Steady-state dosing

Why does a drug that acts within an hour still take days to reach its full effect?

A drug reaches its target within the hour; you can measure it working the same afternoon. And yet the instruction is to give it a week before judging whether it suits you.

Nothing about the molecule is slow, so the delay is not in the drug's action — it is arithmetic. Ask what "full effect" means here and the answer is: the amount in the body has finally stopped rising. Why should that take days?

Begin with how a body gets rid of a drug. For most drugs at ordinary amounts, what is removed per unit time is a constant fraction of what is present, not a constant quantity — because enzymes and kidneys act on whatever is presented to them, and while working below capacity, twice as much presented means twice as much removed. That is first-order elimination, and the half-life restates it: the time for the amount present to fall by half, whether you start high or low.

Now give a dose every so often. Before the second arrives some of the first is still there, so it starts from a floor rather than nothing, and that floor rises with each dose.

Does it rise forever? Here is the reason it cannot. A fixed fraction is removed each interval, so the more that has accumulated, the more is removed. The amount leaving grows with the amount present; the amount arriving does not. One quantity growing, one fixed — they must meet, and where they meet is steady state: a balance between two rates, not a ceiling the chemistry runs into.

The arithmetic answers more than it looks. Let f be the fraction of a dose still present one interval later. The peak after the first dose is D; after the second, D + fD; after the n*th, the geometric sum D(1 - f^n)/(1 - f), approaching D/(1 - f). So the fraction of the plateau reached after *n intervals is exactly 1 - f^n. Put the interval at one half-life, so f = 1/2, and read off: 50%, then 75%, 87.5%, 93.75%, 96.875%.

Now the step that matters. That sequence is indexed in half-lives, not doses. Against elapsed time it is 1 minus one-half raised to the power (time divided by half-life), and the dosing interval has dropped out entirely. Time to plateau is set by the half-life alone: for a drug with a six-hour half-life that is a matter of a day, for one with a half-life of days a fortnight, however often you dose.

Two consequences follow straight from the formula. First, double the dose and you double the plateau without shortening the time to reach it: same shape, different ceiling, because doubling everything doubles both sides of the balance.

Second, the two dials do different jobs. The dose rate — amount per unit time — fixes the average level at plateau; the interval fixes how far the level swings between peak and trough. The same daily amount in four small portions or one large settles to the same average with a very different ride, which is why the two can be chosen almost separately.

Which also shows what a loading dose is. A larger first dose fills the space the drug spreads into at once, ordinary doses thereafter holding it there — the half-life has not changed, you have started at the plateau rather than climbing to it. The cost is that a dose cannot be taken back, and whether that trade is worth making is a clinical judgement this account cannot help with.

The same arithmetic runs backwards on stopping: the fall away takes the same half-lives as the climb.

The nearest neighbour is first-pass-metabolism.md, and it is easy to fuse the two. The fixed point of difference: that piece is about how much of each dose arrives, decided by absorption and the liver's first pass; this one about what happens to arriving doses stacked over time. Bioavailability helps set the plateau's height and has no bearing on how many half-lives it takes to get there. drug-tolerance.md borders it too, and the two make different predictions: kinetics says the effect stops changing once the level plateaus, tolerance says it keeps falling at an unchanged level. dose-response.md supplies the curve that turns whatever level you reach into an effect.

One misconception the name invites: steady state does not mean the concentration is constant. It means every interval now looks like the last — still a peak after each dose, still a trough before the next. The pattern is steady, not the number.

The caveats each break something. The account assumes first-order elimination, one effective compartment, and an unchanging clearance. Some drugs saturate their own elimination, and then there is no plateau and a modest increase in dose produces a disproportionate rise. Some induce the enzymes that clear them, so the clearance defining the plateau moves while the drug climbs toward it. Active metabolites bring their own half-lives. And the effect can lag the concentration when the drug turns over something slow, so the effect may plateau well after the drug.

Which gives the test. Give the same drug at two dose sizes and follow the time course: first-order kinetics predicts the curves are scalar multiples, same shape and same time to plateau, different heights. The refuting observations are sharp. If the larger dose took longer to level off, or never levelled off, elimination is saturating and this arithmetic does not apply. If the effect went on rising after the concentration had stopped, the slow step is downstream of the drug.

• What changes when a drug saturates the enzymes clearing it, so no plateau exists at all.

Every term the collection defines is gathered in the glossary.