dialysis-time-hemodialysis-is-a-process-by-which-a-machine-is-used-to-filter-urea-and-other-waste-pr

Question

Dialysis time Hemodialysis is a process by which a machine is used to filter urea and other waste products from an individual's blood when the kidneys fail. The concentration of urea in the blood is often modeled as exponential decay. If $K$ is the mass transfer coefficient $($ in $\mathrm{mL} / \mathrm{min}), c(t)$ is the urea concentration in the blood at time $t(\mathrm{in} \mathrm{mg} / \mathrm{mL})$ and $V$ is the blood volume, then $c(t)=c_{0} e^{-K t / V}$ where $c_{0}$ is the initial concentration at time $t=0.$
(a) How long should a patient be put on dialysis to reduce the blood urea concentration from an initial value of 1.65 $\mathrm{mg} / \mathrm{mL}$ to 0.60 $\mathrm{mg} / \mathrm{mL}$ , given that $K=340 \mathrm{mL} / \mathrm{min}$ and $V=32,941 \mathrm{mL} ?$
(b) Derive a general formula for the dialysis time $T$ in terms of the initial urea concentration $c_{0}$ and the target urea concentration $c(T)$ .

Chris Trentman · Accepted Answer

were asked to answer a question about dialysis time using logarithms including the semi log and log log plots. So I told hemodialysis is a prominent process by which machine is used to filter Yuria and other waste products from individuals blood When the kidneys fail, staff, he told the concentration of your urea in the blood is modeled as exponential decay. If K. Is the mass transfer coefficient in units of milliliters per minutes. C. Of T. Is the area concentration in the blood at time. T. And units of milligrams per milliliter and V. Is the blood volume I&#x27;m assuming is in no leaders. Then the urea concentration of blood at time T satisfies the equation. C. F. T equals C. Not times E. To the negative Katie over V. Where CNN is the initial concentration In the blood at time T equals zero. In part, a police show up. We&#x27;re totally patient as right. An initial blood urea concentration of 1.65 mg grams from millimeter mm. We&#x27;re told that K. is equal to 340 no L per minute and the volume V. Is 32,941 millimeters. And were asked how long should a patient be put on dialysis to reduce the blood urea concentration To the final value .60 mg from the leader. Well, simply says stewed into our formula. You have that are desired Concentration 60 is equal to our initial concentration 1.65 times each of the negative K. is 340 times T. Over just this V. Which is 32,000 941. The estrogen pills. Mm. How to soul. I&#x27;m gonna isolate the exponential turn by itself. Each of the -340 T. Over 32,941 equals 0.60 Over 165. What? Yeah. And now we&#x27;ll take natural algorithm of both sides. And we get that negative. 340 T Over 32,941 equals the natural algorithm of 60 over 165 premature isolating for tea. We get that T. is equal to negative 32,941 over 340 times the natural log of .60 Over 1.65. Plugging this into a calculator. This is approximately 98.0. And the unit is in minutes. Yeah. You actually yeah enjoy your fat baby. Then in part V. Were asked to derive a general formula for the dialysis time. T. In terms of the initial Yuria concentration C. Not any target Yuria concentration CFT. Yeah. Well all we have to do is simply solve are given equation. So we know that target your concentration of sea of big T. And we know from our equation this is equal to the initial you&#x27;re a concentration C. Not times E. To the negative K. Times the dialysis time big T. V over V. Once again we can solve for big T. By cursed, isolating the financial term, eat the negative K. Big T. V over V equals see a big T oversee not and therefore negative K comes big T over V equals the natural log of see if big T oversee, knots. And so we find a T big T. Is equal to negative the over K. Times the natural log of cFB T oversee. Not. So in other words we have that dialysis time T. Is the opposite of the blood volume V divided by the mass transfer coefficient, times the natural log of the ratio of the target urea concentration to the initial Yuria concentration.

Fuzail Shakir · Answer

Okay, so we know that the rate at which Yuria is removed from from the blood is describing the equation of Cft, where Kay is the rate of flow through the diary to the dialyzers and me, the volume of the patient zeros, the amount of Yuria and the blood that Time t Z zero. So we had to evaluate the that status. The limit as be approaches infinity of the integral from zero to be off K over v times C zero Eat the power of negative Katie divided by V, which is equal to K over V. C. Zero times that limit. As he approaches infinity off Beat the power of negative Katy over over V divided by native Katy over v So this is equal to K me. C zero divided by Katia Feast Looked like that. Be divided by dated Katy Times the limit A And then we&#x27;ll have to evaluate e to the power of naked and Katy over V from zero to be so simplify things down. This castle&#x27;s with this, uh, we and hey, we can&#x27;t love the T, can we? No, it&#x27;s lovely tea stuck in there. All right, this cake is against love with this cake. So we end up negative C zero times eat the power of negative G&#x27;s. That&#x27;s one over won over E to the power of Katy over V multiplied by t. Evaluated from zero to be. This is equal to negative. C zero times one over E to the power off K V over V. Times be minus one over zero. It turns out, yeah, minus one over zero toe one over. Real small numbers. A really big number like infinity. So this is infinity. So therefore, this whole thing is equal to infinity. So it&#x27;s time Urgent.

Eduard Sanchez · Answer

Hi there. So for this problem we have some kind of bashing U. Of T. When she is following our older B is zero. The exponential of minus error ties the over the volume. So we know that art is the rate of law of glass through the dialyzers B is the volume and center to is the amount of urea in the blood at time T equals to zero. Dad when we need to calculate first is the integral from zero to positive infinity of this function. So we&#x27;re going to do that. You&#x27;re the infinity or the function UFT which is simply the discussion here and now we need to determine what of this. What are the of these quantities? Depends on the time Banks County. And we can see that the only turn to depends on T. Is the exponential function which has in its argument depends on T. So we can just we&#x27;re all the rest outside the integral in which is simply need to integrate this afforded. We are integrated this in the body old Bt which is the only valuable in here D. T. To make easier the ism integral. We can&#x27;t that that w we can we can do what we call a change memorials and then call w minus R. T over B. So that if we degree initiate this w we will obtain minus are over because that is a constant. The of the differential heat. No, we are going to put this information home so we can see from here. Uh and deal teams is my news the over our nine don&#x27;t live you. So putting that we can see that that factor that will came from these differential goes away where they want outside injured role. Um they create this way we can see that this goes with this and this with this we all just off this and we will have the exponential of value. This integral of course is not defining these intervals. We could do that but which is simply going to solve the interval and later on um return to to the national volleyball team. So we know that um the integral of the exponential is just the exponential function. So it&#x27;s that easy. So we will obtain the U. S. Financial on W. And we need to know who is W. So w is this in here? So we just put that and we need to evaluate in the initial Interpol&#x27;s which is zero to positive infinity. So we&#x27;re going to do that uh w is minus hard times the over B. And this is from zero to cool ass. Um infinity from zero. We know that the exponential function is just want so you know there&#x27;s we need to, you know when you have all played an integral, you first emulated in the upper limit which is infinity. But when you put black community in here you can see that it is now the financial, it is the exponential of minus ability. And when you see the graph of the exponential function which is this when we go to minus infinity, Biggs function tends to be zero. So yes, that&#x27;s zero and that is minus. We know that if the revelation in the upper limit minus the evolution in the and lower in it, so we&#x27;ll take minus Heat of zero because that&#x27;s the the lower limit and this science council and we will attain C. O. Um the o the exponential of zero. And you know the the exponential zero is want because in here it is one and will obtain the constant. These is zero. So where we are to obtain it is that the interval of this function give us the amount of urea in the blood at time zero. So that&#x27;s they&#x27;re not seeming amount of syria you can structure for block. So that is the interpretation of this office in the room, mounds of yuri, um our time, The Evil zero. So that&#x27;s the amount of period, the initial period. So we have extracted all of that amount of furia when we enhance the time to infinity because that&#x27;s the maximum we can strive from block. Thanks

Michael Cooper · Answer

in this exercise we&#x27;re asked to consider a situation of dialysis. And we are given a function. You represents the amount of your area in a patient&#x27;s blood in milligrams at a given time t where time is minute. Sorry. It took me a lot to verify. And the equation or in the problem but it says that. Uh huh. The rate is milligrams per minute. So time is in minutes. R. Is the rate of flow of blood through the dialogue. This machine. Mhm. Yeah. Oh mm B. Is volume of the patient&#x27;s blood in milliliters and Cease of zero is equal to the initial concentration of Yuria and blood. So looking at this R. Is the rate of flow divided by the volume. So that gives me the amount of blood that can how much of the blood flows through the ratio of the rate to the volume of the patient&#x27;s blood. So what percentage of the patient&#x27;s blood or what ratio of the patient&#x27;s blood is flowing through the machine each minute times the initial concentration times he raised to the power of negative are times time divided by volume. So um the more time that passes, the smaller this exponential. Um Yes exponential expression is the less the higher percentage or the less of the initial concentration remains. And then we are asked to evaluate Yeah you can&#x27;t a girl. Mhm. From zero. Yeah. Uh huh. Yeah. Yeah. 30. I believe this big away the integral from time equals 02 tiny those 30 minutes of this dialysis dialysis from shen with respect to T. Yeah and explain what I mean. So this integral written in terms of the actual equation actually gets simplified a little bit because our and the and see subzero which is the initial concentration are all constants in this situation. The only thing that is changing is time, time is my variable of integration. And so all of that can fall outside. Uh huh. The actual integral. And then I&#x27;m evaluating the furniture of 2 30. The definite it&#x27;s a girl E. To the power of negative R. T over B. With respect to T. Yeah. Yeah. Well and when I do that that is going to require us to do some substitution. So everybody to switch to just being able to draw wrong. Yeah. I&#x27;m gonna change my color to black. It&#x27;s so the problem is I&#x27;ve got this weird funky exponents in my exponential situation. I know how to find the integral of E. To the X power dx is just itself. Right? The exponent here has more than just the variables. I am going to do a substitution. I&#x27;m going to call function you that whole exponent negative. RT divided by V. And in order to evaluate an integral I need to know what the derivative of that is. So do you is just going to be negative are over being DT to make the substitution a little easier because I have a D. T. Here. I am going to find out what D. T. Equals by rewriting that. Mhm. So if I divide both sides by negative are over B. I get the DT equals negative B. Over our. Do you? And now I am going to substitute negative Rt over be for you. And I am going to substitute. No. Yeah T. T. With negative V. Over R. D. Use put the DT here and change that. I&#x27;m going to put to you here. I&#x27;m gonna rewrite all of this. Mm. Yeah. So I still have the R. I still have to be I still need the initial concentration. I am still evaluating an integral. Yeah I&#x27;m going to come back to the limits of integration in just a second and I still have but instead of raising it to the power of negative RT over mean I&#x27;m going to raise it to the U. Power. And instead of D. T. I now have negative. This is multiplied by negative being over R. D. T. Yeah simplifying that a little bit negative. The over our is still a constant negative B. Over our times are over. B cancels each other out. So. Uh huh. Look what happens to all of that. So this are going to cancel how this are because our divided by R. Is one. This baby is going to cancel how the Mississippi because a divided by B. Is one the negative is a constant. So I can go outside and can rewrite all this as. Uh huh negative. I still have my initial concentration of the definite and to grow you know where that came from. Get rid of it. The definite integral of he to new power. Okay that should be D. You because I rewrote D. T. As negative V. Over R. D. Use this was negatively over hard to do instead of D. T. That makes this. Yeah. But then I&#x27;ve got these limits. I have changed my variable of integration from T. T. You. So I need to change the limits from T. T. You as well. If T equals zero you is going to equal negative. Are times T divided by a Z. Are divided by the negative are times zero is zero divided by via zero. So when T equals zero U. Equals zero. When T equals 30 you is going to equal negative are times 30 divided by the so my upper limit of integration is going negative. 30? Mm hmm. Over. Okay. Got Okay give us me definitely integral that I can that I can find the anti derivative for I know that the anti derivative of each of the U. D. U. Is just each of the U. Power. So this all becomes negative. Initial concentration keeps Sorry about that initial concentration times. Yeah. Uh huh. Yeah. To the youth power Evaluated over the integral from zero To -34. Right? Yes. And then evaluating that I still have initial consultation tom. Uh huh. Okay. Very quickly being as one else today, times Plugging the negative 30 are over be into. Each of the power gives me me to the negative. Uh huh. There are over B. Yeah Plugging zero into the eat the U. power anything to visit all. Power is one. I can multiply the initial concentration through. And given the initial concentration of syria and the blood minus. These are the power of the amount of time. 30 minutes times are divided by the total volume of blood. And this will give me the total milligrams of syria that remain in the patient&#x27;s blood. The initial amount of area and the patient&#x27;s blood minus the amount of your area that has been removed. Which is based on the rate Of the patient&#x27;s blood flowing through. As a ratio to the total amount of patients blood. Multiplied by the amount of time that has passed 30 minutes. Oh okay. Uh huh.

Fuzail Shakir · Answer

Okay, so we know the population&#x27;s blood volumes vi milliliters on that the blood is diverted through the allies dialyzers at a rate of K milliliters per minute. So therefore, we can say that at the start of the treatment, the patient&#x27;s blood contained CF zero is equal to see zero milligrams per milliliter Furia. So if we formulate the process of dialysis as an initial value probable, we thesis the initial value problem. And if we do so for the concentration, we can take the integral of both sides that plug for C a time tease equal to zero. We can plug and see not to sell for a final last year.

Fuzail Shakir · Answer

So this is our integral. We end up with K divided by V. C zero multiplied by e to the power Negative. Katie divided. Be divided by t over V and we&#x27;re gonna evaluate this from 0 0 to 30 and that&#x27;s your final answer.

Problem

Dialysis treatment adequacy (a) The quantity $K …

Problem 53 Easy Difficulty

Answer

(a) $t \approx 98.0$ minutes
(b) $T=-\frac{V}{K} \ln \left(\frac{c(T)}{c_{0}}\right)$

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(a) $t \approx 98.0$ minutes(b) $T=-\frac{V}{K} \ln \left(\frac{c(T)}{c_{0}}\right)$

Biocalculus Calculus for the Life Sciences

Grace H.

Caleb E.

Samuel H.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartBiocalculus Calculus for the Life Sciences

Grace H.

Caleb E.

Samuel H.

Michael J.

(a) $t \approx 98.0$ minutes
(b) $T=-\frac{V}{K} \ln \left(\frac{c(T)}{c_{0}}\right)$

James Stewart

Biocalculus Calculus for the Life Sciences