chapter 10, cbs
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COMPUTER-ASSISTED OPTIMIZATION OFPHARMACEUTICAL FORMULATIONS AND
PROCESSESBhupinder Singh, R. K. Gupta and Naveen Ahuja
Chapter 10.
INTRODUCTION
Pharmaceutical product and process design problems are normally characterized by multiple objectives
(Fonner et al., 1970; Banker & Anderson, 1987). In an attempt to achieve such objectives, a pharmaceutical
scientist has to fulfill various control limits for a formulation. Some characteristics for ascertaining the control
limits, common for all dosage forms include unit cost, physico-chemical stability and physiological availability
of the active ingredient. Apart from meeting these common traits, a particular dosage form must also satisfy
certain "individual" quality performance characteristics. In case of tablets, for instance, hardness, friability,
disintegration test, dissolution rate, etc., would be most appropriate to control. As most of the objectives of
a formulation are often differing, accepting a suitable compromise between one or more properties (e.g.,
dissolution rate at the expense of hardness) usually becomes unavoidable. Thus the primary aim of theformulator is to find a suitable compromise under the given set of restrictions rather than designing the best
formulation (Fonner et al., 1970; Shekh et al., 1980; Banker & Anderson, 1987; Podczeck, 1996).
Since decades, drug formulations are being developed by trial and error. The previous experience,
knowledge and wisdom of the formulator have been the key factors in formulating new dosage forms or
modifying the existing ones. At times, when the developer is intuitive, skilled and "fortunate", such
nonsystematic approach may yield surprisingly successful outcomes. Invariably however, when skill, wisdom
or luck is not in his favour, it leads to squandering remarkable volume of time, energy and resources (Lewis,
2002). Though a new product may be developed, yet it may retain any defects or problems inherent in the old
product. The modification of the formulation or the process is carried out by studying the influence of
composition and process variables on dosage form characteristics, changing one separate/single factor at a
time (COST), while keeping others as constant. Using this 'COST' approach, the solution of a specific problematic
property can be achieved somehow, but attainment of the true optimum composition or process is never
guaranteed (Tye, 2004). This may be ascribed to the presence of interactions, i.e., the influence of one or morefactors on others. The final product may be satisfactory but mostly sub-optimal, as a better formulation might
still exist for the studied conditions. Therefore, the conventional 'COST' approach of drug formulation
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development suffers from several pitfalls (Fonner et al., 1970; Schwartz et al., 1973; Belloto Jr. et al., 1985;
Stetsko, 1986; Lewis et al., 1999; Lewis, 2002; Myers, 2003; Singh & Ahuja, 2004). These drug product
inconsistencies are generally due to inadequate knowledge of causal factor and response relationship(s).
The said approach is quite:
time consuming,
energy utilizing,
uneconomical,
unpredictable,
unsuitable to plug errors,
ill-suited to reveal interactions, and
yielding only workable solutions.
Computer-based systematic design and optimization techniques, on the other hand, have widely been
practiced to alleviate such inconsistencies (Irvin & Notari, 1991; Singh et al., 2005a; Tye, 2004). Such
techniques are usually referred to as 'computer-aided dosage form design' (CADD). Their implementation
invariably encompasses the statistical design of experiments (DoE), generation of mathematical equationsand graphic outcomes, thus depicting a complete picture of variation of the response(s) as a function of the
factor(s) (Doornbos & Haan, 1995; Schwartz & Connor, 1996; Lewis, 2002). Optimization techniques possess
much greater benefits, as they surmount several pitfalls inherent to the traditional approaches (Lewis et al.,
1999; Tye, 2004). The meritorious features that such techniques offer include:
best solution in the presence of competing objectives,
fewer experiments needed to achieve an optimum formulation,
significant saving of time, effort, materials and cost,
easier problem tracing and rectification,
possibility of estimating interactions,
simulation of the product or process performance using model equation(s), and
comprehension of process to assist in formulation development and subsequent scale-up.
Thus, the trial and error COST approach requires many experiments for little gain in information about
the system under investigation. In contrast, systematic optimization methodology offers an organized approach
that connects experiments in a rational manner, giving more precise information from fewer experiments. (Tye,
2004). Hence of late, DoE optimization techniques have become a regular practice globally in the design and
development of an assortment of dosage forms. However, implementation of such rational approaches usually
involves a great deal of mathematical and statistical complexities. Manual calculation of such optimization
data being quite cumbersome, often calls for the indispensable help of an apt computer interface (Banker &
Anderson, 1987). With the advent of the pertinent computer software coupled with the powerful hardware,
the erstwhile arduous task has grossly been simplified and streamlined. The computational hiccups involved
during optimization of pharmaceutical products have greatly been reduced by the availability of comprehensive
and user-interactive software (Lewis et al., 1999; Singh, 2003). Conduct of systematic DoE studies using
computers usually obviates the requirement of an in-depth knowledge of statistical and mathematical precepts.
Nevertheless, the comprehension of varied concepts underlying these methodologies is certainly a must for
the successful conduct of optimization studies.
The information on such rational techniques, however, lies scattered in different books and journals, and
complete description on variegated vistas of optimization is not available from a single textual source. The
current chapter is an attempt to acquaint the reader with the fundamental principles and precepts of systematic
optimization methodologies, and to present a concise and lucid account on the use of its methodologies in
the computer-assisted design and development of wide-ranging pharmaceutical formulations and processes.
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Chapter 10. Computer-Assisted Optimizat ion of Pharmaceutical Formulat ions and Processes 275
1.2 OPTIMZATION: BASIC CONCEPTS AND TERMINOLOGY
The word, optimize simply means to make as perfect, effective or functional as possible (Schwartz & Connor,
1996). The term optimized has been used in the past to suggest that a product has been improved to accomplishthe objectives of a development scientist (Singh & Ahuja, 2004). However, today the term implies that
computers and statistics have been utilized to achieve the objective(s). With respect to drug formulations or
pharmaceutical processes, optimization is a phenomenon of finding "the best" possible composition or
operating conditions (Lewis, 2002). Accordingly, optimization has been defined as the implementation of
systematic approaches to achieve the best combination of product and/or process characteristics under a
given set of conditions (Tye, 2004).
1.2.1 Variables
Design and development of drug formulation or pharmaceutical process usually involve several variables
(Lewis, 2002). The input variables, which are directly under the control of the product development scientist,
are known as independent variables, e.g., compression force, excipient amount, mixing time, etc. Such variables
can either be quantitative or qualitative. Quantitative variables are those that can take numeric values (e.g.,
amount of disintegrant, suspending agent, temperature, time, etc.) and are continuous. Instances of qualitative
variables, on the other hand, include the type of emulgent, solubilizer or tabletting machine. Their influence
can be evaluated by assigning dummy values to them.
The independent variables, which influence the formulation characteristics or output of the process, are
labeled as factors. The values assigned to the factors are termed as levels, e.g., 30 and 50 are the levels for
the factor, temperature. The restrictions placed on the factor levels are known as constraints (Bolton, 1990;
Schwartz & Connor, 1996).
The characteristics of the finished drug product or the in-process material are known as dependent
variables, e.g., drug release profile, friability, size of tablet granules, disintegration time, etc. (Box et al., 1960;
Bolton, 1990; Doornbos & Haan, 1995; Lewis et al., 1999; Montgomery, 2001). Popularly termed as response
variables, these are the measured properties of the system to estimate the outcome of the experiment. Usually
these are the direct function(s) of any change(s) in the independent variables.
Accordingly, a drug formulation (product) with respect to optimization techniques can be considered as
a system, whose output (Y) is influenced by a set of controllable (X) and uncontrollable (U) input variablesvia a transfer function (T). Fig. 10. 1 depicts the same graphically (Cochran & Cox, 1992; Doornbos & Haan,
1995).
The nomenclature of T depends upon the predictability of the output as an effect of change of the input
variables. If the output is totally unpredictable from the previous studies, T is termed as black box. The term,
white box is used for a system with absolutely true predictability, while the term, gray box is used for moderate
Fig. 10. 1. System with controllable input variables (X), uncontrollable input variables (U), transfer function (T) andoutput variables (Y).
X Y
U
T
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predictability. Using optimization methods, the attempt of the formulator is to attain a white box or nearly
white box status from the erstwhile black or gray box status observed in the traditional studies (Lewis et al.,
1999; Montgomery, 2001). The more is the number of variables in a given system, the more complicated
becomes the job of optimization. Nevertheless, regardless of the number of variables, there exists a distinctrelationship between a given response and the independent variables (Schwartz & Connor, 1996).
1.2.2 Effect, Interaction and Confounding
The magnitude of the change in response caused by varying the factor level(s) is termed as an effect. The
main effect is the effect of a factor averaged over all the levels of other factors (Bolton, 1990; Cochran & Cox,
1992).
However, an interaction is said to occur, when there is "lack of additivity of factor effects". This implies
that the effect is not directly proportional to the change in the factor levels (Bolton, 1990). In other words, the
influence of a factor on the response is nonlinear (Doornbos & Haan, 1995; Lewis et al., 1999). Also, an
interaction may be said to take place when the effect of two or more factors is dependent on each other, e.g.,
effect of factor A depends on the level given to the factor B (Montgomery, 2001; Stack, 2003; Tye, 2004). The
measured property of the interacting variables not only depends on their fundamental levels, but also on the
degree of interaction between them. Fig. 10. 2 illustrates the concept of interaction graphically.
The term orthogonality is used, if the estimated effects are due to the main factor of interest and are
independent of interactions (Box et al., 1960; Bolton, 1990). Conversely, lack of orthogonality (or independence)
is termed as confounding or aliasing (Cochran & Cox, 1992). When an effect is confounded (or aliased), one
cannot assess how much of the observed effect is due to the factor under consideration. The effect is
influenced by other factors in a manner that cannot easily be explored. The measure of the degree of confounding
is known as resolution (Tye, 2004). Confounding is a bias that must be controlled by suitable selection of the
design and data analysis. Interaction, on the other hand, is an inherent quality of the data, which must be
explored. Confounding must be assessed qualitatively; while interaction may be tested more quantitatively
(Stack, 2003).
1.2.3 Code transformation
The process of denoting a natural variable into a dimensionless coded variable Xi such that the central value
of experimental domain is zero is known as coding or normalization (Bolton, 1990; Schwartz & Connor, 1996;
Lewis et al., 1999; Montgomery, 2001). Various salient features of the transformation include:
Fig. 10. 2. Diagrammatic depiction of interaction. The unparallel lines in the figure 2(b) describe the phenomenon ofinteraction between drug and polymer levels affecting drug dissolution. Linear (); nonlinear lines (...).
NO INTERACTION INTERACTION
DRUG DRUG
Low polymer level
Low High
DISSOLUTION
Low High
Low Polymer level
High Polymer level
High Polymer level
DISSOLUTIO
N
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Chapter 10. Computer-Assisted Optimizat ion of Pharmaceutical Formulat ions and Processes 277
depiction of effects and interaction using signs (+) or (-),
allocation of equal significance to each axis,
easier calculation of the coefficients, easier calculation of the coefficient variances,
easier depiction of the response surfaces, and
orthogonality of the effects.
Generally, the various levels of a factor are designated as -1, 0 and +1, representing the lowest, intermediate
(central) and the highest factor levels investigated, respectively. For instance, if starch, a disintegrating
agent, is studied as a factor in the range of 5 to 10% (w/w), then codes -1 and +1 signify 5% and 10%
concentrations, respectively. The code 0 would represent the central point at the mean of the two extremes,
i.e., 7.5% w/w.
1.2.4 Factor Space
The dimensional space defined by the coded variables is known as factor space (Lewis et al., 1999). Fig. 10. 3
illustrates the factor space for two factors on a bidimensional (2-D) plane during a typical tablet compressionprocess. The part of the factor space that is investigated experimentally for optimization is the experimental
domain (Doornbos & Haan, 1995; Lewis et al., 1999). Also known as the region of interest, it is enclosed by the
upper and lower levels of the variables. The factor space covers the entire figure area and extends even
beyond it, whereas the design space of the experimental domain is the square enclosed by X1 = 1, X2 = 1.
1.2.5 Experimental Design
Conduct of an experiment and subsequent interpretation of its experimental outcome are the twin essential
features of the general scientific methodology (Cochran & Cox, 1992; Lewis, 2002). This can be accomplished
only if the experiments are carried out in a systematic way and the inferences are drawn accordingly. An
Fig. 10. 3. Quantitative factors and the factor space. The axes for the natural variables, Ethyl cellulose:Drug and Span80 are labelled as U1 and U2 and those of the corresponding coded variables as X 1 and X2.
Span 80 (% w/v)
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experimental design is the statistical strategy for organizing the experiments in such a manner that the
required information is obtained as efficiently and precisely as possible (Kettaneh-Wold, 1991; Cochran &
Cox, 1992). Runs or trials are the experiments conducted as per the selected experimental design (Bolton, 1990;
Doornbos & Haan, 1995). Such DoE trials are arranged in the design space in such a way that reliable andconsistent information is achievable with minimum experimentation. The layout of the experimental runs in a
matrix form, as per the experimental design, is known as design matrix (Lewis et al., 1999). The choice of the
design depends upon the proposed model, shape of the domain and the objective of the study. Primarily, the
experimental (or statistical) designs are based on the principles of randomization (the manner of allocations of
treatments to the experimental units), replication (the number of units employed for each treatment) and error
control or local control (grouping of specific type of experiments to increase the precision) (Das & Giri, 1994;
Montgomery, 2001).
1.2.6 Response Surfaces
Conduct of DoE trials, as per the chosen statistical design, yields a series of data on response variables
explored. Such data can be suitably modeled to generate mathematical relationship between the independent
variables and the dependent variable. Graphical depiction of the mathematical relationship is known as
response surface (Lewis et al., 1999; Myers, 2003). A response surface plot is a 3-D graphical representationof a response plotted between two independent variables and one response variable. The use of 3-D response
surface plots allows understanding of the behaviour of the system by demonstrating the contribution of the
independent variables.
The geometric illustration of a response, obtained by plotting one independent variable versus another,
while holding the magnitude of response level and other variables as constant, is known as a contour plot
(Singh & Ahuja, 2004). Such contour plots represent the 2-D slices of 3-D response surfaces. The resulting
curves are called contour lines. Fig. 10. 4 depicts the response surface and contour lines for the response
variable of percent drug entrapment in liposomal vesicles of nimesulide (Singh et al., 2005b). For complete
response depiction amongst 'n' independent variables, a total of nC2 number of response surfaces and
contour plots may be required. In other words, 1, 3, 6 or 10 number of 3-D and 2-D plots are needed to provide
depiction of each response for 2, 3, 4 or 5 number of variables, respectively.
Fig. 10. 4. (a) A typical response surface plotted between a response variable, percent drug entrapment and twofactors, cholesterol (CHOL) and phospholipid (PL) in case of vesicular systems; (b) the corresponding contour lines.
(a) (b)
-1
0
1
-1
0
150
65
80
95
PLC H OL
50-65 65-80 80-95
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1.3 MATHEMATICAL MODELS
Mathematical model, simply referred to as the model, is an algebraic expression defining the dependence
of a response variable on the independent variable(s). Mathematical models can either be empirical or theoretical(Doornbos & Haan, 1995; Lewis, 2002). An empirical model provides a way to describe the factor-response
relationship. It is most frequently, but not invariably, a set of polynomial equations of a given order (Box &
Draper, 1987; Myers & Montgomery, 2002). Most commonly used linear models are shown in equations 1-3:
22110 XX)y(E ++= (1)
211222110 XXXX)y(E +++= (2)
2222
2111211222110 XXXXXX)y(E +++++= (3)
where, E(y) represents the measured response, Xi, the value of the factors, and 0, i, ii, and ij are theconstants representing the intercept, coefficients of first-order terms, coefficients of second-order quadratic
terms and coefficients of second-order interaction terms, respectively. Equations 1 and 2 are linear in variables,
representing a flat surface and a twisted plane in 3-D space, respectively. Equation 3 represents a linear
second-order model that describes a twisted plane with curvature, arising from the quadratic terms.
A theoretical model or mechanistic model may also exist or be proposed. It is most often a nonlinear model,
where transformation to a linear function is usually not possible. However, theoretical relationships are rarely
employed in pharmaceutical product development.
1.4 FACTOR STUDIES
Systematic screening and factor influence studies are usually carried out as a prelude to DoE optimization
(Lewis, 2002). These are often sequential stages in the development process. Screening methods are used to
identify important and critical effects (Murphy, 2003). Factor studies aim at quantitative determination of the
effects as a result of a change in the potentially critical formulation or process parameter(s). Such factor
studies usually involve statistical experimental designs, and the results so obtained provide useful leads for
further response optimization studies.
1.4.1 Screening of Influential Factors
As the term suggests, "screening" is analogous to separating "rice" form "rice husk", where "rice" is a group
of factors with significant influence as response, and "husk" is a group of the rest of the noninfluential
factors. A product development scientist normally has numerous possible input variables to be investigated
for their impact on the response variables. During initial stages of optimization, such input variables are
explored for their influence on the outcome of the finished product to see if they are factors (Lewis et al., 1999;
Myers, 2003). The process, called as screening of influential variables, is a paramount step. An input variable,
identified as a factor increases the chance of success, while an input variable that is not a factor has no
consequence. Further, an input variable falsely identified as a factor unduly increases the effort and cost,
while an unrecognized factor leads to wrong picture and a true optimum may be missed (Lewis, 2002).
The entire exercise aims at selecting the active factors and excluding the redundant variables, but not at
obtaining complete and exact numerical data on the system properties. Such reduction in the number of
factors becomes necessary before the pharmaceutical scientist invests the human, financial and industrial
resources in more elaborate studies. This phase may be omitted if the process is known well enough from the
analogous studies. Even after elimination of the noninfluential variables, the number of factors may still be
too large to optimize in terms of available resources (time, manpower, equipment, etc.). Generally, more
influential variables are optimized, keeping the less influential ones as constant at their best levels. The
number of experiments is kept as small as possible to limit the volume of work carried out during the initial
stages (Singh et al., 2005a). The experimental designs employed for the purpose are commonly termed as
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screening designs (Murphy, 2003; Myers, 2003). Usually these designs are first-order and low-resolution
designs.
1.4.2 Factor Influence Study
Having screened the influential variables, a more comprehensive study is subsequently undertaken to quantify
the effect of factors, and to determine the interactions, if any (Bolton, 1990; Lewis et al., 1999; Montgomery,
2001). Herein, the studied experimental domain is less extensive, as quite fewer active factors are studied. The
models used for this study are neither predictive nor capable of generating a response surface. The number
of levels is usually limited to two (i.e, at the extremes). However, sufficient experimentation is carried out to
allow for the detection of interactions amongst factors. The experiments conducted at this step may often be
reused during optimization or response modeling phase by augmenting with additional design points.
Central points (i.e., at the intermediate level), if added at this stage, are not included in the calculation of
model equations (Doornbos & Haan, 1995; Lewis, 2002). Nevertheless, they may prove to be useful in
identifying the curvature in the response, in allowing the reuse of the experiments at various stages; and if
replicated, in validating the reproducibility of the experimental study.
1.5 OPTIMIZATION METHODOLOGIES
Broadly, DoE optimization methodologies can be categorized into two classes, i.e., simultaneous optimization,
where the experimentation is completed before the optimization takes place and sequential optimization,
where experimentation continues sequentially as the optimization study proceeds (Doornbos & Haan, 1995;
Schwartz & Connor, 1996). The whole optimization endeavour is attempted in several steps, commencing from
the screening of influential factors, factor influence studies, and applying one or more of the various techniques
to reach an optimum (Lewis et al., 1999; Singh & Ahuja, 2004).
1.5.1 Simultaneous Optimization Methodology
Generally termed as response surface methodology (RSM), simultaneous optimization approach is a model-
dependent technique (Doornbos & Haan, 1995). The key elements in its implementation encompass, the
experimental designs, mathematical models and the graphic outcomes. One or more selected experimental
response(s) is (are) recorded for a set of experiments, carried out in a systematic way, to predict an optimumand the interaction effects. This is followed by the determination of the mathematical model for each response
in the zone of interest, i.e., the experimental domain. Rather than estimating the effects of each variable
directly, RSM involves fitting the coefficients into the model equation of a particular response variable and
mapping the response, i.e., studying the response over whole of the experimental domain in the form of a
surface (Lewis et al., 1999; Myers & Montgomery, 2002; Myers, 2003).
Principally, RSM is a group of statistical techniques for empirical model building and model exploitation
(Box & Draper, 1987; Myers, 2003). By careful design and analysis of experiments, it seeks to relate a response
to a number of predictors affecting it by generating a response surface. A response surface is an area of space
defined within upper and lower limits of the independent variables depicting the relationship of these variables
to the measured response.
1.5.1.1 Experimental designs
The designs used for simultaneous methods are frequently referred to as response surface designs. Variousexperimental designs frequently involved in the execution of RSM can broadly be classified as:
A Factorial design and modifications
B Central Composite design and modifications
C Mixture designs
D D-optimal designs
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A. Factor ial design and mod if ications
Factorial designs (FDs; full or fractional) are the most frequently used response surface designs. These
are generally based upon first-degree mathematical models (Bolton, 1990; Myers & Montgomery, 2002; Li,2003). Full FDs involve studying the effect of all the factors (n) at various levels (x), including the interactions
amongst them, with the total number of experiments as xn. The simplest FD involves study of two factors at
two levels, with each level coded suitably. FDs are said to be symmetric, if each factor has same number of
levels, and asymmetric, if the number of levels differs for each factor (Lewis et al., 1999). Besides RSM, the
design is also used for screening of influential variables and factor influence studies. Fig. 10. 5 represents a
22 and 23 FD pictorially, where each point represents an individual experiment.
The mathematical model associated with the design consists of the main effects of each variable plus all
the possible interaction effects, i.e., interactions between the two variables, and in fact, between as many
factors as are there in the model.
The mathematical model generally postulated for FDs is given as Equation 4.
++++= ...XXX...XX...XY 3211232112110 (4)
where, i, ij and represent the coefficients of the variables and the interaction terms, and the randomexperimental error, respectively. The effects (coefficients) in the model are estimated usually by multiple linear
regression analysis (MLRA). The topic is discussed in greater detail later under section 1.5.1.2, 'Model
selection'. Their statistical significance is determined and then a simplified model can be written.
In a full FD, as the number of factors or factor levels increases, the number of required experiments
exceeds the manageable levels. Moreover with a large number of factors, it is plausible that the highest-order
interactions have no significant effect. In such cases, the number of experiments can be reduced in a systematic
way, with the resulting design called as fractional factorial designs (FFD). An FFD is a finite fraction (1/x r) of
a complete or full FD, where r is the degree of fractionation and xn-r is the total number of experiments
required (Doornbos & Haan, 1995; Lewis et al., 1999; Li, 2003). However, by reducing the number of experiments,
the ability to distinguish some of the factor effects is partly sacrificed, i.e., the effects can no longer be
uniquely estimated. The degree of fractionation should not be large because this leads to confounding of
factor effects not only with the interactions but also with other factor effects.
Table 10. 1 illustrates the layout of the experiments as per an FD. Lines 1-4 of columns 1 and 2 show a 2 2design for two factors, lines 1-8 of columns 1-3 a 2 3 design for three factors and lines 1-16 of columns 1-4 a 2 4
design for four factors. Lines 1-16 of columns 1-5 describe a 2 5-1 FFD with degree of fractionation, r, as 1
(Menon et al., 1996).
Fig. 10. 5. (a) 22 full factorial design (b) 23 full factorial design.
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Table 10. 1. Experimental layout as per full and fractional factorial designs for two to five factors
Experiment run X1 X2 X3 X4 X5
1 -1 -1 -1 -1 +12 +1 -1 -1 -1 -1
3 -1 +1 -1 -1 -1
4 +1 +1 -1 -1 +1
5 -1 -1 +1 -1 -1
6 +1 -1 +1 -1 +1
7 -1 +1 +1 -1 +1
8 +1 +1 +1 -1 -1
9 -1 -1 -1 +1 -1
10 +1 -1 -1 +1 +1
11 -1 +1 -1 +1 +1
12 +1 +1 -1 +1 -1
13 -1 -1 +1 +1 +1
14 +1 -1 +1 +1 -1
15 -1 +1 +1 +1 -1
16 +1 +1 +1 +1 +1
Table 10. 2. A Plackett-Burman design for 8 experiments
Experiment run X1 X2 X3 X4 X5 X6 X7
1 +1 +1 +1 -1 +1 -1 -1
2 -1 +1 +1 +1 -1 +1 -1
3 -1 -1 +1 +1 +1 -1 +1
4 +1 -1 -1 +1 +1 +1 -1
5 -1 +1 -1 -1 +1 +1 +1
6 +1 -1 +1 -1 -1 +1 +1
7 +1 +1 -1 +1 -1 -1 +1
8 -1 -1 -1 -1 -1 -1 -1
Plackett-Burman Design (PBD) is a special two-level FFD used generally for screening of (K = N-1)
factors, where N is a multiple of 4 (Plackett & Burman, 1946). Also known as Hadamard design or symmetrically
reduced 2k-r FD, the design is easily constructed. Table 10. 2 presents the PBD layout for 8 experiments.
Star designs can be used to provide a simple way to fit a quadratic model (Cochran & Cox, 1992; Doornbos
& Haan, 1995). These designs alleviate the problem encountered with FFDs, which do not allow detection of
curvature unless more than two levels of a factor are chosen. The number of experiments required in a star
design is given by 2n + 1. A central experimental point is located, from which other factor combinations are
generated by moving the same positive and negative distance (= step size). For two factors, the star design
is simply a 22 FD rotated over 45 with an additional center point. The design is invariably orthogonal and
rotatable (Lewis, 2002).
In general, the first-order experimental designs must enable estimation of the first-order effects, preferably
free from interference by the interactions between factors and other variables. These designs should also
allow the testing for the goodness of fit of the proposed model (Myers & Montgomery, 2002; Lewis, 2002).
Even if they are able to determine the existence of the curvature of the response surface, they should normally
be used only in the absence of curvature of the response surface.
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Run Variable factors Response variables
X1 X2 X3 Y1 Y2 Y31 30 6/1 500 20.0 0.8 27.5 1.1 38.0 1.2
2 30 2/1 500 33.0 1.0 45.4 1.1 65.2 1.1
3 10 6/1 500 42.4 0.9 58.7 1.5 80.5 0.9
4 10 2/1 500 66.1 1.3 85.6 1.2 94.1 2.0
5 30 4/1 700 15.4 1.1 21.1 1.6 29.5 1.1
6 30 4/1 300 53.9 1.4 71.7 1.8 85.1 1.0
7 10 4/1 700 32.9 0.8 46.5 1.3 68.5 1.2
8 10 4/1 300 82.4 2.0 91.0 2.0 93.8 2.0
9 20 6/1 700 10.8 1.0 14.8 1.1 20.3 1.3
10 20 6/1 300 47.4 1.1 62.7 1.3 80.3 1.5
11 20 2/1 700 22.1 1.2 30.4 1.2 42.8 1.7
12 20 2/1 300 75.3 0.9 87.1 2.0 94.0 1.913 20 4/1 500 26.5 1.0 36.4 1.5 50.8 1.3
14 20 4/1 500 24.0 1.5 32.9 2.0 47.0 2.0
15 20 4/1 500 25.0 1.2 34.9 1.5 49.3 2.0
Factors Levels Response variables
-1 0 1
X1: Plasticizer concentration (%) 10 20 30 Y1: Cumulative percent drug release after 3 h
X2: Polymer ratio 2/1 4/1 6/1 Y2: Cumulative percent drug release after 4 h
X3: Quantity of coating dispersion (g) 3 00 5 00 7 00 Y3: Cumulative percent drug release after 6 h
* data taken from Kramar et al., 2003.
Table 10. 3. Design layout as per Box-Behnken design.*
Hence, such designs are also known as uniform shell designs. The total number of experiments is given as
n2
+n+1. For two factors, the design is geometrically shaped in the form of a regular hexagon with a centerpoint, thus requiring a total of 7 experiments. This design has the advantage that the experimental domain can
be shifted in any direction by adding experiments on one side of the domain and eliminating them at the other.
The design is highly recommended for pharmaceutical product development (Lewis, 1999).
C. Mixture designs
In FDs and the CCDs, all the factors under consideration can simultaneously be varied and evaluated at all the
levels. This may not be possible under many situations. Particularly, in pharmaceutical formulations with
multiple excipients, the characteristics of the finished product usually depend not so much on the quantity of
each substance present but on their proportions. Here, the sum total of the proportions of all excipients is
unity and none of the fractions can be negative. Therefore, the levels of the various components can be
varied with the restriction that the sum total should not exceed one. Mixture designs are highly recommended
in such cases (Cornell, 1990; Lewis et al., 1999; Lewis, 2002; Singh & Ahuja, 2004). In a two-component
mixture, only one factor level can be independently varied, while in a three-component mixture only two factor
levels, and so on. The remaining factor level is chosen to complete the sum to one. Hence, they have oftenbeen described as experimental designs for the formulation optimisation (Cornell, 1990; Schwartz & Connor,
1996). For process optimisation, however, the designs like FDs and CCDs are preferred.
There are several types of mixture designs, the most popular being the simplex designs. A simplex is the
simplest possible n-sided figure in a (n-1) dimensional space. It is represented as a straight line for two
components, as a 2-D triangle for three components, as a 3-D tetrahedron for four components and so on.
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Fig. 10. 7. Simplex mixture designs a) linear model; b) quadratic model; c) special cubic model.
Scheffs designs, also at times referred as simplex mixture designs (SMD), can either be centroid or lattice
designs (Scheff, 1958; Doornbos & Haan, 1995). Both of these are identical for first and second-order
models, but differ from third-order onwards. The design points are uniformly distributed over the factor space
and form the lattice. The design point layout for three factors using various models is shown in Fig. 10. 7,
where each point refers to an individual experiment.
Scheffs polynomial equations are used for estimating the effects. General mathematical models for 3
components are given as under:
Linear : 332211 XXXY ++= (6)
Quadratic : 322331132112332211 XXXXXXXXXY +++++= ...(7)
Special cubic model: 321123322331132112332211 XXXXXXXXXXXXY ++++++= (8)
The mathematical model of mixture designs does not have the intercept in its equations. As a consequence,
these Scheff models are not calculated by linear regression. Special regression algorithms are required
(Doornbos & Haan, 1995). Table 10. 4 shows the design matrix for a simplex lattice design generated for
optimization of dissolution enhancement of an insoluble drug (prednisone) with the physical mixtures of
superdisintegrants (Ferrari et al., 1996).
Extreme vertices design, another type of a mixture design, is used when there are restrictions on the levels
of the factors (Doornbos & Haan, 1995; Lewis et al., 1999). For instance, in a study involving a tablet
Formulation X1 X2 X3 Percent drug dissolved in 10 min
1 1 0 0 15.2
2 0 1 0 2.8
3 0 0 1 23.1
4 0.5 0.5 0 55.3
5 0.5 0 0.5 59.5
6 0 0.5 0.5 20.6
7 0.33 0.33 0.33 82.4
8 0.667 0.167 0.167 44.7
9 0.167 0.667 0.167 45.5
10 0.167 0.167 0.667 71.6
X1: Croscarmellose Sodium
X2: Dicalcium Phosphate Dihydrate
X3: Anhydrous -Lactose
* data taken from Ferrari et al., 1996.
Table 10. 4. Design layout for simplex lattice design
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formulation for direct compression, use of more than 2% of lubricant or more than 30% of disintegrant is
meaningless. Usually, there are restrictions on both the lower and upper limits of the factors. In such designs,
the observations are made at the corners of the bounded design space, at the middle of the edges, and at the
center of the design space, which can be evaluated only by regression.
D D-optimal designs
If the experimental domain is of a definite shape, e.g., cubic or spherical, the standard experimental designs are
normally used. However, in case the domain is irregular in shape, D-optimal designs can be used (Lewis et al.,
1999). These are non-classical experimental designs based on the D-optimum criterion, and on the principle of
minimization of variance and covariance of parameters (de Aguiar et al., 1995; Doornbos & Haan, 1995). The
optimal design method requires that a correct model is postulated, the variable space defined and the number
of design points fixed in such a way that will determine the model coefficients with maximum possible
efficiency. One of the ways of obtaining such a design is by the use of exchange algorithms using computers
(Chariot et al., 1988; Lewis et al., 1999). These designs can be continuous, i.e., more design points can be
added to it subsequently, and the experimentation can be carried out in stages. D-optimal designs are also
used for screening of factors. Depending upon the problem, these designs can also be used along with
factorial, central composite and mixture designs.Table 10. 5 gives a comparative account of important experimental designs employed for RSM, listing their
advantages and disadvantages.
1.5.1.2 Model selection
"All models are wrong. But some are useful." This assertion of Box & Draper (1987) characterizes the situation
that a formulation scientist faces while optimizing a system. Accordingly, the success of optimization study
depends substantially upon the judicious selection of the model. In general, a model has to be proposed
before the start of the DoE optimization study (Myers & Montgomery, 2002). Model selection depends upon
the type of the variables to be investigated and the type of the study to be made, i.e., factor screening,
description of the system, or prediction of the optima or feasible regions. The choice also depends on the a
priori knowledge of the experimenter about possible interactions and quadratic effects (Doornbos & Haan,
1995). If the model chosen is too simple, higher-order interactions and effects may be missed because the
relevant terms are not part of the model. If the model selected is too complicated, over fitting of the data may
occur. The effect is a larger variance in the predictions, and reliability of the predicted optimum would be too
low.
The models mostly employed to describe the response are first, second and very occasionally, third order
polynomials. A first-order model is initially postulated. If a simple model is found to be inadequate for
describing the phenomenon, the higher order models are followed.
After hypothesizing the model, a series of computations are performed subsequently to calculate the
coefficients of polynomials and their statistical significance to enable the estimation of the effects and
interactions.
A Calculation of the coefficients of polynomial equations
Regression is the most widely used method for quantitative factors (Bolton, 1990; Myers, 1990). It cannot be
used for qualitative factors, because interpolation between discrete (dummy) factor values is meaningless. In
ordinary least-squares regression (OLS), a linear model, as shown in Equation 9, is fitted to the experimental
data, i.e., in estimating the values of in such a way that the sum of squared differences between predictedand observed responses is minimized.
110 X)y(E += or2111110 XX)y(E ++= ... (9)
Multiple nonlinear regression analysis (MLRA) can be performed for more factors, X i, interactions, XiXj,
and higher order terms, as depicted in Equation 10.
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Table 10. 5. Popular experimental designs for response surface optimization with merits and limitations (Doornbos &Haan, 1995; Schwartz & Connor, 1996; Lewis et al., 1999; Montgomery, 2001; Singh & Ahuja, 2004)
DesignFactorial
Fractional factorial
Plackett-Burman
Star
Central composite
Simplex lattice
Extreme-vertices
D-Optimal
MeritsEfficient in estimating main effects and
interactions
Maximum usage of data
Used for screening of factors, factor influence
studies
Suitable for large number of factors or factor
levels
Suitable for very large number of factors, where
even FFDs require a large number of experiments
Study keeps a central point, hence, suitable for
second-order effects
Allows the work to proceed in stages, i.e., if
linear design does not adequately fit the data,
suitable number of experiments can be added to
run a CCD and determine the quadratic effects.
Combines the advantages of FDs and star designs
Requires fewer experiments
Suitable for formulations in which a constraint is
imposed on the combination of factor levels
Suitable for formulations in which a constraint is
imposed on levels of the factors and/or on the
combination of factor levels
Can be employed even if experimental domain is
irregular in shape
LimitationsReflection of curvature not possible in a 2 level
design
Large number of experiments required
Prediction outside the region is not advisable
Effects cannot be uniquely estimated, as are
confounded with interaction terms. Difficult to
construct
Fixed designs in which runs are predetermined
and are limited to 16 experimentsEffects confounded as suitable for two levels only
Does not reveal interactions
Difficult to practice with fractional values of
As the numbers of coefficients in the model are
exactly equal to the number of design points, it is
not possible to estimate residual error. Even
replication allows only the estimation of
experimental error
Interactions and quadratic effects are not
estimated
Calculations can only be performed by regression
Involves a relatively complex model
...XXXX)y(E 212122110 ++++= (10)
MNLRA may also be performed in certain situations, wherein the factor-response relationship is nonlinear.
Regression analysis can only be performed on the coded data or the original values after one or several
models have been postulated, the choice being based on some expectation of the response surface.
B Estimation of the significance of coefficients and model
Significance of coefficients can be estimated using ANOVA followed by Student's t-test (Box et al., 1960;
Bolton, 1990). ANOVA computation can be performed using Yates algorithm to find the significance of each
coefficient. It is always advisable to retain only significant coefficients in the final model equation. This
ANOVA helps in determining the significance of the model as well as of the lack of fit. The values of
Pearsonian coefficient of determination (r2) and that adjusted for degrees of freedom (r2adj) of the polynomial
equation are also compared. The value of r2 is the proportion of variance explained by the regression according
to the model, and is the ratio of the explained sum of squares to that of the total sum of squares. The closer the
value of r2 to unity, the better is the fit and better apparently is the model (Myers, 1990; Lewis et al., 1999).
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288 Pharmaceutical Product Development
However, there are limitations to its use in MLRA, especially in comparing the models with different number
of coefficients fitted to the same data set. A saturated model will inevitably give a perfect fit, and a model with
almost as many coefficients as data is likely to yield a higher value for r2. In such cases, r2adj is preferred,
which corrects the r2 value for the number of degrees of freedom (Bolton, 1990; Myers, 1990). The value ofr2adj is calculated using equivalent mean squares in place of sum of squares, and has value usually less than
r2. Finally, all these parameters are assessed to help in choosing the most appropriate model for a particular
response. The final polynomial equation is subsequently used to calculate the magnitudes of effects and
interactions.
C Model diagnostic plots
One or more of the model diagnostic plots can be plotted to investigate the goodness of fit of the proposed
model:
Actual vs predicted: A graph is plotted between the actual and the predicted response values (Montgomery,
2001; Singh & Agarwal, 2002). It helps in detecting a value, or group of values, that are not easily predicted
by the model. Ideally, such plots passing through origin should be highly linear, i.e., with r2 values close to
unity. These plots are simple to construct and comprehend. They reveal the most pragmatic information of
prognosis, i.e., whether the experimentally observed values of responses are analogous with those predictedusing optimization methodology. Fig 8 (a) illustrates the same.
Residuals vs predicted: Residuals (or error) is the magnitudinal difference between the observed and the
predicted response(s). Studentized residuals is the residuals converted to their standard deviation units
(Bolton, 1990; Singh & Ahuja, 2002). The residuals (or studentized residuals) are plotted versus the predicted
values of the response parameters. It tests the assumption of constant variance. The plot should have a
random and uniform scatter with points close to zero axis and a constant range of residuals across the graph
(Fig.8 (b)). Distinct patterns like expanding variance (megaphone pattern) in the plots are indicative of the
need for a suitable data transformation (like logarithmic, exponential, square root, inverse, etc.).
Residuals vs run: This is a plot of the residuals versus order of the experimental run (Montgomery, 2001).
It checks for lurking variables that may have influenced the response during the experiment. The plot should
show a random and uniform scatter as in Fig. 10. 8(c). Trends indicate a time-related variable lurking in the
background.
Residuals vs factor: This is a plot of the residuals versus any selected factor (Myers & Montgomery,2002). It checks whether the variance not accounted for by the model is different for different levels of a factor.
Ideally, the plot should exhibit a random scatter. Pronounced curvature may indicate a systematic contribution
of the independent factor that is not accounted for by the model.
Normal probability plot: The plot indicates whether the residuals follow a normal probability distribution,
in which case the points will follow a straight line when plotted on a probit scale (Fig 8(d)). Definite patterns
like an "S-shaped" curve, suggest that transformation of the response data may provide a better analysis
(Lewis, 2002).
Outlier T: This is a measure of how many standard deviations the actual value deviates from the value
predicted after deleting the point in question. Many a times, this is referred to as an "externally studentized
residual", since the individual case is not used in computing the estimate of variance (Montgomery, 2001).
Outliers should be investigated to find out if a special cause can be assigned to them. If a cause is found, then
it may be acceptable to analyze the data without that point. If no special cause is identified, then the point
probably should remain in the data set. The graphical plots provide a better perspective on whether a case (or
two) grossly deviates from the others or not. Fig. 10. 8(e) depicts the same with one distinct outlier.
Cook's distance: It provides measures of the influence, potential or actual, of the individual runs
(Montgomery, 2001). This is a measure of the effect that each point has on the model. A point that has a very
high distance value relative to the other points may be an outlier, as shown in Fig. 10. 8 (f).
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a. b. c.
d. e . f.
g. h .
Fig 10.8. Various types of diagnostic plots for selecting suitable model(s). a) predicted vs. actual; b) Studentizedresiduals vs. predicted; c) Studentized residuals vs. run; d) normal probability plots; e) outlier T plot; f) Cook's distanceplot; g) leverage plot; h) Box-Cox plot.
Leverage: This is a measure of degree of influence of each point on the model fit (Montgomery, 2001). If
a point has a leverage of 1, then the model must go through that point (Fig. 10. 8 (g)). Verily, such a point
controls the model. The point with leverage near one, should be reduced by adding or replicating points.
Box-Cox plot for power transforms: The Box-Cox plot is a tool to help in determining the most appropriate
power transformation for application to response data (Lewis et al., 1999; Montgomery, 2001). Most data
transformations can be described by the power function, = fn(), where is the standard deviation, isthe mean and is the power. If the standard deviation associated with an observation is proportional to themean raised to the power, then transforming the observation by the (1 - ) (or) power gives a scalesatisfying the equal variance requirement of the statistical model.
Ac tu a l
Predicted
0 . 8 1
1 . 0 4
1 . 2 6
1 . 4 9
1 . 7 1
0 . 8 1 1 .0 4 1 .2 6 1 .4 9 1 .7 1
R u n N u m b e r
StudentizedR
esiduals
- 3 . 0 0
- 1 . 5 0
0 . 0 0
1 . 5 0
3 . 0 0
1 2 3 4 5 6 7 8 9
P r e d i c t e d
Studentized
Residuals
- 3 .0
-1 .5
0 .0
1 .5
3 .0
0 .8 1 1 .0 4 1 .2 6 1 .4 9 1 .7 1
R e s i d u a l
Norm
al%
Probability
-0 .0 1 3 -0 .0 0 8 -0 .0 0 3 0 .0 0 2 0 .0 0 7
1
5
1 0
2 0
3 0
5 0
7 0
8 0
9 0
9 5
9 9
Outlier T
R u n N u m b e r
OutlierT
- 1 0 . 1
-6 .7
-3 .3
0 .1
3 .5
1 2 3 4 5 6 7 8 9
R u n N u m b e r
Cook's
Distance
0 . 0 0
1 . 0 9
2 . 1 9
3 . 2 8
4 . 3 8
1 2 3 4 5 6 7 8 9
R u n N u m b e r
Leverage
0 .0
0 .2
0 .3
0 .5
0 .7
0 .8
1 .0
1 2 3 4 5 6 7 8 9
.. I . .
. I . .
m :
L a m b d a
Ln(ResidualSS)
- 6 .92
-3 .41
0 . 1 0
3 . 6 2
7 . 1 3
-3 -2 -1 0 1 2 3
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290 Pharmaceutical Product Development
Fig 8 (h) shows a typical Box-Cox plot plotted between Ln(Residuals) and . Here, the value of near 0suggests no power transformation.
1.5.1.3. Search for an o ptim um
Optimization of one response, or the simultaneous optimization of multiple responses can be accomplished
either graphically or numerically.
A Graphical optimization (Response surface analysis)
Graphical optimization displays the area of feasible response values in the factor space. For this, the graphical
optimization criterion is set (Schwartz & Connors, 1996; Lewis et al., 1999; Myers, 2003). Selection of optima
in graphical methods is not based upon minimization or maximization of any function. Hence, graphical
methods require only computability but not continuity or differentiability of the function(s) as in the classical
techniques. The experimenter has to make a choice, 'trading off' one objective for other(s), according to
acceptability, i.e., the relative importance of the objectives considered. The success in locating an optimum
lies in the sagacious interpretation and/or comparison of the resulting plots, leading to attainment of the best
compromise. One or more of the following techniques may be employed for the purpose:
1. Search methods
These methods are employed for choosing the upper and lower limits of the responses of interest (Schwartz
& Connors, 1996). In these search methods, the response surfaces, as defined by the appropriate equations,
are searched to find the combination of independent variables yielding the optimum. Two major steps are
used viz. feasibility search and grid search. Together, these techniques are also referred to as brute force
method (Bolton, 1990; Doornbos & Haan, 1995). The feasibility search method is used to locate a set of
response constraints that are just at the limit of possibility. One selects several values for the responses of
interest and a search of the response surface is made to determine whether a solution is feasible. The
feasibility search method yields the possibilities satisfying the constraints. Subsequently, the exhaustive
grid search is applied, wherein the experimental range is divided into a grid of specific size, and searched
methodically. Grid search method can provide a list of possible formulations and the corresponding response
values.
2. Overlay plotsThe response surfaces or contour plots are superimposed over each other to search for the best compromise
visually. Minimum and maximum boundaries are set for acceptable objective values. The region is highlighted
where all the responses are acceptable. Within this area, an optimum is located, trading off the different
responses. The use of overlay diagrams is limited only to three or four response variables (Doornbos & Haan,
1995; Lewis, 2002). Fig. 10. 9 depicts an instance of overlay plots used for locating optimum formulation with
response values of release till 18 h, (Rel18h) between 80-85% and bioadhesive strength (F) between 24-28 g
(Singh et al., 2003a).
3. Pareto optimality charts
In order to find the most optimum factor combinations satisfying various objectives of a formulation, a pareto
optimality approach may also be used (Doornbos & Haan, 1995). In this method, a graph is plotted between
the predicted values of the objectives and the variables. These are also called multiple criterion decision
making plots. The space occupied by the resulting cloud of points is called the feasible criterion space.
Special subsets of the points (forming a shell partly around the cloud) are the pareto-optimal (PO) points. APO point is a point in the feasible criterion space, when there exists no other point in that space which yields
an improvement in one criterion without causing degradation in the other.
Graphical analysis is usually preferred in case of single response. However, in case of multiple responses,
it is usually advisable to conduct numerical or mathematical optimization first to uncover a feasible region.
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Fig 10.9. A contour overlay plot, plotted between two excipients X1 : HPMC and X2 : Sod. CMC shows the regionbetween the two set criterions, i.e., Release till 18 h, Rel18h should be between 80 - 85% and bioadhesive strength, Fshould be between 24 to 28 g.
B. Mathematical optimization methods
1. Desirability functions
This technique involves a way of overcoming the difficulty of multiple, sometimes opposing responses
(Derringer & Suich, 1980). Each response is associated with its own partial desirability function. If the value
of the response is optimum, its desirability equals 1, and if it is totally unacceptable, its value is zero. Thus the
desirability for each response can be calculated at a given point in the experimental domain. An overalldesirability function can be calculated by multiplying all of the r partial functions together and taking its rth
root. The optimum is the point with the highest value for the desirability. The contour plots of desirability
surface around the optimum should be studied along with the contour plots of the other responses, as
described in overlay plots.
2. Objective functions
These methods are used to seek an optimum formulation by solving the equation (objective function) either
for a maximum or a minimum in the presence of equality and/or inequality constraints (Benkerrour et al., 1984;
Das & Giri, 1994; Schwartz & Connor, 1996). Objective function may be expressed as Equation 11, and the
inequality and equality constraints as Equation 12 and 13.
)XX(fY 21= (11)
i.e., inequality constraint0)X,X(f)X(G
211=
(12)
i.e., equality constraint 0)X,X(f)X(H 212 == (13)
If the objective function is expressed as a function of a single variable, i.e., Y = f(X), calculus based
mathematical approach is applied to find the maximum or minimum of a function. First derivative of the
function can be taken and by setting it equal to zero, the value of X can be solved to obtain the maximum or
H P M C
Sod.
CM
C
-1 .0 -0 .5 0 .0 0 .5 1 .0
-1 .0
-0 .5
0 .0
0 .5
1 .0
Rel18h : 80
Re l18h : 85
F: 24
F: 28
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292 Pharmaceutical Product Development
minimum. When the relationship for the response Y (objective function) is given as a function of two or more
independent variables, as in Equation 11 for X1 and X2, the problem is slightly more involved. Mathematically,
appropriate manipulations with partial derivatives of the function can locate the necessary pair of X values
for the optimum. This type of optimization is known as classical optimization and is applicable only tounconstrained problems. Particularly, these techniques find limited use in the optimization of pharmaceutical
dosage forms, where the problems generally are the constrained ones (Fonner et al., 1970; Schwartz &
Connor, 1996).
3. Sequential unconstrained minimization technique (SUMT)
The above-mentioned technique can also be used for solving the objective function for a maximum or a
minimum (Takayama et al., 1985). In this method, the constrained optimization problem is transformed to an
unconstrained one by adding a penalty function, with the resulting function called as transformed
unconstrained objective function. However, as different starting points may lead to different optimum solutions,
application of a suitable random number technique like Monte Carlo approach can be used.
4. Lagrangian method
The method can be used for optimization of functions expressed in Equations 11-13 using a series of steps viz.
determining objective functions and constraints, changing the inequality constraint to equality constraint by
introducing a slack variable (q) for each inequality constraint (Schwartz & Connor, 1996). Several equations
are combined into a Lagrange function (F) with one Lagrange multiplier (l) for each constraint. Lagrange
function is then partially differentiated for each variable and a set of simultaneous equations are solved by
setting derivatives equal to zero.
1.5.2 Sequential Optimization Methodology
Despite the numerous meritorious visages of simultaneous approaches, there are situations where there is
hardly any a priori knowledge about the effects of variables (Schwartz et al., 1973; Doornbos & Haan, 1995;
Araujo & Brereton, 1996). Such situations call for the application of the sequential methods. In sequential
approach, optimization is attempted in a step-wise fashion. Experimentation is started at an arbitrary point in
the experimental domain and responses are evaluated. Subsequent experiments are designed based upon the
results of these studies, according to an algorithm that directs newer experiments towards the optimum.
Whether the chosen optimum is a maximum or a minimum, the general term used for this approach is "hillclimbing" (Doornbos & Haan, 1995; Lewis et al., 1999). An important aspect of sequential designs is to know
when the goal has been accomplished. There are many different 'stopping criteria' to choose from. Nonetheless,
sometimes the best method involves the experimenter's skill in judging the true optimum, which generally is
a local maximum and minimum.
There are two main model-based methods for extrapolating outside the domain, steepest ascent or steepest
descent (first-order model) and optimum path (second-order). In addition, there is another model-independent
sequential-simplex method.
The inherent advantages of these methods are:
no need of planning all the experiments simultaneously,
a priori knowledge of the response surface not essential, and
interactive.
However, various disadvantages encompass: number of experiments to reach an optimum can not be predicted,
optimum found may not be the global optimum,
robustness is not known,
unsuitable for multiple objective problems,
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attainment of optimum is judged only by the expert developmental scientist,
mathematical model and complete response surface is not generated,
yields unreliable results when multiple optima exist, and applicable only when response surface is continuous.
1.5.2.1. Steepest ascent (descent) meth ods
These methods are direct optimization methods for first-order designs (Lewis et al., 1999; Myers, 2003). They
are good choice when the optimum is outside the domain and is to be arrived at rapidly. These approaches are
an amalgamation of model-independent and model-dependent methods. The direction of the steepest increase
of the response in terms of coded variables is determined, and then experiments are carried out along this line
(Lewis, 2002). This is followed by measurement of the response and is continued until an optimum is reached.
1.5.2.2. Optimum path m ethod
This method is just analogous to steepest ascent method, where the optimum is also searched outside the
experimental domain by extrapolation. Such situations arise when choosing a very extensive experimental
domain is difficult or the possible experimental domain is not known at the beginning of the study. However,this method is used for searching the optimum by extrapolation from a second-order design along a curved
path.
1.5.2.3. Sequent ial sim plex techn iques
The technique consists of first generating data from n + 1 experiments, where n is the number of independent
variables or factors (Shekh et al., 1980; Bolton, 1990; Araujo & Brereton, 1996). Based on n + 1 responses and
predetermined rules, one result is eliminated and a new experiment is performed. A decision is made as a result
of experimentation, eventually terminating the study at an optimal response. Fig. 10. 10 illustrates various
steps involved under the approach using an arbitrary example. A simplex is constructed by selecting three
combinations (A, B and C) of two variables (X1 and X2). Three experiments are carried out and evaluated, and
the worst response illustrated as point A is identified. The next experiment is conducted for a combination
moving away from point A. This is achieved by reflecting the triangle ABC around BC axis. The experiment at
point D is performed, and the response is compared with the response at point A, B and C. The next movedepends on the relative values of the four responses:
If the response at point D is greater than the responses at A, B and C, the next experimental point is E.
If the response at point D is greater than the response at B but smaller than the response at C, thetriangle BCD is reflected about CD axis and the next experimental point is F.
If the response at point D is lower than the responses at B and C, but greater than at point A, the nextexperimental point is G.
If the response at point D is lower than the responses at A, B and C, the next experimental point is H.
1.5.2.4. Evolu tion ary op eration s (EVOP)
It is a popular technique in several industrial processes (Schwartz & Connor, 1996; Lewis et al., 1999). The
underlying basis for this approach is that the production procedure (formulation and process) is allowed to
evolve to the optimum by careful planning and constant repetition. The process is run in such a way that it
produces a product that meets all the specifications and at the same time, generates information on productimprovement. Generally, these involve factorial and simplex designs requiring a large number of experiments.
In a typical industrial process, this extensive experimentation is usually not a problem, since the process will
be run repeatedly over and over again. However, in the most complex situations involving product development,
it is not so because there is often insufficient freedom in the formula or process to allow necessary
experimentation. In pharmaceutical product development setup, however, more efficient methods are desirable.
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Fig. 10. 10. Schematic diagram illustrating the stages of optimization using a simplex method.
1.5.3 Artificial Neural Networks
Of late, the application of artificial neural networks (ANNs) in the field of pharmaceutical development and
optimization of dosage forms has become a blown out topic of discussion in the pharmaceutical literature
(Takayama et al., 1999; Takayama et al., 2003). The ANNs are model-independent computational paradigms
that can simulate the neurological processing ability of the human brain. The neural networks, consisting of
inter-connected adaptive processing units, so-called neurons, are able to discern complex and latent patterns
in the information presented to them. ANN is a computer-based learning system that can be applied to
quantify a nonlinear relationship between causal factors and pharmaceutical responses by means of iterative
training of data obtained from a designed experiment (Achanta et al., 1995; Bourquin et al., 1997). The results
obtained from implementation of an experimental design are used as input information for learning. Once
trained, the neurons of an ANN may be used to forecast outputs from new sets of input conditions (Peck etal., 1989; Achanta et al., 1995; Zupancic Bo ic et al., 1997; Bourquin et al., 1997).
A typical ANN must have one input layer and one output layer, and may contain one or more hidden
layers as depicted in Fig.11. The information is passed from input layer to the output layer through hidden
layer(s) by the network connections or synapses. Modeling starts with a random set of synaptic weights and
proceeds in iterations. During each iteration, connection weights are adapted via selected modeling. The
basis of such modeling technique is to minimize the error, i.e., the difference between the momentarynetwork signal and the aimed signal based on the experimental results. When the minimal " error" isobtained, learning is completed and connection weights become the memory units. After this, the test set of
values can be applied on a learned ANN to evaluate it. Subsequently, it can be used for output prediction on
the basis of the new input values. The modeling is invariably done via a suitable computer software.
The prediction ability (PA) or reliability of an ANN output depends heavily on the training data (So &
Karplus, 1996). Two problems that tend to diminish PA are overfitting (i.e., few data points per network
connection) and overtraining (long network training period). Thus, ANN does not work well with many
variables and few formulations. Further, the results from ANN cannot be treated statistically and no definitive
reasons can be given for the same. In an attempt to improve PA and to reduce training efforts, genetic neural
networks (GNN), and generalized regression networks (GRN) have been used with fruition, respectively.
While the former employs a combination of genetic algorithms with ANN, the latter utilizes the modelization
of the function more or less directly from the training data. Since ANNs require a great deal of iterative
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Chapter 10. Computer-Assisted Optimizat ion of Pharmaceutical Formulat ions and Processes 295
Fig. 10. 11. Schematic diagram illustrating various parts of an Artificial Neural Network. X 1 - X3 represent the inputfactors; Y as the response variable connected to the input layer via various nodes of hidden layer (H1-H9). W11 andW93 represent the connections between the corresponding input factors and the nodes of the hidden layer while W1y,W5y and W9y denote the connections between the corresponding respective hidden nodes and output layer, Y.
computations, the use of versatile computer software dedicated for the purpose becomes almost obligatory
for their execution (Bourquin et al., 1997).
1.5.4 Choosing an Optimization Methodology
In case of single response, graphical analysis is opted for (Lewis et al., 1999). However, in case of multiple
response variables, certain responses can oppose one another. Accordingly, changes in a factor that improve
one response may have a negative effect on another. Since it is not usually possible to obtain the best values
for all the responses, optimization principally embarks upon finding experimental conditions where different
responses are most satisfactory, over all. Nevertheless, there is a certain degree of subjectivity in weighing
up their relative importance.
1.6 COMPUTER USE IN OPTIMIZATION
Development of the principles behind optimization, now known as DoE, dates back to the 1920s with its
Table 10. 6. Suitability of various optimization methods under variegated situations
Optimization method
GRAPHICAL ANALYSIS
DESIRABILITY FUNCTION
STEEPEST ASCENT
OPTIMUM PATH
SEQUENTIAL SIMPLEX
EVOLUTIONARY OPERATIONS
Model situations for use
Mathematical model of any order, Normally no more than 4 factors,
Preferably in single response
Mathematical model of any order, Number of factors between 2 and 6, Multipleresponses
First-order model, Optimum outside the domain, Single response
Second-order model, Optimum outside the domain, Single response
No mathematica l model, Direct optimiza tion, Single or multiple responses,
Industrial situation, Little variation possible
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discovery by British statistician, Ronald Fisher. Optimization, however, lay virtually dormant due to the
complex and tedious hand calculations it required. Software that automates the designed-experiment optimization
studies was invented in the early days of mainframe computers (Potter, 1994). Mainframes, requiring
programming skill s beyond most sta tis ticians' scope, chugged through complica ted DoE equations.Nevertheless, it wasn't until those room-sized computers became desktop PCs, that affordable DoE software
for the non-statisticians first appeared. Now a days, computer use is considered almost indispensable in the
design and optimization methods, as a great deal of intricate statistical and mathematical calculations are
involved (Doornbos & Haan, 1995; Podczeck, 1996; Tye, 2004). Particularly, the ANN optimization is based
totally upon the computer interface, tailor-made for the purpose (Bourquin et al., 1997).
The computer software have been used almost at every step during the entire optimization cycle ranging
from selection of design, screening of factors, use of response surface designs, generation of the design
matrix, plotting of 3-D response surfaces and 2-D contour plots, application of optimum search methods,
interpretation of the results, and finally the validation of the methodology (Potter, 1994). Verily, many software
packages lead the user through the data analysis even without a mathematical model or statistical equations
in sight. Use of pertinent software can make the DoE optimization task a lot easier, faster, more elegant and
economical (Singh, 1997; Singh, 2003; Tye, 2004). Specifically, the erstwhile impossible task of generating
varied kinds of 3-D response surfaces manually is accomplished with phenomenal ease using appropriatesoftware (Bolton, 1987; Potter, 1994).
1.6.1 Choice of Computer Software Package:
Many commercial software packages are also available, which are either dedicated to a set of experimental
designs or are of a more general statistical nature with modules for select experimental design(s). The dedicated
computer software is frequently better as the user pays only for the DoE capabilities (Potter, 1994). In
contrast, the more powerful, comprehensive and expensive statistical packages like SPSS, SAS, BBN, BMDP,
MINITAB, etc. are geared up for larger enterprises offering diverse facilities for statistical computing, support
for networking and client-server communication, and portability with a variety of computer hardware (Potter,
1994; Singh, 2003, Singh et al., 2005a). When selecting a DoE software, it is important to look for not only a
statistical engine that is fast and accurate but also the following:
A simple graphic user interface (GUI) that's intuitive and easy-to-use.
A well-written manual with tutorials to get you off to a quick start. A wide selection of designs for screening and optimizing processes or product formulations.
A spreadsheet flexible enough for data entry as well as dealing with missing data and changed factorlevels.
Graphic tools displaying the rotatable 3-D response surfaces, 2-D contour plots, interaction plots andthe plots revealing model diagnostics
Software that randomizes the order of experimental runs. Randomization is crucial because it ensuresthat "noisy" factors will spread randomly across all control factors.
Design evaluation tools that will reveal aliases and other potential pitfalls.
After-sales technical support, online help and training offered by manufacturing vendors
Table 10. 7 lists some commonly used computer software for optimization along with their salient features.
Today, these off-the-shelf software packages commonly sell for divergent prices, varying widely from $99 to
$2500, depending upon the features provided with these software. The actual number of computer systems,
however, is much more as the field is still rapidly growing.
1.7 PLAN TO IMPLEMENT OPTIMIZATION: AN OVERVIEW
The overall approach for conduct of computer-assisted optimization studies in the development of drug
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Chapter 10. Computer-Assisted Optimization of Pharmaceutical Formulations and Processes 297
Table 10. 7. Important computer software for optimization and their salient features
Software
Design Expert
MINITAB
JMP
CARD
DoE PRO XL &
DoE KISS
MATREX
Cornerstone
ECHIP
GRG2
DoE PC IV
STATISTICA
NEMROD@
MODDE
SPSS
Omega
DoE WISDOM
COMPACT
OPTIMA
XSTAT
FACTOP
Salient features
Powerful, comprehensive and popular package used for
optimizing pharmaceutical formulations and processes; allows
screening and study of influential variables for FD, FFD, BBD,
CCD, PBD and mixture designs; provides 3D plots that can be
rotated to visualize the response surfaces and 2D contour maps;
numerical and graphical optimization
Powerful DoE software for automated data analysis, graphic and
help features, MS-Excel compatibility, includes almost all designs
of RSM
DoE software for automated data analysis of various designs of
RSM, graphic and help features
Powerful DoE software for automated data analysis, includes
graphic and help features
MS-Excel compatible DoE software for automated data analysis
using Taguchi, FD, FFD and PBD. The relatively inexpensive
software, DoE KISS is, however, applicable only to singleresponse variable.
Excel compatible optimization software with facilities for various
experimental designs and Taguchi design.
DoE software with features for executing various experimental
designs
Used for designing and analyzing optimization experiments
Mathematical optimization program to search for the maximum
or minimum of a function with or without constraints
Used for designing the optimization experiments
ANN-based software based on GRN technique
Suitable for FDs and CCDs, has features for numerical
optimization and graphic outputs
Suitable for response surface modeling and evaluation of fitting ofmodel
Comprehensive statistical software with facilities for
implementing experimental designs
Only for mixture designs; only program that supports multi-
criterion decision making by Pareto- optimality, upto six
objectives and has various statistical functions
Supports designs for screening, D-optimal, Taguchi and user
defined designs, also options are available for pareto optimality
charts
Optimization software for systematic DoE and response surface
methodology studies with state-of-art mathematical search
techniques
Generates the experimental design, fits a mathematical equations
to the data and graphically depicts response surfacesAids in selection of an experimental design, has modules for
numerical optimization and graphic outcomes
Aids in the optimization of formulation using various FDs, and
other designs through development of polynomials and grid
search; includes computer-aided-education module for
optimization
Source
www.statease.com
www.minitab.com
www.jmp.com
www.s-matrix.com
www.sigmazone.com
http://www.rsd-associates.com/
matrex.htm
www.brooks.com
www.echip.com
www.fp.mcs.anl.gov/otc/Guide/
SoftwareGuide/Blurbs?grg2.html
http://www.adeptscience.co.uk/as/
products/qands /qasi/doepciv/
www.statsoftinc.som
www.umt.ciw.uni-karlsruhe.de/
22713
www.umetrics.com
www.spss.com
www.winomega.com
www.launsby.com
www-fp.mcs.anl.gov/otc/ guide/
SoftwareGuide/Blurbs/
compact.html
www.optimasoftware.co.uk
www.amazon.com
www.puchd.ac.in/uips/bhoop.html
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298 Pharmaceutical Product Development
General DoE software with features for implementation of
Taguchi, CCDs and FDs.Optimization software for linear and nonlinear problems with
state-of-art mathematical programs
Aids in optimization based on simplex and D-optimal designs
www.engenious.com/
release1_11isightenhance.htmlwww.solver.com
www.multisimplex.com
iSIGHT
SOLVER
Multisimplex AB
Software Salient features Source
product systems can be described by a DoE optimization strategy. Although there is no infallible plan, yet its
choice depends on the diverse characteristics of the problem at hand, the required quality of remedy and the
quantum of experimental effort to gain the information (Lewis, 2002; Myers, 2003; Singh, 2003; Singh &
Ahuja, 2004). The salient steps involved in an optimization plan encompass:
1. Defining the objective: The optimization objective, i.e., the property of interest is clearly defined (e.g.,
drug release from a compressed tablet). Selection of the response variables should be made with
dexterity. Selected response variables should be such that they provide maximum information with
minimal experimental effort and time.
2. Choice of appropriate computer interface: Since the use of computers is nearly obligatory for
implementing an optimization plan, the choice of apposite software is vital. The computer package
selected for the purpose should ideally encompass the facilities of executing several experimental
designs for screening as well as response surface optimization, generating design matrices and response
surfaces, and conducting statistical analysis and graphics for model diagnostic analysis.