Effect of intercellular collisions on red blood cell membrane damage

Abstract

Modelling blood flow and particularly cellular damage induced by supra- and non-physiological blood-flow conditions is crucial when developing novel blood-exposed biomedical devices and treatments. Blood is composed of 30–50% red blood cells (RBCs) by volume, yet the mechanisms and effects of intercellular collisions are frequently neglected in red blood cell damage models. These effects are investigated by employing fully coupled 3D fluid structure-interaction simulations to simulate the collision processes in a Couette shear flow and to gauge their effect on the strain experienced by the RBC membrane as well as the transmembrane hemoglobin diffusion rate. Intercellular collisions are found to nearly double the membrane strain at hemolytic shear rates, with declining effect as the shear rate increases, and have a similar effect on sublethal hemoglobin diffusion. Viscoelastic simulations were conducted to examine the effect of incorporating membrane viscosity on the strain experienced by red blood cell membrane during collisions, and find minimal impact of incorporating viscoelasticity at high shear. Incorporating the effect of intercellular collisions is found to be a crucial factor for predicting stress-induced cellular damage under dynamic conditions.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-025-01344-0.

Introduction

The modeling of blood rheology, especially under the supra- and non-physiological dynamic conditions imposed by blood-wetted biomedical devices, poses significant complexity due to the unique physiological and biomechanical properties of red blood cells (RBCs), which comprise 40–50% of blood volume. The unique structure of the RBC membrane, comprising a lipid bilayer reinforced by a spectrin mesh bound by actin, imparts remarkable toughness and deformability. Under non-physiological flow conditions, however, such as those encountered in heart valves and ventricular assist devices, RBCs are susceptible to damage and rupture, known as hemolysis, which poses severe risks to patient safety^1–4. Despite advances in hemodynamic modeling and the corresponding RBC dynamics, a comprehensive understanding of the effects of RBC collisions on membrane stress on RBCs remains elusive, and consequently, this phenomenon is frequently overlooked as a modeling consideration. This paper aims to elucidate these aspects by simulating the RBC collision process.

RBC membrane modelling and hemolysis

The foundation of hemodynamic modeling rests on the complex behavior and dynamics of RBCs. Their anucleate nature, ellipsoid biconcave and biconvex shape, and the inextensibility of the hyperelastic membrane, as detailed by numerous canonical works^5–11, underscore the complexity inherent in accurately modeling blood flow. Advancements in RBC membrane modeling have progressed from simple strain-energy-based hyperelastic formulations to more sophisticated mathematical descriptions incorporating spectrin network stretching and viscoelasticity^12–15, such models remain impractical for fluid-structure-coupled modelling and no model to our knowledge has been made yet that captures RBC dynamics at high deformation and shear.

Hemolysis modeling spans empirical to constitutive approaches, from empirical relationships between hemoglobin release as a function of stress and time^16–21, to integrating complex relationships that track RBC membrane deformation^22–24. Such approaches often either underpredict sublethal hemolysis or overpredict hemolysis due to rapidly fluctuating loads. Constitutive modeling approaches can account for complex and dynamic conditions found inside of devices such as Left Ventricular Assist Devices (LVADs). These consitutive strain-based models oversimplify the RBC shape as an ellipsoid and underpredict loading²⁵. Multiscale approaches that capture complex membrane mechanics under dynamic loading²⁶, neglect the effect of collisions and simplify the effect of turbulence as the application of the Reynolds stress. The effect of RBC collisions or turbulence on membrane damage, however, remains underexamined and is contentious, leading it to be often neglected. To the best of our knowledge, no constitutive model considers the effect of RBC collisions on membrane strain.

Relationship between RBC collisions, hematocrit, and turbulence on hemolysis

There is a strong connection between the effect of turbulence on hemolysis, and the effect of RBC collisions, and an appropriate discussion of these effects requires consideration of both phenomena. For example, the work of Antiga and Steinman²⁷ emphasizes the critical role of intercellular interactions, which intensify shear stress at minuscule scales in densely packed RBC suspensions, as primary drivers of hemolysis under turbulent conditions. Indeed, the impact of turbulence on hemolysis extends the complexity due to the dynamic interaction between RBCs and small-scale turbulent eddies. Such interactions, which have never been adequately examined either experimentally or numerically at high shear, may have a potent effect on hemolysis rates and effective stress experienced by RBC membranes. Antiga and Steinman go on to suggest that RBC collisions and turbulence are intrinsically linked, as the interactions between RBCs become the smallest-scale recipient of any fluctuating small-scale velocity component. This assertion would help explain why experimental in-vivo evidence shows that turbulence in blood has non-Kolmogorov decay characteristics and decays faster than in hydrodynamic flows²⁸. Jones²⁹ and subsequent studies by Quinlan³⁰ and Faghih and Sharp¹⁹, further corroborate this narrative and underscore the insufficiency of the Reynolds stress and Kolmogorov scale in accurately quantifying the effects of turbulence on RBC damage, indicating a connection between intercellular interactions and small scale turbulent fluctuations due to the heavily particle-laden nature of blood. These studies highlight both the insufficiency of our current understanding of turbulence as applied to blood and also highlight the need to examine how cellular interactions affect red blood cell membrane damage. Indeed, although turbulence is often neglected in blood flow under physiological conditions, such flows have been shown to be turbulent³¹, and there does not exist presently a model that accounts for the unique characteristics of turbulence in blood. Furthermore, presently many blood damage models apply laminar stress or Reynolds stress as the input, but the turbulent viscous stress alone, has been found as a more accurate predictor in highly turbulent cases^32,33.

Yet, as intercellular collisions are a driving factor behind turbulence and hemolysis and this is partly due to the high volume fraction of RBCs, it is somewhat perplexing that early experimental studies, including those by Fok and Schubothe³⁴ and Leverett et al.³⁵, suggest a minimal impact of hematocrit variations on hemolysis under constant shear stress in controlled environments like a Couette rheometer. These findings have been contested in the context of complex flow conditions, such as those encountered in LVADs, where there is a strong effect of hematocrit on hemolysis^36,37. Indeed, recent low-shear simulation studies of capsules³⁸ show that hemolysis index increases as a function of hematocrit. Vexingly, low-shear simulations of capsule suspensions show a slight decrease, or increase in deformation index, depending on whether the capsule membrane was purely elastic or viscoelastic³⁹. Takeishi⁴⁰ on the other hand see no effect of changing hematocrit (between 0.11 and 0.41) on deformation parameter for a viscoelastic suspension of RBCs. Furthermore, Rydquist⁴¹ found that RBC deformation parameter decreased in a RBC suspension in a microchannel when transitioning from Re of 180 to Re = 320, indicating increased turbulence decreased RBC deformation. In another study, Gou et al.⁴² compute transmembrane release of ATP in a suspension of RBCs in a microchannel. Notably, although we know that deformation parameter in fact does not indicate membrane strain in complex loading²⁵, none of these studies report membrane strain (except Porcaro et al.³⁸ and Gou et al.⁴² who report membrane dammage indirectly but do not quantify the effect of collisions). Furthermore, with the exception of the study by Gou⁴² and Rydquist⁴¹, who simulate RBC suspensions with shear rates up to Ca < 100, few studies examine the effect at the hemolytic shear rates induced by biomedical devices.

Although the interaction between RBCs in blood flow, and by extension hematocrit, are intrinsically linked to turbulence and influence hemolysis, a continued misapprehension that these phenomena do not affect blood damage results in a pervasively simplistic understanding of cellular damage mechanisms that neglects both inter-cellular collisions and turbulence. This gap in understanding is detrimental to our ability to model blood damage, and this work aims to address this by quantifying and examining the mechanics of how RBC collisions affect RBC membrane damage at high shear.

Effect of viscoelasticity on intercellular collisions and hemolysis

While turbulence and collisions exacerbate hemolysis, RBCs with greater viscoelasticity are believed to be more resistant to these high-frequency small-scale deformations, as indicated by their decreased deformation⁴¹. Single-cellular studies have consistently shown that viscoelasticity tends to decrease the peak deformation in both capsules and RBCs in a shear flow^43–45. At steady state, the effect of viscoelasticity is dependant on the flow condition, as in a Couette shear flow, steady-state capsule deformation is altered by viscosity⁴³ yet counterintuitively, in a straight microcapillary, the steady-state deformation of capsules is not materially altered⁴⁶ between viscoelastic and elastic cases.

There is limited literature directly addressing the effect of viscoelasticity on membrane stress or biaxial strain, especially under high shear and unsteady loading or collisional conditions in RBC or capsule suspensions. This is important because the deformation parameter and trends observed in single-cell studies do not have a direct relationship to membrane stress or strain²⁵. In colliding RBCs, loading arises from both shear and collisions, but as collision rates rise with hematocrit and shear rate, it remains unclear which dominates^38,47. Consequently, the effect of viscoelasticity in high-shear, collisional flows is poorly understood. Indeed, to our knowledge no study examines viscoelastic suspensions of RBCs or capsules at high shear. At low shear, in a suspension, the situation is somewhat complicated⁴⁸. showed that viscoelasticity strongly affects the deformation of RBC suspensions and dynamics such as cell migration, a result supported by³⁹, though no strong effect of viscoelasticity was found for on rotation angles and periods, or the load application response time. Notably, however, the results of Guglietta et al.³⁹ demonstrated that the deformation difference due to viscoelasticity decreases with increased hematocrit, indicating a reduction in the effect of viscoelasticity with increasing hematocrit (i.e.: collision rate or load application frequency). Indeed their results indicate that although initially high at low capillary numbers, the proportion of viscous stress to total stress in the cell membrane begins to decline as capillary number increases. This study also imposes a distance-based repulsive force, independent of membrane stiffness or shear rate, and therefore does not replicate the realistic lubrication forces that separate capsules or RBCs in suspension. Another analysis of RBC suspensions in unsteady flows found that as loading frequency increased, the imaginary (oscillatory or elastic) component of complex viscosity increased, while the real (viscous) component decreased, indicating a shift toward elastically dominated deformation behavior at high frequency loading⁴⁰. More classical viscoelastic finite-element RBC simulations⁴⁹ similarly found that after a critical loading rate, energy dissipation of the viscoelastic RBC membrane declined indicating a shift toward elastic behavior. Clearly, however, the effect of viscoelasticity is complex and its effect in suspensions of RBCs or capsules is not well understood. Indeed to the best of our knowledge, no simulation study directly quantifies the effect of viscoelasticity on membrane strain at high shear under complex loading conditions such as in a capsule suspension.

Materials and methods

Unsteady incompressible finite-volume computational fluid dynamics simulations are executed using Ansys Fluent™. We select the fully-Eulerian structural methodology for membrane modelling developed by Pozrikidis^50,51 and extended by several others^31,52–54. The methodology uses the volume-of-fluid (VOF) multiphase numerical scheme⁵⁵ to model plasma and cytoplasm fluid phases and is implemented in ANSYS Fluent™.

Fluid and structural model

The momentum conservation equation is solved for the mixture velocity and density field common to both phases, given below.

1

Where is the fluid stress tensor, which is modelled as laminar and Newtonian, is the cytoplasmic volume fraction, and are the mixture fluid velocity vector and density, respectively, and is an immersed-boundary force vector that transfers momentum between the structural model and the fluid. is an indicator term to locate the interface between plasma and hemoglobin phases, which is given in the supplementary material.

Membrane force is applied as a source term to the momentum conservation equation, , and coupled semi-implicitly to the structural equations which are solved in a segregated manner⁵⁵. An elastic membrane is defined at the interface between the plasma and cytoplasm phases. The deformation of the membrane, relative to the deformed frame, is represented by the left Cauchy-Green deformation tensor, , which is conserved in the Eulerian reference frame.

2

In this work both Einstein tensor notation and bolded quantities are used to denote rank-two tensors. Here is the velocity gradient tensor, is the surface-projected left Cauchy-Green deformation tensor, and is the material derivative. is the surface projection tensor, computed from the surface normal vector, , at the plasma-hemoglobin interface. The structural model is implemented by solving user-defined scalar (UDS) equations to model the transport of the deformation tensor in ANSYS Fluent⁵⁵. Specific details about how this is executed are given in the supplementary material.

The source, , is computed from the in-plane tension, .

3

In-plane Piola-Kirchoff stress, , can be obtained by differentiating the material strain energy, and projecting it onto the surface. The final relation for stress is given below^31,50,53

4

and are the first and second invariants of the deformation tensor, . is material strain energy given as a function of and . Bending moments are neglected, for the simple reason that the momentum source resultant of the area inextensibility term is always normal to the surface (manifested by ) when there is a spatial gradient to , and this term is at least one order of magnitude greater than the bending stress, as shown in the supplementary material. Consequently, the bending moment has minimal effect on the dynamics or elastic state of the RBC membrane and can be safely neglected in all simulated cases that include area inextensibility.

Computational domain and boundary conditions

RBCs are simulated in a planar Couette flow in a cubic microchannel. This flow regime is preferred as it permits control of shear rate independently from hematocrit and prevents RBC exclusion near boundaries. Figure 1 illustrates the computational domain and boundary conditions. No-slip walls are specified at the upper and lower boundaries with opposing velocities so that the net velocity is zero. Translation periodicity is applied in the streamwise and spanwise boundaries.

Fig. 1.

Schematic illustration of a red blood cell (tonicity of 130 mOsm, r = 3.91 ) initialized in the computational domain used for RBC collision simulations, and a cutting plane showing the volume fraction and shape of the RBC as well as the grid refinement scheme.

The domain, measuring 20 μm on each side, is divided into 40 cube-hexahedral elements in each direction. Further grid refinement is achieved by employing adaptive mesh refinement at regions near the cellular boundaries that define the cell membrane. The Polyhedral Unstructured Mesh Adaption (PUMA) algorithm, which halves the element size at each refinement level, is used to avoid gradient discontinuities from hanging nodes. Three adaptive refinement levels are employed in non-viscoelastic simulations, and four in viscoelastic simulations, using volume-fraction gradient mesh refinement criteria, which ensures that the boundary is sufficiently refined. Two flow regimes are simulated for numerical validation. Firstly, we simulate an RBC parachuting in a cylindrical microchannel reproducing the experiment of Tomaiuolo et al.⁵⁶, shown in the supplementary material, and secondly, we reproduce the experiments of Mills et al.¹¹ in our Couette flow domain.

A convective Courant number of 0.5 ensures the RBC interface does not cross more than one cell per timestep. Time steps, varied with shear rate, generally maintain a maximum Courant number of 0.4, with simulations running for 10,000 to 20,000 timesteps. Beyond >200, excessive numerical diffusion was observed (exceeding 2% of the mean cellular value given in Fig. 7), and the simulations were terminated.

Numerical solution scheme and initialization

In this study, the volume fraction discretization utilizes a second-order accurate implicit compressive scheme with sharp interface formulation and a relative implicit cutoff value of 10^− 8. The gradient is approximated using a node-based, third-order accurate approach. This scheme averages the gradients computed at each of the nodes in a cell, which produces a larger gradient stencil than the cell-based scheme and thus reduces noise and numerical error⁵⁵. Pressure-velocity coupling is achieved through the scalable Pressure-Implicit Splitting Operator (PISO) scheme, while momentum discretization employs a second-order upwind scheme for enhanced stability. Structural convection-diffusion equations are discretized with the 3rd-order Monotonic Upstream-Centred Scheme for Conservation Laws (MUSCL) scheme to minimize the numerical diffusion of the deformation tensor. For grid convergence, the refinement depth of the adaptively refined grid is gradually increased, with each depth level corresponding to a halving of spacing, and four levels attempted in total. Two levels were necessary to sufficiently replicate the strain and RBC dynamics observed by Mills et al.¹¹, offering double the refinement of the grid used by Takagi et al.³¹ to conduct fully Eularian FSI simulations of RBCs. Biconcave RBCs are initialized with a 130 mOsm state⁷. Cells are initialized with random orientation and uniformly separated depending on the desired hematocrit. The Y-component of velocity is set in the cell equal to 10% of the max streamwise velocity to perturb each cell vertically and facilitate the collisions.

RBC membrane material modelling and properties

Following Skalak et al.⁹, this study uses the strain energy approach for developing constitutive hyperplastic material models. The first and second volumetric invariants are given as and , respectively. Surface invariants are computed by applying the plane stress condition to the volumetric invariants. Here volumetric invariants are used in the constitutive model to reflect that the bilipid membrane stretching in fact has a thinning and hardening effect^57,58.

5

In general, the s subscript is dropped to denote the zeroed invariants, i.e.: and The first and second invariants of the Cauchy deformation tensor, and respectively, represent the (square of the) principle uniaxial strain , and area (biaxial) strain, , respectively. Consequently constitutive laws having proportionality to uniaxial tension or compression can be expressed in terms of , and proprotionality to area strains are expressed in terms of II_s.

Existing models such as Skalak’s area-incompressible or Mill’s et al.¹¹ Yeoh model fail to reproduce the experimentally observed RBC lengths at high shear rate (45 000 s^− 1) due to insufficient stiffness. To overcome this difficulty, we follow Skalak et al.⁹ and introduce area incompressibility to the Yeoh model by adding the (integral of) the area stretch . In addition, a 7th -order polynomial term is added to further stiffen the model at very high shear rates.

6

Mills et al. Suggests a range of surface elasticity moduli, , that are between and the high-strain modulus to be between . In the present study and , are chosen to reproduce the rupture condition, which is 6.4% average area strain at 45 000 s^-1. Furthermore, are chosen to reproduce the extensions observed by optical tweezer stretching during the Mills et al.¹¹.

Computation of transmembrane hemoglobin diffusion rate

Transmembrane hemoglobin diffusion due to mechanoporation of the RBC membrane is simulated. Hemoglobin transport is simulated by solving a UDS equation in ANSYS Fluent⁵⁵. The diffusivity coefficients inside of the cell and in plasma are computed using self-diffusion as a function of hemoglobin concentration, . Generally, the procedure of Sohrabi and Liu²⁶ and Nikfar et al.⁵⁹ is followed to compute transmembrane diffusion, we deviate from this procedure only in that we use the expected pore radius, and apply Tolpekina’s thermodynamic scaling precedure appropriately bridge the gap between simulations of done at varying scales. Transmembrane hemoglobin diffusivity rate, , is a function of the nanopores formed on the membrane when the membrane is stretched. The model consists of computing the reduced diffusivity of hemoglobin through nanopores, , computed following Davidson and Deen⁶⁰ and multiplying it by the expected pore area. Self-diffusion rates are obtained from Longeville et al.⁶¹. Tolpekina et al.^62,63 developed a 2D cavitation-based free energy model to predict the radius of a pore-forming on a bilipid membrane during mechanoporation by finding the equilibrium between line energy around a pore’s edge and the strain energy of the stretched membrane. This free energy model for pore radius is modified by incorporating the probability of pore formation from Koshiyama and Wada⁵⁸, to obtain the expected radius, Koshiyama and Wada also found that quasi-equilibrium membrane stretching led to single-pore formation in small membrane patches, and thus it is assumed that one pore can form per membrane patch whose size is on the order of the spectrin fibre length, (80 nm)². The equilibrium assumption can be easily justified as cellular time scales are several orders of magnitude slower than molecular time scales^58,64. Consequently, pore density is a function of the probability of pore formation, .

7

8

9

10

11

Here, , and are the instantaneous biaxial (area) strain of the membrane, the cutoff area strain, and the standard deviation used in the probabilistic model. and , are obtained from Koshiyama and Wada⁵⁸’s equilibrium case, as 1.08, and 0.08, respectively. is obtained by Tolpekina⁶² by fitting the pore radius obtained from Course- Grained Molecular Dynamics (CGMD) simulation data, to their free-energy model, finding that line-energy , and compressibility modulus was calculated following den Otter and Briels⁶³.The pore radius and pore density are used to compute the ratio of pore area, , to the membrane area in the cell, . The nanopore-restricted hemoglobin diffusion rates, and , are computed based on Davidson and Deen⁶⁰, and enable diffusion after the pore is larger than the minimum length of the hemoglobin molecule (See Valtchanov⁶⁵ for more details).

Rezazadeh et al.^66,67 find that pore diameter is limited by the nominal spectrin fibre length of the RBC cytoskeleton, and thus deviating from Sohrabi and Liu²⁶, the thermodynamic scaling procedure suggested by Tolpekina et al.⁶³ was applied to scale Koshiyama and Wada⁵⁸ and Tolpekina et al.’s⁶² correlations to a patch size of (80 nm)², which effectively limits pore diameter to the nominal spectrin fibre length of 80 nm.

Figure 2 shows that the distributions of porosity (a) and pore radius (b) resulting from this simple constitutive formulation have similar orders of magnitude to what was observed in the much larger and spectrin-limited CGMD simulations of Razizadeh et al.⁶⁷, and reproduce important features such as a critical threshold of pore formation around 6–8% area strain, which we consider an improvement on Tolpekina’s model.

Fig. 2.

Illustration of pore density and pore radius over the expected range of distributions (top), and comparison against CGMD simulations^66,67 of mechanoporation of a bilipid membrane without (WO) a cytoskeleton, and a bilipid membrane reinforced by an 80 nm (D80) and 40 nm (D40) spectrin cytoskeleton (Patch Area = 0.1218 )

Fluid properties and modelling viscoelasticity

Two fluids are considered in our simulations, plasma and cytoplasm, which are treated as incompressible Newtonian fluids with densities of 1.050 and 1.118 [kg m^− 3], respectively, and viscosities of 1.003 and 5.015 [cP], respectively. In the majority of cases, the fluid model does not consider viscoelasticity, which is acknowledged as a limitation. Material properties in computational cells containing both fluids are computed simply as an average weighted by each fluid’s respective volume fraction.

Two viscoelastic cases are attempted wherein viscoelasticity is modelled by directly specifying the membrane viscosity of in computational cells at the membrane interface, as computed by Mills⁶⁸. In the current study, the surface viscosity, , is divided by the membrane thickness, , which is 0.125 μm, to obtain the volumetric viscosity specified on the membrane, , is given by the equation below.

12

Where is a boundary indicator function, described in the supplementary material. The application of Newtonian viscosity in the fluid momentum equation when coupled with the fully Eulerian structural modelling approach is functionally equivalent to a Kelvin-Voigt damper, with the exception that in the present approach the viscoelastic resistance has a membrane-normal component.

A value in the lower range of membrane viscosities found in the literature is purposefully chosen because of the need to maximize numerical stability. Preliminary studies were executed to examine the effect of membrane viscosity, however, it was found that high viscosities necessitated very small grid sizes and rendered viscoelastic simulation of multiple colliding cells infeasible. Although membrane viscoelasticity is generally acknowledged to influence the collision process, and will increase membrane stiffness, in most of our cases it is not included in the mathematical formulation as it drastically increases the computational resources required. Two simple viscoelastic cases are considered and simulated to examine this effect in isolation.

Objectives and executed test cases

A series of numerical experiments, summarized in Table 1 below, are executed to examine the effect of collisions on RBC damage in shear flow, varying hematocrit, and shear rate. The objectives of our experiments are to validate our numerical method and to disentangle the effect of shear rate, collisions, hematocrit, and viscoelasticity on RBC membrane strain. Notably, the Capillary number in our flow is in the range of for multicellular cases.

Table 1.

Summary of simulations executed to examine the effect of RBC collisions.

Case	Parameters varied	Objectives
RBC in a microchannel	U = 0.27 cm s− 1	Validation against Tomaiuolo et al. 57
Deformation of a RBC		Validation against Mills et al.11
Effect of Shear Rate		Examine the interaction of shear rate with simple collisions on transmembrane Hb diffusion.
Effect of Hematocrit		Examine the interaction of hematocrit with collisions on transmembrane Hb diffusion.
Effect of Viscoelasticity		Examine the effect of the interaction of membrane viscosity with collisions on membrane strain.

Results

Model validation

Simulation results compare favorably with the optical tweezer experiments conducted by Mills et al.¹¹ shown in Fig. 3. Force is computed by integrating the shear stress over the RBC membrane. Length is measured at peak extension (shown at the bottom), which occurs shortly after initializing the simulation with an RBC that is aligned with the principle axis of the shear stress tensor.

Fig. 3.

Quantitative comparison of the length of an RBC stretched by shear in the present simulations at peak stretch (blue dots), with the optical tweezers experiments of Mills et al.¹¹ (black and white diamonds). Solid-red, dotted-red, and dash-dot blue lines represent models fit by Mills et al. with a range of shear moduli, to their optical tweezers measurements of membrane stress.

As expected, the modifications to the Yeoh hyper-elastic model integrate additional higher-order terms that minimally influence the membrane response at low shear rates so that the model behaves as developed by Mills et al.¹¹. Although the modified model exhibits increased stiffness — due to these higher-order terms and the adoption of surface over volumetric invariants—the simulation outcomes are consistent with Mills et al.‘s experimental measurements within their error margins. Not depicted in Fig. 3, the RBC length stabilizes at roughly 18–19 μm under shear rates ranging from 20,000 to 45,000 s^− 1, a consequence of introducing these high-order terms in the updated material model equation .

Dispersed flow mechanics and hemoglobin diffusion

Figure 4 illustrates the relationships between time and membrane-averaged area strain, maximum area strain, and uni-axial strain with shear rate, calculated between the first and last peak of membrane strain. Both linear and area strains display threshold behavior induced by the high-order stiffening terms introduced. Interestingly, the maximum area strain remained generally below the 23% threshold computed from Li et al.¹⁴, which is based on the strain at which spectrin-actin bonds dissociate. The goal of maintaining an average area strain near 6.4% at rupture shear rates between 42,000 and 45,000 s^− 1, was achieved by adjusting the C7 stiffness coefficients in equation .

Fig. 4.

(a) Membrane-averaged and membrane-maximum area strain and (b) membrane-averaged (principal component of linear) strain for a single red blood cell in a Couette shear flow.

The maximum area strain peaked between shear rates of 10,000 and 20,000 s^− 1 at nearly 14%. Membrane-averaged uni-axial strain likewise peaked around 35%.

Figure 5 illustrates the pore diameter and density at a shear rate of 20,000 s^− 1, both at peak stretch and during steady tank treading.

Fig. 5.

Distribution of area strain, , pore density, , and pore diameter, , on the RBC surface during (a) peak stretch and (a) tank treading at time under high shear rate of 20 000 s^-1.

Fig. 6.

Visualization of collision in a Couette shear flow at 20 000 s^-1 between (a) two RBCs experiencing a glancing collision, and (b) a head-on collision involving 4 cells. Collisions begin at , , and , respectively.

Length of the RBCs are 18.3 μm and 14.3 μm during peak extension and during tank treading, respectively. The membrane material and mechanoporation models effectively set a critical membrane area strain for pore initiation and a minimum radius for diffusion, and we find significant local transmembrane hemoglobin diffusion rate for . Consequently, nanopore formation is primarily at the RBC membrane’s stretched peripheral regions, with significant diffusion occurring for a small portion of the cell membrane. Interestingly, we find increased overall strain and both pore diameter and density during tank treading, despite a shorter overall length. As a final note, our mechanoporation model encouragingly predicts pore diameters in the same order as observed by Ohta et al.⁶⁹, on the surface of a sheared RBC.

Collision dynamics

We observe that in colliding cases, RBCs are generally shorter and more ellipsoidal compared to their solitary counterparts at similar shear rates. Hydrodynamic lubrication, inversely proportional to cellular spacing, prevents direct cell contact, observable as the highly deformable cells maintain uniform separation during the collision. We identify two primary collision types: glancing collisions in dispersed flows with unconstrained cellular paths, and head-on collisions in constrained, high hematocrit environments, illustrated in Fig. 7a) and b), respectively. Head-on collisions cause RBCs to contract, leading to mutual rotation. Glancing collisions, with greater intercellular spacing, experience less shear and no significant path alteration. This transfer of angular momentum may be viewed as a generation of small-scale fluctuations by the collision processes. The high shear imposed by the intercellular collisions is also highly dissipative, and indeed the Kolmogorov scale (computed by subtracting the mean velocity field) in the two-cell and four-cell cases at = 20,000 s^− 1 is between 14.4 and 10.2 , respectively, hinting that it is related to intercellular spacing.

Fig. 7.

Comparison of membrane-averaged area strain variation over time at (a) a single cell with varying shear rate (b) two-cells under varying shear rate (c) varying number of cells at constant shear rate (20 000 s^-1) (effective hematocrit is between 3 and 36%, computed as the ratio of cytoplasmic to total volume).

Effect of RBC collisions on membrane strain and sublethal hemolysis

Figures 2a) and 7 b) illustrate the effect of collisions by comparing a single-cell case to a simple two-cell case in which varying shear rates are applied. Collisions significantly strain RBCs, with membrane-averaged strain deviating by 10–20% from the time-averaged strain. Membrane-averaged area strain peaks in the two-cell case correspond to times when the cells are in closest proximity (collision frequency, in this case, is approximately ), whereas in the single-cell case, peaks are related to the elastic response of the red blood cell in the flow. Between the single-cell and two-cell cases, there is an increase in mean and peak membrane-averaged area strains.

Table 2 shows the time average of the membrane-averaged area strains and hemoglobin diffusion coefficients for the single and two-cell cases at shear rates ranging between 4,820 and 45,000 s^− 1, which covers hemodynamic conditions found in devices such as stents and LVADs. There is a significant increase in both membrane-averaged area strain, which saturates at approximately 14%, as well as the membrane-averaged effective diffusion coefficient. Time averaging is done between the first and last peak of each simulation, ranging from .

Table 2.

Time- and membrane-averaged values of area strain and effective transmembrane hemoglobin diffusion coefficients for one and two cells in a couette shear flow. Diffusion rates are normalized by the zero-concentration self-diffusion rate.

1	4820	0.0160	0.0721	3.88·10− 4
2	4820	0.0471	0.1376	0.045
1	10,000	0.0320	0.1090	0.0429
2	10,000	0.0559	0.1516	0.1636
1	20,000	0.06474	0.1395	0.2734
2	20,000	0.0843	0.2059	0.4685
1	30,000	0.06175	0.1125	0.3305
2	30,000	0.0919	0.1726	0.3986
1	45,000	0.08494	0.139	0.4582
2	45,000	0.136	0.204	0.7628

The effect of hematocrit on membrane strain is tied to the shear rate by the collision frequency. As hematocrit increases, the collision frequency () increase and so does area strain. The average area strain increases with both shear rate and with the number of cells, reaching saturation at around 14%. This trend is consistently seen both in Fig. 2c) and in Table 3. The high-cellular cases do not reach a steady state, as the closer spacing between RBCs in the higher hematocrit cases resulted in numerical instability resulting in early termination of the simulations. Nevertheless, the trend is clear, and a consistent increase in membrane-averaged area strain is observed with increasing collision rate, which is visualized in Fig. 8.

Table 3.

Comparison of time and membrane-averaged area strain and effective transmembrane hemoglobin diffusion coefficients with increasing hematocrit.

1	3.02	0.0647	0.1395	0.273
2	6.0	0.0843	0.2059	0.469
4	12.1	0.0844	0.2110	0.407
6	18.1	0.0777	0.2129	0.435
8	24.1	0.1343	0.2845	0.752
12	36.2	0.1379	0.3324	0.733

Fig. 8.

The relationship between membrane-averaged area strain (black), Hb diffusion rate (red) and mean collision frequency.

In all cases, elevated membrane-averaged area strain results in elevated hemoglobin diffusion coefficients. Transmembrane Hb diffusion rate is three orders of magnitude lower for the single-cell case than the two-cell case at 4820 s^− 1 in Table 1, showing that collisions result in lowering the minimum shear rate threshold for significant sublethal hemoglobin diffusion, which is known to occur at shear rates as low as 1000 s^− 170. Additionally, the hemoglobin diffusion seems to reach a saturation point at approximately 0.7, which is consistent with a maximum pore diameter of 90 nm. Furthermore, average area strain increases with both shear rate and cell count, with peak values at about 14% area strain, significantly greater than the single-cell counterpart at equivalent shear rate.

Instantaneous membrane-averaged transmembrane hemoglobin diffusion rate correlates remarkably well to an erf surface of membrane-averaged and membrane-maximum area strain, shown for the two-cell case in Fig. 9. All cases, in general, have this trend, though the best fit occurs in the colliding cases because complex membrane mechanics like membrane wrinkling and buckling, which tend to apply compressive loads and do not promote pore formation, are suppressed by cellular collisions.

Fig. 9.

Fitting of effective transmembrane hemoglobin diffusion coefficient with membrane-averaged area strain and membrane-maximum area strain to an erf surface.

Effect of viscoelasticity

In this section, the effect of membrane viscoelasticity on the dynamics of red blood cell collisions and their subsequent impact on area strain and transmembrane hemoglobin diffusion coefficient are examined. As the viscoelastic response time in single-cell cases requires prohibitively long simulation times, the focus will be on the relatively short timescales of collisions in the presence of 2 and 4 cells, corresponding to 6% and 12.1% hematocrit levels. The viscoelastic properties of the membrane are modelled by applying a Newtonian viscosity multiplied by the membrane boundary indicator function (detailed in the supplementary material) wherein the viscosity inside of the membrane is set to 0.3 [µPa s m]. By analyzing the dynamics of collisions between two cells with realistic membrane viscosities in Fig. 10, the influence of viscoelasticity on red blood cell behavior and hemolysis is ascertained.

Fig. 10.

Illustration of glancing collision between red blood cells wherein viscosity inside of the boundary indicator function where the structural model is set to 0.3 [µPa s m].

In Fig. 10, it is observed that the dynamics of red blood cell collisions and the shape of the red blood cells are significantly different when viscoelasticity is incorporated into the red blood cell membrane. The inclusion of membrane viscosity, which is nearly three orders of magnitude higher than that of the nearby plasma or cytoplasm, adds considerable stiffness to the membrane on the timescale of the collision, exacerbated by the high shear rate.

Indeed, Fig. 10 shows that during the collision, there is a much slower response of the membranes due to the proximity of the other RBC. Deformation seems to occur locally only for the portions of the RBC that are near the other RBC, and the response of the rest of the membrane is delayed. The increased stiffness causes the spacing between the membranes to decrease, and spacing appears much smaller than in the non-viscoelastic case. Because of this, significant shear is applied to the red blood cells, and they begin to rotate around one another due to their collision.

Figure 11 below shows the membrane-averaged, -maximum, and -minimum strains experienced by the membrane with 2 and 4 cells, as well as membrane-averaged effective diffusion coefficients. Most notably, it appears the overall magnitude of the area-averaged stress is similar between the viscoelastic and non-viscoelastic 2- and 4-cell cases. Both viscoelastic cases were terminated early due to numerical instability induced by the viscoelastic collision, and thus further analysis is required to examine this effect. This is a somewhat counter-intuitive result as it was expected that the smaller spacing and more localized impact zone would result in more damage to the red blood cells. Membrane-maximum strains, shown in Fig. 11, show a small decrease of maximum area strain for the 4-cell viscoelastic case, though overall maximum strain levels are similar for both viscoelastic and non-viscoelastic cases.

Fig. 11.

Comparison of (a) membrane-averaged area strain and (b) membrane-averaged Hb diffusion rate during collisions at 20,000 s^-1 in the viscoelastic and non-viscoelastic cases for 2 and 4 cells.

Our preliminary results indicate that while viscoelasticity has a large impact on the dynamics of the cells, it does not appear to have a strong impact on the time- and membrane-averaged area strain experienced by red blood cells during collisions.

Discussion

Material modelling and sublethal hemolysis

In this study, we modified Mill’s et al.¹¹ hyperelastic Yeoh material model, making it area inextensible and able to limit extension at high shear rates, which was critical for accurately replicating the experimentally observed RBC shapes. The 7th-order polynomial terms had minimal influence at lower shear rates, allowing retention of the original coefficients and successful replication of optical tweezer experiments at high shear rates. These coefficients effectively constrained the membrane-averaged area strain to around 8% at a 45,000 s^− 1 shear rate, with maximum strains limited to 23–25%.

In the single-cell case, shear rates below 5,000 s^− 1 did not induce sufficient membrane strain to permit sublethal hemolysis, as relative transmembrane hemoglobin diffusion coefficients stayed below 10^− 3. As shear rate increases so does transmembrane hemoglobin diffusion, plateauing at 70–74% of the self-diffusion rate at near-complete membrane poration. The current work is distinguished from previous studies in the membrane-poration model, where we compute the expected pore radius given the strain-based probability of pore formation. In our model, we asserted that pore size is limited by the nominal spectrin fibre length, which limits area extensibility and thus maximum pore area. As we model pore opening probabilistically with a normal distribution, the distribution of the local transmembrane diffusion coefficient closely fits an erf surface varying with average and maximum area strain (in Fig. 9) with an R² = 0.94. The close fit of pore parameters to an erf surface indicates that pore parameters are normally distributed, which is also found in nanopore distributions found during electroporation experiments⁷¹. This may simply be an artefact, however, of the probability of pore formation being modelled using an erf function of membrane strain here and by Koshiyama et al.⁵⁸, it follows that the distribution of the nanopore parameters on the hemolyzing cell is also normally distributed.

Effect of collisions on area strain and sublethal hemolysis`

Data derived from Table 2 shows that collisions consistently elevate the average strain on RBC membrane in a Couette shear flow, more so at lower shear rates, where the strain at 10,000 s^− 1 nearly doubles. This effect is prominent even at elevated shear rates of 30,000 s^− 1 and 45,000 s^− 1, where strain experienced by colliding cells is 60% greater. We note that while the average area strain in scenarios involving collisions reached a plateau at 14%, the maximum strain was generally near 23% but increased to 33% in high hematocrit cases. This is reconciled with the traditional rupture criterion offered by Leverett^35,72,73 (between 2% and 6%) for long-term load application, by noting that in our case load application is rapid (collision frequency is a function of shear rate) and RBCs experiencing high-frequency impact loading rupture at much greater global area strains between 30–50%⁷⁴. We see a similar trend in Table 3 when increasing the number of cells in the domain while keeping the shear rate constant, where membrane-averaged area strain generally increases with hematocrit and reaches a plateau of 14%. It remains difficult, however, to make a definitive conclusion regarding the effect of hematocrit on membrane area strain from the data in this study, as many of the high hematocrit simulation cases diverged before reaching a steady state due to numerical stability issues resulting from the semi-implicit fluid-structure coupling. Although there is a tendency for area strain to increase more quickly in high hematocrit cases, more simulation time is necessary to observe the steady value. Nevertheless, a consistent trend is seen between increased collision frequency and increased area strain, which contradicts early studies that found little to no relationship between hematocrit and hemolysis, and indicated that collisions do not significantly to membrane damage. Leverett et al.³⁵, for example, examined the effect of hematocrit at high shear rates between 10⁵ and 3·10⁵ s^− 1, at which point the cell membrane will experience lysis regardless of the effect of collisions. Indeed, we find that the effect of collisions is pronounced at lower shear rates in the range of physiological flow, and less pronounced near rupture strains. Furthermore, the collision rate ( is so high that even in Leverette’s most dispersed case, at Hct = 0.3%, at , the collision rate is significant, and equivalent to our two-cell case, indicating that significant degree of cellular collisions were occurring.

During a collision, the cells pass close to each other and experience high shear with a lubricating film between the cells. The effective increase in shear stress is proportional to the collision frequency, the shear rate, and the thickness of this lubricating film between colliding cells. Increasing both hematocrit and shear rate has the effect of increasing collision frequency and decreasing the lubricating film thickness.

As RBC collisions substantially increase the strain experienced by the RBC membrane, they correspondingly result in a proportionally greater rate of sublethal hemolysis. Furthermore, it was also found that RBC collisions lower the threshold for sublethal hemolysis. In the cases with and without collisions, the onset of sublethal hemolysis begins at a shear rate below 1000 s^− 1 and 4820 s^− 1, respectively. This is not surprising as the collision event causes a localized increase of membrane area strain near the collision site, and it is generally difficult to attain the high level of area strain required (> 6%) to permit transmembrane hemoglobin diffusion without collision events. This result is corroborated by experiments showing that sublethal hemolysis can occur at shear rates as low as 1800 s^− 169, which is consistent with our colliding cases but not the dispersed cases.

Preliminary viscoelastic simulations were also executed for 2- and 4-cell cases at 20,000 s^− 1 and revealed that although collision dynamics were appreciably altered, evidently the trend in mean area strain was similar between viscoelastic and non-viscoelastic cases. Notably, these simulations suffered from numerical instability and were terminated early, and thus further work is necessary to validate the effect. More studies are necessary, however, to fully characterize the influence of viscoelasticity on intercellular collisions and hemolysis.

Implications of collision mechanics to turbulence in blood

In the present study, it was found that turbulent dissipation increases with hematocrit and collision frequency, and in fact, the Kolmogorov scale was computed (from time- and spatially-averaged velocity field) to be on the order of the intercellular spacing, at 14.4 and for the 2- and 4-cell cases. This result while, interesting, may simply be a coincidence due to the small domain size and consequently low system Reynolds number. It is also noted, however, that larger-scale structures and thus instability-generating environments that would create small-scale turbulent fluctuations are not present. This perspective also helps explain the observation of non-Kolmogorov turbulent decay found in physiological flows of patients in vivo^28,75, as it implies an increased turbulence decay rate due to enhanced dissipation. We find preliminary indications that collisions between RBCs generate disorderly motion on the order of the RBC diameter, during ‘head-on’ collisions when inter-cellular spacing is sufficiently low for cells to transfer angular momentum onto each other. This process can be viewed as a kind of production from the mean flow to cellular-scale vorticity, effectively supporting the hypothesis of Antiga and Steinman²⁷ who suggest that the smallest turbulent length scale in blood should be on the order of the RBC diameter, or specifically, the mean inter-cellular spacing. It is difficult to make conclusions regarding turbulence characteristics due to the limited size of the computational domain in this study, however. Yet, there is likely a strong effect of RBC collisions on the production and dissipation of turbulence, which should be followed up in future work.

Study limitations and validity of the computational method

This study has numerous limitations arising from the computational method and simplistic flow conditions. A major limitation is the occurance of numerical diffusion during transport of the deformation tensor, which confined our simulation time to approximately < 200 to ensure that the diffused deformation components outside the cellular membrane region were below 2% of the mean membrane value. Furthermore, the numerical scheme requires at least two elements between cellular membranes, and in the case of viscoelastic cases, the lubricating film between the cells is quite thin and a very fine grid spacing was required. This, when paired with multiphase stability criteria requiring Courant numbers below one, resulted in long compute times and high numerical diffusion that limited the duration of time that could be simulated. Also of note is that it was not possible to directly validate the strain or the mechanoporation characteristics of our model with experimental data, as strain distributions on RBC membranes have yet to be quantified in dynamic states due to the difficulty in observing transient phenomena in a shear flow. Nevertheless, the model has been used^26,59,76 to predict sublethal hemolysis in biomedical flows to the same precision as experimental shear-based correlations.

Furthermore, although domain size was increased when increasing hematocrit so that RBCs were one RBC diameter away from the wall, the domain may influence the dynamics of the collision process. Specifically, in a less constrained flow regime, head-on collisions may not occur and the severity of these types of collisions may be diminished. Moreover, the small number of cells and small domain size prohibit us from performing appropriate turbulence analysis. It is noted however that velocity gradients and their spatial variation near the walls are two orders of magnitude lower than near the site of the RBC collision, indicating that their effect is minimal, and the presence of nearby RBCS themselves will prevent cross-stream motion, and moving the wall further from the RBC collision site simulates a dispersed flow. Indeed, it is more likely that if the effect of moving RBCS above and below were accounted for addition collisions would result, and thus the effect of collisions would be exacerbated. Given this and that the effective hematocrit in these cases was relatively low, it is expected that at least the trend observed in the present work is valid.

Finally, the simulation approach generally neglects viscoelasticity, which is an important aspect of quantifying the true response of the RBC to collisions. Preliminary viscoelastic investigations were conducted for two- and four-cell cases and found reduced transient peak strain, but similar average strains in two- and four-cell colliding cases as the non-colliding cases. Simulation times for the viscoelastic cases were however short due to high computational cost and numerical instability, and furthermore, this section of the model is not experimentally validated through transient relaxation experiments, as such validation is not feasible using our current approach. Although these results should be viewed as preliminary and a more detailed and comprehensive analysis of the effect of viscoelasticity on collisions is warranted, we nevertheless expect that the general trend of our results to hold with or without the addition of membrane viscosity.

Therefore, opportunities for future research are to verify the interesting mechanics found in this study that could benefit from more data, expand the dynamic conditions of testing, and include a more comprehensive study of viscoelasticity. In general, however, this study finds that collisions are an important, if not, dominant mechanism of RBC membrane damage in blood. Future work should also investigate constitutive models to incorporate collision mechanics in blood damage assessment.

Conclusion

This study examined the effects of red blood cell (RBC) collisions on the strain accumulated on the RBC membrane, as well as the diffusion of transmembrane hemoglobin. To this end, a battery of cellular-scale computational studies examined the collision process under varying shear rates and hematocrit. A novel formulation of the Yeoh model was proposed for a hyperelastic membrane that integrated area inextensibility and high-order terms to increase stiffness at high shear rates. Validation against experimental data derived from optical tweezer studies showed that the model reproduced the force-displacement curves shown in experiments at lower shear rates.

The study’s findings revealed that RBC collisions indeed had a significant and perhaps dominant impact on membrane area strain, in some instances, increasing membrane-averaged area strain by up to a factor of two. The overall dynamics and strain experienced by the RBC membrane in a shear flow appeared in fact to be largely driven by these collisions. Preliminary viscoelastic simulations were also conducted and indicated that viscoelasticity diminished the peak strain experienced by the membrane but did not significantly alter the average strain. This relative effect was amplified at lower shear rates yet persisted at higher shear rates. Likewise, elevating the hematocrit, at a constant shear rate, seemed to intensify the area strain developed on the membrane. The influence on transmembrane hemoglobin diffusion was noticeable and proportional to the rise in area strain. These findings are in general contradiction to the commonly accepted knowledge that collisions between RBCs do not significantly influence hemolysis or the development of area strain.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Author contributions

Conceptualization, R.M. , H.V; methodology, H.V.; software, H.V.; validation, H.V.; formal analysis, H.V.; investigation, H.V.; resources, R.M and R.C.; data curation, H.V.; writing—original draft preparation, H.V.; writing—review and editing, H.V, and R.M.; visualization, H.V.; supervision, R.M.; project administration, R.M.; funding acquisition, R.M. All authors have read and agreed to the published version of the manuscript.” Please turn to the CRediT taxonomy for the term explanation. Authorship must be limited to those who have contributed substantially to the work reported.

Funding

This research was generously funded by NSERC and McGill University Health Centre.

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.