Analysis of Response Surface Methodology (RSM)

Analysis of receipt Surface methodological analysis (RSM)1.3 DESIGN OF EXPERIMENTS 34, 35Trans-Pacific Partnership economic role model agreement has clearly defined that m any(prenominal)(prenominal) studies must be manoeuvreed to take on a face. anatomy of experiments (DOE) has proven to be an effective tool for cooking scientists throughout the many stages of the ca-caulation off rophy. At every step of face development, DOE skunk aid in making intelligent decisions. These move include excipient compatibility studies, process feasibility studies, formulation optimisation, process optimization, scale-up and manufacturing process characterization. Lastly, the harvest-home and manufacturing process must be validated before it is on the market.The book of account optimize is defined as, making as perfect, effective or operative as possible. Optimization may be interpret as to key out the value of controllable item-by-item inconstant quantity, that gives the most de sired value of dependent variables. The application of formulation optimization techniques is relatively new to the praxis of pharmacy when used intelligently, with the common sense, these statistical methods go away broaden the eyeshot of the formulation process.At the Preformulation stage, before any experiment is conducted certain difficulty arises, it is often non cognise before hand which variable will signifi crouptly influence the reply. Screening aspirations and ANOVA helps to solve this problem.A arcminute serious complication may arise with new excipients and new process compute, for which qualitative or quantitative cause argon not known and ar unpredictable. The fol firsting questions must be answered before choosing any purport of experiment.The third complication is that formulated products, in particular dosage form has to confirm to several requirements, very often competing. The formulator has to trade off objectives and ask a compromise.A fourth prob lem is the lack of insight to arrange an adequate optimization studies.Above altogether in the performance of an optimization weigh, the formulation development scientist crumb also be a grammatical constituent as personal variation.1.3.1 Terms used in Design of experimentsVariablesThese atomic moment 18 the measurements, value, which be characteristics of the information. There atomic tally 18 two types of variables dependent variables and independent variables. mugwump variables(X) are set in advance, which are not influenced by any opposite determine e.g., Lubri bottomlandts concentration, drug to polymer ratio, etc. Dependent variables(Y) are the expiry variables, influenced by the independent variables e.g., bad weather, dissolution rate, etc.FactorFactor is an charge variable such as concentration, temperature, lubricant agent, drug to polymer ratio, polymer to polymer ratio or polymer grade. A cipher can be qualitative or quantitative. A quantitative factor h as a numerical value to it for example, concentration (1%, 2% so on), drug to polymer ratio (11, 12etc). Qualitative factors are the factors, which are not numerical value, for example, the polymer grade, humidity condition, type of equipment, etc. these are decided in nature.LevelsThe levels of a factor are the values or targetation assigned to the factor. For e.g. in concentration (factor) 1 % will be one level, while 2% will be another level. twain contrastive plasticizers are levels for grade factor. Usually levels are indicated as low, core or high level. Normally for ease of calculation the numeric and discrete levels are converted to 1 (low level) and +1 (high level).The cosmopolitan formula for this variety isWhere X is the numeric valueResponseResponse is mostly interpreted as the outcome of an experiment. It is the effect, which we are going to evaluate i.e. Disintegration beat, time of buoyancy, etc.EffectThe effect of a factor is the change in response caused b y varying the levels of the factor. This describes the relationship surrounded by various factors and levels.fundamental interactionInteraction is also similar to effect, which gives the overall effect of two or more variables (factors) on a response. For example, the combined effect of lubricants (factor) and glidants (factor) on insensibility (response) of a chit.In the trial and error method, a lot of formulations turn in to be prepared to get a conclusion, which involves lots of money, time and energy. These can be minimized by the use of optimization technique.1.3.2 Optimization act uponGenerally optimization process involves the following steps.Based on the preliminary knowledge or experience or from literature, the independent variables are unflinching and set in the beginning.Selection of a suitable model, based on the results of the factor, screening is done.The experiments are marked and conducted.The responses are analyzed by ANOVA, tryout on lack of fit, to get an information-based numeral model for individually individual response.The responses are screened, by using double criteria to get the values of independent variables.Experimental DesignExperimental project is a statistical externalize that prescribes or advises a set of combination of variables. The fall and layout of these design compass razes within the observational sphere, depends on the number of effects that must be approximationd. Depending on the number of factors, their levels, possible interactions and set up of the model, various observational designs are chosen. Each experiment can be stand for as a show up within the experimental domain, the point organism defined by its co-ordinate (the value given to the variables) in the space.1.3.3 Response Surface MethodologyResponse fall out methodology (RSM) is an experimental strategy that was developed in the 1950s36. RSM is comprised of a group of numeral and statistical techniques that are based on fitti ng experimental selective information graveld from studies established using an experimental design, to empirical models and that are subsequently used to define a relationship between the responses observed and the independent introduce variables37, 38. RSM is able to define the effect of independent variables alone and in combination with the manufacturing processes infra investigation.A typical RSM study begins initially with the definition of a problem to be investigated and involves establishing which variables and associated responses are to be study, monitored, and measured and how these will be measured. A summary of the subsequent RSM address includes36Performance of the relevant DOE.Estimation of the coefficient in the relevant response draw close equation.Checking of the adequacy of the equation to describe the fit.Studying the response show to get a line and evaluate the region(s) of interest.The term RSM originates from the graphical perspective generated after fitness of the mathematical model has been established 37, 38 with a graphical representation of the data presented primarily as a three-dimensional (3D) image and/or as contour plots39.The relationship between a response and an stimulant variable can be described by Equation 1.1y= f(x1, x2, x3xn) + Where,y = relevant responsef = unknown function of a responsex1, x2,..xn = independent variablesn= number of independent variables = statistical error that represents other sources of variability not accounted for by fContour plot can be described asi. Mound-shaped that has elliptical contours with a nonmoving point at the position of a maximum response.ii. Saddle-shaped that has a hyperbolic trunk of contours with a stationary point that is neither a maximum nor tokenish point.iii. Constant (stationary) ridge response surface in which the contours are presented as concentric elongated ellipses with a stationary point in the region of the design region.iv. A rising (or falling) ridge response surface with a stationary point that is away(p) the design region 39.The stationary point is a combination of design variables where the surface presents as either a maximum and/or a token(prenominal) in all directions. If the stationary point is a maximum in one direction and minimum in another direction, the stationary point is termed a saddle point. When the surface is curved in one direction but is fairly perpetual and this is considered a ridge response 40.By plotting a response, y, against one or two insert variables a surface, known as the response surface can be generated in two or three dimensions. In general the form of the function, f, is unknown and may be very tangled depending on the effect of the scuttlebutt variables on the response. Therefore RSM aims at approximating f by use of a suitable, separateed polynomial equation in some region(s) of the values for the independent process variables41. The mathematical or polynomial equations that describe the relationship(s) between the independent and dependent variables may be first gear, second base or third order, depending on how the output variables or responses react to changes in the insert variables.If the response is a linear function of the independent variables, because the function can be written as a first order model (Equation 1.2). In this model the response variables that fit a linear model are generally variables that are significantly alter by a small change in the value of the input factors and that process little(a) or no interaction(s) between the input variable terms.y= 0+ 1x1+ 2x2+..+ Second order equations are used to generate linear and quadratic response equations that exhibit interactions between the input factors and can be represented by Equation 1.3.y= 0+ 1x1+ 2x2+ 12x12++ It has been report that second order models are also applicable to input factors that exhibit extensive variability over an experimental domain and these relationships are li ft out described using Equation 1.4y= 0+ 1x1+ 2x2+ 12x12+ 11x12+ 22x22+..+ Wherey= responsex1, x2,..xn = input factors0= constant that represents the intercepti= coefficient of first order termii= coefficient of second order termij= coefficient of second order interactionThe values of the coefficients in the model are generated through multiple linear regression analysis of the data that has been collected. A coefficient with a tyrannical value points to an agonistic effect of the input factor on the response, whereas coefficients with contradict values indicate an antagonistic effect.1.3.4 Choice of Response Surface DesignCentral Composite Design (CCD)A CCD was originally presented by stripe and Wilson and is based on a factorial design with additional points to estimate the curvature of that design. CCD encompasses a full factorial or fractional factorial onrush which can be represented, as shown in Figure 1.1, as the eight corners of a cube.There are the six points, known as the axile or star points, located in the centre of all(prenominal) face of the cube with a final point located in the middle of the cube that is known as the centre point 37. The axial points are experimental runs where all but one of the factors to be investigated is set at the intermediate level under consideration. The axial points are all equidistant from the centre point and are denoted using the symbol, alpha (). The factors under consideration are usually investigated at five different levels and are always represented by coded values viz., -, -1, 0, +1 and +.Figure 1.1Schematic plat representing the levels studied in a Central Composite DesignThe surmount of the axial points from the centre point is dependent on the number of factors investigated in the design and is established using Equation 1.5. =2k/4Where,k= the factor number = axial pointThe number of experiments required for a CCD approach is figure using Equation 1.6N= k2+ 2k+ C0Where,N= the experiment numberk= th e factor numberC0= the replicate number of the central pointThe number of experiments required in an experimental study is important as it determines how much data will be generated, in addition to being an indicator of the amount of time that will be required to conduct the study.Types of central composite designCentral composite design can be divided into three types.Table 1.2 Types of central composite designBox-Behnken Design (BBD)The BBD describes a class of second-order designs based on a three-level unelaborated factorial approach which are also represented as coded values viz., -1, 0 and +1 42 . In this design approach, the treatment combinations are located at the midpoint(s) of the edge of the process space and at the centre, as represented in Figure 1.2.Figure 1.2 Schematic diagram representing the levels studied in a Box-Behnken DesignThe number of experiments for Box-Behnken Designs can be calculated using Equation 1.7.N= 2k (k-1) +C0Where,N= the number of experimentsk = the factor numberC0= the replicate number of the central pointFor experiments in which at that place are three or less input variables the BBD design offers some advantage over the CCD approach, in that a fewer number of experimental runs are required. However this advantage does not exist when four or more parameters are to be investigated. A advertize advantage of BBD is that it does not include the need to evaluate situations in which all factors are at the same time held at their highest and lowest levels. The use of a BBD therefore allows a formulation scientist to avoid undertaking experiments that are to performed under extreme conditions and that may urinate substandard results due to the inclusion of data generated from these extreme high and low levels 37.Doehlert DesignThe Doehlert design is an experimental design approach in which different factors can be studied at different levels simultaneously43. This aspect of the Doehlert design is an important characteristic when using some input variables that may be subject to restrictions such as for example cost or experimental constraints (limited amounts of raw material or limited amount of time available) thereby making it a practical and economic alternative to other, second-order experimental design approaches37.This design describes a circular domain of two input variables, a spherical domain for three input variables and a hyper-spherical space for situations in which more than three input variables are to be investigated and which highlights the uniformity of the input variables to be studied in the experimental domain 37.The schematic design space of a Doehlert design for two variables is shown in Figure 1.3, and is represented by a central point and six points of a mend hexagon.An interesting feature of the Doehlert design is that new factors may be introduced during the class of a study without losing relevant and/or valuable information from the data already generated from the experim ental runs that have already been completed.Figure 1.3 Schematic diagram representing the levels studied in a Doehlert DesignThe number of experiments required for a Doehlert design is determined using Equation 1.8 37N= k2+ k+ C0Where,N= the number of experimentsk= the factor numberC0= the replicate number of the central point1.3.5 Mathematical OptimizationOptimization is a mathematical method used to determine an optimal response and is defined as the most advantageous state of existence of the carcass under investigation44. septuple linear regression equations generated from statistically designed experiments provide a translation of the change of a response with a change in input factors and further, allows for the determination of input variables that will produce an optimized response.A difficulty that occurs in optimization procedures is the need to establish a compromise between the anticipate response variables. This challenge is often encountered in the process of opti mization of tablets where the optimum tablet may be one that has superior strength and little or no friability, yet must also have a short disintegration time. Often an increase in tablet hardness results in an increase in the disintegration time of a tablet and therefore a compromise between these contradictory response variables is required to achieve an optimized formulation.1.3.6 Advantages of RSMThe principal(a) advantage of RSM in relation to classical experimental methods and approaches of data evaluation in which only one variable is investigated at a time, is that a large amount of information can be generated from a relatively small number of experiments 38. RSM is therefore less time and cost consuming than the classical approach that requires a large number of experiments to be conducted to be able to explain the behavior of a system 38, 39.A further advantage, with the use of RSM is that it is possible to observe interaction effects of the independent input parameters on the response(s) being monitored 38. The model equation that is generated from the data is able to be used to explain the effect of combinations of independent input variables on the outcome of a process or product.1.3.7 Disadvantages of RSMA primary disadvantage of RSM is that fitting data to a second order polynomial for systems that contain some curvature is often not well accommodated by the second order polynomials that are produced. If the system cannot be explained by a first or second order polynomial, it may be indispensable to reduce the range of independent input variables under consideration as this may then increase the accuracy of the model being considered38. other disadvantage is that although RSM has the potential to evaluate interaction effects of the independent input parameters, it is unable to be used to explain why an interaction(s) has occurred (210). A further disadvantage is that RSM is poor at predicting the potential outcomes for a system operated outs ide the range of study under consideration451.3.8 Software for Design of experimentsMany commercialized software packages are available which are either dedicated to experimental design alone or are of a more general statistical type.Softwares dedicated to experimental designsDESIGN rightECHIPMULTI-SIMPLEXNEMRODWSoftware for general statistical natureSASMINITAB