Numerical simulation of steady flow in a model of the aortic bifurcation.

The governing equations of steady flow of an incompressible viscous fluid through a 3-D model of the aortic bifurcation are solved with the finite element method. The effect of Reynolds number on the flow was studied for a range including the physiological values (200 < or = Re < or = 1600). The symmetrical bifurcation, with a branch angle of 70 degrees and an area ratio of 0.8, includes a tapered transition zone. Secondary flows induced by the tube curvature are observed in the daughter tubes. Transverse currents in the transition zone are generated by the combined effect of diverging and converging walls. Flow separation depends on both the Reynolds number and the inlet wall shear.     

Numerical simulations of steady flow inside a three dimensional aortic bifurcation model

A finite element formulation of the Navier-Stokes equations for three dimensional flow is presented. The equations are solved using the finite element method. The model is constructed from a cast of a human aortic bifurcation. The numerical problems introduced by solving the equation system are discussed and special attention is paid to the selection of the linear equation solver. The simulations of the steady blood flow patterns in an aortic bifurcation is shown. The results of the numerical analysis are presented as three dimensional plots of velocity vectors, wall shear vectors, streamlines and pressure isobars. The flow simulations are done for Reynolds number 10. The flow patterns found in the bifurcation model are discussed in connection with proposed theories to explain the event of early atherosclerosis.     

Three-dimensional steady flow through a bifurcation

Steady flow of an incompressible, Newtonian fluid through a symmetric bifurcated rigid channel was numerically analyzed by solving the three-dimensional Navier-Stokes equations. The upstream Reynolds number ranged from 100 to 1500. The bifurcation was symmetrical with a branch angle of 60 deg and the area ratio of the daughter to the mother vessel was 2.0. The numerical procedure utilized a coordinate transformation and a control volume approach to discretize the equations to finite difference form and incorporated the SIMPLE algorithm in performing the calculation. The predicted velocity pattern was in qualitative agreement with experimental measurements available in the literature. The results also showed the effect of secondary flow which can not be predicted using previous two-dimensional simulations. A region of reversed flow was observed near the outer wall of the branch except for the case of the lowest Reynolds number. Particle trajectory was examined and it was found that no fluid particles remained within the recirculation zone. The shear stress was calculated on both the inner and the outer wall of the branch. The largest wall shear stress, located in the vicinity of the apex of the branch, was of the same order of magnitude as the level that can cause damage to the vessel wall as reported in a recent study.     

Stress analysis in a layered aortic arch model under pulsatile blood flow

Background  

Many cardiovascular diseases, such as aortic dissection, frequently occur on the aortic arch and fluid-structure interactions play an important role in the cardiovascular system. Mechanical stress is crucial in the functioning of the cardiovascular system; therefore, stress analysis is a useful tool for understanding vascular pathophysiology. The present study is concerned with the stress distribution in a layered aortic arch model with interaction between pulsatile flow and the wall of the blood vessel.     

A mathematical model of flow through a collapsible tube--I. Model and steady flow results

A one-dimensional model is presented to describe the flow through a collapsible tube whose ends are tethered to rigid tubes and which is enclosed in a pressurized chamber. Results are presented for the special case of steady flow. Predicted pressure drop versus flow rate (delta P-Q) characteristics agree qualitatively with available experimental data. The significance of the model and of various physical parameters, in regard to the shape of these characteristics, is discussed.     

Unsteady and three-dimensional simulation of blood flow in the human aortic arch

A three-dimensional and pulsatile blood flow in a human aortic arch and its three major branches has been studied numerically for a peak Reynolds number of 2500 and a frequency (or Womersley) parameter of 10. The simulation geometry was derived from the three-dimensional reconstruction of a series of two-dimensional slices obtained in vivo using CAT scan imaging on a human aorta. The numerical simulations were obtained using a projection method, and a finite-volume formulation of the Navier-Stokes equations was used on a system of overset grids. Our results demonstrate that the primary flow velocity is skewed towards the inner aortic wall in the ascending aorta, but this skewness shifts to the outer wall in the descending thoracic aorta. Within the arch branches, the flow velocities were skewed to the distal walls with flow reversal along the proximal walls. Extensive secondary flow motion was observed in the aorta, and the structure of these secondary flows was influenced considerably by the presence of the branches. Within the aorta, wall shear stresses were highly dynamic, but were generally high along the outer wall in the vicinity of the branches and low along the inner wall, particularly in the descending thoracic aorta. Within the branches, the shear stresses were considerably higher along the distal walls than along the proximal walls. Wall pressure was low along the inner aortic wall and high around the branches and along the outer wall in the ascending thoracic aorta. Comparison of our numerical results with the localization of early atherosclerotic lesions broadly suggests preferential development of these lesions in regions of extrema (either maxima or minima) in wall shear stress and pressure.     

Pressure drop and flow rate measurements in a human aortic bifurcation cast for steady and pulsatile flow

Pressure drop and flow rate measurements in a rigid cast of a human aortic bifurcation under both steady and physiological pulsatile flow conditions are reported. Integral momentum and mechanical energy balances are used to calculate impedance, spatially averaged wall shear stress and viscous dissipation rate from the data. In the daughter branches, steady flow impedance is within 30% of the Poiseuille flow prediction, while pulsatile flow impedance is within a factor of 2 of fully developed, oscillatory, straight tube flow theory (Womersley theory). Estimates of wall shear stress are in accord with measurements obtained from velocity profiles. Mean pressure drop and viscous dissipation rate are elevated in pulsatile flow relative to steady flow at the mean flow rate, and the exponents of their Reynolds number dependence are in accord with available theory.     

Numerical investigation of the three-dimensional flow in a human lung model

The flow field at inspiration and expiration in the upper human airways consisting of the trachea down to the sixth generation of the bronchial tree is numerically simulated. The three-dimensional steady flow at a hydraulic diameter-based Reynolds number Re(D)=1250 is computed via a lattice-Boltzmann method (LBM). The simulation is validated by the experimental data based on particle-image velocimetry (PIV) measurements. The good agreement between numerical and experimental results is evidenced by comparing velocity contours and distributions in a defined reference plane. The results show the LBM to be an accurate tool to numerically predict flow structures in the human lung. Using an automatic Cartesian grid generator, the overall process time from meshing to a steady-state solution is <12h. Moreover, the numerical simulation allows a closer analysis of the secondary flow structures than in the experimental investigation. The three-dimensional streamline patterns reveal some insight on the air exchange mechanism at inspiration and expiration. At inspiration, the slower near-wall tracheal flow enters through the right principal bronchus into the right upper lobar bronchus. The bulk mass flux in the trachea is nearly evenly distributed over the left upper, center and lower lobar bronchi and the right center and lower bronchi. At expiration, the air from the right upper lobar bronchus enters the right center of the trachea and displaces the airflow from the lower and center right bronchi such that the tracheal positions of the streamlines at inspiration and expiration are switched. The flow in the left bronchi does not show this kind of switching. The findings emphasize the impact of the asymmetry of the lung geometry on the respiratory air exchange mechanism.     

Entry flow in a circular tube of aortic arch dimensions

Steady flow within a uniform circular curved tube formed by two 90-deg elbows was studied as a function of psi, the angle between the planes of curvature of the two elbows. Boundary layer separation was found at two locations. The sites of these separation zones were observed to be essentially independent of psi while the Reynolds number at which separation was first detected was found to decrease as psi increased. The relation between separation and the pathogenesis of atherosclerosis is discussed. Secondary flow pattern was found to depend on psi and in some instances on Reynolds number as well.     

One-dimensional steady inviscid flow through a stenotic collapsible tube

A one-dimensional inviscid solution for flow through a compliant tube with a stenosis is presented. The model is used to represent an artery with an atherosclerotic plaque and to investigate a range of conditions for which arterial collapse may occur. The coupled equations for flow through collapsible tubes are solved using a Runge-Kutta finite difference scheme. Quantitative results are given for specific physiological parameters including inlet and outlet pressure, flow rate, stenosis size, length and stiffness. The results suggest that high-grade stenotic arteries may exhibit collapse with typical physiological pressures. Critical stenoses may cause choking of flow at the throat followed by a transition to supercritical flow with tube collapse downstream. Greater amounts of stenosis produced a linear reduction of flow rate and a shortening of the collapsed region. Changes in stenosis length created proportional changes in the length of collapse. Increasing the stiffness of the stenosis to a value greater than the nominal tube stiffness caused a greater amount of flow limitation and more negative pressures, compared to a stenosis with constant stiffness. These findings assist in understanding the clinical consequences of flow through atherosclerotic arteries.     

Pulsatile flow past aortic valve bioprostheses in a model human aorta

Pulsatile flow development past tissue valve prostheses in a model human aorta has been studied using qualitative flow visualization and quantitative laser-Doppler techniques. Experiments were conducted both in steady and physiological pulsatile flow situations and the measurements included the pressure drop across the valve, the instantaneous flow rate as well as the velocity profiles and turbulent stresses downstream to the valves. Our study shows that the velocity profiles with pericardial valves are closer to those measured past natural aortic valves. The porcine valves with a smaller valve opening area produce a narrower and stronger jet downstream from the valve with relatively larger turbulent axial stresses in the boundary of the jet. Our study suggests that the pericardial valves with turbulent stresses comparable to those of caged ball and tilting disc valves are preferable from a hemodynamic point of view.     

Computation of steady flow in a two-dimensional arterial model

The Navier-Stokes equations are solved numerically for steady flow through a double-branched two-dimensional section of a three-dimensional model of a canine aorta for which experimental data is available. The numerical scheme involves transforming the physical coordinates to a curvilinear boundary-fitted coordinate system and performing finite-difference computations in the transformed system. Shear stress at the wall is calculated for a Reynolds number of a 1,000 with branch-to-main aortic flow rate ratios as a parameter. The results are compared with the aforementioned experimental data and show reasonable qualitative agreement.     

On the flow field distortions due to the aortic arch twist

The aortic arch has, on average, an angle of twist of 15 degrees. The purpose of the research conducted was to ascertain if this twist angle has any effect on the flow field of the aortic arch and the distribution of the flow amongst the branches. It was found that the blood flow distribution is practically independent of the angle of twist of the aortic arch. However, the destination of a fluid particle located at a specific point in the cross-section at the entrance to the aortic arch does depend on the angle of twist.     

Computation of steady three-dimensional flow in a model of the basilar artery

The flow in the basilar artery arises from the merging of the flows from the two vertebral arteries. To study the flow phenomena in the basilar artery, computations have been performed using a finite element (FE) method. We consider steady flow in a geometrically symmetric confluence. For simplicity, channels with a rectangular cross-section have been used. Both symmetric and asymmetric flow cases have been considered. The results show that for the Reynolds number of interest the flow downstream of the junction is highly three-dimensional, and that the flow at the end of the basilar artery, where it splits again, will not be fully developed. The computed phenomena have been confirmed by laser Doppler velocity measurements.     

Wall shear stress distribution in a model human aortic arch: assessment by an electrochemical technique

Wall shear stress (WSS) distribution in a human aortic arch model is studied using 130 cathode electrodes flush-mounted on the model walls. Flow visualizations are made in a transparent geometry model to identify the regions of fluid mechanical interests, e.g. regions of flow separation, eddy formation and flow stagnancy. The 130 electrodes are strategically positioned in the arch based on information obtained from the flow visualizations. The measured data indicate that the aortic arch may be categorized into eight regions: three along the inner wall of the arch (A,B,C); and five near the outer wall (D,E,F,G,H). (1) The regions of low WSS are distributed along the inner wall of the ascending aorta A; the inner wall of the descending aorta C; and the upstream inner wall of the innominate and the common carotid branchings F. (2) The high WSS regions are distributed along the outer wall of the arch E; and the inner wall in the arch opposite to the left subclavian branching B. (3) In certain regions, high and low WSS may be found next to each other (e.g. G and H) without a definable boundary in between; and (4) as the Reynolds number increases, the areas of low WSS decrease, while the high WSS areas increase with no obvious change in magnitude of the stress along the inner wall of the arch. At the branchings, the WSS distribution is not affected by the Reynolds number within the range of observations. The measured WSS distribution is compared with Rodkiewicz's map of early atherosclerotic lesions in the aortic arch of cholesterol fed rabbits.     

In vitro hemodynamic investigation of the embryonic aortic arch at late gestation

This study focuses on the dynamic flow through the fetal aortic arch driven by the concurrent action of right and left ventricles. We created a parametric pulsatile computational fluid dynamics (CFD) model of the fetal aortic junction with physiologic vessel geometries. To gain a better biophysical understanding, an in vitro experimental fetal flow loop for flow visualization was constructed for identical CFD conditions. CFD and in vitro experimental results were comparable. Swirling flow during the acceleration phase of the cardiac cycle and unidirectional flow following mid-deceleration phase were observed in pulmonary arteries (PA), head-neck vessels, and descending aorta. Right-to-left (oxygenated) blood flowed through the ductus arteriosus (DA) posterior relative to the antegrade left ventricular outflow tract (LVOT) stream and resembled jet flow. LVOT and right ventricular outflow tract flow mixing had not completed until approximately 3.5 descending aorta diameters downstream of the DA insertion into the aortic arch. Normal arch model flow patterns were then compared to flow patterns of four common congenital heart malformations that include aortic arch anomalies. Weak oscillatory reversing flow through the DA junction was observed only for the Tetralogy of Fallot configuration. PA and hypoplastic left heart syndrome configurations demonstrated complex, abnormal flow patterns in the PAs and head-neck vessels. Aortic coarctation resulted in large-scale recirculating flow in the aortic arch proximal to the DA. Intravascular flow patterns spatially correlated with abnormal vascular structures consistent with the paradigm that abnormal intravascular flow patterns associated with congenital heart disease influence vascular growth and function.     

Comparison of steady and pulsatile flow near the ventral and dorsal walls of casts of human aortic bifurcations

Steady and pulsatile flows were passed through casts of human aortic bifurcations and, by means of a laser Doppler anemometer, fluid velocities were measured at selected sites near the ventral and dorsal walls. At these sites, in the vicinity of the bifurcation, the influence of secondary flow is significant and therefore an appreciation of the phasic variation of secondary flow patterns is important. Results are presented comparing the flow direction in both steady and pulsatile flow at sites in three casts. The common features of the flow at these sites were the persistence of the flow direction during the accelerating and decelerating phases of the pulsatile cycle, and the consistently smaller angle (measured from the inlet centerline) of the pulsatile flow direction as compared to the angle of the flow direction in steady flow.