A drug with a volume of distribution of 500 L and elimination half-life of 40 hours is given as a continuous IV infusion. Approximately how long does it take to reach 90% of steady-state plasma concentration?

• A Approximately 40 hours (one half-life)
• B Approximately 200 hours (5 half-lives = 5 × 40 hours)
• C Approximately 133 hours (~3.3 half-lives = 3.3 × 40 hours) ✓
• D It depends on the infusion rate; the time to 90% steady-state cannot be calculated without knowing the infusion rate

Explanation

For first-order kinetics, steady-state during continuous infusion is approached as an exponential function: Css × (1 - e^(-kt)). The fraction of steady-state achieved after n half-lives follows: 50% at 1 t½, 75% at 2 t½, 87.5% at 3 t½, 93.75% at 4 t½, and 96.9% at 5 t½. To achieve 90% steady-state, approximately 3.32 half-lives are required (since 1 - e^(-0.693 × 3.32) = 0.9). With t½ = 40 hours: 3.32 × 40 ≈ 133 hours. Importantly, neither the volume of distribution nor the infusion rate affects the TIME to reach steady state — only the elimination half-life determines this. The infusion rate determines the LEVEL (concentration) of steady state (Css = R0/CL).

Reference: KD Tripathi, Essentials of Medical Pharmacology, 8th ed.

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