Problem 96 - A Graphical Approach to Precalculus with Limits

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

The steady-state drug amount is 3 mg after a long period.

Step 1: Define Initial Conditions

Let's begin by understanding the problem. Each day a person receives 2 mg of the drug. This means at the end of day 0, the drug amount is 0 mg since they haven't started the treatment yet.

Step 2: Understand Daily Drug Dynamics

Each day, the new dosage adds to the leftover amount from the previous day. Additionally, the body utilizes 40% of the drug present at the start of the day. Therefore, 60% of the drug remains for the next day.

Step 3: Establish Drug Formula

Let \( A_n \) be the amount of drug at the end of day \( n \). At the end of the first day, \( A_1 = 2 \) mg, since no drug was present beforehand to be utilized. At the end of the second day, \( A_2 = 2 + 0.6 \times A_1 = 2 + 0.6 \times 2 = 2 + 1.2 = 3.2 \) mg.

Step 4: Derive General Expression for Day n

Continue this reasoning, each subsequent day receives 60% of the previous day's drug amount plus 2 mg from the new dose. Thus, \( A_n = 2 + 0.6 \times A_{n-1} = 2(1+0.6+0.6^2+\cdots+0.6^{n-1}) \). Using the formula for a geometric series, this becomes \( \sum_{i=1}^{n} 2(0.6)^{i} \).

Step 5: Calculate Steady State Quantity

Over a long period, the series sums to its limit as \( n \to \infty \). This is a geometric series with a common ratio less than one, summing to \( \frac{2(0.6)}{1-0.6} = 3 \), so the drug stabilizes around 3 mg as \( n \to \infty \).

Key Concepts

Geometric Series in Drug Modeling

In the study of drug dosage calculations, geometric series play a crucial role when determining how the amount of drug in the body builds up over time. Geometric series occur when each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

For example, if you take a new dose every day, and only 60% of the current drug amount remains in the body for the next day, then the amount of drug present forms a geometric series. This occurs because each day's leftover drug is a fixed percentage of the amount from the previous day.

• The first dose just adds a fixed amount, e.g., 2 mg.
• The second day adds the new dose plus 60% of the previous day's drug amount.
• This pattern continues, forming the series: \(2, 2 \times 0.6, 2 \times 0.6^2, \ldots\)

The formula to sum up such a series, when the common ratio is denoted by \(r\) and is less than one, becomes: \( \sum_{i=1}^{n} a \cdot r^i \), where \(a\) is the initial amount at each increment, which in our context represents the new daily dose.

Drug Dosage Calculation

Drug dosage calculation involves understanding how an administered drug accumulates in the body over time, and how it reaches a steady state after multiple dosages. The key factors in solving these problems include initial dose, body utilization rate, and frequency of administration.

In medical practice, getting these calculations correct is crucial for patient safety and efficacy of the treatment. Using the geometric series formula as previously established, we can summarize the continual drug build-up in the body.

• Each new dose adds a specific amount daily (e.g., 2 mg in our example).
• Drug metabolism or utilization in the body is represented as a percentage (e.g., 40% utilization, so 60% remains).
• Summing the series helps to predict how much drug remains in the body over time.

Thus, determining the formula \( A_n = 2 + 0.6 \times A_{n-1} \) allows for accurate predictions on the quantity of the drug left in the body at any given day, ensuring proper medical monitoring and adjustments if necessary.

Exponential Decay and Steady State

Exponential decay appears in drug metabolism because the body eliminates a certain percentage of the drug over time, characterized by a decreasing pattern. This is why we see the decay factor (e.g., 0.6) repeating in our formulas.

Eventually, a balance or steady state is reached when the amount added equals the amount being utilized. This is when the body achieves a consistent level of drug, indicated by the formula result approaching a specific number as days continue.

• Because the series we are summing is converging, over a long period, the body reaches a steady state of approximately 3 mg.
• The steady state ensures that the amount of drug stabilizes, preventing overdoses while still ensuring efficacy.
• This concept is crucial for adjusting dosages in treatments, to maintain steady therapeutic levels in the patient.

So, using the idea of exponential decay in conjunction with geometric series, caregivers can determine optimal dosage intervals and amounts for their patients.