Diffusion in a Cellular Medium: A (1+1)-Dimensional Model

We consider the diffusion of a material in a one-dimensional medium consisting of a large number of cells separated from the extracellular space by permeable membranes. The extracellular space is completely connected and allows unrestricted diffusion of the material. Furthermore, the material can diffuse within a given cell, i.e., the intracellular space; however, direct diffusion from one cell to another cell cannot occur. There is a movement of material across the permeable membranes between the intraand extracellular spaces. Material from one cell can cross the permeable membrane into the extracellular space, the diffuse through the extracellular space, and eventually enter the intracellular space of a second cell. Here we develop a simple set of model equations to describe this phenomenon and obtain the solutions using an eigenfunction expansion. We show that the solutions obtained using this method are particularly convenient for interpreting data from experiments that use techniques from Nuclear Magnetic Resonance Imaging.