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within a range of Y
T1
to Y
T2
. Let us assume that a mathematical model has been developed
for each response variable, yielding model H and model T.
Now the calculations can be done, using a computerized program, and the only independent
variable values, (X i ), output from the program produces the responses that meet the
following constraints;
Lower < Y H < Upper, and
Lower < Y T < Upper
If the shape of response surface is to be examined, another data analysis option is to produce
graphics. These graphics can be looked like the contour plots or three-dimensional plots.
However, care should be taken if the number of variables studied is more than two. Because
some of the variables must be kept constant and all effects may not be clear.
Response Surface Method
This is a group of statistical experimental designs can be applied to pharmaceutical
development problems. As defined within the upper and lower limits of the independent
variables, a response surface is an area and it is a function of the relationship of these
variables to the measured response (dependent variables). The objective of response surface
studies is to get a regression model that gives a mode of changing of mathematical
evaluation in the response due to changes in the independent variables. When a minimum or
maximum of a surface is required, the process is called optimization. In the early 1950’s this
group of designs was introduced
23,24
. In different areas, these methods were accepted and
implemented by the researchers such as agriculture
25-28
, engineering
29,30
chemistry
31,32
,
veterinary science
33
, and other disciplines. Currently these methods have been applied to
pharmaceutical fields
34-40
.
Selection of the variables
Once the problem is correctly described for solution, for example the hardness and
dissolution rate of a tablet or modification of the rate of release of a drug from its dosage
form, appropriate variables should be selected. The selected independent variables should be
quantifiable and easily controlled. In these designs, the qualitative variables such as mixer X
or mixer Y cannot be used. The examples of quantifiable variables are compression force
and speed exerted by a tablet press, amounts of ingredients or their ratios, duration of
mixing, and temperature, humidity of the environment, etc. The range for each variable
should be recognized from the experience of the experimenter or from the results of small
pilot experiments. Each variable should be simply controlled to give suitable increments
within the selected range. It is important to set up proper range so that the product made in
experiment according to the design should be reliable for testing (measurement of a
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response). For example, setting the range of lubrication in a tablet formulation at 0 – 10%
may probably avoid the formation of suitable tablets within this range. A better suitable
range might be 0.25 – 1.0%. This depends on the properties of the formulation and the
previous experience of the formulator. The ranges selected should not be too wide or too
narrow. In case of too wide range, the model selected may not be sufficient to estimate
response surface, and the products produced in some experiments may not be suitable for
testing. If the range is too marrow, the scope for optimization of the formulation would be
limited.
Selection of a Model
A model, as mentioned earlier, should be selected during planning stage, before the
experiment is started. The model should be used to describe the relationship between
independent and dependent variables. This is necessary, because the selection of an
appropriate experimental design depends partly on the type and number of regression
coefficients to be judged.
Statistical theory is applied sufficiently to distribute the experimental points within the space
for the experiment. Each of the coefficients would be estimated with the same degree of
confidence. Accordingly, the developed model would estimate all regions of the surface with
similar consistency.
Some common regression models are shown below:
Where,
Y = measured response (physical characteristic)
X i = values of an independent variable
B o = constant
B i = coefficient of the linear terms
B
ii
= coefficient of the quadratic terms
B
ij
= coefficient of the cross product or interaction terms
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Where, i = (1 to n) and j = (2 to n) n = number of variables
Equation7 is a simple linear model with B o which represents the Y-intercept and B
1
represents the slope. Estimates of B o and B can be calculated for various values of X and the
corresponding response, Y. This can be performed by using a statistical software package
and a computer.
The same process can be used to estimate the coefficients in equations 8 – 10.The equation 8
is another linear model but it has three independent variables, X 1 , X 2 , and X 3 .
Equation 9 represents another linear model which includes cross product or interaction terms
that requires the estimates of cross product coefficients, B
12
, B
13
, and B
23
.
Thus, the equation 7, 8, and 9 can be used to describe linear relationship or planes (flat
surface).
The equation 10 is used in many response-surface designs because it utilizes quadratic terms,
Xi 2 that accounts for curvature in response surface. This model is called a second order
polynomial. It is useful because many response surfaces encountered have some degree of
curvature.
Based on an estimate of the type of response expected, an appropriate model should be
selected. Sometimes, such information may not be available. Thus, an equation of the type
represented by equation 10 is used. However, based on theory or previous empirical data the
experimenter can use a different model.
Experimental Error and Lack of fitness
Providing an estimate of experimental error is another important part of the planning stage.
This error is a measure of the intrinsic variability in the system under study. If a system
possesses a large number of variability, a reliable estimate becomes necessary. It may be
difficult to get a suitable mathematical model. Complete experiments should be repeated to
get an estimate of this type of error. By repeating the measurements, the precision of the
measurement operation can be easily checked. Repetition of the measurement can compare
the results of two exactly similar (identical) experiments. This comprises all errors present
during the whole experiment. An adequate number of repetitions should be determined
during the planning stage. In a design, it is not required to repeat all of the experiments in the
design.
On the other hand, lack of fitness indicates how well a regression model fits the generated
data. This is achieved by subtracting the experimental error from the total error listed after
performing a computerized regression analysis. The statistical significance of the lack of
fitness can be tested by use of an F- Test
40
.
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Contour Designs
It has been considered that quality of pharmaceutical products should be designed and builtin during manufacture of a product. Most of the quality problems have been found to be
related to the design of a pharmaceutical product
41
. A poorly designed pharmaceutical
product is likely to show poor safety and therapeutic efficacy even many tests or analyses
have been performed to establish its quality. Thus, the concept of quality by design (QbD)
was introduced. Quality cannot be improved simply by increasing the number of analyses of
pharmaceutical products. Thus, the quality must be built into the product. It is a systematic
approach to pharmaceutical development. It starts with predefined objectives, highlighting
the product, process understanding and in-process control. This is based on sound science
and quality risk management. The concept of QbD can be applied based on the knowledge
and scientific understanding to support pharmaceutical development
42
. The objectives of
pharmaceutical QbD are:
To get meaningful product quality specifications,
To increase process capability and decrease variability,
To improve pharmaceutical development and manufacturing efficiencies, and
To improve cause-effect analysis and regulatory flexibility
The implementation of risk-based approaches and pharmaceutical QbD has been supported
by most of regulatory agencies throughout the world
43
. The concept Pharmaceutical QbD
has been used to improve the manufacture of pharmaceutical products in terms of six-sigma
approach. Six-sigma is a system of practices to achieve process improvement; as a result, the
chance of out-of-specification (OOS) products can be significantly reduced. The results
showing out-of specification (OOS) indicate that this is an issue to pharmaceutical industries
44
.
However, due to poor reliability of analytical methods, the quality problems occur. Thus,
analytical QbD is useful in the development and optimization of robust and economic
analytical methods. In fact, analytical QbD, if implemented, can provide a better solution to
OOS results and can reduce the risk of method failure.
The development and optimization of pharmaceutical products and analytical methods have
been carried out traditionally by analyzing one factor at time (OFAT approach)
45-47
. Within
an appropriate range or level one factor is changed keeping other factors constant. The OFAT
approach does not allow evaluating the existence of interaction between the factors. This
may result in an inadequate conduction of the development and optimization. These
limitations can be overcome by design of experiments (DoE). DoE is a set of statistical tools
that provide better results with less number of experiments. The DoE include screening
designs and optimization designs.
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Steps involved
QbD consists of all elements of pharmaceutical development. This will allow designing a
quality product and its manufacturing process so that it can consistently deliver the intended
performance related to its safety and efficacy. With the help of QbD concept, pharmaceutical
development can provide a complete understanding of the product and its manufacturing
process
48
.Analytical methods are considered an integral part of pharmaceutical
development. If the QbD approach is applied to develop the analytical method, it would be
To attain regulatory flexibility,
To reduce out-of-specification results,
To achieve a high degree of robustness and a cost-effective analytical method
Establishment of Quality Target Profile (QTPP)/ Analytical Target Profile (ATP)
For summing up of quality characteristics of pharmaceutical products the Quality Target
Profile (QTPP)/ Analytical Target Profile (ATP) is established to ensure safety and efficacy.
The definition of QTPP consists of the expectations in final pharmaceutical product. It
includes
Use in clinical setting,
Route of administration,
Dosage form,
Delivery system,
Dosage strength,
Container closure system,
Factors affecting pharmacokinetic properties, and
Product quality criteria such as stability, purity, sterility, etc.
Analytical Quality by Design (AQbD) begins with the definition of Analytical Target Profile
(ATP). ATP defines the objectives of the analytical method. This will force
The selection of the method,
Design, and
Development activities
In other words,, it can be stated that ATP is a statement of what is to be measured, say API;
in which it should be measured, say in pharmaceutical dosage form; when and why it should
be measured say during final testing. A required level of confidence that is, target
measurement uncertainty – generally estimated from accuracy and precision, should be
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mentioned in the definition to ensure the quality of analytical results. A well-defined QTPP
or ATP avoids wasting of time and resource.
Identification of Critical Quality Attributes (CQA)/Analytical Method
Performance Characteristics (AMPC)
It is the next step in pharmaceutical QbD and CQA is chemical, physical, biological or
microbiological properties of pharmaceutical inprocess or finished product. These
characteristics must be within appropriate specifications so that the quality is ensured. CQA
include identity, assay, content, uniformity, degradation, products, residual solvents, drug
release or dissolution, moisture content, microbial limits, and physical properties such as
color, shape, size, and friability. To guide the product and process development, potential
CQA are derived from QTPP. Thus, the Critical Material Attributes (CMA) and Critical
Process Parameters (CPP) should be identified consequently to achieve the CQA and QTPP.
The physical, chemical, biological, or microbiological properties (CMA) complied by the
input materials can ensure the desired CQA. Similarly, the mixing time, stirring speed,
temperature, air flow, etc. (CPP) should be monitored before or during the process to ensure
the desired CQA.
The Analytical Method Performance Characteristics (AMPC) should be defined to meet the
needs of ATP. AMPC may be classified into two categories as per the source of error–
Systematic (bias) variability such as accuracy, specificity, and linearity
Random variability such as precision, limit of detection, and limit of quantification
Fig. 2.6 Schematic diagram of implementation of QbD 1. Pharmaceutical QbD 2. Analytical QbD
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•
Moreover, the definition of AMPC should include range and robustness. It is always
suggested to include a joint criterion of method characteristics (at least, accuracy and
precision) in ATP. Selection of analytical technique such as chromatographic,
spectrophotometric, or microbiological assays must be determined by ATP and AMPC
definitions. A schematic diagram of the steps for implementation of pharmaceutical QbD is
showed in Fig.2.6(1) . It includes the relationship among CMA, CPA, CQA, Design Space,
and QTPP. Moreover, the steps for implementation of analytical QbD are shown in Fig. 2.6
(2) . This includes the relationship CAP, AMPC, MODR, and ATP.
Risk assessment
A systematic process of organizing knowledge information to support decision is called Risk
assessment. There are three essential elements in risk assessment:
Identification of risk: Systematic use of information to identify potential sources of hazard
from historical data, theoretical analysis, and concerns to stakeholder,
Risk analysis: Estimation of risk associated with the identified hazards; and
Evaluation of risk: Comparison of the estimated risks using quantitative or qualitative scale
to determine their significance.
Ishikawa or fishbone diagram and failure mode and effects analysis (FMEA) is the risk
assessment tool wide used. With the help of Ishikawa diagram one can answer to this
question, “What might be wrong?” sometimes the FMEA method is used to carry out a
quantitative risk assessment. This provides a risk priority number [RPN = P×S×D] which is
calculated on the basis of occurrence probability (P), severity (S), and likelihood of detection
(D). FMEA helps one to answer the questions, “What is the probability that it will be
wrong?” and “What are the consequences (severity)?”
Design of Experiments (DoE)
For determination of the relationships between input factors ( x i – independent variables)
affecting one or more output responses (y – dependent variables), through the establishment
of mathematical models [y = f ( x i )], DoE is used. DoE is a structured and organized
method. In this method, the controlled input factors are systematically varied to determine
their effects on the output responses; this is used to determine the most important input
factors. The identification of input factors setting leading to optimized output responses, and
the clarification of interactions between input factors.
Selection of experimental design
For selection of best experimental design one should consider the following aspects:
Defined objectives,
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(a)
(b)
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Number of input factors,
Interactions to be studied,
Statistical validity
Effectiveness of each design
To provide a better understanding of application of DoE, experimental designs may be
divided into two types:
Screening designs; and
Optimization designs.
The Table 2.1 indicates a summary of screening and optimization design characteristics,
such as number of required, number of levels of input factors, and numbers of factors to be
studied.
Screening designs
Because of the cost-effective advantages, the most used screening designs are:
Two-level full factorial designs,
Fractionate factorial designs, and
Placket -Burman designs
Table 2.1 Summary of screening and optimization designs characteristics, number of experiments,levels, and factors
A wide number of input factors with reduced numbers of experiments can be studied by
these experimental designs. To provide a better understanding of the effects of input factors
on output responses, these should be considered; however, these have some limitations.
Two-level full factorial designs are the most powerful screening designs. These can be used
to estimate the main effects of input factors and their interactions on output responses. The
principal limitation of these designs is the requirement of large number of experiments
compared to fractionate factorial designs and Plackett-Burman designs. The number of
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experiments required for two-level full factorial designs may be calculated as 2k, where k is
the number of input factors to be studied.
The fractionate factorial designs are most widely used for screening purposes. These designs
allow to evaluate large number of input factors with a reduced number of experiments
required. This may be achieved by fractionating a full factorial 2k design as shown in Fig. 2.
7.
This may be obtained by fractionating a full factorial 2k design into a 2k–p design, where p
is the number of generators chosen to fractionate the design as shown in Fig. 2.7. For
example, four input factors when examined, a half-fraction factorial design (2
4–1
= 8
experiments) may be taken on. Similarly, a quarter fraction factorial designs (2
5–2
= 8
experiments) may be conducted to study five input factors. Thus, the number of experiments
remains same.
Fig. 2.7 Illustration of a half-fractionate two-level factorial design for three input factors (a) (2
3-1m
) matrix (b)
Complementary to A, matrix (c) Two-level full factorial design (2 3 ) matrix
Placket-Burman designs are special type of two-levels fractionate factorial designs
(resolution III). These allow to study up to N – 1 input factors with N experiments; where N
is the multiple of 4. An example of Placket-Burman design matrix used to study 11 input
factors with 12 experiments is shown in Table 2.2.
Table 2.2 Plackett-Burman design matrix to study 11 input factors (X1 to X11) with 12 experiments
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Fig. 2.8 Pareto charts representing the effects of main factors (a) (a, b, and C of input factors (b) Main effects (a, and b) and
interation effects (a*c) of input factors
Screening designs are sometimes used in the first step to DoE to select the most important
input factors and reject the insignificant ones. Pareto charts are useful tools to achieve this
purpose ( Fig. 2.8 ). These allow to put the input factors and their interactions according to
importance. For example, based on Pareto chart shown in Fig.2.8A , it can be concluded that
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