Reducing the influence of geometry-induced gradient deformation in liquid chromatographic retention modelling

Rapid optimization of gradient liquid chromatographic (LC) separations often utilizes analyte retention modelling to predict retention times as function of eluent composition. However, due to the dwell volume and technical imperfections, the actual gradient may deviate from the set gradient in a fashion unique to the employed instrument. This makes accurate retention modelling for gradient LC challenging, in particular when very fast separations are pursued. Although gradient deformation has been addressed in method-transfer situations, it is rarely taken into account when reporting analyte retention parameters obtained from gradient LC data, hampering the comparison of data from various sources. In this study, a response-function-based algorithm was developed to determine analyte retention parameters corrected for geometry-induced deformations by specific LC instruments. Out of a number of mathematical distributions investigated as response-functions, the so-called “stable function” was found to describe the formed gradient most accurately. The four parameters describing the model resemble the statistical moments of the distribution and are related to chromatographic parameters, such as dwell volume and flow rate. The instrument-specific response function can then be used to predict the actual shape of any other gradient programmed on that instrument. To incorporate the predicted gradient in the retention modelling of the analytes, the model was extended to facilitate an unlimited number of linear gradient steps to solve the equations numerically. The significance and impact of distinct gradient deformation for fast gradients was demonstrated using three different LC instruments. As a proof of principle, the algorithm and retention parameters obtained on a specific instrument were used to predict the retention times on different instruments. The relative error in the predicted retention times went down from an average of 9.8% and 12.2% on the two other instruments when using only a dwell-volume correction to 2.1% and 6.5%, respectively, when using the proposed algorithm. The corrected retention parameters are less dependent on geometry-induced instrument effects. © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )


Introduction
The majority of methods in liquid chromatography (LC) utilize gradient elution, where the fraction of strong solvent ( e.g. the organic modifier in reversed-phase LC) ϕ is gradually increased. Ana-lyte retention depends on the mobile-phase composition and, thus, on the applied gradient when the analyte moves through the column. Consequently, models that describe the retention of analytes when using a gradient must accurately account for the true shape of the programmed gradient. To automate and accelerate the development of effective gradient-elution methods, computer-aided optimization tools, such as ChromSword [1] , PEWS [2] , Drylab [3] and MOREPEAKS (formerly PIOTR) [4] , employ scanning experiments to establish the required retention parameters for each analyte [ 5 , 6 ]. The gradient delay arising from the dwell volume ( V D ) of the LC system [ 7 , 8 ] is generally taken into account during retention prehttps://doi.org/10.1016/j.chroma.2020.461714 0021-9673/© 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ) diction [3] . The dwell volume is different between instruments, but it is generally assumed that apart from this delay the actual gradient delivered to the column is identical to the programmed gradient. However, other instrument-related factors, such as errors in temperature and flow rate, will also influence the separation [ 9 , 10 ]. Gritti et al. have extensively investigated gradient deformation in reversed-phase LC and the effects thereof on the separation [11][12][13] . They were able to improve retention prediction for fast gradients on a single instrument by taking the adsorption isotherms of individual analytes into account [11] . In the same study it was shown that for less-retained compounds the resolution would collapse when fast gradients are applied and the authors proposed to modify the gradient to prevent this behaviour.
Gradient deformations can be caused, for example, by flow imperfections caused by a mixer or by regular dispersion in the connection tubing. Modest gradient deformation may be of limited concern when the retention parameters obtained using a specific instrument are exclusively used for optimization of gradients on the same instrument. However, because deformation of the gradient is dependent on the mobile-phase delivery assembly, the installed injection devices and the (pre-column) connectors of the instrument, the obtained retention parameters cannot be used to accurately predict analyte retention on other LC systems. A correct comparison of (reported) retention parameters acquired on various LC gradient instruments is only possible after accounting for the differences in the actual gradient shapes [14] .
Geometry-induced deviations from programmed eluent compositions are relatively most prominent in very fast gradients, such as those encountered in ultra-high-performance LC (UHPLC) or in the second dimension of comprehensive two-dimensional liquid chromatography (LC × LC). Quarry et al. showed already in the 1980s that the actual shape of fast gradient programs in particular can be significantly deformed [15] . Deformations can be induced by the specific (mixing) properties and interactions of the two solvents forming the gradient, as well as by the geometrical features of the LC instrument. Retention-scanning experiments can in principle be conducted using isocratic elution. However, when applying the retention parameters thus obtained for predicting gradient separations, correction for gradient deformation is still required [ 15 , 16 ].
The deformation of a linear gradient depends on the flow rate ( F ) and the slope of the gradient, which is the change in the volume fraction of modifier ( ϕ) divided by the duration of the linear segment of the gradient ( t G ) [17] . The most-accurate experimental method to reveal the true gradient profile is through detection of a chromophoric agent dissolved in one of the gradientforming solvents [15] . Another approach is through interpretation of isocratically acquired retention parameters [18] , but this requires a large number of runs [19] . In silico accounting for the gradient deformation arising from the LC system would be an attractive next step, as it can potentially be automated and requires a minimal number of measurements. Ideally, it would improve the accuracy of predicted optimal gradient separations.
In this paper, we present a novel computational strategy to establish the effects of gradient deformations caused by the geometry of the instrument, yielding geometry-independent retention parameters from a limited number of gradient experiments. We demonstrate that the influence of gradient deformation is potentially significant and that it is worthwhile to correct for this. As input data, our algorithm employs a measured gradient delay in a water-water system. For our algorithm, multiple response functions were tested to determine the most accurate and best interpretable model. To incorporate the gradient deformation into retention modelling, new models were derived that support any number of gradient steps. The used experimental setup exclusively provides information on the geometry-induced gradient deformation, but excludes any effects of the solvent and mixtures, such as viscosity, density and miscibility effects. Solvent adsorption [ 20 , 21 ] and solvatochromic effects were not studied.

Theory
In this paper we employ the log-linear ("linear solvent strength", LSS) model for retention prediction, but other retention models may be used as well.

Retention time in LSS gradient elution with linear gradient
In the log-linear model ( Eqn. 1 ), k 0 represents the extrapolated retention factor at ϕ = 0 and S represents the magnitude of change in ln k with increasing eluent strength ( ϕ).
This model is often referred to as the linear solvent strength (LSS) model in combination with linear gradients [23] .
In the event that an analyte elutes before a programmed gradient, the retention time ( t R, before ) is given by where k init is the analyte retention factor at the start of the gradient and t 0 depicts the column dead time. By incorporating the LSS model into the gradient equation it follows that is the slope of a gradient running from ϕ init to ϕ final , t init is the initial isocratic time, t D the dwell time, k ϕ the retention factor at a certain fraction of strong solvent and τ = t D + t init + t 0 . Schoenmakers et al. derived equations to predict retention times during a linear gradient [5] t R, gradient = 1 SB as well as retention times in the event that the analyte elutes after the gradient, with retention factor at the final conditions k final , and gradient time t G .

Describing the shape of the geometry-corrected gradient
Characterizing the shape of the geometry-corrected gradient (GCG) starts by finding a model and related parameters that accurately describes how the programmed gradient is influenced by the instrument. This response of the system can be expressed in the form of a distribution or a so-called response function. The whole gradient experiences the same geometrical effects as expressed through this response function. Summing all prior signals resulting from the response function at any point in time results in the GCG. Examples of a programmed gradient, the corresponding response function and the GCG are depicted in Fig. 1 .
The response function can be expressed using a mathematical distribution, the properties of which can be described using its statistical moments [24] . A graphical overview of the moments [25] and their parametrized symbols, which are used in this paper as instrument parameters, is shown in Fig. 2 . The corresponding equations can be found in Supplementary Material section S-1.
The zeroth moment ( A ) is the area. In our case, this moment is adjusted to be identical to the composition at a certain time point.
The first moment ( μ) is normalized for the area and gives the centre of gravity of the distribution (mean), which is equal to the dwell time of the setup. The variance ( σ 2 ) or scale is the centralized second moment (i.e. corrected to the first moment) and which  is correlated to the width of the distribution, which captures some of the flow profile of the instrument. The skewness (i.e. magnitude of tailing/fronting) is the standardized and centred third moment (i.e. corrected to the variance and the first moment) and is instrument dependent and correlated to the kurtosis (i.e. degree of flattening). The kurtosis (K) is the standardized and centred fourth moment (i.e. corrected to the variance and the first moment). The skewness and the kurtosis together describe the degree of tailing and the shape of the distribution.The response curve thus can describe the deviation from the programmed gradient arising from any possible source, such as the dwell volume, flow and imperfections therein ( e.g. flow turbulence caused by the mixer or sharp bends in tubing) .

Sample preparation
All test-compound solutions were prepared in ACN and contained 100 mg •L −1 of thiourea as t 0 marker. The approximate concentrations of the solutions were: EMOD, 250 mg •L −1 ; TOL, 1500 mg •L −1 ; SUD, PHEN and ANT, each 500 mg •L −1 .

Chromatographic method
Measurements of the actual gradient shape were performed on all LC instruments without a column at flow rates of 0.25, 0.5 and 0.75 mL •min −1 . Solvent A was water and solvent B was water containing 0.1 vol% acetone. An initial isocratic time (100% A) of 0.25 min was used. The gradient ran from 0 to 100% B in 0.5, 1.0 or 1.5 min. All gradient measurements were performed in triplicate. UV detection was performed at 210 nm with a bandwidth of 4 nm; the reference wavelength was 360 nm with a bandwidth of 100 nm and a slit size of 4 nm. The sampling rate for Instruments 1 and 3 was 160 Hz, while for Instrument 2 it was 20 Hz. Column ovens were set to 30 °C.
Retention-time measurements of the test compounds were performed on all instruments with the LC column installed using a flow rate of 0.5 mL •min −1 with an initial isocratic time (100% solvent A) of 0.25 min. The gradient ran from 0 to 100% B in 0.5, 1.0 or 1.5 min, followed by a 10-min isocratic hold. Solvent A was ACN-water (5:95, v/v) and solvent B was ACN-water (95:5, v/v). Between measurements, 10 min of equilibration time was allowed. For all analyte measurements, the same two bottles of solvent mixtures (A (5:95, v/v) and B (95:5, v/v)) were used which were ultrasonicated before use. The injection volume was 5 μL. All testcompound solutions contained a t 0 marker and were measured individually. Measurements were repeated 4 times and thus 5 measurements per solution. Detection was at 254 nm with a bandwidth of 4 nm; the reference wavelength was set to 360 nm with a bandwidth of 100 nm and a slit size of 4 nm. The sampling rate was 160 Hz for instruments 1 and 3 and 20Hz for instrument 2. Column ovens were set to 30 °C.

Data treatment
All algorithms were written in MATLAB 2019b update 3 (Mathworks, Natick, MA, USA). Measurements of the actual gradient shape were first normalized between 0 and 1, after which three identical measurements were averaged to minimize noise. The recorded gradient measurements were truncated to a period of 6 minutes for establishing the response-function parameters to reduce the computation time. Retention times and t 0 values were averaged before fitting the retention model.

Results and Discussion
Our strategy encompasses three steps to correct the measured retention parameters for the actual gradient. Firstly, the instrument-specific response function that determines the shape of Table 1 Obtained sum-of-squared errors (SSE) values for the regression experiments determined on Instrument 1 with flow rates of 0.25, 0.5 and 0.75 mL •min −1 and gradients times of 0.5, 1 and 1.5 min. Color scale from red through yellow (50%) to green representing high to low SSE values. the actual gradient is determined. Secondly, this response function is used to predict the corrected shape (GCG) of a gradient of interest. The GCG can then be used to more accurately determine the LSS model parameters describing analyte retention. Finally, these latter retention parameters are used to predict the retention on a different instrument, using its specific response function.

Selecting the optimum response function
Generally, two methods exist to describe the gradient deformation. One relies on a direct fit of the gradient curves. The other method, describes how every timepoint of the initial gradient passes through the detector. The first approach should allow a description of the start (quick bend), middle (linear), and end of the gradient (slow bend). This may, for example, be achieved with an alternative-skew exponential power distribution [26] , which was slightly altered for this purpose, resulting in an accurate description of the profile. However, this first approach works only for linear gradients and no chromatographically meaningful correlations between the parameters describing different gradients could be found, resulting in large errors when predicting new gradients. Therefore, in this paper the second approach is followed. Which is not limited to linear gradients.
Multiple mathematical distribution functions were tested for their suitability to describe the response function to construct the GCG by applying the function to all time points. The gradientprofile measurements on Instrument 1 were used for the initial exploration of the different response functions. The sum of squared errors ( SSE) of the fitted distributions are reported in Table 1 and Fig. 3 . The four-parameter stable function is referred to as "4p-Stable". To test the influence of the asymmetry and tailing factor in the stable function, two stable functions with one of these parameters fixed to its "Gaussian" state were tested. These were referred to as "3p-Stable". Besides the stable and Gaussian functions, other distributions that can express asymmetrical tailing were tested. These specific distributions were tested because of their ability to describe significant tailing [22] , which is required to cover the slow rounding at the end of the actual gradient. Equations of the tested distributions can be found in Supplementary Material section S-2. Two examples per fitted distribution can be found in Supplementary Material section S-3. The four-parameter stable function was found to yield the smallest SSE and is referred to as the stable distribution in the rest of the paper.
The selected stable distribution contains four parameters ( δ, γ , β, α) which respectively resemble the four statistical moments ( μ, σ, S, K), although they are defined differently [22] , as indicated in Fig. 2 . The response functions tested all are distribution functions, based on the same underlying mathematics, but their unique properties can be described through their characteristic function expressed as ϑ X (u ) . This way of describing the stable function is the most concise way to cover all possible interpretations of the stable function [22] . The ϑ X (u ) is defined, while u is in the real domain, where u is the x domain up to the upper limit of x , i is the imaginary unit and e is Euler's number. E(x ) is the expected value, F (x ) is the cumulative distribution function and f (x ) is the probability density function. X states the variables in the equation. In case of the stable distribution these equal ( δ, γ , β, α) . The tailing parameter (α) is restricted between 0 < (α) ≤ 2 and the asymmetry parameter (β ) is restricted between -1 ≤ β ≤ 1. When α equals 2, the distribution is Gaussian. When β is positive the distribution is tailing and when β is negative the distribution is fronting. δ is the mean parameter and γ the scale parameter. Both are positive real numbers for the application as response function.
The ϑ X (u ) of the stable distribution is defined as Eqn. 8 , where α does not equal 1 and X equals ( δ, γ , β, α) . In this equation, the sign logic is defined as follows:  Table 3 Response function parameters obtained by fitting the gradient profiles obtained on three different instruments with the four-parameter stable function. Parameters were obtained by fitting a series of gradient profiles (see Table 2 ) simultaneously. The variance and mean parameters were normalized by multiplying by the flow rate.
Application of the Fourier-inversion theorem to the ϑ X (u ) yields the following equation for the probability density function of the stable distribution, where t is the time.
The above results ( Table 1 , Fig. 3 ) show that the stable distribution function describes the data best. An additional advantage is that its parameters resemble the statistical moments, allowing the user to explain differences between instruments in a less-abstract way than with some of the other distributions.

Determining instrumental parameters
Fitting the selected response function to each experimental gradient profile obtained with Instrument 1 yielded scale (γ ) and mean (δ) parameters that appeared inversely correlated to the flow rate. Table 2 provides the best-fit response-function parameters for each setting. Moreover, the asymmetry factor ( β) was found to be always positive and a tailing factor ( α) lower than 2, indicating tailing of the distribution function. This was expected since any additional instrument component of the flow-delivery system may induce flow imperfections which is expected to result in a gradient delay, and not an acceleration. This would primarily be observable in the mean parameter ( δ) , but if this delay is not uniform it leads to tailing. Flow imperfections can occur due to irregular flow in the mixer and other parts of the LC instrument. Interaction between the solvent and the LC instrument is expected to be minimal in a well-designed and well-maintained system. In the present experiments (mixing water with water containing acetone) only interaction of acetone with system components would be reflected in the gradient profile, which was assumed to be minimal. Parameters for each instrument were obtained by fitting all the gradient-profile measurements simultaneously with the response curve (i.e. the stable function), so as to incorporate the effect of the flow rate which introduces an additional error between measurements. In case of an random error the response function can be defined in less detail and thus results in a more Gaussian shape of the response function. The obtained parameters for the individual response curves are shown in Table 3 . In Fig. 4 the resulting response functions are plotted for a flow rate of 0.5 mL •min −1 . For Instrument 2 the response function is a Gaussian distribution since the tailing parameter (α) is 2. This was caused by the downward drift in the detector response, which the regression model attempted to compensate as seen in Supplementary Material section S-4. While this effect was caused by the detector and thus has no influence of the actual gradient, the measured data is influenced. However, this resulted in a Gaussian response function which means that the deformation in the form of bending at the end is less well described and thus making the correction less accurate. These response curves were used to describe the GCG. An example is shown in Fig. 5 , where the measured gradient of Instrument 1 ( F = 0.5 mL •min −1 , t G = 1 min) is plotted (blue curve) with the reproduced gradient (GCG) overlayed (dashed red curve).

Retention modelling using n gradients
In order to improve retention modelling, the method to compute retention times had to be adapted to accommodate for the established GCG shapes. The equations for the retention time Eqns. 4 and (5) were derived by incorporating the retention model ( Eqn. 1 ) in the general gradient equation ( Eqn. 3 ) for the case of a linear gradient. The GCGs as shown in Fig. 5 were approximated numerically as a series of short linear segments. To facilitate this, an adjusted gradient equation was derived to handle any number ( n ) of gradient steps with duration t n , slope B n and initial composition ϕ n (with ϕ n + 1 = ϕ n + t n B n ). The retention factor at ϕ n is denoted by k n . The derived formulas are shown in Eqn.s 10 and 11 (See Supplementary Material section S-5 for a detailed derivation).
At this stage, retention models can be included, as shown for the LSS model in Eqns. 12 and 13 . The derivation of the Eqns. 12 and 13 can be found in Supplementary Material section S-5.

t R, d uring n th grad ient
t R, a f ter n gradients = k n +1 t 0 − Eqn.s 12 and 13 can be used to numerically approximate the retention time for elution under GCG conditions. However, since, t D is already included in the GCG it should be set to zero.

Without correction for gradient deformation
For comparison, retention parameters for the LSS model were first established assuming a perfect linear gradient, only considering the dwell time of the instruments ( i.e. not correcting for instrument-induced deformation, using Eqns. 4 and 5 ). In essence, Eqn.s 12 and 13 were applied to the theoretical gradient profile to eliminate the error due to the model, but these equations reduce to Eqn.s 4 and 5 when there is no change in slope during the gradient. The dwell time was taken as the time difference between the midpoint of the programmed gradient and that of the measured gradient. The results are shown in Table 4 . The measured retention times for each test compound and the accompanying t 0 values can be found in Supplementary Material section S-6. While the retention parameters obtained for a specific instrument allow computation of retention times of the test compounds on the same instrument with good accuracy, as expected, it is clear that the LSS parameters established using different instruments deviate dramatically. This is particularly evident for Instrument 2. These results confirm that only adjusting for the dwell time does not suffice to obtain correct retention parameters, and clearly illustrate the adverse effects of the gradient deformation.

Incorporating correction for geometry-induced gradient deformation
The LSS parameters of the test compounds were also established using Eqns. 13 and 14, i.e. correcting for gradient deformation employing the established GCGs for each LC instrument. The GCGs were calculated using the parameters from Table 3 and approximated by 100 linear-gradient steps. The results are listed in Table 5 . Gradient correction yields higher ln k 0 and S values as compared to those obtained by correcting only for the dwell time (see Section 4.3.1 , Table 4 ). In all cases but ANT on Instrument 1, the model showed a much better fit to the retention times, as indicated by the SSE values. The obtained retention parameters ( ln k 0 and S ) still vary significantly. Although geometry-induced gradient deformation have been eliminated still, solvent-related deformations remain, which may be significant, due to the differences in pump technology between the LC instruments.
In order to test whether the retention parameters obtained using the GCG correction yield more consistent retention prediction Table 4 LSS parameters ( Eqn. 1 ) determined for the test compounds on each LC instrument using uncorrected gradient parameters and Eqns. 4 and 5 . Colour scale from red through yellow (50% of Table 4 and 5 ) to green representing high to low SSE values of the predicted versus the experimental retention times respectively.

Table 5
LSS parameters Eqn. 1 ) as determined for the test compounds on each LC instrument using corrected gradient parameters and Eqns. 12 and (13) . Colour scale from red through yellow (50% of Table 4 and 5 ) to green representing high to low SSE values of the predicted versus the experimental retention times respectively. among the LC systems, the error in predicted retention times was calculated. The retention parameters determined on Instrument 1 were used to predict retention times on Instruments 2 and 3, using uncorrected gradient data (dwell volume only) or incorporating gradient correction (using GCGs). The obtained results are summarized in Fig. 6 (see Supplementary Material section S-7 for a detailed overview of each compound and instrument combination). Fig. 6 indicates that after GCG correction for the influence of the instrumentation the relative error decreases in all situations in comparison with correcting for the dwell volume only. Especially the retention time prediction for instrument 2 is approaching acceptable errors. It was not expected that this prediction would perform best since the response function was influenced by the downward drift resulted in a Gaussian response function which should result in a less accurate defined GCG.
The error for many compounds is still in the high single digits. This may be due -at least in part -to solvent-related deformations. The fact that we aim to correlate a binary system with two quaternary systems may contribute to different deformations. Quaternary pumps tend to give rise to composition errors due to volume contraction during proportioning at the multi-channel gradient valve, while the same effect causes flow errors for binary pumps.
Additionally, the measurements of the gradient profiles may be improved. In this research, acetone was used, which is a volatile compound. Non-constant losses in the online degasser of the flowdelivery module may cause errors. This may be verified by running a backward gradient (from 100% to 0% B). In that case the acetonecontaining solvent will spend less time in the degasser when channel B has no flow. A non-volatile UV-absorbing analyte, which does not adsorb to the degasser membrane or other surfaces, may improve the accuracy of the measured gradient profiles and the de-rived instrument parameters. Another limitation of our approach is the assumption that the response of the UV detector is linear for acetone and that solvatochromic effects do not occur. However, the effect of the latter is not expected in the water-water and acetone system used in this study. Finally, the results obtained for Instrument 2 showed that detector drift can affect the obtained instrumental parameters (See example in Supplementary Material section S-4). Additional studies on all of the above points may further refine the corrections.
Although not all instrument influences could be corrected for, a significant reduction in prediction errors was achieved, which may improve retention modelling and method transfer and may contribute to determining instrument-independent retention parameters.

Concluding remarks and outlook
We have developed an algorithm to correct retention modelling for gradient deformation induced by instrument geometry. Several mathematical distributions were evaluated for their ability to describe the response function associated with the gradient deformation. The four-parameter stable distribution was found to be most suitable for this purpose. Using this response function, the geometry-corrected gradient (GCG) shape for water-based systems could be accurately described. Both the variance and the mean of the response function proved inversely proportional to the flow rate.
For retention prediction the deformed ( i.e. non-linear) gradient profile was approximated by a hundred small linear segments. Equations were derived to compute retention times for a compound eluting during and after such complex multi-step gradients. This allowed correcting for the GCG shape and resulted in Fig. 6. Relative errors (%) in the predicted retention times of the test compounds on Instruments 2 (top) and 3 (bottom) obtained when using retention parameters determined for the test compounds on Instrument 1 at different flow rates. Relative errors obtained using uncorrected gradients and Eqns. 4 and 5 (left) and using GCG correction and Eqns. 12 and 13 (right). more comparable retention parameters between instruments. Most importantly, we found that correcting the retention parameters geometry-induced deformations significantly improved the prediction of retention times on other instruments. The average reduction of the prediction error depended on the instrument. Using data obtained on one instrument (Instrument 1) and the newly developed algorithm improved the average relative error in retention time from 9.8% and 12.2% down to 2.1% and 6.5 % for two other instruments (Instrument 2 and 3, respectively) in comparison with a conventional approach (only correcting for the dwell volume).
However, while prediction accuracy could be improved, a large spread remained between retention parameters for various analytes obtained using different instruments. Thus, such parameters should not be interpreted as the true retention parameters. In our proof-of-principle study we corrected for gradient deformation measured with water and water containing a tracer (acetone) as the gradient-forming solvents. This allowed correction for geometry-induced deformation of the gradient. While the response function described this water-water system adequately, additional effects due to viscosity and density differences and possible volume contraction or expansion are expected when mixing different solvents. Taking these solvent effects on the gradient shape into account may potentially improve the accuracy of the retention parameters and thus further reduce the effect of the instrumentation on the obtained retention parameters. However, this correction may be more complex, as more variables related to the solvents used, additives, temperature, pressure, etc. may need to be considered. When using different solvents solvatochromic effects may also occur, which may affect the measured gradient profile. Therefore, methods to account for changes in the absorption coefficient may also need to be explored. Furthermore, it should be noted that our measurements were exclusively conducted using fast gradients. The results show that the extent of deformation depends on the employed flow rate. Lower flow rates may be expected to yield improved prediction accuracies. Moreover, the present study was limited to the log-linear ("linear-solvent-strength") retention model. Further improvements may be obtained by investigating other models.
Nevertheless, correction using the current algorithm yielded significantly improved prediction accuracies across different instruments.

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. 8