Problem 13 - Calculus for Biology and Medicine

Question

Ibuprofen in Blood You are modeling the concentration of the drug ibuprofen (Advil) in a person's blood after they take one pill. We assume that after they take the pill the drug enters their blood effectively instantaneously. Ibuprofen has first order elimination kinetics. (a) Explain why the concentration of drug in their blood satisfies a differential equation: $$ \frac{d c}{d t}=-k_{1} c \quad \text { with } \quad c(0)=c_{0} $$ and explain what the constants \(k_{1}\) and \(c_{0}\) represent. (You do not need to solve the differential equation.) (b) You measure the following data for the concentration of ibuprofen in a patient's blood $$ \begin{array}{cc} \hline t \text { (hrs) } & c(t)(\mathrm{mg} / \text { liter }) \\ \hline 0 & 40 \\ 1 & 30.3 \\ \hline \end{array} $$ Write down the solution to the differential equation from part (a). Then calculate the parameters \(c_{0}\) and \(k_{1}\) that fit the model to this data.

Step-by-Step Solution

Answer

The constants are \(c_0 = 40\) mg/L and \(k_1 \approx 0.2786\) hr\(^{-1}\).

Step 1 - Understanding the Differential Equation

The differential equation \( \frac{d c}{d t} = -k_{1} c \) represents a first-order kinetics model for the rate of change of concentration \(c\). Here, the rate of elimination of ibuprofen from the blood is proportional to its current concentration. The constant \(k_{1}\) is the elimination rate constant, and \(c_0\) is the initial concentration of ibuprofen at time \(t=0\).

Step 2 - Writing the Solution of the Differential Equation

The general solution to the differential equation \( \frac{d c}{d t} = -k_{1} c \) is \( c(t) = c_0 e^{-k_1 t} \), where \(c_0\) is the initial concentration at \(t=0\).

Step 3 - Applying Initial Condition

Given \(c(0) = 40\), we substitute into the solution to confirm \(c_0 = 40\), so the equation becomes \(c(t) = 40 e^{-k_1 t}\).

Step 4 - Using Data to Find k1

We use the data point \((t=1, c=30.3)\). Substituting into the equation, we have \(30.3 = 40 e^{-k_1}\). Solving for \(k_1\), we get \(-k_1 = \ln\left(\frac{30.3}{40}\right) = \ln(0.7575)\), thus \(k_1 \approx 0.2786.\)

Key Concepts

First Order Kinetics

First-order kinetics describes a process where the rate of reaction or change of concentration is directly proportional to the concentration itself. In the context of pharmacokinetics, this means that the rate at which a drug is eliminated from the body is proportional to the drug's current concentration in the bloodstream.

For ibuprofen, this relationship is expressed as a differential equation:

• \(rac{d c}{d t} = -k_{1} c\)

Here,

• \(c(t)\) is the concentration of the drug at time \(t\).
• \(k_{1}\) is the elimination rate constant, reflecting how quickly the drug is metabolized or removed.

This equation implies that the rate of change of the drug concentration decreases as the concentration decreases, explaining why drugs often need regular dosing to maintain therapeutic levels. This concept is foundational in pharmacokinetics, allowing for the prediction of drug concentrations over time.

Drug Concentration Modeling

Modeling the concentration of a drug in the blood involves deriving equations that predict how the drug concentration changes over time. For many drugs, like ibuprofen, we use first-order kinetics to model this behavior. The goal is to predict the concentration at any given time, which in turn helps in making decisions for dosing and scheduling.

The main equation used here is:

• \(c(t) = c_0 e^{-k_1 t}\)

In this formula:

• \(c_0\) is the initial concentration just after administration.
• \(e\) is the base of the natural logarithm, indicating exponential decay.

By measuring the concentration at different times, we can estimate parameters like \(k_1\), the elimination rate constant. This modeling aids in ensuring that drug levels remain effective and safe within the patient's system, avoiding under or overdosing.

Elimination Rate Constant

The elimination rate constant, denoted as \(k_1\), is essential for understanding how quickly a drug is removed from the bloodstream. It emerges from the first-order kinetics model as described in the differential equation.

To determine \(k_1\), we need:

• Initial concentration \(c_0\).
• Concentration data at various times.

For example, using the data point of ibuprofen concentration \(c(1) = 30.3\) mg/L when \(c(0) = 40\) mg/L, we apply the formula:

• \[30.3 = 40 e^{-k_1}\]

Solving this gives \[k_1 = 0.2786\].

This constant is a measure of the rate at which the drug concentration decreases over time. Understanding \(k_1\) helps tailor dosing schedules so that drug levels remain therapeutically effective.