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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 $c(T)$ .

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

Calculus 1 / AB

Chapter 1

Functions and Sequences

Section 5

Logarithms; Semilog and Log-Log Plots

Functions

Baylor University

University of Nottingham

Idaho State University

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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'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're totally patient as right. An initial blood urea concentration of 1.65 mg grams from millimeter mm. We'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'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'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'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.

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