The Rheology of Blood Flow in a Branched Arterial System

Abstract

Blood flow rheology is a complex phenomenon. Presently there is no universally agreed upon model to represent the viscous property of blood. However, under the general classification of non-Newtonian models that simulate blood behavior to different degrees of accuracy, there are many variants. The power law, Casson and Carreau models are popular non-Newtonian models and affect hemodynamics quantities under many conditions. In this study, the finite volume method is used to investigate hemodynamics predictions of each of the models. To implement the finite volume method, the computational fluid dynamics software Fluent 6.1 is used. In this numerical study the different hemorheological models are found to predict different results of hemodynamics variables which are known to impact the genesis of atherosclerosis and formation of thrombosis. The axial velocity magnitude percentage difference of up to 2 % and radial velocity difference up to 90 % is found at different sections of the T-junction geometry. The size of flow recirculation zones and their associated separation and reattachment point's locations differ for each model. The wall shear stress also experiences up to 12 % shift in the main tube. A velocity magnitude distribution of the grid cells shows that the Newtonian model is close dynamically to the Casson model while the power law model resembles the Carreau model.

Zusammenfassung

Die Rheologie von Blutströmungen ist ein komplexes Phänomen. Gegenwärtig existiert kein allgemein akzeptiertes Modell, um die viskosen Eigenschaften von Blut wiederzugeben. Jedoch gibt es mehrere Varianten unter der allgemeinen Klassifikation von nicht-Newtonschen Modellen, die das Verhalten von Blut mit unterschiedlicher Genauigkeit simulieren. Die Potenzgesetz-, Casson und Carreau-Modelle sind beliebte nicht-New-tonsche Modelle und beeinflussen die hämodynamischen Eigenschaften in vielen Situationen. In dieser Studie wurde die finite Volumenmethode angewandt, um die hämodynamischen Vorhersagen dieser Modelle zu untersuchen. Um die finite Volumenmethode zu implementieren, wurde die Fluiddynamiksoftware Fluent 6.1 verwendet. In dieser numerischen Studie wurde gefunden, dass die unterschiedlichen hämorheologischen Modelle unterschiedliche Resultate für die hämodynamischen Größen vorhersagen, von denen bekannt ist, dass sie die Entstehung von Arteriosklerose und die Bildung von Thrombose beeinflussen. Es wurde gefunden, dass die relative Differenz der axialen Geschwindigkeit bis zu 2% und die der radialen Geschwindigkeit bis zu 90% in unterschiedlichen Abschnitten der T-Verbindung beträgt. Die Größe der Strömungszirkulationszonen und ihrer dazugehörigen Trennungs- und Vereinigungspunkte differieren für jedes Modell. Die Scherspannung an der Wand erfährt ebenfalls eine Verschiebung im Hauptrohr von bis zu 12%. Der Verlauf der Geschwindigkeit auf den Gitterzellen zeigt, dass das Newtonsche Modell mit Bezug auf die Dynamik dem Casson-Modell nahe ist, während das Potenzgesetzmodell dem Carreau-Modell ähnlich ist.

Résumé

La rhéologie de l'écoulement sanguin est un phénomène complexe. Présentement, il n'y a pas de consensus universel sur le modèle qui représente la propriété visqueuse du sang. Cependant, parmi la classification générale des modèles non-Newtoniens qui simulent le comportement du sang avec différents degrés de précision, il y a plusieurs différences. Les lois de puissance, les modèles de Casson et Carreau sont des modèles non-Newtoniens populaires et ont un effet sur les quantités hémodynamiques sous plusieurs conditions. Dans cette étude, la méthode de volume fini est utilisée pour explorer les prédictions hémodynamiques de chacun de ces modèles. Pour implémenter la méthode de volume fini, le logiciel de calcul de dynamique des fluides Fluent 6.1 a été utilisé. Dans cette étude numérique, les différents modèles hémorhéologiques tendent à prédire des résultats différents pour les variables hémodynamiques qui sont reconnues comme ayant un impact sur la genèse de l'artériosclérose et de la thrombose. Une différence jusqu'à 2% dans l'amplitude de la vélocité axiale et une différence jusqu'à 90% dans la vélocité radiale sont découverts dans différentes sections d'une géométrie de type jonction en T. La taille des zones de re-circulation d'écoulement et les localisations des points de séparation et de rattachement qui leur sont associées, diffèrent pour chacun des modèles. La contrainte de cisaillement aux parois présente également un déplacement de 12% dans le tube principal. La distribution de l'amplitude de vitesse dans les cellules du maillage montre que le modèle Newtonien est dynamiquement proche du modèle de Casson tandis que le modèle en loi de puissance ressemble au modèle de Carreau.

1 INTRODUCTION

Whole blood consists of formed elements that are suspended in plasma. The majority of the formed elements are red blood cells (RBCs) as a result they are important components in determining the flow characteristics of blood. At low shear rates, that is, values less than 100 s⁻¹, the RBCs aggregate and form rouleaux. Rouleaux aggregation disperses as the shear rate increases, reducing the viscosity of blood. The resulting shear-thinning behavior caused by rouleaux disaggregations in blood plasma is the principal cause of the non-Newtonian behavior of blood. However, with further increment of the shear rate beyond the low shear rate region, the shear-thinning characteristics disappear and blood demonstrates Newtonian behavior [1 - 6].

White blood cells and platelets have concentrations that are much below that of RBCs and contribute negligibly to the viscosity of blood. Besides the influence of cellular components in plasma flow conditions, the size of the flow domain impacts the shear-thinning behavior of blood. Both in medium and large size arteries, non-Newtonian viscosity influence hemodynamics factors [7 - 8]. In some capillaries, small vessels with a diameter close to the size of cells, the non-Newtonian property of blood is more predominant [7, 10, 11].

For a complete description of hemodynamics phenomena, defining the appropriate viscous model, which takes into account the low and high shear rate behavior of blood, is essential. With the choice of appropriate empirically determined parameters, which fit experimental results for each non-Newtonian model, the dependence of viscosity with shear rate is typically determined. There are many empirically derived constitutive equations to represent the viscous property of blood. In general these models divided into Newtonian and non-Newtonian models. Among the non-Newtonian models there are different variants. The power law, Casson and Carreau models are the most widely used [12 - 15] and give a variable viscosity. For a Newtonian fluid, in contrast, a constant which is referred to as the high shear rate limit viscosity of blood, with a value of 0.035 P, is commonly used.

These different constitutive equations differ in their prediction of the flow behavior of blood. Previous numerical studies of blood flow in a conduit show that non-Newtonian models generally predict a flow profile in a tube with plug centerline distribution differing from the parabolic Poiseulle flow profile [7, 16]. Another experimental and numerical study on blood flow establishes the same behavior in a 90 degree curved tube and carotid artery bifurcation using a blood analog sample [8, 9]. Non-Newtonian behavior also affects the wall shear stress (WSS) predicting larger values than the Newtonian models [7, 17 - 19]. In our study, we compare simulation results of the velocity profile in axial and radial direction, velocity distribution in the entire grid cells, wall shear stress and vortex length for all major aforementioned rheological models of blood in a two-dimensional T-junction. We also present their implication in the genesis of atherosclerosis and thrombos formation.

In general the rheology of blood demonstrates viscoelastic behavior due to disaggregation and deformation of red blood cells [20 - 25]. In addition blood flow is pulsatile because of the nature of the heart pumping actions. However, in this study we considered only the viscous and the steady state flow behavior of blood neglecting the elastic and pulsatile nature. This simplification makes computational and analysis effort more manageable in comparative study like this, where many constitutive models are considered, while providing meaningful and relevant description about the dynamics of the system [26 - 28].

The T-junction geometric model has been serving as an ideal simplified model to study hemodynamics phenomena both experimentally and theoretically. It has been a geometrical model of choice because in addition to its simplicity, its flow features demonstrate the most common flow behavior at arterial bifurcations. Flow disturbances characterized by recirculation zones and low WSS are the most notable common features with real arteries [29 - 31].

Initiation of atherosclerosis and thrombosis are influenced by hemodynamics variables. Altered flow patterns characterized by flow recirculation are known flow behavior which regulates local transfer of atherosclerotic plasma and cellular components [29, 32]. Also low and oscillating WSS modulates the permeability of the arterial wall to blood components and influences the atherosclerosis process through activation of different cell signaling pathways and gene expressions [33, 34]. The difference in the hemodynamic variables of the different hemorheological models used commonly in theoretical and experimental studies can affect hemodynamics description of mechanisms which contribute to atherothrombogenesis. A more exact and more universal model is needed to characterize blood rheology in order to obtain consistent hemodynamic variables and investigate disease processes more accurately.

Accurate depiction of blood rheology ultimately will help us to accurately describe the flow dynamics in the cardiovascular system, design superior prostheses and extra-corporeal flow devices. This will also contribute to the improvement of clinical and surgical procedures.

2 METHODS

Diverse forms of constitutive equations have represented the shear-thinning behavior of blood. The most common constitutive equations characterizing this rheological behavior are divided into two general categories namely Newtonian and non-Newtonian models. There are the variants of the non-Newtonian model characterizing shear thinning behavior due to rouleaux dispersion at low shear rate. There is also the higher shear rate approximation single value viscosity, Newtonian model, which takes the shear independent constant viscosity of 0.0035 Pl at all shear rates.

The power law model, Casson model and Carreau-Yasuda model are commonly used non-Newtonian models [7 - 11, 16, 30, 35]. The power law model of blood viscosity takes the form of

$$\mu =k{\dot{\gamma }}^{(n-1)}$$ (1)

where μ is viscosity, k is the flow consistence index and n is the power law index. The power law index n specifies the extent of the non-Newtonian behavior. The consistency index k and the Power law index n are dependent on the constituents of blood such as hematocrit, fibrinogen, cholesterol, etc.

Experiment shows blood at rest requires a yield stress to start flowing. The power law does not take into account this characteristic feature. The Casson model, however, takes into consideration this behavior of blood and is given by the equation

$$\mu =\frac{{\tau }_o}{\dot{\gamma }}+\frac{\sqrt{\eta {\tau }_o}}{\sqrt{\dot{\gamma }}}+\eta$$ (2)

where τ_O is the yield stress and η is the Casson rheological constant. The values of τ_O and η depends on hematocrit H. The third non-Newtonian model considered in this article is the Carreau model given by

$$\mu ={\mu }_o+({\mu }_o+{\mu }_o){\left[1+{\left(\lambda \dot{\gamma }\right)}^2\right]}^{(n-1)∕2}$$ (3)

where μ_O and μ_∞ are the zero and infinite shear rate limit viscosities respectively, λ is the relaxation time constant and n is the power law index. The relaxation time constant and the power law index control the respective transitions and slope in the Power law region. The values of the parameters for blood for each constitutive equation are given in Tab.1. The mathematical modeling of flow phenomena is accomplished by employing the momentum balance equation

$$\rho \frac{\partial \overrightarrow{v}}{\partial t}+\rho (\overrightarrow{v}\cdot \nabla )\overrightarrow{v}=-\nabla p+\nabla \cdot \tilde{\tau }$$ (4)

where $\overrightarrow{\tau }$ is velocity, p is pressure, $\tilde{\tau }$ is the stress tensor and ρ is the density of the flowing fluid. The shear stress $\tilde{\tau }$ is related to viscosity μ and shear rate tensor D according to the relation:

$$\tilde{\tau }=\mu \left(\dot{\gamma }\right)D$$ (5)

where $\dot{\gamma }$ is the shear rate. The relation between $\dot{\gamma }$ and the rate of deformation tensor D is expressed as

$$\dot{\gamma }=\sqrt{\frac{1}{2}\sum _i\sum _j{D}_{\mathit{ij}}{D}_{\mathit{ji}}}$$ (6)

Explicitly D is simplified to give:

$$D=\left[\begin{array}{cc}\hfill \frac{\partial u}{\partial x}\hfill & \hfill \frac{1}{2}\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)\hfill \\ \hfill \frac{1}{2}\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)\hfill & \hfill \frac{\partial v}{\partial y}\hfill \end{array}\right]$$ (7)

To make the mathematical model a complete system, the equation of continuity is solved together with the momentum balance equation. The continuity equation is given by:

$$\frac{\partial \rho }{\partial t}+\nabla \cdot \left(\rho \overrightarrow{v}\right)=\mathrm{○}$$ (8)

where ρ is the density of fluid.

Table 1.

Parametric values for the non-Newtonian constitutive equations

Parameters	Values
Flow consistency index (k)	0.017 Pan
Power law index (n)	0.708
Yield stress (τ0)	0.005 N
Casson rheological constant (η)	0.0035 Pa·s
Zero shear rate limit (μ0)	0.056 Pa·s
Infinite shear rate limit (μ∞)	0.0035 Pa·s
Relaxation time constant (λ)	3.313 s
Power low index in Carreau model (n)	0.3568

The above system of equations leads to nonlinear partial differential equations, which are not amenable to analytic solutions. We used finite volume method to obtain solutions. In this study we investigated the flow fields and WSS in a T-junction by using different flow rheology models. To carry out the numerical simulation proprietary software Fluent 6.1 developed by Fluent Inc. is used. Fluent 6.1 makes use of finite volume method in its solver. The flow domain is discretized with quadrilateral elements using automatic mesh generation software GAMBIT 2.0. The total number of nodes in the grid is 3,256. The computation is conducted in a one node four-processor environment. The total computational effort expended is four hours.

The sketch of the T-junction geometry is shown in Fig. 1. In our model the flow is assumed steady and laminar with Reynolds number about 480. At the inlet, a parabolic velocity profile of maximum velocity 0.32 m/s is assumed, and at the exit, the gradients of the flow variables are kept at zero. At the wall, a no-slip boundary condition is imposed.

Figure 1.

Sketch of the flow geometry showing relevant dimensions and radial lines L1, L2, L3, etc. their axial velocity distribution is discussed.

In Fluent the complete set of the nonlinear partial differential equations is discretized in each control volume via the divergence theorem to construct algebraic equations for the discrete dependent variables and conserved scalars. A Gauss-Seidel linear equation solver is used in conjunction with an algebraic multigrid (AMG) method to solve the resultant scalar system of equations for the dependent variable in each cell. We employed the segregated solver to solve sequentially the momentum balance and mass conservation equations. The coupling between velocity and pressure is achieved through the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm.

3 RESULTS

The viscosity versus shear rate relationship as predicted by the rheological equations of blood based on the parametric values given in Tab. 1 is indicated graphically in Fig. 2. These parametric values are obtained from previous studies [7, 10, 36]. The figure shows the Carreau model gives the highest viscosity in all shear rate ranges. In comparison, at high shear rate the power law model and at low shear rate the Newtonian model give the lowest viscosity. The Carreau model coincides with the power law model at low shear rates, but at large shear rate ranges, it coincides with the Casson and Newtonian models.

Figure 2.

Comparison of viscosity Vs shear rate relationship as predicted by different blood constitutive equations.

The simulation of the flow fields in the T-junction (Figs. 3, 4, 5 and 6) indicates that the different constitutive equations predict different velocity profiles. The contrast is more pronounced for the radial velocity than the axial velocity. For the axial velocity, maximum velocity exhibited a percentage variation of up to 2 % as indicated in Fig. 3 and 4. For the radial velocity profile as Figs. 5 and 6 shows a dramatic difference which ranges up to 90 % is found. This variation of the flow fields is also different at different sections of the flow domain. In general maximum velocity is predicted by Newtonian model and minimum velocity is predicted by Carreau model. This is acceptable in light of the fact that the Carreau model has the highest viscosity in all shear rate ranges and the Newtonian model has the smallest viscosity in the significant shear rate ranges in our problem domain. The difference in the velocity profile is pronounced at and around the center of the tube and the difference diminishes away from the centerline towards the wall.

Figure 3.

Axial component velocity profiles in the main tube along different lines: (A) line L1, (B) line L2, (C) line L3, and (D) line L4.

Figure 4.

Axial component velocity profiles in the side tube along different lines: (A) line L5 and (B) line L6.

Figure 5.

Radial component velocity profiles in the main tube along different lines: (A) line L1, (B) line L2, (C) line L3, and (D) line L4.

Figure 6.

Radial component velocity profiles in the side tube along different lines: (A) line L5 and (B) line L6.

The vortex lengths predicted by the models vary significantly as visual inspection of the streamlines given in Fig. 7 indicates. The degree of such variations is higher in the main tube than in the side tube. In the main tube, the vortex length of the Newtonian model (Fig. 7d) is almost two times larger than that of the Carreau model (Fig. 7a) in contrast to a much smaller proportion in the daughter tube. The Newtonian model (Fig. 7d) provides the largest vortices and the Carreau model (Fig. 7a) gives the smallest size vortices. This is acceptable in light of the fact that the larger the viscosity the smaller the flow disturbance effects and the Carreau model has the largest viscosity in all shear rate ranges and the Newtonian model has the least except in low shear rate ranges as depicted in Fig. 2. Such variation of recirculation zones impact the local rate of platelet adhesion, monocytes migration and infiltration of lipid bearing macromolecules to the artery wall which can influence in atherosclerosis and therombogenesis.

Figure 7.

Flow recirculation zones showing variation in vortex size as predicted by: (A) Carreau model, (B) power law model (C) Casson model, and (D) Newtonian model

In addition to affecting the velocity profile the different viscous models provide varying WSS values (Fig. 8) with this variation is seen predominantly in the upper wall (Fig. 8a). The Carreau model shows the largest WSS ranging up to 12 % percentile difference with other models. This behavior is acceptable in light of highest viscosity values of the model in all shear rate values. Newtonian blood on the other hand generally exhibits the least WSS. This is again a consequence of its least viscosity except in shear rate values approximately greater than 250 s⁻¹ where the power law demonstrates the least value.

Figure 8.

Wall shear stress (WSS): (A) the main tube upper wall, (B) at the main tube lower wall, (C) at the daughter tube left side wall, and (D) at the daughter tube right side wall.

To further clarify the role of rheological constitutive equations on flow fields, a histogram of the velocity distribution of the flow fields (Fig. 9) at the cells of the flow domain grids are given. The Newtonian and Casson models are comparable in their distribution while the power law and Carreau model shows similar patterns. More number of cells attached to the maximum velocity of 0.32 m/s in the flow domain in Newtonian and Casson models than the other models.

Figure 9.

Histogram of the velocity magnitude distribution across the flow domain as predicted by rheological models: (A) Casson, (B) Carreau, (C) Newtonian, and (D) power law.

4 DISCUSSION

As illustrated in Fig. 2, none of the hemorheological models are agreeable in the whole shear rate ranges. This indicates the fact that the rheological characteristic of blood is an intricate property and it has been a difficult task to obtain a well-defined universal model. The complexity is resulted from the diversity as well as the shear dependent deformation of the constituents. This nature of blood makes both experimental and theoretical works a formidable task to get a universal model.

Though the models are not agreeable in whole shear rate ranges they are comparable to one another. In general the power law, Casson and Carreau models represent the shear thinning behavior and at certain shear rates these models do coincide with each other. The power law model and the Carreau model conform well at low shear rate ranges, while at higher shear rate ranges the Casson model converges with the Carreau model.

The variability of the rheological models could be reflected in velocity profiles, wall shear stress, mass transfer, residence time, pressure distribution, vortex length, etc. This study only investigated the impact of the models on velocity profiles, shear stress and vortex length. Each of these hemodynamic factors in turn can influence cardiovascular disease. The velocity profile can influence the convective flux of atherogenic plasma components from the lumen to the wall. The other hemodynamic factor WSS is known to regulate the rate-limiting behavior of the endothelial cell layer and influencing macromolecule diffusion to the wall of the artery via activation of cell signaling pathways and gene expressions [32, 33].

The numeric simulation indicates a difference between the different rheology models in there prediction of flow fields and WSS. This difference depends on the location in the flow fields. The radial velocity affected more than the axial velocity. A variation in axial velocity profile is also found similar to that in previous numerical and experimental studies on flow in carotid bifurcation and in a ninety degree curved tube under unsteady flow condition using the Newtonian and Carreau-Yasuda models for blood analog fluid [8, 9]. A pronounced variation in the axial velocity fields were found in an experimental study using a laser-Doppler anemometer in a three-dimensional T-junction for a blood analog fluid and in numerical study of blood flow in a carotid artery bifurcation [17, 19, 31]. Radial flow velocities can augment the transmural mass transfer of macromolecules which initiate atherosclerosis [30]. Previous studies seldom consider the effect of non-Newtonian rheology on radial velocity and the comparisons that we can draw.

Previous numerical studies of arterial blood flow and low-density lipoprotein (LDL) transport in T-junction geometry shows the recirculation zone size increases with increase of Reynolds number [32]. The same study also shows that the peak of the mass transfer distribution of LDL from the lumen to the wall is the largest at lower Reynolds number and the peaks are observed at reattachment points. This shows that the observed variation in vortex size of flow recirculation zones indicated in Figure 7 between Newtonian and non-Newtonian models is resulted from variation in shear rate dependence of the models and can affect hemodynamics description of atherotherombogenesis.

The WSS, the other important hemodynamics physical quantity, also depends on constitutive equations (Fig. 8). The WSS variation on blood flow rheology models has different patterns at different wall sections of the T-junction model. The WSS at the upper wall of the main tube shows the largest contrast showing a maximum percentage difference of 12 % (Fig. 8A). The side branch walls however show the minimal contrast (Figs. 8C and D). Previous numerical study on the effect of non-Newtonian viscosity of blood on flows in a human coronary artery casting shows that the non-Newtonian model yields a larger WSS than a Newtonian model [7]. Another flow simulation study in a forty five degree end-to-side anastomoses model and study at aortic and carotid artery bifurcation reveals that non-Newtonian blood rheology has a significant effect on steady and unsteady flow WSS [16 - 18]. Numerical study on non-Newtonina blood rheology in different coronary artery shows that different models render varying prediction of WSS magnitude. However the patterns of WSS are similar [26].

5 CONCLUSIONS

Blood rheology is a complex property and there is no agreeable single model for its representation. These diverse models affect hemodynamics descriptions to different degrees. The axial velocity change of about 2 % is observed both in the main and branch tubes. The radial velocity profile which controls the convective flux of atherogenic cellular and plasma components to the artery wall experience the largest contrast that ranges up to 90 %. The size of the recirculation zone and the location of separation reattachment point's prediction vary for each model. The Newtonian model predicts the largest vortex length and the Carreau model predict the least which half in size. The variation in vortex strength can affect local mass transfer of atherogenic plasma protein and cellular blood components which regulate atherotherombogenesis. The wall shear stress which is known to modulate the behavior of the artery wall permeability also shows significant variation which ranges up to 12 %. Both the convective flux and WSS play roles in initiating arterial diseases. Therefore, the observed discrepancies can influence hemodynamics description of atherotherombogenesis and this signifies that for future and more profound hemodynamics studies of cardiovascular diseases a more universal and profound hemorheological model is essential.