# What Shape is your Method In? A Tutorial on the Application of Experimental Designs to Development of Chromatographic Methods By Dr Jeff Hughes School.

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What Shape is your Method In? A Tutorial on the Application of Experimental Designs to Development of Chromatographic Methods By Dr Jeff Hughes School of Applied Science, RMIT University Melbourne, Australia

RMIT UniversitySlide 2 The aim of this tutorial is to demonstrate how the methods of experimental design can be used to investigate chromatographic procedures In this tutorial we will look at using Factorial Designs to answer the following questions: Which factors significantly influence the of separation of peaks in our chromatogram? Which factor has the greatest influence on separation?

RMIT UniversitySlide 3 Data for this tutorial is taken from Chemometrics: Experimental Design by Ed Morgan Calculations are demonstrated using Excel, but can be carried out using the commercial program Minitab ( a demo version can be downloaded from http://minitab.com )http://minitab.com

RMIT UniversitySlide 4 Factorial Designs The most suitable type of design for screening Each variable (factor) has a set number of possible levels or values If there are k variables, each set at 2 possible levels (high and low) then there are 2 k possible combinations These designs are called two-level factorial designs. If all combinations are used they are called full factorial designs

RMIT UniversitySlide 5 Screening Aim – identify significant factors (variables) A factor is significant if its influence is greater than the noise level (experimental error) Usually carry out screening using reduced designs such as factorial or Plackett- Burman designs

RMIT UniversitySlide 6 The trials in a factorial design can be represented as points on an n-dimensional cube (n=3 in this case) 1,1,1 -1,1,1 -1,1,-1 1,-1,1 - 1,-1,1 -1,-1,-1 1,-1,-1 11,-1

RMIT UniversitySlide 7 Case Study – HPLC method Aim: to optimise the separation of peaks in a HPLC analysis

RMIT UniversitySlide 8 Define the Response The CRF (chromatographic response function) is used to quantify separation of peaks. This function thus gives a single number to the quality of a chromatogram. The aim is thus to maximise the CRF

RMIT UniversitySlide 9 Define the Factors The factors studied in this study were levels in the eluent of:- Acetic Acid Methanol Citric Acid

RMIT UniversitySlide 10 Experimental Domain LowHigh Acetic Acid (mol/L) 0.0040.01 % Methanol7080 Citric Acid (g/L)26

RMIT UniversitySlide 11 Factorial design (Coded form) Run Number Acetic Acid MethanolCitric AcidCRF 1--- 2+-- 3-+- 4++- 5--+ 6+-+ 7-++ 8+++ This design gives all combinations of the factors at 2 levels +, high -, low

RMIT UniversitySlide 12 Factorial design (Uncoded) Run Number Acetic Acid MethanolCitric AcidCRF 10.004702 20.01702 30.004802 40.01802 50.004704 60.01704 70.004804 80.01804 This table shows the actual levels of the variables used in the experiments. Normally the order of experiments is randomised but we will keep it in this structured forms so you can see the patterns Results are inserted here when the experiments are performed

RMIT UniversitySlide 13 Factorial design (Uncoded) Run Number Acetic Acid MethanolCitric AcidCRF 10.00470210 20.017029.5 30.00480211 40.0180210.7 50.0047049.3 60.017048.8 70.00480411.9 80.0180411.7 The CRF values are now inserted after the experiments (chromatographic runs) are carried out

RMIT UniversitySlide 14 Analysis of the results - Excel Calculate Main Effects – this calculates the effect on the response solely due to one factor Main effects are the difference between average response at high level of the factor – average response at low level

RMIT UniversitySlide 15 Acetic Acid MethanolCitric Acid CRF 10 +1 9.5 +111 +1 10.7 +19.3 +1+18.8 +1 11.9 +1 11.7 10.1811.3310.43 10.559.4010.30 -0.371.930.13 Average of the high values of CRF for each variable e.g for AA = (9.5+10.7+8.8+11.7)/4 Average of low values of CRF for each variable e.g for AA = (10+11+9.3+11.9)/4 The Main Effect is the difference between the high and low average e.g for AA = (10.19-10.55) Calculation of Main Effects The main effects can also be calculated by multiplying the variable column by the CRF column pairwise, adding up the column and then dividing by 4

RMIT UniversitySlide 16 Calculation of Interactions Acetic Acid MethanolCitric Acid AA*MAA*CAM*CACRFAA*M*CRF +1 10+10 +1 +19.5-9.5 +1 +111-11 +1 +1 10.7+10.7 +1 9.3+9.3 +1+1+18.8-8.8 +1 +111.9-11.9 +1 11.7+11.7 Sum = 0.5 0.1250.0250.825 0.5/4 = 0.125 Interactions coefficients –found by multiplying the appropriate variable columns Interactions calculated by multiplying the CRF column and the appropriate variable interactions column. To get the interaction effect add up the column and divide by 4

RMIT UniversitySlide 17 Factorial Calculations using Minitab The program Minitab can be used to carry out calculations as follows:- To set up the design: Stat > DOE > Factorial > Create Factorial Design Type of Design: 2 level factorial design 9default generators) Number of factors: 3 Designs: Full Factorial Factors see screen dump on next slide Options: do not randomize ( normally should randomize but for the tutorial not randomizing makes it easier to see patterns in the layout)

RMIT UniversitySlide 18

RMIT UniversitySlide 19 The generated FFD design as it should appear in Minitab Type the CRF responses here after performing the experiments

RMIT UniversitySlide 20

RMIT UniversitySlide 21 Factorial Calculations using Minitab To analyse the design: Stat > DOE > Factorial > Analyse factorial Design Click on C8 CRF as the response. Accept the default values

RMIT UniversitySlide 22 Factorial Calculations - Minitab 31/07/2007 11:26:30 Welcome to Minitab, press F1 for help. Results for: Worksheet 2 Full Factorial Design Factors: 3 Base Design: 3, 8 Runs: 8 Replicates: 1 Blocks: 1 Center pts (total): 0 All terms are free from aliasing. Design Table Run A B C 1 - - - 2 + - - 3 - + - 4 + + - 5 - - + 6 + - + 7 - + + 8 + + + Minitab Output

RMIT UniversitySlide 23 Factorial Fit: CRF versus Acetic Acid, Methanol, Citric Acid Estimated Effects and Coefficients for CRF (coded units) Term Effect Coef Constant 10.3625 Acetic Acid -0.3750 -0.1875 Methanol 1.9250 0.9625 Citric Acid 0.1250 0.0625 Acetic Acid*Methanol 0.1250 0.0625 Acetic Acid*Citric Acid 0.0250 0.0125 Methanol*Citric Acid 0.8250 0.4125 Acetic Acid*Methanol*Citric Acid 0.0250 0.0125 Main Effects Interactions Coefficients – from fitting to a second order equation Y = b o +b 1 *x 1 +b 2 *x 2 +b 3 *x 3 +b 12 *x 1 *x 2 +b 13 *x 1 *x 3 +b 23 *x 2 *x 3 +b 123 *x 1 *x 2 *x 3 Where x 1 is acetic acid, x 2 is methanol and x 3 is citric acid Note, however, coefficients are simply twice the effects so no new information

RMIT UniversitySlide 24 What do the results tell us? The main effects tell us which variable has the strongest effect on the response (CRF) – in this case methanol has the strongest effect on CRF A negative effect means the response is reduced as the variable increases. The negative effect for acetic acid means that as we increase the concentration of acetic acid, the CRF gets smaller (and hence our separation is worse)

RMIT UniversitySlide 25 What about interactions? An interaction effect is where the effect on the response of one variable depends on the level of another variable. In this study methanol and citric acid seem to have the largest interaction.

RMIT UniversitySlide 26 Main Effects and Interactions Plots Main effects plots help to visually display the variable effect. They graph the average response at the high and low levels. The steeper the graph, the stronger the effect The plots can be drawn in Miniab as follows:- Stat > DOE >Factorial > factorial Plots Tick Main Effects Plots Setup > Select CRF as the response and choose all 3 variables (>>) The Interactions plots can be produced similarly (just select Interactions instead of Main Effects Plots)

RMIT UniversitySlide 27 Note that Methanol has the steepest slope, indicating the strongest effect CRF average at high methanol CRF average at low methanol

RMIT UniversitySlide 28 The plots show there is an interaction effect with methanol and citric acid at high methanol CA has a Positive effect but at low methanol it has a negative effect on CRS

RMIT UniversitySlide 29 Conclusions Methanol has the largest effect on CRF The Methanol effect strongly depends on the Citric Acid level. Citric acid has a positive effect at high Methanol but a negative effect at low Methanol All 3 variables do seem to affect the result. Citric acid has the smallest main effect but large interaction effect Hence probably cant screen out any of these variables from further study

RMIT UniversitySlide 30 Significance – Normal Probability Plots Normal Probability Plots are used to test whether data is normally distributed. In our case, we can use such a plot to test for significance of the effects/coefficients If the effects are not significant we expect variations just to be due to random error and this can be tested with the plots. It is only a guide, however, as we have no real estimate of the experimental error In Minitab the plot can be generated: Stat > DOE >factorial > Analyse Factorial Design > Graphs and Select Effects Plots (Normal)

RMIT UniversitySlide 31 Effects due to random errors should be on a straight line. This plot indicates Methanol and the Methanol/Citric Acid interaction are significant effects

RMIT UniversitySlide 32 Problems We have not replicated any experiments so no determination of error. We cannot tell if the coefficients (effects) overall are significant (although normal probability plots help). We can only compare them to see which is the most significant We also cannot test for curvature – i.e are the effects of the variables linear. A non-linear effect can be when the response at the high and low levels is similar but at intermediate values is much higher or lower. pH effects are often non- linear

RMIT UniversitySlide 33 Solution? Add centre points!! Centre points are experiments with all variables set at 0 (coded) i.e. mid values Replication of the centre point allows determination of error CodedUncoded Acetic Acid 00.007 Methanol075 Citric Acid04

RMIT UniversitySlide 34 Acetic AcidMethanolCitric Acid CRF 10.0 1 9.5 1 11 1110.7 19.3 118.8 1111.9 11111.7 00010.2 00010.4 Results added for the centre points

RMIT UniversitySlide 35 Estimated Effects and Coefficients for CRF (coded units) Term Effect Coef SE Coef T P Constant 10.362 0.05000 207.25 0.003 Acetic Acid -0.3750 -0.1875 0.05000 -3.75 0.166 Methanol 1.9250 0.9625 0.05000 19.25 0.033 Citric Acid 0.1250 0.0625 0.05000 1.25 0.430 Acetic Acid*Methanol 0.1250 0.0625 0.05000 1.25 0.430 Acetic Acid*Citric Acid 0.0250 0.0125 0.05000 0.25 0.844 Methanol*Citric Acid 0.8250 0.4125 0.05000 8.25 0.077 P is the probability a coefficient is not significantly different from zero i.e no effect on CRF. A low probability (< 0.05 at the 5% level) indicates high significance. The methanol effect is the only significant one at the 5% level although the methanol-citric acid effect is just above the 5% level

RMIT UniversitySlide 36 The centre point responses are all on the linear response line. Thus no curvature is indicated.

RMIT UniversitySlide 37 The Normal Plot shows also that Methanol is the only significant effect but Methanol/CA interaction is probably also significant

RMIT UniversitySlide 38 Next Phase? Factorial designs give indication of significant effects and interactions Designs such as the Central Composite Design (CCD) can be used to find the best (optimal) settings of the variables and plot Response Surfaces CCD designs involve adding extra points (trials) to the Factorial Designs

RMIT UniversitySlide 39 CCD design for 3 factors. The 8 factorial points are corners of the cube. In this study the cube points correspond to the FFD. 6 axial points are added to form a CCD Cube (factorial) points Axial points

RMIT UniversitySlide 40 This is an example of a response surface plot for the optimization of capillary electrophoresis separation of a mixture of ranitidine-related compounds. Ranitidine is a drug used in the treatment of gastric and duodenal ulcers. The two variables being optimized are pH and applied voltage. The response variable used in the optimization is the logarithm of the CEF function, a parameter devised to assess the quality of a chromatogram (this is an alternative response function to the CRF). This function takes into account peak separation and total elution time (J. Chrom. A 766 245-54. Optimization of the capillary electrophoresis separation of ranitidine and related compounds. V.M. Morris, C. Hargraeves, K. Overall, P.J.Marriott, J.G.Hughes) Example of a CCD design and the generated response surface

RMIT UniversitySlide 41 Extra Information Notes on factorial and central composite designs can be found at the authors website: Chemometrics in Australia This site also has an Excel spreadsheet which sets out the calculations for the FFD design used in this tutorial

RMIT UniversitySlide 42 Author: Dr Jeff Hughes School of Applied Science, RMIT University Melbourne, Australia http://www.rmit.edu.au/staff/hughes j

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