[2] MATHEMATICAL CONCEPTS, RATES, HALF-LIFE

zero order reaction in ordinary rectangular graph

linear

zero order reaction in semilog graph

curved

Concentration (C)

zero order

C=-kt + C0

Elimination Constant (k0)

zero order

k0= C1-C2 / t1-t2

Half-life (t½)

zero order

t1/2 = 0.5C0 /k0

First Order Reaction in ordinary rectangular graph

curved

First Order Reaction in semilog graph

lnear

Concentration (lnC)

first order

lnC= -kt + lnC0

Rate Constant (k1)

first order

k0= lnC1-lnC2 /t1-t2

Half-life (t½)

first order

t1/2 = 0.693/ k1

Pseudo-First Order Reaction

If the number of points on the line and number of outliers are equal, it is pseudo first order.

Pseudo-First Order Reaction

ex. Given eight values, four are on the line, four are outliers.

RULER METHOD (BACK EXTRAPOLATION)

-not reliable; most likely erroneous

-can only acquire ESTIMATE

-extrapolate until x or y-intercept

LEAST SQUARE METHOD (LINEAR REGRESSION)

-more accurate (in statistics)

-table of values: x, y, x2, xy

Exponents

N= b^x

Logarithms

The logarithm of a positive number N to a given base b is the exponent x to which the base must be raised to equal the number N.

If N = b^x, then

logb N = x

APPLICATIONS

-pH of biological fluids can affect LADME

-pH scale is a logarithmic scale:

Most drugs are either

weak bases or weak acids.

The pH of the biological fluid

determines the degree of ionization of the drug which influences the pharmacokinetic profile.

pH

-log[H+]

log(1/H+)

[H+]

10^-pH

GRAPHS

important method of visualizing relationships between variables

GRAPHS

in pharmacokinetics: time

-independent variable

-on the abscissa (x-axis)

GRAPHS

in pharmacokinetics: Drug concentration

-dependent variable

-on the ordinate (y-axis)

Curve Fitting

Implies that there is some sort of relationship between variables x and y (ie. drug vs. pharmacologic effect)

Curve Fitting

Continuous function of x and y

Equation of a Straight Line

y = mx + b

Straight lines are very useful for __

accurately predicting values for which there are no experimental observations.

Linear Regression

This method is often encountered and used in clinical pharmacy studies to construct a linear relationship between an independent variable and a dependent variable.

Linear Regression

It can be done using an MS Excel, Scientific Calculators and Graphical Method (Semilog & Graphing Paper)

Coefficient of determination (r2)

It expresses how much variability in the outcome is explained by the input factor.

RATE PROCESSES

The rate of chemical reaction is the velocity with which the reaction occurs.

In pharmacokinetics, rate is used to

describe processes such as drug absorption or drug elimination.

rate equation

dC/ dt= -(k)(C)^n

ORDER OF REACTION

The way in which the concentration of the drug or reactant in a chemical reaction affects the rate of chemical reaction or process.

ORDER OF REACTION

In pharmacokinetics, two orders are of importance, the zero order and the first order.

ZERO - ORDER KINETICS

-Proceeds over time (t) independent from the concentration of the drug (C)

-Fixed amount of drug is removed per unit time

ZERO - ORDER KINETICS

rate eq

dC/dT= -k0

FIRST - ORDER KINETICS

-The rate of a first-order process is dependent upon the concentration of the drug (C) .

-Fixed percentage of drug is removed per unit time

FIRST - ORDER KINETICS

rate eq

dC/ dT= -k1(C)

HALF-LIFE

time required for one-half of the drug concentration to disappear (t½ )

Half-life (t½)

zero order

Dependent on:

-Concentration at time zero

-Rate constant

Half-life (t½)

first order

Independent of: Concentration at time zero

DETERMINATION OF ORDER

-Plot the data on a rectangular graph

-If the data appear to be a curve rather than a straight line, the reaction rate for the data is non-zero order

-Plot the data on a semi-log graph

-If the data appear to form a straight line with good correlation using linear regression, then the data likely follow first-order kinetics