Model‐based optimization of an ion exchange chromatography process for the separation of von Willebrand factor fragments and human serum albumin

Column liquid chromatography plays a vital role in the downstream processing of biopharmaceuticals, where the goal is to extract and purify a target protein from a mixture. In this work, we examine a real‐world ion exchange chromatography process where electrostatic interactions between proteins and the chromatographic medium are the predominant forces being exploited for the separation of the participating species. More specifically, we successfully separate a recombinant dimeric fragment of the von Willebrand factor, used for the half‐life prolongation of certain proteins, from human serum albumin. Compared to the originally conducted experiments, the optimized experiment achieves a higher product purity in less time and with lower consumption of buffer salts, hence being favorable both in economical and ecological terms.

As chromatography steps account for up to 70% of the total costs in the downstream processing of biopharmaceuticals [3], we identify significant potential for process optimization.Not only can we expect economical benefits but also ecological benefits, as process optimization can be performed to ensure an economical use of chemicals involved in chromatographic separation processes, such as chromatography resins, buffer salts, and detergents.Process optimization can therefore have a positive impact both on economical and ecological aspects.Of course, at all times we have to ensure that the product satisfies imposed purity requirements.
Our contribution builds upon a framework for optimization of chromatographic processes previously proposed in [4].Therein, a hypothetical multimodal chromatographic separation process has been optimized.In this work, we investigate a real-world ion exchange chromatography (IEX) process by considering a two-component system, where the product is a recombinant dimeric fragment of the von Willebrand factor (VWF-12) and the impurity is human serum albumin (HSA).Mathematical optimization is employed to compute an optimized control strategy that leads to both less consumption of chemicals and reduced operating time.

MATHEMATICAL MODELS
In column liquid chromatography, a liquid solvent (mobile phase) is pumped through a column that is packed with porous particles (particle/stationary phase).The longer a substance remains in the stationary phase, the longer its retention time will be.Separation of a mixture is then achieved by exploiting different retention times of the respective substances.Multiple phenomena occur in column liquid chromatography processes, such as transport of the substances within the column, axial dispersion, external and internal mass transfer resistances, and adsorption processes.We employ the so-called transport-dispersive model (TDM), compare [5, Section 6.2.5.1], that incorporates all of the above mentioned phenomena, except for external and internal mass transfer resistances, which are "lumped" into an effective mass transfer resistance.
The TDM describes the concentration profiles of all components i along the axial position  ∈ (0,   ) of the column for a given time  ∈ (0,   ), The states in the TDM (1) are the concentrations in the mobile phase  , , the liquid particle phase  , , and the adsorbed particle phase   .The first Equation (1a) is an advection-diffusion-reaction type partial differential equation (PDE) model describing the transport of a component within the column, as well as the mass transfer between mobile phase and liquid particle phase.The second Equation (1b) is a spatially distributed ordinary differential equation (ODE) incorporating the adsorption process and the transition back into the mobile phase.The last Equation (1c) is a kinetic version of the actual adsorption process and is given below in (2).With the ⋆ symbol we signify that the adsorption process may generally depend on all components in the stationary phase, both liquid and adsorbed.
Model parameters occurring in (1) are the bed and particle porosity,   and   , respectively, the particle radius   , the axial dispersion coefficient  ax , and the component-dependent mass transfer coefficient  eff .Lastly,  sup is the superficial flow velocity, which is typically computed from a prescribed volumetric flow rate V.
Generally speaking, proteins bind in IEX at low salt concentrations, whereas high salt concentrations are employed to remove the proteins from the column.A widely applied model to describe the adsorption behavior in IEX processes is the so-called steric mass action isotherm model [6] Here,  in, is the inlet concentration of component i, a time-dependent control degree of freedom.Later in our case study,  in,salt will be the main control to optimize the chromatographic separation process.The initial values correspond to an equilibrated column: no proteins are present in the beginning and an initial salt concentration  salt,init chosen by the experimenter is realized.We hence have for  ∈ [0,   ]  , (0, ) = 0,  , (0, ) = 0,   (0, ) = 0, for  ∈ {1, … ,  comp },  ,salt (0, ) =  salt,init ,  ,salt (0, ) =  salt,init ,  salt (0, ) = Λ IEX .
Since the volume of the chromatography column used in our experiments is comparably small, compare Section 4, it is not sufficient to consider the column alone.Rather, the whole chromatography system has to be taken into account, including additional tubing, mixing chambers, and detectors.To model mixing chambers and detectors, we use a continuous stirred tank (CST) model given by the initial value problem where  cst init is an experimenter-defined initial substance concentration, V is the volumetric flow rate, and  cst is the tank volume.Additional tubing with length  dpf > 0 is modeled by a dispersed plug flow (DPF) model given by the one-dimensional PDE With this, all employed models are introduced.The actual modeling of the chromatography system used in our case study is schematically depicted in Figure 1.

MIXED-INTEGER OPTIMAL CONTROL PROBLEMS FOR CHROMATOGRAPHIC PROCESSES
Multiple performance criteria exist for chromatographic processes, compare [5,Chapter 7], but we focus on yield, purity, and batch-cycle time.To define these quantities, we first introduce the injected, eluted and collected amounts of substance that are for  ∈ [0,   ] and  ∈ {1, … ,  comp } given by Here,  out is the concentration of a component leaving the chromatography system and coll ∈  ∞ ([0,   ], {0, 1}) is a binary control that signifies whether the eluate is collected or not.Based on the amounts of substance we can now introduce yield and purity of a component i via .
F I G U R E 1 Schematic depiction of the employed chromatography system model.In parentheses we state the type of model being used.Furthermore, we summarize selected model parameters used in our case study, compare Section 4, primarily for extra column equipment.
In our case study, we optimize the batch-cycle time of the process, while also requiring a certain yield and purity of the product.Using a rather high-level problem formulation, for example, by "hiding" the actual PDE and ODE constraints, we solve The first and second constraint in (4) are the purity and yield requirement imposed on the product component * , respectively, whereas the last constraint ensures that a certain fraction   of component i hast left the chromatography system at the end of the process.We aim to perform so-called bind-and-elute experiments that are composed of two phases: In the first phase, the mixture is injected into the system for a prescribed amount of time  inj ∈ (0,   ) and with given sample concentrations  load, .Operational conditions are chosen such that the product binds on the column.Since the sample injection is carried out on a different flow path in our case, compare Figure 1, the salt inlet  in,salt remains fixed during the first phase.In the second phase, sample injection is stopped and the salt inlet is free for optimization to eventually elute the product.We hence arrive at the control constraints where  in,salt is a lower bound and  in,salt is an upper bound on the salt inlet concentration.
We highlight that (4) in combination with ( 5) is mixed-integer optimal control problem (MIOCP) with PDE and ODE constraints, as the amounts of substance (3) are computed by using states of the chromatography system model, and since the collected amounts of substance are affected by the binary eluate collection control coll.
A strategy to numerically solve the resulting MIOCP by means of a direct approach is exhaustively described in [4].We therefore only briefly summarize the fundamental steps.The PDE model is spatially discretized, thus replacing all PDEs by systems of ODEs.Subsequently, we transform the resulting MIOCP, now solely constrained by ODEs, into a continuous optimal control problem by means of partial outer convexification and relaxation [8].Here, as the only discrete control is the binary-valued eluate collection that enters the ODE right-hand sides linearly, a simple relaxation coll() ∈ [0, 1] a.e.yields the convex hull with respect to the controls [8].
In a subsequent step, we use the direct single shooting approach, compare [9], to transform the continuous optimal control problem into a nonlinear program, the latter being solved with the nonlinear interior-point method IPOPT [10].Lastly, as the optimized relaxed controls may not be feasible for the original MIOCP formulation, we have to apply a suitable reconstruction scheme.We choose the so-called sum-up rounding procedure [8], which has provable approximation properties that lead to -feasible and -optimal controls, compare [11].

RESULTS
We now present results obtained when optimizing the IEX separation process.To this end, we first state the used chromatography equipment and chemical substances, followed by a note on the calibration procedure, where we also summarize the employed model parameters.Finally, we present data of the actual original and optimized chromatographic experiments.

Apparatus, column, chemical substances
An ÄKTA pure 150 chromatography system is used to perform chromatographic experiments.The chromatography column is a Tricorn 5/50 column with 5 mm inner diameter and a height of 50 mm that is shortened to 25 mm by adjusting the height of the movable adapter.The ion exchanger is a Capto Q ImpRes strong anion exchanger.Information for all the aforementioned equipment can be obtained from data sheets provided by the manufacturer Cytiva (Marlborough, MA, USA).Buffer components, such as NaCl (sodium chloride) and tris (tris(hydroxymethyl)aminomethane) are purchased from Merck KGaA (Darmstadt, Germany).
In our case studies we examine two substances.The product is VWF-12 (Octapharma Biopharmaceuticals GmbH, Heidelberg, Germany), a recombinant dimeric fragment of the so-called von Willebrand factor that aims to prolong the half-life of proteins, such as factor VIII, which is used for the treatment of hemophilia A. For more information we refer to [12].As an impurity HSA (human serum albumin) is used (Sigma-Aldrich, St. Louis, MO, USA).HSA elutes at similar salt concentrations and is therefore challenging to remove.The sample itself is composed of a 1:1 (w/w) mixture of VWF-12 and HSA; the buffer solutions are composed of 0.05 M Tris-HCl (pH 8) with either 0.1 M or 1.0 M NaCl, the latter being the salt component.Intermediate salt concentrations are realized by mixing the aforementioned two possible buffer solutions.

A note on model calibration
Before using a model for optimization, we have to determine suitable model parameters for the TDM (1), as well as for the respective DPF and CST models for extra column equipment.The actual procedure is rather involved and employs a newly developed method whose description is subject of a future publication.We nonetheless remark that a general approach for model calibration for chromatographic processes is described in [5,Section 6.5].The determined model parameters for our case study are summarized in Figure 1 for extra column equipment and in Table 1 for the remaining TDM parameters.

Optimization of the separation process
We solve (4) in combination with (5), VWF-12 being the product component.We set Yield min,VWF-12 = Purity min,VWF-12 = 0.99 and   = 0.95 for  ∈ {HSA, VWF-12}.Based on the employed chromatography system, compare Figure 1, we set  out = Note: Model parameters for extra column equipment are summarized in Figure 1.

𝑐 cst
detector .The eluting concentration is also used for the computation of the model response, as measurement data is given in UV absorbance.According to the Lambert-Beer law, the (UV) absorbance  () for a given wavelength  can be computed via The attenuation coefficients  () for HSA and VWF-12 for  = 280 nm and the cell path length  cp of the detector are stated in Table 1.The sample concentration  load, will be treated as a model parameter as neither HSA nor VWF-12 are 100% pure; the corresponding estimates are also summarized in Table 1.The volumetric flow rate is fixed at V = 0.5 mL min −1 and we furthermore set  in,salt = 0.1 M and  in,salt = 1.0 M as lower and upper bounds on the salt inlet, respectively.We solve the resulting MIOCP with the direct approach outlined in Section 3.For the spatial discretization of the PDE models we use a higher-order linear finite volume method; partial outer convexification and relaxation is realized in our case simply by setting coll() ∈ [0, 1].The equidistant control grid applied in the direct single shooting method consists of intervals with length Δ = 1 min.The salt inlet concentration is parameterized as a piecewise linear and continuous control, whereas the relaxed eluate collection is a piecewise constant control.An  2 -regularization is used for the salt inlet  in,salt and an  1 -regularization is used for the relaxed eluate collection coll; both controls are regularized to their respective lower bounds.
Model responses (predictions) and actual measurements of the original and optimized experiment are depicted in Figure 2. The original experiment is a bind-and-elute experiment with  inj = 20 min, which we also use for the optimized experiment, followed by a so-called clean-after-binding phase for 5 min.Afterwards, the elution phase begins by application of a linear salt gradient, that is, by linearly increasing the salt concentration from 0.1 M to 1.0 M in 20 min.We observe that our model response captures the resulting chromatogram well (note that this experiment has not been used for calibration).
Before we discuss the optimized experiment in detail, we first emphasize that we do not directly apply the computed optimized salt inlet (dashed line), but an altered trajectory (solid green line).Our rationale is that we want to perform experiments that are as similar as possible to the experiments performed for calibration, as we can put such results into practice very quickly.However, the crucial parts of the salt inlet with regard to the separation process, namely the initial salt concentration and the slope of the salt gradient, are approximately the same.We therefore do not expect significant changes in the respective results.
We observe that the initial salt concentration  salt,init is higher in the optimized experiment than in the original experiment, which leads to parts of HSA (the impurity) leaving the chromatography system already during sample injection.We can furthermore see that the predicted response of the optimized experiment results in a baseline separation of HSA and VWF-12.The eluate collection coll is already binary feasible and the horizon of collection is indicated by the colored area in Figure 2.
F I G U R E 2 Measurements and model predictions of the original and optimized experiment.For the real-world data, only a subset of data points is shown.Model predictions are obtained via (6).The colored section within the chromatogram signifies where the eluate is collected.
When evaluating the optimized real-world experiment, we first and foremost observe that VWF-12 and HSA are indeed well separated.Furthermore, particularly the peak positions are reasonably well predicted by our model.This result is particularly remarkable as the applied salt gradient is twice as steep as the steepest salt gradient used in experiments for calibration, hence we achieve a good predictability with the employed calibrated models.
However, we also observe that the UV signal obtained from the real-world experiment is generally lower than predicted.We could not track down the reason for this, but one possible explanation is based on the fact that the optimized experiment was performed almost one and a half years later than the bind-and-elute experiments used for model calibration.Hence, a loss of UV absorption due to the long time and storage may be a possible explanation.We are nonetheless confident that the calibrated model can predict the real-world behavior well.

CONCLUSION AND OUTLOOK
Our main result is the successful separation of VWF-12 and HSA.After setting up models to describe IEX processes and determination of suitable model parameters, we aimed to minimize the process duration (batch-cycle time) while also ensuring a high quality of the product.Indeed, the impurity (HSA) could be well separated from VWF-12, while we also reduced the process duration by approximately one third.Furthermore, in spite of the higher initial salt concentration in the optimized experiment, a comparison of the areas under the respective salt inlets depicted in Figure 2 lets us conclude that the overall consumption of salt could be reduced by approximately 37%, which is both economically and ecologically favorable.Despite the encouraging results, we identify room for improvements and extensions.First of all, we investigated a mixture of two known components having a purity of well more than 90%, respectively.In practical applications, the composition of a mixture is typically not clear at all, hence strategies of robust optimization should be employed to adequately deal with this kind of uncertainty, for example, with a polynomial chaos approach [13] or an approach based on scenario trees [14,15].Moreover, depending on the actual examined chromatographic separation process, further chemical quantities could be considered as additional process controls, most notably the pH.Lastly, further optimal control problem formulations can be investigated, dependent on the actual needs and requirements of the experimenter.

A C K N O W L E D G M E N T S
Analogously to the TDM (1),  dpf is the superficial flow velocity and  dpf ax is the axial dispersion coefficient.The DPF model is completed with Danckwerts' boundary conditions for  ∈ [0,   ] and initial values for  ∈ [0,   ], Dominik H. Cebulla, Christian Kirches, and Andreas Potschka acknowledge funding by the German Ministry for Education and Research (BMBF) under grant agreement no.05M17MBA (MOPhaPro).Open access funding enabled and organized by Projekt DEAL.C O N F L I C T O F I N T E R E S T S TAT E M E N TThe authors declare no conflicts of interest.O R C I DDominik H. Cebulla https://orcid.org/0000-0002-3025-8673Christian Kirches https://orcid.org/0000-0002-3441-8822Andreas Potschka https://orcid.org/0000-0002-6027-616XR E F E R E N C E S [7]hich reads for component  ∈ {1, … ,  comp } and for the salt component  ads and  des are adsorption and desorption rates, respectively,  p is the binding charge,  is a parameter to account for shielding behavior of the proteins, and Λ IEX is the total ionic capacity.The TDM (1) is completed with Danckwerts' boundary conditions[7], which read, for  ∈ {1, … ,  comp } ∪ {salt} and  ∈ [0,   ], Summary of model parameters.
TA B L E 1