Question

You sprain your ankle and your doctor prescribes 325 milligrams of an anti- infl ammatory drug every 8 hours for 10 days. Sixty percent of the drug is removed from the bloodstream every 8 hours. a. Write a recursive rule for the amount of the drug in the bloodstream after \(n\) doses. b. The value that a drug level approaches after an extended period of time is called the maintenance level. What is the maintenance level of this drug given the prescribed dosage? c. How does doubling the dosage affect the maintenance level of the drug? Justify your answer.

Short answer

For part a, the recursive formula representing the amount of drug in the bloodstream after \(n\) doses is \(D_n = 0.4D_{n-1} + 325\). The maintenance level for the originally prescribed dosage is approximately 541.67 milligrams (part b). If the dosage is doubled, the maintenance level also doubles to 1083.33 milligrams (part c).

Step-by-step solution

Recursive Formula

First, let's denote the dosage as \(D_n\) where \(D_n\) is the amount of the drug in the bloodstream after \(n\) doses. We know that the drug dosage is 325 milligrams every 8 hours, and 60% of the drug is removed every 8 hours. Therefore, we can express \(D_n\) as: \(D_n = 0.4D_{n-1} + 325\) This is a recursive formula with \(D_n\) (the dosage after \(n\) doses) being dependent on \(D_{n-1}\) (the dosage after \(n-1\) doses). For \(n=0\), \(D_0=325\).

Maintenance Level

The maintenance level of a drug is defined as the equilibrium concentration of the drug in the bloodstream over time. This level is reached when the input (the new dose) equals the output (the drug lost). This implies that \(D_n=D_{n-1}\). In mathematical terms, \(D=0.4D+325\). Thus, the equilibrium drug concentration, denoted by \(D\), can be found by solving this equation for \(D\). To find \(D\): First, move the term \(0.4D\) to the left side of the equation: \(D-0.4D=325\), which simplifies to: \(0.6D=325\) Then, divide both sides of this equation by 0.6 to solve for \(D\): \(D = \frac{325}{0.6} = 541.67\) milligrams.

Effect of Doubling the Dosage

We'll find out how doubling the dose to 650 milligrams affects the maintenance level. Again, we'll find the equilibrium level by setting the input (the new dose) equal to the output (the drug lost). This gives: \(D=0.4D+650\). To find the new maintenance level: First, move the term \(0.4D\) to the left side of the equation: \(D-0.4D=650\), which simplifies to: \(0.6D=650\) Then, divide both sides of this equation by 0.6 to solve for \(D\): \(D = \frac{650}{0.6} = 1083.33\) milligrams. This means that if the dosage is doubled, the maintenance level doubles as well, affirming that the maintenance level directly depends on the dosage amount.

Key concepts

Maintenance Level

The maintenance level of a drug refers to the steady state concentration that is achieved in the bloodstream when a drug is administered consistently over time. When you take medication, your body begins to process and eliminate it at a certain rate. Simultaneously, new doses are added at regular intervals. The maintenance level is reached when the amount of drug being added equals the amount being metabolized and removed.
To find this balanced state, we use the recursive formula where each new dose (\(D_n = 0.4D_{n-1} + 325\)) considers how much of the drug is left after the last dose. We solve for the maintenance level by setting the current dose equal to the previous dose: \(D = 0.4D + 325\). Simplifying this gives us 541.67 milligrams as the concentration at which the drug stabilizes.

Equilibrium Concentration

Equilibrium concentration is reached when the levels of drug in the bloodstream stop fluctuating significantly over time, implying stability. At this point, the system inputs and outputs are balanced. The recursive sequence of administering the drug leads us to this equilibrium state.
Mathematically, the equilibrium can be identified by setting the rate of incoming drug equal to the rate at which it is metabolized and excreted. In this scenario, the equilibrium concentration can be calculated by solving the equation \(D = 0.4D + 325\), resulting in a steady concentration of 541.67 milligrams. This concentration represents the equilibrium state, also known as the steady state, of the drug dosages over the extended period.

Dosage Adjustment

Dosage adjustment involves changing the quantity of a drug administered to achieve a desired effect in the bloodstream. This exercise demonstrates how tweaking the dose can alter the maintenance level. For example, if the dosage is doubled from 325 mg to 650 mg, the established equilibrium shifts.
To find the new maintenance level with this increased dose, we adjust the recursive equation to \(D = 0.4D + 650\). Solving this results in a new equilibrium concentration of 1083.33 milligrams. This example illustrates how an increment in dosage leads to a direct increase in the maintenance level, showcasing the importance of dosage adjustment in pharmacokinetics for therapeutic effectiveness.