Problem 25 - Calculus

Question

A drug is administered intravenously at a constant rate of \(r\) mg/hour and is excreted at a rate proportional to the quantity present, with constant of proportionality \(\alpha>0\) (a) Solve a differential equation for the quantity, \(Q,\) in milligrams, of the drug in the body at time \(t\) hours. Assume there is no drug in the body initially. Your answer will contain \(r\) and \(\alpha .\) Graph \(Q\) against \(t\) What is \(Q_{\infty},\) the limiting long-run value of \(Q ?\) (b) What effect does doubling \(r\) have on \(Q_{\infty} ?\) What effect does doubling \(r\) have on the time to reach half the limiting value, \(\frac{1}{2} Q_{\infty} ?\) (c) What effect does doubling \(\alpha\) have on \(Q_{\infty} ?\) On the time to reach \(\frac{1}{2} Q_{\infty} ?\)

Step-by-Step Solution

Answer

\(Q_{\infty} = \frac{r}{\alpha}\). Doubling \(r\) doubles \(Q_{\infty}\), time \(t_{1/2}\) unchanged. Doubling \(\alpha\) halves \(Q_{\infty}\) and \(t_{1/2}\).

Step 1: Understand the Problem

We need to model the rate of change of the drug quantity, \( Q(t) \), in the body. The drug is administered at a constant rate \( r \) and is excreted proportionally to its current amount with proportionality constant \( \alpha \). This can be expressed as a differential equation.

Step 2: Formulate the Differential Equation

The change in the drug quantity over time, \( \frac{dQ}{dt} \), is the rate of administration minus the rate of excretion. This is formulated as: \[ \frac{dQ}{dt} = r - \alpha Q. \] This is a first-order linear differential equation.

Step 3: Solve the Differential Equation

To solve \( \frac{dQ}{dt} = r - \alpha Q \), we use an integrating factor method. The integrating factor is \( \mu(t) = e^{\alpha t} \). Multiply through by this factor: \[ e^{\alpha t} \frac{dQ}{dt} + \alpha e^{\alpha t} Q = re^{\alpha t}. \] Integrating both sides with respect to \( t \) gives: \[ Q(t) = \frac{r}{\alpha} + Ce^{-\alpha t}, \] where \( C \) is the constant determined by initial conditions.

Step 4: Determine Initial Condition and Particular Solution

Given that initially, \( Q(0) = 0 \), substitute into the solution: \[ 0 = \frac{r}{\alpha} + C, \] which gives \( C = -\frac{r}{\alpha} \). Thus, the particular solution is: \[ Q(t) = \frac{r}{\alpha}(1 - e^{-\alpha t}). \]

Step 5: Long-term Behavior of \( Q(t) \)

As \( t \to \infty \), the term \( e^{-\alpha t} \to 0 \). Hence, the limiting value is \( Q_\infty = \frac{r}{\alpha} \). Graphing this, \( Q(t) \) approaches \( \frac{r}{\alpha} \) as \( t \) increases.

Step 6: Effect of Doubling \( r \)

When \( r \) is doubled, \( Q_\infty = \frac{2r}{\alpha} \). Thus, doubling \( r \) doubles the long-term value of \( Q \). To find the time to reach half of this value: Solve \( \frac{r}{\alpha} = \frac{2r}{\alpha} \left(1 - e^{-\alpha t_{1/2}}\right) \) yielding \( t_{1/2} = \frac{\ln 2}{\alpha} \). Thus, doubling \( r \) doesn't change \( t_{1/2} \).

Step 7: Effect of Doubling \( \alpha \)

When \( \alpha \) is doubled, \( Q_\infty = \frac{r}{2\alpha} \). Thus, \( Q_\infty \) is halved. For the time to reach half \( Q_\infty \): \( t_{1/2} = \frac{\ln 2}{2\alpha} \), meaning the time is halved as well.

Key Concepts

Drug Administration

When a drug is administered intravenously, it enters the body at a steady pace, represented by the rate \( r \) in milligrams per hour. This steady influx is crucial for maintaining a consistent drug level in the bloodstream. Intravenous administration ensures that the drug bypasses the digestive system, allowing it to act more rapidly in the body.

The rate \( r \) is pivotal in determining the body's drug level over time. This is because it directly affects how quickly the drug concentration rises, influencing the therapeutic effectiveness.

In the equation \( \frac{dQ}{dt} = r - \alpha Q \), the term \( r \) is the constant addition of the drug into the system, impacting the overall quantity \( Q(t) \), the amount of drug in the body at any given time \( t \). An understanding of this administration rate helps in accurately predicting the drug's behavior, especially when combined with other factors like excretion.

Excretion Rate

The excretion rate of a drug is an essential factor in pharmacokinetics, represented by \( \alpha \), which is the constant of proportionality. This rate is important because it defines how quickly the drug leaves the body. The proportional relationship means that the more drug present, the faster it is excreted, which can influence treatment schedules and dosages.

In the differential equation \( \frac{dQ}{dt} = r - \alpha Q \), the \( \alpha Q \) term represents this loss. The excretion rate \( \alpha \) gives insight into the drug's half-life, which is the time it takes for half of the drug to be eliminated from the bloodstream.

A higher \( \alpha \) means quicker excretion, requiring a faster or higher dosage administration to maintain effective drug levels. Thus, the balance between \( r \) and \( \alpha \) is key to ensuring the drug stays within the therapeutic window without reaching toxic levels.

Long-term Behavior

Understanding the long-term behavior of drug concentration in the body is crucial, particularly in chronic treatments. As time \( t \) increases, the exponential decay component \( e^{-\alpha t} \) in the solution \( Q(t) = \frac{r}{\alpha}(1 - e^{-\alpha t}) \) fades away.

This results in the drug quantity approaching a stable limit, the long-term behavior denoted by \( Q_\infty = \frac{r}{\alpha} \). This value represents the equilibrium concentration achievable as time goes to infinity, assuming constant rate of administration and excretion.

The equilibrium level tells us about drug efficiency and safety on a long-term basis. It's crucial for medications that need to be maintained at a constant level, such as hormones or antibiotics. Monitoring \( Q_\infty \) helps adjust dosage strategies to suit prolonged use, ensuring that the drug remains effective without reaching detrimental concentrations.

Initial Conditions

The initial conditions of a differential equation define the starting circumstances, crucial for determining the particular solution to an equation. For the drug equation \( Q(t) = \frac{r}{\alpha}(1 - e^{-\alpha t}) \), the initial condition given is \( Q(0) = 0 \), indicating no drug present initially in the body.

By substituting this condition, we find the constant of integration \( C \), confirming the specific solution that fits real-world situations.

Initial conditions are significant for predicting the system's future behavior from a known starting point. In the context of drug administration, they establish the timeline and rate at which therapeutic levels are attained.

Adjusting the initial conditions can simulate various scenarios, like starting medication after a drug-free period, giving pharmacologists and medical professionals valuable insights for patient care.