To create a vascular graft, vascular smooth muscle cells are uniformly seeded throughout a collagen hydrogel scaffold $($blue$)$ in the shape of a cylindrical shell. The shell has a length $L=10cm,$ inner radius $R_1=2mm,$ and outer radius $R_2=3mm.$ A flow of nutrient solution containing PDGF$,$ a molecule which is necessary for cell proliferation, is supplied to the inside surface of the hydrogel shell. For simplicity, consider the cell$-$hydrogel mixture to be a homogenous phase. Also assume the following: $1)$ convection occurs only in the z$-$direction, while diffusion occurs in the $r$ direction; and $2)$ PDGF is consumed in the cell$-$hydrogel phase at a uniform and constant rate of $R_P=-1{10}^{-15}mo\frac{l}{s}\frac{?}{L}.$ To provide stability, the outer surface is coated with a hard plastic that is impermeable to PDGF $($no flux at $r=R_2).$ You can assume diffusive flux into the scaffold does not change with $z,$ and the system is at steady state.

a$)$ What are the boundary conditions for this problem statement?

b$)$ Starting with the cylindrical form of the differential mass balance equation, find an equation for the radial diffusive flux of PDGF$,N_P(r),$ in the hydrogel $($variables only$).$

c$)$ Find the release rate of PDGF from the nutrient solution into the scaffold at $r=R_1.$