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

Mechanical differences of sickle cell trait (SCT) and normal red blood cells

Sickle cell trait (SCT) is a condition in which an individual inherits one sickle hemoglobin gene (HbS) and one normal beta hemoglobin gene (HbA). It has been hypothesized that under extreme physical stress, the compromised mechanical properties of the red blood cells (RBCs) may be the underlying mechanism of clinical complications of sickle cell trait individuals. However, whether sickle cell trait (SCT) should be treated as physiologically normal remains controversial. In this work, the mechanical properties (i.e., shear modulus and viscosity) of individual RBCs were quantified using a microsystem capable of precisely controlling the oxygen level of RBCs' microenvironment. Individual RBCs were deformed under shear stress. After the release of shear stress, the dynamic cell recovery process was captured and analyzed to extract the mechanical properties of single RBCs. The results demonstrate that RBCs from sickle cell trait individuals are inherently stiffer and more viscous than normal RBCs from healthy donors, but oxygen level variations do not alter their mechanical properties or morphology.

Multi-scale biological and physical modelling of the tumour micro-environment

Paced by advances in high performance computing, and algorithms for multi-physics and multi-scale simulation, a number of groups have recently established numerical models of flowing blood systems, where cell-scale interactions are explicitly resolved. To be biologically representative, these models account for some or all of: (1) fluid dynamics of the carrier flow, (2) structural dynamics of the cells and vessel walls, (3) interaction and transport biochemistry, and, (4) methods for scaling to physiologically representative numbers of cells. In this article, our interest is the modelling of the tumour micro-environment. We review the broader area of cell-scale resolving blood flow modelling, while focusing on the particular interactions of tumour cells and white blood cells, known to play an important role in metastasis.

Fluctuations of red blood cell membranes: The role of the cytoskeleton

We theoretically investigate the membrane fluctuations of red blood cells with focus laid on the role of the cytoskeleton, viewing the system as a membrane coupled to a sparse spring network. This model is exactly solvable and enables us to examine the coupling strength dependence of the membrane undulation. We find that the coupling modifies the fluctuation spectrum at wavelengths longer than the mesh size of the network, while leaving the fluid-like behavior of the membrane intact at shorter wavelengths. The fluctuation spectra can be markedly different, depending on not only the relative amplitude of the bilayer bending energy with respect to the cytoskeleton deformation energy but also the bilayer-cytoskelton coupling strength.

Viscoelastic transient of confined red blood cells

The unique ability of a red blood cell to flow through extremely small microcapillaries depends on the viscoelastic properties of its membrane. Here, we study in vitro the response time upon flow startup exhibited by red blood cells confined into microchannels. We show that the characteristic transient time depends on the imposed flow strength, and that such a dependence gives access to both the effective viscosity and the elastic modulus controlling the temporal response of red cells. A simple theoretical analysis of our experimental data, validated by numerical simulations, further allows us to compute an estimate for the two-dimensional membrane viscosity of red blood cells, η(mem)(2D) ∼ 10(-7) N ⋅ s ⋅ m(-1). By comparing our results with those from previous studies, we discuss and clarify the origin of the discrepancies found in the literature regarding the determination of η(mem)(2D), and reconcile seemingly conflicting conclusions from previous works.

Multiple stiffening effects of nanoscale knobs on human red blood cells infected with Plasmodium falciparum malaria parasite

Our coarse-grained molecular dynamics (CGMD) simulations show that the deposition of nanoscale knobs, rather than spectrin network remodeling, is the primary cause of the dramatically increased stiffness of the Plasmodium falciparum (Pf)-infected red blood cell (RBC) membranes. Our analyses further reveal that the knobs stiffen the RBC membrane in a unique manner by simultaneously harnessing composite strengthening, strain hardening, and knob density-dependent vertical coupling effects. In addition to providing a fundamental understanding of the stiffening mechanism of Pf-infected RBCs, our simulation results suggest potential targets for antimalarial therapies.

During its asexual development within the red blood cell (RBC), Plasmodium falciparum (Pf), the most virulent human malaria parasite, exports proteins that modify the host RBC membrane. The attendant increase in cell stiffness and cytoadherence leads to sequestration of infected RBCs in microvasculature, which enables the parasite to evade the spleen, and leads to organ dysfunction in severe cases of malaria. Despite progress in understanding malaria pathogenesis, the molecular mechanisms responsible for the dramatic loss of deformability of Pf-infected RBCs have remained elusive. By recourse to a coarse-grained (CG) model that captures the molecular structures of Pf-infected RBC membrane, here we show that nanoscale surface protrusions, known as “knobs,” introduce multiple stiffening mechanisms through composite strengthening, strain hardening, and knob density-dependent vertical coupling. On one hand, the knobs act as structural strengtheners for the spectrin network; on the other, the presence of knobs results in strain inhomogeneity in the spectrin network with elevated shear strain in the knob-free regions, which, given its strain-hardening property, effectively stiffens the network. From the trophozoite to the schizont stage that ensues within 24–48 h of parasite invasion into the RBC, the rise in the knob density results in the increased number of vertical constraints between the spectrin network and the lipid bilayer, which further stiffens the membrane. The shear moduli of Pf-infected RBCs predicted by the CG model at different stages of parasite maturation are in agreement with experimental results. In addition to providing a fundamental understanding of the stiffening mechanisms of Pf-infected RBCs, our simulation results suggest potential targets for antimalarial therapies.

A phenomenological particle-based platelet model for simulating filopodia formation during early activation

We developed a phenomenological three-dimensional platelet model to characterize the filopodia formation observed during early stage platelet activation. Departing from continuum mechanics based approaches, this coarse-grained molecular dynamics (CGMD) particle-based model can deform to emulate the complex shape change and filopodia formation that platelets undergo during activation. The platelet peripheral zone is modeled with a two-layer homogeneous elastic structure represented by spring-connected particles. The structural zone is represented by a cytoskeletal assembly comprising of a filamentous core and filament bundles supporting the platelet's discoid shape, also modeled by spring-connected particles. The interior organelle zone is modeled by homogeneous cytoplasm particles that facilitate the platelet deformation. Nonbonded interactions among the discrete particles of the membrane, the cytoskeletal assembly, and the cytoplasm are described using the Lennard-Jones potential with empirical constants. By exploring the parameter space of this CGMD model, we have successfully simulated the dynamics of varied filopodia formations. Comparative analyses of length and thickness of filopodia show that our numerical simulations are in agreement with experimental measurements of flow-induced activated platelets. Copyright © 2015 John Wiley & Sons, Ltd.

A Multiple Time Stepping Algorithm for Efficient Multiscale Modeling of Platelets Flowing in Blood Plasma

We developed a multiple time-stepping (MTS) algorithm for multiscale modeling of the dynamics of platelets flowing in viscous blood plasma. This MTS algorithm improves considerably the computational efficiency without significant loss of accuracy. This study of the dynamic properties of flowing platelets employs a combination of the dissipative particle dynamics (DPD) and the coarse-grained molecular dynamics (CGMD) methods to describe the dynamic microstructures of deformable platelets in response to extracellular flow-induced stresses. The disparate spatial scales between the two methods are handled by a hybrid force field interface. However, the disparity in temporal scales between the DPD and CGMD that requires time stepping at microseconds and nanoseconds respectively, represents a computational challenge that may become prohibitive. Classical MTS algorithms manage to improve computing efficiency by multi-stepping within DPD or CGMD for up to one order of magnitude of scale differential. In order to handle 3-4 orders of magnitude disparity in the temporal scales between DPD and CGMD, we introduce a new MTS scheme hybridizing DPD and CGMD by utilizing four different time stepping sizes. We advance the fluid system at the largest time step, the fluid-platelet interface at a middle timestep size, and the nonbonded and bonded potentials of the platelet structural system at two smallest timestep sizes. Additionally, we introduce parameters to study the relationship of accuracy versus computational complexities. The numerical experiments demonstrated 3000x reduction in computing time over standard MTS methods for solving the multiscale model. This MTS algorithm establishes a computationally feasible approach for solving a particle-based system at multiple scales for performing efficient multiscale simulations.

Microvascular blood flow resistance: Role of red blood cell migration and dispersion

Microvascular blood flow resistance has a strong impact on cardiovascular function and tissue perfusion. The flow resistance in microcirculation is governed by flow behavior of blood through a complex network of vessels, where the distribution of red blood cells across vessel cross-sections may be significantly distorted at vessel bifurcations and junctions. In this paper, the development of blood flow and its resistance starting from a dispersed configuration of red blood cells is investigated in simulations for different hematocrit levels, flow rates, vessel diameters, and aggregation interactions between red blood cells. Initially dispersed red blood cells migrate toward the vessel center leading to the formation of a cell-free layer near the wall and to a decrease of the flow resistance. The development of cell-free layer appears to be nearly universal when scaled with a characteristic shear rate of the flow. The universality allows an estimation of the length of a vessel required for full flow development, lc ≲ 25D, for vessel diameters in the range 10 μm < D < 100 μm. Thus, the potential effect of red blood cell dispersion at vessel bifurcations and junctions on the flow resistance may be significant in vessels which are shorter or comparable to the length lc. Aggregation interactions between red blood cells generally lead to a reduction of blood flow resistance. The simulations are performed using the same viscosity for both external and internal fluids and the RBC membrane viscosity is not considered; however, we discuss how the viscosity contrast may affect the results. Finally, we develop a simple theoretical model which is able to describe the converged cell-free-layer thickness at steady-state flow with respect to flow rate. The model is based on the balance between a lift force on red blood cells due to cell-wall hydrodynamic interactions and shear-induced effective pressure due to cell-cell interactions in flow. We expect that these results can also be used to better understand the flow behavior of other suspensions of deformable particles such as vesicles, capsules, and cells.

Adhesive dynamics

Adhesive dynamics (AD) is a method for simulating the dynamic response of biological systems in response to force. Biological bonds are mechanical entities that exert force under strain, and applying forces to biological bonds modulates their rate of dissociation. Since small numbers of events usually control biological interactions, we developed a simple method for sampling probability distributions for the formation or failure of individual bonds. This method allows a simple coupling between force and strain and kinetics, while capturing the stochastic response of biological systems. Biological bonds are dynamically reconfigured in response to applied mechanical stresses, and a detailed spatio-temporal map of molecules and the forces they exert emerges from AD. The shape or motion of materials bearing the molecules is easily calculated from a mechanical energy balance provided the rheology of the material is known. AD was originally used to simulate the dynamics of adhesion of leukocytes under flow, but new advances have allowed the method to be extended to many other applications, including but not limited to the binding of viruses to surface, the clustering of adhesion molecules driven by stiff substrates, and the effect of cell-cell interaction on cell capture and rolling dynamics. The technique has also been applied to applications outside of biology. A particular exciting recent development is the combination of signaling with AD (so-called integrated signaling adhesive dynamics, or ISAD), which allows facile integration of signaling networks with mechanical models of cell adhesion and motility. Potential opportunities in applying AD are summarized.

Two-dimensional simulation of red blood cell motion near a wall under a lateral force

The motion of a red blood cell suspended in a linear shear flow adjacent to a fixed boundary subject to an applied lateral force directed toward the boundary is simulated. A two-dimensional model is used that represents the viscous and elastic properties of normal red blood cells. Shear rates in the range of 100 to 600 s^{-1} are considered, and the suspending medium viscosity is 1 cP. In the absence of a lateral force, the cell executes a tumbling motion. With increasing lateral force, a transition from tumbling to tank-treading is predicted. The minimum force required to ensure tank-treading increases nonlinearly with the shear rate. Transient swinging motions occur when the force is slightly larger than the transition value. The applied lateral force is balanced by a hydrodynamic lift force resulting from the positive orientation of the long axis of the cell with respect to the wall. In the case of cyclic tumbling motions, the orientation angle takes positive values through most of the cycle, resulting in lift generation. These results are used to predict the motion of a cell close to the outer edge of the cell-rich core region that is generated when blood flows in a narrow tube. In this case, the lateral force is generated by shear-induced dispersion, resulting from cell-cell interactions in a region with a concentration gradient. This force is estimated using previous data on shear-induced dispersion. The cell is predicted to execute tank-treading motions at normal physiological hematocrit levels, with the possibility of tumbling at lower hematocrit levels.

Multiscale Particle-Based Modeling of Flowing Platelets in Blood Plasma Using Dissipative Particle Dynamics and Coarse Grained Molecular Dynamics

We developed a multiscale particle-based model of platelets, to study the transport dynamics of shear stresses between the surrounding fluid and the platelet membrane. This model facilitates a more accurate prediction of the activation potential of platelets by viscous shear stresses - one of the major mechanisms leading to thrombus formation in cardiovascular diseases and in prosthetic cardiovascular devices. The interface of the model couples coarse-grained molecular dynamics (CGMD) with dissipative particle dynamics (DPD). The CGMD handles individual platelets while the DPD models the macroscopic transport of blood plasma in vessels. A hybrid force field is formulated for establishing a functional interface between the platelet membrane and the surrounding fluid, in which the microstructural changes of platelets may respond to the extracellular viscous shear stresses transferred to them. The interaction between the two systems preserves dynamic properties of the flowing platelets, such as the flipping motion. Using this multiscale particle-based approach, we have further studied the effects of the platelet elastic modulus by comparing the action of the flow-induced shear stresses on rigid and deformable platelet models. The results indicate that neglecting the platelet deformability may overestimate the stress on the platelet membrane, which in turn may lead to erroneous predictions of the platelet activation under viscous shear flow conditions. This particle-based fluid-structure interaction multiscale model offers for the first time a computationally feasible approach for simulating deformable platelets interacting with viscous blood flow, aimed at predicting flow induced platelet activation by using a highly resolved mapping of the stress distribution on the platelet membrane under dynamic flow conditions.

Probing red blood cell mechanics, rheology and dynamics with a two-component multi-scale model

This study is partially motivated by the validation of a new two-component multi-scale cell model we developed recently that treats the lipid bilayer and the cytoskeleton as two distinct components. Here, the whole cell model is validated and compared against several available experiments that examine red blood cell (RBC) mechanics, rheology and dynamics. First, we investigated RBC deformability in a microfluidic channel with a very small cross-sectional area and quantified the mechanical properties of the RBC membrane. Second, we simulated twisting torque cytometry and compared predicted rheological properties of the RBC membrane with experimental measurements. Finally, we modelled the tank-treading (TT) motion of a RBC in a shear flow and explored the effect of channel width variation on the TT frequency. We also investigated the effects of bilayer-cytoskeletal interactions on these experiments and our simulations clearly indicated that they play key roles in the determination of cell membrane mechanical, rheological and dynamical properties. These simulations serve as validation tests and moreover reveal the capabilities and limitations of the new whole cell model.

Effect of N-ethylmaleimide, chymotrypsin, and H₂O₂ on the viscoelasticity of human erythrocytes: experimental measurement and theoretical analysis

The physiological functions of erythrocytes depend critically on their morphology, deformability, and aggregation capability in response to external physical and chemical stimuli. The dynamic deformability can be described in terms of their viscoelasticity. We applied jumping optical tweezers to trap and stretch individual red blood cells (RBCs) to characterize its viscoelasticity in terms of the Young's modulus and viscosity by analyzing the experimental data of dynamic deformation using a 2-parameter Kelvin solid model. The effects of three chemical agents (N -ethylmaleimide, Chymotrypsin, and Hydrogen peroxide) on RBC's mechanical properties were studied by comparing the Young's modulus and viscosity of RBCs with and without these chemical treatments. Although the effects of each of these chemicals on the molecular structures of RBC may not be exclusive, based on the dominant effect of each chemical, we attempted to dissect the main contributions of different constituents of the RBC membrane to its viscosity and elasticity.

Deformation and dynamics of red blood cells in flow through cylindrical microchannels

The motion of red blood cells (RBCs) in microcirculation plays an important role in blood flow resistance and in the cell partitioning within a microvascular network. Different shapes and dynamics of RBCs in microvessels have been previously observed experimentally including the parachute and slipper shapes. We employ mesoscale hydrodynamic simulations to predict the phase diagram of shapes and dynamics of RBCs in cylindrical microchannels, which serve as idealized microvessels, for a wide range of channel confinements and flow rates. A rich dynamical behavior is found, with snaking and tumbling discocytes, slippers performing a swinging motion, and stationary parachutes. We discuss the effects of different RBC states on the flow resistance, and the influence of RBC properties, characterized by the Föppl-von Kármán number, on the shape diagram. The simulations are performed using the same viscosity for both external and internal fluids surrounding a RBC; however, we discuss how the viscosity contrast would affect the shape diagram.

White blood cell margination in microcirculation

Proper functioning of white blood cells is not possible without their ability to adhere to vascular endothelium, which may occur only if they are close enough to vessel walls. To facilitate the adhesion, white blood cells migrate toward the vessel walls in blood flow through a process called margination. The margination of white cells depends on a number of conditions including local hematocrit, flow rate, red blood cell aggregation, and the deformability of both red and white cells. To better understand the margination process of white blood cells, we employ mesoscopic hydrodynamic simulations of a three-dimensional model of blood flow, which has been previously shown to capture quantitatively realistic blood flow properties and rheology. The margination properties of white blood cells are studied for a wide range of hematocrit values and flow conditions. Efficient white blood cell margination is found in an intermediate range of hematocrit values of Ht ≈ 0.2-0.4 and at relatively low flow rates, characteristic of the venular part of microcirculation. In addition, aggregation interactions between red blood cells lead to enhanced white-blood-cell margination. This simulation study provides a quantitative description of the margination of white blood cells, and is also highly relevant for the margination of particles or cells of similar size such as circulating tumor cells.

Margination of micro- and nano-particles in blood flow and its effect on drug delivery

Drug delivery by micro- and nano-carriers enables controlled transport of pharmaceuticals to targeted sites. Even though carrier fabrication has made much progress recently, the delivery including controlled particle distribution and adhesion within the body remains a great challenge. The adhesion of carriers is strongly affected by their margination properties (migration toward walls) in the microvasculature. To investigate margination characteristics of carriers of different shapes and sizes and to elucidate the relevant physical mechanisms, we employ mesoscopic hydrodynamic simulations of blood flow. Particle margination is studied for a wide range of hematocrit values, vessel sizes, and flow rates, using two- and three-dimensional models. The simulations show that the margination properties of particles improve with increasing carrier size. Spherical particles yield slightly better margination than ellipsoidal carriers; however, ellipsoidal particles exhibit a slower rotational dynamics near a wall favoring their adhesion. In conclusion, micron-sized ellipsoidal particles are favorable for drug delivery in comparison with sub-micron spherical particles.

Stretching and relaxation of malaria-infected red blood cells

The invasion of red blood cells (RBCs) by malaria parasites is a complex dynamic process, in which the infected RBCs gradually lose their deformability and their ability to recover their original shape is greatly reduced with the maturation of the parasites. In this work, we developed two types of cell model, one with an included parasite, and the other without an included parasite. The former is a representation of real malaria-infected RBCs, in which the parasite is treated as a rigid body. In the latter, where the parasite is absent, the membrane modulus and viscosity are elevated so as to produce the same features present in the parasite model. In both cases, the cell membrane is modeled as a viscoelastic triangular network connected by wormlike chains. We studied the transient behaviors of stretching deformation and shape relaxation of malaria-infected RBCs based on these two models and found that both models can generate results in agreement with those of previously published studies. With the parasite maturation, the shape deformation becomes smaller and smaller due to increasing cell rigidity, whereas the shape relaxation time becomes longer and longer due to the cell's reduced ability to recover its original shape.

Molecular modeling of ErbB4/HER4 kinase in the context of the HER4 signaling network helps rationalize the effects of clinically identified HER4 somatic mutations on the cell phenotype

In the ErbB/HER family of receptor tyrosine kinases, the deregulation of the EGFR/ErbB1/HER1, HER2/ErbB2, and HER3/ErbB3 kinases is associated with several cancers, while the HER4/ErbB4 kinase has been shown to play an anti-carcinogenic role in certain tumors. We present molecular and network models of HER4/ErbB4 activation and signaling in order to elucidate molecular mechanisms of activation and rationalize the effects of the clinically identified HER4 somatic mutants. Our molecular-scale simulations identify the important role played by the interactions within the juxtamembrane region during the activation process. Our results also support the hypothesis that the HER4 mutants may heterodimerize but not activate, resulting in blockage of the HER4-STAT5 differentiation pathway, in favor of the proliferative PI3K/AKT pathway. Translating our molecular simulation results into a cellular pathway model of wild type versus mutant HER4 signaling, we are able to recapitulate the major features of the PI3K/AKT and JAK/STAT activation downstream of HER4. Our model predicts that the signaling downstream of the wild type HER4 is enriched for the JAK-STAT pathway, whereas downstream of the mutant HER4 is enriched for the PI3K/AKT pathway. HER4 mutations may hence constitute a cellular shift from a program of differentiation to that of proliferation.

Three-dimensional dynamics of oblate and prolate capsules in shear flow

We study computationally the dynamics of oblate and prolate spheroidal capsules in simple shear flow with small inertia for a range of dimensionless shear rates. The capsule is modelled as a liquid droplet enclosed by a hyperelastic membrane, and its equatorial plane is initially tilted out of the plane of shear. We find, at low shear rates, the well-accepted tumbling motion is not always stable for both oblate and prolate capsules. For an oblate capsule, the dominant stable modes for increasing dimensionless shear rate are as follows: rolling with the equatorial plane staying in the plane of shear, precessing following Jeffery's orbit [Proc. R. Soc. London A 102, 161 (1922)], and tumbling. Interestingly, the order of modes is reversed for a prolate capsule: tumbling, precessing, and rolling with increasing dimensionless shear rate. At transitional regimes, we find the stable motion of a capsule can depend on its initial titled angle, even at the same shear rate. At high dimensionless shear rates, a spheroidal capsule undergoes a complicated oscillating-swinging motion: Its major axis oscillates about the plane of shear in addition to the swinging about a mean angle with flow direction found previously, and the amplitudes of both oscillations decrease when increasing the dimensionless shear rate towards a steady tank treading motion asymptotically. We summarize the results in phase diagrams and discuss the reorientation of both oblate and prolate capsules in a wide range of dimensionless shear rates.

Computational biorheology of human blood flow in health and disease

Hematologic disorders arising from infectious diseases, hereditary factors and environmental influences can lead to, and can be influenced by, significant changes in the shape, mechanical and physical properties of red blood cells (RBCs), and the biorheology of blood flow. Hence, modeling of hematologic disorders should take into account the multiphase nature of blood flow, especially in arterioles and capillaries. We present here an overview of a general computational framework based on dissipative particle dynamics (DPD) which has broad applicability in cell biophysics with implications for diagnostics, therapeutics and drug efficacy assessments for a wide variety of human diseases. This computational approach, validated by independent experimental results, is capable of modeling the biorheology of whole blood and its individual components during blood flow so as to investigate cell mechanistic processes in health and disease. DPD is a Lagrangian method that can be derived from systematic coarse-graining of molecular dynamics but can scale efficiently up to arterioles and can also be used to model RBCs down to the spectrin level. We start from experimental measurements of a single RBC to extract the relevant biophysical parameters, using single-cell measurements involving such methods as optical tweezers, atomic force microscopy and micropipette aspiration, and cell-population experiments involving microfluidic devices. We then use these validated RBC models to predict the biorheological behavior of whole blood in healthy or pathological states, and compare the simulations with experimental results involving apparent viscosity and other relevant parameters. While the approach discussed here is sufficiently general to address a broad spectrum of hematologic disorders including certain types of cancer, this paper specifically deals with results obtained using this computational framework for blood flow in malaria and sickle cell anemia.

Elastic behavior of a red blood cell with the membrane's nonuniform natural state: equilibrium shape, motion transition under shear flow, and elongation during tank-treading motion

Direct numerical simulations of the mechanics of a single red blood cell (RBC) were performed by considering the nonuniform natural state of the elastic membrane. A RBC was modeled as an incompressible viscous fluid encapsulated by an elastic membrane. The in-plane shear and area dilatation deformations of the membrane were modeled by Skalak constitutive equation, while out-of-plane bending deformation was formulated by the spring model. The natural state of the membrane with respect to in-plane shear deformation was modeled as a sphere ([Formula: see text]), biconcave disk shape ([Formula: see text]) and their intermediate shapes ([Formula: see text]) with the nonuniformity parameter [Formula: see text], while the natural state with respect to out-of-plane bending deformation was modeled as a flat plane. According to the numerical simulations, at an experimentally measured in-plane shear modulus of [Formula: see text] and an out-of-plane bending rigidity of [Formula: see text] of the cell membrane, the following results were obtained. (i) The RBC shape at equilibrium was biconcave discoid for [Formula: see text] and cupped otherwise; (ii) the experimentally measured fluid shear stress at the transition between tumbling and tank-treading motions under shear flow was reproduced for [Formula: see text]; (iii) the elongation deformation of the RBC during tank-treading motion from the simulation was consistent with that from in vitro experiments, irrespective of the [Formula: see text] value. Based on our RBC modeling, the three phenomena (i), (ii), and (iii) were mechanically consistent for [Formula: see text]. The condition [Formula: see text] precludes a biconcave discoid shape at equilibrium (i); however, it gives appropriate fluid shear stress at the motion transition under shear flow (ii), suggesting that a combined effect of [Formula: see text] and the natural state with respect to out-of-plane bending deformation is necessary for understanding details of the RBC mechanics at equilibrium. Our numerical results demonstrate that moderate nonuniformity in a membrane's natural state with respect to in-plane shear deformation plays a key role in RBC mechanics.

Erythrocyte hemodynamics in stenotic microvessels: a numerical investigation

This paper presents a two-dimensional numerical investigation of deformation and motion of erythrocytes in stenotic microvessels using the immersed boundary-fictitious domain method. The erythrocytes were modeled as biconcave-shaped closed membranes filled with cytoplasm. We studied the biophysical characteristics of human erythrocytes traversing constricted microchannels with the narrowest cross-sectional diameter as small as 3 μm. The effects of essential parameters, namely, stenosis severity, shape of the erythrocytes, and erythrocyte membrane stiffness, were simulated and analyzed in this study. Moreover, simulations were performed to discuss conditions associated with the shape transitions of the cells along with the relative effects of radial position and initial orientation of erythrocytes, membrane stiffness, and plasma environments. The simulation results were compared with existing experiment findings whenever possible, and the physical insights obtained were discussed. The proposed model successfully simulated rheological behaviors of erythrocytes in microscale flow and thus is applicable to a large class of problems involving fluid flow with complex geometry and fluid-cell interactions. Our study would be helpful for further understanding of pathology of malaria and some other blood disorders.

Dynamic simulation and metabolome analysis of long-term erythrocyte storage in adenine-guanosine solution

Although intraerythrocytic ATP and 2,3-bisphophoglycerate (2,3-BPG) are known as direct indicators of the viability of preserved red blood cells and the efficiency of post-transfusion oxygen delivery, no current blood storage method in practical use has succeeded in maintaining both these metabolites at high levels for long periods. In this study, we constructed a mathematical kinetic model of comprehensive metabolism in red blood cells stored in a recently developed blood storage solution containing adenine and guanosine, which can maintain both ATP and 2,3-BPG. The predicted dynamics of metabolic intermediates in glycolysis, the pentose phosphate pathway, and purine salvage pathway were consistent with time-series metabolome data measured with capillary electrophoresis time-of-flight mass spectrometry over 5 weeks of storage. From the analysis of the simulation model, the metabolic roles and fates of the 2 major additives were illustrated: (1) adenine could enlarge the adenylate pool, which maintains constant ATP levels throughout the storage period and leads to production of metabolic waste, including hypoxanthine; (2) adenine also induces the consumption of ribose phosphates, which results in 2,3-BPG reduction, while (3) guanosine is converted to ribose phosphates, which can boost the activity of upper glycolysis and result in the efficient production of ATP and 2,3-BPG. This is the first attempt to clarify the underlying metabolic mechanism for maintaining levels of both ATP and 2,3-BPG in stored red blood cells with in silico analysis, as well as to analyze the trade-off and the interlock phenomena between the benefits and possible side effects of the storage-solution additives.

Lipid bilayer and cytoskeletal interactions in a red blood cell

We study the biomechanical interactions between the lipid bilayer and the cytoskeleton in a red blood cell (RBC) by developing a general framework for mesoscopic simulations. We treated the lipid bilayer and the cytoskeleton as two distinct components and developed a unique whole-cell model of the RBC, using dissipative particle dynamics (DPD). The model is validated by comparing the predicted results with measurements from four different and independent experiments. First, we simulated the micropipette aspiration and quantified the cytoskeletal deformation. Second, we studied the membrane fluctuations of healthy RBCs and RBCs parasitized to different intraerythrocytic stages by the malaria-inducing parasite Plasmodium falciparum. Third, we subjected the RBC to shear flow and investigated the dependence of its tank-treading frequency on shear rate. Finally, we simulated the bilayer-cytoskeletal detachment in channel flow to quantify the strength of such interactions when the corresponding bonds break. Taken together, these experiments and corresponding systematic DPD simulations probe the governing constitutive response of the cytoskeleton, elastic stiffness, viscous friction, and strength of bilayer-cytoskeletal interactions as well as membrane viscosities. Hence, the DPD simulations and comparisons with available independent experiments serve as validation of the unique two-component model and lead to useful insights into the biomechanical interactions between the lipid bilayer and the cytoskeleton of the RBC. Furthermore, they provide a basis for further studies to probe cell mechanistic processes in health and disease in a manner that guides the design and interpretation of experiments and to develop simulations of phenomena that cannot be studied systematically by experiments alone.

Probing vasoocclusion phenomena in sickle cell anemia via mesoscopic simulations

Vasoocclusion crisis is a key hallmark of sickle cell anemia. Although early studies suggest that this crisis is caused by blockage of a single elongated cell, recent experiments have revealed that vasoocclusion is a complex process triggered by adhesive interactions among different cell groups in multiple stages. However, the quantification of the biophysical characteristics of sickle cell anemia remains an open issue. Based on dissipative particle dynamics, we develop a multiscale model for the sickle red blood cells (SS-RBCs), accounting for diversity in both shapes and cell rigidities, to investigate the precise mechanism of vasoocclusion. First, we investigate the adhesive dynamics of a single SS-RBC in shear flow and static conditions, and find that the different cell groups (SS2: young-deformable SS-RBCs, ISCs: rigid-irreversible SS-RBCs) exhibit heterogeneous adhesive behavior due to the diverse cell morphologies and membrane rigidities. We quantify the observed adhesion behavior (in static conditions) in terms of a balance of free energies due to cell adhesion and deformation, and propose a power law that relates the free-energy increase as a function of the contact area. We further simulate postcapillary flow of SS-RBC suspensions with different cell fractions. The more adhesive SS2 cells interact with the vascular endothelium and trap ISC cells, resulting in vasoocclusion in vessels less than 12-14 μm depending on the hematocrit. Under inflammation, adherent leukocytes may also trap ISC cells, resulting in vasoocclusion in even larger vessels.

Effect of deformability difference between two erythrocytes on their aggregation

In this study, we investigated the rheology of a doublet that is an aggregate of two red blood cells (RBCs). According to previous studies, most aggregates in blood flow consist of RBC doublet-pairs and thus the understanding of doublet dynamics has scientific importance in describing its hemodynamics. The RBC aggregation tendency can be significantly affected by the cell's deformability which can vary under both physiological and pathological conditions. Hence, we conducted a two-dimensional simulation of doublet dynamics under a simple shear flow condition with different deformability between RBCs. To study the dissociation process of the doublet, we employed the aggregation model described by the Morse-type potential function, which is based on the depletion theory. In addition, we developed a new method of updating fluid property to consider viscosity difference between RBC cytoplasm and plasma. Our results showed that deformability difference between the two RBCs could significantly reduce their aggregating tendency under a shear condition of 50 s(-1), resulting in disaggregation. Since even under physiological conditions, the cell deformability may be significantly different, consideration of the difference in deformability amongst RBCs in blood flow would be needed for the hemodynamic studies based on a numerical approach.

Simulation of platelets suspension flowing through a stenosis model using a dissipative particle dynamics approach

Stresses on blood cellular constituents induced by blood flow can be represented by a continuum approach down to the μm level; however, the molecular mechanisms of thrombosis and platelet activation and aggregation are on the order of nm. The coupling of the disparate length and time scales between molecular and macroscopic transport phenomena represents a major computational challenge. In order to bridge the gap between macroscopic flow scales and the cellular scales with the goal of depicting and predicting flow induced thrombogenicity, multi-scale approaches based on particle methods are better suited. We present a top-scale model to describe bulk flow of platelet suspensions: we employ dissipative particle dynamics to model viscous flow dynamics and present a novel and general no-slip boundary condition that allows the description of three-dimensional viscous flows through complex geometries. Dissipative phenomena associated with boundary layers and recirculation zones are observed and favorably compared to benchmark viscous flow solutions (Poiseuille and Couette flows). Platelets in suspension, modeled as coarse-grained finite-sized ensembles of bound particles constituting an enclosed deformable membrane with flat ellipsoid shape, show self-orbiting motions in shear flows consistent with Jeffery's orbits, and are transported with the flow, flipping and colliding with the walls and interacting with other platelets.

Multiscale modeling of blood flow: from single cells to blood rheology

Mesoscale simulations of blood flow, where the red blood cells are described as deformable closed shells with a membrane characterized by bending rigidity and stretching elasticity, have made much progress in recent years to predict the flow behavior of blood cells and other components in various flows. To numerically investigate blood flow and blood-related processes in complex geometries, a highly efficient simulation technique for the plasma and solutes is essential. In this review, we focus on the behavior of single and several cells in shear and microcapillary flows, the shear-thinning behavior of blood and its relation to the blood cell structure and interactions, margination of white blood cells and platelets, and modeling hematologic diseases and disorders. Comparisons of the simulation predictions with existing experimental results are made whenever possible, and generally very satisfactory agreement is obtained.

Blood flow in small tubes: quantifying the transition to the non-continuum regime

In small vessels blood is usually treated as a Newtonian fluid down to diameters of ~200 μm. We investigate the flow of red blood cell (RBC) suspensions driven through small tubes (diameters 10-150 μm) in the range marking the transition from arterioles and venules to the largest capillary vessels. The results of the simulations combined with previous simulations of uniform shear flow and experimental data show that for diameters less than ~100 μm the suspension's stress cannot be described as a continuum, even a heterogeneous one. We employ the dissipative particle dynamics (DPD) model, which has been successfully used to predict human blood bulk viscosity in homogeneous shear flow. In tube flow the cross-stream stress gradient induces an inhomogeneous distribution of RBCs featuring a centreline cell density peak, and a cell-free layer (CFL) next to the wall. For a neutrally buoyant suspension the imposed linear shear-stress distribution together with the differentiable velocity distribution allow the calculation of the local viscosity across the tube section. The viscosity across the section as a function of the strain rate is found to be essentially independent of tube size for the larger diameters and is determined by the local haematocrit (H) and shear rate. Other RBC properties such as asphericity, deformation, and cell-flow orientation exhibit similar dependence for the larger tube diameters. As the tube size decreases below ~100 μm in diameter, the viscosity in the central region departs from the large-tube similarity function of the shear rate, since H increases significantly towards the centreline. The dependence of shear stress on tube size, in addition to the expected local shear rate and local haematocrit, implies that blood flow in small tubes cannot be described as a heterogeneous continuum. Based on the analysis of the DPD simulations and on available experimental results, we propose a simple velocity-slip model that can be used in conjunction with continuum-based simulations.

A review of numerical methods for red blood cell flow simulation

In this review, we provide an overview of the simulation techniques employed for modelling the flow of red blood cells (RBCs) in blood plasma. The scope of this review omits the fluid modelling aspect while focusing on other key components in the RBC-plasma model such as (1) describing the RBC deformation with shell-based and spring-based RBC models, (2) constitutive models for RBC aggregation based on bridging theory and depletion theory and (3) additional strategies required for completing the RBC-plasma flow model. These include topics such as modelling fluid-structure interaction with the immersed boundary method and boundary integral method, and updating the variations in multiphase fluid property through the employment of index field methods. Lastly, we summarily discuss the current state and aims of RBC modelling and suggest some research directions for the further development of this field of modelling.

Effect of chain chirality on the self-assembly of sickle hemoglobin

We present simulation results on the self-assembly behavior of sickle hemoglobin (HbS). A coarse-grained HbS model, which contains hydrophilic and hydrophobic particles explicitly, is constructed to match the structural properties and physical description of HbS. The hydrophobic interactions are shown to be necessary with chirality being the main driver for the formation of HbS fibers. In the absence of chain chirality, only small self-assembled aggregates are observed whereas self-assembled elongated steplike bundle microstructures appear when we include chain chirality. We also investigate the effect of confinement on self-assembly, and find that elongated fibers-similar to open-space ones-can be obtained in hard confinement domains but cannot be formed within compliant red blood cell (RBC) domains under the same assumptions. We show, however, that by placing explicitly HbS fibers inside the RBCs and subjecting them to linear elongation and bending, we obtain different types of sickle-shaped RBCs as observed in sickle cell anemia.

Nanocarrier Hydrodynamics and Binding in Targeted Drug Delivery: Challenges in Numerical Modeling and Experimental Validation

This review discusses current progress and future challenges in the numerical modeling of targeted drug delivery using functionalized nanocarriers (NC). Antibody coated nanocarriers of various size and shapes, also called functionalized nanocarriers, are designed to be injected in the vasculature, whereby they undergo translational and rotational motion governed by hydrodynamic interaction with blood particulates as well as adhesive interactions mediated by the surface antibody binding to target antigens/receptors on cell surfaces. We review current multiscale modeling approaches rooted in computational fluid dynamics and nonequilibrium statistical mechanics to accurately resolve fluid, thermal, as well as adhesive interactions governing nanocarrier motion and their binding to endothelial cells lining the vasculature. We also outline current challenges and unresolved issues surrounding the modeling methods. Experimental approaches in pharmacology and bioengineering are discussed briefly from the perspective of model validation.

Continuum- and particle-based modeling of shapes and dynamics of red blood cells in health and disease

We review recent advances in multiscale modeling of the mechanics of healthy and diseased red blood cells (RBCs), and blood flow in the microcirculation. We cover the traditional continuum-based methods but also particle-based methods used to model both the RBCs and the blood plasma. We highlight examples of successful simulations of blood flow including malaria and sickle cell anemia.

Spring-network-based model of a red blood cell for simulating mesoscopic blood flow

We developed a mechanical model of a red blood cell (RBC) that is capable of expressing its characteristic behaviors in shear flows. The RBC was modeled as a closed shell membrane consisting of spring networks in the framework of the energy minimum concept. The fluid forces acting on RBCs were modeled from Newton's viscosity law and the conservation of momentum. In a steady shear flow, the RBC model exhibited various behaviors, depending on the shear rate; it tumbled, tank-treaded, or both. The transition from tumbling to tank-treading occurred at a shear rate of 20 s( - 1). The simulation of an RBC in steady and unsteady parallel shear flows (Couette flows) showed that the deformation parameters of the RBC were consistent with experimental results. The RBC in Poiseuille flow migrated radially towards the central axis of the flow channel. Axial migration became faster with an increase in the viscosity of the media, qualitatively consistent with experimental results. These results demonstrate that the proposed model satisfies the essential conditions for simulating RBC behavior in blood flow. Finally, a large-scale RBC flow simulation was implemented to show the capability of the proposed model for analyzing the mesoscopic nature of blood flow.

Simulation of malaria-infected red blood cells in microfluidic channels: Passage and blockage

Malaria-infected red blood cells (iRBCs) become less deformable with the progression of infection and tend to occlude microcapillaries. This process has been investigated in vitro using microfluidic channels. The objective of this paper is to provide a quantitative basis for interpreting the experimental observations of iRBC occlusion of microfluidic channels. Using a particle-based model for the iRBC, we simulate the traverse of iRBCs through a converging microfluidic channel and explore the progressive loss of cell deformability due to three factors: the stiffening of the membrane, the reduction of the cell's surface-volume ratio, and the growing solid parasites inside the cell. When examined individually, each factor tends to hinder the passage of the iRBC and lengthen the transit time. Moreover, at sufficient magnitude, each may lead to obstruction of narrow microfluidic channels. We then integrate the three factors into a series of simulations that mimic the development of malaria infection through the ring, trophozoite, and schizont stages. These simulations successfully reproduce the experimental observation that with progression of infection, the iRBC transitions from passage to blockage in larger and larger channels. The numerical results suggest a scheme for quantifying iRBC rigidification through microfluidic measurements of the critical pressure required for passage.

How malaria parasites reduce the deformability of infected red blood cells

The pathogenesis of malaria is largely due to stiffening of the infected red blood cells (RBCs). Contemporary understanding ascribes the loss of RBC deformability to a 10-fold increase in membrane stiffness caused by extra cross-linking in the spectrin network. Local measurements by micropipette aspiration, however, have reported only an increase of ∼3-fold in the shear modulus. We believe the discrepancy stems from the rigid parasite particles inside infected cells, and have carried out numerical simulations to demonstrate this mechanism. The cell membrane is represented by a set of discrete particles connected by linearly elastic springs. The cytosol is modeled as a homogeneous Newtonian fluid, and discretized by particles as in standard smoothed particle hydrodynamics. The malaria parasite is modeled as an aggregate of particles constrained to rigid-body motion. We simulate RBC stretching tests by optical tweezers in three dimensions. The results demonstrate that the presence of a sizeable parasite greatly reduces the ability of RBCs to deform under stretching. With the solid inclusion, the observed loss of deformability can be predicted quantitatively using the local membrane elasticity measured by micropipettes.

Simulation and modelling of slip flow over surfaces grafted with polymer brushes and glycocalyx fibres

Fabrication of functionalized surfaces using polymer brushes is a relatively simple process and parallels the presence of glycocalyx filaments coating the luminal surface of our vasculature. In this paper, we perform atomistic-like simulations based on dissipative particle dynamics (DPD) to study both polymer brushes and glycocalyx filaments subject to shear flow, and we apply mean-field theory to extract useful scaling arguments on their response. For polymer brushes, a weak shear flow has no effect on the brush density profile or its height, while the slip length is independent of the shear rate and is of the order of the brush mesh size as a result of screening by hydrodynamic interactions. However, for strong shear flow, the polymer brush is penetrated deeper and is deformed, with a corresponding decrease of the brush height and an increase of the slip length. The transition from the weak to the strong shear regime can be described by a simple 'blob' argument, leading to the scaling γ̇₀ ∝ σ^3/2, where γ̇₀ is the critical transition shear rate and σ is the grafting density. Furthermore, in the strong shear regime, we observe a cyclic dynamic motion of individual polymers, causing a reversal in the direction of surface flow. To study the glycocalyx layer, we first assume a homogeneous flow that ignores the discrete effects of blood cells, and we simulate microchannel flows at different flow rates. Surprisingly, we find that, at low Reynolds number, the slip length decreases with the mean flow velocity, unlike the behaviour of polymer brushes, for which the slip length remains constant under similar conditions. (The slip length and brush height are measured with respect to polymer mesh size and polymer contour length, respectively.) We also performed additional DPD simulations of blood flow in a tube with walls having a glycocalyx layer and with the deformable red blood cells modelled accurately at the spectrin level. In this case, a plasma cell-free layer is formed, with thickness more than three times the glycocalyx layer. We then find our scaling arguments based on the homogeneous flow assumption to be valid for this physiologically correct case as well. Taken together, our findings point to the opposing roles of conformational entropy and bending rigidity - dominant effects for the brush and glycocalyx, respectively - which, in turn, lead to different flow characteristics, despite the apparent similarity of the two systems.

Role of red blood cell elastic properties in capillary occlusions

The shape transformations of red blood cells stuck in capillary narrowings with radii close to the critical radius where the maximum deformations occur are analyzed. The membrane skeleton deformations are studied within the effective network model and the continuum elastic model, whereas the area-difference elasticity model is applied to describe the phospholipid bilayer. A minimization of the total free energy is performed to determine the cell shapes in a stopped flow, which are calculated by a triangulated representation of the membrane surface. The shapes are asymmetric, characterized by a single invagination, which decreases with decreasing radii of the narrowing and vanishes at its critical radius. The largest stretching deformations of the skeleton are at the ends of the elongated shape, and remarkable shear deformations appear around the invagination. The membrane's mechanical energy increases with the decreasing radius of the narrowing, predominantly due to the deformation of membrane skeleton. The increase in the shear energy is significantly larger than any other energy contribution within both models. The pressure differences needed for the penetration into the narrowing are strongly coupled with the membrane's mechanical energy. Their values were found to be of the order of 10 Pa. Both models correspond well.

Forces on a wall-bound leukocyte in a small vessel due to red cells in the blood stream

As part of the inflammation response, white blood cells (leukocytes) are well known to bind nearly statically to the vessel walls, where they must resist the force exerted by the flowing blood. This force is particularly difficult to estimate due to the particulate character of blood, especially in small vessels where the red blood cells must substantially deform to pass an adhered leukocyte. An efficient simulation tool with realistically flexible red blood cells is used to estimate these forces. At these length scales, it is found that the red cells significantly augment the streamwise forces that must be resisted by the binding. However, interactions with the red cells are also found to cause an average wall-directed force, which can be anticipated to enhance binding. These forces increase significantly as hematocrit values approach 25% and decrease significantly as the leukocyte is made flatter on the wall. For a tube hematocrit of 25% and a spherical protrusion with a diameter three-quarters that of the vessel, the average forces are increased by ~40% and the local forces are more than double those estimated with an effective-viscosity-homogenized blood. Both the enhanced streamwise and wall-ward forces and their unsteady character are potentially important in regard to binding mechanisms.

Determination of critical parameters in platelet margination

An investigation of margination dependence on hematocrit, platelet shape, and viscosity ratio of plasma to cytoplasm is presented. Whole blood is modeled as a suspension of deformable red blood cells (RBCs) and rigid platelets in a viscous liquid. The fluid phase is simulated using the lattice-Boltzmann method, the RBC membranes are modeled with a coarse-grained spectrin-link method, and the dynamics of rigid particles are updated using Newton's equations of motion for axisymmetric shapes. The results emphasize that an increase in hematocrit increases the rate of margination. The viscosity ratio between the interior cytoplasm and suspending fluid can considerably alter the rate of margination. The aspect ratio of surrogate platelet particles influences the rate of margination as well. Spherical particles tend to migrate more quickly than disks. Highly viscous or rigid RBCs slow down margination.

Quantifying the rheological and hemodynamic characteristics of sickle cell anemia

Sickle erythrocytes exhibit abnormal morphology and membrane mechanics under deoxygenated conditions due to the polymerization of hemoglobin S. We employed dissipative particle dynamics to extend a validated multiscale model of red blood cells (RBCs) to represent different sickle cell morphologies based on a simulated annealing procedure and experimental observations. We quantified cell distortion using asphericity and elliptical shape factors, and the results were consistent with a medical image analysis. We then studied the rheology and dynamics of sickle RBC suspensions under constant shear and in a tube. In shear flow, the transition from shear-thinning to shear-independent flow revealed a profound effect of cell membrane stiffening during deoxygenation, with granular RBC shapes leading to the greatest viscosity. In tube flow, the increase of flow resistance by granular RBCs was also greater than the resistance of blood flow with sickle-shape RBCs. However, no occlusion was observed in a straight tube under any conditions unless an adhesive dynamics model was explicitly incorporated into simulations that partially trapped sickle RBCs, which led to full occlusion in some cases.

Predicting the morphology of sickle red blood cells using coarse-grained models of intracellular aligned hemoglobin polymers

Sickle red blood cells (SS-RBCs) exhibit heterogeneous cell morphologies (sickle, holly leaf, granular, etc.) in the deoxygenated state due to the polymerization of the sickle hemoglobin. Experimental evidence points to a close relationship between SS-RBC morphology and intracellular aligned hemoglobin polymers. Here, we develop a coarse-grained (CG) stochastic model to represent the growth of the intracellular aligned hemoglobin polymer domain. The CG model is calibrated based on the mechanical properties (Young's modulus, bending rigidity) of the sickle hemoglobin fibers reported in experiments. The process of the cell membrane transition is simulated for physiologic aligned hemoglobin polymer configurations and mean corpuscular hemoglobin concentration. Typical SS-RBC morphologies observed in experiments can be obtained from the current model as a result of the intracellular aligned hemoglobin polymer development without introducing any further ad hoc assumptions. It is found that the final shape of SS-RBCs is primarily determined by the angular width of the aligned hemoglobin polymer domain, but it also depends, to a lesser degree, on the polymer growth rate and the cell membrane rigidity. Cell morphologies are quantified by structural shape factors, which agree well with experimental results from medical images.