Quantification of red blood cell deformation at high-hematocrit blood flow in microvessels
Graphical Abstract
A snapshot of red blood cells in a microvessel with a diameter of 50 μm, hematocrit of 45%, and pseudo-shear rate of 90 s−1, and corresponding stretching ratio of red blood cells as a function of radial position.
Introduction
Blood is a dense suspension of highly deformable red blood cells (RBCs) in plasma. An RBC is a biconcave cell with a high surface-to-volume ratio, in which a Newtonian solution of hemoglobin is enclosed by a thin membrane. The membrane consists of a lipid bilayer underlined by a spectrin network (Mohandas and Evans, 1994), exhibiting small resistances to shear and bending (Evans, 1983, Evans, 1989). Hence, RBCs deform significantly in blood flow. The deformation of RBCs greatly affects the mechanics of blood flow, especially in the microcirculation. Interesting features of blood flow include the Fåhræus effect (Fåhræus, 1929), Fåhræus–Lindqvist effect (Fåhræus and Lindqvist, 1931), and formation of a cell-free layer (CFL) (Goldsmith, 1971, Tateishi et al., 1994, Kim et al., 2007). Recently, it was found that RBC deformation triggers the release of adenosine triphosphate (ATP) (Fischer et al., 2003, Moehlenbrock et al., 2006, Wan et al., 2008, Forsyth et al., 2011), which acts as a signaling molecule in various physiological processes. Some diseases such as malaria (Cooke et al., 2001, Dondorp et al., 2000, Suresh, 2006), type II diabetes (Tsukada et al., 2001), and sickle cell anemia (Higgins et al., 2007) are also linked to RBC deformability. Hence, to better understand the physiological and pathological conditions of the cardiovascular system, it is crucial to quantify the deformation of RBCs.
Recent confocal microscopy with microfluidics has improved experimental measurements of the behavior of RBCs in microvessels. For example, studies have examined the dispersion of RBCs (Lima et al., 2009) and tracer particles (Saadatmand et al., 2011) in 50- to 100-μm vessels using confocal micro-particle tracking velocimetry (Lima et al., 2007). However, owing to light scattering by RBCs and light absorption by hemoglobin, RBCs can be observed only at hematocrits (Hcts) of <20% with this method (Lima et al., 2009, Saadatmand et al., 2011). Thus, previous experiments have failed to quantify the deformation of RBCs in blood flow at physiologically relevant Hcts.
Numerical modeling can provide information for various Hcts. However, numerical simulations of blood flow in microvessels are challenging because of problems related to coupling membrane mechanics and fluid mechanics as well as computational costs. A few studies have examined three-dimensional simulations of blood flow with multiple RBCs. Zhao et al. (2010) developed a numerical model based on a boundary integral method and analyzed the shapes of RBCs and viscosity for vessels up to 16.9 μm in diameter, involving O(101) RBCs. Freund and Orescanin (2011) further investigated the deformation and motion of RBCs, the blood viscosity, and the local Hct in an 11.3-μm vessel using this method. Dupin et al., (2007) proposed a lattice-Boltzmann-based method and simulated O(102) RBCs in a rectangular channel. Clausen et al. (2010) developed a lattice-Boltzmann method for simulating O(103) RBCs on the IBM Blue Gene/P, but they focused on the performance of their method. Dissipative particle dynamics has also been applied successfully to investigate the apparent viscosity and CFL for vessels up to 40 μm in diameter (Fedosov et al., 2010, Fedosov et al., 2011a, Fedosov et al., 2011b).
Thus, the deformation of RBCs in microvessels is not well understood. Given that RBCs are approximately 8 μm in diameter, their flow characteristics in vessels with a few tens of micrometers in diameter may differ from those in smaller microvessels. To efficiently simulate blood flow in large microvessels, thousands of RBCs must be involved. Previously, we developed a numerical model of micro-scale blood flow based on a particle method (Kondo et al., 2009, Imai et al., 2010). This method has been applied successfully to study the microcirculation in malaria infection (Kondo et al., 2009, Imai et al., 2010, Imai et al., 2011) and thrombogenesis (Kamada et al., in press). More recently, we have developed a highly scalable parallel implementation of this method for large-scale studies and we confirmed that our model predicted well the CFL thickness and Fåhræus effect (Alizadehrad et al., 2012). The objective of the present paper was to investigate the deformation of RBCs in microvessels for a variety of vessel diameters, Hcts, and shear rates. We simulated blood flow in circular channels for diameters of 8–50 μm, Hcts of 20–45%, and shear rates of 20–150 s−1. First, our model was further validated by comparing the apparent viscosities between our simulation and experimental results. Then, we quantified the deformation of RBCs for these conditions.
Section snippets
Numerical model
The details of the model can be found in Imai et al. (2010), and we provide a brief review here. All blood components, including plasma, cytoplasm, and membranes, are modeled using a finite number of particles. Assuming that plasma and cytoplasm are incompressible viscous fluids, the motion of particles is governed by the conservation laws of mass and momentum aswhere t refers to the time; ρ, the density; u, the velocity; p, the pressure; μ, the dynamic viscosity; D/Dt,
Apparent viscosity
The relative apparent viscosity obtained from the simulations is shown in Fig. 1 for vessel diameters ranging from 9 to 50 μm and Hcts of 20%, 30%, and 45%, with a pseudo-shear rate of around 90 s−1. Pries et al. (1992) provided an empirical description of in vitro experimental data for the apparent viscosity as a function of the diameter and Hct. Our results agreed very well with their description. Our model correctly reproduced a nonlinear increase in the apparent viscosity with increases in
Conclusions
A parallel simulation of a particle-based model has been applied to study RBCs deformation in microvessels. The predicted apparent viscosity was in good agreement with experimental data, and this result supports the validity of our model. To our knowledge, this is the first quantitative study of the deformation of RBCs in vessels with a few tens of micrometers in diameter. Our simulation demonstrated that because of the shape transition in response to local shear stress and the wall effect, the
Conflicts of interest statement
There is no conflict of interest.
Acknowledgments
This research was supported by a Grant-in-Aid for Scientific Research (S) (No. 23220012), by a Grant-in-Aid for Young Scientists (A) (No. 24680048) from the JSPS.
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