With IV infusion, a drug reaches 90% of its final steady state in 10 hours. The elimination half-life of the drug must be approximately
## **Core Concept**
The core concept being tested here involves the pharmacokinetics of drugs, specifically the relationship between the time required to reach a certain percentage of steady state and the elimination half-life ((t_{1/2})) of a drug. At steady state, the rate of drug administration equals the rate of drug elimination.
## **Why the Correct Answer is Right**
To reach 90% of the steady state, a drug must undergo a certain number of elimination half-lives. The formula to calculate the fraction of steady state achieved after time (t) is:
[ f = 1 - e^{-lambda t} ]
where (lambda) is the elimination rate constant, and (t) is time. The elimination half-life ((t_{1/2})) is related to (lambda) by the equation:
[ t_{1/2} = frac{0.693}{lambda} ]
It is known that:
- After 1 (t_{1/2}), the drug reaches 50% of steady state.
- After 2 (t_{1/2}), the drug reaches 75% of steady state.
- After 3 (t_{1/2}), the drug reaches 87.5% of steady state.
- After 4 (t_{1/2}), the drug reaches 93.75% of steady state.
Given that the drug reaches 90% of its final steady state in 10 hours, we can approximate this to being around 3 to 4 half-lives, as 3 half-lives achieve 87.5% and 4 half-lives achieve 93.75%. Therefore, 10 hours corresponds roughly to 3.3 half-lives (an interpolation between 3 and 4 half-lives for 87.5% to 93.75% steady state).
## **Why Each Wrong Option is Incorrect**
**Option A:**
This option suggests a much longer half-life than what is estimated from the information provided. Given that 3 half-lives equal approximately 87.5%, and assuming 10 hours corresponds to about 3.3 half-lives for 90%, option A overestimates the half-life.
**Option B:**
This option suggests a half-life of 3.3 hours, which would fit with 10 hours being approximately 3 half-lives. However, precise calculation yields that 3 half-lives would be 9 hours for a drug with a 3-hour half-life, making 3.3 hours a plausible but let's see other options.
**Option D:**
This option suggests a half-life that would result in the drug reaching steady state much quicker than 10 hours, which does not align with reaching 90% of steady state in that timeframe.
## **Clinical Pearl / High-Yield Fact**
A useful clinical pearl is that it generally takes about 4 to 5 elimination half-lives for a drug to reach steady state. Additionally, knowing that after 3 half-lives a drug is approximately 87.5% of its steady state, and after 4 half-lives it's about 93.75%, helps in estimating drug accumulation.
## **Correct Answer:** .