Modeling of reaction-diffusion transport into a core-shell geometry

Fickian diffusion into a core-shell geometry is modeled. The interior core mimics pancreatic Langerhan islets and the exterior shell acts as inert protection. The consumption of oxygen diffusing into the cells is approximated using Michaelis-Menten kinetics. The problem is transformed to dimensionless units and solved numerically. Two regimes are identified, one that is diffusion limited and the other consumption limited. A regression is fit that describes the concentration at the center of the cells as a function of the relevant physical parameters. It is determined that, in a cell culture environment, the cells will remain viable as long as the islet has a radius of around $142 \mu m$ or less and the encapsulating shell has a radius of less than approximately $283 \mu m$. When the islet is on the order of $100 \mu m$ it is possible for the cells to remain viable in environments with as little as $4.6\times10^{-2} mol/m^{-3}$ $O_2$. These results indicate such an encapsulation scheme may be used to prepare artificial pancreas to treat diabetes.