Drug Half-Life Formula
Calculate drug concentration over time using half-life: C(t) = C₀ × (1/2)^(t/t½).
Includes elimination rate constant and worked examples.
The Formula
t½ = 0.693 / k_e
The drug half-life (t½) is the time required for the concentration of a drug in the body to fall to half its original value. After one half-life, 50% remains. After two half-lives, 25%. After five half-lives, about 3% — considered effectively eliminated for clinical purposes.
C₀ is the initial concentration (after distribution), t is the elapsed time, and t½ is the half-life. The elimination rate constant k_e = 0.693/t½ (0.693 = ln 2).
Half-life varies enormously between drugs. Aspirin: ~3–6 hours. Ibuprofen: ~2 hours. Diazepam (Valium): 20–100 hours. Amiodarone (cardiac drug): 40–55 days. Digoxin: 36–48 hours.
Half-life determines dosing intervals. For consistent drug levels, drugs are typically dosed every half-life to every two half-lives. It takes approximately 4–5 half-lives to reach steady-state concentration when starting a drug, and the same time for the drug to be essentially eliminated after stopping.
This formula assumes first-order kinetics (most drugs), where elimination rate is proportional to concentration. Some drugs (alcohol, phenytoin at high doses) follow zero-order kinetics at therapeutic concentrations — they eliminate a fixed amount per hour regardless of concentration, which changes the math significantly.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| C(t) | Drug concentration at time t | mg/L or ng/mL |
| C₀ | Initial drug concentration | mg/L or ng/mL |
| t | Time elapsed | hours |
| t½ | Half-life of the drug | hours |
| k_e | Elimination rate constant | per hour (h⁻¹) |
Example 1
A drug has t½ = 6 hours and initial concentration 80 mg/L. What is the concentration after 18 hours?
18 hours = 3 half-lives; C = 80 × (1/2)³ = 80 × 0.125
C = 10 mg/L after 18 hours
Example 2
How many hours until a drug with t½ = 8h falls below 5% of its starting level?
5 half-lives = 95% eliminated → remaining = 3.125%
After 4.32 half-lives = 5% exactly; practical answer: 5 half-lives
5 × 8 = 40 hours to reach ~3% (effectively eliminated)
When to Use It
- Calculating dosing intervals for maintenance therapy
- Estimating time to drug clearance before surgery or procedures
- Predicting drug interactions from timing of doses
- Toxicology: estimating when a drug will clear after overdose