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As defmed in Chapter I, the study of pharmacokinetics involves the application of principles of kinetics. The principles of kinetics, in most cases, involve rather extensive use of logarithms, as well as construction of graphs with the data in order to obtain a clear and comprehensive picture of the data.
Even in a tabular form, the raw data do not present a clear picture. When it is very difficult to comprehend raw data, any conclusions drawn from the raw data are a gross approximation at best.
In constructing graphs, an effort is made to obtain a linear relationship between the variables. If it is not possible to obtain a straight line, one may settle for a smooth curve, because even a smooth curve permits a much better understanding of relationship between the variables than looking at raw data in a tabular form.
Constructing a graph to obtain a linear relationship between variables under study offers several advantages. For example, a graph expressing the concentration of drug or the amount of drug as a function of time can be used to predict the value of one variable e. More importantly, a linear relationship affords the opportunity to express the relationship between variables as a mathematical equation, which can then be used to predict the connection between variables in a scientific manner.
Logarithms were invented more than 3 centuries ago. At that time, calculators were not available to carry out difficult calculations and logarithms provided a relatively simple means of conducting time-consuming lengthy and complicated calculations.
Logarithms were also used as a convenient means of expressing very large or very small numbers in a simple fashion. For example, a small number, such as concentration of hydronium ions in a dilute solution may be expressed as a decimal fraction, e. Because logarithms convert a relatively very small number into a more conveniently expressible number, logarithms have no units, and are considered as real numbers. Logarithms are exponential functions.
Logarithms play an important role in pharmacokinetic calculations, because the pharmacokinetics of various rate processes, such as absorption, metabolism, distribution, and elimination, etc. For example, in the equity:. Chapter 2. Therefore, by definition, this equity is expressed in the logarithmic notation as:. The exponent b to which the base a is raised to give x in equation is referred to as the logaritlun of x. Since the logaritlun of x is b, therefore, b is known as the antilogarithm of x.
For example, the equity 10 2 is expressed in the logaritlunic notation as logJO 2, and as before, the exponent 2. Since the. The exponent 3 to which the base 4 is raised to give the value 64 is therefore the logarithm of As above, since the logarithm of 64 is 3, therefore 3 is the antilogaritlun of Logarithms were invented by John Napier of Scotland more than years ago. He used Natural.
Log Number, 2. A few years later, Henry Briggs used Napier's discovery, and instead of using the Natural Log Number, Briggs introduced 10 as the base for his logarithms. Napier's system is favored by physical chemists and is used extensively in pharmacokinetics. Briggs' system is favored by biological scientists and is widely used for computation purposes. It is known as the Briggsian. The Natural Log Number is the quantity a in. Similarly, the equity 4 3.
Logaritluns or Common Logarithms. Common logarithms, also known as Briggsian logarithms, use 10 as the base. The base 10 is the quantity a in equation Therefore, equation in common logarithms is:.
As a general practice, when common logarithms are written as log, it is understood that the term log represents common logarithms, and therefore base 10 is not written.
Equation is generally written as:. Natural logarithms are also known as Napierian logarithms. Since the base used in natural. The base e must be written if natural logarithms are abbreviated as log.
If one does not want to write the base e, then instead of writing log, one writes In. Thus, equation may be written as:. The Natural Log Number, the quantity e used as the base in natural logarithms , is defmed as the.
The value of this series, as the value of n. Therefore, 2. When n is equal to 1,. Table shows how the value 2. Table Numerical Value. Logarithms provided a relatively simple means of lengthy, and complicated calculations. Using logarithms, one could convert multiplications into simple additions, calculations dealing with divisions into calculations dealing with exponential terms into simple multiplication of terms.
These rules are similar for natural logarithms The more important rules are given in Table Common Logarithms. Natural Logarithms. In alb. In f' In e-Q. Calculate the common logarithm of. Calculate the cornmon loganthm of. The relationship between cornmon and natural logarithms is as follows:. Now, if we take natural logarithms on both sides of equation then,. Substituting the value of b from equation into equation , we have.
What is the cornmon logarithm of , if natural logarithm of is 5. According to equation ,. If cornmon logarithm of 40 is 1. From equation ,. In Data in some studies are presented in a tabular form in order to comprehend salient points being presented. In pharmacokinetic studies, data in a tabular form may not necessarily provide important and vital information sought from the study. Therefore, the data must invariably be plotted on a graph paper so as to provide a better understanding of information desired from the study.
Graphs are convenient and important way of picturing functions geometrically by means of a rectangular coordinate system.
Since a linear relationship between variables being investigated provides a more meaningful picture, the data are often plotted in such a manner that a linear relationship between variables can be obtained. The equation of the straight line thus obtained also enables express! The horizontal line is called x-axis, x-coordinate or abscissa, and the vertical line is called y-axis, y-coordinate or ordinate.
Convenient units of length are selected to layoff distances to the right and left, and upwards and downwards from the point of origin. Distances to the right and upward of origin are called positive distances, and those to the left and downward are called negative distances. The independent variable is measured along the horizontal coordinate scale, and dependent variable is measured along vertical coordinate scale.
It is important to select units of measurement along the axes in order to maximize graph on graph paper. To draw the graph of a linear function equation of a straight line , it may be sufficient to plot two of its points and draw a straight line through them.
However, plotting more than two points serves as a check against errors. The axes of a two-dimensional graph divide the plane into four segments or quadrants. Second Quadrant. First Quadrant. Fourth Quadrant. The four quadrants of a two-dimensional graph are counted counter clock-wise starting with top right-hand quadrant and represent the following. This is the upper right-hand quadrant, and is also known as the first quadrant. This is the upper left-hand quadrant, and is also known as the second quadrant.
This is the lower left-hand quadrant, also known as the third quadrant, and. This is the lower right-hand quadrant, and is also known as the fourth quadrant. In a two-dimensional graph, any point in any of the four quadrants may be located by its distance from the two axes.
Any point on the x-axis has its ordinate equal to 0, and any point on the y-axis has its abscissa equal to O. In all pharmacokinetic studies, negative values are not encountered. The most conunon situation is when both x and y have positive values. The simplest relationship between the two variables is described by a first-degree equation. A first-degree equation is an equation in which the value of exponent of the variables is equal to one. A plot of first-degree equation on a rectangular two-dimensional graph yields a straight line described by the following equation, known as the equation of a straight line:.
If y-intercept is positive, the point of intersection is above the x-axis.
ISBN 13: 9789350909393
As defmed in Chapter I, the study of pharmacokinetics involves the application of principles of kinetics. The principles of kinetics, in most cases, involve rather extensive use of logarithms, as well as construction of graphs with the data in order to obtain a clear and comprehensive picture of the data. Even in a tabular form, the raw data do not present a clear picture. When it is very difficult to comprehend raw data, any conclusions drawn from the raw data are a gross approximation at best. In constructing graphs, an effort is made to obtain a linear relationship between the variables. If it is not possible to obtain a straight line, one may settle for a smooth curve, because even a smooth curve permits a much better understanding of relationship between the variables than looking at raw data in a tabular form. Constructing a graph to obtain a linear relationship between variables under study offers several advantages.
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Jaypee Biopharmaceutics and Pharmacokinetics by PL Madan for Medical Pharmacology 2 Edition 2014
Biopharmaceutics and Pharmacokinetics deals with what body does to the drug. In this book along with the fundamentals of absorption distribution metabolism and excretion current topics such as in vitro systems have also been introduced. In case of Pharmacokinetic Models authors have tried to keep mathematical part simpler and minimal. This book also introduces Non-compartmental models which are gaining increasing importance in the analysis of pharmacokinetic data. In the chapter Applications of Pharmacokinetics newer concepts such as chronopharmacokinetics have been introduced.