Question

A certain drug is being administered intravenously to a hospital patient, Fluid containing \(5 \mathrm{mg} / \mathrm{cm}^{3}\) of the drug enters the patient's bloodstream at a rate of \(100 \mathrm{cm}^{3} \mathrm{hr}\). The drug is absorbed by body tissues or otherwise leaves the bloodstream at a rate proportional to the amount present, with a rate constant of \(0.4(\mathrm{hr})^{-1}\). (a) Assuming that the drug is always uniformly distributed throughout the bloodstream, write a differential equation for the amount of the drug that is present in the bloodstream, at any time. (b) How much of the drug is present in the bloodstream after a long time?

Short answer

Answer: 1250 mg

Step-by-step solution

(Step 1: Write down the given parameters)

The parameters we have are: 1) Drug concentration in the fluid: \(5\frac{mg}{cm^3}\) 2) Rate at which the fluid enters the bloodstream: \(100\frac{cm^3}{hr}\) 3) Rate constant of the drug leaving the bloodstream: \(0.4(hr)^{-1}\)

(Step 2: Setup the differential equation)

Let A(t) be the amount (in mg) of the drug in the bloodstream at time t (in hours). The rate of the drug entering the bloodstream is given by the product of the drug concentration and the fluid inflow rate: Rate in: \(5\frac{mg}{cm^3} \times 100\frac{cm^3}{hr} = 500\frac{mg}{hr}\) The rate of the drug leaving the bloodstream is proportional to the amount present at a given time, with the rate constant \(0.4(hr)^{-1}\): Rate out: \(0.4(\mathrm{hr})^{-1} \times A(t)\) Now we can write the differential equation representing the rate of change of the drug amount in the bloodstream with respect to time: \(\frac{dA(t)}{dt} = \text{Rate in} - \text{Rate out}\)

(Step 3: Write the differential equation)

Plugging the expressions for the rate in and rate out into the equation, we get: \(\frac{dA(t)}{dt} = 500 - 0.4A(t)\)

(Step 4: Solve the differential equation)

In order to solve the differential equation, we can use an integrating factor in the form of \(e^{\int -0.4dt} = e^{-0.4t}\). Multiplying both sides of the equation by the integrating factor, we get: \(e^{-0.4t}\frac{dA(t)}{dt} + 0.4e^{-0.4t}A(t) = 500e^{-0.4t}\) The left-hand side of the equation is now the derivative of \(e^{-0.4t}A(t)\). So we can write: \(\frac{d}{dt}(e^{-0.4t}A(t)) = 500e^{-0.4t}\) Integrate both sides with respect to t: \(e^{-0.4t}A(t) = -\frac{500}{0.4}e^{-0.4t} + C\)

(Step 5: Find the particular solution)

Divide both sides by \(e^{-0.4t}\), we get: \(A(t) = -\frac{500}{0.4} + Ce^{0.4t}\) Now, we need to find the drug amount in the bloodstream after a long time. As time goes to infinity, the exponential term will approach zero: \(A(\infty) = -\frac{500}{0.4}\) Therefore, the amount of the drug in the patient's bloodstream after a long time is 1250 mg.

Key concepts

Intravenous Drug Administration

Intravenous drug administration directly introduces medication into a patient's bloodstream. This method is fast and efficient because it bypasses the digestive system, allowing the drug to act almost immediately. In the given scenario, the medication is mixed in a liquid form at a concentration of \(5 \ \text{mg/cm}^3\) and infused at a steady rate of \(100 \ \text{cm}^3/ ext{hr}\). By controlling the infusion rate, doctors can manage the drug's dosage and maintain desired plasma drug levels.

Intravenous administration is especially useful for delivering precise doses of medication, which is critical here to ensure the patient receives a full therapeutic effect without delays. However, because the drug's presence in the bloodstream is regulated by both the infusion rate and how the body processes and absorbs it, understanding other concepts, like the rate of change and steady state solution, becomes essential to predict and control dosing outcomes effectively.

Exponential Decay

Exponential decay describes how quantities diminish over time at a rate proportional to their current value. In the context of drug distribution, exponential decay is crucial in understanding how the drug leaves the bloodstream. The rate constant given in the problem, \(0.4 (hr)^{-1}\), indicates that the amount of drug being removed is proportional to the current concentration of the drug in the bloodstream.

This mathematical construct ensures that the greater the quantity present, the faster it will deplete, reflecting real-life physiological processes where the body metabolizes or eliminates drugs at rates dependent on their concentration. Using exponential decay, healthcare professionals can estimate how long it takes for a certain percentage of the drug to clear from the system. This is important for planning dosage intervals and preventing potential toxicity from drug accumulation.

Rate of Change

The rate of change in this context is crucial to modeling how the drug concentration alters in the bloodstream over time. It is the difference between the inflow rate, \(500 \ \text{mg/hr}\), and the outflow rate, proportional to the present amount \(A(t)\), expressed in the differential equation:
\[ \frac{dA(t)}{dt} = 500 - 0.4A(t) \]
This equation depicts that the amount of drug will increase as long as the infusion rate exceeds the elimination rate. Over time, this helps determine when the rate of elimination will balance the rate of administration.

Understanding the rate of change provides a foundation for predicting how long it will take to reach a therapeutic level of the drug, or when it might drop below that level, requiring a renewed dose. This underscores the importance of mathematical modeling in pharmacology, providing insights into the kinetics of drug distributing across the patient's system.

Steady State Solution

A steady state solution is a key concept showing when the system reaches a point where the inflow of the drug is balanced by the outflow, and hence the amount stays constant over time. This is described mathematically by setting the derivative in the differential equation to zero, which depicts no change over time:
\[ 0 = 500 - 0.4A(t) \]
Solving for \(A(t)\), we find that \(A(t) = 1250 \ \text{mg}\). This indicates that, eventually, the drug amount stabilizes at 1250 mg without further increase or decrease, assuming no changes in the administration or elimination parameters.

The steady state is a valuable concept in pharmacology and medicine, because it reflects the effectiveness of the drug as well as its safety profile over an extended period. Calculating this balance allows healthcare providers to understand the long-term effects of continuous drug administration and adjust dosages as necessary to maintain efficacy and minimize potential side effects.