Question

A physiological model A common assumption in modeling drug assimilation is that the blood volume in a person is a single compartment that behaves like a stirred tank. Suppose that the blood volume is a four-liter tank that initially has a zero concentration of a particular drug. At time \(t=0,\) an intravenous line is inserted into a vein (into the tank) that carries a drug solution with a concentration of \(500 \mathrm{mg} / \mathrm{L} .\) The inflow rate is \(0.06 \mathrm{L} / \mathrm{min}\). Assume that the drug is quickly mixed thoroughly in the blood and that the volume of blood remains constant. a. Write an initial value problem that models the mass of the drug in the blood, for \(t \geq 0\) b. Solve the initial value problem and graph both the mass of the drug and the concentration of the drug. c. What is the steady-state mass of the drug in the blood? d. After how many minutes does the drug mass reach \(90 \%\) of its steady-state level?

Short answer

Question: Write an initial value problem to model the mass of the drug in the blood, solve the initial value problem and graph the mass and concentration of the drug, determine the steady-state mass of the drug, and find the time it takes for the drug mass to reach 90% of its steady-state level. Answer: The initial value problem for the mass of the drug in the blood is given by the first-order linear differential equation: dy/dt = 30 for t ≥ 0, with the initial condition y(0) = 0. Solving this equation, we get y(t) = 30t. The concentration of the drug in the blood is given by c(t) = y(t)/V = (30t)/4, where V = 4 L. In this case, there is no steady state as there is no outflow of the drug. Assuming that the maximum concentration is 500 mg/L (the concentration of the drug solution in the intravenous line), it takes 60 minutes for the drug mass to reach 90% of the maximum concentration (450 mg/L).

Step-by-step solution

Identifying the independent and dependent variables

Let \(y(t)\) represent the mass of the drug in the blood at time \(t\). Since this problem involves drug assimilation over time, the independent variable is time \(t\) and the dependent variable is mass \(y(t)\).

Writing the problem

We must consider the inflow and outflow of the drug into the blood. The inflow rate is 0.06 L/min, and the drug mass coming in per minute is given by the concentration multiplied by the inflow rate (\(500\;\text{mg/L}\times 0.06\;\text{L/min} = 30\;\text{mg/min}\)). Since there is no outflow of the drug, the initial value problem can be represented by the following first-order linear differential equation: $$ \frac{dy}{dt} = 30 \quad \text{for} \quad t\geq 0, $$ with the initial condition \(y(0) = 0\), since there is initially no drug in the blood. #b- Solving the initial value problem and graphing results#

Solving the differential equation

This is a simple first-order linear differential equation with constant coefficients. We can solve it by integrating both sides: $$ \int \frac{dy}{dt} dt = \int 30 dt, $$ which gives us $$ y (t) = 30t + C, $$ where \(C\) is the integration constant. Using the initial condition \(y(0) = 0\), we find that \(C = 0\), so $$ y(t) = 30t. $$

Calculating the concentration

To find the concentration of the drug in the blood, we divide the mass of the drug by the volume of the blood: $$ c(t) = \frac{y(t)}{V} = \frac{30t}{4}, $$ where \(V =4\) L is the volume of blood.

Graphing the results

Now we can create graphs of the mass of the drug (\(y(t)\)) and the concentration of the drug (\(c(t)\)) as functions of time. The mass of the drug will increase linearly over time, while the concentration will also increase linearly. #c- Steady-state mass of the drug#

Finding the steady state

Steady state refers to the point where the mass of the drug in the blood no longer changes. However, the problem states that there's only inflow and no outflow of the drug from the blood. Therefore, there is no steady state in this case, and the mass of the drug will keep increasing linearly over time. #d- Finding the time to reach 90% of steady state#

Problem clarification

Since it was established that there's no steady state in this case, we cannot determine when the drug mass will reach 90% of its steady-state level. However, it's possible that there's a misunderstanding, and the problem is asking for the time it takes for the drug mass to reach 90% of its maximum concentration given by the drug dosage. In this case, let's assume that the maximum concentration is 500 mg/L (the concentration of the drug solution in the intravenous line).

Finding the time

Ninety percent of the maximum concentration is \(0.9\times 500 = 450\) mg/L. Using the previously determined concentration expression \(c(t) = \frac{30t}{4}\), we can write: $$ \frac{30t}{4} = 450. $$ Solving for \(t\), we get: $$ t = \frac{450\times 4}{30} =60\;\text{minutes}. $$ So, the mass of the drug will reach 90% of its maximum concentration (450 mg/L) after 60 minutes.

Key concepts

Initial Value Problem

An initial value problem is a type of differential equation that comes with an extra condition, known as the 'initial condition'. This condition gives us a starting value for the dependent variable we are trying to find. In the context of drug assimilation, the initial value problem is set up to describe how the mass of the drug changes over time.
For this scenario, the initial value problem consists of a first-order linear differential equation for the mass of the drug, denoted as \( y(t) \), with the initial condition \( y(0) = 0 \). This tells us that at time \( t=0 \), the mass of the drug in the blood is zero, meaning the drug had not been administered yet.
To summarize:

• The differential equation represents the relationship between the change in drug mass over time and the inflow rate of the drug.
• The initial condition helps to find the particular solution that fits the real-world scenario being modeled.

Steady State

The steady state in the context of a drug assimilation model is the point at which the mass of the drug in the system doesn't change anymore because the rates of inflow and outflow are equal. However, in our exercise, the model does not allow for a steady state. Why? Because there is no outflow; the drug is only entering the system at a constant rate.
This means the mass of the drug will continue to increase as long as the intravenous line feed continues. It can be understood that for a real steady state, the system must have outflow equal to the inflow, ensuring the system's balance. As our model is missing this outflow component, no steady-state mass is achievable. It's important in modeling to consider both inputs and outputs for a complete understanding.

Drug Assimilation Model

A drug assimilation model helps us understand how a drug moves through a body or a specific part of it. In this exercise, the model assumes a single compartment (the blood) in which the drug is mixed thoroughly.
The model needs to account for several factors:

• The rate at which the drug enters the blood system - in this case, 0.06 L/min with a concentration of 500 mg/L.
• The constraint of a fixed volume, here modeled as 4 liters of blood.
• The absence of drug outflow, unique to this model's assumptions but critical to note for understanding real-world limitations or adjustments.

This simplified model helps in predicting how the drug concentration in the blood will change over time, aiding in potentially adjusting dosages or understanding physiological impacts after administration.

Linear Differential Equation

A linear differential equation describes a relationship between a function and its derivatives. The model given in this problem is characterized by a simple first-order linear differential equation.
The equation is \( \frac{dy}{dt} = 30 \), which tells us how the mass of the drug in the blood changes over time. The number 30 is derived from multiplying the inflow rate by the concentration (0.06 L/min ⨉ 500 mg/L = 30 mg/min), representing the drug's rate of mass inflow.
The solution to this differential equation, given the initial condition \( y(0) = 0 \), is \( y(t) = 30t \). This tells us that the mass of the drug increases linearly with time—an elegant way to predict outcomes and inform dosing schedules.