optimization final

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 OUTPUTS REAL SYSTEM

INPUT FACTOR LEVELS

MATHEMATICAL MODELOF

SYSTEM

OPTIMIZATION PROCEDURE

RESPONSE

SURAJ C. AACP PAGE 1

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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. SURAJ C. AACP PAGE 6

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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

LOW

HIGH

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,


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,


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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