Chapter 10, CBS

7/27/2019 Chapter 10, CBS

1/46

273

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

PDF created with FinePrint pdfFactory trial version http://www.fineprint.com
http://www.fineprint.com/http://www.fineprint.com/


2/46

274 Pharmaceutical Product Development

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.



3/46

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



4/46


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



5/46


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)



6/46


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



7/46


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



8/46


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



9/46


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.



10/46


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.



11/46


12/46


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.



13/46


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



14/46


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.



15/46


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).



16/46


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).



17/46


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



18/46


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.



19/46


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



20/46


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,



21/46


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.



22/46


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



23/46


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



24/46


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



25/46

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



26/46


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.

Chapter 10, CBS

Documents