Problem 86 - A Graphical Approach to College Algebra

Question

(Modeling) Drug Dosage Certain medical conditions are treated with a fixed dose of a drug administered at regular intervals. Suppose that a person is given 2 milligrams of a drug each day and that during each 24-hour period the body utilizes \(40 \%\) of the amount of drug that was present at the beginning of the period. (a) Show that the amount of the drug present in the body at the end of \(n\) days is $$ \sum_{i=1}^{n} 2(0.6)^{i} $$ (b) What will be the approximate quantity of the drug in the body at the end of each day after the treatment has been administered over a long period?

Step-by-Step Solution

Answer

(a) \( \sum_{i=1}^{n} 2(0.6)^{i} \); (b) 3 mg.

Step 1: Understand the Problem

We need to model the amount of drug present in the body at the end of each day. The drug dose is 2 mg per day, and the body retains only 60% of the drug from the previous day as it utilizes 40% each day. We need to express this as a series and find the long-term behavior.

Step 2: Express the Series

Each day, the retained drug amount from the previous days diminishes to 60%.. The amount of drug present at the end of the first day is simply 2 mg after receiving the initial dose. For the second day, we will have 60% of the 2 mg from the first day plus 2 mg from the second day. This results in:\[\text{Amount at end of day } n = \sum_{i=1}^{n} 2(0.6)^{i}\]This series accounts for the initial 2 mg dose every day, multiplied by the 60% retention from each subsequent day, with each term representing a diminishing portion of the original doses as time passes.

Step 3: Calculate the Infinite Sum for Long-term Quantity

In the long-term, the series becomes an infinite geometric series as \(n\) approaches infinity. The sum of an infinite geometric series \( \sum a r^i \) where \(|r| < 1\) is \( \frac{a}{1-r} \). Here, \(a = 2(0.6)\) and \(r = 0.6\):\[\text{Long-term amount} = \frac{2 \times 0.6}{1 - 0.6} = \frac{1.2}{0.4} = 3\]Thus, the amount of the drug in the body at the end of each day stabilizes to 3 mg after a long duration of treatment.

Key Concepts

Geometric Series

In the context of drug dosage modeling, a geometric series is essential for understanding how the drug accumulates and diminishes over time in the body. When we administer a constant daily dose of a drug, like 2 milligrams in this case, and know that the body retains 60% of the drug each day, we can express this situation using a geometric series.

A geometric series takes the form \[\sum_{i=1}^{n} ar^i\] where \(a\) is the initial amount, and \(r\) is the common ratio, representing the proportion of the drug retained daily. For drug dosage, each daily dose represents an additional term in the series, continuously adding to previous retained amounts.

This results in the mathematical expression \(2(0.6)^i\) for the drug received on day \(i\) and retained by that percentage. The power of the geometric series is that it allows a complex real-world process to be understood using straightforward mathematical principles.

Key features:

• Every term in the series diminishes as it relates to prior doses.
• The common ratio \(r\), less than 1, ensures that the sum converges, representing diminishing contributions of prior doses as time goes on.

Long-term Behavior

Understanding the long-term behavior of a drug in the body invites analysis of how the series behaves as the number of terms grows very large. This is particularly important in medical scenarios where treatments may be ongoing for extended periods, often indefinitely.

The long-term behavior of the drug accumulation can be analyzed by considering the series as an infinite geometric series. For an infinite series of the form \[\sum ar^i\] where \(|r| < 1\), the sum can be calculated with the formula \[\frac{a}{1-r}\].

In our drug model:

• The first term \(a = 2(0.6)\) accounts for the drug dealt with on the first day.
• The common ratio \(r = 0.6\) reflects retention daily.
• Substituting these into the formula gives \(\frac{1.2}{0.4} = 3\) mg.

This elucidates that, over many days, the drug concentration stabilizes at 3 mg, meaning the initial effects of new doses and declining retention balance out to a consistent daily body concentration. This understanding aids doctors in prescribing appropriate dosages that consider long-term patient needs.

Mathematical Modeling

Mathematical modeling is a powerful tool for interpreting real-world phenomena and forecasting future outcomes. In the context of drug dosage, it allows us to predict the behavior of drugs in the human body over time using mathematical expressions and logical reasoning.

To model drug dosage, we start by interpreting how the drug is processed — 40% is used daily, leaving 60% retained. This creates a recurring process ideal for modeling with a geometric series, reflecting accumulation in both short and long terms. Applying mathematical modeling:

• Provides a structured way to predict drug levels day-by-day.
• Includes calculating initial dose effects and the impact of biological retention rates.
• Ensures patient safety through anticipating how drug levels stabilize, avoiding overdoses or underdosing.

This approach not only sharpens our comprehension of drug behavior but also informs clinical decisions, ensuring effective and safe treatment over sustained treatment courses. Modeling bridges the gap between theoretical mathematics and practical healthcare applications.