Multiple Dosing: Drug Accumulation to Steady State

Dosing Regimen

Dosing interval τ 8 h

Elimination half-life t½ 6 h

Dose size (C added per dose) 4 mg/L

Current C(t)

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Steady-state peak

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Accumulation ratio R = 1/(1−e^(−kτ))

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How it works

Each dose behaves like an independent one-compartment bolus: it lands instantly and then decays exponentially at rate k = ln(2)/t½. When doses arrive faster than the drug clears, the tails of earlier doses haven't decayed away before the next one lands, so the concentrations simply add — the superposition principle for a linear system. The curve climbs dose after dose until the amount cleared between doses equals the amount added by each dose: that balance point is the steady state.

C(t) = Σₙ Dₙ · e^(−k·(t − n·τ)),  summed over doses n·τ ≤ t
k = ln(2) / t½
Css,max = D / (1 − e^(−kτ))     (steady-state peak)
R = 1 / (1 − e^(−kτ))            (accumulation ratio)

Dosing interval τ — time between doses; a shorter interval relative to the half-life leaves less time to clear each dose, so the curve accumulates to a higher steady state.
Half-life t½ — time for the body to clear half the drug; a longer half-life means slower decay between doses and more overlap.
Dose size — concentration added instantly by each dose (dose amount ÷ volume of distribution); scales the whole curve linearly.
Loading dose — a first dose sized to R× the maintenance dose, so the very first peak already sits at the eventual steady-state peak instead of climbing toward it over several half-lives.

Real-world relevance: this is the exact math behind why antibiotics, anticonvulsants and cardiac drugs are dosed on a fixed schedule — clinicians pick τ and dose size so Css,max and Css,min both land inside the drug's therapeutic window (shown as the translucent band), and use a loading dose whenever waiting several half-lives for the level to build up would leave a serious infection under-treated.

Concentration C(t) Individual doses Target range band Now marker

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