Question

Suppose that you take 200 mg of an antibiotic every 6 hr. The half-life of the drug is 6 hr (the time it takes for half of the drug to be eliminated from your blood). Use infinite series to find the long-term (steady-state) amount of antibiotic in your blood.

Short answer

Answer: 400 mg

Step-by-step solution

Setup the Infinite series

Let's denote the amount of antibiotic in the blood after each dose taken as A_n, where n represents the number of doses taken. An infinite geometric series can be set up to represent the total amount of antibiotic in the blood after taking multiple doses: A = A_1 + A_2 + A_3 + A_4 + ... The first term, A_1, is the initial amount of antibiotic after taking the first dose (200 mg). For each subsequent term (A_2, A_3, ..., A_n), we need to consider that the previous amount of antibiotic in the blood has decreased by half because of the 6-hour half-life of the drug.

Calculate the common ratio

To find the common ratio (r) of the geometric series, we can divide the second term (A_2) by the first term (A_1). Since the half-life of the antibiotic is 6 hours, and the person takes another dose every 6 hours, the amount in the body decreases by half before the next dose is taken: r = A_2 / A_1 = (1/2 A_1) / A_1 = 1/2

Apply the formula for the sum of an infinite geometric series

The formula for the sum of an infinite geometric series is: Sum = A_1 / (1 - r) Here, A_1 = 200 mg and r = 1/2. Plugging these values into the formula, we can find the steady-state amount of the antibiotic in the blood: Sum = 200 / (1 - 1/2)

Calculate the steady-state amount

Now, let's calculate the sum of the series: Sum = 200 / (1/2) = 200 / (0.5) = 400 Therefore, the long-term (steady-state) amount of antibiotic in the blood is 400 mg.

Key concepts

Half-Life of a Drug

Understanding the half-life of a drug is crucial in pharmacology and medicine, as it determines the duration a drug stays active in the body. The half-life is the period it takes for the concentration of the drug in the bloodstream to be reduced by half. This concept is particularly relevant when determining dosing schedules to maintain therapeutic levels without causing toxicity.

In the given exercise, the antibiotic has a half-life of 6 hours, meaning that every 6 hours, the concentration of the drug in the body decreases to 50% of its previous value. This exponential decay of drug concentration is important for calculating the steady-state concentration, especially when drugs are taken repeatedly.

To illustrate, if a person takes a 200 mg dose of an antibiotic with a 6-hour half-life, after 6 hours, there would theoretically still be 100 mg of the drug active in the system. Before the drug can fully diminish, another dose is taken, and the cycle continues, creating an overlap of drug concentrations leading to a steady-state.

Steady-State Concentration

The steady-state concentration is a pharmacological term referring to the scenario where the overall intake of a drug balances out with its elimination from the body. Essentially, the amount of drug being absorbed is equal to the amount being metabolized and excreted. For many medications, achieving a steady-state concentration is vital for ensuring consistent therapeutic effects.

In our example, the patient is taking regular doses of an antibiotic every 6 hours, which is equal to the drug's half-life. This timing allows the drug to reach a point where the amount added to the system with each dose is roughly the same amount that has been eliminated since the previous dose, resulting in a stable concentration in the blood.

The steady-state is significant as it helps avoid concentrations that are too high, causing toxicity, or too low to be effective. Health professionals use this understanding to design dosing regimens that optimize drug efficacy and safety.

Sum of an Infinite Series

The sum of an infinite series is a foundational concept in calculus, particularly in solving problems involving repetitive processes that approach a fixed limit. Specifically, the sum of an infinite geometric series can be determined when the series converges, that is, when the absolute value of the common ratio (r) is less than one.

The general formula for the sum of an infinite geometric series is \( S = \frac{A_1}{1 - r} \), where \( A_1 \) is the first term and \( r \) is the common ratio between successive terms. When \( |r| < 1 \) the series converges to a definite sum, while if \( |r| \geq 1 \) the series diverges and does not sum to a finite number.

Applying this to the exercise, the medication's half-life creates a series with a common ratio of \( \frac{1}{2} \) because the amount of drug is halved with each half-life. As this ratio is less than one, the series converges. Using the formula provided, the patient can expect to approach a steady-state concentration of 400 mg. This calculation is an excellent tool to predict the maximum amount of drug that will build up in the patient's system over time under a consistent dosing schedule.