Problem $1$: Oxygenation of Cell Aggregates, Adapted from Deen $3-3(40$ POINTS$)$

Oxygen is consumed in body tissues or by cells in vitro at a rate that is nearly independent of the $O_2$

concentration. As a model for either a small tissue region without blood vessels or a spheroidal aggregate of

isolated cells, consider steady $O_2$ diffusion in a sphere of radius $r_0$ that consists of tightly packed cells. Because

cell membranes offer negligible resistance to $O_2$ diffusion, the sphere may be viewed as a homogeneous region

in which $O_2$ is consumed at a constant volumetric rate $k_0.$ The $O_2$ concentration at the edge of the sphere

is $C_0.$

$($a$)$ Determine the $O_2$ concentration profile $C(r)($when $O_2$ is present throughout the sphere$).$ Nondimen$-$

sionalize your solution such that only the Damk$$hler number $Da=k_0\frac{r_0^2}{D}{C}_0$ appears. Note, this is $Da$ for

zeroth$-$order reactions.

$($b$)$ It it possible for enough $O_2$ in the cell to be consumed that an anoxic region develops where $C=0.$ Using

your solution from $($a$),$ determine the $C_0$ value in terms of the Damk$$hler number above which enough oxygen

is supplied to the cell to avoid anoxia. $($Hint: In $($a$),$ the consumption rate is assumed to be $k_0$ everywhere.

However, in a region where $C=0,$ no $O_2$ can be consumed. Part $($a$)$ assumes consumption occurs anyway,

leading to something weird.$)$

$($c$)$ Determine the $O_2$ concentration profile $C(r)$ when there is an anoxic core of radius $r_c$ where $C=0.$

Use the same nondimensionalization from $($a$).r_c$ isn't known, so also show how to calculate the value of $r_c.$

$($Hint: Expressing $r_c$ as an implicit equation is fine. There is no need to solve for $r_c$ explicitly$).$

$($d$)$ Oxygen consumption and diffusivity values representative of mammalian cells are $k_0=30\frac{M}{s}$ and

$D=2{10}^{-9}\frac{m^2}{s}.$ What is the maximum radius $r_0$ that will be fully oxygenated if $C_0=50M?$ What

radius will oxygenated if $C_0=200M?($The lower value of $C_0$ is representative of body tissues and the

higher one is approximately that for cells in contact with air.$)$