Problem 55 - Calculus and its applications

Question

Average drug dose. The concentration, \(C(t),\) of phenylbutazone, in micrograms per milliliter \((\mu g / m L)\), in the plasma of a calf injected with this anti-inflammatory agent is approximately $$ C(t)=42.03 e^{-0.01050 t} $$ where \(t\) is the number of hours after the injection and \(0 \leq t \leq 120 .\) (Source: A. K. Arifah and P. Lees, "Pharmacodynamics and Pharmacokinetics of Phenylbutazone in Calves," Journal of Veterinary Pharmacology and Therapeutics, Vol. \(25,299-309\) (2002).) a) Given that this model is accurate for \(0 \leq t \leq 120\), what is the initial dosage? b) What is the average amount of phenylbutazone in the calf's body for the time between 10 and 120 hours?

Step-by-Step Solution

Answer

a) Initial dosage: 42.03 μg/mL. b) Average concentration (10 to 120 hours): 22.45 μg/mL.

Step 1: Finding the Initial Dosage Step

To find the initial concentration (dosage) of phenylbutazone in the calf's plasma, we need to evaluate the concentration function \(C(t)\) at the start of the given range. This is done by plugging in \(t = 0\) into the function \(C(t) = 42.03 e^{-0.01050 t}\). Thus, \(C(0) = 42.03 e^{0} = 42.03\, \mu g / m L\). This represents the initial dosage.

Step 2: Understanding Average Concentration Step

The average concentration of phenylbutazone in the calf between 10 and 120 hours can be calculated using the formula for the average value of a continuous function over an interval \([a, b]\). This formula is \(\frac{1}{b-a} \int_{a}^{b} C(t)\, dt\). Here, \(a = 10\) and \(b = 120\). This means we will evaluate and integrate the concentration function from 10 to 120 hours.

Step 3: Set Up Integral for Average Concentration Calculation Step

First, set up the definite integral \(\int_{10}^{120} 42.03 e^{-0.01050 t} \, dt\). We will evaluate this integral to find the total concentration over 10 to 120 hours. The average will be this integral divided by \( (120 - 10) = 110 \).

Step 4: Calculating the Definite Integral Step

The integration of the function \(42.03 e^{-0.01050 t}\) involves a basic u-substitution, where \(u = -0.01050 t\) making \(du = -0.01050 \, dt\). Thus, the integral becomes \(42.03 \int e^{u} \left(-\frac{1}{0.01050}\right) \, du\). This simplifies to \(-\frac{42.03}{0.01050} e^{-0.01050 t}\). Evaluate this from 10 to 120.

Step 5: Evaluating the Integral and Calculating the Average Step

Now, substitute and evaluate the integral: \[-\frac{42.03}{0.01050} [e^{-0.01050 \times 120} - e^{-0.01050 \times 10}]\]Calculate the numerical values and the average concentration: \[= -4002.857 (e^{-1.26} - e^{-0.105})\] \[= 4002.857 (0.283 - 0.900)\] \[= 4002.857 \times (-0.617)\]Calculate this to find the total plasma concentration, and divide by 110 to find the average.

Step 6: Final Calculation of Average Concentration Step

Calculate the numerical value from the integral:\[= -4002.857 \times (-0.617) \approx 2470 \,\mu g \, h / mL\]Now, divide by 110 hours to find the average concentration:\[\text{Average Concentration} = \frac{2470}{110} \approx 22.45 \,\mu g / mL\]

Key Concepts

Exponential Decay

Exponential decay is a process where a quantity decreases at a rate proportional to its current value. It's widely used in calculus applications to model phenomena like radioactive decay, population decline, and in the context of our exercise, drug concentration over time. The mathematical model for exponential decay is expressed as \[ C(t) = C_0 \cdot e^{-kt} \]where:

• \( C(t) \) is the concentration at time \( t \)
• \( C_0 \) is the initial concentration
• \( k \) is the decay constant

In the exercise, the given formula \( C(t) = 42.03 e^{-0.01050 t} \) represents the decay of drug concentration in a calf's plasma. As time progresses, the concentration diminishes due to the exponent \(-0.01050t\). This exponential function captures the natural process by which chemical substances break down or dissipate in an organism.

Definite Integral

The definite integral is a concept in calculus used to determine the accumulation of quantities, which is particularly helpful in calculating areas under curves. It is represented as:\[ \int_{a}^{b} f(x) \, dx \]In the context of pharmacokinetics, the definite integral can be used to find the total concentration of a drug over a specified period. For example, to find the total concentration of phenylbutazone from 10 to 120 hours, we evaluate the integral:\[ \int_{10}^{120} 42.03 e^{-0.01050 t} \, dt \]This equation calculates the accumulated amount of drug in the system, accounting for its decay over the given interval. After integrating, the result gives the total microgram-hours per milliliter over the time period.

Average Value of a Function

The average value of a function over an interval gives a single value that represents the typical height of the function on that interval. It is calculated by taking the definite integral of the function over the interval and dividing by the range of the interval. The formula is:\[ \text{Average} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx \]In the scenario with phenylbutazone, the average concentration of the drug from 10 to 120 hours is found using:\[ \frac{1}{120 - 10} \int_{10}^{120} 42.03 e^{-0.01050 t} \, dt \]By evaluating this, we can determine the typical concentration of the drug in the calf's plasma over that period. This numerical average provides a useful summary for understanding how the drug remains present in the bloodstream.

Pharmacokinetics

Pharmacokinetics is the branch of pharmacology dedicated to determining the fate of substances that enter an organism. This involves studying the time course of drug absorption, distribution, metabolism, and excretion. In this exercise, the focus is on the drug phenylbutazone in calves.

The concentration model \( C(t) = 42.03 e^{-0.01050 t} \) describes how the drug disperses and diminishes over time post-injection. Pharmacokinetics provides important insights into:

• How quickly a drug takes effect
• The duration of its action
• The time it takes for a drug to decrease to a safe or non-effective level

These insights ensure proper dosage regimens and help in predicting how interventions may affect the overall drug action within the body. Understanding these aspects is crucial for optimizing therapeutic strategies.