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Optimization techniques in formulation Development- Plackett Burmann Design and D-Optimal

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D.R. Chandravanshi

The document discusses optimization techniques in pharmaceutical formulation, focusing on the Plackett-Burman design and D-optimal designs. It outlines the importance of experimental design principles, including randomization, replication, and local control, and details various types of optimization techniques used to enhance drug formulation processes. Additionally, it explains the relationships between independent and dependent variables, providing insights into effective experimental methodologies in pharmacy.

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING: PLACKETT BURMAN DESIGN AND D-OPTIMAL Presented by:- Dau Ram Chandravanshi M.Pharma. 1st sem.
Contents  Introduction General optimization techniques  Optimization parameter  Design of experiments  Basic Principles of experimental design. • Randomization • Replication • Local control Types of optimization techniques. Application Software for optimization and design References
Introduction It is the process of finding the best way of using the existing resources while taking in to the account of all the factors that influences decisions in any experiment. The objective of designing quality formulation is achieved by various optimization techniques. In Pharmacy word “optimization” is found in the literature referring to study of the formula. In formulation development process generally experiments by a series of logical steps, carefully controlling the variables and changing one at a time until satisfactory results are obtained.
REAL SYSTEM MATHEMATICAL MODEL OF SYSTEM RESPONSE OPTIMIZATION PROCEDURE INPUT FACTOR LEVEL INPUT OUTPUT General Optimization Technique
Optimization Parameters PARAMETERS PROBLEM TYPE CONSTRAINED UNCONSTRAI NED VARIABLE DEPENDENT INDEPENDENT
Problem Types Unconstrained • In unconstrained optimization problems there are no restrictions. • For a given pharmaceutical system one might wish to make the hardest tablet possible. • The making of the hardest tablet is the unconstrained optimization problem. Constrained • The constrained problem involved in it, is to make the hardest tablet possible, but it must disintegrate in less than 15 minutes.
(a) Variables The development procedure of the pharmaceutical formulation involve several variables. The independent variables are under the control of the formulator. These might include Ingredients, compression force or the die cavity filling or the mixing time. The dependent variables are the responses or the characteristics that are developed due to the independent variables. The more the variables that are present in the system the more the complications that are involved in the optimization (b) Factor:- It is Assigned and Independent variables, which affect the product or output of the process. It is an assigned quantitative and qualitatively like this
Quantitative: Numerical factor assigned to it. Ex; Concentration- 1%, 2%, 3% etc. Qualitative: These are not numerical. Ex; Polymer grade, humidity condition etc. (c) Level:- Levels of a factor are the values or designations assigned to the factor. (d) Response surface:- Response surface representing the relationship between the independent variables X1 and X2 and the dependent variable Y. (e) Run or trials:- Experiments conducted according to the selected experimental design (g) Contour Plot:- Geometric illustration of a response obtained by plotting one independent variable against another, while holding the magnitude of response and other variables as constant. (h) Interaction:- It gives the overall effect of two or more variables means lack of additivity of factor effects Ex: Combined effect of lubricant and glidant on hardness of the tablets.
Basic Principles of experimental design  Randomization- Randomization is the random process of assigning treatments to the experimental units. The random process implies that every possible allotment of treatments has the same probability.  Replication- By replication we mean that repetition of the basic experiments. For example, If we need to compare the grain yield of two varieties of wheat then each variety is applied to more than one experimental units. The number of times these are applied to experimental units is called their number of replication. It has two important properties: • It allows the experimenter to obtain an estimate of the experimental error. • The more replication would provide the increased precision by reducing the standard errors.
It has been observed that all extraneous source of variation is not removed by randomization and replication, i.e. unable to control the extraneous source of various. Thus we need to a refinement in the experimental technique. In other words, we need to choose a design in such a way that all extraneous source of variation is brought under control. For this purpose we make use of local control, a term referring to the amount of (i) balancing, (ii) blocking.  Local control
Types of optimization techniques 1. Plackett-burman designs. 2. Factorial designs. 3. Fractional factorial design (FFD). 4. Response surface methodology. 5. Central composite design (box-wilson design). 6. Box-behnken designs.
1. Plackett Burman Design In 1946, R.L. Plackett and J.P. Burman published their now famous paper "The Design of Optimal Multifactorial Experiments" in Biometrika (vol. 33). This paper described the construction of very economical designs with the run number a multiple of four (rather than a power of 2). Plackett- Burman designs are very efficient screening designs when only main effects are of interest. These designs have run numbers that are a multiple of 4. Plackett- Burman (PB) designs are used for screening experiments because, in a PB design main effects are in general heavily confounded with two- factor interactions. The PB design in 12 runs for example may be used for an experiment containing up to 11 factors.
Plackett-Burman Design in 12 Runs for up to 11 Factors Pattern X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 1 +++++++++++ +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 2 -+-+++---+- -1 +1 -1 +1 +1 +1 -1 -1 -1 +1 -1 3 --+-+++---+ -1 -1 +1 -1 +1 +1 +1 -1 -1 -1 +1 4 +--+-+++--- +1 -1 -1 +1 -1 +1 +1 +1 -1 -1 -1 5 -+--+-+++-- -1 +1 -1 -1 +1 -1 +1 +1 +1 -1 -1 6 --+--+-+++- -1 -1 +1 -1 -1 +1 -1 +1 +1 +1 -1 7 ---+--+-+++ -1 -1 -1 +1 -1 -1 +1 -1 +1 +1 +1 8 +---+--+-++ +1 -1 -1 -1 +1 -1 -1 +1 -1 +1 +1 9 ++---+--+-+ +1 +1 -1 -1 -1 +1 -1 -1 +1 -1 +1 10 +++---+--+- +1 +1 +1 -1 -1 -1 +1 -1 -1 +1 -1 11 -+++---+--+ -1 +1 +1 +1 -1 -1 -1 +1 -1 -1 +1 12 +-+++---+-- +1 -1 +1 +1 +1 -1 -1 -1 +1 -1 -1
20-Run Plackett-Burman Design X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19 1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 2 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 3 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 4 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 5 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 6 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 7 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 8 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 9 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 10 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 11 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 12 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 13 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 14 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 15 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 16 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 17 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 18 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 19 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1
24-Run Plackett-Burman Design X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19 X20 X21 X22 X23 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 3 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 4 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 5 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 6 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 7 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 8 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 9 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 10 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 11 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 12 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 13 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 14 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 15 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 16 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 17 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 18 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 19 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1
20 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 21 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 22 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 23 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 24 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 These designs do not have a defining relation since interactions are not identically equal to main effects. With the 2 k=pIII designs, a main effect column Xi is either orthogonal to XiXj or identical to plus or minus Xi Xj. For Plackett-Burman designs, the two-factor interaction column XiXj is correlated with every Xk (for k not equal to i or j). Economical for detecting large main effects. However, these designs are very useful for economically detecting large main effects, assuming all interactions are negligible when compared with the few important main effects.
D- Optimal design D-optimal designs are one form of design provided by a computer algorithm. These types of computer-aided designs are particularly useful when classical designs do not apply. Unlike standard classical designs such as factorials and fractional factorials, D-optimal design matrices are usually not orthogonal and effect estimates are correlated. These types of designs are always an option regardless of the type of model the experimenter wishes to fit (for example, first order, first order plus some interactions, full quadratic, cubic, etc.) or the objective specified for the experiment (for example, screening, response surface, etc.). D-optimal designs are straight optimizations based on a chosen optimality criterion and the model that will be fit. The optimality criterion used in generating D-optimal designs is one of maximizing |X'X|, the determinant of the information matrix X'X.
Cont.. This optimality criterion results in minimizing the generalized variance of the parameter estimates for a pre-specified model. As a result, the 'optimality' of a given D-optimal design is model dependent. That is, the experimenter must specify a model for the design before a computer can generate the specific treatment combinations. Given the total number of treatment runs for an experiment and a specified model, the computer algorithm chooses the optimal set of design runs from a candidate set of possible design treatment runs. This candidate set of treatment runs usually consists of all possible combinations of various factor levels that one wishes to use in the experiment. In other words, the candidate set is a collection of treatment combinations from which the D-optimal algorithm chooses the treatment combinations to include in the design. The computer algorithm generally uses a stepping and exchanging process to select the set of treatment runs.
Cont.. D-optimal designs are useful when resources are limited or there are constraints on factor settings. The reasons for using D-optimal designs instead of standard classical designs generally fall into two categories: 1. Standard factorial or fractional factorial designs require too many runs for the amount of resources or time allowed for the experiment. 2. The design space is constrained (the process space contains factor settings that are not feasible or are impossible to run). Example: Suppose an industrial process has three design variables (k = 3), and engineering judgment specifies the following model as an appropriate representation of the process.
Cont.. The levels being considered by the researcher are (coded) X1: 5 levels (-1, -0.5, 0, 0.5, 1) X2: 2 levels (-1, 1) X3: 2 levels (-1, 1) Due to resource limitations, only n = 12 data points can be collected. Create the candidate set - Given the experimental specifications, the first step in generating the design is to create a candidate set of points. The candidate set is a data table with a row for each point (run) to be considered for the design, often a full factorial. For our problem, the candidate set is a full factorial in all factors containing 5*2*2 = 20 possible design runs.
Candidate Set for Variables X1, X2, X3 X1 X2 X3 -1 -1 -1 -1 -1 +1 -1 +1 -1 -1 +1 +1 -0.5 -1 -1 -0.5 -1 +1 -0.5 +1 -1 -0.5 +1 +1 0 -1 -1 0 -1 +1 0 +1 -1 0 +1 +1 0.5 -1 -1 0.5 -1 +1 0.5 +1 -1 0.5 +1 +1 +1 -1 -1 +1 -1 +1 +1 +1 -1 +1 +1 +1
Generating a D-optimal design D-optimal designs maximize the D-efficiency, which is a volume criterion on the generalized variance of the parameter estimates. The D-efficiency values are a function of the number of points in the design, the number of independent variables in the model, and the maximum standard error for prediction over the design points. The best design is the one with the highest D-efficiency. Other reported efficiencies (e.g. A, G, I) help choose an optimal design when various models produce similar D-efficiencies. The D-optimal design (D=0.6825575, A=2.2, G=1, I=4.6625) using 12 runs is shown in Table in standard order. The standard error of prediction is also shown. The design runs should be randomized before the treatment combinations are executed.
Final D-optimal Design X1 X2 X3 OptStdPred -1 -1 -1 0.645497 -1 -1 +1 0.645497 -1 +1 -1 0.645497 -1 +1 +1 0.645497 0 -1 -1 0.645497 0 -1 +1 0.645497 0 +1 -1 0.645497 0 +1 +1 0.645497 +1 -1 -1 0.645497 +1 -1 +1 0.645497 +1 +1 -1 0.645497 +1 +1 +1 0.645497
Applications Formulation and Processing. Clinical Chemistry. Medicinal Chemistry. High Performance Liquid Chromatographic Analysis. Study of Pharmacokinetic Parameters. Allow large number of variables to be investigated in compact trial.
Software for Designs and Optimization Software’s dedicated to experimental designs  DESIGN EXPERT  ECHIP  MULTI-SIMPLEX  NEMRODW  Software for general statistical nature  SAS  MINITAB  SYSTAT  GRAPHPAD PRISM
References 1. Badawi MA, El-Khordagui LK. A quality by design approach to optimization of emulsions for electrospinning using factorial and D-optimal designs. Eur J Pharm Sci. 2014;58:44-54. 2. Garg Rahul Kumar* and Singhvi Indrajeet, Optimization techniques: an overview for formulation development, Asian J. Pharm. Res. Vol 5, Issue 3, 217-221, 2015. 3. Fukuda I.M, Pinto C.F.F, Moreira C.S, Saviano A.M, Lourenço F.R. Design of experiments (DOE) applied to pharmaceutical and analytical quality by design (QBD) Braz. J. Pharm. Sci. 2018;54(Special):e01006. 4. https://www.itl.nist.gov/div898/handbook/index.htm 5. Sahira N. Muslim, Alaa N. Mahammed, Hadeel K. Musafer, Israa M. S. AL_Kadmy, and Shatha A. Shafiq, Sraa N. Muslim “Detection of the Optimal Conditions for Tannase Productivity and Activity by Erwinia Carotovora”, Journal of Medical and Bioengineering Vol. 4, No. 3, June 2015.
Optimization techniques in formulation Development- Plackett Burmann Design and D-Optimal

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Optimization techniques in formulation Development- Plackett Burmann Design and D-Optimal

  • 1. OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING: PLACKETT BURMAN DESIGN AND D-OPTIMAL Presented by:- Dau Ram Chandravanshi M.Pharma. 1st sem.
  • 2. Contents  Introduction General optimization techniques  Optimization parameter  Design of experiments  Basic Principles of experimental design. • Randomization • Replication • Local control Types of optimization techniques. Application Software for optimization and design References
  • 3. Introduction It is the process of finding the best way of using the existing resources while taking in to the account of all the factors that influences decisions in any experiment. The objective of designing quality formulation is achieved by various optimization techniques. In Pharmacy word “optimization” is found in the literature referring to study of the formula. In formulation development process generally experiments by a series of logical steps, carefully controlling the variables and changing one at a time until satisfactory results are obtained.
  • 6. Problem Types Unconstrained • In unconstrained optimization problems there are no restrictions. • For a given pharmaceutical system one might wish to make the hardest tablet possible. • The making of the hardest tablet is the unconstrained optimization problem. Constrained • The constrained problem involved in it, is to make the hardest tablet possible, but it must disintegrate in less than 15 minutes.
  • 7. (a) Variables The development procedure of the pharmaceutical formulation involve several variables. The independent variables are under the control of the formulator. These might include Ingredients, compression force or the die cavity filling or the mixing time. The dependent variables are the responses or the characteristics that are developed due to the independent variables. The more the variables that are present in the system the more the complications that are involved in the optimization (b) Factor:- It is Assigned and Independent variables, which affect the product or output of the process. It is an assigned quantitative and qualitatively like this
  • 8. Quantitative: Numerical factor assigned to it. Ex; Concentration- 1%, 2%, 3% etc. Qualitative: These are not numerical. Ex; Polymer grade, humidity condition etc. (c) Level:- Levels of a factor are the values or designations assigned to the factor. (d) Response surface:- Response surface representing the relationship between the independent variables X1 and X2 and the dependent variable Y. (e) Run or trials:- Experiments conducted according to the selected experimental design (g) Contour Plot:- Geometric illustration of a response obtained by plotting one independent variable against another, while holding the magnitude of response and other variables as constant. (h) Interaction:- It gives the overall effect of two or more variables means lack of additivity of factor effects Ex: Combined effect of lubricant and glidant on hardness of the tablets.
  • 9. Basic Principles of experimental design  Randomization- Randomization is the random process of assigning treatments to the experimental units. The random process implies that every possible allotment of treatments has the same probability.  Replication- By replication we mean that repetition of the basic experiments. For example, If we need to compare the grain yield of two varieties of wheat then each variety is applied to more than one experimental units. The number of times these are applied to experimental units is called their number of replication. It has two important properties: • It allows the experimenter to obtain an estimate of the experimental error. • The more replication would provide the increased precision by reducing the standard errors.
  • 10. It has been observed that all extraneous source of variation is not removed by randomization and replication, i.e. unable to control the extraneous source of various. Thus we need to a refinement in the experimental technique. In other words, we need to choose a design in such a way that all extraneous source of variation is brought under control. For this purpose we make use of local control, a term referring to the amount of (i) balancing, (ii) blocking.  Local control
  • 11. Types of optimization techniques 1. Plackett-burman designs. 2. Factorial designs. 3. Fractional factorial design (FFD). 4. Response surface methodology. 5. Central composite design (box-wilson design). 6. Box-behnken designs.
  • 12. 1. Plackett Burman Design In 1946, R.L. Plackett and J.P. Burman published their now famous paper "The Design of Optimal Multifactorial Experiments" in Biometrika (vol. 33). This paper described the construction of very economical designs with the run number a multiple of four (rather than a power of 2). Plackett- Burman designs are very efficient screening designs when only main effects are of interest. These designs have run numbers that are a multiple of 4. Plackett- Burman (PB) designs are used for screening experiments because, in a PB design main effects are in general heavily confounded with two- factor interactions. The PB design in 12 runs for example may be used for an experiment containing up to 11 factors.
  • 13. Plackett-Burman Design in 12 Runs for up to 11 Factors Pattern X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 1 +++++++++++ +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 2 -+-+++---+- -1 +1 -1 +1 +1 +1 -1 -1 -1 +1 -1 3 --+-+++---+ -1 -1 +1 -1 +1 +1 +1 -1 -1 -1 +1 4 +--+-+++--- +1 -1 -1 +1 -1 +1 +1 +1 -1 -1 -1 5 -+--+-+++-- -1 +1 -1 -1 +1 -1 +1 +1 +1 -1 -1 6 --+--+-+++- -1 -1 +1 -1 -1 +1 -1 +1 +1 +1 -1 7 ---+--+-+++ -1 -1 -1 +1 -1 -1 +1 -1 +1 +1 +1 8 +---+--+-++ +1 -1 -1 -1 +1 -1 -1 +1 -1 +1 +1 9 ++---+--+-+ +1 +1 -1 -1 -1 +1 -1 -1 +1 -1 +1 10 +++---+--+- +1 +1 +1 -1 -1 -1 +1 -1 -1 +1 -1 11 -+++---+--+ -1 +1 +1 +1 -1 -1 -1 +1 -1 -1 +1 12 +-+++---+-- +1 -1 +1 +1 +1 -1 -1 -1 +1 -1 -1
  • 14. 20-Run Plackett-Burman Design X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19 1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 2 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 3 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 4 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 5 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 6 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 7 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 8 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 9 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 10 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 11 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 12 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 13 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 14 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 15 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 16 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 17 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 18 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 19 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1
  • 15. 24-Run Plackett-Burman Design X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19 X20 X21 X22 X23 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 3 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 4 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 5 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 6 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 7 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 8 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 9 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 10 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 11 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 12 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 13 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 14 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 15 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 16 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 17 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 18 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 19 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1
  • 16. 20 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 21 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 22 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 23 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 24 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 These designs do not have a defining relation since interactions are not identically equal to main effects. With the 2 k=pIII designs, a main effect column Xi is either orthogonal to XiXj or identical to plus or minus Xi Xj. For Plackett-Burman designs, the two-factor interaction column XiXj is correlated with every Xk (for k not equal to i or j). Economical for detecting large main effects. However, these designs are very useful for economically detecting large main effects, assuming all interactions are negligible when compared with the few important main effects.
  • 17. D- Optimal design D-optimal designs are one form of design provided by a computer algorithm. These types of computer-aided designs are particularly useful when classical designs do not apply. Unlike standard classical designs such as factorials and fractional factorials, D-optimal design matrices are usually not orthogonal and effect estimates are correlated. These types of designs are always an option regardless of the type of model the experimenter wishes to fit (for example, first order, first order plus some interactions, full quadratic, cubic, etc.) or the objective specified for the experiment (for example, screening, response surface, etc.). D-optimal designs are straight optimizations based on a chosen optimality criterion and the model that will be fit. The optimality criterion used in generating D-optimal designs is one of maximizing |X'X|, the determinant of the information matrix X'X.
  • 18. Cont.. This optimality criterion results in minimizing the generalized variance of the parameter estimates for a pre-specified model. As a result, the 'optimality' of a given D-optimal design is model dependent. That is, the experimenter must specify a model for the design before a computer can generate the specific treatment combinations. Given the total number of treatment runs for an experiment and a specified model, the computer algorithm chooses the optimal set of design runs from a candidate set of possible design treatment runs. This candidate set of treatment runs usually consists of all possible combinations of various factor levels that one wishes to use in the experiment. In other words, the candidate set is a collection of treatment combinations from which the D-optimal algorithm chooses the treatment combinations to include in the design. The computer algorithm generally uses a stepping and exchanging process to select the set of treatment runs.
  • 19. Cont.. D-optimal designs are useful when resources are limited or there are constraints on factor settings. The reasons for using D-optimal designs instead of standard classical designs generally fall into two categories: 1. Standard factorial or fractional factorial designs require too many runs for the amount of resources or time allowed for the experiment. 2. The design space is constrained (the process space contains factor settings that are not feasible or are impossible to run). Example: Suppose an industrial process has three design variables (k = 3), and engineering judgment specifies the following model as an appropriate representation of the process.
  • 20. Cont.. The levels being considered by the researcher are (coded) X1: 5 levels (-1, -0.5, 0, 0.5, 1) X2: 2 levels (-1, 1) X3: 2 levels (-1, 1) Due to resource limitations, only n = 12 data points can be collected. Create the candidate set - Given the experimental specifications, the first step in generating the design is to create a candidate set of points. The candidate set is a data table with a row for each point (run) to be considered for the design, often a full factorial. For our problem, the candidate set is a full factorial in all factors containing 5*2*2 = 20 possible design runs.
  • 21. Candidate Set for Variables X1, X2, X3 X1 X2 X3 -1 -1 -1 -1 -1 +1 -1 +1 -1 -1 +1 +1 -0.5 -1 -1 -0.5 -1 +1 -0.5 +1 -1 -0.5 +1 +1 0 -1 -1 0 -1 +1 0 +1 -1 0 +1 +1 0.5 -1 -1 0.5 -1 +1 0.5 +1 -1 0.5 +1 +1 +1 -1 -1 +1 -1 +1 +1 +1 -1 +1 +1 +1
  • 22. Generating a D-optimal design D-optimal designs maximize the D-efficiency, which is a volume criterion on the generalized variance of the parameter estimates. The D-efficiency values are a function of the number of points in the design, the number of independent variables in the model, and the maximum standard error for prediction over the design points. The best design is the one with the highest D-efficiency. Other reported efficiencies (e.g. A, G, I) help choose an optimal design when various models produce similar D-efficiencies. The D-optimal design (D=0.6825575, A=2.2, G=1, I=4.6625) using 12 runs is shown in Table in standard order. The standard error of prediction is also shown. The design runs should be randomized before the treatment combinations are executed.
  • 23. Final D-optimal Design X1 X2 X3 OptStdPred -1 -1 -1 0.645497 -1 -1 +1 0.645497 -1 +1 -1 0.645497 -1 +1 +1 0.645497 0 -1 -1 0.645497 0 -1 +1 0.645497 0 +1 -1 0.645497 0 +1 +1 0.645497 +1 -1 -1 0.645497 +1 -1 +1 0.645497 +1 +1 -1 0.645497 +1 +1 +1 0.645497
  • 24. Applications Formulation and Processing. Clinical Chemistry. Medicinal Chemistry. High Performance Liquid Chromatographic Analysis. Study of Pharmacokinetic Parameters. Allow large number of variables to be investigated in compact trial.
  • 25. Software for Designs and Optimization Software’s dedicated to experimental designs  DESIGN EXPERT  ECHIP  MULTI-SIMPLEX  NEMRODW  Software for general statistical nature  SAS  MINITAB  SYSTAT  GRAPHPAD PRISM
  • 26. References 1. Badawi MA, El-Khordagui LK. A quality by design approach to optimization of emulsions for electrospinning using factorial and D-optimal designs. Eur J Pharm Sci. 2014;58:44-54. 2. Garg Rahul Kumar* and Singhvi Indrajeet, Optimization techniques: an overview for formulation development, Asian J. Pharm. Res. Vol 5, Issue 3, 217-221, 2015. 3. Fukuda I.M, Pinto C.F.F, Moreira C.S, Saviano A.M, Lourenço F.R. Design of experiments (DOE) applied to pharmaceutical and analytical quality by design (QBD) Braz. J. Pharm. Sci. 2018;54(Special):e01006. 4. https://www.itl.nist.gov/div898/handbook/index.htm 5. Sahira N. Muslim, Alaa N. Mahammed, Hadeel K. Musafer, Israa M. S. AL_Kadmy, and Shatha A. Shafiq, Sraa N. Muslim “Detection of the Optimal Conditions for Tannase Productivity and Activity by Erwinia Carotovora”, Journal of Medical and Bioengineering Vol. 4, No. 3, June 2015.

Editor's Notes

  • #10: For e.g. if number of treatment = 3 (say, A, B, and C) and replication = r = 4, then the number of elements = t * r = 3 * 4 = 12 = n. Replication means that each treatment will appear 4 times as r = 4.
  • #11: Balancing: Balancing means that the treatment should be assigned to the experimental units in such a way that the result is a balanced arrangement of treatment. Blocking: Blocking means that the like experimental units should be collected together to far relatively homogeneous groups. A block is also a replicate.