Repeat-interval arithmetic

Two figures in, both yours: a half-life and the spacing between repeats. What comes out is a geometric series, which is a statement about those two numbers and about nothing else.

Educational arithmetic only. Not medical advice and not a dosing protocol.

Enter the figures above.

This worksheet computes in the browser, so it needs JavaScript switched on. The ten half-life pages and six interval pages carry the same figures pre-computed across a grid.

Where that comes from

Let r be the fraction of one amount still present after exactly one interval: r = 0.5^{interval ÷ half-life}. When the interval equals the half-life, r is exactly a half.
Immediately after the first amount, one amount is present. Immediately after the second, 1 + r is. After the third, 1 + r + r². That is a geometric series, and after n repeats it sums to (1 - rⁿ) ÷ (1 - r).
As n grows, rⁿ goes to zero and the peaks approach 1 ÷ (1 - r). At r = 0.5 that limit is exactly 2: the peaks approach twice one amount and never reach it.
The number of intervals to get within one percent of the limit is the smallest n with rⁿ ≤ 0.01, which is ln(0.01) ÷ ln(r) rounded up. At r = 0.5 that is seven.
Everything above is computed in floating point rather than exactly, because the exponent is fractional and the value is irrational for most inputs. Every figure the worksheet prints is therefore labelled approximate.

What this figure is not

This is educational arithmetic on figures entered by the reader. It is not medical advice, not a dosing protocol, and not a recommendation that anything be prepared, obtained or administered.

Nothing here states what quantity is appropriate for anyone, and there is no function in it that produces an amount to administer. A question about anyone’s health belongs with a licensed clinician who can examine the person asking.

Compounds of the kind this arithmetic is used on are largely research chemicals that are not approved by the FDA for human use. An exact calculation is not an accurate measurement: what is actually in a syringe depends on the preparation and the device, neither of which this page can see.

A half-life belongs to the population, formulation and route it was measured in. Carrying one anywhere else is an assumption rather than a calculation, and this site holds no half-life for any compound: the figure is whatever was typed into the form.

An interval on this site is a parameter, never a proposal. No page here suggests how often anything should happen, and a geometric series evaluated at a spacing somebody chose is not a schedule.

No qualified clinical reviewer is recorded against this site, and no credential is claimed anywhere on it. the full version.

What the series is a statement about

Two numbers. It says that if a quantity decays by a constant proportion over each interval, and an identical quantity is added at the start of each interval, the total present at the start of each interval climbs towards a limit set by the ratio of the interval to the half-life. That is a mathematical fact about exponential decay and addition.

It is not a statement about a body. A real concentration-time curve rises before it falls, because a compound has to be absorbed before it can be cleared, and this model has no rising limb at all. It also has one compartment, no distribution, no protein binding, no active metabolites and no variation between one person and another.

Why the ratio matters more than either figure alone

Only the ratio of the interval to the half-life appears in r, so a 24-hour half-life at a 24-hour interval and a 7-day half-life at a 7-day interval produce identical arithmetic. Both give r = 0.5 and a limit of two. The absolute figures change how long the approach takes in clock time; they do not change the shape.

Which is why this site has a page family for each axis rather than one combined table. The half-life pages hold the half-life fixed and sweep the interval; the interval pages hold the interval fixed and sweep the half-life. Both are projections of the same function.

The honest reading of a large limit

An interval much shorter than the half-life gives a limit well above one: at a seven-day half-life and a daily interval, r is about 0.906 and the limit is about 10.7. That number is often misread as a prediction. It is not one. It says that on this model, with these two figures, the peaks approach about eleven times one amount - and the model excludes everything that would make that figure wrong about a person.

What it is good for is the shape of the arithmetic: why a short interval relative to a half-life accumulates, why the approach to the limit takes many intervals rather than two or three, and why halving an interval does not halve anything in a straight line.