Compartmental Modelling & Dosing

Pharmacokinetic compartmental models provide mathematical frameworks to describe how drug concentrations change over time. The one-compartment model assumes the body behaves as a single, homogeneous space. After an IV bolus, drug concentration declines monoexponentially: C(t) = C0 × e^(-kel × t), where kel is the elimination rate constant. On a semi-logarithmic plot (log C vs time), this appears as a straight line with slope = -kel/2.303. This linearity underpins back-extrapolation to estimate C0 and hence Vd.

Half-life (t½) is the time for concentration to fall by 50%: t½ = ln2/kel = 0.693/kel. Because CL = kel × Vd, it follows that t½ = 0.693 × Vd / CL. Half-life is thus determined by both Vd and CL — a drug with large Vd or low CL will have a long t½. Time to reach steady state during repeated dosing is approximately 4–5 half-lives, regardless of dose or dosing frequency. This is a fixed biological property of the drug. Increasing dose raises the steady-state concentration but does not alter the time to reach it.

Clearance (CL) is the most important pharmacokinetic parameter relating dose rate to steady-state concentration: CL = Dose / AUC (after IV) = kel × Vd. The area under the concentration-time curve (AUC) reflects total drug exposure. For oral dosing: AUC = F × Dose / CL.

Loading dose and maintenance dose calculations are essential for clinical practice. Loading dose (LD) achieves the target steady-state concentration (Css) rapidly: LD = Vd × Css_target. Maintenance dose (MD) replaces drug eliminated per dosing interval (τ): MD = CL × Css_target × τ. For oral drugs: MD = (CL × Css × τ) / F. These formulae underpin the rational initiation of drugs with long half-lives where waiting 4–5 t½ for steady state would be clinically unacceptable (e.g. digoxin, loading with IV and oral).

The two-compartment model recognises a rapid distribution phase (alpha phase) — drug moving from plasma to peripheral tissues — followed by a slower elimination phase (beta phase). Aminoglycosides exhibit this behaviour; post-distribution trough sampling (>18–24 h after dose for once-daily dosing) ensures accurate reflection of elimination, not distribution, for therapeutic drug monitoring.

Non-linear (Michaelis-Menten) kinetics occur when drug concentrations approach the capacity of elimination enzymes. Phenytoin is the paradigmatic example: at high therapeutic concentrations, CYP2C9 is nearly saturated. The rate of elimination follows Vmax × C / (Km + C); when C >> Km, elimination becomes zero-order (constant amount per unit time regardless of concentration). Clinically, small dose increments cause disproportionately large rises in plasma concentration, making phenytoin notoriously difficult to dose. Monitoring of free phenytoin levels is essential.

Therapeutic drug monitoring (TDM) is indicated for drugs with narrow therapeutic windows, unpredictable pharmacokinetics, or serious toxicity. Aminoglycosides: once-daily extended-interval dosing — aim for trough <1 mg/L at 18–24 h using Hartford nomogram; peak (1 h post-dose) typically >10 mg/L for Gram-negative infections. Vancomycin: AUC/MIC-guided dosing (target AUC 400–600 mg·h/L) has replaced trough-only monitoring to minimise nephrotoxicity. Digoxin: narrow therapeutic window 0.5–2.0 ng/mL; sample >6 h post-dose to avoid distribution phase interference.

Dose adjustment in renal impairment uses Cockcroft-Gault to estimate creatinine clearance (CrCl): CrCl = [(140 − age) × weight × (0.85 if female)] / (72 × serum creatinine in mg/dL). Reduce dose proportionally to CrCl for renally cleared drugs or extend dosing interval. In the elderly, expect reduced CYP activity, reduced renal function, increased Vd for lipophilic drugs (higher body fat), reduced albumin, and polypharmacy — all increasing drug sensitivity and adverse event risk.