A hybrid one-dimensional/Womersley model of pulsatile blood flow in the entire coronary arterial tree

Abstract

Huo Y, Kassab GS. A hybrid one-dimensional/Womersley model of pulsatile blood flow in the entire coronary arterial tree. Am J Physiol Heart Circ Physiol 292: H2623-H2633, 2007. First published January 5, 2007; doi:10.1152/ajpheart.00987.2006.-Using a frequency-domain Womersley-type model, we previously simulated pulsatile blood flow throughout the coronary arterial tree. Although this model represents a good approximation for the smaller vessels, it does not take into account the nonlinear convective energy losses in larger vessels. Here, using Womersley's theory, we present a hybrid model that considers the nonlinear effects for the larger epicardial arteries while simulating the distal vessels (down to the 1st capillary segments) with the use of Womersley's Theory. The main trunk and primary branches were discretized and modeled with one-dimensional Navier-Stokes equations, while the smaller-diameter vessels were treated as Womersley-type vessels. Energy losses associated with vessel bifurcations were incorporated in the present analysis. The formulation enables prediction of impedance and pressure and pulsatile flow distribution throughout the entire coronary arterial tree down to the first capillary segments in the arrested, vasodilated state. We found that the nonlinear convective term is negligible and the loss of energy at a bifurcation is small in the larger epicardial vessels of an arrested heart. Furthermore, we found that the flow waves along the trunk or at the primary branches tend to scale (normalized with respect to their mean values) to a single curve, except for a small phase angle difference. Finally, the model predictions for the inlet pressure and flow waves are in excellent agreement with previously published experimental results. This hybrid one-dimensional/Womersley model is an efficient approach that captures the essence of the hemodynamics of a complex large-scale vascular network. The present model has numerous applications to understanding the dynamics of coronary circulation. coronary flow; pulse wave; admittance; impedance THE FLUID DYNAMIC APPROACH uses the geometry and mechanical properties of the vasculature and the principles of conservation of mass and momentum to obtain the blood flow-pressure relation. The solution of such equations yields information on instantaneous velocity and pressure distributions. The fluid dynamic approach was first formulated in 1755 by Euler. Womersley (32,33) and McDonald [as reported by Nichols and O'Rourke (22)] linearized Euler's equation and presented the theory and solution in great detail. Because of the complexity of the vascular architecture, however, this approach recognizes that it is necessary to idealize the geometry of the vasculature.The frequency-domain Womersley-type approach was recently applied to the entire coronary arterial tree down to the first segment of (arterial) capillaries in a vasodilated, potassiumarrested heart (7). Inasmuch as this approach represents a linearization of the fluid dynamic (time-domain) approach, it ign...

Scalability and in vivo validation of a multiscale numerical model of the left coronary circulation

“…Huo and Kassab (25,26) validated pulsatile flow simulations from a 1D coronary network model against measurements in isolated, arrested, and vasodilated pig hearts, but the waveforms were not representative of the in situ beating heart and did not contain high-frequency transients. Rumberger and Nerem (54) and Guiot et al (21) compared measured flow waveforms with (71) compared measured and 1D modelsimulated flow waveforms in a single patient; however, the focus was on gross changes to the flow waveform that could likely also be predicted by a 0D model.…”

“…That these effects play a role in shaping the coronary arterial flow waveform was first suggested by Rumberger et al (54,55) on the basis of flow transients ("oscillations") observed in early systole and diastole in the left anterior descending artery of horses, features that are also evident in published waveforms from humans and other species (5,14,38,76). More recently, therefore, a number of one-dimensional (1D) or multiscale (i.e., combined 0D/1D) models of the coronary circulation, which are best suited to the study of wave propagation effects, have been described (25,45,52,57,58,71). Validation of such models requires direct comparison of simulated and experimental waveform shapes, including any high-frequency flow transients.…”

A two-level time step technique for the partitioned solution of one-dimensional arterial networks

“…In particular, the value of the Young modulus E S for each arterial segment has been taken from [2] (for the body and the cerebral parts) and from [28] (for the coronaries). The Poisson ratio m S has been set equal to 0:5, as the arterial wall is assumed to be incompressible.…”

A 1D arterial blood flow model incorporating ventricular pressure, aortic valve and regional coronary flow using the locally conservative Galerkin (LCG) method

“…The classic linear analytical solution known as Womersley flow [42] is a Fourier technique and enables calculation of axi-symmetric velocity profiles by treating each harmonic separately and adding the results together. This technique is commonly used [1,29,36] along with 1D electrical analogs [15] but these do not account for non-linear effects, which are important particularly in the large arteries [5,21,43]. The non-linear equations have been discretized using the finite difference method [2,7,44], as well as various other methods such as the characteristic [16], Galerkin least-squares [6] and discontinuous Galerkin methods [3][4][5].…”