Chapter 4

One Compartment IV Bolus

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Half-life of Elimination, t_1/2

Another important property of first order kinetics is the half-life of elimination, t_1/2.

Defining Equation

Equation 4.10.1

Equation 4.10.2

Equation 4.10.3

Equation 4.10.4

Equation 4.10.5

OR

Equation 4.10.6

Note: Independent of concentration. This a property of first order processes

These equations can be used as an approximate method of calculating kel. If we look at a plot of Cp versus time on semi-log graph paper.

Figure 4.10.1 Semi-log Plot of Cp versus Time Illustrating t_1/2 Calculation

The steps to take are:

1. Draw a line through the points (this tends to average the data)
2. Pick any Cp and t₁ on the line
3. Determine Cp/2 and t₂ using the line
4. Calculate t_1/2 as (t₂ - t₁)

And finally kel = 0.693/t_1/2 (Equation 4.10.5)

You might also consider determining Cp/4 or Cp/8 after two half-lives or three half-lives, respectively. This should provide a more accurate answer as the differences in Cp and t will be larger.

The line smooths out the bumps. There may be less accurate data points, so by putting in a line you average the data. The half-life is the same whether going from 40 to 20 or from 10 to 5 mg/L. This is a property of the first order process.

Note:

Go from:

Cp - > Cp/2 in 1 half-life i.e. 50.0 % lost 50.0 %
Cp - > Cp/4 in 2 half-lives i.e. 25.0 % lost 75.0 %
Cp - > Cp/8 in 3 half-lives i.e. 12.5 % lost 87.5 %
Cp - > Cp/16 in 4 half-lives i.e. 6.25 % lost 93.75 %
Cp - > Cp/32 in 5 half-lives i.e. 3.125 % lost 96.875 %
Cp - > Cp/64 in 6 half-lives i.e. 1.563 % lost 98.438 %
Cp - > Cp/128 in 7 half-lives i.e. 0.781 % lost 99.219 %

Thus over 95 % is lost or eliminated after 5 half-lives. Typically, with pharmacokinetic processes, this is considered the completion (my definition unless told otherwise) of the process [Although in theory it takes an infinite time]. Others may wish to wait 7 half-lives where over 99% of the process is complete. Others have suggested that three half-lives are sufficient.

In the pharmacokinetic area of study the half-life of a drug usually refers to the biological or terminal half-life. These terms have different meaning for some people. I tend to view them both as referring to half-life measured for the terminal or slowest slope on the semi-log drug concentration versus time plot. At low concentration more processes tend to follow first order kinetics. However, at later times with lower concentrations assay sensitivity can be a serious problem. Also, if absorption is very slow the slowest slope may refer to the absorption process instead of drug disposition.

• Ritschel, W.A. and Kearns, G.L. 2004 Handbook of Basic Pharmacokinetics ... including Clinical Applications, 6th ed., American Pharmaceutical Association, Washington, DC ISBN 1-58212-054-4 pp 369-401
• Half-life has been discussed on the PharmPK listserv.
• Other Pharmacokinetic textbooks

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