A Self-Consistent, High-Fidelity Adsorption Model for Chromatographic Process Predictions: Low-to-High Load Density and Charge Variants in a Preparative Cation Exchanger

Abstract

The development of ion exchange chromatography to polish biopharmaceuticals requires extensive experimental benchmarking. As part of the Design of Experiments (DoE), statistical models increased efficiency somewhat and are still state of the art; however, the capability to predict process conditions is limited due to their nature as interpolating models. Applying the DoE still requires numerous experiments and is constrained to the design space, posing a risk of missing the potential optimum. To make a leap in model-based process development, applying extrapolating models can tremendously extend the design space and also allow for process understanding and knowledge transfer. While existing chromatography modeling software explains experimental data, it often lacks predictive power for new conditions. In academic–industrial cooperation, we demonstrate a new high-fidelity model based on biophysics for developing ion-exchange chromatography in biomanufacturing, making it a general tool in rationalizing process development for the present demand of recombinant proteins and monoclonal antibodies and the emerging demand of new modalities. Using the new computational tool, we achieved predictability and attained high accuracy; with minimal experimental effort to calibrate the system, the mathematical model predicted sensitive process conditions, and even described product-related impurities, antibody charge variants. Thus, the computational tool can be deployed for process-by-design and material-by-design approaches.

1. Introduction

1.1. Biopharma at a Crossroads

Healthcare systems face mounting pressure from aging populations, chronic diseases, personalized medicine demands, and rising pandemic threats. These challenges require innovation to control costs and accelerate drug development, which currently averages 8.5 years [1,2]. While biopharmaceutical development has advanced—driven by artificial intelligence (AI) in drug discovery for proteins and novel modalities like viral vectors, exosomes, and mRNA therapies [1]—downstream processing (DSP) remains a bottleneck. DSP, particularly chromatography, accounts for 40% of downstream costs and up to 30% of total manufacturing expenses in the trillion-dollar biotech industry [3,4,5,6]. Emerging modalities like mRNA and viral vectors face even greater hurdles, with low yields (averaging 20%) and high impurity levels [7]. AstraZeneca’s (Cambridge, UK) COVID-19 vaccine production underscored DSP limitations, where impurity-related failures led to toxic doses and severe risks [8].

1.2. Chromatography: A Cornerstone with Bottlenecks

Ion-exchange (IEX) chromatography is a cornerstone of DSP valued for its robustness, selectivity, and mild operating conditions, especially in purifying recombinant proteins and emerging modalities [9]; however, it faces two key bottlenecks, as outlined below:

• High material and time consumption [10]: Scaling requires extensive, time-consuming laboratory experiments. Limited biopharmaceutical quantities further restrict feasible trials, and unsuccessful purification can lead to significant financial losses;
• Knowledge gaps for new bio-therapeutics [7,11]: Emerging modalities like RNA/DNA vaccines, plasmid DNA, and viral vectors often require entirely new purification strategies. Established techniques fall short, leading to labor-intensive, trial-and-error approaches.

1.3. Why Digital Transformation in DSP Lags Behind

Despite advances in upstream processing (USP) through AI and modeling technologies, downstream processing (DSP) has been slow to adopt these innovations. DSP, particularly IEX chromatography, still relies heavily on labor-intensive methods like the Design of Experiments (DoE), where early-stage development can require numerous parameter variations—an effort that is both time-consuming and resource-intensive. Mechanistic models, combining fluid dynamics with adsorption mechanisms, offer a promising solution by enabling in silico simulations that accelerate development, improve scalability, and conserve resources [7,12,13]. While the mathematical description of fluid mechanics is on a sufficiently high level of accuracy, existing binding models like Steric Mass Action (SMA) and Colloidal Particle Adsorption (CPA) have critical shortcomings [11,14,15]:

• Limited biophysical understanding: current models fail to capture the complexity of binding interactions for proteins, particularly for novel modalities, under typical buffer conditions, and even more so under more complex buffer conditions, especially in the case of anion-exchange chromatography and strong pH dependency;
• High calibration effort: extensive empirical data and complex optimization algorithms are required for calibration;
• Limited prediction capability: calibration parameters lack physical relevance, restricting predictive power to narrow datasets;
• Inability to generalize: current models struggle to adapt to the diversity and complexity of proteins and new therapeutic modalities.

1.4. The Evolution of IEX Modeling

The development of IEX adsorption models has aimed to better describe protein retention and adsorption mechanisms. Early stoichiometric displacement (SD) models explained protein adsorption as the reversible displacement of counter-ions but were limited to linear behavior at low protein loads [16,17,18,19]. The SMA model improved on this by accounting for nonlinear adsorption, ionic capacity, and steric hindrance effects [20]. However, it struggled to capture intermolecular interactions, limiting its accuracy for adsorption isotherms and elution profiles [21,22,23,24,25,26,27]. Non-stoichiometric models, incorporating electrostatic interactions, further advanced IEX modeling. Formulations like those by Oberholzer [28] and Guelat [29] included surface coverage and lateral interactions but introduced calibration challenges due to pH and ionic-strength dependencies. The CPA model combined steric surface-blocking effects (scaled-particle theory) with electrostatic forces (Yukawa lattice), enabling a more detailed description of linear and nonlinear adsorption [30]. Despite its mechanistic strengths, CPA faces significant challenges for industrial use [13], as follows: parameter estimation complexity, experimental and computational burden, sensitivity and validation challenges, limited practical usability (e.g., future research is necessary for multicomponent and complex elution behaviors), and the need for complex and robust calibration frameworks.

1.5. A New Digital Contribution in IEX Modeling

This study introduces a next-generation mechanistic modeling framework, addressing the limitations of existing IEX adsorption models like SMA and CPA. The new adsorption model integrates deeper biophysical insights through distinct input parameters, including protein charge, protein size, ionic strength, salt ion valency, salt ion size, temperature, resin properties, maximal load density, ionic capacity, and the phase ratio. These parameters enable a precise, mechanistic description of adsorption dynamics. Coupled with established fluid transport models (e.g., transport-dispersive models), the binding model effectively captures adsorption dynamics, including protein interactions and elution profiles, with minimal calibration effort. Validated through an academic–industrial collaboration, the tool reduced experimental effort to two simple calibration experiments, with little material consumption and less labor required compared to the CPA model, while accurately predicting process outcomes, such as the separation of product-related impurities and antibody charge variants at high loading conditions.

Concretely, the experimental dataset is rationalized with the new adsorption model, which is then compared to a previous study, utilizing the CPA model, regarding calibration efficiency and predictive modeling power. Beforehand, the estimation of key parameters of the new binding model was enabled by inverse fitting the model output to experimental chromatograms. In addition, a sensitivity analysis was conducted to assess the model’s accuracy toward salt concentration and maximal load density, the physical integrity of the maximal load density, and whether this parameter can be directly obtained from the experimental data of material specifications.

This framework represents a major advancement in model-based process development. While validated for monoclonal antibody isoforms under typical buffer composition in preparative cation-exchange chromatography, its robust biophysical foundation suggests applicability to emerging modalities and complex buffer compositions in various separation processes, leveraging universal principles of molecular interactions. These results underscore the transformative potential of predictive, high-fidelity models to accelerate biopharmaceutical development, shorten timelines, enhance process robustness, and reduce failure risks in downstream processing.

2. Materials and Methods

2.1. Materials

The materials and experiments in this study were already published in the work of Briskot et al. [12], and specifically related to the dataset of “mAb4” in their study. Below, we highlight the essential data and the primary differences between our study and the original work by Briskot et al.

2.1.1. Protein

The retention behavior of the monoclonal antibody (mAb) with a molecular weight of approximately $M_{w,i}=150$ kDa was investigated. Analytical data identified several isoforms of the mAb, which comprised one main isoform, four acidic charge variants, and two basic charge variants. The exact ratio between the main isoform and the grouped acidic and basic variants was determined through fractional analysis of the chromatograms. Within the acidic and basic groups, we assumed equal ratios for the individual charge variants as their specific distribution could not be resolved from the chromatograms. The key distinction between this study and Briskot’s work lies in the detailed inclusion of a finite number of isoforms within the acidic and basic charge variants. In contrast, Briskot simplified the system to three isoform clusters (one main, one acidic, and one basic) [12].

2.1.2. Adsorber System

Experiments were conducted using a prepacked column with the strong cation exchanger Capto S ImpAct ($L_c=100 \mathrm{mm}, V_c=4.66 \mathrm{mL}$) (Cytiva, Uppsala, Sweden). The Capto S ImpAct resin features a highly cross-linked agarose base matrix and its dynamic binding capacity (DBC) ranges from 84 to 122 mg mL⁻¹ for six different antibodies at a residence time of 5.4 min in a Tricorn 5/50 column at pH 5.25 to 5.5 [31]. Static binding capacity (SBC) with 50 mM sodium acetate (NaAc) at pH 5 and an additional 50 mM sodium chloride (NaCl) lay around 100 mg mL⁻¹ for four tested antibodies, and the total ionic capacity of the resin was reported as 37 to 63 µmol (H⁺) mL⁻¹ medium [31].

2.1.3. Experiments

All preparative chromatographic experiments were performed at a constant linear velocity of 100 cm h⁻¹. The following linear-gradient experiments (LGEs) and step-gradient experiments (SGEs) were conducted as detailed below:

• Low load density (low LD): Performed at low column load densities of 1 g L⁻¹ by injecting a small protein pulse onto the equilibrated column, followed by a wash step. Proteins were eluted using linear salt gradients of 20, 40, or 60 column volumes (CV);
• Medium and high LD: step elution and gradient elution (20 CV) were conducted at protein load densities of 20 g L⁻¹ and 93 g L⁻¹, respectively.

The buffer system consisted of NaAc combined with NaCl to control ionic strength. On-line, ionic strength was measured in conductivity. Equilibration and post-load washing steps were performed using a NaAc/NaCl buffer at pH = 5.0. The antibody isoforms’ pI values range from 7.5 to 9.5. Therefore, selecting buffer conditions at pH 5 ensured a sufficient difference from these values, ultimately enabling robust binding conditions. Elution was achieved using linear and step gradients with tailored ionic strength and buffer composition. On-line, protein concentration was measured at 280 nm in absorption units.

To conduct the in-depth analysis of the LGE at high LD, further information from the fraction analysis was deployed since the protein absorption data signaled an oversaturation of the detector. Besides identifying and classifying the isoforms, the fractions were also analyzed with size-exclusion chromatography (SEC)—the absorption signals did not indicate oversaturation and, therefore, were considered as the sum signal of the isoforms in the preparative chromatographic process. In addition, the high-molecular-weight (HMW) impurities were identified but were not regarded in particular—to some extent in this work while not at all in the original work. In those cases, analytical SEC of the fractions enabled the determination of the ratio of monomeric species to trace amounts of HMW species and their contribution to the absorption signal in the preparative process, respectively.

Guaranteeing the conservation of mass and tracking mass balances of the monomeric species enabled the calculation of conversion factors from protein concentration in g L⁻¹ to mAU, either from the UV signal in the preparative process or from the SEC analytics. The conversion factors allow the comparison between simulations and experiments.

2.2. Methods

2.2.1. Model

The transport-dispersive model (TDM), also referred to as the Lumped Rate Model with Pores (LRMP), was used to simulate the chromatography process. This well-established mechanistic framework [32] captures mass transport dynamics in packed chromatography beds, considering both column-wide and adsorber-specific transport phenomena. The mass balance within the interstitial column volume, where the solute concentration—$c_i^l$ [g L⁻¹] of species $i$—evolves over time and space is described as follows [12,32]:

$${\displaystyle \frac{\partial c_i^l}{\partial t}}=-u_{int}{\displaystyle \frac{\partial c_i^l}{\partial x}}+D_{ax}{\displaystyle \frac{{\partial }^2{c}_i^l}{\partial x^2}}-{\displaystyle \frac{1-{\epsilon }_V}{{\epsilon }_V}}{\displaystyle \frac{6}{d_p}}{k}_{eff,i}(c_i^l-c_i^p)$$

where ${\epsilon }_V$ is the void fraction of the packing; $d_p$ is the adsorber particle diameter; $D_{ax}$ is the axial dispersion coefficient; and $k_{eff,i}$ is the effective mass transfer coefficient.

Within the liquid phase of the porous beads, the solute concentration $c_i^p$ is as follows [12,31]:

$${\displaystyle \frac{\partial c_i^p}{\partial t}}={\displaystyle \frac{6}{{\epsilon }_p{d}_p}}{k}_{eff,i}\left(c_i^l-c_i^p\right)-{\displaystyle \frac{1-{\epsilon }_p}{{\epsilon }_p}}{\displaystyle \frac{\partial q_i^{ads}}{\partial t}}$$

where ${\epsilon }_p$ is the particle porosity and $q_i^{ads}$ [g L⁻¹] is defined as the load density of protein bound on the adsorber—implicitly involving information of the adsorber, phase ratio $A_s$ [m⁻¹], ionic capacity ${\Lambda }_{IEX}$ [M], maximal load density $q_{max}$ [g L⁻¹], and information of the modalities and the mobile phase.

The adsorption term, ${\displaystyle \frac{\partial q_i^{ads}}{\partial t}}$, is governed by the new binding model provided by the first author, and by ONYX (Onyx Biotech GmbH, Vaterstetten, Germany) in the future. This proprietary mechanistic framework relies on the following input parameters:

• Modality (net) charge [−];
• Modality size [nm];
• Specific adsorption parameter [−];
• Salt ion concentrations [M];
• Salt ion valencies [−];
• Size of hydrated salt ions [Å];
• Maximal load density [mg mL⁻¹];
• Ionic capacity [M];
• Phase ratio [m² mL⁻¹];
• Temperature [K].

By coupling the new binding model with the TDM, adsorption dynamics including protein interactions and elution profiles can be effectively captured.

2.2.2. Software

The system of partial differential equations was numerically solved using MATLAB 2024Rb. Time discretization employed the variable time-stepping scheme ODE15s, while spatial discretization was performed using the WENO scheme [33].

2.2.3. Parameter Estimation

Parameter estimation involved two main categories: (1) the fluid dynamic parameters described by the TDM framework; and (2) the input parameters required for the new adsorption model. For the established fluid dynamic parameters—including the axial dispersion coefficient $D_{ax}$ and column properties such as void fraction ${\epsilon }_V$ and particle porosity ${\epsilon }_p$—we relied on the values determined in the work of Briskot et al. [12]. These parameters were derived using well-known tracer pulse injection experiments with a high-salt buffer system and dextran. The effective mass transfer coefficient ($k_{eff,i}$) for the mAb was adopted from Briskot’s study, where it was calibrated experimentally across all isoforms [12]. For consistency, we applied the same value across all isoforms in our study.

In contrast to Briskot’s study [12], which used the literature data for the phase ratio, $A_s$, of a different resin, Capto S, we performed inverse size exclusion chromatography (iSEC) measurements specifically for Capto S ImpAct to determine the phase ratio and potentially calculate the ligand surface density ${\Gamma }_L={\Lambda }_{IEX} {A_S}^{-1}$. The measurements to determine the phase ratio were analyzed using a statistical model developed by DePhillips and Lenhoff [34], ensuring a precise characterization of the resin.

The new binding model introduces the estimation of key adsorption parameters. The charge of each isoform was calibrated using an inverse method by fitting the model output to experimental chromatograms. Additionally, a sensitivity analysis of the maximal load density, $q_{max}$, was required to systematically investigate the physical integrity of the parameter and the model’s accuracy; in doing so, the sensitivity analysis rationalized the maximal LD with the SBC/DBC specified by the manufacturer. Notably, the maximal LD was determined using the same buffer system as reported in Cytiva’s application note involving SBC [31] and DBC [35]. The mAb size was adopted from Briskot’s work, which was based on the established literature data [12]. Overall, the new adsorption model requires the calibration of one parameter, the charge per isoform, and an additional parameter across all isoforms, the system-specific adsorption parameter, and the accurate implementation of the physics-based parameter $q_{max}$.

Table 1. Predetermined and calibrated input parameters for the chromatography model consisting of the TDM in combination with the new adsorption model.

	Parameter		Unit	Value	Determined
TDM	$V_c$		mL	4.7	[12]
	$L_c$		mm	100	[12]
	$D_{ax}$		mm2 s−1	0.32	[12]
	$d_p$		µm	50	[12]
	${\epsilon }_V$		-	0.38	[12]
	${\epsilon }_p$		-	0.83	[12]
	$k_{eff,i}$		µm s−1	1.96	[12]
	Conversion factor, low LD		mAU L g−1	382–439	*
	Conversion factor		mAU L g−1	294.5	*
ADSORPTION	$q_{max}$ (Maximal LD)		mg mL−1	95	[31,35] **
	${\Lambda }_{IEX}$ (Ionic capacity)		µmol(H+) mL−1	63	[31]
	$A_s$ (Phase ratio)		m2 mL−1	160	iSEC
	Temperature		K	298	-
	Counterion size		Å	3.6	[36]
	Co-ion size		Å	3.3	[36]
	Counterion valency		e (1.6 × 10−19 C)	+1	-
	Co-ion valency			−1	-
	mAb charge	Main		+84.1	***
		Basic 1		+84.6	****
		Basic 2		+85.1	****
		Acidic 1		+83.6	****
		Acidic 2		+83.2	****
		Acidic 3		+82.7	****
		Acidic 4		+82.2	****
	System-specific adsorption parameter		-	−9.051	***
	mAb size		nm	5.5	[12]

* via mass balances, in mAU L g⁻¹: 382 (40 CV), 439 (20, 60 CV), 294.5 (all others); ** additional sensitivity analysis of manufacturer specifications; *** calibrated on 1 LGE with low LD; **** calibrated on 1 LGE with high LD.

summarizes all key data relevant to our study, including the input parameters required for the TDM and the new adsorption model. It indicates whether each parameter was determined through experiments, calibrated, or directly taken from prior studies, which performed conventional experiments.

In Briskot’s study [12], four parameters per isoform cluster and two parameters across all isoforms—totaling fourteen parameters for three clusters—were fitted to experimental chromatograms using a combination of global and local optimization techniques. While this method ensures comprehensive parameter estimation, it requires significant computational effort and relies heavily on advanced optimization algorithms.

In contrast, and for the sake of comparison in terms of isoform clusters, our approach capitalized on the low number of fitting parameters, one per isoform cluster and two parameters across all isoforms—totaling five parameters in the new model—and their distinct physical meaning, allowing for simple manual fitting. This simplified the estimation process significantly as parameters like the charge of the protein and its actual seven isoforms could be quickly and intuitively approximated based on their well-understood physical properties. This approach not only reduced complexity but also streamlined the parameter estimation process without compromising accuracy. In the future, implementing local and global optimization algorithms can automate and enhance this process.

3. Results

In the first part, the monomeric species as one component are investigated—as in the original work—with low-load-density LGEs (Section 3.1.1) and medium- and high-load-density LGEs and SGEs (Section 3.1.2). In addition, the first part demonstrates how the new adsorption model captures the high sensitivity of the elution behavior toward salt concentration (Section 3.1.3). The second part explores the dataset in depth and goes beyond the investigation of the original work; herein, the monomeric isoforms are studied in detail (Section 3.2). Demonstrating the capability of the new adsorption model to describe seven antibody isoforms (Section 3.2.1) led to the investigation of the sensitivity of the elution behavior toward the parameter maximal load density (Section 3.2.2). Moreover, self-consistency with the low-LD LGE used for initially calibrating the monomeric species altogether is shown (Section 3.2.3). Lastly, an optimal process run through model-based process development is highlighted (Section 3.2.4).

3.1. Results of Monomeric Species

3.1.1. Three LGEs with Low Load Density

As the phase ratio is specified and the size of the ions and the modality are accounted for, the calibration involves only one LGE to determine the charge and the system-specific adsorption parameter of the monomeric species. Figure 1A compiles three chromatograms involving process runs at low load densities of 20, 40, and 60 CV. Different salt gradients lead to different elution behaviors; therefore, the run at 60 CV was used to calibrate the adsorption model and to determine the charge and the system-specific adsorption parameter—and while retention is perfectly described, there is visually some deviation between the simulation and the experimental data concerning the peak form. In this regard, it is important to note that only two parameters are calibrated on a single process run, and all the other parameters are predetermined. That said, it is remarkable that the calibrated model facilitated the prediction of the runs at 20 CV and 40 CV.

3.1.2. SGE and LGE at Medium LD, and SGE at High LD

The calibrated model accurately predicted different operating conditions and load densities (Figure 1B–D). By doing so, the retention time at medium and high LDs of the three experiments is predicted correctly—especially the elution behavior of the two SGEs is predicted with high precision. In Figure 1D, it is shown how well the simulation agrees with the asymmetrical elution behavior for the majority of the monomeric species. At high LD in Figure 1B, the UV detector is oversaturated; hence, the signal is cut at 3000 mAU, and only the peak’s sharp rise and smooth fall can be compared to the simulation. Nonetheless, it is noteworthy how precisely the model predicts the rear part of the peak. Eventually, the model predicts SGEs even at high LD with high sophistication based on only one LGE with low LD (Section 3.1.1).

3.1.3. Sensitivity Analysis at Medium and High LDs

The sensitivity analysis showcases how the high-fidelity model can assess the robustness of the operating conditions. Figure 2A,B show simulations of salt variations in the elution step at 0.210 M ± 0.008 M (±4%) compared to the corresponding SGE at 0.210 M and medium LD. Such slight variations already change the elution behavior and set off the good agreement of simulation and experimental data, as illustrated in Figure 1D. Simulating a concentration 8% below the operating conditions (−0.017 M) significantly changes the peak form and stretches it out so that the elution of a small proportion of the antibody is predicted at the second elution step (Figure 2C). Eventually, the artificial drop in the salt concentration, simulated in the pursuit of the sensitivity analysis, can explain the experimental data at the second elution step; the analysis of the corresponding fraction indeed indicates 54% monomeric species compared to 46% HMW species, while only the former ones are simulated in this work.

A similar trend is observed at high LD when the salt step is reduced by 8% (Figure 2D); however, the agreement between the experiment and simulation is not as disrupted as in the case of medium LD (Figure 2C). Intriguingly, trace amounts of mAb are also predicted at the second elution step, adding the same unexpected element to the findings. Concretely, measurements of the corresponding fraction indicated 47% monomeric species compared to 53% HMW species.

3.2. Results of Isoforms

The characterization of the isoforms has not been investigated in-depth in the original work but is the second part of this work. To understand the complex elution behavior, additional data from the fraction analysis is included when studying the LGE at high LD.

3.2.1. Valid Description of the Complex Elution Behavior in an LGE at High LD

Fraction analysis involving purities of the main species between 41 and 65% enabled the construction of the chromatogram and pseudo-chromatograms. In Figure 3, experimental data and simulation results are shown. The experimental data are obtained by partitioning the sum signal into pseudo-chromatograms with the acidic, basic, and main isoforms. For instance, the fraction at 120 mL contains 26% acidic, 64% main, and 10% basic species, which roughly translates to 1000, 2600, and 400 mAU once accounting for the sum signal of 4000 mAU. To complete the description of the experiment, trace amounts of the HMW species have been identified to elute at roughly 138 mL, and since the impurities elute after the basic isoforms, these are neglected in the analysis of the charge variants in this work—as they have been in the original work.

The pseudo-chromatograms of the simulation are constructed similarly to the experimental ones. However, one more layer is added underneath as an assumed number of isoforms of each cluster is taken for simulation; thereby, the chromatographic process is simulated with four acidic, one main, and two basic species that each hold slightly different charges (Table 1). Moreover, the four acidic species and the two basic species are each lumped together to build the sum of the acidic species and basic species. In the last layer, those two sum signals and the main species build the sum signal of all monomeric species.

Eventually, the sum signal of all the isoforms and the sum signal of the pseudo-chromatograms of the lumped acidic, main, and lumped basic isoforms are used to assess the agreement between simulation and experimental data. In Figure 3, it can be evaluated how well the simulation explains the complex elution behavior seen in the experimental data by the following:

• Describing the asymmetrical elution profile of the isoforms altogether, the overall sum signal, over more than ten column volumes;
• Describing the asymmetrical elution profile of the main and basic species;
• Describing the occurrence of the plateau of the acidic species;
• Describing the sharp bend in the sum signal of all isoforms at the beginning of the elution;
• Emphasizing that the operating conditions are indeed at high LD as the simulation indicates a small breakthrough of the acidic and main species.

3.2.2. Sensitivity Analysis of Maximal LD ($q_{max}$)

The sensitivity analysis emphasizes the tremendous impact of the maximal LD on the process under high-LD conditions. Concretely, it showcases how sensitive the elution behavior reacts to different ratios of LD to maximal LD. In Figure 3, the simulation involves $q_{max}$ = 95 mg mL⁻¹ while the LD is 93 mg mL⁻¹. In Figure 4A–D, slight changes in the maximal LD are shown with 90, 93, 97, and 100 mg mL⁻¹. Two significant things can be observed, as outlined below:

• At 90 mg mL⁻¹, a significant breakthrough seems evident from the LD being 93 mg mL⁻¹ (Figure 4A). There is some breakthrough at the edge case where maximal LD and LD are the same (Figure 4B). At 97 mg mL⁻¹ (Figure 4C) and 100 mg mL⁻¹ (Figure 4D), the simulation indicates a breakthrough that is negligibly low;
• The more significant the breakthrough, the smoother the peak front of the main species becomes and the lower its peak maximum. Such as, the peak front of the lumped acidic species becomes smoother. On the other end, at 100 mg mL⁻¹, the form of the elution profile changes, and the plateau of the acidic species is lifted up and in less favorable agreement with the experimental data. The elution behavior of the basic species seems not to change.

As pointed out in Section 4.2.2, these quantitative results of the sensitivity analysis provide strong evidence for the high fidelity of the adsorption model.

3.2.3. Re-Engineering of Calibration by Accounting for Charge Variants

Closing the loop and re-engineering the chromatographic process used for calibration is the ultimate test to verify the integrity of the adsorption model. The question is whether incorporating the details on the charge variants provides the same result as the chromatographic run used for calibration at the beginning. Figure 5A shows the simulation results of the seven isoforms and the sum signal at 60 CV, both at low-LD conditions. Comparing the sum signal to the signal of Figure 1A shows that the simulation results are visually congruent.

3.2.4. Process Optimization

After the various and intense scientific analyses of the model quality in the previous subsections, the prospect of conducting process optimization with the new adsorption model is promising. Based on the parameter calibration of the charge variants on the LGE at high LD (Section 3.2.1), two new elution steps were identified to increase the purity of the main species. In Figure 5B, the first step facilitates the elution of the majority of the acidic species while simultaneously more acidic species are eluted than the main species. Further, only a tiny amount of the acidic species elutes in the second step, while the main species contributes the majority to the sum signal of all the monomeric species.

4. Discussion

This section elucidates how, in what way, and why the proposed adsorption model is superior to current adsorption models. Briefly, it reduces the calibration effort significantly, which goes hand-in-hand with the capability to predict very accurately—no matter if only the entirety of the monomers is the subject of interest (Section 4.1) or the very fine distinction between the monomeric isoforms (Section 4.2). Both studies are based on coherent data provided at one pH condition.

How is the model better in describing monomeric species?

The high-fidelity adsorption model only requires one process to be run to calibrate the adsorption model of the monomeric species; thereby, the load density of the calibration run can be low (Section 4.1.1). In the end, the model can predict other gradient experiments with low load density (Section 4.1.1), and gradient and step experiments with moderate and high load densities (Section 4.1.2 and Section 4.2.1). The prediction of the linear-gradient experiments is especially remarkable as the elution behavior is described with an accuracy of a few millimolars.

In what way does the model offer advantages over the CPA model in describing monomeric species?

The high-fidelity model better describes the adsorption mechanism and holds only two degrees of freedom: the charge of the protein species and the system-specific adsorption parameter (more precisely, it is three parameters in total when including the effective mass transfer that the original work has already determined). When describing only the monomeric species altogether at low-to-moderate LD, there are 50% (3/6) fewer degrees of freedom in the new adsorption model compared to the CPA model so the calibration effort is reduced.

On the other hand, a direct comparison for the different antibody isoforms can hardly be made as the CPA model’s efficiency in calibrating with its capability in predicting is poorly discussed in the original work. In fact, attaining high calibration efficiency with one species is just a sliver of light for a much larger problem as some degrees of freedom in the models actually scale with the number of species—a significant challenge in model-based process development, which we discuss in detail.

Remarkably, differentiating between the isoforms in the new adsorption model is performed by only one parameter—the charge (while the system-specific adsorption parameter and the effective mass transfer can be neglected for such slight deviations in the molecule and are considered to be constant across all isoforms). Calibrating the adsorption model only on one parameter per isoform is the physical and mathematical minimum to describe the invariance between product and impurities. In comparison, the CPA model involves four parameters per isoform at one pH condition. When simplifying and only considering the sum of the acidic and basic species besides the main species, this already scales up to 12 (+2) degrees of freedom for the three clusters of isoforms (see Table 1, columns “mAb4” in Briskot et al. [12], when neglecting two additional degrees of freedom that do not scale with the number of isoforms: the effective mass transfer, and the specific adsorber surface area). As the original work did not discuss the calibration and validation of the model parameters with this dataset in depth, one may only guess the effort and effectiveness ranging from 4 (+2) degrees of freedom for the lumped monomeric species at one pH condition up to 18 (+2) for the clustered acidic and basic species at three pH values (see Table 1, columns “mAb4” in Briskot et al. [12]). Once the actual number of isoforms with four acidic and two basic species besides the main species is considered, there are actually 28 (+2) degrees of freedom in the model—simultaneously, the new adsorption model with 7 (+2) degrees of freedom reduces calibration efforts by 70% (9/30) when accounting for independent experiments like various chromatographic runs and fraction analysis required for calibration—and once two more pH values are considered, two more parameters per isoforms are potentially totaling 42 (+2) degrees of freedom in the CPA model.

Nevertheless, calibrating multiple degrees of freedom makes model-process development labor-intensive in the wet lab—contradicting its intended purpose. More problematically, the high number of degrees of freedom leads to significant overfitting risk and reduced predictive capability.

Overfitting is the use of models that include more terms than are necessary or use more complicated approaches than are necessary. The goal is to build a model with deliberate degrees of freedom and key predictors that can also be used by anyone operating with the model. On the other hand, adding degrees of freedom that perform no useful function means that in future use of the model to make predictions, the operator needs to measure and record these predictors to estimate their values in the model [37]. This procedure unnecessarily increases efforts in model-based process development, in silico through complicated numerical solution strategies and uncertainty assessments, and in the lab through the requirement of numerous experiments and the waste of materials. The high intellectual and materialistic resource consumption is relevant for one dataset in the short term and scales with future datasets in the long term.

Further, it introduces additional sources of error, as every predictor is also prone to error when conducting the estimating procedure, expanding the model’s compound error and leading to prediction mistakes. Adding irrelevant degrees of freedom can worsen predictions because the coefficients fitted to them add random variation to the subsequent predictions [37].

Why is the model better in describing monomeric species?

The reason why the new adsorption model requires minimal efforts to calibrate and is capable of predicting lies in its high fidelity characterized by the following:

• Being closer to reality and very well describing the actual adsorption mechanism;
• Incorporating biophysical properties with physical integrity as predictors: in addition to the size of the modality, the model involves the maximal load density, the valences and sizes of the salt ions, the temperature, and the phase ratio of the material.

While in the CPA and the SMA model, parameters with fitted values aim to capture the effects at high load densities—an operating condition that leads to complex elution behaviors as noted in the original work [12]—the new adsorption model incorporates the maximal load density with physical integrity (Section 4.2.2). Considering this critical parameter provides not only more information a priori but also very sensitive information, significantly lowering the number of degrees of freedom; hence, significantly less calibration is needed. On the other hand, the sensitivity analysis of the maximal LD demonstrates the high fidelity of the model. It proves its excellent capability to predict, which is also demonstrated by the sensitivity analysis for the slightest salt concentration changes (Section 4.1.3).

Lastly, other models may struggle with self-consistency; once the analysis of the charge variants has been conducted, the reconstruction of the sum signal of the monomeric species fails to provide the same result as in the initial calibration run. As in Section 4.2.3, the new adsorption model is self-consistent and provides back the calibration run.

4.1. Discussion of Monomeric Species as a Whole

The charge and the system-specific adsorption parameter are the only two parameters to calibrate and ultimately describe the elution behavior for the monomeric species, and provide a good guess for the main isoform. As both parameters are distinctive in the prospect of separating via chromatography, it is best practice to calibrate them at the operating conditions where the elution behavior is impacted most sensitively—ideally, through a very long gradient or under isocratic conditions. At these conditions, the relationship between elution salt and elution behavior can be resolved at its finest. In this dataset, an LGE of 60 CV has been identified to fulfill this criterion.

4.1.1. Only One Low-LD LGE Enables Prediction

While in the original work [12], the simulation describes somewhat better the peak form, it matches the experimental retention time somewhat worse. While this work considers absorption data at 280 nm, the original work used a time series of the detector signal at 290 nm; and as the conversion factor to relate the concentration of injected protein to the absorption unit is not provided by those authors, a valid comparison between the elution profiles cannot be made.

However, there is a far more important distinction worth noting: the original work does not claim to predict any chromatographic runs of the dataset but rather assesses the model’s capability to describe elution behaviors to be decent by stating that there is good agreement of experimental and simulated data. In this work, all the experimental data, gradients, and steps, at low or high LD, were predicted based on one calibration at 60 CV at low LD (Section 4.1.2 and Section 4.1.3). Further, it gives a good guess for the main isoform (Section 4.2.1) and will prove the self-consistency of the new adsorption model (Section 4.2.3). In the beginning, the capability to predict the elution behavior at 40 and 20 CV is showcased in Figure 1A. It is also shown how the simulated elution peak becomes wider in relation to the corresponding experimental elution peak the steeper the gradient—something that can be related to an inaccurate value for axial dispersion. In this work, we focus on adsorption and, therefore, simply selected the value for axial dispersion determined in the original work [12]. To find a better agreement, fine-tuning of the axial dispersion could be performed. Despite this, the two adsorption parameters play out to be calibrated very accurately as we will discuss in the following subsections.

As salt gradients dominate the elution behavior over the fluid dynamics (dispersion and mass transfer) the longer they are, these become better operating conditions compared to SGEs to assess the accuracy of an adsorption model in regards to predictability. In other words, the elution behavior depends more on the contributions by the adsorption and is somewhat more decoupled from fluid dynamic effects (mainly convection contributes when salt conditions change very smoothly).

In what way is the model close to the theoretical and mathematical optimum in describing monomeric species?

The efficacy of the new model by describing the chromatographic process with two adsorption parameters becomes more striking when compared to the absolute minimum that is required to describe a symmetrical peak since mathematically, a symmetrical peak follows a Gaussian distribution that in turn holds two degrees of freedom: the mean and the variance. Describing the chromatographic process by two adsorption parameters and convection is almost at this mathematical minimum (when effectively reducing the TDM to the ideal chromatography model for this theoretical comparison by neglecting dispersion and mass transfer, which connects adsorption and convection in the TDM). It becomes particularly illustrative when you compare the cases of loading and eluting. During loading, there obviously is no elution peak to evaluate as the buffer conditions favor adsorption of the applied species over desorption—further speaking in mathematical terms, this is why the initial condition of the injected pulse, including pulse height and width, is forgotten to some degree; hence, convection alone does not lead to a peak at the end of the column. Only when an eluting agent is applied, a peak is obtained. In that manner, retention time and peak width are described by at least one more parameter, the adsorption parameter. For sure, convection is required for the evolution of the species through the column; however, there exists only a retention time when desorption occurs and, therefore, convection intermingles somewhat with adsorption. Concretely, the minimal way to describe the interplay requires one adsorption parameter that perfectly relates the terms ${\displaystyle \frac{\partial c_i^l}{\partial t}}$ and ${\displaystyle \frac{\partial q_i^{ads}}{\partial t}}$ (Section 2.2.1) to each other, at any operating condition, while guaranteeing an accurate description of the symmetrical elution peak. The new adsorption model is very close to this and eventually holds two parameters that accurately intermingle with convection (and also dispersion and mass transfer) to obtain the retention time and peak width. In light of this theoretical comparison, the description and the prediction of the physical chromatographic process can be performed with only three parameters, convection and two adsorption parameters, which is outstandingly close to the absolute minimum of two parameters for a Gaussian distribution.

4.1.2. The Single Low-LD LGE Also Enables Prediction of Elution Profiles at Medium and High LDs

The accurate prediction of the elution behavior at 40 and 20 CV at low LD already provides a sneak peek at the new adsorption model’s high capabilities (Section 4.1.1). In this subsection, three out of four theoretical hypotheses on how transport of the monomeric species occurs in the column are showcased as LD increases: the LGE and SGE at medium LD; and the SGE at high LD.

Based on the single LGE with low LD, the LGE with medium LD is predicted sophisticatedly and provides further validation to the new model once a greater mass of the antibody is applied. Next, the SGE at medium LD demonstrates how the adsorption model predicts the elution behavior at the operating condition, where it correctly rationalizes the interplay between adsorption and fluid dynamics (all the terms in the two equations of Section 2.2.1 contribute to the transport once the salt conditions change suddenly). Lastly, at high LD, more complexity is added to this interplay as the applied antibody mass is potentially limited by a maximal load density. Eventually, the adsorption model captures the interplay of adsorption and fluid dynamics of the SGE at high LD in a sophisticated manner. Based on the calibration of that one LGE at low LD, it predicts the sharp rise and smooth tailing with high precision (Figure 1B).

From a logical standpoint, breakthrough of the antibody would occur once the column is overloaded. However, the maximal LD can also lead to complex elution behaviors once adsorption more significantly impacts the transport in the column in the LGE, the fourth scenario, where the elution behavior cannot be described by the entirety of the monomeric species anymore (Section 4.2.1).

4.1.3. High Salt Sensitivity of the High-Fidelity Adsorption Model

As the prediction of SGEs and especially LGEs demonstrates high accuracy of the adsorption model, the sensitivity analysis showcases its remarkable precision. The sensitivity analysis reveals that the process in this elution step is highly responsive to changes in the salt concentration of a few to several millimolars, as evidenced by a significant change in the elution behavior at medium LD once the ionic strength drops by 0.017 M (Figure 2C). This precision is further underscored by the prediction of a small proportion of monomeric species eluting in the second step, which aligns with the experimental data. In essence, the simulated result suggests that deviations in the salt concentration might have occurred in the experiment—to be precise, the drop in salt concentration in the SGE potentially occurred some time after the elution front since the simulation actually predicts an elution peak that is slightly higher than the experimental peak in the medium-LD SGE (Figure 1D). Conversely, the simulation provides valuable insights and suggests specifications regarding mobile phase preparation and effective mixing capabilities in the technical process. The model’s precision in prediction can be used to analyze operating conditions concerning their robustness, being useful for scaling from lab to industrial scale and for planning experiments.

Sensitivity analysis has proven to be an effective method to scientifically showcase that the new adsorption model is highly accurate. Furthermore, the approach reinforces the evidence that the model is of high fidelity since the model even captures the slightest concentration fluctuations and vice versa, enabling the development of robust processes.

4.2. Discussion of Isoforms

The characterization of the charge variants has not been investigated in depth in the original work [12] but is a major part of this work. Therefore, additional data from fraction analysis from the LGE at high load density are included to understand the complex elution behavior. In the following subsections, it is pointed out how the high-fidelity model with only one degree of freedom per isoform very sensitively describes the complex elution of various charge variants.

4.2.1. The High-Fidelity Model Is Able to Describe Very Complex Elution Behaviors at High LD, Rationalizing with Charge Variants

As the ionic capacity, the phase ratio, the sizes of ions and antibodies, and the maximal LD were predetermined, accounting for four acidic, one main, and two basic species with slightly different charges describes the complex elution behavior in the LGE with high LD in a highly sophisticated manner; therein, one parameter—the charge—distinguishes between the charge variants. Eventually, the binding conditions of the antibody isoforms, ensured by a sufficient difference between the buffer pH and their pI values, are quantified by the model, with the net charges of the relevant isoforms ranging narrowly from +82.2 to +85.1. Consistency is given as the calibrated values of the net charge grow incrementally from the most acidic to the most basic species, each by roughly +0.5 (Table 1).

The very good agreement of the simulation with the experimental data incorporates all relevant information, including the exact number of charge variants identified. With different information or in the case of other impurities, there are further strategies to obtain a good agreement between simulations and experiments. In general, it can be a good approach to either lump together some impurities to limit the degrees of freedom or to take into account the mass ratio of the individual species when given.

Based on only one degree of freedom per isoform, the adsorption model can eventually describe the complex elution behavior, ultimately fostering its high fidelity. With the calibrated model, the full potential of the simulation can finally be unlocked by performing robustness tests (Section 3.1.3) and optimizations (Section 3.2.4).

By having demonstrated the high fidelity of the adsorption model and its capability to describe charge variants at high LD, it provides strong process design capabilities while demanding minimal experimental effort. In this sense, one low-LD LGE to calibrate the adsorption parameters of the monomeric species (Section 4.1.1) in combination with the strong theoretical assumption of incremental changes in the charge of a reasonable number of isoforms can be used to separate charge variants a priori, without conducting labor-intensive fraction analysis. Performing this and adding one more calibration experiment to determine the effective mass transfer to that one low-LD LGE reduces the experimental effort to two simple experiments—with little material consumption, minimal labor and a reduction by one order of magnitude compared to the CPA model, whether isoform clusters are surrogated (2/18) or seven individual species (2/30) are incorporated (Section 4: In what way does the model offer advantages over the CPA model in describing monomeric species?).

4.2.2. The High Fidelity Lies in the Physical Integrity of the Parameter Values Considered in the Model Like $q_{max}$ Specified by Manufacturers

Besides changes in the salt concentration (Section 4.1.3), changes in the process parameter $q_{max}$ also play out to have a decisive impact on the chromatographic process. Considering this process parameter when designing a chromatographic process can be a decisive element. While variations in the column parameters, phase ratio, and ionic capacity also influence the process, they do not as sensitively; that said, the upper value of the ionic capacity specification is considered acceptable, as the typical range for ion-exchange resins lies between 100 and 200 µmol mL⁻¹ [38]. Eventually, the discussion is limited to the critical parameter $q_{max}$.

At first, it is evident that there is a breakthrough at 90 mg mL⁻¹ as the LD is 93 mg mL⁻¹ (Figure 4A). Secondly, this result elucidates a lot about the quality of the adsorption model once $q_{max}$ is compared to the measurements by the manufacturer, SBC [31] and DBC [35]. The manufacturer only provides rough measurement values of four antibodies’ SBC with around 100 mg mL⁻¹; interestingly, the experiments have been conducted under the same buffer and salt conditions as in the dataset of this work, 50 mM NaAc at pH 5 and an additional 50 mM NaCl. On the other hand, the DBC value for a mAb, one of the four antibodies—also measured under the same buffer and salt conditions as in the dataset—is given. The DBC allows for a quantitative comparison between $q_{max}$ in the simulation and an exact DBC_10% value; although this value of 109 mg mL⁻¹ already indicates a breakthrough of the antibody where 10% of the concentration loaded onto the column is reached, it provides a first good guess to simulate the process. In other words, the maximal capacity of the column to bind an antibody must be somehow lower than the DBC_10% value—something that can also be derived from its and three other antibodies’ rough SBC values. Indeed, a good agreement between the simulation and experimental data is given at $q_{max}$ = 95 mg mL⁻¹ (Section 3.2.1).

Eventually, not only can DBC measurements be used to provide a good guess for the maximal load density as an input parameter for the simulation, but more importantly, they provide strong evidence of how accurately the adsorption model describes reality. A deeper discussion of the sensitivity of $q_{max}$ further underscores the validation:

Bringing together both significant observations of Section 3.2.2 elucidates that lowering $q_{max}$ better describes the forms of the pseudo-chromatograms, despite leading to a breakthrough. On the other hand, increasing $q_{max}$ prevents the breakthrough and indicates a bend in the signal of the acidic species, contributing to the formation of the bend in the sum signal at the beginning of the elution; however, the forms of the pseudo-chromatograms do not agree as well anymore. Consequently, it is fair to conclude that the reality lies somewhere in between. As the actual value of $q_{max}$ is considered, it is remarkable to note how closely the simulation matches reality while physical integrity is conserved and the adsorption model is, in turn, validated. Furthermore, it is pointed out how sensitive the process is to the maximal LD, and specifically to the ratio of LD to maximal LD, highlighting the importance of proper analysis of this critical parameter when designing a chromatographic process or developing new resins.

4.2.3. The High-Fidelity Adsorption Model Is Self-Consistent

The re-engineering of the LGE with low LD has been successful; in so doing, the process condition in the dataset involving the longest gradient and highest sensitivity toward adsorption (Section 4.1) has been selected as it also provides the most sensitive condition to test the self-consistency of the model. That said, the adsorption model is indeed self-consistent—a remarkable result.

In this sense, the initial calibration run provided a good guess for describing the complex behavior in the LGE with high LD; then, additional information about seven isoforms was incorporated, and the sensitive impact of the maximal load density was included. Closing the loop and eventually describing the LGE with low LD while incorporating all the information ultimately proves the self-consistency of the model.

To be very accurate, the guess of the charge in the main species, obtained in the low LGE, is a pseudo-charge as it describes all monomeric species together. Consequently, the charge of the main species in the LGE with high LD theoretically has to be different from the pseudo-charge of all the monomeric species in the LGE with low LD. Technically, it demands an iterative process where the good guess (one pseudo-charge) of all the monomeric species must agree, to a sufficient degree of accuracy, with the weighted average of the seven charge variants (seven charges). In this study, the charge of the main species is finally set as equal to the pseudo-charge—a fair approach since most of the mass is the main species while holding a higher charge than acidic and a lower charge than basic species—somewhat forming a weighted average. Comparing Figure 1A and Figure 5A shows that the simulations are visually indistinguishable; therefore, the approach is already sufficient, and further iterations and fine-tuning to increase overall accuracy can be discarded.

4.2.4. The High-Fidelity Model Potentially Unfolds the Actual Power of Simulations: Optimizing Processes (Beyond Plausibly Describing Experimental Data)

Using the adsorption model unfolds the actual power of simulations and identifies a better process than the original LGE with fractions, which obtained the main species at only 41–65% purity. In performing this, the simulation of an SGE predicts an optimized process with the purity of the main component above 65% during the entire second elution step. Further fine-tuning of the optimization result may lead to an even better trade-off between purity and yield, while limiting the loss of main species during the first elution step. That said, this simulation result seems promising in optimizing the chromatographic process by a mathematical model that has been proven capable of rationalizing the biophysical information of charge variants (Section 4.2.1, Section 4.2.2 and Section 4.2.3). In the next step, fine-tuned simulation results could be validated by experiments.

At a broader level, when combined with robustness tests, the high-fidelity model emerges as a potent tool to mitigate risks during process development since maintaining consistent charge profiles throughout development and commercialization is not a given. It requires the implementation of appropriate control strategies, which are necessary to mitigate the safety and efficacy risks of the product [39]. By rationalizing the biophysical properties of the charge variants through the high-fidelity model, we potentially ensure that the charge profile is maintained at an optimal and controlled level, thereby enhancing product safety through the process. Exploring this through both simulations and experiments could be valuable in future work.

5. Conclusions

Based on the calibration of only one LGE at low LD (Section 4.1.1), the high-fidelity model successfully predicts monomeric species in LGEs and SGEs at medium and high LDs (Section 4.1.2) and provides a good guess for the main species when analyzing charge variants (Section 4.2) while being self-consistent (Section 4.2.3).

In Section 4.1.3, the sensitivity analysis has proven to be a suitable scientific approach to demonstrate the high fidelity of the adsorption model by showcasing its high accuracy in predicting the slightest changes in the salt concentration. In addition, the sensitivity analysis of the maximal LD (Section 4.2.2) hardens the evidence that the adsorption model is of high fidelity as it incorporates the parameter of maximal LD with physical integrity; thus, the parameter is directly comparable to experimental DBC and SBC values. On the other hand, the adsorption model rationalizes the significant impact of the two parameters and can be used to determine the salt concentration or ratio of LD to maximal LD in the pursuit of designing the process; or to determine $q_{max}$ values for a broad range of applications in the pursuit of designing the material. Due to their sensitivity, the two critical process parameters are also well-suited for robustness testing.

Overall, the model-based process development is demonstrated. Not only is the plausibility of the complicated elution behavior of an LGE at high LD based on the biophysical properties of the chromatographic process (Section 4.2.1) but also an improved separation of charge variants through two elution steps is presented (Section 4.2.4), leading to higher purity of the main antibody species than in the LGE. The high-fidelity model facilitates optimizations of the chromatographic process and robustness tests, and may assist in maintaining acidic and basic species within a controlled range, crucial in ensuring safety and efficacy throughout early-phase and late-phase development and commercialization [39]. Having demonstrated high fidelity, the process design and risk assessment in terms of charge variants can also be performed a priori, without labor-intensive fraction analysis. The calibration effort reduces to two simple experiments—with little material consumption and less labor required—and drops by one order of magnitude compared to the CPA model.

Since the high-fidelity adsorption model in combination with the fluid dynamics model is based on biophysical properties of the target modality and relevant impurities, explicitly their charges and sizes, the separation system, the adsorbent, and the mobile phase, it makes it a promising tool in rationalizing process development for recombinant proteins, monoclonal antibodies, and the emerging demand of new modalities including the myriad of drug candidates discovered with artificial intelligence.