Problem 3 - Applied Calculus

Question

Every morning, a patient receives a \(50-\mathrm{mg}\) injection of a drug. At the end of a 24 -hour period, \(60 \%\) of the drug remains in the body. What quantity of drug is in the body (a) Right after the \(3^{\mathrm{rd}}\) injection? (b) Right after the \(7^{\text {th }}\) injection? (c) Right after an injection, at the steady state?

Step-by-Step Solution

Answer

(a) 98 mg, (b) 121.50 mg, (c) 125 mg.

Step 1: Understand Dynamics of Drug Retention

Each day, a patient gets a 50-mg injection. By the end of the day, 60% of the drug remains. So the amount remains while introducing new ones each day on top of it. If \( x_n \) represents the drug amount after \( n \) injections, the recurrence relation is given by \( x_{n} = 0.6x_{n-1} + 50 \).

Step 2: Calculate Right After the 3rd Injection

To find the quantity of the drug after the 3rd injection, calculate using the recurrence relation. Start with \( x_1 = 50 \) because the first injection adds 50 mg:\[x_2 = 0.6(50) + 50 = 80 \x_3 = 0.6(80) + 50 = 98 \\]So, the drug quantity after the 3rd injection is 98 mg.

Step 3: Calculate Right After the 7th Injection

Continue using the recurrence relation for each subsequent day:\[x_4 = 0.6(98) + 50 = 108.8 \x_5 = 0.6(108.8) + 50 ≈ 115.28 \x_6 = 0.6(115.28) + 50 ≈ 119.17 \x_7 = 0.6(119.17) + 50 ≈ 121.50 \\]Thus, right after the 7th injection, the drug quantity is approximately 121.50 mg.

Step 4: Analyze Steady State Behavior

In the steady state, the amount remains constant daily, so set up the equation for steady state \( x = 0.6x + 50 \). Solving for \( x \):\[0.4x = 50 \x = \frac{50}{0.4} \x = 125 \\]Therefore, in the steady state, the quantity of drug is 125 mg.

Key Concepts

Recurrence Relation

When we talk about recurrence relation in the context of a drug retention model, it's essentially about understanding the pattern in which the drug accumulates in the body.
In our situation, a person receives a 50 mg injection of a drug every day, but not all of it leaves the body right away. Instead, 60% of the drug stays in the system for the next day.
To predict how much of the drug is in the body after a certain number of doses, we use a recurrence relation.

• Begin with the first day: After the first injection, there's 50 mg in the body.
• The next day, 60% of that stays, and another 50 mg is added.
• We express this mathematically as: \( x_n = 0.6x_{n-1} + 50 \).

This means that the current amount of drug depends on a percentage of what was there before plus the new injection. It's a simple yet powerful tool to model ongoing processes.

Steady State Analysis

Steady state analysis revolves around identifying a point at which the system reaches equilibrium. For drug retention, it refers to when the quantity of drug in the body levels out over time, assuming regular injections continue.
This is important because it helps understand at what point the drug's concentration stabilizes, avoiding either ineffective low doses or potentially harmful high doses. In our example, we determine the steady state by examining when the amount of drug introduced equals the amount retained. Here’s how it works:

• We start by assuming the amount of drug after a given dose must be the same before and after the next dose.
• Set up the equation: \( x = 0.6x + 50 \).
• Solve for \( x \) which represents the steady state amount, yielding \( x = 125 \) mg.

This means that no matter how many doses are given, the amount of drug stabilizes at 125 mg. It simplifies the planning of long-term medication regimes.

Calculus in Medicine

Calculus in medicine is a fundamental part of analyzing and predicting how different substances behave in the body over time. Through mathematical models, we can simulate how drug levels increase, decrease, or steady. This is immensely helpful when trying to manage effective dosing and ensuring safety in treatments. For our drug retention example, calculus helps:

• Identify how drug concentrations fluctuate between doses.
• Determine the impact of factors like metabolism, absorption, and injection frequency.
• Predict when a drug reaches its steady state and how it behaves thereafter.

Calculus allows healthcare professionals to maintain optimal drug levels, minimizing adverse effects and maximizing therapeutic benefits. Such models, enabled by calculus, are invaluable tools in modern medicine, providing quantitative backing to pharmacological decisions.