Kinetics of rouleau formation. II. Reversible reactions

Cellular Blood Flow Modeling with HemoCell

Many of the intriguing properties of blood originate from its cellular nature. Bulk effects, such as viscosity, depend on the local shear rates and on the size of the vessels. While empirical descriptions of bulk rheology are available for decades, their validity is limited to the experimental conditions they were observed under. These are typically artificial scenarios (e.g., perfectly straight glass tube or in pure shear with no gradients). Such conditions make experimental measurements simpler; however, they do not exist in real systems (i.e., in a real human circulatory system). Therefore, as we strive to increase our understanding on the cardiovascular system and improve the accuracy of our computational predictions, we need to incorporate a more comprehensive description of the cellular nature of blood. This, however, presents several computational challenges that can only be addressed by high performance computing. In this chapter, we describe HemoCell ( https://www.hemocell.eu ), an open-source high-performance cellular blood flow simulation, which implements validated mechanical models for red blood cells and is capable of reproducing the emergent transport characteristics of such a complex cellular system. We discuss the accuracy and the range of validity, and demonstrate applications on a series of human diseases.

Anisotropic short-range attractions precisely model branched erythrocyte aggregates

Homogeneous suspensions of red blood cells (RBCs or erythrocytes) in blood plasma are unstable in the absence of driving forces and form elongated stacks, called rouleaux. These erythrocyte aggregates are often branched porous networks - a feature that existing red blood cell aggregation models and simulations fail to predict exactly. Here we establish that alignment-dependent attractive forces in a system of dimers can precisely generate branched structures similar to RBC aggregates observed under a microscope. Our simulations consistently predict that the growth rate of typical mean rouleau size remains sub-linear - a hallmark from past studies - which we also confirm by deriving a reaction kernel taking into account appropriate collision cross-section, approach velocities, and an area-dependent sticking probability. The system exhibits unique features such as the existence of percolated and/or single giant cluster states, multiple coexisting mass-size scalings, and transition to a branched phase upon fine-tuning of model parameters. Upon decreasing the depletion thickness we find that the percolation threshold increases but the morphology of the structures opens up towards an increased degree of branching. Remarkably the system self-organizes to produce a universal power-law size distribution scaling irrespective of the model parameters.

Supercluster states and phase transitions in aggregation-fragmentation processes

We study the evolution of aggregates triggered by collisions with monomers that either lead to the attachment of monomers or the break-up of aggregates into constituting monomers. Depending on parameters quantifying addition and break-up rates, the system falls into a jammed or a steady state. Supercluster states (SCSs) are very peculiar nonextensive jammed states that also arise in some models. Fluctuations underlie the formation of the SCSs. Conventional tools, such as the van Kampen expansion, apply to small fluctuations. We go beyond the van Kampen expansion and determine a set of critical exponents quantifying SCSs. We observe continuous and discontinuous phase transitions between the states. Our theoretical predictions are in good agreement with numerical results.

Relationship between red blood cell aggregation and dextran molecular mass

The aim of this study was to investigate the aggregation of red blood cells (RBCs) suspended in dextran solution at various levels of molecular mass. Dextran solutions at molecular mass 40, 70, 100 and 500 kDa at concentration from 2 to 5 g/dL were used to suspend the RBCs. The radius and velocity of sedimenting RBC aggregates were investigated using image analysis. The radius and sedimentation velocity of aggregates increased initially, then decreased after achieving maxima. The maximal velocity of RBC aggregates showed a bell-shaped dependence on dextran molecular mass and concentration, whereas maximal radius showed monotonic increase with both factors. Difference between aggregate and solution density was estimated using aggregate radius and sedimentation velocity and dextran solution viscosity, and was consistent across most molecular mass and concentration levels. This allowed to calculate the porosity of aggregates and to show that it monotonically decreased with the increase in the solution density, caused by the increase in the dextran concentration. The results provide insight into the RBC aggregation process in solutions of proteins of different size, reflecting various pathological conditions. The currently reported data can be potentially applied to specific pathophysiological conditions giving an interpretation that is not yet fully discussed in the literature.

A Deadly Embrace: Hemagglutination Mediated by SARS-CoV-2 Spike Protein at Its 22 N-Glycosylation Sites, Red Blood Cell Surface Sialoglycoproteins, and Antibody

Rouleaux (stacked clumps) of red blood cells (RBCs) observed in the blood of COVID-19 patients in three studies call attention to the properties of several enveloped virus strains dating back to seminal findings of the 1940s. For COVID-19, key such properties are: (1) SARS-CoV-2 binds to RBCs in vitro and also in the blood of COVID-19 patients; (2) although ACE2 is its target for viral fusion and replication, SARS-CoV-2 initially attaches to sialic acid (SA) terminal moieties on host cell membranes via glycans on its spike protein; (3) certain enveloped viruses express hemagglutinin esterase (HE), an enzyme that releases these glycan-mediated bindings to host cells, which is expressed among betacoronaviruses in the common cold strains but not the virulent strains, SARS-CoV, SARS-CoV-2 and MERS. The arrangement and chemical composition of the glycans at the 22 N-glycosylation sites of SARS-CoV-2 spike protein and those at the sialoglycoprotein coating of RBCs allow exploration of specifics as to how virally induced RBC clumping may form. The in vitro and clinical testing of these possibilities can be sharpened by the incorporation of an existing anti-COVID-19 therapeutic that has been found in silico to competitively bind to multiple glycans on SARS-CoV-2 spike protein.

A constitutive hemorheological model addressing the deformability of red blood cells in Ringer solutions

Red blood cells (RBCs) can deform substantially, a feature that allows them to pass through capillaries that are narrower than the largest dimension of an undeformed RBC. Clearly, to understand how they transport through our microcirculation, we need a constitutive model able of accurately predicting the deformability of RBCs, which seems currently unavailable. To address this void, we herein propose a new model that accounts for the deformability of RBCs by modeling them as deformed droplets with a constant volume. To make sure the model is by construction thermodynamically admissible we employ non-equilibrium thermodynamics as our tool. Since RBCs are merely droplets with the inner fluid exhibiting a higher viscosity than that of the outer one, RBCs are described by a conformation tensor constrained to have a constant determinant (volume). The model predicts the second normal stress coefficient in steady-state simple shear flow to first shear thicken and then shear thin, which is an unexpected behavior; however, we cannot judge whether such a prediction is aphysical or not due to unavailable experimental rheological data in the literature. We show that the new model is capable of addressing the deformability of isolated (very low hematocrit) RBCs in simple shear and the shear viscosity of non-aggregating blood. As derived the model addresses only non-aggregating blood, but can very easily be generalized to account for aggregating blood.

Aggregation of red blood cells in suspension: study by light-scattering technique at small angles

Red blood cells (RBCs) in the presence of plasma proteins or other macromolecules have a tendency to form aggregates. Light-scattering technique was used to investigate the RBC aggregation process. A highly diluted suspension of RBCs was illuminated with a 632.8-nm HeNe laser. Angular-resolved measurements of light intensity scattered by an RBC suspension from a 200-microm thick optical glass cuvette during 10 min of their aggregation process were performed at 1 to 4 off-axis deg with a very high angular resolution, at hematocrits in the range of 3.5 x 10(-2) to 10(-1). The angular spreading of forward-scattered light at small angles during the RBC aggregation process was described in terms of a new, effective phase function model that has been used for fitting the experimental data. The aggregated RBCs' optical properties, such as effective scattering anisotropy and scattering cross section, were determined. The results were compared with prediction of Mie theory for equivolumetric spherical particles. The time dependence of the aggregates mean radius and of the mean number of cells per aggregate was also calculated. Last, the potential of the proposed technique (forward-scattering light technique) as a new quantitative investigation of cellular aggregation process was estimated.

Plasma preparation from whole blood using ultrasound

A technique to efficiently separate plasma from human whole blood is described. Essentially, 3-mL samples are held on the axis of a tubular transducer and exposed for 5.7 min to an ultrasonic standing wave. The cells concentrate into clumps at radial separations of half wavelength. The clumps grow in size and sediment under gravity. A distinct plasma/cell interface forms as the cells sediment. The volume of clarified plasma increases with time. The separation efficiencies of transducers of 29-mm and 23-mm internal diameters driven, by test equipment, at radial resonances of 3.4 and 1.5 MHz, respectively, were compared. The average efficiency of separation was 99.6% at 1.5 MHz and 99.4% with the 3.4-MHz system. The cleared plasma constituted 30% of the sample volume at 1.5 MHz and 25% at 3. 4 MHz. There was no measurable release of haemoglobin or potassium into the suspending phase, indicating that there was no mechanical damage to cells at either frequency. A total of 114 samples from volunteers and patients were subsequently clarified in a 1.5-MHz system driven by an integrated generator. The average efficiency of clarification of blood was 99.76% for the latter samples. The clarification achieved is a significant improvement on that previously reported (98.5%) for whole blood exposed to a planar ultrasonic standing wave field (Peterson et al. 1986). We have, therefore, now achieved a six-fold reduction of cells in plasma compared to previous results.

On the relation between aggregation, packing and the backscattered ultrasound signal for whole blood

Previous studies have shown that the backscattered ultrasound (US) power from blood depends on the manner in which red blood cells (RBCs) are packed and, in particular, on spatial variations in the red blood cell number density (i.e., the RBC concentration variance). Experimental measurements have also shown that the backscattered US power depends on the degree of RBC aggregation, and it has been hypothesized that this is primarily due to the effect of RBC aggregation on the concentration variance. An initial simulation study of the relationship between RBC aggregation and packing statistics is presented, in which the effects of hematocrit, aggregate size, shape and size distribution on concentration variance are investigated. Both two-dimensional (2-D) and 3-D samples of aggregated and disaggregated RBCs were simulated; these enabled the concentration variance to be calculated. In agreement with theoretical predictions and experimental US results, the concentration variance for disaggregated RBCs is shown to be lowest at low and high hematocrits, and to peak at intermediate hematocrits. The concentration variance is shown to be particularly sensitive to changes in aggregate size and size distribution, and less sensitive to the shape of small aggregates. The results of this study provide a foundation for relating the state of aggregation in a blood sample to the manner in which RBCs are packed and, therefore, to the backscattered US power.

Power Doppler ultrasound scan imaging of the level of red blood cell aggregation: an in vitro study

PURPOSE

The purpose of this study was to evaluate the effect of the shear rate on red blood cell (RBC) aggregation with power Doppler ultrasound scanning (PDU), pulsed-wave Doppler scanning, and color Doppler flow imaging.

METHODS

Equine and porcine blood were circulated with a steady flow in a phantom with a diameter of 9.52 mm. The color Doppler flow imaging mode was used to estimate the velocity profile and the shear rate across the tube. A transfer function that related the Doppler scan power, measured in gray level with the PDU method, to the power, measured in decibels with the pulsed-wave Doppler scan technique, was used to estimate the echogenicity of blood and the level of aggregation.

RESULTS

For the four experiments reported, the power peaked at low shear rates probably because of increased RBC collisions and aggregation and then decreased thereafter because of disaggregation. The largest power variations were measured at shear rates of less than 40 seconds -1. At flow rates that varied between 75 and 500 mL/min, the echogenicity was low near the wall of the tube, increased toward the middle, and decreased at the tube center. The Doppler scan power was uniform across the tube at flow rates of 750 and 1000 mL/min.

CONCLUSION

PDU is reliable to quantify the echogenicity of blood and the level of RBC aggregation. In comparison with other methods proposed to measure RBC aggregation, ultrasound scanning is applicable in vivo and may help to improve our basic understanding of the relationship between the hemodynamic of the circulation and RBC aggregation in human vessels.

A model for the kinetics of homotypic cellular aggregation under static conditions

We present the formulation and testing of a mathematical model for the kinetics of homotypic cellular aggregation. The model considers cellular aggregation under no-flow conditions as a two-step process. Individual cells and cell aggregates 1) move on the tissue culture surface and 2) collide with other cells (or aggregates). These collisions lead to the formation of intercellular bonds. The aggregation kinetics are described by a system of coupled, nonlinear ordinary differential equations, and the collision frequency kernel is derived by extending Smoluchowski's colloidal flocculation theory to cell migration and aggregation on a two-dimensional surface. Our results indicate that aggregation rates strongly depend upon the motility of cells and cell aggregates, the frequency of cell-cell collisions, and the strength of intercellular bonds. Model predictions agree well with data from homotypic lymphocyte aggregation experiments using Jurkat cells activated by 33B6, an antibody to the beta 1 integrin. Since cell migration speeds and all the other model parameters can be independently measured, the aggregation model provides a quantitative methodology by which we can accurately evaluate the adhesivity and aggregation behavior of cells.

Power Doppler ultrasound evaluation of the shear rate and shear stress dependences of red blood cell aggregation

The use of power Doppler ultrasound at 10 MHz is evaluated as a method to study the shear rate and the shear stress dependences of red blood cell aggregation. This evaluation was based on six in vitro experiments conducted in a 1.27-cm diameter tube under steady flow conditions. Porcine whole blood was circulated in the flow model at flow rates ranging between 125 to 1500 ml/min (mean shear rate across the tube ranging between 6 and 74 s-1). For each flow condition, the variation of the Doppler power across the tube and the velocity profile were measured by moving the Doppler sample volume across the tube diameter. For each radial position, the shear rate within the Doppler sample volume was also determined by considering the radial power pattern of the ultrasound beam. To estimate the shear stress within the Doppler sample volume, the apparent viscosity of blood samples withdrawn from the flow model was measured for each experiment. The variation of the Doppler power as a function of the shear rate within the sample volume showed a rapid reduction of the power between 1 and 5 s-1, a transition region between 5 and 10 s-1, and a very slow reduction beyond 10 s-1. Little variation of the Doppler power was measured for shear stress higher than 2 dyn/cm2. The maximum Doppler power for all flow rates was usually found near the center of the tube. Based on the ultrasonic scattering models, which predict that the Doppler power is related to the volume square of the scatterers, the method described in the present study showed a very high sensitivity to the presence of red blood cell aggregation for shear rates below 10 s-1.

A mathematical model for concentration of blood affecting erythrocyte sedimentation

The rate at which red blood cells fall in vitro is used as a common clinical test for a number of pathological conditions. But this test becomes unreliable when one finds flaws in the model under consideration and these doubts raise the issue of aggregation of the red blood cells in concentration whose mathematical analysis is relatively unknown. However Huang et al. (Biorheology 8, 157-163, 1971) made some efforts and their model resembles certain moving boundary problems. In the present work the modifications to this model have been suggested so far as the concentration of the blood is concerned, it being one of the important factors to decide ESR (Erythrocyte Sedimentation Rate). In the equation for nutrient concentration, to be more realistic, we have taken account of transfer of nutrient to the tissue from the blood. The exact solution has been obtained using Laplace transform. A finite element technique has been suggested which provides results closer to that of the exact solution. Our results for the blood concentration may be useful for conducting the ESR tests. A special case for an emergent patient when nutrient concentration falls down considerably and glucose is provided to improve the condition, has been shown graphically.

Mechanisms of successive modes of erythrocyte stability and instability in the presence of various polymers

Upon examination in real time of the adhesion of human erythrocytes by observing cells suspended by ultrasonic radiation force in solutions of dextran, polylysine, and polyethylene glycol, it was reported earlier that concave-ended cell pairs and rouleaux are seen in low (0.5-2.0% w/v) concentrations of Dextran T500. At concentrations of 5-7%, dextran spherical cell doublets and convex-ended cell agglutinates are formed. When adhesion occurs in polylysine (MW 14,000) or in polyethylene glycol (MW 8,000) only spherical cell doublets or convex-ended cell clumps occur. The final cell movement completing the formation of these adhesion products takes place over time scales of the order of 1s. In this work, quantitative consideration is given to the extent to which repulsion between adhesion-inducing macromolecules associated with the glycocalyx and those free in solution can influence adhesion through a phase separation effect. It is shown for cells in dextran and in polylysine that the forces associated with this repulsion are of the same order of magnitude as the electrostatic interactions between cells.

Interaction of transport phenomena and surface reactions

Phenomena that occur when blood contacts a foreign surface are controlled by transport processes in the bloodstream immediately above the surface and by reactions at the surface itself. All constituents of blood that attach to a surface may leave the surface and be replaced by other constituents. Some of the constituents may attach in superposition. This turnover of constituents at the surface is influenced by the nature of the substrate and the shear stress between the blood and the surface. The turnover process is easy to demonstrate with platelets. When platelets depart from a surface, the site they vacate has a higher attractiveness for adhesion of other platelets than the surrounding surface that has not had a cell attached to it, but this enhanced attractiveness diminishes in a few minutes if a new occupant does not occupy the site. As data accumulate, it is beginning to appear that platelet turnover differs between different blood donors but remains relatively constant for each individual unless dietary or pharmacological changes that affect the turnover rate are introduced.

Real time observations of polylysine, dextran and polyethylene glycol induced mutual adhesion of erythrocytes held in suspension in an ultrasonic standing wave field

A technique which enables cells to be observed in suspension for times of the order of minutes (employing acoustic radiation forces in a 1 MHz ultrasonic standing wave field) is described. Video recordings of the mutual adhesion of human erythrocytes in suspension have been analysed. Concave-ended cell doublets and linear rouleaux developed in 0.5-1.5% w/v Dextran T500 by a gradual (2.5-17 s) increase in the area of cell contact over the cell cross-section. The concave-ended rouleaux form was not seen in polylysine or in polyethylene glycol. In 5-7% dextran and in 20 micrograms/ml polylysine mutual adhesion was a two stage process. Cells first form a strong local contact which persists (without apparently growing in area) for a number of seconds following which the cell surfaces move suddenly to form a spherical doublet. The average initial contact time and engulfment time for cells in 7% Dextran T500 are 18 and 2.7s, respectively. The corresponding values for cells in 20 micrograms/ml, 14 kDa, polylysine are 2.7 and 0.3s. There was no initial contact delay during spherical doublet formation in 1 mg/ml polylysine. Electron microscopy showed that the intercellular seam for spherical doublets formed with all three agglutinating molecules was bent in a wavy lambda approximately equal to 4 micron) profile. The thickness of the intercellular space varied in a spatially periodic way (lambda approximately equal to 0.8 microns) for cells in polylysine. Examples of periodic intercellular spaces were seen by light microscopy in polyethylene glycol induced clumps. The role of interfacial instability in the adhesion processes is discussed.