Numerical simulation of red blood cell distributions in three-dimensional microvascular bifurcations
Introduction
Red blood cells (RBCs) occupy approximately 40–45% of the total blood volume and are therefore the principal component of blood. Hence, the rheological properties of RBCs have significant influence on the microvascular flow when diameters of RBCs and vessels are of the same order of magnitude (Goldsmith, 1986, Fung, 1997, Popel, 2005). RBCs are biconcave disc-shaped and contain hemoglobin, which is an oxygen-transporting protein, and exhibit a deformability that depends on the enveloping of the cell membrane. A characteristic of deformable RBCs is their axial concentration. The RBCs near the vessel wall migrate to the axial center with deformation. As a result, a plasma layer forms near the wall where no RBCs exist. The formation of a plasma layer decreases the flow resistance and increases the flow velocity. Consequently, the hematocrit of microcirculation, which is the volume fraction of RBCs, decreases with decreasing vessel diameter, a phenomenon called the Fåhræus effect. Moreover, the formation of the plasma layer leads to a nonproportional distribution of RBCs and plasma at the bifurcation, which can result in a complete absence of RBC flow passing through the bifurcation. This phenomenon is called plasma skimming. As a result, heterogeneity of oxygen in microvascular networks occurs. Therefore, it is very important to investigate the flow behavior of RBCs in microvascular flows.
Several recent numerical studies have focused on the behaviors of deformable RBCs in microvascular flows (Boryczko et al., 2003, Dupin et al., 2007, Sugiyama et al., 2010, Zhao et al., 2010, Tsubota and Wada, 2010, Imai et al., 2010, Alizadehred et al., 2012). Numerical studies on RBC aggregation (Liu et al., 2004, Liu and Liu, 2006, Zhang et al., 2008, Zhang et al., 2009) and low deformability (Zhang et al., 2009) in abnormal RBC states have also been reported. RBC behaviors indicated in these studies are well described by existing numerical techniques.
Other studies have investigated the distribution of cells at microvascular bifurcation. Chien et al. (1985) experimentally investigated the distribution of spherical and disk-shaped particles at a symmetric T-bifurcation at low Reynolds number. Pries et al. (1990) examined the plasma separation effect in vivo and proposed an empirical relationship describing the distribution of RBC flows at microvascular bifurcations. Several studies numerically investigated RBC motion, trajectories, and partitioning through two-dimensional (2D) microvascular bifurcation. (Hyakutake et al., 2006, Barber et al., 2008, Chesnutt and Marshall, 2009, Xiong and Zhang, 2012, Xu et al., 2012, Yin et al., 2013). Hyakutake et al. (2008) investigated flow behaviors of RBCs and liposome-encapsulated hemoglobin at 2D microvascular bifurcations using the lattice Boltzmann method (LBM) and clarified that the partial replacement of RBCs by liposome-encapsulated hemoglobin reduces the bias of oxygen flux. Moreover, an extension to a deformable RBC model was conducted (Hyakutake et al., 2010). However, these studies conducted simulations using 2D models, but there are no studies employing a three-dimensional (3D) microvascular bifurcation model. In this study, in order to demonstrate behaviors of flowing RBCs at a microvascular bifurcation in more detail, we constructed 3D microvascular bifurcation models using a parent vessel of diameter 10 μm and investigated the effect of the bifurcation angle and shape on the distributions of RBCs. These results provide valuable insights into the mechanism of RBC behavior through microvascular bifurcations.
Section snippets
Numerical model
For the present simulation, we constructed 3D microvascular bifurcation models and investigated flow behaviors of RBCs, especially the distributions of RBCs. Fig. 1 shows schematics of microvascular bifurcation models employed. We considered two types of models, that is, a symmetric model (Model A) and asymmetric models (Models B and C). In the case of Models A and B, diameters of the parent vessel (Dp) and daughter vessels (D1 and D2) are 10 μm whereas, in the case of Model C, Dp is 10 μm and D1
Results
We first investigated the flow behavior of a single RBC in a microvascular bifurcation. This RBC was arranged at the center of the parent vessel upstream of the bifurcation and gradually deformed into a parachute shape. We previously confirmed that the length of the parent vessel was sufficiently long to ensure that the RBC deformation was in a steady state. Therefore, we ensured a parent vessel length of 15 μm. We increased the fractional flow (Q1/Qp), beginning at 0.0 in increments of 0.05,
Discussion
We simulated the flow behavior of RBCs through 3D microvascular bifurcations consisting of a parent vessel of diameter 10 μm and two daughter branches. We focused on the effect of the bifurcation geometry on the RBC distribution. The blood flow was computed using LBM in conjunction with the IBM for incorporating fluid–membrane interactions between the flow field and deformable RBCs. The RBC model used in this simulations was based on our previous 2D study (Hyakutake et al., 2010) and the 2D
Conclusions
We constructed 3D symmetric and asymmetric microvascular bifurcation models using a parent vessel whose diameter is 10 μm and investigated the flow behaviors of RBCs through the three bifurcations considered. The blood flow was computed using LBM in conjunction with the IBM for incorporating fluid–membrane interactions between the flow field and deformable RBCs. First, we investigated the flow behaviors of single RBCs through the three microvascular bifurcations. In the case of the symmetric
References (38)
- et al.
Effect of particle collisions and aggregation on red blood cell passage through a bifurcation
Microvasc. Res.
(2009) - et al.
Improved measurements of the erythrocyte geometry
Microvasc. Res.
(1972) The microrheology of human blood
Microvasc. Res.
(1986)- et al.
Lattice Boltzmann simulation of blood cell behavior at microvascular bifurcations
Math. Comput. Simul.
(2006) - et al.
Modeling of hemodynamics arising from malaria infection
J. Biomech.
(2010) - et al.
Rheology of red blood cell aggregation by computer simulation
J. Comput. Phys.
(2006) - et al.
Red blood cell partitioning and blood flux redistribution in microvascular bifurcation
Theor. Appl. Mech. Lett.
(2012) - et al.
Multiple red blood cell flows through microvascular bifurcations: Cell free layer, cell trajectory, and hematocrit separation
Microvasc. Res.
(2013) - et al.
Red blood cell aggregation and dissociation in shear flows simulated by lattice Boltzmann method
J. Biomech.
(2008) - et al.
Effects of erythrocyte deformability and aggregation on the cell free layer and apparent viscosity of microscopic blood flows
Microvasc. Res.
(2009)
A spectral boundary integral method for flowing blood cells
J. Comput. Phys.
Quantification of red blood cell deformation at high-hematocrit blood flow in microvessels
J. Biomech.
Computational fluid dynamic simulation of aggregation of deformable cells in a shear flow
J. Biomech. Eng.
Simulated two-dimensional red blood cell motion, deformation, and partitioning in microvessel bifurcations
Ann. Biomed. Eng.
A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems
Phys. Rev.
Dynamical clustering of red blood cells in capillary vessels
J. Mol. Model.
Model studies on distributions of blood cells at microvascular bifurcations
Am. J. Physiol. Heart Circ. Physiol.
Computational model of whole blood exhibiting lateral platelet motion induced by red blood cells
Int. J. Numer. Method Biomed. Eng.
Modeling the flow of dense suspensions of deformable particles in three dimensions
Phys. Rev. E.
Cited by (46)
Analysis and prediction of hematocrit in microvascular networks
2023, International Journal of Engineering ScienceRed blood cell lingering modulates hematocrit distribution in the microcirculation
2023, Biophysical JournalIn vitro study on the partitioning of red blood cells using a microchannel network
2022, Microvascular ResearchCitation Excerpt :If the channel width is smaller, which is less than the RBC itself, the RBC tends to flow in a parachute style in the parent channel. Therefore, partitioning bias becomes larger than the current results and the empirical model (Hyakutake and Nagai, 2015). In this experimental condition, the flow rate in the parent channel was 0.1 nL/s for both the single and network models.
Advances in mathematical models of the active targeting of tumor cells by functional nanoparticles
2020, Computer Methods and Programs in BiomedicineParallel modeling of cell suspension flow in complex micro-networks with inflow/outflow boundary conditions
2020, Journal of Computational PhysicsCitation Excerpt :However, this method is in large part limited to 2D simulations, primarily because the existence of both meshes and particles leads to high computational cost. More recently, Hykutake et al. [45] presented a 3D simulation of about 10 RBCs in a bifurcated microchannel, to analyze the RBC distribution. Apart from these simple configurations, several efforts have been made to model the RBCs in very complicated micro-networks.
ATP Release by Red Blood Cells under Flow: Model and Simulations
2018, Biophysical JournalCitation Excerpt :During their travel in the microcirculation, they experience a cascade of branching vessels. Many numerical studies have been performed in single or multiple bifurcations regarding more or less complex models of blood flow (36–41). The arterioles are wrapped by smooth muscle cells and are well innervated, so in principle they are capable of controlling their pressure via vasomotion.