Problem 54 - Calculus for Biology and Medicine

Question

A drug has zeroth order elimination kinetics. At time \(t=0\) an amount \(a_{0}=20 \mathrm{mg}\) is present in the blood. One hour later, at \(t=1\), an amount \(a_{1}=14 \mathrm{mg}\) is present. (a) Assuming that no drug is added to the blood between \(t=0\) and \(t=1\), calculate the amount of drug that is removed from the blood each hour. (b) Write a recursion relation for the amount of drug \(a_{t}\) that is present at time \(t\). Assume no extra drug is added to the blood. (c) Find an explicit formula for \(a_{t}\) as a function of \(t\). (d) When does the amount of drug present in the blood first drop \(\operatorname{to} 0 ?\)

Step-by-Step Solution

Answer

(a) 6 mg/hour. (b) \(a_{t+1}=a_t-6\). (c) \(a_t=20-6t\). (d) 3.33 hours.

Step 1: Calculate the Rate of Drug Elimination

Given that the drug follows zeroth order elimination kinetics, the rate of elimination is constant. The amount present at \(t=0\) is 20 mg, and at \(t=1\) is 14 mg. Calculate the rate \(k\) of drug elimination per hour using: \[k = a_0 - a_1 = 20 \text{ mg} - 14 \text{ mg} = 6 \text{ mg/hour}.\] Thus, 6 mg of the drug is eliminated each hour.

Step 2: Write the Recursion Relation

The amount of the drug present at the next time point is the current amount minus the rate of elimination. The recursion relation for the amount of drug \( a_t \) at time \(t\) is given by: \[ a_{t+1} = a_t - 6. \] This equation captures the constant removal of 6 mg from the amount present.

Step 3: Derive the Explicit Formula for \(a_t\)

Starting from the initial amount at \(t=0\), the explicit formula for the amount of drug at any time \(t\) can be derived using the rate found: \[ a_t = a_0 - 6t = 20 - 6t. \] This formula calculates the amount of drug remaining after \(t\) hours.

Step 4: Determine When the Drug Amount Reaches Zero

Set the equation from Step 3 to zero to find when the drug amount will first drop to zero: \[ 0 = 20 - 6t. \] Solve for \(t\): \[ 6t = 20 \ t = \frac{20}{6} = \frac{10}{3} \approx 3.33 \text{ hours}. \] Thus, the drug amount will first reach zero at approximately 3.33 hours.

Key Concepts

Rate of Drug Elimination

When discussing zeroth order kinetics, it's essential to understand that the rate of drug elimination remains constant over time. In this scenario, the drug's concentration decreases consistently by a fixed amount each hour. The difference in drug concentration between two time points determines this constant rate.

For example, if you start with a concentration of 20 mg at time zero and observe a concentration of 14 mg one hour later, the rate of drug elimination can be calculated. Subtract the latter amount from the initial amount:

• Initial amount, \( a_0 = 20 \) mg
• Amount after one hour, \( a_1 = 14 \) mg

Using the formula:

• Rate, \( k = a_0 - a_1 = 20 \, \text{mg} - 14 \, \text{mg} = 6 \, \text{mg/hour} \)

This constant rate of elimination means that every hour, 6 mg of the drug will consistently be eliminated from the bloodstream. This elimination pattern is pivotal in understanding how the drug's concentration changes over time.

Recursion Relation

The concept of a recursion relation is pivotal for systematically understanding how a sequence evolves step-by-step. In zeroth order kinetics, it represents the repetitive and consistent decrease in drug concentration over time.

Starting with a given amount of drug in the blood, you can predict subsequent amounts by applying the rate of elimination. For our example, the recursion relation can be established as follows:

• \( a_{t+1} = a_t - 6 \)

This recursion relation effectively describes how at each time step \(t\), the amount of drug decreases by 6 mg. It's a simple and efficient way to project future drug concentrations, assuming no additional drug is introduced into the system. As such, recursion relations are an essential part of modeling drug kinetics, helping predict how and when a substance will exit the body entirely.

Explicit Formula

An explicit formula provides a direct way to determine the amount of drug in the bloodstream at any given time. It offers a more accessible method than iteration, as it allows for the computation of the drug's concentration at any time \(t\) without needing to calculate each preceding time point.

For a zeroth order kinetic process, the explicit formula is derived from the initial amount and the steady rate of elimination observed. Starting with an initial amount, \( a_0 \), and using the rate of elimination, \( k \), the formula for the drug amount at any future time \(t\) becomes:

• \( a_t = a_0 - kt \)
• In our specific scenario: \( a_t = 20 - 6t \)

This formula demonstrates a linear decline in the drug concentration, accurately capturing the constant rate of 6 mg per hour previously calculated. With this formula, it's possible to easily determine the precise moment when the drug concentration in the blood will reach zero, marking the end of its pharmacological effect.