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simultaneously a reference product, and to compare the results during formulation
development. However, there are various established methods or parameters for comparison
of the in-vitro dissolution profiles. These methods are:
Exploratory data analysis method,
Statistical method,
Model dependent method, and
Model independent method
Similarity factor (f 2 ) is one of the model independent methods and the similarity factor (f 2 )
has been gaining huge importance since the method is recommended by various regulatory
authorities throughout the world. The USFDA has defined the similarity factor as the
logarithmic reciprocal square root transformation of one plus sum of the squared differences
between the dissolution values of test and reference products over all time-points . The f
2
value of more than 50 on a scale ranging from 0 to 100 indicates the close similarity between
two in-vitro dissolution profiles. It can be considered as a basis for performing the in-vivo
bioequivalence study, if necessary. The similarity factor can be easily calculated. It uses only
a single number to express the difference between the in-vitro dissolution profiles. Similarity
factor has also received some criticisms for its application due to its conceptual and
statistical limitations. For example, (1) A strong statistical justification is required for the
basic criteria to declare and accept the similarity between the in-vitro dissolution profile (f
2
profile). (2) No statistical hypothesis can be made because it is not possible to examine the
probability and rate of false positive or negative results. (3) For a particular dissolution
profile, the value of f 2 is sensitive to the number of time-points selected, and (4) The f 2 is
not sensitive to shape of the dissolution curves since its results do not demonstrate the extent
and degree of deviation between the in-vitro dissolution profiles. Due to these limitations,
during formulation development of generic tablet, capsule, or pellet, the similarity factor
alone as a parameter is not used to compare the in-vitro dissolution profiles. Because
application of similarity factor alone does not produce accurate and reliable comparison of
the in-vitro dissolution profiles of innovator and generic product.
As per the suggestion of US FDA and EMEA, the two dissolution profiles can be declared
similar if f 2 remains within 50 – 100. The value of f 2 can be calculated by using the
following equation:
img
Where, n is the number of dissolution sample times,
R t is the individual or mean % dissolved at each time point t for the reference dissolution
test profiles,
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T t is the individual or mean % dissolved at each time point t for test dissolution test profiles.
Similarity factor should be between 0 – 100.
It would be 100 when comparative groups of reference and test are identical, and it
approaches 0 as the dissimilarity increases. This factor is approved by the FDA as
acceptable, and the method is preferred for comparison of dissolution profiles.
The major advantages of the above equation for f 2 are that (1) it is easy to calculate,and (2)
it provides a single number to describe the comparison of dissolution profile data.
The similarity between dissolution profiles can be evaluated based on following
parameters:
There should be minimum of three dissolution time points measured.
The number of drug products to be tested for dissolution would be 12 for both test and
reference.
Not more than one mean value of more than 85% dissolved for each product.
Standard deviation of mean of any product should not be more than 10% from the
second to last dissolution time points.
In case of similar profiles f 1 should approach to zero and f 2 should approach 100.
Generally, the equivalence is ensured when f 1 remains within 0 – 15 and f 2 within 50
– 100.
The limits of f 2 varies from to 100; because when
Both the profiles are identical, i.e., (R t – T t ) = 0; so, f 2 = 50×log100 = 50×2 =100.
Both the profiles are not identical to the extent that the dissolution of any one of the
products completes before other one starts; that is, (R t – T t ) = 100.
img
So, the range of f 2 would vary from 0 – 100.
Average difference of not more than 10% at any sampling time point between reference and
test may be acceptable. When 10% average absolute difference is substituted in the initial
equation for f 2 , the value of f 2 becomes 50. Thus, the average value of f 2 stands within 50
– 100.
The following recommended points should be kept in mind:
Dissolution measurement of both products should be done under exactly same
conditions and sample withdrawal timing should also be same.
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Dissolution time points recommended for immediate release product are 15, 30, 45,
and 60 minutes and for extended-release products are to be 1, 2, 3, 5, and 8 hours.
The value of f 2 is sensitive to the number of dissolution time points, so only one
measurement should be considered after 85% dissolution of the product.
For rapidly dissolving products, i.e., more than 85% release takes place within 15
minutes or less, comparison of the profile is not necessary.
The mean dissolution value for Reference should be derived preferably from the last
pre changed (Reference) batch.
To allow the use of mean data, %coefficient of variation (%CV) at earlier time points,
such as 15 min, should not be more than 20% and other time points should not be
more than 10%.
Advantages of pair wise procedure
The methods are easy to calculate,
The methods provide a single number to describe the comparison of dissolution profile
data.
Disadvantages of the method
The f 1 and f 2 equations do not consider the variation or correlation structure in the
data.
The values of f 1 and f 2 are sensitive to the number of dissolution time point used.
If the test and reference formulation are inter-changed, f 2 remains unchanged but f 1 is
not yet differences between the two mean profiles remain the same.
The basis of the criteria for deciding the difference or similarity between dissolution profiles
is not clear.
The similarity factor (f 2 ) depends on sampling scheme from apparatus means selection and
determination of number of dissolution time points. When there would be the same reference
and test products with a difference in the number and time of dissolution time points, they
show different results
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.
Higuchi and Peppas Plot
In 1963 Higuchi
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first proposed a method of drug is released from a matrix system. This
model can speak about the release of water soluble, poorly water-soluble drugs incorporated
in the semisolid and solid matrices. At the beginning it was considered for planar system,
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(a)
(b)
(c)
(d)
(e)
(f)
subsequently it was extended to various geometrics and porous systems
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.This model is
based on the following hypotheses:
Accordingly, the Initial concentration of drug in the matrix is much higher than
solubility of the drug.
Diffusion of drug occurs only in one dimension (edge effect must be negligible).
The sizes of the drug particles are much smaller than the thickness of the system.
Swelling and dissolution of matrix are insignificant.
Diffusivity of drug is constant, and
Perfect sink conditions are always attained in the release environment.
model can be expressed as:
img
Where,
Q = the amount of drug released in time t per unit area, A
C = initial drug concentration,
C S = solubility of drug in the matrix media, and
D = diffusivity of the drug molecules (diffusion coefficient) in the matrix substance.
This relation remains valid during all the time, except when the total depletion of the drug in
the therapeutic system can be attained. The dissolution of the drug from a planar
heterogeneous matrix system can be studied with two conditions: (1) the concentration of
drug in the matrix is lower than its solubility, and (2) the release occurs through pores in the
matrix. The release of the drug in two situations is expressed as:
img
Where,
D = diffusion coefficient of the drug molecules in the solvent,
= the porosity of the matrix,
= tortuisity of the matrix,
Q = the amount of drug released in time t per unit area, A
C = initial drug concentration,
C S = solubility of drug in the matrix media, and
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D = diffusivity of the drug molecules (diffusion coefficient) in the matrix substance.
Tortuisity is defined as the dimensions of radius and branching of the pores and canals in
the matrix . In general, the Higuchi model can be simplified as:
img
Where, K H is the Higuchi dissolution constant
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,
The data obtained can be plotted as cumulative percentage drug release vs. square root of
time
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. This relationship can be used to describe the drug dissolution from several types of
modified release pharmaceutical dosage forms, as in the case of some transdermal systems
and matrix tablets with water soluble drugs
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In 1983 Korsemeyer-Peppasderived a simple relationship. According to the derivation,
release of drug from a polymeric system follows the equation.
img
Where,
f t = fraction of the dose of drug released at time t,
K = release rate constant, and
n = release exponent.
The diffusion of drug from a controlled release polymeric system with the form of a plane
sheet, of thickness, δ can be represented by;
img
Where,
D, the diffusion coefficient, it is concentration independent.
If the release of drug occurs under perfect sink conditions, the following initial and boundary
conditions can be assumed as:
img
Where,
c o is the initial concentration of drug in the device,
c 1 is the concentration of drug at the polymer-water interface.
A relatively accurate equation for small values of t can be expressed as:
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img
If the main mechanism of release of drugis the diffusion, a graph that represents the amount
of drug released in the referred condition versus the square root of time, would produce a
straight line. Under certain experimental conditions the release mechanism deviates from the
Fick’s equation, it showsa anomalous behavior (non-Fickian)
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.In such cases a generic
equation can be used:
img
Where n is the release exponent. The value of ‘n’ is used to characterize different release for
cylindrical shaped matrices as mentioned in the table 6.1.
Table 6.1 Interpretation of diffusional release mechanisms from polymeric films
img
To study the release kinetics, the data generated from in vitro drug release studies are to be
plotted as log cumulative percentage drug released versus log time.
This equation can be used to the linearization of release data from various formulations of
microcapsules or microspheres.
Significance and Linearity Concept
Significance
Regression analysis is used to find out a relationship between a known and an unknown
variable to determine the unknown one. The term regression was first used by Sir Francis
Galton in the year 1877. He studied that the heights of children born to tall parents tend to
move back or regress towards the mean height of the population. He used the word
‘regression’ to name the process of predicting one variable from another
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. Then to
describe the process by which several variables are used to predict one another, the term
‘multiple regression’ came
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. The F-test of overall significance in a regression is a test to
know whether a given regression model provides a better fit to a dataset than a model with
no predictor variables.
An integral part of statistics in many areas of science is testing of hypothesis. The level of
significance is an important input into hypothesis testing. It controls the critical value and
power of the test; thus, it provides a consequent impact on the inference. It is the probability
of rejecting the true null hypothesis that represents the degree of risk that someone wishes to
take for Type I error. It is a convention to set the level at 0.05, although 0.01 and 0.10 levels
are widely used. As a result, the question comes immediately: “how can the level of
significance be chosen?” or “can we always choose 0.05 under all circumstances?” Most
unfortunately there is no book on statistics that can give in-depth answer to this fundamental
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question. It is only convention that the level is to be set at 0.05 based on R.A. Fisher’s
argument. According to this argument one out of twenty chances represents an unusual
occurrence of sampling
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. There is no scientific basis for this choice
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. When setting the
level of significance, some important factors must be carefully considered. For example, the
level of significance should be set as a decreasing function of sample size, and with a full
consideration of the implications of Type I and Type II errors
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Statistical significance helps to quantity whether the result occurs possibly by chance or due
to some causative factor. When a result is found significant, it simply means that it has really
taken place, not just by a chance. Hence, to conduct an experiment, a survey should be
conducted, poll should be taken, and analysis of the data set should be made. For example, if
a sample of population of interest is taken for a study, no single data point should be
considered, although possibly could be done. Similarly, during marketing campaign of a new
concept, it should be considered whether the present activity has been working better than
the immediately previous one. No single targeted customer in the sample group should be
considered individually. The result of an experiment can be said to have statistical
significance, if it happens is not by chance for a given statistical significance level.
Statistical significance level reflects the risk tolerance and confidence level. For example, an
A/B testing experiment is being run with a significance level of 95%. This means that to
determine a winner, one must be 95% confident that the observed results are real and not an
error caused by randomness. It also means that there is a 5% chance of wrongness.
Statistical significance is a mathematical method to prove that a particular static is reliable.
When a decision is made based on the results of experiments being run, it is necessary to be
sure that there is a genuine relationship between the results and the decision taken. In fact,
the statistical significance indicates the risk of tolerance and confidence level.
For understanding the concept some related terms are defined to remind the reader.
The p-value is the probability value of observing an effect from a sample. A p-value of <
0.05 is conventionally threshold for declaring statistical significance.
Sample size means how large is a sample in the experiment. If the sample size is larger, it
provides better confidence in inferring the results. For example, in running a test on a
website, if there is more traffic the site receives, the sample size is more that is large number
of data will help to make a good and reliable decision with least error. But if there is very
less traffic to the site, a smaller number of data would be available and deciding on that the
chance of error would be more.
The effect size means the difference in results between two sets of samples and indicates
practical significance. If there is a small effect size (say a 0.1% increase) a very large sample
size would be required to determine whether that difference is significant or just a chance. In
fact, if a very large effect on large sample size is observed, it can be used to validate against
smaller sample size with a higher degree of confidence.
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The confidence interval around the effect size means the difference between the upper and
lower boundaries of what is observed in the experiment.
The confusions regarding statistical significance can be summarized mainly in the following
seven points
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Significance involves an important, real effect size,
No significance involves a small effect size,
Significance disproves the tested hypothesis,
Significance proves an alternative hypothesis,
Significance clears the method used,
No significance can be explained by bad method, and
No significance in a follow up study means a replication failure.
Let us assume that α is the level of significance. That is the probability of rejecting the true
null hypothesis (Type I error) and β is the probability of accepting the false hypothesis (Type
II error). 1 – β is the power of the test. Simply, we can assume that the losses expected from
Type I and Type II errors are identical. It is justified to set the level of significance as
decreasing function of sample size.
Suppose, X 1 ,….X n are random samples taken from a normal distribution with the
population mean μ known as standard deviation of 2. Let us test for Ho: μ=0 against H1: μ ˃
0. The test statistic is
Where, 𝑋 is the mean sample. Now, at the 5% level of significance, H o is rejected, if Z is
grea t er than the critical value of 1.645 or X is greater than 2(1.645)/ √𝑛 . This may be noted
that the Z statistic is an increasing function of sample size. This means that when the level of
significance is fixed, the null hypothesis is more possibly to be rejected as the sample size
increases. Suppose μ = 0.5 is a value of substantive importance under H1. The Table 6.2
presents β = P(Z < 1.645|μ = 0.5, σ = 2) along with the power and critical values for a range
of sample sizes.
Table 6.2 Sample Size, Probabilities of Type I and II errors, Power, and Critical Values
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The upper panel resents the case where α remains fixed at 0.05 irrespective of sample size;
while the lower panel presents the case where α decreases with increasing sample size but
remains in balance with the value of β. The upper panel shows that when the sample size is
small, the value of β is unreasonably high compared to 0.05; this results in a low power of
the test. When the sample size is large, the power of test is high, but it appears that α is
unreasonably high compared to β. For example, when sample size is 300, α is about 12.5
times higher than β. In this case, a negligible deviation from the null hypothesis may appear
to be statistically significant.
Linear Regression
When a scatter plot relationship is found, it can be made a line to summarize the relationship
in the data and necessary predictions can also be made in these data. This process is called
linear regression. Presently there are more advanced methods to fit a line to data; generally,
we draw a line that pass through the middle of the points as shown in the Fig.6.10 . The line
A is the average line both sides of which have almost equal numbers of data.
Once an average line is constructed, a suitable equation fitting the line is found out and by
using the equation necessary predictions are to be made.
The percent of adults smoked since 1965 were surveyed and based on data observed an
average plot has been drawn and shown in Fig 6.11 shows a relationship between the percent
of smokers and the year. The line passes through (0,40) and (30, 20). Hence, the slope of line
would be
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Fig.6.10 Linear graph in scater plot
The intercept point on Y-axis is (0, 40); so the y-intercept is 40. Now, the equation can be
written as y = m x + c; where the value of m = – 0.667 and the value of intercept is 40.
Fig. 6.11 Linear plot for developing equation
Thus, the equation can be written as;
y = mx + c
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