Calculation of effective diffusivities and reactivities in immobilized cell systems using finite difference methods

Immobilized cell systems typically consist of a single cell type encased in a semi-rigid polymer support. The rates at which nutrient molecules diffuse and react in these materials determine the feasible longevity and the amount of desirable product generated by the cells. Finite difference techniques were developed to calculate effective diffusivities and rates of reaction of small molecules in such immobilized cell systems. The structures analyzed consist of multiple cellular inclusions distributed in a continuous phase where molecules diffuse more slowly in the cells than in the continuous phase. Diffusivities are in excellent agreement with available theoretical bounds. Under typical reactive conditions, the depth to which oxygen can penetrate ranges from 24-200 p,m, depending on the cell volume fraction, oxygen supply, and cellular uptake kinetics. Increases in the cell fraction beyond 0.55 yield minimal increases in the oxygen consumption rate, suggesting that such materials are limited by the diffusive supply of oxygen.