Problem 33 - Biocalculus Calculus for the Life Sciences

Question

Salicylic acid pharmacokinetics In the article cited in Example 5 the authors also studied the formation and concentration of salicylic acid in the bloodstream of 10 volunteers. A model for the concentration is $$C(t)=11.4 t e^{-t}$$ where \(t\) is measured in hours and \(C\) in \(\mu \mathrm{g} / \mathrm{mL} .\) Calculate \(\int_{0}^{4} C(t) d t\) and include the units in your answer.

Step-by-Step Solution

Answer

7.924 \(\mu\text{g mL}^{-1}\).

Step 1: Understand the problem

The given problem requires you to find the definite integral of the function \(C(t) = 11.4 t e^{-t}\) from \(t = 0\) to \(t = 4\). This integral represents the total amount of salicylic acid in the bloodstream within these 4 hours.

Step 2: Setup the integral

The definite integral to solve is \(\int_{0}^{4} 11.4 t e^{-t} \, dt\). This represents calculating the area under the curve of \(C(t)\) between \(t = 0\) and \(t = 4\).

Step 3: Solve using integration by parts

Integration by parts is needed here. Set \(u = 11.4t\) and \(dv = e^{-t} dt\). Then, \(du = 11.4 \, dt\) and \(v = -e^{-t}\). The integration by parts formula is \(\int u \, dv = uv - \int v \, du\).

Step 4: Apply the integration by parts formula

Substitute \(u\), \(du\), \(v\), and \(dv\) into the formula: \(11.4t(-e^{-t}) - \int -e^{-t}(11.4) \, dt\), which simplifies to \(-11.4t e^{-t} + 11.4 \int e^{-t} \, dt\).

Step 5: Integrate the remaining function

Integrate \(\int e^{-t} \, dt\) to get \(-e^{-t}\). Substitute back into the equation to find the full antiderivative: \(-11.4t e^{-t} - 11.4(-e^{-t})\), simplifying to \(-11.4t e^{-t} + 11.4 e^{-t}\).

Step 6: Evaluate the definite integral

Now compute \([-11.4t e^{-t} + 11.4 e^{-t}]\) from 0 to 4. Substitute \(t = 4\) and \(t = 0\) into the expression and compute the difference.

Step 7: Calculate the final values

Compute \(-11.4 \times 4 \times e^{-4} + 11.4 \times e^{-4}\) and \(-11.4 \times 0 \times e^{0} + 11.4 \times e^{0}\). Simplify both to \(-45.6 e^{-4} + 11.4 e^{-4}\) and \(11.4\), then the result is approximately \(11.4 (1 - e^{-4}) - 45.6 e^{-4}\).

Step 8: Simplify and include units

The result, when calculated, evaluates to approximately 7.924 \(\mu \text{g mL}^{-1}\). This is the total concentration of salicylic acid in the bloodstream from \(t = 0\) to \(t = 4\) hours.

Key Concepts

Definite Integral

A definite integral is a fundamental concept in calculus used to calculate the total accumulation of a quantity. In the context of this problem, it helps determine the total amount of salicylic acid in the bloodstream over a specified time period.

The goal is to compute the definite integral of the function \(C(t) = 11.4t e^{-t}\) from \(t=0\) to \(t=4\). This process is akin to finding the area under the curve of the concentration function within those time limits.

Definite integrals are bounded by two limits, which, in this case, are the beginning and end times of interest (0 to 4 hours). The solution \(\int_{0}^{4} C(t) \, dt\) provides a numerical value, representing how much of a substance is present over that time frame. The outcome of integrating over this interval gives insight into the total mass or concentration of a compound, measured in micrograms per milliliter (\(\mu g/mL\)) in this scenario.

Integration by Parts

Integration by parts is a technique derived from the product rule for differentiation, useful for solving integrals involving products of functions. In the given problem, the concentration function \(C(t) = 11.4t e^{-t}\) is a product of two functions: a polynomial \(11.4t\) and an exponential function \(e^{-t}\).

To use integration by parts, identify components of the product as \(u = 11.4t\) and \(dv = e^{-t} dt\). From this setup:

• The derivative \(du = 11.4 \, dt\)
• The integral of \(dv\), which is \(v = -e^{-t}\)

Integration by parts follows the formula: \(\int u \, dv = uv - \int v \, du\). By substituting in our identified parts, calculate the integral by evaluating \(-11.4t e^{-t} - 11.4 \int e^{-t} \, dt\), leading to the simplified integration outcome.

This technique effectively simplifies calculation when dealing with complex integrals formed by the product of functions, making it a powerful method in solving problems in pharmacokinetics and beyond.

Concentration Model

A concentration model in pharmacokinetics describes how the concentration of a drug or compound varies within the bloodstream over time. The model allows for predictions and analysis of drug behavior in the body, crucial for understanding dosing and therapeutic effects.

In this instance, the concentration model is defined by \(C(t) = 11.4t e^{-t}\), where \(t\) is the time in hours and \(C(t)\) is the concentration in \(\mu g/mL\). This specific model considers how the concentration initially increases—due to the term \(t\)—and then decreases exponentially due to \(e^{-t}\), reflecting biological elimination processes.

Using this model, integration enables the calculation of total concentration over a period, providing a quantitative value for overall exposure to the compound. This helps in determining appropriate dosage and understanding the drug's efficacy and safety over time, aligning with the objectives of pharmacokinetic studies.