Instrument II · three clocks
How first-order elimination shapes a curve.
This instrument draws in your browser, and it needs JavaScript to do it. Nothing is sent anywhere. The same curve is set out as a table in the notes below.
Ten half-lives on each slip, one large square to each, every slip on its own clock. The step at the start is the whole starting amount.
At this instant
Population curves from published parameters. This tool takes no one’s data and computes no one’s therapy.
A beta blocker given into a vein for short-term control of heart rate, for example around surgery. Enzymes inside red blood cells break it down rapidly. Its volume of distribution, 3.4 litres per kilogram, is more than a kilogram of body could hold: most of the drug sits outside the plasma.
From the US prescribing information for Brevibloc and the Irish summary of product characteristics.
The stimulant in coffee. A US prescribing label for caffeine citrate gives 5 hours for adults; in a 1983 study of ten healthy young men it ranged from 2.7 to 9.9 hours. Taken by mouth, it is absorbed essentially completely, which is why its curve can start from the whole dose.
From the US prescribing information for caffeine citrate and Blanchard and Sawers, 1983.
An antibody used to treat cancer, a protein of about 149 kDa. A healthcare provider gives it, as an infusion into a vein or, in a newer form, an injection under the skin. The body clears it slowly: about 252 mL of plasma a day after the first dose.
From the US prescribing information for Keytruda and Keytruda Qlex.
The dial is a clock that turns once every half-life. Its hand starts at the top with the whole amount on the outer ring; with every turn the point it carries comes halfway to the centre, so the spiral it draws is the curve of what is left, wound round. After five turns about 3% is left, after ten about 0.1%. The numbers on the rim are the clock of the substance chosen: minutes for esmolol, hours for caffeine, days for pembrolizumab. When the clock changes the hand keeps its pace; only the numbers change.
The three slips are the same story drawn flat, one per substance, each on its own clock and ruled one large square to a half-life. Each curve follows one rule, called first-order elimination: in every equal stretch of time, the body removes the same fraction of whatever is left, so the amount removed shrinks as the amount left shrinks. That rule draws a curve that halves, then halves again, and never quite reaches zero. The time it takes to halve is the half-life.
Write what is left as a share of the starting amount. After one half-life it is a half, after two a quarter, after n half-lives a half multiplied by itself n times. The half-life is the only place where the substance appears, so counting time in half-lives takes it out of the sum: what remains is the same curve for all three. That is why the three slips, each on its own clock, draw one curve, and why the dial needs only one spiral.
On a logarithmic scale each curve becomes a straight line, because every half-life takes off the same distance; on the dial the turns of the spiral fall evenly, for the same reason. The slope of the line is the elimination rate constant, the natural logarithm of 2 divided by the half-life, which is why a short half-life draws a steep line.
| Half-lives | Left | Esmolol | Caffeine | Pembrolizumab |
|---|---|---|---|---|
| 0 | 100% | 0 | 0 | 0 |
| 1 | 50% | 9 min | 5 h | 22 days |
| 2 | 25% | 18 min | 10 h | 44 days |
| 3 | 12.5% | 27 min | 15 h | 66 days |
| 4 | 6.25% | 36 min | 20 h | 88 days |
| 5 | 3.13% | 45 min | 25 h | 110 days |
| 6 | 1.56% | 54 min | 30 h | 132 days |
| 7 | 0.781% | 63 min | 35 h | 154 days |
| 8 | 0.391% | 72 min | 40 h | 176 days |
| 9 | 0.195% | 81 min | 45 h | 198 days |
| 10 | 0.0977% | 90 min | 50 h | 220 days |
The volume of distribution is not a place in the body. It is a ratio: the amount of drug in the body divided by its concentration in the plasma, written as the volume that would hold the whole amount at that concentration. A kilogram of body cannot hold 3.4 litres, so esmolol’s figure says that most of it sits outside the plasma.
For curves like these, the half-life follows from two numbers printed on the same documents: the natural logarithm of 2, times the volume, divided by the clearance. Esmolol’s 3.4 litres per kilogram and its clearance of 285 millilitres per kilogram per minute give about 8 minutes, against the 9 on its label. Pembrolizumab’s 6.0 litres and its clearance of 195 millilitres a day, once doses have settled, give about 21 days, against 22.
Each curve starts at the moment the whole dose has entered the body and spread through it, and shows elimination only. Real curves also rise while a drug is absorbed, and drop fast early on while it spreads into tissues: esmolol’s early drop has its own half-life of about 2 minutes. The vertical axis is a percentage of the starting amount, not a concentration. No doses appear on this page, on purpose.
Every value is copied from the documents below: US prescribing information through DailyMed, summaries of product characteristics from Ireland’s Health Products Regulatory Authority, and one paper. The caffeine label is for caffeine citrate, a medicine used in premature babies; it gives adult values for comparison. All were read on 6 October 2026.
Brevibloc (esmolol hydrochloride) injection, US prescribing information, Baxter, DailyMed. The half-lives, about 9 minutes for elimination and about 2 for distribution.
Brevibloc Premixed, summary of product characteristics, revised March 2019, HPRA. The volume of distribution, 3.4 l/kg, and the clearance, 285 ml/kg/minute.
Caffeine citrate oral solution, US prescribing information, Sagent, DailyMed. The adult half-life, 5 hours, and the adult volume of distribution, 0.6 L/kg.
J. Blanchard, S. J. Sawers, The absolute bioavailability of caffeine in man, European Journal of Clinical Pharmacology 24 (1983), 93–98, PubMed. Absorption by mouth, 108.3 ± 3.6%, and half-lives from 2.7 to 9.9 hours.
Keytruda (pembrolizumab) injection, US prescribing information, Merck Sharp & Dohme, DailyMed. The half-life, 22 days; the volume of distribution, 6.0 L; the clearance, 252 mL/day after the first dose and 195 mL/day at steady state; the molecular weight, 149 kDa.
Keytruda Qlex (pembrolizumab and berahyaluronidase alfa) injection, US prescribing information, Merck Sharp & Dohme, DailyMed. The form given under the skin, approved in the United States in 2025.
It explains a mechanism. It is not for deciding anything about any medicine anyone takes: not when, not how much, not whether to stop. There is no field here for a dose, a weight or a time, on purpose. If you came with a question about your own medicine, or someone else’s, this page can’t answer it. A doctor or a pharmacist can.
Made by Vincenzo Lambiase, medical student. It is drawn in your browser and sends nothing anywhere. To hear when the next instrument is out, write to me.