, viscoelastic matrix of chromatography particles is coupled to the ﬂuid ﬂow via anaxial force balance. This approach was followed by severalauthors
16,24,25
and was originally proposed by Janssen in1895.
26
The one-dimensional, cross-sectionally averagedaxial force balance neglecting the inﬂuence of gravity is
@
r
@
z
52
@
p
@
z
2
4
d
col
l
f
m
1
2
m
r
(6)
Hence, an increase of solid stress
r
acting on the chroma-tography matrix along the axial direction
z
is caused by aloss of ﬂuid pressure along the
z
-direction. The second term
Figure 2. Depiction of Kelvin-Voigt viscoelastic behavior.
The numbers refer to the different phases of the strain responseto an applied rectangular stress step. (A) Applied rectangular stress function. (B) Resulting strain function. (C) Strain/stresshysteresis. The time trajectory is depicted by the arrows.
960
Biotechnol. Prog.,
2013, Vol. 29, No. 4
on the right-hand side of this equation accounts for wall fric-tion.
16
d
col
is the column diameter,
g
f
is a friction coefﬁcient,and
m
is Poisson’s ratio (deﬁned as the ratio of transverse toaxial strain when an object is compressed). Wall frictionleads to a reduction of solid stress and allows for higher ﬂow rates. Therefore, this effect is called wall support.According to Stickel and Fotopoulos, the relation of bedporosity and bed compression is
27
e
5
e
0
2
k
1
2
k
(7)
where
e
0
is the gravity settled bed porosity. The continuityequation of a deformable packing of porous particles with avolume
V
was formulated according to Shadday
25
V
@
u
@
z
52
@
ð
e
V
Þ
@
t
2
e
p
@
½ð
1
2
e
Þ
V
@
t
(8)
The left-hand side of Eq. 8 is the convective term. Whenthe packing is compressed, the incompressible ﬂuid isremoved from the decreasing interstitial volume of the pack-ing and from the decreasing volume of the particle pores.For reasons of simplicity, the intraparticle porosity
e
p
isassumed to be constant and independent of bed stress. Whenthe packing is relaxed, ﬂuid accumulates in the increasinginterstitial volume and in the increasing volume of the parti-cle pores. Accordingly, the two terms on the right-hand sideof Eq. 8 are the interstitial and pore volume accumulationterms, respectively.The boundary conditions of the mathematical model arethe transient time functions of the applied stress, i.e. thepressure drop, on the chromatography packing. The modelcomprises up to eight model parameters which are: meanparticle diameter
d
p
, gravity settled height of the packing
h
0
,gravity settled bed porosity
e
0
, the parameters
E
and
g
describing the viscoelasticity of the chromatography par-ticles, the wall friction coefﬁcient
l
f
, the Poisson’s ration
m
,and the intraparticle porosity
e
p
. The mathematical modelingapproach distinguished between: (1) A homogeneous basicmodel which underlies the assumption that the bed compactsand relaxes uniformly along the height. Wall support andintraparticle porosity were not taken into consideration. Thehomogeneous model can be envisioned as a simple ﬁnite dif-ference scheme with only one cell and two nodes. For thenumerical implementation, Eqs. 1 and 6 without the secondright-hand term were discretized. (2) An inhomogeneousmodel that is able to calculate characteristic bed variableslike stress, strain, and bed porosity as a function of bedheight. Optionally, wall support or intraparticle porosity wastaken into consideration. For inhomogeneous modeling, thegoverning equations have to be discretized spatially in theaxial direction. The packed bed domain was divided intonodes and cells for spatial discretization (the space betweentwo nodes is the cell). Hundred nodes were chosen to ensurea stable numerical solution. An
Euler forward
scheme wasused in the spatial domain, the time domain was treatedexplicitly. A deformable mesh was used for the ﬁnite differ-ence models. A similar procedure was previously applied byShadday.
25
Initially, i.e. prior to any deformation, all cellshave uniform widths. Local solid phase deformation leads toan equal deformation of individual cells and no solid phasecrosses the surface of a control volume. Therefore, the masswithin each control volume stays constant over time. Thisprocedure had the advantage that the solid phase did nothave to be balanced by an additional equation. The modelswere implemented in Matlab (version: R2012a, supplier:MathWorks, Natick, MA, USA).
Results and Discussion
Characterization of the stationary phases
The measured particle size distributions of both media aregiven in Figure 3. The particle size distribution of resin CAPwas narrower than the one of resin SEP. The measured parti-cle ranges and mean particle sizes of both media wereslightly below the data provided by the vendor.The results of the relaxation experiments in order to deter-mine the viscoelastic parameters Young’s modulus
E
anddamping constant
g
for both media are given in Figure 4.From the ﬁtted curves of Figure 4, the viscoelastic parame-ters given in Table 1 were obtained directly without the needof a regression analysis as it was the case in previouswork.
18
Other parameters that characterize the resins are Poisson’sratio
m
, wall friction coefﬁcient
l
f
, and intraparticle porosity
e
p
. These parameters were taken from literature and averaged(Table 2). For Poisson’s ratio
m
, a mean value of 0.3 wasdetermined.
12,15,25
The wall friction coefﬁcient
l
f
was set to0.2.
15
Intraparticle porosity
e
p
values vary from 0.5 to 0.85,depending on the type of resin and the ionic strength of themobile phase.
1,28
A mean value of 0.7 was chosen as wasassumed by Shadday.
25
The pressure drop as a function of the superﬁcial velocityis given in Figure 5 for the empty column and the columnpacked with each of the two chromatography media with
h
d
col
(data for empty column and resin SEP extracted fromHekmat et al.
8
. As can be seen, the pressure drop course of
Figure 3. Particle size distributions of the chromatographymedia.
(A) Resin SEP. (B) Resin CAP.
Biotechnol. Prog.,
2013, Vol. 29, No. 4
961
the newly developed resin CAP is almost linear and com-pared to resin SEP, no critical superﬁcial velocity isapproached within the pressure drop range of up to 6 bar.Hence, as expected, the newly developed resin CAP witha smaller mean diameter exhibits somewhat larger pressuredrops at lower ﬂow rates compared to resin SEP. However,low ﬂow rates are disadvantageous in order to obtain highseparation yields. On the other hand, resin CAP enablesadvantageously high ﬂow rates at higher pressure drops.
Comparison of modeling results with experimental data
Modeling results were compared with experimental datathat were taken from the ﬁrst simulated separation cycles of long-term experiments. For all model calculations, the vis-cosity
l
of the mobile phase was kept constant at 1 mPa sand the column inner diameter
d
col
was 140 mm. The error because of the fact that the hydraulic head was not consid-ered in the modeling was estimated to be less than 0.5% of the applied pressure drop. Therefore, the inﬂuence of thehydraulic head was neglected.
Homogeneous Model
. The results for resin SEP using bothmodeling approaches assuming the resins have viscoelasticor predominantly elastic properties are given in Figure 6.The parameters of the viscoelastic model were:
E
5
7.0
3
10
6
N m
2
2
,
g
5
2.0
3
10
7
N s m
2
2
,
d
p
5
90
l
m,
e
0
5
0.26, and
h
0
5
133 mm. The parameters of the elastic modelwere identical except for the damping constant
g
, which wasreduced by a factor of 500. The relatively low value of
e
0
5
0.26 was attributed to the fact that the resin SEP has a broadparticle size distribution. Therefore, the gravity settled poros-ity was expected to be lower than the estimated value of
e
0
5
0.32 using the mean particle diameter.
8
This is because of the fact that small particles are likely to occupy the intersti-tial space between larger particles and therefore, the porosityis reduced.
23
According to Yang,
29
the value of
e
0
5
0.26was approximated from the consideration of a binary mixtureof spheres from the lower end (
d
p
5
50
l
m) and the upper end (
d
p
5
130
l
m) of the Gaussian-type distribution givenin Figure 3A.As can be seen, a fair agreement between model calcula-tions and experiment was obtained. The viscoelastic modelpredicted an overshoot of the ﬂow rate, which was larger than the measured course (Figure 6B). Similar ﬂow rateovershoots were obtained in previous experiments.
8
Theovershoots can be explained by the viscoelastic behavior of the chromatography bed. The ﬂow-induced compression of the packed bed lags behind the change of pressure drop.This leads to a situation where the applied pressure dropalready is at the ﬁnal value whereas the bed is not yet fullycompressed. Hence, the pressure drop is temporarily toohigh for the current state of the chromatography bed result-ing in an elevated ﬂow rate compared to the equilibrated bedstate. As expected, the elastic model did not predict an over-shoot of the ﬂow rate. A fair agreement between model cal-culations and experiment of the transient bed height wasobtained (Figure 6C). The viscoelastic model gave a rather smooth course which was closer to the experimental data.The elastic model gave a more trapezoidal course. The cal-culated compression–relaxation hysteresis using the visco-elastic model was less bulky and the slope of the hysteresiswas smaller compared to the experiment (Figure 6D). How-ever, considering the simplicity of the present model, the
Table 1. Viscoelastic Parameters Young’s Modulus
E
and DampingConstant
g
and Corresponding Conﬁdence Intervals for Both Media
Young’s Modulus
E
(N m
2
2
)Damping Constant
g
(N s m
2
2
)Resin SEP 7.0
3
10
6
6
5% 2.0
3
10
7
6
5%Resin CAP 1.4
3
10
7
6
7% 4.0
3
10
7
6
7%
Figure 4. Time course of bed compression during relaxation experiments.
Gray data points: averaged experimental data (from 5 to 10 relaxation curves). Black curves: least squares curve-fit. (A) Resin SEP,
r
0
5
2 bar,coefficient of determination
R
2
5
0.967. (B) Resin CAP,
r
0
5
6 bar, coefficient of determination
R
2
5
0.931.
Table 2. Average Parameters Poisson’s Ratio
m
, Wall Friction Coefﬁ-cient
l
f
, and Intraparticle Porosity
e
p
Poisson’s Ration
m
Wall FrictionCoefﬁcient
l
f
IntraparticlePorosity
e
p
0.3 0.2 0.7
Figure 5. Column pressure drop as a function of the superfi-cial velocity for the 140 mm column packed with
h
d
col
.
The critical superficial velocity of resin SEP is 570 cm h
2
1
(dashed line).
962
Biotechnol. Prog.,
2013, Vol. 29, No. 4

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