Various optimization techniques and their role in pharmaceutical

sciences.

Aakanksha guptaroll no: 04

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optimization

“An art, process, or methodology of making something (a design, system, or decision) as perfect, as functional, as effective as possible.”

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Seminar Outline:

• Introduction• Key term used in in optimization• Applied Optimization Methods• Applications• Conclusion• References

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Objectives of Pharmaceutical Optimization

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ADVANTAGES

• Yield the “best solution” within the domain of study.

–Require fewer experiments to achieve an optimum formulation.

• Can trace and rectify “problem”in a remarkably easier manner

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Key term used in optimization process

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Type of optimization techniques

Classical method Applied methodA. Factorial designs and

Modifications a. Full Factorial Design b. Fraction Factorial Design

i. Homogenous fractionalii. Mixed level fractionaliii. Box-Hunteriv. Plackett-Burmanv. Taguchivi. Latin square

B. Central composite design and modificationsC. Mixture designD. D-optimal design

A.Evolutionory

Operation (EVOP)

B.Simplex Lattice

C.Lagrangian Method

D.Search Method

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Applied optimization

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Cont………..

• The classic calculus methods apply basically to unconstrained problems . but in pharmacy all problem are constrained

• Deming and king presented a general flowchart.• Involve the effect on a real system of changing some input

(some factor or variable) is observed directly at the output (one measures some property), and that set of real data is used to develop mathematical models.

• The responses from the predictive models are then used for optimization

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Flow line of applied optimization

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1. EVOP METHOD

make very small changes in formulation repeatedly. The result of changes are statistically analyzed. If there is improvement, the same step is repeated until

further change doesn’t improve the product. Where we have to select this technique?This technique is especially well suited to a production situation.The process is run in a way that is both produce a product thatmeets all specifications and (at the same time) generates

information on product improvement.11

Advantages: • generates information on product development.• predict the direction of improvement.• Help formulator to decide optimum conditions for the formulation

and process.Limitation More repetition is required Time consuming Not efficient to finding true optimum Expensive to use.

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• Example: In this example, A formulator can changes the concentration of binder (no of experiment is done) and get the desired hardness.

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2.SIMPLEX METHOD It was introduced by Spendley et.al. A simplex is a geometric figure, defined by no. of points or

vertices equal to one more than no. of factors examined. Once the shape of a simplex has been determined, the

method can employ a simplex of fixed size or of variablesizes that are determined by comparing the magnitudes ofthe responses after each successive calculation

It is of two types:A. Basic Simplex MethodB. Modified Simplex Method.

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Cont…..

The simplex method is especially appropriate when:• Process performance is changing over time.• More than three control variables are to be changed.• The process requires a fresh optimization with each new lot of

material. The simplex method is based on an initial design of k+1,

where k is the number of variables. A k+1 geometric figure in a k-dimensional space is called a simplex. The corners of this figure are called vertices.

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Basic Simplex Method:

It is easy to understand and apply. Optimization begins with the initial trials.

Number of initial trials is equal to the number of control variables plus one.

These initial trials form the first simplex. The shapes of the simplex in a one, a two and a three variable

search space, are a line, a triangle or a tetrahedron respectively.

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Rules for basic simplex: The first rule is to reject the trial with the least favorable

value in the current simplex The second rule is never to return to control variable

levels that have just been rejected.

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Modified simplex method

It was introduced by Nelder-Mead in 1965. It can adjust its shape and size depending on the response in

each step. This method is also called the variable-size simplex method.

Rules:1. Contract if a move was taken in a direction of less favorable

conditions2. Expand in a direction of more favorable conditions.

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• Advantage • This method will find the true optimum of a response with

fewer trials than the non-systematic approaches or the one-variable-at-a-time method.

• Disadvantage :• There are sets of rules for the selection of the sequential

vertices in the procedure.• Require mathematical knowledge.

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Example

• Special cubic simplex design for a three component mixture. Each point represent a different formulation SA = stearic acid; DCP= dicalcium phosphate, ST= starch

• Constraint : with the restriction that the sum of their total weight must equal to 350 mg, 50 mg = active ingredient

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example• Development of an analytical method (a continuous flow

analyzer) by Deming and king.• The two independent variable show the pump speeds for the

two reagents required in the analysis reaction.• The initial simplex is represented by the lowest triangle; the

vertices represent the Spectrophotometric response.• The strategy is to move toward a better response by moving

away from the worst response 0.25, conditions are selected at the vortex 0.6 and indeed, improvement is obtained.

• One can follow the experimental path to the optimum 0.721.

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Spectrophotometric response at given wavelength

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3.LAGRANGIAN METHOD It represents mathematical techniques.

It is an extension of classic method.

applied to a pharmaceutical formulation and processing.

This technique follows the second type of statistical design

This technique require that the experimentation be completed before optimization so that the mathematical models can be generates.

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• Steps involved:1. Determine the objective function.2. Determine the constraints.3. Change inequality constraints to equality constraints.4. Form the Lagrange function F:a. one Lagrange multiplier λ for each constraintb. one slack variable q for each inequality constraint.5. Partially differentiate the Lagrange function for each variable and set derivatives equal to zero6. Solve the set of simultaneous equations.7. Substitute the resulting values into objective function

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• Where we have to select this technique?This technique can applied to a pharmaceutical formulation and processing.

• Advantages :lagrangian method was able to handle several responses or dependent variables

• Disadvantages:Although the lagrangian method was able to handle several responses or dependent variables, it was generally limited to two independent variables

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Example Optimization of a tablet. phenyl propranolol(active ingredient)-kept constant X1 – disintegrate (corn starch) X2 – lubricant (stearic acid) X1 & X2 are independent variables. Dependent variables include tablet hardness,

friability ,volume, invitro release rate e.t.c.., It is full 32 factorial experimental design. Nine formulation were prepared

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Cont………..

PPolynomial models relating the response variables to independents were generated by a backward stepwise regression analysis program.

Y= B0+B1X1+B2X2+B3 X12 +B4 X2

2 +B5 X1 X2 +B6 X12X2

+ B7X1 X2 2+B8X1

2X22....................(1)

Y – Response Bi – Regression coefficient for various terms containing

the levels of the independent variables. X – Independent variables. one equation is generated for each response or dependent variable.27

Formulationno,.

Drug Dicalciumphosphate

Starch Stearic acid

1 50 326 4(1%) 20(5%)

2 50 246 84(21%) 20

3 50 166 164(41%) 20

4 50 246 4 100(25%)

5 50 166 84 100

6 50 86 164 100

7 50 166 4 180(45%)

Tablet formulation

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Tablet formulations

Constrained optimization problem is to locate the levels of stearic acid(x1) and starch(x2).

This minimize the time of invitro release(y2),average tablet volume(y4), average friability(y3)

To apply the lagrangian method, problem must be expressed mathematically as follows

Y2 = f2(X1,X2)-invitro release…………(2)

Y3 = f3(X1,X2)<2.72 %-Friability………..(3)

Y4 = f4(x1,x2) <0.9422 cm3 …………..(4)

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Cont………

5≤X1 ≤45………(5) 1≤X2 ≤41……….(6) Equation (5) and (6) serve to keep the solution within the

experimental range. Inequality constraints must be converted to equality constrained

by introducing slack variable. Introduce of langrage multiplier λ to each equality constraint Several equation are then combined into Lagrange function. Partial differentiation of the Lagrange function and solving the

resulting set of six simultaneous equation. Value are obtained for the appropriate levels of x1 and x2, to yield

and optimum in vitro time of 17.9 min (t50%).30

Cont………

The solution to a constrained optimization program may depend heavily on the constraints applied to the secondary objectives.

Graphical representation.

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Contour plot

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Hardness of tablet Dissolution time (t50%)

Contour plot

• C) Feasible solution space indicate by crosshatched area

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Unlike the Lagrangian method, do not require differentiability of the objective function.

used for more than two independent variables. The response surface is searched by various methods to find

the combination of independent variables yielding an optimum.

It take five independent variables into account and is computer assisted.

4.SEARCH METHOD

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Example: optimization of tablet formulation

The experimental design used was a modified factorial

Independent Variables Dependent Variables X1 = Diluents ratio Y1 = Disintegration time

X2= Compression force Y2= Hardness

X3= Disintegrant levels Y3 = Dissolution

X4= Binder levels Y4 = Friability

X5 = Lubricant levels Y5 = Porosity

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• The first 16 trials are represented by +1 and -1.

• The remaining trials are represented by a -1.547, zero or 1.547

• The data were subjected statistical analysis, followed by multiple regression analysis

• The type of predictor equation used in this design is a second-order polynomial:

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Cont………

for optimization itself , two major steps were used: Feasibility search grid search

Feasibility search used to locate a set of response constraints that are just at the

limit of possibility. Select several response ones wishes to constrain search of the response surface is made to determine whether a

solution is feasible. For e.g the constraints in table were fed into the computer and

were relaxed ones at a time until a solution was found.38

Specification of feasibility search

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Cont……..

the first feasible solution was found at disintegration time time =5 min, hardness=10 kg, and dissolution = 100 % at 50 min.

The next step, the grid search, Experimental range is divided into a grid of specific size

and divided into a grid of specific size and methodically searched.

From an input of the desired criteria, the program prints out all points (formulation) that satisfy the constraints.

Thus , the best or most acceptable formulation is selected from the grid search printout to complete the optimization.

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Cont………

graphic approaches are also available and graphic output is provided by a plotter from computer tapes.

The output includes plots of a given responses as a function of a single variable as in figure (a) & (b) or as a function of all five variables.

The abscissa for both types is produced in experimental units, rather than physical units, so that it extends from -1.547 to +1.547.

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Cont………

An infinite number of this plot is possible, since for each curve represented, four of the five variables must remain constant at some level.

This is analogous to a partial derivative situation, and the slope of any one graph does indeed represent a partial derivative of the response for one of the independent variables.

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Step summarized as follows:

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Advantages:• Takes five independent variables in to account• Person unfamiliar with the mathematics of optimization and

with no previous computer experience could carry out an optimization study.

• It do not require continuity and differentiability of functionDisadvantage : • One possible disadvantage of the procedure as it is set up is

that not all pharmaceutical responses will fit a second-order regression model.

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Literature reviewKanani R. et al.,Development and characterization of antibiotic

orodispersible tablets» To formulated oro-dispersible tablet of

Azithromycin that is intended to disintegrate rapidly into the oral cavity and form a stabilized dispersion.

» Preliminary study: different superdisintegrant croscarmllose sodium (CCS) , sodium starch glycolate (SSG), crosspovidone (CPVP), were evaluate for weight variation, content uniformity, hardness, disintegrantion time, and friability .

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Cont…..

• Simplex lattice design:• Independent variable : this design utilised using amount of

intragranular concentration of superdisintegrant, sodium starch glycolate(A), cros-carmellose sodium (B), crospovisone(c)

• Dependent variable: hardness(R1), disintegration time(R2), Friability (R3), wetting time(R4).

• A total of 11 formulation with 4 replica was obtained and optimized .

• From response surface plot of disintegration time, wetting time, friability and hardness were found.

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Formulation using simplex lattice design

Ingredient(mg) F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11

Intragranular

Azithromycin 100 100 100 100 100 100 100 100 100 100 100

Sodium starch glycolate 5 - 30 10 30 - 30 - - 5 20

Crosscarmellose sodium 20 30 - 10 - 30 - - - 5 5

Crospovidone 5 - - 10 - - - 30 30 20 5

Avicel 38 38 38 38 38 38 38 38 38 38 38

Sodium lauryl sulphate 2 2 2 2 2 2 2 2 2 2 2

Extragranular

Aerosil 5 5 5 5 5 5 5 5 5 5 5

Magnesium stearate 5 5 5 5 5 5 5 5 5 5 5

Aspartame 20 20 20 20 20 20 20 20 20 20 20

Total 200 200 200 200 200 200 200 200 200 200 20047

Design summary response data

Run SSG CCS CPVP Hardness

kg/cm2

DT(second) Friability(%

)

%CPR

1 5 20 5 0.33±0.5773 20.33±0.5773 0.43±0.0264 19.33±0.5773

2 - 30 - 3.66±0.5773 22.26±0.5773 0.34±0.0173 18±0.0000

3 30 - - 3.66±0.5773 30.33±0.5773 0.49±0.0173 24.66±0.5773

4 10 10 10 4±0.0000 19±1 0.47±0.0100 14.33±0.5773

5 30 - - 3.66±0.5773 30.33±0.5773 0.49±0.0173 24.66±0.5773

6 - 30 - 3.66±0.5773 22.66±0.5773 0.34±0.0173 18±0.0000

7 30 - - 3.66±0.5773 30.33±0.5773 0.49±0.0173 24.66±0.5773

8 - - 30 3.66±0.5773 17±1 0.34±0.0173 15.33±0.5773

9 - - 30 3.66±0.5773 17±1 0.34±0.0173 15.33±0.5773

10 5 5 20 3±0.0000 12.66±0.5773 0.30±0.0157 11.33±0.5773

11 - 5 5 3.33±0.5773 27.66±0.5773 0.51±0.0057 21±1

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equation• Hardness R1= +3.66*A+3.66*B+3.77*C-4.68*A*B-8.64*A*C-

8.64*B*C+75.04*A*B*C

• Disintegration time R2 = +30.66*A+22.66*B+17.00*C+41.32*A*B-16.76*A*C-58.70*B*C-14.49*A*B*C

• Friability R3= +0.49*A+0.34*B+0.34*C+0.86*A*B-0.70*A*C-0.76*B*C-3.96*A*B*C

• Wetting time R4= +24.66*A+18.00*B+15.33*C+15.36*A*B-28.62*A*C-8.70*B*C-177.12*A*B*C

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Result• Using simplex lattice design from the regression analysis

and 3‐D surface plot it is obtained thato Crospovidone with combination of other two super‐disintegrants is

showing good decrease in hardness.o In case Disintigration time, Crospovidone with combination of

croscarmellose is very effective to decrease the Disintrigation time which is desirable.

o While in case of friability crospovidone with combination of croscarmellose sodium and sodium starch glycolate very effective to decrease the Friability which is desirable.

o And in case of wetting time Crospovidone and Croscarmellose sodium are effective to decrease the Wetting Time which is desirable. (P<0.0001).

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Conclusion of this review

• Amongst the various combinations of diluents and disintegrants used in the study, tablets that were formulated (wet granulation) using Crospovidone (10%), crosscarmelose sodium and sodium starch glycolate ( each 5%) exhibited quicker disintegration of tablets than compared to those other combination of disintegrants in different concentration.

• The effectiveness of super‐disintegrants was in order of Crospovidone>Croscarmellose sodium>sodium starch glycolate.

• Formulation F10 was the optimized formulation having least disintegration time as well as other parameters were in acceptable range.

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conclusion• Optimization techniques are a part of development process.• The levels of variables for getting optimum response is

evaluated.• Different optimization methods are used for different

optimization problems.• Optimization helps in getting optimum product with desired

bioavailability criteria as well as mass production.• More optimum the product = More $$ the company earns in

profits!!!

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References:• Schwarts J. B., et al., Optimization techniques in pharmaceutical

formulation and processing, in: Banker G. S., et al. (eds), Modern pharmaceutics, Marcel Dekker Inc., 4th edition (revised and expanded), vol- 121, 607-620, 2005.

• Jain N. K., Pharmaceutical product development, CBS publishers and distributors, 1st edition,297-302, 2006.

• Cooper L. and Steinberg D., Introduction to methods of optimization, W.B.Saunders, Philadelphia, 1970.

• Bolton. S., Stastical applications in the pharmaceutical science, Varghese publishing house,3rd edition, 223

• Deming S.N. and King P. G., Computers and experimental optimization, Research/Development, vol-25 (5),22-26, may 1974.

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Cont…….

• Rubinstein M. H., Manuf. Chem. Aerosol News,30, Aug 1974.• Digaetano T.N., Bull.Parenter.Drug Assoc., vol-29,183, 1975.• Spendley, W., et al., Sequential application of simplex

designs in optimization and evolutionary operation, Technometrics, Vol- 4 441–461, 1962.

• O’connor R.E., The drug release mechanism and optimization of a microcrystalline cellulose pellet system, P.h.d.Dissetation, Philadelphia College of Pharmacy & Science, 1987.

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Cont……….

• Forner, D.E., et al., Mathematical optimization techniques in drug product design and process analysis, Journal of pharmaceutical sciences. , vol-59 (11),1587-1195, November 1970.

• Shirsand SB., et al.,Formulation and optimization of mucoadhesive bilayer buccal tablets of atenolol using simplex design method. Inetnational journal of pharmaceutical Investigation. January 2012,volume 2,issue 1,34-40.

• Kanani R., et al., Development and characterization of antibiotic orodispersible tablets. International Journal of Current Pharmaceutical Research. 2011,vol3,issue 3,27-32

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