Problem 20 - Calculus for Biology and Medicine

Question

Drug Elimination Suppose that a drug is administered to a person in a single dose, and assume that the drug does not accumulate in body tissue, but is excreted through urine. Denote the amount of drug in the body at time \(t\) by \(x_{1}(t)\) and in the urine at time \(t\) by \(x_{2}(t) .\) If \(x_{1}(0)=6 \mathrm{mg}\) and \(x_{2}(0)=0\), find a system of differential equations for \(x_{1}(t)\) and \(x_{2}(t)\) if it takes 20 minutes for the drug to be at one- half of its initial amount in the body.

Step-by-Step Solution

Answer

The system of differential equations is: \(\frac{dx_1}{dt} = -\frac{\ln 2}{20}x_1\) and \(\frac{dx_2}{dt} = \frac{\ln 2}{20}x_1\).

Step 1: Define the decay rate of the drug

The drug is eliminated exponentially, and it takes 20 minutes for the drug concentration to halve. Therefore, the rate at which the drug is eliminated from the body is defined. The half-life formula is: \ \(\frac{1}{2}x_1(0) = x_1(0) e^{-kt} \) Solve for decay constant \(k\): \ \(-kt = \ln\left(\frac{1}{2}\right) \iff k = \frac{\ln 2}{20}\) The decay constant \(k = \frac{\ln 2}{20} \approx 0.0347\).

Step 2: Write the differential equation for the drug in the body

Given \(x_1(0) = 6\), the rate of change of \(x_1(t)\) is determined by the decay constant. Thus, the differential equation is: \ \(\frac{dx_1}{dt} = -kx_1 = -\frac{\ln 2}{20}x_1\).

Step 3: Determine the differential equation for the amount of drug in the urine

The rate the drug moves from the body to the urine is the same as it leaves the body. Therefore, the rate of change of \(x_2(t)\) is the negative of \(\frac{dx_1}{dt}\): \ \(\frac{dx_2}{dt} = \frac{\ln 2}{20}x_1 \).

Step 4: System of differential equations

Combining cell 2 and cell 3, we have the system: \ \(\frac{dx_1}{dt} = -\frac{\ln 2}{20}x_1\) \ \(\frac{dx_2}{dt} = \frac{\ln 2}{20}x_1\) \ This system models the drug elimination from the body to the urine.

Key Concepts

Drug Elimination

When a drug is administered to the body, it undergoes a process known as drug elimination. This is the body's way of removing the drug, mainly through metabolic processes and excretion. In this context, the drug does not remain in the body tissues but is excreted in the urine as the primary pathway. Let's break this down a bit more: - **Metabolism**: The liver often transforms the drug into metabolites, which are then more easily excreted. - **Excretion**: The kidneys filter the blood, removing the drug and its metabolites from the body through urine. In mathematical terms, drug elimination can be modeled using differential equations to track the amount of drug in the body over time, focusing on how the drug transitions from the bloodstream to being expelled from the body. This helps understand the dynamics of the drug's presence in the system and informs dosing regimens.

Half-life

The half-life of a drug is a fundamental concept in pharmacokinetics. It represents the time required for the drug's concentration in the body to decrease by half. This is a crucial index because it helps determine how often a drug should be administered to maintain therapeutic effect.In the exercise we're discussing, the half-life is given as 20 minutes. This means that every 20 minutes, the drug amount is reduced to 50% of its previous concentration.The formula for half-life in exponential decay is:- \[\frac{1}{2}x_1(0) = x_1(0)e^{-kt}\]This relationship allows us to solve for the decay constant, an important factor that affects how quickly the drug is metabolized and excreted.

Exponential Decay

Exponential decay refers to the decrease of a quantity at a rate proportional to its current value. In drug elimination, this explains how the concentration of the drug reduces in the body over time. - **Key Characteristics**: The rate of elimination is faster at higher concentrations and slows down as the concentration decreases. This results in a smooth, downward-sloping curve when plotted against time.To represent this mathematically, we use the differential equation:- \[\frac{dx_1}{dt} = -kx_1\]Here, \(k\) is the decay constant, representing the proportionality factor in the rate of reduction. With a decay constant of approximately 0.0347 (as calculated from the half-life), this equation shows how the drug level in the body diminishes exponentially.

Rate of Change

The rate of change is a concept used to describe how quickly the amount of the drug in the body is changing over time. This is critical for understanding the drug's kinetics to ensure its effectiveness and safety.In the context of drug elimination:- The rate at which the drug leaves the body can be described using the differential equation for \(x_1(t)\): - \[\frac{dx_1}{dt} = -\frac{\ln 2}{20}x_1\]- Likewise, the rate at which it accumulates in the urine is expressed by: - \[\frac{dx_2}{dt} = \frac{\ln 2}{20}x_1\]These equations tell us how fast the drug is decreasing in the body and increasing in the urine. Understanding this rate helps in planning dosing intervals to achieve the desired therapeutic outcomes without causing toxicity or sub-therapeutic dosing.