Problem 59 - Algebra and Trigonometry with Analytic Geometry

Question

Drug dosage A certain drug has a half-life of about 2 hours in the bloodstream. The drug is formulated to be administered in doses of \(D\) milligrams every 4 hours, but \(D\) is yet to be determined. (a) Show that the number of milligrams of drug in the bloodstream after the \(n\)th dose has been administered is $$ D+\frac{1}{4} D+\cdots+\left(\frac{1}{4}\right)^{n-1} D $$ and that this sum is approximately \(\frac{4}{3} D\) for large values of \(n\). (b) A level of more than 500 milligrams of the drug in the bloodstream is considered to be dangerous. Find the largest possible dose that can be given repeatedly over a long period of time.

Step-by-Step Solution

Answer

For large \(n\), the approximation of the geometric series is \(\frac{4}{3}D\) and the largest safe dose is 375 mg.

Step 1: Understanding Drug Half-life

The drug's half-life is the time it takes for its quantity in the bloodstream to reduce by half. Here, the half-life is 2 hours, so in 2 hours the amount of drug will be halved. This characteristic will help us understand how drug concentration decreases over time.

Step 2: Setting Up Drug Decay Equations

When a dose of \(D\) mg is administered, in 4 hours (because we administer another dose every 4 hours), the amount remaining from this dose will be \(\left(\frac{1}{2}\right)^2 D = \frac{1}{4}D\). This pattern follows after each dose due to the repetitive half-life reductions.

Step 3: Constructing the Summation Formula

When the \(n\)th dose is administered, the total amount of drug is the sum of remaining amounts from all previous doses: \(D + \frac{1}{4}D + \left(\frac{1}{4}\right)^2D + \cdots + \left(\frac{1}{4}\right)^{n-1}D\). It forms a geometric series with the first term as \(D\) and the common ratio \(\frac{1}{4}\).

Step 4: Solving the Geometric Series

The geometric series \(D + \frac{1}{4}D + \left(\frac{1}{4}\right)^2D + \cdots + \left(\frac{1}{4}\right)^{n-1}D\) has the sum formula \(S_n = D \frac{1 - \left(\frac{1}{4}\right)^n}{1 - \frac{1}{4}}\). For large \(n\), \(\left(\frac{1}{4}\right)^n\) approaches 0, so the series sum is approximately \(\frac{4}{3}D\).

Step 5: Determining Safe Dosage Level

Given the danger threshold is 500 mg, we use \(\frac{4}{3}D = 500\) mg to find the largest safe dose \(D\). Solving this gives \(D = \frac{3}{4} \times 500 = 375\) mg.

Key Concepts

Drug Half-life

The concept of a drug's half-life is fundamental to understanding medication dosing schedules. The half-life is the time required for the concentration of the drug in the bloodstream to reduce to half its original amount. For example, if a drug has a half-life of 2 hours, then every 2 hours, the amount of the drug present in the body is reduced by 50%.

This knowledge is crucial because it allows medical professionals to predict how long a drug will exert its effect and help in deciding the proper dosing intervals. Understanding half-life ensures that the drug remains at a therapeutic level in the bloodstream without reaching toxic levels. If a drug's concentration is maintained at a stable state within the body, it helps in optimizing the therapeutic effects while minimizing the chance of adverse reactions.

Half-life Calculations

Calculating the remaining concentration of a drug over time using its half-life can often be tricky. In the scenario provided, a drug is administered at regular intervals, while the remaining concentration diminishes due to its half-life.

Let's dive into how the mathematical calculations are made for this drug with a 2-hour half-life, which gets a fresh dose every 4 hours. Starting with an initial dose of \(D\) milligrams, after the first 4-hour interval, only \(\frac{1}{4}D\) milligrams of the initial dose remain because in each 2-hour period, the drug undergoes a half-life process twice. After each subsequent dose, the remaining amounts from previous doses and the new dose together form a geometric series:
- The first term is \(D\)
- The common ratio is \(\frac{1}{4}\)

The sum of this series, which represents the total drug concentration in the bloodstream, is calculated using the formula for geometric series:\[S_n = D \frac{1 - \left(\frac{1}{4}\right)^n}{1 - \frac{1}{4}}\]As \(n\) tends to a very large number, particularly with consistent dosing over time, \(\left(\frac{1}{4}\right)^n\) approaches zero. This simplifies the sum to approximately \(\frac{4}{3}D\). This approximation helps to predict the long-term effect of dosing the drug over time.

Drug Dosage Calculation

Calculating the correct drug dosage is vital for ensuring safety and effectiveness in medical treatments. In the case of drugs with a known half-life, the dosage calculation must account for the maximum safe concentration in the bloodstream. Let's explore how it's done.

Given that the dangerous threshold is above 500 milligrams, we use the geometric series summation to maintain safety. Using the simplified formula for a large \(n\), we have \(\frac{4}{3}D = 500\) mg to estimate doses. From this approximation:\[D = \frac{3}{4} \times 500 = 375 \text{ mg}\]
This result implies that a dose of 375 milligrams can be administered safely every 4 hours without exceeding the hazardous limit of 500 milligrams in the bloodstream. Adjusting the dosage in this manner allows for maintaining an effective therapeutic level while minimizing the risk of toxicity.

Thus, understanding both the calculation and practical application of drug half-lives not only empowers healthcare providers to administer medications safely but also helps in optimizing patient care.