Problem 49 - Calculus and its applications

Question

Suppose an oral dose of a drug is taken. Over time, the drug is assimilated in the body and excreted through the urine. The total amount of the drug that has passed through the body in T hours is given by $$ \int_{0}^{T} E(t) d t $$ where \(\mathrm{E}(t)\) is the rate of excretion of the drug. A typical rate-of- excretion function is \(\mathrm{E}(t)=t e^{-k t},\) where \(k>0\) and \(t\) is the time, in hours. Use this information. Find \(\int_{0}^{\infty} E(t) d t,\) and interpret the answer. That is, what does the integral represent?

Step-by-Step Solution

Answer

The integral \( \int_{0}^{\infty} t e^{-k t} \, dt = \frac{1}{k^2} \) represents the total drug excretion over time.

Step 1: Writing the Integral

We begin by writing the integral for the excretion rate function over an infinite time period. The expression becomes: \[ \int_{0}^{\infty} t e^{-k t} \, dt \] where \(E(t) = t e^{-k t}\).

Step 2: Integration by Parts Setup

To solve the integral, we'll apply the integration by parts formula: \[ \int u \, dv = uv - \int v \, du. \] We choose \( u = t \) and \( dv = e^{-kt} \, dt \). Thus, \( du = dt \) and \( v = -\frac{1}{k} e^{-kt} \).

Step 3: Apply Integration by Parts

Using the integration by parts formula: \[ \int_{0}^{\infty} t e^{-kt} dt = \left[ -\frac{t}{k} e^{-kt} \right]_{0}^{\infty} + \frac{1}{k} \int_{0}^{\infty} e^{-kt} \, dt. \]

Step 4: Evaluate the Boundary Terms

The first term evaluates as follows: \[ \lim_{T \to \infty} -\frac{T}{k} e^{-kT} = 0 \] since \(e^{-kT}\) approaches 0 faster than \(T\) grows, and for \(t = 0\), the expression is 0. Thus, \[ \left[ -\frac{t}{k} e^{-kt} \right]_{0}^{\infty} = 0. \]

Step 5: Evaluate the Remaining Integral

The remaining integral is: \[ \frac{1}{k} \int_{0}^{\infty} e^{-kt} \, dt = \frac{1}{k} \left[ -\frac{1}{k} e^{-kt} \right]_{0}^{\infty} = \frac{1}{k} \left( 0 - \left(-\frac{1}{k} \right) \right) = \frac{1}{k^2}. \]

Step 6: Final Result

Combining results gives the full integral evaluation as: \[ \int_{0}^{\infty} t e^{-kt} \, dt = \frac{1}{k^2}. \] This value represents the total amount of the drug excreted over an infinite time span across the entire administration period.

Key Concepts

Excretion Rate Function

When we talk about the excretion rate function in the context of pharmacokinetics, we are essentially describing how a drug exits the body over time. The function \(E(t) = t e^{-kt}\) is a model that represents the rate at which a drug is excreted through urine. In this function, \(t\) stands for time in hours, and \(k\) is a constant that reflects the rate of excretion. The function \(e^{-kt}\) represents exponential decay; it's what causes the excretion rate to decrease over time as the drug is gradually removed from the body.

The product \(t e^{-kt}\) suggests that initially, the rate increases because of the \(t\) term, peaking at a certain time, after which the drug's removal ramps down. This behavior is typical for many substances, where initially, there is rapid excretion that slows as the concentration of the drug decreases.

Definite Integration

Definite integration is a powerful tool in calculus used to find the total or accumulated value from a rate of change. In this exercise, definite integration helps us calculate the total amount of drug excreted from the body over an infinitely long period. This is expressed through the integral \(\int_{0}^{\infty} E(t) \, dt\).

By setting the limits from 0 to \(\infty\), we aim to capture the entire span of the drug's presence in the system. This means looking at not just the initial stages of excretion but observing how the body continues to excrete the drug over time till it's fully cleared out. The solution of \(\frac{1}{k^2}\) as the result of this definite integration shows us how much of the drug has been thoroughly processed in the long term. This outcome is vital for understanding drug disposition in a clinical and pharmacological perspective.

Pharmacokinetics

Pharmacokinetics involves studying how drugs move within the body from the point of administration, through absorption, distribution, metabolism, and finally, excretion. This exercise particularly focuses on the excretion phase, using the integration by parts technique to solve an integral that symbolizes this process.

There are several processes and equations involved in pharmacokinetics, but the heart of the concept is understanding how drugs are removed from the system, which is crucial for determining dosage and frequency. By calculating the integral of the excretion rate function over an indefinite period, healthcare providers can predict how much of a drug remains in the body at any given time. This information is critical for ensuring the drug's safety and effectiveness, preventing overdose, and managing side effects. Thus, such calculations form a backbone for making informed decisions in drug therapy management.