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MultiSimplex and experimental design as chemometric MultiSimplex and experimental design as chemometric tools to optimize a SPE-HPLC-UV method for the tools to optimize a SPE-HPLC-UV method for the

determination of eprosartan in human plasma samplesdetermination of eprosartan in human plasma samples

FromJournal of Talanta

By:Samereh Ranjbar

The MultiSimplex is designed as a true multivariate non-linear optimization tool that combines the modified simplex method with the fuzzy set theory by means of the membership functions providing an efficient and flexible tool for handling different and conflicting optimisation criteria (maximisation, minimisation and target values). Different response variables with separate scales and optimisation objectives can then be combined into a joint response measure called the aggregated value of membership. The most outstanding feature of MultiSimplex is that it allows simultaneous optimization of several response signals.

MultiSimplex suggests a k + 1 number of experiments, where k is the number of variables to be studied. Once the experiments are carried out, the result of the experiments are introduced and MultiSimplex suggests one new experiment. And the process goes on until the optimum conditions are reached.

A simplex is a geometric figure having a number of vertices equals to one more than the number of dimensions in a space. A simplex is defined by k+1 points in a k-dimensional space. Each vertex corresponds to a set of experimental condition.

The basic simplex algorithm follows a few rules:The first rule is to reject the trial with the least favorable response value in the current simplex.The second rule is never to return to control variable levels that have just been rejectedTrials retained in the simplex for a specified number of steps are reevaluatedCalculated trials outside the effective boundaries of the control variables are not made.

simplexsimplex

The modified simplexThe modified simplex

The modified simplex method has much in common with the basic method but can adjust its shape and size depending on the response in each step.

New rules:Expand in a direction of more favorable conditionsContract if a move was taken in a direction of less favorable conditions.

- The modified simplex will usually reach the optimum region quicker than with the basic method and pinpoint the optimum levels more closely.

The Modified Simplex Algorithm

The concept of "fuzzy sets", introduced by zadeh in 1965, is a method that allows individual response variables with separate scales and optimization objectives to be combined into a joint response measure called the aggregated value of membership.The fuzzy set membership function are the means for handling multiple responses in the multisimplex.

Creating a joint responseCreating a joint response There is no simple procedure for combining the different response variables into one generate performance measure because:

The response variables are measured with different scalesThe relative significance of different response variables differsFor some response variables the objective is maximization, but for others it is minimization or a specific target.

In fuzzy set theory the term ‘‘target’’ can be represented with a characteristic function varying with the response variable. This function, varying between 0 and 1, is the membership function of the variable in question. The higher the membership value is, the closer to the optimum the simplex is.

Evaluation of the response in the simplex method is carried out using the desirability functions, also known as membership functions.

MembershipMembership

For example: When the aim of the optimization is to maximize (or minimize) the response variables, the simplex method assigns yi to each variable involved, which is a membership function m(yi) defined as:

where:Ymin is the low limit for acceptable values of yi.Ymax is the value above, which a further increase is without significance.R is a constant.

The aggregated value of membershipThe aggregated value of membership

After identifying all the membership functions they are combined (aggregated) in a weighted geometric mean, to form the response of the whole system. The relative importance of each response variable in the optimization process is considered by assigning a value βi, called the degree of influence, to each variable, with values going from 0 to 1.

The aggregated value of membership is expressed as:

The performance of MultiSimplex is fairly easy to follow. The first step consists on the definition of the optimisation project which includes the control the control variablesvariables (variables to be optimised), their reference , maximum, minimum and step size value for each control variable; the response variables the response variables (responses to be optimised), their relative influence, the optimization goal and the shape (this parameter specifies the form of individual membership function) for each defined variable.

What are the reference value and step sizeWhat are the reference value and step size?? The range of variation in a control variable (the difference between high

and low level), in the first simplex, is called step size. The reference value is used to position the simplex in the variable space.

The levels for each control variable in the first simplex are allowed to vary between the reference value plus half the step size and the reference value minus half the step size.

The defined response variables were the corrected area (analyte’s area/internal standard’s area), the separation of eprosartan chromatographic peak against interference chromatographic peaks and the retention time of the analyte and the internal standard.

The experiments proposed by the MultiSimplex program and the membership value of each of the experiments decided to stop the optimization after the 30th run.

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Spiked sample

Experimental design has been applied to the optimization of different clean-upclean-up procedures. In this work, the optimization of the extraction procedure was made by means of two different experimental designs: a fractional factorial designfractional factorial design and aa central composite designcentral composite design

some variables were optimized each one at a time to avoid extending the number of variables included in the experimental domain. the measured response is only one (the higher corrected area or the recovery). These variables were: the extraction cartridge, the elution liquid composition and the washing liquid.

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OVAT MethodologyOVAT Methodology

Effect of cartridge characteristics on the recovery percentage of SPEEffect of the elution solvent on chromatographic peak area Effect of the elution profile methanol–phosphate buffer 0.1M pH 2

Fractional Factorial DesignFractional Factorial Design

The experimental variables considered in the FFD were: buffer solution concentration (x1), washing liquid volume (x2), drying time (x3) and elution liquid volume (x4).

Corrected areas obtained from the 24−1 proposed fractional factorial

design

Central Composite DesignCentral Composite Design From the four proposed variables for the FFD, the volume of elution solution has shown lack of significance, so it has been fixed (2mL). Thus, the CCD was a 23 +(3×2) + k design, k being the number of replicates of the center point (k = 2).

A sum of the Optimized SPE procedure for

plasma samples containing eprosartan.

Chemometric approach allowed us to reduce the number of experiments needed for optimization of extraction procedure and chromatographic separation.

The performance of the MultiSimplex optimization is very satisfactory. the MultiSimplex if compared with other multivariate optimisation tools such as experimental designs the amount of experiments required to define the response surface is lower.