Mathematical modeling of vasomotion in supplying the nutrients in clogged blood vessels: an analytical approach

This paper develops a theoretical model to examine the role of vasomotion and its effect on mass transfer processes. Vasomotion, consisting of periodic wave-like oscillations of blood vessel walls, is frequently observed in microcirculation and demonstrates distinct features in arterioles. The advection-diffusion equation governs the mass transport. This analysis can be used to model pathological conditions of blood flow, such as the formation of fatty cholesterol plaques and artery-clogging blood clots within the vessel lumen. These obstructions are modeled as a fictitious porous medium with permeability K. The fluid flow is described using the Darcy-Brinkman equation to quantify the effect of vasomotion. The novelty of the proposed framework is that it integrates the understanding of coupled vasomotion-driven flow and mass transfer in clogged arterioles using a lumped-parameter resistance approach. The developed model combines both blood flow and mass transfer resistances under pathological conditions and establishes their interaction via the Chilton-Colburn J-factor analogy. The present study employs the Frobenius method to solve the governing flow equations and obtain an expression for the velocity field. Furthermore, in the limiting case (absence of hematocrit and vasomotion), an approximate resistance formula for blood flow through a porous medium is derived and validated against previously obtained results. The mass transport equation is solved using the method of separation of variables, within the framework of a Sturm-Liouville problem. The governing equations are analyzed in a wave frame of reference moving in the positive axial direction with wave speed c to capture the effects of vasomotion. The influence of hematocrit level, Darcy number, Schmidt number, and power-law index is illustrated through three-dimensional surface plots and corresponding contour plots. The simulation results show that resistance increases with higher hematocrit levels and diffusivity effects, and decreases with increasing Darcy number. Compared with healthy conditions, fatty mass particles significantly increase resistance by altering viscosity and permeability.