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Chapter 6: Problem 60

BIOMEDICAL: Drug Absorption To determine how much of a drug is absorbed into the body, researchers measure the difference between the dosage \(D\) and the amount of the drug excreted from the body. The total amount excreted is found by integrating the excretion rate \(r(t)\) from 0 to \(\infty\). Therefore, the amount of the drug absorbed by the body is $$ D-\int_{0}^{\infty} r(t) d t $$ If the initial dose is \(D=200\) milligrams \((\mathrm{mg})\), and the excretion rate is \(r(t)=40 e^{-0.5 t} \mathrm{mg}\) per hour, find the amount of the drug absorbed by the body.

Short Answer

Expert verified
The amount of the drug absorbed by the body is 120 mg.

Step by step solution

01

Recognize the Problem Elements

The problem involves calculating how much of a drug is absorbed into the body based on the initial dosage and excretion rate over time. You are given that the total amount excreted is found by integrating the excretion rate from 0 to infinity.
02

Set Up the Integral for Excretion

Given the excretion rate is \( r(t) = 40e^{-0.5t} \) milligrams per hour, you need to integrate this function from 0 to infinity to find the total amount of drug excreted. The integral is: \[ \int_{0}^{\infty} 40e^{-0.5t} \, dt \]
03

Solve the Integral

To evaluate the integral \( \int_{0}^{\infty} 40e^{-0.5t} \, dt \), we start by finding the antiderivative of \( 40e^{-0.5t} \). The integral becomes: \[ \int 40e^{-0.5t} \, dt = \frac{40}{-0.5}e^{-0.5t} + C = -80e^{-0.5t} + C \] Evaluate from 0 to infinity: \[ [-80e^{-0.5t}]_{0}^{\infty} \] At infinity, \( e^{-0.5t} \) becomes 0, and at zero, \( e^{-0.5 \times 0} = 1 \). Therefore, the result is: \(-80(0) - (-80)(1) = 80 \) mg. This is the total excreted amount.
04

Calculate the Absorbed Amount

Subtract the excreted amount from the initial dosage to find the absorbed amount using the formula: \[ D - \int_{0}^{\infty} r(t) \, dt = 200 - 80 \] Therefore, the absorbed amount is \( 120 \) mg.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integral Calculus and Drug Absorption
The process of determining how much of a drug is absorbed by the body involves integral calculus. Integral calculus allows us to calculate the area under a curve, which in this case represents the drug's excretion rate over time. By integrating the excretion rate, researchers can determine the total amount of the drug excreted by the body. This involves solving an improper integral, as the excretion rate function is defined from time zero to infinity. To illustrate this with our specific example, we are given the excretion rate function: \[ r(t) = 40e^{-0.5t} \] By integrating this function from 0 to \( \infty \), we can find the total amount of the drug excreted over time. This integral gives insight into the dynamics of how the drug leaves the body, which is crucial for understanding the overall absorption process.
Understanding Excretion Rate
The excretion rate of a drug is the rate at which it is eliminated from the body over time. It usually decreases as time progresses, often following an exponential decay pattern. In our example, the excretion rate is defined as: \[ r(t) = 40e^{-0.5t} \] This equation tells us several things:
  • The initial rate of excretion at \( t = 0 \) is 40 mg per hour.
  • The rate decreases over time, due to the \( e^{-0.5t} \) term, which models the exponential decay.
To calculate the total excretion, we integrate this function over all time (from 0 to \( \infty \)). The result, 80 mg, is the total drug excreted. This information is vital in calculating the absorbed drug amount, reflecting the body's retention.
Biomedical Applications of Integral Calculus
Integral calculus is invaluable in biomedical fields, particularly in pharmacokinetics, where understanding drug absorption and excretion is critical. By modeling drug behavior through equations and integrals, researchers can predict how a drug behaves in the body. This helps in dosing decisions and ensuring therapeutic efficacy. Some key applications include:
  • Calculating the total absorbed dose of a drug after accounting for excretion.
  • Designing sustained release formulations that ensure drugs are released over a desired period.
  • Understanding the concentration-time profile of drugs within the bloodstream.
In our specific problem, using integral calculus to find the absorbed drug amount allows for a better interpretation of how much of the initial 200 mg dose remains effective in the patient's body. This is crucial for effective medical treatment and personalized medicine approaches.

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