As we discussed in class, if we accept some assumptions (not strictly met for bl

Question

As we discussed in class, if we accept some assumptions (not strictly met for blood flow through blood vessels, but close enough for our purposes), the flow of a fluid through a tube is proportional to the 4th power of the radius of the tube—or, equivalently, to the square of the cross-sectional area. Thus, if cross-sectional area is reduced by 10%, the new flow will be (0.9)2 = 81% of unobstructed flow. In humans, occlusion (blockage) of the coronary arteries (which feed oxygenated blood to the heart tissue itself) is usually symptomless even with a 50% reduction in cross-sectional area. Significant symptoms are typically not experienced until the reduction in cross-sectional area reaches about 70% (above 75-80% serious problems often develop).

Consider whether the non-linear relationship between flow and degree of reduction of cross-sectional area can explain why a 50% blockage typically does not produce symptoms whereas a 70% blockage does:

What is the predicted new flow rate given a 50% reduction in cross-sectional area? Show your work.

What is the predicted new flow rate given a 70% reduction in cross-sectional area? Show your work.

Given these results, does it seem plausible to you that no symptoms would be felt until 70% occlusion? Explain.

In reality, the analysis above overestimates the effect of a given reduction in effective artery cross section on blood flow through an artery. The artery itself provides resistance to flow, as considered above, but other blood vessels with which the artery in question is in series contribute resistance to flow as well. Not until occlusion of the coronary arteries exceeds around 75% does the increased resistance in the coronary arteries themselves become the dominant factor limiting blood delivery to the heart muscle.

Suppose the artery in question is just one of ten blood vessels contributing equally to the total resistance to flow. If resistance doubles in one of these blood vessels, by what percentage does total resistance increase?

Explanation / Answer

Answer:

1. For 50% blockage, the flow rate also becomes 50%.

So, 0.52 = 0.25. The predicted new flow rate is 25% of the unblocked flow rate.

2. For 70% blockage, the flow rate becomes:

0.32 = 0.09. The predicted new flow rate is 9% of the unblocked flow rate.

3. In case of 70% blockage, the flow rate becomes 9% of the unblocked flow rate. So, it is very likely that this flow rate will show up (symptomatic) in the patient.

4. The arteries are in series. So, the total resistance is a combination of the 10 resistances in series.

So, RT = R1 + R2 +.......+ R9 + R10 = 10R (RT = Total resistance)

Here, R1 = 2R

So, RT' = 2R + 9R = 11R

Therefore, percentage increase in total resistance = {(11-10)R / 10R} * 100

= 10%

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