The decay curve
One half-life, plotted. The curve shows the shape of the arithmetic, and the table under it gives the only points where the answer is exact.
Educational arithmetic only. Not medical advice and not a dosing protocol.
Enter the figures above.
| Half-lives elapsed | Remaining |
|---|---|
| 1 | 50% |
| 2 | 25% |
| 3 | 12.5% |
| 4 | 6.25% |
| 5 | 3.125% |
| 6 | 1.5625% |
| 7 | 0.78125% |
Exact at the whole multiples, approximate everywhere else
The fraction
remaining is 0.5t ÷ t½. At whole multiples of the half-life
the exponent is a whole number and the answer is exactly 1 ÷ 2n
- a rational number this site computes as one, which is why the table above has no rounding
in it.
Between those points the exponent is fractional and the value is irrational. There is no exact decimal for it, so the curve is computed in floating point and every figure derived from it is labelled approximate. That distinction runs through this whole site: the vial arithmetic is exact, the kinetic arithmetic is not, and each says which.
What the curve leaves out
Absorption, distribution between tissues, protein binding, active metabolites and the variation between one person and another. A real concentration-time curve after a subcutaneous injection rises before it falls, because the compound has to get in before it can be cleared, and this model has no rising limb.
What the curve is good for is the shape: how quickly a given half-life brings an amount down, why five half-lives is the usual rule of thumb for "most of it has gone", and why doubling a half-life does not double the time to clear.
Why no half-life is supplied
A half-life is a measurement, and it belongs to the study that measured it: the species, the formulation, the route, the dose range and the assay all travel with the number. The same compound can have several published half-lives that differ by more than the precision anyone would quote them to.
So this page asks for one rather than holding a table. The half-life pages are a sweep across plausible values of the parameter, chosen to span the range this arithmetic is useful over, and each one says plainly that it names no compound.