Preload and Pressure - Chemical Engineering Video Lecture

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.