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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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There may be a mathematical model, formulated for a given situation. For implementation of
the model, the values of the number of parameters must be mentioned or known. The set of
data is known through the experiments or observations. The job is to determine the values of
parameters that fit best with the data. Using the term ‘best’ requires some references that
have been used to some criteria for optimization and just one interpretation should always be
used. Applications have been found in statistics (regression, maximum probability),
econometrics, and virtually every area of science.
Problem Types
In general, during optimization two types of optimization problem are encountered –
Constrained, and
Unconstrained.
The term, constraints, means restrictions placed on the system. Restrictions may refer to
physical limitations or probably by simple practicality, for example, economic
considerations. Within the area of physical reality, most importantly and commonly the
constrained problem is observed in pharmaceutical operations, unconstrained problems are
very rarely found. In pharmaceutical systems, the formulator may always experience
restrictions which are of competitive in nature; for example, it would not be reasonable to
assume that the hardest tablets can be prepared with least compression and ejection forces,
the tablets will disintegrate fast and will have good dissolution profile. During optimization,
it is sometimes become necessary to forgo one characteristic for another. Hence, the primary
objective of optimization may not be to find out the absolute maxima or minima, but to find
out an overall preselected or expected result for each parameter. Drug products are
developed to get the best formulation by an effective compromise between competing
parameters. Within a given set of restrictions the process is finalized.
Moreover, in pharmaceutical manufacturing the difficulty is that the formulations are not
simple systems. Since these are produced by using many ingredients, they have many
variables also. The ingredients may interact with one another and produce unexpected
results.
Constrained optimization problems may have more than two approaches, out of which two
are being discussed here. The first approach is the Sequential Unconstrained Minimization
Technique (SUMT) and the second approach is the direct or constrained methods. The
general constrained optimization problem can be solved initially by converting it to a similar
unconstrained problem and solved by using any one of appropriate algorithms. The SUMT
approach is equivalent to unconstrained problem by sacrificing the original objective
function for any constraint violation.
In unconstrained optimization problems there are no restrictions. For a given pharmaceutical
system one might wish to make the hardest tablet possible.
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Variables
There are many variables associated with formulation development and with the process of
manufacturing. Mathematically these can be classified into two groups:
Independent variables, and
Dependent variables
The variables related to development and manufacturing of formulations are called
independent variables; for example, amount of an ingredient to be added, mixing time,
temperature, etc. These are directly controlled by the formulation scientists.
The dependent variables are the responses to or characteristics of the in process materials or
finished drug products. This is to be indicated that the process and formulation variables are
independent variables. These variables are directly controlled by the formulation scientist.
These variables include the amount of an ingredient to be added or the duration of mixing in
a particular step of process. The dependent variables are the responses or the parameters of
the materials in progress or the finished drug product. Generally, these occur directly as
result of any change in the formulation or process. If in a particular process the numbers of
variables are more, the optimization process becomes more complicated. Despite of the
number of variables, a given response must be related to independent variables. Once for a
given response such relationship becomes known, it will define a response surface as shown
in Fig.2.1. To find out the values of independent variables, X 1 and X 2 , the surface needs to
be evaluated and the most desirable level of response, Y can be achieved. Any number of
variables can be considered mathematically in a complicated way, but only two variables can
be considered graphically as indicated by the Fig 2.1.
Fig. 2.1 Response surface representing the relationship between independent variablesm X 1 and X 2 and dependent
variable, Y
1
Optimization Techniques in Pharmaceutical Formulation
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For optimization there may be many methods such as classical and others. These methods
have been described very nicely in the literature. A general flowchart’ has been shown in Fig
2.2 which can describe the general optimization methods. If some factors or variables are
changed on a real system, the effect on the real system can be directly observed at the output
in forms of property. These real data can be used to develop mathematical model. The
responses from the predictive models can then be used to optimize.
Fig. 2.2 Flowchart for optimation
Simplex Method
This model generally is most used in pharmaceutical experimental optimization. This model
was originally proposed by Spendley et al.
10
. This method can be widely used in areas other
than formulation and processing. A very good and appropriate example to explain the
principle of this method is the application to the development of analytical method using a
continuous flow analyzer.
A simplex is a geometric figure having one more point than the number of factors. Thus, for
two factors or independent variables, the simplex can be represented by a triangle. After
determination of the shape of the simplex, the method can use a simplex of fixed size or of
variable sizes that are determined by comparing the magnitudes of the responses after each
successive calculation. The Fig 2.3 shows a set of simplex movements to the optimum
conditions using a variable size method. The Fig shows that two independent variables
represented by the axes show the speeds of the pumps for the two reagents required in the
analysis. The lowest triangle represents the initial simplex; the vertices show the
spectrophotometric response. The plan is to move toward a better response by moving away
from the worst response. Since the poorest response is 0.25, the conditions are selected at the
vortex, 0.66, and certainly, development is obtained. One can follow the experimental path
to the optimum, 0.721.
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Fig. 2.3 Schematic diagram of simplex method for optimization (Response is absorbance at a definite wavelength)
For pharmaceutical formulations, the simple method was used to search for an optimum
capsule formula. This report also describes the necessary methods of reflection, expansion,
and contraction for the appropriate geometric figures. The same laboratories applied this
method to study a solubility problem involving butoconazole nitrate in a multicomponent
system
11
.
An alteration of the simplex technique
12
was reported to a TI-59 calculator and applied
successfully to a direct compression tablet of acetaminophen (paracetamol). When applied
this method
13
to a liquid system (a pharmaceutical solution) it could able to optimize
physical stability of the product. Later on this method was applied to analytical works.
Deming observed
14
that initially in absence of complete knowledge of the response, the
simplex method was considered probably the most appropriate type.
Lagrangian Method
This method represents the mathematical optimization method. The method is an extension
of Classical method and is most probably first time used in pharmaceutical formulation and
process optimization. This method was selected and applied to formulation of tablet having
two independent variables
15
. A constant amount of the drug, phenyl propanolamine
hydrochloride was taken and the amounts of disintegrant (corn starch) and lubricant (stearic
acid) were considered as independent variables and varied as X 1 and X 2 respectively. The
hardness, friability, volume, in-vitro release rate, and urinary elimination rate in human
beings were considered as dependent variables.
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This method requires that before optimization experimentation should be completed to
generate the mathematical model. Nine formulations were prepared and the experimental
design was kept 3 2 factorial. Polynomial models relating the response variables to the
independent variables were generated by a backward stepwise regression analysis program.
The analyses were performed on a polynomial of the form:
According to standard stepwise regression method, the terms were retained or eliminated. In
the above equation y stands for any given response and Bi stands for the regression
coefficient for the various terms containing the levels of the independent variables. For each
response or variable, one equation is created.
From the polynomial equations, a graphic method may be obtained. To apply the Lagrangian
method, this problem should be expressed mathematically as follows:
To maintain the solution within the experimental range the equation 5 and 6 are used. Before
the operation starts, the previous inequality constraints need to be converted to equality
constraints. This can be done by introducing a slack variable q, for each. Then a variety of
equations are combined into a Lagrange function, F, and this requires an introduction of
Lagrange multiplier, λ for each constraint.
A technique called sensitivity analysis can give information and the formulator can further
exchange one property for another. For sensitivity analysis the formulator explains the
constraint optimization problem for systematic changes in the secondary objectives.
Several steps in the Lagrange method is summarized below:
Determination of the function of the objective
Determination of the constraints
Changing the inequality constraints to equality constraints
Fixing or deciding the Lagrange function, F as mentioned below:
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One Lagrange multiplier, λ for each constraint,
One slack variable, q for each inequality constraint,
Partial differentiating the Lagrange function for each variable and setting the
derivatives equal to zero.
Solution of the set of simultaneous equation,
Replacement of the resulting values into objective functions.
The computer performs all these steps of the Lagrange method; however manually this
method requires application of significant mathematical information (input) from the person
concerned. This method was modified by Buck and his associates
16
. They modified the
philosophy into four phases and defined as;
Preliminary planning phase,
Experimental phase,
Analytical phase, and
Verification phase.
The concept was demonstrated by them using a tablet and a suspension formulation for
optimization as reference examples.
Search method
On the contrary, search methods do not require continuity or differentiability of the function
as required by the mathematical optimization method. It is to be computable. In these
methods, the response surfaces are searched to find the combination of independent
variables by various methods to get the optimum by the appropriate equation. The
Lagrangian method can handle several responses or dependent variables. Generally, it is
limited to two independent variables. However, Schwartz et al.
17
applied a search method of
optimization to a pharmaceutical system. It is a computer assisted method and can handle
five independent variables.
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Fig. 2.4 Graphic location of optimum (maximum and minimum)
Extreme vertices design
Anderson and McLean first described the method
18
. This is a fixed design, developed for
pharmaceutical mixtures with only one independent variable, the amount of each of the
ingredients in the mixture. There is no process variable such as temperature, pressure, time,
etc. in the design. The response variable is commonly some direct measure of each mixture
prepared, such as viscosity, dispersibility and solubilization capacity. Concentration ranges
for each ingredient in the mixture are set by the experimenter. An algorithm is used to
determine the experimental points. Snee and Marquardt
19,20
improved the algorithm
developed initially by Anderson and McLean. Currently, the Extreme Vertices design was
applied to a pharmaceutical solubility problem
21
.
Classical method
Classic optimization method has been resulted from application of calculus to the basic
problem of finding the minimum and maximum of a function. The method is useful for
problems which is not too complex and does not have more than few variables. However, the
concept is important. The graph in Fig 2.4 represents the relationship between a response (Yaxis) and a single independent variable (X-axis). It is a hypothetical system, the entire graph
is visible, and the highest point (maximum) and the lowest point (minimum) can be marked.
Unnecessary use of the calculus in plotting of data using equation is not desired. If the
relationship between X and Y is expressed as: Y = f (X)
Y becomes the function of X. The first derivative can be taken, set equal to zero, and X can
be solved to obtain the values of maximum and minimum. When the functions of X are more
than one, there will be more than one solution subject to the first derivative is set equal to
zero. All the various solutions may be maxima or minima, or a mixture of both.
There are other methods also to find out whether a maximum or minimum is being
determined; that is, using second derivative. There are methods to determine the maximum
or minimum; for example, by examining the many local peaks. These methods are explained
well in the elementary calculus. Cooper and Steinberg have explained the methods of
optimization.
When for the response Y, the relationship is given as a function of two independent
variables, X 1 and X 2 , it can be expressed as
Y = f (X 1 , X 2 )
Graphically, there are contour plots as shown in Fig 2.5 on which the axes represent how two
independent variables X 1 and X 2 and the contours represent a specific level of Y. Moreover,
an optimum can be selected graphically. The pair of X values for the optimum can be located
by calculating mathematically with partial derivatives of the function. In case of more than
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two variables, it becomes impossible calculate graphically. Mathematically it is still possible
by using partial derivatives,matrices, determinants, and so on.
Fig. 2.5 Contour plot representing relationship between X 1 and X
2
Statistical Design
The scientists have been used their prior knowledge for development of formulations
satisfactorily without considering critical aspects, such as high dose, poor aqueous solubility,
poor compressibility or other physicochemical properties. Sometimes, adequate formulation
is developed, but there is no time to spend in developing the best possible formulation or
process. By applying the principles of statistics efficient and economical information can be
understood by the relationships between the variables. However, it is doubtful that the
optimum formulation or process can be determined without the use of a mathematical model.
To get the maximum information using the least number of experiments, statistical designs
were developed. Of course, the interactions between variables as well as experimental errors
would remain. The planning must be carefully designed and the statistical rules must be
adhered. The experimenter should accurately define why the experiment is being done and
the steps to be followed. When an entirely new process or formulation is required to be
developed, using a high speed mixer or a controlled release product, this scientist should
realize that a sequential approach is necessary. This method needs to carry out small pilot
experiments to generate information for making decision to provide some estimate of
experimental error. When studies on planning using statistical experimental designs are
being done following steps are to be followed:
Describe the problem carefully to be solved,
Select the independent variables, and
Decide the upper and lower limits for each,
Identify the dependent variables to be measured,
Identify the mathematical model to be used,
Supply an estimate of experimental error,
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Determine how the data will be collected,
Decide how the data will be analyzed,
Decide how the results will be implemented.
Most used optimization methods can be broadly divided into two categories:
Experimentation and optimization study move simultaneously,
Optimization starts after completion of experimentation.
The first type stands for evolutionary operations and is the simple method. The second one is
by more classic mathematical and search methods. For the methods of the second type, it is
necessary to relate between any dependent variable and one or more independent variables.
The necessary relationship can be obtained by two possible methods:
Theoretical, and
Empirical.
If the formulator knows beforehand the theoretical equation for the required properties of
formulation, no experimentation would be necessary. However, much of the work in
pharmaceutical formulations has been carried out in search of such relationships. It is the
responsibility of the formulator to generate the relationships between the variables for the
particular formulation and process. According to Davis
22
, the behavior of chemical
reactions or any system is directed by ascertainable laws. This was a theoretical statement.
Optimum conditions could be possibly determined by applying such laws. Practically, the
underlying mechanisms of the system are frequently so complicated that an empirical
approach becomes necessary. The empirical or experimental approach can be applied for a
system with single independent variable. The formulator performs the experiment at various
levels, measures the desired property, and finds out a relationship, commonly by simple
regression analysis or by the leastsquares method. Generally, there is more than one
important variable. The experimenter must start the work of statistical design of experiments
and multiple linear analyses. Statistical design and multiple regression analysis are separate
and relatively large fields. There are methods available for selecting experimental points so
that (1) entire area of interest is covered, and (2) analysis of the results would allow
separation of variables. That is, statistical analysis should be performed in such a way that
‘which variable is responsible for which specific result’ can be understood.
Factorial design is the most widely used experimental plan. By multiple regression methods,
it is possible to generate the relationships between the variables from the experimental data.
The resulting equations become the basis of optimization.
Processing (Analysis of Data)
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After completion of appropriate experiments for a fixed design, the values for each response
and the corresponding values for each independent variable are to be paired. In a statistical
software package on a computer, the data is entered. The selected model should be used in a
regression analysis for each set of independent variable/ response variable data. Coefficients
of regression model could be calculated for each data set and the effects of the independent
variables on each response could also be estimated. The model, thus completed, should be
examined to see whether response surface based on statistical tests have been adequately
described or not. This can be done by following the method indicated below:
Compare the absolute values of the regression coefficient with its standard error. The
standard error should be less than 50% of the value of the coefficient.
Examine the residuals indicating the difference between the actual responses measured
and those predicted by the regression model. The residuals include variation due to
experimental error and variations of the model from actual values.
By subtracting pure experimental error from the total error, the lack of fitness is to be
measured. The result of the calculation indicating the amount of error due to lack of fitness
should be tested using the F-test. Lack of fitness shows how well a model can fit the data,
independent of experimental error. Lack of fitness indicates how well a model fits the data,
independent of experimental error. This procedure is most useful for models based on theory.
The value of correlation coefficient to be examined to see whether it is close to 1.0. If the
value lies within 0.9 – 1.0, generally it is accepted for most of the developmental works. If a
coefficient is found to be more than 0.95, it is considered as very good.
The ultimate test of a model is to predict a response; if possible, by setting the levels of the
independent variable to some level not used in the experiments. The prediction is then tested
by performing an experiment at the selected conditions and comparing the measured
response to the prediction.
Once a model is found suitable, it can be used to optimize the system under study or it can
help to characterize the system.
Utilization of Model
The aim of optimization is to find out the levels of each independent variable that can
produce the best response. To search for the optimum, the range of values must be less than
or equal to the range used in the experimental design. The method used to find out the
optimum point is called a grid search. The method forces the calculation of all responses
within the ranges of the independent variables.
The output of the search could be all calculated responses (Y values), or a listing of values of
independent variables that yield Y values within selected constraints. For example, a tablet is
expected to have hardness (kp) within a range of Y
H1
to Y
H2
and a dissolution time (T80)
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