Harmonic solutions of a mixed boundary problem arising in the modeling of macromolecular transport into vessel walls

The earliest events leading to atherosclerosis involve the transport of lpw density lipooprotein (LDL) cholesterol from the blood across endothelial cells that line the artery wall. Laplace’s equation describes the steady state diffusion profile of a tracer through the vessel wall. This gives rise to a boundary value problem with mixed Dirichlet and Robin conditions. We construct a linear system of integral equations that approximate the coefficients of the series expansion of the solution. We prove the existence of the solution to this problem analytically by using Gershgorin’s theorem on the location of the eigenvalues of the corresponding matrix. We give a uniqueness proof using Miranda’s theorem [C. Miranda, P.D.E. of Elliptic Type, Springer-Verlag, Berlin, 1970]. The analytical construction method forms the basis for a numerical calculation algorithm. We apply our results to the transport problem above, and use them to interpret experimental observations of the growth of localized tracer leakage spots with tracer circulation time.