Biological Half-life - Rate Equations - First-order Elimination

First-order Elimination

There are circumstances where the half-life varies with the concentration of the drug. Thus the half-life, under these circumstances, is proportional to the initial concentration of the drug A₀ and inversely proportional to the zero-order rate constant k₀ where:

This process is usually a logarithmic process - that is, a constant proportion of the agent is eliminated per unit time. Thus the fall in plasma concentration after the administration of a single dose is described by the following equation:

• C_t is concentration after time t
• C₀ is the initial concentration (t=0)
• k is the elimination rate constant

The relationship between the elimination rate constant and half-life is given by the following equation:

Half-life is determined by clearance (CL) and volume of distribution (V_D) and the relationship is described by the following equation:

In clinical practice, this means that it takes 4 to 5 times the half-life for a drug's serum concentration to reach steady state after regular dosing is started, stopped, or the dose changed. So, for example, digoxin has a half-life (or t_½) of 24–36 h; this means that a change in the dose will take the best part of a week to take full effect. For this reason, drugs with a long half-life (e.g. amiodarone, elimination t_½ of about 58 days) are usually started with a loading dose to achieve their desired clinical effect more quickly.