B. Solve the differential equation using the boundary conditions: C = C 1 at r =

Question

B. Solve the differential equation using the boundary conditions:
C = C1 at r = R1 and C = 0 at r = R2.

Encapsulated liposomes for long-term drug delivery Liposomes are small spherical lipid bilayer membranes which can be used to enclose high-molecular weight drugs in aqueous solution (MW 1000mKT) 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. poly-L-lysine dru lipid membrane C- C This device is a spherical system with the lipid membrane at radius R1 and the encapsulant between R1 and R2

Explanation / Answer

the mass balance equation becomes for a differential element

NAr*4PIr2|r- NAr*4PIr2|r+dr =0 (1)( under steady state, rate of mass in = rate of mass out)

NAr= Flux = -DAB* dCA/dr

Eq.1 can be written in differential form

d/dr(NAr*r2) =0

when integrated, NAr*r2= C', Where C' is integration constant

Nr = C'/r2,

-D*dC/dr = C'/r2, D is the diffusivity , dC/dr is concentration gradfient

-D dC= C'dr/r2

D*dC= -C'dr/r2

D*C = C'/r +C'', where C'' is another integration constant

at r= R1, C= C1 boundary condition (1) and r= R2, C2=0 boundary condition (2)

D*C1= C'/R1 + C'' (1) and D* 0 = C'/R2 +C" (2)

from Eq.2, C'/R2= -C''

from Eq.1

D*C1= C'/R1-C"/R2

C"(1/R1-1/R2)= D'C1

C'' = DC1 (1R1-1/R2)

C'= -DC1R2/(1R1-1/R2)

D*C = -DC1R2/ (1/R1-1/R2)*r+DC1/(1/R1-1/R2)

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