Non-Linear (Michaelis-Menten) Pharmacokinetics

Most drugs follow first-order (linear) kinetics, where the rate of elimination is proportional to concentration and a constant fraction of drug is eliminated per unit time. However, when elimination pathways become saturated — typically enzymatic metabolism or active transport — kinetics become dose-dependent, non-linear, and far more clinically unpredictable. This saturable behaviour is described by Michaelis-Menten kinetics.

The Michaelis-Menten equation describes the rate of elimination: dC/dt = −Vmax × C / (Km + C), where Vmax is the maximum rate of elimination (concentration units per time) and Km is the Michaelis constant, equal to the concentration at which the elimination rate is half-maximal. Two limiting behaviours emerge from this equation. When C << Km (drug concentration much lower than Km), the denominator approaches Km and the equation simplifies to dC/dt ≈ −(Vmax/Km) × C — first-order kinetics with apparent rate constant ke = Vmax/Km. When C >> Km (drug concentration much higher than Km), the denominator approaches C and the equation simplifies to dC/dt ≈ −Vmax — zero-order kinetics, where a constant amount of drug (not a constant fraction) is eliminated per unit time regardless of concentration.

Phenytoin is the most clinically important example of Michaelis-Menten kinetics. Phenytoin is hydroxylated by CYP2C9 and CYP2C19 to the inactive metabolite 5-(4-hydroxyphenyl)-5-phenylhydantoin (HPPH). These enzymes operate near saturation at therapeutic plasma concentrations (10–20 mg/L). For phenytoin, Km ≈ 4–8 mg/L and Vmax ≈ 7 mg/kg/day, though both vary substantially between individuals due to genetic polymorphisms. The consequence is a disproportionate rise in plasma concentration with small dose increases: a dose increase from 300 to 350 mg/day might raise the concentration from 12 to 24 mg/L — a 100% increase in concentration from a 17% dose increase. This extreme non-linearity makes phenytoin dose titration hazardous and unpredictable.

Practical consequences of Michaelis-Menten kinetics for phenytoin include: (1) Half-life is not constant — it increases as concentration rises, because ke = Vmax/(Km + C) falls as C rises. Time to steady state is therefore unpredictable and cannot be estimated from a simple formula. (2) Small dose changes have large effects near saturation. (3) Drug interactions are particularly dangerous — inhibitors (fluconazole, amiodarone, valproate) push concentrations into the toxic range far more steeply than predicted for a first-order drug. (4) Inter-patient variability in Vmax and Km means Bayesian dosing software (e.g., DoseMeRx) or the Vozeh-Sheiner nomogram should be used to individualise doses based on measured concentrations.

Other examples of clinically relevant non-linear kinetics include ethanol (CYP2E1 saturated at low BAC — zero-order kinetics throughout the relevant range), high-dose aspirin (saturation of salicylate conjugation), fluorouracil at high doses, and some monoclonal antibodies that saturate target-mediated drug disposition (TMDD) — a receptor-mediated process analogous to Michaelis-Menten enzyme kinetics. Zero-order kinetics also occur with some controlled-release formulations and transdermal patches that release drug at a constant rate independent of concentration gradient.