Numerical simulation of mass transfer in porous media of blood vessel walls

Abstract

The tunica media of a blood vessel wall is modeled as a heterogeneous medium composed of a periodic array of cylindrical smooth muscle cells and a continuous interstitial fluid phase of proteoglycan and collagen fibers. By applying Brinkman's model to describe the behavior of the interstitial flow, we obtain an analytical solution for the transmural flow field through the periodic array of smooth muscle cells in the form of a power series, making it possible to compute the convection of solutes in the interstitial phase. With reaction of solutes at the surface of smooth muscle cell membranes being treated as boundary conditions and the diffusion of species being limited to the interstitial fluid phase only, mass transfer in the media of blood vessel walls is simulated numerically using Cray supercomputers. It is found that the Sherwood number (the dimensionless mass-transfer coefficient) is not only constant for all interior smooth muscle cells but also minimally sensitive to changes of parameters controlling the relative rates of diffusion and convection in the interstitial fluid phase and the reaction on the smooth muscle cell surface. In addition, the Sherwood number is not very sensitive to changes in the volume fraction of smooth muscle cells. A homogeneous, one-dimensional model (effective-medium model) is also developed to predict the bulk concentration profile in the media, based on the equivalent properties of the effective medium derived from the heterogeneous medium. A comparison of results from the one-dimensional model and two-dimensional simulation is quite satisfactory for all practical ranges of parameters. It is also determined that, for a small molecule such as ATP, the mass transfer to the surface of smooth muscle cells is “reaction limited” as assumed previously in the literature, whereas, for a large molecule such as low-density lipoprotein, the mass transfer might not be reaction limited.