Duality

The term “duality” refers to the ability to treat two different types of problems identically as a result of their differential equations being identical. Consider, for example, the equation for the simple LRC circuit:

Now consider the equation for velocity of a mass on a spring, with damping:

Mathematically, the two equations are the same. The only difference is in the names of the parameters.

Damping corresponds to resistance. V=iR corresponds to F=cv.

The spring corresponds to a capacitor. q=CV corresponds to x=F/k.

The mass behaves like an inductor. V=Ldi/dt corresponds to F=m dv/dt.

It is not surprising, then, that inductance is sometimes called “inertance,” in analogy to the inertia of a mass. Some other things to consider are:

1. Regardless of the force at time t=0, a mass initially at rest will have a velocity of zero at t=0. Similarly, the current in an inductor cannot change instantaneously. This is what inertia really means; an object will keep moving unless it is operated on by a force over some time period.

2. In contrast, the charge on a capacitor or the displacement of a spring can, in theory, change instantaneously if the force changes instantaneously.

3. For a damper, velocity can change instantaneously, but displacement cannot.

A third example of duality is in Fluid Flow. Consider a tube, as shown to the right, and the relationship between pressure and flow rate.

There are three phenomena that affect the relationship between flow and pressure. The first is a resistance caused by viscosity. You will learn about this in fluid mechanics. For this phenomenon, pressure is directly proportional to flow rate.

The second phenomenon is the expanding of the tube as a result of the pressure. This actually obeys a non-linear relationship, but if the tube doesn’t expand too much, it can be approximated linearly:

This simply says that an increase in pressure will cause the walls to stretch and thus make the volume inside the tube larger. This is just like a capacitor, with volume being charge and pressure being voltage, that is . Note that for the capacitance, the is across the tube wall, not from one end of the tube to the other. Thus, this is analogous to having a capacitor shunted to ground, like the figure to the right.

If this is an aorta, then implies that . In other words, the “current” in the capacitor is the rate at which blood is being stored in the aorta as a result of the increased vessel diameter.

The value of the capacitance can be deduced from the definition , analogous to . Lets assume that a pressure change from systole to diastole changes the radius of the aorta by 10%. Then (i.e. the volume at higher pressure minus the volume at lower pressure). The change in pressure is just , which might be 120 mmHg – 80 mmHg = 40 mmHg for a normal human. (Note that 1 mmHg is equal to 1330 dynes/cm²). It is thus an easy matter to solve for C with:

The third phenomenon is the inertia of the fluid. Balancing forces we get that:

But since ,

So the inductance in this case is .

It is legitimate to ask where the capacitor and inductor really belong in the circuit. The tube does not consist of a resistor followed by a capacitor to ground followed by an inductor. The resistance, capacitance, and inductance are all distributed throughout the tube. In circuit theory, this can be modeled as a transmission line, which ends up providing wave solutions. In vascular fluid mechanics, wave phenomena occur as well, and these have been heavily studied. However, if the overall interest is to understand what the general flow rates are in the arterial system, one can use the model pictured above for each segment of the artery tree, combine these elements as appropriate, and solve the simple circuit equations.

You may also hear about “acoustic” impedance. There is an analogy between acoustics, which deals with pressure and velocity within a medium, and the RLC circuit. However, the analogy is not as strong as that with mechanical impedance. For a plane wave, it can be shown that the pressure is directly proportional to rc u, where r is the density of the medium, c is the speed of sound in the medium (assumed constant), and u is the velocity of the air disturbance. One can, of course, talk about the mass of the medium (inertia) and about the compressibility of the medium (capacitance). However, the acoustic impedance does not relate to a pressure drop across some element. In fact, classical acoustics leaves out damping, which would correspond to resistance. Acoustic impedance relates more to the reflection of acoustic waves from surfaces. This is exactly the phenomenon exploited in ultrasound. Any time a sound wave hits an interface within the body where the sound speed changes, a wave is reflected back. The time required for the wave to reach the receiver indicates the depth of the interface. The larger the impedance mismatch, the stronger the reflected signal. The interface between tissue and blood, for example, generally provides a strong ultrasonic signal.

Exercises

Exercise 1:

Poiseuille’s equation describes the pressure drop in a long pipe as follows:

Where is the cross-sectional averaged velocity equal to , r₀ is the radius of the pipe, m is the dynamic viscosity, and L is the length of the pipe. Use this relationship to derive a formula for flow resistance, R, as defined by the equation . Based on this approach, what is the resistance of a) two pipes of lengths L₁ and L₂ and radii r₀₁ and r₀₂ in series, and b) two pipes of lengths L₁ and L₂ and radii r₀₁ and r₀₂ in parallel.

Exercise 2:

Sometimes it is not possible to make a direct comparison between fluid mechanical and electrical elements. Consider the case of flow in a venturi (a narrowing in a tube—medically this is called a stenosis if it is in the artery). The equation looks like , where is the blood density and C_L depends on the geometry of the stenosis. The ratio now depends on the value of Q. The best approach in this case is to pick a point around which you expect Q to vary and define the resistance as the rate of change of P with respect to Q (i.e. ).

A. Take the derivative of the expression for above to obtain a relationship for this definition of R.

B. How does it compare with what you would get if you simply divided the equation by Q?

C. Which method is “correct” and why?

D. Draw a plot of the relationship between pressure and flow, and draw two straight lines that indicate the two possible definitions of R. You may pick whatever values you wish for , C_L, and Q.

Steven A. Jones

BIEN 402, Biomedical Senior Design

Louisiana Tech Unversity