Width Based Characterization of Chromatographic Peaks: Beyond

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Width Based Characterization of Chromatographic Peaks. Beyond Height and Area. Akinde Florence Kadjo, Hongzhu Liao, Purnendu K. Dasgupta, and Karsten G. Kraiczek Anal. Chem., Just Accepted Manuscript • DOI: 10.1021/acs.analchem.6b04858 • Publication Date (Web): 28 Feb 2017 Downloaded from http://pubs.acs.org on March 4, 2017

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Analytical Chemistry

Width Based Characterization of Chromatographic Peaks. Beyond Height and Area. Akinde F. Kadjo, Hongzhu Liao, Purnendu K. Dasgupta∗ Department of Chemistry and Biochemistry, University of Texas at Arlington, Arlington, TX 76019-0065, USA

Karsten G. Kraiczek Agilent Technologies, Hewlett-Packard Strasse 8, D 76337 Waldbronn, Germany

*Email:

[email protected] Fax: (817) 272-3808

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ACS Paragon Plus Environment


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ABSTRACT The preceding paper (Anal. Chem. 2017, 89, yyyy-zzzz) introduced width-based quantitation (WBQ). The present paper focuses on: (1) Situations where WBQ is effective while height/area-based linear calibrations fail, e.g., when (a) the detector is in a nonlinear response region, (b) the detector/data system is saturated, causing clipping/truncation of the signal, or (c) the detector signal is not a single-valued function of concentration, as when a fluorescence signal goes into the self-quenched domain. (2) Utilization of WBQ in post-column reagent addition methods where the reagent produces a significant detector background. WBQ can minimize added reagent without sacrificing the upper determination limit; a limited reagent amount truncates peaks from high analyte concentrations but does not hamper WBQ at a low height. (3) A description of peak asymmetry via leading/trailing half-widths vs. relative height (fraction of maximum height) plots. (4) A holistic description of chromatographic peaks through 6 parameters describing the two independent generalized Gaussian distributions that underlie the WBQ chromatographic peak model. (5) Characterization of shape by widths at multiple heights and shape-based impurity detection.

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The companion paper1 introduced width based quantitation (WBQ) and its merits vs. height- and area-based quantitation. At high concentrations, the signal may enter a nonlinear response region or even be truncated from detector/data system limitations, causing area- and especially height-based quantitation to fail. However, if measured at signal levels before the onset of nonlinearity/saturation, the width may still be unaffected. A related situation arises in detection after post-column reagent (PCR) addition. Often the PCR itself exhibits a finite detector response, contributing to the background and noise.2,3 One would like to limit the amount of the added reagent. But this restricts the upper limit of measurement: Insufficient PCR would truncate the analyte peak. WBQ is also applicable in this case. Even a combination of such situations that produce unusual (W- or M-shaped) peak responses can be successfully quantitated. The preceding paper focused on width measurement at a single height.Error! Bookmark not defined.

Width can obviously be measured at many heights. Width data at

multiple heights can be exploited in many ways: First, peak asymmetry is often specified as an index of non-ideality. The simplest expression is b/a where a and b are respectively the leading and trailing half-widths of the peak at some specific values of the relative height 1/ , ( being defined as hmax/h),Error! Bookmark not defined. typically 0.05 or 0.10. The U. S. Pharmacopeia defines symmetry factor or tailing factor as (a + b)/2a at 1/

0.05.4 The skewness from the third central moment is another asymmetry index.

But no single numeric index can adequately describe peak symmetry. Pápai and Pap5 discussed many alternatives and suggested a complex five-step method to assess peak

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symmetry. We propose a scalable (concentration-independent) depiction of peak variance as a function of 1/ to provide a holistic picture of peak asymmetry. Second, describing a chromatographic peak goes beyond symmetry. A more complete and numerically tractable description of shape is needed. It is recognized that the shape is generally analyte concentration independent: without column overloading, a wellbehaved analyte peak is scalable. Impurity perception is based on shape change, but quantitative diagnostic criteria are elusive. A numerical description of peak shape allows any departure from the pure analyte benchmark to be statistically identified, without requiring the analyte and impurity to behave differently in some multidimensional detection scheme (e.g. differences in optical absorption spectrum, etc.). We first describe the principles below.

PRINCIPLES We proposed1 that within 1/ = 0.05-0.95, each side of a chromatographic peaks follow a Generalized Gaussian Distribution model (GGDM) that share a common apex at t = 0. | |

,

...(1)

, A true Gaussian is a subclass of a GGD where the exponent m = n = 2 and the standard deviation s = s’. Let us individually consider the leading (Wh(l)) and trailing halfwidth (Wh(t)). As one end of Wh(l) or Wh(t) at any height h lies at t = 0 and the other end at t, these leading/trailing half widths are simply t = t. Transposition of eq 1 gives:

4


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/

/

...(2)

and in a linear form:

A plot of the leading or trailing

.

.

vs.

...(3)

then provides useful information:

the slope (0.5 for a true Gaussian) and s (computed from the intercept) for each half together defines the peak shape while the coefficient of determination r2, or root mean square relative error (RMSRE) for the fit in eq. 3 provides the degree of appropriateness of the model. Impurity Diagnostics when Pure Standards are Unavailable. Even without a known standard shape, an inability to fit the GGDM may potentially signal the presence of an impurity. Fits to Eq. 1- 3 may be tested. While these cannot be used for truncated peaks because the apex cannot be located, the approximation represented by eq. 1-9a (reproduced below in a different form) is applicable

...(4)

or

...(5)

a

References to figures or equations in ref. (1) are simply preceded by a prefix 1‐, including those in the SI.

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where =

...(6)

and thence

= r being ∗

and

...(7)

. Unlike eq 3, eq 5 does not require

to be known.

Best fit values for , , can be obtained by nonlinear least square routines, e.g., Microsoft Excel SolverTM. The goodness of fit (linear r2 or better, RMSRE) is an indication of the validity of the model. If chromatographic data are available for multiple concentrations (even if some peaks at higher concentrations are truncated), and

is

within the linear response range, eq 7 states that the respective intercepts ( ) for each fit at individual concentrations will be linearly related to

in the absence of impurities.

Impurity Diagnostics When a Pure Standard is Available. Later we examine quantitatively how the presence of an impurity changes the shape. Qualitatively, by definition, an impurity contributes a smaller portion of the overall peak response compared to the main component. An impurity therefore affects the peak-width differently at the bottom vs. the top. Therefore, WBQ applied to an impure peak that is based on calibration by pure standards will predict different concentrations using lower vs. higher h calibrations. Impurity diagnostics can be based on a significant increase in variance (e.g., F-test) of the predicted concentrations from width-based calibrations at multiple heights of the suspect peak vs. that for standards at the same heights.

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Also, ratios of widths at various heights can be used as shape indices. The change in this shape index may indicate the presence of impurities. For a Gaussian peak, it is readily derived (see eq. 1-3):

Wh1/Wh2 =

...(8)

For example, W0.2/W0.8 for a Gaussian peak is readily calculated to be 2.686. For a peak defined by two independent GGDs, as in eq. 4, widths at minimally three heights are needed to compute a numerical constant.

For any peak obeying eq 4, the terms

the specific values of

i

or

are constants readily derivable from

chosen. We show in the SI for example,

. . . .

is expected to be

~0.4, a shape criterion. From three Wh measurements, j and k (eq 5) can also be computed. Even when a peak does not exactly follow eq. 4, we find that

. . . .

or a similar

parameter involving some other combination of Wh values of the examined peak can be compared with the corresponding value of a standard for impurity diagnostics, with conclusions based on statistical criteria.

EXPERIMENTAL SECTION Caffeine chromatographic data were originally generated to validate the performance of a high dynamic range photodiode array spectrometer;6 the experimental details were described.1 7



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Ion Chromatography (IC) was conducted on a IC-25 (Figure 1) or a ICS 5000 system (Figure 2) at 30 C using an ASRS-Ultra II anion suppressor in external water mode. An AD20 absorbance detector was used for absorbance measurement. All components were from www.thermofisher.com. For IC determination of formate and silicate, a permeative amine introduction system (PAID)8 which introduces diethylamine to detect acids, even very weak acids, was used prior to conductivity detection.

RESULTS AND DISCUSSION WBQ with Peak Maximum in Nonlinear Response Regime. With nonlinear/saturated detector response, area- or height-based quantitation has obvious limitations. Figure 1 shows nitrate chromatograms with absorbance extending to the nonlinear region.

could be expressed as: . ∗

, min

.

. , r2 = 0.9935 ...(9)

with 11.4% RMSRE in comparison to 373% and 1130% RMSRE for area and height based unweighted regression. The quantitation errors in the three paradigms (heightarea- and width-based) are shown in Table S1A in the Supporting Information (SI); WBQ outperforms area and especially height-based quantitation in both unweighted and 1/x2weighted regression. Height-based quantitation, most affected by nonlinearity, shows the highest error. WBQ is not significantly affected by weighting,Error! outperforms the others, especially in the unweighted mode.

8


Bookmark not defined.

it

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Although an RMS relative error of 11.4% may seem high, this needs to be judged in perspective to errors that result in conventional methods. In part the error arises from a change in the peak shape; this is readily apparent at the highest level. It is instructive to consider instead caffeine, a well-behaved model analyte in reverse-phase liquid chromatography and also as an absorbance standard. Caffeine chromatograms are shown in Figures S1-S3 for injected amounts ranging from 0.2 ng to 100 g, a 5.5 order of magnitude span. At the low end, consideration of results below 2 ng are omitted due to noise. Over a 50-fold range of 2-100 g (n=6) at the high end, the top three concentration responses are truncated and the fourth is in the nonlinear response domain. Only the bottom two are in the linear domain. RMSRE for WBQ at 250 mAU for the 2-100 g range is 3.2% compared to 368 (33.2)% and 915 (50.5)% respective errors for area- and height-based unweighted (1/x2-weighted) linear regression. See Table S1B for detailed analysis of the caffeine data Response Non-Monotonic with Concentration and/or Truncated. Many fluors undergo self-quenching at higher concentrations. An interesting use has involved a fluorescent eluent in the quenched domain to generate positive signals in indirect fluorescence detection9 but mostly it limits the use of fluorescence detection at higher concentrations. If the peak apex of a fluorophore is in the quenched domain, an Mshaped response results. In altogether other situations, a W-shaped response is possible. For example, IC relies on conductometry. But very weak acids barely ionize and are undetectable. We have been exploring determination of strong to very weak acids by adding a base post-suppression as a PCR.7-15 Following base addition, the principal conductive species is OH-. An eluting acid reacts as HX + OH-  H2O + X-, the 9



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highly mobile OH- is replaced by less conductive X-, producing a negative response. Reducing the PCR concentration decreases the background and associated noise, improving LODs.7 However, when the analyte concentration exceeds the base background, the response is truncated. As in a strong acid-strong base conductometric titration, on the leading edge of the eluite acid peak, conductivity decreases until the base is neutralized, then conductivity increases until the peak apex (unless truncated by detector/data system) and the process reverses on the trailing edge creating a W(Figure S4) or center-truncated W (Figure 2a). For a very weak acid like silicate (pKa 9.3), the response is considerably more complicated due to multiple factors: insufficiently high pH to cause complete ionization of silicate, buffer formation and poor to no ionization of the excess silicic acid (Figure 2b). In performing WBQ for formate and silicate in Fig. 2, the parameters in eq 5 are fit to obtain the lowest RMSRE; over a 0.1-6 and 0.1-10 mM concentration span the RMSRE were 5.0 and 5.7%, respectively. Given the unusual peak shapes, this is remarkable. Depiction of Peak Shapes and Asymmetry. Asymmetry at 1/ = 0.05 and 0.10 are typically cited. These cannot describe overall symmetry. Our love for symmetry16,17 affects visual perception; this is rarely objective (Figure S5). Plots of left- or right halfwidth vs. time have been beautifully explored by Baeza-Baeza et al.18 for many purposes. However, for any peak, it would be obvious that Wh,l = tR‐t and Wh,r=t‐tR. As such, plots of Wh(l,t) vs. t are merely inverted triangles version of the peaks. Compare to a plot of Wh(l,t) vs. t for citrate and acetate along with the original chromatogram (Figure S6, S7). plots of Wh(l,t) vs. 1/ offers a different perspective, as shown in Figure S5 for simulated Gaussian peaks and in Figure 3 for several previously discussed1real 10


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chromatographic peaks (Figures 1-S6-1-S9) with large variations in symmetry. These depict not only how each half behaves comparatively, but also fronting and tailing. If chromatographic peaks were perfectly Gaussian, a single parameter, specifying the SD will be sufficient. As this is not the case, one has the choice of remaining within the strict Gaussian paradigm and assuming that is not a constant. It can be readily computed at all points in the peak from s t/ 2 ln

and examined as a function of

the relative height or time. Figure 4 plots s vs. 1/ for caffeine over a 300-fold range of concentration. Figure S8 shows s vs. t, providing slightly different information. Whereas s varies considerably across a peak, its constancy as a function of concentration is remarkable. Similar constancy of shape vs. concentration for bromide (original chromatograms in Figure S9) appear in Figure S10 where the leading and trailing halves are further apart than with caffeine. Similar single-concentration plots for chloride, nitrate, citrate, acetate and formate which exhibit different degrees of symmetry are shown in Figures S11-S15. Figure 3 may provide a holistic view of variance across the peak but a complete description still requires the dependence of s with 1/ or t to be specified. If we adopt the GGDM (eq 3) instead of the strict Gaussian paradigm, such a description is selfcontained. Figure 5 shows ln Wh(l) or ln Wh(t) vs. ln (ln ) plots for chloride nitrate and citrate including the slope, SD (calculated from the intercept), and r2. For all three, the plots are essentially linear (except for the tail of the chloride peak. The slope and the SD then (for each half of the peak) provide a more complete description than any presently available. The r2 values serve as a quality assurance for the model. Conventional

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asymmetry can be estimated at any value of 1/ from these specifications (reliability depending on r2). WBQ relies on peak shape being concentration-independent. A change in shape brought about by concentration-dependent ionization or chromatographic overloading is not accommodated. Plots for formate and acetate are shown in Figure S16; unlike the fully/uniformly ionized situation in Fig. 5, except for the leading edge of the acetate peak, these are obviously nonlinear. However, it is still possible to specify 1/ limits within which the peak fits the GGDM. Impurity-Induced Shape Change. A Numerical Simulation. If the impurity and analyte differ in exact retention and/or peak SD, the eluting band shape will differ in principle from that of the pure analyte. Consider both the analyte (concentration C) and impurity (concentration 0.1C) to have Gaussian profiles (Figure 6) eluting at identical retention times (tR). We assume analyte peak SD to be unity. If the impurity peak SD were the same, the overall response would be the same as the response of an analyte peak of concentration 1.1C.Chromatographic theory assumes column efficiencies to be analyte-independent but in reality SDs are rarely identical even when retention times are. Consider impurities with SDs of 1.41 and 0.71. Applying WBQ at 1/ = 0.9 in either case will result in 1.1 C as concentration (Cpred). But applying WBQ at 1/ = 0.2 will lead for s=0.71 to 1.1C>>Cpred ~>C and for s=1.41 to Cpred >>1.1 C. For identical analyte and impurity tR’s, Cpred at high 1/ > Cpred at low 1/ if SDanalyte>SDimpurity. The opposite holds when SDanalyte>1.1C. Even when the analyte does not exactly obey the GGDM, the WBQbased Cpred,low 1/ /Cpred,high 1/ ratio will increase in the presence of an impurity, increasingly so with increasing tR. Impurity Detection in Real chromatograms. Pure standard Available. Consider calibration curves are made with pure standards; during analysis of real samples, an impurity is suspected in the analyte peak. For a calibration curve of 200-2000 µM bromide, both height/area display excellent linear correlation with concentration but an intercept is present (Figures S19, S20). Best practice of WBQ utilizes peaks that scale with concentration; this is nearly seen (Figure S20). Although we have not taken advantage of this in impurity analysis below, there is discernible improvement in scalability after proportional intercept correction (Figure S21). Figure 7 shows the same bromide samples each now containing 20 µM nitrate. The relative impurity amount is thus 1-10%. Except perhaps for the 10% case, the impurity is not readily discernible. We had outlined two approaches to impurity diagnosis: (a) checking equivalence of quantitation using calibration at multiple heights (at least one high, one low, within the realm of the examined peak) and (b) examination of ratio of log

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(width ratio) parameters for a putatively impure peak vs. standards. Both are examined below. The data in Figure S9 were used to generate nine separate calibration curves at up to 9 different heights ranging from 2 to 43 µS/cm. The various Wh values for the standards were processed by the corresponding calibration equations to generate the Cpred values (blue triangles, Figure 8). The best fit through these are shown as dashed lines. The Cpred values are essentially height-independent (RMSRE 0.7 -2.2%). In contrast, in all cases for the impurity containing peaks, Cpred at the various heights differed much more. Cpred variances for all but the 200 µM case are higher for the impure samples at the p

Width Based Characterization of Chromatographic Peaks: Beyond

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