Hemodialyzer clearance $(2$ pts$)$

One major goal of hemodialysis is to remove urea among other wastes. Blood urea concentration before and after hemodialysis over time $T$ are referred as $c_0$ and $c_r,$ respectively. The blood urea concentration is measured by blood urea nitrogen $($BUN$)$ assay. The urea reduction ratio $($URR$)$ is defined as $\frac{c_r}{c_o}.$

Let's consider a single compartment model, in which urea enter from blood to dialysate as a first order reaction with reaction constant $k_1.$

$c_{\text{b}\text{l}\text{o}\text{o}\text{d}}(t)k_1{c}_{\text{d}\text{i}\text{a}\text{l}\text{y}\text{s}\text{a}\text{t}\text{e}}(t)$

$(1)$ Derive an expression for $\frac{c_T}{c_0}$ using this model.

$(2)$ Define urea clearance $K=k_{1}V,$ where $V$ is the total blood volume of the patient. Derive an equation to measure $K.$

$(3)$ For a BNU measurement curve below, assuming the patient blood volume $V$ is $5$ liter, calculate urea clearance $K.$