A comprehensive complete note on Optimization related to my last uploaded ppt. Covers the whole topic.

1. SURAJ C. | P.P.M. | February 24, 2014
OPTIMIZATION TECHNIQUES
A REVIEW

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INTRODUCTION
• It can be defined as “to make perfect”.
• OPTIMIZATION is an act, process, or methodology of making design, system or
decision as fully perfect, functional or as effective as possible.
• Optimization of a product or process is the determination of the experimental
conditions resulting in its optimal performance.
• In Pharmacy, the word “optimization” is found in the literature referring to “any study
of formula.”
• In developmental projects, pharmacist generally experiments by
 A series of logical steps,
 Carefully controlling the variables and
 Changing one at a time until satisfactory results are obtained.
• This is how the optimization done in pharmaceutical industry.
• 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.
NOTE: It is not a Screening technique.
INPUTS OUTPUTSREAL
SYSTEM
INPUT FACTOR
LEVELS
MATHEMATICAL
MODELOF
SYSTEM
OPTIMIZATION
PROCEDURE
RESPONSE
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OBJECTIVE
• The major objective of the product optimization stage is to ensure the product selected
for further development is fully optimized & complies with the design
specification & critical quality parameters described in the product design report.
• The key outputs from this stage of development will be:-
 A quantitative formula defining the grade & quantities of each excipient & the
quantity of candidate drug,
 Defined pack,
 Defined drug, excipient & component specification &, defined product
specifications.
IMPORTANCE
• For the formulation of drug products in various forms this optimization technique
is mainly used.
• It is the process of finding the best way of using the existing resources while taking
in to the account of all the factors which will affect the experiments.
• Final product will definitely meet the bio-availability requirements.
• This will also help in understanding the theoretical formulations.
OPTIMIZATION PROCESS
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1. DOE:
 Strategy for setting up experiments in such a manner that the required
information is obtained as efficiently as precisely possible.
 It indicates the no. of experiments to be conducted with a given no. of variables
& their levels.
 It includes the outputs → Response.
 Experimental designs are available viz.
a) Factorial designs,
b) Central composite designs etc.
 For large number of process variables screening designs are mainly used. Example:
Fractional factorial designs etc.
 BENEFITS of experimental design:
 Saving time, money & drug substance.
 Identification of interactions effects.
 Characterization of response surface.
2. Analysis of Results – Modelling:
 The results obtained are analyzed by this step.
 Conclusion can be drawn for the best possible product.
 Modeling is necessary because the operating conditions employed in the
experiments are far from the actual optimum.
 Variables & responses are correlated for the quantitative relationship.
 Examples:
a) Liner (mathematical experiments) &
b) Non-linear (graphs, response curves etc.).
3. Simulation & Search:
 In this case, the models are used for predicting the theoretical formulations.
 It can be achieved by
a) Systematic or
b) Random procedure.
 Reliable parameters are identified for satisfying the quality constraints.
 Eg: Response surface methods, contour plots etc.
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OPTIMIZATION PARAMETERS
• It includes –
• VARIABLES:
1. Independent Variables:
 These factors are controlled by the experimenter.
 A reasonable idea is already available on important variables & their effective
ranges.
 Still it is needed because it does not allow the missing of the important
variables.
 It can classified further as:
a. Quantitative: Measurable factors, time, temperature, concentration etc.
b. Qualitative: Type of solvent, type of catalyst, brands of materials etc.
 Another classification includes :
Formulation
variables
Process variables
Drug (API) Granulation time
Diluent Drying inlet temperature
Binder Mill speed
Disintegrating
agents
Blending time
Glidant Compression force
Optimization Parameters
Variables Problems
Independent
Dependent Constrained
Unconstrained
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a. Process Variables &
b. Formulation Variables
2. Dependent Variables:
 These responses are resulted from the independent variables and obtained
from the experimentations.
 It is important to have the knowledge of the responses.
 Classified as:
a. Quantitative: Yield, % of purity etc.
b. Qualitative:
 Appearance, luster, lumpiness, odour, taste etc.
 These are evaluated on a number scale (5- 10).
 Example: 0: standard
 -1 or + 1 -> Slight difference from the standard
 -2 or + 2 -> Moderate difference from the standard
 -3 or + 3 -> Extreme difference
c. Quantal:
 Pass or fail, ‘go’ or ‘no go’, ‘clear’ or ‘turbid’ etc.
 These could be expressed as percentage of response.
 This is actually a quality control tool.
• PROBLEMS:
1. Constrained:
 A tablet can be hardest possible, but it must disintegrate in less than 5
minutes.
 In tablet production three components can be varied, but together the
weight should be restricted to 350mg only. Amount of active ingredient will
be also fixed.
 Some ingredients must be present in the minimal quantity to produce an
acceptable product. This is called Design of Constraints.
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 Ex: 3 variable components: stearic acid, starch & dibasic calcium
phosphate.
*(Further the lower limit for varying ingredient is often not equal to zero.)
2. Unconstrained:
 A tablet can be hardest possible in case of chewable tablets.
 If there are no constraints an ingredient can be used as 0% level as well as
100%.
 In pharmaceutical formulations, restrictions are always placed on the
systems.
 Ex: Hardest tablet is needed to be produced at lowest compression
pressure & ejection force, but disintegration & dissolution must be faster.
FUNDAMENTAL CONCEPTUAL TERMS
• FACTOR:
 A factor is an assigned variable such as concentration, temperature, pH etc
• LEVELS:
 The levels of the factor are the values or designations assigned to the factor.
 Examples of levels are 30˚ and 50˚ for the factor temperature, 0.1M and 0.3M
for the factor concentration.
 Higher level can be denoted by ‘+’ and the lower level by ‘-’ signs.
• EFFECTS:
 The effect of the factor is the change in response caused by varying the levels
of the factor.
 The main effect is the effect of a factor averaged over all levels of the other
factors.
• RESPONSE:
 Response is mostly interpreted as the outcome of an experiment.
 It is the effect, which we are going to evaluate i.e., disintegration time,
duration of buoyancy, thickness, etc.
• INTERACTIONS:
 It is also similar to the term effect, which gives the overall effect of two or
more variables (factors) of a response.
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 For example,
 The combined effect of lubricant (factor) and glidant (factor) on
hardness (response) of a tablet.
 From the optimization we can draw conclusion about.
 Effect of a factor on a response i.e., change in dissolution rate as the
drug to polymer ratio changes.
CLASSICAL OPTIMIZATION
• Involves application of calculus to basic problem for maximum/minimum function.
• One factor at a time (OFAT).
• Restrict attention to one factor at a time.
• Not more than 2 variables.
• Using calculus the graph obtained can be solved.
Y = f (x)
• When the relation for the response y is given as the function of two independent
variables,X1 & X2
Y = f(X1, X2)
• The above function is represented by contour plots on which the axes represents the
independent variables X1 & X2
Response
Variable
Independent Variable
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OFAT vs DOE
Properties OFAT DOE
Type
Classical- Sequential one factor method Scientific – simultaneous with
multiple factor method
No. of experiments
High – Decided by experimenter Limited – Selected by design
Conclusion
Inconclusive – Interaction unknown Comprehensive – Interactions
studied too.
Precision & Efficiency
Poor – sometimes misleading result with
errors (4 exp.)
High – Errors are shared evenly (2
exp.)
Consequences
One exp. Wrong… all goes wrong -
Inconclusive
Orthogoanl design – Predictable &
conclusive
Information gained
Less per experiment High per experiment
STATISTICAL DESIGN
• STATISTICAL TECHNIQUES:
 Techniques used divided in to two types:
1. Experimentation continues as optimization proceeds
(Represented by evolutionary operations (EVOP), simplex methods.)
2. Experimentation is completed before optimization takes place.
(Represented by classic mathematical & search methods.)
2. Experimentation is completed before optimization takes place:
 Theoretical approach: If theoretical equation is known, no
experimentation is necessary.
Independent
Variable - X2
Independent
Variable - X1
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 Empirical or experimental approach: With single independent variable
formulator experiments at several levels.
• STATISTICAL TERMS:
 Relationship with single independent variable –
1. Simple regression analysis or
2. Least squares method.
 Relationship with more than one important variable –
1. Statistical design of experiment &
2. Multi linear regression analysis.
 Most widely used experimental plan – Factorial design.
• STATISTICAL METHODS:
1. Optimization: helpful in shortening the experimenting time.
2. DOE: is a structured , organized method used to determine the relationship
between –
 the factors affecting a process &
 the output of that process.
3. Statistical DOE: planning process + appropriate data collected + analyzed
statistically.
MATHEMATICAL MODELS
• Permits the interpretation of RESPONSES more economically & becomes less
ambiguous.
1. First Order: 2 Levels of the factor – Linear relationship.
 LCL (Lower control limit) - {-ve or -1}
 UCL (Upper control limit) - {+ve or +1}
2. Second Order: 3 Levels (Mid-level) – coded as “0” – Curvature effect.
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OPTIMIZATION TECHNIQUES
Parametric Non-Parametric
Factorial
Central
Composite
Mixture
Lagrangian
Multiple
Fractional
Factorial
Plackett-
Burman
Evolutionary
methods
EVOP REVOP
X
Response
LOW HIGH
Predictable Response at X1
FIRST ORDER
X1
Response
LO
W
HIG
H
True
Respons
SECOND ORDER
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1. FULL FACTORIAL DESIGN: (FFD)
N = LK
• Where,
K = number of variables
L = number of variable levels
N = number of experimental trials
• For example, in an experiment with three factors, each at two levels, we have eight
formulations, a total of eight responses.
• Table 1 (shows levels of the ingredients) and Table 2 (shows 23
full factorial design.)
• The optimization procedure is facilitated by the fitting of an empirical polynomial
equation to the experimental results.
Y = B0 + B1X1 + B2X2 + B3X3 + B12X1X2 + B13X1X3 + B23X2X3 + B123X1X2X3 ----- (1)
• The eight coefficients in above equation will be determined from the eight responses
in such a way that each of the responses will be exactly predicted by the polynomial
equation.
• For example,
 In formulation 1, X1 = X2 = X3 = 0
 Substituting it in equation,
Y = B0 = 5
 In formulation 2, X2 = X3 = 0
 Substituting it in equation
Y = B0 + B1X1
9 = 5 + B1 (2)
B1= 2.
Table-1
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• Similarly, we can calculate other coefficients. Substituting it in the equation (1) we get
the polynomial equation from which the response can be obtained for any level of
ingredients.
2. CENTRAL COMPOSITE DESIGN: (CCD)
• Central composite design was discovered in 1951 by Box and Wilson hence also called
as Box-Wilson design.
• Central composite design is comprised of the combination of two-level factorial
points 2K-F
, axial or star points 2K, and a central point C.
• Thus the total number of factor combinations in a CCD is given by:
N = 2K-F
+ 2K + C
• Where,
K = number of variables
F = fraction of full factorial
C = number of center point replicates
• The major advantage of designs of this type is the reduction in the number of
experimental trials.
• Table 3 shows number of experimental trials required for 3K-F
designs and a typical
composite design with a single center point 2K-F
+2K+1 for up to four independent
variables.
Table-2
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3. SIMPLEX LATTICE DESIGN:
• The simplex lattice design was discovered by Spendley.
• This procedure may be used to determine the relative proportion of ingredients
that optimizes a formulation with respect to a specified variable(s) or outcome.
• In the present example, three components of the formulation will be varied-
 stearic acid,
 starch and
 dicalcium phosphate
 with the restriction that the sum of their total weight must equal 350 mg.
• The active ingredient is kept constant at 50 mg, the total weight of the formulation is
400 mg.
NOTE: For the sake of convenience, only one effect, dissolution rate, is measured.
• The arrangement of three variable ingredients in a simplex is shown in Figure 1.
• The simplex is generally represented by an equilateral figure, such as
 triangle for the three component mixture and
 tetrahedron for a four component system.
• Each vertex represents a formulation containing either
 a pure component or
 the maximum percentage of that component, with the other two components
absent or at their minimum concentration.
• In this example, the vertices represent mixtures of all three components, with each
vertex representing a formulation with one of the ingredients at its maximum
concentration.
NOTE: The reason for not using pure component is that a formulation containing only
one component would result in an unacceptable product.
Table-3
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• In this case, the lower and upper limits are
 stearic acid 20 to 180 mg (5.7 to 51.4 %),
 starch 4 to 164 mg (1.1 to 46.9 %) and
 dicalcium phosphate 166 to 326 mg (47.4 to 93.1 %).
• Various formulations can be studied in this triangular space.
• One basic simplex design includes formulations at each vertex, halfway between the
vertices, and at one center point as shown in below figure.
NOTE: A formulation represented by a point halfway between two vertices contains
the average of the min and max concentrations of the two ingredients represented by
the two vertices.
Table – 4: Composition of seven formulas with their responses:
• If the vertices in the design are not single pure substance (100 %), as in the case in
this example, the computation is made easier if a simple transformation is initially
Fig. 1
Table - 4
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performed to convert the maximum percentage of a component to 100 %, and the
minimum percentage to 0 % as follows,
Transformed % = (Actual %-minimum %) / (Maximum % - minimum %)
• Then the required empirical formula is concluded.
4. LAGRANGIAN METHOD:
• This optimization method was the first to be applied to a pharmaceutical
formulation and processing problems.
• In below example,
 the active ingredient, phenyl propalamine HCl, was kept at a constant level,
and
 the levels of disintegrant (starch) and lubricant (stearic acid) were selected as
the independent variables, X1 and X2.
• The dependent variables include
 tablet hardness,
 friability,
 volume,
 in vitro release rate and
 urinary excretion in human subjects.
• Table 5: shows possible compositions of nine formulations.
Table - 5
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• Fig.2: Counterplots of the effect of different levels of ingredients (independent
variables) on the measured response (dependent variables.)
• As represented in figure 2:
 2(a) shows the contour plots for tablet hardness as the levels of independent
variables are changed.
 2(b) shows similar contour plots for the dissolution response, t50%. If the
requirements on the final tablet are that hardness is 8-10 kg and t50% is 20-33
min,
 2(c) the feasible solution space is indicated in figure, this has been obtained by
superimposing figure 2(a) and 2 (b) and several different combinations of X1 and
X2 will suffice.
5. FRACTIONAL FACTORIAL DESIGN:
N = LK –F
• Where,
L = Number of variable levels
K = Number of variables
F = Fraction of full factorial (F=1, Fraction is 1/2 F=2, Fraction is 1/4)
N = Number of experimental trials
Fig. 2
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• In an experiment with a large number of factors and/or a large number of levels for the
factors, the number of experiments needed to complete a factorial design may be
inordinately large.
• If the cost and time considerations make the implementation of a full factorial design
impractical, fractional factorial design can be used in which a fraction of the original
number of experiments can be run.
6. Plackett – Burmann Design: (PBD)
N = K+1
• Where,
K = number of variables
N = number of experimental trials
• Placket Burman Design (PBD) is a special two-level FFD used generally for screening
of factors, where N is as a multiple of 4.
• Placket Burman Design also is known as Hadamard design.
• In Plackett and Burman design the low level is always denoted as -1 and the high level
as +1.
• In the table 4 the three factors are at two levels so total eight combinations are possible.
• The remaining four factors represent the interaction between individual factors.
• So there are seven factors in total, i.e. one less than total number of experiment.
Formulation X1 X2 X3 X1X2 X1X3 X2X3 X1X2X3 Y
1. -1 -1 -1 +1 +1 +1 -1 5
2. +1 -1 -1 -1 -1 +1 +1 9
3. -1 +1 -1 -1 +1 -1 +1 8
4. +1 +1 -1 +1 -1 -1 -1 10.8
5. -1 -1 +1 +1 -1 -1 +1 10
6. +1 -1 +1 -1 +1 -1 -1 10
7. -1 +1 +1 -1 -1 +1 -1 16.5
8. +1 +1 +1 +1 +1 +1 +1 16.5
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SIMULATION & SEARCH METHODS
• INTRODUCTION:
 Search method does not requires CONTINUITY or DIFFERENTIALITY function.
 Search methods also known as - “Sequential optimization”.
NOTE: Simulation involves the computability of a response.
 A simple inspection of experimental results is sufficient to choose the desired
product.
 If the independent variable is Qualitative – Visual observation is enough.
 Computer aid not required, but if it there, then added advantage.
 Even 5 variables can be handled at once.
• TYPES:
1. Steepest Ascent Method
2. Response Surface Methodology (RSM)
3. Contour Plots
1. STEEPEST ASCENT METHOD:
 Procedure for moving sequentially along the path (or direction) in order to
obtain max. ↑ in response.
 Applied to analyze the responses obtained from:
a) Factorial Designs
b) Fractional Factorial Designs
NOTE: Initial estimates of DOE are far from actual, so this method chosen for
optimum value.
2. RESPONSE SURFACE METHODOLOGY:
 A 3-D geometric representation that establishes an empirical relationship
between responses & independent variables.
 For:
a) Determining changes in response surface
b) Determining optimal set of experimental conditions
NOTE: Overlap of plots for complete info is possible.
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3. CONTOUR PLOTS:
 Are 2-D (X1 & X2) graphs in which some variables are held at one desired level &
specific response noted.
 Both axes are in experimental units.
 Sometimes both the contour & RSM plots are drawn together for better
optimum values.
REFERENCES
1. Subhramanium C V S, Thimmasetty J; Industrial Pharmacy, Selected Topics,
2013; 1st
Edition: 188- 276.
2. Pingale P L, et.al. Optimization techniques for pharmaceutical product
formulation. World J Pharm Pharmaceuti Sci. 2013; 2(3): 1077-89.
3. Dumbare A S, et.al. Optimization: A Review. Intl J Univ Pharm Life Sci. 2012;
2(3): 503-15.
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