Next Generation Peak Fitting for Separations

Separation scientists frequently encounter critical pairs that are difficult to separate in a complex mixture. To save time and expensive solvents, an effective alternative to conventional screening protocols or mathematical peak width reduction is called iterative curve fitting. This method does not sharpen the peaks to enhance the chromatographic resolution, but extracts the original shape from overlapping peaks in a complex separation, as if an isolated compound were injected. The generalized family of Haarhoff-van der Linde of peak functions accounts for most chromatographic peak shapes under analytical, isocratic or gradient elution, and mass-overloaded conditions. Four illustrative examples are discussed: i) subsecond separation of five compounds; ii) area extraction from 30 partially resolved peaks separated in under a minute; (iii) iterative curve fitting and baseline correction for a nicotine containing E-liquid; and (iv) advantages of fitting an overloaded peak shape for preparative separations. The large F-statistic, and R² near to 1.0 in all cases, shows excellent modeling of the data’s variance.

Many real-world chromatographic separations have difficult critical pairs that fail to resolve under most experimental conditions (1,2). Standard advice in such instances is to employ a different column chemistry, design a better gradient, or select a different mobile phase system. Screening conditions to produce a chromatogram without peak overlap consumes costly time and solvents in a busy laboratory. More sophisticated and expensive options include two-dimensional liquid chromatography (2D-LC), two-dimensional gas chromatography (2D-GC), and information-rich detectors such as mass spectrometers, or photodiode array detectors. Also, the optimal choices of mobile and stationary phases are not always sustainable, and greener solvents often produce skewed and low-efficiency peaks (3,4). With a current emphasis on “greening” separation methods and improving workflow in terms of saving time and solvents (3,5,6), it is the right time for separation scientists to resort to digital signal processing (DSP) approaches. DSP techniques are essential in medical imaging, astronomy, and spectroscopy. If physicians can trust digitally enhanced medical images for life-altering health decisions, separation scientists should also fully utilize the benefits that DSP could bring to their research work. Currently, and largely unbeknownst to most users, all mass spectrometers and nuclear magnetic resonance (NMR) signals are highly processed mathematically. Although not the focus of this work, multiple DSP techniques in the literature reduce the peak widths and increase chromatographic resolution (7). Herein, we discuss one compelling alternative DSP technique that offers digital resolution without altering peak widths or shapes called iterative curve fitting. This approach, also referred to as peak fitting, allows for the mathematical modeling of a given peak shape to subsequently calculate peak areas, heights, statistical moments, and other chromatographic figures of merit in a noise-free environment. The term iterative implies that the curve fitting process is repeatedly improved until the residuals (the difference between the data and the model) are minimized, usually through least squares formulation and non-linear optimization algorithms. With this protocol, the profile of each peak, overlapping or non-overlapping, is obtained as if a single component were injected instead of a mixture.

Almost all chromatographic peaks have a degree of asymmetry (8,9), yet the most common methods to calculate figures of merit (such as resolution, efficiency, and retention factor) are based on Gaussian distribution approximations. The interest in resolving partially overlapping chromatographic signals with mathematical models dates back to the early 1970s, when it was realized that the simple Gaussian model for fitting peaks could not capture the variance of the data effectively (10). Despite this understanding and the development of over 200 different versions of chromatographic peak functions (11), two models have remained consistently popular: the Gaussian and the exponentially modified Gaussian (EMG) distributions. The current need to address the demands posed by complex separations in today’s laboratories is to develop new powerful models that can fit or consider most experimental isocratic or gradient peak shapes under analytical or overload conditions. Secondly, the peak fitting procedure should automatically attain the global minimum of the least square objective function through iterative non-linear optimization without manual intervention or requiring the user’s initial estimates. The key advantage of curve fitting in chromatography is that the extracted peak data is drift and noise free. This information can be obtained with high statistical confidence even when signals from single channel detectors such as ultraviolet-visible (UV-Vis) spectroscopy evaporative light scattering, charged aerosol, and refractive index detectors coincide to a significant degree. The same concepts apply to gas chromatography, with single-channel data from flame ionization, barrier discharge, or thermal conductivity detectors.

Popular chromatography data systems (CDS) currently do not extract peak information from modeling, and can only do the rudimentary perpendicular drop or tangents method to obtain crude area information (9,12). The proposed peak functions here are new theoretical models derived from the Haarhoff-van der Linde peak function (HVL)(13), extended by generalizing the underlying statistical Gaussian distribution and incorporating instrument response functions (I) (14). These statistical innovations add third and fourth-moment adjustments to the core peak shape (the probability density function [PDF]), and the signal processing innovations add the I as a convolution to the core peak PDF (15). These models are supersets of the HVL that manage the non-idealities of tailed and fronted shapes that increase with non-linear chromatographic concentration effects. Consequently, the results computed from these fitted models are accurate and precise for single and overlapping peaks. The estimated and derived parameters include accurate peak areas, retention times, resolutions, means (t_r), variances (σ²), skews, and kurtosis values while mapping nearly all the data’s variance. Here, a new peak model called the twice-generalized HVL model will be abbreviated as Gen2HVL, and when convolved with a half-Gaussian and exponential decay, I term, as Gen2HVL<ge>. The mathematical documentation of these functions is available elsewhere (14), and can be applied using built-in curve fitters combined with custom-made Python or MATLAB scripts. Eventually, we believe that advanced signal processing will be embedded into all commercial chromatography data systems, just as it is for NMR, mass spectrometry (MS), Raman, and Fourier transform infrared spectroscopy (FT-IR).

Experimental

Columns

C18 columns were purchased from Supelco (Ascentis) in the dimensions of 150 mm x 3.0 mm (i.d.) and from Agilent (EC-C18) in the dimensions of 50 mm x 3.0 mm (i.d.), both packed with 2.7 μm superficially porous particles (SPPs). The HILIC separation shown was conducted using a custom prepared bare silica column in the dimensions of 10 mm x 3.0 mm (i.d.) with 1.9 μm SPPs as fully described elsewhere (16). The analytical overloaded separation shown was conducted using a NicoShell column in the dimensions of 100 mm x 4.6 mm (i.d.) with 2.7 μm SPPs acquired from AZYP, LLC.

Instruments

Example chromatograms displayed here were collected on two systems: Thermo Vanquish UHPLC and Agilent 1220 HPLC. The Vanquish contained a quaternary pump (VF-P20A), a split sampler (VF-A10-A), and a variable wavelength detector (VF-D40-A). The Agilent 1220 HPLC contained a G4290B UV detector. Separations were conducted at ambient temperature unless otherwise noted to avoid the additional volume of tubing required to use a column oven. Mobile phases are reported as volume-to-volume ratios.

Software

All chromatograms were analyzed using PeakLab (v. 1.05.02) from AIST Software. Baseline correction (when noted) was done in PeakLab using the non-parametric linear model or constrained cubic splines (Figure 3) with manual piecewise sectioning. Peak fitting was achieved with the “local maxima peaks,” “hidden peaks/second derivative,” or “hidden peaks residuals” options. All figures were made by exporting the fits from PeakLab and plotting them in MATLAB (2023a, The MathWorks, Inc.).

Results and Discussion

To demonstrate the power of advanced peak fitting, we have curated four vastly different case studies discussed below that we believe will be of interest to a practicing chromatographer. These examples include three modes of chromatography (reversed-phase liquid chromatography [RPLC], hydrophilic-interaction chromatography [HILIC], and normal phase liquid chromatography [NPLC]) in both chiral/achiral and analytical/overload settings. It should be noted that iterative curve fitting is not a peak deconvolution process, where the goal is to reduce peak width and increase analytical resolution by reversing the mathematical process of convolution. In a loose sense, scientists use deconvolution when they mean the decomposition of a signal, which refers to the breakdown of a complex signal into its constituent signals. For example, in liquid chromatography-mass spectrometry (LC–MS), the total ion chromatogram (TIC) can be decomposed into extracted ion chromatograms for each monitored m/z, and then all extracted ion chromatograms can be summed to regain the TIC. Curve fitting mathematically allows for a similar decomposition, but only uses one data channel. Here, each peak is transformed into its own fully resolved signal as if it were injected into the chromatograph alone and in its pure form, even if the original chromatogram contains overlap.

Sub-Second Separation of Five Nucleosides Using HILIC

Current instrumentation technologies now allow for the separation of components on the sub-second scale in both gas (17) and liquid (16) chromatography. To achieve this level of speed in LC, small columns with high flow rates must be used with a pump capable of delivering flow rates as high as 8 mL/min at ≥ 800 bar. Figure 1 shows an example of HILIC separation where five nucleosides (6-aza-2-thiothymine, thymine, 5-(hydroxymethyl)uracil, 6-aminouracil, and isocytosine) were separated in just ~1 s using a short 10 mm × 3.0 mm custom made bare silica superficially porous particle (SPP) column (16). Ultrafast separations are excellent case studies for peak fitting, as the high number of analytes present in a short time window and low column efficiencies frequently produce peak overlap in the chromatogram. These peaks are also subject to further significant shape deformation from extra-column effects due to their low retention factors and the relatively large system volumes compared to the small volumes of the narrow and short columns (15,18).

Figure 1: Sub-second HILIC separation of five nucleosides (6-aza-2-thiothymine, thymine, 5(hydroxymethyl)uracil, 6-aminouracil, and isocytosine) fit with five Gen-2HVL<ge> functions. (Conditions: Vanquish UHPLC, 10 mm x 3.0 mm (i.d.), 1.9 μm bare SPP silica, 250 Hz, 8.0 mL/min, UV detection at 254 nm). Raw data (black, modeled peaks (red). Data reproduced from Patel et al. (16).

The coefficient of determination (R²) is commonly used within chemistry and known to be equal to one minus the ratio of the residual sum of squares (SS_rs or SSE) to the total sum of squares (SS_tot or SSM). The F-statistic is another metric widely used by statisticians but less so by chemists. When statistically generalizing models, one must use caution in adding parameters to the function since the model with more parameters will fit the given data at least as well and often better with a higher R², than the one with fewer parameters. However, this increase is not always statistically significant. The F-statistic can be calculated to see if the improvement in the fit is significant using the same SS_rs and SS_tot data, accounting for the number of parameters used in each model (14). Therefore, we consider the function that produces the highest F-value to be the best model to estimate the experimental data. Note that no R² or F value can indicate satisfactory goodness-of-fit (GOF) alone, but they help compare functions when modeling the same data set (19,20). Visual inspection of the residuals is of the highest importance in assessing GOF, and unfortunately, no GOF determination within classical statistics is better than human interpretation. Ideally, the residuals produce a flat line with random fluctuations from the noise in the experimental data and are not modeled by the peak functions. Any deviation from this flat randomly oscillating line indicates misfit between the data and the model.

Figure 1 shows Gen2HVL convolved with the half Gaussian and exponential instrument response, Gen2HVL<ge>, fit to the ultrafast chromatogram (red) discussed above. The modeled data (black) describes the experimental peak profiles (red) with an excellent GOF. As expected, when using this effective model, the R² and F values were 0.9996 and 23,582, respectively. The residuals here are ideal and reflect the variance from the noise of the experimental data not being represented by the model. This result represents a greater than ten-fold increase in GOF compared to the classical Gaussian function (F = 2,983). With the data modeled through the iterative curve fitting process in 1D, each analyte signal can now be treated as if it were a separate chromatogram. Figure 1 shows the individual extracted peaks (blue), which can be integrated, quantified, and statistically described accurately in the absence of noise. The sum of all extracted peaks recreates the original chromatogram with the added advantages of no drift, noise, or overlap.

Separation of 30 Peaks in Just 1 Minute

Giddings applied statistical overlap theory to chromatography, predicting that a column has a finite peak capacity roughly equal to

(valid only with isocratic elution considering a resolution of one and moderate retention) (2). Furthermore, it was predicted that only 18% of a column’s peaks would contain a single component in a random chromatogram (2). Fast separations, with samples containing many components, will often result in peak overlap due to the limited number of plates provided by small columns used at high linear velocities. Peak capacity can be physically improved by gradient elution as the column’s apparent efficiency increases or by 2D separations, which increase time and expense. Alternatively, peak fitting can be used when operating above peak capacity without changing the experimental conditions. Figure 2a presents a reversed-phase linear gradient separation of 30 components in just 1 min on a 5 cm by 3 mm (i.d.) C18 column. The large amount of peak overlap in this separation makes it an appropriate application to evaluate the limits of peak fitting science. In Figure 2a, the chromatogram is fitted with the Gen2HVL function. The resultant fit is highly accurate with an R² of 0.9997 and a F-statistic of 249,725. More importantly, the residuals observed in this separation are ideal, showing a nearly flat line with random fluctuations due to the noise in the experimental data. This example shows an ideal outcome from peak fitting and provides an application for fast separations with high peak density. As demonstrated in Figure 2b, each fitted peak can now be individually treated as its own fully resolved signals without noise. Integrating the resolved fitted peaks (Figure 2b) to gain their individual areas is highly accurate and reproducible, whereas it is difficult or impossible with the raw data (Figure 2a).

Figure 2: Ultrafast separation of 30 components in under one minute (a) fit with the Gen2HVL function and (b) extracted peak profiles. Conditions: Vanquish UHPLC, 50 mm x 3.0 mm (i.d.) EC-C18 with 2.7 μm SPPs, ambient temperature, 250 Hz, 2.5 mL/min, UV detection at 254 nm.

Analysis of a Commercial E-Liquid with Gradient Elution RPLC

E-liquids are fluids used in electronic cigarettes, which are popular amongst today’s youth (21). A recent study using LC paired with high-resolution mass spectrometry (LC–HRMS) found over 2,000 components in some commercial e-liquids (22). Of these components, most of their identities are unknown, meaning their human activity and toxicology are also unknown. Methods to separate, identify, and quantify components in these liquids are extremely important for public health and policy making. When short wavelengths (~190–220 nm) are used to analyze these samples, molecules that do not contain or have weak chromophores can be detected. This wavelength region is problematic when the chromatographer uses alcohols (such as methanol) in the mobile phase as they absorb deep UV (3). In isocratic elution, the mobile phase absorbing deep UV light results in a raised baseline and reduced sensitivity. This absorption becomes more problematic when using gradient elution as the baseline drifts from the low absorbing water to the stronger absorbing methanol. Chemically, the drifting baseline can be reduced by using acetonitrile as the organic solvent, but this is more expensive and far less environmentally benign (3).

Figure 3a (purple) shows a cogent example of the analysis of a commercial e-liquid using gradient elution RPLC with two key issues: a heavily drifting baseline and severe signal overlap. Combining UV detection at 210 nm with methanol as the organic solvent in the gradient creates an undesirable drift in the baseline (Figure 3a, purple). Even with the reduced sensitivity from the high background signal, this chromatogram provided significantly more detectable components than when the sample was identically analyzed at 254 nm. To solve this problem with post-analysis calculations, the baseline was first extracted (Figure 3a, blue) using a constrained cubic spline with user-controlled piecewise sectioning of the baseline in PeakLab. The baseline correction can also include undesirable peaks and features. The result of the baseline subtraction is presented in Figure 3a (black), showing a zeroed baseline with preservation of the peaks, their location, and their intensity. The area of these peaks and their retention time can be accurately determined by fitting the baseline corrected chromatogram. Figure 3b shows a “zoomed in” portion of this chromatogram fit with 20 Gen2HVL functions. The R² from this fit is 0.9997, the F-statistic is 324,597, and the residuals (Figure 3b, orange) are random with low relative intensity. These merits indicate a robust and accurate fit; peak area and location information are now readily available from this problematic chromatogram.

Figure 3: (a) Reversed phase gradient separation of a commercial e-liquid (purple), modeled baseline (blue), and subsequent baseline corrected chromatogram (black). (b) Fitting of a section of the baseline corrected chromatogram using Gen2HVL functions. Conditions: Vanquish UHPLC, 150 mm x 3.0 mm (i.d.) Supelco C18 with 2.7 μm SPPs, 25 ºC, 50 Hz, 0.425 mL/min, UV detection at 210 nm.

Mass Overload of a Drug's Enantiomers in NPLC

Preparative liquid chromatography revolves around overloading the column to collect the largest amount of desired components per mass of stationary phase per unit of time (productivity) (23). The band profiles and retention times are challenging to understand and predict when overloading columns (24). To obtain this information experimentally, a loading study is often completed on smaller columns with the same packing and then directly scaled to the larger, more industrial sized column (5,25). Overloading the analytical sized column can also be helpful for “semi-preparative” chromatography when only a small amount (μg to mg scale) of a highly pure compound is needed in environments where a larger column is unavailable. Semi-preparative chromatography is often used to isolate enantiomers of a compound when small masses are needed for stereospecific activity testing. This process involves using a chiral stationary phase, overloading the phase, and then collecting the fractions for each enantiomer.

Figure 4 contains an example of part of a mass loading study for R/S-alprenolol using a semi-synthetic macrocyclic glycopeptide chiral stationary phase (NicoShell) in the normal phase. Even though the column is of analytical size (100 mm x 4.6 mm i.d.), up to ~0.5 mg of the drug can be loaded on the column per injection. When doing loading studies, it is beneficial to overload the column to the point where the peaks begin to overlap, known as touching bands, to maximize productivity (23). This overlap, however, does introduce some uncertainty since the start and end of the overlapping peaks cannot be seen (Figure 4, black). To ensure maximum enantiomeric purity (> 99.9%) of the isolated compound, the area of overlap should be separately collected and recycled. To avoid the trial-and-error method when determining where to collect the enantiomers, the peaks can be fit using a model like the Gen2HVL. The Gen2HVL can also manage non-linear peak shapes. Figure 4 shows the fitting of both enantiomers (green and pink) with the Gen2HVL function. The region of enantiomeric overlap that must be recycled is now readily seen between the dotted lines. Note that the residuals here are larger in magnitude than the analytical examples presented above due to the overload profile. Still, using this technique makes it immediately apparent to the chromatographer where to collect the fractions.

Figure 4: Overloaded separation of 587 μg of R/S-alprenolol using normal phase chiral chromatography with HPLC-UV fit with the Gen2HVL function. Conditions: Agilent 1220 HPLC, 100 mm x 4.6 mm (i.d.) NicoShell with 2.7 SPPs, collected at 20 Hz, down sampled at 6.67 Hz. Data reproduced from Aslani et al. (26).

Pitfalls in Peak Fitting

As a note of caution, peak fitting should not be used when two or more peaks overlap entirely. As a rule of thumb, the peak maximum or a shoulder must be visually visible before proceeding with the iterative curve fitting process. To confirm the presence of a shoulder, the first or second derivative can be analyzed, looking for splits in the derivatives in the region of interest. The higher the degree of overlap, the lower the statistical confidence will be in the fits. When analyzing overlapping peaks produced by mass spectrometric detection, there is a danger of ion suppression in the region of co-elution, which can give erroneous results. Note that peak fitting is still helpful for LC-MS when there is partial overlap of isobars, as seen in the cases of enantiomers, epimers, and structural isomers. When using generalized functions with several parameters (6 for Gen2HVL, 9 for Gen2HVL<ge>), each parameter may not be statistically significant, meaning a specific parameter is zero. The statistical significance of function parameters can be important for theoretical modeling but has trivial effect on the empirical information (peak area, location, and efficiency), which was the focus of this study.

Conclusions

This article demonstrates a powerful peak model applied to real chromatographic data under analytical (isocratic/gradient) and overload conditions. The newly introduced twice-generalized peak function based on Haarhoff van der Linde (HVL) offers the separation community a versatile function that can solve the frequent peak overlap problems and help them save time, solvents, and costs. Human judgment of the residuals is necessary when assessing GOF and ensuring the fit received is adequate. The F-statistic is a robust and reliable metric when deciding what function should be used to model the data and allows the number of parameters in the function to be considered for statistical significance. Applying iterative curve fitting to your workflow will save time, money, and solvent while producing high-quality and statistically confident analytical results for the separation scientist.

Data Availability Statement

Raw data for this manuscript is publicly available in the Harvard Dataverse Repository under a CC0 1.0 DEED license. In the data file, the first column represents time in minutes, and the second column represents signal in mAU. Please see DOI: 10.7910/DVN/UZX46O

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About the Authors

M. Farooq Wahab is a senior Research Engineering Scientist in the Department of Chemistry and Biochemistry at the University of Texas at Arlington. Direct correspondence to: mfwahab@uta.edu

Troy T. Handlovic is a third-year Ph.D. candidate in the Department of Chemistry and Biochemistry at the University of Texas at Arlington.

Daniel W. Armstrong is the Robert A. Welch Distinguished Professor in the Department of Chemistry and Biochemistry at the University of Texas at Arlington.