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# High blood pressure and Bernoulli's Law

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When there is high blood pressure, is the velocity of blood low, as predicted by Bernoulli's Law?

My Princeton Review instructor says that Bernoulli's Law (P1V1 = P2V2) is directly applicable to the circulatory system: when blood pressure is high, velocity is low; when blood pressure is low, velocity is high.

However, this reasoning contradicts my intuitions. The effects of high blood pressure include bursting of blood vessels, stroke, and heart attack. Blood would would have to travel very quickly to dislodge any fat deposits or burst any blood vessels. So, this means that high blood pressure is associated with high velocity, which is not what Bernoulli's Law states. Furthermore, if blood pressure is inversly proportional to velocity, then your pressure would be highest when you're heart stops pumping, which doesn't make any sense at all. When you're in a stressed or excited state, your blood pressure is elevated, isn't your blood being pumped at a faster rate?

What's the deal, hommies?

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Ok. Bernoulli's equation states that when the height above the ground (or some other common denominator) is constant, then pressure is inversely related to fluid velocity. (The full equation is P1 +0.5(density)v1^2 +density(g)(h) = P2 + 05.(denisty)v2^2 + density(g)(h)). Your instructor may have explained the example of a tornado causing your windows to burst outwards. The high speed winds rush past your house, and consequently the pressure outside your house decreases relative to the pressure inside your house. The difference in pressure causes your windows to explode outward.

For some of the conditions you are talking about, you also have to consider the continuity equation: that the rate of flow of a volume of fluid is constant. Or the area of a pipe is inversely related to the velocity of the fluid.

If you have a clot or plaque in your subclavian artery, you are narrowing that vessel and thus increasing the velocity through that vessel. If the velocity increases, the pressure in that part of the vessel decreases. If this clot or plaque is at the point where the carotid artery branches off of the subclavian to feed the brain, you will have less blood being pushed up the carotid via Bernoulli's equation (since there is a difference in height now too). This can lead to transient ischemic attacks or, in severe/prolonged cases, a stroke. (wow, this is much easier to describe with a diagram!)

Remember that as your arteries divide into arterioles and capillaries, your cross-sectional area increases tremendously!! As a result, the overall velocity through each individual capillary bed is MUCH slower than through your aorta. Remember too, that your heart increases its rate and force of contraction to supply the blood that your body needs/demands. Those changes to cardiac output are NOT the long-term determinants of blood pressure. Things like vasoconstriction, vascular structure (the size of your lumen), and blood volume are and much of this is regulated by the kidney.

This is getting beyond the scope of the MCAT. I'm not sure I've answered your questions, but hopefully you can make sense of it or someone else can come up with a better explanation.

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hey dopetown,

the fluid mechanics of the circulatory system is actually really, really complex - there are a lot of different elements that need to be taken into account for a complete description of what's really going on in a blood vessel. so it can be confusing to try to relate simplified descriptions (like Bernoulli's eqn) to actual flow conditions in a real person.

i don't have a great physiology background, but i would imagine that you are correct in thinking that when blood pressure increases, blood moves faster, which seems to be in contradiction with Bernoulli's equation.

i've actually never seen Bernoulli's equation as you have written it, and i'm 99% sure that it's not right, to be honest. in the form that 007 has stated it, bernoulli's eqn says that energy is conserved. the really important thing to remember here is that Bernoulli's equation is only valid if there is no energy added to (or taken away from) the fluid. if energy is added or removed, then low pressure, low velocity flow can suddenly become high pressure, high velocity flow or vice versa, and Bernoulli's equation is no longer valid. a perfect example of this is flow out of a barrel (velocity zero, gauge pressure zero) through a pump into a tube. in this case, the flow is low pressure, low velocity on one side of the pump, and high pressure, high velocity on the other - the pump adds the required energy. much like the heart. so your blood pressure will not be highest when your blood velocity is zero, and blood pressure does increase with stress - because the heart adds energy, and Bernoulli's equation is no longer valid

here's an example of how the pressure/flow relationship as described by bernoulli's eqn is (sort of) correct: consider flow in a pipe through a constriction - much like blood flowing past a clot. let's assume that the flow is steady (i.e. flow in must equal flow out). when the blood flows past the clot, the velocity must increase since there is a smaller cross sectional area for the blood to flow through, as 007 said. here we have NOT added any energy to the flow, so pressure will drop as the fluid passes the clot. this is the really simplified version that you might see on the MCAT. Note that the change in pressure will not be inversely proportional to the change in velocity - if pressure drops by 2 fold, velocity will NOT increase by 2 fold. take a look at the equation that 007 has written to see why this is.

now, since this is simplified, it is not what's actually happening in a person. at all.

in reality, friction between the flowing blood and the vessel walls cause the blood to lose energy as it flows through the circulatory system (clot or no clot). A constriction (like a clot) increases this loss in energy, making it necessary for the heart to add more energy to the blood to ensure that it can make it past the clot and to the arterioles. assuming that the blood requirements of the body are the same with and without the clot, the added energy when the clot is present does not go to increasing blood velocity. it goes to... you guessed it - increasing the blood pressure upstream of the clot. so now by the continuity equation (flow in equals flow out) you have increased velocity at the clot, but since the clot increases the energy requirements for circulation, the pressure of the blood flowing out of the heart must also rise.

even this is really, really simplified, but there's no sense in getting into the rest of it - the math is from hell.

other mechanisms can explain the other situations you describe (bursting blood vessel, etc). there are a lot of assumptions in bernoulli's equation that aren't *actually* true in the circulatory system, like inviscid flow, stationary vessel walls, steady flow, no heat added, etc. but i would guess that the most important for the MCAT is no energy can be added or taken away when you use this equation.

lastly, in case you're wondering, blood is incompressible, so density will remain constant in any question you encounter.

hope this helps,

s.

edited twice for clarity... it's hard to explain without math 8o

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Thanks for the added explanation skofu...I knew there were points I was missing, but I was writing my post so late at night.

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Thanks a lot for the explaination. It really helped :hat

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no problem. i'd be happy to take a crack at any other physics questions you have - engineering degrees are great prep for the PS section of the MCAT... heh heh.

s.

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Great answer to the whole hemodynamic question. I was bored and I though I just mention one small thing that I didn’t agree with.

While blood is incompressible (which you stated) its density is NOT constant. Blood is a heterogeneous mixture and there can be changes to its composition e.g. hematocrit (and there often are, ie capillary flow, unequal flow distribution in branches, etc.) which might change its density.

Just being nit picky

InuvikGirl

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A couple of things:

Disclaimer: I am post-call so I tend to overanalyze things. I also tend to buy a lot of stuff on ebay post-call.

1. This knowledge won't be necessary for the MCAT.

2. Bernoulli's Law CAN be applied in the human body, but from as much as I understand it, you can't apply it to the circulatory system because the pressure is generate IN the vessels. Therefore blood pressure and flow are directly related, but there are better equations to explain it(I'll explain later). Bernoulli's equation works better with things like air flow in the lungs. In end expiration, your bronchioles will be small and the speed of air rushing through them will cause a suction effect(ie low pressure) that will collapse the airway. This is why you can never expire ALL of the air in your lungs. However if you can slow down your respiration or increase the end expiratory pressure by pursing your lips, you can prevent the collapse of your airways and exhale a greater amount of air.

3. Instead of Bernoulii's law for the circulation, a better one would be comparisons to the electrical system - V=IR which becomes (delta)P=QR(flow times resistance) or Q=P/R. P is the pressure difference between two points in your vessels and the flow(Q) depends on the intrinsic resistance of the vessel. R is determined by vessel size, viscosity, length, etc. With high pressure differences, there will be greater flow through your vessels. When you exercise, your blood vessels dilate and you recruit more vessels so your resistance decreases, further increasing flow.

So to answer the original question, this is the OPPOSITE of what your MCAT tutor is apparently describing. Bernoulli's equation can only be applied to very small parts of a blood vessel, as has been described above.

3. The part about the blood being heteronegous is very true. The most vivid example of this is in the capillary circulation. Let's assume that your blood is made of either cells or plasma, plasma being a free fluid. The fluid near the vessel wall will be much slower because it obeys the laws of laminar flow. In fact in theory, the fluid right next to the wall has to be stopped. If cells were at the wall, they would roll and cause lots of turbulence, etc. Therefore, they tend to be in the middle of the vessel, where the flowrate is greatest. If you take a snapshot of blood in capillaries, you will see that all the red blood cells are lined up neatly in the middle of capillary and are going very rapidly, actually they are going faster than the plasma in your blood. But I'm getter off topic and my focus is waning...

Anyway, you don't need to know this stuff for the MCAT....and I miss biophysics.

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This is one of my favourite posts in a long time.

I agree with the whole density issue for blood...however, I'd just like to remind the original poster that unless otherwise stated in the passage or a particular question, assume the density of a fluid is constant on the MCAT.

Best of luck on Test Day!

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to the original poster:

Rouleaux Formation (red blood cell grouping in blood flow) is way beyond the scope of the MCAT - no question would ask for this level of detail. nor would a question ask about the compressibility of blood as 007 mentioned.

the dP/dx = QR description of circulation will probably not be on the MCAT, since differential equations aren't on it. if something like this shows up, it'll be really, really simplified and probably explained in the passage, so i would not worry at all about this - only worry about what's actually going to be in the PS section. many biological applications you might see in MCAT questions could seem counter-intuitive, but this is because the physics is so simplified.

as much as i'd love to get into a discussion about the validity of using various analysis techniques to describe biological systems (and i'm not kidding), without a good (i.e. huge) chalkboard, a computer running Matlab, and in-person interaction, it's just not really going to work. a couple of cool articles with medical applications on some of the most recent fluid mechanics analysis techniques used to describe flow in the circulatory system can be found here and here

my \$0.02

s.

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Great PDF on physiological fluid mechanics from an MIT course

I guess I'll add my 1 cent...due to the presence of laminar flow everywhere in the normal circulation except for the aorta in systole, nearly all the energy lost due to friction occurs as a result of friction between adjacent lamina, not between RBCs and the vessel walls (the plasma in contact with the vesses wall doesn't slip). In areas of constriction, the turbulence can occur and a lot of energy goes into the kinetic energy of the eddies.

I hope there are some engineers in the 0T9 class, from skofu's posts it looks like they'd be a great resource!

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thanks studentz! engineering gave me a pretty good handle on physics. as for remembering huge amounts of information... ummm.... well, let's hope i learn how to do that in the fall.

s.

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Some stuff not quite right.... some comments.

Bernoulli's Law (P1V1 = P2V2)

P1V1=P2V2 - that's from the ideal gas law -- n=const... T=const, remember that from thermodynamics.

Boyle's law is the name.

when blood pressure is high, velocity is low; when blood pressure is low, velocity is high.

That is true (if one ignores frictional losses and pumps)... BUT only along a stream line.*

Furthermore, if blood pressure is inversly proportional to velocity, then your pressure would be highest when you're heart stops pumping
Not inverse... but the high-low relationship holds; P is highest for V=0. They use this phenomenon to measure airspeed-- it is called a Piot tube flowmeter.

Bernoulli can be derived from the energy equation... so if you want to think of it in terms of a pendulum---

pressure = potential energy

velocity = kinetic energy

If you have a clot or plaque in your subclavian artery, you are narrowing that vessel and thus increasing the velocity through that vessel. If the velocity increases, the pressure in that part of the vessel decreases.

In the ideal situation you get 100% pressure recovery (i.e. the pressure increases again after the fluid has passed the narrow spot). Pressure recovery is actually very important practically--- it is pretty much why all cars decrease in cross-sectional area toward the rear.... and it is why airplane taper at the rear. The idea is to increase pressure recovery (reduce drag). Pressure recovery can be understood conceptually with Bernoulli's and conservation of mass i.e. rho1*V1*A1= rho2*V2*A2.

Any case, ... to make things clear... one cannot use Bernoulli to explain pressure drop in a stenosed vessel. One can only throw in a fiddle factor-- that has nothing to do with Bernoulli. If one has a stenosis and one wants to calculate the pressure drop across it one has to use the Navier-Stokes equations-- and while I've published on solving 'em numerically I'm not going there... 'cause one needs to know some serious math to understand it.

so your blood pressure will not be highest when your blood velocity is zero, and blood pressure does increase with stress - because the heart adds energy, and Bernoulli's equation is no longer valid
Heart is okay to include-- you just have to break apart the calculation. Actually... if one does pipe calcs that's what is interesting -- 'cause you have to solve for the intersection of the pump characteristic and the resistance of the piping system--messy iterative calculation... but fun stuff if you use a computer. Any case, definitely not MCAT material.

here we have NOT added any energy to the flow, so pressure will drop as the fluid passes the clot. this is the really simplified version that you might see on the MCAT. Note that the change in pressure will not be inversely proportional to the change in velocity - if pressure drops by 2 fold, velocity will NOT increase by 2 fold. take a look at the equation that 007 has written to see why this is.
Correct.

same with and without the clot, the added energy when the clot is present does not go to increasing blood velocity. it goes to... you guessed it - increasing the blood pressure upstream of the clot.
Actually... generally the pressure upstream stays pretty much the same. The pressure downstream of the stenosis is less and the flow through the stenosed vessel is less (think V=I*R -- Ohm's law... analogy P=Q*R). The decreased flow is what really is the problem... as decreased blood ->decreased oxygen delivery ->hypoxia ->cells unhappy ->bad things happen.

there are a lot of assumptions in bernoulli's equation that aren't *actually* true in the circulatory system, like inviscid flow, stationary vessel walls, steady flow, no heat added, etc.

inviscid flow? we're not talking Euler equations (i.e. Reynolds number is not very high) -- inviscid flow is mostly useless... unless you're patching boundary layer equations to Euler equation solutions for external supersonic flows.

stationary wall... no biggie-- friend did a PhD on that for coronary flow

steady-- that's the biggie (that's also why a calc I did took 3 months of computational time... but that's another story)

heat transfer -- no biggie

... there other wacky things... shear-time dependence--- practically irrelvant for blood; non-newtonian... ignore; two phase flow...ignore-- unless you're talking vessels less than 100 micrometers.

While blood is incompressible (which you stated) its density is NOT constant.
It is more-or-less constant... unless you're dealing with small vessels. If you want to nitpick... no fluid is truly of constant density if one goes to the molecular level. Practically it doesn't matter that much for most calculations--one can still use continous equations.

P=Q*R
IMHO is fair game. Don't need any calculus... it is just like Ohm's law.

I guess I'll add my 1 cent...due to the presence of laminar flow everywhere in the normal circulation except for the aorta in systole,
wondering... CO=5 L/min, diameter=2.5 cm Q=VA -\-> V = Q/A = 5*1./1000/pi/0.025^2 = 2.55 m/min

with kinematic visc. = 3.5e-6 m^2/s Re = 303, thus peak Re=600 or 700 ... sort of what I remember for the abdominal aorta. I wonder a bit about that statement. Laminar is typically Re <2300 for a pipe. I wonder if you're confusing recirculating flow with turbulent flow.

Any case... if you want to test your Bernoulli knowledge look at the good 'ol Venturi meter.

I have to think back to my MCAT studying... I had a Kaplan book that *no sh!!tting here* applied Bernoulli's equation across the flow in a pipe (i.e. across streamlines)! Funny is that Poiseuille flow actually makes the assumption that pressure isn't a function of the radius.

* there is one trivial exception where it doesn't have to be along a stream line... but I'm not going there.

My bit of trivia...

If you want to twist your head around something--- the hematocrit decreases as the vessels get smaller-- known as the Fahraeus effect.

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My bit of trivia...

If you want to twist your head around something--- the hematocrit decreases as the vessels get smaller-- known as the Fahraeus effect

It had to do with what I explained earlier about RBCs moving towards the centre of the capillary, maximizing its speed through the capillary and minimizing its time in the capillary. Plasma therefore spends more time in the capillary(conservation of mass) and therefore there is most plasma than RBC left in the smaller tubes.

The best analogy I heard was from my biophysics prof about cars and people crossing a bridge. They enter the bridge at the same time but the car, being faster, leaves the bridge first. (I wish I had a diagram). When you do this many times, you find that the people on the sidewalk are packed closer together and the cars are farther apart. The ratio of cars:p eople on the bridge decreases.

Anyone have a good picture of this?

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strider 2004, did you happen to study biophysics with Dr. McDonald at UWO? The explanation was memorable.

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Yeah I did a few years ago. That year(bio 302,303,330a,331g) was still my favourite year of schooling through all of university including med school.

Are you in the program now?

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No I took it a few years ago too. I did 330a and 331g in 2001/2002 and took 302 and 303 in 2002/2003 (my third year). I graduated 2004 in biophysics.

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I graduated in 2000. You probably had Laura or Sean as TAs at some point? Laura was in my class, Sean was a couple of years above me. I did all the 300s in my 2nd year(the fun year) then did the biopysics prereqs(applied math, phys chem) in my 3rd year(the bad year).

I do find I still miss biophysics....you get a bit of it in medicine but not much. There isn't much time to go through the basic principles, or much use, really. Despite how Dr Steinman made us go through the rigors of calculating cardiac output to maintain a certain cerebral flow rate, noone actually does that.

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I had Sean as a TA and did my 302 project with him. I think that he is in England now working on his post doc. Not to worry though, Steinman is still around and taking pleasure in torturing poor undergrads with his exams and disposition.

What school are you doing meds at? What year? I'm stating this year at Ottawa.

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