Problem 8 - Applied Calculus

Question

Morphine is administered to a patient intravenously at a rate of 2.5 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

The differential equation is \( \frac{dM}{dt} = 2.5 - 0.347M. \)

Step 1: Define the change in morphine

We need to account for both the administration of morphine and its metabolization. The rate of change of morphine in the body, \(\frac{dM}{dt}\), consists of an increase from the intravenous administration and a decrease due to metabolism.

Step 2: Express morphine administration as an equation

Morphine is being added to the body at a constant rate. This can be modeled by the equation \(\text{Rate of administration} = 2.5 \ \text{mg/hr}.\)

Step 3: Express morphine metabolism as a proportionality equation

Since \(34.7\%\) of the current morphine in the body, \(M\), is metabolized every hour, the rate of decrease is proportional to the amount of morphine present. This can be described with the equation \(\text{Rate of metabolism} = -0.347M.\)

Step 4: Combine both rates into a differential equation

The differential equation for the rate of change of morphine in the body is given by the sum of the rates from administration and metabolism. Thus, we have: \[ \frac{dM}{dt} = 2.5 - 0.347M. \]

Key Concepts

Rate of Change

When dealing with differential equations, understanding the concept of the rate of change is crucial. A rate of change refers to how a quantity changes over time. It is the derivative in the context of differential equations. In the problem with morphine, the rate of change of morphine in the body combines two aspects: the rate of administration and the rate of metabolization.

For the morphine administered intravenously, the rate of change is straightforward. It’s a constant value added to the body, which in this case is 2.5 mg per hour. This constant affects how quickly or slowly the amount of morphine increases. Meanwhile, the morphine being metabolized leaves the body, decreasing the amount present. By understanding these aspects, we can construct a differential equation that accurately models the situation.

In general, it's essential to determine all contributing factors to the rate of change in any situation involving differential equations. This includes identifying both increases and decreases in the quantity of interest, which helps in creating a holistic model of the process being studied.

Proportionality

Proportionality is a key idea used in differential equations to model real-world scenarios where a quantity changes in relation to another. In the morphine metabolism problem, the rate at which morphine leaves the body is directly proportional to the current amount of morphine in the body. This means that the more morphine present, the more will be metabolized each hour.

Expressed mathematically, this is the equation:

• Rate of metabolism = \(-0.347M\),

where \(M\) is the amount of morphine in the body. The negative sign indicates a decrease, and 0.347 is the proportionality constant, reflecting that \(34.7\%\) of morphine is metabolized.

Proportional relationships are fundamental in forming realistic models in various fields, from biology to economics. They help us understand how changes occur in response to varying conditions. By leveraging proportionality, we can predict how quickly or slowly changes happen, providing insight into dynamic systems.

Mathematical Modeling

Mathematical modeling involves creating mathematical expressions or equations to describe a real-world scenario accurately. In this problem with morphine, a differential equation is used as the mathematical model to depict how morphine levels in a patient change over time.

The model here incorporates both the constant inflow of morphine and its proportional outflow due to metabolism:

• The administration of morphine is a constant input, represented simply as 2.5 mg/hr.
• The metabolism of morphine is a proportional output, shown as \(-0.347M\).

Combining these gives the differential equation:\[ \frac{dM}{dt} = 2.5 - 0.347M \] Modeling like this is crucial, as it distills the problem to its mathematical essence, allowing predictions and solutions to be derived systematically. Mathematically modeling how morphine levels behave under specific conditions is an example of applying calculus to drug administration, helping healthcare professionals make informed decisions.