Optimization through statistical response surface methods

basics of design of experiment DoE

1. “OPTIMIZATION THROUGH
STATISTICAL RESPONSE
SURFACE METHODS”
1
BHARATI VIDYAPEETH UNIVERSITY
POONA COLLEGE OF PHARMACY,PUNE
PRESENTED BY:
CHRISTY P GEORGE
M . Pharmacy Pharmaceutics
1
st
year

2. CONTENTS
INTRODUCTION TO OPTIMIZATION
PLACKETT BURMEN
CENTRAL COMPOSITE DESIGNS
BOX BEHNKAN
REFERENCE
2

3. INTRODUCTION
“ RESPONSE SURFACE METHODOLOGY (RSM) IS A COLLECTION OF STATISTICAL AND
MATHEMATICAL TECHNIQUES USEFUL FOR DEVELOPING, IMPROVING, AND
OPTIMIZING PROCESSES.” It is used in the design, development, and formulation of
new products, as well as in the improvement of existing product designs.
What is optimization ?
Merriam Webster dictionary defines optimization as
“an act, process, or methodology of making something (as a design, system, or decision)
as fully perfect, functional, or effective as possible;specifically:the mathematical
procedures (as finding the maximum of a function) involved in this”
Optimization characterizes the activities involved to find “the best”.
Inpharmaceuticsoptimizationisrelativetoprocessingandformulationofdrug
products

account all the factors that influences decisions in any experiment.

of
CRITERIA
Inpharmaceuticsoptimizationisrelativetoprocessingandformulationofdrug
productsinvariousforms.
process of finding the best way of using the existing resources while taking into
account all the factors that influences decisions in any experiment.
Pharmaceutical formulator aims to produce product not only meets the requirements
of BIO-AVAILABILITYbut also from the PRACTICAL MASS PRODUCTION
CRITERIA 3

4. IDENTIFICATION
OF FACTORS
SCREENING
OF FACTORS
RESPONSE
SURFACE
METHODS
OPTIMIZATION
Factors that
define a
process,
property is
identified
Screening to
identify most
influential
factors
Optimization
and
identification
of the design
space
4

5. •Pharmaceutical formulating in research and industrial level involves large number
•
•Pharmaceutical formulating in research and industrial level involves large number
of attributes. Any process or product under study may have various critical factors
or attributes defining it. Optimization studies are aiming at identifying and
quantifying these factors to obtain a design space where an optimized result is
expected.
•For addressing these critical factors and its effects different terms are coined in
experimental designs
BACKGROUND
Independent variables
Independent or input variables are those
factors which are at control for the
experimenter. These factors are varied in
an accepted limits to identify the response
it makes on the dependant variable
Independent variables
Independent or input variables are those
factors which are at control for the
experimenter. These factors are varied in
an accepted limits to identify the response
it makes on the dependant variable
Dependant variable
Dependant variables are those
factors which are studied in an
experiment. Factors are
described to define this
independent variable
Dependant variable
Dependant variables are those
factors which are studied in an
experiment. Factors are
described to define this
independent variable
While defining the independent variables a confidence level must be
specified. This is obtained by the extensive literature survey and analysis of
the data
In pharmaceutics there are two type of variables PROCESS AND
FORMULATION VARIABLES
5

6. •Consider anydependent variable “y”and there is a set ofinput variables
(dependant variable) x
1, x
2, …, x
k( For e.g. y might be the viscosity of a
polymer and x
1, x
2, and x
3might be the reaction time, the reactor
temperature, and the catalyst feed rate in the process).
•When the underlying mechanism is not fully understood, and the
experimenter must approximate the unknown function f with an appropriate
empirical model (εis error factor),
y = f(x, x, …, x) + εy = f(x
1, x
2, …, x
k) + ε
HOW TO GENERATE A MODEL ?
DESIGN OF EXPERIMENTS(DOE)
MULTIPLE LINEAR REGRESSION
TECHNIQUE
SURFACE RESPONSE PLOTS
This
empirical
model is
called
RESPONSE
SURFACE
MODEL
This
empirical
model is
called
RESPONSE
SURFACE
MODEL
6

7. Y= β+βX+βX+βXX
2Y= β
0+β
1X
1+β
2X
2+β
12X
1X
2
B0 –CONSTANT
B1, B2, B12
Y
X1, X2
X1X2
B0 –CONSTANT
B1, B2, B12 –REGRESSION COEFFICIENT
Y-DEPENDANT VARIABLE
X1, X2-INDEPENDENT VARIABLES
X1X2-INTERACTION
+ SIGN INDICATES
INCREASE IN Y
-
DECREASE IN Y
+ SIGN INDICATES
INCREASE IN Y
-SIGN INDICATES
DECREASE IN Y
IT CAN BE 1
ORDER OR 2
ORDER
POLYNOMIALS
IT CAN BE 1
ST
ORDER OR 2
ND
ORDER
POLYNOMIALS
BIVARIATE -MULTIVARIATE
7

8. SURFACE RESPONSE PLOTS
•Graphical perspective of the information is plotted in 3 dimensional
•
•Graphical perspective of the information is plotted in 3 dimensional
responsesurface plots.
•It is also convenient to view the response surface in the two-
dimensional. This type of plot is called contour plot
Design
space
8

9. TYPES OF SECOND ORDER RESPONSE
SURFACES AND THEIR CONTOUR PLOTS.
TYPES OF SECOND ORDER RESPONSE
SURFACES AND THEIR CONTOUR PLOTS.
(a) shows the surface with a maximum point,
(b) shows the surface with a minimum point
(c) shows the surface with a saddle point.
9

10. HOW TO
KNOW
THE
DATA
MODEL
EXACTLY
FITS THE
DATA?
STATISTICAL METHODS
P VALUE -The p-value for each
term tests the null hypothesis that
the coefficient is equal to zero (no
effect). A low p-value (< 0.05)
indicates that the independent
variable has significant effect on
response variable.
R SQUARE VALUE-Accounts for
the variation of the data. Higher the
R square the curve fits
better(max=1).
F-VALUE-TheF ValueorF ratiois
thetest statistic used to decide
whether the model as a whole has
statistically significant predictive
capability. The null hypothesis is
rejected if the F ratio is large. If the
F ratio is high model is therefore
accepted.
EXPERIMENTAL
METHOD
Validation of the
model can be done
by the
experimental
method. Testing of
the sample in the
design space
matching the
observed value
and the predicted
value
10

11. SCHEME OF OPTIMIZATION
11

12. WHY TO DESIGN EXPERIMENT?
TO OBTAIN MAXIMUM INFORMATION
FROM MINIMUM NUMBER OF
EXPERIMENTS.
TO SCREEN FACTORS WHEN LARGE
NUMBER OF FACTORS ARE PRESENT
TO GET A CURVE THAT EXACTLY FITS
THE ACTUAL MODEL. EXPERIMENTS HAS
TO BE DESIGNED IN SUCH A WAY THAT
OBTAINED DATA EXACTLY SIMULATES
ORIGINAL MODEL
HOW TO CONDUCT EXPERIMENTS?
DIFFERENT SOFTWARES ARE AVAILABE
Design expert
Statistica
MODDE
DOE++
StatSoft
Experstat
12

13. EXPERIMENTAL DESIGNS FOR OPTIMIZATION
SCREENING DESIGNS
•TWO LEVEL FACTORIAL
DESIGN
•FRACTIONAL FACTORIAL
DESIGNS
•PLACKETT BURMAN DESIGN
•MIXED LEVEL SCREENING
DESIGN
•D OPTIMAL SCREENING
DESIGN
RESPONSE SURFACE
DESIGN
•THREE-LEVEL FULL
FACTORIAL
•CENTRAL COMPOSITE
•BOX–BEHNKEN
•DOEHLERT DESIGNS.
13

14. POINTS ESTIMSTED
BY DOE
14

15. PLACKETT BURMAN DESIGN
• in 1946 by statisticians Robin L. Plackett and J.P. Burman, it is
an efficient screening method to identify the active factors using as few
•
•Only main effects are analyzed in this design. Interaction effects are not
•
•Developed in 1946 by statisticians Robin L. Plackett and J.P. Burman, it is
an efficient screening method to identify the active factors using as few
experimental runs as possible.
•Placket and Burman showed how full factorial design can be
fractionalized in a different manner, to yield saturated designs where
the number of runs is a multiple of 4, rather than a power of 2.
•Only main effects are analyzed in this design. Interaction effects are not
included
•General polynomial obtained plackett burmen screening is
Y = b
0+ b
1x
1 + b
2x
2+ ...................... + b
kx
k
b(0-k) is the regression coefficients
Y is the dependant variable
K number of independent variable
X (1-k) independent variables
15

16. Possible to neglect higher order
When to use plackett burman?
•Carried out at the early stages
for screening
•Possible to neglect higher order
interactions
•2-level multi-factor
experiments.
•When number of factors
involved is higher.
•To economically detect large
main effects.
•Particularly useful for number
of runs(N)= 12, 20, 24, 28 and
36
ASSUMPTIONS
•Fractional factorial designs for studying k
= N–1 variables in Nruns, where Nis a
multiple of 4.
•Only main effects are of interest.
•No defining relation since interactions
are not identically equal to main effects.
16

17. APPLICATION OF PLACKETT–BURMAN SCREENING DESIGN FOR PREPARING
GLIBENCLAMIDE NANOPARTICLES FOR DISSOLUTION ENHANCEMENT
Sunny R. Shah, Rajesh H. Parikh, Jayant R. Chavda, Navin R. Sheth; Bhagvanlal
Kapoorchand Mody Government Pharmacy College, Rajkot, India, Ramanbhai
Patel College of Pharmacy, CHARUSAT, Changa, India; Department of
Pharmaceutical Sciences, Saurashtra University, Rajkot, India
OBJECTIVE: -To improve the dissolution characteristics of a poorly water-soluble drug
glibenclamide (GLB), by preparing nano particles through liquid anti solvent precipitation.
A Plackett
and process variables.
OBJECTIVE: -To improve the dissolution characteristics of a poorly water-soluble drug
glibenclamide (GLB), by preparing nano particles through liquid anti solvent precipitation.
A Plackett–Burman screening design was employed to screen the significant formulation
and process variables.
INDEPENDENT
VARIABLES
DEPENDANT
VARIABLES
amount of
poloxamer 188 (X1),
amount of PVP S
630 D (X2),
solvent to anti
solvent volume ratio
(S/AS) (X3),
amount of GLB(X4)
speed of mixing (X5).
Mean particle size
(Y1),
Saturation solubility
(Y2)
% DE 5min
dissolution efficiency
after 5 min (Y3)
GLB was dissolved in acetone at definite
concentration and sonicated for 20 s. The
solution was filtrated through a 0.22 μ
Whatman filter paper. The prepared GLB
solution was injected by syringe onto the
tip of the anti solvent water containing
each specific concentration of polymer
and/or surfactant with stirring.
Precipitation took place immediately
upon mixing and formed a suspension
with bluish appearance. Then centrifuged
and dried to get nanoparticles 17

18. A total of 12 experimental trials involving 5 independent variables were generated by
Minitab® 16 (USA).
DISSOLUTION EFFICIENCY AFTER 5 MINUTUS(% DE
5min)=13.4 +0.156 PX−0.578
PD+0.513S/AS−0.0721 Drug concentration +0.0387 Speed
SATURATION SOLUABILITY(SS) = 9.87 + 0.0893 PX−0.193 PD+ 0.253S/AS −0.0352 Drug
concentration + 0.0151 Speed
MEAN PARTICLE SIZE(PS) = 830−8.14 PX + 12.8 PD−11.1S/AS+1.42 Drug
concentration−0.676 Speed
POLYNOMIAL EQUATIONS DERIVED
18

19. All the predetermined independent variables except drug concentration were
found to affect the dependent variables. The optimized formulation maintained
the crystallinity of GLB and released almost 80% drug as compared to pure GLB
which showed 8.5% dissolution within 5 min. The improved formulation could
offer an improved drug delivery strategy, which still needs to be correlated with
in vivo studies.
All the predetermined independent variables except drug concentration were
found to affect the dependent variables. The optimized formulation maintained
the crystallinity of GLB and released almost 80% drug as compared to pure GLB
which showed 8.5% dissolution within 5 min. The improved formulation could
offer an improved drug delivery strategy, which still needs to be correlated with
in vivo studies.
RESULTS AND CONCLUSIONS
19

20. CENTRAL COMPOSITE DESIGN
A Box-Wilson Central Composite Design, commonly called `a central composite design is
an important experimental design which gives a response surface curve that suites a 2
nd
order polynomial(quadratic) empirical model.
A second-order model can be constructed efficiently with central composite designs
(CCD). It is a combination of
It contains an imbedded factorial or fractional factorial design with center points that is
enlarged with a group of `star points' that allow estimation of curvature and formation of
second order model
FIRST-ORDER (2) DESIGNS FIRST-ORDER (2
N
) DESIGNS
(FACTORIAL DESIGNS)
+
CENTRE AND AXIAL POINTS
20

21. SALIENT FEATURES OF CENTRAL COMPOSITE DESIGNS
The design consists of three distinct sets
of experimental runs:
A factorial(perhaps fractional)
design in the factors studied, each
having two levels;
A set of center points, experimental
runs whose values of each factor are
the medians of the values used in
the factorial portion.
A set of axial points (star point),
experimental runs identical to the
centre points except for one factor,
which will take on values both
below and above the median of the
two factorial levels, and typically
both outside their range.
If the distance from the center of the
design space to a factorial point is ±1
unit for each factor, the distance from
the center of the design space to a star
point is|α| > 1. The precise value
ofαdepends on certain properties
desired for the design and on the
number of factors involved.
The value of αis given by the following
equations
First equation is for full factorial cases and
second one for fractional factorial
α=*2
k
]
1/4
α = *number of factorial runs+
1/4
21

22. Thestarpointsrepresentnewextremevalue(low&high)foreach
factorinthedesign
Topicturecentralcompositedesign,itmustimaginedthatthereare
severalfactorsthatcanvarybetweenlowandhighvalues.
Centralcompositedesignsareofthreetypes
CIRCUMSCRIBED(CCC)
DESIGNS-Cube points at
the corners of the unit cube
,star points along the axes
at or outside the cube and
centre point at origin
INSCRIBED (CCI)
DESIGNS-Star points take
the value of +1 & -1 and
cube points lie in the
interior of the cube
FACED(CCI) –star points
on the faces of the cube.
22

23. Development and optimization of baicalin-loaded solid lipid nanoparticles
prepared by coacervation method using central composite design
Jifu Hao , Fugang Wang, Xiaodan Wang; Department of Pharmaceutics, College
of Pharmacy, Shandong University, 44 Wenhua Xilu, China
OBJECTIVE : -To design and optimize a novel baicalin-loaded solid lipid nano particles (SLNs)
carrier system composed of a stearic acid alkaline salt as lipid matrix and prepared as per the
coacervation method in which fatty acids precipitated from their sodium salt micelles in the
presence of polymeric nonionic surfactants. A two-factor five-level central composite design
(CCD) was introduced to perform the experiments
ENCAPSULATION EFFICIENCY,
PARTICLE SIZE AND
POLYDISPERSITY INDEX (PDI) ARE
THE DEPENDANT VARIABLES
WHICH ARE EVALUATED

DRUG/LIPID RATIO AS
INDEPENDENT VARIABLE

SELECTED AS A LIPID MATRIX

ARE SELECTED
ENCAPSULATION EFFICIENCY,
PARTICLE SIZE AND
POLYDISPERSITY INDEX (PDI) ARE
THE DEPENDANT VARIABLES
WHICH ARE EVALUATED
AMOUNT OF LIPID AND THE
DRUG/LIPID RATIO AS
INDEPENDENT VARIABLE
STEARATE SODIUM WAS
SELECTED AS A LIPID MATRIX
AXIAL AND CENTRAL POINTS
ARE SELECTED
23

24. CCD matrix was generated by Design-Expertsoftware.
A total of 13 experiments, including four factorial points, four axial points and five
replicated center points were selected.
A quadratic polynomial model was generated to predict and evaluate the
independent variables with respect to the dependent variables.
Yi = A0 + A1X1 + A2X2 + A3X1X2 + A4X
2
1 + A5X
2
2Yi = A0 + A1X1 + A2X2 + A3X1X2 + A4X
2
1 + A5X
2
2
24

25. Response surface analyses were plotted
in three-dimensional model graphs for
optimization of nano particles with
suitable and satisfied physicochemical
properties.
It describe the interaction and quadratic
effects of two independent variables on
the responses or dependent variables.
25

26. RESULTS
Optimization of an SLN formulation is a complex process, which requires to consider a
large number of variables and their interactions with each other.
This study conclusively demonstrates that the optimal formulations may be successfully
obtained using the central composite design.
The derived polynomial equations and response surface plots aid in predicting the values
of selected independent variables for preparation of optimum formulations with desired
properties
The composition of optimum formulation was determined as 0.69% (w/v) lipid and
26.64% (w/w) drug/lipid ratio, which fulfilled the requirements of optimization. At
these levels, the predicted values of Y1 (EE), Y2 (particle size), and Y3 (PDI) were
84.13%, 356.6 nm, and 0.178, respectively.

formulation factors levels was prepared. The observed optimized formulation had EE of
(86.29
were in good agreement with the predicted values.
The composition of optimum formulation was determined as 0.69% (w/v) lipid and
26.64% (w/w) drug/lipid ratio, which fulfilled the requirements of optimization. At
these levels, the predicted values of Y1 (EE), Y2 (particle size), and Y3 (PDI) were
84.13%, 356.6 nm, and 0.178, respectively.
To confirm the predicted model, a new batch of SLN according to the optimal
formulation factors levels was prepared. The observed optimized formulation had EE of
(86.29 ±1.43)%, particle size of (343.7 ±7.07) nm, and PDI of (0.169 ±0.0036), which
were in good agreement with the predicted values.
CONCLUSION
26

27. BOX-BEHNKEN DESIGN
INTRODUCTION
Behnken designs have treatment combinations that are at the midpoints of
INTRODUCTION
This is a second order design introduced by Box and Behnken in 1960.They
do not contain embedded factorial or fractional factorial design. Box-
Behnken designs have treatment combinations that are at the midpoints of
the edges of the experimental space and require at least three continuous
factors. The following figure shows a three-factor Box-Behnken design.
Points on the diagram represent the experimental runs that are done:
Yo = bo + b1X1 + b2X2 + b3X3 + b12X1X2 + b13X1X3 +
b23X2X3 + b11X
2
1+ b22X
2
2+ b33X
2
3
The general quadratic equation obtained. It gives
three type of information
(1)main effects for factorsx
1, ...,x
k,
(2)their interactions (x
1*x
2, x
1*x
3, ... ,x
k-1*x
k),
(3)quadratic components (x
1**2, ..., x
k**2).
27

28. Number of
experiments in a
Box Behnken design
is determined by the
following equation
N =(2 f ( f − 1)) + c 0
F is the number factors
C0 is the central point
Levelsareusually
expressedas(coded–
1,0,+1)i.e.lower
higherandmiddle
Thethirdlevelfora
continuousfactor
facilitatesinvestigation
ofa quadratic
relationshipbetween
theresponseandeach
ofthefactors
28

29. WHY BBD IS USED?
It is rotatable or nearly rotatable second-order


It is rotatable or nearly rotatable second-order
designs based on three-level.
More efficient compared to CCD and other
designs.(number of coefficients in the model
divided by the number of experiments). More
data in less number of runs.
BBD does not contain combinations for which all
factors are simultaneously at their highest or
lowest levels. So these designs are useful in
avoiding experiments performed under extreme
conditions.
29

30. STATISTICAL OPTIMIZATION AND CHARACTERIZATION OF PH-INDEPENDENT
EXTENDED-RELEASE DRUG DELIVERY OF CEFPODOXIME PROXETIL USING
BOX–BEHNKEN DESIGN
Ali Mujtaba, Mushir Ali, Kanchan Kohli. Department of Pharmaceutics,
Faculty of Pharmacy, Hamdard University, New Delhi 110062, India
OBJECTIVE: -To develop and optimize the pH-independent extended-release (ER)
formulations of cef
by employing a 3
OBJECTIVE: -To develop and optimize the pH-independent extended-release (ER)
formulations of cef-podoxime proxetil (CP) using response surface methodology
by employing a 3-factor, 3-level Box–Behnken statistical design.
INDEPENDENT
VARIABLES
DEPENDANT
VARIABLES
AMOUNT OF RELEASE
RETARDANT POLYMERS
HPMC K4M
(X1),
sodium
alginate (X2)
MCC (X3).
Cumulative
percentage
release of drug at
2, 4, 8, 14 and 24
hours to detect
the burst effect
and to ensure
complete drug
release.
30

31. A total of 15 experimental runs are conducted to obtain results. Statistical analysis
were done for the fitting of the curve.
31

32. Evaluation of main effects, interaction effects and quadratic effects of the
formulation ingredients on the in vitro release of CP extended-release formulations.
Second order nonlinear polynomial models with Design Expert®(Version 8.0.7.1, Stat-
Ease Inc.)
Yo = bo + b1X1 + b2X2 + b3X3 + b12X1X2 + b13X1X3 + b23X2X3
+ b11X
2
1 + b22X
2
2 + b33X
2
3
Yois the dependent variable
bois an intercept
b1–b33are regression
coefficients computed from the
observed experimental values of Y
X1–X3are the coded levels of
independent variables.
X1X2and Xi(i= 1, 2 or 3)
represent the interaction and
quadratic terms
32

33. RESULTS AND CONCLUSIONS
Two-dimensional contour plot and three-
dimensional response surface plot are
presented.

was kept at a 20% (w/w) level.

nearly linear relationship of factor X1with
factors X2 on the response Y2 h.

relationship on the responses Y4 h, Y8 h, Y14
hand Y24 h.
Two-dimensional contour plot and three-
dimensional response surface plot are
presented.
In all the presented figures, the third factor
was kept at a 20% (w/w) level.
The almost straight lines in plots predicted
nearly linear relationship of factor X1with
factors X2 on the response Y2 h.
However, factors X1and X2 have non-linear
relationship on the responses Y4 h, Y8 h, Y14
hand Y24 h.
The optimum values of the variables were obtained by
graphical and numerical analyses using theDesign-
Expert®software. The optimum response was found with
Y2 h(19.89%), Y4 h(42.15%), Y8 h(68.32%), Y14
h(83.02%) and Y24 h(98.4%) at X1, X2and X3values of
25, 15.14,20.14% respectively. To verify these values, the
optimum formulation was prepared according the above
values of the factors at X1, X2and X3and subjected to the
dissolution test. 33

34. REFERENCE
•BiekeDejaegher, YvanVander Heyden; Experimental designs and their recent advances in set-up, data
interpretation,andanalytical applications; Journal of Pharmaceutical and Biomedical Analysis 56 (2011)
141–158
•Raymond H. Myers, Douglas C. Montgomery, Christine M. Anderson-Cook;3
RD
Edition; Response surface
methodology : process and product optimization using designed experiments.
•Pharmaceutical statistics; practical and clinical application 5
th
edition. Sanford Bolton; Charles Bon
34

35. 35