Peak-valley ratio

Simplex Optimization Criteria. For chromatographic optimization, it is necessary to assign each chromatogram a numerical value, based on its quality, which can be used as a response for the simplex algorithm. Chromatographic response functions (CRFs), used for this purpose, have been the topics of many books and articles, and there are a wide variety of such CRFs available (33,34). The criteria employed by CRFs are typically functions of peak-valley ratio, fractional peak overlap, separation factor, or resolution. After an extensive (but not exhaustive) survey, we... 

Like CRF-1, CRF-2 also favors short analysis times and well resolved peaks. However, there is no threshold value for resolution, and the compromise between resolution and analysis time is not as well defined as in CRF-1. Inclusion of analysis time in the denominator of an objective function may result in the loss of some resolution, compensated for by a rapid analysis time (51). This is likely to occur to some extent when the resolution threshold is defined in excess of 1 to 1.25, since the peak-valley ratio utilized by CRF-2 does not diminish to an appreciable extent until the resolution falls below this range. Note that as the resolution drops below 1 to 1.25, however, CRF-2 decreases rapidly, and it is unlikely that a short analysis time will compensate for such poor resolution (33). Nevertheless, if a user-specified minimum resolution is an absolute requirement, it is probably better to use a threshold criterion such as CRF-1 in which the desired resolution is stipulated by the user. 

Figure 2. Measurement of the peak valley ratio used in several chromatographic response functions (equations 6 and 7).

Three definitions of peak-valley ratios are illustrated in figure 4.2. All of them express the extent of separation as some measure of the depth of the valley between two peaks divided by some measure of the peak height. The first criterion (P) measures the depth of the valley relative to the interpolated peak height as shown in figure 4.2.a. The corresponding expression is ... 

The second peak-valley ratio was suggested by Schupp [403] and is illustrated in figure 4.2.b. In this criterion, which we will refer to as the median peak-valley ratio (PJ, thedepth of the valley at a point midway between two successive peaks (fm) is measured relative to the average peak height (gj which equals the interpolated peak height at that point. The corresponding equation is ... 

Figure 4.2 Three definitions for peak-valley ratios as elemental criteria to quantify the extent of separation between a pair of adjacent peaks in a chromatogram, (a) Peak-valley ratio (P eqn.4.3) according to Kaiser, (b) median peak-valley ratio (Pm eqn.4.4) according to Schupp and (c) (opposite page) the valley-to-top ratio (P eqn.4.5) according to Christophe. 

The peak-valley ratios vary from zero for separations where no valley can be detected, to unity for complete separation. It ought to be noticed that a P value equal to zero does not necessarily imply that two solutes elute with exactly the same retention time (or k value). There is a threshold separation below which the presence of two individual bands in one peak only leads to peak broadening or deformation, without the occurrence of a valley. In these cases Rs values are indeed not equal to zero, because by definition (eqn.1.14) Rs is proportional to the difference in retention times. 

For Gaussian peaks of equal height the value of the peak-valley ratio (then the same according to all three definitions) can readily be expressed in terms of Rs. This can be done by relating the parameters f, g and v (see figure 4.2) to the parameters that describe a Gaussian peak (trand h). For the first of a pair of Gaussian peaks (peak A) we can write... 

A comparison of various elemental criteria has been reported by Knoll and Midgett [412] and by Debets et al. [413]. Figure 4.4 shows the variation of some of the criteria for the separation of pairs of chromatographic peaks as a function of the time difference between the peak tops (At = t2 — t,). By definition, Rs (and hence S) varies linearly with At. The peak-valley ratios (P) and the fractional overlap both increase rapidly with increasing At at first, but level off towards At 4 ct to reach the limiting value of 1. At high values of At, Rs and S will keep increasing, while the other criteria will not. 

For practical evaluation FO is a very unattractive criterion. Its variation with At and with the peak area ratio A is similar to that of the peak-valley ratio Pv. Pand Pmare similar to each other in all respects. Pm may be obtained from the chromatogram slightly more easily than P, because it only requires location of the peak tops, and not of the valleys. To calculate Rs from the chromatogram an estimate of N is required. S can be estimated very easily, using only the retention times of individual peaks. 

Below a certain threshold resolution, no valley can be observed between two adjacent peaks in a chromatogram. In that case the value for any of the peak-valley ratios would equal zero. In theory, the value for Rs and S would exceed zero for any two peaks that have different retention times (At > 0). In practice, this difference vanishes if the presence of two peaks cannot be discerned from the chromatogram. However, the occurrence of ill-resolved peaks in a chromatogram may be recognized visually at resolutions well below 0.6 (the threshold value below which P equals zero for Gaussian peaks of equal height) (see ref. [401], figure 2.11, p.38). Moreover, there are several techniques which may be of... 

Figure 4.6 Variation of the sum of peak-valley ratios as a function of the number of plates for the two chromatograms (a and b) shown in figure 4.1, and for a third chromatogram (c), shown in figure 4.8. P was calculated from eqn.(4.10). Negative values for P were set equal to zero. The sum of resolution values is shown as a dashed line for chromatogram a only.

The use of the sum of logarithms may have a slight disadvantage in the case where a value of zero occurs for one of the pairs of peaks. If any of the peak-valley ratios (P, Pm or Pv) is used, then this problem is aggravated because these criteria take on a value of zero below a certain threshold resolution. The obvious way around this problem, however, is to set the sum of logarithms equal to minus infinity or to a large negative number once a value of zero occurs. 

All product criteria will be zero if any single pair of peaks is completely unresolved. For FO, Rs and S this situation theoretically only occurs if the retention times of two peaks are equal. For peak-valley ratios a value of zero is estimated from the chromatogram below a certain threshold separation, which for Gaussian peaks corresponds to Rs < 0.59... 

Data for capacity factors, elemental criteria and for criteria judging the extent of separation in the entire chromatograms. Chromatograms are shown in figure 4.8. Criteria for pairs of peaks separation factor (S, eqn.4.15), resolution (Rs, eqn.4.14) and peak-valley ratio (P, eqn.4.10). 

The square root should now be incorporated in all product criteria, i.e. also in the case in which IIP or IIPm is used instead of T1PV. This is because it is now now a sensible convention to incorporate two values for a peak-valley ratio (be it P, Pm or Pv) into the criterion for every peak of interest. If we did not follow this convention, then a different situation would exist if two relevant peaks were adjacent (yielding one combined value for the peak-valley ratio) or separated by an irrelevant peak (which would lead to two different... 

If peak-valley ratios are used as elemental criteria, then the separation between the first peak and the (imaginary) preceding one, as well as the separation between the last peak and the (imaginary) following one, may readily be characterized by a P value of one. The retention time of the last peak, which may be used in combination with a product of P values (see table 4.13) refers to the last appearing peak in the chromatogram, no matter whether or not this is a relevant peak. 

If a product of peak-valley ratios is used as the optimization criterion, then two values would need to be used for every peak, one describing its resolution from the preceding peak in the chromatogram and the other one describing its resolution from the following peak. Because a product criterion is used, the weighting factors (g) will appear as powers in the product. If we assume the weighting factors to be positive, we may write... 

This product of peak-valley ratios can be normalized to the sum of weighting factors, so that the true value of the resolution product is less obscured by the arbitrarily selected values for g ... 

Figure 4.14 A typical chromatogram containing a solvent peak followed by three small [teaks, h, and v, can be used to estimate the peak-valley ratio of the first peak (see figure 4.2.C).

The sum of peak-valley ratios was used as the resolution term in a composite optimization criterion, which otherwise corresponds to eqn.(4.30). Berridge also added a term to describe the contribution of the number of peaks (n). With this, the complete optimization criterion became... 

For criteria based on the peak-valley ratio (P) no modification of the criterion used for isocratic optimization may be necessary (see section 4.6.2). 

It can be seen in the chromatogram of figure 6.11 that four peaks (the three antioxidants plus an unknown impurity) are amply resolved to the baseline. This implies that all values for the peak-valley ratio P are equal to 1 and that the criterion has become very insensitive to (minor) variations in the resolution between the different peak pairs. In the area of the parameter space in which four well-resolved peaks are observed, the only remaining aim of the optimization procedure is to approach the desired analysis time of 4 minutes. The irrelevance of the minimum time tmin is illustrated by the occurrence of the first peak in figure 4.9 well within the value of 1.5 min chosen for this parameter. 

Fig. 6... Two elemental p criteria (a) peak-valley ratio. P = f/g (b) valley-to-peak ratio, />,.= — where represents either /ii or /ij depending on the substance of main interest.

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