Estimation of increased flow resistance in a narrow catheterized artery — a theoretical model

Abstract

The changed flow pattern in a narrow catheterized artery is studied and an estimate of the increased flow resistance is made. The anomalous behaviour of blood in small blood vessels has been taken into account by modelling blood as a Casson fluid possessing some finite yield stress. Both the cases of steady and pulsatile flow situations are studied. The pulsatile flow is analysed by considering the pressure gradient as a periodic function of time with small inertial effects. The resulting quasi-steady non-linear coupled implicit system of differential equations governing the flow are solved using a perturbation analysis, where it is assumed that the Womersley frequency parameter is small (α < 1) which is reasonable for physiological situations in small blood vessels as well as in coronary arteries. The effect of pulsatility, catheter radius and yield stress of the fluid on the yield plane locations, velocity distribution, flow rate, shear stress and frictional resistance are investigated. Because of the yield stress θ, two yield surfaces are found to be located in the flow field. Depending on the ratio k (catheter size/vessel size) ranging from 0.3 to 0.7 (which is widely used in coronary angioplasty procedures), the frictional resistance to flow in large blood vessels, where the effect of yield stress can be neglected (i.e. θ = 0), increases by a factor ranging from 3 to 33. In small blood vessels with the same range of catheter size and an unit pressure gradient, frictional resistance increase was by a factor of 7-21 when θ = 0.05 and 11-294 when θ = 0.1. For small values of k and θ, the frictional resistance increase was relatively less, but still appreciable. For comparative large values of k and θ, the frictional resistance increases to several hundred times thus implying that the combined effect of increased catheter radius and yield stress is to obstruct the fluid movement considerably.