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Problem 46 Hard Difficulty

Dialysis removes urea from a patient's blood by diverting
some blood flow externally through a dialyzer. The rate at
which urea is removed from the blood (in mg/min) is
modeled by the equation
$$c(K, t, V)=\frac{K}{V} c_{0} e^{-K t / V}$$
where $K$ is the rate of blood flow through the dialyzer $($ in $\mathrm{mL} / \mathrm{min}$ ), $t$ is the time (in min), $V$ is the volume of the patient's blood (in $\mathrm{mL} ),$ and $c_{0}$ is the amount of urea in the blood (in mg ) at time $t=0 .$ Calculate $\partial c / \partial K, \partial c / \partial t,$ and $\partial c / \partial V$ and interpret them.

Answer

$$
-\frac{K}{V^{2}} c_{0} e^{\frac{-K t}{V}}+\frac{K^{2} t}{V^{3}} c_{0} e^{\frac{-K t}{V}}
$$

Related Courses

Calculus 3

Biocalculus Calculus for the Life Sciences

Chapter 9

Multivariable Calculus

Section 2

Partial Derivatives

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Video Transcript

Hello. So this is a nice problem. So let's read it. Dialysis removes Yuria from a patient's blood by diverting some blood flow externally through a duct. Violet dialyzers The rate at which your he is removed from the blood and milligrams permanent leader is modeled by this formula right here where K is the rate of blood flow and Mel leaders per minute T is the time in minutes and V is the volume of the patient's blood and mill leaders. And that would be in milliliters cute and then see not is the amount of Yuria present in the blood in milligrams at time. Zero. So this problem asked us first to you calculate the partial derivative of tea. Then it's asks us, Teoh, calculate the partial derivative with respect to um Oops, Sorry. The first one have a typo. So the first one is the partial derivative with respect. Okay, So with respect, Teoh, the rate of the blood flow then the second one is with respect to TV. So with respect to time and then the third one is the concentration on Syria with respect to the patient's blood volume. So I'm gonna do these in a color coded fashion, starting with the partial derivative of that function with respect. Okay, so the partial derivative of sea with respect to Okay, well, we have two K's in this. And first, let's just rewrite the function so that all of the V's are, uh, actually, no, let's just let's just go straight at it. Let's separate the constants. So see, not be from the variable of interest, which is K So we have K and we have e, um to the negative TV Overbey. That's also a constant times. Okay, So as a reminder, when we're confronted with a variable times the variable again, we have to do the derivative by parks. So this is what I like to do is, say, the left side times the derivative of the right side, plus the derivative of the left side times the right side retained. And I think that's the chain rule product rule of differentiation. So when you use this method, then you mean that constant out front and then so then this becomes K. So the lefts entertained times. So first will treat this as something like either the acts. So what's the derivative of the X Well, I just needed the X. So eat to the negative T for the times K and then times the derivative of that quote unquote X, which will It's a constant give to Ruby Times. So the variable of interest cake so than that negative t Overbey comes down and then we add it. So the derivative of K, that's just one times that right side retained. So then we can supply that out into our final form. So we have that partial derivative of R. C of our concentration of Yuria with respect to the blood flow is can be written as c not, um, Overbey times you to the negative, Katie. Over. B. So I'm gonna pull that out because we have that here and here and then times Negative, Katie Overbey plus one that would be my most simplified version of this partial derivative. So then it asks us to interpret what this is. So this is the change in concentration of, uh, Yuria. So the solid of urine with respect to you, the patient's blood flow Oops, too, on the nose in there. And that's what this new expression is telling us. Then we can move on to the second one. We're taking me partial derivative. Come on. The partial derivative of this formula with respect to t So looking at this, we only have one team, the equation, So okay, the sea, not her all constants that could be pulled out. And then we basically air left with e to the quote unquote x, where t is the variable interest. So this becomes e to the negative k Overbey, and I'll just isolate that times t. So it makes it really straightforward. So then what's the derivative of X? Because the derivative even the access tv x times the derivative of the X So are quote unquote X here is going to be our t. So then that brings us down a negative K Overbey. So the final form here is going to be negative. K squared over b squared with that initial concentration times e to the negative k t over b. And that's gonna be the partial derivative with respect to T. And what this is saying is the change in a concentration of Yuria with respect to time, and then finally will take that final partial derivative. So we have the partial derivative of our function with respect to, uh, be. And I'm gonna rewrite it so that we have all of our bees. So first we pull out of Constance R k and R C. Not. That's a constant in the first part of the expression, and then we have one were V so I'll just regret that is e to the negative one times e to the negative k t. That's the constant times e to the negative one again because we have that one of her V. So again, we're going to use that product. Real differentiation. So we have our the negative one times the derivative of our quote you. Yes. So we have e to the negative k t over b times the derivative of that quote unquote x. So that's going to be our negative, Katie. And then we're gonna add Well, what's the derivative of you tonight? If one well at spey to the negative two times negative one and then we have our right side retained so e to the negative k t Overbey again. So let's rewrite this. Make it a little more straightforward. So we've got so I'm gonna back up a step, so I actually made a mistake right here. So this should be the derivative of negative Katie beat in the negative one. As we just saw, The derivative of even the negative one is redid the negative two times. Negative one. So this becomes a positive Katie times. I'll just add the negative, too. Right there. So since we have a veto, the negative too, in both parts of this expression. And I'll just put there that we have our case, you know? Then you can pull out that needed the negative to you. So we have kay, not Kate. Times the initial concentration divided by V squared on. Then we also have this in both expressions, so e to the negative k t Overbey, and then this will be multiplied by Katie over B, which is what's left in that expression. You pull out this components and that component and then minus, uh, one. So that would be the partial derivative of sea with respect to be and well, what does this mean? So this again means the change and the concentration of the area in patients blood with respect to the patients. What volume and zooming out so you can see all the different partial derivatives. We don't care. So the reason why we would be interested in these is we might be interested in how the if the patients in critical condition, how that you're in the area and their patients blood is changing with time. So if it's changing quickly with time, that's good. We're pulling it out of the system. Maybe we're interested in the changing concentration of that Yuria with respect to the patient's blood volume. So maybe the patient has a large maybe a small volume of blood. So having even a low concentration of Yuria would be severely impactful to that particular patient and then this final one. So we wanted to look at the changing concentration of your re ever with respect some patients blood flow, so is the is a high concentration of Yuria, causing the patient's blood to slow down. What a lower concentration cause they're paid their blood speed up. That's kind of thing you can look at by using this type of equation here. So that's why we would be interested in taking all these partial derivatives, and this is sort of what each one implies

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