Effect of nonaxisymmetric hematocrit distribution on non-Newtonian blood flow in small tubes

Continuum microhaemodynamics modelling using inverse rheology

Modelling blood flow in microvascular networks is challenging due to the complex nature of haemorheology. Zero- and one-dimensional approaches cannot reproduce local haemodynamics, and models that consider individual red blood cells (RBCs) are prohibitively computationally expensive. Continuum approaches could provide an efficient solution, but dependence on a large parameter space and scarcity of experimental data for validation has limited their application. We describe a method to assimilate experimental RBC velocity and concentration data into a continuum numerical modelling framework. Imaging data of RBCs were acquired in a sequentially bifurcating microchannel for various flow conditions. RBC concentration distributions were evaluated and mapped into computational fluid dynamics simulations with rheology prescribed by the Quemada model. Predicted velocities were compared to particle image velocimetry data. A subset of cases was used for parameter optimisation, and the resulting model was applied to a wider data set to evaluate model efficacy. The pre-optimised model reduced errors in predicted velocity by 60% compared to assuming a Newtonian fluid, and optimisation further reduced errors by 40%. Asymmetry of RBC velocity and concentration profiles was demonstrated to play a critical role. Excluding asymmetry in the RBC concentration doubled the error, but excluding spatial distributions of shear rate had little effect. This study demonstrates that a continuum model with optimised rheological parameters can reproduce measured velocity if RBC concentration distributions are known a priori. Developing this approach for RBC transport with more network configurations has the potential to provide an efficient approach for modelling network-scale haemodynamics.

Continuum Scale Non Newtonian Particle Transport Model for Hæmorheology

We present a continuum scale particle transport model for red blood cells following collision arguments, in a diffusive flux formulation. The model is implemented in FOAM, in a framework suitable for haemodynamics simulations and adapted to multi-scaling. Specifically, the framework we present is able to ingest transport coefficient models to be derived, prospectively, from complimentary but independent meso-scale simulations. For present purposes, we consider modern semi-mechanistic rheology models, which we implement and test as proxies for such data. The model is verified against a known analytical solution and shows excellent agreement for high quality meshes and good agreement for typical meshes as used in vascular flow simulations. Simulation results for different size and time scales show that migration of red blood cells does occur on physiologically relevany timescales on small vessels below 1 mm and that the haematocrit concentration modulates the non-Newtonian viscosity. This model forms part of a multi-scale approach to haemorheology and model parameters will be derived from meso-scale simulations using multi-component Lattice Boltzmann methods. The code, haemoFoam, is made available for interested researchers.

Assessment of surface roughness and blood rheology on local coronary haemodynamics: a multi-scale computational fluid dynamics study

The surface roughness of the coronary artery is associated with the onset of atherosclerosis. The study applies, for the first time, the micro-scale variation of the artery surface to a 3D coronary model, investigating the impact on haemodynamic parameters which are indicators for atherosclerosis. The surface roughness of porcine coronary arteries have been detailed based on optical microscopy and implemented into a cylindrical section of coronary artery. Several approaches to rheology are compared to determine the benefits/limitations of both single and multiphase models for multi-scale geometry. Haemodynamic parameters averaged over the rough/smooth sections are similar; however, the rough surface experiences a much wider range, with maximum wall shear stress greater than 6 Pa compared to the approximately 3 Pa on the smooth segment. This suggests the smooth-walled assumption may neglect important near-wall haemodynamics. While rheological models lack sufficient definition to truly encompass the micro-scale effects occurring over the rough surface, single-phase models (Newtonian and non-Newtonian) provide numerically stable and comparable results to other coronary simulations. Multiphase models allow for phase interactions between plasma and red blood cells which is more suited to such multi-scale models. These models require additional physical laws to govern advection/aggregation of particulates in the near-wall region.

Experimental and Numerical Study of Blood Flow in μ-vessels: Influence of the Fahraeus–Lindqvist Effect

The study of hemodynamics is particularly important in medicine and biomedical engineering as it is crucial for the design of new implantable devices and for understanding the mechanism of various diseases related to blood flow. In this study, we experimentally identify the cell free layer (CFL) width, which is the result of the Fahraeus–Lindqvist effect, as well as the axial velocity distribution of blood flow in microvessels. The CFL extent was determined using microscopic photography, while the blood velocity was measured by micro-particle image velocimetry (μ-PIV). Based on the experimental results, we formulated a correlation for the prediction of the CFL width in small caliber (D < 300 μm) vessels as a function of a modified Reynolds number (Re∞) and the hematocrit (Hct). This correlation along with the lateral distribution of blood viscosity were used as input to a “two-regions” computational model. The reliability of the code was checked by comparing the experimentally obtained axial velocity profiles with those calculated by the computational fluid dynamics (CFD) simulations. We propose a methodology for calculating the friction loses during blood flow in μ-vessels, where the Fahraeus–Lindqvist effect plays a prominent role, and show that the pressure drop may be overestimated by 80% to 150% if the CFL is neglected.

Influence of erythrocyte aggregation at pathological levels on cell-free marginal layer in a narrow circular tube

Human red blood cells (RBCs) were perfused in a circular micro-tube (inner diameter of 25 μm) to examine the dynamic changes of cell-free marginal region at both physiological (normal) and pathophysiological (hyper) levels of RBC aggregation. The cell-free area (CFA) was measured to provide additional information on the cell-free layer (CFL) width changes in space and time domains. A prominent enhancement in the mean CFL width was found in hyper-aggregating conditions as compared to that in non-aggregating conditions (P <  0.001). The frequent contacts between RBC and the tube wall were observed and the contact frequency was greatly decreased when the aggregation level was increased from none to normal (P <  0.05) and to hyper (P <  0.001) levels. In addition, the enhanced aggregation from none to hyper levels significantly enlarged the CFA (P <  0.01). We concluded that the RBC aggregation at pathophysiological levels could promote not only the CFL width (one-dimensional parameter) but also the spatiotemporal variation of CFA (two-dimensional parameter).

Erythrocyte aggregation may promote uneven spatial distribution of NO/O2 in the downstream vessel of arteriolar bifurcations

This study examined the effect of red blood cell (RBC) aggregation on nitric oxide (NO) and oxygen (O2) distributions in the downstream vessels of arteriolar bifurcations. Particular attention was paid to the inherent formation of asymmetric cell-free layer (CFL) widths in the downstream vessels and its consequential impact on the NO/O2 bioavailability after the bifurcations. A microscopic image-based two-dimensional transient model was used to predict the NO/O2 distribution by utilizing the in vivo CFL width data obtained under non-, normal- and hyper-aggregating conditions at the pseudoshear rate of 15.6±2.0s(-1). In vivo experimental result showed that the asymmetry of CFL widths was enhanced by the elevation in RBC aggregation level. The model demonstrated that NO bioavailability was regulated by the dynamic fluctuation of the local CFL widths, which is corollary to its modulation of wall shear stress. Accordingly, the uneven distribution of NO/O2 was prominent at opposite sides of the arterioles up to six vessel-diameter (6D) away from the bifurcating point, and this was further enhanced by increasing the levels of RBC aggregation. Our findings suggested that RBC aggregation potentially augments both the formation of asymmetric CFL widths and its influence on the uneven distribution of NO/O2 in the downstream flow of an arteriolar bifurcation. The extended heterogeneity of NO/O2 downstream (2D-6D) also implied its potential propagation throughout the entire arteriolar microvasculature.

Two-dimensional transient model for prediction of arteriolar NO/O2 modulation by spatiotemporal variations in cell-free layer width

Despite the significant roles of the cell-free layer (CFL) in balancing nitric oxide (NO) and oxygen (O2) bioavailability in arteriolar tissue, many previous numerical approaches have relied on a one-dimensional (1-D) steady-state model for simplicity. However, these models are unable to demonstrate the influence of spatiotemporal variations in the CFL on the NO/O2 transport under dynamic flow conditions. Therefore, the present study proposes a new two-dimensional (2-D) transient model capable of predicting NO/O2 transport modulated by the spatiotemporal variations in the CFL width. Our model predicted that NO bioavailability was inversely related to the CFL width as expected. The enhancement of NO production by greater wall shear stress with a thinner CFL could dominate the diffusion barrier role of the CFL. In addition, NO/O2 availability along the vascular wall was inhomogeneous and highly regulated by dynamic changes of local CFL width variation. The spatial variations of CFL widths on opposite sides of the arteriole exhibited a significant inverse relation. This asymmetric formation of CFL resulted in a significantly imbalanced NO/O2 bioavailability on opposite sides of the arteriole. The novel integrative methodology presented here substantially highlighted the significance of spatiotemporal variations of the CFL in regulating the bioavailability of NO/O2, and provided further insight about the opposing effects of the CFL on arteriolar NO production.

Effect of erythrocyte aggregation on spatiotemporal variations in cell-free layer formation near on arteriolar bifurcation

OBJECTIVE

To investigate how red blood cell aggregation could modulate the spatial variations in cell-free layer formation in the vicinity of an arteriolar bifurcation.

METHODS

Visualization of blood flow was performed in upstream and downstream vessels of arteriolar bifurcations in the rat cremaster muscles under reduced flow conditions before and after induction of red blood cell aggregation to both physiological normal- and pathological hyperlevels seen in humans.

RESULTS

Large asymmetries of layer widths on opposite sides of the downstream vessel were attenuated along the vessel and this effect could be prominently enhanced by the hyperaggregation due to a higher formation rate of the layer which was greater on one side than the other of the vessel. The proportion of downstream layer formation constituted by the smaller downstream vessel generally increased with a thicker layer width at the wall of the upstream vessel adjacent it. A greater tendency of the layer formation in the smaller downstream vessel was found under the hyperaggregating condition than normal-aggregating and nonaggregating conditions.

CONCLUSION

Red blood cell aggregation could attenuate the asymmetry in cell-free layer formation on opposite sides of the downstream vessel, but enhances the heterogeneity of the layer formation between downstream vessels.

Effect of uneven red cell influx on formation of cell-free layer in small venules

This study examined how the uneven influx of red blood cells (RBCs) from feeding vessels influences formation of cell-free layer (CFL) in the downstream vessel of a venular bifurcation. Spatio-temporal variations of the CFL width along the downstream vessel (19-41-μm inner diameter, D) were determined at 0.5D intervals from 0.5D to 3.0D away from the bifurcation. Upstream flow conditions were quantified by the ratio of volume flow rates (Q*=Q(High)/Q(Low)) between high flow (Q(High)) and low flow feeding (Q(Low)) vessels. The RBC aggregation level in the rats was adjusted to be at healthy human levels by infusing Dextran 500. Our results suggested that the CFL formation process could be seen only from 2.0D away from the bifurcating point. The mean CFL width at the wall adjacent to the feeding vessel with a higher flow rate was consistently greater than that at the opposite wall, leading to an asymmetric CFL formation in the vessel. A positive relation (P<0.05) between the asymmetry of the CFL width and the volume flow rate ratio (Q*) was found. Our numerical prediction showed that flow resistance in the venular network could be significantly increased by the asymmetric formation of CFL downstream and this effect might become more pronounced under pathological flow conditions such as hyper-aggregating and/or low shear conditions.

The effect of red blood cell aggregation on velocity and cell-depleted layer characteristics of blood in a bifurcating microchannel

Red blood cell (RBC) aggregation is a multifaceted phenomenon, and whether it is generally beneficial or deleterious remains unclear. In order to better understand its effect on microvascular blood flow, the phenomenon must be studied in complex geometries, as it is strongly dependent on time, flow, and geometry. The cell-depleted layer (CDL) which forms at the walls of microvessels has been observed to be enhanced by aggregation; however, details of the characteristics of the CDL in complex regions, such as bifurcations, require further investigation. In the present study, a microchannel with a T-junction was used to analyze the influence of aggregation on the flow field and the CDL. Micro-PIV using RBCs as tracers provided high resolution cell velocity data. CDL characteristics were measured from the same data using a newly developed technique based on motion detection. Skewed and sharpened velocity profiles in the daughter branches were observed, contrary to the behavior of a continuous Newtonian fluid. RBC aggregation was observed to increase the skewness, but decrease the sharpening, of the velocity profiles in the daughter branches. The CDL width was found to be significantly greater, with a wider distribution, in the presence of aggregation and the mean width increased proportionally with the reciprocal of the fraction of flow entering the daughter branch. Aggregation also significantly increased the roughness of the interface between the CDL and the RBC core. The present results provide further insight into how RBC aggregation may affect the flow in complex geometries, which is of importance in both understanding its functions invivo, and utilizing it as a tool in microfluidic devices.

Large deforming buoyant embolus passing through a stenotic common carotid artery: a computational simulation

Arterial embolism is responsible for the death of lots of people who suffers from heart diseases. The major risk of embolism in upper limbs is that the ruptured particles are brought into the brain, thus stimulating neurological symptoms or causing the stroke. We presented a computational model using fluid-structure interactions (FSI) to investigate the physical motion of a blood clot inside the human common carotid artery. We simulated transportation of a buoyant embolus in an unsteady flow within a finite length tube having stenosis. Effects of stenosis severity and embolus size on arterial hemodynamics were investigated. To fulfill realistic nonlinear property of a blood clot, a rubber/foam model was used. The arbitrary Lagrangian-Eulerian formulation (ALE) and adaptive mesh method were used inside fluid domain to capture the large structural interfacial movements. The problem was solved by simultaneous solution of the fluid and the structure equations. Stress distribution and deformation of the clot were analyzed and hence, the regions of the embolus prone to lysis were localized. The maximum magnitude of arterial wall shear stress during embolism occurred at a short distance proximal to the throat of the stenosis. Through embolism, arterial maximum wall shear stress is more sensitive to stenosis severity than the embolus size whereas role of embolus size is more significant than the effect of stenosis severity on spatial and temporal gradients of wall shear stress downstream of the stenosis and on probability of clot lysis due to clot stresses while passing through the stenosis.

Temporal and spatial variations of cell-free layer width in arterioles

Separation of red blood cells and plasma in microcirculatory vessels produces a cell-free layer at the wall. This layer may be an important determinant of blood viscosity and wall shear stress in arterioles, where most of the hydraulic pressure loss in the circulatory system occurs and flow regulatory mechanisms are prominent. With the use of a newly developed method, the width of the cell-free layer was rapidly and repeatedly determined in arterioles (10- to 50-μm inner diameter) in the rat cremaster muscle at normal arterial pressure. The temporal variation of the cell-free layer width was non-Gaussian, but calculated mean and median values differed by <0.2 μm. The correlation length of the temporal variations downstream (an indication of mixing) was ∼30 μm and was independent of pseudoshear rate (ratio of mean velocity to vessel diameter) and of vessel diameter. The cell-free layer width was significantly different on opposite sides of the vessel and inversely related. Increasing red blood cell aggregability reduced this inverse relation but had no effect on correlation length. In the diameter range studied, the mean width of the cell-free layer increased from 0.8 to 3.1 μm and temporal variations increased from 30% to 70% of the mean width. Increased aggregability did not alter either relationship. In summary, the cell-free layer width in arterioles is diameter dependent and shows substantial non-Gaussian temporal variations. The temporal variations increase as diameter increases and are inversely related on opposite sides of the vessel.

Numerical investigation of non-Newtonian microcirculatory blood flow in hepatic lobule

The circulation in the liver is unique at macroscopic and microscopic levels. At the macroscopic level, there is an unusual presence of portal and arterial inputs rather than a single arterial input. At the microscopic level, a series of microenvironments in the acinar system is essential in controlling the functional characteristics of hepatic parenchymal cells. Since the hemodynamics is much less studied in the multifunctional liver, an attempt is made to study the hepatic hemodynamics in a segment of a hepatic lobular structure, that is made up of high-pressure oxygenated arteriole, low-pressure nutrient-rich portal venule, fenestrated sinusoidal space and hepatic venule. Our goal is to dispel some of the myths of this complex vascular bed by means of finite volume blood flow simulation. Flow features like high-velocity gradients near the fenestrations, flow reversal and Dean vortices in the sinusoidal space are analyzed within the non-Newtonian framework. Since no distinct exact or numerical solutions are available for this complex vascular bed, the present simulated results are compared with the available clinical observations. Results revealed that the pressure plays a key role in hepatic blood flow.

Finite element modelling and simulations in cardiovascular mechanics and cardiology: A bibliography 1993–2004

The paper gives a bibliographical review of the finite element modelling and simulations in cardiovascular mechanics and cardiology from the theoretical as well as practical points of views. The bibliography lists references to papers, conference proceedings and theses/dissertations that were published between 1993 and 2004. At the end of this paper, more than 890 references are given dealing with subjects as: Cardiovascular soft tissue modelling; material properties; mechanisms of cardiovascular components; blood flow; artificial components; cardiac diseases examination; surgery; and other topics.

Microcirculation and Hemorheology

Major experimental and theoretical studies on microcirculation and hemorheology are reviewed with the focus on mechanics of blood flow and the vascular wall. Flow of the blood formed elements (red blood cells (RBCs), white blood cells or leukocytes (WBCs) and platelets) in individual arterioles, capillaries and venules, and in microvascular networks is discussed. Mechanical and rheological properties of the formed elements and their interactions with the vascular wall are reviewed. Short-term and long-term regulation of the microvasculature is discussed; the modes of regulation include metabolic, myogenic and shear-stress-dependent mechanisms as well as vascular adaptation such as angiogenesis and vascular remodeling.

Numerical analysis of flow through a severely stenotic carotid artery bifurcation

The results of computational simulations may supplement MR and other in vivo diagnostic techniques to provide an accurate picture of the hemodynamics in particular vessels, which may help demonstrate the risks of embolism or plaque rupture posed by particular plaque deposits. In this study, a model based on an endarterectomy specimen of the plaque in a carotid bifurcation was examined. The flow conditions include steady flow at Reynolds numbers of 300, 600, and 900 as well as unsteady pulsatile flow. Both dynamic pressure and wall shear stress are very high, with shear values up to 70 N/m2, proximal to the stenosis throat in the internal carotid artery, and both vary significantly through the flow cycle. The wall shear stress gradient is also strong along the throat. Vortex shedding is observed downstream of the most severe occlusion. Two turbulence models, the Chien and Goldberg varieties of k-epsilon, are tested and evaluated for their relevance in this geometry. The Chien model better captures phenomena such as vortex shedding. The flow distal to stenosis is likely transitional, so a model that captures both laminar and turbulent behavior is needed.

Effect of erythrocyte aggregation on velocity profiles in venules

A recent whole organ study in cat skeletal muscle showed that the increase in venous resistance seen at reduced arterial pressures is nearly abolished when the muscle is perfused with a nonaggregating red blood cell suspension. To explore a possible underlying mechanism, we tested the hypothesis that red blood cell aggregation alters flow patterns in vivo and leads to blunted red blood cell velocity profiles at reduced shear rates. With the use of fluorescently labeled red blood cells in tracer quantities and a video system equipped with a gated image intensifier, we obtained velocity profiles in venous microvessels (45-75 microm) of rat spinotrapezius muscle at centerline velocities between 0.3 and 14 mm/s (pseudoshear rates 3-120 s(-1)) under normal (nonaggregating) conditions and after induction of red blood cell aggregation with Dextran 500. Profiles are nearly parabolic (Poiseuille flow) over this flow rate range in the absence of aggregation. When aggregation is present, profiles are parabolic at high shear rates and become significantly blunted at pseudoshear rates of 40 s(-1) and below. These results indicate a possible mechanism for increased venous resistance at reduced flows.