various applied optimization techniques and their role in pharmaceutical science.

Various optimization techniques
and their role in
pharmaceutical sciences .

aakanksha gupta
roll no: 04

1

optimization

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

2

seminar outline:

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

3

objectiVes of
pharmaceutical
optimization

4

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 5

Key term used in optimization process

6

Type of optimization techniques
Classical method
A. Factorial designs and
Applied method
Modifications
a. Full Factorial Design A.Evolutionory
b. Fraction Factorial
Design Operation (EVOP)
i. Homogenous fractional
ii. Mixed level fractional B.Simplex Lattice
iii. Box-Hunter C.Lagrangian Method
iv. Plackett-Burman
v. Taguchi D.Search Method
vi. Latin square
B. Central composite design and
modifications
C. Mixture design
D. D-optimal design 7

applied optimization

8

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

9

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 that
meets 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. 12

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

13

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 variable
sizes that are determined by comparing the magnitudes of
the responses after each successive calculation
It is of two types:
A. Basic Simplex Method
B. Modified Simplex Method.

14

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.

15

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.

17

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
conditions
2. 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.

19

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.

21

Spectrophotometric response at given wavelength

22

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.
23

• 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 constraint
b. one slack variable q for each inequality constraint.
5. Partially differentiate the Lagrange function for each
variable and set derivatives equal to zero
6. Solve the set of simultaneous equations.
7. Substitute the resulting values into objective function
24

• 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
25

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

26

Cont………..

 Polynomial models relating the response variables to
independents were generated by a backward stepwise
regression analysis program.
 Y= B0+B1X1+B2X2+B3 X12 +B4 X22 +B5 X1 X2 +B6 X12X2

+ B7X1 X2 2+B8X12X22....................(1)

Y – Response
Bi – Regression coefficient for various terms containing
the levels of the independent variables.
X – Independent variables. 27



Tablet formulation

Formulation Drug Dicalcium Starch Stearic acid
no,. phosphate
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%)

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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) 29

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.

31

Contour plot

Hardness of tablet Dissolution time (t50%)

32

Contour plot
• C) Feasible solution space indicate by crosshatched area

33

4.SEARCH METHOD

 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.

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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

35

• 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:

37

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

39

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.

40

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.

41

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.

42

Step summarized as follows:

43

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 function
Disadvantage :
• One possible disadvantage of the procedure as it is set up is
that not all pharmaceutical responses will fit a second-order
regression model.

44

Literature review
Kanani 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 . 45

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.

46

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 200 200 200 200 200 200 200 200 200 200 200
47

Design summary response data

Run SSG CCS CPVP Hardness DT(second) Friability(% %CPR
kg/cm2 )
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

48

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 49

Result
• Using simplex lattice design from the regression
analysis and 3‐ D surface plot it is obtained that
o 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
50
Time which is desirable. (P<0.0001).

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.
51

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!!!

52

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
53
1974.

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.

54

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

55

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