Problem 29 - Calculus for Biology and Medicine

Question

Consider a two-compartment model where, instead of having a separate reservoir feeding into tank 1, the two tanks are separated by two pipes, one of which carries water from tank 1 to tank 2 , at rate \(q\), and the other carries water from tank 2 to \(\operatorname{tank} 1\), at the same rate \(q\). A schematic and diagram of the flows is given in Figure \(8.59\). (a) Explain why, although there is no net flow between the tanks, we would expect the concentrations in the tanks to change over time. (b) Explain why the change in concentrations over time can be modeled using differential equations: $$ \begin{array}{l} \frac{d C_{1}}{d t}=\frac{q}{V_{1}}\left(C_{2}-C_{1}\right) \\ \frac{d C_{2}}{d t}=\frac{q}{V_{2}}\left(C_{1}-C_{2}\right) \end{array} $$ (c) To solve the differential equations in \((8.95)\) start by assuming that \(V_{1}=V_{2} .\) Then define \(C(t)=\frac{1}{2}\left(C_{1}+C_{2}\right)\) and by deriving a differential equations for \(d C / d t\) and explain why \(C(t)\) is constant. (d) Using the fact that \(C(t)\) is a constant, eliminate \(C_{2}(t)\) from the equation for \(\frac{d C_{1}}{d t}\). Solve the equation you then obtain, and write down expressions for \(C_{1}(t)\) and \(C_{2}(t)\). (e) Use the expression from part (d) to explain why, no matter what the starting values for \(C_{1}(0)\) and \(C_{2}(0)\) are, we expect \(C_{1}(t)\) and \(C_{2}(t)\) to converge to the same limit as \(t \rightarrow \infty\). (f) To solve the differential equations in \((8.95)\) in the most general case \(\left(V_{1} \neq V_{2}\right)\), let \(C(t)=\frac{V_{1} C_{1}+V_{2} C_{2}}{V_{1}+V_{2}}\) (the weighted average of the concentrations in the two tanks). Explain why \(C(t)\) is a constant. (g) Using the fact that \(C(t)\) is a constant, eliminate \(C_{2}(t)\) from the equation for \(\frac{d C_{1}}{d t} .\) Solve the equation you then obtain, and write down expressions for \(C_{1}(t)\) and \(C_{2}(t)\). (h) Use the expression from part (g) to explain why, no matter what the starting values for \(C(t)\) and \(C_{2}(t)\) are, we expect \(C_{1}(t)\) and \(C_{2}(t)\) to converge to the same limit as \(t \rightarrow \infty\).

Step-by-Step Solution

Answer

The concentrations change due to mixing despite no net flow. Using differential equations, we show concentrations converge to a common value, influenced by the initial conditions and the tanks' volumes.

Step 1: Understanding Non-Net Flow and Concentration Change

Even though there is no net flow of water between the tanks, the concentrations change because water with potentially different concentrations is continually being exchanged between the two tanks. As a result, the chemical makeup in each tank is altering over time as they mix.

Step 2: Using Differential Equations to Model Concentration Changes

The change in concentration can be described using the differential equations: \( \frac{d C_{1}}{d t}=\frac{q}{V_{1}}(C_{2}-C_{1}) \) and \( \frac{d C_{2}}{d t}=\frac{q}{V_{2}}(C_{1}-C_{2}) \). These equations represent the rates of change of concentrations based on the flow of water and the difference in concentrations between the tanks.

Step 3: Assuming Equal Volumes and Defining the Average Concentration

Assume \( V_1 = V_2 \) and define \( C(t) = \frac{1}{2}(C_1 + C_2) \). Differentiating \( C(t) \) with respect to \( t \) yields \( \frac{dC}{dt} = \frac{1}{2} \left( \frac{dC_1}{dt} + \frac{dC_2}{dt} \right) \) which, given the rate equations from Step 2, results in zero. This shows that \( C(t) \) is constant over time.

Step 4: Solving the Differential Equation for \( C_1(t) \)

Using the constancy of \( C(t) \), replace \( C_2(t) \) in \( \frac{dC_1}{dt} = \frac{q}{V_1}(C_2 - C_1) \) with \( 2C(t) - C_1(t) \). This modifies the equation to \( \frac{dC_1}{dt} = \frac{2q}{V}(C - C_1) \), a standard first-order linear differential equation which can be solved to obtain an exponential decay solution.

Step 5: Long-term Behavior of \( C_1(t) \) and \( C_2(t) \)

The solution from Step 4 shows that both \( C_1(t) \) and \( C_2(t) \) approach the constant \( C \) as \( t \to \infty \). Thus, regardless of initial concentrations, the concentrations in both tanks will converge to this average value in the long-term.

Step 6: General Case with Different Volumes

Let \( C(t) = \frac{V_1 C_1 + V_2 C_2}{V_1 + V_2} \). Differentiating \( C(t) \) shows that the weighted average remains constant due to mass conservation. This follows from a similar argument as in Step 3, considering the individual differential equations for \( C_1 \) and \( C_2 \).

Step 7: Solving for \( C_1(t) \) with General Volumes

Using the constancy of \( C(t) \), express \( C_2 \) in terms of \( C(t) \) and \( C_1(t) \). Substitute this into the equation for \( \frac{dC_1}{dt} \) which yields a solvable first-order linear differential equation. Solving this equation gives expressions for \( C_1(t) \) and \( C_2(t) \) that also show exponential convergence.

Step 8: Convergence of \( C_1(t) \) and \( C_2(t) \) in General Case

Regardless of initial conditions, \( C_1(t) \) and \( C_2(t) \) converge to a common limit given by the constant \( C(t) \). The exponential terms in their solutions decrease over time, causing the concentrations in the two tanks to equalize as \( t \to \infty \).

Key Concepts

Two-Compartment Model

The two-compartment model is a fundamental concept used in biology to understand how substances like drugs, nutrients, or chemicals move between two areas or compartments. Imagine having two tanks of water that are partially connected at the bottom; that's similar to the compartments here. In the exercise, we have two tanks with water that can flow between them in both directions. This setup mimics many real-world biological systems, where substances need to be distributed evenly across different areas.

While it may seem odd that there is no net flow of water—meaning the same amount of water enters tank 1 from tank 2 as leaves tank 1 to tank 2—the concentrations of any dissolved substances can still change. Why? Because the solution in each tank could start with different concentrations, and as water flows, the tanks exchange these concentrations, altering the chemical makeup of each tank over time. Understanding this model is essential for students looking to apply math to biological systems, as it illustrates the dynamics of transport and distribution in a simplified manner.

Rate of Change

In the context of differential equations, the rate of change is the way we describe how one quantity changes with respect to another, typically over time. For our two-compartment model, we focus on how the concentrations, or the amounts of dissolved substances, within each compartment change as water moves between them.

The differential equations used here are given by:

• \( \frac{d C_1}{d t} = \frac{q}{V_1} (C_2 - C_1) \)
• \( \frac{d C_2}{d t} = \frac{q}{V_2} (C_1 - C_2) \)

Each equation tells you how the concentration in one tank changes by relating it to the flow rate \( q \), the volume of the tank \( V \), and the difference in concentrations between the two tanks.

This change arises because there's an exchange of water that effectively mixes the concentrations of each tank, causing them to either increase or decrease over time until they eventually stabilize. Differential equations like these help predict how long it will take for the tanks to reach a stable state where concentrations no longer change significantly.

Convergence of Solutions

When we talk about the convergence of solutions in the context of differential equations, we refer to how and when the values, like concentrations in our tanks, settle at a common point over time. This concept helps understand the long-term behavior of the system.

In our model, you'll find that despite any initial differences in concentration, as time \( t \) progresses, the concentrations \( C_1(t) \) and \( C_2(t) \) in both tanks tend to become equal. This is what we mean by convergence: the solutions of the differential equations move towards a common balance point.

The mathematical solution involves considering an average concentration \( C(t) \), which remains constant due to mass conservation. Both tanks are influenced by exponential decay functions that describe how the concentration in each tank evolves. Consequently, irrespective of the starting concentrations, the system stabilizes, and the concentrations converge to a median level determined by the initial setup. Understanding convergence is crucial for predicting how biological and chemical systems will behave in the long term.