Under certain assumptions, the release of a drug in the polymer coating of a drug eluting stent can be calculated using the Higuchi equation:

$\frac{M_t}{A}=\sqrt[\phantom{2}]{c_s(2c_{\text{i}\text{n}\text{i}}-c_s)Dt}$

where $D$ is the diffusivity of the drug in the polymer day$),C_{\text{i}\text{n}\text{i}}$ is the amount of drug dispersed per unit volume in the polymer coating $(\frac{g}{c}{m}^3),C_s$ the solubility of the drug in the polymer $(\frac{g}{c}{m}^3).$

a$.$ What are the assumptions that Higuchi equation is valid? Do you think drug eluting stent meet these assumptions?

b$.$ Let's say this equation can be used, determine $24$ hour of drug release if drug eluting stent is modeled as a cylinder of $1mm$ in diameter, $10cm$ long, $D=0\mathrm{.}01c\frac{m^2}{?}$ day, $C_{\text{i}\text{n}\text{i}}$ is $0\mathrm{.}2\frac{g}{c}{m}^3,C_s$ is $0\mathrm{.}001\frac{g}{c}{m}^3.$

c$.$ In the classical derivation of the Higuchi equation, a pseudo$-$steady$-$state approach was used to draw the conclusion that the drug concentration$-$distance$-$profile between the surface of the film and the "diffusion front" is linear. The pseudo$-$steady$-$state is valid for the ointment$/$polymer containing initially a large excess of drug compared with the drug solubility and the dissolution is much faster than the diffusion. Use Fick's second law to make an argument for the linear drug concentration$-$distance$-$profile.