Arterial elastance (Ea) and preload

Let's talk about the pressure-volume curve. And we're actually going to use this to figure out how preload fits into the story. So let's do volume going that way and pressure going up. And I'm going to use our normal lines, our end-systolic pressure-volume relationship, something like this. And I'll also use the same color to show the end-diastolic pressure-volume relationship. And I wanted to keep an eye on this line because this is the one that's going to be of interest to us right now. This is the end-diastolic pressure-volume relationship. And I'm also going to sketch over all of this the arterial elastance. Remember, we talked arterial elastance? That was E with a little a, and the formula was pressure at the end of systole divided by stroke volume. And I'm going to quickly sketch out where those points would be. This would be our end-systolic pressure. And our stroke volume would be basically-- if we drop this dot, something like this, right? So this would be our stroke volume right here. So these two numbers kind of help make our slope or arterial elastance. And we can actually also use these to help sketch out our pressure-volume loop. So it basically would go down just like that and at some point kind of pick up our curve. And we would pass over the purple line, keep going. And right at the same mark where the volume is, we go up, and we have contraction. So the heart is contracting here, and it continues contracting. And then, finally, blood gets ejected, knocked out, and that goes into the aorta. So that's our pressure-volume loop. Now, I want to point out to you a couple things. This one, in particular, this is our end diastole point. And remember, when we talk about end diastole, it should remind you of preload. In fact, I'm going to jot the preload equation down here. Remember, preload we said is-- and this is from Mr. Laplace. This is wall stress at the end of diastole. And I'm going actually go ahead and write out the whole equation, just to make sure that we're completely on the same page and that we remember this equation. This is pressure times the radius at the end of diastole divided by 2 times the wall thickness at the end of diastole. So all this stuff is happening at that point, the yellow arrow point. So here's the question-- what would happen if we actually increased the pressure? In other words, at the end of diastole. Or in other words, increased the preload. So the pressure right now, if I was to kind of figure it out, it'd be somewhere around there. That's the pressure at the moment at the end of diastole. But let's say I increased it. I wanted to increase it to, let's say, something like this. And I can kind sketch out that that would be, let's say, right here. And immediately, you can see what would happen. You'd have an increase-- or it seems like you'd have an increase in stroke volume, right? Because now you have extra blood entering, so contraction would happen only at this point right here. And I'm going to start contraction. I'll finish off the loop in a moment, but I want to actually jump back to our elastance equation. I want to make sure we follow the math on this one because it's going to be helpful and make sure we don't get stuck in any kind of mental traps. So you remember that's the equation for elastance, arterial elastance. And then we also said previously that there's heart rate times resistance, and that should all equal each other, right? That's how we wrote it out. Now, I did not mention changing the heart rate, and I didn't mention changing resistance. I left all that the same. And if all of that's the same, and now I'm increasing stroke volume, I want you to take a guess as to what would happen to pressure. Now, if it's going to have to remain the same, the overall equation has to be the same, then you know the only way to do that is to also increase the pressure, right? There's no other way to do that. And that's the math. That's the beauty of the numbers, and it helps us kind of figure out what's going on. So if the pressure is going to go up, then I can actually draw that on my equation. I can say, well, my equation's going to have maybe a point right here, where the new pressure is. So this is my new pressure at the end of systole. So let me go over that one more time, why I knew that had to happen. I first said my stroke volume was going up. And so if my stroke volume goes up and I don't change the heart rate or the resistance, then I know that the pressure at the end of systole has to go up as well. So that's why it mathematically makes sense. But I also want to make sure it intuitively makes sense. In other words, if you put more blood in the aorta, then it makes sense that the pressure at the end of systole, which is the same as the pressure in the aorta at that point, would be higher. So more blood in the aorta it will create more pressure, and that makes sense, right? Because you have more molecules kind of bouncing around, more blood molecules. And so, of course, pressure would go up. So this is really interesting because I know there's a temptation to say, well, if you have an increase like this-- and this is why I didn't finish drawing it-- there's a temptation to kind of say, well, then that just means that your stroke volume is going to go up. And that's the end of the story. In fact, I see it drawn like this quite a lot. And it's really confusing. And to put it bluntly, it's actually wrong. It should not be drawn like this. The truth is that the stroke volume goes up, and it also increases the end-systolic pressure. So the right way to draw this is to say, OK, well, it goes up, up, up, up, up. And then, you have ejection, something like that. And then, of course, it goes down from that point. And it meets up like that. Now, you probably know something right away, which is that if I'm saying stroke volume is going up because of all this, then isn't stroke volume going down because you've lost a little bit a stroke volume on this side? And the answer is yes. Yes, you do have a little bit less stroke volume because of this loss over here. But because you have so much gain on this side, you actually make up for it. So overall, stroke volume still goes up. So the old stroke volume, which is right here, is still less than the new stroke volume, which is right here. So that's something you should definitely keep in mind. And you also have a couple other changes, which is that you have an increase in the afterload right here. So afterload is actually going up as well. And so here I want to show you that if you have a change that's driven by just preload, if that's the main change, that the loop entirely changes. But the slope of the line-- this purple line that we've been drawing-- actually stays about the same. I'm just going to try to make sure I draw it about the same, and it looks like that. So this is what the new line would look like for preload. And now you can actually get an appreciation for the fact that if you increase preload, you're basically shifting the line this way. And if you decrease preload, I could show you the exact opposite, and it would go the other way. So this is what increasing and decreasing preload does to our line. It does not change the slope of the line. So the elastance or the slope stays the same, but where the line is located on the volume axis actually shifts over.