Problem 8 - Applied Calculus

Question

(a) An allergy drug with a half-life of 18 weeks is given in \(100-\mathrm{mg}\) doses once a week. At the steady state, find the quantity of the drug in the body right after a dose. (b) The drug does not become effective until the quantity in the body right after a dose reaches \(2000 \mathrm{mg}\). How many weeks after the first dose does the drug become effective?

Step-by-Step Solution

Answer

(a) 95948 mg. (b) 14 weeks.

Step 1: Understanding Half-life

The half-life of a drug is the time it takes for the drug quantity in the body to reduce by half. In this case, it's specified as 18 weeks.

Step 2: Formula for Drug Quantity in Steady State

When a drug is given repeatedly, the quantity after each dose stabilizes at the 'steady state.' The expression for the steady state with doses given once a week is: \[ Q = \frac{D}{1 - \left(\frac{1}{2}\right)^{T/\text{half-life}}} \] where \(Q\) is the drug quantity at steady state, \(D\) is the dose (100 mg), and \(T\) is the dosing interval (1 week).

Step 3: Substitute Values into Formula

Substitute the given values into the formula: \[ Q = \frac{100}{1 - \left(\frac{1}{2}\right)^{1/18}} \]

Step 4: Calculate Quantity at Steady State

Calculate the expression: \[ Q \approx \frac{100}{1-0.999039} \approx 95948.08 \text{ mg} \] So, the quantity right after a dose at steady state is approximately 95948 mg.

Step 5: Understanding Drug Effectiveness Condition

The drug is only effective when the post-dose quantity reaches 2000 mg. The quantity increases exponentially each week from the initial dose to the steady state.

Step 6: Use Exponential Growth Formula

The quantity of drug after the nth dose is given by \[ Q_n = Q_{n-1} \times \left( \frac{1}{2} \right)^{1/18} + 100 \]. Iterate from the first dose until the quantity is at least 2000 mg.

Step 7: Iteration Until Effectiveness

Starting from 0 mg, iterate each week: substitute into the formula for \(Q_n\) until \(Q_n \geq 2000\) mg is achieved.

Step 8: Determine Week Number

Use the iterative calculations or a computing tool to find the week number when \(Q_n \geq 2000\). It takes about 14 weeks to reach 2000 mg.

Key Concepts

Steady State

When administering medication at regular intervals, the body reaches a point where the amount of drug entered equals the amount leaving the body over a specific period. This is known as the steady state.
The quantity of the drug in the body stabilizes at this steady state because periodic dosing compensates for the drug eliminated through metabolism and excretion.
Understanding steady state is crucial for ensuring that the drug remains effective and safe over a long period.To calculate the steady state, use the formula:

• \[ Q = \frac{D}{1 - \left(\frac{1}{2}\right)^{T/\text{half-life}}} \]
• Where:
• \(Q\) is the drug quantity at steady state,
• \(D\) is the dose administered (100 mg in our example), and
• \(T\) is the dosing interval (1 week for our exercise).

By substituting these values, we find that at steady state, the quantity of the drug reaches a significant level allowing for consistent therapeutic effects.

Exponential Growth

Exponential growth is a fundamental concept in pharmacokinetics, as it regards how drug levels in the body increase over time when doses are administered at regular intervals.
When we start with an initial dose, the drug concentration in the body rises gradually, following an exponential curve until it reaches the steady state.During each dosing interval, the drug concentration builds up as new doses are added, compounded with the remaining portions of previously administered doses.
This is articulated through a formula representing consecutive doses:

• \[ Q_n = Q_{n-1} \times \left( \frac{1}{2} \right)^{1/18} + 100 \]
• Where:
• \(Q_n\) is the concentration after the nth dose, and
• \(Q_{n-1}\) is the concentration before the nth dose.

The importance of understanding this concept is to predict the timeline for a drug to become effective, which means reaching the minimum required concentration.

Dosing Interval

The dosing interval is the time between the administration of successive drug doses. Proper scheduling of this interval is critical to maintaining desired drug levels in the body, achieving steady state, and ensuring therapeutic effectiveness. In pharmacological treatments, the dosing interval is meticulously calculated based on the drug's half-life, ensuring that the drug remains above the effective concentration threshold.
In the exercise provided, a dosing interval of 1 week is used, which fits the drug's 18-week half-life for consistent dosing and effective build-up.

• Advantages of perfect dosing intervals include:
• Maintaining drug levels within a therapeutic range,
• Reducing the risk of side effects, and
• Facilitating patient adherence to the treatment schedule.

Thus, understanding dosing intervals helps in planning treatments that optimize patient outcomes and build towards achieving the steady state efficiently.