Fully-developed one-dimensional Casson flow through a single vessel of varying radius is proposed as a model of low Reynolds number blood flow in small stenosed coronary arteries. A formula for the resistance-to-flow ratio is derived, and results for yield stresses of tau 0 = 0, 0.005 and 0.01 Nm-2, viscosities of mu = 3.45 x 10(-3), 4.00 x 10(-3) and 4.55 x 10(-3) Pa.s and fluxes of 2.73 x 10(-6), x 10(-5) and x 10(-4) m3 s-1 are determined for segment of 0.45 mm radius and 45 mm length, with 15 mm abnormalities at each end where the radius varies by up to +/- 0.225 mm. When tau 0 = 0.005 N m-2, mu = 4 x 10(-3) Pa.s and Q = 1, the numerical values of the resistance-to-flow ratio vary from lambda = 0.525, when the maximum radii of the two abnormal segments are both 0.675 mm, to lambda = 3.06, when the minimum radii are both 0.225 mm. The resistance-to-flow ratio moves closer to unity as yield stress increases or as blood viscosity or flux decreases, and the magnitude of these alterations is greatest for yield stress and least for flux.

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This paper presents a mathematical model of biological structures in relation to coronary arteries with atherosclerosis. A set of equations has been derived to compute blood flow through these transport vessels with variable axial and radial geometries. Three-dimensional reconstructions of diseased arteries from cadavers have shown that atherosclerotic lesions spiral through the artery. The theoretical framework is able to explain the phenomenon of lesion distribution in a helical pattern by examining the structural parameters that affect the flow resistance and wall shear stress. The study is useful for connecting the relationship between the arterial wall geometries and hemodynamics of blood. It provides a simple, elegant and non-invasive method to predict flow properties for geometrically complex pathology at micro-scale levels and with low computational cost.

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