about the validity of the LOD concept in chromatography. To eliminate any .... MD, in contrast, is measured (without a column) under specified conditions [ 1-4] ...
Clarification of the Limit of Detection in Chromatography 1)
J. P. Foley 21 / J . G. D o r s e y * Department of Chemistry, University of Florida, Gainesville, F L 32611, USA
developed by the American Society for Testing and Materials (ASTM) [1-4]. The limit of detection characterizes an overall trace analytical procedure consisting of one or more steps, whereas the minimum detectability describes only one step in a chromatographic analysis: detection. (We further differentiate these concepts in a later section,
Key Words Chromatography
Limit of detection Calculation of limit
Eliminating Mistaken Identities.) Summary Current problems with the limit of detection (LOD) concept in chromatography are reviewed. They include the confusion of the LOD with other separate, distinct concepts in trace analysis such as the minimum detectability (MD); the use of arbitrary, unjustified models for the calculation of the LOD; the use of concentration units instead of units of amount; and the failure to account for differences in chromatographic conditions when comparing LODs. Solutions to these LOD problems are discussed. Two models are proposed for calculating the chromatographic LOD. A new concept, the standardized chromatographic LOD, is introduced to account for differences in chromatographic bandwidths of experimentally measured LODs. The standardized chromatographic LOD is shown to be a more reliable parameter than the conventional (non-standardized) chromatographic LOD.
Introduction fhe limit of detection (LOD) is generally defined as the smallest concentration or amount of analyte that can be detected with reasonable certainty for a given analytical procedure. Though arguably the most important figure of merit in trace analysis, the LOD remains an ambiguous quantity in the field of chromatography. Detection limits differing by orders of magnitude are frequently reported for very similar (sometimes identical!) chromatographic systems. Such huge discrepancies raise serious questions about the validity of the LOD concept in chromatography. To eliminate any confusion at the outset, we must emphasize that the "limit of detection" (or "detection limit") concept ~'e have just described is vastly different from the univer~Uy recognized minimum detectability (MD) concept
1) This paper is published as a discussion material. We welcome commentsfrom our readers. 2) Present Address: National Bureau of Standards, Building 222, RoomAll3, Washington, D.C. 20234 ChromatographiaVol. 18, No. 9, September 1984 0009-5893/84/9 0503--09 $ 03.00/0
As part of our continuing effort to accurately measure and interpret chromatographic figures of merit [5], our goal in this report is to transform the LOD in chromatography from an ill-defined, ambiguous concept to a reliable, meaningful parameter. We attempt to accomplish this by first identifying the current problems with the chromatographic LOD concept and then discussing possible solutions to these problems, with emphasis given to those problems which are the major sources of the observed discrepancies in chromatographic detection limits. Especially significant is a method we propose which accounts for the different amount of chromatographic dilution that an analyte undergoes in different chromatographic systems (or more rigorously, for differences in the chromatographic bandwidths - see Converting to chromatographic reference
conditions). It is beyond the scope of this report to introduce or review the limit of detection (LOD) from a historical or theoretical point of view; such discussions and references to additional discussions may be found elsewhere [6-9]. We assume some prior knowledge of the LOD and focus on solving the problems associated with this concept in chror[aatography. Most of our discussion is written from the perspective of peak height as the analytical signal, although two sections are equally pertinent to analyses using peak area (Eliminating mistaken identities and Using the correct units). Gaussian peak profiles are assumed throughout.
Identifying Current Problems Literature survey results Initially, to determine the sources of discrepancy in chromatographic detection limits, we conducted a limited survey of analytical textbooks, chromatographic monographs, and the primary chromatographic literature. This survey revealed two mistakes of omission, as well as four major sources of discrepancies. All six are summarized in Table I. The first two problems are omissions which discredit the work reported, at least to some degree. They can be eliminated, however, if more attention is given
Originals 9 1984 Friedr. Vieweg & Sohn VerlagsgesellschaftmbH
503
Table I. Problems with the limit o f detection (LOD) concept in chromatography mistakes of omission 1. Detection limits were not reported, although trace analysis was stressed. 2. Detection limits were reported, but no f o r m u l a f o r calculating the LOD was given.
during manuscript preparation, and thus may be dismissei without further discussion. The remaining four probler: in Table I comprise the major sources of the discrepancie~ in observed chromatographic detection limits and are th~ topics to be addressed in this report.
Numerical example
sources of discrepancies 3. The LOD concept was confused with other concepts used in trace analysis, particularly the m i n i m u m detectability (MD). 4. A r b i t r a r y LOD models were used with little or no justification. 5. Detection limits were reported as concentrations instead o f amounts. 6. Differences in chromatographic conditions were not taken into account.
Table II. Example o f widely differing detection limits A ssump tions 1. 2. 3. 4. 5.
Conventional liquid chromatograph with U V detector Beer's Law applies, i.e., A = ebc. Analytical sensitivity. S = eb = 1 0 , 0 0 0 A U Lmo1-1 Peak-to-peak noise, N p . p = 2 X 1 0 - S A u Root-mean-square noise, Nrm s = 1/5 Np,p
Varl'ables Experiment A Vin j LO D clef n VM(mL) k N (plates) ov(mL) a
Experiment B
5uL 10 N p.p/S 2.5 10 1000 0.87
20uL 3 N r ms/S 0.75 3 10,000 0.03
8.7 X 1 0 - 6 M
4.5 X 1 0 - 9 M
Results LODS reported b
log ( L O D A / L O D B) = 3.3 orders o f magnitude difference! a the bandwidth o f the analyte peak, calculated f r o m o v = V M ( 1 + k)/N 1/2 b reported to more than one significant figure f o r use in Table Vl.
Before proceeding, however, a numerical example whic! incorporates problems 4 - 6 in Table I will be given. TI~ example will demonstrate the magnitude of these probler~ and will facilitate later discussion. Although designed wit~ liquid chromatography in mind, the points made by tl~ example apply equally well to gas and supercritical flui~ chromatography. The initial assumptions, experimental conditions, an~ results of this example are shown in Table II. To avoi~ the confusion which it would certainly have caused, problez 3 of Table I was not incorporated into this example. Ifi'. had been, the results might have been even more discordan~ Nevertheless, as seen in the bottom row of Table II, th~ LODs for the two systems employing identical detectors differ by 3.3 orders of magnitude! Although a detailed explanation of this example is bey0ni the scope of the present discussion, the huge discrepan~ in the two LODs will be reconciled after solutions to tk.~ last three problems in Table I are discussed. This examp',~ should awaken the reader to the seriousness of the~ problems and demonstrate why they must be elkninate~ if the chromatographic LOD is to be a meaningful figure of merit.
Solving the Problems Eliminating mistaken identities The limit of detection has unfortunately been confu~ with three other concepts - particularly the minimu~ detectability (MD) - which are also used in characterizir,i chromatographic trace analyses. Table III includes syrnb01~l and definitions for all four of these concepts.
Table III. Trace analysis concepts in chromatography terms applicable to other multi-step analytical procedures 1. limit o f detection ( L O D ) pseudo-synonyms b 2. (analytical) sensitivity (S)
-- smallest concentration o r amount o f analyte that can be detected with reasonable certainty for a give~ analytical procedure a --
m i n i m u m absolute quantity, m i n i m u m detectable amount, sensitivity
-
slope o f the calibration curve - signal o u t p u t per unit concentration o r a m o u n t o f analyte introducec in a given analytical procedure c
terms applicable only to the detection step in chromatography 3. (minimum) detectability (MDI
pseudo-synonyms b 4. detector sensitivity (S d)
--
m i n i m u m concentration or mass f l u x o f analyte (in a solvent) which gives a detector signal that can be discerned from the noise with reasonable certainty, generally recognized to be twice the peak-t0-peak noise [1--4]. m i n i m u m detectable level, m i n i m u m detectable concentration slope o f the detector response curve -- signal o u t p u t per unit concentration or mass f l u x of analyte introduced to a detector
a In the section Using the Correct Units, we show that concentration units are inappropriate f o r the LOD in chromatography. b We recommend that the use o f these pseudo-synonyms be discontinued immediately. c In Using the Correct Units, we show that the independent variable o f a chromatographic calibration curve is the a m o u n t o f analyte injects: and not the concentration o f analyte injected.
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Originals
One reason that the LOD is confused with the other concepts in Table III, particularly the MD, is the redundant nomenclature for the LOD and the MD, as evidenced by the partial, but representative list of pseudo-synonyms for these concepts (given in Table III) which appear frequently in the literature. This redundant terminology has 0nly confused the chromatographic community, particularly since the pseudo-synonyms for the two different concepts are themselves quite similar. We recommend that the use ofall such pseudo-synonyms be discontinued immediately. Even if the confusion resulting from the redundant termin01ogycould be eliminated, the LOD might still be confused with the MD by the apprentice chromatographer because their definitions, as shown in Table III and in eqs. (1) and (2), are so similar in appearance (cf. meaning, however). LOD = arbitrary detector signal level/ (analytical) sensitivity MD = arbitrary detector signal level/ detector sensitivity
(1) (2)
] Furthermore, as can be inferred from eqs. (1) and (2), the L0D and MD can be related mathematically. Assuming an analytical signal in terms of peak height, the relationship fora concentration sensitive chromatographic system is [9] L0D = (27r) 1/2 [VM(1 + k)/N 1/2] bMD
(3)
where VM represents the corrected gas holdup volume in GC and the column void volume in LC; k is the capacity factor; N is the number of theoretical plates; and b is a unitless parameter which permits the LOD and the MD to be defined independently of one another. For a mass sensitive chromatographic system, the relationship between the LOD and the MD is [ 10] L0D = (2rr) l/z [tM (1 + k)/N l/z ]bMD
(4)
where tM is the retention time of an unretained solute, corrected for gas compressibility in GC. Yet despite their similarities, the LOD and MD are distinct concepts, as a closer scrutiny of Table III shows. The LOD isa general concept characterizing any overall trace analytical procedure consisting of one or more steps, whereas the MD is a specific term characterizing only one step in a chromatographic analysis: detection. For example, the LOD must, by det'mition, include the chromatographic dilution of the analyte, whereas the MD cannot. Furthermore, a chromatographic LOD is measured experimentally with a complete chromatographic system (including a column) under the specific conditions of a given trace analysis; the MD, in contrast, is measured (without a column) under specified conditions [ 1-4] that in general do not correspond to those of the analysis. Finally, one last difference should be noted: Referring to eqs. (1) and (2), an arbitrary detector signal level of twice the peak-to-peak detector noise [1-4] has been universally agreed upon for MD calculations. No such consensus has beenreached for the LOD.
ChromatographiaVol. 18, No. 9, September 1984
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Choosing a model
As discussed earlier, the LOD may be defined as the quotient of an arbitrary detector signal and the analytical sensitivity (DS/S). In our survey of the chromatographic trace analysis literature, we found a multitude of arbitrary DS/S levels used, ranging from 2 to 10. To further complicate matters, neither the measures of the signal (peak height, peak area, etc.) or the noise (peak-to-peak, root mean square, etc.) were specified in many instances. These inconsistencies and ambiguities are not surprising since (to our knowledge) no standard model for the LOD has ever been proposed, much less adopted, by any recognized organization for the field of chromatography!* We note specifically the omission of an LOD definition in chromatography by the American Society for Testing and Materials (ASTM) and by the International Union of Pure and Applied Chemistry (IUPAC) in their respective publications on gas and/or liquid chromatography nomenclature [ I - 4 , 13-16]. The omission by these and other organizations is also substantiated in two reviews [17,18]. More importantly, however, the inconsistencies and ambiguities in the chromatographic LOD can be eliminated completely if a clearly stated LOD model is adopted. We therefore propose the adoption of, with minor reinterpretation, the IUPAC model for spectrochemical analysis [11] or a model based on first order error propagation [8]. These models are given in eqs. (5a) and (5b), respectively, LOD = 3SB/S
(5a)
LOD = 3[s 2 + s2 + (ilS)2s2]1121S
(5b)
where S, i, Ss, and si are the analytical sensitivity (slope), intercept, and their respective standard deviations of the calibration curve obtained via linear regression; and sB is the standard deviation of the spectroscopic blank signal, calculated from 20 or more measurements. The factor of 3 in the numerator of the right-hand expressions of eqs. (5a) and (5b) gives a practical confidence level of 90% to 99.7%, depending on the probability distribution of the blank signal and the accuracy of SB [8, 11, 19]. Though smaller or larger factors could be used instead of 3, the resulting confidence levels would be too low or too high for practical use in most cases. Both the original proponents of these models [8, 11] and others [19] strongly recommend the use of the factor 3. We concur. Both models are proposed for adoption because it seems preferable to let the chromatography community judge their respective merits. Indeed, strong arguments can be made for each. The IUPAC model, on the one hand, is computationaUy simpler and has already been employed, though infrequently, in the chromatographic literature. * The model which the International Union of Pure and Applied Chemistry (IUPAC) adopted in 1975 [111 was chosen specifically for spectrochemical analysis. Though the ACS Subcommittee on Environmental Chemistry reaffirmed this model in 1980 [12], it did so for the area of environmental chemistry and not specificaUy for the field of chromatography.
505
On the other hand, the error propagation model is not really all that complicated; many pocket calculators with linear regression capability can be easily programmed for the error propagation model. Furthermore, the error propagation LOD model takes uncertainties of the slope and intercept of the calibration curve into account, resulting in a more realistic numerical estimate.
Interpretation. The IUPAC and error propagation LOD models were developed originally for spectroscopic trace analysis. Nearly all the associated concepts, [e.g., the calibration curve, the sample (analyte + matrix), etc.] have identical, straightforward interpretations in chromatography. One aspect, however, does not: the interpretation and measurement of sB. Intuitively it is clear that the chromatographic baseline is analogous to the blank signal in spectroscopy. Moreover, just as SB represents a measure of the noise in spectroscopy (the standard deviation of the blank), sB must also represent an analogous measure of the noise (baseline fluctuations) in chromatography. A useful mnemonic would be to refer to Su as the "standard deviation" of the baseline. Given this chromatographic interpretation of Sn, how should Su be measured? One possible procedure would be to estimate SB from 20 or more measurements of only that portion of the noise observed at the analyte's retention time in the absence of the analyte (when a blank solution is injected). This is directly analogous to the measurement of the blank signal (at the analytical wavelength) in spectroscopy. But this procedure would require 20 or more injections of blank solution (~> 20 blank chromatograrns!) and is obviously impractical:
can be estimated from one-fifth of the peak-to-peak n0i~ i.e., SB = NR~/5
(6cl
which is dearly more practical than the procedure involve! 20 or more injections of blank solution. Two additional comments regarding the practical procedure for measuring SB should be noted: 1. It is beyond the scope of this report to discuss the measurement of Np.p in detail, except to recommene that the "sufficiently wide region of the chromatogram over which Np_p is measured be at least as wide as 3 base widths of the analyte peak, and that this regi0" contains the analyte peak. Various procedures have already been described for measuring Np_p in the presence of long-term noise or drift [ 1-4]. 2. If periodic fluctuations in the baseline are present, z value (of r) less than 5 should be used in eq. (6b). The values of r may be determined from previously derive~ relationships between Nrms and Np_p for various periodic signals [21]. If, for example, a triangular baseline observed (possibly resulting from flow pulsations in the detector cell due to an insufficiently dampened solvent delivery system), then r = 3.5 and SB = Np.p/3.5. Using the correct units
Furthermore, if the noise is sufficiently random and normally distributed (Gaussian), then r = 5 (ref. 20) and SB
A common misconception about chromatographic detec tion limits is that it is equally correct to report them ~, relative values with units of concentration as it is to rep0n them as absolute quantities with units of amount. Wha~ follows in this section is an attempt to clarify this misc0n ception. Dimensional analysis. One way of deducing the correcl units of the chromatographic LOD is by dimensi0na' analysis. Referring to the definition of the MD in Table li it is clear that if the appropriate units of concentration and mass flux are used for the MD in eqs. (3) and (4j for the concentration sensitive and mass sensitive detector cases, respectively, the units for the LOD in eqs. (3)an~ (4) must be in terms of an amount (i.e., moles, grams or some fraction thereof). Another approach via dimensional analysis is to consider the right-hand expression of eq. (1). Given the defmiti0n for the analytical sensitivity, it is clear that the units for this term (denominator of eq. (1) should be the quotient of the units of the measured signal and the units of the independent variable. Therefore, since the units for the noise expression (numerator of eq. (1)) are the same those for the measured signal, the units for the detecti0r limit should be the same as the units of the independen~ variable of the calibration curve. Thus, to decide whict units are correct for the LOD, we need only to identif! the independent variable of the calibration curve, i.e to determine whether the chromatographic signal depends on the concentration or amount of analyte injected. Equations (7) [9, 10] and (8) (inferred by analogy) bel0~ show that the signal (peak height, hp) is directly proportional to the maximum concentration, C m a x , d e t , or the
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Chromatographia Vot. 18, No. 9, September 1984
1. Most importantly, a true blank solution (sample analyte) would be difficult if not impossible to make! 2. It would be much too time consuming. 3. Variables which affect retention would require strict control. 4. The retention time of the analyte would need to be known very precisely. A much more practical procedure for measuring SB becomes apparent if one remembers that the standard deviation (root mean square) of a random (periodic) signal can be closely approximated by the quotient o f the range of signal values observed and a parameter, r, dependent on the type of signal, i.e., Ssignal = range/r
(6a)
Assuming that the signal of interest is the short-term noise (baseline fluctuations) on a chromatogram, the (short-term) peak-to-peak noise, Np_p, if measured over a sufficiently wide region of the chromatogram, is equivalent to the range in eq. (6a). Thus, the standard deviation of the noise, SB (equivalent to the root mean square noise, N r m s ) , is given by SB = N r m s =
Np.p/r
(6b)
Originals
maximum mass flux, Fmax, det, of the chromatographic peak flowing through the detector, which in turn are directly proportional to the amount of analyte injected, qinj:
hp oc Cmax ,det = qinj NI/2/( 27"01/2 VR
(7)
hp cc Fmax, det = qiniNl/2/(21r) 1/2 tR
(8)
Thus the independent variable for a chromatographic calibration curve is the amount (not concentration) of analyte injected and once again the same conclusion is reached: Chromatographic LODs should be reported as amounts, (i.e., in moles, grams, or some fraction thereof) andnot as concentrations! Identifying faulty logic. Despite the above arguments, some researchers insist on reporting their chromatographic LODs as concentrations. Assuming splitless injection, theirrationale might be as follows: 1. The amount of analyte injected, qini, is the product of the concentration of analyte in the sample, Cinj, and the volume of sample injected, Vinj : qinj = Cinj Vinj
(9)
2. The concentration of analyte injected, Cinj, is directly proportional to the amount of analyte injected (via rearrangement of eq. (9)): Cinj = qinj/Vinj
(10)
3. Conclusion: the relative LOD (in units of concentration), CL, is proportional by 1/Vinj to the absolute LOD, qL (in units of amount): CL = qL/Vinj
(11)
Though reached in a straightforward manner, the above conclusion is nevertheless false. The error in reasoning is best described as an improper or incomplete analogy. In going from a true expression, eq. (10), to a false statement, eq. (11), Cinj and qinj were replaced by two limiting quantities CL and qL, respectively. No analogous substitution was made for Vini, however, and therein lies the error. Vinj may vary continuously over 0 < Vinj < Vinj, max, where Vinj,max is a limiting, maximum injection volume to be discussed momentarily. Unless Vin j = Vinj,max, eq. (11) is false. A numerical example will help demonstrate the absurdity of eq. (11). Suppose the absolute LOD (qL) for an analyte had been determined independently by two scientists using the same LC system to be 1 x 10 -~2 mol. If the scientists had used different injection volumes of 5pL and 50pL, then according to eq. (11) the relative LODs (CL'S) for the same chromatographic system would be 2 x 10 -6 M and 2 x 10 -7 M, respectively. Clearly eq. (11) is inappropriate. The correct expression, eq. (12) below, is obtained by using Vinj,max in place of Winj. This expression is of little value, however, because of the difficulty in obtaining an accurate, precise estimate of Vinj, max : EL = qL/Vinj, max
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(12)
Originals
Problems with estimating the maximum injection volume. Experimentally, Vini,max can be determined by increasing Vini until column overloading or some other adverse phenomenon is observed. This operational definition of Vinj,max is unsatisfactory, however, because the injection volume at which these events occur is too dependent on the experimental conditions, such as the sample matrix, the percent loading of the column, etc. Alternatively, numerous theoretical expressions may be developed for the estimation of Vinj,max. Although many are only applicable for special cases, a few are completely general. Perhaps the best is one which relates Vinj,ma x to the maximum tolerable loss in resolution [22]. Our extension of this expression is reported below as eq. (13), and is derived in the Appendix to this Report. Vini, max = 2 (3) qz (y2 + 2 y) t/2 VR/N l/z
(13)
where y is the maximum tolerable loss in resolution due to a finite injection volume. Despite the appeal of eq. (13) (or other theoretical approaches), all theory-based estimations of Vinj,ma x have some serious shortfalls which may be summarized as follows: First, it seems unlikely that expressions based on differing criteria will predict the same or even similar values for Vinj,max- We would not expect theoretical expressions based on such divergent criteria as resolution loss and column overloading, for example, to yield convergent values for Vinj, maxSecond, the assumptions made for a given approach may not always be valid. The constant "2(3)q2" in eq. (13) resulted from the assumption of plug injection. Experiments have demonstrated, however, that this assumption is frequently invalid and that the value of this numerical "constant" varies considerably [9].
Finally, Vini, max may be highly sensitive to changes in a parameter related to the criterion itself. Referring again to eq. (13), if y = 0.04 (4 % loss in resolution), eq. (13) be comes Vinj, max ~ VR/N 1/2
(14)
If y were chosen to be 0.01 or 0.10, then Vinj,ma x would be approximately equal to 0.5 or 1.5 times the value predicted by eq. (14). Conclusion. Although the calculation of a relative detection limit from an absolute detection limit may be rationalized by eq. (12), the experimental and/or theoretical difficulties in obtaining an accurate estimate of the maximum injection volume preclude the use of this approach. In addition, dimensional analysis shows that chromatographic LODs are properly reported only as amounts (absolute quantities) and not as concentrations (relative quantities). From these perspectives it is clearly more meaningful to report chromatographic LODs as amounts rather than as concentrations. Converting to chromatographic reference conditions
As seen from eqs. (3) and (4), the LOD depends not only on the detector characteristics (MD) but also on three
507
chromatographic parameters [k, N, and VM (or tM) ] which characterize the column and the solute. In this section a method is proposed for taking the effects of these parameters into account, thereby solving the final problem associated with the chromatographic LOD (see Table I). For chromatographs with concentration-sensitive detectors, the relationship between the LOD and the chromatographic parameters is given by qL = dVM (1 + k)/N 1/2
(15)
where terms of eq. (3) [(2rr) 1/2, b, MD] have been incorporated into a single proportionality constant, d. Alternatively, since Va = VM (1 + k) and N = (VR/OV) 2, we may write qL = d V R / N 1 / 2
(16)
qL = dov
(17)
or
where VR is the retention volume (corrected for compressibility in GC) and Ov is the bandwidth of the chromatographic peak, in volume units. Equations (15)-(17) are equivalent, and any one of them may be used to derive an expression which accounts for the effects of these chromatographic parameters on the LOD. For simplicity, we shall use eq. (17). Consider an LOD, qL1, obtained under one set of chromatographic conditions, i.e., crv = oVl. By analogy with eq. (17), QL1 = d o v t
(18)
Likewise, for qL2, an LOD obtained under a second set of chromatographic conditions, (19)
qL2 = d O v 2
Dividing eq. (19) by eq. (18) and solving for qL2 yields qL2 = [OV2/OV 1 ] q L l
(20a)
which can also be written as qL2 = [VR2/VR1 ] IN1/N2 ] 1/2 qLl
(20b)
qL2 = [VM2/VM1 ] [(k2 + 1)/(k1 + 1)1
(20c)
or
[N1/N2 ]1/2 qLl
Eqs. (20a)-(20c) are important for two reasons: First, they can be used to predict the change in the detection limit when switching from one set of chromatographic conditions to another. Thus-they are useful to the analyst interested in lowering the detection limits via improvements in the chromatographic (rather than in the detection) aspects of the trace analysis. It should be noted, however, that the detection limits cannot be lowered ad infinitum by such improvements. Eventually the analysis will be optimized to the point where further improvements in the chromatography will necessitate a reduction in sample volume which offsets the chromatographic improvements [9]. Of course, in situations where very little sample is available, this reduction in sample size while maintaining a constant detection limit will be helpful.
508
Second, eqs. (20a)-(20c) can be used to compare detecti0: limits obtained under different chromatographic conditi0r. (bandwidths). Consider two such detection limits obtaine: using different concentration sensitive detectors. "[he~ detection limits can be compared only after one of t~ two detection limits is converted from its present yak:. obtained under a certain chromatographic bandwidth (z of conditions) to a value corresponding to the bandwidt! of the other LOD, or after both LODs are converted t: values corresponding to a ttdrd bandwidth. Whichever way is chosen, it should be recognized that t~,e final bandwidth chosen in the LOD conversion procedur~ serves as a reference state and that the initial bandwidth(s is(are), by definition, (an) experimental state(s). Th~ from eq. (20a) we may write, for chromatographs empl0! ing concentration sensitive detectors, qL, ref = [ OV,ref/Ov, exp ] qL, exp
(2h
where the subscripts "ref" and "exp" refer to reference and experimental, respectively. Similar equations resultiri from the incorporation of this notation into eqs. (20b) ani (20c) are easily inferred. The above derivation can be repeated in an analog0~ fashion for chromatographs with mass sensitive detectors The result is qL, tee = [ O't, ref/O't, exp ] qL, exp
(22,
where ot is the bandwidth of the peak in time units. Given eqs. (21) and (22), the analyst now has a method for comparing detection limits obtained under different ckr~ matographic conditions (states). Given two or more LOI)~ measured under different chromatographic states, the analyst merely converts all of them to values correspondini to a single, arbitrary, reference state. Though the choice of the reference state is arbitrary, thi selection of the reference state is nonetheless important For example, it would be counterproductive for an analys: to employ a different chromatographic reference state each time a new group of experimental LODs (measurei under different experimental states) were to be compared since LODs in separate groups could not be compared 1 this was done. For similar reasons, it would also be counteI productive if each analyst or group of analysts used differeni reference states. On the other hand, the reference states by definition mus~ be different for concentration sensitive and mass sensitive chromatographic systems. And in addition, typical chr~ matographic states (bandwidths) vary considerably depen6 ing on the physical state of the mobile phase (gas, super. critical fluid, or liquid) and on the type of column used (packed or open tubular). The logical compromise which we suggest is a fixed refer. ence state (bandwidth) to be used exclusively for each0! four chromatographic areas - conventional liquid chr~ matography (LC), microbore liquid chromatograph! (MLC), packed column gas chromatography (PGC), an~ open tubular column gas chromatography (OTGC)with each of the two types of detectors [concentrati0:
Chromatographia Vol. 18, No. 9, September 1984
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sensitive (eq. (21)) or mass sensitive (eq. (22))], giving a total of eight fixed references states. Furthermore, we propose that the LODs resulting from the conversion to these fixed reference states be referred to as standardized chromatographic LODs. To emphasize this, eqs. (21) and (22) may be rewritten as
Third, the standardized LOD facilitates the prediction of the value of the LOD under experimental chromatographic conditions. Rearranging eqs. (23) and (24), one obtains
qL, exp
=
[OV,exp/~ ref] qL,std
(25)
and
qL,std = [ OV,ref/OV, exp ] qL, exp
(23)
and qL,std = [Or,ref/Ot, exp ] qL, exp
(24)
respectively, where the standardized LODs are indicated bythe subscript "std". Note that we have purposely omitted supercritical fluid chromatography (SFC) and open tubular liquid chromatography (OTLC) from this discussion both for the purpose of simplicity and because these areas have not matured sufficiently to lend themselves to this reference state treatment. We also chose not to have a separate category for high-speed liquid chromatography because this area employs concentration-sensitive detectors almost exclusively, and the volume bandwidths important for these detectors will usually be within a factor of two of the conventional LC reference volume bandwidth. Differences in opinion regarding these omissions may exist, of course. It may become necessary to develop chromatographic reference states for these and other areas we have chosen to omit. Our focus, however, is not on the number of chromatographic reference states that should be employed, but on the reference state concept itself and the resulting standardized chromatographic LODs, the benefits 0fwkich will be discussed presently.
Advantages o f the chromatographic reference state concept and the resulting standardized chromatographic LODs. If the values for the reference states are chosen judiciously, the resulting standardized LODs will be superior to experimental (hereafter termed conventional) LODs in several ways: First, standardized LODs generally represent a more realistic measure of the trace analysis capabilities of a given chromatographic system, thus permitting an analyst to decide whether or not a particular application reported in the literature will be feasible with his or her system. Conventional LODs, on the other hand, may be very misleading. As evidenced in the literature, many trace analyses have been performed using chromatographic systems which have only been partially optimized, if at all. As a result, unusually pessimistic (very high) LODs are obtained. In contrast, some overly optimistic (very low) LODs have been reported in some instances where the particular chromatographic systems have been optimized more than the typical system could have been. Second, the use of standardized LODs permits a fair comparison of trace analysis chromatographic systems in different areas, e.g., OTGC vs. LC, or with different types of detectors (mass vs. concentration sensitive). Such a comparison is not valid if conventional LODs are used, evenwithin a given area using the same type of detector.
Chr0matographia Vol. 18, No. 9, September 1984
Originals
qL,exp = [Ot, exp/Ot,ref] qL, stcl
(26)
Assuming qL, sta has been reported, one merely needs to substitute values for Ov, exp(Or Ot.exp) in eq. (25) or (26) in order to calculate qL,exp. Fourth, the standardized LOD can serve not only as a figure of merit for the overall chromatographic analysis, but as a figure of merit for the detector as well. The need to measure the minimum detectability (i.e., to independently characterize the detector) would therefore be eliminated, resulting in a considerable savings of time. Numerically defining the reference states. Values for the proposed chromatographic reference states (bandwidths) were determined by assigning reference values to the component parameters [VM (or tM), k, and N] and then calculating the reference bandwidths using eq. (27) or (28):
OV,ref
=
VM,ref(1
+
kref)/(Nref) 1/2
(27)
Ot, ref = tM, ref(1 + kref)/(Nref) 1/2
(28)
The suggested values for kref, Nref, VM,ref, and tM,re f are listed in Table IV. They were selected according to one or more of the following criteria: 1. The value represents an intermediate chromatographic performance which is easily achieved except under unusual circumstances. 2. The value falls within a range of values reported directly in the scientific and trade literature. 3. The value falls within a range of values calculated from column manufacturer specifications. 4. The value fails within a previously recommended range (k only - see ref. [23]). Despite our best attempts to use logical criteria in choosing appropriate values for kref, Nref, VM,ref, and tM.ref, we recognize that differences in opinion concerning the criteria or the values selected may exist. We welcome suggestions for adjustments to these values, particularly for those changes which significantly affect the values of the overall chromatographic reference states (bandwidths), i.e., for those changes in kref, Nref, VM,ref, and tM,re f which do not offset themselves in eq. (27) or (28). Table IV. Suggested values for the component parameters of the chromatographic reference states a parameter
LC
MLC
PGC
OTGC
kre f
4
4
4
4
Nre f
10,000
10,000
10,000
90,000
V M , re f (mL)
1.5
0.2
3
2
tM,ref (min)
1
1
0.5
1
LC = conventional liquid chromatography, MLC = microbore liquid chromatography, PGC = packed column gas chromatography, and OTGC = open tubular column gas chromatography.
509
Table V. Proposed standardized LOD equations and the suggested chromatographic reference states
q L,std = [GV,ref/UV ,exp] q L,exp
concentration sensitive detector mass sensitive detector
qL,std = [at,ref/Ot,exp] q L , e x p
OV,ref(mL) b Ut,ref (min.) c
The reconciliation is summarized in Table Vl, th0u~ readers who wish to perform the calculations will needl refer to conditions specified in Table II. The progress oftL LOD reconciliation in Table VI may be noted by inspect~ either the LOD values themselves in the second and tt/ columns, or their ratio in the fourth column, given: orders of magnitude. The degrees to which the gi~: problems are responsible for the initial discrepancy betwee: the LODs are shown in the fifth column; they are 0~ rained by subtracting successive values in the fourth colun~ Three steps were performed in the reconciliation. First, ti LOD values reported incorrectly in units of concentratic: (molL-1 ) are converted to the appropriate units of am0u: (mol) by multiplying by the corresponding sample injectic: volumes (Vinj'S). Second, these unit-corrected LODs a: then converted to values consistent with the IUPAC m0d~ previously discussed (qL = 3sB/S). For case A, since Np( 5s B (as discussed earlier in Choosing a Model), the t02, must be reduced by a factor of 50/3. For case B, sine: Nrms = SB, no adjustment is necessary. Finally, using tk: appropriate equation and reference state in Table V a~: the experimental bandwidths given in Table II, the IUPAI consistent LOD values are converted to standardize: LODs, thereby adjusting for differences in the experimentz chromatographic conditions. As seen in Table VI, the discrepancy between the LODs: reduced significantly with every successive stage. It sh0~: be noted that the largest source of discrepancy is due t differences in the experimental chromatographic conditi0: (bandwidths). This demonstrates the need for a standardize! chromatographic limit of detection. Finally, the LODst the bottom row of Table VI are identical, indicating th the reconciliation has been completely successful.
LC a
MLC a
PGC a
OTGC a
0,075 0.050
0,010 0.050
0.150 0.025
0.033 0.017
key to abbreviations given in Table IV. b calculated via equation (27) using values from Table IV. c calculated via equation (28) using values from Table IV.
The proposed chromatographic reference states and corresponding standardized LOD equations are shown in Table V. Table V and the advantages enumerated earlier conveniently summarize the bulk of the material presented in this section. Only one additional point needs to be made: experimental bandwidths are best measured directly using equations such as Oexp = Wb/4
(29)
O~xp = Woa/4.292
(30)
or
where Wb and Wo. 1 represent the base width and the width at 10% peak height, in units of volume or time, whichever is appropriate. Alternatively, the bandwidths can be calculated from their component parameters iN, k, and V M (or tM)], but this is disadvantageous in two respects. First, it requires at least three measurements instead of one. Second, different expressions and/or different interpretations are required for gradient (temperature or mobile phase) elution conditions.
Conclusion
The numerical example - revisited
Our comments and reconunendations for obtaining meanir! ful chromatographic detection limits are summarized beI0'~ We hope that some, if not all, of these guidelines willb. put into practice in order to make the limit of detecti0: a more reliable figure of merit in chromatography.
We return to our hypothetical LOD example (Table 1I and associated text) to reconcile the large differences in the reported detection limits. Recall that since identical chromatographic detectors were employed, the huge discrep. ancies were attributed solely to problems 4 - 6 of Table I. These problems will now be eliminated one at a time using solutions proposed in this report. Identical detection limits will be obtained at the conclusion of this reconcilation, thus validating quantitatively our LOD example and demonstrating the efficacy of our solutions.
1. The limit of detection (LOD) should not be confu~: with the minimum detectability (MD), the (analytic/ sensitivity (S), or the detector sensitivity (Sd).
Table V l . Reconciling the differences in the detection limits from the numerical example in Table II a
Step 0
Initial
1
Amount
2
IUPAC def'n
3
qL,std
[LOOA7 &10~
LOD A
LOD B
log L L O D B j
8.7 X 1 0 - 6 M
4.5 X 1 0 - 9 M
3.3
44
pmoi
90 fmoi
2.7
2.6 pmol
90 fmol
1.5
225 fmol
0.0
225
fmol
0,6 1,2 1.5
a Values in this table are reported to more than one significant figure for purposes of illustration only. Detection limits are normally reported to only one significant figure.
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Chromatographia Vol. 18, No. 9, September 1984
Originals
2. Chromatographic detection limits should be calculated using the IUPAC and/or the error propagation model(s). The calibration curve should be constructed from a plot of signal versus amount (not signal versus concentration) of analyte injected, thus ensuring that the resulting LODs will be reported properly as amounts (and not as concentrations)! 3. If obtained under non-standard chromatographic conditions, the detection limits should be standardized using the equations in Table V. Standardized chromatographic LODs are superior to their conventional (nonstandardized) counterparts for several reasons, in particular because they permit the valid comparison of trace analysis chromatographic systems in different areas (conventional LC, microbore LC, packed column GC, and open tubular column GC) and/or with different types of detectors (mass and concentration sensitive).
Appendix - D e r i v a t i o n o f Vinj, ma x
We begin with a previously derived expression [16] which relates the peak volume observed for a finite size sample, Vw, to the volume of injected sample, Vinj, and to the peak volume observed for a very small sample, Vp Vw 2 = V~ + 4/3(Vini) 2
(A.1)
Let Vw = (y + 1)Vp
(A.2)
where y = loss in resolution due to a finite injection volume. Substituting eq. (A.2) i n t o eq. (A.1) and solving for Vinj yields Vinj = [ 3 / 4 ( y 2 + 2y)] 1/2 Vp
(A.3)
We now express Vp in terms of N, k, and VM using eqs. (A.4)-(A.6), assuming Vp = Vb, the volume corresponding to the base width of the peak. N = (VR/oV) 2
(A.4)
v~ -- 4 o v
(A.5)
VR
(a.6)
=
VM (1 + k)
lho result is Vp = 4VM(1 + k)/N u2
(1.7)
Substitution of eq. (A.7) into eq. (A.3) yields the expression Vinj = [3/4(Y 2 + 2y] u2 4VM (1 + k)/N u2
(A.8)
If y is re-designated to be the maximum tolerable loss in resolution, then eq. (A.8) becomes, upon rearrangement, Vinj,max = 2 ( 3 ) 1/2 (y2 + 2y)1/2 VM (1 + k)/N 1/2
(A.9)
which is the desired expression (eq. (13)).
Chromatographia Vol. 18, No. 9, September 1984
OriginaLs
Acknowledgement JPF gratefully acknowledges partial support of this work by a 1983 ACS Analytical Division Fellowship sponsored by the Society of Analytical Chemists of Pittsburgh. JGD gratefully acknowledges Eli Lilly and Company and the Alcoa Foundation for support of this research. This work was presented in part at the 186th National Meeting of the American Chemical Society, Washington, DC, September 1, 1983.
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