Fd

1. Factorial Designing – An Essential Tool in Pharmaceutical
Optimization
Alka Singh1
* & C.S. Chauhan2
1. Research Scholar, B.N Institute of Pharmaceutical Science, B.N. University Udaipur,
Rajasthan, India
2. Professor, B.N Institute of Pharmaceutical Science, B.N. University Udaipur, Rajasthan,
India
1
Email- alkasingh1790@gmail.com
2
Email- bhuraj@rediffmail.com
*Corresponding Authors
Alka Sing
B.N Institute of Pharmaceutical Science, B.N. University Udaipur, Rajasthan
E.mail: Alkasingh1790@gmail.com Contact no- 8279960272
Abstract
With the ever-increasing techniques of research, the traditional experimental design is not
sufficient as well as satisfying to contribute enough to the experiment's robustness.
Formulation development by utilising factorial design is a smarter way of experimentation
which act as a crucial tool for optimization. Factorial design is more efficient for the type of
experiments having effect of two or more factors. Factorial design is more adaptable and
provides more significant information about the process and product, as well as a variety of
software. This manuscript provides the reader with a thorough understanding of factorial
experimental design, its theoretical foundation, and how to utilise the statistical tool in real-
world research.
Keywords: factorial design, optimization, research
INTRODUCTION:
Factorial design is a statistic type research methodology that recognize the interactive effect of
all possible combination variables in a set of experiment. A full factorial experiment is a
statistical design that includes two or more factors, each having a discrete possible value or
"level," and whose experimental units include all potential combinations of these levels across
all such factors. A fully crossed design is another name for a full factorial design. An
experiment gave the researcher the chance to examine both the individual components' and
their interactions' effects on the response variable.
For the vast majority of factorial experiments, each factor has only two levels. For example
with two factors each taking two levels, a factorial experiment would have four treatment
combinations in total and is usually called a 2x2 factorial design.
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2. A fractional factorial design may be used, in which part of the possible combinations are
excluded (often at least half), if the number of combinations in a full factorial design is too
high to be logistically practical.
History of statistics application in research
The world has never experienced so many scientific and technological breakthroughs and inv
entions
in its recorded history of human civilization. When it comes to production now a days both
quality and quantity are equally important. Thus standard operation procedures (SOP), in
process quality control (IPQC) and Total quality management (TQM) came into existence. All
of these advancements are based on the idea of reducing errors and saving time, energy as well
as money. Reliable projections based on scientific ideas lend credibility and robustness to a
product or process. In a traditional experimental approach, quality is ensured through physical
testing and inspection, standards are based on batch history, and alterations are discouraged
and thus freezing the process, specifications. Quality by design QbD is more reliable approach
for processing data. In this regard, design of experiments (DOE) has proven to be an effective
strategy for identifying specific characteristics that influence product defect levels. DOE has a
long history dating back to the 1920s, when Prof. Ronald Fisher, a British statistician,
pioneered ground breaking uses in scientific research. Engineers and researchers can now use
this technology as a universal software tool.
Common terminologies used in factorial design are given below-
Optimization- It is the process of determining the most effective approach to use the already
available resources while taking into account every factor that affects a decision in every
experiment. Perfect, effective or functional as possible by choosing best elements from some
set of available alternatives. Therefore, what determines an experiment's best performance is
the choice of experimental conditions. The methodical design of tests used in modern
pharmaceutical optimization aims to reduce formulation inconsistencies.
Objective- Used to specify the objective of an optimization experiment or the property of
interest (criteria).
Variables- Development of a product/ process involves several influential factors, at various
% of influence. Variations are independent, dependent, quantitative and qualitative variables.
Independent variable are under direct control of the investigator (drug concentration, polymer
composition etc.) Dependent variable are the response of the finished product (tablet/
microsphere) based on the influence of dependent variables eg.,drug release profile
etc. Quantitative variables are those can take up numerical values. Eg. Temperature, pressure,
concentration. Qualitative variables include type of carrier/ polymer etc.
Factor- A factor is an assigned variable such as concentration, temperature etc. Quantitative
factor has a numerical value (1%,2%,3%), qualitative factors include batch of
materials, excipients, treatment, diet, labs, analysts etc. and they are not numerical.
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3. Levels- A factor's levels are values or labels that have been ascribed to it. Descriptions of
factor and levels is given in table no. 1.
Response- The association between the change in each factor's level and the change in reaction
is measured by an objective variable that is calculated.
Effect- Is the change in response by a varying level of a particular factor and give relationship
between various factors and level.
Table 1. Description of factor and level
Factor Level
Concentration 1%, 2%
Temperature 20º,30º
pH 4.6, 6.8
Design space- It is the recognised range of process variables that has been shown to offer QA.
Generally speaking, modifying the allowed ranges for process parameters and formulation
qualities is not regarded as working within the design space. Movement out of space is seen as
a change and ordinarily starts a post-approval regulatory change process. 1,2
.
Factorial experimental design:3, 4
Utilizing the design space allows for reliable and consistent information to be obtained with
the least amount of tests possible while adopting QbD.
Factorial Designs and modifications
The experimental design's mathematical model is made up of two or more "factors" that operate
on two or more "levels."
These are of two types Full factorial design or Fractional factorial design
Full factorial design: YX
:22
,23
,32,
33
X=Factors and Y= levels
Factorial Designs First-degree mathematical models serve as the foundation for factorial
designs (full or fractional).
Full Factorial Designs: The influence of every factor (n) at different levels (x), as well as their
interactions, with the total number of experiments as xn
.
Symmetric: Similar numbers of levels exist for each factor.
Asymmetric: For each factor, there are varying numbers of levels.
Merits of factorial designing:
1. Quality incorporated into product as well as process, is based on scientific understanding.
2. Specifications will be based on performance of the product.
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4. 3. Focus on robustness, understanding and controlling.
4. Process is flexible within the design space.
5. Submission of the report will be explained by the product and process knowledge.
Number of variables influences the results of a research product and process development.
Variables are called as “factors” in factorial design. Some factors never been considered or it
might have discovered during the process. Byrne and Taguchi divided potential experiment-
influencing elements into controllable (noise) and non-noise categories. Additional categories
for noise concerns include expensive, difficult, and impossible to control. Researchers mainly
use controllable factors. Thus random factors and co-variants are seldom used. Factorial
techniques are the foundation of ANOVA and multiple regression analysis. Software is
available to assist with the calculations. 5, 6
.
Factorial design (FD) is also known as experimental designs for the first degree models, are
the most common technique. The simplest way to set up a design of experiments (DOE) is to
take 2 or more variables (n) and test at different levels. In a full factorial approach all factors
are combined with each other on all levels and the number of experiments
becomes f n
where f is the factor and n is the level. 32
full factorial design involves nine
experiments, 42
involves 16 and 52
involves 25. if the level becomes 3 then the number of
experiments becomes 33
, 43
and 53
. Naturally the number of experiments becomes more and
exceeds manageable levels. Therefore, the levels considered are usually 2 to minimize the
number of experiments. If each factor has the same number of levels, the design is said to be
symmetric eg. 22
, 33
etc. If the number of level differs from the factor the design is called
asymmetric, eg. 23
, 32
etc. But if it is essential to conduct the design for all required
experiments, one can consider fractional factorial design (FFD). Here the experiments are cut
short systematically. FFD is a fraction (1/ xp
) of the complete FD, were p is degree of
fractionation. The total number of experiments for FFD is given by f n-p
. Each
experimentation is called as “trial” or “run”. Standard symbols, data interpretation,
experimental design representation and interaction are mentioned in table no. 2, 3 respectively.
Table 2. Standard symbols for particular ratio of drug: excipients
Formulation Standard symbols Effect (%drug release)
Low drug- low excipient 1 10 %
Low drug- high excipient a 10%
High drug- low excipients b 20%
High drug- high excipient ab 30%
Note: low and high value refers to low and high concentration presented for the drug
and excipients.
Interaction = [ab-b]- [a-(1)] / 2 = 5%
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5. Table 3. Experimental Matrix
Experiment f1 f2 f3 Interpretation
1 -1 -1 -1 Zero level interaction
2 -1 +1 -1 Main factor effect f2
3 +1 -1 -1 Main factor effect f1
4 -1 -1 +1 Main factor effect f3
5 +1 +1 +1 Interaction between f1, f2, f3.
22
design (4 experiments can be conducted )and 23
design (8 experiments can be carried out).
Low (-1) and high (+1) levels are combined together.
Central composite design (CCD):
Box et al. initially described Central Composite Design (CCD), a unique advanced type of full
factorial design. Instead of square and cube, 22
and 23
are represented by circular or spherical
respectively. A centroid experiment (axial points) and a group of experiments (star points) are
also included in the 2n full factorial design. The star points are placed at specific intervals along
the axis from the centroid in order to create circular or spherical domains.
Fractional factorial design is opposite to Taguchi model. To avoid the above problems
fractional factorial designs were introduced. Here large multi fractional studies should be
divided into blocks. Within each block experiments were undertaken randomly. Blocks must
be performed one at a time. Correction in the factorial design can be made between the blocks.
Advantage of such study is that first block may reveal all information required. Some free
software downloads are also available regarding factorial designing7
Demonstration of factorial designing:8
Adetogun GE et al. has reported the implementation of factorial designing, in his experiment.
Experiments were conducted using a factorial design and formulation statistics to examine the
effects of the gum type used as a binding agent (B), its concentration (C), and the relative
density (D) of the tablet on the tensile strength (TS), brittle fracture index (BFI), disintegration
time (DT), and crushing strength-friability/disintegration time ratio (CSFR/DT) of paracetamol
tablets. Here each of high variables were utilized as “high level” ( subscript H)and “low level”
( subscript L). Number of experiments were 23
ie, 8. Thus the combinations were:
BLCLDL, BLCLDH, BLCHDL, BLCHDH
BHCHDH, BHCHDL, BHCLDH, BHCLDL
BL : represents formulation with binding agents Delonix regia seed gum + tragacanth or acacia
gum + tragacanth.
BH : represents formulation with binding agents tragacanth + acacia gum or delonix regia seed
gum + acacia gum.
CH and CL represents high (5% w/w) and low concentration (2% w/w) of gum binding agent
respectively.
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6. Tablets with relative densities of 0.80 and 0.90, respectively, are represented by DL and DH.
It was feasible to evaluate the effects that each of the three variables (B, C, and D) had on the
mechanical/disintegration qualities of the tablets and ascertain if the variables were interacting
independently of one another by combining the data from the combinations into a number of
sets. By adding up all "high" values of B and deducting the sum of all "low" levels of B, it was
possible to determine the impacts of increasing B on the mechanical/disintegration
characteristics.
¼ {(BHCHDH + BHCHDL + BHCLDH + BHCLDL) – ( BLCLDL + BLCLDH + BLDHDL + BLCHDH)}
The same procedure was used for C and D. Then the results of combinations n which they
appear “high” and “low” levels were summed up and the sum of the combinations was
subtracted to obtain the interaction coefficient. Eg. For B and C:
¼ {(BLCLDL + BLCLDH + BLCHDL + BLCHDH) – ( BHCHDH + BHCHDL + BHDLDH + BHCLDL)}
A zero result indicates no interaction, but if the interaction coefficient differs from zero, then
the two variables concerned were interacting with each other. The more the coefficient differs
from zero, the higher the interaction. The results were subjected to the analysis of variations
(ANOVA) at a 5 % probability level.
Demerits of factorial designing:
1. Insignificant variables may not be able to detect on time.
2. Not possible to separate the aliased effects from each other.
3. Later if the result indicates undesired effects and then all the experiments undertaken at
that level are of no use.
4. At worse the full experimental plan has to be redesigned and repeated.
5. May not be economic and consumes more time.
Conclusion
Factorial design and statistical analysis have been used in the formulation development by
optimization of pharmaceutical process and product development by systematic approach
which facilitate the best results with minimum efforts and maximum efficacy. By lowering the
number of experimental trials during formulation development, optimization strategies can
help reduce product costs. The improved experimental designs are also useful for examining
precise quality assurances for regulatory bodies in regards to improved product quality.
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