Problem 15 - Calculus for Biology and Medicine

Question

Find the corresponding compartment diagram for each system of differential equations. \(\frac{d x_{1}}{d t}=-1.2 x_{1}\) \(\frac{d x_{2}}{d t}=0.3 x_{1}-0.2 x_{2}\)

Step-by-Step Solution

Answer

The compartment diagram consists of two compartments; \(x_1\) has an outflow rate of 1.2, \(x_1\) transfers to \(x_2\) at rate 0.3, and \(x_2\) has an outflow rate of 0.2.

Step 1: Understand Compartment Models

Compartment models are used to describe how substances are transferred between different parts (compartments) of a system over time. Each compartment represents a part of the system that has the same concentration or amount of a substance.

Step 2: Identify the Variables and Their Changes

The system of differential equations consists of two variables, \(x_1\) and \(x_2\). \(\frac{d x_{1}}{d t}=-1.2 x_{1}\) describes the change in compartment 1. There is no input, as indicated by the absence of other terms; the only change is a removal (with rate 1.2) from compartment 1. \(\frac{d x_{2}}{d t}=0.3 x_{1}-0.2 x_{2}\) shows the change in compartment 2: it gains from compartment 1 at a rate of 0.3 and loses at a rate of 0.2.

Step 3: Draw the Diagram

In the compartmental diagram, draw two boxes labeled \(x_1\) and \(x_2\) representing each compartment. For \(x_1\), draw an arrow leaving the box to indicate outflow with rate 1.2. Draw an arrow from \(x_1\) pointing towards \(x_2\) to represent the transfer with rate 0.3. Lastly, draw an arrow leaving \(x_2\) to represent the outflow from compartment 2 with rate 0.2.

Key Concepts

Compartment Models

Imagine dividing a system into multiple parts, called compartments, each of which can hold a substance. Compartment models are a mathematical representation used to describe the movement and transformation of substances within a system over time. These models are crucial in areas such as pharmacokinetics, ecology, and chemical engineering.

Each compartment can be thought of as a distinct segment that has its own concentration of the substance. The interactions between these compartments are often governed by rates, which can indicate how fast the substance is being transferred from one compartment to another or how quickly it is leaving a compartment altogether. Think of compartments as watertight containers connected by pipes. The rates at which fluid flows through these pipes can vary, and they dictate the changes occurring in the compartments.

By understanding the rates and connections between compartments, we can predict how the amount of substance will evolve over time across the entire system. This understanding helps in formulating solutions to real-world problems where distribution and conversion processes are involved.

System of Differential Equations

A system of differential equations is a set of mathematical equations that describe how variables change in relation to each other over time. This system is often used when variables are interconnected, meaning the change in one affects the change in another.

Let's explore the system given in the exercise:

• The equation \(\frac{d x_{1}}{d t} = -1.2 x_{1}\) illustrates how the quantity in compartment 1 decreases at a rate proportional to its current value. The negative sign indicates a loss, showing that this substance is being removed from compartment 1.
• The equation \(\frac{d x_{2}}{d t} = 0.3 x_{1} - 0.2 x_{2}\) describes compartment 2. Here, there is an interaction: compartment 2 gains substance from compartment 1 at a rate of 0.3 and loses it at a rate of 0.2. This adds a layer of complexity as both gain and loss occur.

By using differential equations, we can model dynamic systems and predict their future states, given their current conditions. This is powerful for understanding complex systems in various scientific and engineering fields.

Compartmental Diagram

A compartmental diagram is a visual representation of a compartment model. It helps in understanding and interpreting the movement of substances within the compartments.

In the diagram, each compartment is typically depicted as a box, and the flows between them, such as transfers or losses, are shown as arrows. These arrows have labels that indicate the rate of flow.

Consider the diagram for the given system of differential equations:

• Draw a box labeled \(x_1\) to represent compartment 1.
• From \(x_1\), draw an arrow pointed outward, labeled with the rate 1.2, to show the outflow or removal from this compartment.
• Next, draw a box labeled \(x_2\) for compartment 2, and connect \(x_1\) to \(x_2\) with an arrow labeled 0.3 to indicate the transfer from compartment 1 to 2.
• Finally, from \(x_2\), draw another arrow pointing outward, labeled with the rate 0.2, indicating the outflow from compartment 2.

This visual tool provides a clear and intuitive way to see how substances move within the system, making it easier to predict behavior and changes over time.