Getting Ea (arterial elastance) from the PV loop

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.