$(20$p$)$ Congestive heart failure is a common cause of death in the

elderly. The heart muscle gets progressively weaker due to a variety

of underlying diseases. The heart compensates by building more

muscle mass. However, in the late stages of the disease, this has the

unfortunate consequence of reducing the volume of blood that can be

pumped on each cycle. Eventually the cardiac output is reduced to the

level needed to sustain life.

In a new mode of treatment, it is proposed to insert a drug$-$releasing

sleeve inside the coronary artery, to release digoxin at a constant rate, $),$ into the

coronary blood flow that feeds the heart. The blood from the heart mixes with that in the body after it

passes through the body. The blood flow rate through the coronary artery $(3mm$ inside diameter$)$ is

approximately $120m\frac{L}{m}in.$ Digoxin is excreted from the kidneys with a first order rate constant, $k_e$

$=128m\frac{L}{m}in.$ Assume the volume of the heart tissue is $1L$ and the distribution volume of the drug in

the rest of the body is $420L.$

Calculate the area of the sleeve that is required for $N=1\mathrm{.}5m\frac{g}{c}{m}^{2}hr,$ if the therapeutic concentration

in the heart should be $2n\frac{g}{m}L,$ at steady state.