Univariate optimization

The complexity of the response surface is what makes the optimization of chromatographic selectivity stand out as a particular optimization problem rather than as an example to which known optimization strategies from other fields can be readily applied. This is illustrated by the application of univariate optimization. In univariate optimization (or univariate search) methods the parameters of interest are optimized sequentially. An optimum is located by varying a given parameter and keeping all other parameters constant. In this way, an optimum value is obtained for that particular parameter. From this moment on the optimum value is assigned to the parameter and another parameter is varied in order to establish its optimum value . 

Univariate optimization is a common way of optimizing simple processes, which are affected by a series of mutually independent parameters. For two parameters such a simple problem is illustrated in figure 5.3a. In this figure a contour plot corresponding to the three-dimensional response surface is shown. The independence of the parameters leads to circular contour lines. If the value of x is first optimized at some constant value of y (line 1) and if y is subsequently optimized at the optimum value observed for x, the true optimum is found in a straightforward way, regardless of the initial choice for the constant value of y. For this kind of optimization problem univariate optimization clearly is an attractive method. 

Drawn lines indicate the course of the optimization procedure. In figure b it is illustrated that for dependent parameters several reinitiations of the procedure are required to approach the optimum. 

Unfortunately, for the optimization of chromatographic selectivity we will have to deal with response surfaces that correspond to figure 5.3c or usually figure 5.3d [503], often with a considerably higher degree of complexity than shown there (compare figure 5.1). 

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Failure of Univariate Optimization. In the univariate approach to optimization, all variables but one are held constant at arbitrary values while the remaining variable is changed until an "optimum" response is found. The process is then repeated for each successive variable, using the "optimum" value for any variables that have now been "optimized" and arbitrary values for all the remaining variables except the one that is presently being investigated. 

Figure 5.3 Univariate optimization of individual parameters, (a) A simple response surface with two mutually independent parameters, (b) A simple surface with dependent parameters, (c) (opposite page) A complex surface with independent parameters, (d) (opposite page) A complex surface with dependent parameters. 

Univariate optimization (table 5.7a) is not a good method for the optimization of chromatographic selectivity. This will be clear from the table, since despite a fairly high number of experiments only a local optimum will be located on the kinds of response surfaces typically encountered in chromatography (see figure 5.3). Moreover, the local optimum may be of little value, because no overall impression of the response surface is obtained during the process. Once this has been established, the other (favourable) characteristics of this method are no longer relevant. 

In an attempt to use milder acidic conditions the prereduction of Se(IV) to Se(VI) was carried out with a mixture of HC1 and HBr (10 percent v/v each) instead of HC1 alone (>50 percent v/v) [52]. Experimental parameters were selected by a univariate optimization method. The main advantage of the MW heating was that it allowed for a strict control over the heating power as well as over the time the heating was applied. Seven samples of orange juice were analyzed. Selenium was present in five of them as a mixture of Se(IV) and Se(VI), Se(IV) being the predominant species with concentrations ranging from 5.20 0.08 to 9.50 0.09 pg 1 1. 

Table 3.5 shows the three electrophoretic factors and levels selected in which experimental optimization, in terms of overall response (% conversion), could be performed. A design matrix was then generated for the Box-Behnken study (Table 3.6). It was found that voltage and mixing time, when combined, had a significant effect on % conversion. Here, the extent of contact between substrate and enzyme is dictated by the difference in electrophoretic mobilities, which is in turn dictated by mixing time and voltage. Such an interaction would not have been possible by use of classical univariate optimization methods.

Whenever a new CE method is being developed, optimization strategies are usually applied to improve analysis speed, sensitivity, and resolution, using these three parameters or a combination of them as the monitored output (also called response or performance criteria). Very frequently, a step-by-step approach in which each factor is varied sequentially is followed. In this case, all parameters are kept constant, while the parameter of interest is varied and the response is measured. Depending on the problem (especially when the number of factors to optimize is very low) and on the performance criteria, univariate optimization can be useful, that is, the analysis of a single compound with only one component of the BGE. However, in most cases, a step-by-step optimization is laborious and tedious because it typically requires a high number of experiments. Furthermore, and more important, it does not consider possible interactions between factors. 

FIGURE 12.1. (a) Representation of a univariate optimization scheme. The concentric circles represent a surface response and the center is the maximum response. (1) The x-variable (or factor) value is fixed and variable y is optimized (2) y is fixed at best response while x is varied (3) during optimization of x, a better value is found, thus requiring new experiments varying y. According to this experimental setup, intersection of (2) and (3) would be the best response, (b) Representation of a bidimen-sional simplex BNW and the reflection R of the worse value W. Reprinted with permission from Reference 4. 

Fast, D.M. Culbreth, P.H. Sampson, E.J. Multivariate and univariate optimization studies of liquid-chromatographic separation of steroid mixtures. Clin.Chem., 1982, 28, 444-448 [also estriol, hydrocortisone, progesterone, testosterone]... 

There exist several simplex methods. In this chapter, we will discuss three of them, in increasing order of complexity the basic simplex, the modified simplex and the super-modified simplex. The more sophisticated methods are able to adapt themselves better to the response surface studied. However, their construction requires a larger number of experiments. In spite of this, the modified and super-modified simplexes normally are able to come closer to the maximum (or minimum if this were of interest) with a total number of experiments that is smaller than would be necessary for the basic simplex. In this chapter, we will see examples with only two or three variables, so that we can graphically visualize the simplex evolution for instructive purposes. However, the efficiency of the simplex, in comparison with univariate optimization methods, increases with the number of factors. 

Felhofer, J., Hanrahan, G., and Garcia, C.D. (2009) Multivariate versus univariate optimization of separation conditions in micellar electrokinetic chromatography. Talanta, , 1172-1178. 

Despite the multivariate nature of experimental designs, univariate optimization strategies can be applied when there is no interaction between factors. An interaction can be better understood with the help of Figure 2. 

The optimization of the variables is a critical step in the design of new analytical methods. Optimization involves the selection of the chemical and instrumental factors which may affect the analytical signal, and the choice of the values of the variables to obtain the best response from the chemical system. For this purpose, two different strategies can be used. In the traditional univariate optimization, all values of the different factors except one are constant, and this one is the object of the examination. The alternative to this strategy is the use of chemometric techniques based mainly on the use of experimental designs (Tarley, et al. 2009). 

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