Effects of non-Newtonian blood and metabolic states of the blood and vessel wall on the optimum design of single vessels and the vascular bifurcation

Employing the optimality principle, we attempted to predict the effects of non-Newtonian blood and the metabolic states of individual vessel segments on the optimum vascular design. Our results implied that irrespective of the vessel caliber, the optimum flow rate of non-Newtonian blood through a cylindrical vessel is less than that of Newtonian blood by not more than some 12-13%, even though the non-Newtonian nature is within the pathologically-realistic highest range. Non-Newtonian blood does not exert the slightest degree of influence on the optimum geometry of the vascular bifurcation. In contrast, as the metabolic state of the vessel wall overwhelms that of the blood, the optimum flow through the cylindrical vessel becomes markedly increased: the optimum relative caliber of the branch of the bifurcation decreases and the optimum branching angle increases.     

Predicted wall shear rate gradients in T-type arteriolar bifurcations

The purpose of this study was to examine the theoretical impact of the local bifurcation geometry on the shear rate gradient in a divergent arteriolar-type bifurcation. Newtonian flow through an arteriolar bifurcation was modeled using 3-dimensional computational fluid dynamics (CFD). Branching angles of 30 degrees, 50 degrees, 70 degrees, 90 degrees, 110 degrees, 130 degrees, and 150 degrees were studied at a Reynolds number (Re) of 0.01 in seven separate models. Both the flow split (30%) and the branch to main vessel diameter ratio (4/5) were held constant. Velocity profiles were predicted to deviate significantly from a parabolic form, both immediately before and after the branch. This deviation was shown to be a function of the local bifurcation geometry of each model, which consisted of a branching angle and associated feed-branch intersection shape. Immediately before and after the branch, the shear rate along the lateral branching wall was predicted to exceed (5-fold) that calculated for fully developed flow in the feed. In vivo data were from the anesthetized (pentobarbital, 70 mg/kg) hamster cremaster muscle preparation. Red blood cells were used as flow markers in arteriolar branch points (n = 74) show that a significant gradient in shear rate occurs at the locations and branch shapes predicted by the computational model. Thus, for low Re divergent flow, the gradient in shear rate measured for non-Newtonian conditions, is approximated by a finite element fluid dynamics model of Newtonian flow.     

Numerical Analysis of Blood Flow through COVID-19 Infected Arteries

Computational Fluid Dynamics has become relevant in the study of hemodynamics, where clinical results are challenging to obtain. This paper discusses a 2-Dimensional transient blood flow analysis through an arterial bifurcation for patients infected with the Coronavirus. The geometry considered is an arterial bifurcation with main stem diameter 3 mm and two outlets. The left outlet (smaller) has a diameter of 1.5 mm and the right outlet (larger), 2 mm. The length of the main stem, left branch and right branch are fixed at 35 mm, 20 mm and 25 mm respectively. Viscosity change that occurs in the blood leads to different parametrical changes in blood flow. The blood flow towards the smaller branch is significantly affected by the changed blood viscosity. Extended regions of high pressure and increased velocity towards the larger outlet are obtained. The Time Averaged Wall Shear Stress (TAWSS) for the corona affected artery is found to be 10.4114 Pa at a 90° angle of bifurcation as compared to 2.45002 Pa of the normal artery. For varying angles of bifurcation, an angle of 75° was found to have a maximum Time Averaged Wall Shear Stress of 2.46076 Pa and 10.42542 Pa for normal and corona affected artery, respectively.     

Numerical investigation of the non-Newtonian pulsatile blood flow in a bifurcation model with a non-planar branch

The pulsatile flow of non-Newtonian fluid in a bifurcation model with a non-planar daughter branch is investigated numerically by using the Carreau-Yasuda model to take into account the shear thinning behavior of the analog blood fluid. The objective of this study is to deal with the influence of the non-Newtonian property of fluid and of out-of-plane curvature in the non-planar daughter vessel on wall shear stress (WSS), oscillatory shear index (OSI), and flow phenomena during the pulse cycle. The non-Newtonian property in the daughter vessels induces a flattened axial velocity profile due to its shear thinning behavior. The non-planarity deflects flow from the inner wall of the vessel to the outer wall and changes the distribution of WSS along the vessel, in particular in systole phase. Downstream of the bifurcation, the velocity profiles are shifted toward the flow divider, and low WSS and high shear stress temporal oscillations characterized by OSI occur on the outer wall region of the daughter vessels close to the bifurcation. Secondary motions become stronger with the addition of the out-of-plane curvature induced by the bending of the vessel, and the secondary flow patterns swirl along the non-planar daughter vessel. A significant difference between the non-Newtonian and the Newtonian pulsatile flow is revealed during the pulse cycle; however, reasonable agreement between the non-Newtonian and the rescaled Newtonian flow is found. Calculated results for the pulsatile flow support the view that the non-planarity of blood vessels and the non-Newtonian properties of blood are an important factor in hemodynamics and may play a significant role in vascular biology and pathophysiology.     

Numerical investigation of the non-Newtonian blood flow in a bifurcation model with a non-planar branch

The non-Newtonian fluid flow in a bifurcation model with a non-planar daughter branch is investigated by using finite element method to solve the three-dimensional Navier–Stokes equations coupled with a non-Newtonian constitutive model, in which the shear thinning behavior of the blood fluid is incorporated by the Carreau–Yasuda model. The objective of this study is to investigate the influence of the non-Newtonian property of fluid as well as of curvature and out-of-plane geometry in the non-planar daughter vessel on wall shear stress (WSS) and flow phenomena. In the non-planar daughter vessel, the flows are typified by the skewing of the velocity profile towards the outer wall, creating a relatively low WSS at the inner wall. In the downstream of the bifurcation, the velocity profiles are shifted towards the flow divider. The low WSS is found at the inner walls of the curvature and the lateral walls of the bifurcation. Secondary flow patterns that swirl fluid from the inner wall of curvature to the outer wall in the middle of the vessel are also well documented for the curved and bifurcating vessels. The numerical results for the non-Newtonian fluid and the Newtonian fluid with original Reynolds number and the corresponding rescaled Reynolds number are presented. Significant difference between the non-Newtonian flow and the Newtonian flow is revealed; however, reasonable agreement between the non-Newtonian flow and the rescaled Newtonian flow is found. Results of this study support the view that the non-planarity of blood vessels and the non-Newtonian properties of blood are an important factor in hemodynamics and may play a significant role in vascular biology and pathophysiology.     

Development of a general method for designing microvascular networks using distribution of wall shear stress

In the present study, theoretical formulations for calculation of optimal bifurcation angle and relationship between the diameters of mother and daughter vessels using the power law model for non-Newtonian fluids are developed. The method is based on the distribution of wall shear stress in the mother and daughter vessels. Also, the effect of distribution of wall shear stress on the minimization of energy loss and flow resistance is considered. It is shown that constant wall shear stress in the mother and daughter vessels provides the minimum flow resistance and energy loss of biological flows. Moreover, the effects of different wall shear stresses in the mother and daughter branches, different lengths of daughter branches in the asymmetric bifurcations and non-Newtonian effect of biological fluid flows on the bifurcation angle and the relationship between the diameters of mother and daughter branches are considered. Using numerical simulations for non-Newtonian models such as power law and Carreau models, the effects of optimal bifurcation angle on the pressure drop and flow resistance of blood flow in the symmetric bifurcation are investigated. Numerical simulations show that optimal bifurcation angle decreases the pressure drop and flow resistance especially for bifurcations at large Reynolds number.     

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.     

Combined effects of pulsatile flow and dynamic curvature on wall shear stress in a coronary artery bifurcation model

A three-dimensional model with simplified geometry for the branched coronary artery is presented. The bifurcation is defined by an analytical intersection of two cylindrical tubes lying on a sphere that represents an idealized heart surface. The model takes into account the repetitive variation of curvature and motion to which the vessel is subject during each cardiac cycle, and also includes the phase difference between arterial motion and blood flowrate, which may be nonzero for patients with pathologies such as aortic regurgitation. An arbitrary Lagrangian Eulerian (ALE) formulation of the unsteady, incompressible, three-dimensional Navier-Stokes equations is employed to solve for the flow field, and numerical simulations are performed using the spectral/hp element method. The results indicate that the combined effect of pulsatile inflow and dynamic geometry depends strongly on the aforementioned phase difference. Specifically, the main findings of this work show that the time-variation of flowrate ratio between the two branches is minimal (less than 5%) for the simulation with phase difference angle equal to 90 degrees, and maximal (51%) for 270 degrees. In two flow pulsatile simulation cases for fixed geometry and dynamic geometry with phase angle 270 degrees, there is a local minimum of the normalized wall shear rate amplitude in the vicinity of the bifurcation, while in other simulations a local maximum is observed.     

Branch angle and flow into a symmetric bifurcation

Arterial branches are found to be a major site for formation of arterial plaque. In this study, we investigate the role of the bifurcation angle on the flow into a symmetric bifurcation. Specially, how the changes in the bifurcation angle influences the distribution of axial wall shear in the bifurcation model. The flow in a range of branch opening half-angle of pi/25< or =theta< or =pi/4 are numerically simulated. The flow in the above models is calculated for the inlet flow Reynolds numbers of 250, 500, 1000, and 2000. It is found that at higher values of the opening angle of the bifurcation, the possibility and severity of flow separation at the appropriate wall location increases.     

Computer simulation of growth of anastomosing microvascular networks

Stochastic growth of polygonal microvascular networks was simulated on computer by dichotomous terminal branching and bridging (anastomosing with an existing segment). The model was applied to describe microvascular growth into a rectangular plane from the sides when vessels bifurcate in a probabilistic manner. The angle of bifurcation was drawn from a normal distribution, the mean of which was varied between 40 degrees and 80 degrees. The resulting networks contained an average of 88-104 nodes of which 30-38% were due to bridging. Number of nodes, number of branches, number of vascular polygons and a fractal dimension representing the density of nodes were calculated for each simulated network. Capillary density increased when mean angle of bifurcation was increased between 40 degrees and 80 degrees. Distributions of normalized vessel lengths and polygon shapes were compared with those of a mesenteric vascular network. The distributions were not found to be significantly different (p less than 0.05) for most values of the mean angle of bifurcation, matching best for the mean bifurcation angle of 50 degrees. Vascular polygons had an average shape between pentagonal and hexagonal for the mesenteric network as well as for all values of the mean bifurcation angle used in this study.     

Blood flow distribution in the branches of frog mesentery arterial microvessels]

The blood flow distribution in 49 arterial branchings of the mesentery (R. temporaria) was investigated (D of the trunk = 25.7 + 0.0 mum). Linear rate was measured by the impulse digital chronometry of the intervals of the erythrocyte transit time. The geometric characteristics of the branching was determined in vivo, on photographs. An asymmetric structure of the investigated branching was shown; branch 1 had the inner initial cross-section which was 2.2 times greater than that of branch 2 and lesser turning angles (29 and 59 degrees). The blood flow in branch 1 was three times greater than the blood flow in branch 2; this was due to its greater inner initial cross-section and a higher linear rate. According to calculations, the blood flow resistance of the branch-turn was insignificant in the general blood flow resistance of branches; therefore the turning angle of the branches could not serve as an important regulator of the volume of the blood flowing in them. An experimentally revealed association between the blood flow in the branches, their radius and their turning angles is well described by equations of the "optimal" model of the vessel branching.     

A Novel Tram Stent Method in the Treatment of Coronary Bifurcation Lesions – Finite Element Study

A novel stent was designed for the treatment of coronary bifurcation lesion, and it was investigated for its performance by finite element analysis. This study was performed in search of a novel method of treatment of bifurcation lesion with provisional stenting. A bifurcation model was created with the proximal vessel of 3.2 mm diameter, and the distal vessel after the side branch (2.3 mm) was 2.7 mm. A novel stent was designed with connection links that had a profile of a tram. Laser cutting and shape setting of the stent was performed, and thereafter it was crimped and deployed over a balloon. The contact pressure, stresses on the arterial wall, stresses on the stent, the maximal principal log strain of the main artery and the side-branch were studied. The study was performed in Abaqus, Simulia. The stresses on the main branch and the distal branch were minimally increased after deployment of this novel stent. The side branch was preserved, and the stresses on the side branch were lesser; and at the confluence of bifurcation on either side of the side branch origin the von-Mises stress was marginally increased. The stresses and strain at the bifurcation were significantly lesser than the stresses and strain of the currently existing techniques used in the treatment of bifurcation lesions though the study was primarily focused only on the utility of the new technology. There is a potential for a novel Tram-stent method in the treatment of coronary bifurcation lesions.     

Changes of zero-stress state of rat pulmonary arteries in hypoxic hypertension

Zero-stress state of the main pulmonary arteries, from the main trunk to a vessel with a lumen diameter approximately 60 microns, was determined in 25 normal control and 38 hypoxic pulmonary hypertensive rats. Pulmonary hypertension was induced by placing the rats in a hypoxic chamber with 10% O2-90% N2 at atmospheric pressure. The zero-stress state of each vessel was obtained by first cutting the vessel transversely into a series of rings and then cutting each ring radially, whereupon the ring opened into a sector, which is characterized by an opening angle defined as the angle subtended between two lines originating from the midpoint of the inner wall (endothelium) to the tips of the inner wall. Whereas the pulmonary blood pressure increased monotonically during the development of pulmonary hypertension, the opening angle followed a different course; e.g., the values (means +/- SD) of the opening angle at the pulmonary trunk at times 0 (control) and 2, 12, 28, 96, 144, 240, 480, and 720 h after exposure to hypoxia are, respectively, 294 +/- 30 degrees, 378 +/- 24 degrees, 385 +/- 12 degrees, 374 +/- 11 degrees, 246 +/- 63 degrees, 267 +/- 49 degrees, 193 +/- 19 degrees, 195 +/- 83 degrees, and 239 +/- 38 degrees. Trends at other places on the artery are similar, but the magnitudes differ. In this period of time, intimal edema and thickening were found. The intima media thickened rapidly from 48 to 240 h and then more slowly from 240 to 720 h. Adventitia thickened later; its thickness exceeded that of the intima media at approximately 96 h. Thus the changes of zero-stress state of the pulmonary arteries are seen to be related to the nonuniform remodeling of the vessel wall as revealed by the edema, blebs, and thickening of different layers.     

A numerical study on hemodynamics in the left coronary bifurcation with normal and hypertension conditions

In this study, a three-dimensional analysis of the non-Newtonian blood flow was carried out in the left coronary bifurcation. The Casson model and hyperelastic and rigid models were used as the constitutive equation for blood flow and vessel wall model, respectively. Physiological conditions were considered first normal and then compliant with hypertension disease with the aim of evaluating hemodynamic parameters and a better understanding of the onset and progression of atherosclerosis plaques in the coronary artery bifurcation. Two-way fluid–structure interaction method applying a fully implicit second-order backward Euler differencing scheme has been used which is performed in the commercial code ANSYS and ANSYS CFX (version 15.0). When artery deformations and blood pressure are associated, arbitrary Lagrangian–Eulerian formulation is employed to calculate the artery domain response using the temporal blood response. As a result of bifurcation, noticeable velocity reduction and backflow formation decrease shear stress and made it oscillatory at the starting point of the LCx branch which caused the shear stress to be less than 1 and 2 Pa in the LCx and the LAD branches, respectively. Oscillatory shear index (OSI) as a hemodynamic parameter represents the increase in residence time and oscillatory wall shear stress. Because of using the ideal 3D geometry and realistic physiological conditions, the values obtained for shear stress are more accurate than the previous studies. Comparing the results of this study with previous clinical investigations shows that the regions with low wall shear stress less than 1.20 Pa and with high OSI value more than 0.3 are in more potential risk to the atherosclerosis plaque development, especially in the posterior after the bifurcation.     

Modelling of the provisional side-branch stenting approach for the treatment of atherosclerotic coronary bifurcations: effects of stent positioning

The most common approach to treat atherosclerosis in coronary bifurcations is the provisional side-branch (PSB) stenting, which consists sequentially of the insertion of a stent in the main branch (MB) of the bifurcation and a dilatation of the side branch (SB) passing through the struts of the stent at the bifurcation. This approach can be followed by a redilatation of the MB only or by a Final Kissing Balloon (FKB) inflation, both strategies leading to a minor stent distortion in the MB. The positioning of the stent struts in the bifurcation and the stresses generated in the stent and vessel wall are worthy of investigation for a better understanding of the mechanobiology of the system. For this purpose, a computer model of an atherosclerotic coronary bifurcation based on the finite element method was developed; the effects of performing the final redilatation with the two strategies utilising one or two balloons and those created by a different stent strut positioning around the SB were investigated. Results correlate well with previous experimental tests regarding the deformation following balloon expansion. Furthermore, results confirm firstly that the re-establishment of an optimal spatial configuration of the stent after the PSB approach is achieved with both strategies; secondly, results show that case of stent positioning with one cell placed centrally (with regard to the SB) should be preferred, avoiding the presence of struts inside the vessel lumen, which may reduce hemodynamic disturbances. The central positioning also resulted in a better solution in terms of lower stresses in the stent struts and, more importantly, in the vascular tissues.     

Blood Vessel Adaptation with Fluctuations in Capillary Flow Distribution

Throughout the life of animals and human beings, blood vessel systems are continuously adapting their structures – the diameter of vessel lumina, the thickness of vessel walls, and the number of micro-vessels – to meet the changing metabolic demand of the tissue. The competition between an ever decreasing tendency of luminal diameters and an increasing stimulus from the wall shear stress plays a key role in the adaptation of luminal diameters. However, it has been shown in previous studies that the adaptation dynamics based only on these two effects is unstable. In this work, we propose a minimal adaptation model of vessel luminal diameters, in which we take into account the effects of metabolic flow regulation in addition to wall shear stresses and the decreasing tendency of luminal diameters. In particular, we study the role, in the adaptation process, of fluctuations in capillary flow distribution which is an important means of metabolic flow regulation. The fluctuation in the flow of a capillary group is idealized as a switch between two states, i.e., an open-state and a close-state. Using this model, we show that the adaptation of blood vessel system driven by wall shear stress can be efficiently stabilized when the open time ratio responds sensitively to capillary flows. As micro-vessel rarefaction is observed in our simulations with a uniformly decreased open time ratio of capillary flows, our results point to a possible origin of micro-vessel rarefaction, which is believed to induce hypertension.     

Sex differences in intracranial arterial bifurcations

Background: Subarachnoid hemorrhage (SAH) is a serious condition, occurring more frequently in females than in males. SAH is mainly caused by rupture of an intracranial aneurysm, which is formed by localized dilation of the intracranial arterial vessel wall, usually at the apex of the arterial bifurcation. The female preponderance is usually explained by systemic factors (hormonal influences and intrinsic wall weakness); however, the uneven sex distribution of intracranial aneurysms suggests a possible physiologic factor—a local sex difference in the intracranial arteries.Objective: The aim of this study was to explore sex variation in the bifurcation anatomy of the middle cerebral artery (MCA) and internal carotid artery (ICA), and the subsequent hemodynamic impact.Methods: Vessel radii and bifurcation angles were measured in patients with MCA and ICA bifurcations. Data from a previously published study of 55 patients undergoing diagnostic cerebral digital subtraction angiography at Dalcross Private Hospital in Sydney, Australia, between 2002 and 2003, were available for analysis. The measurements were used to create idealized, averaged bifurcations of the MCA and ICA for females and males. Computational fluid dynamics simulations were performed to calculate hemodynamic forces in the models.Results: The vessel radii and bifurcation angles of 47 MCA and 52 ICA bifurcations in 49 patients (32 females, 17 males; mean age, 53 years; age range, 14–86 years) were measured. Statistically significant sex differences were found in vessel diameter (males larger than females; P < 0.05), but not in bifurcation angle. Computational fluid dynamics simulations revealed higher wall shear stress in the female MCA (19%) and ICA (50%) bifurcations compared with the male bifurcations.Conclusions: This study of MCA and ICA bifurcations in female and male patients suggests that sex differences in vessel size and blood flow velocity result in higher hemodynamic forces acting on the vessel wall in females. This new hypothesis may partly explain why intracranial aneurysms and SAH are more likely to occur in females than in males.     

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.     

Carotid geometry effects on blood flow and on risk for vascular disease

It has been widely observed that atherosclerotic diseases occur at sites with complex hemodynamics, such as artery bifurcations, junctions, and regions of high curvature. These regions usually have very low or highly oscillatory wall shear stress (WSS). In the present work, 3D pulsatile blood flow through a model of the carotid artery bifurcation was simulated using a finite volume numerical method. The goal was to quantify the risk of atherogenesis associated with different carotid artery geometries. A risk scale based on the average WSS on the sinus wall of the internal carotid artery was proposed-a scale that can be used to quantify the effect of the carotid geometry on the relative risk for developing vascular disease. It was found that the bifurcation angle and the out-of-plane angle of the internal carotid artery affect the formation of low stress regions on the carotid walls. The main conclusions are: (a) larger internal carotid artery angles (theta(IC)) generally increase the frequency and the area of blood recirculation and lower the WSS on the sinus wall, hence increasing the risk of plaque build-up; (b) off-plane angles were found to lower the WSS on the sinus for geometries with theta(IC)25 degrees . Larger off-plane angles generally increase the danger of plague build-up; (c) for theta(IC) < 25 degrees , the off-plane angle does not have an obvious effect on the hemodynamic WSS; (d) symmetric bifurcations were found to increase the WSS on the sinus wall and ease the risk of vascular disease.     

Relationship between hypertension, hypertrophy, and opening angle of zero-stress state of arteries following aortic constriction

Examination of changes occurring in the zero-stress state of an organ provides a way to study cellular growth in the organ due to change of physical stresses. The zero-stress state of the aorta is not a tube. It is a sector with an opening angle that varies with the location on the aorta and changes with cellular remodeling. Blood vessel remodeling can be induced by imposing a constriction on the abdominal aorta by a metal clip (aortic banding), which causes an increase of blood pressure, hypertrophy of the aortic wall, and large change of opening angle. The correlation of the opening angle with the blood vessel wall thickness and blood pressure changes in rat's aorta due to aortic banding is presented in this report. The opening angle changes daily following the aortic banding. Blood pressure rises in vessels of the upper body, but that in the lower body decreases at first and then rises to an asymptotic value. Blood vessel wall thickness increases in rough proportion to blood pressure. Vessel diameter changes also. But the most dramatic is the course of change of the zero-stress state. Typically, the time to reach 50 percent of asymptotic hypertrophy of blood vessel wall thickness is about 3-5 days. The corresponding time for blood pressure is about 7 days. The opening angle of the zero-stress state, however, increases rapidly at first, reaches a peak in about 2 to 4 days, then decreases gradually to a reduced asymptote. The exact values of the time constants depend on the location along the aortic tree. In general, the course of change of residual strain is very different from those of the blood pressure and the blood vessel wall thickness.