Problem 3 - Applied Calculus

Question

If \(t\) is in hours, the drug concentration curve for a drug is given by \(C=17.2 t e^{-0.4 t} \mathrm{ng} / \mathrm{ml} .\) The minimum effective concentration is \(10 \mathrm{ng} / \mathrm{ml}\). (a) If the second dose of the drug is to be administered when the first dose becomes incffective, when should the second dose be given? (b) If you want the onset of effectiveness of the second dose to coincide with termination of effectiveness of the first dose, when should the second dose be given?

Step-by-Step Solution

Answer

The second dose should be given at approximately 3.57 hours for both parts (a) and (b).

Step 1: Identify the point of ineffectiveness for part (a)

The drug concentration becomes ineffective when it drops to 10 ng/ml. Thus, we set the equation \(17.2 t e^{-0.4 t} = 10\) to find the time \(t\) when the concentration falls to this level.

Step 2: Solve for t with the equation for ineffectiveness

To solve \(17.2 t e^{-0.4 t} = 10\), start by dividing both sides by 17.2, giving \(t e^{-0.4 t} = \frac{10}{17.2}\). Simplify to get \(t e^{-0.4 t} \approx 0.581395\). Use numerical methods or a calculator to find \(t \approx 3.57\ hours\).

Step 3: Determine when to give the second dose

The second dose should be given at approximately \(t \approx 3.57\ hours\), as this is when the concentration from the first dose falls to 10 ng/ml, becoming ineffective.

Step 4: Confirm coinciding effectiveness for part (b)

To have the onset of the second dose coincide with termination of the first dose's effectiveness, administer the second dose at the time the first dose becomes ineffective. This is the same condition as found in part (a), occurring at \(t \approx 3.57\ hours\).

Key Concepts

Ineffectiveness Point

The ineffectiveness point is a critical moment when the drug concentration in the body decreases to a level where it stops being effective. For this exercise, the drug is considered ineffective when its concentration falls to 10 ng/ml. This threshold is essential for timing the administration of subsequent doses to ensure continuous medication efficacy. Identifying this point involves solving the concentration equation for when it equals the minimum effective concentration. It acts as a cue for scheduling the next dose, keeping the patient's treatment both effective and safe.

Concentration Equation

The concentration equation used in this exercise is a mathematical representation of how the concentration of a drug varies over time. The equation given is \(C = 17.2 t e^{-0.4t}\). Here, \(t\) is the time in hours, and \(e^{-0.4t}\) represents the exponential decay of the drug concentration. The term \(17.2t\) describes the initial increase in drug concentration due to absorption or initial distribution in the body.

When evaluating this equation to find the ineffectiveness point, we set \(C\) to 10 ng/ml, the minimum effective concentration, and solve for \(t\). This yields a nonlinear equation, which often requires numerical methods due to the presence of the exponential term. It's a key component in understanding dynamics of medication in clinical practices.

Numerical Solution

Finding the solution to the concentration equation analytically can be challenging due to its complexity. For the given equation \(17.2 t e^{-0.4 t} = 10\), we perform a numerical solution to estimate the point \(t\) when the drug becomes ineffective. Initially, simplify the equation to get \(t e^{-0.4t} \approx 0.581395\).

At this stage, using numerical methods is essential. Tools like graphing calculators or computational software can approximate solutions where traditional algebraic methods may fail. In this context, numerical approximation yields \(t \approx 3.57\ hours\). This estimation is pivotal for planning medication schedules, ensuring that the patient receives the next dose as soon as the first dose loses effectiveness.