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# Getting Ea (arterial elastance) from the PV loop

Video transcript

We've been talking a lot
about pressure volume loops, but we haven't talked about this
formula, delta P equals Q times R, in quite some time. And you may be wondering, what
is the relationship, if any, between the two? And so actually, there is
a really nifty relationship between them. And I'm going to kind of
try to point out to you, and see if you don't
think so as well. It's pretty awesome. If you stick with me, you'll
see that it's pretty amazing. So delta, what does that mean? Delta means change. And here we're talking
about pressure. So I'm going to
write it all out. Make sure we're very
clear, because as I said, it's been awhile since we've
had to think about this stuff. So I thought it would be kind
of neat to go over it again, and kind of review it. And I'm going to
write out the formula in a little different way. We have Pa and Pv, right? And so a is arterial. And that just means, what is
the pressure in the arteries. And then you subtract out
the pressure in the veins. So in another way of
thinking about it, you're basically saying
there's a lot of pressure in the arteries when the
blood leaves the heart. And then it goes through the
capillaries and the veins, and by the time it
gets back to the heart, the pressure is almost gone. And so that pressure
drop is going to be equal to
two things, right? This first one is blood flow. Blood flow we measure
in volume over time, or you could say
volume per minute. And the other one is resistance. So this is resistance. So the amount of
pressure that falls is going to be
equal to the blood flow times the resistance. And blood flow is
actually also something you can kind of break
down a little bit. We could say blood
flow is stroke volume. And stroke volume is
volume in one heartbeat. And you multiply that by
the number of heartbeats in a minute, and we
call that heart rate. So if I say, hey,
what's your heart rate? You'd say, well, let's
say 60 beats per minute. Or 100 beats per minute. Some number, right? So that's the heart rate. And take all of that and
multiply it by resistance, and you get the
change in pressure. So this is our formula, right? Delta P equals Q times R.
Kind of spelled out for us. Now I'm going to make
a little bit of space. And now what I want to do is go
back to a pressure volume loop. And I want to show
you exactly how it relates to this cool equation. So this is our pressure
volume loop, let's say. I'm going to draw
it nice and big so you can see
everything clearly. And we have volume on the
bottom axis, going this way. And pressure on
this axis, going up. Going higher as you go up. And this is of course from the
left ventricle's perspective, so I'm just going
to write LV here, just so we don't forget,
of course, that it's from the left
ventricle's perspective. And two lines, right? So we have one line, like this. Let's call this our end systolic
pressure volume relationship. I want you to pay special
attention to that line. And this other line, that
kind of goes down here, and we call this our end
diastolic pressure volume relationship. And I'm going to quickly sketch
out a pressure volume loop. So we've got something like. This is our pressure
volume loop. Let's say the loop goes
down, and then goes up, something like this. I'm just kind of
quickly sketching it, so I apologize if it
doesn't look too pretty. But this is our pressure volume
loop, something like that, right? Now how can we get
information for our equation from this loop? Can it provide any
information for us? And the answer is
it can, but we have to make some
assumptions if we're going to use our
pressure volume loop. And for starters, keep
in mind that we're looking at the left
ventricle's pressure, but our equation up above
was about arterial pressure. But of course, one assumption
I can make right away is that, well, the pressure
in the left ventricle is about equal to the
pressure in the arteries during ejection. So remember, when the blood
is kind of squirting out of the left ventricle,
there is a continuous space between the left
ventricle the aorta, which is one of the larger arteries. And that's during ejection. So between this part of our
loop, our pressure volume loop, this is ejection. I could say, well, the
pressure is about the same in the arteries as it is
and the left ventricle. And so, taking it one
step further, I could say, well, I don't really
want a bunch of numbers. I don't need, like, 50
numbers, or really, I guess an infinite
number of numbers here. I need one number. I need one number to
plug into this equation. How am I going to
get that one number? Well, I guess we have to
do another assumption. We would say, well, the
arterial pressure-- remember I'm kind of assuming that
during ejection, the two are about the same--
during ejection then, arterial pressure,
I could say, well, isn't that kind of
summarized, or can I just use the mean or the
average pressure? So can't I just take the
mean arterial pressure? And we've done that
in the past, right? We've said, well, OK. We've got two values here. We've got our systolic
value here at the top. This is our systolic. And we've got our diastolic
value at the bottom. Down here is our diastolic. And if you use these
numbers and pull out some average mean
arterial pressure, then that's good enough. That tells you a
little bit about what the overall average pressure is. And that would be fine, that
would be fine, actually. But let's say I'm having
kind of a lazy day. And I don't really feel
like doing any math there. I don't want to
take these numbers, and then plug them into
some other formula. I don't want to do
any of that stuff. I just want one number
from this PV loop that's going to help me
get a rough sense for what the arterial pressure is. Remember, this is what I'm
after, this arterial pressure, because that's what's
in the equation. Well I could, I
suppose I could just go ahead and peak at this guy. This is our pressure
at the end of systole. I could say, well, what is the
pressure at the end of systole? And I know that that number
is going to be somewhere between systolic pressure
and diastolic pressure. So it's going to be
in the right range. And that is exactly what we do. So that kind of overall
end systolic pressure is what people use in
this equation sometimes. They say, well, let's
just assume-- again, this is our list
of assumptions-- let's assume that the
end systolic pressure is good enough to
give us information about the arterial pressure. That's what we do. We're going to use that number. Just because it's easy to get. Now a third assumption,
and I promise I won't make a big
long, long list, just a few assumptions here, the
third assumption is about PV. It's about this number. The venous pressure. Now remember, the end
systolic pressure, guys, is really somewhere around here. It's like 90, maybe it's
100, somewhere pretty high for most of us, right? In comparison, the venous
pressure is going to be what? Let's say it's
three, maybe five. It's going to be
some small number. Something very low. So if the venous
pressure so low compared to the arterial pressure, I
could just assume it's zero. I could say, well, the other one
is so darn big that subtracting a tiny little number
like three or five or whatever the
number is, is not going to make a huge difference. So let me just kind
of assume it zero. And if it's zero, then I can
kind of forget about it, right? Because a number minus
zero is just a number. So that's my third assumption. And these assumptions,
again, are just there to make our lives a
little bit simpler. So let's use these
assumptions, and I'm going to rewrite
that equation now. I'm going to just
keep it like that. And so our equation rewritten
would be something like this. It would say, OK,
well we have pressure at the end of
systole minus zero-- so I'm going to leave that
away-- equals stroke volume times heart rate
times resistance. And I'm going to go ahead
and divide both sides of the equation
by stroke volume. And this cancels. So my final equation
here-- I'm going to write in a different color,
so it's nice and bold for you-- my final equation is the end
systolic pressure divided by stroke volume equals
heart rate times resistance. So that's fine. What have I done really? I've just kind of moved things
around, simplified things, maybe. But you're probably
still looking at this and thinking a big so what? Who cares, right? What does this do for me? Well let me take a moment, I'm
just going to erase some stuff. And while I erase, I want
you to take a good, hard look at this graph, and see if
you can notice anything. And it's a little
bit of a riddle. So I challenge you
to see if you can see how this new equation
that we've written out could, in some way, be useful. So let me just be very
careful and erase all this. And I'll give you a moment
then to think about it. So go ahead and see if you
can come up with anything. And now I've cleaned up
my graph pretty well. So I'm going to
re-label stuff, and I guess that'll buy you a few
more moments to keep thinking. So let's say this point
was my end systolic pressure, same as before. And you remember
that this point was kind of where my pressure
volume loop ended. So I'm going to use
two different colors. I'm going to say OK,
well there's this line. And that's just my
end systolic pressure. That's this number. And then I've got this line. And that's between
these two right here. This is my stroke volume. So that's this number. So I've got my end systolic
pressure and my stroke volume that now, you can see,
they're both on my graph. And if I divided
one over another, what does that mean exactly? If I have pressure over
volume, you might be thinking, bells might be going
off in your head, that this sounds
awfully familiar. Pressure divided by volume
sounds like elastance. And if I'm going
to draw the line, this is what the
line would look like. And this elastance
is actually, it's actually called exactly that. It's called arterial elastance. And the reason we use
the word elastance is because we're
taking a pressure and dividing it by volume,
so it has the same units. And we actually call
this E with a little a. E sub a. So there's a line
here, and the point where our two lines
cross, this line and this line, the
point where they cross, is our end systolic pressure. So we've found a
relationship now between what's going
on in the arteries, namely if we're thinking
about the arteries having a certain flow and resistance
and a change in pressure. We have started, now,
seeing that you can actually use our pressure volume
loop, which we've always thought of as being kind
of a left ventricle story. And that it actually tells you
a little bit more than just what's going on
the left ventricle. We can actually use
it to figure out what's also kind of going
on in the arteries as well. And so we're going to keep
coming back to this line. This E-A line. We're going to revisit
it in the future. And you're going to
see how powerful it is to have all this
information on one graph.