Question

Figure 15.3 shows the quantity of the drug atenolol in the body as a function of time, with the first dose at time \(t=0 .\) Atenolol is taken in \(50 \mathrm{mg}\) doses once \(a\) day to lower blood pressure. (a) If the half-life of atenolol in the body is 6 hours, what percentage of the atenolol \(^{9}\) present at the start of a 24 -hour period is still there at the end? (b) Find expressions for the quantities \(Q_{0}, Q_{1}, Q_{2},\) \(Q_{3}, \ldots,\) and \(Q_{n}\) shown in Figure \(15.3 .\) Write the expression for \(Q_{n}\) (c) Find expressions for the quantities \(P_{1}, P_{2}, P_{3}, \ldots,\) and \(P_{n}\) shown in Figure \(15.3 .\) Write the expression for \(P_{n}\)

Short answer

Answer: 6.25%

Step-by-step solution

Find the remaining percentage of atenolol after 24 hours

Since the half-life of atenolol is 6 hours, we need to determine how many half-life periods there are in 24 hours. We have: Number of half-life periods in 24 hours = 24 hours / 6 hours = 4. As there are 4 half-life periods, the remaining percentage can be calculated as: Remaining percentage = \((\frac{1}{2})^4 \times 100 \% = 6.25 \%\) So after 24 hours, 6.25% of the atenolol taken at the start of the day remains in the body.

Find expressions for the quantities \(Q_{0}, Q_{1}, Q_{2}, \ldots,\) and \(Q_{n}\)

Since the atenolol is taken once a day, we can find the remaining amount of atenolol in the body every 24 hours as follows: \(Q_{0} = 50\) \(Q_{1} = 50(1 - 0.0625)\) \(Q_{2} = 50(1 - 0.0625)(1 - 0.0625)\) \(Q_{3} = 50(1 - 0.0625)(1 - 0.0625)(1 - 0.0625)\ \ldots\) \(Q_{n} = 50(1 - 0.0625)^n\) This gives us the expression for the remaining amount of atenolol at each day's end.

Find expressions for the quantities \(P_{1}, P_{2}, P_{3}, \ldots,\) and \(P_{n}\)

Since the atenolol is taken once a day, 50 mg is added at each day's beginning. Therefore, for each day, the total amount of atenolol in the body is equal to the sum of the amount remaining from the previous day and the newly added 50 mg. We can find the expressions according to this using the expressions for \(Q_n\): \(P_{1} = Q_{0} + 50 = 50 + 50\) \(P_{2} = Q_{1} + 50 = 50(1 - 0.0625) + 50\) \(P_{3} = Q_{2} + 50 = 50(1 - 0.0625)(1 - 0.0625) + 50 \ \ldots\) \(P_{n} = Q_{n} + 50 = 50(1 - 0.0625)^n + 50\) These expressions give us the total amount of atenolol in the body at the beginning of each day (right after taking the dose).

Key concepts

Half-life

The concept of half-life is fundamental when discussing how quickly substances, like drugs, decay in the body. A half-life is the amount of time it takes for half of a substance to decrease through natural processes. In pharmacology, understanding the half-life of a medication helps determine dosage schedules to maintain therapeutic levels of the drug in the body.
For atenolol, the half-life is 6 hours. This means that every 6 hours, the concentration of atenolol in the body is reduced by half. If you want to know how much of the drug remains in the body after a certain period, you can calculate it using the formula: \(\text{Remaining} = (\frac{1}{2})^n \times 100\%\). Here, \(n\) is the number of half-life periods that have elapsed.
In the given exercise, after 24 hours, there are 4 such periods because 24 divided by 6 equals 4. The calculation shows that 6.25% of the drug remains after a day, illustrating the exponential decay characteristic of half-life.

Drug Dosage Calculations

Drug dosage calculations are crucial to ensure a medication remains effective and safe. Medicines like atenolol need to be dosed correctly based on the body's elimination rate to maintain their therapeutic effect. An understanding of half-life helps in planning when and how much of a drug should be administered.
In the problem, atenolol is dosed once a day at 50 mg. After the first dose, the remaining amount of the drug decreases. At each 24-hour interval, the remaining drug should ideally relate to the dosage plan drawn from the half-life calculation.
The expressions \(Q_n = 50(1 - 0.0625)^n\) help define the remaining quantity of atenolol at the end of each day. Having this plan allows for accurate dose adjustments in real situations, ensuring beneficial drug levels are sustained throughout therapy.

Sequences and Series

Sequences and series are mathematical concepts that represent ordered lists and sums of numbers, often utilized in various calculations, including those related to pharmacokinetics for drug levels.
In this example, we discuss a series of dose calculations combined with the remaining drug in the body. The sequence defined by \(Q_n\) represents each dose episode's remaining atenolol amount. Even more, \(P_n = Q_n + 50\) represents a new series where each day’s total is the leftover from prior medication plus a new dose.
This method of series and sequences aids in visualizing how drugs build up or reduce over time, crucial for understanding long-term medication management. By mastering these mathematical tools, predicting and adjusting drug levels for patient needs becomes efficient and clear.