Problem 84 - Calculus for Biology and Medicine

Question

Suppose that a drug is eliminated so slowly from the blood that its elimination kinetics can be essentially ignored. Then according to Section \(5.9\) the total amount of drug in the blood is given by a differential equation: $$ \frac{d M}{d t}=A(t) $$ where \(A(t)\) is the rate of absorption. We will show in Chapter 8 that if the drug is absorbed into the blood from a pill in the patient's gut, then \(A(t)\) is given by a function $$ A(t)=C e^{-k t} $$ where \(C>0\) and \(k>0\) are constants that depend on the type of the drug being administered. Assume that at \(t=0\) there is no drug present in the patient's blood (i.e., \(M(0)=0\) ). Solve this initial value problem, and, using the methods from Section \(5.6\), sketch the graph of \(M(t)\) against \(t\).

Step-by-Step Solution

Answer

The solution is \( M(t) = \frac{C}{k}(1 - e^{-kt}) \) and it approaches \( \frac{C}{k} \) as \( t \to \infty \).

Step 1: Express the Differential Equation

We begin by expressing the differential equation as given: \( \frac{dM}{dt} = A(t) = Ce^{-kt} \).

Step 2: Integrate the Differential Equation

To find the solution to this differential equation, we need to integrate both sides. The integral of the left side is \( \int \frac{dM}{dt} \, dt = M(t) \) and the integral of the right side is \( \int Ce^{-kt} \, dt \).

Step 3: Compute the Integral

Performing the integration on the right side, we use the formula \( \int e^{-kt} \, dt = -\frac{1}{k}e^{-kt} + C_1 \). Thus, the integral becomes \( M(t) = -\frac{C}{k}e^{-kt} + C_1 \).

Step 4: Apply Initial Condition

We know from the problem statement that \( M(0) = 0 \). Substituting into the equation gives: \( 0 = -\frac{C}{k}e^{0} + C_1 \) which simplifies to \( 0 = -\frac{C}{k} + C_1 \). Solving for \( C_1 \) gives \( C_1 = \frac{C}{k} \).

Step 5: Write the General Solution

Substituting \( C_1 \) back into the equation for \( M(t) \), we have \( M(t) = -\frac{C}{k}e^{-kt} + \frac{C}{k} \). This simplifies to \( M(t) = \frac{C}{k}(1 - e^{-kt}) \).

Step 6: Sketch the Graph

The graph of \( M(t) = \frac{C}{k}(1 - e^{-kt}) \) against \( t \) starts at zero and approaches \( \frac{C}{k} \) as \( t \) becomes very large. This is an exponential growth curve, leveling off at \( \frac{C}{k} \).

Key Concepts

Drug Absorption in Differential Equations

Drug absorption in the bloodstream is a complex but fascinating topic. When we talk about drug absorption, particularly in the context of differential equations, we are often interested in understanding how a drug enters the blood over time and affects the body. Scientists model this process using mathematical functions. In this case, when a drug is absorbed at a rate given by the function \( A(t) = Ce^{-kt} \), it describes how quickly the drug concentration changes. Both \( C \) and \( k \) are constants that are crucial in determining the specific absorption rate for different drugs.
The exponential term \( e^{-kt} \) shows how the absorption rate decreases exponentially over time, simulating the behavior of many actual drugs. This model gives us a clear mathematical representation of how drugs enter the bloodstream, offering valuable insights for medical applications, including dosing schedules and delivery methods.

Understanding Initial Value Problems

An initial value problem is a way of describing a system governed by a differential equation with known values at the start. In this exercise, the differential equation \( \frac{dM}{dt} = Ce^{-kt} \) models the drug absorption, and \( M(0) = 0 \) is our initial condition. This tells us that at time zero, there is no drug in the bloodstream.
The goal of solving an initial value problem is to find a specific function that satisfies both the differential equation and initial condition. By integrating the differential equation and applying the initial condition, we work step-by-step to find the exact behavior of the system over time. This allows us to predict how the drug amount will change, which is essential in ensuring that patients receive the correct dosage.

Exponential Functions in Modeling

Exponential functions, like \( A(t) = Ce^{-kt} \), are fundamental in modeling decay processes, such as drug absorption. These functions feature specific characteristics that make them ideal for representing decreasing phenomena. Here’s why:

• The base of the natural logarithm, \( e \), is used in these functions, which allows us to describe continuous growth and decay.
• The negative exponent \(-kt\) indicates an exponential decay process, which is typical for how a drug's absorption rate decreases over time.

By analyzing the function \( A(t) = Ce^{-kt} \), we can see how the absorption rate diminishes. The constant \( k \) controls the speed of this decay; higher values mean a quicker decrease. Together, exponential functions offer a powerful tool for depicting real-world scenarios like drug absorption, making it easier for scientists and clinicians to understand and predict the effects of medication.