Dialysis time Hemodialysis is a process by which a machine...

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 $c(T)$ .

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 $c(T)$ .

Key Concepts

Derivation of a General Time Formula

Deriving a formula for the time required to reduce a substance to a target level involves rearranging the exponential decay equation and applying logarithmic operations. This provides a universal expression that relates the initial concentration, target concentration, and system-specific parameters such as volume and mass transfer coefficient.

Mass Transfer Coefficient

The mass transfer coefficient quantifies the efficiency with which a substance is removed from a system in processes like dialysis. It is a critical parameter in modeling the rate at which waste products are cleared, and it appears in the exponent of the decay equation, influencing the overall speed of the process.

Exponential Decay Model

This model describes processes in which the rate of decrease of a quantity is proportional to the current amount. It is used in various fields such as pharmacokinetics, nuclear decay, and hemodialysis, where the concentration of a substance decreases rapidly at first and then levels off over time.

Logarithmic Transformation

Solving problems involving exponential decay often requires taking the natural logarithm of both sides of an equation. This technique allows one to isolate variables that are in the exponent, such as time, thereby making it possible to derive formulas that relate initial and final values.

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