¹D-Guided Differential Rescaling of a Contour Plot in Comprehensive 2D Gas Chromatography

Abstract

A ¹D-guided differential rescaling algorithm for a contour plot is developed based on our recently proposed comprehensive two-dimensional gas chromatography (GC × GC) system with a first-dimensional (¹D) detector added. Chromatograms obtained from ¹D and second-dimensional (²D) detectors are both incorporated during the data processing. As compared to the conventional contour plot methods using only ²D data, our algorithm can significantly improve precision and consistency of GC × GC results in terms of retention times, peak widths, and peak areas or volumes, regardless of modulation time selection, modulation phase shift fluctuations, and modulation duty cycle. The peak identification, quantification, and capacity can therefore be enhanced. Furthermore, the ¹D-guided differential rescaling method is shown to better handle the coelution and missing peak issues often encountered in the conventional methods. Finally, the new method exhibits high versatility in ¹D and ²D detector selection, which greatly broadens GC × GC utility. Our method can easily be adapted to other two-dimensional chromatography systems that have direct access to ¹D chromatograms.

Introduction

Comprehensive two-dimensional gas chromatography (GC × GC, Figure 1A) uses two columns of different stationary phases so that vapor analytes are subjected to two independent separations to achieve a peak capacity higher than the corresponding one-dimensional GC.¹⁻⁶ A modulator is placed between the first-dimensional (¹D) and second-dimensional (²D) columns and periodically sends a portion of the ¹D effluent to the ²D column. Usually, a pneumatic modulator or thermal modulator is used to achieve so-called snapshot modulation or accumulative modulation.^7,8

Figure 1.

(A) Conventional comprehensive two-dimensional GC (GC × GC) configuration, where only a ²D detector (pink) is used at the end of the ²D column. (B) New GC × GC configuration, where an additional ¹D detector (blue) is used at the end of the ¹D column.

In a conventional GC × GC (Figure 1A), only a ²D detector is used at the end of the ²D column to detect the effluents coming out of the system. A two-dimensional (2D) chromatogram (i.e., contour plot) is constructed by using the ²D chromatograms sequentially measured by the ²D detector. ¹D information (such as retention time ¹t_r, peak profiles, and intensities) can only be indirectly extracted from the ²D data (along with the timing information provided by the modulator). There are a few major drawbacks in the conventional GC × GC. First, there is a dilemma in modulation time, P_M.^6,9,10 On one hand, accurate reconstruction of ¹D peaks requires a shorter P_M. On the other hand, ²D separation prefers to have a longer P_M to improve ²D separation and avoid a potential wrap-around issue. A long P_M (i.e., a low ¹D sampling rate) often results in distorted peak profiles, additional peak broadening, inaccurate peak retention times, and missing peaks in ¹D, which in turn reduces the overall peak capacity for GC × GC and adversely affects peak identification and quantification.^8,11 The modulation time dilemma exacerbates when asymmetric peaks (tailing or fronting) and/or narrower peaks are present in ¹D, and when the analyte quantity is so low that may not be detectable by the ²D detector.^6,10,12,13 In addition, the normal randomness of modulation timing produces varied displays of ²D peaks (i.e., varied intensities and ¹D timing of the ²D slices), and hence alters the reconstructed contour profiles (position, shape, and amplitude) from run to run. This further affects the precision, reproducibility, and reliability in 2D contour plots, especially the information projected along ¹D.

Marriott and his coworkers have been pioneering in improving the precision of ¹D information extraction based on the conventional GC × GC setup. Experimental efforts included mapping out a series of GC × GC analyses by varying the start time of the modulator (i.e., only varying the phase shift, whereas other conditions remain the same).⁶ Theoretically, this approach is likely to obtain the true peak maximum (i.e., ¹t_r) and peak area in ¹D. However, practical implementation is very challenging due to the substantially prolonged turn-around time, inaccurate control of phase shift, and fluctuations in other chromatographic conditions such as ambient temperature, and flow rate, etc. Recently, the same team introduced a novel data processing approach based on conventional GC × GC with mass spectrometry (MS) to improve compound identification.¹⁴ The workflow involves curve-fitting and considering modulation phase shift. Improvement in resolution and reproducibility was successfully demonstrated. However, this method relies heavily on a precise control of phase shift, which is infeasible for many exiting modulators. Prior knowledge of the number of peaks is also needed from MS data, which limits its application for the detectors without identification capabilities. Additionally, use of various fitting techniques that depend on peak profiles complicates automation in data analysis and could impair its performance in the presence of narrow peaks that disallow for sufficient modulation events.

Instead of relying solely on ²D data, one intuitive way, and probably the only precise way, to obtain the true ¹D information is to add a ¹D detector at the ¹D column outlet to probe effluents before they enter a modulator (Figure 1B). This configuration has several advantages. First, the ¹D information can be obtained directly from the ¹D detector with a much higher ¹D sampling rate (equal to data acquisition frequency of the ¹D detector), instead of being indirectly reconstructed from the ²D (sampling rate is equal to modulation frequency 1/P_M). The ¹t_r, peak profile, and intensity along ¹D can be precisely and reliably extracted regardless of modulation time selections and phase shift fluctuations. Second, peak capacity (n_c = ¹n_c × ²n_c) of GC × GC can be improved. In conventional GC × GC, ¹n_c is extracted from the ²D chromatogram and can be deteriorated due to insufficient modulation frequency (or ¹D sampling rate). Finally, it improves precisions for trace compounds analysis when analytes can be picked by the ¹D detector but fail at the ²D detector after several times of modulations. All these result in more reliable and reproducible resolution, identification, and quantifications of analytes in GC × GC.

For the proposed configuration, the requirements for such a ¹D detector are as follows: (1) non- or minimally destructive to the analytes and (2) introducing negligible extra band broadening (i.e., zero dead volume). Unqualified detector examples include MS, flame ionization detector (FID), and conventional thermionic ionization detector (TID). In contrast, the microfluidic photoionization detector (μPID) previously developed by our group^15,16 serves as a desired candidate for this role.

In our hardware work,^17,18 we have demonstrated the feasibility of adding a μPID at the end of the ¹D column in GC × GC systems. Meanwhile, we have also introduced a curve-fitting based method in an attempt to incorporate the information obtained by both ¹D and ²D detectors.¹⁷ However, some guess work and curve-fitting for various case enumerations are involved in the previous algorithm, which compromises its reliability when there are serious coelutions and thus hinders its applicability. Here, we propose a completely new method, the ¹D-guided differential rescaling (abbreviated as ¹D-DR) method, which eliminates the aforementioned guess work and curve-fitting process. The algorithm developed here makes full use of the ¹D and ²D information and is broadly applicable to various 2D systems (such as GC × GC and LC × LC). In the following sections, the theory and algorithm will be first discussed, followed by systematic evaluation of the performance related to modulation times, modulation phase fluctuations, modulation duty cycles, and ¹D/²D detector responsivities. Data generated both artificially and experimentally will be utilized for verification and discussion.

Theory and Algorithm Description

We start by briefly describing the conventional procedures (Figure 2A) to obtain a 2D chromatogram (contour plot) using a conventional GC × GC configuration (Figure 1A).

Figure 2.

(A) Conventional method to obtain a 2D chromatogram (contour plot). (B) ¹D-DR method to obtain a 2D chromatogram (contour plot). Red solid lines: major grids. Black dashed lines: minor grids.

Step 1 (Conventional)

Stack modulated ²D chromatogram slices (i.e., those obtained from the ²D detector) sequentially side-by-side along the ¹D time axis. The modulated ²D chromatogram slices are depicted on the major grids in red in Figure 2A. The distance between the neighboring major grids is equal to the modulation time (P_M).

Step 2 (Conventional)

More ²D chromatogram slices are generated through data interpolation based on the neighboring modulated ²D chromatograms and are stacked along the ¹D axis. The interpolated ²D chromatograms are plotted on the minor grids in black dashed lines in Figure 2A. The distance between the neighboring minor grids depends on the data interpolation density, which is usually chosen arbitrarily. In this work, we use 0.01 s so as to compare the conventional method with the ¹D-DR method.

Five different interpolation methods were recently compared by Allen and Rutan,¹⁹ showing nearly equal performance. In the current work, the modified Akima piecewise cubic Hermite interpolation²⁰ (Makima for short) as implemented in the “interp2” command in MATLAB was used to avoid the overshooting problem found in spline fitting.

Step 3 (Conventional)

Merge all ²D chromatograms (both modulated ones on the major grids and interpolated ones on the minor grids) and export a contour plot for data visualization.

As can be seen from above, the direct input of conventional method is a limited number of discrete snapshots (often 1–4 per peak) with a relatively large interval determined by P_M (Figure 2A, step 1). To obtain a visually continuous contour plot, significantly more data are generated to enrich the gaps between the modulations (i.e., between the adjacent major grids). Conventionally, these newly generated data can only be predicted through mathematical interpolation, which is mostly monotonical, between neighboring ²D chromatograms on major grids (Figure 2A, step 2). However, the evolution of the ¹D effluent from one modulation to the next is often nonmonotonical or is at varied slopes of change. The deviations between the true ¹D effluent evolution and that predicted via interpolation fundamentally account for the reduced precisions of ¹D information in the conventional method.

In contrast to the conventional method described above, adding a ¹D detector in GC × GC (Figure 1B) allows us to obtain nearly continuous ¹D chromatogram [¹S(t)]. Therefore, on top of the interpolation described above, the access to ¹S(t) contributes an additional confinement to all ²D chromatograms [²S(t)]. That is, ²S(t) should not be interpolated “arbitrarily” in the conventional method, rather it needs to be “reshaped” by ¹S(t). Figure 2B describes the workflow of our ¹D-DR method for the GC × GC configuration in Figure 1B and how to use ¹S(t) to confine or reshape ²S(t). Similar to the conventional procedures, there are three major steps involved. Step 1 is to stack modulated ²D chromatograms along the ¹D time axis as the major grids. Steps 2 and 3 are to interpolate data and export a contour plot. The major difference between our method and the conventional one is in step 2, where areas of all stacked ²D chromatograms (on both major grids and minor grids) are rescaled based on the area under the ¹D chromatogram during the corresponding loading session (i.e., ¹D-guided rescaling).

The details of our ¹D-DR method are described as follows (Figure 2B).

• Step 1 (¹D-DR) is the same as step 1 in the conventional procedures.

• Step 2 (¹D-DR). Data interpolation and rescaling.

• Step 2A is to rescale the modulated ²D chromatograms based on the ¹D chromatogram.

Due to mass conservation, each modulated ²D chromatogram (red major grid in Figure 2B) is produced by part of ¹S (t) loaded into the ²D column in a modulation period. Therefore, for a ²D chromatogram, ²S_n(t), on a red major grid at t_n, where t_n is the start time of the nth modulation event, the total quantity of the analytes loaded into the ²D column is proportional to the area under ¹S(t) measured by the ¹D detector, within this modulation period, i.e.

1

where t_L is the ¹D-to-²D loading time. 0 < t_L < P_M for modulators with a duty cycle of <1 (e.g., most pneumatic modulators) and t_L = P_M for ones with a 100% duty cycle (e.g., most thermal modulators). Similarly, assuming no wrap-around issue in ²D separation, the total quantity of analytes detected by the ²D detector should be proportional to the total area under ²S_n(t), i.e.

2

where ²S_n(t) is the ²D chromatogram measured by the ²D detector during the nth modulation period. A rescaling factor, R_n, for this modulation period can be obtained as

3

Due to mass conservation, ²S_n(t) for this modulation can be rescaled to

4

where ²S^′_n is the rescaled ²D chromatogram.

Step 2B is to interpolate more ²D chromatograms along the ¹D time axis using the rescaled ²D slices (²S^′_n) obtained from step 2A. As illustrated in Figure 2B, these rescaled ²D chromatograms are depicted on the major grids (red solid lines) with a time interval of P_M. The interpolated ones are depicted on the minor grids (black dashed lines), with an interval equal to data acquisition time of the ¹D detector (¹τ = 1/¹f seconds for ¹f Hz ¹D data acquisition frequency). Note that ¹τ should be much smaller than the P_M, since ¹τ is generally well below 1 s (¹τ = 0.01 s in this work) whereas P_M is usually on the order of several seconds. For the sake of convenience, we use the Makima interpolation in this work as described previously, though any interpolation method should work. Note that this step is similar to step 2 in the conventional method, except that the modulated ²D chromatograms (on the major grids) that have been rescaled using ¹D data in step 2A (²S^′_n) are used for interpolation instead of the original modulated ones (²S_n).

As a result, a series of ²D chromatograms (on both major grids and minor grids), ²E_k(t), are obtained and each grid is ¹τ apart, where k represents the kth grid that has a ¹D time stamp of k·¹τ.

Step 2C is to rescale all ²D chromatograms from step 2B [²E_k(t)] based on the ¹D chromatogram. Under the similar logic of step 2A, each ²D chromatogram on a major or minor grid is supposed to be produced by a corresponding slice of the ¹D chromatogram loaded into the ²D column. However, except those on the major grids, which are obtained from the ²D detector and have been rescaled in step 2A, all ²D chromatograms on the minor grids are produced by interpolation in step 2B. Therefore, we need to rescale all ²D chromatograms using the ¹D chromatogram as guidance. This way is equivalent to creating a series of virtual ²D injections with a pseudoloading time (t_L,pseudo) and a pseudomodulation time of ¹τ.

We can apply the same technique introduced in step 2A to all ²D chromatograms.

5

where t_r,k = k·¹τ, is the kth grid (major and minor grids are counted together).

6

Note that eq 6 is the same as eq 2, except that ²E_k(t) includes the chromatograms on both major and minor grids. Accordingly, ²E_k(t) for the kth grid can be rescaled to

7

where R_k is

8

For pneumatic modulators (duty cycle ≪ 1), where generally t_L ≪ P_M < ¹W_1/2 (peak width at half-maximum in ¹D), t_L,pseudo can be set to be t_L, which is the case for all examples presented in section “Results and Discussion—Influence of Modulation Time and Phase Shift, Coelution, and Versatility in Detector Responsivity”. However, for thermal modulators, the duty cycle is close or equal to 1 (i.e., t_L = P_M). We can still set t_L,pseudo to be t_L, which can still yield excellent consistency and accurate extraction of ¹t_r. However, the resolution (i.e., the reconstructed ¹D peak width using our method) might not be satisfactory, especially when t_L = P_M is comparable to actual ¹D peak widths or to the distance between the neighboring coeluted peaks. In this case, we suggest setting t_L,pseudo to be a much smaller value to improve the resolution. Detailed discussions can be found in section “Results and Discussion—Duty Cycle”. Note that the major grids that have been rescaled in step 2A may also be rescaled once more in step 2C, since t_L in eq 1 of step 2A may be different from t_L,pseudo in eq 5 of step 2C.

It is worth emphasizing that for each round of rescaling (step 2A and 2C), only the total area under the ²D chromatograms (major grids in step 2A, and both major and minor grids in step 2C) are changed. In other words, only the intensity ordinates of ²D chromatogram data points are rescaled whereas the abscissas (²D time axis) remain the same. The chromatogram profiles (e.g., the peak number, peak width, symmetry, ²t_r, and relative peak height among different peaks in each ²D chromatogram slice) are preserved. It is highly recommended to correct the potential issues of major grids (e.g., peak height reduced due to mass loss in the modulator) first to accurately anchor the subsequent interpolation in step 2B. Therefore, we do not suggest deferring the rescaling in step 2A to step 2C; the two rescaling procedures in step 2A and step 2C should be carried out separately. To further help understand step 2C in our ¹D-DR method and why we used the name “¹D-DR”, we add an illustration in Figure S1.

Step 3 (¹D-DR) is the same as step 2 in the conventional procedure, which is to merge all as-obtained ²D chromatograms and export a contour plot for data visualization.

Experiments

Artificial Peak Generation

An in-house MATLAB program is used to generate the sampled Gaussian and exponentially modified Gaussian (EMG) peaks in ¹D with known elution times, peak standard deviations (i.e., peak width), peak areas, and data acquisition frequency (100 Hz or ¹τ = 0.01 s). The ²D peaks are also generated using Gaussian functions with different modulation times (P_M), phase shifts, and different ¹D and ²D detector responsivity ratios. ¹D-to-²D loading periodically starts at φ + P_M × n, (n = 0, 1, 2, 3, ...). φ (<P_M) is the phase shift within a modulation period P_M and describes the relative positions of the modulation events with respect to the position of a ¹D chromatographic peak (or band). The elution time along ¹D and ²D is arbitrarily chosen and does not affect the conclusion. The features of the generated peaks are discussed in detail in section “Results and Discussion”.

Experimental Data

Experimental chromatograms are generated on a portable comprehensive 2D μGC system constructed in-house, which consists of a DB-1 ms commercial column from Agilent J&W (nonpolar, length 10 m, i.d. 250 μm, film thickness 0.25 μm, P/N: 122-0162), a WAX coated ²D microfabricated column (polar, length 0.5 m, cross section: 250 μm × 250 μm), a microfabricated flow-restricted pneumatic modulator, a helium cartridge (for carrier gas), and two flow-through μPIDs at the ¹D and ²D outlets, respectively. The current 2D μGC device is very similar to the one reported in ref (18), except that the microfabricated ¹D column in ref (18) is now replaced by a commercial column. A breath sample from a lab member is collected as the model system. During the chromatographic separation, the helium flow rate is 2 mL/min in the ¹D column, column temperature is kept at 25 °C for 2 min, then first ramped to 80 °C at a rate of 10 °C/min, and next ramped to 120 °C at a rate of 40 °C/min. The operation on breath sampling and ¹D separation is the same as reported in our previous work.^21,22 The settings on pneumatic modulation and ²D separation are the same as in the μGC × μGC system described in ref (18).

Results and Discussion

Influence of Modulation Time and Phase Shift

We first evaluate the influence of the modulation time and phase shift on symmetric peaks. In Figures 3 and S2, different ²D chromatograms (red curves) with varied P_M (1, 2, and 3 s) are artificially generated for the same ¹D symmetric Gaussian peak (blue curves). For each modulation time, four ²D chromatograms with different phase shifts are presented, two with in-phase modulation (φ = 0.4 and 0.9 s for P_M = 1 s; 1.9 and 0.9 s for P_M = 2 s; 0.4 and 1.9 s for P_M = 3 s), the other two with out-of-phase modulation (φ = 0.2 and 0.6 s for P_M = 1 s; 0.3 and 1.3 s for P_M = 2 s; 1 and 2.8 s for P_M = 3 s). “In-phase” and “out-of-phase” modulations refer to the modulations whose starting times are symmetrically and asymmetrically located with respect to the ¹D peak apex.

Figure 3.

Artificially generated ¹D and ²D chromatograms with reconstructed contour plots using the conventional and the ¹D-DR method, illustrating the influence of the modulation time (P_M) and phase shift (φ). The blue and red curves represent the ¹D and ²D chromatograms, respectively. The blue bars mark the areas under the ¹D peaks during ¹D-to-²D loading sessions. The ¹D and ²D peak profiles are the same for all cases, with ¹t_r = 35 s, ²t_r = 0.3 s, ¹W_1/2 = 2.6 s, and ²W_1/2 = 0.035 s. Modulation time and phase shift are varied. Loading times (t_L) are kept at 0.15 s. t_L,pseudo is set to be the same as t_L (i.e., 0.15 s). The x-axes for all figures represent the ¹D retention time ranging from 30 to 40 s. More results can be found in Figure S2.

As can be seen from the contour plots using the conventional method (left panels in Figures 3 and S2), the in-phase modulation produces symmetric peaks while the out-of-phase modulation produces asymmetric ones (tailing in Figure 3 and fronting in Figure S2). The increased modulation time (i.e., lower modulation frequency or decreased modulation ratio, defined as the ratio between the ¹D peak width and P_M(23)) significantly impairs the reliability of the as-obtained contour plot with a distorted peak profile, broadened peak widths (hence increased peak volumes), and deviated ¹t_r (i.e., peak apex) from its original position. Such a ¹D peak broadening effect can be explained by the sampling theory.⁸ Therefore, using the conventional method with only the ²D data inevitably reduces the overall GC × GC peak capacity and peak quantification accuracy. The detailed features (symmetry level, ¹D retention time ¹t_r, peak volume, and peak width) of each reconstructed pattern in the 2D contour plots are summarized in the “Conv.” columns in Table 1.

Table 1. Features (Symmetry Level, ¹D Retention Time, Peak Volume, and Peak Width) of Reconstructed Contour Plots and the ¹D Peak Obtained from the ¹D Detector Corresponding to the Artificially Generated Peaks with Different Modulation Times P_M and Phase Shifts φ^a.

PM (s)	φ (s)	symmetry level			1tr (s)			peak volume		1W1/2 (s)
		Conv.	1D-DR	1D detector signal	Conv.	1D-DR	1D detector signal	Conv.	1D-DR	Conv.	1D-DR	1D detector signal
1	0.6	lightly asymmetrical (L)	highly symmetrical	highly symmetrical	34.93	35	35	0.758	0.7	2.63	2.6	2.6
	0.4	symmetrical			35.01			0.758		2.62
	0.2	lightly asymmetrical (R)			35.07			0.758		2.64
	0.9	symmetrical			34.99			0.758		2.67
2	0.3	highly asymmetrical (L)			34.35			0.752		3.23
	1.9	symmetrical			35.07			0.750		3.72
	1.3	highly asymmetrical (R)			35.40			0.755		2.67
	0.9	symmetrical			34.97			0.757		2.47
3	1	highly asymmetrical (L)			34.05			0.715		3.47
	0.4	symmetrical			35.19			0.649		5.46
	2.8	highly asymmetrical (R)			35.89			0.726		3.40
	1.9	symmetrical			34.97			0.856		3.08

^a“L” and “R” indicate that the peak apex leans towards the left and right side, respectively. ¹t_r: ¹D retention time at the peak apex. Peak volume: volume under a peak in the contour plot. ¹W_1/2: peak width at half-maximum of the ¹D projection of the peak in the contour plot.

In contrast, by rescaling the sequential ²D data using the same ¹D chromatogram, the ¹D-DR method generates consistent contour plots with identical ¹t_r, peak volume, and peak width, as shown in the right panels of Figures 3 and S2 and summarized in the “¹D-DR” columns in Table 1. It should be noted that a ¹D detector was previously adopted in 2D assisted liquid chromatography (2DALC), along with a ²D detector.²⁴ In that arrangement, the ²D data was only used to help better resolve and hence quantify the ¹D peaks. In contrast, our ¹D-DR method is to use the ¹D data to fully rescale the ²D chromatogram, thus correcting the entire 2D chromatogram (i.e., contour plot).

More artificial experiments with asymmetric ¹D peaks are presented in Figure S3 and Table S1, simulating the tailing scenarios. The ¹D asymmetric peaks are generated using EMG and ²D peaks are still symmetric Gaussian. Again, as compared to the conventional method, the ¹D-DR method produces more robust and consistent results.

It is worth mentioning that, in this work, when describing the analysis results as “reproducible” or “consistent”, we refer in particular to the scenarios when only modulation settings vary or fluctuate, whereas all other chromatographic conditions (injections, hardware settings, and ambient environment) remain the same. Under this assumption, the effluent from the ¹D column before entering the modulator remains the same from run to run.

Coelution

Artificial Data

Figure 4 presents the artificially generated ¹D and ²D chromatograms with serious coelution issues, as well as contour plots obtained from the conventional and the ¹D-DR method. Peak information in both ¹D and ²D is detailed in the caption of Figure 4. It is shown that the reconstructed contour plot by the conventional method using the ²D chromatograms alone identifies only a single peak, failing to deconvolute the coeluted peaks in the ¹D chromatogram. This is because the modulation (blue bars) misses to sample the peak apex or valley in the coeluted peaks, as demonstrated in Figure 4A. To further separate coeluted peaks in the contour plots, complicated deconvolution algorithms, such as the parallel factor analysis (PARAFAC)^25,26 or the Multivariate Curve Resolution-Alternating Least Squares (MCR-ALS) method,²⁷⁻²⁹ are adopted in previous studies. But they may still not be able to completely resolve the coelution issue. In contrast, the contour plot in Figure 4C using the ¹D-DR method clearly shows two separated peaks, demonstrating its superior ability in handling ¹D coelution, which helps recover the missed peaks and hence improves the overall GC × GC peak capacity as well as peak quantification accuracy.

Figure 4.

(A) Artificially generated ¹D and ²D chromatograms with serious coelution issues. The blue and red curves represent the ¹D and ²D chromatogram, respectively. The blue bars mark the areas under the ¹D peaks during ¹D-to-²D loading sessions. The pink and green semiopaque areas represent two coeluted peaks eluted from ¹D, respectively. For the pink peak, ¹t_r = 3.6 s, ²t_r = 0.4, ¹W_1/2 = 1.0 s, and ²W_1/2 = 0.12 s. For the green peak, ¹t_r = 4.8 s, ²t_r = 0.46 s, ¹W_1/2 = 1.5 s, and ²W_1/2 = 0.15 s. t_L = 0.2 s and P_M = 1.0 s. The two analytes are partially coeluted in ¹D and are almost completely coeluted in ²D. (B) Reconstructed contour plot with the conventional method. (C) Reconstructed contour plot using the ¹D-DR method.

Experimental Data

In this work, we use an exhaled breath sample from a lab member as the model system. The portions of the ¹D and ²D chromatograms and corresponding contour plots of the sections with noticeable performance enhancement over the conventional method are plotted in Figure 5. As compared to the conventional contour plots (Figure 5E–H), the respective plots using the new method (Figure 5I–L) produce much better resolution and sharper peaks. For example, the modulations in Figure 5A,C,D inevitably miss the peak apexes or valleys of the partially coeluted peaks, therefore the contour plots using only ²D data in the conventional method waste the resolutions achieved along ¹D column (hence reduced resolution). Regarding the examples in Figure 5B, the conventional contour plot shows one missing peak at ¹t_r = 346 s, resulting from mass loss during the modulation. For example, the peak height ratio between the middle peak (¹t_r = 346 s) and the neighboring peaks are ∼2:7 (the peak on the left, ¹t_r = 338 s) and ∼2:5 (the peak on the right, ¹t_r = 353 s), whereas the corresponding ratios of the ²D slices after modulation decreased down to ∼1:30 and 1:20. This makes the smaller peaks between 335 and 352 s buried in the noise, and therefore, disappear in the conventional contour plot (Figure 5F). The ¹D-DR method is able to rescale the ²D slices total area back based on the corresponding slices in ¹D data, thus producing one additional distinct peak (Figure 5J).

Figure 5.

Portions of exemplary chromatograms of an exhaled breath sample. (A–D) ¹D (blue lines) and ²D (red lines) chromatograms. (E–H) 2D contour plots using the conventional method [only ²D data in (A–D)]. (I–L) 2D contour plots using the ¹D-DR method [both ¹D and ²D data in (A–D)].

Additional work where a mixture of 40 known chemical compounds were used as the model system can be found in ref (18), showing significant improvement in resolution and quantification capabilities. All 40 peaks were separated using the ¹D-DR method, whereas only 32 peaks were counted in the conventional contour plot. Evaluation of individual analyte analysis was also performed with benzene and alkanes (C_7–9). As compared to the conventional method, values including ¹t_r, ¹W_1/2, and peak capacities extracted from contour plot using ¹D-DR method are in line with those obtained in the ¹D chromatogram. See detailed discussions regarding the experimental conditions in ref (18).

Duty Cycle

So far, all artificial and experimental examples in the above discussion utilize a duty cycle much smaller than 100% (mainly for pneumatic modulators). In this section, we will evaluate the performance of the ¹D-DR method for a duty cycle of 100% (mainly for thermal modulators, t_L = P_M). In Figure S4 and Table S2, we generate artificial examples with a 100% duty cycle using two different modulation times (P_M = 2 and 3 s), each having four different phase shifts (including in-phase, out-of-phase, symmetric, tailing, and fronting), φ. ¹D peak profiles, ²W_1/2, and ²t_r, are kept the same as in Figures 3 and S2, and Table 1. As shown in Figure S4, the contour plots using the conventional method deliver varied peak symmetry levels, ¹t_r, and ¹W_1/2 (hence varied peak capacity). Regarding the ¹D-DR method with t_L,pseudo set as t_L (i.e., P_M), identical contour plots, ¹t_r, peak volumes, and ¹W_1/2 are obtained regardless of the phase shift (Table S2). The ¹t_r values obtained from the ¹D-DR contour plot are identical to the values extracted directly from the ¹D detector signal.

¹W_1/2 from a contour plot using ¹D-DR is generally smaller than that from the conventional method (hence improving peak capacity). However, we do notice that ¹W_1/2 from the ¹D-DR contour plot, though consistent among various phase shift values, are higher than the value directly extracted from the ¹D detector signal. This is because as compared to a pneumatic modulator (t_L ≪ ¹W_1/2), t_L is much wider (the duty cycle is close or equal to 100%) in a thermal modulator and the increased t_L leads to the peak broadening in step 2C in the ¹D-DR method. If a better resolution is desired (that is, ¹W_1/2 decreases to the value that is the same as obtained directly from the ¹D detector signal), one can set the value of t_L,pseudo in eq 5 to be much smaller than t_L and ¹W_1/2. For example, in Figure S5 when we decrease t_L,pseudo in step 2C from 2 or 3 s to 0.15 s (same values as in Figures 3 and S2 and Table 1), identical contour plots are obtained regardless of P_M and φ. The as-obtained contour plots, together with ¹t_r, ¹W_1/2, symmetric level, are the same as in Figures 3 and S2 and Table 1 and as the corresponding values extracted from the ¹D signal, although a 100% duty cycle is used.

Figures S6 and S7 systematically investigate the influence of the phase shift on a challenging case with a duty cycle = 100%, where two sharp peaks are seriously coeluted in both dimensions. A large modulation time P_M (=2 s) is intentionally selected, which is bigger than ¹W_1/2 of both coeluted peaks (1 and 1.2 s) and the distance between the two peak apexes (1.8 s). All possible phase shifts distanced by 0.1 s (φ = 0–1.9 s) are enumerated in Figure S6(A1–T1). Figure S6(A2–T2) show the corresponding contour plots using the conventional method. These 20 contour plots vary drastically and half of them fail to resolve the coeluted peaks (Figure S6(A2–F2,J2,K2)). Although Figure S6(M2–T2) completely resolve the coeluted peaks, the retention time, ¹t_r, for both peaks shifts substantially, which may affect the subsequent identification of these peaks.

The contour plot using the ¹D-DR method, with t_L,pseudo set as P_M = 2 s, continues to exhibit high consistency by yielding identical contour plots (Figure S7A). However, it does not show better performance in terms of resolution, that is, the two ¹D peaks are not well resolved. This is because t_L,pseudo, being set as 2 s (=P_M) is too wide, which inevitably covers the entire peak apex or even both peak apexes and the peak valley. Consequently, changes in a less distinct way when shifting the pseudoloading session from k to k + 1. Therefore, the resolution achieved in ¹D (by using a ¹D detector) is not fully utilized. In contrast, when we reduce t_L,pseudo to 0.1 s (≪peak width and distance between the peak apexes), a contour plot (Figure S7B) with well-resolved ¹D peaks, and correct and consistent ¹t_r and peak volumes are produced, regardless of the phase shift. The above discussion shows that the ¹D-DR method has performance superior to the conventional method, even for a duty cycle of 100%, in which case one can improve the resolution by setting t_L,pseudo to be much smaller than the ¹D peak width (e.g., ∼10 times smaller than ¹W_1/2), especially when the modulation time is large compared the peak widths and distance between neighboring peak apexes.

Versatility in Detector Responsivity

Another advantage of using the ¹D chromatogram data to rescale the contour plot is that it makes the reconstructed contour plots immune to detector responsivity changes. Up until now, we have assumed that ¹D and ²D detectors are identical. In real-world scenarios, the ¹D and ²D detectors may have different responsivities even to the same analyte (such as photoionization detector—PID for ¹D, and flame ionization detector—FID for ²D). To demonstrate the robustness of the ¹D-DR method with varying responsivity ratios between the ¹D and ²D detectors, three scenarios with different ¹D and ²D detector responsivity ratios were generated as shown in Figure 6 and Table S3. Other than the responsivity ratios, all the other peak features were kept the same for the three scenarios.

Figure 6.

Artificially generated ¹D and ²D chromatograms with different ¹D and ²D detector responsivity ratios and their corresponding reconstructed contour plots using the conventional method and the ¹D-DR method. The x-axes for all figures represent the ¹D retention time ranging from 30 to 50 s.

In our modeling, the ¹D detector has a responsivity 5 and 3 to species A and B, respectively. In the first row of Figure 6, we assume that the ²D detector is of the same type as the ¹D detector and therefore has the responsivity of 5 and 3 to species A and B, too. Note the absolute responsivity of the ²D detector changes proportionately (for example, responsivity to species A and B increases to 10 and 6), it does not change the 2D contour plot and the results due to the ratiometric nature between the ¹D and ²D detection of our method (see eqs 3 and 8). This row is used as a control, where both the conventional and our method can see two peaks in the 2D contour plot with accurate quantification (or peak volume).

The second row simulates the scenarios where the ¹D and ²D detectors have different responsivities to species A and B, but both ¹D and ²D have a higher responsivity to species A than to species B (i.e., ¹D detector has responsivity of 5 and 3 to species A and B, respectively, whereas ²D detector has a responsivity of 3 and 2 to species A and B, respectively). The third row simulates the scenario where the ²D detector has a responsivity of 2 to species A, lower than its responsivity (=3) to species B. In both scenarios, the conventional method yields peaks with different peak volumes and width, whereas our method yields consistent peak features. This is because all the changes in the contour plots due to the ²D detector ratio change are rescaled according to the original ¹D detector responsivity ratio. Our method enables high versatility in detector selection. For example, we can choose PID/PID as the ¹D and ²D detectors, or PID/FID as the ¹D and ²D detectors.

Furthermore, the above discussion also suggests that we can afford some mass loss in the ¹D detector. In our previous discussion, we had assumed that the ¹D detector is nondestructive. But now, based on our results in Figure 6, we can use a ¹D detector that may have some mass loss (i.e., minimally destructive detector rather than nondestructive detector). The mass loss scenario can be understood as ¹D and ²D showing different responsivities. For example, with a 10% mass loss, the absolute responsivity of ¹D and ²D detector becomes 5:3 vs 4.5:2.7 (for the first row in Figure 6) and 5:3 vs 2.7:1.8 (for the second row in Figure 6). The scaling factor introduced by the mass loss can be easily corrected after rescaling with the ¹D data, as long as the mass loss is not too high to hide all ²D signals below the baseline noise level.

Conclusions

In conclusion, by leveraging a ¹D detector, we have developed a novel ¹D-guided differential rescaling algorithm using the ¹D data that can significantly improve GC × GC performance in terms of chromatographic resolution (or peak capacity), retention time accuracy, and quantification precision. We showed that our new method exhibits high stability and consistency against the influence of modulation time selection and phase shift fluctuation, the presence of ¹D coelution, and ¹D/²D detector responsivity changes. The method can easily be extended to other multidimensional instruments such as LC × LC. One of the major drawbacks of the proposed method is added complexity in instruments due to the extra detector, which may be overcome by integrating the detector with the column during future device development.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.analchem.4c00202.

• ¹D-guided differential rescaling method and the influence of modulation time, phase shift, duty cycle, and pseudoloading time on 2D contour plots, figures compare the 2D contour plots generated with the conventional method and the ¹D-guided differential rescaling method, and tables summarize the parameters and results used in the aforementioned figures. (PDF)

The authors declare no competing financial interest.