Accumulation Factor Calculator

The result is the closed form of an infinite sum, which is worth seeing derived rather than quoted. Each dose contributes a full amount and then leaves behind a fraction r for the next one, which leaves behind r squared, and so on indefinitely. Adding every one of those contributions gives the level once the process has settled. Divide that total by a single dose and what remains is the accumulation factor. The whole expression collapses to one line only because the ratio between successive terms is constant, and that constancy is exactly the first-order assumption.

That ratio r is fixed entirely by the interval measured against the half-life, and that is the only place the half-life enters. Written as one half to the power of the interval over the half-life, it is precisely the fraction left behind by the previous dose at the moment the next one arrives. Nothing about absorption, formulation or individual variation touches it. The guide to steady state and accumulation works through the same derivation more slowly, including why it is the peak rather than the average that this series describes.

A geometric series has a closed form only when its ratio is smaller than one, and here the ratio is a fraction of material remaining, so it always is, provided the interval and the half-life are both positive. That sounds like a formality, but it is the condition the method rests on. The series is a claim about infinitely many past doses each contributing progressively less. With an interval of zero the terms would not shrink at all, and the sum would not exist.

The behaviour near the far boundary is where intuition fails hardest, and it is better computed than assumed. As the interval grows relative to the half-life the ratio shrinks toward zero and the factor approaches one, meaning no accumulation, and it gets there quickly. A factor of 1.07 at an interval of one tenth of a half-life already means a peak barely above a single dose, and by ten half-lives the factor is indistinguishable from one at any precision worth quoting. Nothing here is a threshold; it is simply how fast a small exponent becomes a negligible number.

The interval in this calculation is a mathematical input. It is a number you supply, and neither this page nor this site tells you what interval to use for anything. That is not a hedge and it is not a refusal to be useful. There is no interval that follows from a half-life alone, no interval that is correct across substances, and no interval that is correct across routes of administration, formulations or individuals. Publishing one would mean inventing a fact, and inventing a fact here is the one failure mode worth avoiding at any cost.

What the page does offer is a sensitivity curve, which is genuinely useful without being a recommendation. The factor is gentle near one and steep near two, so modest changes in the interval produce very different factors. Feed the same interval and half-life to the time to steady state calculator and it returns the number of half-lives needed to come within a chosen tolerance of that same settled level, which is the companion figure for the same pair of inputs.

Be exact about what is being compared. This is a ratio of peak concentrations under the model, setting a settled peak against a single-dose peak. It is not a ratio of effects, of durations, of thresholds crossed or of anything else. Turning it into one of those is a jump from arithmetic to a claim about a body, and no calculator can make that jump safely. The full context of what the family omits is set out on the original half-life page , and it is a longer list than the formula suggests.

The model also treats removal as a single first-order rate. If material leaves by more than one route at once, the rate that actually governs the interval is the combined one, which is shorter than either route alone, and the effective half-life calculator combines them. Feeding that combined figure back in here is the honest version of the calculation. Feeding in a single route half-life instead overstates the accumulation, because it makes removal look slower than the system can deliver.