Half-lives, decay and steady state

Simulation only. Educational — not a medical device. Not for diagnosis, treatment, monitoring or dosing. The drug is invented and so is its window band; figures are not settings, doses or targets.

Companion to chapter 17 (treatment and life support) of Human Biology for the Medical Maker — the one-compartment simulator the chapter suggests building, with units carried in the types — and to the PET section of chapter 16 (diagnostic devices) and the endocrine section of chapter 10.

Every input is typed with its unit — 6 h, 110 min, 400 mg, 40 L, 100 %, 5 mg/L — and converted explicitly. A bare number is refused, as is a unit of the wrong kind (6 mg for a half-life), and a value outside the tool’s bounds is rejected rather than clamped.

How to use it

Repeated dosing — an invented drug with first-order elimination: dose, bioavailability F, volume of distribution, half-life, dosing interval τ and how long to show. The chart shows concentration (mg/L) against time with an illustrative window band you set (invented — not a therapeutic range), the steady-state peak and trough, and lines at 4 and 5 half-lives. Cards give the first-dose peak F·D/Vd, k = ln 2 ÷ t½, the accumulation factor R = 1 ÷ (1 − 2^−τ/t½), steady-state peak, trough and average, how far to steady state the plot has got, and the share of the last interval below, within and above the band. Loading dose adds a first dose of D × R that starts at the steady-state peak. Single bolus shows plain decay and how long the level stays above the band’s lower edge — double the dose and that grows by one half-life, not twice as long.
Fraction remaining — 2⁻ⁿ for n = 0–10 half-lives, with times for a half-life you give and the matching fraction of steady state on regular dosing (1 − 2⁻ⁿ).
Radioactive decay — F-18 (≈110 min), Tc-99m (≈6 h) or any half-life. Enter the activity at calibration and the calibration, injection and scan times (24-hour clock) to get the fraction remaining and activity at each, and what decay correction is worth to an FDG SUV: forgetting it, or processing a late scan as if it were on time.
Hormone time constants — the book’s blood half-lives for adrenaline, insulin, cortisol and thyroxine as ranges, their time constants τ = t½ ÷ ln 2, how long a new level takes to settle (about 4–5 half-lives), and how much of the old level is left after a time you choose — on a logarithmic ruler from minutes to weeks.

The book’s exercises are buttons: repeated doses every 8 h with a 6 h and then a 24 h half-life (ch17:1040-1042 — steady state in about a day, then in four to five days, only 87.5 % of the way after three); Q15’s 8 h half-life (32–40 h to steady state, 1/8 left after 24 h); 400 mg with a 6 h half-life leaving 25 mg after a day (ch17:771-776); and FDG calibrated at 07:30, injected at 09:00 and scanned at 10:00 (fraction remaining ≈ 0.567 and ≈ 0.389; a 10:30 scan processed as 10:00 under-reads SUV by about 17 %), and 371 MBq scanned an hour later (≈ 254 MBq; forgetting decay correction is a ×1.46 error).

Links: Share link copies a URL with every tab’s inputs; each tab has its own address — for example half-lives/#decay (or #dosing, #table, #hormones).

Limits

A textbook model, not a prediction: one compartment, first-order elimination, instantaneous absorption of the bioavailable fraction, identical doses on a fixed schedule. Real drugs absorb over time, distribute into several compartments, may saturate their elimination, and vary from person to person. Nothing here is a dose or a regimen; prescribing and dosing belong to clinicians and pharmacists (ch17:1183-1185).
The window band is invented for illustration. It is not a therapeutic range for any drug.
Isotope half-lives are the book’s rounded figures (F-18 110 min, Tc-99m 6 h; NNDC NuDat gives 109.77 min and 6.0067 h). Times are on one day; crossing midnight is refused rather than guessed. For real imaging data to practise on, use public, de-identified sets such as The Cancer Imaging Archive (TCIA).
Hormone half-lives are approximate textbook figures (ch10:385-394; app-c:392) — orientation only.

Related: unit-safe clinical arithmetic (the infusion error drill and unit conversions), isotope patterns and exact mass, ODE solver (for multi-compartment models), growth curves.