In this problem we consider a tissue in the form of a sphere consisting of a

matrix of cells in which molecular oxygen is diffusing from outside the tissue

surface where the concentration is $C_0($maintained as a constant$).$ As oxygen

diffuses through the matrix, the cells use the oxygen in various metabolic

pathways that we model as an overall "zeroth" order chemical reaction $($i$.$e$.$ the

rate is independent of the concentration of the oxygen$).$ Thus the mass balance

of the oxygen, assuming spherical symmetry so there is only a flux of oxygen in

only the $r$ direction $(N_r)$ which is given in spherical coordinates by:

$0=-\frac{1}{r^2}\frac{d[r^2{N}_r]}{dr}-k$

The radius of the tissue mass is given by ro$.$ We can distinguish two cases. In

the first, the concentration of oxygen $C(r)$ throughout the sphere is greater than

zero; particularly at the core $r=0,$ the concentration is greater than zero.

Determine the concentration distribution. In the second case, because of the

reaction there is a central region in which there is no oxygen; due to the zeroth

order reaction, all the oxygen is consumed. The radius of the inner core is

denoted by $r_c.$ Find this radius and a criterion for when the oxygen free zone will

exist.