a pharmacokinetics model
/ FAR-ma-co-kuh-NET-ics /
When you swallow a pill, the drug does not stay at a fixed level — it floods in, peaks, then slowly clears as your liver and kidneys break it down and flush it out. Too little and it does nothing; too much and it is toxic. A pharmacokinetics model is the differential-equation description of this rise and fall, the tool that tells a doctor how big a dose to give and how often. It is, quite literally, the mathematics of keeping a drug in its safe and effective window.
The simplest version treats the body as a single well-mixed compartment of volume V holding drug at concentration C. The body eliminates the drug at a rate proportional to how much is present, giving C' = -k C: pure first-order decay, so the concentration falls as C0 times e^(-kt), exactly like radioactive decay, with a characteristic half-life (ln 2)/k. A two-compartment model adds a tissue box exchanging with the blood, capturing the faster initial drop as the drug distributes, then the slower decline as it is eliminated.
The real work is dosing. Because elimination is exponential, a single dose decays away, so repeated doses are given at fixed intervals; each new dose stacks on what remains of the previous ones, and the peaks and troughs climb until inflow per interval balances outflow — the concentration settles into a steady oscillating band. The model lets you choose the dose size and spacing so this steady band sits inside the therapeutic window: high enough to work, low enough to be safe. This is precisely why labels say 'every 8 hours' and not 'whenever'.
Honesty matters in medicine. The clearance rate k varies from person to person — with age, kidney and liver function, weight, and other drugs — so a 'standard dose' is an average that can be wrong for an individual; this is why some drugs need blood-level monitoring. And the one-compartment exponential picture is an idealization: absorption, multiple tissues, and saturable enzymes can all bend the curve, which fuller models address at the cost of more parameters.
A drug with half-life 6 hours has elimination rate k = (ln 2)/6 ≈ 0.116 per hour. After one dose the blood level falls by half every 6 hours. Dosing every 6 hours, the level builds over a few doses to a steady sawtooth oscillating between a peak and a trough that both stay inside the safe range.
Repeated dosing builds to a steady band; dose and interval are chosen to keep it in the therapeutic window.
The clearance rate k is patient-specific, not universal — it shifts with age, organ function, weight, and drug interactions. A single 'standard dose' is an average that can overshoot or undershoot in any given person, which is why narrow-window drugs need blood-level monitoring.