Drug Half-Life Simulator: Elimination & Steady-State Accumulation

simulator intermediate ~10 min
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Css_max = 28.4 mg/L, accumulation factor = 1.42

A drug with 6 h half-life dosed every 8 h reaches steady-state peak of 28.4 mg/L (1.42× the single-dose peak) and trough of 8.4 mg/L.

Formula

C(t) = C0 × e^(-ke·t), where ke = ln(2)/t½
Css_max = C0 / (1 - e^(-ke·tau))
Css_min = C0 × e^(-ke·tau) / (1 - e^(-ke·tau))

Exponential Decay

After a drug enters the bloodstream, the body eliminates it through metabolism (primarily hepatic) and excretion (primarily renal). For most drugs at therapeutic concentrations, elimination follows first-order kinetics: a constant fraction is removed per unit time. This produces exponential decay of plasma concentration, characterized by the half-life — the single most cited pharmacokinetic parameter.

The Sawtooth Pattern

With repeated dosing, each new dose adds to the drug remaining from previous doses. Plasma concentration follows a sawtooth pattern — sharp rises at each dose, exponential declines between doses. The peaks and troughs gradually climb until reaching steady state, where the amount administered per interval exactly equals the amount eliminated. This simulation visualizes the sawtooth accumulation in real time.

Reaching Steady State

Regardless of dose size or frequency, steady state is reached in approximately 5 half-lives. A drug with a 6-hour half-life reaches steady state in about 30 hours; one with a 24-hour half-life takes about 5 days. This principle governs loading dose calculations — when immediate therapeutic levels are needed, a loading dose bypasses the slow accumulation phase.

Clinical Dosing Decisions

The relationship between half-life and dosing interval determines fluctuation and accumulation. Dosing at one half-life intervals produces 2× accumulation with moderate fluctuation. Dosing much more frequently than the half-life produces high accumulation with minimal fluctuation (approaching a constant infusion). These trade-offs guide clinicians in selecting dosing regimens that maintain drug levels within the therapeutic window.

FAQ

What is drug half-life?

Half-life (t½) is the time required for plasma drug concentration to decrease by 50%. For first-order elimination, t½ is constant regardless of concentration. It determines how long a drug stays active, how often it must be dosed, and how long until steady state is reached (approximately 5 half-lives).

What is steady-state concentration?

Steady state is reached when the rate of drug administration equals the rate of elimination. At steady state, peak and trough concentrations repeat identically each dosing interval. It takes approximately 4-5 half-lives to reach 94-97% of steady state regardless of dose or interval.

What is the accumulation factor?

The accumulation factor R = 1/(1 - e^(-ke·tau)) tells you how much higher steady-state levels are compared to a single dose. When the dosing interval equals the half-life, R = 1/(1-0.5) = 2, meaning steady-state peak is twice the single-dose peak.

Why does it take 5 half-lives to reach steady state?

After each half-life, 50% of the drug is eliminated. After 5 half-lives, the fraction remaining from the first dose is (0.5)^5 = 3.1%. The cumulative input and output are within 3% of equilibrium — close enough to be considered 'at steady state' for clinical purposes.

Sources

Embed

<iframe src="https://homo-deus.com/lab/pharmacology/half-life-elimination/embed" width="100%" height="400" frameborder="0"></iframe>
View source on GitHub