Problem 4 - Brenner and Stevens’ Pharmacology

Question

What dose of a drug should be injected intravenously every 8 hours to obtain an average steady-state plasma drug concentration of \(5 \mathrm{mg} / \mathrm{L}\) if the drug's volume of distribution is \(30 \mathrm{~L}\) and its clearance is \(8 \mathrm{~L} / \mathrm{h}\) ? (A) \(40 \mathrm{mg}\) (B) \(80 \mathrm{mg}\) (C) \(160 \mathrm{mg}\) (D) \(320 \mathrm{mg}\) (E) \(400 \mathrm{mg}\)

Step-by-Step Solution

Answer

The answer is (D) 320 mg.

Step 1: Understand the Pharmacokinetic Terms

Before solving, we must understand the key terms. **Steady-state concentration (Css) ** is the constant drug level in the bloodstream achieved when the rate of drug administration equals the rate of elimination. **Volume of Distribution (Vd)** represents the degree to which a drug is distributed in body fluids and tissues. **Clearance (Cl)** measures the body's efficiency in eliminating the drug. These properties affect dosing.

Step 2: Calculate the Required Dose

Use the formula: \[Dose = Css imes Cl imes au\]where Css is the target steady-state concentration (\(5 \, \mathrm{mg/L}\)), Cl is the clearance (\(8 \, \mathrm{L/h}\)), and \(\tau\) is the dosing interval (every 8 hours). Insert the known values:\[Dose = 5 \, \mathrm{mg/L} imes 8 \, \mathrm{L/h} imes 8 \, \mathrm{h}\]

Step 3: Perform the Calculations

Calculate the dose by multiplying the values: \[Dose = 5 \, \mathrm{mg/L} imes 8 \, \mathrm{L/h} imes 8 \, \mathrm{h} = 320 \, \mathrm{mg}\]

Step 4: Match with Options

Compare the calculated dose of \(320 \, \mathrm{mg}\) with the given options: - (A) 40 mg - (B) 80 mg - (C) 160 mg - (D) 320 mg - (E) 400 mg Option (D) matches the calculated dose.

Key Concepts

Steady-state concentration

Steady-state concentration, often represented as \( C_{ss} \), refers to the condition where the amount of drug entering the bloodstream is equal to the amount being eliminated. This results in a constant effective concentration of the drug in the body. Achieving a steady-state is crucial for maintaining therapeutic effects without causing toxicity.
A steady-state concentration is typically reached after multiple dosing intervals, usually around 4 to 5 half-lives of the drug. Once this point is reached, the drug's effects are consistent, allowing for better control of the patient's condition. In the context of dosing calculations, establishing the steady-state concentration helps determine the necessary dose to maintain effective treatment.

• Controlled dosing: Ensures that drug levels remain consistent and effective.
• Critical for long-term treatments: Useful for drugs that are continually administered, such as antibiotics or chronic condition medications.

Volume of Distribution

The volume of distribution (\( V_d \)) is a theoretical concept that provides insight into how a drug disperses throughout the body. It is calculated by dividing the amount of drug in the body by the plasma concentration of the drug.
This parameter indicates the extent to which a drug permeates into body tissues compared to remaining in the bloodstream. A high volume of distribution implies that the drug extensively enters body tissues, whereas a low volume indicates that it largely remains in the blood.

• An indicator of drug distribution: High \( V_d \) suggests extensive distribution into tissues.
• Helps predict the loading dose: Crucial for determining how much drug is needed to quickly achieve therapeutic concentrations.

Understanding \( V_d \) is key in calculating the initial loading dose required and adjusting doses for individuals based on specific characteristics, such as body weight or presence of disease states that may affect distribution.

Drug Clearance

Drug clearance (\( Cl \)) represents the body's ability to eliminate a drug or a foreign substance. It is usually measured in volume per unit time, such as liters per hour (L/h). Clearance accounts for the efficiency of the kidneys, liver, and other organs involved in drug metabolism and excretion.
Considered a vital pharmacokinetic parameter, drug clearance is pivotal in influencing both the dosing intervals and the amount of drug administered. It ensures that drug levels do not accumulate to toxic levels and helps maintain steady-state concentrations at safe and effective levels.

• Determines dose and frequency: Affects how frequently doses need to be administered to maintain efficacy.
• Varies with organ function: Changes in liver or kidney function can significantly alter drug clearance rates.

Adjustments in drug clearance can be necessary for patients with impaired organ function, to prevent drug accumulation and possible adverse effects.

Dosing Calculations

Dosing calculations in pharmacokinetics are crucial for determining the appropriate amount of medication to administer at specific intervals, ensuring effective treatment and avoiding toxicity. One fundamental formula used for these calculations is:\[\text{Dose} = C_{ss} \times Cl \times \tau\]where \( C_{ss} \) is the desired average steady-state concentration, \( Cl \) is the clearance, and \( \tau \) is the dosing interval. Understanding and applying this formula helps in achieving the desired therapeutic effect while minimizing overdose potential.
This is particularly useful with intravenous drugs, where precision in dosing is critical due to direct entry into the bloodstream.

• Essential for therapeutic effectiveness: Accurate dosing ensures that drug remains within a therapeutic range.
• Prevents adverse effects: Proper calculations prevent high plasma concentrations that can cause side effects.
• Tailored to patient needs: Takes into account individual pharmacokinetic variables, such as metabolism and organ function.

This approach helps customize drug therapy for optimal efficacy and safety across varied patient conditions.