Problem 81 - Calculus and its applications

Question

Chemotherapy. The dosage for Carboplatin chemotherapy drugs depends on several parameters for the particular drug as well as the age, weight, and sex of the patient. For female patients, the formulas giving the dosage for such drugs are $$ D=0.85 A(c+25) \text { and } c=(140-y) \frac{w}{72 x} $$ where \(A\) and \(x\) depend on which drug is used, \(D\) is the dosage in milligrams \((\mathrm{mg}), c\) is called the creatine clearance, \(y\) is the patient's age in years, and \(w\) is the patient's weight in kilograms (kg). (Source: U.S. Oncology.) a) Suppose a patient is a 45 -year-old woman and the drug has parameters \(A=5\) and \(x=0.6 .\) Use this information to write formulas for \(D\) and \(c\) that give \(D\) as a function of \(c\) and \(c\) as a function of \(w\). b) Use your formulas from part (a) to compute \(d D / d c\). c) Use your formulas from part (a) to compute \(d c / d w\). d) Compute \(d D / d w\). e) Interpret the meaning of the derivative \(d D / d w\).

Step-by-Step Solution

Answer

The dosage increases by approximately 9.35 mg for each 1 kg increase in weight.

Step 1: Write Formula for Dosage

First, we'll express the dosage formula. Given \( A = 5 \) for the drug and using the formula \( D = 0.85 A (c + 25) \), we can substitute \( A \) to get \( D = 0.85 \times 5 \times (c + 25) \). This simplifies to \( D = 4.25(c + 25) \).

Step 2: Write Formula for Creatine Clearance

Using the formula \( c = (140 - y) \frac{w}{72 x} \), and given \( y = 45 \) and \( x = 0.6 \), we substitute these values to get \( c = (140 - 45) \frac{w}{72 \times 0.6} \). This simplifies to \( c = 95 \frac{w}{43.2} \).

Step 3: Compute Derivative of Dosage with Respect to Clearance

Using the formula from Step 1, \( D = 4.25(c + 25) \), we compute the derivative \( \frac{dD}{dc} = 4.25 \), as the only variable in the expression is \( c \).

Step 4: Compute Derivative of Clearance with Respect to Weight

Using the formula from Step 2, \( c = 95 \frac{w}{43.2} \), we compute the derivative \( \frac{dc}{dw} = \frac{95}{43.2} \), which simplifies to approximately \( 2.1991 \).

Step 5: Compute Derivative of Dosage with Respect to Weight

We know from the chain rule that \( \frac{dD}{dw} = \frac{dD}{dc} \times \frac{dc}{dw} \). Hence, \( \frac{dD}{dw} = 4.25 \times 2.1991 \approx 9.35 \).

Step 6: Interpret the Meaning of the Derivative

The derivative \( \frac{dD}{dw} \approx 9.35 \) mg/kg indicates that for every 1 kg increase in weight, the dosage \( D \) increases by approximately 9.35 mg, assuming other factors remain constant.

Key Concepts

Chain Rule

In differential calculus, the chain rule is a powerful tool used to compute the derivative of a composite function. Suppose you have two functions, where one function's output is fed into another, like the formulas provided for chemotherapy dosage where the creatinine clearance, \( c \), is part of the dosage equation.

The chain rule helps us find how a change in one variable affects another, especially when these variables are not directly connected. Think of it as a way to "chain" the effects of these changes together.

• To use the chain rule, consider each relationship. First, calculate the derivative of the dosage \( D \) with respect to clearance \( c \), noted as \( \frac{dD}{dc} \).
• Next, compute the derivative of clearance \( c \) with respect to weight \( w \), noted as \( \frac{dc}{dw} \).
• Apply the chain rule: \( \frac{dD}{dw} = \frac{dD}{dc} \times \frac{dc}{dw} \). This gives the change in dosage \( D \) based on a change in weight \( w \).

This approach allows us to see how a small change in the patient's weight impacts the overall dosage they receive, completing the chain between these two seemingly separate variables.

Derivative Interpretation

Derivatives in calculus are fundamental to understanding how variables change with respect to one another. In the context of the chemotherapy dosage example, the derivative \( \frac{dD}{dw} \) tells us how the weight \( w \) of the patient affects the dosage \( D \).

This particular derivative gives insight into the sensitivity of the dosage to changes in weight.

• If \( \frac{dD}{dw} \) is large, even a small change in weight causes a significant change in dosage.
• If it is small, the dosage is not highly sensitive to changes in weight.

In this exercise, calculating \( \frac{dD}{dc} \) allows us to understand how creatinine clearance impacts dosage, and \( \frac{dc}{dw} \) provides insight into how the patient's weight affects clearance.

By piecing these relationships together with the chain rule, \( \frac{dD}{dw} \) reflects the combined sensitivity of the dosage to changes in patient weight, showcasing the interconnected nature of these physiological factors.

Chemotherapy Dosage Calculation

Calculation of chemotherapy dosage is crucial, as it ensures the treatment is both safe and effective. Dosage calculations often rely on several patient-specific parameters, like age, weight, and kidney function, represented by creatinine clearance in the exercise formula.

The dosage formula \( D = 0.85 A(c + 25) \) adjusts the general drug dosage \( A \) with specific patient data to cater to individual needs.

• Here, the creatinine clearance \( c = (140-y) \frac{w}{72x} \) represents how well a patient's kidneys are functioning, indirectly indicating how the patient's body processes the drug.
• Weight \( w \) is a direct patient measurement affecting \( c \), hence dosage \( D \), by reflecting the patient's overall body mass. The heavier the patient, the higher the creatinine clearance, and subsequently, the dosage.

Correctly implementing these calculations guarantees the dosage is neither too low to be ineffective nor too high to cause toxicity. Understanding the derivatives' role in these calculations can give medical professionals extra tools for precise medication delivery, personalized for each unique patient condition.