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Edited by: Timothy W. Secomb, University of Arizona, United States

Reviewed by: Panagiotis Dimitrakopoulos, University of Maryland, College Park, United States; Dmitry A. Fedosov, Forschungszentrum Jülich, Germany

*Correspondence: Gábor Závodszky

This article was submitted to Computational Physiology and Medicine, a section of the journal Frontiers in Physiology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Many of the intriguing properties of blood originate from its cellular nature. Therefore, accurate modeling of blood flow related phenomena requires a description of the dynamics at the level of individual cells. This, however, presents several computational challenges that can only be addressed by high performance computing. We present Hemocell, a parallel computing framework which implements validated mechanical models for red blood cells and is capable of reproducing the emergent transport characteristics of such a complex cellular system. It is computationally capable of handling large domain sizes, thus it is able to bridge the cell-based micro-scale and macroscopic domains. We introduce a new material model for resolving the mechanical responses of red blood cell membranes under various flow conditions and compare it with a well established model. Our new constitutive model has similar accuracy under relaxed flow conditions, however, it performs better for shear rates over 1,500 ^{−1}. We also introduce a new method to generate randomized initial conditions for dense mixtures of different cell types free of initial positioning artifacts.

On the cellular level, blood is a dense suspension of various types of cells. Red blood cells (RBC) form the primary component with an approximate volume fraction of 42% (Davies and Morris,

In the solutions targeting these questions the mechanical responses of the RBCs and PLTs are often expressed with constitutive models applied through their membranes accounting for the responses of the various structural elements (Ye et al.,

In this paper, our framework called Hemocell (High pErformance MicrOscopic CELlular Libary)^{1}

The solvent (blood plasma) in Hemocell is modeled as an incompressible Newtonian fluid using the lattice Boltzmann method implemented in the Palabos library (Lagrava et al., _{v} vertices which are connected by _{e} edges yielding _{t} surface triangles (see Figure _{v} = 642 vertices, _{e} = 1, 920 edges, and _{t} = 1, 280 faces. The mechanical behavior of a cell is expressed using this discrete membrane structure. The response to deformations is formulated as a set of forces acting on the cell membrane, which is coupled to the plasma flow through a validated in-house immersed-boundary implementation (Mountrakis et al., ^{6} cells executing on 8,192 cores without significant loss of parallel efficiency.

Visualization of the membrane meshes of an RBC

In Hemocell, two distinct constitutive model have been implemented for RBCs to act on the membrane mesh. The first one is based on the systematic coarse-graining of the model of Dao et al. (

The free-energy of a cell is described as

The location _{i} of each vertex on the membrane mesh is updated according to the force:

The total free-energy is composed of the following elements:

The in-plane potential models the compression response of the underlying cytoskeletal network along the membrane surface. The edges of the surface triangles represent the cumulative behavior of the local spectrin links using the wormlike chain (WLC) nonlinear spring description:
_{m} are the persistence length and the maximum length of the spectrin links, _{i} = _{i}/_{m} ∈ [0, 1), _{0} is the average length of links, _{0} = _{0}/_{m} and

The potential to account for bending rigidity:
_{i} is the instantaneous, θ_{0} is the equilibrium angle between neighboring faces sharing an edge, and κ is the bending constant.

The volume conservation energy is a fictitious potential which accounts for the forces arising from the change of volume:
_{0} is the equilibrium volume of the cell.

The area conservation potential is similarly a non-physical term representing the inextensible nature of the bilipid layer:
_{0} are the global and _{k}, _{0,k} are the local actual and equilibrium surface areas, respectively.

The additional term to correct membrane viscosity:
_{m,n} denotes the relative velocity of the vertices

The free parameters of this model (κ = 100 _{B}_{V} = 6, 000, _{A} = 5, 900, _{Al} = 100) were adopted from (Fedosov et al.,

We propose a new material model in the form of a set of forces acting on the same triangulated cellular membrane. The initial assumption for this model is that during small deformations all these forces present a linear regime with different slopes as the response types correspond to different components of the cell and are independent of each other. However, for large enough deformations the cytoskeleton adds contribution to all of them, resulting in qualitatively similar behavior. For instance, a response for small bending is assumed to be dominated by the curvature rigidity of the bilipid membrane resulting in a term linear in angle for the DEM membrane, while for larger deformation the underlying cytoskeleton deforms as well yielding an additional quickly diverging term. In the following we describe this model in two steps by separating the phenomenological description and the implementation.

The link force acts along links between surface points and represents the response to stretching and compression of the underlying spectrin-network beneath the representative link. The formulation of the force is similar in spirit to the worm-like-chain potential model often used to mimic the mechanical properties of polypeptide chains. It presents a linear part which corresponds to smaller deformations and a fast-diverging non-linear part which represents the limits of the material toward this type of deformation by quickly increasing the force response as the stretch approaches the persistence-length.
_{0} with τ_{l} = 3.0 is chosen based on the assumption that the represented spectrin-network reaches its persistence length at the relative expansion ratio of 3. The persistence-length of a spectrin filament was taken as

The bending force acts between two neighboring surface elements representing the bending response of the membrane arising primarily from the non-zero thickness of the spectrin-network. On each surface it points along the normal direction of that surface. As opposed to the previous model in which bending is expressed by modeling the bending rigidity of the bilipid membrane (Helfrich, _{i} − θ_{0}. From simple geometric considerations it follows that the limiting angle τ_{b} scales with the discretization length of the surface elements (_{0}). We fix the smallest representable curvature _{0} = 0.5 μ

The local surface conservation force acts locally on surface elements (i.e., triangles) and has the same form. It represents the combined surface response of the supporting spectrin-network and the lipid bilayer of the membrane to stretching and compression. This force is applied to all three vertices of each face and it points toward the centroid of the corresponding surface triangle.
_{a} = 0.3, thus prohibiting surface area changes larger than 30%.

The volume conservation force is the only global term. It is used to maintain quasi-incompressibility of the cell. It is applied at each node of each surface element and it points toward the normal of the surface.
_{v} = 0.01 and κ_{v} = 20 _{B}

This constitutive model has three free parameters for RBC modeling : κ_{l}, κ_{b}, and κ_{a}. These are chosen to satisfy mechanical single-cell experimental results.

The proposed forces can be realized in multiple ways on the given DEM structure, thus the implementation method is an inseparable part of the model. Figure

For each edge

The bending force is applied for each edge

The local surface conservation force acts on each

The volume conservation force is applied on the three nodes of each surface element:
_{j} is the surface area of the j-th element and _{avg} is the average surface area.

The free parameters of our mechanical RBC membrane model are fit to match the results of the optical-tweezer stretching experiments (Mills et al.,

The mechanical properties of the RBCs are validated using experiments measuring these two basic deformations.

In the optical-tweezers experiment small silica beads are attached on the opposing sides of the RBC. One is then fixed to the wall of the experimental container while the other is moved away by a focused laser beam. The arising forces result in a stretching of the RBC along the longitudinal axis and contraction along the transversal axes. In our simulation the same force magnitudes are used as in the experiment. They are applied on five percent of the membrane area on the opposing ends of the RBC. These areas represent the attachment surfaces of the silica beads.

The stretching curves of the two material models implemented in Hemocell are compared to the experiment of Mills et al. (

The results of the RBC stretching simulations. The upper arm of the curve denotes axial diameter expansion due to the stretching force, while the lower arm depicts the transversal contraction of the cell.

In the wheeler experiment performed by Yao et al. (^{−1}, in accordance with the experiment. The deformation index of the RBC is defined as given in Yao et al. (

where _{0} is the original diameter of the RBC (7.82 μ_{max} is the maximal diameter during the deformation at a constant shear rate value. The results are compared to the experimental results and to simulated results of MacMeccan et al. (

Results of the wheeler RBC shearing simulations. Both constitutive models show good agreement with the measurements. Inset image in the lower right corner: RBC shapes at an increased shear rate of 1, 500 ^{−1} for the two implemented material models—top, Dao/Suresh model; bottom, our model.

The fit to the experimental results yield κ_{l} = 15 _{B}_{a} = 5 _{B}_{b} = 80 _{B}_{B}_{b}. Additionally, with the selected κ_{a} value the local surface extensions under physiological flow conditions are smaller than the set limit of 30%, typically below 7%, which agrees with the literature (Fung, _{s} = 27.82 μ_{s} = 25 − 50 μ

From the unique material properties an emergent ability of RBCs traveling in small, confined flows is their deformation to parachute-like shapes (Noguchi and Gompper,

Transition of an RBC toward parachute shape while traveling in a micro-channel of

An important component of simulating blood flows on a cellular level is the selection of initial conditions for the cells, such as position and orientation. These are far from trivial since, due to the biconcave shape of RBCs, their volume (≈ 71 μ^{3}) compared to the volume of their enclosing box (≈ 224 μ^{3}) is low. Using the densest possible packing along a regular grid thus yields a hematocit of 32% which is often inadequate as it does not reach the level of physiologic hematocrit of human blood. A further issue is the need for a randomized distribution to avoid initial artifacts originating from the regular positioning and orientations. To circumvent these difficulties an additional kinetic simulation was developed to compute realistic initial distributions even at high hematocrit values. Instead of the real biconcave shapes, the enclosing ellipsoid of the RBCs were used to execute a simple kinetic process for hard ellipsoid packing. The so-called the force-bias model (Mościński et al., ^{in} represents the possible largest scaling in the system without any overlap between the cells. While ^{out} is initially set so that the merged volume (counting overlapping volumes only once) of all the ellipsoids scaled with it equals the total volume of the enclosing ellipsoids corresponding to the desired hematocrit level. Then, a repulsive force is applied between overlapping ellipsoids, proportional to the volume of the overlapping regions:

where _{ij} equals 1 if there is an overlap between particle _{ij} is a chosen potential function. In our case, the potential function was selected to be proportional to the overlapping volume of the ^{out} scaled particles. The positions are updated following Newtonian mechanics where mass is proportional to the particle scaling radius. This ensures that larger particles will move slower than smaller ones (i.e., an RBC will push away a platelet rather than the other way around). The rearrangement of the cells have a tendency of increasing ^{in}. As a final step the size of ^{out} is reduced every iteration according to a chosen contraction rate τ. The computation stops when ^{out} ≤ ^{in} at which point the system is force-free, since there are no overlaps. Using this method, we were able to push the initial hematocrit value up to 46% covering the physiological range. Additionally, we can fix the orientation of the cells by only allowing translation of their center of mass during this computation, thus predefining the alignment of the particles. This is beneficial for initializing higher velocity flows where the cells are expected to be lined up with the bulk flow direction. Figure

Randomized packing of a 50 μ^{3} cubic domain with RBCs (red) and PLTs (yellow).

It is possible to initialize simulations of up to 10^{6} cells efficiently this way. These simulations are free from regular-grid positioning artifacts from the beginning, which in turn reduces the computational time significantly. Though the actual computational cost it saves varies by geometry, hematocrit, flow velocity, etc., in our simulations the warm-up phase needed to allow the initially regularly placed cells to arrange more realistically amounted to 10-30% of the total computational time, while with the randomized initialization this whole phase could be omitted.

Our ultimate goal of accurate mechanical modeling of cellular membranes in blood flows is to allow for the resolution of the collective transport dynamics and coupling this to relevant biochemical processes. In the following, these transport properties are explored using the new constitutive model in the cases of a straight vessel sections of varying diameters. A snapshot from the simulation of the _{l}, κ_{a}, κ_{b} are multiplied by 10. These simulations also benefit from the above mentioned randomized initialization of the cells.

Blood flow simulation in a straight vessel section of ^{4} cells and it was computed on 512 cores.

The first fundamental transport property examined is the apparent viscosity. The results are compared to the experimental results collected by Pries et al. (

Relative apparent viscosity as a function of diameter at the hematocrit levels of 20, 30, and 45%. Please note that the curve of 30% originates from the fitted empirical formula of Pries et al. (

The results show good agreement with the measurements. For the simulations of

Another distinctive feature of cellular suspension flows is the formation of a cell-free layer (CFL) close to the walls as a result of lift force acting on the cells. The width of the appearing cell-free zones are defined using the density distribution of cells. It is the distance from the wall at which point the density distribution averaged along the vessel section reaches 5%. The results are compared to the

The width of the cell-free layer as a function of vessel diameter in a straight section.

While our simulated diameter range surpasses the bounds of the experimental range, the overlapping region shows good agreement for the hematocrit level of 30 and 45%. The level of 20% does not have a directly corresponding measurement, however, it is situated between between the experimental results of 16% and 30%, as expected. For a given diameter the CFL decreases with the increase of hematocrit as more RBCs are packed into the same domain volume.

Finally, to validate the flow profile in stationary flow a straight, rectangular channel was set up to recreate the flow environment of the experimental work of Carboni et al. (

Velocity profile arising in Hemocell simulation (red line) and the particle tracking results of Carboni et al. (

The simulated profile fits the measurement well and has the same plug-shape along with similar widths of high-shear regions at the sides of the channel.

It is a well-known phenomenon that toward low shear-rate values the viscosity of blood increases steeply (Chien,

Viscosity drops as clusters are breaking up with the increase of flow velocity and shear.

The relation appears to be logarithmic (see the fitted exponential decay), which is in agreement with the literature (Baskurt and Meiselman,

Due to the elastic deformable nature of RBCs, blood can exhibit yield-stress behavior if the hematocrit level is high enough (Picart et al.,

Elastic behavior of dense RBC clusters cause an initial viscosity peak.

During the first 3 ms the relative viscosity rises from the value of 1 steeply while the plasma flow slows down. At this stage, the RBCs do not flow but deform. The local velocity in the fluid corresponds to the deformation velocity of the cells. Around 4 ms, the relative viscosity reaches its peak value and the clusters start to break up, i.e., the relative positions of the RBCs start to change and the suspension no longer displays solid-like features. After this point blood quickly settles back to its stable final relative viscosity. The same viscosity pattern is observable for all simulations with ^{−10} − 10^{−9} Ns/m for the bilipid membrane membrane (Waugh,

The novel material model produces results in good agreement with several experiments targeting both single-cell mechanics and collective transport behavior. It also performs well for higher shear rate values where the other investigated model might fall short. It is capable of capturing the emerging solid-like behavior of dense RBC suspensions under low shear-rates. Furthermore, since our RBC material model is able to handle strong deformations coupling it with the LBM method for the plasma flow which operates at very small time-steps (in the order of 10^{−8}

The framework itself is structured to be easy to extend with additional material models and cell types, e.g., white blood cells, and with other fields, such as concentrations of different chemical components as well as with new biophysical processes, for instance bond formations. The efficient highly parallel implementation is capable of handling large domain sizes, thus it is able to cover the range between cell-based micro-scale and macroscopic domains.

The demonstrated capabilities make this framework in combination with our constitutive model an ideal environment for exploring the transport effects of blood flows

GZ conceived the research, designed the model and wrote 50% the paper; BvR collected and analyzed the experimental data, validated the Dao/Suresh model in our implementation, revised the new material model, and wrote 50% of the paper; VA contributed to the technical realization of Hemocell and revised the final version of the paper; AH conceived and supervised the research, and revised the manuscript. All authors read and approved the final version of the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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