The model

k = ln(2) ÷ t½ · C(t) = Σ dose × e^(−k(t − tᵢ))

Elimination is treated as first-order: a constant fraction of what is present is cleared per unit time, giving the familiar exponential decay. The elimination rate constant k is the natural logarithm of 2 divided by the half-life.

For repeated dosing the model uses superposition — each dose decays independently and the concentrations add. That is why the curve climbs across the first several doses before settling: you are adding a new dose before the previous ones have fully cleared.

Steady state, peak and trough

Css,max = dose ÷ (1 − e^(−kτ)) · Css,min = Css,max × e^(−kτ)

Steady state is reached when the amount eliminated between doses equals the amount administered. As a rule of thumb it takes roughly four to five half-lives, and this is independent of dose — a larger dose reaches a higher steady state, not a faster one.

The dosing interval τ relative to half-life sets the peak-to-trough ratio. Dosing at intervals much shorter than the half-life produces a smooth, flat curve with high accumulation. Dosing at intervals much longer produces sharp peaks and near-complete washout between doses, with almost no accumulation.

This is the practical reason a peptide with a very short half-life dosed once daily behaves nothing like the same peptide dosed three times daily, even at matched total daily amounts.

Reading the curve

The peak-to-trough ratio displayed alongside the chart is the single most useful number. A ratio near 1 means near-constant exposure. A large ratio means exposure is dominated by brief peaks, which matters when the effect you want tracks peak concentration rather than average concentration — or when side effects track peaks.

Concentrations here are in arbitrary units relative to dose. The shape, the timing, and the ratios are the informative parts, not the absolute values.

Frequently asked questions

How long does it take to reach steady state?

Approximately four to five half-lives, regardless of dose size or dosing interval. After four half-lives you are at about 94% of steady state; after five, about 97%. Increasing the dose raises the steady-state concentration but does not shorten the time taken to reach it.

How do you calculate the elimination rate constant from half-life?

Divide the natural logarithm of 2 (about 0.693) by the half-life. A peptide with a 24-hour half-life has an elimination rate constant of roughly 0.0289 per hour. That constant drives the exponential decay curve for every dose.

Why does dosing frequency change the curve so much at the same total dose?

Because accumulation depends on the ratio between dosing interval and half-life. Dosing more often than the half-life means each dose lands before the previous has cleared, so concentrations build to a higher, flatter steady state. Dosing far less often than the half-life allows near-complete washout, producing sharp peaks and minimal accumulation.

What is the peak-to-trough ratio and why does it matter?

It's the steady-state maximum divided by the steady-state minimum — how much concentration swings between doses. A ratio near 1 means steady exposure; a high ratio means exposure is concentrated in brief peaks. It matters because some effects and some side effects track peak concentration rather than average exposure.