Analog computer model of the liver

Analog computer and analog method

An analog computer is a computing device that represents each variable by a continuously varying physical quantity (usually voltage) and solves equations by connecting electronic units that behave like the mathematical operations. The analog method solves a differential equation by building a circuit whose voltages obey the same equation (an analogue of the system) and recording the output.

The main units are:

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.

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

Equations

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.

 +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.

Pros and cons of the analog computer

Advantages of the analog computer

• Operates in parallel, so the solution is fast, even real time.
• Continuous solution; no step size or numerical round-off error.
• Interactive: parameters can be changed by a knob and the result seen immediately.
• Shows the physical meaning of the equation; good for control studies.

Disadvantages

• Accuracy is limited (about 0.1 to 1 percent) by component tolerance, noise and drift.
• Amplitude scaling and time scaling are needed, and are tedious.
• The size of the problem is limited by the number of units available.
• Hard to implement nonlinearities, logical decisions and data storage.
• Re-patching for a new problem takes time; poor repeatability and documentation.