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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5670_Библиотеки_им_академика_М_И_Перельмана
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input factor ‘a’ is the most important, then the factor c and then b. When a fractionate
factorial design was used, attention should be paid to analyze Pareto chart.
For example, let us consider that the resolution between two peaks as functions of the pH of
mobile phase (a), amount of organic solvent in the mobile phase (b), and flow rate (c) are
being developed. On the basis of Pareto chart shown in Fig 2.8A , one can conclude that the
flow rate the flow rate is a critical analytical parameter. However, the flow rate is
confounded with the interaction between pH and amount of organic solvent in the mobile
phase (c = a×b). This conclusion was wrongly assumed because Pareto charts showed in Fig
2.8A was obtained using a fractionate factorial design with resolution III. When using a full
factorial, flow rate will not be confounded with the interaction between pH and amount of
organic solvent in mobile phase. This is evidenced in Pareto chart shown in Fig 2.8B.
Pareto charts does not provide information of how the output responses are affected by
varying input factor level. Such information may be achieved by the main effects and
interactions plots shown in Fig.2.9. For example, output response increased by varying the
input factor ‘a’ from low (–1) to high (+1) level; while output response decreases by varying
the input factor ‘b’ from low (–1) to high (+1) level as shown in Fig 2.9A.
Fig. 2.9 Plots of input factors (a) Main effects (b) Interaction
The advantage of DoE approach over the OFAT experimentation depends on the elucidation
of interactions between input factors. For example, in Fig 2.9B , the output response
increases significantly by varying the input factor ‘a’ from low (–1) to high (+1) level with
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(a)
(b)
(c)
‘b’ fixed at low level, while output response remains almost constant by varying the input
factor ‘a’ from low (–1) to high (+1) level with ‘b’ fixed at high level (+1). Plots of
interaction effects are used to identify the synergism or antagonism between input factors on
output responses. Central composite designs (CCD) are one of the most used optimized
design. They use 5 level of each input factor with a reduced number of experiments required
compared to 3 level full factorial design. A 3D spatial representation of CCD for three input
factors has been given in Fig 2.10. This design consists of the following parts:
The factorial design points (black dots);
The axial points (grey dots); and
The center point (white dots).
Fig. 2.10 Schematic diagram of three-level of cental composite design for 3 input factors
Optimization designs
Three-level full factorial designs, central composite designs, and Box-Behnken designs are
the most used optimization designs. They allow modeling complex response surface. One of
the most important drawback of screening designs depend on the fact that they only allow
modeling 1
st
order (linear) response surface. They have only two level for each input factor.
Optimization design uses 3 to 5 levels of each input factors. These allow modeling 2
nd
order
(quadratic) response surface
45
. Due to increased number of experiments required, they are
usually used to study a reduced number of numbers of input factors. Sometimes, three-level
full factorial design is used when two or three input factors are studied because an increased
number of experiments will be required. The number of experiments necessary may be
calculated as 3k, where k is the number of input factors to be studied. For example, a threelevel full factorial design for three input factors requires 3 3 = 27 experiments.
The mathematical model can be selected based on the application of Analysis of Variance
(ANOVA). The principal concept of ANOVA is to compare by varying the level of input
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factors with the variability due to residual error. From this comparison, it becomes possible
to evaluate the significance of the regression model as shown in Table 2.3.
Table 2.3 Three level full factorial design for three input factors (a, b, and c)
Regression analysis is valid when the residues (square of the difference between response
predicted by mathematical model and experimental response) present normal distribution
and homoscedasticity. If necessary, a Box-Cox transformation (for example, in
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transformation – λ = 0) of output response may be used to improve normality and
homoscedasticity of residues.
On the basis of ANOVA, it can be decided to include or to exclude the coefficients of linear
terms (e.g., a, b, and c), interaction terms (e.g. ab, ac, and bc), and quadratic terms. This
decision is based on p-values for each coefficient regression term. Commonly, a coefficient
regression term should be included in regression model when its p-value < 0.05. In other
words, the coefficient regression term is significantly different from 0. When the regression
coefficient term is different from 0 (p-value ˃ 0.05), it indicates that the output response is
not affected by varying the input factor levels. Thus, from regression model, this coefficient
regression term may be excluded. Adjustment of multiple regression models should be done
on the basis of determination coefficients, R 2 , R 2 -adj, and R 2 -pred. Determination
coefficient (R 2 ) is the proportion of the variance in the output response that is predicted
from the input factors. However, R 2 will always increase by adding new terms to the
regression model. The adjusted R 2 (R 2 -adj) is a modified version of R 2 . The R 2 -adj
increases if the new term improves the regression model and it decrease when the term does
not improve the regression model. Therefore, R 2 -adj can be used to compare the
explanatory power of regression models containing different number of terms. Predictive R
2
(R 2 -pred) indicates how well a regression model predicts output responses for new
observations. R 2 -pred can be calculated systematically by removing each observation from
the data set, estimating the regression equation, and determining how well the model
predicts the removed observation. R 2 -adj and R 2 -pred are always lower than R 2 .
Implementing control strategy and Continuous improvement
To ensure that critical material attributes (CMA) and critical process parameters (CPP) are
within the expected limits. Apparently, control space should be within the design space
42
.
img
Fig. 2.11 Additivity of effects (no Interaction)
In control strategy implementation, Analytical Process Technology (PAT) is an important
tool. Once it enables real-time release testing and provides an increased level of quality
assurance compared to conventional end product testing. PAT is not the only way to
implement real-time release. Predictive models can also be used as alternative to
conventional release testing
41
. The control strategy can be obtained from the data collected
during development of the method and validation for analytical QbD. This helps to predict
the ability of method to meet analytical target profile (ATP). Continuous monitoring of the
performance of method allows detecting, identifying, and dealing with out-of-trend
performance of the analytical method. The pharmaceutical product manufacturing and its
analytical method can be improved continuously throughout the lifecycle of the product by
the QbD concept. This includes
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Reduction of product variability,
Improvement of process performance,
Reduction of out-of-specification results,
Improvement of analytical performance, etc.
Applications of Design of Experiments in QbD and AQbD
Quality by design concept was accepted by FDA in 2004 and described in ‘pharmaceutical
cGMPs for 21
st
century – a risk-based approach’. International conference on harmonization
(ICH) Q8 pharmaceutical development, Q9 quality risk assessment, and Q10 pharmaceutical
quality system provide detailed requirements regarding pharmaceutical product quality. The
QbD and DoE approaches help to implement ICH/Q8 and ICH/Q9.
Since QbD concept has been accepted by FDA, DoE has been widely used to provide a
complete understanding of the product and its manufacturing process. With respect to
screening and optimization of pharmaceutical products and their manufacturing processes,
many applications can be found in the literature. Various input factors (independent
variables), such as excipient concentrations, stirring time, stirring speed, temperature,
pressure, etc. may be screened and optimized by using DoE. The output responses
(dependent variables) studied are particle size, entrapment efficiency, dissolution rate, etc.
Some examples of application of DoE have been given in table 2.4.
Application of screening designs in pharmaceutical QbD permits to identify the CMAs and
CPPs (independent variables) that can affect the critical quality attributes (CQAs)
(dependent variables) and, therefore, the quality target profile (QTPP).
Table 2.4 Some applications of DoE for pharmaceutical development, experimental design adopted. [independent variables
(input factors – Is) and dependent variables – Ds)]
img
img
img
Moreover, optimizing design, surface response methodology and multiple response
optimizations define a design space region in which CQAs and QTPP are attended. The
implementation of a design space region based on product and process understanding permit
regulatory flexibility. The changes within the design space region do not require prior
approval of regulatory department. Currently, DoE has been used in the rational
development and optimization of analytical methods. Compositions of culture media,
composition of mobile phase, flow rate, time of incubation are the examples of input factors
(independent variables). These may be screened and optimized using DoE. Various output
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responses such as retention time, resolution between peaks, microbial growth, are the
examples of dependent variables.
Factorial Designs
In experiments where the effects of different factors, or conditions, on experimental results
are to be elucidated, factorial designs are used. For example, to determine the effect of
compression pressure and lubricant on the hardness of a tablet formulation, to determine the
effect of disintegrant and amount of lubricant on tablet dissolution, or to determine the
efficacy of a combination of two active ingredients in an over-the-counter cough preparation,
factorial designs are used. For simultaneous determination of the effects of various factors
and to find out their interactions factorial designs are of choice. To understand the concept of
factorial design, following terms are to be known.
Factor
It is an assigned variable such as concentration, temperature, lubricating agent, drug
treatment, or diet. In an experiment, the choice of factor depends on experimental objectives
and is predetermined by the experimenter. A factor may be qualitative or quantitative. A
quantitative factor possesses a numerical value assigned to it; while a qualitative factor does
not possess the value, it possesses only the name. For example, the term, concentration is a
quantitative factor because it may have the values of 0.5%, 1.0%, 2.0%, etc.; while
treatment, diet, laboratories, analysts, tablet diluents, etc. are qualitative factors because each
of these has a specific name, not any number can be assigned.
Levels
Levels are associated with a factor. They indicate the values or designations assigned to the
factor. For example, 20 o C, 30 o C are the levels to the factor temperature, similarly, drug and
placebo are the levels for the factor, drug-treatment, 0.5molar and 1.0 molar are the levels
for the factor, concentration. The factorial experiments of trials include all combinations of
all levels of all factors. The investigation of the effects of concentration/amount of drug and
concentration/amount of lubricant on dissolution time of a tablet is an exactly two-factor
experiment. When two levels for each factor become necessary, that is, four runs (four
formulations, F 1 – F 4 ) would be required as shown below.
Symbol Formulation
F1 Lower amount of drug and lower amount of lubricant
F2 Lower amount of drug and higher amount of lubricant
F3 Higher amount of drug and lower amount of lubricant
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F4 Higher amount of drug and higher amount of lubricant
Effects
The change in response caused by varying the level(s) of the factor is called the effect of that
factor. When the effect of a factor is averaged over all levels of the other factors is called the
main effect. In the above example, a two-factor experiment with two levels each of drug and
lubricant, the main effect due to drug would be the difference between the average response
when drug is at the high level (runs F 3 and F 4 ) and the average response would be when
drug is at the low level (runs F 1 and F 2 ). In case of above example, the main effect can be
characterized as a linear response ( Fig 2.12 ), since the effect is the difference between the
two points, such as
Main effect of the drug = F 4 + F 3 – F 2 – F 1 = (F 4 + F 3 )– (F 2 + F 1 )
img
Fig. 2.12 Linear effect of drug
To define more clearly the nature of the response as a function of the factor (amount of
drug), more than two points would be required. If the response is plotted against the level of
a quantitative factor, it is nonlinear as shown in Fig 2.13 , the curved response of a factor at
three levels. In most cases, an important objective of a factorial experiment is to characterize
the effect of changing the levels of a factor or combination of factors on the response
variable.
Interaction
img
Fig. 2.13 Nonlinear (quadratic) effect
img
Fig. 2.14 Illustration of Interaction
It may be considered as a lack of additivity of factor-effects. Ina two-factor experiment, if
factor A and factor exerts an effect equal to 5 and the factor B exerts an effect of 10, the
additivity would be 5+10=15, when both A and B are at their high levels (two-level
experiment). If the effect is more than 15 when factors are at high levels, the result is called
synergistic with respect to two factors. If the effect is more than 15 and both factors (A and
B) are at their high levels, the result is called synergistic with respect to the two factors. If
the effect is less than 15 and the factors A and B are at their high levels, the effect is called
antagonistic with respect to two factors. In language of statistics, interaction is the lack of
additivity. In the above said example where two factors are at two levels, interaction would
be considered as the difference between the effects of concentration of drug at the two
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lubricant levels. Similarly, interaction is also the difference between the effects of lubricant
at two levels of drug. As shown in Fig 2.14 , if the effects of drug remain same in presence
of both high and low levels of lubricant, the system is additive, and no interaction would
take place.
If the lines nteraction represent the effect of amount of drug at each level of lubricant are
parallel, there will be no interaction as shown in Fig 2.14A ; when the lines are not parallel
as shown in Fig 2.14B , interaction would take place. In other words, lack of parallelism
indicates interaction. The effect of amount of drug on dissolution depends on the amount of
lubricant present in the formulation.
Factorial designs have various advantages
When there is no interaction, factorial designs become most efficient in estimating
main effects.
If there is interaction, factorial designs become necessary to disclose and identify the
interactions.
Since the effects of factor are measured over varying levels of other factors,
conclusions become applicable to a wide range of conditions.
Maximum use is made of the data since all main effects and interactions are calculated
from all of the data.
Factorial designs are orthogonal; all estimated effects and interactions do not depend
on the effects of other factors.
An illustration
The Table 2.5 presents the data generated in an experiment with three factors each at two
levels. There is no repetition in this experiment. Repetition would be repeating each of the
eight runs one or more times. The results in Table 2.5 have been presented in standard order.
Recording the results in this form is useful when the data would be analyzed by hand or for
input into computers where software packages need data to be entered in a specified or
standard format. The standard order for a 22 experiment consists of the first four factor
combinations in Table 2.5. The experiment has been conducted to analyze and designed to
investigate the effects of three components (factors)—stearate, drug, and starch on the
thickness of a tablet formulation. In this example, two levels have been chosen for each
factor. If there is financial constraint, use of more than two levels would result in too large an
experiment. For example, if one of the three factors were to be studied at three levels, 12
formulations would have to be tested for a 2 × 2 × 3 factorial design. Because only two
levels have been investigated, nonlinear responses could not be elucidated. However, the
pharmaceutical scientist can feel that the information from this two-level experiment would
be sufficient to identify the effects that would help in designing and formulating the final
product. The levels of the factors in this experiment were as follows:
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Table 2.5 Effect of Stearate, Drug, and Starch Concentration on Tablet Thickness (Results of 23 Factorial Experiments)
img
By hand in simple designs the main effects, and ANOVA as well as the interaction can be
computed. Usually readily available computer programs are available for use for more
complex analyses. Generally, for n factors, an n-way of analysis of variance is suitable. In
typical factorial designs, the factors are commonly considered to be set.
For two-level experiments, the effects can be calculated by applying the signs (+ or – )
arithmetically for each of the eight responses as shown in Table 2.6 which is made by
placing a + or – in column A, B, C, depending on whether or not the appropriate factor is at
the high or low level in a particular run. When the letter appears in the column for factor
combination, a + appears in the column corresponding to that letter.
Table 2.6 Signs to Calculate Effects in a 2 3 Factorial Experiment
img
img
Fig. 2.15 Main effect of the factor, Stearate
For example, for the product combination ab, a (+) come in the column A and B; while a (–)
comes in the column C. Since, the column A represents the high level, the column runs for a,
ab, ac and abc with (+); while for it (1), b, c, and bc run with (–), because the column runs at
low level. The other columns in the Table 2.6 representing AB, AC, BC, and ABC indicate
the interactions. For example, AB represents the interactions of the factors A and B, etc. The
sign in a particular column such as in AC, is the product of individual factors such as A and
C [(– )×(–) = (+)]; similarly, the sign in the column ABC for factor combination, a is (+) is
the result of multiplication of signs in three columns, i.e. A×B×C =(+)×(–)×(–) = ABC = (+).
However, the average effect can be obtained by multiplying the signs of response times for
each of the eight runs in a column, divided by 2
n–1
; where, n is the number of factors (here,
three). Thus, for main effect of A (Stearate) can be demonstrated as
img
This is noted that average main effect of A
= average effect at high level – average effect at the low level.
Considering the results of experiment for each of the eight runs, the average main effect of
stearate on the tablet thickness can be calculated as
img
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(a)
(b)
This can be interpreted that after increasing the level of stearate from low to high, the
thickness of the tablet increases by 0.022cm and this is obtained after averaging all the levels
of two other factors. This has been shown in the Fig 2.8.
The effects of interaction, such as AC, can be calculated in the same way as the main effects
have been done. The signs would be applied all the eight responses in the eight columns as
has been done above and the total responses would be divided by (2
n–1
) where, n is the total
number of factors. Thus, the interaction, AC can be defined as one-half the difference
between the effect of A when C is at the high level and the effect of A when C is at the low
level as shown in Fig. 2.9. Applying the signs as noted above, the AC interaction is
estimated as
img
The Fig 2.16 shows the interaction. As shown by the Fig 2.9 , starch (factor C) at high level
(50mg) with increasing concentration of stearate from 0.5mg to 1.5mg increases the
thickness by 0.0355 cm. At low level (30mg) starch with increasing concentration of stearate
from 0.5 to 1.5mg increases the thickness by 0.0085cm. Thus, stearate has a greater effect
when the concentration of starch increases and results a possible starch × stearate interaction.
On the other hand, lack of interaction can be shown by the same effect of stearate at both
high and low concentrations of starch. In an actual experiment, in the absence of interaction,
the effect of stearate may not be identical at both high and low concentrations of starch due
to the experimental error. The statistical tests described below show how the significance of
observed nonzero effects can be determined.
img
Fig. 2.16 Starch-Stearate interaction
The interaction described is symmetrical. The interaction, AC can be described in two
equivalent ways:
The effect of stearate is greater when the concentration of starch is high, or
The effect of starch concentration is greater when the concentration of stearate is high
(1.5 mg) compared to its effect at low stearate concentration (0.5 mg). The effect of
starch at low stearate concentration is 0.051. The effect of starch at high stearate
concentration is 0.078.
Application in Formulation
The optimization technique supported with statistically valid experimental design is an
efficient and economical method.
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