Anal. Chem. 1997, 69, 2976-2979
Critical Peak Resolution in Multicomponent Chromatograms Attila Felinger
Department of Analytical Chemistry, University of Veszpre´ m, Egyetem utca 10, Veszpre´ m, H-8200 Hungary
The critical degree of peak overlap is determined for multicomponent chromatograms. As the heights of adjacent peaks significantly differ, no single value of critical peak resolution can be used in multicomponent separations; one has to determine the distribution of the critical peak resolutions, taking into account the peak height distribution. An analytical expression is derived for the critical peak resolution, from which one can obtain the average or maximum probable peak resolution by integration. The theoretical model can be applied to any peak height distribution with ease.
exponential peak height distribution. Very recently, Davis8 has used a Monte Carlo simulation to generate multicomponent chromatograms with exponential peak height distribution, and obtained an estimated histogram for the distribution of the critical peak resolution. He found that the critical resolution is bound between 0.5 and 1.2, the average resolution being Rs ) 0.726. In this study, a theoretical justification is given for the distribution of the critical peak resolution. The expressions are derived for uniform and exponential peak height distributions, but the model can rather simply be extended to any otherseven empiricalspeak height distribution.
Peak overlap in multicomponent chromatograms gained a lot of attention recently. The statistical theory of peak overlap developed in the early 1980s by Davis and Giddings1 was a landmark step to draw attention to the previously unexpected degree of peak overlap one may witness during a multicomponent separation. The statistical theory of peak overlap has been further developed, and, at present, it is able to take into account different models of interval distributions between adjacent peaks,2 peak density inhomogeneities,3 superposition effects,4 etc. In spite of the extensive progress in the theory peak overlap, one basic issue has remained unanswered: the critical resolution below which one is unable to recognize the fact of peak overlap. Peak capacity and other chromatographic properties are usually defined assuming that the critical resolution at which the valley between the two peaks disappears is Rs ) 0.5. That value is the lowest possible resolution, and it is sufficient for peak resolution only if the component peaks are of identical height. Obviously, much higher resolution is required when the concentrations of the adjacent components differ, because small peaks can easily hide below the “skirt” of the higher peaks. The minimum retention time increment to achieve separation can be determined as the function of peak height ratios. Based on numerical simulations, El Fallah and Martin5 concluded that the average resolution Rs ) 0.71 is required when the concentration of the individual components present in the sample follows an exponential distribution. Creten and Nagels6 found a slightly higher average peak resolution, Rs ) 0.8, with abundant numerical simulations, based on an empirically determined peak height distribution. Schure7 determined the critical resolution as Rs ) 0.678 for uniform peak height distribution and Rs ) 0.725 for
THEORY The critical resolution is searched for in the case of overlapping chromatographic peaks. The chromatographic peaks are assumed to be symmetrical and modeled with Gaussian functions given by
(1) Davis, J. M.; Giddings, J. C. Anal. Chem. 1983, 55, 418-424. (2) Pietrogrande, M. C.; Dondi, F.; Felinger, A.; Davis, J. M. Chemom. Intell. Lab. Syst. 1995, 28, 239-258. (3) Davis, J. Anal. Chem. 1994, 66, 735-746. (4) Felinger, A. Anal. Chem. 1995, 67, 2078-2087. (5) El Fallah, M. Z.; Martin, M. Chromatographia 1987, 24, 115-122. (6) Creten, W. L.; Nagels, L. Anal. Chem. 1987, 59, 822-826. (7) Schure, M. R. J. Chromatogr. 1991, 550, 51-69.
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(
fi(t) ) hi exp -
)
(t - mi)2 2σi2
(1)
For the sake of simplicity, it is assumed that the width of adjacent peaks is constant (σ ) σ1 ) σ2), and the first peak of the pair is located at the origin (m1 ) 0). As only the relative peak heights determine the critical resolution, the equation for the two peaks to be investigated can be formulated as
( )
f(t) ) f1(t) + f2(t) ) exp -
(
)
2
(t - m) t2 + h exp 2 2σ 2σ2
(2)
where h ) h2/h1 is the ratio of the two peak heights. With the above notation, the peak resolution is expressed as
Rs )
m2 - m1 2(σ1 + σ2)
)
m 4σ
(3)
Figure 1 helps to illustrate how the critical resolution can be determined. The left-hand side column of the figure shows two fused peaks, as well as the first and the second derivatives of the signal. A “bump” on the descending part of the peak indicates the presence of the second peak, but the resolution is insufficient for quantitative analysis. The first derivative possesses a maximum at the inflection point of the signal, and accordingly, the second derivative is zero at that point. If we increase the resolution up to the critical one, a “shoulder” appears on the signal. (8) Davis, J. M. Chromatographia 1996, 42, 367-377. S0003-2700(97)00241-2 CCC: $14.00
© 1997 American Chemical Society
t)
m ( xm2 - 4σ2 ) 2Rsσ ( σx4Rs2 - 1 2
(10)
Substituting the positive root for t into the first derivative of the overlapping peaks, and rearranging that expression for h, we can express the relative peak height for the critical resolution as
h ) (x4Rs2 - 1 + 2Rs)2 exp(-4Rsx4Rs2 - 1)
Figure 1. Illustration of different resolutions. Two overlapping peaks (upper row) and their first (middle row) and second derivatives (lower row) are shown. Worse than critical (first column), critical (second column), and better than critical (third column) resolutions are shown, respectively.
That singular resolution is the transition between fused peaks and valley resolution. At that particular resolution, both the first and the second derivatives of the signal are zero. When the resolution is further increased, the valley emerges between the two peaks, and only the first derivative is zero at the valley point. Accordingly, the critical resolution, i.e., the shoulder limit, is found when both the first and the second derivatives of the signal are equal to zero.
d[f1(t) + f2(t)] df1(t) df2(t) ) + )0 dt dt dt d2[f1(t) + f2(t)]
d2f1(t) )
dt2
dt2
d2f2(t) +
dt2
)0
(4)
(5)
The first and the second derivative of the Gaussian peak are
t - mi dfi(t) )fi(t) dt σ2
(6)
i
Equation 11 relates the relative peak heights and the critical resolution. The plot of this function is given in Figure 2 as the thick solid line. Since the peak widths are identical, the critical resolution is the same for the relative peak heights h and 1/h. Accordingly, it is sufficient to regard the relationship in the h ) 0-1 region or, alternatively, in the h ) 1-∞ region. We shall utilize this symmetry and restrict ourselves to the former domain; i.e., we determine the relative peak height as
h)
d2fi(t) dt2
(t - mi)2 - σi2 ) fi(t) σi4
(7)
respectively. We can combine eqs 4 and 5 into
d2f1(t)/dt2
df1(t)/dt df2(t)/dt
)
d2f2(t)/dt2
(8)
When the derivatives of the overlapping Gaussian peaks are substituted into eq 8, we get a simple equation to solve:
t2 - mt + σ2 ) 0 Solving the above equation for t yields
(9)
min(h1,h2) max(h1,h2)
(12)
In order to calculate the resolution for a multicomponent chromatogram, we have to know the peak height distribution. The distribution of the relative peak heights must be determined from the peak height distribution. The simplestsbut not very realisticsassumption is that each component has equal chance to take any concentration in the sample; i.e., the peak heights follow a uniform distribution. It can be shown that the relative peak height also follows a uniform distribution in that case.9 In this instance, the relative peak heights are distributed uniformly. The distribution of the critical resolutions can be determined graphically by projecting the points of the vertical axis of Figure 2 onto the horizontal axis via the h(Rs) function. The density of the points on the horizontal axis will determine the distribution of the critical resolutions. For the analytical derivation of the distribution for the case of uniform peak height distribution, we have to differentiate eq 11 according to Rs, and the distribution will be given by
ru(Rs) ) -
and
(11)
dh ) dRs
8x4Rs2 - 1(x4Rs2 - 1 + 2Rs)2 exp(-4Rsx4Rs2 - 1) (13) The negative sign in the above equation is required because h(Rs) is a decreasing function, and being such it may be considered as a survival function rather than a cumulative distribution function. In Figure 3, ru(Rs) is plotted, and the result of a Monte Carlo simulation is also shown to confirm the validity of eq 13. The Monte Carlo simulation was carried out by generating five million uniformly distributed random number pairs for the peak heights, and the critical resolution was calculated for those peak heights. Finally, a histogram was calculated from the results of the simulation. In a real multicomponent chromatogram, however, peak heights are distributed according to an exponential distribu(9) Feller, W. An Introduction to Probability Theory an its Applications; John Wiley & Sons: New York, 1966; Vol. II.
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Figure 2. Plot of the relative peak height against the critical resolution, as given by eq 11.
Figure 4. Plot of the distribution of the critical resolution for the case of exponential peak height distribution. Theoretical model of eq 18 (line), and the results of numerical calculations (dots).
that smaller relative ratios, i.e., large peak height differences, have higher probabilities, and the distribution of relative peak heights should be remapped in order to determine the distribution of the critical resolutions. The purpose of the remapping is to have uniformly distributed relative peak heights when projecting onto the peak resolution axis. After this scale transformation, we will have uniformly distributed peak height ratios, and the probability density function of the critical resolutions can be determined with the same manner as in the case of uniform peak height distribution. For the scale transformation of the relative peak heights, we have to determine the inverse of the cumulative distribution function given in eq 15
P-1(h) )
Figure 3. Plot of the distribution of the critical resolution for the case of uniform peak height distribution. Theoretical model of eq 13 (line), and the results of numerical calculations (dots).
tion.5,7,10 That means that only a few components have high concentration in a sample, and the number of low-concentration components is increasingly higher. A great number of components have rather small amounts present in the sample, and they are lost in the baseline noise. As the relative peak heights determine the critical resolution, we have to find the distribution of the ratio of exponentially distributed random numbers. It can be shown7,9 that the probability density function of the ratio of exponentially distributed random numbers is
p(h) )
1 (1 + h)2
(14)
∫
2h 2 du ) P(h) ) 0 2 h +1 (1 + u)
(15)
It is obvious from the probability density function given in eq 14 (10) Dondi, F.; Pietrogrande, M. C.; Felinger, A. Chromatographia 1997, 45, 435-440.
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(16)
Then, eq 11 should be replaced by the following equation in the case of exponentially distributed peak heights, in order to have a uniform relative peak height density along the vertical axis of Figure 2.
h ) (x4Rs2 - 1 + 2Rs)2 exp(-4Rsx4Rs2 - 1) 2-h
(17)
Equation 17 is solved for h then, and, similarly to the process followed in the case of uniform peak height distribution, the distribution of the critical resolutions is determined by differentiation.
re(Rs) )
Since we are concerned about the h ) 0-1 region only, p(h) must be multiplied by 2 for our calculations, as half of the relative peak heights are lower than 1. The cumulative distribution function of the relative peak heights is h
h 2-h
16x4Rs2 - 1(x4Rs2 - 1 + 2Rs)2 exp(4Rsx4Rs2 - 1) [(x4Rs2 - 1 + 2Rs)2 + exp(4Rsx4Rs2 - 1)]2
(18)
Equation 18 is the relationship that determines the distribution of the critical peak resolution in a multicomponent chromatogram. re(Rs) is plotted in Figure 4. The function perfectly matches the numerical results of the Monte Carlo simulation given by dots. DISCUSSION Equation 18 allows the numerical calculation of some important values for the peak resolution in multicomponent chromatograms.
The median of re(Rs) is found at Rs ) 0.711; i.e., 50% of the peaks will require higher critical resolution, and for the other half of the peaks, that resolution will suffice. That value is identical to the average resolution calculated by El Fallah and Martin.5 The distribution of Rs is, however, asymmetrical, and accordingly the mean valuesor expected valuesdiffers from the median, Rs ) 0.725. That latter value, on the other hand, equals the critical resolution indicated by Schure7 and Davis.8 The calculation of the area below the distribution shows that 99% of the critical resolutions are above Rs ) 0.515, or conversely below Rs ) 1.069. That is, on one hand it is practically very improbable that overlapping peaks have identical heights; accordingly, it is very unlikely (less than 1%) that adjacent peaks will resolve below Rs ) 0.52. On the other hand, as there is also a limited probability that the heights of adjacent peaks differ by several orders of magnitude: Rs ) 1.07 will resolve 99% of the individual peaks, provided that they are equidistantly distributed along the time axis. The theoretical model derived in this study allows the definition of peak capacity according to the aims of a chromatographer. Obviously, the use of Rs ) 0.5 is an impractical lower limit. Using the average resolution Rs ) 0.725, derived for exponential peak height distribution, the space occupied by one peak is 2.9σ. A surprising result is the value of Rs ) 1.07, which is required for the resolution of 99% of the peaks. At that value, when peaks are 4.28σ apart, we have almost baseline separation for peaks of equal (11) Davis, J. M. Chromatographia 1997, 44, 81-90.
height. In a multicomponent chromatogram, however, 1% of the peaks will still be fused, even if the retention time increments are constant. Using the above procedure, the critical peak resolution can be determined for any peak height distributionseither theoretical or empiricalsprovided that one is able to define the inverse cumulative distribution of the relative peak heights. In a very recent study, Davis extended the analysis of peak overlap to groups of several fused peaks and determined the critical resolution between adjacent clusters.11 The critical resolutions derived by that approach are different from the ones given here. The model derived in this study, however, can be utilized to further enhance the study of cluster overlap. ACKNOWLEDGMENT The author acknowledges Joe M. Davis (Southern Illinois University at Carbondale) for many stimulating discussions. This work was supported in part by Grants F15700 from the Hungarian National Science Foundation (OTKA) and MKM 332/1996 from the Hungarian Ministry of Culture and Education.
Received for review March 4, 1997. Accepted May 9, 1997.X AC970241Y X
Abstract published in Advance ACS Abstracts, June 15, 1997.
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