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Text of BASIC PHARMACOKINETICS-Ch2: Mathematics Review

Basic Pharmacokinetics REV. 00.1.6 2-1Copyright 1996-2000 Michael C. Makoid All Rights Reserved 2 Mathematics ReviewAuthor: Michael Makoid, Phillip Vuchetich and John CobbyReviewer: Phillip VuchetichBASIC MATHEMATICAL SKILLS OBJECTIVES1. Given a data set containing a pair of variables, the student will properly construct (III) various graphs of the data.2. Given various graphical representations of data, the student will calculate (III) the slope and intercept by hand as well as using linear regression.3. The student shall be able to interpret (V) the meaning of the slope and intercept for the various types of data sets. 4. The student shall demonstrate (III) the proper procedures of mathematical and algebraic manipulations.5. The student shall demonstrate (III) the proper calculus procedures of integration and differentiation. 6. The student shall demonstrate (III) the proper use of computers in graphical simu-lations and problem solving. 7. Given information regarding the drug and the pharmacokinetic assumptions for the model, the student will construct (III) models and develop (V) equations of the ADME processes using LaPlace Transforms.8. The student will interpret (IV) a given model mathematically.9. The student will predict (IV) changes in the final result based on changes in vari-ables throughout the model.10. The student will correlate (V) the graphs of the data with the equations and mod-els so generated. Mathematics ReviewBasic Pharmacokinetics REV. 00.1.6 2-2Copyright 1996-2000 Michael C. Makoid All Rights Reserved Concepts of MathematicsPharmacokinetcs is a challenging field involving the application of mathematicalconcepts to real situations involving the absorbtion, distribution, metabolism andexcretion of drugs in the body. In order to be successful with pharmacokinetics, acertain amount of mathematical knowledge is essential.This is just a review. Look it over. You should be able to do all of these manipulations.This chapter is meant to review the concepts in mathematics essential for under-standing kinetics. These concepts are generally taught in other mathematicalcourses from algebra through calculus. For this reason, this chapter is presented asa review rather than new material. For a more thorough discussion of any particu-lar concept, refer to a college algebra or calculus text.Included in this section are discussions of algebraic concepts, integration/differen-tiation, graphical analysis, linear regression, non-linear regression and the LaPlacetransform. Pk Solutions is the computer program used in this course. Something new - LaPlace transforms. Useful tool.A critical concept introduced in this chapter is the LaPlace transform. The LaPlacetransform is used to quickly solve (integrate) ordinary, linear differential equa-tions. The Scientist by Micromath Scientific Software, Inc.1 is available for work-ing with the LaPlace transform for problems throughout the book. 1. MicroMath Scientific Software, Inc., P.O. Box 21550, Salt Lake City, UT 84121-0550, Mathematics ReviewBasic Pharmacokinetics REV. 00.1.6 2-3Copyright 1996-2000 Michael C. Makoid All Rights Reserved Mathematical Preparation2.2.1 ZERO AND INFINITYAny number multiplied by zero equals zero. Any number multiplied by infinity equals infinity. Any number divided by zero is mathematically undefined.Any number divided by infinity is mathematically undefined.2.2.2 EXPRESSING LARGE AND SMALL NUMBERSLarge or small numbers can be expressed in a more compact way using indices.How Does Scientific Notation Work?Examples: 316000 becomes 0.00708 becomes In general a number takes the form:Where A is a value between 1 and 10, and n is a positive or negative integerThe value of the integer n is the number of places that the decimal point must bemoved to place it immediately to the right of the first non-zero digit. If the decimalpoint has to be moved to its left then n is a positive integer; if to its right, n is anegative integer.Because this notation (sometimes referred to as Scientific Notation) uses indi-ces, mathematical operations performed on numbers expressed in this way are sub-ject to all the rules of indices; for these rules see Section 2.2.4.A shorthand notation (AEn) may be used, especially in scientific papers. This maybe interpreted as , as in the following example:2.28E4 = ( )3.16 1057.083 10A 10nA 10n2.28410 22800 =Mathematics ReviewBasic Pharmacokinetics REV. 00.1.6 2-4Copyright 1996-2000 Michael C. Makoid All Rights Reserved SIGNIFICANT FIGURESA significant figure is any digit used to represent a magnitude or quantity in theplace in which it stands. The digit may be zero (0) or any digit between 1 and 9.For example:How do I determine the number of significant figures?Examples (c) to (e) illustrate the exceptions to the above general rule. The value 10raised to any power, as in example (c), does not contain any significant figures;hence in the example the four significant figures arise only from the 10.65. If oneor more zeros immediately follow a decimal point, as in example (e), these zerossimply serve to locate the decimal point and are therefore not significant figures.The use of a single zero preceding the decimal point, as in examples (d) and (e), isa commendable practice which also serves to locate the decimal point; this zero istherefore not a significant figure.What do significant fig-ures mean?Significant figures are used to indicate the precision of a value. For instance, avalue recorded to three significant figures (e.g., 0.0602) implies that one can reli-ably predict the value to 1 part in 999. This means that values of 0.0601, 0.0602,and 0.0603 are measurably different. If these three values cannot be distinguished,they should all be recorded to only two significant figures (0.060), a precision of 1part in 99.After performing calculations, always round off your result to the number of sig-nificant figures that fairly represent its precision. Stating the result to more signifi-cant figures than you can justify is misleading, at the very least!TABLE 2-1 Significant FiguresValueSignificant FiguresNumber of Significant Figures(a) 572 2,5,7 3(b) 37.10 0,1,3,7 4(c)10.65 x 1040,1,6,5 4(d) 0.693 3,6,9 3(e) 0.0025 2,5 2Mathematics ReviewBasic Pharmacokinetics REV. 00.1.6 2-5Copyright 1996-2000 Michael C. Makoid All Rights Reserved RULES OF INDICESWhat is an index? An index is the power to which a number is raised. Example: where A is anumber, which may be positive or negative, and n is the index, which may be pos-itive or negative. Sometimes n is referred to as the exponent, giving rise to thegeneral term, Rules of Exponents. There are three general rules which applywhen indices are used.(a) Multiplication (b) Division (c) Raising to a PowerThere are three noteworthy relationships involving indices:(i) Negative Index As n tends to infinity then .(ii) Fractional Index(iii) Zero IndexAnAnAmA = n m + AnBm AB---. ,| `nB n m +=AnAm------- An m = AnBm------- AB---. ,| `nB n m =An( )mAnm=A n 1An------ = n ( ) A n 0 A1n---An=A01 =Mathematics ReviewBasic Pharmacokinetics REV. 00.1.6 2-6Copyright 1996-2000 Michael C. Makoid All Rights Reserved LOGARITHMSWhat is a logarithm? Some bodily processes, such as the glomerular filtration of drugs by the kidney,are logarithmic in nature. Logarithms are simply a way of succinctly expressing anumber in scientific notation.In general terms, if a number (A) is given by then where log signifies a logarithm to the base 10, and is the value of the logarithmof (A).Example: 713000 becomes , and , thus 713000 becomes and Logarithms to the base 10 are known as Common Logarithms. The transformationof a number (A) to its logarithm (n) is usually made from tables, or on a scientificcalculator; the reverse transformation of a logarithm to a number is made usinganti-logarithmic tables, or on a calculator.What is the characteris-tic? the mantissa?The number before the decimal point is called the characteristic and tells the place-ment of the decimal point (to the right if positive and to the left if negative). Thenumber after the decimal is the mantissa and is the logarithm of the string of num-bers discounting the decimal place.2.2.6 NATURAL LOGARITHMSWhat is a natural loga-rithm?Instead of using 10 as a basis for logarithms, a natural base (e) is used. This naturalbase is a fundamental property of any process, such as the glomerular filtration of adrug, which proceeds at a rate controlled by the quantity of material yet to undergothe process, such as drug in the blood. To eight significant figures, the value of thetranscendental function, e, is... Strictly speaking, Where is an inte-ger ranging from 1 to infinity , A 10n= A ( ) log n =n7.13 1057.13 100.85= 100.85105 105 0.85 + ( )105.85= =713000 ( ) log 5.85 =e 2.7182818 = e 11x!----x 1 =+ = x ( )Mathematics ReviewBasic Pharmacokinetics REV. 00.1.6 2-7Copyright 1996-2000 Michael C. Makoid All Rights Reserved denotes the summation from , and! is the factorial (e.g., 6! = 6x5x4x3x2x1= 720)In general terms, if a number (A) is given by , then by definition,Where, ln signifies the natural logarithm to the base , and is the value of thenatural logarithm of .Natural logarithms are sometimes known as Hyperbolic or Naperian Logarithms;again tables are available and scientific calculators can do this automatically. Theanti-logarithm of a natural l