Biological half-life

The biological half-life or elimination half life of a substance is the time it takes for a substance (for example a metabolite, drug, signalling molecule, radioactive nuclide, or other substance) to lose half of its pharmacologic, physiologic, or radiologic activity, as per the MeSH definition. In a medical context, half-life may also describe the time it takes for the blood plasma concentration of a substance to halve ("plasma half-life") its steady-state. The relationship between the biological and plasma half-lives of a substance can be complex, due to factors including accumulation in tissues, active metabolites, and receptor interactions.^[1]

Biological half-life is an important pharmacokinetic parameter and is usually denoted by the abbreviation t_½.^[2]

While a radioactive isotope 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. See the article rate equation.

1 Examples of biological half-lives
2 Rate equations
- 2.1 First-order elimination
3 Sample values and equations
4 See also
5 References

Examples of biological half-lives

Water

The biological half-life of water in a human is about 7 to 14 days. It can be altered by behavior. Drinking large amounts of alcohol 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 at Harwell) is to increase the rate at which the water in the body is replaced with new water.

Alcohol

The removal of ethanol (drinking 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 an appropriate amount of ethanol to a person who has ingested methanol. 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

Substance	Half-life	Notes
Amiodarone	25 days
Cisplatin	30 to 100 h
Chlorambucil	1.53 h
Digoxin	24 to 36 h
Fluoxetine	1 to 6 days	The active metabolite of fluoxetine is lipophilic and migrates slowly from the brain to the blood. The metabolite has a biological half-life of 4 to 16 days.
Methadone	15 to 60 h, in rare cases up to 190 h.^[3]
Oxaliplatin	14 min^[4]
Salbutamol	1.6 h

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 substances, it is 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 (physiologically-based pharmacokinetic modelling). 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.
Caesium 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.

Rate equations

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:

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

This process is usually a logarithmic process - that is, a constant proportion of the agent is eliminated per unit time.^[5] Thus the fall in plasma concentration after the administration of a single dose is described by the following equation:

$C_{t} = C_{0} e^{-kt} \,$

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:

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

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

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

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.

Sample values and equations

Variable	Description	Example value	Abbreviation(s)	Formula
Dose	loading dose (LD), or steady state/maintenance dose (MD)	1000 mg	$\textstyle D$	$\textstyle = V_{d} \times C_{0}$
Volume of distribution	The apparent volume in which a drug is distributed immediately after it has been injected intravenously and equilibrated between plasma and the surrounding tissues.	25 L	$\textstyle V_d$	$\textstyle = D / C_0$
Concentration	initial or steady-state concentration of drug in plasma	40.0 mg/L	$\textstyle C_{0} \ or \ C_{ss}$	$\textstyle = D / V_d$
Biological half-life	The time required for the concentration of the drug to reach half of its original value.	14 hr	$\textstyle T_{1/2}$	$\textstyle = ln (2) / K_{e}$
Elimination rate constant	The rate at which drugs are removed from the body.	0.05 /hr	$\textstyle K_e$	$\textstyle = ln (2) / T_{1/2} = Cl / V_{d}$
Elimination rate	rate of infusion required to balance elimination	50 mg/hr	$\textstyle K_{in}$	$\textstyle = C_{ss} \times Cl$
Area under the curve	The integral of the plasma drug concentration (C_p) after it is administered.	0.1 mg/mL×hr	$\textstyle AUC_{0\ \infty}$	$= \int_{0}^{\infty}C_p\, dt$
Clearance	The volume of plasma cleared of the drug per unit time.	1.25 L/hr	$\textstyle Cl$	$\textstyle= V_{d} \times K_{e} = D/AUC$
Bioavailability	The fraction of drug that is absorbed.	1	$\textstyle F$	$= \frac{[AUC]_{oral}\div dose_{oral}}{[AUC]_{IV}\div dose_{IV}}$
C_max	The peak plasma concentration of a drug after oral administration.	40.0 mg/L	$\textstyle C_{max}$	direct measurement
C_min	The lowest (trough) concentration that a drug reaches before the next dose is administered.	1.0 mg/L	$\textstyle C_{min}$	direct measurement
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References

↑ Lin VW; Cardenas DD (2003). Spinal cord medicine. Demos Medical Publishing, LLC. p. 251. ISBN 1888799617. http://books.google.co.uk/books?id=3anl3G4No_oC&pg=PA251&lpg=PA251&source=bl&ots=DsuJ9YCfzo&sig=5VzoJK5BqWMoMm7T3DzAKvpG_Xo&hl=en&ei=45X8SZGWAd3TjAe3pIiSAw&sa=X&oi=book_result&ct=result&resnum=5.
↑ IUPAC Gold Book internet edition: "biological half life".
↑ Manfredonia, John (March 2005). "Prescribing Methadone for Pain Management in End-of-Life Care". JAOA—The Journal of the American Osteopathic Association. http://www.jaoa.org/cgi/content/full/105/3_suppl/18S. Retrieved 2007-01-29.
↑ Ehrsson, Hans et al. (Winter 2002). "Pharmacokinetics of oxaliplatin in humans". Medical Oncology. http://journals.humanapress.com/index.php?option=com_opbookdetails&task=articledetails&category=humanajournals&article_code=MO:19:4:261. Retrieved 2007-03-28.
↑ Birkett DJ (2002). 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 eliminationPharmacokinetics Made Easy (Revised Edition). Sydney: McGraw-Hill Australia. ISBN 0-07-471072-9.

Medication > Pharmacology

Pharmacokinetics	ADME: Absorption • Distribution • Metabolism • Excretion (Clearance) Loading dose • Volume of distribution (Initial) • Rate of infusion Compartment • Bioequivalence • Bioavailability Onset of action • Biological half-life • Plasma protein binding Phase 1 reaction • Phase 2 reaction Therapeutic index (LD₅₀/ED₅₀)

Pharmacodynamics	Toxicity (Neurotoxicology) • Dose-response relationship (Efficacy, Potency) Antimicrobial pharmacodynamics: Minimum inhibitory concentration/Bacteriostatic • Minimum bactericidal concentration/Bactericide

Agonism and antagonism	Agonist: Inverse agonist • Irreversible agonist • Partial agonist • Superagonist • Physiological agonist Antagonist: Competitive antagonist • Irreversible antagonist • Physiological antagonist Other: Binding • Affinity • Binding selectivity • Functional selectivity

Other	Drug tolerance: Tachyphylaxis Drug resistance: Antibiotic resistance • Multiple drug resistance

Related fields/subfields	Pharmacogenetics • Pharmacogenomics • Neuropsychopharmacology (Neuropharmacology, Psychopharmacology)