The steady state accumulation factor is the fixed multiplier, R = 1 / (1 - e^(-k x tau)), that tells you how much a repeated dose builds up on top of itself once dosing at a constant interval has continued long enough for the buildup to level off. This post is a mathematics explainer, not medical or dosing guidance. Every equation below is superposition applied to the same exponential decay model covered in first-order elimination and half-life, the math, the same model the half-life calculator uses when it reports a steady-state peak and trough for a dosing interval. None of it is a claim about how any real substance behaves in a body, and none of it is a recommended dosing interval. The calculators these equations drive state the same thing on their own pages.
Repeated doses, added together
If a single dose decays as A(t) = A0 x e^(-kt), then a series of equal doses given at a fixed interval tau produces an amount at any time that is the sum of every dose given so far, each decayed by however long it has had to decay. This is superposition: because the underlying equation is linear, the combined effect of several doses is just the sum of each one’s individual decay curve. Early doses have decayed more by the time a later dose is given, so each new dose adds to whatever is left of the ones before it, rather than starting from zero.
What is the steady-state accumulation factor formula?
If dosing continues indefinitely at interval tau, the total amount present right before each new dose approaches a fixed value: the accumulation factor. It comes from summing an infinite geometric series of decayed doses:
R = 1 / (1 - e^(-k x tau))R is a multiplier: at steady state, the amount just before a dose is R times a single dose’s amount at that same point in its own decay. A short interval relative to the half-life (frequent dosing relative to how fast the model decays) produces a large R, meaning substantial buildup between doses; a long interval relative to the half-life keeps R close to 1, meaning very little carries over. The accumulation factor calculator evaluates this formula directly from a half-life (converted to k) and a dosing interval.
How long until steady state
Accumulation approaches its asymptote but never mathematically reaches it in finite time; the model asks instead how many half-lives it takes to reach a chosen fraction of that asymptote. Solving 1 - 0.5^n = target fraction for n gives the number of half-lives needed:
n = log2(1 / (1 - target fraction))The time to steady state calculator solves this for a chosen target fraction (commonly a figure like 90 or 95 percent of the asymptote) and converts the resulting number of half-lives into actual time using the half-life you supply.
Steady-state fraction after N doses
A related but distinct question is not how much time has passed, but how close the accumulation is after a specific, finite number of doses, rather than in the limit of infinite dosing. This is a partial sum of the same geometric series used for R, normalized against the full asymptote:
fraction after N doses = (partial sum of N decayed doses) / (full asymptotic sum)The steady-state fraction calculator computes this directly: given a half-life, a dosing interval, and a dose count, it reports what fraction of the eventual steady-state level that many doses have produced so far.
Time to reach a target amount
A separate question from accumulation across doses: given a single starting amount decaying on its own, how long until it drops to some specific target amount, not a fraction. Rearranging the decay equation for time:
t = half-life x log2(A0 / target amount)The time to target amount calculator solves this directly, which is the same algebra as time to percent remaining from the first-order elimination post, just expressed against an absolute target amount instead of a fraction of the start.
Effective half-life for two elimination pathways
When a model includes two independent first-order elimination pathways acting on the same amount simultaneously (for example, two parallel routes each with their own rate constant), the combined decay is still a single exponential, but with a combined rate constant that is the sum of the two individual ones. Expressed as half-lives, that combination is a reciprocal sum:
1 / t_effective = 1 / t_a + 1 / t_bThe effective half-life calculator solves this for t_effective given the two pathway half-lives. The effective half-life is always shorter than either individual pathway’s half-life on its own, since two simultaneous elimination routes always clear faster together than either one alone.
What none of this is
Every quantity above, R, time to steady state, steady-state fraction, and effective half-life, is a property of the mathematical model given the numbers you supply, not a prediction this site makes about any specific substance’s actual behavior. Nothing here recommends a dosing interval, a target fraction, or an amount; those are all inputs the model takes, not outputs it asserts.
In short: the steady-state accumulation factor is R = 1 / (1 - e^(-k x tau)), the fixed multiple a repeated dose builds toward once dosing at interval tau continues long enough to level off.