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Computers and Chemical Engineering 55 (2013) 148– 157

Contents lists available at SciVerse ScienceDirect

Computers and Chemical Engineering

jo u r n al homep age : www.els evier .com/ locate /compchemeng

ffects of uncertainties in experimental conditions on the estimationf adsorption model parameters in preparative chromatography

iklas Borga, Karin Westerberga, Niklas Anderssona, Eric von Lieresb, Bernt Nilssona,∗

Department of Chemical Engineering, Lund University, Lund, SwedenInstitute of Bio- and Geosciences, IBG-1: Biotechnology, Forschungszentrum Jülich, Jülich, Germany

a r t i c l e i n f o

rticle history:eceived 8 October 2012eceived in revised form 13 March 2013ccepted 2 April 2013vailable online 6 May 2013

a b s t r a c t

Model-based process design is increasingly popular when designing pharmaceutical purification pro-cesses. The effect of uncertainties in concentration measurements on the estimation of model parametersis analyzed for two cases of non-isocratic adsorption chromatography. A model, calibrated to experiments,is used to generate data by adding a Monte Carlo sampled error in the inlet concentrations. New modelparameters are estimated by minimizing the deviation between the synthetic data and the model. Thefirst case is a separation of rare earth elements by ion-exchange chromatography and the second case is

eywords:odeling

on-exchange chromatographyeversed-phase chromatographyodel calibration

a purification of insulin from a product-related impurity by reversed-phase chromatography. It is shownthat normally distributed errors in the concentrations result in deviations in the UV-signal that are notnormally distributed. With the applied method, known concentration distributions can be translated intoprobability distributions of the model parameters, which can be taken into account in the model-based

arameter estimationarameter uncertainty

process design.

. Introduction

Preparative chromatography is an important separation tech-ique in the pharmaceutical, biopharmaceutical and food indus-ries (Guiochon, 2002). It is essential to find suitable operatingonditions to ensure the purity of the product and to achieveigh yields. Finding such operating conditions experimentally isime-consuming and labor-intensive, even with the available high-hroughput techniques that can increase the amount of availablexperimental data (Treier, Lester, & Hubbuch, 2012).

Modeling is an effective tool for the analysis and designf preparative chromatography processes (Guiochon & Beaver,011) and several examples of model-based process design foriopharmaceutical applications have been published, see for exam-le Kaltenbrunner, Giaverini, Woehle, and Asenjo (2007), Shene,ucero, Andrews, and Asenjo (2006), and Karlsson, Jakobsson,xelsson, and Nilsson (2004). The models applied here consist ofartial differential equations describing the most important phys-

cal and chemical properties of the system. The concentrationrofiles in the axial column dimension and in the radial particleimension are the driving forces for the mass transport, while the

dsorption is driven by the equilibrium between the concentrationn the dissolved phase and on the solid phase inside the parti-les. Common assumptions made when modeling chromatography

∗ Corresponding author. Tel.: +46 462228088.E-mail address: bernt.nilsson@chemeng.lth.se (B. Nilsson).

098-1354/$ – see front matter © 2013 Elsevier Ltd. All rights reserved.ttp://dx.doi.org/10.1016/j.compchemeng.2013.04.013

© 2013 Elsevier Ltd. All rights reserved.

are, for example, that there is no concentration gradient in theradial column dimension, and that all stationary phase particles areidentical in morphology and size. These assumptions are generallyconsidered to introduce negligible errors to the models for columnsat lab-scale. Adsorption on the stationary phase is described by anequilibrium or kinetic model including parameters for the adsorp-tion’s dependency on aqueous concentrations and the propertiesof the solvent and the stationary phase.

Aside from any error caused by the model assumptions, theparameters in the model have to be estimated and are thusattributed with an uncertainty. The process model is normally cal-ibrated to lab-scale experimental data, to reproduce the behaviorin the studied system. A popular method to estimate the modelparameters is the inverse method (Felinger, Alberto Cavazzini, &Guiochon, 2003; James, Sepúlveda, Charton, Quinones, & Guiochon,1999; Seidel-Morgenstern, 2004; Zhang, Selker, Qu, & Velayudhan,2001). In the inverse method, experimental elution profiles areused to estimate the model parameters by minimizing the discrep-ancies between simulated elution profiles and the experimentaldata (Dose, Jacobson, & Guiochon, 1991). A least squares minimizer,using a quasi-Newton approach is often used to perform this esti-mation. While it is sensitive to the initial guess due to local minima,it requires relatively few simulations.

The inverse method is useful for fitting a simulation model

which reproduces experimental data within the calibrated range. Itdoes however require a priori selection of the model structure andthat the properties of the experimental system are well known.Erroneous assumption of the experimental system properties may

N. Borg et al. / Computers and Chemical

Nomenclature

Methode experimentmp model with true parameters pmpi model with estimated parameters pin normalizing functionoe operating conditions in experimentp true parameterspi estimated parameters, sample ix true concentrations [mol m−3]x estimated concentrations [mol m−3]εx,i error in estimated concentration [mol m−3]ye experimental dataym,p model data using true parameters pym,pi

model data using estimated parameters pi

ys,i synthetic data, sample i

State variablesci desorbed concentration of component i [mol m−3]cs desorbed concentration of salt ion [mol m−3]ct,i concentration of component i in tank [mol m−3]ct,i,in inlet concentration of component i in tank

[mol m−3]qi adsorbed concentration of component i [mol m−3]qmax,j maximum adsorbed concentration of competing

component j [mol m−3]t time [s]xε volume fraction of ethanolz axial coordinate [m]

Model parametersDax dispersion coefficient [m2 s−1]Hi Henry coefficient in MPM of component iKeq,i equilibrium coefficient in SMA of component iNc number of competing componentskkin,i kinetic coefficient in of component i

[m3� mol−� s−1]SMA [s−1]MPMui linear velocity of component i [m s−1]� i solvophobicity coefficient of component iεc interstitial porosityεp,i apparent porosity of component ivi characteristic charge of competing component j�j shielding factor of competing component j� ligand concentration in particle [mol m−3]� residence time in mixing tank [s]

Calibration methodRSS residual sum of squaresNexp number of experimentsNd,i number of data points in experiment iUVexp,j,i experimental UV data point j in experiment i [mAU]UVsim,j,i simulated UV data point j in experiment i [mAU]||UVexp,i|| norm of experimental UV trace in experiment i

[mAU]

Retention correlationVR,i retention volume of component i [m3]

iaSt

VNA,i dead volume of component i [m3]Vcol volume of column [m3]

ntroduce errors in the estimated model parameters. Sajonz (2004)nd Samuelsson, Zang, Murunga, Fornstedt, and Sajonz (2008);amuelsson, Fornstedt, and Sajonz (2008) have analyzed the quan-itative effect of an error in the assumed hold-up time in the

Engineering 55 (2013) 148– 157 149

column on the Langmuir adsorption model parameters. By cal-ibrating a model with erroneous hold-up time to a simulationwith a true hold-up time, it was found that the erroneous hold-uptimes resulted in errors in estimated adsorption model parameters.Samuelsson, Fornstedt, et al. (2008) also showed the qualitativeeffect that a different adsorption model could give a better fit thanthe true adsorption model when an erroneous hold-up time wasused. Other uncertainties in the model parameters can be an effectof variations in the experimental data. Joshi, Kremling, and Seidel-Morgenstern (2006) used synthetic adsorption equilibrium datato quantitatively determine the effects of concentration measure-ment error on adsorption model parameters. The synthetic datawas generated by adding a Monte Carlo sampled absolute error toexperimental equilibrium data. In Zhang et al. (2001), the quan-titative effect of detector disturbance on the parameters of themulti-component Langmuir adsorption model was studied. Thenumber of replicates needed to estimate the adsorption parameterswithin a specific confidence interval was determined by calibrat-ing the model to data from simulations with an added Monte Carlosampled relative error on the detected UV-absorbance signal.

The variations in experimental data do not necessarily appearas a bias in the detector. Hibbert, Jiang, and Mulholl (2001) ana-lyzed the deviation of the UV-signal in analytical chromatographyby adding a Monte Carlo sampled error to the experimental con-ditions in simulations. This showed that deviations caused bythe experimental conditions, and not a noisy detection, can nei-ther be modeled by an absolute, nor a relative random error. Inorder to understand the effect experimental errors have on theestimation of model parameters, this kind of the error must beanalyzed.

Although the studies mentioned above are performed for iso-cratic conditions, preparative chromatography processes are oftennon-isocratic. Vivó-Truyols, Torres-Lapasió, & Garcııa-Alvarez-Coque (2003) studied the effect of concentration uncertainty onexplicit linear and quadratic retention model parameters for non-isocratic analytical conditions, but the effect of errors in theconcentration measurements on the fitted model parameters ofcolumn models and for preparative conditions has not been studiedbefore.

In this study we analyze the quantitative effect of uncertaintyin the experimental conditions on the estimated chromatographymodel parameters for a non-isocratic, preparative process. A sim-ulation model is used to generate synthetic data with Monte Carlosampled experimental conditions, resulting in non-normally dis-tributed errors in the elution profiles. The inverse method is thenused to estimate the model parameters by fitting the model to thesynthetic data, resulting in a distribution of the estimated modelparameters.

Two cases are studied: Case 1 is the separation of rare earthelements (REEs) by ion-exchange chromatography. The model wascalibrated to experimental data that has been presented by Ojala,Max-Hansen, Kifle, Borg, & Nilsson (2012). The experimental con-ditions studied are the total concentration of rare earth elements,the concentrations of acid in the buffers, and ligand density in thestationary phase. Case 2 is the removal of a product-related impu-rity from insulin by reversed-phase chromatography. Westerberg,Borg, Andersson, & Nilsson (2012) have previously described thismodel. The experimental variables studied in Case 2 are the con-centration of ethanol, the concentration of protein, and the purityof the feed. The general outline of the method is presented first,followed by the experimental and computational details of the twocases. The effects of the experimental errors on the simulated chro-

matogram and on the estimated parameters are presented next,followed by a discussion of the implications for model calibra-tion and the application of models in the design of preparativechromatography.

1 emical Engineering 55 (2013) 148– 157

2

2

(((

Wfuip

tymeteebilm

e

m

m

ebtt

Msamea

m

m

m

etc

2

tww

Table 1Experiments in Case 1, separation of REE.

Experiment 1 2 3 4

∂ci

∂t= −ui

∂ci

∂z+ Dax

∂2ci

∂z2− 1 − εc

εc + (1 − εc)1 − εp,i

∂qi

∂t(7)

Table 2Experiments in Case 2, purification of insulin.

50 N. Borg et al. / Computers and Ch

. Method

.1. Estimation of model parameter uncertainties

Model-based process design usually involves three main steps:

1) Performing experiments2) Calibrating the model to the experimental data3) Application of the model to for example process analysis or

design.

The uncertainties in model parameters are estimated in Step 2.e present here a method to estimate such uncertainties that arise

rom uncertainties in the experimental conditions. The methodses the estimated model parameters and the estimated exper-

mental uncertainties to determine the uncertainties in modelarameters.

Experiments, e, are performed under certain operating condi-ions, oe, and at certain experimental conditions, x, to give a result,e. We define operating conditions as settings sent to our equip-ent during the experiments, such as volumes and flow rates, and

xperimental conditions as measured properties: concentrations,emperature and pH. We use the model, m, with a set of param-ters, p, at the same operating conditions, oe, and the measuredxperimental conditions, x, to model this. The model is calibratedy estimating a set of p such that the difference between the exper-

mental data and the model result, ym, is minimized in the sense ofeast squares after normalization, n, which is known as the inverse

ethod.

(x, oe) = ye (1)

p(x, oe) = ym,p (2)

inpi

((n(ye) − n(ym,p))T (n(ye) − (n(ym,p))) (3)

We assume that the measured experimental conditions and thestimated model parameters represent the actual experimentalehavior accurately, and this allows us to determine the uncer-ainties in estimated parameters, εp, based on the uncertainties inhe experimental conditions.

Other possible experimental conditions are generated using theonte Carlo method. These conditions are centralized at the mea-

ured conditions, x, and are subject to an additional error, εx,i. Theyre simulated by the calibrated model to give synthetic data, ys,i. Theodel is calibrated again by estimating new p such that the differ-

nce between the synthetic data and the model result is minimizedfter normalization.

p(x + εx,i, oe) = ys,i (4)

pi(x, oe) = ym,pi

(5)

inpi

((n(ye) − n(ym,pi))T (n(ye) − (n(ym,pi

))) (6)

Each sample, i, of the experimental conditions gives a sample ofstimated of the parameters. The distribution of model parameterhat is obtained in this way from a sufficient number of samplesan be correlated with the experimental uncertainty.

.2. Experimental setup, Case 1: separation of REE

The experiments were performed on an Agilent 1200 HPLC sys-em (Santa Clara, CA, USA) connected to a UV detector. The columnas a 5 �m Eclipse XDB-C18 (Hewlett-Packard, Newport, DE, USA),ith a length of 150 mm, an inner diameter of 4.6 mm, and packed

End of elution gradient 25% 50% 75% 100%

with particles with a diameter of 5 �m. To the column, 0.336 mmolion-exchange ligands were added. We assumed that 95% of theligands were bound, giving a ligand concentration of 200 mM inthe particles. A solution of arsenazo III of concentration 0.1 g/l wasadded at a flow rate of 0.8 ml/min between the column and theUV detector. Arsenazo III creates metal–ion complexes that canbe detected with the UV detector. Two solutions of nitric acidwere prepared at different concentrations, 7 mM (Solution A) and1000 mM (Solution B). A mixture of REEs dissolved in acid (7 mMnitric acid, 3.47 mM Nd, 3.33 mM Sm, 3.29 mM Eu and 3.18 mM Gd)was prepared and 50 �l of this was injected into the column. An elu-tion gradient of nitric acid (20 ml) obtained by mixing Solution Awith Solution B was applied to elute the components. Four exper-iments were performed. The elution gradient was ended at 25%Solution B in the first experiment, at 50% in the second, at 75% inthe third, and at 100% in the fourth, Table 1. The experiments aredescribed in detail in Ojala et al. (2012).

2.3. Experimental setup 2: purification of insulin

These experiments were performed on an ÄKTA Purifier 100 (GEHealthcare, Uppsala, Sweden) connected to a UV detector. The col-umn used was a Kromasil C8-100-10 (Eka Nobel, Bohus, Sweden)reversed-phase column, with a length of 250 mm and an innerdiameter of 4.6 mm, packed with particles of diameter 10 �m. Twobuffers were prepared consisting of 200 mM ammonium acetateadjusted to pH 4 with acetic acid and ethanol (EtOH) to give a massfraction (w/w) of 0.1 EtOH (buffer A) and 0.4 w/w EtOH (buffer B). Amixing tank with the volume 0.1 ml was installed before the columnto ensure a stable concentration of EtOH after a step-wise increaseof buffer B.

A protein solution of concentration 6 mg/ml, composed of 93.7%insulin and 6.3% impurity, was injected into the column. The molarmass of both insulin and the impurity was 5808 g/mol. The compo-nents were eluted by a step-wise increase in the concentration ofethanol, achieved by mixing buffer A and buffer B. Six experimentswere performed with different injection volumes and differentstep-wise increases in ethanol concentration, Table 2. The experi-ments are described in detail in Westerberg et al. (2012).

2.4. Model formulation

2.4.1. Mass transferThe two processes studied are described by different isotherm

models since the adsorption mechanisms differ. However, weassume an infinitely fast diffusion into the particles in both cases.The transport of components through the column is thereforedescribed by the same convective–dispersive model (Guiochon,Felinger, Shirazi, & Katti, 2006) in both cases, Eq. (7).

Experiment 1 2 3 4 5 6

Injection volume (ml) 0.5 0.5 0.5 1 3 5Elution (%B) 64 61 59 59 59 59

N. Borg et al. / Computers and Chemical Engineering 55 (2013) 148– 157 151

Table 3Experimental uncertainties in Case 1.

Process parameter Total concentration ofREE in sample

Acid concentration inSolution A

Acid concentration inSolution B

Ligandconcentration

Set point 13.28 mM 7.0 mM 1000 mM 200 mM

to 21

a

c

2

cb

fi

c(2

2

wApchuc

2

tn

TE

rsd 1% 5%

a The bound ligands were sampled with a uniform probability function from 189

∂ct,i

∂t= (ct,i,in − ct,i)

�(8)

In Case 2, purification of insulin, a buffer mixing tank is describeds ideally mixed with known inlet concentrations, Eq. (8).

The concentration in the mixing tank is then used as the inletoncentration in column.

.4.2. AdsorptionThe adsorption of the rare earth elements in the ion-exchange

olumn is described by a steric mass action model (SMA), proposedy Brooks and Cramer (1996), Eq. (9).

∂qi

∂t= kkin,i

⎛⎝keq,ici

⎛⎝ −

∑j=1:Nc

(vj + �j)qj

⎞⎠

vi

− qicvis

⎞⎠ (9)

In the model presented here, the shielding factor, �, is set to 0or all components. The model also requires that electro-neutralitys maintained, Eq. (10).

∂cs

∂t= − 1 − εc

εc + (1 − εc)εp,s

∑i=1:Nc

vi∂qi

∂t(10)

The adsorption of insulin and the impurity in the reversed-phaseolumn is described by the Langmuir mobile phase modulatorsMPM) model (Karlsson, Jakobsson, Brink, Axelsson, & Nilsson,004), Eq. (11).

∂qi

∂t= kkin,i

⎛⎝Hie

�ixe ci

⎛⎝1 −

∑j=1:Nc

qj

qmax,j

⎞⎠− qi

⎞⎠ (11)

.4.3. Initial and boundary conditionsAt initial time t = 0, all mobile phase concentrations in Case 1

ere the same as that of Solution A, and in Case 2, that of buffer. The stationary phase concentrations were set to 0 for all com-onents except the salt ions, which were set to the maximumoncentration, �. The outlet of the column was simulated by aomogeneous von Neumann condition, while the inlet of the col-mn was simulated by a Robin condition in Case 1, and a Dirichletondition (the concentration in the tank) in Case 2.

.5. Calibration of the model to experimental data

In Case 1, separation of REE, the inverse method was used to findhe model parameters, v and Dax, which are common to all compo-ents and Keq, which is specific to each component, that minimize

able 4xperimental uncertainties in Case 2.

Process parameter Protein concentrationin sample

Impurity insample

Set point 6 ml 6.3%

rsd 2.5% 7.7%

1% 189–211 mMa

1 mM.

the residual sum of squares (RSS) between the experimental dataand the model.

RSS =Nexp∑i=1

Nd,i∑j=1

(UVexp,j,i − UVsim,j,i

||UVexp,i||√

Nexp

)(12)

The reversed-phase model parameters H, � and qmax used in Eq.(11) in Case 2, purification of insulin, were estimated by Westerberget al. (2012).

2.6. Creation of synthetic data sets

Samples of experimental conditions were generated using theMonte Carlo method. These values were used to simulate syntheticexperiments. The number of samples was set to 100, in order toobtain sufficient data points to provide a distribution of the modelparameters estimated from these synthetic datasets.

In Case 1, the acid concentrations and the total amount of REEwere sampled from normal distributions, while the distribution ofthe amounts of bound ligands was assumed to be uniform. Theexperimental parameters included in the study were: total concen-tration of REEs in the sample, acid concentration in Solution A, acidconcentration in Solution B and the fraction of ligands that werebound. Table 3 gives mean values and relative standard deviations(rsds). The relative concentrations of the REEs were assumed to beconstant.

In Case 2, purification of insulin, the total concentration of pro-teins in the sample and the ethanol concentration in the sample, inbuffer A and in buffer B were sampled with normal distributions.Mean values and rsds in the experiments are given in Table 4.

One hundred datasets were simulated for both Case 1 and Case2 using the model parameter values which were obtained in thecalibration to experimental data. The experimental error was thesame for all experiments in each data set.

2.7. Calibration of model to synthetic data sets

The model parameters were estimated to each synthetic datausing the original values of the assumed concentrations. Theinverse method was used to minimize a weighted residual squaresum for both cases and the residual was calculated using Eq.(12), previously used for the calibration to experimental data. Theparameters Keq for each component, and v and Dax for all compo-nents were estimated in Case 1. The parameters H and � for eachcomponent and qmax for both components were estimated in Case 2.The mean value and the variance of each parameter were calculated

from the resulting distributions of model parameter values.

When the model is calibrated to simulated data, all lack of fitdue to an erroneous model structure is removed, and the effectsof the experimental errors on the estimated parameters can be

EtOH insample

EtOH inbuffer A

EtOH inbuffer B

0.1 w/w 0.1 w/w 0.1 w/w0.5% 0.5% 0.5%

152 N. Borg et al. / Computers and Chemical Engineering 55 (2013) 148– 157

Table 5Parameter values and standard deviations in Case 1.

Parameter Keq,Nd Keq,Sm Keq,Eu Keq,Gd v Dax

Original model 1.68 × 10−3 3.37 × 10−2 0.14 −7

Mean 1.78 × 10−3 3.58 × 10−2 0.15� 3.1 × 10−4 6.2 × 10−3 2.6 ×

Table 6Parameter values and standard deviations in Case 2.

Parameter H1 H2 �1 �2 qmax

Original model 3.547 4.31 −40.78 −41.80 35.0

sgeicoa

3

3

wmmfm(

Ft

Mean 3.560 4.32 −40.77 −41.72 35.1� 0.091 0.12 0.21 0.69 1.2

tudied. By adjusting the initial values of the minimization, conver-ence to a point very close to the true minima could be obtained forach sample. Given that the model reproduces the studied behav-or well enough and that the minimization converges to a pointlose to the true minima, the effect of concentration uncertaintyn estimated model parameters in a calibration to experiments isccurately reproduced.

. Results and discussion

.1. Calibration of the model to experimental data

In Case 1, separation of REEs, the parameters in the SMA modelere adjusted to give a simulated UV absorbance signal thatatched the experimental one as closely as possible. The fitted

odel parameter values are listed in Table 5. The model parameters

or Case 2, purification of insulin, were not estimated from experi-ental data in this work. The parameters found in Westerberg et al.

2012) were used, see Table 6.

ig. 1. Results of calibration. Solid lines show simulations, and dashed lines experimentao 75% Solution B. (d) Gradient to 100% Solution B.

1 0.359 2.902 1.741 × 100 0.381 2.904 1.741 × 10−7

10−2 6.6 × 10−2 8.3 × 10−2 2.5 × 10−10

Fig. 1 shows the fit of the model calibrated for Case 1. The firstand last peaks have noticeable tails in all experiments, the secondpeak has a less pronounced tail, and the third peak may or maynot have such a tail. These tails were not reproduced by the chosenmodel. The imperfect fit may be due to errors in the experimentsor simplifications made in constructing the model, as discussed inthe Introduction. Further, the model predicts wider peaks than theexperimental peaks. This is a systematic error that is expected fromleast squares calibration, since the sum of squares of many lowresidual points is lower than the sum of squares of a few high resid-ual points, even when the area under the UV absorbance graph isthe same. The least squares method causes the peaks predicted bythe model to be wider than those in the experimental data if thepositions do not match perfectly.

3.2. Creation of synthetic data sets

Fig. 2 shows the UV absorbance signal from all the syntheticdata sets with Monte Carlo sampled experimental errors togetherwith the nominal operating point. The peak heights in the syntheticdata sets for Case 1 match the experimental peak heights well, butthe retention volumes of the components differ. This is similar to

results presented by Hibbert et al. (2001). In an individual data set,the retention volumes of all components are changed alike relativeto the nominal value as an effect of the experimental error. Thepeaks seem to overlap when all UV traces are presented together,

l results. (a) Gradient to 25% Solution B. (b) Gradient to 50% Solution B. (c) Gradient

N. Borg et al. / Computers and Chemical Engineering 55 (2013) 148– 157 153

Fig. 2. Predictions for Case 1, separation of REE, from the 100 synthetic data sets. The bold black line shows the prediction of the model with the set point concentrations.( olutio

bcsstwt

iecttsswi

3

oCebp�

r(2c

a) Gradient to 25% Solution B. (b) Gradient to 50% Solution B. (c) Gradient to 75% S

ut the peaks in any one UV trace do not, in fact, overlap. We con-lude that the degree of separation achieved is not affected by thetudied experimental errors. The results for Case 2 are similar, aseen in Fig. 3. Here too the retention volume was affected more thanhe height of the peaks, but since the product and impurity peaksere closer together the separation was also changed, in contrast

o Case 1.The simulations presented in Figs. 2 and 3 show that the pos-

tions of the peaks depend more strongly on the experimentalrrors than do the widths and heights of the peaks. The simulatedhromatograms were normalized to give the same total area ashose from the experiments, and thus the widths and heights ofhe peaks are correlated. The synthetic data sets used here differignificantly from those obtained by adding noise to the detectorignal (Joshi et al., 2006 and Zhang et al., 2001). The signal coincidesith the base line far from the peaks, which would not be the case

f noise had been added to the signal.

.3. Calibration of the model to synthetic datasets

The model was recalibrated to the synthetic data sets using theriginal values of the assumed concentrations in the two cases. Inase 1, common v and Dax and individual Keq for each REE, werestimated as in the original calibration. Table 5 presents the cali-rated parameter values and their standard deviations and Table 6resents the parameter values and their standard deviations of H,

and qmax for Case 2.Calibration to the experimental data gave much larger

esiduals (RSS = 0.16) than calibrations to the synthetic datamax(RSS) = 3.6 × 10−5 in Case 1 and max(RSS) = 5.3 × 10−4 in Case). The larger residuals when calibrating to the experiments areaused by a non-perfect model structure. The very low residuals

n B. (d) Gradient to 100% Solution B.

show that the effect of the error in the concentrations is compen-sated for by the estimated model parameters. Experimental errorsof this type are therefore not detected as a lack of fit in the param-eter estimation procedure.

3.4. Estimated model parameter correlations

The correlations between uncertainties in experimental condi-tions and uncertainties in model parameters were calculated fromthe estimated parameter samples, as were parameter to param-eter correlations. Fig. 4 shows the estimated model parametersplotted against the experimental parameters in each data set andagainst each other for Case 1. The correlation coefficients betweenthe ligand concentration (�) and the value of each Keq are high(greater than 0.88). This correlation can be expected, as the reten-tion with low load is a function of Keq, v and �, as shown by Eq. (13)(Pedersen, Mollerup, Hansen, & Jungbauer, 2003).

VR,i − VNA,i = Vcol(1 − εc)εp,iKeq,i

(

cs

)vi

(13)

All components behave in the same way when the ligand con-centration and the hydrogen ion concentrations vary, which givesa very high correlation coefficient (greater than 0.99) betweeneach pair of Keq. The SMA model for ion exchange chromatogra-phy includes the effect of overloading of the column. However, allexperiments were performed at low loads of metal ions why thiseffect did not influence the results.

Three additional observations can be drawn from Fig. 4: firstly,

v was strongly correlated to the concentration of acid in Solution A.We expected to see a strong correlation with the acid concentra-tion in Solution B, not A, since the absolute uncertainty of the acidconcentration in this solution was much higher. However, since

154 N. Borg et al. / Computers and Chemical Engineering 55 (2013) 148– 157

Fig. 3. Predictions for Case 2, purification of insulin, from the 100 synthetic data sets. The bold black line shows the prediction of the model with the set point concentrations.(a) 0.5 ml load, elution with 64% buffer B. (b) 0.5 ml load, elution with 61% buffer B. (c) 0.5 ml load, elution with 59% buffer B. (d) 1 ml load, elution with 59% buffer B. (e) 3 mlload, elution with 59% buffer B. (f) 5 ml load, elution with 59% buffer B.

acoAbswwvTmrd

dcuibibaEiil

ll of the components elute while the elution buffer was mostlyomposed of Solution A, the effect of this solution was greater. Sec-ndly, Dax was correlated to the concentration of acid in the Solution. However, Dax is highly correlated to v, and thus the correlationetween it and the concentration of acid in Solution A is a con-equence of this, rather than a direct correlation. The peaks areider when v is low, Dax thus decreases to compensate for thisidening, sharpening the peaks to the correct width. Finally, the

alues of Keq for the different components are highly correlated.his shows that, even if the exact values of Keq cannot be deter-ined with a relative standard deviation of less than 10%, their

elation to each other, and thus the separation, can be accuratelyetermined.

Fig. 5 shows the correlations between the experimental con-itions and parameter values, and the parameter to parameterorrelations for Case 2. Five conclusions can be drawn from the fig-re: Firstly, the uncertainty in the concentration of EtOH in buffer B

s more important than the uncertainty in concentration of EtOH inuffer A. This can be expected, as the absolute uncertainty is greater

n buffer B, although the relative uncertainty is the same, and moreuffer B than buffer A is used in the elution step. Secondly, H and �re the parameters that are most sensitive to the concentration of

tOH. This allows us to conclude that the elution step is the mostmportant step in the calibration of these parameters. Thirdly, qmax

s affected most strongly by the total amount of protein that wasoaded, which is expected, since qmax determines the capacity of

the column. Fourthly, the correlation between H1 and H2 is high.This correlation is similar to the correlations between Keq for theREEs in Case 1; both are correlated to the ethanol concentration inthe same way. Finally, the correlation between qmax and �2 is muchlower than the correlation between qmax and �1.

The estimation of the adsorption model parameters is influ-enced by the uncertainty in experimental concentrations, butwhich concentration is most sensitive will depend on the case athand and the experimental design. In a well-designed experimentthe retention volume and consequently the retention parame-ters will be most sensitive to the properties of the elution buffer.The parameter describing the overloading will be sensitive to theamount of protein or ions loaded on the column and thus the con-centration of the sample.

The method used in this study can be used to find the corre-lations between measurement uncertainty and model parameteruncertainty, but if the purpose is limited to identifying the mostsensitive experimental parameters the computational effort maybe too large. However, the non-linear distribution of the modelparameters is found which is useful if the purpose of the model is tobe used for process design. The robustness of a design with respectto the model parameter uncertainty can be found from the model

parameter distributions, and new samples of model parameters canbe drawn if the distributions are approximated by for example amultivariate normal distribution. This was demonstrated in Borg,Westerberg, Schnittert, von Lieres, & Nilsson (2012).

N. Borg et al. / Computers and Chemical Engineering 55 (2013) 148– 157 155

Fig. 4. Correlations between experimental samples and model parameters correlations, and parameter-toparameter correlations with 99% confidence regions for the sepa-ration of REE in Case 1. A high correlation results in a single line, with either positive or negative slope, such as the parameter-to-parameter correlation for when there is nocorrelation the data form ellipses with no slope.

156 N. Borg et al. / Computers and Chemical Engineering 55 (2013) 148– 157

F lationp

4

tuadn

ig. 5. Correlations between experimental samples and model parameters correurification of insulin in Case 2.

. Conclusions

The model could be calibrated with a very low residual, evenhough the assumed concentrations were erroneous. A low resid-

al therefore does not imply that the model adsorption parametersre accurately determined. Results obtained with the syntheticatasets show that uncertainties in experimental conditions doot result in the same errors as noisy signals, the same has been

s, and parameter–toparameter correlations with 99% confidence regions for the

shown for analytical conditions by Hibbert et al. (2001). A fail-ure to determine these experimental conditions accurately willaffect the estimated values of the model parameters when themodel is calibrated to experimental data using the inverse method.

The use of synthetic data eliminated errors due to the detectorand those arising from assumptions made to simplify the model,and allowed the simulation of exact and known experimentalerrors.

emical

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N. Borg et al. / Computers and Ch

Uncertainties in the experimental conditions, once they haveeen determined, can be translated into uncertainties in the esti-ated model parameters and considered in model-based process

esign. In Case 1, the separation of REEs, the uncertainty in themount of ligands bound to the column was strongly correlated tohe equilibrium constant of each element. This correlation showshat the ligand density must be precisely measured in order tobtain accurate estimates of the equilibrium constants. In Case, the purification of insulin, the uncertainty in the ethanol con-entrations in the buffers had the greatest effect on the estimatedarameter values.

Two cases of chromatography are described here, but we expecthat the method employed in this study can be applied to manyynamic process models to determine model parameter distri-utions from experimental uncertainties, which can be used fornalyzing different operating conditions and for designing optimalrocesses in a more robust manner.

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