Problem 32 - Thomas Calculus

Question

Drug concentration An antibiotic is administered intravenously into the bloodstream at a constant rate \(r .\) As the drug flows through the patient's system and acts on the infection that is present, it is removed from the bloodstream at a rate proportional to the amount in the bloodstream at that time. Since the amount of blood in the patient is constant, this means that the concentration \(y=y(t)\) of the antibiotic in the bloodstream can be modeled by the differential equation $$\frac{d y}{d t}=r-k y, \quad k>0$$ \begin{equation} \begin{array}{l}{\text { a. If } y(0)=y_{0}, \text { find the concentration } y(t) \text { at any time } t} \\ {\text { b. Assume that } y_{0}<(r / k) \text { and find } \lim _{y \rightarrow \infty} y(t) . \text { Sketch the }} \\\ {\text { solution curve for the concentration. }}\end{array} \end{equation}

Step-by-Step Solution

Answer

The concentration at any time is \( y(t) = \frac{r}{k} + (y_0 - \frac{r}{k})e^{-kt} \), and it approaches \( \frac{r}{k} \) as \( t \rightarrow \infty \).

Step 1: Set up the differential equation

We start with the given differential equation \( \frac{dy}{dt} = r - ky \). This describes the rate of change of the concentration \( y(t) \) with respect to time \( t \). \( r \) is the constant rate the drug is added, while \( ky \) represents the removal rate, proportional to the concentration present.

Step 2: Solve the differential equation

We rearrange the equation to separate variables: \( \frac{dy}{r - ky} = dt \). Integrating both sides, we get \( \int \frac{1}{r - ky} \, dy = \int dt \). This integration gives us \( -\frac{1}{k} \ln|r - ky| = t + C \), where \( C \) is the constant of integration. Solving for \( y \), we find \( y(t) = \frac{r}{k} + (y_0 - \frac{r}{k})e^{-kt} \).

Step 3: Apply initial condition

Using the initial condition \( y(0) = y_0 \), we have \( y(0) = \frac{r}{k} + (y_0 - \frac{r}{k})e^{0} = y_0 \). This confirms the derived formula. Thus, the expression for \( y(t) \) is \( y(t) = \frac{r}{k} + (y_0 - \frac{r}{k})e^{-kt} \).

Step 4: Analyze behavior as \( t \rightarrow \infty \)

Assuming \( y_0 < \frac{r}{k} \), as \( t \rightarrow \infty \), the exponential term \( e^{-kt} \rightarrow 0 \). Therefore, \( \lim_{t \rightarrow \infty} y(t) = \frac{r}{k} \). This shows that the concentration approaches \( \frac{r}{k} \) over time.

Step 5: Sketch the solution curve

The solution curve starts at \( y_0 \), increases or decreases depending on whether \( y_0 \) is less than or greater than \( \frac{r}{k} \), and will asymptotically approach \( \frac{r}{k} \). The curve is exponential with a horizontal asymptote at \( \frac{r}{k} \).

Key Concepts

Drug Concentration Model

Understanding how drugs behave in the body is essential, especially for ensuring they're effective and safe. In this model, we're looking at the concentration of an antibiotic in the bloodstream over time, known as the drug concentration model. This model helps predict how the level of medication changes, which is key to managing dosages.

The main equation here is a differential equation, which mathematically describes how quantities change. For the drug concentration, the equation is \(\frac{dy}{dt} = r - ky\). Here, \(r\) represents how fast the drug is added to the system (the administration rate), while \(ky\) indicates how the drug is removed, which depends on how much drug is currently in the bloodstream. This setup captures the balance of adding and subtracting the drug concentration over time.

Exponential Growth and Decay

This model shows us a combination of exponential growth and decay. The addition of the drug at a constant rate attempts to increase the concentration, while the removal of the drug at a rate proportional to its current concentration attempts to decrease it.

When you solve the equation \(\frac{dy}{dt} = r - ky\), you'll notice that the solution \(y(t) = \frac{r}{k} + (y_0 - \frac{r}{k})e^{-kt}\) involves an exponential decay term \(e^{-kt}\). This represents how the drug concentration decreases over time if no additional drug is added.

The solution stabilizes over time because the exponential term diminishes, and the concentration approaches a steady-state level, which is the amount that balances the addition and removal processes.

Solution Curve Analysis

To really understand how the drug concentration changes, we analyze its solution curve. This curve illustrates how \(y(t)\), the concentration of the drug, changes as time \(t\) progresses.

In the case where \(y_0 < \frac{r}{k}\), the initial concentration is below the steady-state value. The solution curve will start lower, gradually rise, and level off near \(\frac{r}{k}\) as \(t\) goes toward infinity. If \(y_0 > \frac{r}{k}\), the curve will decrease and approach \(\frac{r}{k}\). If \(y_0 = \frac{r}{k}\), the curve is a flat line since the system is already in equilibrium.

The key points about the solution curve are:

• The starting point is determined by the initial condition \(y_0\).
• The curve asymptotically approaches \(\frac{r}{k}\).
• Its shape is dictated by the exponential decay term \(e^{-kt}\).

Initial Conditions in Differential Equations

Initial conditions are crucial in differential equations because they help pinpoint the specific solution that fits a real-life situation. In our drug concentration model, we use the initial condition \(y(0) = y_0\), indicating the drug concentration at the start.

This condition allows us to solve the differential equation uniquely and determine how the drug's concentration changes from the initial state. It ensures that our mathematical description of \(y(t)\) takes into account the starting scenario of the system.

Initial conditions are essential when we want predictive accuracy. They ensure that the model adheres to the practical aspects of the problem, such as how much drug was present at the start of administration. This backdrop builds a realistic understanding of how systems evolve over time based on where they begin.