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We're going to talk about preload. And what I'm going to do is I'm going to take the equation that we came up with-- the equation for preload-- which had to do, if you remember, with wall stress. And I'm going to actually kind of tweak it in a way that will, hopefully, teach us interesting things about understanding where preload comes from. So wall stress, you remember, is actually at a certain point in time. So all stress-- let's say, at the end of contraction-- would not be preload. Really, preload is at the beginning of contraction. Or really, I can rephrase that as "end of diastole." And the equation is, of course, the wall stress equation. So you're just looking at three major variables-- the pressure at the end of diastole, the radius at the end of diastole, and actually two times the wall thickness at the end of diastole. You're going to get really sick of me saying "end of diastole," so let me just write it out here once. And then, from then on, I'll just keep writing "ED." And so you'll know that every time I write "ED," I'm talking "end of diastole." And the reason I want to keep emphasizing the point is that you don't forget that this is a specific point in time on that kind of pressure-volume loop in the left ventricle. So let's get started. How are we going to tweak this equation? Well, the first thing I want to do is actually of draw out-- do you remember we talked about the cross-section of the left ventricle? If we were to actually kind of draw it out, it would look maybe something like this. And I'm going to take different colors now to highlight different parts of this. So you've got the pressure kind of pushing out like this on this left ventricle. This is the pressure. And then, you've got, let's say, the radius here. I'm going to write r prime. And I know before I wrote r in, I think was the term I'd used. I'm going to write r prime. And there's also this radius, and this I'm going to call just r. So there's r and r prime. And I'm going to make it really clear where this is all headed because I'm going to draw this third part-- let's do it in orange-- from here to here. And this is w. So I basically am setting up for you a nice little math equation. Right? You've got r at the end of diastole equals r prime at the end of diastole plus w at the end of diastole. This is a little math equation. Right? And I can take this equation, I can immediately just plug it in right there. This, I'm going to write as a number 1 because this is the first change we're going to make to our equation. I could say, well, all this preload stuff, then, equals pressure times this quantity. And I'm going to write the quantity as r prime-- just borrowing from that equation I just wrote out to the left-- plus the wall thickness at end of diastole divided by 2 times w at the end of diastole. So that's our new equation, and just kind of borrowing from the picture. Now, I'm going to do one more picture. And let's do the second picture in, let's say, a blue color. Remember, there's a sphere. Kind of looking at the doughnut-shape above, and now thinking about it as a sphere. And I've got something like this. Almost like a little ball. Right? And you remember that the left ventricle is not a sphere, but it's actually quite close. And so when we talk about spheres and volumes, it's not unreasonable to also kind of think about the left ventricle. And the relationship here is 4/3 pi r cubed. And when I say "r," am I really talking about this r or this r? Well, that's a good question. Actually, I'm really talking about the second one-- the one with the r prime. And the reason I know that is because the volume is the volume of blood, and the blood is hanging out on the inside of the ventricle in this space right here. Right? And it's not the total ventricle because you can't really include the walls. I can't really include this space when I'm thinking about the volume of the left ventricle. So let me actually now make sure I include the ED part of this so I don't change my lettering around. And I'm going to actually reorganize this equation. So I actually reorganize this equation to look like this. I could say, well, r prime equals-- and now I'm just going to take the cube root of all of this and flip around the whole thing. And it will look like that. This is my new equation. Right? I've got end-diastolic radius on the inside equals what I've written out there. So now I'm going to just do the same thing. I'm going to plug that equation in. This is our second step. And just to forewarn you, we're going to take three steps total. And in our second step, we get to something like this. Now, I know you're probably thinking, why did I take something that was so simple and make it so confusing-looking? And in a moment you'll see why this is actually not-- maybe I'll do a double parentheses here. It's actually going to, in the long run, help us out a lot. So I realize right now it might seem like I'm taking a step backwards in terms of simplicity. But in the long run, it's definitely going to help us out. So we've got our wall thickness still. And I've got to divide all of this by 2 times w ED. Now, I've taken two big steps. Right? Two big changes. But if you look at it, really, the equation is still not that different from how it was when we first started. So let's now think about one final step to take. And you remember, we talk about pressure and volume. Literally, we talk about this all the time now, don't we? We have pressure and volume. We actually can, we said, follow the left ventricle's pressure-volume curve. We said it actually kind of tends to go up at the end, but it stays pretty constant for most of it. And we said that, if you actually-- if you recall-- divide pressure by volume, if we actually take P divided by V, that gives us the slope of the line. And we call the slope of the line "elastance." Right? Do you remember that term "elastance"? And so, for example, if I say, what's the elastance at this purple point? Well, the elastance would be this purple line. Right? Whereas, if I say, what's the elastance at this other point? Let's say this green point over here. Well, then the elastance is much greater. Right? So really, just the slope of the line at a given point is the elastance. And interestingly, elastance actually stays pretty constant. It doesn't really change much for this whole period of time. And then, finally, at this part of the curve, it starts to rise quickly. So you can see that elastance is constant for a while, but then it goes up pretty quickly after that. Now the reason that I wanted to put up the equation of elastance is that I wanted to also show you that, if I reorganize the equation, I get something like this. I could say, well, volume-- again, at end-diastole-- equals pressure divided by elastance, which I'm just going to write as an "E." OK? Something like that. So this is my new equation. I'm just going to do the exact same thing. I'm going to plug that equation in. This is my third and final step. Right? So my third step gets me to something like this. I'm going to write it down here. And I'm just going to rewrite "preload" on this side just to remind us what all this equals on the side. Preload. It equals pressure at end-diastole times-- and this is a big parentheses, and I'm going to make a little parentheses here-- the cube root of everything on the inside. So it'd be 3. And then, I'm going to replace the volume now with pressure at end-diastole divided by elastance at end-diastole. Right? Maybe I have to extend that out. Divided by 4 times pi. And take all of this, multiply it-- or actually, sorry. My mistake. We're not multiplying it now. Adding it to the wall thickness. And then, closing that up and dividing everything by 2 times the wall thickness. So this is our final equation now. I'm going to give a little bit of space, and now let's think about what the implications are. Well, first, let's try to simplify this. Because I realize that this is starting to look a little bit scary, but I want to just kind of box the variables. So where are the variables? Well, I've got one variable there, pressure. And I've also got-- in green, I've got elastance right there. And then, let's do in blue, I've got wall thickness. So these are my three variables. I started with three variables, and I still have three variables. And I'm going to write them out just so we have it very clear what we're dealing with. So these are our variables. And our first one is going to be pressure. That's our first one. And this is pressure-- of course, when I write all of these, I'm talking about end-diastolic pressure. So I'll write that here. Our second variable is going to be elastance. And again, it's elastance, not at all times, but at a specific time. And of course, it depends on where we are on that curve-- whether we're at the kind of beginning bit or the last bit. And the third variable is wall thickness. Now, thinking about these three variables, you might be wondering exactly how we calculate preload if we want to do it quickly. I mean, look at this. This is a pretty complicated formula. And a lot of times, you may not have time to sit there and actually crunch through all the numbers. So is there a quick way to kind of do this? The answer is yes, there is. And people have kind of used this shortcut all the time. And the shortcut is basically to say, OK. Well, look, the wall thickness-- this and this-- is really not going to change from heartbeat to heartbeat. Right? It's not like your left ventricle is going to grow and get thicker from moment to moment. So this one-- even though it's a variable in our equation-- it's pretty steady over short periods of time. Right? It's not going to be changing much. So I'm going to write that here. I'm just going to say it's basically constant. And when I say "constant," I mean-- let me write that out. Let's say "short term." Over the short term. Right? Now, what about elastance? Well, elastance, it does change a little bit. Right? Elastance does change, let's say, from this area up here to this area. Right? So it does change, but it's also not going to change dramatically-- especially if you're in that purple zone. So I'm going to write something in between. I'm going to write it's "variable and constant" really depending on where you are on the curve. Right? So somewhere in between. It's either going to be variable if you're in the green part or it's going to be constant if you're in the purple part. Now, pressure, this one-- this variable-- is variable. This is completely variable. Right? It depends on when you look at the heart beat. But for example, you could be at this point and be very low or at this part and be very high. So pressure's going to change dramatically. Now if you look at that equation, you can see that, if I'm trying to make an estimate of preload-- if I want to make a guess as to whether preload is going to go up or down-- the main thing affecting preload out of this equation-- if we assume that this part is constant, and this is constant-- and we look at this whole term and we realize that there's a cube root there-- so it's going to be a very small number-- well, then, the only part left is this. Right? This is the biggest part of our equation that's going to affect our preload. A lot of times people will kind of shorten all this and say, well, if the pressure in the left ventricle goes up or the pressure on the left ventricle goes down, this variable-- if it goes up or down, then my preload has gone up or down. So a lot of times, people don't actually calculate preload, but they know if it's changed just by looking at pressure.