Drug Diffusion & Pharmacokinetics

Two compartments, one dose

A two-compartment pharmacokinetic model treats the body as two well-mixed tanks: a central compartment (blood plasma and highly perfused organs, where drug concentration is measured) and a peripheral compartment (less-perfused tissue that the drug diffuses into and back out of more slowly). Drug moves between them by first-order kinetics — the rate is proportional to the concentration difference — and leaves the body permanently from the central compartment by elimination, mostly hepatic metabolism and renal clearance.

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dC1/dt = -(k10 + k12)·C1 + k21·(V2/V1)·C2 + dose_input(t)
dC2/dt = k12·(V1/V2)·C1 - k21·C2

  C1, C2 = concentration in central / peripheral compartment
  k10    = elimination rate constant (clearance from central compartment)
  k12,k21 = distribution rate constants between compartments
  V1, V2 = volumes of distribution of each compartment

IV bolus: the classic biexponential decline

An intravenous bolus deposits the whole dose into the central compartment instantly, so C1 starts at its maximum and falls immediately — no absorption phase to wait through. Plotted on a log scale, the concentration-time curve for a two-compartment model is not a single straight line but the sum of two exponential decays: a fast initial distribution phase as the drug equilibrates into the peripheral compartment, followed by a slower, shallower elimination phase once the two compartments are roughly in equilibrium and the decline is governed mainly by clearance. Mistaking the fast distribution phase for the true elimination half-life is a classic dosing-interval error.

Oral dosing: an absorption phase changes everything

Oral dosing adds a third process: the drug must first be absorbed from the gut into the central compartment, usually modelled as its own first-order rate constant ka. This produces the familiar rise-then-fall curve — concentration climbs while absorption outpaces elimination, peaks at Cmax at time Tmax (the moment absorption and elimination rates exactly balance, dC/dt = 0), then declines as elimination dominates once most of the dose has been absorbed. A drug with fast absorption (large ka) relative to elimination produces a sharp, early, high peak; slow absorption produces a flatter, delayed, lower peak for the same total dose — the basis of extended-release formulations, which deliberately shrink ka to smooth the curve out.

IV bolus:   C1(0) = Dose/V1, then biexponential decline (distribution + elimination)
Oral dose:  dC1/dt includes + ka·A(t),  dA/dt = -ka·A(t)   (A = drug remaining in gut)
            Cmax, Tmax found where dC1/dt = 0
Infusion:   constant input rate R, approaches steady state  Css = R / CL
            (CL = clearance = k10 · V1)

The numbers a PK curve is boiled down to

Four summary numbers drive almost every real dosing decision. Cmax and Tmax describe the peak — how high the concentration gets and how fast — and matter most for drugs where toxicity or efficacy depends on a peak threshold. AUC (area under the concentration-time curve) is proportional to total drug exposure and to bioavailability; comparing AUC between an oral dose and an IV dose of the same drug is literally how bioavailability is measured. The elimination half-life, t½ = ln(2)/k10, sets the dosing interval — a drug is considered to reach steady state after roughly 4-5 half-lives of repeated dosing, and it clears the body to a negligible level after roughly the same number of half-lives once dosing stops.

Infusion and steady state

A continuous IV infusion at constant rate R does not spike and decay — concentration rises asymptotically toward a steady-state plateau Css = R / CL, where clearance CL is the product of the elimination rate constant and the central volume of distribution. This is the standard way critical-care drugs (sedatives, some antibiotics, vasopressors) are dosed when a stable, sustained concentration matters more than a peak — the infusion rate is chosen backward from the desired Css, and reaching it still takes the same 4-5 half-lives regardless of the rate, only the final plateau height changes.

Frequently asked questions

Why does an IV bolus dose show two different decay rates on a log plot?

Because the drug is doing two things at once early on: rapidly equilibrating into the peripheral (tissue) compartment, and being permanently eliminated. The fast initial drop is mostly redistribution into tissue; once the two compartments are near equilibrium, the slower, shallower second slope reflects the true elimination rate — mistaking the fast phase for the real half-life leads to under-dosing errors.

What determines Cmax and Tmax for an oral drug?

They're set by the balance between the absorption rate constant ka and the elimination rate constant k10. Concentration rises as long as absorption outpaces elimination and peaks (Tmax) exactly when the two rates are equal; a larger ka relative to k10 produces a higher, earlier Cmax for the same dose, while extended-release formulations deliberately lower ka to flatten the peak.

What is AUC used for in pharmacokinetics?

AUC, the area under the concentration-time curve, is proportional to total systemic drug exposure. It's the standard way to measure and compare bioavailability — for instance, comparing the AUC of an oral dose to the AUC of an IV dose of the same drug tells you what fraction actually reaches the bloodstream intact.

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Pharmacokinetics: How a Two-Compartment Model Predicts Drug Levels