# Biological half-life

The biological half-life of a substance is the time required for half of that substance to be removed from an organism by either a physical or a chemical process. Biological half-life is an important pharmacokinetic parameter and is usually denoted by the abbreviation t1/2.

While a radioactive substance decays perfectly according to first order kinetics where the rate constant is fixed, the elimination of a substance from a living organism follows more complex kinetics.

## Examples of biological half-lives

###  Water

The biological half-life of water in a human is about 7 to 10 days. It can be altered by behaviour. Drinking large amounts of beer will reduce the biological half-life of water in the body. This has been used to decontaminate humans who are internally contaminated with tritiated water (tritium). Drinking the same amount of water would have a similar effect, but many would find it difficult to drink a large volume of water. The basis of this decontamination method (used as Harwell) is to increase the rate at which the water in the body is replaced with new water.

###  Alcohol

For instance, the removal of ethanol (alcohol) through oxidation by alcohol dehydrogenase in the liver from the human body is limited. Hence the removal of a large concentration of alcohol from blood may follow zero order kinetics. Also the rate limiting steps for one substance may be in common with other substances. For instance, the blood alcohol concentration can be used to modify the biochemistry of methanol and ethylene glycol. In this way the oxidation of methanol to the toxic formaldehyde and formic acid in the human body can be prevented by giving a person who has ingested methanol an alcoholic drink. Note that methanol is very toxic and causes blindness and death. A person who has ingested ethylene glycol can be treated in the same way.

###  Prescription medications

####  Prozac

Some substances migrate slowly from the brain to the blood. The active metabolite of Prozac, a prescription antidepressant, remains a long time in the brain because it is lipophilic. biological half life of 4 to 16 days. The biological half life of the parent drug is 1 to 6 days.

####  Salbutamol

Salbutamol 1.6 hours

####  Digoxin

Digoxin 24 to 36 hours

####  amiodarone

amiodarone 90 days

####  Cis-platin

Cis platin 20 to 30 minutes

####  Chlorambucil

Chlorambucil 1.5 hours

####  Vincristine

Vincristine 19 to 155 hours

###  Metals

The biological half-life of caesium in humans is between one and four months. This can be shortened by feeding the person prussian blue. The prussian blue in the digestive system acts as a solid ion exchanger which absorbs the caesium while releasing potassium ions.

For some substance, is it important to think of the human or animal body as being made up of several parts, each with their own affinity for the substance, and each part with a different biological half-life. Attempts to remove a substance from the whole organism may have the effect of increasing the burden present in one part of the organism. For instance, if a person who is contaminated with lead is given EDTA in a chelation therapy, then while the rate at which lead is lost from the body will be increased, the lead within the body tends to relocate into the brain where it can do the most harm.

• Polonium in the body has a biological half-life of about 30 to 50 days.
• Cesium in the body has a biological half-life of about one to four months.
• Lead in bone has a biological half-life of about ten years.
• Cadmium in bone has a biological half-life of about 30 years.
• Plutonium in bone has a biological half-life of about 100 years.
• Plutonium in the liver has a biological half-life of about 40 years.

A substance can have an effect on the health of a person long after the substance has left the body. For example, a car crash under the influence of chemicals may have consequences long into the future. A carcinogenic substance may cause cancer cells to appear, which may continue to multiply even after exposure to the carcinogen has stopped.

## Maths

### Zero-order elimination

There are circumstances where the half-life varies with the concentration of the drug. For example, ethanol may be consumed in sufficient quantity to saturate the metabolic enzymes in the liver, and so is eliminated from the body at an approximately constant rate (zero-order elimination). 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:

$t_{1/2} = \frac{0.5 A_{0}}{k_{0}} \,$

### First-order elimination

This process is usually a first-order logarithmic process - that is, a constant proportion of the agent is eliminated per unit time (Birkett, 2002). Thus the fall in plasma concentration after the administration of a single dose is described by the following equation:

$C_{t} = C_{0} e^{-kt} \,$
• Ct 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:

$k = \frac{\ln 2}{t_{1/2}} \,$

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

$t_{1/2} = \frac{{\ln 2}.{V_D}}{CL} \,$

In clinical practice, this means that it takes just over 4.7 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 t½) of 24-36 hours; 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 t½ of about 90 days) are usually started with a loading dose to achieve their desired clinical effect more quickly.

• Half-life, pertaining to the general mathematical concept in physics or pharmacology.
• Kinetics

## References

Birkett DJ (2002). Pharmacokinetics Made Easy (Revised Edition). Sydney: McGraw-Hill Australia. ISBN 0-07-471072-9.