Two-Compartment Model & Kinetics of Multiple Dosing
This unit advances the pharmacokinetic models to the two-compartment open model — accounting for the distribution phase observed with many drugs after IV bolus. It covers the bi-exponential equation, the concepts of α (distribution) and β (elimination) phases, macro/micro rate constants, and parameter calculation using the method of residuals. The second half covers the critical clinical application of multiple dosing — steady-state drug levels, the accumulation factor, and the calculation of loading and maintenance doses to rapidly achieve and maintain therapeutic concentrations.
Syllabus & Topics
- 1Two-Compartment Model – Concept: Many drugs do NOT distribute instantaneously — after IV bolus, Cp declines in TWO phases: (1) Rapid distribution phase (α phase): drug distributes from blood → tissues. (2) Slower elimination phase (β phase): drug eliminated from body while redistribution continues. Semi-log plot: bi-exponential decline (two straight-line segments). Model: Central compartment (compartment 1: blood + highly perfused organs) ↔ Peripheral compartment (compartment 2: slowly perfused tissues like fat, muscle). Drug enters/exits only via central compartment.
- 2Two-Compartment IV Bolus – Equation: Cp(t) = A·e^(-αt) + B·e^(-βt). Where: A, B = hybrid coefficients (y-intercepts of the two lines). α = distribution rate constant (faster, steeper slope). β = elimination rate constant (slower, terminal slope). α > β always. The equation is bi-exponential — sum of two first-order processes. Semi-log plot: (1) Terminal linear portion: slope = −β/2.303, y-intercept = B (extrapolated back). (2) Residual line: subtract B·e^(-βt) from observed Cp → plot gives slope = −α/2.303, y-intercept = A.
- 3Micro Rate Constants: Transfer rate constants between compartments: k₁₂ = rate constant central → peripheral. k₂₁ = rate constant peripheral → central. k₁₀ = elimination rate constant from central compartment. Relationships: α + β = k₁₂ + k₂₁ + k₁₀. α × β = k₂₁ × k₁₀. k₂₁ = (Aβ + Bα)/(A + B). k₁₀ = αβ/k₂₁. k₁₂ = α + β − k₂₁ − k₁₀. These micro constants describe the actual transfer processes, while α and β are ‘hybrid’ constants representing overall decline rates.
- 4Two-Compartment – Parameters: Vd(central) = Vc = Dose/(A + B) → volume of central compartment. Vd(area) = Vd(β) = Dose/(β × AUC) → most commonly used Vd for two-compartment. Vd(ss) = Vc(1 + k₁₂/k₂₁) → at steady state. AUC = A/α + B/β. t½(α) = 0.693/α (distribution half-life). t½(β) = 0.693/β (elimination half-life — the one reported clinically). CLt = Dose/AUC = β × Vd(β). Example: Gentamicin — distribution phase (t½α ≈ 15-30 min) → elimination phase (t½β ≈ 2-3 h).
- 5When to Use Two-Compartment: Use two-compartment model when semi-log plot of Cp vs time shows distinct bi-exponential decline (curve initially, then straight line). Common drugs: Aminoglycosides (Gentamicin, Amikacin), Digoxin, Theophylline, Lidocaine, many cytotoxic drugs. Clinical significance of distribution phase: drug effect may not correlate with plasma levels during distribution → wait until post-distribution (pseudo-equilibrium) → measure trough levels. Example: Digoxin — measure levels 6-8 hours post-dose (after distribution is complete).
- 6Multiple Dosing – Concept: Most drugs given as multiple doses (not single). With repeated dosing at fixed intervals (τ): drug accumulates until INPUT per interval = OUTPUT per interval → Steady State. Accumulation factor: Rac = 1/(1 − e^(-KE·τ)). If τ = t½: Rac = 1/(1 − 0.5) = 2 → drug level at steady state is 2× that after single dose. At steady state: Css(max) = (F·Dose)/(Vd · (1 − e^(-KE·τ))) and Css(min) = Css(max) · e^(-KE·τ). Average steady state: Css(avg) = (F·Dose)/(CL·τ).
- 7Steady-State Drug Levels: Time to reach steady state: 4-5 × t½ (same as IV infusion — independent of dose or interval). Css(avg) = F·Dose/(CL·τ). This is equivalent to IV infusion Css = R₀/CL where R₀ = F·Dose/τ (average dosing rate). At steady state: Css fluctuates between Css(max) and Css(min). Fluctuation = (Css(max) − Css(min))/Css(avg) × 100%. To ↓fluctuation: ↓dose + ↓interval (give smaller doses more frequently) or use sustained-release formulations. Therapeutic window: Css(max) should be below toxic concentration (MTC), Css(min) should be above minimum effective concentration (MEC).
- 8Loading Dose (DL): Purpose: rapidly achieve therapeutic Css without waiting 4-5 half-lives. DL = Css(desired) × Vd / F. Or: DL = Maintenance Dose × Rac. Example: Digoxin — t½ ≈ 36 h → steady state in 6-7 days without loading dose → unacceptable delay for heart failure → give loading dose (‘digitalization’). For IV: DL = Css × Vd. For oral: DL = Css × Vd/F. Loading dose depends on Vd (NOT on clearance). Maintenance dose depends on clearance (NOT on Vd). This distinction is clinically crucial.
- 9Maintenance Dose (DM): Purpose: maintain Css(avg) at the target therapeutic concentration. DM = Css(avg) × CL × τ / F. Or: DM = DL / Rac = DL × (1 − e^(-KE·τ)). At steady state: rate in = rate out → F·DM/τ = CL × Css(avg). Dose adjustment: in renal impairment (↓CLR → ↓CLt → ↓DM or ↑τ). Methods: (1) Reduce dose, keep interval (↓fluctuation). (2) Keep dose, extend interval (↑fluctuation). (3) Combination approach. Aminoglycosides (concentration-dependent): extend interval. β-lactams (time-dependent): reduce dose or use continuous infusion.
- 10Clinical Significance: (1) Narrow therapeutic index drugs (Phenytoin TI: 10-20 μg/mL, Digoxin TI: 0.5-2 ng/mL, Lithium TI: 0.6-1.2 mEq/L): small DM changes → big Css changes → toxicity. (2) Therapeutic Drug Monitoring (TDM): measure Css(min) (trough — just before next dose). MUST be at steady state (wait 4-5 t½ after starting or dose change). Peak levels measured for aminoglycosides (concentration-dependent efficacy). (3) Dose individualization: use measured Cp to calculate patient’s own KE, Vd → personalized dosing.
Learning Objectives
Exam Prep Questions
Q1. Why does the loading dose depend on Vd, not CL?
The loading dose aims to FILL the body’s ‘volume’ to the target concentration immediately: DL = Css × Vd. It’s like filling a container to a certain level — the volume of the container (Vd) determines how much drug you need, not how fast it drains (CL). Conversely, the maintenance dose replaces what’s eliminated: DM = Css × CL × τ — it depends on how fast drug is cleared (CL). So: Vd determines how much drug goes IN initially, CL determines how much needs to be REPLACED.
Q2. What is the Accumulation Factor and when is it used?
The accumulation factor Rac = 1/(1 − e^(-KE·τ)) tells you how much drug accumulates with repeated dosing compared to a single dose. If τ = t½: Rac = 2 (drug level doubles). If τ = 2t½: Rac ≈ 1.33 (minimal accumulation). If τ = 0.5t½: Rac ≈ 3.41 (significant accumulation). Used to: (1) Predict Css from single-dose data. (2) Calculate loading dose: DL = DM × Rac. (3) Determine if a drug will accumulate significantly with a given dosing interval.
Q3. Why measure trough (Css(min)) levels in TDM?
Trough levels (measured just before the next dose) represent the LOWEST drug concentration in the dosing interval. For most drugs, the trough must stay above the MEC (minimum effective concentration) throughout the interval for therapeutic efficacy. Trough is preferred because: (1) Most reproducible time point (not affected by absorption variability). (2) Easiest to time (just before next dose). (3) For narrow TI drugs, if the trough is safe, the peak is likely safe too. Exception: aminoglycosides — peak levels also measured (concentration-dependent killing).