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we're going to talk about preload and what I'm going to do is I'm going to actually 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 and so wall stress let's say at the end of I don't know 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 the wall thickness 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 edie and so you'll know that every time I write IDI I'm talking about end of diastole and the reason I want to keep kind of 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 kind of draw out 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 and doing an 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 I could say and this I'm going to write as the number one because is the first change we're going to make to our equation I could say well all this preload stuff then equals pressure right 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 and let's say a blue color so we've got remember there's a sphere kind of looking at the the doughnut shape above but now thinking about it as a sphere and I've got something like this almost like a little ball right and 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 wall so I can't really include this space when I'm thinking about the volume of the left ventricle so let me actually know make sure I include sorry the edie part of this so I don't change my lettering around and I'm going to actually reorganize this equation so I can actually reorganize this equation to look like this I could say well R prime equals now I'm just going to take the cube root right of all of this and flip around the whole thing the whole kit and caboodle and it will look like that this is my new equation right I've got n 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 and this is our second step and this de for warn you we're going to take three steps total and in our second step we get to something like this we actually 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 parenthesis here is 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 sickness still and I've got to divide all this by 2 times W edie 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 kind of step to take and you remember we talked about pressure and volume literally we talked about this all the time now don't we we have pressure and volume and we actually can we said follow the left ventricles 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 right 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 all right remember that term elastance and so for example if I say what's the elastance at this point in this purple point well the allowance 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 the elastance actually stays pretty constant doesn't really change much for this whole period of time and then finally at this 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 it and 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 steps get you know gets me to something like this I mean right down here and I'm just going to rewrite preload on this side just to remind us what all this equals on this side preload and equals pressure at end diastole times and this is a big parenthesis and I'm going to make a little parenthesis here the cube root of everything on the inside so it'd be three and then I'm going to replace the volume now with pressure add in diastole divided by elastance and then diastole right maybe I have to extend that out divided by four times pi and take all this multiply it Oh actually sorry my mistake not multiplying it now adding it right adding it to the wall thickness and then closing that up and dividing everything by two times the wall thickness so this is our kind of final equation now I'm going to give it 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 but I've got one variable there pressure and I've also got let's do 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 this is 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 these I'm talking about end diastolic pressure I'll write that here our second one our second variable is going to be elastance right elastance and again it's elastance not it at all times but at a specific time and of course it depends on where we are on that curve whether or the the kind of beginning bid or the last bit and the third variable is wall thickness so now think 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 through and actually crunch through all the numbers so is there a quick way to kind of do this and in the end the answer is yes there is and people kind of use this shortcut all the time and the shortcut is basically to say okay 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 sicker for a 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 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 the elastance does change you know let's say from this area up here to this area right so it does change but it's it but it's also not going to change dramatically especially if you're in that kind of purple zone right so I'm going to write something in between so I'm going to write its 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 kind of 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 heartbeat but you could be for example you could be at this point and be very low or this part would be very high so pressure is going to change dramatically now if you look at that equation you can see that if I'm trying to make an estimate of pressure of preload rather and 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 the only part left is this right this is the the biggest part of our equation that's going to affect our preload so 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 in 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