### Compartment Analysis: Drug Concentration in Blood and Cerebrospinal Fluid#### Problem Statement:A patient initially has a concentration of 50 mg/L of morphine in her blood and 70 mg/L in her cerebrospinal fluid. Suppose that the drug moves into the cerebrospinal fluid at a per-capita rate of 0.4 and that the amount that is moved back from the cerebrospinal fluid to the bloodstream is 0.2. #### Metabolism Assumptions:Further suppose that the metabolism of medicine in the bloodstream is 0.1 and the metabolism of the drug in the cerebrospinal fluid is 0.2.#### Compartment Diagram:The diagram is divided into two compartments:1. **Blood**2. **Cerebrospinal Fluid (Brain)**(Note: The provided image contains two empty boxes here that students need to fill as part of the exercise.)#### Variables:Let $ x(t) $ represent the concentration of the drug in the blood.Let $ y(t) $ represent the concentration of the drug in the cerebrospinal fluid (brain area).Complete this compartment diagram.#### Explanation of the Mathematical Model:The given differential equations represent the rates of change in drug concentration in the blood and cerebrospinal fluid based on the given metabolism and transfer rates.##### Differential Equations:\[\frac{dx}{dt} = -0.5x + 0.2y\]\[\frac{dy}{dt} = 0.4x - 0.4y\]#### Task:Solve for $ x(t) $ and $ y(t) $ given the initial conditions stated above. Simplify each expression as much as possible and write them in the blanks below.- **Initial Concentrations:** - Blood: $ x(0) = 50 $ mg/L - Cerebrospinal Fluid: $ y(0) = 70 $ mg/L##### Solutions:X(t) = __________________________________________________________Y(t) = __________________________________________________________**Note:** On the next page, please show all your calculations.---By analyzing and solving these differential equations, we can understand how the morphine concentration changes over time in the patient's bloodstream and cerebrospinal fluid, taking into account both metabolism and inter-compartment transfer rates.

5. A patient initially has a concentration of 50mg/L of morphine in her blood and 70mg/L in her cerebrospinal fluid. Suppose that the drug moves into the cerebrospinal fluid at a per-capita rate of .4 and that the amount that is moved back from the cerebrospinal fluid to the bloodstream is .2 . Suppose further that the metabolism of medicine in the bloodstream is .1 and the metabolism of the drug in the cerebrospinal fluid is .2 Blood cerebrospinal fluid ( brain ) Let x(t) represent the concentration in the blood and y(t) represent the concentration in the

5. A patient initially has a concentration of 50mg/L of morphine in her blood and 70mg/L in her cerebrospinal fluid. Suppose that the drug moves into the cerebrospinal fluid at a per-capita rate of .4 and that the amount that is moved back from the cerebrospinal fluid to the bloodstream is .2 . Suppose further that the metabolism of medicine in the bloodstream is .1 and the metabolism of the drug in the cerebrospinal fluid is .2 Blood cerebrospinal fluid ( brain ) Let x(t) represent the concentration in the blood and y(t) represent the concentration in the

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

6th Edition

ISBN:9781337111348

Author:Bruce Crauder, Benny Evans, Alan Noell

Publisher:Bruce Crauder, Benny Evans, Alan Noell

Chapter5: A Survey Of Other Common Functions

Section5.1: Logistic Functions

Problem 17E: World Population The following table shows world population N, in billions, in the given year. Year...

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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

### Compartment Analysis: Drug Concentration in Blood and Cerebrospinal Fluid

#### Problem Statement:
A patient initially has a concentration of 50 mg/L of morphine in her blood and 70 mg/L in her cerebrospinal fluid. Suppose that the drug moves into the cerebrospinal fluid at a per-capita rate of 0.4 and that the amount that is moved back from the cerebrospinal fluid to the bloodstream is 0.2.

#### Metabolism Assumptions:
Further suppose that the metabolism of medicine in the bloodstream is 0.1 and the metabolism of the drug in the cerebrospinal fluid is 0.2.

#### Compartment Diagram:
The diagram is divided into two compartments:
1. **Blood**
2. **Cerebrospinal Fluid (Brain)**

(Note: The provided image contains two empty boxes here that students need to fill as part of the exercise.)

#### Variables:
Let $ x(t) $ represent the concentration of the drug in the blood.
Let $ y(t) $ represent the concentration of the drug in the cerebrospinal fluid (brain area).

Complete this compartment diagram.

#### Explanation of the Mathematical Model:
The given differential equations represent the rates of change in drug concentration in the blood and cerebrospinal fluid based on the given metabolism and transfer rates.

##### Differential Equations:
\[
\frac{dx}{dt} = -0.5x + 0.2y
\]

\[
\frac{dy}{dt} = 0.4x - 0.4y
\]

#### Task:
Solve for $ x(t) $ and $ y(t) $ given the initial conditions stated above. Simplify each expression as much as possible and write them in the blanks below.

- **Initial Concentrations:**
- Blood: $ x(0) = 50 $ mg/L
- Cerebrospinal Fluid: $ y(0) = 70 $ mg/L

##### Solutions:
X(t) = __________________________________________________________

Y(t) = __________________________________________________________

**Note:** On the next page, please show all your calculations.

---

By analyzing and solving these differential equations, we can understand how the morphine concentration changes over time in the patient's bloodstream and cerebrospinal fluid, taking into account both metabolism and inter-compartment transfer rates.

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