Gerhard Rappitsch

Computer simulation of local blood flow and vessel mechanics in a compliant carotid artery bifurcation model

To investigate the effect of the distensible artery wall on the local flow field and to determine the mechanical stresses in the artery wall, a numerical model for the blood flow in the human carotid artery bifurcation has been developed. The wall displacement and stress analysis use geometrically non-linear shell theory where incrementally linearly elastic wall behavior is assumed. The flow analysis applies the time-dependent, three-dimensional, incompressible Navier-Stokes equations for non-Newtonian inelastic fluids. In an iteratively coupled approach the equations of the fluid motion and the transient shell equations are numerically solved using the finite element method. The study shows the occurring characteristics in carotid artery bifurcation flow, such as strongly skewed axial velocity in the carotid sinus with high velocity gradients at the internal divider wall and with flow separation at the outer common-internal carotid wall and at the bifurcation side wall. Flow separation results in locally low oscillating wall shear stress. Further strong secondary motion in the sinus is found. The comparison of the results for a rigid and a distensible wall model demonstrates quantitative influence of the vessel wall motion. With respect to the quantities of main interest, it can be seen, that flow separation and recirculation slightly decrease in the sinus and somewhat increase in the bifurcation side region, and the wall shear stress magnitude decreases by 25% in the distensible model. The global structure of the flow and stress patterns remains unchanged. The deformation analysis shows that the tangential displacements are generally lower by one order of magnitude than the normal directed displacements. The maximum deformation is about 16% of the vessel radius and occurs at the side wall region of the intersection of the two branches. The analysis of the maximum principal stresses at the inner vessel surface shows a complicated stress field with locally high gradients and indicates a stress concentration factor of 6.3 in the apex region.

VALIDATED COMPUTATION OF PHYSIOLOGIC FLOW IN A REALISTIC CORONARY ARTERY BRANCH

The pulsatile flow field in an anatomically realistic model of the bifurcation of the left anterior descending coronary artery (LAD) and its first diagonal branch (D1) was simulated numerically and measured by laser Doppler anemometry. The inlet velocity profiles used in the computer simulation and in the physical experiments were physiologically realistic. The computational geometric model was developed on the basis of a digitized arterial cast. The curvature of the LAD over the cardiac surface leads to axial velocity profiles which are slightly skewed towards the epicardial wall. Downstream of the bifurcation, a strong skewing occurs towards the flow divider walls as a result of branching. Locally, the wall shear stress component caused by the complex secondary velocity can be as high as the axial component. The wall shear stress representation from a cell-based perspective exhibits low shear stress and large deviation from the time-averaged shear stress direction during systole. In diastole, the instantaneous wall shear stress direction nearly corresponds to the mean direction. The comparison of computed and measured axial velocity results shows generally good agreement. In contrast to computed flow patterns in simpler geometries constructed from cylindrical tubes, the flow field is found to be smoother, presumably reflecting the adaptation of the vascular contour to the contained flow.

Hemodynamics in the carotid artery bifurcation: a comparison between numerical simulations and in vitro MRI measurements

The presence of atherosclerotic plaques has been shown to be closely related to the vessel geometry. Studies on postmortem human arteries and on the experimental animal show positive correlation between the presence of plaque thickness and low shear stress, departure of unidirectional flow and regions of flow separation and recirculation. Numerical simulations of arterial blood flow and direct blood flow velocity measurements by magnetic resonance imaging (MRI) are two approaches for the assessment of arterial blood flow patterns. In order to verify that both approaches give equivalent results magnetic resonance velocity data measured in a compliant anatomical carotid bifurcation model were compared to the results of numerical simulations performed for a corresponding computational vessel model. Cross sectional axial velocity profiles were calculated and measured for the midsinus and endsinus internal carotid artery. At both locations a skewed velocity profile with slow velocities at the outer vessel wall, medium velocities at the side walls and high velocities at the flow divider (inner) wall were observed. Qualitative comparison of the axial velocity patterns revealed no significant differences between simulations and in vitro measurements. Even quantitative differences such as for axial peak flow velocities were less than 10%. Secondary flow patterns revealed some minor differences concerning the form of the vortices but maximum circumferential velocities were in the same range for both methods.

Numerical study of wall mechanics and fluid dynamics in end-to-side anastomoses and correlation to intimal hyperplasia

In order to analyse the wall mechanics and the flow dynamics in compliant vascular distal end-to-side anastomoses, computer simulation has been performed. In a model study the effect of compliance mismatch on the wall displacements and on the intramural stresses as well as the influence of wall distensibility on the flow patterns are demonstrated applying two distensible models with different graft elasticity. In addition, the flow in a rigid model simulating a vein graft without adaption of the venous lumen has been investigated. The geometries for these models were obtained from a concurrent experimental study, where the formation of distal anastomotic intimal hyperplasia (DAIH) was studied in untreated and externally stiffened autologous venous grafts in sheep. In the flow study the time-dependent, three-dimensional Navier-Stokes equations describing the motion of an incompressible Newtonian fluid are applied. The vessel wall is modelled using a geometrically non-linear shell structure. In an iteratively coupled approach the transient shell equations and the governing fluid equations are solved numerically using the finite element method. In both compliant models maximum displacement and areas of steep stress gradients are observed in the junction region along the graft-artery intersection. The comparison of the normal deformations and the distribution and magnitude of intramural stress shows quantitative differences. The graft elasticity acts as a regulating factor for the deformability and the stress concentration in the junction area: In the model with high graft-elasticity maximum normal deformation at the side wall is 17%. This is twice as large as in the stiff graft model and maximum principle stress at the inner surface differs by one order of magnitude. The numerical results concerning the flow patterns indicate strongly skewed axial velocity profiles downstream of the junction, large secondary motion, flow separation and recirculation on the artery floor opposite the junction and at the inner wall downstream of the toe. In these regions a correlation between the time-averaged fluid wall shear stress and intimal thickening found in the animal experiment can be observed, whereas the pronounced formation of DAIH at the suture line seems to be mainly dependent on wall mechanical factors such as intramural stress and strain.

COMPUTER SIMULATION OF CONVECTIVE DIFFUSION PROCESSES IN LARGE ARTERIES

A numerical analysis of flow and convection-dominated diffusion processes in an axi-symmetric tube with a local constriction simulating a stenosed artery is carried out. The primary aim of this study is to demonstrate the effect of wall shear stress and recirculating flow on the concentration distribution in the vessel lumen and on wall mass transfer. The applied physical parameters describe the convective-diffusive transport of oxygen in the human abdominal aorta. The flow dynamics is described applying the incompressible Navier-Stokes equations for Newtonian fluids, the mass transport is modelled by the convection diffusion equation. For the solute flux at the wall a model with shear-dependent permeability and a model with constant wall permeability are compared. The results demonstrate a strong influence of the mural permeability characteristics on the shape of the flux and interfacial concentration profiles along the wall. The numerical solution of the flow equations and the coupled mass transport equation uses the finite element method. The application of a streamline upwind procedure for the transport equation and a special subelement technique enable a stable solution in the convection-dominated diffusion process. The analysis illustrates an essential influence of the flow patterns on the mass transport. In the reversed flow region downstream of the stenosis the oxygen concentration is decreased to 75% of the inlet concentration value.

Numerical modelling of shear-dependent mass transfer in large arteries

SUMMARY A numerical scheme for the simulation of blood flow and transport processes in large arteries is presented. Blood flow is described by the unsteady 3D incompressible Navier‐Stokes equations for Newtonian fluids; solute transport is modelled by the advection‐diffusion equation. The resistance of the arterial wall to transmural transport is described by a shear-dependent wall permeability model. The finite element formulation of the Navier‐Stokes equations is based on an operator-splitting method and implicit time discretization. The streamline upwind=Petrov‐Galerkin (SUPG) method is applied for stabilization of the advective terms in the transport equation and in the flow equations. A numerical simulation is carried out for pulsatile mass transport in a 3D arterial bend to demonstrate the influence of arterial flow patterns on wall permeability characteristics and transmural mass transfer. The main result is a substantial wall flux reduction at the inner side of the curved region. # 1997 John Wiley & Sons, Ltd.

Mathematical modeling of arterial blood flow and correlation to atherosclerosis

The importance of arterial flow phenomena with regard to atherosclerosis motivates detailed studies of local cardiovascular flow dynamics. The quantitative analysis of flow characteristics can contribute to the understanding of fluid dynamic induced and favored mechanisms in atherogenesis. Numerical methods are very useful in supporting experimental methods and often enable the determination of flow variables which are difficult to obtain in experiments. Due to the development of improved numerical procedures for the blood specific flow equations and due to the application of modern computer technology, the calculations can be carried out under conditions describing the physiological situation in a realistic manner including essential effects. Here various aspects of arterial flow simulation are presented and discussed. The possibilities and difficulties of numerical simulation of arterial flow using finite element approximations are illustrated with the aid of the human carotid artery bifurcation and a distal end-to-side anastomosis model. Further convective-diffusive mass transport in a curved tube model is considered.

Flow dynamic effect of the anastomotic angle: a numerical study of pulsatile flow in vascular graft anastomoses models

Pulsatile non-Newtonian blood flow in three distal end-to-side anastomoses models of prosthetic grafts has been analysed applying computer simulation. The anastomotic angles of the three-dimensional geometric models vary over 30°, 45° and 60°. The pulsatile inflow conditions in the graft remain unchanged. Numerical results are presented for the axial and the secondary flow velocity and the wall shear stress distribution. Further, the paths of fluid particles and the particle residence times are demonstrated by means of numerical flow visualization. The quantitative study shows complex flow and wall shear stress patterns with flow separation and strong secondary motion in the host vessel and a weakly migrating flow stagnation point on the arterial floor during the entire pulse cycle in the models under consideration. The flow dynamic effects, which seem to be important in the development of intimal hyperplasia, are dependent on the anastomotic angle and increase with increasing angle.

Analysis of Hemodynamic Fluid Phase Mass Transport in a Separated Flow Region

The mass transfer behavior in the recirculation region downstream of an axisymmetric sudden expansion was examined. The Reynolds number, 500, and Schmidt number, 3200, were selected to model the mass transfer of molecules, such as ADP, in the arterial system. In a first step the transient mass transport applying zero diffusive flux at the wall was analyzed using experiments and two computational codes. The two codes were FLUENT, a commercially available finite volume method, and FTSP, a finite element code developed at Graz University of Technology. The comparison of the transient wall concentration values determined by the three methods was excellent and provides a measure of confidence for computational mass transfer calculations in convection dominated, separated flows. In a second step the effect of the flow separation on the stationary mass transport applying a permeability boundary condition at the water-permeable wall was analyzed using the finite element code FTSP. The results show an increase of luminal ADP surface concentration in the upstream and in the downstream tube of the sudden expansion geometry in the range of six and twelve percent of the bulk flow concentration. The effect of flow separation in the downstream tube on the wall concentration is a decrease of about ten percent of the difference between wall concentration and bulk concentration occurring at nearly fully developed flow at the downstream region at a distance of 66 downstream tube diameters from the expansion. The decrease of ADP flux into the wall is in the range of three percent of the flux at the downstream region.