$1.$ Encapsulated liposomes for long$-$term drug delivery $(35)$

$$Liposomes$$ are small spherical lipid bilayer membranes which can be used to enclose high$-$molecular

weight drugs in aqueous solution $($MW $>1000$ g

mol $)$ and deliver them at a controlled rate. Unfortunately,

they interact with the immune system, and so must be surrounded with a protective $$encapsulant$$

material, such as poly$-$L$-$lysine.

This device is a spherical system with the lipid membrane at radius R$1$ and the encapsulant between

R$1$ and R$2.$ Transport of the drug will be in the radial direction and limited by one of two mechanisms:

$$ Transport through the very thin lipid membrane, which we$$ll model using a mass transfer coefficient:

Jr $=$ hD$($C$1$ C$2)$ where C$1$ is the concentration inside the membrane and C$2$ is the

concentration between the membrane and the encapsulant.

$$ Diffusion through the encapsulant material with diffusivity D$.$

You may assume the concentration outside the encapsulant is zero, and that the $$equilibrium constant$$

keq for the drug entering the encapsulant is $1.$

Data: R$1=0\mathrm{.}01$ cm$,$ R$2=0\mathrm{.}02$ cm$,$ D $=107$ cm$2$

s $.$ Note that as the drug leaves the device, C$1$ and C$2$

will both slowly fall $($very slowly, so we can assume quasi$-$steady$-$state below$),$ so they are not constant.

$($a$)$ Approximately how long does it take to establish a steady$-$state diffusion profile across the encapsulant?

$($b$)$ Sketch the concentration profile assuming transport is completely controlled by the membrane,

and make a separate sketch assuming complete control by diffusion through the encapsulant.

$($c$)$ Starting from a mass balance in the radial direction, derive the differential equation for steadystate

diffusion of the drug through the encapsulant.

$($d$)$ Solve the differential equation from part $1$c$,$ and using the boundary conditions r $=$ R$1)$ C $=$ C$2$

and r $=$ R$2)$ C $=0,$ express the concentration of the drug in the encapsulant as a function of r$.$

$($e$)$ Use the solution from part $1$d to write expressions for the flux Jr and sphere area A as a function

of r$.$ The product Jr $$ A should be independent of r$.$ Suppose the drug transport is limited by

diffusion through the encapsulant, so C$2=$ C$1.$ In this case, what is that product Jr $$A$,$ in terms

of the concentration C$1?$

$1$

$($f$)$ If the drug delivery is purely limited by transport through the membrane, so C$2=0,$ what will

be the product Jr $$ A in terms of the concentration C$1?$ Use hD $=1\mathrm{.}4\backslash$times $106$ cm$/$s $.$

$($g$)$ Based on your answers to parts $1$e and $1$f$,$ is the rate$-$limiting mechanism here the membrane or

the encapsulant? Which part therefore gives the more correct rate of drug delivery Jr $$ A$?$