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3. When appropriate retention models are available, it is possible to predict chromatograms within a given range of parameter values. In this way, laboratory experiments are replaced by computer simulations that require only a fraction of the time to perform.
4. When it is known how the separation is affected by variations in the experimental conditions, the stability or ruggedness of the separation under the selected conditions can be estimated. In addition, it is often possible to predict how to change in instrumentation will affect the separation, thus assisting in technology transfer.
The foregoing considerations will be illustrated in the following sections. In Section IV. A we will emphasize the general principles and terminology underlying most computer-assisted approaches. One specific issue, the evaluation of chromatograms, will be dealt with in Section IV. B. Algorithms and applications involving gradient elution are discussed in Section IV. C, and the optimization of isocratic conditions is explored in Section IV. D.
A. General Considerations
During every optimization process, three aspects need to be considered: (1) the definition of the parameter space, (2) the definition of the response, and (3) the approach followed to search the parameter space for the conditions that result in the maximum response. These three elements are also encountered during the optimization of selectivity in liquid chromatography, as illustrated in Fig. 7 and 8.
The first step in the optimization process is the selection of the appropriate parameters. Some of the parameters typically considered in liquid chromatography are listed in Table 1 . Typically, the selection is based on some knowledge of the sample. This may include analyte polarity, presence of ionizable groups, and such, as well as constraints imposed by the results of the method-selection step (Fig. 6). For instance, if the separation will be based on differences in the hydrophobicity of nonionizable compounds, the selected parameters for a reversed-phase separation will most likely be the type and percentage of organic modifier. If the presence of ionized compounds is expected, the type and concentration of an ion-pairing
Figure 7 Example of a one-parameter optimization using an interpretive approach. By using the retention times observed in two chromatograms (numbered 1 and 2) and applying a linear model, retention times at other mobile-phase compositions are predicted. The quality of the predicted chromatograms is quantified using a criterion (minimum resolution in the chromatogram). This results in the response surface shown in the top panel. (From Ref. 155.)
reagent are appropriate parameters. The choice is ultimately made on the basis of whatever knowledge is available about the sample composition, combined with the experience of the chromatographer. A first-guess expert system may also be used.
The number of optimization parameters considered is an important characteristic because the amount of experimentation required to describe the retention behavior in a given system often increases exponentially with that number. Figure 7 illustrates an example of a one-parameter optimization. It shows how the retention of a set of compounds is expected to vary as a function of the mixing ratio of two mobile phases. Note that the dimensionality of this problem is 1, the only parameter being the fraction of mobile-phase 2 in the mixture. Figure 8 shows an example of a two-parameter optimization, taken from the area of micellar liquid chromatography (MLC) . The two parameters are the concentration of the surfactant and the amount of organic modifier, 2-propanol. Because these two concentrations can be selected independently from each other, the dimensionality is 2.
The selection of the parameters is followed by the definition of a range of values that will be considered. The limits will be defined based on the expected behavior of the sample and practical considerations. Again, expert systems can assist less-experienced chromatogra-phers. The set of all possible combinations of the parameter values evaluated in the optimization is generally referred to as the parameter space. Every point in the parameter space uniquely defines a set of chromatographic conditions that correspond to a chromatogram.
For example, the limits in Fig. 7 for the fraction of mobile phase 2 are 0 and 1, corresponding to pure mobile-phase 1 and pure mobile-phase 2, respectively. The parameter space
Computers and Liquid Chromatography
Figure 8 Example of (part of) a two-parameter optimization using an interpretive approach. The parameter space is defined by the concentrations of 2-propanol (%PrOH) and that of a cationic surfactant ([ÑÒÀÂ]). The bottom panel shows the expected retention behavior of two components in a section of the parameter space, based on three observations (dashed lines). The top panel shows the resulting response surface defined by a criterion (minimum resolution in the predicted chromatogram).