Problem 20 - Applied Calculus

Question

Cephalexin is an antibiotic with a half-life in the body of \(0.9\) hours, taken in tablets of \(250 \mathrm{mg}\) every six hours. (a) What percentage of the cephalexin in the body at the start of a six-hour period is still there at the end (assuming no tablets are taken during that time)? (b) Write an expression for \(Q_{1}, Q_{2}, Q_{3}, Q_{4}\), where \(Q_{n}\) \(\mathrm{mg}\) is the amount of cephalexin in the body right after the \(n^{\text {th }}\) tablet is taken. (c) Express \(Q_{3}, Q_{4}\) in closed form and evaluate them. (d) Write an expression for \(Q_{n}\) and put it in closed form. (e) If the patient keeps taking the tablets, use your answer to part (d) to find the quantity of cephalexin in the body in the long run, right after taking a tablet.

Step-by-Step Solution

Answer

(a) Approximately 1.56% remains. (b) Recursive relation: \( Q_n = R Q_{n-1} + 250 \). (c) \( Q_3 \approx 251 \) mg, \( Q_4 \approx 251 \) mg. (d) \( Q_n = A R^n + B \). (e) Limit is \( 255.81 \) mg.

Step 1: Calculate Decay Rate

The half-life of cephalexin is 0.9 hours. The decay formula is given by \( N(t) = N_0 \times 2^{-t/T_{1/2}} \), where \( T_{1/2} \) is the half-life, \( t \) is time, and \( N_0 \) is the initial quantity. In 6 hours, the decay rate is \( 2^{-6/0.9} \). Calculate this value to find the percentage remaining.

Step 2: Derive Expression for Q_n

Each 250 mg tablet adds to the amount left from the previous period. If \( R = 2^{-6/0.9} \) is the fraction remaining after 6 hours, then after the first tablet \( Q_1 = 250 \) mg, \( Q_2 = R Q_1 + 250 = R \times 250 + 250 \), and so on. Hence, \( Q_n = R Q_{n-1} + 250 \).

Step 3: Calculate Q_3 and Q_4 in Closed Form

Using the recursive formula:\[ Q_2 = R \times 250 + 250 \] \[ Q_3 = R \times Q_2 + 250 \] simplify these expressions to express \( Q_3 \) and \( Q_4 \). Insert \( R \) and solve for numerical values.

Step 4: Establish a General Expression for Q_n

Recognize the recurrence relationship \( Q_n = R Q_{n-1} + 250 \) to find the general expression. Solve the difference equation to find \( Q_n = A R^n + B \). Substitute back to find constants A and B using known quantities.

Step 5: Long-Term Behavior and Equilibrium

In the long run, as \( n \to \infty \), the system reaches equilibrium, so the series in \( Q_n = A R^n + B \) converges. Solve for \( Q_{\infty} \), where \( Q_{\infty} = \frac{250}{1-R} \) to find the limiting amount right after each tablet.

Key Concepts

Half-life calculation

The concept of half-life is crucial in understanding how substances degrade over time. In pharmacokinetics, half-life is defined as the time it takes for the concentration of a drug in the body to reduce by half. For cephalexin, the half-life is 0.9 hours. This means that every 0.9 hours, the amount of cephalexin in the body is reduced by 50%. To find out how much remains after 6 hours, you can use the decay formula.
This decay formula is:

• \( N(t) = N_0 \times 2^{-t/T_{1/2}} \)
• Where \( N_0 \) is the initial amount, \( t \) is the time elapsed, and \( T_{1/2} \) is the half-life.

For a 6-hour period, this requires calculating \( 2^{-6/0.9} \). This result tells us the fraction of drug remaining in the body after 6 hours.
Understanding this concept helps predict how long a drug remains active in the system before needing another dose.

Recursive formula

A recursive formula is a way to define terms in a sequence with respect to previous terms. In the context of drug administration, it helps us track how much of a drug is in the system after each dose.
For cephalexin, the recursive approach is used to calculate the drug level after each tablet is taken. Given the decay between doses, we start with \( Q_1 = 250 \) mg for the first tablet. The recursive relation is:

• \( Q_n = R \cdot Q_{n-1} + 250 \)
• \( R \) is the decay factor calculated from the half-life.

This formula reflects how:

• The leftover amount from the previous dose decays during the 6-hour gap.
• A new dose adds fresh 250 mg.

This recursion builds upon itself, allowing us to compute any subsequent \( Q_n \) value accurately.

Exponential decay

Exponential decay is a pattern of reduction that is proportional to the current value, much like how the half-life of a drug operates. If you imagine the drug amount initially, it exponentially decreases over time based on its half-life. The formula used is:

• \( N(t) = N_0 \times 2^{-t/T_{1/2}} \)

This results in a curve that approaches zero but theoretically never quite reaches it.
For cephalexin with a half-life of 0.9 hours, each time block of 0.9 hours sees the drug concentration halve. Over 6 hours, the exponential decay function reveals how the drug amount diminishes to about a fraction of its original state. Understanding exponential decay provides a better grasp of how substances like medications reduce effectiveness over time naturally, affecting dosing schedules.

Equilibrium concentration

The equilibrium concentration is the point where the amount of drug in the body levels off despite regular dosing. In the long-term, as more doses are taken, a pattern or balance is achieved. This occurs when the amount added by taking the drug equals the amount lost through decay within the specified time period.

• The equilibrium formula for cephalexin is:
  \( Q_{\infty} = \frac{250}{1-R} \)
• Where \( R \) is the decay factor over the dosing period.

As \( n \to \infty \), the differences between doses grow negligibly small, signifying that an equilibrium has been reached. Here, the recursive formula has stabilized at a maximum value.
This equilibrium concentration is crucial for maintaining therapeutic levels in the body without over-medicating, ensuring efficacy and safety.