Leveraging Multidimensional Modeling to Resolve Frequent Separation Challenges in HPLC

Starting with the pioneering work of Laub and Purnell in the 1970s, gas chromatography (GC) and high performance liquid chromatography (HPLC) modeling has evolved into a key research area aimed at supporting industry practitioners by providing a deeper understanding and helping to rationalize time and resource use (1).

Today, analysts can choose from a variety of method development tools, some with distinct modeling strategies and areas of application. The advantages and limitations of these tools—as well as when and how to apply the most appropriate approach—should be carefully evaluated. For example, fully empirical design-of-experiment (DoE) tools rely on larger experimental data sets to establish complex factor–response relationships and are extremely useful in scenarios where no supporting theoretical background is available. Other tools use molecular descriptors—either computationally derived or empirically determined—to predict chromatographic behavior, but these may lack accuracy, robustness, or the quick responsiveness required in real-world applications.

A different modeling approach has been implemented that prioritizes the application of fundamental chromatographic principles (first‑principles) to correlate experimental parameters with modeling responses. One of the main advantages of this approach is its efficiency: it requires only a limited number of initial experiments (typically two or three per factor) to calibrate a highly flexible and descriptive model. Once calibrated, the model can accurately depict complete separation patterns, making it valuable not only for method development but also for system characterization and comparative analysis.

Comparative Design Space (DS) Modeling Studies

When performing any type of modeling, it is imperative to select meaningful parameters, define appropriate experimental ranges, and conduct experiments in a controlled and reproducible manner. In this context, multidimensional separation models—such as gradient time (t_G) and temperature (T), along with their three-dimensional (3D) extensions incorporating ternary organic composition (t_C), pH, or additive concentration (a_C)—have proven particularly valuable in method development in pharmaceutical research. For example, a t_G-T model only requires a 2 × 2 = 4-run DoE. When a third non-linear parameter is introduced—such as pH, t_C, or a_C—the experimental design expands to 2 × 2 × 3 = 12 input runs (Figure 1).

The selection of parameter ranges is guided by both software recommendations and fundamental chromatographic principles, including compatibility between the stationary phase, mobile phase, and elution mode. The modeling process typically begins by defining the separation challenge. This is followed by outlining suitable experimental setups, which are then subjected to in-depth modeling to capture the inherent dynamics of separation (Steps 1–3).

While this approach effectively accounts for all possible parameter combinations within a given separation system, the impact of categorical variables—such as different stationary phases or instrument-to-instrument variability between LC systems—has not yet been systematically addressed. To bridge this gap, recent research has focused on the development of the design space comparison (DSC) tool (Step 4), which has expanded the practical utility of multidimensional modeling in method development, robustness assessment, and column selection.

Application Examples

Study 1: Column Interchangeability Studies (2,3)

Unquestionably, selectivity is the most influential term in Purnell’s fundamental resolution equation (4). Therefore, understanding and controlling selectivity is crucial in HPLC separations. During the early stages of method development, exploring stationary phases with distinct (“orthogonal”) selectivities can facilitate the detection and control of unknown impurities in a pharmaceutical sample. However, during method validation, it becomes essential to identify backup or replacement columns with identical or similar selectivities to prevent delays in drug release due to potential supply interruptions of the primary column.

In a recent study five common 5 × 0.21 cm USP L1-type C18 stationary phases with varying physicochemical properties were directly compared. Following 12 calibration experiments per column, red-colored method operable design regions (MODRs) indicating baseline separations (R_s,crit.≥ 1.5) were identified, albeit with certain changes in the overall chromatographic landscape, such as elution windows, changing critical peak-pairs, and even some elution order alterations.

In Figure 2, the two most distinct stationary phases are compared, while maintaining identical column formats and modeling conditions. The HSS C18 phase is characterized by high surface coverage (3.2 µmol/m²), whereas the HSS C18 SB phase represents its residual silanol-rich counterpart (1.6 µmol/m²). The reasonably orthogonal selectivity contributions of these phases were also confirmed using the Tanaka test, as well as by evaluating their large difference in similarity factors (F_s >> 5) based on the Snyder-Dolan Hydrophobic Subtraction Model (HSM) database. Despite these pronounced physicochemical differences, alignment of the modeled MODRs revealed a shared region where equivalent separation conditions could be established (t_G = 12 min, T = 30 °C, pH = 2.5). Under these conditions, both phases demonstrated interchangeable chromatographic performance, not only with respect to selectivity and resolution but also in terms of comparable elution windows.

Study 2: Batch-to-Batch Reproducibility Study (5)

Column manufacturing, like any chemical process, begins with the proprietary high-purity synthesis of the base material (usually silica), followed by surface modification and subsequent treatments such as end-capping and thermal processing. In the early days of chromatography, columns were packed manually by chromatographers, leading to significant variations in packing bed density (void volumes) and surface ligand (energetic) homogeneity, consequently, inconsistent selectivities and broadened peaks.

Today, pre-packed columns from specialized vendors are readily available, with strict quality controls in place to minimize production variability. This is achieved through a deep understanding of material science, the use of standardized reagents, and controlled manufacturing procedures. Despite these efforts, minor batch-to-batch variations may still occur and stationary phases with identical specifications from different vendors can exhibit different selectivity profiles. In addition, column aging, particularly through the hydrolysis of bonded ligands and in case of base silica material, their gradual exchange with residual silanol groups, can further impact performance.

In this published example (5), out‑of‑the-box separation performance of 12 identical bridged ethylene‑hybrid (BEH) ultrahigh-pressure liquid chromatography (UHPLC) columns from different batches were systematically tested and compared. To achieve this, 3D t_G-T-pH models were constructed, incorporating the suggestion for an optimal implementation order by taking into account different time needs for column equilibration and varying method conditions. As a result, the entire experimental set could be completed within just 12 h of laboratory work.

Overall, strong agreement was observed in the trajectories and development of the MODR regions across the tested column batches. However, as shown in Figure 3, notable differences were identified in certain subset regions, particularly in terms of their exact location and volume within their respective DSs. Selecting a method from these divergent areas may result in inconsistent performance across different column batches. In contrast, choosing a set point within the shared intercolumn MODR region ensures robust baseline separation, independent of the specific column batch in use.

Study 3: Comparing HPLC Systems (6)

HPLC instrumentation has seen remarkable advancements over the past 60 years. Today, there are many prominent vendors, each offering their own HPLC and UHPLC products. As a result, over the life cycle of a developed method, it is common for analyses to be performed on different brands and types of instruments. This makes method transfer an essential aspect of routine application and long-term method sustainability.

Geometrical transfer between various UHPLC and HPLC systems can be straightforwardly performed, either through manual calculations or by using specific calculator tools. Accounting for differences in column geometry as well as system volumes, such as dwell volume (V_D) and extracolumn volume (V_ECV)is also possible using software (7,8). However, technological differences between systems—if overlooked—can also significantly impact the success of method transfer. For instance, solvent delivery variations may arise due to differences in pump mechanisms, mixing configurations, and mixing techniques. Likewise, thermal control within the column compartment may vary depending on preheating methods or in-column temperature regulation. In addition, thermosensor sensitivity and control settings may also differ between systems, potentially affecting peak retention times and thus, selectivity—even when using instruments with identical catalog specifications.

Bearing this in mind, using t_G-T-t_C DS-models, two UHPLC systems were systematically compared: a binary system with a 200 µL dwell volume and a quaternary system with a 380 µL dwell volume. In both cases, the same column was installed, and the same test sample containing 14 peaks was injected under identical conditions. While the differences between the MODRs of the two systems were subtle (Figure 4), the matching irregular MODR regions were shifted and their volumes slightly varied, resembling the differences experienced in previous column batch-to-batch studies.

Here, although a commonly applied experimental compensation (initial hold) may account for V_D-differences, a closer examination of the chromatograms under identical working conditions revealed a more intriguing result. As expected, the first peak eluted earlier in the lower dwell volume system; however, unexpectedly, the last peak eluted later, highlighting a complex system-specific chromatography. Nevertheless, this can be effectively mitigated by selecting robust method conditions from the shared instrument-specific MODRs, with or without prior compensation of system volumes.

Study 4–5: CEX Elution Modes (9) and RPLC Buffer Studies (10)

In this example, two COVID-19-relevant IgG-type monoclonal antibodies (mAbs), specifically casirivimab and imdevimab, were comprehensively analyzed using prototype ultra-short column formats (1–2 cm in length) across multiple chromatographic modes, including reversed-phase and cation-exchange chromatography (CEX). The use of ultra-short columns has been recommended in earlier studies, particularly for large biomolecules. Belonging to this group, monoclonal antibodies usually exhibit an “on-off” retention mechanism, in which the effective bed length is relatively short when running common linear mobile phase gradients (11).

Within this experimental framework, two elution modes for CEX were also systematically tested and compared using t_G-T modeling heat maps. Namely, pH gradient was evaluated against the traditional sodium chloride elution (often referred to as “salt gradient”) technique. The pH gradient approach demonstrated superior separation and selectivity, particularly for the various charge variants of the antigen-binding fragment F(ab’)2, while the separation of charge variants for the single-stranded crystallizable region (sFc) showed slightly better results with salt-gradient (Figure 5).

Mobile phase effects play an important role not only in large molecule CEX analysis but also in small molecule reversed-phase liquid chromatography (RPLC). The definition and understanding of pH effects, as one of the most influential parameters for selectivity, are therefore crucial. However, unlike CEX, which typically uses purely aqueous eluents, reversed-phase systems always contain a significant proportion of organic modifier (eluent B), complicating accurate pH determination and thus, its practical use. In addition, analyte molecules may possess ionizable functional groups and the residual silanols on the stationary phase can also become charged depending on the pH, further influencing retention and selectivity.

In another demonstrative study, a non-volatile buffer system (sodium phosphate) and a volatile buffer (ammonium acetate) was compared within the same nominal pH range of 6.0–8.0, using t_G-T-pH modeling DSs. Surprisingly, under the selected working conditions shown in Figure 6, a notable difference in elution order was observed. Although separation dynamics becomes more complicated when factoring in the specific selection of gradient time and T conditions, modeling results demonstrated that, depending on the applied pH, the two buffer systems can either be conveniently replaced in the lower pH region (pH < 7) or behave as non-alternatives to each other under alkaline conditions.

Summary

The presented examples highlight the added value of applying systematic modeling approaches to effectively and elegantly address practical challenges commonly encountered in routine analytical work. By following a similarly structured workflow, modeling fingerprints can be constructed and compared across a range of real-world separation scenarios, offering valuable insights into specific multidimensional separation behavior—supporting both characterization and comparative HPLC analysis.

Trademark notice: Waters, BEH, UPLC, and BioResolve are trademarks of Waters Technologies Corporation.

References

(1) Laub, R.J.; Purnell, J. H. Criteria for the Use of Mixed Solvents in Gas—Liquid Chromatography. J. Chrom. A 1975, 112, 71–79. DOI: 10.1016/S0021-9673(00)99943-6

(2) Kormány, R.; Soós, B.; Horváth, K. Updating the European Pharmacopoeia Impurity Profiling Method for Cetirizine and Suggesting Alternative Column, Using Design Space Comparison. J. Pharm. Biomed. Anal. 2024, 237, 115776. DOI: 10.1016/j.jpba.2023.115776

(3) Zöldhegyi, A.; Horváth, K.; Kormány, A. Revisiting Column Selectivity Choices in Ultra-high Performance Liquid Chromatography–Using Multidimensional Analytical Design Spaces to Identify Column Equivalency. J. Chrom. A 2024, 1719, 464738. DOI: 10.1016/j.chroma.2024.464738

(4) Purnell, J. H. The Correlation of Separating Power and Efficiency of Gas-chromatographic Columns. J. Chem. Soc. 1960, 1268–1274.

(5) Rácz, N.; Kormány, R.; Fekete, J.; Molnár, I. Establishing Column Batch Repeatability According to Quality by Design (QbD) Principles Using Modeling Software. J. Pharm. Biomed. Anal. 2015, 108, 1–10. DOI: 10.1016/j.jpba.2015.01.037

(6) Zöldhegyi, A.; Jürgen, H.-J.; Molnár, I. Comparing Multivariate Eluent Design Spaces for Systematic Characterization of (U)HPLC Columns. The Column 2021, 17 (1), 13–21.

(7) Kormány, R.; Fekete, J.; Guillarme, D.; Fekete, S. Reliability of Computer-assisted Method Transfer Between Several Column Dimensions Packed with 1.3–5 µm Core–shell Particles and Between Various Instruments. J. Pharm. Biomed. Anal. 2014, 94, 188–195. DOI: 10.1016/j.jpba.2014.01.037

(8) Enesei, D.; Kapui, I.; Fekete, S.; Kormány, R. Updating the European Pharmacopoeia Impurity Profiling Method for Terazosin and Suggesting Alternative Columns. J. Pharm. Biomed. Anal. 2020, 187, 1–10. DOI: 10.1016/j.jpba.2020.113371

(9) Duivelshof, B.; Zöldhegyi, A.; Guillarme, D.; Lauber, M.; Fekete, S. Expediting the Chromatographic Analysis of COVID-19 Antibody Therapeutics with Ultra-short Columns, Retention Modeling and Automated Method Development. J. Pharm. Biomed. Anal. 2022, 221, 115039. DOI: 10.1016/j.jpba.2022.115039

(10) Zöldhegyi, A.; Soós, B.; Horváth, K.; Molnár, I.; Kormány, R. Extended Multidimensional Design Space Studies: Comparing Volatile and Non-volatile Buffer Systems in UPLC. J. Chrom. A 2025, 1751, 465951. DOI:10.1016/j.chroma.2025.465951

(11) Fekete, S.; Bobály, B.; Nguyen, J. M.; et al. Use of Ultrashort Columns for Therapeutic Protein Separations. Part 1: Theoretical Considerations and Proof of Concept. Anal. Chem. 2021, 93, 1277–1284. DOI: 10.1021/acs.analchem.0c04082

Arnold Zöldhegyi is a PhD candidate at Pannonia University, supervised by Krisztián Horváth and Róbert Kormány. He developed his expertise in high performance liquid chromatography at the Budapest University of Technology under renowned chromatographer Jenő Fekete. Since joining the Molnár-Institute for Applied Chromatography in 2017, he has worked under the leadership of Imre Molnár on design space modeling solutions and advanced research. His work focuses on supporting scientists in R&D and QC to tackle complex HPLC separation challenges.

Róbert Kormány is a chemist and special analyst in chromatography. He graduated from the University of Debrecen, then completed his education with an additional degree in chromatography and a Ph.D. in chemical sciences from the Department of Inorganic and Analytical Chemistry of the Budapest University of Technology and Economics in the laboratory of Jenõ Fekete. He has worked at Egis Pharmaceuticals PLC for nearly 20 years, where he primarily deals with the development of (U)HPLC methods. His main scientific field is the study of retention in reversed-phase liquid chromatography and retention modeling.

Szabolcs Fekete worked in the pharmaceutical industry at analytical R&D for 10 years, then moved to the University of Geneva in Switzerland and worked as a scientific collaborator for a decade. In April 2021, he joined Waters Corporation and now works as a consulting scientist. His current interests include separations of new chemical modalities, fundamentals of chromatography, column technology, and new method development approaches.

Krisztián Horváth is associate professor of analytical chemistry at the University of Pannonia. His work centers on HPLC method development and optimization, retention mechanisms, mass transfer, and DoE. He has authored 50+ papers and leads projects advancing chromatographic instrumentation and modeling, including a current food-allergen quality initiative. He serves the separation science community in several leadership roles, including as secretary of the Hungarian Society for Separation Sciences and as the Hungarian representative to the Central European Group for Separation Science.

Imre Molnár is founder and president of Molnár-Institute for Applied Chromatography. Receiving a doctorate degree from German Saarland University in analytical chemistry, Molnár joined Csaba Horváth’s research team at Yale, resulting in numerous publications such as the Theory of Solvophobic Interactions and on the fundamentals of reversed phase chromatography. After returning to Europe he founded the Institute for Applied Chromatography in Berlin in 1981. Since 1984, joining the team of Lloyd Snyder and John Dolan, he has focused on in-silico method development in HPLC, resulting in the software DryLab.