---
title: "Significance of differential equations; liver model"
subject: "Simulation and Modeling"
chapter: 3
key: "semester_8/simulation_and_modeling:diff-eq-significance-liver"
source: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/diff-eq-significance-liver
site: BCT Engineering
---

# Significance of differential equations; liver model

Simulation and Modeling · Chapter 3: Continuous system simulation

- Asked in: 2074 Bhadra, 4+6 marks
- Video: [Analog Computer Model of Liver | Mathematical Model & Differential Equations](https://www.youtube.com/watch?v=STVwmytFlR8) — Engivault
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/diff-eq-significance-liver
- Whole chapter: https://www.bctengineer.com/md/semester_8/simulation_and_modeling/chapter-3

## Question

Write down the significance of differential equations. Explain the analog computer model of the liver with necessary equation and diagrams.

## Short answer

Differential equations express continuous systems' rates of change, enabling prediction and simulation study. The liver model uses three compartments—blood, liver, bile—with equations dx1/dt = -k1x1+k2x2, dx2/dt = k1x1-(k2+k3)x2, dx3/dt = k3x2. Its analog circuit employs integrators and potentiometers set to k1, k2, k3, producing outputs -x1, -x2, -x3 from initial X0.

## Answer

#### Significance of differential equations
- Most physical, biological, electrical and economic systems change continuously with time, and the rate of change is naturally expressed as a differential equation.
- They give the mathematical model of a continuous system, from which the response to any input can be predicted.
- Examples: `M x'' + D x' + K x = F(t)` for a suspension, `L q'' + R q' + q/C = V` for a circuit, `dP/dt = kP` for a population.
- Most real equations have no closed-form solution, so they are solved by simulation: numerically on a digital computer or with an analog computer.
- They help in studying stability, transient response and effect of parameter changes without building the real system.
- They are the basis of control, signal processing and system analysis.

#### Liver model
The liver removes substances from the blood and passes them into bile. A common test injects a small amount of a radioactive-tagged dye into the blood and measures how the amount in blood, in liver and in bile varies with time. The system is modelled by three compartments (a compartmental model):

- `x1(t)`: amount of tracer in the blood.
- `x2(t)`: amount of tracer in the liver.
- `x3(t)`: amount excreted into bile (and intestine).

Assumptions: each flow is proportional to the amount in the compartment it leaves (first-order kinetics); the compartments are well mixed; the tracer injected at t = 0 gives `x1(0) = X0`, `x2(0) = x3(0) = 0`.

```text
  Blood x1 --k1--> Liver x2 --k3--> Bile x3
           <--k2--
```

#### Equations
```text
dx1/dt = -k1 x1 + k2 x2
dx2/dt =  k1 x1 - (k2 + k3) x2
dx3/dt =  k3 x2
```
- k1: rate of uptake from blood by the liver.
- k2: rate of return from liver to blood.
- k3: rate of excretion from liver to bile.

The total `x1 + x2 + x3 = X0` stays constant (mass balance), which is a useful check on the model.

#### Analog computer model
Each integrator gives the output with sign inverted, so the integrator outputs are `-x1`, `-x2`, `-x3`. Coefficient potentiometers are set to k1, k2, k3, and inverters supply the positive signs where needed.

```text
 +x2 -[k2]--\
             +-->[Int 1]--> -x1
 -x1 -[k1]--/
 +x1 -[k1]--\
             +-->[Int 2]--> -x2
 -x2 -[k2+k3]/
 +x2 -[k3]---->[Int 3]--> -x3
```
(+x1 and +x2 are obtained from -x1 and -x2 by inverters.)

Connections:
1. Integrator 1 input = `-k1 x1 + k2 x2` (the first from the output `-x1` through pot k1, the second from `+x2` through pot k2). Output `-x1`; its initial condition is set to `-X0`.
2. Integrator 2 input = `k1 x1 - (k2 + k3) x2`. Output `-x2`; initial condition zero.
3. Integrator 3 input = `k3 x2`. Output `-x3`; initial condition zero.
4. Outputs are plotted on a recorder against time.

The recorded curves show `x1` falling, `x2` rising to a peak and then falling, and `x3` rising towards X0. The values of k1, k2, k3 are adjusted till the curves match clinical data, and a diseased liver is shown by a smaller k3.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Why are most real differential equations solved by simulation?
   A. They have no closed-form solution
   B. They are always linear
   C. They contain no time variable
   D. They are too simple
   Answer: A. Most real equations cannot be solved in closed form, so numerical or analog simulation is used. (Key point: Why simulation for differential equations)

2. In the liver model, dx2/dt equals:
   A. -k1 x1 + k2 x2
   B. k1 x1 - (k2 + k3) x2
   C. k3 x2
   D. k1 x1 + k2 x2 + k3 x2
   Answer: B. The liver gains from the blood (k1 x1) and loses to the blood (k2 x2) and to the bile (k3 x2). (Key point: Compartment equation)

3. In the analog liver model, what initial condition is set on integrator 1?
   A. Zero
   B. +X0
   C. -X0, because the integrator output is -x1
   D. -k1
   Answer: C. Integrator 1 outputs -x1 and x1(0) = X0, so its initial condition is -X0; the others start at zero. (Key point: Integrator initial condition)

4. What do the recorded curves of the liver model show?
   A. x1 rises, x2 falls, x3 constant
   B. x1, x2, x3 all fall
   C. x2 constant and x3 falls
   D. x1 falls, x2 rises to a peak then falls, x3 rises toward X0
   Answer: D. Tracer leaves the blood, passes through the liver, and accumulates in bile approaching X0. (Key point: Expected curves)

## Related in Simulation and Modeling

- [Analog computer model of the liver](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Aanalog-computer-liver-model) — Simulation and Modeling, chapter 3, asked 5×
- [Significance of differential equations](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Adiff-eq-significance) — Simulation and Modeling, chapter 3
- [Continuous system simulation; analog computer example](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Acontinuous-simulation-analog-example) — Simulation and Modeling, chapter 3

## Related in other subjects

- [Least squares significance, normal equations](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Aleast-squares-significance-normal) — Numerical Method, chapter 4
- [Maxwell's equations, integral to differential](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Amaxwell-integral-to-differential-explain) — Engineering Physics, chapter 9, asked 3×