Three-dimensional flows in a hyperelastic vessel under external pressure

We study the collapsible behaviour of a vessel conveying viscous flows subject to external pressure, a scenario that could occur in many physiological applications. The vessel is modelled as a three-dimensional cylindrical tube of nonlinear hyperelastic material. To solve the fully coupled fluid–structure interaction, we have developed a novel approach based on the Arbitrary Lagrangian–Eulerian (ALE) method and the frontal solver. The method of rotating spines is used to enable an automatic mesh adaptation. The numerical code is verified extensively with published results and those obtained using the commercial packages in simpler cases, e.g. ANSYS for the structure with the prescribed flow, and FLUENT for the fluid flow with prescribed structure deformation. We examine three different hyperelastic material models for the tube for the first time in this context and show that at the small strain, all three material models give similar results. However, for the large strain, results differ depending on the material model used. We further study the behaviour of the tube under a mode-3 buckling and reveal its complex flow patterns under various external pressures. To understand these flow patterns, we show how energy dissipation is associated with the boundary layers created at the narrowest collapsed section of the tube, and how the transverse flow forms a virtual sink to feed a strong axial jet. We found that the energy dissipation associated with the recirculation does not coincide with the flow separation zone itself, but overlaps with the streamlines that divide the three recirculation zones. Finally, we examine the bifurcation diagrams for both mode-3 and mode-2 collapses and reveal that multiple solutions exist for a range of the Reynolds number. Our work is a step towards modelling more realistic physiological flows in collapsible arteries and veins.     

Low Reynolds number equi-bifurcation flow in a two-dimensional channel

The fluid flow in some physiological vessels such as the blood flow in blood vessels and the air flow through bronchi and bronchioles in the lungs undergoes a large number of bifurcations. The understanding of the bifurcation flow is of importance for a better comprehension of its effect in the blood and the air circulatory systems of the living body. The Reynolds number of flow in large blood vessels and bronchi is high and fluid inertia plays a dominant role in the bifurcation flow in such vessels. In small caliber blood vessels such as arterioles and capillaries, and bronchioles, the Reynolds number of flow is quite low and the effect of fluid inertia is negligible compared to the pressure and shear forces. In order to have a quantitative understanding of the bifurcation flow at low Reynolds numbers, the low Reynolds number equi-bifurcation flow in a two-dimensional channel at zero bifurcation angle is studied based on the Stokes approximation. The solution of the problem is posed as an infinite series, where the truncated version is used in numerical calculations. The results of this analysis is discussed in connection with the bifurcation flow of blood in small caliber blood vessels and that of the air in bronchioles in the lung.     

Flow in t-bifurcations: Effect of the sharpness of the flow divider

The local geometry of a bifurcation has been hypothesized to be a potential geometrical risk factor for the development of atherosclerosis. While flow division and branch area ratios clearly affect the flow field, the importance of the flow divider shape is not as clear. A fast spectral element computational fluid mechanics (CFD) solver was used to simulate flow through 90 ° T-bifurcations with three different flow divider shapes. Other factors, such as flow partition, area ratio, and bifurcation angle, were kept constant. A Reynolds number range of 15 to 350 was studied to bracket experimental results in the literature. The variation in the sharpness of the corners had a dramatic effect on both the flow field and wall shear stress distribution in the side branch, but little effect on the flow in the main tube. The magnitude of reverse velocities and wall shear stress in the side branch increased linearly over a physiological range of Reynolds number and corner shape. This paper verifies the accuracy and usefulness of spectral element CFD in studying three-dimensional hemodynamics.     

The effects of non-Newtonian viscoelasticity and wall elasticity on flow at a 90 degrees bifurcation

To study the fundamentals of hemodynamics in arteries, the flow parameters: pulsatility, elasticity and non-Newtonian viscoelasticity were considered in detail in a 90 degrees-T-bifurcation of a rigid and elastic model. The velocity distribution 2.5 mm behind the bifurcation in the straight tube was measured with a laser-Doppler-anemometer. The fluid used was an aqueous glycerine solution and a viscoelastic Separan mixture. Flow visualization studies were done with a sheet of laser light in the plane of the bifurcation. The velocity distribution was measured for both steady and pulsatile flows with a laser-Doppler-anemometer in a backward scattered way. From the velocity measurements the shear gradients were calculated. Substantial differences were found in the flow behavior of Newtonian and non-Newtonian fluids, especially behind the bifurcation in the main tube, where secondary flows and flow separation started. Also, differences due to the elastic and rigid wall could be seen. Very high shear gradients were found in the flow between main flow and the separation zone which can lead to a damage of the blood cells.     

Coupled fluid–structure interaction hemodynamics in a zero-pressure state corrected arterial geometry

Hemodynamic conditions in large arteries are significantly affected by the interaction of the pulsatile blood flow with the distensible arterial wall. A numerical procedure for solving the fluid–structure interaction problem encountered in cardiovascular flows is presented. We consider a patient-specific carotid bifurcation geometry, obtained from 3D reconstruction of in vivo acquired tomography images, which yields a geometrical representation of the artery corresponding to its pressurized state. To recover the geometry of the artery in its zero-pressure state which is required for a fluid–structure interaction simulation we utilize inverse finite elastostatics. Time-dependent flow simulations with in vivo measured inflow volume flow rate in the 3D undeformed artery are performed through the finite element method. The coupled-momentum method for fluid–structure interaction is adopted to incorporate the influence of wall compliance in the numerical computation of the time varying flow domain. To demonstrate the importance in recovering the zero-pressure state of the artery in hemodynamic simulations we compute the time varying flow field with compliant walls for the original and the zero-pressure state corrected geometric configurations of the carotid bifurcation. The most important resulting effects in the hemodynamic environment are evaluated. Our results show a significant change in the wall shear stress distribution and the spatiotemporal extent of the recirculation regions.     

A study of the bifurcation behaviour of a model of flow through a collapsible tube

Most of the elastic tubes found in the mammalian body will collapse from a distended circular cross section and when collapsed may undergo flow-induced oscillations. A mathematical model describing fluid flow in a collapsible tube is analysed using the software package AUTO-86. AUTO-86 is used for continuation and bifurcation problems in systems of non-linear ordinary differential equations. The model is a third-order lumped-parameter type and is based on the classical “Starling resistor”; it describes the unsteady flow behaviour and, in particular, the experimentally observed self-excited oscillations, in a way which is simple enough to give physical understanding, yet still firmly based on fluid mechanical principles. Some of the bifurcation types found in this model bear close resemblance to the types suggested by experimental observations of self-excited oscillations in collapsible tubes; they thus shed some light on the various topological changes which occur in practice, particularly in view of the fact that some of the points found numerically are diffcult to achieve experimentally, while the existence of others can only be inferred indirectly and uncertainly from experiment.     

Characteristics of secondary flow in steady and pulsatile flows through a symmetrical bifurcation

Steady and pulsatile flow in a glass model simulating an arterial bifurcation was investigated by flow visualization techniques. Secondary flow generated at the bifurcation has a similar pattern to a vortex, called the horseshoe vortex, produced around a wall-based protuberance in a circular tube. The same flow disturbance was clearly observed during the decelerating phase of pulsatile flow. The vortex produces a stagnation point on the top and bottom wall just upstream from the bifurcation apex. When aluminium dust was suspended in the test fluid perfusing the blood vessel model, particles deposited over an area spreading from the stagnation point to the lateral corners of the bifurcation. Comparison between the present results and topographical patterns of atherosclerosis reported in the literature suggests that it is in such low shear regions that lipid deposition tends to occur most.     

Analysis of non-Newtonian effects within an aorta-iliac bifurcation region

The geometry of the arteries at or near arterial bifurcation influences the blood flow field, which is an important factor affecting arteriogenesis. The blood can act sometimes as a non-Newtonian fluid. However, many studies have argued that for large and medium arteries, the blood flow can be considered to be Newtonian. In this work a comprehensive investigation of non-Newtonian effects on the blood fluid dynamic behavior in an aorta-iliac bifurcation is presented. The aorta-iliac geometry is reconstructed with references to the values reported in Shah et al. (1978); the 3D geometrical model consists of three filleted cylinders of different diameters. Governing equations with the appropriate boundary conditions are solved with a finite-element code. Different rheological models are used for the blood flow through the lumen and detailed comparisons are presented for the aorta-iliac bifurcation. Results are presented in terms of the velocity profiles in the bifurcation zone and Wall Shear Stress (WSS) for different sides of the bifurcation both for male and female geometries, showing that the Newtonian fluid assumption can be made without any particular loss in terms of accuracy with respect to the other more complex rheological models.     

Local geometry of arterial branching

A model of the geometrical structure of arterial bifurcations is proposed in the context of optimality of the bifurcation as a fluid conducting system. Optimality is considered both globally, in terms of the cardiovascular system as a whole, and locally, in terms of the orderliness of the flow in the bifurcation region. It is shown that a bifurcation can be optimal both globally and locally. Typical examples of such bifurcations are given.     

Computational analysis of flow structure and particle deposition in a single asthmatic human airway bifurcation

This paper aims to improve current understanding of flow structure and particle deposition in asthmatic human airways. A single, symmetric airway bifurcation, corresponding to generations 10–11 of Weibel’s model, is investigated through validated numerical simulations. The parent airway segment is modelled as a smooth circular tube. The child segments are considered asthmatic and their cross-section is modelled as a constricted tube with sinusoidal folds uniformly distributed along the circumference. The flow structure and particle deposition pattern for normal (i.e., healthy) and asthmatic airway bifurcations are compared and discussed. The numerical results reveal that the secondary flow in the asthmatic airway bifurcation is much stronger than in the healthy one, resulting in higher particle deposition. The effects of size of the lumen area and number of folds on particle deposition and pressure drop are also investigated. It is found that particle deposition efficiency is significantly affected by lumen area of the asthmatic segment (the smaller the lumen area, the higher the particle deposition efficiency). The effect of number of folds is small. Particle deposition efficiency also increases with Reynolds number. The pressure drop in the asthmatic airway bifurcation depends mainly on size of the lumen area. The effect of number of folds becomes important for strongly collapsed airways.     

Pulsatile non-newtonian blood flow in three-dimensional carotid bifurcation models: a numerical study of flow phenomena under different bifurcation angles

Flow and stress patterns in human carotid artery bifurcation models, which differ in the bifurcation angle, are analysed numerically under physiologically relevant flow conditions. The governing Navier-Stokes equations describing pulsatile, three-dimensional flow of an incompressible non-Newtonian fluid are approximated using a pressure correction finite element method, which has been developed recently. The non-Newtonian behaviour of blood is modelled using Casson's relation, based on measured dynamic viscosity. The study concentrates on flow and stress characteristics in the carotid sinus. The results show that the complex flow in the sinus is affected by the angle variation. The magnitude of reversed flow, the extension of the recirculation zone in the outer sinus region and the duration of flow separation during the pulse cycle as well as the resulting wall shear stress are clearly different in the small angle and in the large angle bifurcation. The haemodynamic phenomena, which are important in atherogenesis, are more pronounced in the large angle bifurcation.     

Influence of bifurcations on forced oscillations in an airway model.

Forced oscillations is a technique to determine respiratory input impedance from small amplitude sinusoidal pressure excursions introduced at the airway opening. Models used to predict respiratory input impedance typically ignore the direct effect of bifurcations on the flow, and treat airway branches as individual straight tubes placed appropriately in parallel and series. The flow within the individual tubes is assumed equivalent to that which would occur in infinitely long tubes. In this study we examined the influence of bifurcations on impedance for conditions of the forced oscillatory technique. We measured input impedance using forced oscillations in straight tubes and in an anatomically-relevant, four generation physical model of a human airway network. The input impedance measured experimentally compared well to that obtained theoretically using model predictions. The predictive scheme was based on appropriate parallel and series combinations of theoretically computed individual tube impedances, which were computed from solutions to oscillatory flow of a compressible gas in an infinitely long rigid tube. The agreement between experimental measurements and predictions indicates that bifurcations play a relatively minor direct role on the flow impedance for conditions of the forced oscillations technique. These results are explained in terms of the small tidal volumes used, whereby the axial distance traveled by a fluid particle during an oscillation cycle is appreciably smaller than branch segment lengths. Accordingly, only a small fraction of fluid particles travel through the bifurcation region, and the remainder experience an environment approaching flow in an infinite straight tube. The relevance of the study to the prediction of impedances in the human lung during forced oscillations is discussed.     

Evaluation of a transient,simultaneous, arbitrary Lagrange–Euler based multi-physics method for simulating the mitral heart valve

A transient multi-physics model of the mitral heart valve has been developed, which allows simultaneous calculation of fluid flow and structural deformation. A recently developed contact method has been applied to enable simulation of systole (the stage when blood pressure is elevated within the heart to pump blood to the body). The geometry was simplified to represent the mitral valve within the heart walls in two dimensions. Only the mitral valve undergoes deformation. A moving arbitrary Lagrange–Euler mesh is used to allow true fluid–structure interaction (FSI). The FSI model requires blood flow to induce valve closure by inducing strains in the region of 10–20%. Model predictions were found to be consistent with existing literature and will undergo further development.     

Pulsatile non-Newtonian blood flow simulation through a bifurcation with an aneurysm

Blood flow is analysed by means of computer simulation in an idealized arterial bifurcation model which is pathologically altered by a saccular aneurysm. The theoretical study of the flow pattern and the paths of fluid particles is carried out under pulsatile Newtonian and non-Newtonian flow conditions. The governing equations are solved numerically with the use of the finite element method. The results show the disturbed blood flow in the bifurcation and the relatively low intra-aneurysmal flow circulation. In addition to the study of basic flow patterns in the segment, a comparison of non-Newtonian and Newtonian results is carried out. This comparison proves that for the considered large artery model under physiological flow conditions where the yield number is relatively low there is no essential difference in the results.     

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.     

A newly designed oscillating viscometer for blood viscosity measurements

A newly designed type of oscillating viscometer is described. The viscometer consists of either a tube or a rod oscillating at a resonance frequency with amplitudes in the micro- and nanometer range. A fluid flowing through the tube or surrounding the rod damps the torsional oscillations. The increase in the damping depends on the viscosity of the fluid and is used to determine viscosity. It was found that viscosity measurements are feasible during blood flow. This new type of viscometer may be useful to the study of biophysical properties of blood at the wall surface during flow and give new insights into blood flow. The device allows direct viscosity measurement on blood directly as it is drawn from the vein through the tube without any anticoagulant.     

Blood flow downstream of a two-dimensional bifurcation

Many cardiovascular lesions such as aneurysms, intimal cushions, and atherosclerotic plaques tend to occur near bifurcations. This suggests that hemodynamic factors may be involved. Since measuring devices (such as anemometers) are still too large to allow local measurements of flow disturbances, we have attempted to predict the nature of these factors mathematically. Biological variables include pulsatile flow of a nonNewtonian fluid in distensible branching vessels with different angles and flow rates. Our initial analysis considers the flow in a two-dimensional bifurcation with a symmetrical flow divider perfused with steady flow at variable Reynolds numbers. At all flows, high shear forces develop on either side of the flow divider (i.e. at the apex of the bifurcation). With high flows, regions of sluggish or reverse flow develop near the outer walls of the bifurcation. The analysis confirms that the flow at the apex is quite different from that at the outer angles and that the latter varies more with flow rate than the former.     

Effects of velocity jumps on blood flow in large arteries

When a human being experiences a sudden velocity change, the blood flow is disturbed. A theoretical analysis to predict the effects of sudden velocity changes on blood flow in large arteries is presented. The situations is modelled as a one-dimensional flow problem in a viscoelastic tube where the fluid viscosity convective term in the equation of motion and nonlinearity in the elastic modulus of the tube wall are neglected. The governing equations of the model are solved by Laplace transformation. The computed results show that relatively high blood pressures, capable of harming circulation, are produced even by relatively moderate velocity jumps.