Simulating Blood Cells and Blood Flow
Blood performs a multitude of functions on its way through our body, from the transport of oxygen to the immune response after infections. In addition, the circulatory system may be also affected by injuries which cause bleeding, by the formation of plaques
in arteries which cause coronary heart disease, and it provides the pathway for the organism invasion by bacteria or viruses. Thus, modeling of blood flow and its functions is an important challenge with many medical implications, but also with many interesting physical phenomena.

Figure 1: A scanning electron micrograph of blood cells. From left to right: red blood cell (red), activated platelet (yellow), and white blood cell (cyan). Source: The National Cancer Institute at Frederick (NCIFrederick).

Blood is a complex fluid, which consists of blood plasma – a liquid comprised of water, small molecules and various proteins – and blood cells. There are several types of blood cells, which include red blood cells (RBCs), white blood
cells (WBCs) and platelets, as shown in Fig. 1, with typical sizes of about 8μm, 10μm, and 2μm, respectively. The blood cells together take up about 45% of the blood volume. However, they appear in very different concentrations: one microliter of blood contains of about 5 million RBCs, 5,000 WBCs, and 250,000 platelets. Therefore, the flow properties of blood are dominated by the highly abundant RBCs [1,2].
Mesoscopic Modeling of Blood
The numerical modeling of blood flow in the circulatory system is very challenging due to the large range of length scales involved, ranging from the nanometer size of proteins and the micrometer size of blood cells to the centimeter size of large vessels (arteries and veins) and the decimeter size of the heart. Therefore, no single simulation technique is able to cover all required length scales. On the macroscopic level, appropriate for studying blood flow in the heart and in large vessels, Computational Fluid Dynamics (CFD) – which describes blood as a continuum fluid using constitutive equations for the fluid properties – has been very successful. However, constitutive equations are not always accurate. Moreover, this approach fails when the particulate nature of blood becomes important, for instance, when modeling the blood flow in small vessels (arterioles, capillaries, and venules). Therefore, mesoscopic models and simulation approaches, which describe blood as a suspension of soft, deformable particles or cells have been developed in recent years.

Figure 2: Simulation of blood under shear flow. RBCs are shown in red and in orange, where orange color depicts the rouleaux structures formed due to aggregation interactions between RBCs. The image also displays cut RBCs with the inside drawn in cyan to illustrate RBC shape and deformability.

Two main ingredients are necessary for the mesoscopic modeling of blood cells under flow, the membrane and the embedding fluid. The membrane of a RBC consists of a fluidic lipid bilayer, to which a biopolymer (spectrin) network is attached. This composite membrane can be described by a triangulated network of springs with bending and stretching elasticity [3,4]. For the fluid, mesoscale hydrodynamics simulation techniques – such as Dissipative Particle Dynamics (DPD) [5], MultiParticle Collision Dynamics (MPC) [6], and the Lattice Boltzmann method (LBM) – have proven to be very advantageous, because these hydrodynamics techniques contain thermal fluctuations and can be easily coupled to the membrane. The mesoscale hydrodynamics approaches are also very suitable for running on massively parallel supercomputers, such as JUROPA and JUGENE at the Jülich Supercomputing Centre, due to the simplicity of parallelization of the above methods. In order to highlight the strengths of mesoscale modeling to study blood flow, we focus further on the two recent examples: blood rheology and the margination of WBCs in microchannel flows.
Blood Rheology
Rheology is the science of the behavior and properties of fluids under flow. The aim of modern rheology is the prediction of the macroscopic flow properties, such as the fluid viscosity, from the sizes, deformability, and interactions of the constituent molecules and particles. For complex fluids, the rheological properties can be very rich. A characteristic feature is that under shear flow, the fluid viscosity may not be constant, but depend on the shear rate. Such fluids are called “nonNewtonian”. Blood is nonNewtonian and exhibits shearthinning, i.e. with increasing shear rate, the viscosity of blood decreases.

Figure 3: nonNewtonian relative viscosity (the cell suspension viscosity normalized by plasma viscosity) of whole blood as a function of shear rate at H_{t} =0.45 and 37°C. Experimental data are shown for whole blood (green crosses, black circles, and black squares) and for nonaggregating RBC suspension (red circles and red squares). Simulation results [7] are shown for the both cases with black dashed and solid lines, respectively.

In order to predict the shearthinning behavior, we study RBC suspensions in shear flow illustrated in Fig. 2, which are characterized by the RBC volume faction Ht, called the “hematocrit”,
and the shear rate γ. The simulation results for the relative viscosity η/η (where η is the plasma viscosity without cells) are displayed in Fig. 3. The two curves in Fig. 3 are for whole blood,
for which proteins in the blood plasma (e.g. fibrinogen) mediate attractive interactions which at low shear rates lead to the formation of cylindrical stacks of RBCs called “rouleaux”, and for RBCs in Ringer solution, where these proteins have been removed and the interactions are purely repulsive.
In both cases, blood is found to be a shearthinning fluid, an important property to reduce the load on the heart. However, the shear thinning is much more pronounced for whole blood due to the formation of rouleaux structures at low shear rates, which break up into smaller aggregates and finally single cells with increasing shear rates. Above a shear rate of approximately 5s1, the viscosity curves for whole blood and Ringer solution merge, and therefore we can conclude that rouleaux are absent at higher shear rates. In Fig. 3, the comparison of simulation and experimental results shows excellent agreement.
What can we now learn from simulations, which has not been already known from experiments? First, the good agreement between simulations and experiments for the viscosity of whole blood is obtained by tuning the strength of the aggregation interactions between RBCs. Thus, we are able to predict the attractive force between RBCs to be in the range of 3 to 7 picoNewton, 10 million times smaller than the weight of a mosquito! This force
is below the resolution of current experimental techniques. Second, Fig. 3 shows a second shearthinning regime above a shear rate of about 5s^{1}, where aggregation interactions are not relevant. This regime is due to deformations and orientations of RBCs, as revealed by a more detailed analysis of the simulated RBC shapes. In the simulations, the shapes are characterized by the eigenvalues λ of the gyration tensor, from which the asphericity α = (Σ (λ_{i}  λ_{j})^{2}) / (2 Σ_{i}λ_{j})^{2}
can be calculated [which yields α = 0 for a sphere, α = 1 for a long and thin rod] The simulation results for the asphericity distributions, displayed in Fig. 4, show that at low shear rates all RBCs maintain their equilibrium discocyte shape; at higher shear rates, in the second shearthinning regime, the distribution broadens and shifts to smaller α values, indicating strongly deformed and more rounded RBCs; finally, at very high shear rates, the distribution shifts to large values of α, indicating prolateshaped RBCs, which are elongated and oriented by the flow. Thus, the simulations provide new insights into the deformation and dynamics of single cells, but also their correlations and interactions, under flow.
White Blood Cell Margination
WBCs in our organism take part in the defense against various infections. Experimental observations revealed an interesting behavior of WBCs in blood flow such that they migrate towards the vessel walls, a process called “margination". This phenomenon appears to be biomedically very important, since WBCs need to firmly adhere to vessel walls to perform their function, and thus first they have to closely approach or touch the walls, which is facilitated by their margination.

Figure 4: RBC asphericity distributions to describe cell deformations trough the deviation from a spherical shape. The asphericity for a RBC in equilibrium is α =0.154.

WBC margination is governed by hydrodynamic interactions of cells with the vessel walls and with each other in blood flow. In fact, RBCs play a major role in WBC margination, since RBCs experience a stronger hydrodynamic lift force [8] away from the walls – due to their discoidal shape and deformability – than WBCs, which are nearly spherical in shape and resistant to deformations. As a consequence, RBCs migrate to the center of a vessel and effectively push out WBCs to the vessel walls.
WBC margination strongly depends on
blood cell deformability, hematocrit H_{t} local flow rate, and RBC aggregation. Therefore, numerical modeling of blood flow provides an excellent opportunity
to study WBC margination in detail
for various flow and cell properties. To
explore the effect of many parameters
on WBC margination, we employ a
twodimensional (2D) blood flow model
shown in Fig. 5, which significantly re
duces computational cost, however,
captures essential physics. In 2D, the
cells are modeled using a beadspring
representation illustrated in Fig. 5 [9]. The flow rate in the channel is characterized by a dimensionless shear rate γ = γτ, where γ is the average shear rate and τ is a characteristic RBC relaxation time
Fig. 6 shows a WBC margination diagram for various hematocrits and flow rates. The margination probability is defined as a probability for a WBC to be closer than 500nm to the channel walls. Clearly, efficient WBC margination occurs only* within an intermediate range of hematocrits H_{t}=0.20.5 and flow rates γ = 110.
The simulation results are consistent
with experimental observations that
WBC adhesion is found mainly in venules (but not in arterioles) in the organism. The characteristic value of γ* in venules are low enough, while in arterioles γ* >= 30 [2]. Thus, efficient WBC margination and consequent adhesion are mainly expected in the venular part of microcirculation. Our simulations also showed that RBC aggregation enhances WBC margination, while WBC deformability attenuates this effect [9].

Figure 5: Simulation snapshot of the flow (from left to right) at H^{t} =0.35 and dimensionless shear rate γ*= 1.59. The large quasispherical particle is the WBC.

Conclusions
The above models and results demonstrate a promising role of numerical modeling to understand and predict blood flow and its related processes. The potential of these models extends not only to RBC suspensions under various conditions, but also to modeling of other cell and capsule suspensions, blood flow in diseases (e.g., malaria, sicklecell anemia), biomedical applications (e.g., labonchip, microfluidics), etc. The main advantages of accurate modeling of such suspensions in comparison to experimental tests are robustness and low cost of numerical simulations. The future growth of supercomputing resources and further advances in cell modeling will make blood simulations a versatile and widely available tool in biophysical and biomedical research and applications.
Acknowledgement
The simulations reported in this article have been performed on the supercomputers JUROPA and JUGENE at the Jülich Supercomputing Centre. We thank FZJ and GCS for generous computing time grants.

Figure 6: Probability diagrams of WBC margination with respect to γ* and H_{t} Symbols (o) indicate the values of H_{t} and γ* for which simulations were performed.

• Dmitry A. Fedosov^{*}
• Gerhard Gompper
Theoretical Soft Matter and Biophysics Institute of Complex Systems and Institute for Advanced Simulation
Forschungszentrum Jülich
* Dmitry Fedosov has recently received a Sofia Kowalewskaja Award by the Alexander von Humboldt Foundation for numerical studies of blood flow in health and disease.
top

