A multiscale red blood cell model with accurate mechanics, rheology, and dynamics

Multiscale systems biology and physics of thrombosis under flow

Blood clotting under hemodynamic conditions involves numerous multiscale interactions from the molecular scale to macroscopic vessel and systemic circulation scales. Transmission of shear forces to platelet receptors such as GPIbα, P-selectin, α(2)β(1), and α(2b)β(3) controls adhesion dynamics. These forces also drive membrane tether formation, cellular deformation, and mechanosignaling in blood cells. Blood flow results in red blood cell (RBC) drift towards the center of the vessel along with a near-wall plasma layer enriched with platelets. RBC motions also dramatically enhance platelet dispersion. Trajectories of individual platelets near a thrombotic deposit dictate capture-activation-arrest dynamics as these newly arriving platelets are exposed to chemical gradients of ADP, thromboxane, and thrombin within a micron-scale boundary layer formed around the deposit. If shear forces are sufficiently elevated (>50 dyne/cm(2)), the largest polymers of von Willebrand Factor may elongate with concomitant shear-induced platelet activation. Finally, thrombin generation enhances platelet recruitment and clot strength via fibrin polymerization. By combination of coarse-graining, continuum, and stochastic algorithms, the numerical simulation of the growth rate, composition, and occlusive/embolic potential of a thrombus now spans multiscale phenomena. These simulations accommodate particular flow geometries, blood phenotype, pharmacological regimen, and reactive surfaces to help predict disease risk or response to therapy.

Multiscale models of thrombogenesis

To restrict the loss of blood follow from the rupture of blood vessels, the human body rapidly forms a clot consisting of platelets and fibrin. However, to prevent pathological clotting within vessels as a result of vessel damage, the response must be regulated. Clots forming within vessels (thrombi) can restrict the flow of blood causing damage to tissues in the flow field. Additionally, fragments dissociating from the primary thrombus (emboli) may lodge and clog vessels in the brain (causing ischemic stroke) or lungs (resulting in pulmonary embolism). Pathologies related to the obstruction of blood flow through the vasculature are the major cause of mortality in the United States. Venous thromboembolic disease alone accounts for 900,000 hospitalizations and 300,000 deaths per year and the incidence will increase as the population ages (Wakefield et al. J Vasc Surg 2009, 49:1620-1623). Thus, understanding the interplay between the many processes involved in thrombus development is of significant biomedical value. In this article, we first review computational models of important subprocesses of hemostasis/thrombosis including coagulation reactions, platelet activation, and fibrin assembly, respectively. We then describe several multiscale models integrating these subprocesses to simulate temporal and spatial development of thrombi. The development of validated computational models and predictive simulations will enable one to explore how the variation of multiple hemostatic factors affects thrombotic risk providing an important new tool for thrombosis research.

Margination of white blood cells in microcapillary flow

Margination of white blood cells (WBCs) towards vessel walls is an essential precondition for their efficient adhesion to the vascular endothelium. We perform numerical simulations with a two-dimensional blood flow model to investigate the dependence of WBC margination on hydrodynamic interactions of blood cells with the vessel walls, as well as on their collective behavior and deformability. We find WBC margination to be optimal in intermediate ranges of red blood cell (RBC) volume fractions and flow rates, while, beyond these ranges, it is substantially attenuated. RBC aggregation enhances WBC margination, while WBC deformability reduces it. These results are combined in state diagrams, which identify WBC margination for a wide range of flow and cell suspension conditions.

Contractile network models for adherent cells

Cells sense the geometry and stiffness of their adhesive environment by active contractility. For strong adhesion to flat substrates, two-dimensional contractile network models can be used to understand how force is distributed throughout the cell. Here we compare the shape and force distribution for different variants of such network models. In contrast to Hookean networks, cable networks reflect the asymmetric response of biopolymers to tension versus compression. For passive networks, contractility is modeled by a reduced resting length of the mechanical links. In actively contracting networks, a constant force couple is introduced into each link in order to model contraction by molecular motors. If combined with fixed adhesion sites, all network models lead to invaginated cell shapes, but only actively contracting cable networks lead to the circular arc morphology typical for strongly adhering cells. In this case, shape and force distribution are determined by local rather than global determinants and thus are suited to endow the cell with a robust sense of its environment. We also discuss nonlinear and adaptive linker mechanics as well as the relation to tissue shape.

Multiscale modeling of red blood cell mechanics and blood flow in malaria

Red blood cells (RBCs) infected by a Plasmodium parasite in malaria may lose their membrane deformability with a relative membrane stiffening more than ten-fold in comparison with healthy RBCs leading to potential capillary occlusions. Moreover, infected RBCs are able to adhere to other healthy and parasitized cells and to the vascular endothelium resulting in a substantial disruption of normal blood circulation. In the present work, we simulate infected RBCs in malaria using a multiscale RBC model based on the dissipative particle dynamics method, coupling scales at the sub-cellular level with scales at the vessel size. Our objective is to conduct a full validation of the RBC model with a diverse set of experimental data, including temperature dependence, and to identify the limitations of this purely mechanistic model. The simulated elastic deformations of parasitized RBCs match those obtained in optical-tweezers experiments for different stages of intra-erythrocytic parasite development. The rheological properties of RBCs in malaria are compared with those obtained by optical magnetic twisting cytometry and by monitoring membrane fluctuations at room, physiological, and febrile temperatures. We also study the dynamics of infected RBCs in Poiseuille flow in comparison with healthy cells and present validated bulk viscosity predictions of malaria-infected blood for a wide range of parasitemia levels (percentage of infected RBCs with respect to the total number of cells in a unit volume).

One of the most severe forms of cerebral malaria is caused by the parasite Plasmodium falciparum. During its development inside the host red blood cell it causes major mechanical and biochemical changes to the cell and can even alter its biconcave shape. The two main mechanical modifications are significant stiffening of the cell and increased cytoadherence to the arterial wall and other cells. These two effects can lead to dramatic flow modifications of infected blood, especially in the smallest vessels where occlusions are possible. In this work we combine a first-principles mathematical approach together with single-cell measurements to model seamlessly the infected and healthy red blood cells as well as the plasma flow, and predict their collective behavior in blood flow. Through systematic parallel simulations we quantify the mechanical and rheological properties of infected blood in malaria for a wide range of parasitemia levels, investigating in particular the effect of temperature.

Wall shear stress-based model for adhesive dynamics of red blood cells in malaria

Red blood cells (RBCs) infected by the Plasmodium falciparum (Pf-RBCs) parasite lose their membrane deformability and they also exhibit enhanced cytoadherence to vascular endothelium and other healthy and infected RBCs. The combined effect may lead to severe disruptions of normal blood circulation due to capillary occlusions. Here we extend the adhesion model to investigate the adhesive dynamics of Pf-RBCs as a function of wall shear stress (WSS) and other parameters using a three-dimensional, multiscale RBC model. Several types of adhesive behavior are identified, including firm adhesion, flipping dynamics, and slow slipping along the wall. In particular, the flipping dynamics of Pf-RBCs observed in experiments appears to be due to the increased stiffness of infected cells and the presence of the solid parasite inside the RBC, which may cause an irregular adhesion behavior. Specifically, a transition from crawling dynamics to flipping behavior occurs at a Young's modulus approximately three times larger than that of healthy RBCs. The simulated dynamics of Pf-RBCs is in excellent quantitative agreement with available microfluidic experiments if the force exerted on the receptors and ligands by an existing bond is modeled as a nonlinear function of WSS.

Predicting human blood viscosity in silico

The viscosity of blood has long been used as an indicator in the understanding and treatment of disease, and the advent of modern viscometers allows its measurement with ever-improving clinical convenience. However, these advances have not been matched by theoretical developments that can yield a quantitative understanding of blood's microrheology and its possible connection to relevant biomolecules (e.g., fibrinogen). Using coarse-grained molecular dynamics and two different red blood cell models, we accurately predict the dependence of blood viscosity on shear rate and hematocrit. We explicitly represent cell-cell interactions and identify the types and sizes of reversible rouleaux structures that yield a tremendous increase of blood viscosity at low shear rates. We also present the first quantitative estimates of the magnitude of adhesive forces between red cells. In addition, our simulations support the hypothesis, previously deduced from experiments, of yield stress as an indicator of cell aggregation. This non-Newtonian behavior is analyzed and related to the suspension's microstructure, deformation, and dynamics of single red blood cells. The most complex cell dynamics occurs in the intermediate shear rate regime, where individual cells experience severe deformation and transient folded conformations. The generality of these cell models together with single-cell measurements points to the future prediction of blood-viscosity anomalies and the corresponding microstructures associated with various diseases (e.g., malaria, AIDS, and diabetes mellitus). The models can easily be adapted to tune the properties of a much wider class of complex fluids including capsule and vesicle suspensions.

A multiscale modeling approach to investigate molecular mechanisms of pseudokinase activation and drug resistance in the HER3/ErbB3 receptor tyrosine kinase signaling network

Multiscale modeling provides a powerful and quantitative platform for investigating the complexity inherent in intracellular signaling pathways and rationalizing the effects of molecular perturbations on downstream signaling events and ultimately, on the cell phenotype. Here we describe the application of a multiscale modeling scheme to the HER3/ErbB3 receptor tyrosine kinase (RTK) signaling network, which regulates critical cellular processes including proliferation, migration and differentiation. The HER3 kinase is a topic of current interest and investigation, as it has been implicated in mechanisms of resistance to tyrosine kinase inhibition (TKI) of EGFR and HER2 in the treatment of many human malignancies. Moreover, the commonly regarded status of HER3 as a catalytically inactive 'pseudokinase' has recently been challenged by our previous study, which demonstrated robust residual kinase activity for HER3. Through our multiscale model, we investigate the most significant molecular interactions that contribute to potential mechanisms of HER3 activity and the physiological relevance of this activity to mechanisms of drug resistance in an ErbB-driven tumor cell in silico. The results of our molecular-scale simulations support the characterization of HER3 as a weakly active kinase that, in contrast to its fully-active ErbB family members, depends upon a unique hydrophobic interface to coordinate the alignment of specific catalytic residues required for its activity. Translating our molecular simulation results of the uniquely active behavior of the HER3 kinase into a physiologically relevant environment, our HER3 signaling model demonstrates that even a weak level of HER3 activity may be sufficient to induce AKT signaling and TKI resistance in the context of an ErbB signaling-dependent tumor cell, and therefore therapeutic targeting of HER3 may represent a superior treatment strategy for specific ErbB-driven cancers.

Time-dependent and outflow boundary conditions for Dissipative Particle Dynamics

We propose a simple method to impose both no-slip boundary conditions at fluid-wall interfaces and at outflow boundaries in fully developed regions for Dissipative Particle Dynamics (DPD) fluid systems. The procedure to enforce the no-slip condition is based on a velocity-dependent shear force, which is a generalized force to represent the presence of the solid-wall particles and to maintain locally thermodynamic consistency. We show that this method can be implemented in both steady and time-dependent fluid systems and compare the DPD results with the continuum limit (Navier-Stokes) results. We also develop a force-adaptive method to impose the outflow boundary conditions for fully developed flow with unspecified outflow velocity profile or pressure value. We study flows over the backward-facing step and in idealized arterial bifurcations using a combination of the two new boundary methods with different flow rates. Finally, we explore the applicability of the outflow method in time-dependent flow systems. The outflow boundary method works well for systems with Womersley number of O(1), i.e., when the pressure and flowrate at the outflow are approximately in-phase.

Predicting dynamics and rheology of blood flow: A comparative study of multiscale and low-dimensional models of red blood cells

We compare the predictive capability of two mathematical models for red blood cells (RBCs) focusing on blood flow in capillaries and arterioles. Both RBC models as well as their corresponding blood flows are based on the dissipative particle dynamics (DPD) method, a coarse-grained molecular dynamics approach. The first model employs a multiscale description of the RBC (MS-RBC), with its membrane represented by hundreds or even thousands of DPD-particles connected by springs into a triangular network in combination with out-of-plane elastic bending resistance. Extra dissipation within the network accounts for membrane viscosity, while the characteristic biconcave RBC shape is achieved by imposition of constraints for constant membrane area and constant cell volume. The second model is based on a low-dimensional description (LD-RBC) constructed as a closed torus-like ring of only 10 large DPD colloidal particles. They are connected into a ring by worm-like chain (WLC) springs combined with bending resistance. The LD-RBC model can be fitted to represent the entire range of nonlinear elastic deformations as measured by optical-tweezers for healthy and for infected RBCs in malaria. MS-RBCs suspensions model the dynamics and rheology of blood flow accurately for any vessel size but this approach is computationally expensive for vessel diameters above 100μm. Surprisingly, the much more economical suspensions of LD-RBCs also capture the blood flow dynamics and rheology accurately except for small-size vessels comparable to RBC diameter. In particular, the LD-RBC suspensions are shown to properly capture the experimental data for the apparent viscosity of blood and its cell-free layer (CFL) in tube flow. Taken together, these findings suggest a hierarchical approach in modeling blood flow in the arterial tree, whereby the MS-RBC model should be employed for capillaries and arterioles below 100μm, the LD-RBC model for arterioles, and the continuum description for arteries.

Three-dimensional vesicles under shear flow: numerical study of dynamics and phase diagram

The study of vesicles under flow, a model system for red blood cells (RBCs), is an essential step in understanding various intricate dynamics exhibited by RBCs in vivo and in vitro. Quantitative three-dimensional analyses of vesicles under flow are presented. The regions of parameters to produce tumbling (TB), tank-treating, vacillating-breathing (VB), and even kayaking (or spinning) modes are determined. New qualitative features are found: (i) a significant widening of the VB mode region in parameter space upon increasing shear rate γ and (ii) a robustness of normalized period of TB and VB with γ. Analytical support is also provided. We make a comparison with existing experimental results. In particular, we find that the phase diagram of the various dynamics depends on three dimensionless control parameters, while a recent experimental work reported that only two are sufficient.

Blood flow and cell-free layer in microvessels

Blood is modeled as a suspension of red blood cells using the dissipative particle dynamics method. The red blood cell membrane is coarse-grained for efficient simulations of multiple cells, yet accurately describes its viscoelastic properties. Blood flow in microtubes ranging from 10 to 40 μm in diameter is simulated in three dimensions for values of hematocrit in the range of 0.15-0.45 and carefully compared with available experimental data. Velocity profiles for different hematocrit values show an increase in bluntness with an increase in hematocrit. Red blood cell center-of-mass distributions demonstrate cell migration away from the wall to the tube center. This results in the formation of a cell-free layer next to the tube wall corresponding to the experimentally observed Fahraeus and Fahraeus-Lindqvist effects. The predicted cell-free layer widths are in agreement with those found in in vitro experiments; the results are also in qualitative agreement with in vivo experiments. However, additional features have to be taken into account for simulating microvascular flow, e.g., the endothelial glycocalyx. The developed model is able to capture blood flow properties and provides a computational framework at the mesoscopic level for obtaining realistic predictions of blood flow in microcirculation under normal and pathological conditions.

Tank-treading of erythrocytes in strong shear flows via a nonstiff cytoskeleton-based continuum computational modeling

We develop a computationally efficient cytoskeleton-based continuum erythrocyte algorithm. The cytoskeleton is modeled as a two-dimensional elastic solid with comparable shearing and area-dilatation resistance that follows a material law (Skalak, R., A. Tozeren, R. P. Zarda, and S. Chien. 1973. Strain energy function of red blood cell membranes. Biophys. J. 13:245-264). Our modeling enforces the global area-incompressibility of the spectrin skeleton (being enclosed beneath the lipid bilayer in the erythrocyte membrane) via a nonstiff, and thus efficient, adaptive prestress procedure which accounts for the (locally) isotropic stress imposed by the lipid bilayer on the cytoskeleton. In addition, we investigate the dynamics of healthy human erythrocytes in strong shear flows with capillary number Ca =O(1) and small-to-moderate viscosity ratios 0.001 ≤ λ ≤ 1.5. These conditions correspond to a wide range of surrounding medium viscosities (4-600 mPa s) and shear flow rates (0.02-440 s(-1)), and match those used in ektacytometry systems. Our computational results on the cell deformability and tank-treading frequency are compared with ektacytometry findings. The tank-treading period is shown to be inversely proportional to the shear rate and to increase linearly with the ratio of the cytoplasm viscosity to that of the suspending medium. Our modeling also predicts that the cytoskeleton undergoes measurable local area dilatation and compression during the tank-treading of the cells.

Quantifying the biophysical characteristics of Plasmodium-falciparum-parasitized red blood cells in microcirculation

The pathogenicity of Plasmodium falciparum (Pf) malaria results from the stiffening of red blood cells (RBCs) and its ability to adhere to endothelial cells (cytoadherence). The dynamics of Pf-parasitized RBCs is studied by three-dimensional mesoscopic simulations of flow in cylindrical capillaries in order to predict the flow resistance enhancement at different parasitemia levels. In addition, the adhesive dynamics of Pf-RBCs is explored for various parameters revealing several types of cell dynamics such as firm adhesion, very slow slipping along the wall, and intermittent flipping. The parasite inside the RBC is modeled explicitly in order to capture phenomena such as "hindered tumbling" motion of the RBC and the sudden transition from firm RBC cytoadherence to flipping on the endothelial surface. These predictions are in quantitative agreement with recent experimental observations, and thus the three-dimensional modeling method presented here provides new capabilities for guiding and interpreting future in vitro and in vivo studies of malaria.

Combined simulation and experimental study of large deformation of red blood cells in microfluidic systems

We investigate the biophysical characteristics of healthy human red blood cells (RBCs) traversing microfluidic channels with cross-sectional areas as small as 2.7 × 3 μm. We combine single RBC optical tweezers and flow experiments with corresponding simulations based on dissipative particle dynamics (DPD), and upon validation of the DPD model, predictive simulations and companion experiments are performed in order to quantify cell deformation and pressure-velocity relationships for different channel sizes and physiologically relevant temperatures. We discuss conditions associated with the shape transitions of RBCs along with the relative effects of membrane and cytosol viscosity, plasma environments, and geometry on flow through microfluidic systems at physiological temperatures. In particular, we identify a cross-sectional area threshold below which the RBC membrane properties begin to dominate its flow behavior at room temperature; at physiological temperatures this effect is less profound.

A low-dimensional model for the red blood cell

The red blood cell (RBC) is an important determinant of the rheological properties of blood because of its predominant number density, special mechanical properties and dynamics. Here, we develop a new low-dimensional RBC model based on dissipative particle dynamics (DPD). The model is constructed as a closed-torus-like ring of 10 colloidal particles connected by wormlike chain springs combined with bending resistance. Each colloidal particle is represented by a single DPD particle with a repulsive core. The model is able to capture the essential mechanical properties of RBCs, and allows for economical exploration of the rheology of RBC suspensions. Specifically, we find that the linear and non-linear elastic deformations of healthy and malaria-infected cells match those obtained in optical tweezers experiments. Through simulations of some key features of blood flow in vessels, i.e., the cell-free layer (CFL), the Fahraeus effect and the Fahraeus-Lindqvist effect, we verify that the new model captures the essential shear flow properties of real blood, except for capillaries of sizes comparable to the cell diameter. Finally, we investigate the influence of a geometrical constriction in the flow on the enhancement of the downstream CFL. Our results are in agreement with recent experiments.