In ultrasonography, the difference between emitted and observed frequencies varies proportionally with vessel pressure. On that basis, it can be concluded that a doppler shift of the lowest magnitude would be observed in:

A. a wide occluded vessel

B. a wide unobstructed vessel

C. a narrow occluded vessel **correct answer**

D. a narrow unobstructed vessel

It says the Pressure is proportional to the frequency shift, so a low frequency shift means low pressure. From here I used the formula P = F/A and concluded that P would decrease if A increased. However, the solution takes another route and uses Bernoulli's equation to conclude that a low pressure would result in a high velocity.

Further, it shows with A1*v1 = A2*v2, an increased velocity would then require a lowered Area. I'm curious as to why these 2 answers are opposite of another. Does P = F/A not apply in this case?

The problem with using pressure = force / area is that you are assuming the force is constant. That's fine for stationary/non-moving/static fluids, but it's not true when the fluid (e.g. blood) is actively moving. In cases of moving fluids, you need to think of two things: flow and energy. In the case of a closed system like blood circulation, flow can be generally considered constant, so the continuity equation applies. Energy is generally viewed to be conserved in ideal cases (it's usually fine to approximate blood as an ideal fluid because the math is easier to deal with), and so Bernoulli's equation applies.

The question stem states that blood vessel pressure is directly proportional to emitted - observed frequencies = Doppler shift magnitude. So the lowest Doppler shift magnitude is seen in regions of lowest pressure. To do this, Benoulli's equation becomes useful and we can treat the problem as energy conservation.

Bernoulli's equation is useful because it uses the concept of energy density (energy per unit volume). This is nice because the formulas for kinetic and potential energies can be expressed in terms of fluid densities (because density = mass / volume). So this means:

Kinetic energy density = (kinetic energy) / volume = (1/2 * m * v^2) / volume = (1/2) * density * v^2

Potential energy density = (potential energy) / volume = (m * g * h) / volume = density * g * h

Pressure = force / area and from unit conversions, we see: 1 Pa = 1 N / m^2 = 1 (J / m ) / m^2 = 1 J / m^3 (note that 1 J = 1 N * m). So pressure is basically another type of fluid energy density.

Bernoulli's equation states that the sum of all these energy densities is constant because energy is conserved. Source:

Pressure
The height in this problem isn't changing, so the change in potential energy density is zero. So this means regions of lowest pressure are regions of highest kinetic energy density. Since the fluid density is constant, fluid velocity is highest at regions of lowest pressure.

Because fluid flow is constant, fluid velocity is inversely related to cross sectional area of the vessel. Combining these concepts, we see that regions of lowest Doppler shifts --> regions of lowest pressure --> regions of highest fluid velocity --> regions of lowest vessel areas. Lowest vessel areas are ones that are narrowest and disturbed/blocked (like plaques building up on blood vessel walls). So that's why narrow occluded vessels are regions corresponding to lowest Doppler shifts.

It's an excellent problem that shows the importance of fluid dynamics in medicine, especially in cardiology!