Problem 19 - Applied Calculus

Question

Each morning, a patient receives a \(25 \mathrm{mg}\) injection of an anti- inflammatory drug, and \(40 \%\) of the drug remains in the body after 24 hours. Find the quantity in the body: (a) Right after the \(3^{\text {rd }}\) injection. (b) Right after the \(6^{\text {th }}\) injection. (c) In the long run, right after an injection.

Step-by-Step Solution

Answer

(a) 39 mg; (b) 41.496 mg; (c) 41.67 mg

Step 1: Understand Drug Accumulation

Each day, the patient receives a 25 mg injection. After 24 hours, 40% of this drug remains. This means 10 mg (which is 40% of 25 mg) stays in the body, while the new 25 mg dose is added.

Step 2: Quantity After First Injection

After the first injection, the quantity in the body is simply 25 mg, as no previous injections have been administered.

Step 3: Calculate After Second Injection

Before the second injection, 10 mg of the first dose remains. After the second 25 mg injection, the total is \( 10 + 25 = 35 \text{ mg} \).

Step 4: Calculate After Third Injection

Before the third injection, 40% of 35 mg remains. Thus, \( 0.4 \times 35 = 14 \text{ mg} \) of drug remains from previous doses. With the third injection, \( 14 + 25 = 39 \text{ mg} \) is present right after the third injection.

Step 5: Calculate After Fourth Injection

Before the fourth injection, 40% of 39 mg (about 15.6 mg) remains. After adding the new 25 mg dose: \( 15.6 + 25 = 40.6 \text{ mg} \).

Step 6: Calculate After Fifth Injection

Before the fifth injection, 40% of 40.6 mg remains: \( 16.24 \text{ mg} \). After adding the new 25 mg dose: \( 16.24 + 25 = 41.24 \text{ mg} \).

Step 7: Calculate After Sixth Injection

Before the sixth injection, 40% of 41.24 mg remains: \( 16.496 \text{ mg} \). After adding the new 25 mg dose: \( 16.496 + 25 = 41.496 \text{ mg} \).

Step 8: Calculate Long-Run Quantity in Body

In the long run, the quantity reaches an equilibrium where each consecutive injection results in the same remaining drug. This equilibrium can be calculated using the formula \( \frac{25}{1-0.4} \), which gives approximately \( 41.67 \text{ mg} \) as the long-term quantity right after an injection.

Key Concepts

Anti-inflammatory Drug

An anti-inflammatory drug is designed to reduce inflammation, which is the body's response to irritation, injury, or infection. These drugs can help alleviate symptoms such as redness, swelling, pain, and heat often associated with inflammatory responses. Inflammatory conditions treated by these drugs include arthritis, tendinitis, and bursitis. Some common types of anti-inflammatory drugs include ibuprofen and aspirin, which may be taken orally or administered through injections for more severe cases.

The function of these drugs is to block certain enzymes and proteins involved in the inflammation process. This helps in soothing the affected area and reducing the overall discomfort experienced by patients. The effectiveness and duration of the effect can vary based on the specific drug and its administration method.

Injection Dosage

The concept of injection dosage refers to the amount of drug administered to the patient through an injection. In this case, a patient receives 25 mg of the anti-inflammatory drug each morning. Dosing is crucial in maintaining therapeutic drug levels in the body.

When calculating dosage, healthcare professionals consider various factors:

• The patient's age and weight
• The severity of the condition
• Other medications the patient may be taking

This ensures that the drug is effective without overtly elevating the risks of side effects or drug toxicity. The goal is to keep the drug’s concentration within the therapeutic window to maximize benefits.

Equilibrium Quantity

The equilibrium quantity of a drug in the body refers to the level reached after repeated doses, where the amount of drug eliminated is equal to the amount administered. For this anti-inflammatory drug, the equilibrium is determined by the balance of daily injections and the drug's retention rate.

Mathematically, this can be found using the formula for equilibrium in this scenario: \[ \text{Equilibrium Quantity} = \frac{\text{Daily Dose}}{1 - \text{Retention Rate}} \]Substituting the known values, the equilibrium quantity becomes \[ \frac{25}{1 - 0.4} \approx 41.67 \text{ mg} \]. This result implies that, in the long run, the drug levels in the body will stabilize at approximately 41.67 mg right after each injection.

Drug Retention Rate

The drug retention rate refers to the percentage of the initial drug dose that remains in the body after a certain period. In this exercise, 40% of the drug remains 24 hours post-injection. This retention rate significantly influences how frequently and in what dosage future injections are given.

Understanding the retention rate is vital for determining how a drug accumulates over time. A higher retention rate could mean less frequent dosing might be necessary since the drug stays in the system longer. Conversely, a lower retention rate might necessitate more frequent dosing. In clinical settings, accurate knowledge of the retention rate helps tailor therapies to maximize drug efficacy while minimizing unwanted side effects.