# Biological half-life

Biological half-life (also known as Elimination half-life, Pharmacologic half-life and ${\displaystyle T_{\frac {1}{2}}}$) of a biological substance such as medication is the time it takes from its maximum concentration (Cmax) to half maximum concentration in human body,[1][2][3][4][5][6] and is denoted by the abbreviation ${\displaystyle t_{\frac {1}{2}}}$. [2][5]

This is used to measure the removal of things such as metabolites, drugs, and signalling molecules from the body. Typically, the biological half-life refers to the body's natural cleansing through the function of the liver and through the excretion of the measured substance through the kidneys and intestines. This concept is used when the rate of removal is roughly exponential.[clarification needed][7]

In a medical context, half-life explicitly describes the time it takes for the blood plasma concentration of a substance to halve (plasma half-life) its steady-state when circulating in the full blood of an organism. This measurement is useful in medicine and pharmacology because it helps determine how much of a drug needs to be taken and how frequently it needs to be taken if a certain average amount is needed constantly. In contrast, the stability of a substance direct in plasma is described with plasma stability that is essential to ensure accurate analysis of drugs in plasma and for Drug discovery.

The relationship between the biological and plasma half-lives of a substance can be complex depending on the substance in question, due to factors including accumulation in tissues (protein binding), active metabolites, and receptor interactions.[8]

## Examples

### 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.[9][10] This has been used to decontaminate humans who are internally contaminated with tritiated water (tritium). The basis of this decontamination method (used at Harwell)[citation needed] 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. Half life is also relative to the subjective metabolic rate of the individual in question.

### Common prescription medications

Substance Biological half-life
Adenosine Less than 10 seconds[citation needed]
Norepinephrine 2 minutes[citation needed]
Oxaliplatin 14 minutes[11]
Salbutamol 1.6 hours[citation needed]
Zaleplon 1–2 hours[citation needed]
Morphine 2–3 hours[citation needed]
Methotrexate 3–10 hours (lower doses),

8–15 hours (higher doses)[12]

Phenytoin 12–42 hours[citation needed]

In rare cases up to 8 days[13]

Buprenorphine 16–72 hours[citation needed]
Clonazepam 30–40 hours[14]
Flurazepam 19–100 hours

Active metabolite (desflurazepam): 1.75–10.4 days[citation needed]

Diazepam 20–100 hours

Active metabolite (nordazepam): 1.5–8.3 days[citation needed]

Donepezil 3 days (70 hours)[15]
Fluoxetine 4–6 days

Active lipophilic metabolite (seproxetine): 4–16 days[citation needed]

Vandetanib 19 days[16]
Amiodarone 25–110 days[citation needed]
Dutasteride 35 days[citation needed]
Bedaquiline 165 days[17]

### 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.[18]

• 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.
• Mercury (as methylmercury) in the body has a half-life of about 65 days.
• Lead in the blood has a half life of 28–36 days.[19][20]
• 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.

### Peripheral half-life

Some substances may have different half-lives in different parts of the body. For example, oxytocin has a half-life of typically about three minutes in the blood when given intravenously. Peripherally administered (e.g. intravenous) peptides like oxytocin cross the blood-brain-barrier very poorly, although very small amounts (< 1%) do appear to enter the central nervous system in humans when given via this route.[21] In contrast to peripheral administration, when administered intranasally via a nasal spray, oxytocin reliably crosses the blood–brain barrier and exhibits psychoactive effects in humans.[22][23] In addition, also unlike the case of peripheral administration, intranasal oxytocin has a central duration of at least 2.25 hours and as long as 4 hours.[24][25] In likely relation to this fact, endogenous oxytocin concentrations in the brain have been found to be as much as 1000-fold higher than peripheral levels.[21]

## Rate equations

### First-order elimination

Half-times apply to processes where the elimination rate is exponential. If ${\displaystyle C(t)}$ is the concentration of a substance at time ${\displaystyle t}$, its time dependence is given by

${\displaystyle C(t)=C(0)e^{-kt}\,}$

where k is the reaction rate constant. Such a decay rate arises from a first-order reaction where the rate of elimination is proportional to the amount of the substance:[26]

${\displaystyle {\frac {dC}{dt}}=-kC.}$

The half-life for this process is[26]

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

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

${\displaystyle t_{\frac {1}{2}}={\frac {{\ln 2}\cdot {V_{D}}}{CL}}\,}$

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 t½) 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 t½ of about 58 days) are usually started with a loading dose to achieve their desired clinical effect more quickly.

### Biphasic half-life

Many drugs follow a biphasic elimination curve — first a steep slope then a shallow slope:

STEEP (initial) part of curve —> initial distribution of the drug in the body.
SHALLOW part of curve —> ultimate excretion of drug, which is dependent on the release of the drug from tissue compartments into the blood.

The longer half-life is called the terminal half-life and the half-life of the largest component is called the dominant half-life.[26] For a more detailed description see Pharmacokinetics § Multi-compartmental models.

## Sample values and equations

Pharmacokinetic metrics

Characteristic Description Symbol Unit Formula Worked example
value
Dose Amount of drug administered. ${\displaystyle D}$ ${\displaystyle \mathrm {mol} }$ Design parameter 500 mmol
Dosing interval Time between drug dose administrations. ${\displaystyle \tau }$ ${\displaystyle \mathrm {s} }$ Design parameter 24 h
Cmax The peak plasma concentration of a drug after administration. ${\displaystyle C_{\text{max}}}$ ${\displaystyle \mathrm {M} }$ Direct measurement 60.9 mmol/L
tmax Time to reach Cmax. ${\displaystyle t_{\text{max}}}$ ${\displaystyle \mathrm {s} }$ Direct measurement 3.9 h
Cmin The lowest (trough) concentration that a drug reaches before the next dose is administered. ${\displaystyle C_{{\text{min}},{\text{ss}}}}$ ${\displaystyle \mathrm {M} }$ Direct measurement 27.7 mmol/L
Volume of distribution The apparent volume in which a drug is distributed (i.e., the parameter relating drug concentration in plasma to drug amount in the body). ${\displaystyle V_{\text{d}}}$ ${\displaystyle \mathrm {m} ^{3}}$ ${\displaystyle {\frac {D}{C_{0}}}}$ 6.0 L
Concentration Amount of drug in a given volume of plasma. ${\displaystyle C_{0},C_{\text{ss}}}$ ${\displaystyle \mathrm {M} }$ ${\displaystyle {\frac {D}{V_{\text{d}}}}}$ 83.3 mmol/L
Absorption half-life The time required for the concentration of the drug to double its original value for oral and other extravascular routes.[citation needed] ${\displaystyle t_{{\frac {1}{2}}a}}$ ${\displaystyle \mathrm {s} }$ ${\displaystyle {\frac {\ln(2)}{k_{\text{a}}}}}$ 1.0 h
Absorption rate constant The rate at which a drug enters into the body for oral and other extravascular routes. ${\displaystyle k_{\text{a}}}$ ${\displaystyle \mathrm {s} ^{-1}}$ ${\displaystyle {\frac {\ln(2)}{t_{{\frac {1}{2}}a}}}}$ 0.693 −1
Elimination half-‍life The time required for the concentration of the drug to reach half of its original value. ${\displaystyle t_{{\frac {1}{2}}b}}$ ${\displaystyle \mathrm {s} }$ ${\displaystyle {\frac {\ln(2)}{k_{\text{e}}}}}$ 12 h
Elimination rate constant The rate at which a drug is removed from the body. ${\displaystyle k_{\text{e}}}$ ${\displaystyle \mathrm {s} ^{-1}}$ ${\displaystyle {\frac {\ln(2)}{t_{{\frac {1}{2}}b}}}={\frac {CL}{V_{\text{d}}}}}$ 0.0578 h−1
Infusion rate Rate of infusion required to balance elimination. ${\displaystyle k_{\text{in}}}$ ${\displaystyle \mathrm {mol/s} }$ ${\displaystyle C_{\text{ss}}\cdot CL}$ 50 mmol/h
Area under the curve The integral of the concentration-time curve (after a single dose or in steady state). ${\displaystyle AUC_{0-\infty }}$ ${\displaystyle \mathrm {M} \cdot \mathrm {s} }$ ${\displaystyle \int _{0}^{\infty }C\,\operatorname {d} t}$ 1,320 mmol/L·h
${\displaystyle AUC_{\tau ,{\text{ss}}}}$ ${\displaystyle \mathrm {M} \cdot \mathrm {s} }$ ${\displaystyle \int _{t}^{t+\tau }C\,\operatorname {d} t}$
Clearance The volume of plasma cleared of the drug per unit time. ${\displaystyle CL}$ ${\displaystyle \mathrm {m} ^{3}/\mathrm {s} }$ ${\displaystyle V_{\text{d}}\cdot k_{\text{e}}={\frac {D}{AUC}}}$ 0.38 L/h
Bioavailability The systemically available fraction of a drug. ${\displaystyle f}$ Unitless ${\displaystyle {\frac {AUC_{\text{po}}\cdot D_{\text{iv}}}{AUC_{\text{iv}}\cdot D_{\text{po}}}}}$ 0.8
Fluctuation Peak trough fluctuation within one dosing interval at steady state. ${\displaystyle \%PTF}$ ${\displaystyle \%}$ ${\displaystyle {\frac {C_{{\text{max}},{\text{ss}}}-C_{{\text{min}},{\text{ss}}}}{C_{{\text{av}},{\text{ss}}}}}\cdot 100\%}$
where
${\displaystyle C_{{\text{av}},{\text{ss}}}={\frac {1}{\tau }}AUC_{\tau ,{\text{ss}}}}$
41.8%

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