Improving retention-time prediction in supercritical-fluid chromatography by multivariate modelling

The prediction of chromatographic retention under supercritical-fluid chromatography (SFC) conditions was studied, using established and novel theoretical models over ranges of modifier content, pressure and temperature. Whereas retention models used for liquid chromatography often only consider the modifier fraction, retention in SFC depends much more strongly on pressure and temperature.

Journal of Chromatography A 1668 (2022) 462909 Contents lists available at ScienceDirect Journal of Chromatography A journal homepage: www.elsevier.com/locate/chroma Improving retention-time prediction in supercritical-fluid chromatography by multivariate modelling Stef R.A Molenaar a,b,1,∗, Mariyana V Savova a,b,c,1,∗, Rebecca Cross d, Paul D Ferguson e, Peter J Schoenmakers a,b, Bob W.J Pirok a,b a Van’t Hoff Institute for Molecular Sciences, Analytical Chemistry Group, University of Amsterdam, Science Park 904, Amsterdam 1098 XH, the Netherlands Centre for Analytical Sciences Amsterdam (CASA), the Netherlands TI-COAST, Science Park 904, Amsterdam 1098 XH, the Netherlands d Chemical Development, Pharmaceutical Technology & Development, Operations, AstraZeneca, Macclesfield, United Kingdom e New Modalities Product Development, Pharmaceutical Technology & Development, Operations, AstraZeneca, Macclesfield, United Kingdom b c a r t i c l e i n f o Article history: Received 29 September 2021 Revised February 2022 Accepted 15 February 2022 Available online 17 February 2022 Keywords: SFC Retention modelling Multivariate Mass fraction Pressure a b s t r a c t The prediction of chromatographic retention under supercritical-fluid chromatography (SFC) conditions was studied, using established and novel theoretical models over ranges of modifier content, pressure and temperature Whereas retention models used for liquid chromatography often only consider the modifier fraction, retention in SFC depends much more strongly on pressure and temperature The viability of combining several retention models into surfaces that describe the effects of both modifier fraction and pressure was investigated The ability of commonly used retention models to describe retention as a function of modifier fraction, expressed either as mass or volume fraction, pressure and density was assessed Using the multivariate surfaces, retention-time prediction for isocratic separations at constant temperature improved significantly compared to univariate modelling when both pressure and modifier fractions were changed The “mixed-mode” model with an additional exponential pressure or density parameter was able to predict retention times within 5%, with the majority of the predictions within 2% The use of mass fraction and density further improves retention modelling compared to volume fraction and pressure These variables however, require extra computations © 2022 The Author(s) Published by Elsevier B.V This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) Introduction In the last decade, the interest in supercritical- and subcriticalfluid chromatography (SFC) has increased considerably following the hardware innovations that have taken place [1–4] These improvements ultimately allowed modern ultra-high-performance SFC instruments to rival the robustness of ultra-high-performance liquid-chromatography (UHPLC) systems on the market [5] Concurrently, today’s practice of employing modified mobile phases with or without additives greatly expanded the polarity range of compounds compatible with SFC This has enabled the use of SFC ∗ Corresponding authors at: Van’t Hoff Institute for Molecular Sciences, Analytical Chemistry Group, University of Amsterdam, Science Park 904, Amsterdam 1098 XH, the Netherlands E-mail addresses: S.R.A.Molenaar@uva.nl (S.R.A Molenaar), M.V.Savova@lacdr.leidenuniv.nl (M.V Savova) These authors contributed equally in a diverse range of areas including natural products [6], pharmaceutical [4], food [7], forensics [8] and bio-analysis [9] Despite the resurgence of SFC in analytical laboratories, the fact that the technique has received little attention for two decades has slowed down progress in fundamental understanding [10] Most fundamental literature focuses on neat-carbon dioxide (neat-CO2 ) separations, which are rarely performed today In contrast, contemporary research aims at broadening the applicability of SFC Consequently, the applicability of retention modelling has remained relatively unexplored The field of retention modelling in chromatography has five fundamentally different main application areas, as recently reviewed by Den Uijl et al [11] Applications such as stationaryphase characterization, understanding retention and lipophilicity determination are of great interest within the field of chromatography These applications however, make use of more complicated retention models such as the linear solvation energy relationship (LSER) or quantitative structure retention relationships (QSRR) Where contributions from, but not limited to, acidity/basicity, mo- https://doi.org/10.1016/j.chroma.2022.462909 0021-9673/© 2022 The Author(s) Published by Elsevier B.V This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) S.R.A Molenaar, M.V Savova, R Cross et al Journal of Chromatography A 1668 (2022) 462909 lar refraction and polarity can be determined by LSER [12–14] and retention for compounds can be predicted when the analyte structure is known by QSRR [15,16] To determine and use these models however, a large amount of input is necessary for the analyte, stationary- and mobile-phase chemistry In the application fields of method transfer and, in the scope of this work, method optimization, typically less complicated retention models are used to find an empirical relationship between retention time and modifier fractions These retention models describe the retention in pure weak solvent and the change in retention due to the strong solvent These parameters differ for different analytes, columns and mobile-phases However it was shown that retention within the same chromatographic mode follow the same retention model [11,17–20] A benefit of these simpler empirical models, is that the relationship can often be determined using a limited number of scouting runs (e.g two to six depending on the retention model used) The development of measurement based retention models can be used for the development of in-silico method optimization approaches such as those already available for one-dimensional [21,22], two-dimensional [23] and even three-dimensional [24] liquid chromatography (LC) The development of such tools for SFC is complicated by the compressibility of sub- and supercritical fluids in comparison to liquids Due to this increased compressibility, many parameters that arguably can be considered constant in LC are not in SFC [25], such as the mobile-phase density, which is a function of pressure and temperature [26,27] Recently, Tyteca et al [28] explored the possibilities of retention modelling in SFC, using non-linear models for the natural logarithm of the retention factor (ln k) as a function of the volume fraction of modifier (ϕ ), namely the quadratic [29], mixed-mode [30] and Neue-Kuss [31,32] models Aiming at optimizing gradient methods, the authors evaluated the possibility of gradient prediction using models built using isocratic, as well as gradient data [28] The authors found that utilizing isocratic retention data for gradient prediction was complicated by pressure effects In this case prediction errors generally were within 10%, irrespective of the conditions of the predicted gradient Models generated with gradient data were more accurate, (prediction errors within 5%) when the starting conditions of the scouting and predicted gradient runs were equal, but were poorer when this was not the case (errors up to 46%) Possible explanations for the reduced accuracy in the latter case were not provided by the authors, but their findings are in agreement with documented issues when constructing retention models using gradient data [33] and the need to consider gradient deformation [34,35] The authors did not evaluate the incorporation of a pressure or density term within the retention models [28] Neither was this the case for other studies in the field [36–38] With pressure impacting retention [39], incorporating the mobile-phase density in the retention models may increase prediction accuracy The recent versions of the Reference Fluid Thermodynamic and Transport Properties Database (REFPROP) of NIST [40] include CO2 -methanol systems, enabling chromatographers to explore the effect of density in such systems [26] In this study we aimed to explore the feasibility of modelling retention of isocratic SFC separations, considering the modifier fraction as well as the pressure and density The retention models evaluated include exponential [41], adsorption [20], quadratic [29], mixed-mode [30] and Neue-Kuss [31,32] models, with an additional term describing either pressure or density Due to the mobile-phase compressibility, the modelling was performed using mass fractions instead of volume fractions, as proposed by de Pauw et al [39] The goodness-of-fit and the ability to predict retention under different experimental conditions were assessed for each model The performance of the models was compared with and without a pressure or density term for nine model analytes with diverse physicochemical properties (e.g acids, bases, neutrals) The ultimate aim of our work is to develop empirical multivariate retention models that can be used in method-optimization software in situations in which both the effects of modifier concentration and pressure need to be considered Experimental 2.1 Chemicals Methanol (MeOH - UPLC/MS grade) was purchased from Biosolve (Valkenswaard, The Netherlands) Ammonium acetate (reagent grade, ≥ 98%), naproxen, salicylic acid, acetaminophen, bisphenol A, hydrochlorothiazide, 4-aminoantipyrine, propranolol hydrochloride, and trimethoprim were obtained from SigmaAldrich (Darmstadt, Germany) Tacrine hydrochloride was obtained from the United States Pharmacopeia (North Bethesda, MD, USA) SFC-grade carbon dioxide (grade 4.8) was obtained from Praxair (Vlaardingen, The Netherlands) Water used during the experiment (18.2 mΩ) was obtained from a Milli-Q purification system (Millipore, Bedford, MA, USA) 2.2 Sample preparation Stock solutions of each analyte were prepared in methanol at a concentration of 100 mg·mL−1 , except for the hydrochlorothiazide and trimethoprim stock solutions, which were prepared at a concentration of 10 mg·mL−1 The compounds that did not readily dissolve in MeOH were ultrasonicated until completely dissolved Standards were stored at °C 2.3 Chromatographic system Experiments were conducted on a Waters Acquity UPC2 system (Waters, Etten-Leur, The Netherlands) equipped with a binary solvent-delivery pump (maximum allowed flow rate and pressure of mL·min−1 and 413 bar, respectively), an autosampler equipped with a 10 μL loop, a column oven, a photodiode-array (PDA) detector fitted with an μL flow-cell, and a two-step (active and passive) backpressure regulator (BPR) The passive component of the BPR kept the pressure at 104.3 bar, whereas the active component adjusted the final pressure until the desired value was reached The data-acquisition rate of the PDA detector was set at 25 Hz The chromatograms were recorded using Empower3 software (Waters) 2.4 Chromatographic conditions A Waters Viridis BEH column (100 × mm i.d., d p = 3.5 μm) was used in the study The mobile phase consisted of CO2 modified with methanol The latter solvent contained 2% (v/v) water and ammonium acetate (20 mM) as additives This additive was shown by Lemasson et al [42] to provide good peaks shapes for a range of analytes of different physico-chemical properties The additive was prepared as a M solution in MilliQ water, and then diluted to 20 mM in MeOH All measurements were carried out in isocratic mode with a fixed volumetric flow rate of 1.5 mL·min−1 Retention was studied by varying the modifier fraction between and 36% (v/v) depending on the analyte studied, to ensure that the analytes were retained on the column (i.e retention factor (k) ≥ 1) and eluted in a reasonable time period (within approximately 30 min) These retention factors were chosen to ensure data was available for modelling, if the analyte does not interact with the column (i.e no retention), no meaningful conclusions about the retention model can be made The analyses were performed at three S.R.A Molenaar, M.V Savova, R Cross et al Journal of Chromatography A 1668 (2022) 462909 backpressure levels of 120, 160 and 200 bar, and two temperatures of 25 and 40 °C The autosampler was set at 25 °C The analytes were dissolved in pure methanol and injected in mixtures, consisting of compounds that were not expected to coelute nor to interact (e.g acidic and basic analytes were not used in the same mixture) The final analyte concentration was 5.0 mg·mL−1 except for hydrochlorothiazide and trimethoprim (4.7 mg·mL−1 ) and tacrine (6.7 mg·mL−1 ) “Partial loop with needle overfill” (PLNO) injection was performed of 0.1 μL volumes using a 10 μL loop, carried out under the default settings of the instrument ACN was used as needle-wash solvent (500 μL) PDA spectra were recorded across the range of 190 to 360 nm, with 210 nm as reference, and a resolution of 1.2 nm The UV spectra were used to confirm identification of the individual components system dead time In SFC, the dead time may change significantly with CO2 -MeOH mixtures as a function of pressure, temperature, and mobile-phase composition In order to account for this variation, the dead time for each individual setting was estimated as follows, unless stated otherwise: (1) the column volume was calculated using the first baseline disturbance of the solvent peak as a marker for the dead time under reference conditions (36% (v/v) of modifier, 25 °C, back-pressure of 200 bar), at which the compressibility of the mobile phase was lowest amongst all the studied settings; (2) the actual volumetric flow rate was calculated by first determining the total mass flow rate (MFtotal ) under experimental conditions This was calculated using the set volumetric flow rate and Eq (3); (3) using MFtotal and the determined mobilephase density, the actual volumetric flow rate was calculated; (4) the dead time was then calculated using the actual volumetric flow rate and the column volume estimated in step 2.5 Software MFtotal = Fv (ϕ (3) Such an complex determination of t0 was undertaken, since the shape of the solvent peak was inconsistent throughout the measurements, under different experimental conditions as well as for replicate measurements, making conventional estimation of t0 prone to visual error [44] Theory 3.1 Mass fraction and density calculations 3.3 Retention modelling For the density calculations, the pressures at the both the pump and the active BPR were recorded throughout the analysis The pressures recorded at the pump and at the active BPR were assumed to be equal to those at the column inlet and outlet, respectively The mean column pressure was approximated as the mean of these two pressures The set column temperature was assumed to equal the fluid’s temperature throughout the column for this project The calculations of the mobile-phase density were carried out using the REFPROP version 10 software curated by NIST [40] As the density of ternary mixtures containing carbon dioxide, methanol and water cannot be calculated using this software, the modifier was assumed to be pure methanol This is a reasonable assumption because the modifier used in this study contained only 2% (v/v) of M aqueous ammonium-acetate solution The density calculations were based on the mixing rules of Kunz et al [43] using the pressure, temperature and mass fraction (χ ) of methanol as input values Due to the compressibility of CO2, χ varies not only with the set volume fraction of modifier, but also with the pumping pressure (Ppump ) The formula used to calculate χ under different conditions in shown in Eq (1) χ= ρMeOH [Tlab , Ppump (Fv , ϕ )] + (1 − ϕ )ρCO2 [TCO2 , Ppump (Fv , ϕ )] ) Retention modelling was performed with an in-house script using the lsqcurvefit function within Matlab-software R2020a (Mathworks, Natick, MA, USA) When utilizing modifiers under sub-critical conditions, contemporary SFC methods often resemble LC For this reason, the retention models evaluated in this study were based on empirical retention models commonly used in LC [11] The exponential model describes partitioning in RPLC separations [41] (Eq (4)) ln k = ln k0 − Sϕ (4) Here, ϕ is the volume fraction of modifier, k0 is the extrapolated retention factor for ϕ = 0, and S is the so-called solventstrength parameter, relating to the strength of the strong solvent (typically 3–5 for small molecules in RPLC) For a broader range of mobile-phase compositions, the relationship between ln k and ϕ is concave, thus the use of a quadratic model (Eq (5)) has been proposed [29], which is given by ln k = ln k0 + S1 ϕ + S2 ϕ (5) Where S1 and S2 are the coefficients that describe the retention under the influence of the modifier The adsorption model (Eq (6)) is typically used for normal-phase separations [20], but has been shown to be robust when using a limited number of input variables in other mixed-mode retention mechanisms, such as HILIC [17] ϕ ρMeOH [Tlab , Ppump (Fv , ϕ )] ϕ ρMeOH [Tlab , Ppump (Fv , ϕ )] + (1 − ϕ )ρCO2 [TCO2 , Ppump (Fv , ϕ )] ln k = ln k0 − n ln ϕ (1) (6) in which Fv is the set volumetric flow rate, ϕ the volume fraction of modifier, Tlab the temperature in the laboratory, assumed to be constant (23 °C) and TCO2 the set accumulator CO2 pump temperature (13 °C), while ρMeOH and ρCO2 are the densities of MeOH and CO2 , calculated using REFPROP software [40] Here, n is the ratio between the molecular area of adsorbed solute (As ) and the number of strong solvent molecules needed to desorb the analyte molecules (nB ) The mixed-mode model, shown in Eq (7), is a combination of the partitioning and adsorption models It has also been used to describe HILIC retention [30] 3.2 Retention factor and column dead time determination ln k = ln k0 + S1 ϕ + S2 ln ϕ The retention factor k for isocratic elution was calculated using Eq (2) k= tR − t0 t0 (7) One disadvantage of the quadratic and mixed-mode models is that they have no exact integral that allows calculation of retention times under gradient conditions The Neue-Kuss model (Eq (8)), which is an empirical model developed to aid integration under gradient conditions, elegantly solves this issue [32] (2) Where tR is the retention time of the analyte, determined using the peak apex – even if the peak was asymmetrical, and t0 is the ln k = ln k0 + ln (1 + S2 ϕ ) − S1 ϕ + S2 ϕ (8) S.R.A Molenaar, M.V Savova, R Cross et al Journal of Chromatography A 1668 (2022) 462909 Fig Boxplots of AIC values for each model AIC scores for the different pressures are binned together From left to right; exponential (Exp, blue, Eq (4)), adsorption (Ads, yellow, Eq (6)), mixed-mode (Mix, green, Eq (7)), quadratic (Quad, purple, Eq (5)) and Neue-Kuss (NK, red, Eq (8)) at 25 °C (left group) and 40 °C (right group) (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Fig Boxplot of prediction errors for each model From left to right; exponential (Exp, blue, Eq (4)), adsorption (Ads, yellow, Eq (6)), mixed-mode (Mix, green, Eq (7)), quadratic (Quad, purple, Eq (5)) and Neue-Kuss (NK, red, Eq (8)) at 25 °C followed by 40 °C (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Due to the compressibility of CO2 -methanol mixtures, the actual value of ϕ does not coincide with the set value and varies along the chromatographic system Meanwhile, χ may depend on pressure but is constant throughout the system [39,45] The work of de Pauw et al published in 2016, showed that retention in SFC is best modelled using χ instead of ϕ [39] Accordingly, in this work the ϕ variable in the above models was replaced by χ It is well-recognized that even under subcritical conditions, pressure or density changes impact retention in SFC For this reason, in addition to the models introduced above where the modifier effect was considered exclusively, two-variable models, including either the pressure or the density, were also studied These two-variable models were constructed by adding a pressure or density term to the existing models (Eqs (4)–(8)) See Eq (9) for an example The additional parameters were equivalent to the second parameter of the exponential and adsorption model (i.e +Sχ and +n ln χ ), with the only difference that instead of χ as a variable, either pressure (P ) of density (ρ ) was used As an example, the addition an exponential pressure parameter to the mixed-mode model describing the effect of modifier entails adding an S3 P term (with S3 as a model coefficient) to the mixed-mode model (Eq (7)), resulting in For a list of all retention equations used, please refer to Supplementary Material Section S-1 ln k = ln k0 + S1 χ + S2 ln χ + S3 P F= 3.4 Goodness of fit and retention-time predictions The goodness-of-fit of the models was determined using the Akaike information criterion (AIC), which takes into consideration the number of parameters used to build the model and, therefore, allows for a fair comparison between models containing different numbers of parameters The AIC is calculated according to [46] AIC = pm + nD ln 2π SSQ nD +1 (10) where pm is the number of parameters in the model, nD is the number of datapoints and SSQ is the sum of squared errors between the fit and the datapoints In order to compare models differing by the addition of a single term, such as the exponential and mixed-mode models, the statistical F-test of regression [47,48] was used with α = 0.05 This test compares two models taking into consideration the residual sum-of-squares of the full model SSres,full with that of the reduced model SSres,red , as show in Eq (11) (9) MSres,diff = MSres,full SSres,full − SSres,red /(νred − νfull ) SSres,full /νfull (11) Fig Retention plots of (A) naproxen, (B) hydrochlorothiazide and (C) tacrine fitted with the mixed-mode model for each different set of pressure and temperature values S.R.A Molenaar, M.V Savova, R Cross et al Journal of Chromatography A 1668 (2022) 462909 MSres,diff and MSres,full are the difference between the residual mean-squares errors of the reduced and full model, and the residual mean-squares error of the full model respectively, while νred and νfull are the respective degrees of freedom of the reduced and full model For example, the exponential model (Eq (4)) can be the reduced model, while the mixed-mode model (Eq (7)) is the full model The prediction error of the models was assessed according Eq (12) Error (% ) = tR,pred − tR,exp · 100% tR,exp between 2.5 and 70 and all models were fitted to these data For all data regarding the experimental retention times, modifier fractions and model parameters, please refer to Supplementary Material Section S-2 The goodness-of-fit of each model was expressed through computation of the AIC Fig shows the spread of the AIC values for each model The AIC value is based on the logarithm of the sum of squared errors Therefore, the smaller or more negative the value, the better the model describes the data The results suggest that the mixed-mode model best describes the data, followed by the adsorption model In contrast, the exponential model appeared not to describe the data accurately To assess the statistical significance of the third parameter in the mixed-mode model compared to the adsorption model, an Ftest was performed following Eq (11) For six out of nine compounds, the addition of the additional term (third parameter) was significant for all six different combinations of backpressures and temperatures, while for the other three compounds the addition was significant for at least three of the combinations Fig shows the retention factors for three analytes, namely naproxen (acidic), hydrochlorothiazide (neutral) and tacrine (basic) with the mixed-mode model fitted to the data As can be seen, the mixed-mode model describes these data well However, the curves for each different combination temperature and backpressure not overlap, indicating that pressure and temperature have a significant effect on retention Besides a comparison of AIC scores, the ability to predict retention times for other mass fractions was investigated Given that pressure and temperature have an effect on retention, as shown in Fig 2, predictions were exclusively assessed for the same pressure and temperature For each analyte, four different mass fractions and corresponding retention times were selected per combination of pressure and temperature and retention models were fitted to these data points The retention times for the remaining mass fractions were then predicted and compared with the experimental data The resulting prediction errors are shown in Fig For a table of datapoints used for each model and the predictions, please refer to Supplementary Material Section S-2 Similar to the AIC values, the mixed-mode model performed best, followed by the adsorption model The exponential model appeared unable to accurately describe the data and was therefore left out of further modelling studies (12) In which tR,pred and tR,exp are the predicted and experimentally observed retention times at the peak maximum, respectively For computing the prediction error, two replicates were used for each point (χ ,ρ ) to construct the model, whereas for computation of AIC values up to three replicate measurements were considered For the latter calculations measurements were performed with and 15% (v/v) of MeOH at 120, 160 and 200 bar Results & discussion For MeOH/CO2 systems the modifier concentration and, to a lesser extent in the practical working range, the density (and thus physical parameters, namely pressure and temperature) govern retention [49] Therefore, accurate retention modelling in SFC requires that the effects of both the modifier concentration and the density be considered These effects were modelled for nine compounds varying in physicochemical properties (e.g acidic, basic and neutral compounds) Prior to modelling both effects simultaneously, the effect of modifier fraction was considered alone 4.1 Modelling the effect of modifier The effect of modifier was assessed at three different backpressures (120, 160 and 200 bar) and two different temperatures (25 and 40 °C) These different conditions were chosen to assess the broader applicability of the retention models As stated in Section 3.1, the modelling was performed against the mass fraction of modifier (χ ), rather than the volume fraction (ϕ ) For each different combination of pressure and temperature the modifier fraction was varied in such a way that the retention factors ranged Fig (A) Changes in modifier mass-fraction with different pressures for a volume fraction of 0.05 Red diamond represent data points for a temperature of 25 °C, green circles represent data for a temperature of 40 °C (B) Changes in modifier mass-fraction with different pressures for a volume fraction of 0.15 Red diamond represent data points for a temperature of 25 °C, green circles represent data for a temperature of 40 °C (C) Changes in mobile phase density when the pressure changes for ϕ = 0.05, 25 °C (Red diamonds), ϕ = 0.15, 25 °C (Green circles), ϕ = 0.05, 40 °C (Purple crosses) and ϕ = 0.15, 40 °C (Yellow squares) (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) S.R.A Molenaar, M.V Savova, R Cross et al Journal of Chromatography A 1668 (2022) 462909 Fig Retention surfaces of (A) naproxen, (B) hydrochlorothiazide and (C) tacrine at 25 °C, fitted with the mixed-mode model with the addition of an exponential pressure parameter Datapoints used to fit the models and are represented by red circles (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Fig Boxplots of prediction errors of each investigated model with different added parameters for the pressure contribution The blue lines indicate prediction errors of ±2.5% From left to right; adsorption (Ads, yellow), mixed-mode (Mix, green), quadratic (Quad, purple) and Neue-Kuss (NK, red) models with no added parameter, followed by the addition of an exponential parameter and adsorption parameter (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) isolated This is shown in Fig 4, with the effect of pressure on the mass fraction of the modifier and mobile-phase density plotted for two different volume modifier fractions (ϕ = 0.05 and ϕ = 0.15) and temperatures (25 and 40 °C) As can be seen in Fig 2, the retention plots of the different pressures are not overlapping This implies that the pressure has a significant effect on retention To compensate for the pressure, extra parameters within the retention models were added to describe the variations due to pressure Instead of creating a retention line as in Fig 2, a retention surface was constructed as shown in Fig Creating the retention surface was achieved using data from different combinations of modifier concentration and pressure For each analyte 12 datapoints were used to construct the surface models, and the remaining datapoints were predicted To perform appropriate modelling, the datapoints used to create the models required elution of the analyte with reasonable retention factors (2.5 < k < 70) For an exact list of datapoints used for modelling and prediction for each analyte please refer to Supplementary Material Section S-3 Fig shows the prediction errors for each model The results suggest that adding an extra parameter for the pressure effect reduced the prediction errors Statistical Ftests were performed to verify the significance of adding an extra parameter to the models The addition of an extra parameter was found to be significant for all of the surface models Furthermore, it can be seen that the prediction errors of the mixed-mode model (based on Eq (7)) with added exponential pressure term, are smallest and most closely centred around 0% Thus, it outperforms Fig Stacked boxplots indicating which models performed (From left to right: adsorption, mixed-mode and Neue-Kuss) best between those with extra pressure (red, bottom) or density (green, top) parameters The black line indicates 50%, i.e where the effects of adding a pressure or density term were comparable The left group has the addition of an exponential term, whereas the right group has the addition of an adsorption term (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 4.2 Modelling the combined effect of modifier and pressure To include the effect of pressure, the actual mass fraction of modifier must be known However, as stated in Section 3.1 (Eq (1)), the system pressure affects the mass fraction of modifier and the effect of pressure can therefore not be entirely S.R.A Molenaar, M.V Savova, R Cross et al Journal of Chromatography A 1668 (2022) 462909 Fig Boxplots of prediction errors of each investigated model (indicated at the bottom) Six (6p) or eight (8p) datapoints were used χ indicates the use of mass fractions in the equation, whereas ϕ indicates the use of volume fractions P and ρ indicate the addition of an exponential pressure or density parameter in the model, respectively The blue lines indicate prediction errors of ±2.5% and 0% From left to right: adsorption (Ads + Exp, yellow), mixed-mode with (Mix + Exp, green), and Neue-Kuss (NK + Exp, red) (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) than ±5%, with more than half of the prediction errors between ±2% Considering density rather than pressure was found to further improve accuracy of the models However, pressure is more practical, requiring less computation, while still improving modelling significantly compared to univariate modelling (e.g modifier fraction only) Similarly, compared to the use of volume fraction, the mass fraction improved the modelling accuracy This modification is worthwhile, despite requiring extra calculations Modelling was performed at two different temperatures and it was apparent that temperature had a significant effect on retention behavior Therefore, the effect of temperature as a third variable could be of interest This, however, would complicate modeling further Retention modeling is of great interest for optimization software and predicting retention under conditions of modifier and pressure gradients is of high relevance Gradient predictions were not investigated in this work To so, gradient equations using partial integrals of both modifier and pressure/density must be derived and will be the focus of future investigations the other models The quadratic model (based on Eq (5)), exhibits a large range of prediction errors, despite the addition of an extra pressure term It was therefore left out of further modelling The addition of a density parameter (i.e exponential and adsorption) instead of a pressure parameter was also investigated using the adsorption, mixed-mode and Neue-Kuss models Calculating the density of the mobile phase required extra computations using the REFPROP software Fig shows stacked boxplots indicating where replacing the pressure parameter for a density parameter yielded better results As can be seen, the errors of only two of the models were improved for more than 50% of the predictions, whereas for the remaining models the predictions performed worse However, for the model that performed best, i.e the mixedmode model with an exponential pressure parameter (+n ln P - see Fig 6), the prediction errors improved for 63% of the measurements Lastly, the performance of the models was tested with fewer datapoints and by varying the input parameters – mass fraction or volume fraction, and pressure or density Volume fractions (at the moment of delivery, i.e at the top of the column) not require extra calculations to set up the models and thus are more straightforward to use For a detailed description of the datapoints used, please refer to Supplementary Material Sections S-4 & S-5 Fig shows the prediction errors of all these models As can be seen, prediction errors are lowest with the additional two datapoints, when mass fraction and the extra density term are used (left-hand side of Fig 8) However, even when points, volume fraction and pressure are used, most of the prediction errors are still within 5% when using the mixed-mode with an extra exponential parameter, which may be acceptable in some cases Such “simple” modelling does not require extra calculations and is simpler to set up than the 8- or 12-point models Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper CRediT authorship contribution statement Stef R.A Molenaar: Conceptualization, Methodology, Software, Validation, Writing – original draft, Visualization, Supervision Mariyana V Savova: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing – review & editing, Visualization Rebecca Cross: Conceptualization, Resources, Supervision Paul D Ferguson: Conceptualization, Resources, Writing – review & editing, Supervision, Project administration, Funding acquisition Peter J Schoenmakers: Conceptualization, Resources, Writing – review & editing, Project administration, Funding acquisition Bob W.J Pirok: Conceptualization, Resources, Writing – review & editing, Supervision, Project administration Conclusions The results of this study underlined that pressure has a significant effect on the retention of analytes in SFC and thus impacts retention modelling Considering pressure when predicting retention as a function of modifier fraction through multivariate modelling yielded a significant improvement compared to univariate modelling The mixed-mode model appeared to predict retention in SFC best, and addition of an exponential parameter for density (ln k = ln k0 + S1 χ + S2 ln χ + S3 ρ ) yielded prediction errors lower S.R.A Molenaar, M.V Savova, R Cross et al Journal of Chromatography A 1668 (2022) 462909 Acknowledgments [20] L.R Snyder, H Poppe, Mechanism of solute retention in liquid—solid chromatography and the role of the mobile phase in affecting separation: competition versus “sorption,”, J Chromatogr A 184 (1980) 363–413, doi:10.1016/ S0 021-9673(0 0)93872-X [21] L.R Snyder, J.W Dolan, D.C Lommen, Drylab® computer simulation for highperformance liquid chromatographic method development I Isocratic elution, J Chromatogr A 485 (1989) 65–89, doi:10.1016/S0021-9673(01)89133-0 [22] 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methanol/carbon dioxide in packed column supercritical fluid chromatography, Anal Chem 62 (1990) 1181–1185, doi:10.1021/ac00210a017 ... retention modelling Considering pressure when predicting retention as a function of modifier fraction through multivariate modelling yielded a significant improvement compared to univariate modelling. .. published in 2016, showed that retention in SFC is best modelled using χ instead of ϕ [39] Accordingly, in this work the ϕ variable in the above models was replaced by χ It is well-recognized that even... within the retention models were added to describe the variations due to pressure Instead of creating a retention line as in Fig 2, a retention surface was constructed as shown in Fig Creating