design of experiments


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design of experiments

  1. 1. Design of Experiments (DOE) G.Gallo 11 marzo 2009
  2. 2. Concept of Quality by Design (QbD) <ul><li>Quality by Design principles have most recently been adopted by the U.S. Food and Drug Administration (FDA) as a vehicle for the transformation of how drugs are discovered, developed, and commercially manufactured. </li></ul><ul><li>Product and process performance characteristics are scientifically designed to meet specific objectives , not merely empirically derived from performance of test batches (Quality by QC inspection) </li></ul><ul><li>Characteristics important to desired performance must be derived from a combination of prior knowledge and experimental assessment during product development . </li></ul><ul><li>From this knowledge and data, a multivariate model linking product and process measurements and desired attributes may be constructed . </li></ul>
  3. 3. Process analytical technology ( PAT ) <ul><li>PAT has been defined by the United States Food and Drug Administration (FDA) as a mechanism to design, analyze, and control pharmaceutical manufacturing processes through the measurement of critical process parameters (CPP) which affect Critical Quality Attributes </li></ul><ul><li>The goal of PAT is to understand and control the manufacturing process, which is consistent with our current drug quality system: “ quality can not be tested into products; it has to be built in by design ”(ICH Q8, Step 2 Document) </li></ul><ul><li>In 2006, Merck & Co.’s Januvia became the first product approved based upon such an application </li></ul>
  4. 4. Different methods <ul><li>COST approach: to change the value of one factor at a time until there no further improvement. </li></ul><ul><li>DOE approach: to construct a carefully selected set of experiments in which all relevant factors are varied simultaneously. </li></ul>Changing a single factor a time doesn’t necessarily provide information about the optimum conditions in particular when there are interactions between factors. BUT
  5. 5. DOE <ul><li>The first statistician to consider a formal mathematical methodology for designing experiments was Sir Ronald A. Fisher, in his landmark The Design of Experiments . (1935) As an example, he described how to test the hypothesis that a certain lady could distinguish by flavour alone whether the milk or the tea was first placed in the cup. While this sounds like a frivolous application, it allowed him to illustrate the most important means of experimental design. </li></ul><ul><li>In 1950, Gertrude Mary Cox and William Gemmell Cochran published the book Experimental Designs which became the major reference work on the design of experiments for statisticians for years afterwards. </li></ul>
  6. 6. Where DOE is used? <ul><li>Design of experiments is a discipline that has very broad application across all the natural and social sciences: </li></ul><ul><ul><li>Screening and identification of important factors </li></ul></ul><ul><ul><li>Optimization of a pharmaceutical formulation </li></ul></ul><ul><ul><li>Optimization of analytical instruments </li></ul></ul><ul><ul><li>Minimization of production costs and pollution </li></ul></ul><ul><ul><li>Robustness testing of products and processes </li></ul></ul>
  7. 7. The main questions <ul><li>Screening </li></ul><ul><ul><li>Which factors are most influential? </li></ul></ul><ul><ul><li>What are their appropriate ranges? </li></ul></ul><ul><li>Optimization </li></ul><ul><ul><li>How can we find the optimum operating conditions? </li></ul></ul><ul><ul><li>Is there a unique optimum or a compromise is necessary? </li></ul></ul><ul><li>Robustness testing </li></ul><ul><ul><li>How should we adjust our factors to guarantee robustness? </li></ul></ul><ul><ul><li>Do we need to change our product specifications prior to claiming robustness? </li></ul></ul>DOE provides an organized approach where the experimenter is guided to perform a set of experiments, that are then evaluated in terms of a local regression model and interpreted to make decisions
  8. 8. Common steps in DOE
  9. 9. Screening <ul><li>You are able to extract a yes or no answer with regard to the influence of a particular factor. </li></ul><ul><ul><li>Pareto principle: 80% of effects are caused by 20% of factors </li></ul></ul><ul><li>Information is gained about how to modify the settings of the important factors to possibly further enhance the results. </li></ul><ul><li>It needs few experiments in relation to the number of factors. </li></ul>
  10. 10. Definition of factors <ul><li>Kind </li></ul><ul><ul><li>Controllable (es. pH) or uncontrollable (es. humidity) </li></ul></ul><ul><ul><li>Process (independent) or mixture (add to 100%) </li></ul></ul><ul><ul><li>Quantitative (continuous) or qualitative (categorical) </li></ul></ul><ul><li>Range </li></ul><ul><ul><li>Quantitative = low (-), high (+) and center point (0) </li></ul></ul><ul><ul><li>Qualitative = up to 12 levels (two to five is the best) </li></ul></ul>Specification of responses <ul><li>Regular = measured and fitted during the investigation </li></ul><ul><li>Derived = computed as function of the factors </li></ul><ul><li>Linked = defined in another project but invoked in this one </li></ul>
  11. 11. N°Experiments X1 X2 X3 1 - - - 2 2 full factorial design 2 + - - replicate experiments 3 - + - 2 3 full factorial design 4 + + - 2 3-1 fractional factorial design 5 - - + 6 + - + 7 - + + 8 + + + 9 0 0 0 10 0 0 0 11 0 0 0
  12. 12. Example 1: Reduction of enamines to saturated amines by formic acid <ul><li>Two factors varied </li></ul><ul><li>Three responses measured </li></ul><ul><ul><li>Yield of side product </li></ul></ul><ul><ul><li>Unreacted starting material </li></ul></ul><ul><ul><li>Yield of desired product </li></ul></ul><ul><li>The response Y3 should be maximized </li></ul><ul><li>A 2 2 full factorial design was applied </li></ul>
  13. 13. Example 1: (continued) <ul><li>Main and interaction effects were considered: </li></ul><ul><ul><li>Yield raises with temperature and decreases with the increase of molar ratio </li></ul></ul><ul><ul><li>Molar ratio effect depends on temperature (mild interaction) </li></ul></ul>
  14. 14. Computation of effects - Least square analysis <ul><li>Seeking the model that minimizes the sum of the squares of the residuals </li></ul><ul><ul><li>Multiple linear regression applied to the modelling of several factors </li></ul></ul><ul><ul><li>Y3= aX1+bX2+c </li></ul></ul><ul><ul><li>Regression coefficients are scaled and centered. </li></ul></ul><ul><ul><li>Low confidence interval depends on the quality of the design. </li></ul></ul><ul><li>Parameters : </li></ul><ul><ul><li>R 2 (green) is a measure of goodness of fit </li></ul></ul><ul><ul><li>Q 2 (blue) is a measure of goodness of prediction ( >0.5) </li></ul></ul><ul><ul><li>difference R 2- Q 2 <0.2-0.3 </li></ul></ul><ul><ul><li>Model validity (yellow) ( >0.25) </li></ul></ul><ul><ul><li>Reproducibility (cyan)( >0.5) </li></ul></ul>
  15. 15. Causes of poor models <ul><li>Curvature </li></ul><ul><ul><li>Quadratic terms to be optimized </li></ul></ul><ul><li>Skew response distribution </li></ul><ul><ul><li>Logaritmic transformation or NegLog </li></ul></ul><ul><li>Bad replicates </li></ul><ul><ul><li>Control of experimental set-up </li></ul></ul><ul><li>Deviating experiments </li></ul><ul><ul><li>Outliers outsite ± 4 standard deviations </li></ul></ul>
  16. 16. Optimization <ul><li>You are able to extract detailed information regarding how the factors combine to influence the responses. </li></ul><ul><ul><li>Positive or negative relation between factor and response </li></ul></ul><ul><ul><li>Linear or quadratic relation </li></ul></ul><ul><li>It requires additional experiments in relation to the number of investigated factors. </li></ul><ul><ul><li>Central composite design (CCC) or face-centered design (CCF) </li></ul></ul><ul><li>A response surface modelling (RSM) allows: </li></ul><ul><ul><li>Prediction of response values for any factor setting in the experimental region </li></ul></ul><ul><ul><li>Identification of the factor setting corresponding to the optimal point </li></ul></ul>
  17. 17. Response contour plot <ul><li>Allows the use of model to make decisions </li></ul><ul><li>Can compare simultaneously more than one response </li></ul><ul><li>Can visualize a conflict in the choice of factor variation </li></ul><ul><li>Allows the choise of further validating experiments </li></ul><ul><ul><li>Gradient techniques </li></ul></ul><ul><ul><li>Optimizer </li></ul></ul>
  18. 18. Robustness testing <ul><li>It is possible to: </li></ul><ul><ul><li>Identify those factors that might have an effect on the results </li></ul></ul><ul><ul><li>Regulate these factors to maintain the results within the specifications </li></ul></ul><ul><ul><li>Determine the sensitivity of the responses to small changes in the factors </li></ul></ul>
  19. 19. Example 2: Quality documentation of a HPLC system <ul><li>Five factors varied </li></ul><ul><ul><li>Four quantitative </li></ul></ul><ul><ul><li>One qualitative </li></ul></ul><ul><li>Three responses measured </li></ul><ul><li>The resolution is constantly maintained = 1.5 or greater </li></ul><ul><li>A linear model is applied </li></ul><ul><ul><li>2 5-2 fractional factorial design </li></ul></ul><ul><ul><li>Eight experiments </li></ul></ul><ul><ul><li>Four repicates, two for each column </li></ul></ul>
  20. 20. Example 2: (continued) <ul><li>To evaluate data distribution of the responses </li></ul><ul><li>To verify an extremely weak relationship between factors and resolution </li></ul><ul><li>To predict the response values of most extreme experiments </li></ul><ul><li>To reformulate the factor settings so that robustness can be obtained </li></ul>
  21. 21. Selection of the model and generation of design <ul><li>Screening </li></ul><ul><ul><li>Linear and interaction models </li></ul></ul><ul><ul><li>Full or fractional factorial design </li></ul></ul><ul><li>Optimization </li></ul><ul><ul><li>Quadratic model </li></ul></ul><ul><ul><li>Composite design </li></ul></ul><ul><li>Robustness testing </li></ul><ul><ul><li>Linear models </li></ul></ul><ul><ul><li>Full or fractional factorial design </li></ul></ul>
  22. 22. Bibliography <ul><li>MODDE software – Umetrics </li></ul><ul><li>Eriksson L. et al. Design of experiments –Umetrics </li></ul><ul><li>Carlson R.&Carlson J. Design and optimization in organic synthesis . Vol 24 Elsevier 2005 </li></ul><ul><li>Andersson M. et al Multivariate methods in tablet formulation suitable for early drug discovery: Predictive models from a screening design of several linked responses, Chemometrics and Intelligent Laboratory System , 87, 2007, 151-156 </li></ul>