Elimination rate constant

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The elimination rate constant K is a value used in pharmacokinetics to describe the rate at which a drug is removed from the system.^[1]

It is often abbreviated K or K_e. It is equivalent to the fraction of a substance that is removed per unit time measured at any particular instant and has units of T⁻¹. This can be expressed mathematically with the differential equation

$C_{{t+dt}}=C_{t}-C_{t}\cdot K\cdot dt$ ,

where $C_{t}$ is the blood plasma concentration of drug in the system at a given point in time $t$ , $dt$ is an infinitely small change in time, and $C_{{t+dt}}$ is the concentration of drug in the system after the infinitely small change in time.

The solution of this differential equation is useful in calculating the concentration after the administration of a single dose of drug:

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

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

Sample values and equations

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

References

↑ Svensén CH, Brauer KP, Hahn RG et al. (September 2004). "Elimination rate constant describing clearance of infused fluid from plasma is independent of large infusion volumes of 0.9% saline in sheep". Anesthesiology 101 (3): 666–674. doi:10.1097/00000542-200409000-00015. PMID 15329591.

v t e Concepts in 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 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

Drug discovery strategies	Classical pharmacology Reverse pharmacology

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

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