A Modeling Approach for Orthogonality of Comprehensive Two

Article pubs.acs.org/ac

A Modeling Approach for Orthogonality of Comprehensive TwoDimensional Separations Zhong-Da Zeng,† Helmut M. Hugel,‡ and Philip J. Marriott*,† †

Australian Centre for Research on Separation Science, School of Chemistry, Monash University, Wellington Rd, Clayton 3800, Australia ‡ School of Applied Sciences, RMIT University, GPO Box 2476, Melbourne 3001, Australia S Supporting Information *

ABSTRACT: A novel method is developed for orthogonality evaluation of comprehensive two-dimensional separations (C2DS). Utilization of efficiency measures such as peak capacity (nc) can be critically evaluated for C2DS analysis to describe an orthogonal separation of the analytes in a 2D plane. Unlike most previous methods focusing on “bin coverage” over 2D space, rather than taking into account the distribution based on accurate peak retention, in the proposed method, the separation orthogonality of C2DS is divided into two parts (i.e., Cpert and Cpeaks). These correspond to peak coverage percent, and 2D distribution correlation of compounds, respectively. Bin occupation and a simple-linear regression model, on the basis of normalized retention times in 2D separation space (1tR and 2tR), are further introduced to quantitatively define the two terms. Orthogonality ranges from 0 to 1 correspond to perfectly correlated and orthogonal separations, respectively, which are presented based on both Cpert and Cpeaks considerations. The advantage of this method is the use of separation properties of C2DS to characterize practical 2D peak distribution and does not rely on assumptions or any imposed limitations. Simulation of comprehensive two-dimensional gas chromatography (GC × GC) was achieved by using the Abraham solvation parameter model, and applied to generate examples for orthogonality assessment. In this work, 225 compounds comprising a range of chemical classes were simulated for separation on two column set pairs comprising low polarity/polar and moderately polar/polar combinations. Results illustrate that the proposed method applied to GC × GC provides a reasonable assessment of 2D separation performance and may be used to derive optimal experimental conditions when used with an experimental design strategy.

T

orthogonal separation of spherical and elliptical spots, using Poisson statistics.7−10 This is attained only for analysis employing two completely independent separation mechanisms.6 Thus, method development leading to a measure of orthogonality (O) has an important role to play, to determine relative 2D separation performance, and then to produce optimized conditions for the C2DS experiment. In this work, the two extreme cases are abbreviated as PCS (perfectly correlated separation) and OS (orthogonal separation) with O = 0 and 1, respectively. The advantages and disadvantages of conventional methods to study orthogonality have been thoroughly reported elsewhere.11,12 For instance, estimation of orthogonality can be attained with determination of correlation coefficients/peak spreading angle, or utilization efficiency of the 2D retention space, since an orthogonal separation should have no correlation from a mathematical point of view.13 But orthogonality of systems with target analytes not diagonally distributed in the C2DS space cannot be ideally calculated by

he multiplicative effects on peak capacity (nc) arising from individual participating column performance is the major and oft-quoted advantage of comprehensive two-dimensional chromatographic separations (C2DS). This has been proven in theory, and through applications to the studies of herbal medicines, drugs, wines, and petroleum, among many other multicomponent samples with a multitude of chemical components.1−3 With the use of comprehensive two-dimensional gas chromatography (GC × GC) as an example, connection of a polarity-complementary column set via a suitable interface, primary (1D) peaks are modulated to isometric 2D fractions, following the concept of the modulation ratio (MR).4 The maximum peak capacity, namely nc,max, ideally is the sum of all separation bins (1nc × 2nc), according to the 1D and 2D peak capacities of the 2D separations, 1nc and 2nc, respectively.5,6 However, practical peak capacity (nc,P) strongly depends on column selection in conjunction with analyte properties, programming temperature, column dimensions and properties, flow rate, and other experimental conditions that affect separation efficiency and therefore utilization of 2D retention space. In terms of the theory of overlap for C2DS, an average of 63% separation space (i.e., 0.63 × nc,max) is occupied for © XXXX American Chemical Society

Received: March 11, 2013 Accepted: June 3, 2013

A

dx.doi.org/10.1021/ac400736v | Anal. Chem. XXXX, XXX, XXX−XXX

Analytical Chemistry

Article

Figure 1. Illustration of C2DS patterns with 4 peaks (1−4) uniformly distributed into 4 rectangular bins in 2D space. (A) Simulated distribution of 4 peaks with very small 1D and 2D retention distance difference (low potential informational entropy). (B) Simulated distribution of 4 peaks but with large 1D and 2D distance difference (high potential informational entropy). B



■

Article

THEORETICAL BASIS The essence of C2DS is to reasonably arrange (and ideally maximize) component retention points (apexes) in a 2D plane in terms of the interactions between solutes and SPs of the 1D and 2D columns. This is achieved by choice of phases and consideration of experimental conditions. The coordinate ranges in the two separation dimensions are different, with one being total retention time in 1D, and the second is usually defined by the modulation period (PM) in 2D (in the absence of wrap-around). In order for magnitudes of 1tR and 2tR to be comparable, such that they achieve equivalence of effects on orthogonality evaluation, the following equation (eq 1) is first applied to normalize 1tR and 2tR. This has been widely applied elsewhere.11,13,17

using the geometric approach with factor analysis. Surface coverage is an intuitive concept to measure orthogonality of 2D separation. Rectangular bins of peaks have been applied to evaluate the utilization of nc,max.7,14 However, this type of method ignores how the 2D peaks are distributed in the separation space. This may overestimate orthogonality metrics for systems with high 2D correlation. In addition, indirect employment of peak information makes orthogonality evaluation insensitive toward separation differences between experimental runs, with some disturbance arising from small changes of conditions. Information-theory-based techniques have potential to evaluate separation efficiency because of the mathematical consistency between chromatographic separation and the fundamental concept of entropy.8,15,16 For instance, mutual information is statistically applied to measure the dependency of data sets and then can be extensively used to evaluate retention correlation in a 2D separation. Methods based on informational entropy and mutual information and conditional entropy were developed to assess informational similarity and orthogonality of C2DS.11,17 Comparison of previous methods for orthogonality metrics has been comprehensively investigated; comparison was achieved on the basis of correlation coefficients, surface coverage, mutual information, and box-counting dimensionality.12 In this study, a model approach is developed to correlate component retention in 1D and 2D separations. This employs the original distribution pattern of the peaks in C2DS space. The R-square statistic of linear modeling is effective to evaluate correlated information obtained from the 1 D and 2D separations. Then, a metric that combines both the contribution of the bin coverage percentage and the quantitative retention correlation is employed for the orthogonality study, by defining a novel evaluation index. This expresses the 2D distribution of peaks in the C2DS plane. It overcomes the disadvantages of previous methods by employing the raw separation data. For example, the situation where many minor differences in peak positions may lead to little or no change in the metric of evaluation has evident limitations to objectively assess separation performance and then determine optimal conditions. This is a consequence of the reluctance to use raw retention data. The proposed method can be applied to all C2DS distributions via component characteristics in the C2DS plane. Here, the Abraham solvation parameter model (ASPM) is used to simulate the separation of compounds on defined stationary phases (SPs) for C2DS analysis.18−20 Accordingly, an orthogonality investigation of the separation of 225 compounds was implemented on two different GC × GC separation systems for the ASPM experiment with two column sets viz. DB-1 and HP-INNOWax and DB-1701 and HP-INNOWax, respectively. This approach allows investigation of the influence of experimental conditions on C2DS. Comparison of results for the proposed method, and five previously reported methods, is achieved by using the same data sets. This approach can generally be applied to the analysis of all other C2DS data, such as comprehensive two-dimensional liquid chromatography (LC × LC). A list of some definitions used here are provided in Table S1 of the Supporting Information. Nomenclature and conventions in C2DS were originally reported in 2003 by Schoenmakers et al. and recently updated.21,22

x

t R,norm =

( xt R − xt R,min) ( xt R,max − xt R,min)

x = 1 or 2 (1)

Notations “min” and “max” represent minimum and maximum tR values in 1D (x = 1) or 2D (x = 2), and “norm” represents the normalized calibration result of each retention element in the equation. As introduced in previous reports, statistical retention independence in each chromatographic dimension is a basis to attain orthogonal separation. If successful, the distribution of the peaks should uniformly occupy the C2DS space of the 2D contour plot with 1tR and 2tR information, since this maximizes entropy of the analytes in 2D space. Of course, maximizing the mutual distance among all the peaks should accordingly achieve the maximum separation under the ideal situation. Thus, a theoretical definition of orthogonality should naturally include two aspects to fully accommodate 2D informational content, such as “distance distribution” (i.e., “coverage percent of bins”) and a notion of “correlation of peak distribution”, as given in eq 2. The former approximately estimates the 2D space occupation that does not consider distribution details of compounds, and the latter further refines the relationship, to quantitatively discover how peaks distribute in the separation space.

O = Cpert × Cpeaks

(2)

Cpert and Cpeaks correspond to the bin coverage percent and correlation distribution pattern of 2D peaks, respectively. They independently range from 0 = PCS to 1 = OS. A previously representative definition of orthogonality is given in eq 3, with an improved version in eq 4,7,14 which were developed in terms of rectangular bins corresponding to the division number, with dimensions 1nc and 2nc. O=

O=

(∑ bins −

nc,max )

(0.63 × nc,max )

(∑ bins − (0.63 ×

(3)

nc,max )

nc2

− nc)

(4)

The notations have been altered from those used in the original reported studies (e.g., nc is used rather than P), but their physical meaning is the same.21,22 As shown in Figure 1, these equations only take into account the first part of eq 2 and essentially produce the same evaluation value for the two separations given in Figure 1, panel A and B, even though they report completely different (but symmetrical) peak distribution patterns. The rectangles C



Article

represent respective bins in which solutes are located. Logically, orthogonality of the two separations should have some difference because of the divergence of statistical overlap probability. In this work, and following previous convention, the orthogonality index O is defined from 0 (PCS) to 1 (OS). In accordance with the definition given in eq 2, orthogonal separation is achieved if and only if Cpert and Cpeaks attain 1 simultaneously, but complete nonorthogonality is achieved should either of them attain 0. This makes the concept of orthogonality ranges reasonable, unlike that in eqs 3 and 4, which artificially subtract the diagonal occupation to force the nonorthogonal separation to zero and then must exclude cases with total bins (∑bins) smaller than nc, respectively (to ensure non-negativity). In this study, the concept of bins is applied to characterize the coverage percent. In eq 5, Cpert is defined according to the fundamental principle of peak distribution in C2DS space. This equation naturally introduces the percent of peak coverage, spanning the 2D separation space. Further, it ideally includes the requirements that the range varies from 0 to 1 for PCS to OS. All bin occupation has equal orthogonality contribution with no forcible subtraction of peaks distributed in the diagonal but take this into account in the Cpeaks term. This is the evident difference with eqs 3 and 4. Cpert =

Therefore, the second part of eq 2 (Cpeaks) can be written as the definition given in eq 10. Since Cpeaks ranges from 0 to 1 for PCS to OS separation, it should have an inverse relation with R2. If 1tR and 2tR have an ideal linear relationship, R2 will be equal to unity, corresponding to PCS. Likewise, R2 is equal to 0 for OS. Further, it should be pointed out that the change of Cpert and Cpeaks approximates to real experimental data. The increase of ∑bins leads to high scatter of peaks in the 2D space; this correspondingly generates a poor linear model with small R2 statistic and accordingly large Cpeaks. It is a critical factor for Cpeaks as a rational refinement and extension of Cpert and then guarantees that eq 1 is an effective estimate of orthogonality. Cpeaks = 1 − R2 =

∑ (yi − yi ̂ )2 ∑ (yi − y)̅ 2

(10)

Figure 2 illustrates the two extreme examples (i.e., PCS and OS with Cpeaks from 0 and 1) (R2 = 1 and 0), resulting in an O

(∑ bins) (0.63 × nc 2)

(5)

The general function to discover correlation between the explanatory (independent) variables and response variables (vectors or matrix) is given in eq 6. For the distribution of peaks for C2DS analysis, with 1tR as x and 2tR as y in this study, a linear relationship should be suitable for the purpose. First, the normalized 1tR is combined with vector 1 (eq 7) to include both slope and intercept for linear modeling. A linear regression model is applied here to fit the equation between 1tR and 2tR and then to investigate variable correlation by comparison of the real and predicted tR by using the model that is derived. This guarantees the objectivity to find 2D separation correlation since no requirement is needed for additional parameter input and further optimization. y = f (X)

(6)

X = [1 1t R,norm ]

(7)

Figure 2. Illustration of correlation and orthogonality of C2DS pattern for nonorthogonal and orthogonal separations. Peak capacities 1nc and 2 nc are equal to 10 in the simulated example. (A) Diagonal distribution of 10 peaks in 10 bins. Peak symbols ( × , *, and +) and line styles (solid, dashed, and dash-dotted) correspond to three different C2DS patterns with the R-square statistic for the linear model of 1.0, 0.93, and 0.98, respectively. Lines 1, 2, and 3 represent the linear models for the peaks distributed in 2D separation space. For details refer to the text. Regression equations: line 1, y = x (solid); line 2, y = 0.92x + 0.04 (dashed); line 3, y = 1.27x − 0.14 (dash-dotted). (B) Simulation of 100 peaks symmetrically distributed over the C2DS plane. Orthogonal separation is attained with the R-square statistic equal to 0. Regression equation: line 1, y = 0.5 (solid).

In eq 8, the calculation of regression coefficient is given to generate the model according to a least-squares fitting approach, where the notations T and −1 represent the transpose and inverse of a vector or matrix, respectively. The R-square value (R2, see eq 9) is a good statistical index to correlate the variable relationships and further evaluate how well the unknown prediction is likely to be tested by the model. Notations ŷ and y̅ correspond to the estimation of 2tR derived from the fitting model and mean of the real 2tR, respectively. Thus, it should be an ideal measure of the correlation of 1tR and 2 tR. b = (XT × X)−1 × XT × y R2 = 1 −

of 0 and 1. In Figure 2A, the three lines 1, 2, and 3 show the linear models to correlate 10 simulated 2D peaks in 3 groups ( × ,*, and +), respectively, but all diagonally distribute in the rectangular bins, as shown with 1nc and 2nc equal to 10. Though the bin coverage percent is the same (10%) in all the 3 separation cases, the R2 values are 1.0, 0.93, and 0.98, respectively, for the linear models. Figure 2B shows vertically symmetrical distribution of all peak apexes with orthogonal

(8)

∑ (yi − yi ̂ )2 ∑ (yi − y)̅ 2

(9) D



Article

divided by 100. Experimental acquisition of I in both 1D and 2D of C2DS has been proposed and reviewed in reported studies.26,27 The findings of eq 12 can be individually applied to simulate the separation in 1D and 2D columns with the temperature-averaged values of SPs. Five temperatures, namely, 60, 80, 100, 120, and 140 °C are investigated in this work. It is found that the 2D GC × GC retention diagram can simulate the real separation, with the equation shown in eq 13.20,28

separation. In accordance with the informational entropy theory introduced above, this maximizes the entropy distribution to give a separation with the lowest potential overlapping statistics. The model mathematically divides the separation space into two parts with R2 = 0 and then Cpeaks = 1. Thus, the advantage of the proposed technique is evident in contrast to the previous studies.

■

EXPERIMENTAL SECTION Several techniques to simulate and predict C2DS analysis have been successfully reported, such as the estimation of 2tR through experimental derivation of 1D elution temperature (Te) values and solute retention factors, and extension of ASPM to C2DS.19,20,23 In this work, the latter strategy developed by Seeley et al. was employed because of the full simulation feature and then to effectively compare the performance of the proposed model and the previous methods with change of 1nc and 2nc. The 1nc and 2nc values are simulated with a series of changing values with consideration of GC × GC separation. The ASPM technique is an effective tool to interpret the retention mechanism of chemical solutes and chromatographic SPs.18−20 In this work, it was employed to simulate a GC × GC separation and peak distribution and then used to generate examples for the study of orthogonality. Two separation systems were theoretically simulated with 225 chemical compositions separated on two column set pairs, namely, columns DB-1 and HP-INNOWax (set 1) and DB-1701 and HP-INNOWax (set 2), respectively.24 They provide different polarity combinations for GC × GC separation, namely, nonpolar (NP)/polar (P) and moderately-P/P, respectively. The Abraham constants of the solutes and SPs are respectively listed in Tables S2 to S4 of the Supporting Information; they were experimentally extracted from a prior study25 and a PhD thesis about ASPM study (Christina Mintz, B.A., 2009, University of North Texas). Abraham Solvation Parameter Model and C2DS Simulation. ASPM was proposed on the basis of the solution cavity theory with three steps, including solvent cavity formation, solvent molecule reorganization, and solute insertion.18 The equation is given in eq 11 for gas phase to a condensed phase-like chromatographic separation.

1.6ΔI = f (I1)

(13)

Here the notation ΔI is the difference of 2I and 1I, which represents the retention indices on the 2D and 1D columns, respectively. Then, distribution of the peaks is employed as examples for the orthogonality study of the proposed method, in comparison with the previously developed methods. Data Analysis. All computer programs involved in this study were coded in-house with the MATLAB environment (version 7.7.0.471, R2008b), such as determination of the division bins, informational entropy, mutual information and conditional entropy, and model establishment. The calculation was implemented on a HP compatible personal computer with Intel(R) Core(TM) 2 Duo CPU and 1.95 GB of RAM memory.

■

RESULTS AND DISCUSSION The selected 225 compounds used to simulate a typical GC × GC separation cover a wide range of component classes. The two systems with different column sets (set 1 and set 2) are applied to compare the 2D separation pattern with different 1tR and 2tR correlations. The results of the proposed strategy are then compared with previous methods. For each separation system, 5 possible combinations of 1nc and 2nc from 15 to 10 are simulated to investigate trends in orthogonality, robustness, and reasonability of the proposed method. In accordance with the Abraham model introduced above for GC × GC separation, 1tR and 2tR values are derived through application of the function given in eq 13. The constants for solutes and SPs listed in Tables S2 to S4 of the Supporting Information are employed for this calculation. Then, normalization of 1D and 2D separation is achieved by using eq 1 for each dimension. In Figure 3, separation based on ASPM of the 225 solutes for each column set is shown as peak apex plots. Corresponding component numbers given in Table S3 of the Supporting Information are included in this Figure, to aid inspection of simulated 2D separation based on ASPM. For clarity, the other 175 compounds given in Table S4 are not labeled. Both 1nc and 2nc are simulated as 10 in Figure 3. Figure 3, panels A and B illustrate GC × GC separations with set 1 and set 2, respectively. Simulated ASPM data appear logical, since all alkanes are located at the lowest 2tR (along with silanes); aromatic compounds are located at much later 2tR positions (nitrotoluene, nitrophenol, and stilbene). The 2D distribution of the peaks in Figure 3A appear greater than that in Figure 3B, due to the larger global separation achieved in Figure 3A. Quantitative correlation coefficients R2 for Figure 3 (panels A and B) (eq 9) are 0.1153 and 0.2510 using linear regression analysis and orthogonality correlation terms, Cpeaks, are 0.8847 and 0.7940, respectively. The normal and robust regression models are plotted in Figure 3 (panels A and B) as solid and dashed lines, respectively. The latter used iteratively reweighted least-squares with the ‘bisque’ weighting function. In terms of the general standard to estimate model outliers for

log SP = c + e × E + s × S + a × A + b × B + l × L (11)

where SP represents the free energy related probe solute properties such as the retention factor (k) and adjusted retention time (tR′). The uppercase letters E, S, A, B, and L represent the parameters to quantify the solute characteristics which may be correlated to polarizability, dipolarity, hydrogen bond donating ability, hydrogen bond accepting ability, and cavity size and dispersion forces, respectively; and the lower case coefficients e, s, a, b, v, and constant c denote the solvent properties acquired from regression analysis with known constants of specific solutes. The extension of ASPM to GC × GC separation has been reported with the new equation written according to eq 12.19,20 I = e′ × E′ + s′ × S′ + a′ × A′ + b′ × B′ + l′ × L′ (12)

where the new constants (both lower and uppercase) are coefficients derived from the original parameters introduced in eq 11, and I is the magnitude of the Kovatś retention index E



Article

of column sets and the 2D visual pattern evident in Figure 3 (panels A and B). This demonstrates the effectiveness of the proposed method for C2DS experimental investigations and should apply to GC × GC, LC × LC, and coupled instruments such as GC × GC hyphenated to time-of-flight mass spectrometry (GC × GC-TOFMS). The drawbacks of previous methods (M1 to M5) can be gleaned by consideration of the data in Tables 1 and 2. First, the results of M1 remain the same with the change of 1nc and 2 nc, and further, the correlation coefficients of set 1 and set 2 vary from 0.3396 to 0.5010 (on a range of 0 for OS and 1 for PCS). It can be found that this method emphasizes discovery of the overall 2D distribution but ignores the utilization efficiency of the 1D and 2D peak capacity. This deviates from the original definition of orthogonality and separations shown in Figure 3. The values of (α + γ) have the same limitations. On the contrary, the critical correlation of peak distribution is not included in the equations of methods M2 and M3 (see eqs 3 and 4), as mentioned in the introduction section. This makes calculation of M2 smaller than the real orthogonality of separation with low 2D retention correlation (larger Cpeaks), as provided by the proposed method (M6). This may be converse with a decrease of Cpeaks. From results introduced in Tables 1 and 2, all the orthogonality metrics of M3 are larger than those obtained from M6 for all 1,2nc. The reason is the forcible subtraction of the term nc and failing to both consider distribution of 2D peaks and employment of raw retention data. It seems that methods M4 and M5 based on informational theory principles provide a reasonable measure of orthogonality, but the change in magnitude of orthogonality is still not sensitive to changes of 1nc and 2nc. For instance, the metrics of M4 varies from 0.1983 to 0.1118 and 0.2265 to 0.1427 for set 1 and set 2 with the changes of 1nc and 2nc from 15 to 10; and orthogonality of M5 varies from 0.8235 to 0.8784 and 0.7913 to 0.8465, respectively. The change of the i.s. results is quite slow with changes to 1,2nc given in Tables 1 and 2. It is easy to see that the nc,max of 1D and 2D has been significantly reduced from 225 to 100 in the simulations. In addition, no clear rules can be found for the metrics of s.p. with the changing of 1,2nc. The orthogonality index is obviously too large for M5 with small 1nc and 2nc. This does not provide a satisfactorily objective assessment of the real experimental separations. The reason is that the random division of 2D separation patterns to obtain bins causes the calculation of entropy and informational theory

Figure 3. Simulated GC × GC separation results using ASPM (Table S2, refer to text) of the 225 compounds on column sets (A) set 1 and (B) set 2 given in Table S3 of the Supporting Information (○, labeled solutes) and Table S4 of the Supporting Information (●; no labels). The linear models with normal and robust regression analysis are plotted by solid and dashed lines, respectively. Set 2 has relatively poorer distribution. Regression equations: (A) y = 0.18x + 0.43 (solid); y = 0.13x + 0.51 (dashed) and (B) y = 0.12x + 0.62 (solid); y = 0.10x + 0.65 (dashed).

regression, all 225 compounds are contained in the robust model for both of the separation systems. In Tables 1 and 2, calculation of orthogonality parameters of the proposed method (M6) is provided with comparison of five other methods. Additional metrics include the peak angle (α + γ) of separation,13 informational similarity (i.s.), % synentropy (s.p.),17 and interim results Cpert and Cpeaks of the proposed approach. In terms of these results, advantages of the proposed method (M6) that provides simultaneous consideration of peak coverage and distribution correlation are evident. For the two GC × GC separations with different column combinations (set 1 and set 2), the gradual change of orthogonality under different peak capacities 1nc and 2nc shows reasonable evaluation of the separation. The orthogonality of set 1 and set 2 with the same 1nc and 2nc are consistent with the separation performance

Table 1. Metric Comparison of the Proposed Approach and Five Previous Methodsa estimation method ncb

1

15 14 13 12 11 10

nc

2

b

15 14 13 12 11 10

M1

c

0.3396 0.3396 0.3396 0.3396 0.3396 0.3396

M2

c

0.5996 0.6398 0.6856 0.7165 0.8002 0.8254

M3

c

0.6706 0.7216 0.7810 0.8257 0.9352 0.9811

M4

c

0.1983 0.1832 0.1655 0.1542 0.1185 0.1118

additional metrics and interim results M5

c

M6

0.8235 0.8342 0.8476 0.8430 0.8806 0.8784

c

0.6241 0.6663 0.7146 0.7509 0.8356 0.8707

α+γ

i.s.

s.p.

Cpert

Cpeaks

0.3464 0.3464 0.3464 0.3464 0.3464 0.3464

0.4342 0.4201 0.4027 0.4094 0.3557 0.3585

66.47 58.57 59.88 60.69 63.98 53.61

0.7055 0.7532 0.8077 0.8488 0.9445 0.9841

0.8847 0.8847 0.8847 0.8847 0.8847 0.8847

a

Six different combinations of 1nc and 2nc are studied for set 1 with a combination of DB-1 and HP-INNOWax with ASPM constants given in Tables S2 to S4 of the Supporting Information. b1nc: 1D peak capacity; 2nc: 2D peak capacity. cM1 to M5 correspond to the methods introduced in refs 13, 7, 14, 17, and 11, respectively. M6 is the proposed method in this study. Among these methods, M1 and M4 were proposed to measure separation correlation and others to measure orthogonality. Equations used to calculate the results of these methods are respectively introduced in the original publications as follows: ref 13, eq 3; ref 7, eq 3 (eq 3 in this work); ref 14, eq 7 (eq 4 in this work); ref 17, eq 6; and ref 11, eq 7. The results of α and γ are computed by eqs 8 and 10 in ref 13 and i.s. by eq 8 in ref 17, respectively. The results of s.p. are obtained by dividing the i.s. values from data diagonally aligned by the total entropy of 2D separation. The interim Cpert and Cpeaks correspond to results from eqs 5 and 10 in this work. F



Article

Table 2. Results Comparison of the Metrics for Set 2 (DB-1701 and HP-INNOWax)a estimation method ncb

1

15 14 13 12 11 10 a

additional metrics and interim results

ncb

M1c

M2c

M3c

M4c

M5c

M6c

α+γ

i.s.

s.p.

Cpert

Cpeaks

15 14 13 12 11 10

0.5010 0.5010 0.5010 0.5010 0.5010 0.5010

0.5362 0.5750 0.5917 0.6173 0.6953 0.7937

0.5996 0.6485 0.6740 0.7114 0.8125 0.9434

0.2265 0.2039 0.2380 0.1822 0.1708 0.1427

0.7913 0.8145 0.7786 0.8201 0.8270 0.8465

0.4809 0.5156 0.5347 0.5614 0.6289 0.7134

0.5247 0.5247 0.5247 0.5247 0.5247 0.5247

0.4753 0.4464 0.4884 0.4392 0.4295 0.4038

51.50 45.49 55.34 41.77 53.23 44.08

0.6420 0.6884 0.7138 0.7496 0.8396 0.9524

0.7490 0.7490 0.7490 0.7490 0.7490 0.7490

2

All the terms and symbols have the same meaning as in Table 1. bSee foonote b in Table 1. cSee footnote c in Table 1.

separation space based on surface coverage, and weak robustness of M1 for data with statistically smaller numbers.12 It should be pointed out that the separation outliers (e.g., components that might be strongly retained) have effects on accurate determination of correlation of peak distribution. They make Cpeaks abnormally large, and this then artificially increases separation orthogonality. However, this also makes Cpert simultaneously a small value, and so balances the manner of obtaining the orthogonality index O by using eq 1. Of course, robust regression can be applied to partially overcome these drawbacks, as well. Weighted means of all the peaks used for regression should be a reasonable index to estimate the distribution correlation under this situation. Thus, it still supports the proposed method as an effective approach for orthogonality evaluation. The reasonability and advantages of the proposed method can be summarized as follows: (1) simultaneous use of information of peak coverage in the 2D plane and mutual distribution correlation are introduced for definition of orthogonality. The former offers the whole separation picture and the latter refines the distribution details. (2) The two terms (Cpert and Cpeaks) and further O range between reasonable values and truly characterize the property of 2D separation. The evolution of the definitions from eqs 3 to 4 validates the application limitation of the methods solely on the basis of “bin occupation”. The proposed method makes each rectangular bin contribute equally to orthogonality (Cpert) and independently introduces the distribution correlation with a new term (Cpeaks). For information theory-based methods, OS cannot be experimentally attainable because of the impossibility of uniform separation and 100% occupation of peaks in the 2D plane. (3) Raw retention data are used for distribution correlation. This overcomes the disadvantages of previous methods with complete utilization of derived data (e.g., based on rectangular bins). Furthermore, it is suitable for the analysis of all 2D separation with no introduction of additional assumptions, such as 10% bin coverage along the diagonal. This makes the proposed method more suited to the C2DS study.

definitions to change, with no obvious defining rules. This has significant effects on determination of the orthogonality calculation. In Figure 4, further comparison of these results is studied to compare, in a relative sense, the proposed method and other

Figure 4. Comparison of orthogonality metrics of the previously developed methods M1 to M5 (see Tables 1 and 2 for details) and the proposed method in this work. The abscissa- and ordinate axis in the two figures correspond to the metrics by M6 (the proposed method) and other methods from M1 to M5, respectively. Each line has six metrics points with 1,2nc to be changed from 15 to 10. (A and B) are the results of set 1 and set 2, which correspond to column sets DB-1 and HP-INNOWax and DB-1701 and HP-INNOWax, respectively.

conventional methods. The present results are plotted along the x axis, with other methods plotted as shown. It can be found that M1 is constant, and M2 and M3 monotonically increase with the 1,2nc change from 15 to 10. However, they may overestimate the orthogonality evaluation, since the metrics of M3 are close to 1 for 1,2nc = 10. This means there is little sensitivity of these methods toward the orthogonality measure with higher utilization of the retention space. The metrics change of M4 and M5 are relatively gentle as 1,2nc changes from 15 to 10. Outliers are found to set 2, with 1,2nc = 13 for these two methods. This means that the division of peak occupation has significant influence on the peak distribution and retention structures. In summary, these results are consistent with the conclusions made in a recent study, including the unsuitability of correlation coefficients to measure separation orthogonality, dependency of discretization of

■

CONCLUSIONS

Decreasing the difference between nc,P and nc,max is one of the most important objectives of an ideal C2DS analysis, to determine optimal experimental conditions, such as column selection/combination and temperature or carrier flow optimization. Orthogonality evaluation helps to calculate the information included in the 2D separation space. The effects of condition changes can be achieved via comparison of orthogonality of different runs. In contrast to previous methods based on employing total bins occupied by 2D peaks, the G



Article

(16) Gong, F.; Liang, Y. Z.; Xie, P. S.; Chau, F. T. J. Chromatogr., A 2003, 1002, 25−40. (17) Slonecker, P. J.; Li, X. D.; Ridgway, T. H.; Dorsey, J. G. Anal. Chem. 1996, 68, 682−689. (18) Kollie, T. O.; Poole, C. F.; Abraham, M. H.; Whiting, G. S. Anal. Chim. Acta 1992, 259, 1−13. (19) Seeley, J. V.; Bates, C. T.; McCurry, J. D.; Seeley, S. K. J. Chromatogr., A 2012, 1226, 103−109. (20) Seeley, J. V.; Libby, E. M.; Edwards, K. A. H.; Seeley, S. K. J. Chromatogr., A 2009, 1216, 1650−1657. (21) Marriott, P. J.; Schoenmakers, P.; Wu, Z. Y. LC GC Europe 2012, 25, 266−273. (22) Schoenmakers, P.; Marriott, P.; Beens, J. LC GC Europe 2003, 16, 335−339. (23) Ryan, D.; Morrison, P.; Marriott, P. J. Chromatogr., A 2005, 1071, 47−53. (24) Poole, C. F.; Poole, S. K. J. Chromatogr., A 2008, 1184, 254− 280. (25) Atapattu, S. N.; Poole, C. F. J. Chromatogr., A 2008, 1195, 136− 145. (26) Bieri, S.; Marriott, P. J. Anal. Chem. 2008, 80, 760−768. (27) von Muhlen, C.; Marriott, P. J. Anal. Bioanal. Chem. 2011, 401, 2351−2360. (28) Seeley, J. V.; Seeley, S. K. J. Chromatogr., A 2007, 1172, 72−83.

proposed method makes full use of the information of rectangular bins and raw (original) retentions. Here, both C2DS peak coverage and internal correlation of the 2D peaks are used for orthogonality calculation. The nature of the 2D peak distribution is characterized through the definitions of two new concepts, namely, Cpert and Cpeaks. Simple linear regression analysis is found to effectively discover quantitative relationships of peaks with 1tR and 2tR data. The evident advantage of the proposed method is the reasonable employment of occupation of bins and further refinement of the distribution of peaks with original retention data information. ASPM of GC × GC separation is utilized to simulate the 2D peak distribution diagrams. Two systems with column sets DB-1 and HPINNOWax and DB-1701 and HP-INNOWax were respectively simulated with separation of 225 chemical components. The results show that the proposed method can be effectively applied for C2DS study.

■

ASSOCIATED CONTENT

* Supporting Information S

Additional nformation as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: + 61 3 99059630. Fax: + 61 3 99058501. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work is financially supported by the Australian Research Council Discovery Grant DP0988656.

■

REFERENCES

(1) Cordero, C.; Rubiolo, P.; Sgorbini, B.; Galli, M.; Bicchi, C. J. Chromatogr., A 2006, 1132, 268−279. (2) Marchetti, N.; Fairchild, J. N.; Guiochon, G. Anal. Chem. 2008, 80, 2756−2767. (3) Mondello, L.; Tranchida, P. Q.; Dugo, P.; Dugo, G. Mass Spectrom. Rev. 2008, 27, 101−124. (4) Khummueng, W.; Harynuk, J.; Marriott, P. J. Anal. Chem. 2006, 78, 4578−4587. (5) Blumberg, L. M. J. Sep. Sci. 2008, 31, 3352−3357. (6) Giddings, J. C. HRC & CC, J. High Resolut. Chromatogr. Chromatogr. Commun. 1987, 10, 319−323. (7) Gilar, M.; Olivova, P.; Daly, A. E.; Gebler, J. C. Anal. Chem. 2005, 77, 6426−6434. (8) Marchetti, N.; Felinger, A.; Pasti, L.; Pietrogrande, M. C.; Dondi, F. Anal. Chem. 2004, 76, 3055−3068. (9) Pietrogrande, M. C.; Marchetti, N.; Dondi, F.; Righetti, P. G. J. Chromatogr., B 2006, 833, 51−62. (10) Pietrogrande, M. C.; Marchetti, N.; Tosi, A.; Dondi, F.; Righetti, P. G. Electrophoresis 2005, 26, 2739−2748. (11) Pourhaghighi, M. R.; Karzand, M.; Girault, H. H. Anal. Chem. 2011, 83, 7676−7681. (12) Gilar, M.; Fridrich, J.; Schure, M. R.; Jaworski, A. Anal. Chem. 2012, 84, 8722−8732. (13) Liu, Z. Y.; Patterson, D. G.; Lee, M. L. Anal. Chem. 1995, 67, 3840−3845. (14) Watson, H. E.; Davis, J. M.; Synovec, R. E. Anal. Chem. 2007, 79, 7924−7927. (15) Eckschlager, K.; Danzer, K. Information Theory in Analytical Chemistry; John Wiley & Sons: New York, 1994. H


A Modeling Approach for Orthogonality of Comprehensive Two

Recommend Documents