Pharmacokinetics — Drug Concentration & Half-life

Four processes, one word: ADME

A drug does not appear everywhere in your bloodstream at therapeutic concentration the instant you take it. It follows a trajectory governed by four processes, conventionally bundled into the acronym ADME: Absorption — moving from the site of administration into the bloodstream. An intravenous injection skips this step entirely (100% bioavailability by definition); an oral tablet has to survive stomach acid and cross the intestinal wall. Distribution — spreading from plasma into tissues, characterised by the volume of distribution V_d = dose / plasma concentration, which can be a few litres for a drug that stays in the blood or hundreds of litres for a lipophilic drug that penetrates deep into fatty tissue. Metabolism — mostly enzymatic transformation in the liver, chiefly by cytochrome P450 enzymes, usually converting the drug into a more water-soluble, less active form. Excretion — removal from the body, mainly through the kidneys and, for some drugs, via bile into the gut.

One consequence worth knowing: an orally absorbed drug travels via the portal vein straight to the liver before it ever reaches general circulation, so a high-clearance drug can lose most of its dose to this first-pass effect before it does anything therapeutic — morphine and propranolol both have oral bioavailability around 25-30% for exactly this reason, which is why IV and oral doses of the same drug are rarely the same number.

The one-compartment model: pure exponential decay

The simplest useful model treats the body as one well-mixed tank. After an IV bolus dose D is placed instantly into the distribution volume V_d, elimination proceeds at a rate proportional to whatever concentration remains — first-order kinetics, the same mathematics as radioactive decay.

dC/dt = −k_e · C            first-order elimination

C(t)  = C₀ · e^(−k_e·t)     C₀ = D / V_d

k_e   = CL / V_d             CL = clearance, volume of plasma cleared per unit time

Plot log(C) against time and this model gives a straight line — a useful sanity check when fitting real blood-level data, and the reason clinical pharmacokinetics traditionally works on a semi-log scale.

Oral dosing: two exponentials racing each other

Swallow the same drug instead of injecting it, and a second first-order process — absorption from the gut, at rate constant k_a — now feeds the compartment while elimination simultaneously drains it. Concentration climbs while absorption dominates and only starts falling once elimination overtakes it.

C(t)   = (F·D·k_a) / (V_d·(k_a − k_e)) · (e^(−k_e·t) − e^(−k_a·t))

t_max  = ln(k_a/k_e) / (k_a − k_e)        peak concentration, typically 0.5-2 h

AUC    = F·D / CL                          total exposure = area under the curve

F, the bioavailability, is the fraction of the oral dose that ultimately reaches systemic circulation intact — 1.0 for IV by definition, well under 1.0 for most oral drugs once absorption losses and the first-pass effect are accounted for. The AUC — the total area under the concentration-time curve — is what regulators actually compare across formulations, because it captures total drug exposure independent of exactly when the peak occurs.

Real drugs often need two compartments

Many drugs, once you look closely at the data, decay biphasically: a fast initial drop as the drug distributes from plasma into peripheral tissue, followed by a much slower decline as it is actually eliminated. A two-compartment model — a central plasma compartment exchanging drug with a peripheral tissue compartment — captures this with a biexponential solution, C₁(t) = A·e^(−αt) + B·e^(−βt), where α is the fast distribution rate and β the slow terminal elimination rate.

This distinction is not academic. If you estimate half-life using only the fast initial drop, you will badly underestimate how long the drug actually persists; for drugs with a large peripheral reservoir like digoxin or amiodarone, dosing on the wrong half-life risks either toxicity from accumulation or therapeutic failure from under-dosing.

Half-life, steady state, and why loading doses exist

The elimination half-life, the time for concentration to drop by 50%, falls straight out of the exponential decay model: t₁/₂ = ln(2)/k_e = 0.693·V_d/CL. It varies enormously — ibuprofen clears in about 2 hours, requiring dosing every 6-8 hours; warfarin runs about 40 hours, so it takes days to reach a stable therapeutic range; amiodarone's half-life is around 58 days, meaning it keeps accumulating in the body for months after starting therapy.

Repeated dosing at a fixed interval τ builds toward a steady-state trough concentration, and it takes roughly five half-lives to get there — for amiodarone that is the better part of a year. A loading dose sidesteps the wait by filling the distribution volume to the target concentration immediately, with smaller maintenance doses afterward simply replacing what clearance removes each interval:

D_L (loading dose)     = C_target · V_d / F
D_M (maintenance dose) = C_target · CL · τ / F

When elimination saturates: Michaelis-Menten kinetics

Most drugs stay comfortably below the concentration at which their metabolising enzymes saturate, so first-order kinetics — elimination rate proportional to concentration — holds throughout the therapeutic range. A handful of drugs, most notoriously phenytoin, do not. Phenytoin's Michaelis constant K_m sits around 5 mg/L, right inside its own 10-20 mg/L therapeutic range, so the enzyme system is already partly saturated at normal doses:

dC/dt = −(V_max · C) / (K_m + C)

C << K_m:  dC/dt ≈ −(V_max/K_m)·C     first-order, k_e = V_max/K_m
C >> K_m:  dC/dt ≈ −V_max              zero-order — constant elimination rate

In the saturating zone, a small increase in dose can produce a disproportionately large jump in steady-state concentration, which is exactly why phenytoin requires close therapeutic drug monitoring rather than the simple linear dose adjustments that work for first-order drugs. The same monitoring logic applies to vancomycin, digoxin, and lithium — all narrow-therapeutic-index drugs where blood levels are checked directly rather than inferred from dose alone. Clearance-based dose adjustment is also standard practice in chronic kidney disease, where renally cleared drugs need proportional reductions as glomerular filtration rate declines, and in paediatrics, where allometric scaling (dose roughly proportional to weight^0.75) approximates how metabolic clearance actually changes from neonate to adolescent.

Frequently asked questions

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