窦部对称狭窄对颈动脉内流场影响的数值研究

以颈动脉分岔血管为例，采用数值方法研究了窦部环缩狭窄之后的流场分布情况，并和正 常血管情况下的流场分布进行了比较. 结果表明，采用环缩方式给颈动脉分岔血管施加对称 的狭窄改变了颈动脉窦内流场，特别是壁面剪应力的分布规律. 低剪应力区出现在狭窄段之 后的窦内，并且沿整个周向均匀分布. 根据低剪应力和动脉粥样硬化的关系，指出： 若人为地给颈动脉窦内施加对称狭窄，则脂质沉积将在狭窄下游的窦内沿周向轴对称 发展. 为了更真实地反映颈动脉窦内的狭窄，建议根据动脉血管中的实际狭窄情况，采用非 对称的狭窄分布模式.     

颈动脉分支的血流动力学数值模拟

采用有限元法数值模拟颈动脉分支的血流动力学。根据在体测量的实际尺寸来构造颈动脉分支的几何模型，以保持模型的解剖精确度；利用在体测量的颈内动脉和颈外动脉流量波形以及主颈动脉的压力波形来确定数值计算的边界条件，以保持数值计算的生理真实性。关注的重点是颈动脉窦内的局部血流形态、二次流和壁面剪应力。在心脏收缩的减速期和舒张期的某些时刻，颈动脉窦中部外侧壁面附近产生了流动分离，形成了一个低速回流区。该流动分离是瞬态的，导致了壁面剪应力的振荡，其振荡范围在-2～6dyn／cm^2之间。同时，颈动脉窦中部横截面内的二次流存在于整个心动周期，最大的二次流速度为同时刻轴向速度平均值的1／3左右。     

Pulsatile flows and wall-shear stresses in models simulating normal and stenosed aortic arches

Pulsatile aqueous glycerol solution flows in the models simulating normal and stenosed human aortic arches are measured by means of particle image velocimetry. Three transparent models were used: normal, 25% stenosed, and 50% stenosed aortic arches. The Womersley parameter, Dean number, and time-averaged Reynolds number are 17.31, 725, and 1,081, respectively. The Reynolds numbers based on the peak velocities of the normal, 25% stenosed, and 50% stenosed aortic arches are 2,484, 3,456, and 3,931, respectively. The study presents the temporal/spatial evolution processes of the flow pattern, velocity distribution, and wall-shear stress during the systolic and diastolic phases. It is found that the flow pattern evolving in the central plane of normal and stenosed aortic arches exhibits (1) a separation bubble around the inner arch, (2) a recirculation vortex around the outer arch wall upstream of the junction of the brachiocephalic artery, (3) an accelerated main stream around the outer arch wall near the junctions of the left carotid and the left subclavian arteries, and (4) the vortices around the entrances of the three main branches. The study identifies and discusses the reasons for the flow physics’ contribution to the formation of these features. The oscillating wall-shear stress distributions are closely related to the featured flow structures. On the outer wall of normal and slightly stenosed aortas, large wall-shear stresses appear in the regions upstream of the junction of the brachiocephalic artery as well as the corner near the junctions of the left carotid artery and the left subclavian artery. On the inner wall, the largest wall-shear stress appears in the region where the boundary layer separates.     

Performance modeling and analysis of blood flow in elastic arteries

Nomenclature　　τ　　wallshearstressγshearrateτy yieldstressηc Cassonviscosityktheconsistencyindexnnon_Newtonianindexτp shearstressofthepthelementωangularvelocityRvessel’sradiusCwavespeedM　　magneticparameter (Hartmannnumber)u,w　velocitycomponentinther_andz_directions,respectivelyP　　pressureα　　unsteadinessparameter k , R meanparametersTp relaxationtimeofthepthelementρ densityIntroductionTheimportancetoatherogenesisofarterialflowphenomenasuchasflowseparation ,recirculationands…     

动脉中非平稳流动壁面剪切力及其Hilbert-Huang变换

本文通过数值方法求解均匀动脉中的非平稳脉动流,给出了通过测量非平稳脉动血流量确定壁面切应力的方法.作为算例,采用实测的大鼠颈总动脉流量信号,求出了均匀动脉壁面切应力波形.进一步对求得的切应力波形进行经验模态分解(EMD),得到了切应力波形的各内在模态(IMF),以及Hilbert幅值谱.从切应力波形经Hilbert-Huang变换得到的IMF和Hilbert谱图可以明显地看出切应力各频率成分的物理意义.所得结果为进一步深入研究非平稳脉动切应力与血管重建的关系提供了一种方法学基础.     

Mathematical modeling and parametric investigation of blood flow through a stenosis artery

In this study, a mathematical model is formulated to examine the blood flow through a cylindrical stenosed blood vessel. The stenosis disease is caused because of the abnormal narrowing of flow in the body. This narrowing causes serious health issues like heart attack and may decrease blood flow in the blood vessel. Mathematical modeling helps us analyze such issues. A mathematical model is considered in this study to explore the blood flow in a stenosis artery and is solved numerically with the finite difference method. The artery is an elastic cylindrical tube containing blood defined as a viscoelastic fluid. A complete parametric analysis has been done for the flow velocity to clarify the applicability of the defined problem. Moreover, the flow characteristics such as the impedance, the wall shear stress in the stenotic region, the shear stresses in the throat of the stenosis and at the critical stenosis height are discussed. The obtained results show that the intensity of the stenosis occurs mostly at the highest narrowing areas compared with all other areas of the vessel, which has a direct impact on the wall shear stress. It is also observed that the resistive impedance and wall shear pressure get the maximum values at the critical height of the stenosis.

     

Numerical simulation of pulsatile non-Newtonian flow in the carotid artery bifurcation

Both clinical and post mortem studies indicate that, in humans, the carotid sinus of the carotid artery bifurcation is one of the favored sites for the genesis and development of atherosclerotic lesions. Hemodynamic factors have been suggested to be important in atherogenesis. To understand the correlation between atherogenesis and fluid dynamics in the carotid sinus, the blood flow in artery was simulated numerically. In those studies, the property of blood was treated as an incompressible, Newtonian fluid. In fact, however, the blood is a complicated non-Newtonian fluid with shear thinning and viscoelastic properties, especially when the shear rate is low. A variety of non-Newtonian models have been applied in the numerical studies. Among them, the Casson equation was widely used. However, the Casson equation agrees well only when the shear rate is less than 10 s-1. The flow field of the carotid bifurcation usually covers a wide range of shear rate. We therefore believe that it may not be sufficient to describe the property of blood only using the Casson equation in the whole flow field of the carotid bifurcation. In the present study, three different blood constitutive models, namely, the Newtonian, the Casson and the hybrid fluid constitutive models were used in the flow simulation of the human carotid bifurcation. The results were compared among the three models. The results showed that the Newtonian model and the hybrid model had verysimilar distributions of the axial velocity, secondary flow and wall shear stress, but the Casson model resulted in significant differences in these distributions from the other two models. This study suggests that it is not appropriate to only use the Casson equation to simulate the whole flow field of the carotid bifurcation, and on the other hand, Newtonian fluid is a good approximation to blood for flow simulations in the carotid artery bifurcation.     

Effect of surface irregularities on unsteady pulsatile flow in a compliant artery

Of concern in the paper is an analytical study of pulsatile blood flow in an irregular stenosed arterial segment through a mathematical model. The model is two-dimensional and axisymmetric with an outline of the stenosis obtained from a three-dimensional casting of a mildly stenosed artery [L. Back, Y. Cho, D. Crawford, R. Cuffel, Effect of mild atherosclerosis on flow resistance in a coronary artery casting of man, J. Biomech. Eng. 106 (1984) 48–53]. The combined influence of an asymmetric shape and surface irregularities of the constriction has been explored in a computational study of blood flow through arterial stenosis with 48% areal occlusion. The moving wall of the artery is included to be anisotropic, linear, viscoelastic, incompressible circular cylindrical membrane shell. The effect of the surrounding connective tissues on the motion of the arterial wall is also paid due attention. Results are also obtained for a smooth stenosis model and also for a stenosis model representative by the cosine curve. An extensive quantitative analysis has been performed in non-uniform non-staggered grids through numerical computations for the effect of surface irregularities on the flow velocity, the flux, the resistive impedance and on the wall shear stress through their graphical representations so as to validate the applicability of such an improved mathematical model.     

Two-phase non-linear model for blood flow in asymmetric and axisymmetric stenosed arteries

The pulsatile flow of a two-phase model for blood flow through axisymmetric and asymmetric stenosed narrow arteries is analyzed, treating blood as a two-phase model with the suspension of all the erythrocytes in the core region as the Herschel-Bulkley material and plasma in the peripheral layer as the Newtonian fluid. The perturbation method is applied to solve the resulting non-linear implicit system of partial differential equations. The expressions for various flow quantities are obtained. It is found that the pressure drop, plug core radius, wall shear stress increase as the yield stress or stenosis height increases. It is noted that the velocity increases, longitudinal impedance decreases as the amplitude increases. For asymmetric stenosis, the wall shear stress increases non-linearly with the increase of the axial distance. The estimates of the increase in longitudinal impedance to flow of the two-phase Herschel-Bulkley material are significantly lower than those of the single-phase Herschel-Bulkley material. The results show the advantages of two-phase flow over single-phase flow in small diameter arteries with stenosis.     

Computer Simulation of Non-Newtonian Flow and Mass Transport Through Coronary Arterial Stenosis

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Multidimensional modeling of the stenosed carotid artery: A novel CAD approach accompanied by an extensive lumped model

This study describes a multidimensional 3D/lumped parameter(LP) model which contains appropriate inflow/outflow boundary conditions in order to model the entire human arterial trees. A new extensive LP model of the entire arterial network(48 arteries) was developed including the effect of vessel diameter tapering and the parameterization of resistance, conductor and inductor variables. A computer aided-design(CAD) algorithm was proposed to effciently handle the coupling of two or more 3D models with the LP model, and substantially lessen the coupling processing time. Realistic boundary conditions and Navier–Stokes equations in healthy and stenosed models of carotid artery bifurcation(CAB) were used to investigate the unsteady Newtonian blood flow velocity distribution in the internal carotid artery(ICA). The present simulation results agree well with previous experimental and numerical studies. The outcomes of a pure LP model and those of the coupled 3D healthy model were found to be nearly the same in both cases. Concerning the various analyzed 3D zones, the stenosis growth in the ICA was not found as a crucial factor in determining the absorbing boundary conditions.This paper demonstrates the advantages of coupling local and systemic models to comprehend physiological diseases of the cardiovascular system.     

An unsteady analysis of non-Newtonian blood flow through tapered arteries with a stenosis

The problem of non-Newtonian and nonlinear blood flow through a stenosed artery is solved numerically where the non-Newtonian rheology of the flowing blood is characterised by the generalised Power-law model. An improved shape of the time-variant stenosis present in the tapered arterial lumen is given mathematically in order to update resemblance to the in vivo situation. The vascular wall deformability is taken to be elastic (moving wall), however a comparison has been made with nonlinear visco-elastic wall motion. Finite difference scheme has been used to solve the unsteady nonlinear Navier-Stokes equations in cylindrical coordinates system governing flow assuming axial symmetry under laminar flow condition so that the problem effectively becomes two-dimensional. The present analytical treatment bears the potential to calculate the rate of flow, the resistive impedance and the wall shear stress with minor significance of computational complexity by exploiting the appropriate physically realistic prescribed conditions. The model is also employed to study the effects of the taper angle, wall deformation, severity of the stenosis within its fixed length, steeper stenosis of the same severity, nonlinearity and non-Newtonian rheology of the flowing blood on the flow field. An extensive quantitative analysis is performed through numerical computations of the desired quantities having physiological relevance through their graphical representations so as to validate the applicability of the present model.     

弯曲狭窄管内血液流的分析

与血管狭窄有关的异常血液动力学特征在血管疾病的发生和发展过程中起着重要的作用,由于血管狭窄和弯曲的综合影响,将会出现一系列有趣的流体力学现象,本文研究具有对称狭窄的弯曲小动脉内定常血液流动,在一定的假设条件下,直接从支配血液流动的Navier-Stokes方程求出问题的摄动解,由此求得弯曲狭窄管內血液流动的轴向速度、二次流速度及压力梯度等分析表达式,并进一步求得轴向和周向血管壁切应力。本文的结果是先前有关狭窄直管和弯曲均匀管流动研究的拓广。     

局部狭窄血管中血液振荡流的速度和切应力

本文建立一种分析局部缓慢狭窄血管中血液振荡流的数学模型，给出了血液的轴向流速，径向流速和切应力的包含压力梯度项的解析表达式，并讨论了血管内由局部狭窄引起的压力梯度沿轴向变化的规律。文章以局部余弦狭窄为例进行数值计算，详细讨论上游均匀管段压力梯度的定常部分和不同次谐波对狭窄管段内流速和切应力的影响。数值结果表明，与均匀管情况相比，在狭窄段内，血液振荡流轴向流速无论平均值还是脉动幅值均明显增大，且径向流速不再为零。但径向流速仍远小于轴向流速。同时，切应力也不再仅由轴向流速梯度提供，径向流速梯度也将产生切应力，但是在计算管壁切向上的切应力时，径向流速梯度的贡献仍相当大。与均匀管管壁切应力沿流运方向保持恒定不同。狭窄管管壁切应力（平均值和脉动值）将随着狭窄高度的增大而增大，在狭窄最大高度处达到最大，因而沿流动方向产生了较大的切应力梯度。     

刚性圆管中血液周期振荡流的切应力分布

本文通过求解圆管内血液振荡流的基本方程，求得圆管内血液流的压力梯度与切应力之间的关系式。在此基础上，详细讲座了圆管中轴向流速和切变率谐波的变化规律，指出流速谐波和切变率谐波的幅值都将随着谐波次数的增大而逐渐减小。为了使所得结果便于应用。文章通过管轴向中心线流速与压力梯度之间的关系式，进一步给出一种利用管轴向中心线流速计算管内切应力分布的简便方法。该方法用于检测活体血管内血液振荡流的切应力分布，具有操作简单，精度较高的优点。最后，以人体颈动脉为例，讨论血液周期振荡流的切应力的分布特性。发现在任意时刻，除了邻近管壁处切应力急剧增大到一定数值之外，沿管截面切应力分布相当均匀且接近于零，呈现出与定常流不同的切应力分布特征。     

局部扩张动脉管血液振荡流的切应力分布

本文求解局部缓慢扩张动脉管中血液振荡流的基本方程，得到血管内血液的流速与压力梯度的关系。通过导出压力梯度沿局部扩张管轴向的变化特性。建立利用扩张段上游血管均匀段中心流速波形确定局部扩张管中血液流的速度和切应力分布的方法，文章以人体颈动脉余弦扩张为例进行分析。详细讨论了局部扩张对血管壁切应力及其梯度分布的影响。数值结果表明，在与刚性均匀管中管壁切应力沿轴向保持不变不同，在局部扩张段，管壁切应力将随着血管半径的增大而减小，因而管壁切应力梯度一般不为零，甚至在某些位置达到相当大的数值。另外，随着血管扩张程度的增加，管壁切应力还将进一步减小，而且管壁切应力梯度也将进一步增大，血管扩张导致管壁切应力的这些变化将直接影响血管壁的结构和功能，使其产生适应性的变化。     

Magnetohydrodynamic approach of non-Newtonian blood flow with magnetic particles in stenosed artery

The non-Newtonian blood flow, together with magnetic particles in a stenosed artery, is studied using a magneto-hydrodynamic approach. The wall slip condition is also considered. Approximate solutions are obtained in series forms under the assumption that the Womersley frequency parameter has small values. Using an integral transform method, analytical solutions for any values of the Womersley parameter are obtained.Numerical simulations are performed using MATHCAD to study the influence of stenosis and magnetic field on the flow parameters. When entering the stenosed area, blood velocity increases slightly, but increases considerably and reaches its maximum value in the stenosis throat. It is concluded that the magnitude of axial velocity varies considerably when the applied magnetic field is strong. The magnitude of maximum fluid velocity is high in the case of weak magnetic fields. This is due to the Lorentz's force that opposes motion of an electrically conducting fluid. The effect of externally transverse magnetic field is to decelerate the flow of blood. The shear stress consistently decreases in the presence of a magnetic field with increasing intensity.     

Effect of couple stresses on the flow in a constricted annulus

The flow of an incompressible couple stress fluid in an annulus with local constriction at the outer wall is considered. This configuration is intended as a simple model for studying blood flow in a stenosed artery when a catheter is inserted into it. The effects couple stress fluid parameters α and σ, height of the constriction (ε), and ratio of radii (k) on the impedance and wall shear stresses are studied graphically. Graphical results show that the resistance to the flow as well as the wall shear stress increases as the ratio of the radii increases and decreases as the couple stress fluid parameters increases.     

Pulsatile flow of Herschel-Bulkley fluid through stenosed arteries—A mathematical model

In this paper, the pulsatile flow of blood through stenosed artery is studied. The effects of pulsatility, stenosis and non-Newtonian behavior of blood, assuming the blood to be represented by Herschel-Bulkley fluid, are simultaneously considered. A perturbation method is used to analyze the flow assuming the thickness of plug core region to be non-uniform changing with axial distance. An expression for the variation of plug core radius with time and axial distance is obtained. The variation of pressure gradient with steady flow rate is given. Also the variation of wall shear stress distribution as well as resistance to flow with axial distance for different values of time and for different values of yield stress is given and the results analyzed.     

Viscoelastic blood flow through arterial stenosis—Effect of variable viscosity

Current theoretical investigation deals with mathematical model of unsteady non-Newtonian flow of blood through a stenosed artery. The flowing blood is considered as a viscoelastic fluid having shear-thinning rheology and characterized by generalised Oldroyd-B model. The arterial wall is considered to be rigid having cosine shaped stenosis in its lumen. The governing equations of motion accompanied by appropriate choice of the initial and boundary conditions are solved numerically by MAC (Marker and Cell) method and the results are checked for numerical stability with desired degree of accuracy. The quantitative analysis has been carried out finally which includes the respective profiles of the flow-field. The key factors like the wall shear stress and flow separation are also examined for further qualitative insight into the flow through arterial stenosis. The present results show quite consistency with several existing results in the literature which substantiate sufficiently to validate the applicability of the model under consideration.