An Anatomically Based Model of Transient Coronary Blood Flow in the Heart

Abstract

An efficient finite difference model of blood flow through the coronary vessels is developed and applied to a geometric model of the largest six generations of the coronary arterial network. By constraining the form of the velocity profile across the vessel radius, the three-dimensional Navier-Stokes equations are reduced to one-dimensional equations governing conservation of mass and momentum. These equations are coupled to a pressure-radius relationship characterizing the elasticity of the vessel wall to describe the transient blood flow through a vessel segment. The two step Lax-Wendroff finite difference method is used to numerically solve these equations. The flow through bifurcations, where three vessel segments join, is governed by the equations of conservation of mass and momentum. The solution to these simultaneous equations is calculated using the multidimensional Newton-Raphson method. Simulations of blood flow through a geometric model of the coronary network are presented demonstrating physiologically realistic flow rates, washout curves, and pressure distributions.

A Hybrid 1D and 3D Approach to Hemodynamics Modelling for a Patient-Specific Cerebral Vasculature and Aneurysm

“…In large arteries, the relative size of red blood cell to vessel diameter is small and blood can be modeled as an incompressible, homogeneous, Newtonian fluid [6,7,8]. If we further assume that the flow in the circumferential direction is negligible and that the radial velocity is small compared to axial velocity, then the governing equations can be reduced to two equations.…”

“…If we further assume that the flow in the circumferential direction is negligible and that the radial velocity is small compared to axial velocity, then the governing equations can be reduced to two equations. By further including a constitutive wall equation [7] we get:…”

“…The hyperbolic set of nonlinear partial differential equations (1-3) is solved numerically using a second order MacCormack finite difference method. Furthermore, a bifurcation model is incorporated to predict flow distribution, velocity and pressure gradient across branches, and thus the whole arterial tree [7].…”

“…On the other hand, blood flow in a vasculature can be modelled with reduced 1D formulations of the governing equations, as used to model flow in cerebral and coronary arterial trees e.g. in [6,7]. However, the 1D models are not capable of capturing complex flow patterns, such as vortices, flow separation and reattachment, or flow reversal as in 3D models.…”

Biofluid Flow and Heat Transfer

“…10 (6) The Nusselt number is the ratio between convective and conductive heat transfer, i.e., the ratio between the product of the convective heat transfer coefficient and the flow characteristic length and the thermal conductivity of the fluid (Nu = h T L/G T ). 11 for a large and small artery and arteriole, respectively.…”