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Pharmacokinetics · One-compartment model

Elimination Rate Constant (ke)

ke is the proportionality constant governing first-order drug elimination. It determines how fast a drug is removed from plasma and is directly related to half-life (ke = 0.693 / t½) and clearance (ke = CL / Vd).

Definition and Key Relationships

Under first-order kinetics, the rate of change of plasma concentration is proportional to the current concentration:

dC/dt = −ke × C

Solving this differential equation gives the monoexponential decay equation used in the one-compartment model:

C(t) = C0 × e−ke·t
ke from half-life
ke = ln(2) / t½ ≈ 0.693 / t½
Units: h⁻¹ or min⁻¹
ke from CL and Vd
ke = CL / Vd
CL in L/h, Vd in L
Half-life from ke
t½ = 0.693 / ke
Same units as 1/ke
Concentration at time t
C(t) = C₀ × e^(−ke × t)
C₀ = initial concentration

Worked Example — Ibuprofen

Ibuprofen has a published half-life of approximately 2 hours. Calculate ke and predict concentration at 4 h after a 400 mg dose (assuming C₀ = 20 mg/L after absorption).

ke = 0.693 / 2 h = 0.347 h⁻¹
C(4h) = 20 × e−0.347 × 4 = 20 × e−1.386 = 20 × 0.25 = 5 mg/L

This confirms that after 2 half-lives (4 h), concentration falls to 25% of C₀. Use the half-life calculator ↗ to visualise this interactively.

C₀ C₀/2 C₀/4 t½ 2t½ 3t½ Time

Clinical Relevance

Renal impairment → ↓ ke

Drugs eliminated by the kidneys have ke proportional to GFR. As GFR falls, ke decreases, t½ increases, and drug accumulates on the same dose interval. Dose reduction or extended interval is required.

Renal dosing guide ↗
Hepatic impairment → ↓ ke (high-extraction drugs)

For high-hepatic-extraction drugs (lidocaine, morphine, propranolol), ke depends on hepatic blood flow and enzyme activity. Liver disease reduces both, substantially extending half-life.

Hepatic dosing guide ↗
Volume of distribution → affects ke

ke = CL / Vd. A high Vd (large tissue distribution) produces a small ke and long t½ even if CL is normal. Example: amiodarone (Vd ~5000 L, t½ 40–55 days) — extensive tissue accumulation dominates PK.

Vd glossary entry ↗
TDM uses ke for dose individualisation

In Bayesian TDM, ke (or its correlate CL) is the primary parameter estimated from measured levels. Accurate ke estimation drives precise dose recommendations for vancomycin, aminoglycosides, and phenytoin.

TDM guide ↗

Frequently asked questions

What is the elimination rate constant (ke)?
The elimination rate constant (ke) is the fraction of drug eliminated per unit time under first-order kinetics: dC/dt = −ke × C. Example: ke = 0.347 h⁻¹ means 34.7% of remaining drug eliminated per hour. It relates directly to half-life: t½ = ln(2) / ke ≈ 0.693 / ke. See NCBI StatPearls on pharmacokinetics ↗ for the first-order derivation.
How is ke related to half-life and clearance?
Three key relationships:
(1) ke = 0.693 / t½ — ke and half-life are inversely related
(2) ke = CL / Vd — ke increases with clearance and decreases with larger Vd
(3) C(t) = C₀ × e−ke·t — plasma concentration falls exponentially at rate ke. Use the half-life calculator ↗ to visualise concentration decay interactively.
What is the difference between first-order and zero-order elimination?
First-order: rate = ke × C — elimination is proportional to concentration. Half-life is constant at all concentrations. Most drugs. Zero-order (saturation kinetics): rate = constant — enzymes are saturated, so a fixed amount (not fraction) is eliminated per time unit. Half-life increases with dose. Examples: phenytoin ↗ (Michaelis-Menten kinetics at therapeutic doses — small dose changes cause disproportionate level changes), aspirin at anti-inflammatory doses, ethanol. For phenytoin, use Vmax/Km PK equations rather than the standard ke formula.
How is ke measured clinically?
ke is estimated from plasma concentrations during the elimination phase: ke = (ln C₁ − ln C₂) / (t₂ − t₁). Plot ln(C) vs time — slope = −ke. For vancomycin TDM, two concentrations (e.g., peak and trough) allow ke calculation. In Bayesian model-informed precision dosing (MIPD) ↗, population PK priors (incorporating renal function ↗, age, weight) are updated with patient-specific concentrations to estimate individualised ke. Software: Pmetrics, InsightRx, DoseMeRx.