How radius affects velocity of blood in blood vessels

[animasian]

I'm having a really difficult time understanding this, despite reading all the previous threads on this topic. My current understanding is:

1) Blood in the body is a non-ideal fluid and therefore volume flow rate, Av, is not constant and we cannot use the continuity equation A1v1 = A2v2.
2) Instead, we model the non-ideal fluid using Poiseville's equation Av = deltaP/R where R is the resistance. We can rewrite this by substituting in the definition of resistance and get Av = deltaP*r^4/nL
3) If you decrease radius (or cross-sectional area) of a blood vessel, volume flow rate of blood decreases
4) If you decrease the radius (or cross-sectional area) of a blood vessel, velocity of blood increases
5) Capillaries have greatest combined cross-sectional area, and therefore they have the greatest volume flow rate and lowest fluid velocity

Ok. So if #1, #2, and #3 are true, then I don't understand how it can possibly follow that #4 is true. If we use Poiseville's equation:

Av = deltaP/R
or
Av = deltaP*r^4/nL

If we DECREASE the radius, as concerned in #4, the right side of the equation decreases by r^4 (resistance of the blood vessel increases by r^4). That means that the left side of the equation should also decrease by r^4. In other words, the product of cross-sectional area and velocity must decrease by r^4. Since cross-sectional area is equal to pi*r^2*v, we see that cross-sectional area will decrease by r^2, but that is not enough of a decrease. The rest of the r^4 decrease must be accounted for by a decrease in velocity.

BUT, it seems that in real life, the opposite happens - as radius decreases, velocity INCREASES in a blood vessel! How can it be that when radius decreases, volume flow rate decreases (as modeled by Poiseville's law) but velocity increases (contradicting Poiseville's law)? How can Poiseville's law be used to model only certain aspects of blood? It just seems so... unscientific. I must be missing something in my logic.

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[labqi]

It's just a different system. I had the same question, and people forwarded me links to help understand. I think it's best to keep the physics separate from bio.


Sent from my iPhone using Tapatalk

 

[Cawolf]

It is important to remember that in the systemic vasculature, a decrease in radius is generally accompanied by an increase in number of vessels (branching).

So even though the radius may decrease from an arteriole to a capillary, there are many times more capillaries - meaning a large cross sectional area - so velocity decreases by the continuity equation.

Maybe you could browse this thread and see if it helps.

http://forums.studentdoctor.net/threads/blood-vessel-diameter-and-pressure.1085923/#post-15466650

Feel free to post a follow up if I missed what you were asking.

 

[justadream]

It is important to remember that in the systemic vasculature, a decrease in radius is generally accompanied by an increase in number of vessels (branching).

So even though the radius may decrease from an arteriole to a capillary, there are many times more capillaries - meaning a large cross sectional area - so velocity decreases by the continuity equation.

Maybe you could browse this thread and see if it helps.

http://forums.studentdoctor.net/threads/blood-vessel-diameter-and-pressure.1085923/#post-15466650

Feel free to post a follow up if I missed what you were asking.

lol my (least) favorite topic on all the MCAT (light slit experiments might be competitive)

I took this from that thread:

Generally:

Dilating blood vessels: more blood flow, decreased pressure, decreased speed

Constricting blood vessels: less blood flow, increased pressure, increased speed

 

[Lighthill-FFT]

There are a lot of problems with what you typed out. But the most important one is this:

"If we DECREASE the radius, as concerned in #4, the right side of the equation decreases by r^4 (resistance of the blood vessel increases by r^4). "

You are right, but if that were strictly the case you would die every time your blood vessels constrict because of the r^4 multiplication factor. Av remains either the same (because the same amount of blood must flow through your body) or slightly slower because of increased surface area to volume ratio. The point being velocity must increase to compensate otherwise you would black out from lack of blood.

Also, a pet peeve. Its not delta. Its the Laplace Operator. Its completely different.

EDIT: A useful rule of thumb (not always true, use it only when you have to) is to know that your body hates changes and will compensate for everything as best it can.

 

[justadream]

@Lighthill-FFT

Do you generally agree with what I had typed out?

 

[Lighthill-FFT]

Er, I'm sorry but are you being sarcastic or actually asking me something? Its really early and I can't tell.

But generally that's the case. I just like giving a reason for things because it makes things easier for me to remember. There are cases where it doesn't happen, but none of them are normal.

EDIT: It also depends on your definition of blood flow.

 

[justadream]

@Lighthill-FFT

lol I was actually asking you.

Can you clarify "It also depends on your definition of blood flow."?

 

[Lighthill-FFT]

Oh sorry.

Blood flow as defined by volume moved per unit time could do anything: increase, decrease, remain the same due to the increase of ratio between SA and Volume, offset by increased velocity. Depending on what set off the constriction in the first place (did you get cold, or did you shoot up epinephrine), it varies. Blood flow as defined by the amount of blood moving instantaneously through a 2D cross-section (formally called blood flux, I guess) should decrease due to the contraction because the space is smaller.

Generally though, your synopsis is what was given in my Guyton and Hall. The math behind it hasn't changed in some odd 100 years, so even though my copy is outdated it should still be reliable in this specific case.

EDIT: I'm sorry again. Ill put a disclaimer. I like nuance, so I always go into nuance. All else held the same (you aren't shooting up epinephrine), your rules are good to go.

 

[deleted480308]

one of the best pieces of advice my ms2 tutor gave me is knowing when to shut off my "why" curiosity.

sometimes it is more efficient to just believe the professor/text and memorize the rule without bothering yourself with all the backstory.......med school will eventually teach you all the backstory you need but it won't always make sense at your stage in the process

 

[Lighthill-FFT]

Very good advice indeed.

 

[The Brown Knight]

I'm having a really difficult time understanding this, despite reading all the previous threads on this topic. My current understanding is:

1) Blood in the body is a non-ideal fluid and therefore volume flow rate, Av, is not constant and we cannot use the continuity equation A1v1 = A2v2.
2) Instead, we model the non-ideal fluid using Poiseville's equation Av = deltaP/R where R is the resistance. We can rewrite this by substituting in the definition of resistance and get Av = deltaP*r^4/nL
3) If you decrease radius (or cross-sectional area) of a blood vessel, volume flow rate of blood decreases
4) If you decrease the radius (or cross-sectional area) of a blood vessel, velocity of blood increases
5) Capillaries have greatest combined cross-sectional area, and therefore they have the greatest volume flow rate and lowest fluid velocity

Ok. So if #1, #2, and #3 are true, then I don't understand how it can possibly follow that #4 is true. If we use Poiseville's equation:

Av = deltaP/R
or
Av = deltaP*r^4/nL

If we DECREASE the radius, as concerned in #4, the right side of the equation decreases by r^4 (resistance of the blood vessel increases by r^4). That means that the left side of the equation should also decrease by r^4. In other words, the product of cross-sectional area and velocity must decrease by r^4. Since cross-sectional area is equal to pi*r^2*v, we see that cross-sectional area will decrease by r^2, but that is not enough of a decrease. The rest of the r^4 decrease must be accounted for by a decrease in velocity.

BUT, it seems that in real life, the opposite happens - as radius decreases, velocity INCREASES in a blood vessel! How can it be that when radius decreases, volume flow rate decreases (as modeled by Poiseville's law) but velocity increases (contradicting Poiseville's law)? How can Poiseville's law be used to model only certain aspects of blood? It just seems so... unscientific. I must be missing something in my logic.

I disagree with some of your points.
1) Blood as all liquids is non-ideal, but remember it's still very viscous, which (according to Reynold's equation) will lower the turbulence and hence it wouldn't be a bad approximation to use continuity equation. Also, most questions pertaining to this on MCAT will ask you to assume it's ideal.
2) you'll want to use BOTH the continuity eqn and the Blood Flow = delta P/R relation.
3) even if you decrease radius, blood flow (in physiology terms: Cardiac Output) will remain constant; think about it, if someone with arterial plaques has thinner space in their arteries, instead of the cardiac output lowering the heart will start working harder and faster; this increased pressure will compensate for the loss of pressure at the thinned arteries
4) hence this holds
5) there are 2 advantages to to branching into smaller blood vessels from larger ones (with the disadvantage being the heart needing to pump more strongly):
a) you can increase blood velocity even if resistance is very high for very thin vessels like capillaries
b) branching allows you distribute blood through different parts of the body

if you halve the radius, resistance increases by factor of 16 as you said. you would expect this to decrease flow rate, but the body really wants to keep cardiac output around 5 L/ min (some tissues will die if this didn't stay constant), so the pressure gradient must also increase. hence vasoconstriction increases blood pressure because the heart is working harder.
in your last paragraph you are excluding impact of pressure gradient in Poiseuille's Law.

Think of it like this: you don't use Poiseuille's Law to find blood flow; since bloodflow is the constant of proportionality, you use it simply to see how much pressure you'll need to account for a change in resistance

 

[animasian]

1) Blood as all liquids is non-ideal, but remember it's still very viscous, which (according to Reynold's equation) will lower the turbulence and hence it wouldn't be a bad approximation to use continuity equation. Also, most questions pertaining to this on MCAT will ask you to assume it's ideal.
2) you'll want to use BOTH the continuity eqn and the Blood Flow = delta P/R relation.

Thanks, your explanation really cleared things up for me. The key for me was realizing that blood flow is approximately constant, since the heart will adjust to changes to keep up the cardiac output. (For some inexplicable reason I had been imagining this isolated blood vessel, when it is in fact not isolated but connected to a living heart - whoops).

5) there are 2 advantages to to branching into smaller blood vessels from larger ones (with the disadvantage being the heart needing to pump more strongly):
a) you can increase blood velocity even if resistance is very high for very thin vessels like capillaries

When moving from arteries to capillaries, isn't the total cross-sectional area increasing? So then wouldn't velocity and pressure both decrease?

 

[The Brown Knight]

Thanks, your explanation really cleared things up for me. The key for me was realizing that blood flow is approximately constant, since the heart will adjust to changes to keep up the cardiac output. (For some inexplicable reason I had been imagining this isolated blood vessel, when it is in fact not isolated but connected to a living heart - whoops).



When moving from arteries to capillaries, isn't the total cross-sectional area increasing? So then wouldn't velocity and pressure both decrease?

You are right, the TOTAL cross sectional area IS increasing when moving into capillaries and hence the pressure IS decreasing. But according to the continuity equation, when cross-sectional area decreases (capillaries obviously are much thinner than arteries) velocity INCREASES. You are probably familiar with the water hose example, where if you squeeze the end of the hose a bit the water shoots farther. Think of the pressure decrease and velocity increase together: you are "using up" the pressure gradient (thereby decreasing pressure) to give the fluid kinetic energy (thereby increasing velocity).

 

[Cawolf]

You are right, the TOTAL cross sectional area IS increasing when moving into capillaries and hence the pressure IS decreasing. But according to the continuity equation, when cross-sectional area decreases (capillaries obviously are much thinner than arteries) velocity INCREASES. You are probably familiar with the water hose example, where if you squeeze the end of the hose a bit the water shoots farther. Think of the pressure decrease and velocity increase together: you are "using up" the pressure gradient (thereby decreasing pressure) to give the fluid kinetic energy (thereby increasing velocity).

I just want to say that this is incorrect.

There is an appreciable decrease in velocity from aorta to capillary bed due to the large increase in cross sectional area.

Flow is constant and velocity decreases.

 

[The Brown Knight]

I just want to say that this is incorrect.

There is an appreciable decrease in velocity from aorta to capillary bed due to the large increase in cross sectional area.

Flow is constant and velocity decreases.

My bad for stating misleading information!
But keeping in mind sb247's advice, is there a reason why velocity is highest in the thickest vessels and lowest in the thinnest? Maybe the increased resistance or increased surface area providing friction?

 

[Cawolf]

@TheBrownKnight

No problem - I felt bad writing it, but I felt it would be best to clear any misinformation.

The thickness of the vessel is not related to the fluid velocity, but the ability of the vessel to withstand pressure, contract, and decrease volume.

The velocity, as you stated, is by continuity. The thickest vessels happen to have the least area because they are closer to the heart - and have the highest pressure within. As the area increases the velocity slows down to almost nothing in the capillary bed (thinnest vessel).

The takeaway is that the velocity and thickness are related by correlation and not causation.