Analysis of Response Surface Methodology (RSM)

Analysis of Response Surface Methodology (RSM) 1.3 DESIGN OF EXPERIMENTS 34, 35 Trans-Pacific Partnership economic framework agreement has clearly defined that many studies must be conducted to develop a formulation. Design of experiments (DOE) has proven to be an effective tool for formulation scientists throughout the many stages of the formulation process. At every step of formulation development, DOE can aid in making intelligent decisions. These steps include excipient compatibility studies, process feasibility studies, formulation optimization, process optimization, scale-up and manufacturing process characterization. Lastly, the product and manufacturing process must be validated before it is on the market. The word optimize is defined as, making as perfect, effective or functional as possible. Optimization may be interpreted as to find out the value of controllable independent variable, that gives the most desired value of dependent variables. The application of formulation optimization techniques is relatively new to the practice of pharmacy when used intelligently, with the common sense, these â€Å"statistical† methods will broaden the perspective of the formulation process. At the Preformulation stage, before any experiment is conducted certain problem arises, it is often not known before hand which variable will significantly influence the response. Screening designs and ANOVA helps to solve this problem. A second serious complication may arise with new excipients and new process factor, for which qualitative or quantitative effects are not known and are unpredictable. The following questions must be answered before choosing any design 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 choose a compromise. A fourth problem is the lack of insight` to perform an adequate optimization studies. Above all in the performance of an optimization study, the formulation development scientist can also be a factor as personal variation. 1.3.1 Terms used in Design of experiments Variables These are the measurements, values, which are characteristics of the data. There are two types of variables; dependent variables and independent variables. Independent variables(X) are set in advance, which are not influenced by any other values e.g., Lubricants concentration, drug to polymer ratio, etc. Dependent variables(Y) are the outcome variables, influenced by the independent variables e.g., hardness, dissolution rate, etc. Factor Factor is an assigned variable such as concentration, temperature, lubricant agent, drug to polymer ratio, polymer to polymer ratio or polymer grade. A factor can be qualitative or quantitative. A quantitative factor has a numerical value to it for example, concentration (1%, 2%†¦ so on), drug to polymer ratio (1:1, 1:2†¦etc). Qualitative factors are the factors, which are not numerical value, for example, the polymer grade, humidity condition, type of equipment, etc. these are discrete in nature. Levels The levels of a factor are the values or designation assigned to the factor. For e.g. in concentration (factor) 1 % will be one level, while 2% will be another level. Two different plasticizers are levels for grade factor. Usually levels are indicated as low, middle or high level. Normally for ease of calculation the numeric and discrete levels are converted to –1 (low level) and +1 (high level).The general formula for this conversion is Where ‘X’ is the numeric value Response Response is mostly interpreted as the outcome of an experiment. It is the effect, which we are going to evaluate i.e. Disintegration time, duration of buoyancy, etc. Effect The effect of a factor is the change in response caused by varying the levels of the factor. This describes the relationship between various factors and levels. Interaction Interaction 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 hardness (response) of a tablet. In the trial and error method, a lot of formulations have 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 Process Generally optimization process involves the following steps. Based on the previous knowledge or experience or from literature, the independent variables are determined and set in the beginning. Selection of a suitable model, based on the results of the factor, screening is done. The experiments are designed and conducted. The responses are analyzed by ANOVA, test on lack of fit, to get an empirical mathematical model for each individual response. The responses are screened, by using multiple criteria to get the values of independent variables. Experimental Design Experimental design is a statistical design that prescribes or advises a set of combination of variables. The number and layout of these design points within the experimental region, depends on the number of effects that must be estimated. Depending on the number of factors, their levels, possible interactions and order of the model, various experimental designs are chosen. Each experiment can be represented as a point within the experimental domain, the point being defined by its co-ordinate (the value given to the variables) in the space. 1.3.3 Response Surface Methodology Response surface methodology (RSM) is an experimental strategy that was developed in the 1950’s36. RSM is comprised of a group of mathematical and statistical techniques that are based on fitting experimental data generated 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 input variables37, 38. RSM is able to define the effect of independent variables alone and in combination with the manufacturing processes under 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 studied, monitored, and measured and how these will be measured. A summary of the subsequent RSM approach includes36 Performance of the relevant DOE. Estimation of the coefficient in the relevant response surface equation. Checking of the adequacy of the equation to describe the fit. Studying the response surface to identify 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 input variable can be described by Equation 1.1 y= f(x1, x2, x3†¦xn) +ÃŽ µ Where, y = relevant response f = unknown function of a response x1, x2,..xn = independent variables n= number of independent variables ÃŽ µ = statistical error that represents other sources of variability not accounted for by f Contour plot can be described as: i. Mound-shaped that has elliptical contours with a stationary point at the position of a maximum response. ii. Saddle-shaped that has a hyperbolic system of contours with a stationary point that is neither a maximum nor minimum 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 outside the design region 39. The stationary point is a combination of design variables where the surface presents as either a maximum and/or a minimum 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 constant and this is considered a ridge response 40. By plotting a response, y, against one or two input 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 complicated depending on the effect of the input variables on the response. Therefore RSM aims at approximating f by use of a suitable, ordered 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, second or third order, depending on how the output variables or responses react to changes in the input variables. If the response is a linear function of the independent variables, then 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 affected by a small change in the value of the input factors and that exhibit little or no interaction(s) between the input variable terms. y= ÃŽ ²0+ ÃŽ ²1x1+ ÃŽ ²2x2+†¦..+ ÃŽ µ 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= ÃŽ ²0+ ÃŽ ²1x1+ ÃŽ ²2x2+ ÃŽ ²12x12+†¦Ã¢â‚¬ ¦+ ÃŽ µ It has been reported that second order models are also applicable to input factors that exhibit extensive variability over an experimental domain and these relationships are best described using Equation 1.4 y= ÃŽ ²0+ ÃŽ ²1x1+ ÃŽ ²2x2+ ÃŽ ²12x12+ ÃŽ ²11x12+ ÃŽ ²22x22+†¦..+ ÃŽ µ Where y= response x1, x2,..xn = input factors ÃŽ ²0= constant that represents the intercept ÃŽ ²i= coefficient of first order term ÃŽ ²ii= coefficient of second order term ÃŽ ²ij= coefficient of second order interaction The 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 positive value points to an agonistic effect of the input factor on the response, whereas coefficients with negative values indicate an antagonistic effect. 1.3.4 Choice of Response Surface Design Central Composite Design (CCD) A CCD was originally presented by Box 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 approach 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 axial or star points, located in the centre of each 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 (ÃŽ ±). The factors under consideration are usually investigated at five different levels and are always represented by coded values viz., -ÃŽ ±, -1, 0, +1 and +ÃŽ ±. Figure 1.1Schematic diagram representing the levels studied in a Central Composite Design The distance 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. ÃŽ ± =2k/4 Where, k= the factor number ÃŽ ± = axial point The number of experiments required for a CCD approach is calculated using Equation 1.6 N= k2+ 2k+ C0 Where, N= the experiment number k= the factor number C0= the replicate number of the central point The 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 design Central composite design can be divided into three types. Table 1.2 Types of central composite design Box-Behnken Design (BBD) The BBD describes a class of second-order designs based on a three-level incomplete 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 Design The number of experiments for Box-Behnken Designs can be calculated using Equation 1.7. N= 2k (k-1) +C0 Where, N= the number of experiments k= the factor number C0= the replicate number of the central point For experiments in which there 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 further advantage of BBD is that it does not include the need to evaluate situations in which all factors are simultaneously 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 produce substandard results due to the inclusion of data generated from these extreme high and low levels 37. Doehlert Design The 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 regular hexagon. An interesting feature of the Doehlert design is that new factors may be introduced during the course of a study without losing relevant and/or valuable information from the data already generated from the experimental runs that have already been completed. Figure 1.3 Schematic diagram representing the levels studied in a Doehlert Design The number of experiments required for a Doehlert design is determined using Equation 1.8 37 N= k2+ k+ C0 Where, N= the number of experiments k= the factor number C0= the replicate number of the central point 1.3.5 Mathematical Optimization Optimization is a mathematical method used to determine an optimum response and is defined as the most advantageous state of existence of the system under investigation44. Multiple linear regression equations generated from statistically designed experiments provide a description 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 anticipated response variables. This challenge is often encountered in the process of optimization 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 necessary to achieve an optimized formulation. 1.3.6 Advantages of RSM The primary 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 RSM A 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 necessary to reduce the range of independent input variables under consideration as this may then increase the accuracy of the model being considered38. Another 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 outside the range of study under consideration45 1.3.8 Software for Design of experiments Many commercial software packages are available which are either dedicated to experimental design alone or are of a more general statistical type. Software’s dedicated to experimental designs DESIGN EXPERT ECHIP MULTI-SIMPLEX NEMRODW Software for general statistical nature SAS MINITAB

Personality Disorders

Compare and Contrast Types of Personality Disorders Written Assignment 5: Compare and Contrast Types of Personality Disorders and Identifying the Three Clusters into Which Most Personality Disorders are Grouped By: Keturah Albright Abnormal Psychology HDV-284074-01X-10FA2 Professor/ Instructor: Jonathan Gibralter Compare and Contrast Types of Personality Disorders Personality disorder is defined as, gradual development of inflexible and distorted personality and behavioral patterns that result in persistently maladaptive ways of perceiving, thinking about, and relating to the world. p. 431Butcher, James N. ) It is important to understand the meaning of personality disorder, in order to properly diagnosing a person. Society tends to place this label on individuals who they feel display characteristics resembling a â€Å"personality disorder’. There are three clusters in which personality disorders are placed in. Cluster A: Consists of paranoid, schizoid, and schizotypal person ality disorders. People with these disorders often seem odd or eccentric, with unusual behavior ranging from distrust and suspicious to social detachment.Cluster B: Includes histrionic, narcissistic, antisocial, and borderline personality disorders. Individuals with these disorders share a tendency to be dramatic, emotional, and erratic. Cluster C: Includes avoidant, dependent, and obsessive-compulsive personality disorders. In Contrast to the other two clusters, people with these disorders often show anxiety and fearlessness. (p 342, Butcher, James N. ) There are some ways in which these three clusters of personality disorders can be compared.All three clusters have personality disorders that would have the person to focus on the needs of themselves and no one else, their needs come first and for most and they may feel justified by their actions. Most of these individuals are said to suffer from, â€Å"chronic interpersonal difficulties and problems with one’s identity or s ense of self. †(p. 341, Butcher, James N. )All three clusters of personality disorders tend to have individuals who lack the ability to function normally in society on a consistent basis.All three clusters mentally have a great false sense of reality that causes them to act in a defensive/protective manner. As well as most of the clusters have symptomatic impulsive behavior. Commonly all three clusters can be treated with some type of therapy. There are also contrasting behavior that defines the different clusters. Cluster A: Feels as though others are mistreating them or taking advantage of them, while Cluster B: bring about physical harm to others with no remorse.Cluster C: Has a need for others to take a lead role in there life for major things, while Cluster B: Seeks to exploit others. Also, Cluster A: Has odd believes and magical thinking that they take confidence in, and Cluster B: Has a fear of being left alone to fend for themselves. In all there are things that can be compared and contrasted with Clusters: A, B, and C, but the main thing to understand is that all of them are real personality disorders that with the proper guidance can be managed with some type of therapy (and depending on the severity, medication as well).Personality disorders if not properly diagnosed could cause great harm to the individual as well as the society in which they live. Reference Butcher, James N. , Mineka, Susan and Hooley, Jill M. (2010). Abnormal Psychology. Boston, MA: Allyn & Bacon Abnormal Psychology, DVD 105, CDL Course 284074 http://emedicine. medscape. com/article/294307 [pic][pic][pic]

How to Discover Your Life Dreams

How To Discover Your Life Dreams I have an assignment for you Do you want to know why you were born? What your sacred mission In life Is? Here's what you do: Schedule a few hours for a personal retreat; a whole day is even better. Get your calendar and cross out that special day with a fat red pen. No one touches that day, It's your date where you discuss your life with God. Take that extended time of personal reflection and bring these questions with you. Here are some â€Å"heart questions† for you to answer†¦ Questions To Answer: What are your greatest strengths?What Is the primary gift that has been given to you to bless the world? What do you enjoy doing the most? What were the happiest moments of your life? What were you doing during those moments? If given an opportunity to live In anyway and do anything?no matter how crazy your choice would be?what kind of life would you choose to live? Imagine yourself on your 75th birthday: What kind of person would you want to be by that time? If family members were to honor you on your 75th birthday party, and thank you and speak about the good things that you have done,what would you want them to ay to you? Ђ Write down a tentative Personal Mission Statement: From your answers above, write a tentative Personal Mission Statement. This one sentence should express what you most deeply believe your life should attain at the end of your days. It should be inspiring and moving as well. And don't worry if it won't be perfect. Your Personal Mission Statement should undergo much revision and editing as the weeks, months, and years go by. You should review your Personal Mission Statement in prayer regularly.Finally, write down your Life-Dreams Now that eve written down a Personal Mission Statement, write down a description of your life twenty years down the road. Picture yourself, your schedule, your work, your house, your relationships, your projects, your character in vivid detail. Don't worry if you fill a few pages for your Life-Dreams. Let your heart speak as you paint in the canvas of your imagination. Create Your Financial Plan Based On Your Life-Dreams After writing your Life-Dreams document, write your financial plan: How much you should be earning next year. Writing down a financial goal is an exercise in faith.And an exercise in personal commitment. Your mind will be forced to open up and think of ways how those goals would be met. When I did this, it was as though power was being released as I wrote my financial goals. But as I scribbled on paper every crazy idea that came to my mind, I repeatedly kept going back to my divinely appointed mission in life? because that was how God was going to provide for me. Let me explain. I believe that when we're faithful to our mission and give ourselves to the world and genuinely help people live a better life, we see the world repaying back.When we give, we receive?and we Just don't receive, but we receive much more than what we have gi ven. Why? Because God has written the Law of reciprocity in the fabric of the universe. Naturally, for this Law of Reciprocity to work, you need to charge what you are worth and not undersell yourself. Earning money doesn't make you less spiritual. If the money that you're going to earn is to support your God-appointed Life-Dreams, it is very spiritual. (It's another issue altogether to be working for a religious ministry or a rather to pick up non-monetary rewards in the after life.So far, Vive been able to attain my financial goals?enough to support my family and more than enough to be generous to those in need? the second part being an intrinsic part of my mission. But more importantly, by creating a financial plan based on my Life-Dreams, I live deliberately, and money is still connected to my mission?not something apart from it. Because success for me isn't having the greatest wealth but the fulfillment of my life-dreams?ninety percent of which aren't financial. May your dreams come true, Boo Sanchez

Causes of the Salem Witch Trials Political, Religious and...

Between the months of June to September of 1692, the infamous witch trials in Salem, Massachusetts resulted in the hanging of 19 men and women; the deaths of five others, including two children, while imprisoned in jail; the pressing to death of an 80-year old man, and the stoning of two dogs for collaborating with the Devil. Hundreds of others faced accusations and dozens more were jailed for months during the progress of the trials. For over three hundred years these events have not only captured the general publics imagination, but that of the academic community. Beginning with Charles Upham, in 1867, historians have attempted to explain the mass hysteria that swept through Salem in 1692. These accounts vary both in their†¦show more content†¦First, the Puritan religion was under direct attack from other religious groups including the Quakers and the Baptists. The Quakers, a religious sect newly founded in England, began coming to Massachusetts in 1656. Laws were enacted to prohibit their coming, but they came in defiance of the laws. Between 1659 and 1661 four people were hanged according to these new laws. Though the Puritans tried to defend themselves by the plea that they were defending the public peace, they were roundly condemned in England. By the mid-1670s, Quakers were protected by the English law and could conduct non-religious business in New England. Moreover, in 1654 the congregation of Cambridge Church was shocked by a statement from Henry Dunster, the highly respected president of Harvard College. While a baptismal service was in progress, he arose to dispute the practice of infant baptism as un-Biblical and proceeded to take each point from the pastors sermon and to answer it with Baptist views. He was silenced, stripped of his Harvard presidency and publicly rebuked. Those perceived as religious dissidents would be accused of witchcraft during the Salem trials. Arguably, the colony had previously been beset by religi ous dissidents, dealing with each episode without incurring a public outcry of a witchcraft conspiracy; however, this time there were other more alarming developments taking place within the colony. A longShow MoreRelatedReasons : The Causes Of The Salem Witch Trials741 Words   |  3 Pagesarrested in Salem, Massachusetts. All but one of these people were believed to be witches (Background Essay). Prior to the hearing in Salem witch trials were carried out in several different towns. â€Å"In 17th century New England witchcraft was a serious crime (Background Essay).† Two girls aged nine and 11, Betty Parris and Abigail Williams, identified a slave name Tituba and two other local women as witches. This led to the accused women being carted off to jail starting the Salem Witch Trials. WithoutRead MoreThe Causes of the Salem Witch Trials (DBQ)1337 Words   |  4 Pagesseem preposterous. Any behavior regarded as strange by fellow citizens was sufficient to hold a trial with a sentence of death. Though such scenarios seem unfathomable in our modern culture, it was a reality for hundreds of New England settlers. The causes of the famous outbreak of witch trials in Salem, Ma ssachusetts are rooted in social, economic, and political aspects of the late 17th century Salem community. Early New Englanders were unable to accept the increase in diversity and the break inRead MoreThe Salem Witch Trials Essay1349 Words   |  6 PagesBefore 1692, the supernatural was a part of people’s everyday normal life. This is so as people strongly believed that Satan was present and active on earth. Men and women in Salem Village believed that all the misfortunes that befell them were the work of the devil. For example, when things like infant death, crop failures or friction among the congregation occurred, people were quick to blame the supernatural. 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The last recorded execution in England of an alleged witch is in 1682, though trials and accusations would still be brought alleged witches right up until the 1800’s both in England and on the continent in most cases /crown rule where witch-hunting/trials and executions took place, probably the most famous and certainly one of the most written about witching episodes is that of Witch Trials of Salem, Massachusetts. The Salem Trials took placeRead MoreImpact of Religion on American History to 1877934 Words   |  4 Pagesweather. Politics and people are widely impacted by religion. Religion is the primary cause of most wars in countries across the world. Many historians believe America was formed on the basis of religion. In this research paper, I will illustrate the impact religion had on American History to 1877. Specifically, it will examine: 1) Major events impacting traditional religious beliefs in America, 2) Religious disputes which impacted land development, and 3) The impact religion had on slavery.Read MoreSalem Witch Trials : Legal Proceedings1338 Words   |  6 PagesThe Salem Witch Trials, also know as the Salem Witchcraft Trials were legal proceedings which took place of course in the Salem Village of Massachusetts. These trials began during the spring of 1692, after a group of young girls in the village, claimed to be possessed by the devil accusing several local women of practicing the craft. Victims were prosecuted and executed for reputedly practicing witchcraft, when little to no evidence of the act itself existed. This historical period resulted in twentyRead MoreWhat Caused The Salem Witch Trials?1331 Words   |  6 PagesWhat Caused the Salem Witch Trials? The Salem Witch Trials are one of the most infamous events in American history, with Salem Massachusetts drawing thousands of visitors and generating millions of dollars every year, regardless of the fact that the trials did not actually take place in modern day Salem. The Salem Witch Trials took place between 1692 and 1693. By the time the court admitted that the trials had been a mistake, more than 200 people had been accused of witchcraft and 20 had been executed