Example 8.5.1: Atherosclerotic Plaque Partially Occludes a Blood Vessel

Humans are not yet evolved to live 70 or more years. Mean live expectancy as little as a hundred years ago was in the mid-50’s, if you only average the people that survived to age 15 – otherwise it was in the 30’s! The average age when a woman bore her first child throughout most of the period we have been considered “human” has been perhaps 14 or 15, and a woman in her thirties was often a grandmother. Because evolution works best if parents don’t hang around too long to compete with their own o spring, we are very likely evolved to die somewhere around the age of 50 or 60, three to four generations (old style) after our own birth. Humans begin to really experience the e ects of aging – failing vision, incipient cardiovascular disease, metabolic slowing, greying hair, wearing out teeth, cancer, diminished collagen production, arthritis, around age 45 (give or take a few years), and it once it starts it just gets worse. Old age physically sucks, I can say authoritatively as I type this peering through reading glasses with my mildly arthritic fingers over my gradually expanding belly at age 56.

One of the many ways it sucks is that the 40’s and 50’s is where people usually show the first signs of cardiovascular disease, in particular atherosclerosis165 – granular deposits of fatty material called plaques that attach to the walls of e.g. arteries and gradually thicken over time, generally associated with high blood cholesterol and lipidemia. The risk factors for atherosclerosis form a list as long as your arm and its fundamental causes are not well understood, although they are currently believed to form as an inflammatory response to surplus low density lipoproteins (one kind of cholesterol) in the blood.

r1

a)

P+ P−

Figure 116: Two “identical” blood vessels with circular cross-sections, one that is clean (of radius r1) and one that is perhaps 90% occluded by plaque that leaves an aperture of radius r2 < r1 in a region of some length L.

Our purpose, however, is not to think about causes and cures but instead what fluid physics has to say about the disorder, its diagnosis and e ects. In figure 116 two arteries are illustrated. Artery a) is “clean”, has a radius of r1, and (from the Poiseuille Equation above) has a very low resistance to any given flow of blood. Because Ra over the length L is low, there is very little pressure drop between P+ and P− on the two sides of any given stretch of length L. The velocity profile of the fluid is also more or less uniform in the artery, slowing a bit near the walls but generally moving smoothly throughout the entire cross-section.

Artery b) has a significant deposit of atherosclerotic plaques that have coated the walls and reduced the e ective radius of the vessel to r2 over an extended length L. The vessel is perhaps

165Wikipedia: http://www.wikipedia.org/wiki/Atherosclerosis. As always, there is far, far more to say about this subject than I can cover here, all of it interesting and capable of helping you to select a lifestyle that prolongs a high quality of life.

90% occluded – only 10% of its normal cross-sectional area is available to carry fluid.

We can now easily understand several things about this situation. First, if the total flow in artery b) is still being maintained at close to the levels of the flow in artery a) (so that tissue being oxygenated by blood delivered by this artery is not being critically starved for oxygen yet) the

fluid velocity in the narrowed region is ten times higher than normal! Since the Reynolds number for blood flowing in primary arteries is normally around 1000 to 2000, increasing v by a factor of 10 increases the Reynolds number by a factor of 10, causing the flow to become turbulent in the obstruction. This tendency is even more pronounced than this figure suggests – I’ve drawn a nice symmetric occlusion, but the atheroma (lesion) is more likely to grow predominantly on one side and irregular lesions are more likely to disturb laminar flow even for smaller Reynolds numbers.

This turbulence provides the basis for one method of possible detection and diagnosis – you can hear the turbulence (with luck) through the stethoscope during a physical exam. Physicians get a lot of practice listening for turbulence since turbulence produced by artificially restricting blood flow in the brachial artery by means of a constricting cu is basically what one listens for when taking a patient’s blood pressure. It really shouldn’t be there, especially during diastole, the rest of the time.

Next, consider what the vessel’s resistance across the lesion of length L should do. Recall that R 1/A2. That means that the resistance is at least 100 times larger than the resistance of the healthy artery over the same distance. In truth, it is almost certainly much greater than this, because as noted, one has converted to turbulent flow and our expression for the resistance assumed laminar flow.

A hundredfold to thousandfold increase in the resistance of the segment means that either the fluid flow itself will be reduced, assuming a constant upstream pressure, or the pressure upstream will increase to maintain adequate flow and perfusion. In practice a certain amount of both can occur – the sti ening of the artery due to the lesion and an increased resting heart rate166 can raise systolic blood pressure, which tends to maintain flow, but as narrowing proceeds it cannot raise it enough to compensate.

At some point, the tissue downstream from the occluded artery begins to su er from lack of oxygen, especially during times of metabolic stress. If the tissue in question is in a leg or an arm, weakness and pain may result, not good, but arguably recoverable. If the tissue in questions is the heart itself or the lungs or the brain this is very bad indeed. The failure to deliver su cient oxygen to the heart over the time required to cause actual death of heart muscle tissue is what is commonly known as a heart attack. The same failure in an artery that supplies the brain is called a stroke. The heart and brain have very limited ability to regrow damaged tissue after either of these events. Occlusion and hardening of the pulmonary arteries can lead to pulmonary hypertension, which in turn (as already noted) can lead to pulmonary edema and a variety of associated problems.

Example 8.5.2: Aneurisms

An aneurism is basically the opposite of an atherosclerotic lesion. A portion of the walls of an artery or, less commonly, a vein thins and begins to dilate or stretch in response to the pulsing of the systolic wave. Once the artery has “permanently” stretched along some short length to a larger radius than the normal artery on either side, a nasty feedback mechanism ensues. Since the crosssectional area of the dilated area is larger, fluid flow there slows from conservation of flow. At the same time, the pressure in the dilated region must increase according to Bernoulli’s equation – the pressure increase is responsible for slowing the fluid as it enters the aneurism and re-accelerating it back to the normal flow rate on the far side.

The higher pressure, of course, then makes the already weakened arterial wall stretch more, which

166Among many other things. High blood pressure is extremely multifactorial.