Problem 1 - Clinical Chemistry

Question

If drug \(\mathrm{X}\) has a half-life \(\left(T_{1 / 2}\right)\) of 2 days (48 hours) and the concentration at 12:00 today was \(10 \mu \mathrm{g} / \mathrm{mL}\), what would the expected concentration of drug \(\mathrm{X}\) be at \(12: 00\) tomorrow? a. \(7 \mu \mathrm{g} / \mathrm{mL}\) b. \(7.5 \mu \mathrm{g} / \mathrm{mL}\) c. \(5 \mu \mathrm{g} / \mathrm{mL}\) d. \(3.5 \mu \mathrm{g} / \mathrm{mL}\)

Step-by-Step Solution

Answer

The expected concentration of drug \mathrm{X} at 12:00 tomorrow is \(7\mu g /ML\).

Step 1: Understand the Half-Life Concept

The half-life of a drug is the time it takes for the concentration of the drug to be reduced by half. For drug \mathrm{X}, the half-life (\(T_{1/2}\)) is given as 2 days or 48 hours.

Step 2: Determine the Time Interval

The problem asks for the concentration of the drug at 12:00 tomorrow, which is 24 hours from the given time (12:00 today).

Step 3: Calculate the Fraction of Time Interval to Half-Life

Calculate the fraction of the 24-hour interval to the half-life of 48 hours: \[ \text{Fraction} = \frac{24\text{ hours}}{48\text{ hours}} = \frac{1}{2} \]

Step 4: Calculate the Remaining Concentration

Using the fraction from the previous step, calculate the concentration reduction: \[ \text{Remaining Concentration} = 10\ \mu \mathrm{g} / \mathrm{mL} \times \left(\frac{1}{2}\right)^{1/2} \approx 10 \times 0.707 = 7.07\ \mu \mathrm{g} / \mathrm{mL} \]

Step 5: Round to the Closest Option

Round 7.07\ \mu \mathrm{g} / \mathrm{mL} to the nearest option provided: \(7\ \mu \mathrm{g} / \mathrm{mL}\).

Key Concepts

Pharmacokinetics

Pharmacokinetics is the study of how drugs move through the body. It involves looking at the processes of absorption, distribution, metabolism, and excretion. When we talk about pharmacokinetics, we're interested in how the body handles a drug over time. For instance, understanding how quickly a drug is absorbed and the path it takes through the body helps in determining the right dosage and frequency of administration. Aspects such as the half-life of a drug are critical to pharmacokinetics because they provide insight into how long the drug stays active in the body.

Drug Concentration

Drug concentration refers to the amount of drug present in a unit volume of blood or plasma, usually expressed in \[ \mu \mathrm{g}/\mathrm{mL} \]. This value is crucial for ensuring therapeutic efficacy while avoiding toxicity. In the context of the given exercise, the initial concentration of drug X is \[10 \mu \mathrm{g}/\mathrm{mL} \]. Over time, the concentration of the drug decreases due to metabolic processes and excretion. Assessing the concentration at different times helps in understanding how effectively the drug is working and how it is being processed by the body.

Half-Life Equation

The half-life of a drug is defined as the time taken for its concentration to reduce to half its initial value. For drug X, the half-life is given as 2 days or 48 hours. The formula used to find the drug concentration after a certain period is: \[ C_t = C_0 \times \bigg(\frac{1}{2}\bigg)^{\frac{t}{T_{1/2}}} \]. Here,

• \( C_t \): Concentration at time \[ t \]
• \( C_0 \): Initial concentration
• \( T_{1/2} \): Half-life of the drug

So, for time intervals that are fractions of the half-life, we adjust the exponent in the formula accordingly. In the exercise, we calculated the remaining concentration for half the half-life duration (24 hours), arriving at the value approximately \[ 7 \mu \mathrm{g}/\mathrm{mL} \].

Exponential Decay

Exponential decay describes the process by which a quantity decreases at a rate proportional to its current value. This concept is crucial in half-life calculations. For any substance undergoing exponential decay, including radioactive isotopes or drug concentration in the body, the quantity decreases by a consistent percentage over each time period. In this exercise, drug X's concentration exhibits exponential decay over time. After 24 hours, which is half of its half-life, its concentration decreased to approximately \[ 7 \mu \mathrm{g}/\mathrm{mL} \], reflecting an exponential decay pattern.