Problem 8 - Applied Calculus

Question

Morphine is administered to a patient intravenously at a rate of \(2.5 \mathrm{mg}\) per hour. About \(34.7 \%\) of the morphine is metabolized and leaves the body each hour. Write a differential equation for the amount of morphine, \(M\), in milligrams, in the body as a function of time, \(t\), in hours.

Step-by-Step Solution

Answer

\( \frac{dM}{dt} = 2.5 - 0.347M \)

Step 1: Understand the problem

We have a system where morphine is entering the body at a constant rate of \(2.5\, \text{mg/h}\) and leaving the body at a rate proportional to the amount present. We need to establish a relationship between these rates and the amount of morphine in the body over time.

Step 2: Establish expressions for rate of change

The rate of change of morphine in the body is influenced by two factors: the rate of morphine entering and the rate leaving the body. The entering rate is \(2.5\, \text{mg/h}\). The leaving rate is \(0.347M\, \text{mg/h}\), where \(M\) is the current amount of morphine.

Step 3: Write the differential equation

The differential equation models the net rate of change of morphine, given by the entering rate minus the leaving rate. This gives us: \[ \frac{dM}{dt} = 2.5 - 0.347M \]. This represents the balance of morphine entering and leaving the system.

Key Concepts

Rate of Change

The concept of "rate of change" is crucial in understanding how quantities like morphine levels in the body vary over time. When we say rate of change, we're talking about how quickly or slowly a variable, such as the amount of morphine, is increasing or decreasing per unit of time. For morphine administered intravenously, the rate at which it enters the body is constant, at 2.5 mg per hour. But as soon as it starts leaving the system, the rate of change depends heavily on two main factors:

• The rate at which it is metabolized and exits the body.
• The current amount of drug present in the system.

In mathematical terms, the rate of change is described using a differential equation. In this case, the formula reflects the balance between the morphine entering the bloodstream and the proportion that is leaving. Understanding this dynamic helps us to predict how the drug is processed over time.

Proportionality

Proportionality in this context deals with the relationship between the amount of morphine in the body and the rate at which it leaves the system. When we say something is proportional, it implies a constant ratio or relation between two quantities. Here, it means the rate of morphine leaving the body is not random but is directly linked to the amount present in the system at any given time.

• The leaving rate is calculated as 34.7% of the current morphine amount, showing a clear proportional relationship.

This is why the term within the differential equation, \(0.347M\), emerges. It combines the percentage rate at which morphine exits with the actual amount present. This proportionality plays a key role in understanding the dynamics of how morphine is gradually reduced in the bloodstream as it gets metabolized and expelled from the body.

Pharmacokinetics

Pharmacokinetics is the branch of pharmacology concerned with the movement of drugs within the body. It covers how medications are absorbed, distributed, metabolized, and eliminated. In our example with morphine, pharmacokinetics helps us to model and understand these processes through mathematics.

• It considers factors like the rate of drug administration, metabolic rates, and the time it takes for the drug to be cleared from the system.
• Through equations, like our differential equation \( \frac{dM}{dt} = 2.5 - 0.347M \), we capture the essence of these pharmacokinetic processes.

The differential equation provides an effective way to visualize and anticipate how the morphine concentration in the body changes over time, integrating both the consistent input and proportional change due to metabolism. Such understanding can guide medical decisions related to dosing and timing of drug administration, ensuring that patient safety and effective treatment are upheld.