ORIGINAL PAPER

Model-based design and integration of a two-stepbiopharmaceutical production process

Bruno Otero • Marcus Degerman • Thomas Budde Hansen •

Ernst Broberg Hansen • Bernt Nilsson

Received: 4 October 2013 / Accepted: 11 March 2014

� Springer-Verlag Berlin Heidelberg 2014

Abstract This paper presents the design of a two-step

process in which the first step is PEGylation of a protein,

and the second step is chromatographic purification of the

target mono-PEGylated protein from the unreacted and the

di-PEGylated impurities. The difference between optimiz-

ing each process step separately and optimizing them

simultaneously is studied. It was found that by optimizing

the steps simultaneously up to a 100 % increase in pro-

ductivity could be obtained without reduction in yield.

Optimizing both steps at the same time makes it possible

for the optimization method to take into account that the di-

PEGylated protein is much easier to separate than the non-

PEGylated protein. The easier separation makes it possible

to get a higher yield and productivity at the same time. The

effect of recycling was also studied and the yield could be

increased by 30 % by recycling the unreacted protein.

However, if maximum productivity is required, batch mode

is preferable.

Keywords Process design � Biopharmaceutical

production � Process integration � Optimization

List of symbols

cA0 Initial concentration of non-PEGylated protein in

the reaction

ci Concentration of component i in the mobile phase

(kmol/m3)

cpool Concentration of mono-PEGylated protein in the

pool (kmol/m3)

cS Concentration of PEG (kmol/m3)

CV Column volumes

Dax Apparent dispersion coefficient (m2/s)

dp Bead diameter (m)

k1 Kinetic constant for side-chain attachment forming

mono-PEGylated protein

k2 Kinetic constant for side-chain attachment forming

di-PEGylated protein

kkin,i Adsorption kinetic rate for component i in the

chromatographic separation (s-1)

Hi Henry’s constant

Pe Peclet number

qi Concentration of component i adsorbed onto the

stationary phase (kmol/m3)

qmax,i The maximum adsorption capacity for component

i (kmol/m3)

rS Ratio between substrate and protein at the start of

the reaction

SP Specific productivity [mg/(h mL)]

t Time from the beginning of the simulation (s)

tR Time for the reaction

v Linear velocity of the bulk flow between the

particles

Vcycle Volume of eluent used in a cycle (L)

Vpool Volume of the pool captured in the

chromatographic purification (L)

VR Reactor volume (L)

x Axial location in the column (m)

B. Otero � M. Degerman � B. Nilsson (&)

Department of Chemical Engineering, Lund University,

Box 124, 221 00 Lund, Sweden

e-mail: Bernt.Nilsson@chemeng.lth.se

M. Degerman

e-mail: marcus.degerman@chemeng.lth.se

T. B. Hansen � E. B. Hansen

Novo Nordisk A/S, Bagsværd, Denmark

123

Bioprocess Biosyst Eng

DOI 10.1007/s00449-014-1174-9

Greek symbols

eC Column void

ep,i Particle porosity for component i

eT Total apparent column porosity

bi Exponential salt coefficient for component i

x Ratio between operating cost and feed cost in the

normalized earnings objective function

Introduction

The biopharmaceutical industry is devoting greater efforts

to understand the processes used, in order to make them

more robust, and to improve yields and overall produc-

tivity. Biopharmaceutical companies were encouraged to

focus on process control and analysis by the PAT initiative

of the US Food and Drug Administration [1]. The ICH has

released several publications demanding a better under-

standing of the products and processes [2, 3], and quality

risk management [4]. A typical biopharmaceutical down-

stream process consists of a number of steps, e.g., chro-

matographic purification and filtration and, in some cases,

modification of the target protein. Chromatographic puri-

fication steps have been extensively studied and optimized

[5–19], but little research has been devoted to optimizing

the process as a whole. When each step is optimized sep-

arately, it is highly probable that a suboptimal operating

space will be found. One of the main challenges in opti-

mizing several steps simultaneously is that an experimental

approach will be very difficult to use, and a model of each

process step is required. However, if the models are com-

plex, the simulation time can be very long, and a balance

must therefore be obtained between model complexity and

simulation time.

Optimization of multiple processing steps and complete

processes has been an active research area in chemical

engineering in decades [20] and has been focused on

steady-state and continuous processes resulting in mixed-

integer nonlinear programming problems, MINLP [21, 22].

Optimization of dynamic operation of process systems has

also been studied [23, 24]. A not so extensively studied

area is optimization of batch processes [25, 26] and recent

studies on integrated batch processes [27].

The formulation of the problem, i.e., defining the

objective function, the decision variables, and constraints,

is the first step in the optimization. It is both the most

important and the most difficult part of the process, as the

decisions made here will have considerable effects on the

final result. In many cases, the aim of process optimization

is to optimize the economy, but it is often difficult to define

an economical objective function. A common solution is,

therefore, to create so-called Pareto fronts between pro-

ductivity and yield [14, 28] or between purity and yield

[29], which provide a graphical tool that can be used to

decide how to run the process.

This study is concerned with the PEGylation of a protein

and its subsequent purification. The target is a mono-

PEGylated protein, which is mixed with low amounts of

unreacted protein and di-PEGylated impurities. The reac-

tion considered is enzymatically catalyzed PEGylation, and

ion-exchange chromatography is used to purify the end

product. The process was optimized with and without

recycling of the non-PEGylated protein in an attempt to

increase the process yield. The optimization of each step

separately, and the optimization of both steps simulta-

neously was compared using Pareto fronts between pro-

ductivity and yield.

Theoretical aspects

Modeling

Reaction model

The reaction model for PEGylation consists of three first-

order irreversible reactions, assuming that interactions only

take place between the substrate and each of the

components:

ocA

ot¼ �k1 � cA � cS � cE ð1Þ

ocD

ot¼ k2 � cB � cS � cE ð2Þ

ocBC

ot¼ k1 � cA � cS � cE � k2 � cB � cS � cE; ð3Þ

where the reaction kinetic constants are denoted k, the

concentrations of the components are denoted c, and the

subscripts A, B, D, S, and E refer to the non-PEGylated

protein, mono-PEGylated protein, di-PEGylated protein,

PEG, and the enzyme, respectively.

Chromatography model

Chromatographic purification is described by the disper-

sion and convection in the mobile phase using a reaction

dispersive model [30]:

oci

ot¼ Dax

o2ci

ox2� v

eT

oc

ox� 1� eC

eT

oqi

ot; ð4Þ

where ci is the concentration of component i in the mobile

phase, qi is the concentration of component i adsorbed onto

the stationary phase, Dax is the dispersion coefficient, eC

the column void, eT the total column porosity, and v the

linear velocity of the bulk flow between the particles.

Bioprocess Biosyst Eng

123

The dispersion coefficient, Dax, is derived from the Pe-

clet number, Pe, [31]:

Pe ¼ v � dp

Dax

ð5Þ

where dp is the bead diameter.

The kinetics governing the adsorption of the particles to

the stationary phase is described by Eqs. 6 and 7 given by

Langmuir’s adsorption model [11, 28]:

oq

ot¼ kkin;i Hi � ci 1�

X

i

qi

qmax i

!� qi

!ð6Þ

Hi ¼Hoi

cbisalt

ð7Þ

where qmax,i is the maximum adsorption capacity for

component i, and Hi is Henry’s constant, which is modified

according to Eq. 7 with the salt concentration, where H0,i is

an experimentally calibrated constant and bi is the protein

binding charge [32].

Optimization

The process is optimized by varying one or more param-

eters, called decision variables, in order to maximize the

objective function. The objective function used in this

study was the normalized earnings (NE) as defined by

Karlsson et al. [28], where x is defined as the ratio between

operating cost and feed cost, SPmax is the maximum spe-

cific productivity obtained when optimizing the NE when

x = 0, and Ymax is assumed to be 100 % (Eqs. 8, 9).

x ¼ FeC

OCþ FeCð8Þ

NE ¼ 1� xð Þ SP

SPmax

þ xY

Ymax

ð9Þ

The definitions of specific productivity (SP) and global

yield (Y) in Eqs. 8 and 9 are given in Eqs. 10 and 11.

SP ¼ cpool � Vpool

Vcycle

ð10Þ

cpool is the concentration of the target component in the pool

obtained during the chromatographic separation, Vpool is the

volume of the pool, and Vcycle is the volume of buffer used in

chromatographic separation cycle; it is calculated as the

volume from the start of the load until the end of the pooling.

SP maximizes how much is produced per how much solvent

is needed for the complete cycle. Vcycle is used instead of time

because minimizing the amount of expensive buffers needed

is often more important than reducing the time of the step.

Y ¼ cpool � Vpool

cA0 � Vr

ð11Þ

cA0 is the concentration of non-PEGylated protein at the

start of the reaction and VR the reactor volume. The yield is

thus defined as the amount in the pool after the purification

divided by how much non-PEGylated protein was added at

the start of the reaction.

Case study

Reaction

The reaction studied was the PEGylation of a model pro-

tein supplied by Novo Nordisk A/S, which has two sites the

PEG molecules can react with. When PEG molecules react

enzymatically with either of these sites, active mono-

PEGylated protein (B/C) is produced, and when PEG reacts

with both sites, di-PEGylated protein (D) is produced,

which has reduced activity [33]:

Aþ S�!k1Bþ S�!k2

D

Aþ S�!k1C þ S�!k2

Dð12Þ

The two mono-PEGylated molecules (B and C) cannot

be distinguished from one another and are considered

analogous. The target product is both B and C and will be

denoted B for the rest of the article.

It is assumed that the reaction is carried out in a batch

reactor containing the native protein, the substrate and the

enzyme. It is assumed that the reactor is loaded with a

batch of 100 g protein, and that the entire contents of the

reactor are loaded onto the column. A typical reaction

profile is shown in Fig. 1.

Fig. 1 Simulation of the enzymatic PEGylation of native protein (A),

mono-PEGylated protein (B) and di-PEGylated protein (D) with

reaction time. A is a solid, B dashed and D a dotted line

Bioprocess Biosyst Eng

123

It can be seen that higher amounts of mono- and di-

PEGylated protein are produced with increasing reaction

time, although the amount of mono-PEGylated protein

decreases slowly if the reaction is allowed to proceed long

enough, due to the reaction of PEG with the second site. The

decision variables varied in the reaction model were initial

protein concentration (cA0), reaction time (tR), and the ratio

between substrate and protein at the start of the reaction (rs).

Purification

The end product is purified by cation-exchange chromatog-

raphy. This method is based on the different charges of the

PEGylated and un-PEGylated proteins. The PEG-molecule

shield the charge on the protein, reducing the adsorption

strength and the proteins elute at a lower salt concentration.

The first to elute is the di-PEGylated protein, followed by the

larger mono-PEGylated protein, and finally the native protein,

which shows the strongest interaction with the cation

exchanger (Fig. 2). Both the mono-PEGylated and the non-

PEGylated protein can be collected where the non-PEGylated

protein can be recycled to increase the yield. This separation is

also the main challenge for the chromatographic step.

The decision variables varied in the purification step are

the length of the salt gradient (VG), the final concentration

of the salt gradient (cSalt,f), and the column load (mLoad).

Recycling

Three different modes of operation were studied.

1. Batch mode A single batch reaction followed by

column separation of the reactor outlet stream.

2. Five-cycle recycling mode Five batch reactions, in

which the unreacted native protein is recycled and

mixed with fresh protein before the reaction step.

After five cycles, the process is stopped and a new

series of reactions is run to avoid the build-up of

degradation products and other impurities. The recy-

cled fraction collected from the chromatography

column is concentrated to the initial feed concentra-

tion prior to mixing with the fresh protein. Ultra-

filtration can be used for concentration and for larger

proteins a higher membrane cut-off will avoid salt

accumulation.

3. Infinite recycling mode In this case, it was assumed

that the process rarely had to be stopped for cleaning.

The batch recycling is run until it has reached steady

state which was then used to compare with the two

other modes.

Simulation

All simulations were performed with a variable-order

differential equation solver (ode15s) in MATLAB [34],

with a relative error tolerance of 0.1 %. The method of

lines [35] was used to discretize the column with 20 grid

points. The first derivative was estimated with a one-for-

ward and three-backward finite difference approximation,

and the second derivative was approximated by a five-

point central finite difference. A Dirichlet boundary con-

dition was used at the column inlet and a homogeneous

Neumann condition at the outlet.

Optimization

The optimization method used was patternsearch in

MATLAB, which is a direct search method [36]. The mesh

tolerance was set to 10-6. To increase the robustness of the

method, MADS (mesh-adaptive search) was activated,

which allowed patternsearch to randomly select the set of

vectors that forms the pattern for each new mesh size.

Pareto front

The optimization problem consists of obtaining the

Pareto front for the three different modes of operation

and optimization approaches. This is done by first

optimizing the maximum specific productivity (x = 0)

and then iterate with increasing values of x, using the

previously found operating point as guess value. The

decision variables are summarized in Table 1, together

with their corresponding bounds, which were used to

ensure that the variables remained between realistic

values.

Fig. 2 Chromatographic purification of di-PEGylated, mono-PEGy-

lated and unreacted protein eluting in this order. The separation was

run at low load at good separation

Bioprocess Biosyst Eng

123

Results and discussion

Process optimization versus step optimization

Normalized earnings were optimized as a function of x, as

explained above. This was done for two cases: the com-

plete process, in which both steps were optimized simul-

taneously, and separate optimization of the reaction and the

chromatographic purification steps. In the first case, the

objective function focused on the output from the purifi-

cation step. In the second case, the reaction was optimized

first, and then the purification step, based on the optimi-

zation of the first step. The results are compared in Fig. 3.

The maximum specific productivity is shown in the figure,

which also gives a minimum yield for the process. Trying

to obtain a lower yield will result in a lower specific pro-

ductivity as well and this part of the graph is not shown as

it is never interesting for design purposes.

Both methods of optimization give yields from about

37 to 67 %, but the specific productivity obtained with

step optimization is, at its worst, 50 % lower than that

obtained by optimizing the whole process. Although the

difference in productivity decreases at higher yields, these

results clearly show that optimizing the two steps sepa-

rately leads to a suboptimal working point. When opti-

mizing the steps separately, the optimal conditions for the

reaction reduce the amount of di-PEGylated produced,

while the amount of unreacted protein is moderate. This

complicates the separation, since the main difficulty lies

in separating the unreacted and mono-PEGylated forms of

the protein, requiring a longer gradient thus a lower

productivity. Optimization of the complete process pro-

vides a balance between the optimal conditions for the

reaction and separation, in which more of the di-PEGy-

lated protein is formed, making purification easier, as the

column yield can be increased giving a higher global

yield. The chromatograms in Fig. 4 obtained with process

optimization (left) and step optimization (right) show the

same global yield, but the productivity was lower with

step optimization. Note the much longer chromatographic

cycle for the step optimal operating point and the lower

concentrations of the peaks.

As explained above, optimization of the complete

process leads to the conversion of almost all the native

protein into either mono- or di-PEGylated protein. This

leads to a decrease in the product yield in the reactor,

but simplifies the removal of the di-PEGylated protein.

The column is loaded so much that the di-PEgylated

goes in flow through and the mono-PEGylated product

requires only a short elution step to elute. With step

optimization, the reaction produces a higher amount of

mono-PEGylated protein, but it is more difficult to

purify, due to the presence of native protein, and a

larger elution volume of 55 L is needed. The decision

variables for the two kinds of optimization can be seen

in Table 2. The ratio of substrate to native protein is

lower in step optimization, which gives a high yield in

the reaction, but makes separation more difficult. The

final salt gradient is lower, the length of the gradient is

increased, and a lower load and thus a larger column are

needed to separate the product.

Batch mode versus recycling mode

A similar analysis was performed to compare the Pareto

fronts obtained from process optimization of the different

recycling modes (Fig. 5).

The maximum productivity obtained was almost

identical in all cases, as can be seen in Fig. 5. This is due

to that to achieve maximum productivity the two recy-

cling modes stopped recycling any non-PEGylated pro-

tein and converted all of it to make the productivity of the

chromatographic purification as high as possible. The

Table 1 Decision variables for the optimization of the enzymatic

reaction and the chromatographic purification step

Decision variable Upper

bound

Reaction Reaction initial concentration (cA0) (g/L) 20

Reaction time (tR) (h) 100

Ratio substrate/protein at start (rS) (-) 2

Column Final salt concentration in gradient (cSalt,f) (mol/L) 1

Gradient length (VG) (CV) 1,000

Load (mLoad) (g/L column) 100

The lower bound for all decision variables is 0

Fig. 3 Pareto front resulting from optimization of the complete

process (solid) and step optimization (dashed) without recycling

Bioprocess Biosyst Eng

123

higher the yield the more recycling is done and at yields

above 60 %, differences become apparent between batch

mode and the recycle modes. The maximum possible

yield obtained with the different processes differs con-

siderably, being 66 % in batch mode and about 86 % in

the recycling modes. At this point almost 50 % of the

non-PEGylated protein is recycled for every batch and at

maximum productivity no recycling is performed at all

making it a batch process again (Fig. 6). Five-cycle

recycling and infinite recycling modes show similar

results, which is due to how quickly the recycling reaches

steady state. This means that a five-cycle recycling will

give the same performance as running infinite recycling

without the risk of build-up of unwanted impurities or

fragmentation of the target protein.

Figure 7 shows chromatograms for optimal cases for

batch mode and five-cycle recycling with similar global

yield and a 10 % difference in productivity. In this case,

the batch mode provides a higher mono-PEGylated protein

yield. It also produces less di-PEGylated protein than the

recycling mode, but this kind of reaction entails slower

separation and, hence, a reduction in productivity. Lower-

yield reactions can be run in recycling mode in order to

obtain better separation, since some of the native protein is

recycled, whereas running only one reaction step and one

Fig. 4 Simulated chromatograms for process optimization (left) and

step optimization (right), at x = 0.2. The vertical black lines show

where the product is pooled. The dotted line shows the salt gradient.

The di-,mono- and non-PEGylated protein is shown with a dashed,

solid and dash–dot line, respectively

Table 2 Decision variables for process optimization and step opti-

mization at 39 % global yield

Process optimization Step optimization

Normalized SP (%) 100.0 49.4

Yield (%) 39.0 39.0

Decision variables

cA0 (g/L) 20.0 20.0

tR (h) 1.9 0.2

rS (-) 1.6 1.1

cSalt,f (mol/L) 1.5 0.8

VG (CV) 0.0 76.0

mLoad (g/L) 85.5 41.9

Fig. 5 Pareto fronts obtained from optimization of batch mode

(solid), five-cycle recycling (dotted) and infinite recycling (dashed)

Bioprocess Biosyst Eng

123

chromatography step gives a better yield, but less efficient

separation.

The values of the decision variables obtained from the

optimization of batch and recycling mode are summarized

in Table 3. It can be seen that batch mode operation

requires a more dilute feed, which means that a larger

volume must be loaded onto the column, and a smaller

substrate:protein ratio, which improves the reaction yield

but entails a longer elution gradient to separate the unre-

acted protein.

Conclusions

This paper presents the optimization of a common

downstream process in the biopharmaceutical industry.

It shows that for this particular case, a simplistic

approach in which the steps are optimized indepen-

dently will lead to suboptimal working points which, in

some cases, can lead to a decrease in productivity of up

to 50 %. The results of this study also show that a

model-based approach is needed to optimize the whole

process, as it would be extremely difficult to optimize

both steps simultaneously using a purely experimental

approach.

Recycling the unreacted protein increases the yield

without significantly affecting the specific productivity. If

maximum productivity is required, batch mode operation

is the preferred choice, while if a compromise can be

accepted between yield and productivity, or if the maxi-

mum yield is required, recycling mode can be used to

improve the performance of the process. Only a small

number of recycling steps (5) is required to obtain a

system working at steady state, which will reduce the

build-up of degradation products and other impurities in

the system.

Fig. 6 For the recycling modes, the amount recycled increases with

increasing global yield. At maximum productivity no recycling is

performed and at maximum yield almost 50 % is recycled during

every batch

Fig. 7 Simulated

chromatograms for optimized

batch mode (left) and for

optimized five-cycle recycling

mode (right) with both 58 %

global yield. The vertical black

lines show where the product is

pooled. The dotted line shows

the salt gradient. The di-,mono-

and non-PEGylated protein is

shown with a dashed, solid and

dash–dot line, respectively

Table 3 Values of decision variables obtained for optimized batch

operation and five-cycle recycling, with global yield at 58 %

Batch mode Recycling

mode

Normalized SP (%) 50.4 64.0

Yield (%) 58.0 58.0

Decision variables

cA0 (g/L) 8.8 20.0

tR (h) 1.8 2.1

rS 1.2 1.4

cSalt,f (mol/L) 0.8 1.0

VG (CV) 84.7 56.0

mLoad (g/L) 49.4 55.0

Bioprocess Biosyst Eng

123

References

1. FDA USFaDA (2004) Guidance for industry PAT—a framework

for innovative pharmaceutical development, manufacturing, and

quality assurance

2. ICH (2009) ICH Q8 pharmaceutical development

3. ICH (2012) ICH Q11 development and manufacture of drug

substances

4. ICH (2005) ICH Q9 quality risk management

5. Degerman M, Jakobsson N, Nilsson B (2007) Modeling and

optimization of preparative reversed-phase liquid chromatogra-

phy for insulin purification. J Chromatogr A 1162(1):41–49.

doi:10.1016/j.chroma.2007.02.062

6. Guelat B, Delegrange L, Valax P, Morbidelli M (2013) Model-

based prediction of monoclonal antibody retention in ion-

exchange chromatography. J Chromatogr A (0). doi:10.1016/j.

chroma.2013.04.048

7. Guelat B, Strohlein G, Lattuada M, Delegrange L, Valax P,

Morbidelli M (2012) Simulation model for overloaded mono-

clonal antibody variants separations in ion-exchange chroma-

tography. J Chromatogr A 1253(0):32–43. doi:10.1016/j.chroma.

2012.06.081

8. Muller-Spath T, Aumann L, Melter L, Strohlein G, Morbidelli M

(2008) Chromatographic separation of three monoclonal antibody

variants using multicolumn countercurrent solvent gradient

purification (MCSGP). Biotechnol Bioeng 100(6):1166–1177.

doi:10.1002/bit.21843

9. Muller-Spath T, Strohlein G, Aumann L, Kornmann H, Valax P,

Delegrange L, Charbaut E, Baer G, Lamproye A, Johnck M,

Schulte M, Morbidelli M (2011) Model simulation and experi-

mental verification of a cation-exchange IgG capture step in

batch and continuous chromatography. J Chromatogr A 1218(31):

5195–5204. doi:10.1016/j.chroma.2011.05.103

10. Carta G, Jungbauer A (2010) Protein chromatography: process

development and scale-up. Wiley, New York

11. Guiochon G, Shirazi DG, Felinger A, Katti AM (2006) Funda-

mentals of preparative and nonlinear chromatography. Elsevier

Academic Press, Amsterdam

12. Degerman M, Jakobsson N, Nilsson B (2008) Designing robust

preparative purification processes with high performance. Chem

Eng Technol 31(6):875–882. doi:10.1002/ceat.200800097

13. Degerman M, Westerberg K, Nilsson B (2009) Determining

critical process parameters and process robustness in preparative

chromatography—a model-based approach. Chem Eng Technol

32(6):903–911. doi:10.1002/ceat.200900019

14. Degerman M, Westerberg K, Nilsson B (2009) A model-based

approach to determine the design space of preparative chroma-

tography. Chem Eng Technol 32(8):1195–1202. doi:10.1002/

ceat.200900102

15. Jakobsson N, Degerman M, Nilsson B (2005) Optimisation and

robustness analysis of a hydrophobic interaction chromatography

step. J Chromatogr A 1099(1–2):157–166. doi:10.1016/j.chroma.

2005.09.009

16. Jakobsson N, Degerman M, Stenborg E, Nilsson B (2007) Model

based robustness analysis of an ion-exchange chromatography

step. J Chromatogr A 1138(1–2):109–119. doi:10.1016/j.chroma.

2006.10.057

17. Westerberg K, Broberg Hansen E, Degerman M, Budde Hansen

T, Nilsson B (2012) Model-based process challenge of an

industrial ion-exchange chromatography step. Chem Eng Technol

35(1):183–190. doi:10.1002/ceat.201000560

18. Westerberg K, Degerman M, Nilsson B (2010) Pooling control in

variable preparative chromatography processes. Bioprocess Bio-

syst Eng 33(3):375–382. doi:10.1007/s00449-009-0335-8

19. Mollerup JM, Hansen TB, Kidal S, Staby A (2008) Quality by

design-thermodynamic modelling of chromatographic separation

of proteins. J Chromatogr A 1177(2):200–206. doi:10.1016/j.

chroma.2007.08.059

20. Chen HS, Stadtherr MA (1985) A simultaneous-modular

approach to process flowsheeting and optimization. Part I: theory

and implementation. AIChE J 31(11):1843–1856. doi:10.1002/

aic.690311110

21. Floudas CA (1995) Nonlinear and mixed-integer optimization.

Oxford University Press, USA

22. Biegler LT, Grossman IE, Westerberg AW (1997) Systematic

methods of chemical process design. Prentice Hall, Upper Saddle

River

23. Biegler LT, Cervantes AM, Wachter A (2002) Advances in

simultaneous strategies for dynamic process optimization. Chem

Eng Sci 57(4):575–593

24. Larsson PO, Akesson J, Carlsson N, Andersson N (2012) Model-

based optimization of economical grade changes for the borealis

borstar polyethylene plant. Comput Chem Eng 46:153–166

25. Srinivasan B, Palanki S, Bonvin D (2003) Dynamic optimization

of batch processes: I. Characterization of the nominal solution.

Comput Chem Eng 27(1):1–26

26. Mendez CA, Cerda J, Grossmann IE, Harjunkoskic I, Fahlc M

(2006) State-of-the-art review of optimization methods for short-

term scheduling of batch processes. Comput Chem Eng 30(6–7):

913–946

27. Nie Y, Bielger LT, Wassick JM (2012) Integrated scheduling and

dynamic optimization of batch processes using state equipment

networks. AIChE J 58(11):3416–3432

28. Karlsson D, Jakobsson N, Axelsson A, Nilsson B (2004) Model-

based optimization of a preparative ion-exchange step for anti-

body purification. J Chromatogr A 1055:29–39

29. Muller-Spath T, Krattli M, Aumann L, Strohlein G, Morbidelli M

(2010) Increasing the activity of monoclonal antibody therapeu-

tics by continuous chromatography (MCSGP). Biotechnol Bioeng

107(4):652–662. doi:10.1002/bit.22843

30. Schmidt-Traub H (2012) Preparative chromatography, 2nd edn

edn. Wiley VCH, New York

31. Sherwood TK, Pigford RL, Wilke CR (1975) Mass transfer, vol

23. McGraw-Hill, New York

32. Melander WR, Rassi Z, Horvath C (1989) Interplay of hydro-

phobic and electrostatic interactions in biopolymer chromatog-

raphy: effect of salts on the retention of proteins. J Chromatogr A

469(3)

33. Harris JM, Martin NE, Mod M (2001) Pegylation, a novel process

for modifying pharmacokinetics. Clin Pharmacokinet 40(7):

539–551

34. Shampine LF, Reichelt MW (1997) The Matlab ODE Suite.

SIAM J Sci Comput 18(1):1–22

35. Davis ME (1984) Numerical methods and modeling for chemical

engineers. Wiley, New York

36. Mathworks (2012) Optimization toolbox—for use with MAT-

LAB. Natick, MA

Bioprocess Biosyst Eng

123