Shape optimization in unsteady blood flow: a numerical study of non-Newtonian effects

This paper presents a numerical study of non-Newtonian effects on the solution of shape optimization problems involving unsteady pulsatile blood flow. We consider an idealized two dimensional arterial graft geometry. Our computations are based on the Navier-Stokes equations generalized to non-Newtonian fluid, with the modified Cross model employed to account for the shear-thinning behavior of blood. Using a gradient-based optimization algorithm, we compare the optimal shapes obtained using both the Newtonian and generalized Newtonian constitutive equations. Depending on the shear rate prevalent in the domain, substantial differences in the flow as well as in the computed optimal shape are observed when the Newtonian constitutive equation is replaced by the modified Cross model. By varying a geometric parameter in our test case, we investigate the influence of the shear rate on the solution.     

Shape optimization in steady blood flow: a numerical study of non-Newtonian effects

We investigate the influence of the fluid constitutive model on the outcome of shape optimization tasks, motivated by optimal design problems in biomedical engineering. Our computations are based on the Navier-Stokes equations generalized to non-Newtonian fluid, with the modified Cross model employed to account for the shear-thinning behavior of blood. The generalized Newtonian treatment exhibits striking differences in the velocity field for smaller shear rates. We apply sensitivity-based optimization procedure to a flow through an idealized arterial graft. For this problem we study the influence of the inflow velocity, and thus the shear rate. Furthermore, we introduce an additional factor in the form of a geometric parameter, and study its effect on the optimal shape obtained.     

A novel solution for fluid flow problems based on the lattice Boltzmann method

Recently, phase separation and fluid flow problems have represented an important development in fluid dynamics, which has many important industrial applications. Lattice Boltzmann method (LBM) is the numerical method that explains the behaviour of fluid dynamics in mesoscopic scale single-component single-phase and multi-component multiphase flows. In this paper, we study the lattice Boltzmann models (LBMs) in two dimensions (2D) with nine directions (Q9), that is the D2Q9 model was used to study the phase separation and observe that the phenomenon of fluid flow in a cylinder has obstacle and square cavity. The simulation results show that fluid flows in the square cavity and in the cylinder, present phase separation of single-component multiphase fluid flow.     

Newtonian and non-Newtonian blood flow in coiled cerebral aneurysms

Endovascular coiling aims to isolate the aneurysm from blood circulation by altering hemodynamics inside the aneurysm and triggering blood coagulation. Computational fluid dynamics (CFD) techniques have the potential to predict the post-operative hemodynamics and to investigate the complex interaction between blood flow and coils. The purpose of this work is to study the influence of blood viscosity on hemodynamics in coiled aneurysms. Three image-based aneurysm models were used. Each case was virtually coiled with a packing density of around 30%. CFD simulations were performed in coiled and untreated aneurysm geometries using a Newtonian and a Non-Newtonian fluid models. Newtonian fluid slightly overestimates the intra-aneurysmal velocity inside the aneurysm before and after coiling. There were numerical differences between fluid models on velocity magnitudes in coiled simulations. Moreover, the non-Newtonian fluid model produces high viscosity (>0.007$>0.007$ [Pa s]) at aneurysm fundus after coiling. Nonetheless, these local differences and high-viscous regions were not sufficient to alter the main flow patterns and velocity magnitudes before and after coiling. To evaluate the influence of coiling on intra-aneurysmal hemodynamics, the assumption of a Newtonian fluid can be used.     

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.     

The effects of slip velocity at a membrane surface on blood flow in the microcirculation

Closed-form solutions are presented for blood flow in the microcirculation by taking into account the influence of slip velocity at the membrane surface. In this study, the convective inertia force is neglected in comparison with that of blood viscosity on the basis of the smallness of the Reynolds number of the flow in microcirculation. The permeability property of the blood vessel is based on the well known Starling's hypothesis [11]. The effects of slip coefficient on the velocity and pressure fields are clearly depicted.     

Computational fluid dynamics in abdominal aorta bifurcation: non-Newtonian versus Newtonian blood flow in a real case study

Hemodynamic in abdominal aorta bifurcation was investigated in a real case using computational fluid dynamics. A Newtonian and non-Newtonian (Walburn-Schneck) viscosity models were compared. The geometrical model was obtained by 3D reconstruction from CT-scan and hemodynamic parameters obtained by laser-Doppler. Blood was assumed incompressible fluid, laminar flow in transient regime and rigid vessel wall. Finite volume-based was used to study the velocity, pressure, wall shear stress (WSS) and viscosity throughout cardiac cycle. Results obtained with Walburn-Schneck’s model, during systole, present lower viscosity due to shear thinning behavior. Furthermore, there is a significant difference between the results obtained by the two models for a specific patient. During the systole, differences are more pronounced and are preferably located in the tortuous regions of the artery. Throughout the cardiac cycle, the WSS amplitude between the systole and diastole is greater for the Walburn-Schneck’s model than for the Newtonian model. However, the average viscosity along the artery is always greater for the non-Newtonian model, except in the systolic peak. The hemodynamic model is crucial to validate results obtained with CFD and to explore clinical potential.     

Numerical simulation of non-Newtonian blood flow dynamics in human thoracic aorta

Three non-Newtonian blood viscosity models plus the Newtonian one are analysed for a patient-specific thoracic aorta anatomical model under steady-state flow conditions via wall shear stress (WSS) distribution, non-Newtonian importance factors, blood viscosity and shear rate. All blood viscosity models yield a consistent WSS distribution pattern. The WSS magnitude, however, is influenced by the model used. WSS is found to be the lowest in the vicinity of the three arch branches and along the distal walls of the branches themselves. In this region, the local non-Newtonian importance factor and the blood viscosity are elevated, and the shear rate is low. The present study revealed that the Newtonian assumption is a good approximation at mid-and-high flow velocities, as the greater the blood flow, the higher the shear rate near the arterial wall. Furthermore, the capabilities of the applied non-Newtonian models appeared at low-flow velocities. It is concluded that, while the non-Newtonian power-law model approximates the blood viscosity and WSS calculations in a more satisfactory way than the other non-Newtonian models at low shear rates, a cautious approach is given in the use of this blood viscosity model. Finally, some preliminary transient results are presented.     

A study of wall shear stress in 12 aneurysms with respect to different viscosity models and flow conditions

Recent computational fluid dynamics (CFD) studies relate abnormal blood flow to rupture of cerebral aneurysms. However, it is still debated how to model blood flow with sufficient accuracy. Common assumptions made include Newtonian behaviour of blood, traction free outlet boundary conditions and inlet boundary conditions based on available literature. These assumptions are often required since the available patient specific data is usually restricted to the geometry of the aneurysm and the surrounding vasculature. However, the consequences of these assumptions have so far been inadequately addressed.     

Numerical simulation of two-phase non-Newtonian blood flow with fluid-structure interaction in aortic dissection

The behavior of blood cells and vessel compliance significantly influence hemodynamic parameters, which are closely related to the development of aortic dissection. Here the two-phase non-Newtonian model and the fluid-structure interaction (FSI) method are coupled to simulate blood flow in a patient-specific dissected aorta. Moreover, three-element Windkessel model is applied to reproduce physiological pressure waves. Important hemodynamic indicators, such as the spatial distribution of red blood cells (RBCs) and vessel wall displacement, which greatly influence the hemodynamic characteristics are analyzed. Results show that the proximal false lumen near the entry tear appears to be a vortex zone with a relatively lower volume fraction of RBCs, a low time-averaged wall shear stress (TAWSS) and a high oscillatory shear index (OSI), providing a suitable physical environment for the formation of atherosclerosis. The highest TAWSS is located in the narrow area of the distal true lumen which might cause further dilation. TAWSS distributions in the FSI model and the rigid wall model show similar trend, while there is a significant difference for the OSI distributions. We suggest that an integrated model is essential to simulate blood flow in a more realistic physiological environment with the ultimate aim of guiding clinical treatment.     

The influence of the non-Newtonian properties of blood on the flow in large arteries: unsteady flow in a 90 degrees curved tube.

A numerical and experimental investigation of unsteady entry flow in a 90 degrees curved tube is presented to study the impact of the non-Newtonian properties of blood on the velocity distribution. The time-dependent flow rate for the Newtonian and the non-Newtonian blood analog fluid were identical. For the numerical computation, a Carreau-Yasuda model was employed to accommodate the shear thinning behavior of the Xanthan gum solution. The viscoelastic properties were not taken into account. The experimental results indicate that significant differences between the Newtonian and non-Newtonian fluid are present. The numerical results for both the Newtonian and the non-Newtonian fluid agree well with the experimental results. Since viscoelasticity was not included in the numerical code, shear thinning behavior of the blood analog fluid seems to be the dominant non-Newtonian property, even under unsteady flow conditions. Finally, a comparison between the non-Newtonian fluid model and a Newtonian fluid at a rescaled Reynolds number is presented. The rescaled Reynolds number, based on a characteristic rather than the high-shear rate viscosity of the Xanthan gum solution, was about three times as low as the original Reynolds number. Comparison reveals that the character of flow of the non-Newtonian fluid is simulated quite well by using the appropriate Reynolds number.     

Transient blood flow in elastic coronary arteries with varying degrees of stenosis and dilatations: CFD modelling and parametric study

In this paper, we have analysed pulsatile flow through partially occluded elastic arteries, to determine the haemodynamic parameters of wall shear stress (WSS), wall pressure gradient and pressure drops (ΔP), contributing to enhanced flow resistance and myocardial ischaemic regions which impair cardiac contractility and cause increased work load on the heart. In summary, it can be observed that stenoses in an artery significantly influence the haemodynamic parameters of wall shear stress and pressure drop in contrast to dilatations case. This deduces that stenosis plays a more critical role in plaque growth and vulnerability in contrast to dilatation, and should be the key element in cardiovascular pathology and diagnosis. Through quantitative analysis of WSS and ΔP, we have provided a clearer insight into the haemodynamics of atherosclerotic arteries. Determination of these parameters can be helpful to cardiologists, because it is directly implicated in the genesis and development of atherosclerosis.     

A study of non-Newtonian aspects of blood flow through stenosed arteries and its applications in arterial diseases

Blood flow through a stenosed artery has been investigated in this paper. Blood has been represented by a non-Newtonian fluid obeying Herschel-Bulkley equation. This model has been used to study the influence of the fluid behaviour index n, shear-dependent nonlinear viscosity K and the yield stress tau H in blood flow through stenosed arteries. The variation of the wall shear stress and the flow resistance with n, K and tau H has been shown graphically. It is observed that the wall shear stress and the flow resistance increase in Herschel-Bulkley fluid in comparison with corresponding Newtonian fluid. It is of interest to note that, in the present model, the thickness of the plug core varies with the axial distance z in the stenotic region. Finally, some biological implications of the present model for some arterial diseases have been briefly discussed.     

Comparative study of flow in right-sided and left-sided aortas: numerical simulations in patient-based models

A right-sided aorta is a rare malformation which may be associated with other various types of congenital heart disease. We utilised haemodynamic, echocardiographic measurements, computerised tomography and image reconstruction software packages that were integrated in a computational fluid dynamics model to determine blood flow patterns in patient-based aortas. In the left-sided aorta, a systolic clockwise rotational component was present, while helical flow was depicted in the aortic arch that was converted in the descending aorta as counter-rotating vortices with accompanying retrograde flow. The right-sided configuration has not altered the orientation of the three-dimensional vortex, but intensification of polymorphic flow patterns, alterations in wall shear stress distribution and development of a lateral pressure gradient at the area of an aneurysmal anomaly was observed. Moreover, increments of Reynolds, Womersley and Dean numbers were evident. These phenomena along with the formation of the aneurysm might influence cardiovascular risk in patients with right-sided aortas.     

A mathematical description of blood spiral flow in vessels: application to a numerical study of flow in arterial bending

Local arterial haemodynamics has been associated with the pathophysiology of several cardiovascular diseases. The stable spiral blood-flows that were observed in vivo in several vessels, may play a dual role in vascular haemodynamics, beneficial since it induces stability, reducing turbulence in the arterial tree, and accounts for normal organ perfusion, but detrimental in view of the imparted tangential velocities that are involved in plaque formation and development. Being a spiral flow considered representative of the local blood dynamics in certain vessels, a method is proposed to quantify the spiral structure of blood flow. The proposed function, computed along a cluster of particle trajectories, has been tested for the quantitative determination of the spiral blood flow in a three-dimensional, s-shaped femoral artery numerical model in which three degrees of stenosis were simulated in a site prone to atherosclerotic development. Our results confirm the efficacy of the Lagrangian analysis as a tool for vascular blood dynamics investigation. The proposed method quantified spiral motion, and revealed the progression in the degree of stenosis, in the presented case study. In the future, it could be used as a synthetic tool to approach specific clinical complications.     

A numerical study of the nonsteady transport of gases in the pulmonary capillaries

A mathematical model is formulated for simulating the unsteady transport of gases in the blood flowing through the pulmonary capillaries. The formulation takes into account the transport mechanisms of molecular diffusion, convection and facilitated diffusion of the species due to haemoglobin. A time dependent situation is created by allowing to vary suddenly the partial pressures of the gases either in the venous blood or in the alveolar air. A numerical technique is described to solve the resulting time-dependent system of nonlinear coupled partial differential equations with the physiologically relevant boundary, entrance and initial conditions. The time required by the gases to achieve equilibrium is computed. It is shown that the dissolved oxygen takes longest in reaching equilibration whereas the carbon dioxide is the fastest. The various physiologically relevant unsteady situations have been examined.     

A study of blood flow in microvessels using biospeckle dynamics

Hydrodynamic properties of blood flows in small microvessels and the patterns of scattering of focused laser beams in such flows were studied. The processes of formation of dynamic biospeckles are considered. The relationship between the optical parameters and hydrodynamic characteristics of blood microflow are analyzed. A new method is proposed for measureming the plasma rate in small microvessels with the use of in vivo microscopy in combination with speckle microscopy.     

A radioangiographic study of the effects of catecholamines on uteroplacental blood flow in the rhesus monkey.

Two placental radioangiographies were performed with a 20-min interval in ten lightly anesthetized rhesus monkeys. Amniotic and aortic pressures, as well as the maternal ECG, were continuously monitored. Norepinephrine or metaproterenol was administered in a low dose which did not cause general cardiovascular effects, prior to the second angiography. Norepinephrine appeared to cause constriction and metaproterenol, dilatation of the uteroplacental vessels. These effects could be diminished or abolished by an alpha- or beta-adrenergic blocking agent, respectively, and thus would appear to be caused by stimulation of adrenergic receptors in the uteroplacental vasculature.     

A microsphere study on the effects of somatostatin and secretin on regional blood flow in anesthetized dogs

The endogenous peptides somatostatin and secretin are effective in the therapy of upper gastrointestinal tract bleeding and acute pancreatitis. The clinical effects may be partly brought about by changes in the regional blood flow. To evaluate the effects of somatostatin (50 and 100 μg/min over 6–8 min) and secretin (0.1 and 0.5 U · kg^?1 · min^?1 over 3–5 min) on tissue blood flow, particularly of the gastrointestinal tract, the tracer microsphere reference sample method was used in anesthetized dogs.Infusion of somatostatin significantly diminished gastric and pancreatic blood flow whereas no changes of duodenal and ileal blood flow could be obtained. Blood flow through spleen, kidneys and adrenal glands was increased but no changes were observed in the blood flow of other tissues. Cardiac hemodynamics remained unchanged.Secretin increased the blood flow of the duodenum, the kidneys and the adrenal glands and diminished gastric blood flow without changing pancreatic, ileal, hepatic, pulmonary and muscle blood flow. Cerebral, pituitary and myocardial blood flow was increased by a higher dose of secretin. It also evoked a slight but significant positive ino- and chronotropic effect. Since secretin and somatostatin differ in their respective effects on gastrointestinal blood flow it is suggested that the previously reported beneficial effects of both peptides on upper gastrointestinal bleeding cannot solely be attributed to changes in regional blood flow.