Problem 12 - Calculus for Biology and Medicine

Question

. Drug Absorption Drug enters a patient's blood by being absorbed from the gut. Assume that the drug enters the patient's blood at a rate that depends on time as \(c e^{-r t}\) where \(c\) and \(r\) are positive constants (the rationale for this formula will be discussed in Section \(8.4\) ) and the drug is eliminated at constant rate \(k\). So: $$ \frac{d M}{d t}=c e^{-r t}-k $$ (a) Assuming \(c>k\) and \(M(0)=0\) (there is no drug present in the patient's blood at the start of the experiment), solve this differential equation. (b) Suppose \(k>0\). What does your solution predict will happen to \(M(t)\) as \(t \rightarrow \infty\) ? Does your answer make sense? (In reality, drug can only be removed at a constant rate until all drug is removed from the blood. That is, the rate of elimination will be \(k\) if \(M>0\) and 0 once \(M\) drops to \(0 .)\) (c) Assume \(k=0\) (i.e., this drug is never eliminated from blood, or is eliminated so slowly that elimination can be neglected). Show that \(\lim _{t \rightarrow \infty} M(t)=\frac{c}{r}\).

Step-by-Step Solution

Answer

(a) \( M(t) = \frac{c}{r}(1-e^{-rt}) - kt \); (b) \( M(t) \to \frac{c}{r} - kt \) as \( t \to \infty \); (c) \( \lim_{t \to \infty} M(t) = \frac{c}{r} \) if \( k = 0 \).

Step 1: Write the Original Differential Equation

The given differential equation is \[ \frac{dM}{dt} = c e^{-rt} - k \] where \( M(t) \) is the amount of drug in the blood at time \( t \), \( c \) and \( r \) are constants specific to the rate of drug absorption, and \( k \) is the constant rate of drug elimination.

Step 2: Integrate the Differential Equation for M(t)

To solve the differential equation for \( M(t) \), integrate both sides with respect to \( t \). For the absorption component:\[ \int c e^{-rt} \, dt = -\frac{c}{r} e^{-rt} + C \]For the elimination component, integrate the constant \(-k\):\[ \int -k \, dt = -kt \]Combine the results:\[ M(t) = -\frac{c}{r} e^{-rt} - kt + C \]

Step 3: Apply Initial Condition for Constant C

Use the initial condition \( M(0) = 0 \) to solve for \( C \). Substitute \( t = 0 \) into the integrated equation:\[ 0 = -\frac{c}{r} e^{0} - k \cdot 0 + C \]\[ 0 = -\frac{c}{r} + C \]\[ C = \frac{c}{r} \]Substituting \( C \) back into \( M(t) \), we have:\[ M(t) = \frac{c}{r} (1 - e^{-rt}) - kt \]

Step 4: Analyze the Limit of M(t) as t Approaches Infinity for k>0

As \( t \rightarrow \infty \), \( e^{-rt} \rightarrow 0 \) due to the nature of exponential decay, so the solution becomes:\[ M(t) \rightarrow \frac{c}{r} - kt \]Since \( c > k \), \( M(t) \) will generally reach a maximum and then decrease without reaching 0 again (because the constant elimination \( k \) outweighs decay at a sustained time frame).

Step 5: Consider k = 0 and Analyze For Limit as t Approaches Infinity

If \( k = 0 \), the elimination term disappears, and the equation simplifies to:\[ M(t) = \frac{c}{r} (1 - e^{-rt}) \]as \( t \rightarrow \infty \), \( e^{-rt} \rightarrow 0 \), therefore:\[ M(t) \rightarrow \frac{c}{r} \]This shows the final amount of the drug in the blood reaches a steady state of \( \frac{c}{r} \).

Key Concepts

Drug Absorption

Drug absorption is a process by which a drug enters the bloodstream from another body compartment like the digestive tract. This process is crucial in pharmacokinetics, which is the branch of pharmacology concerned with how drugs move through the body. In our problem, the drug enters a patient's blood from the gut. This entry is expressed as a mathematical function, with the rate of absorption described by \( c e^{-rt} \), where \( c \) and \( r \) are constants. Here, \( c \) represents the initial rate of absorption, and \( e^{-rt} \) reflects an exponentially decreasing trend over time. The gradual decrease indicates that over time, less of the drug is absorbed because the concentration difference between the gut and blood diminishes. Why? Because as the drug starts getting absorbed into the bloodstream, the driving force decreases, leading to this exponential decay in absorption rate. Recognizing the factors affecting drug absorption can help in predicting how quickly and effectively a drug reaches the bloodstream.

Exponential Decay

Exponential decay describes processes where a quantity decreases at a rate proportional to its current value. In the context of drug absorption, exponential decay captures the waning speed at which the drug enters the bloodstream over time. The formula \( e^{-rt} \) in our differential equation represents this idea. As time \( t \) increases, the value \( e^{-rt} \) approaches zero. This decay behavior is common not just in pharmacology but in many natural and economic processes. For instance, radioactivity and cooling laws obey similar principles. In drug absorption, this rapid initial absorption followed by a slowdown means that, while initially there is a surge of drug entering the system, as equilibrium is reached, the rate slows significantly. Understanding this concept helps evaluate long-term drug availability and effectiveness.

Initial Value Problems

Initial value problems are a type of differential equation that comes with an initial condition. Here, the condition specified is \( M(0) = 0 \), meaning initially the drug amount in the blood is zero. This initial value is crucial because it ensures a unique solution to the differential equation. By setting an initial state, we make precise predictions about how the system behaves over time based on the specified conditions. In our equation, knowing \( M(0) = 0 \) allowed us to solve for the integration constant \( C \), which then informs us of how drug levels evolve. It assures that when we apply the formula for drug concentration, it aligns with realistic scenarios—that is, no drug is present at the start. Initial value problems are pivotal in developing mathematical models in physics, economics, biology, and, as seen here, pharmacokinetics.