Numerical simulation of aortic bifurcation flows: The effect of flow divider curvature

Two dimensional steady flow calculations in computational regions obtained from radiographs of human aortic bifurcations correlate well with unsteady measurements of wall shear in flow-through casts of the same vessels. The results suggest that wall slope may be an important factor affecting the variability of shear along the medial walls of this arterial segment. If extremes of shear stress promote atherogenesis, then variations in the curvature of the proximal iliac arteries may affect the susceptibility of these vessels to vascular disease on their medial aspect.     

Atherosclerosis and flow in carotid arteries with authentic geometries

The influence of blood flow on the depositions and development of atherosclerotic lesions have been observed and described since the 19th century. Observations have shown that depositions correlate with regions of low wall shear stress. However, the exact correlations between depositions, vessel geometry and flow parameters are not yet known. The purpose of this study was the quantification of atherosclerosis risk factors in carotid bifurcation. This artery has attracted particular interest because lesions are often found in this bifurcation. Post mortem, the arteries are excised and vessel casts are produced. Afterwards, the arteries are analyzed morphometrically. The vessel casts are used for the assessment of some geometrical parameters. 31 carotid bifurcations were analyzed in this study. Eight vessel casts were digitized and rendered three-dimensional mathematical models of the arteries. These data were imported by the computational fluid dynamics program FLUENT. Further, the blood flow was reconstructed in a computer model based on the individual vessel geometry. The flow parameters, such as velocity, pressure and wall shear stress were computed. At the same time the geometrical parameters and wall alterations are known. This permits the comparison of the anatomical shape and its flow with the distribution and level of the wall alterations.     

Influence of segmentation on morphological parameters and computed hemodynamics in cerebral aneurysms

This article focuses on the effects of segmentation on cerebral aneurysm's morphological parameters and on blood flow patterns computed using computational fluid dynamics. Segmentation is a non-negligible source of uncertainty that may have a consequent impact on the morphological assessment and the resulting hemodynamical simulations, the latter potentially being a key element in the decision-making therapeutic armamentarium for neuroradiologists and neurosurgeons. From the three patient-specific cases investigated, medical imaging data sets were collected, and four different three-dimensional segmentations were generated by the same senior technician. Morphological parameters were measured, and the aspect ratio was derived. Numerical simulations were performed; flow pattern changes, their impact on wall shear stress (WSS) and their sensitivity within the four reconstructed geometries were analyzed. Aneurysm velocity, vorticity and shear magnitudes were computed and compared. The morphological parameters having the highest variability were the aneurysm lobe dimensions (20 %). The neck length was the second parameter presenting the highest variability (21 %). The neck width variability reached 13.8 %, and the aspect ratio variability reached 14.2 %. The artery height and the artery width presented a variability of 13.7 and 10.8 %, respectively. Finally, the aneurysm depth, aneurysm height and aneurysm width presented variabilities of 12.8, 9.4 and 7.3 %, respectively. Differences in the flow path lines, velocity magnitude, wall shear stress and vorticity are also reported and discussed. The average variability reached 15.6 % for velocity, 25.2 % for vorticity and 25.2 % for shear, these parameters being computed inside the aneurysm. The maximum variability reached 31.0 % for velocity, 54.8 % for vorticity and 58.1 % for shear. A segmentation process reconstructing anatomies that is less sensitive to human intervention would be a future goal worth pursuing.     

Optimising Cell Aggregate Expansion in a Perfused Hollow Fibre Bioreactor via Mathematical Modelling

The need for efficient and controlled expansion of cell populations is paramount in tissue engineering. Hollow fibre bioreactors (HFBs) have the potential to meet this need, but only with improved understanding of how operating conditions and cell seeding strategy affect cell proliferation in the bioreactor. This study is designed to assess the effects of two key operating parameters (the flow rate of culture medium into the fibre lumen and the fluid pressure imposed at the lumen outlet), together with the cell seeding distribution, on cell population growth in a single-fibre HFB. This is achieved using mathematical modelling and numerical methods to simulate the growth of cell aggregates along the outer surface of the fibre in response to the local oxygen concentration and fluid shear stress. The oxygen delivery to the cell aggregates and the fluid shear stress increase as the flow rate and pressure imposed at the lumen outlet are increased. Although the increased oxygen delivery promotes growth, the higher fluid shear stress can lead to cell death. For a given cell type and initial aggregate distribution, the operating parameters that give the most rapid overall growth can be identified from simulations. For example, when aggregates of rat cardiomyocytes that can tolerate shear stresses of up to are evenly distributed along the fibre, the inlet flow rate and outlet pressure that maximise the overall growth rate are predicted to be in the ranges to (equivalent to to ) and to (or 15.6 psi to 15.7 psi) respectively. The combined effects of the seeding distribution and flow on the growth are also investigated and the optimal conditions for growth found to depend on the shear tolerance and oxygen demands of the cells.     

Correlation among shear rate measures in vascular flows

A variety of shear rate measures have been calculated from hemodynamic data obtained by laser Doppler anemometry in flow-through casts of human aortic bifurcations. Included are measures sensitive to the mean and amplitude of the shear rate, its maximum rate of change, the duration of stasis and flow reversal near the wall, and the unidirectionality of the flow. Many of these measures are highly correlated with one another. This suggests that that it will be difficult to identify from in vivo measurements those aspects of the flow field to which the vessel wall is most sensitive. It may be possible to separate the effects of purely temporal factors (e.g., the duration of flow reversal) from those related to wall shear stress.     

A two-dimensional numerical analysis of unsteady flow in the carotid artery bifurcation. A comparison with three-dimensional in-vitro measurements and the influence of minor stenoses

In the present study a two-dimensional finite element model for incompressible Newtonian flow is applicated to the modelling of carotid artery flow. In earlier studies, the numerical model was validated experimentally for several flow configurations. In general the pulsatile flow is characterized by reversed flow regions at the non-divider side walls of both the internal and external carotid arteries. The unsteadiness of the flow is associated with rather complex spatial and temporal velocity distributions and leads to temporal variations of the location and length of the reversed flow regions. As a consequence, pronounced spatial and temporal variations in the wall shear stresses are found. At the non-divider side walls, wall shear stresses are relatively low and exhibits an oscillatory behaviour in space and time. At the divider side walls, wall shear stresses are relatively high and approximately follow the flow rate distribution in time. The aim of this study is not only to present two-dimensional calculations but also to compare the calculated two-dimensional velocity profiles with those from three-dimensional experiments. It is observed that in the common carotid artery and in the proximal parts of the internal and external carotid arteries, the two-dimensional numerical model provides valuable information with respect to the three-dimensional configuration. In the more distal parts of especially the internal carotid artery, deviations are found between the two-dimensional numerical and three-dimensional experimental model. These deviations can mainly be attributed to the neglect of the secondary velocity distribution in the two-dimensional model. In the two-dimensional numerical model the influence of a minor stenosis in the internal carotid artery is hardly distinguishable from a minor geometrical variation without stenosis. Full three-dimensional analyses of the influence of minor stenoses are needed to prove numerically whether in-vivo measurements of the axial velocity distribution are useful in the detection of minor stenoses.     

Spatial comparison between wall shear stress measures and porcine arterial endothelial permeability

A better understanding of how hemodynamic factors affect the integrity and function of the vascular endothelium is necessary to appreciate more fully how atherosclerosis is initiated and promoted. A novel technique is presented to assess the relation between fluid dynamic variables and the permeability of the endothelium to macromolecules. Fully anesthetized, domestic swine were intravenously injected with the albumin marker Evans blue dye, which was allowed to circulate for 90 min. After the animals were euthanized, silicone casts were made of the abdominal aorta and its iliac branches. Pulsatile flow calculations were subsequently made in computational regions derived from the casts. The distribution of the calculated time-dependent wall shear stress in the external iliac branches was directly compared on a point-by-point basis with the spatially varying in vivo uptake of Evans blue dye in the same arteries. The results indicate that in vivo endothelial permeability to albumin decreases with increasing time-average shear stress over the normal range. Additionally, endothelial permeability increases slightly with oscillatory shear index.     

Non-Newtonian Blood Flow in Left Coronary Arteries with Varying Stenosis: A Comparative Study

This paper presents Computational fluid dynamic (CFD) analysis of blood flow in three different 3-D models of left coronary artery (LCA). A comparative study of flow parameters (pressure distribution, velocity distribution and wall shear stress) in each of the models is done for a non-Newtonian (Carreau) as well as the Newtonian nature of blood viscosity over a complete cardiac cycle. The difference between these two types of behavior of blood is studied for both transient and steady states of flow. Additionally, flow parameters are compared for steady and transient boundary conditions considering blood as non-Newtonian fluid. The study shows that the highest wall shear stress (WSS), velocity and pressure are found in artery having stenosis in all the three branches of LCA. The use of Newtonian blood model is a good approximation for steady as well as transient blood flow boundary conditions if shear rate is above 100 s-1. However, the assumption of steady blood flow results in underestimating the values of flow parameters such as wall shear stress, pressure and velocity.     

A mechanism for heterogeneous endothelial responses to flow in vivo and in vitro

Exposure of endothelium to a nominally uniform flow field in vivo and in vitrofrequently results in a heterogeneous distribution of individual cell responses. Extremes in response levels are often noted in neighboring cells. Such variations are important for the spatial interpretation of vascular responses to flow and for an understanding of mechanotransduction mechanisms at the level of single cells. We propose that variations of local forces defined by the cell surface geometry contribute to these differences. Atomic force microscopy measurements of cell surface topography in living endothelium both in vitro and in situ combined with computational fluid dynamics demonstrated large cell-to-cell variations in the distribution of flow-generated shear stresses at the endothelial luminal surface. The distribution of forces throughout the surface of individual cells of the monolayer was also found to vary considerably and to be defined by the surface geometry. We conclude that the endothelial three-dimensional surface geometry defines the detailed distribution of shear stresses and gradients at the single cell level, and that there are large variations in force magnitude and distribution between neighboring cells. The measurements support a topographic basis for differential endothelial responses to flow observed in vivo and in vitro. Included in these studies are the first preliminary measurements of the living endothelial cell surface in an intact artery.     

Numerical flow studies in human carotid artery bifurcations: basic discussion of the geometric factor in atherogenesis

In this study fluid dynamic variables are analysed numerically in different human carotid artery bifurcation models in order to clarify the geometric factor in carotid bifurcation atherogenesis. The geometric variations describe healthy human carotid bifurcation anatomy and concern the shape of the carotid sinus and the angle between the branches. The flow conditions remain unchanged. The governing Navier-Stokes equations describing incompressible, pulsatile, three-dimensional viscous flow are approximated using a pressure correction finite element procedure which has been developed for time-consuming, three-dimensional, time-dependent viscous flow problems. The study concentrates on flow velocity, on detailed analysis of flow separation and flow recirculation, and on wall shear stress distribution. The results show that the extension and the location of the recirculation zone in the sinus as well as the duration of separated flow during the pulse cycle are affected by the geometrical variations. In view of the significance of the reversed flow zones and of the accompanied low shear regions in atherogenesis the geometry-dependent flow separation characteristics in the sinus is of substantial interest.     

Effect of secondary flow on biological experiments in the cone-plate viscometer: methods for estimating collision frequency, wall shear stress and inter-particle interactions in non-linear flow.

We present a theoretical analysis of fluid flow and particle interactions in the cone-plate viscometer under conditions typically applied in biological studies. The analysis demonstrates that at higher shear rates, besides linear primary flow in the rotational direction, prominent non-linear secondary flow causes additional fluid circulation in the radial direction. Two parameters, the cone angle and Reynolds number, characterize flow in the viscometer over all ranges of shear rate. Our results indicate that secondary flow causes positional variations in: (i) the velocity gradient, (ii) the direction and magnitude of the wall shear stress at the plate surface, (iii) inter-particle collision frequency, (iv) magnitude and periodicity of normal and shear forces applied during particle-particle interactions, and (v) inter-particle attachment times. Thus, secondary flow may significantly influence cellular aggregation, platelet activation and endothelial cell mechanotransduction measurements. Besides cone-plate viscometers, this analysis methodology can also be extended to other experimental systems with complex non-linear flows.     

Flow-pressure drop measurement and calculation in a tapered femoral artery of a dog

This study describes the in vivo measurement of pressure drop and flow during the cardiac cycle in the femoral artery of a dog, and the computer simulation of the experiment based on the use of the measured flow, vessel dimensions and blood viscosity. In view of the experimental uncertainty in obtaining the accurate velocity profile at the wall region, the velocity pulse at the center was measured and numerical calculations were performed for the center Une instantaneous velocity and within the two limits of spatial distribution of inlet flow conditions: uniform and parabolic. Temporal and spatial variations of flow parameters, i.e., velocity profile, shear rate, non-Newtonian viscosity, wall shear stress, and pressure drop were calculated. There existed both positive and negative shear rates during a pulse cycle, i.e., the arterial wall experiences zero shear three times during a cardiac cycle. For the parabolic inlet condition, the taper of the artery not only increased the magnitude of the positive and negative shear rates, but caused a steep gradient in shear rate, a phenomenon which in turn affects wall shear stress and pressure. In contrast, for the uniform inlet condition, the flow through the tapered artery was predominantly the developing type, which resulted in reduction in magnitude of wall shear rate along the axial direction.     

Flow in the well: computational fluid dynamics is essential in flow chamber construction

A perfusion system was developed to generate well defined flow conditions within a well of a standard multidish. Human vein endothelial cells were cultured under flow conditions and cell response was analyzed by microscopy. Endothelial cells became elongated and spindle shaped. As demonstrated by computational fluid dynamics (CFD), cells were cultured under well defined but time varying shear stress conditions. A damper system was introduced which reduced pulsatile flow when using volumetric pumps. The flow and the wall shear stress distribution were analyzed by CFD for the steady and unsteady flow field. Usage of the volumetric pump caused variations of the wall shear stresses despite the controlled fluid environment and introduction of a damper system. Therefore the use of CFD analysis and experimental validation is critical in developing flow chambers and studying cell response to shear stress. The system presented gives an effortless flow chamber setup within a 6-well standard multidish.     

Effects of arterial compliance and non-Newtonian rheology on correlations between intimal thickness and wall shear.

A minimally diseased (mean intimal thickness = 56 microns) human aortic bifurcation was replicated in rigid and compliant flow-through casts. Both casts were perfused with physiological flow waves having the same Reynolds and unsteadiness numbers; the pulse pressure in the compliant cast produced radial strains similar to those expected from post-mortem measurements of the compliance of the original tissue. The compliant cast was perfused with a Newtonian fluid and one whose rheology was closer to that of blood. Wall shear rate histories were estimated from near-wall velocities obtained by laser Doppler velocimetry at identical sites in both casts. Intimal thickness was measured at corresponding sites in the original vessel and linear regressions were performed between these thicknesses and several normalized shear rate measures obtained from the histories. The correlations showed a positive slope--that is, the intima was thicker at sites exposed to higher shear rates--consistent with earlier results for relatively healthy vessels, but their significance was often poor. There was no significant effect of either model compliance or fluid rheology on the slopes of the correlations of intimal thickness against any normalized shear rate measure.     

Heterogeneous perfusion is a consequence of uniform shear stress in optimized arterial tree models

Using optimized computer models of arterial trees we demonstrate that flow heterogeneity is a necessary consequence of a uniform shear stress distribution. Model trees are generated and optimized under different modes of boundary conditions. In one mode flow is delivered to the tissue as homogeneously as possible. Although this primary goal can be achieved, resulting shear stresses between blood and the vessel walls show very large spread. In a second mode, models are optimized under the condition of uniform shear stress in all segments which in turn renders flow distribution heterogeneous. Both homogeneous perfusion and uniform shear stress are desirable goals in real arterial trees but each of these goals can only be approached at the expense of the other. While the present paper refers only to optimized models, we assume that this dual relation between the heterogeneities in flow and shear stress may represent a more general principle of vascular systems.     

Spatial variations in endothelial barrier function in disturbed flows in vitro

Hindered barrier function has been implicated in the initiation and progression of atherosclerosis, a disease of focal nature associated with altered hemodynamics. In this study, endothelial permeability to macromolecules and endothelial electrical resistance were investigated in vitro in monolayers exposed to disturbed flow fields that model spatial variations in fluid shear stress found at arterial bifurcations. After 5 h of flow, areas of high shear stress gradients showed a 5.5-fold increase in transendothelial transport of dextran (molecular weight 70,000) compared with no-flow controls. Areas of undisturbed fully developed flow, within the same monolayer, showed a 2.9-fold increase. Monolayer electrical resistance decreased with exposure to flow. The resistance measured during flow and the rate of change in monolayer resistance after removal of flow were lowest in the vicinity of flow reattachment (highest shear stress gradients). These results demonstrate that endothelial barrier function and permeability to macromolecules are regulated by spatial variations in shear stress forces in vitro.     

A population balance model to modulate shear for the control of aggregation in Taxus suspension cultures

Cellular aggregation in plant suspension cultures directly affects the accumulation of high value products, such as paclitaxel from Taxus. Through application of mechanical shear by repeated, manual pipetting through a 10 ml pipet with a 1.6 mm aperture, the mean aggregate size of a Taxus culture can be reduced without affecting culture growth. When a constant level of mechanical shear was applied over eight generations, the sheared population was maintained at a mean aggregate diameter 194 μm lower than the unsheared control, but the mean aggregate size fluctuated by over 600 μm, indicating unpredictable culture variability. A population balance model was developed to interpret and predict disaggregation dynamics under mechanical shear. Adjustable parameters involved in the breakage frequency function of the population balance model were estimated by nonlinear optimization from experimentally measured size distributions. The optimized model predictions were in strong agreement with measured size distributions. The model was then used to determine the shear requirements to successfully reach a target aggregate size distribution. This model will be utilized in the future to maintain a culture with a constant size distribution with the goal of decreasing culture variability and increasing paclitaxel yields.     

Shear rate dependence of free energy barrier height for phenyl ring rotations in polystyrene based on non-equilibrium molecular dynamics simulations

Polystyrene properties are influenced by ring motions in side groups. The main chain conformation and interaction with the surroundings dominate the ring rotations. It is known that shear flow affects linear chain conformation and molecular distribution. However, shear-induced variations in the ring rotations have yet to be studied. This study presents a shear flow system of polystyrene via non-equilibrium molecular dynamics simulations. The free energy barrier of the phenyl ring rotations was obtained from the distribution of angle χ between the ring and main chain based on the Boltzmann distribution law. The results showed that the barrier height approaches a constant value at a shear rate less than 10¹⁰ s^? 1, but decreases with an increase in shear rate higher than 10^10.5 s^? 1. Furthermore, the radial distribution function and potential energies were compared. Remarkably, the shear flow reduced the bond vibrations of the phenyl rings, but increased the separation between intermolecular particles. Hence, a smaller cavity is necessary for the rings to rotate once but more volume is occupied by the rings. The smaller volume obtained via main chain motions needed to construct the cavity lowers the energy barrier height at shear rate higher than 10^10.5 s^? 1.     

Flow distribution through human and canine airways during inhalation and exhalation

Airflow distribution through the tracheobronchial tree is influenced by many factors. In a hollow cast of the central airways, the only factors involved are resistance and inertia of the airflow. Distribution of steady flow during both inhalation and exhalation was measured at different total flow rates in human and canine tracheobronchial casts. The resulting airflow rates in peripheral segments were measured with a sensitive apparatus, which did not disturb the distribution of flow. Inertia of the airflow was found to be small but significant in airways of the human cast and substantially greater in the canine airway cast than in the human cast during inhalation. The influence of airflow inertia during inhalation was largely responsible for the different distributions of flow during inhalation and exhalation through the airway casts. Airflow resistance was found to be considerably greater during exhalation and may have contributed to the redistribution of flow. The forces involved are small but should be considered when modeling the in vivo distribution of airflow.     

Effects of stents under asymmetric inflow conditions

Patient-to-patient variations in artery geometry may determine their susceptibility to stenosis formation. These geometrical variations can be linked to variations in flow characteristics such as wall shear stress through stents, which increases the risk of restenosis. This paper considers computer models of stents in non-symmetric flows and their effects on flow characteristics at the wall. This is a fresh approach from the point of view of identifying a stent design whose performance is insensitive to asymmetric flow. Measures of dissipated energy and power are introduced in order to discriminate between competing designs of stents.