Question

[T] How often should a drug be taken if its dose is \(3 \mathrm{mg}\), it is cleared at a rate \(c=0.1 \mathrm{mg} / \mathrm{h}\), and \(1 \mathrm{mg}\) is required to be in the bloodstream at all times?

Short answer

The drug should be taken every 20 hours.

Step-by-step solution

Determine the Half-life of the Drug

Use the clearance rate to find out how long it takes for half of the drug to be cleared from the bloodstream. Given the clearance rate is a constant value and the initial dose is 3 mg, we need to calculate how long it takes to reduce the dose to 1.5 mg (which is half of 3 mg). Since the clearance rate is 0.1 mg per hour, calculate the time taken to reach 1.5 mg. Let \( T \) be the half-life; hence \( 3 - 0.1T = 1.5 \). Solving this equation gives \( T = 15 \) hours.

Determine Minimum Safe Dose Frequency

The required minimum dose of the drug that must remain in the bloodstream is 1 mg at all times. The drug should be taken often enough so that it doesn't drop below 1 mg before the next dose is administered. Subtract the minimum required amount (1 mg) from the initial dose (3 mg) to determine the buffer that can be cleared (2 mg). Using the clearance rate, calculate how long it takes for 2 mg to be cleared. \( 2 = 0.1t \) implies \( t = 20 \) hours. Therefore, the drug should be taken every 20 hours to maintain at least 1 mg in the bloodstream.

Key concepts

Half-life calculation

The half-life of a drug is a key concept in pharmacokinetics, which refers to the time it takes for the concentration of the drug in the bloodstream to reduce by half. Understanding this time frame is crucial for determining when the next dose of the medication should be administered to maintain its therapeutic effect.

In the provided example, we start with an initial dose of 3 mg. Our task is to calculate how long it takes for this dose to reduce to half, which is 1.5 mg. Given the constant clearance rate of 0.1 mg per hour, we can set up the equation as follows:

• We start with 3 mg in the bloodstream.
• The clearance rate is 0.1 mg per hour.
• The drug decreases to 1.5 mg over time.

The equation becomes: \[ 3 - 0.1T = 1.5 \]Solving this, we find that \( T = 15 \) hours. This means that every 15 hours, the amount of the drug in the body will reduce by half. This information helps in effectively timing when subsequent doses should be administered to avoid levels dropping below the therapeutic requirement.

Clearance rate

The clearance rate is a pharmacokinetic term that refers to the measurement of the efficiency with which a drug is removed from the body. It is expressed as the volume of plasma from which the drug is completely removed per unit time.

In the example problem, the clearance rate is stated as 0.1 mg/hr. This means that every hour, 0.1 mg of the drug is being cleared from the bloodstream. Understanding the clearance rate helps us determine how long a drug stays active and in what time frame we should expect it to diminish.

To ensure that a required concentration of a drug is maintained in the bloodstream, like the 1 mg required in this scenario, it is essential to know how much of the drug will be cleared over a certain time. In this situation, knowing the clearance rate allows us to calculate when the next dose is necessary to maintain effectiveness.

Drug dosage frequency

The frequency at which a drug should be administered depends directly on its clearance rate and the desired therapeutic levels in the bloodstream. This concept is pivotal in making sure a patient gets just enough medication in their system, without allowing it to drop to ineffective levels or rise to toxic levels.

In our specific problem, the initial dose of the drug is 3 mg, and it is necessary to maintain at least 1 mg of the drug in the bloodstream to be effective. After calculating the clearance of the drug, we understand that 2 mg is the buffer we have before the levels fall under the required concentration.

The dose needs to be taken before this buffer is completely cleared, which at a clearance rate of 0.1 mg/hr takes:

• Buffer amount: 2 mg
• Clearance rate: 0.1 mg/hr

The equation is:\[ 2 = 0.1t \]
Solving this gives \( t = 20 \) hours.
This means the drug should be taken every 20 hours to ensure that the bloodstream maintains the minimum concentration of the drug for effective treatment.