# Power Law Approach as a Convenient Protocol for Improving Peak Shapes and Recovering Areas from Partially Resolved Peaks

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## Abstract

Separation techniques have developed rapidly where sub-second chromatography, ultrahigh resolution recycling chromatography, and two-dimensional liquid chromatography have become potent tools for analytical chemists. Despite the popularity of high-efficiency materials and new selectivity columns, peak overlap is still observed because as the number of analytes increases, Poisson statistics predicts a higher probability of peak overlap. This work shows the application of the properties of exponential functions and Gaussian functions for virtual resolution enhancement. A mathematical protocol is derived to recover areas from overlapping signals and overcomes the previously known limitations of power laws of losing area and height information. This method also reduces noise and makes the peaks more symmetrical while maintaining the retention time and selectivity. Furthermore, it does not require a prior knowledge of the total number of components as needed in curve fitting techniques. Complex examples are shown using chiral chromatography for enantiomers, and twin-column recycling HPLC of IgG aggregates and with tailing or fronting peaks. The strengths and weaknesses of the power law protocol for area recovery are discussed with simulated and real examples.

## Keywords

Signal processing Peak area recovery Power laws Exponential functions Peak purity Recycling HPLC## Introduction

One of the fundamental problems in instrumental analysis is to resolve overlapping signals of various analytes, interferences, and matrix components. Partially resolved peaks are routinely encountered in rotational–vibrational spectra, nuclear magnetic resonance, atomic spectroscopy and in a majority of instrumental separation methods [1, 2, 3]. Herein, we focus on the resolution related challenges in instrumental separation techniques. There are at least 62 types of stationary phase chemistries for liquid chromatography, 49 types of gas chromatography phases and 12 different types of supports according to the United States Pharmacopeia [4]. The reason behind developing various phases is that a single mobile phase and stationary phase chemistry combination cannot resolve all compounds of a given mixture. Similarly, the general elution problem in liquid chromatography exists where not all components can be separated in a given time frame using isocratic mobile phases [5]. There is often a critical pair or several pairs which tend to compromise the analysis. This problem is most commonly seen in enantiomeric separations where even broad enantioselectivity chiral selectors cannot separate all chiral compound classes, and mass spectrometry is unable to distinguish them because they have the same molecular weights and fragmentation patterns. An analyte pair may also exist in disproportionate heights, as in column overloading studies [6]. In pharmaceutical analyses, ascertaining peak purity is a requirement, large and small impurity peaks are found in a single chromatogram containing several peaks. In such cases, integration of all major and minor components is necessary for quantification purposes. Problems and their solutions related to chromatographic peak integration are addressed in a detailed monograph by Dyson [7]. Similarly, hyperfast chromatography is becoming an accepted technique after the demonstration of sub-second and millisecond chromatography [8, 9, 10, 11]. Sub-second chromatography is currently limited by the efficiency of the stationary phases hence obtaining a resolution *R*_{s} of 1.5 can be challenging on very short columns due to frit and instrumental dispersion of UHPLC systems and their finite sampling rate [8, 12].

There are at least three possible ways to enhance experimental chromatographic resolution: using small particle size packings, using very long capillaries or columns or by performing alternate pumping recycling chromatography at optimum flow rate under standard columns [13, 14, 15, 16, 17]. However, the selectivity is mainly controlled by the stationary phase–mobile phase chemistry, and the probability of peak overlap will increase as the number of sample components becomes large, as predicted by Davis and Giddings, using Poisson statistics [18]. In this work, a straightforward mathematical protocol is developed that increases resolution to (a) recover peak areas in partially resolved chromatograms and (b) improve peak asymmetry. The method developed here relies on the effect of powers on distribution functions [19, 20, 21]. The calculations do not require any sophisticated software, and most of the work can be carried out in Microsoft Excel or OriginPro. Other resolution-enhancing or area recovery methods such as Fourier self-deconvolution [1], multivariate curve resolution [22], and iterative curve fitting [23] all require significant computational demands, and each approach has its own strengths and weaknesses. The theoretical principles of the normalized power law are given below. The primary advantages and disadvantages of this resolution-enhancing mathematical technique are assessed critically from a practical perspective.

## Theory

### Power Law as a Resolution Enhancement Approach

Equation (1b) is another Gaussian, whose standard deviation has been reduced by a factor of \(\sqrt p\), and the original height has changed to \(H_{{\text{o}}}^{p}\). This property of the peak-like function can be termed as a *power law*. Concurrently, this operation leads to a change in peak area as well which compromises quantitation. The areas of \({G_{\text{o}}}\)and \({G^p}\) correspond to \(\sqrt {2\pi } \sigma {H_{\text{o}}}\) and \(\sqrt {2\pi /p} \sigma H_{{\text{o}}}^{p}\) respectively. The dimensionless relative area change is \(H_{{\text{o}}}^{{p - 1}}/\sqrt{p}\).

### Peak Area Recovery of Partially Resolved Peaks with the Help of a Simple Protocol Using the Properties of Exponential Functions

A simple four-step protocol is developed in this paper to recover the areas of partially unresolved peaks with acceptable accuracy (Table 1) if the peaks are not significantly overlapped, e.g., *R*_{s} ≥ 0.9 as a safe limit with < 1% error (Table 2). In the following derivation, we will assume that the peaks are Gaussian. The quantitative effect of power law on tailing will be addressed in the “Results and Discussion” section.

Protocol for recovering areas from the modified power law from partially resolved peaks

Resolution enhancement and area recovery using the normalized power law | |
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1 | Measure the approximate resolution from a chromatographic data acquisition software, ensure the |

2 | (Optional) If there is excessive noise in the signal, smooth the data by any centered moving average or Savitzky–Golay procedure. The centered moving average will not change the retention time or peak area. Avoid over-smoothing as it may decrease resolution. If needed correct the baseline by subtraction if there is a significant slope or drift by subtracting a blank run |

3 | Normalize the height of the desired peak in the critical pair to unity by dividing by its peak maximum \(({H_o}).\) Apply the same division procedure to the remaining data in the chromatogram |

4 | Raise the signal with a positive power |

5 | Measure the peak area \({A_{\text{N}}}~\)after step 3, and recover the original area \({A_{\text{o}}}~\)by Eq. (5): \({A_{\text{o}}}={H_{\text{o}}}{A_{\text{N}}}\sqrt p\) |

Comparison of Errors in Area Recovery vs. Original Resolution of Gaussian Peaks of Equal Area

Resolution | Power law | % Error in area | |||
---|---|---|---|---|---|

Power ( | Peak 1 area | Peak 2 area | Peak 1 | Peak 2 | |

1.26 | 1.5 | 250.0024 | 249.9977 | 0.0010 | -0.0009 |

1.15 | 2.0 | 250.0224 | 250.0077 | 0.01 | 0.00 |

0.97 | 3.0 | 250.3084 | 250.2077 | 0.12 | 0.08 |

0.84 | 3.8 | 251.8619 | 251.4916 | 0.74 | 0.60 |

0.74 | 6.0 | 256.0671 | 255.2677 | 2.43 | 2.11 |

0.67 | 9.0 | 265.3699 | 264.1958 | 6.15 | 5.68 |

Consider a co-eluting Gaussian peak \({G_{\text{o}}}\), with a height \({H_{\text{o}}}\) units and area \({A_{\text{o}}}\). Assume the peak elutes at time \({t_R}\) with a standard deviation of \(\sigma\). If we normalize the peak height to unity, Eq. (1a) becomes:

*Step 1*:

Apply the desired power \(p\) (> 1) to the normalized peak \({G_{\text{N}}}\), the normalization process keeps the peak height constant while decreasing the peak standard deviation by a factor of \(\sqrt p\), as stated before.

*Step 2*:

The third step involves the determination of the peak area \({A_{\text{N}}}\) of \(G_{{\text{N}}}^{{\text{p}}}\) by integration of Eq. 3 with respect to time

*Step 3*:

*Step 4*:

Equation 5 will be used for recovering peak area information in cases where integration is challenging by standard procedures.

Figure 1 illustrates the concept as a simulation. A simulated critical pair consisting of two perfect Gaussians (*R*_{s} ~ 0.85) with noise is shown. In this case, the two peak areas are 20.0 and 50.0 units. In Fig. 1, assume the area of the peak 1 is of interest. Following the protocol in Table 1, we can recover the area of partially resolved peak 1 with excellent accuracy 20.00 vs. 19.95 units. Similarly, the area of the second peak can be extracted by repeating the same protocol on peak 2. The criterion for choosing *p* is highlighted in Table 1. The question of initial resolution vs. error in the area recovery will be addressed in the "Results and Discussion".

## Materials and Methods

### Software for Data Processing

### Chromatographic Conditions and Hardware

All HPLC grade solvents, reagents, and analytes were purchased from Sigma-Aldrich (St. Louis, MO, USA). Distilled deionized water (18.2 MΩ cm) was obtained from a Milli-Q purification system (EMD Millipore, Billerica, MA, USA). The 2.7 µm superficially porous particles were purchased from Agilent Technologies (Santa Clara, CA, USA). The silica particles were bonded with a modified macrocyclic glycopeptide (NicoShell). The material was packed in-house using a dispersed slurry technique in Isobar column design (Idex Corporation, WA, USA). The detailed packing approaches to achieve this high high-efficiency are described in our recent work [12, 25]. A Vanquish UHPLC instrument (Thermo Fisher Scientific, Waltham, MA) was used for collecting chromatograms. This instrument is custom configured to by-pass the column oven to minimize extra-column effects. The column was connected to the injector and the 2.5 µL UV–Vis detector. The data was sampled at 250 or 200 Hz. The Vanquish system is controlled using Chromeleon 7.2 SR4 software (Thermo Fisher Scientific 2009–2016). Note that there is only one commercially available chromatography data system (Chromeleon) which has incorporated simple power law with a maximum power of 3.

#### Recycling HPLC Experiments and LC Isotopic Separations

The experimental design of the custom-made recycling HPLC system has been given elsewhere [14]. The separation of IgG aggregate/monomer monoclonal antibodies fractionation was obtained using pure water buffered at pH 6.72 with 100 mM phosphate buffer at a flow rate of 0.25 mL/min. The ionic strength of the eluent was further increased by adding NaCl salt (200 mM). The frozen IgG sample, obtained internally from Waters Corporation, was dissolved in 100 µL eluent at a final concentration of 20 g/L. 5.0 µL was injected into a 1.7 µm BEH-200 15 × 0.46 cm. The analytes were detected at 280 nm at 20 Hz sampling frequency.

#### Flow Reversal Experiments

The details of the flow reversal set-up are given previously [26]. In the flow reversal test, uracil was dissolved in 0.5 mL of hot water, and the composition of the sample solvent was matched with the eluent (acetonitrile/water, 80/20, v/v). The concentration of uracil is fixed at 0.05 g/L. The injected sample volume is 2 µL using the flow-through-needle injection mode. The data sampling frequency was 20 Hz (ensuring > 20 points per peak) with detection at 243 nm. The flow rate for these experiments was set at 0.5 mL/min at ambient temperature (296 K).

## Results and Discussion

The protocol for resolution enhancement and area recovery of overlapping peaks is demonstrated using simulated and real experiments. At the very outset, we would like to remark that all mathematical procedures for resolution enhancement do not represent a physical separation inside the column. However, the information obtained from the mathematical processes is real; for example, we can accurately recover areas and peak height from overlapping signals as if the separation were a physical reality with correct retention times.

### Benchmark Testing of Simulated Chromatogram with Partially Resolved/Shouldering Peaks Using the Properties of the Power Law

As stated earlier in the “Theory” section, raising the whole chromatographic signal to a constant power changes the areas as well as the height of all the peaks. If the peak height \({H_{\text{o}}}<1,\) peak height decreases with an application of a positive power. Similarly, if peak height \({H_{\text{o}}}>1\), peak height increases as a power function. However, the virtue of power law approach is that it is the most straightforward approach to enhance the “apparent” resolution of a chromatogram [9, 19, 20, 21]. If a chromatogram contains peaks of disproportionate signal magnitude applying a power to the entire chromatogram yields misleading output as small signals are suppressed (often vanish) relative to the larger peaks. To recover the areas after the application of power law on chromatograms containing peaks of various heights, a four-step procedure is proposed as shown in Table 1. In Table 2, we compare two peaks of equal areas with different resolutions. If the resolution is too low the error in the recovered area using Eq. (5) is large. In the presence of noise and non-zero baseline, we suggest to apply the power law approach with resolution > 0.8 based on moments or resolution based on peak widths at the base. This systematic error in Table 2 arises from the fact that the peak maxima (*H*_{o}) are overestimated as the peaks begin to overlap. Hence the direction of error is positive with higher degrees of overlap.

The outcome of 4-step normalized power law approach (Table 1) is shown on a simulated chromatogram containing five components (Fig. 2). All peaks are all well resolved except peak 3 and 4, as a critical pair (*R*_{s} ~ 0.90, as a limiting case) in Fig. 2A. The area of peak 4 is intentionally chosen to be very small, and it is set to 1.50 signal unit-seconds, whereas peak 3 has an area of 20 signal units-seconds. Thus, the peak area ratio is approximately 13:1. The height of the peak is set to be 0.42. Peak 4 was also made to partially overlap with a peak of the area of 30 units (peak 5). If we integrate peak 4 with the standard approach as a perpendicular drop method, the area of peak 4 is 1.458 units. This leads to − 2.8% error in the area. A tangent skimming approach [7] of peak integration gives an area of 0.539 units, and the error is − 64%. Following guideline 4 in Table 1, a power was chosen to obtain a resolution > 1.5 in the critical pair. Applying the power law without the intermediate normalization step to a power \(~p=\)11, as shown in Fig. 3B, the small peak vanishes. In real chromatograms, peak heights are rarely equal due to differing concentrations and instrument responses, and this approach will give misleading information on the number of components. The disappearance of the small peak 4 in Fig. 2B is expected, because of its peak height is less than unity. When the peak 4 is normalized to unit height (Fig. 2C), and power of \(~p=\) 11 is applied, the peak becomes very well resolved from both peak 4 and peak 5 (Fig. 2D). After measuring the area of the resolved peak 4 as follows: \({A_{{\text{Np}}}}=~1.08925,\) the original area using the Eq. (5), \({A_{\text{o}}}=\left( {0.4223} \right) \,\left( {1.089246} \right)\sqrt {11} =1.52\) is recovered even after the application of a very large power. The error in the recovered peak area found by integration is 1.7% (a difference of + 0.02 units only). Such a large power of 11 may not be required in most cases, but herein we show an extreme example of large powers and very small area values. If we determine the area of each peak in the chromatogram operated upon by a simple power law, it will give rise to a calibration curve which looks like a power function e.g., a parabola with *p* = 11. Thus, the recovery of original area is essential after applying the power law so that the calibration curve is linear.

### Improving Asymmetry with Power Laws

Another advantage of the power law is that it improves the peak shape. It might be satisfactory for the users to employ the perpendicular drop method, but there is a hidden caveat to the perpendicular drop method. The reason is that the perpendicular drop method gives correct peak areas only when the peaks are perfectly Gaussian as well as the same size [7, 27]. If the peaks are not the same size or if they are asymmetric, in such cases, one peak area will be overestimated and the other will be underestimated [28].

*p*. On the other hand, the perfectly Gaussian fronting width will decrease by a factor of \(\sqrt p\). Thus upon application of power law, the asymmetry will decrease by \(\sqrt p\) [21]. This pattern is observed for Fig. 3A, where an exponentially modified Gaussian (EMG) was modeled using the equation. The original peak is shown in red in Fig. 3A.

*H*is the height adjusting factor, and \(\lambda\) is the shape adjusting factor (reciprocal of time constant) [30]. As successive powers are applied after normalization of peak height to unity, the asymmetry at 10% height “roughly” decreases proportionally as \({p^{ - 0.46}}\), which is very close to the theoretical predictions. Figure 3B shows a flow reversal experiment with uracil on a column which produces tailing. Once the flow is reversed very quickly (within seconds), the eluted uracil peak now fronts. The corresponding asymmetries and the areas are shown in the peak labels. The power law protocol (Table 1) recovers the area of the tailing uracil with < 1% error whereas the area error in the severely fronting uracil is less than 2%. Note the areas are extremely small and that there is a significant improvement in the overall peak shape.

### Power Law Applications for Area Recovery

Next, we show two examples of the power law on chromatographic data in the presence of noise and non-zero baselines. One data set is a UHPLC data collected at a very high sampling frequency of 250 Hz and the second set of data is based on recycling HPLC data operated in a twin-column recycling separation process.

Figure 4 shows an interesting application of power law on partially resolved racemic peaks of salbutamol. Both peaks tail very slightly with an asymmetry of ~ 1.21 and 1.24 at 10% height, as is typical of any slow mass transfer kinetics of adsorption and desorption. In the chiral analysis, the area ratio of enantiomers is also of prime importance. Before applying the power law, it was checked that the resolution is 0.88 (Peak Fit). A power of 3 was sufficient for obtaining baseline resolution. After following the protocol given in Table 1 for peak 1 and peak 2 the areas of overlapping peaks were extracted as 1.752 and 1.783 mAU min, respectively. These numbers correspond to an area ratio of 49.55% and 50.44%. As a control, Fig. 4B shows the chromatographically resolved data obtained using optimized chromatographic conditions. The percent areas from the resolved chromatogram is 49.59 ± 0.09% and 50.45 ± 0.09% (triplicate). The area extracted from normalized power law was statistically the same as the chromatographically resolved data. Comparing the data in Fig. 4A, B shows that the perpendicular drop method exhibits highest discrepancy in area ratios of 48.9 ± 0.2 and 51.1 ± 0.2%. It is well known that as tailing increases, the perpendicular drop method gives erroneous results [28]. Further details on the inaccuracies associated with the perpendicular drop method from a theoretical perspective have been conducted [7, 27]. This is understandable from the fact that in this method, the tail of the first is cut off and merged with the area of the second peak. Not surprisingly, the second peak has a slightly higher area. Iterative curve fitting also gives accurate results when the peaks are modeled as exponentially modified Gaussians (see inset in Fig. 4). The best match comes with the power law extracted area vs. chromatographically resolved areas.

In the second application of power law, we deal with a more complex data from a recycling HPLC experiment of IgG [14]. Briefly, two columns are used in the recycling process, hence the name twin-column recycling separation process. The twin-column recycling mode offers an ultrahigh resolution when the chromatography selectivity \((\alpha )\) of desired analytes is extremely small (e.g., 1.02). The only caveat is the time of the separation. It can take several hours for very challenging separations such as the separation of IgG aggregates.

Monoclonal antibodies may decrease over time by either fragmentation and/or self-aggregation. The small amounts of such aggregates are very hard to detect and isolate. In the recycling mode, the switching valves potentially alter the baseline due to temporary flow rate fluctuations during valve switching. Although baseline shift is not a problem in preparative chromatography, baseline oscillations and drifts pose difficulties in quantitation. To estimate the area of small peak 2 in Fig. 5A, there are two problems. We do not know the peak shape of peak 2, and secondly, the baseline is not defined. Hence, the perpendicular drop method will fail here. The effect of the choice of baselines can be significant especially for iterative curve fitting. Thus, the raw data cannot be used as it is for iterative peak fitting before baseline correction. OriginPro™ also offers user-defined baselines such as cubic, a polynomial of degree 5, exponential decay, exponential growth, parabolic, and hyperbolic baselines. Unfortunately, none of the baseline functions could define the baseline in the selected region of Fig. 5A. Figure 5B shows the comparative magnitudes of IgG peak and its high-molecular weight aggregates. Since there is significant noise in the baseline and the peaks, the data were smoothed with a centered moving average function of 101 points (Fig. 5C). In Fig. 5D, we employ the normalized power law with a power of 6 to assist us in estimating the area of the peak 2. The area recovered is ~ 0.33 absorbance-second units. These values can give a reasonable estimate of the area of analytes if pure standards are not available.

### The Question on the Limit of Power *p*

All resolution enhancement methods will have some limitations in terms of distorting peak shape, increasing noise, and failing to resolve peaks. Some caveats for the power are stated here. If the peak maximum has a significant noise, raising data to power will increase the noise if *p* > 1, which is unacceptable for integration purposes. It can be shown from the central limit theorem that a smoothing function will not change peak areas, although the peak heights and shapes may change [31]. Smoothing functions such as Savitzky–Golay, Hamming, RC, central moving average filters which are available in modern HPLC/UHPLC software, do not change the peak area [32]. The normalization step requires an accurate value of *H*_{o}. Therefore, pre-smoothing the data is recommended.

One can apply an infinite power to a chromatogram and make each peak look like a \(\delta\)-function (extremely narrow spikes), this is good for qualitative purposes only-especially if there no overlap and the peaks are not of disproportionate heights. The \(\delta\)-function can be considered as a limit of distributions as the standard deviation approaches zero [33]. For area extraction purposes, the power law is very useful when there is sufficient resolution, and minor resolution enhancement is needed, or peak heights are very different. Similarly, if the resolution is less than 0.9 as calculated by moments or the peak widths at the base, the power might have significant error (Table 2), instead the experimental resolution should be improved by extending the column length or altering the mobile phase composition or temperature.

## Conclusions

Overlapping signals frequently occur in analytical chemistry and hide vital information concerning the sample composition. Herein a simple mathematical method based on the properties of exponential functions was proposed to enhance virtual resolution along with area recovery and make the peaks amenable to integration. This approach is valuable in the sense that it does not require any dedicated or sophisticated software. The proposed methods accurately recover the areas when small peaks elute near large ones or when chromatographic signals overlap. The power law offers an additional advantage by improving peak shapes and reducing the noise. Additionally, this approach does not require knowledge of the number of components as is needed in iterative curve fitting or multivariate curve resolution techniques.

## Notes

### Compliance with Ethical Standards

### Conflict of interest

The authors declare no conflict of interest.

### Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

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