Problem 14 - Applied Calculus

Question

If \(R\) is percent of maximum response and \(x\) is dose in mg, the dose-response curve for a drug is given by $$R=\frac{100}{1+100 e^{-0.1 x}}$$ (a) Graph this function. (b) What dose corresponds to a response of \(50 \%\) of the maximum? This is the inflection point, at which the response is increasing the fastest. (c) For this drug, the minimum desired response is \(20 \%\) and the maximum safe response is \(70 \% .\) What range of doses is both safe and effective for this drug?

Step-by-Step Solution

Answer

The effective range is 46.05 mg for 50% response, and doses between 3 mg and 69 mg are safe.

Step 1: Understand the Dose-Response Curve

The given function is a dose-response curve representing how a body's response to a drug (\( R \)) changes concerning the dose (\( x \)). The curve is given by:\[ R = \frac{100}{1 + 100e^{-0.1 x}} \].This equation shows the percentage of the maximum response as a function of the drug dose, reaching its peak at 100% response.

Step 2: Graph the Dose-Response Function

To graph the function, calculate the value of \( R \) for several doses of \( x \). As \( x \) increases, the exponential term \( e^{-0.1 x} \) decreases, making \( R \) approach 100. Plot these calculated points to visualize how \( R \) varies with \( x \). The graph will show an S-shaped curve characteristic of a logistic function.

Step 3: Identify Dose for 50% Maximum Response

To find the dose corresponding to 50% response, set \( R = 50 \) in the equation:\[ 50 = \frac{100}{1 + 100e^{-0.1 x}} \]. Solve for \( x \):1. Rearrange the equation to isolate the exponential term: \[ 1 + 100e^{-0.1 x} = 2 \].2. Subtract 1 from both sides: \[ 100e^{-0.1 x} = 1 \].3. Divide both sides by 100: \[ e^{-0.1 x} = 0.01 \].4. Take the natural logarithm of both sides: \[ -0.1 x = \ln(0.01) \].5. Solve for \( x \): \[ x = -\frac{\ln(0.01)}{0.1} \approx 46.05 \].

Step 4: Determine Safe and Effective Dose Range

Determine doses for both 20% and 70% response by substituting \( R = 20 \) and \( R = 70 \) into the equation and solving similarly:1. For \( R = 20 \): - Set the equation: \( 20 = \frac{100}{1 + 100e^{-0.1 x}} \). - Solve similarly by substituting and isolating \( x \). - You find \( x \approx 3 \).2. For \( R = 70 \): - Set the equation: \( 70 = \frac{100}{1 + 100e^{-0.1 x}} \). - Solve similarly. - You find \( x \approx 69 \).Thus, the dose range between 3 mg and 69 mg is both safe and effective.

Key Concepts

Logistic Function

A logistic function is a type of mathematical function that creates an S-shaped curve when graphed. It is often used to model situations where growth accelerates to a point and then slows down as it approaches a maximum limit. In the context of the dose-response curve, this function helps illustrate how a drug's effect on the body changes with varying doses. The equation for a logistic function is typically structured to show how close a response gets to a maximum value as the independent variable increases. Understanding logistic functions is essential because they accurately depict many natural processes, including biological responses to drugs. These functions model an initial phase of exponential growth, followed by a slowing pace and eventual plateau.

Drug Safety

Drug safety involves ensuring that the administration of a drug is not only effective but also poses no harm to the patient within a specified dose range. For drug safety assessment, characteristics like the minimum desired response and the maximum safe response are crucial. These parameters help define the effective and safe range of doses a person can take. In the given problem, the safe dose range lies between when the body has a minimal 20% response and when it reaches a maximum 70% response. Identifying this range is vital for preventing adverse effects while ensuring the drug achieves its intended therapeutic effects. Balancing effectiveness and safety is a key component of pharmacology.

Inflection Point

The inflection point on a curve is the point where the function changes curvature, indicating a transition from concave to convex or vice versa. In the context of a dose-response curve, it signifies the dose at which the response rate is increasing most rapidly. For the given equation, this point occurs when the response, R, reaches 50% of its maximum. Mathematically, finding this inflection point involves solving the equation for the dose value where the response rate accelerates fastest. The importance of the inflection point lies in its usefulness in determining optimal drug doses for achieving desired effects quickly, often linking to the maximal therapeutic window.

Graphing Functions

Graphing functions, especially nonlinear ones like the logistic function, allows a visual representation of how variables interact in a given equation. In the exercise, the function R provides the percentage of the maximum response to a drug based on dosage. When graphing this function, you derive a logistic curve that helps visualize how the response escalates sharply and eventually levels off. This S-shaped curve is key in identifying important metrics such as inflection points and safe dose ranges. By plotting various points, such as where responses reach 20%, 50%, and 70%, you can clearly see how the function behaves and determine crucial data for drug application. Graphing is an indispensable tool for understanding and predicting outcomes in science and mathematics.