Problem 12 - Biocalculus Calculus for the Life Sciences

Question

12\. Systemic lupus erythematosus is an autoimmune disease in which some immune molecules, called antibodies, target DNA instead of pathogens. This can be treated by injecting drugs that absorb the offending antibodies. The antibodies are found in both the bloodstream and in organs, and this can be modeled using a two-compartment model: A system of differential equations describing the amount of antibody in each compartment is $$\begin{aligned} \frac{d x_{1}}{d t} &=G+k_{21} x_{2}-k_{12} x_{1}-k x_{1} \\\ \frac{d x_{2}}{d t} &=k_{12} x_{1}-k_{21} x_{2} \end{aligned}$$ where \(G\) is the rate of generation of antibodies, \(k\) is the rate at which the drug treatment removes antibody from the bloodstream, and \(k_{i j}\) is the rate of flow of antibody from compartment \(i\) to \(j .\) The variables \(x_{1}\) and \(x_{2}\) are the amounts of antibody in the bloodstream and organs, respectively, measured in \(\mu \mathrm{g}\) . (See also Exercise 16 in the Review Sec- tion of this chapter.) $$\begin{array}{l}{\text { (a) Use a change of variables to obtain a homogene- }} \\ {\text { ous system of differential equations describing the }} \\\ {\text { situation. }} \\ {\text { (b) What is the general solution to the differential equa- }} \\ {\text { tions in part (a)? }} \\ {\text { (c) What is the general solution obtained in part (b) in }} \\ {\text { terms of the original variables } x_{1} \text { and } x_{2} ?}\end{array}$$

Step-by-Step Solution

Answer

Use the homogeneous change of variables to solve the system and express as general solutions in original variables.

Step 1: Understand the Problem

We have a two-compartment model described by a system of differential equations. The task is to transform this system using a change of variables, solve the homogeneous system, and then express the solution in terms of the original variables.

Step 2: Set Up the Change of Variables

Define new variables to remove the inhomogeneity. Let \( y_1 = x_1 - \frac{G}{k+k_{12}} \) and \( y_2 = x_2 \). This setup helps eliminate the term involving \( G \) in the first equation.

Step 3: Derive the Homogeneous System

Substitute \( x_1 = y_1 + \frac{G}{k+k_{12}} \) and \( x_2 = y_2 \) into the original system.\[\frac{d y_{1}}{dt} = k_{21} y_{2} - (k_{12} + k)y_{1}\]\[\frac{d y_{2}}{dt} = k_{12} y_{1} - k_{21} y_{2}\]This forms a homogeneous system as the constant \( G \) term has been canceled out.

Step 4: Solve the Homogeneous System

Find the characteristic equation by assuming solutions of the form \( y_1(t) = Ae^{\lambda t} \) and \( y_2(t) = Be^{\lambda t} \). The characteristic equation becomes:\[(\lambda + k + k_{12})(\lambda + k_{21}) = k_{12}k_{21}\]Solve this to find the eigenvalues \( \lambda \).

Step 5: General Solution for y_1 and y_2

The general solution of the homogeneous system is given by:\[ y_1(t) = C_1 e^{\lambda_1 t} + C_2 e^{\lambda_2 t} \]\[ y_2(t) = C_3 e^{\lambda_1 t} + C_4 e^{\lambda_2 t} \] where the constants \( C_1, C_2, C_3, \) and \( C_4 \) are determined by initial conditions.

Step 6: Express in Original Variables

Substitute back to express in terms of \( x_1 \) and \( x_2 \):\[ x_1(t) = y_1(t) + \frac{G}{k+k_{12}} = C_1 e^{\lambda_1 t} + C_2 e^{\lambda_2 t} + \frac{G}{k+k_{12}} \]\[ x_2(t) = y_2(t) = C_3 e^{\lambda_1 t} + C_4 e^{\lambda_2 t} \]

Key Concepts

two-compartment model

The two-compartment model is a fundamental concept in biology that's used to understand how substances move between two areas or compartments. In the context of systemic lupus erythematosus (SLE), it helps us study the dynamics of antibodies in the human body. The compartments in this case are the bloodstream and the organs. The model utilizes differential equations to simulate real-life biological processes.

• Compartments: Represent sections where substances, like antibodies, are distributed.
• Interactions: Describe how the substance moves or changes within and between these compartments.

This model is crucial for understanding medication effects, especially for diseases like SLE, where precise targeting is vital.

eigenvalues

Eigenvalues are essential in solving systems of differential equations, like the two-compartment model. They help determine the behavior of the system over time, especially when dealing with complex dynamic systems. Analyzing eigenvalues provides insights into the system's stability and long-term tendencies.

• Finding Eigenvalues: Derived from the characteristic equation of the system.
• System Influence: Different eigenvalues affect how solutions evolve in the two compartments.

In our exercise, eigenvalues help solve the homogeneous system, which reveals how quickly antibodies decay without outside interference.

systemic lupus erythematosus

Systemic lupus erythematosus (SLE) is an autoimmune disease where antibodies, supposed to target pathogens, mistakenly attack the body's own tissues. It can affect multiple organs, causing inflammation and organ damage over time. In the two-compartment model, the antibody movement between the bloodstream and organs is crucial for understanding and managing SLE.

• Antibody Role: Misdirected antibodies are a central feature of SLE.
• Treatment: Involves drugs that reduce harmful antibodies, modeled as differential equations.

This model aids in evaluating treatments that may counteract these antibodies' effects in both compartments.

homogeneous system

A homogeneous system of differential equations has zero as all its non-variable terms, meaning it represents the system's inherent behavior without external influences. By using a change of variables, we transformed our two-compartment model into a homogeneous system, simplifying the problem.

• Transformation: Setting up variables to cancel constant terms helps isolate the system's natural dynamics.
• Simplification: Solving becomes easier, focusing purely on how variables interact.

This step helps understand the natural ebb and flow of antibodies between bloodstream and organs absent new antibodies or treatment.

change of variables

A change of variables is a method used to simplify differential equations by shifting to a new set of variables. This technique allows us to remove constants, making it easier to solve the equations. In our case, it transforms the system into a homogeneous form.

• New Variables: Defined to remove the generation rate of antibodies, focusing on balance.
• Understanding: Simplifies predictions by minimizing complexity.

By choosing appropriate variables in our model, the problem becomes more manageable, allowing us to focus on interactions between compartments.