Elimination rate constant

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 via IV bolus injection:

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.

Concepts in pharmacology

Pharmacokinetics	(L)ADME: (Liberation) 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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