Time to Steady State Calculator

This page returns the time needed to come within a stated tolerance of a level that is never actually attained, and both halves of that sentence matter. Steady state is where input equals removal. Once there, each dose replaces exactly what the interval removed, so the level stops changing, but it stops changing asymptotically: the gap between the present level and the settled level keeps shrinking by the same proportion rather than by the same amount. Asking when you have arrived is therefore ill-posed, while asking how close you are is well-posed. The guide to steady state and accumulation covers the same distinction from the accumulation side.

The fraction reached after a whole number of halvings has a form that can be checked instantly: one minus one half to the power of the count. After one it is a half, after two three quarters, after three seven eighths. Each figure is the previous one plus half of whatever was still missing, which is the same statement as saying the level gains half its remaining distance in every half-life. That geometric approach is why the sequence can never arrive, and also why it covers most of the ground early on.

The range that gets quoted as a rule of thumb, four to five half-lives, is where this page produces it. That range is not a claim about how long anything takes. It is a decision about tolerance. Ninety percent needs about 3.32 half-lives, ninety nine needs about 6.64, and ninety nine point nine needs almost ten. Nothing changes at four or at five. Only the percentage you were willing to accept changes, and that is a choice rather than a discovery. The original half-life page shows the single-dose decline that the same half-life counting describes.

The reason the middle of that range is what people remember is that early gains look dramatic and late gains look like nothing. Moving from a half of the way to three quarters is a fifty percent relative improvement in one step. Moving from 96.9 to 98.4 percent is a relative improvement of about one and a half percent for exactly the same step. The arithmetic is identical in both cases, and the second case is why a figure quoted as four or five half-lives should be read as close enough for the purpose at hand rather than as an endpoint.

The second output is the more portable of the two. Expressed in half-lives the time to reach any chosen percentage is a pure number, carrying no unit and no dependence on which substance is being imagined, so it can be set beside a figure quoted anywhere else with no conversion at all. Expressed in hours it is welded to the particular half-life that was typed in, which in turn is welded to whatever source that came from. If one of these numbers is going to be written down and remembered, write the half-lives. The guide to first-order elimination mathematics explains why that unit is the natural one for the whole family.

It is also the clearest illustration of a point that applies everywhere in this family. None of these calculators produces knowledge about a substance. It produces knowledge about the relationship between two numbers that were already held. The accumulation factor calculator takes the same pair of inputs and reports the companion figure, which is the multiple by which a peak grows once the level has settled.

A percentage of steady state says nothing about whether that level is appropriate, and the two are confused often enough to be worth naming. It is a statement about convergence, not about adequacy. Choosing ninety rather than ninety-four changes the answer by less than a factor of two and changes nothing whatsoever about how the system behaves. The percentage is a tolerance you set, and the page reports the cost of holding to it.

The target must also be below one hundred, and that restriction is mathematical rather than pedantic. At exactly one hundred the logarithm of one over zero is undefined and no finite time exists. Values just below one hundred are perfectly legal and return very large durations, which is the model behaving correctly rather than a fault worth reporting. To see the relationship running forwards instead of backwards, the steady state fraction calculator takes a number of doses where this page takes a target.