Red blood cell dynamics in extravascular biological tissues modelled as canonical disordered porous media

The dynamics of blood flow in the smallest vessels and passages of the human body, where the cellular character of blood becomes prominent, plays a dominant role in the transport and exchange of solutes. Recent studies have revealed that the microhaemodynamics of a vascular network is underpinned by its interconnected structure, and certain structural alterations such as capillary dilation and blockage can substantially change blood flow patterns. However, for extravascular media with disordered microstructure (e.g. the porous intervillous space in the placenta), it remains unclear how the medium’s structure affects the haemodynamics. Here, we simulate cellular blood flow in simple models of canonical porous media representative of extravascular biological tissue, with corroborative microfluidic experiments performed for validation purposes. For the media considered here, we observe three main effects: first, the relative apparent viscosity of blood increases with the structural disorder of the medium; second, the presence of red blood cells (RBCs) dynamically alters the flow distribution in the medium; third, symmetry breaking introduced by moderate structural disorder can promote more homogeneous distribution of RBCs. Our findings contribute to a better understanding of the cell-scale haemodynamics that mediates the relationship linking the function of certain biological tissues to their microstructure.

The dynamics of blood flow in the smallest vessels and passages of the human body, where the cellular character of blood becomes prominent, plays a dominant role in the transport and exchange of solutes. Recent studies have revealed that the microhaemodynamics of a vascular network is underpinned by its interconnected structure, and certain structural alterations such as capillary dilation and blockage can substantially change blood flow patterns. However, for extravascular media with disordered microstructure (e.g. the porous intervillous space in the placenta), it remains unclear how the medium's structure affects the haemodynamics. Here, we simulate cellular blood flow in simple models of canonical porous media representative of extravascular biological tissue, with corroborative microfluidic experiments performed for validation purposes. For the media considered here, we observe three main effects: first, the relative apparent viscosity of blood increases with the structural disorder of the medium; second, the presence of red blood cells (RBCs) dynamically alters the flow distribution in the medium; third, symmetry breaking introduced by moderate structural disorder can promote more homogeneous distribution of RBCs. Our findings contribute to a better understanding of the cell-scale haemodynamics that mediates the relationship linking the function of certain biological tissues to their microstructure.

Introduction
Significant progress in computational modelling has been made over recent years to elucidate the complex behaviour of blood flow in physiological environments, e.g. the small-vessel network in the brain [1][2][3], in the eye [4,5], in tumours [6] and in microaneurysms [7]. However, the flow and transport of blood and solutes in other (e.g. extravascular) types of biological media with vital function, such as the intervillous space (IVS) of the human placenta featuring a highly disordered network of pores and flow passages of size comparable to that of red blood cells (RBCs), remains poorly understood [8,9].
Comprehensive theories of flow and transport in porous media have been established, revealing subtle relationships between pore-scale structural heterogeneity and macroscopic flow properties [10][11][12][13]. However, existing models of flow through porous media often assume a homogeneous fluid and cannot accurately infer the intricate blood rheology in living biological media [14], e.g. the IVS, where the particulate and highly confined character of blood plays a key role, introducing spatio-temporal variability and nonlinearity beyond the description of prevalent continuum models [15,16].
Emerging cell-resolved models of blood flow using advanced mesoscopic methods [4,[17][18][19][20] have been extensively applied since the late 2010s to simulate multiscale haemodynamics in synthetic or realistic vascular networks. These simulations have greatly improved our understanding of microscopic processes in the blood stream mediated by the flowing RBCs within, e.g. biased haematocrit distribution and oxygen transport arising from abnormal branching patterns of the vasculature [21].
Facilitated by robust image segmentation and meshing techniques [22,23], cell-resolved models are now technically applicable to living porous media. These models provide a promising avenue for microscopic characterization of the microstructure-dependent cellular blood flow, which can reproduce haemorheological behaviour based on first principles and inform generalized constitutive relationships needed by more robust continuum models [15,24]. A key aim is the development of effective reduced-order models for efficient simulation of large tissue/organ systems [25]. Additionally, cellular simulations can help design and optimize microfluidic oxygenators that serve as artificial lung assist devices [26].
In this work, we aim to characterize microscopic blood flow in canonical porous medium models constructed to represent simplistic extravascular biological media, for which we have control over the structural characteristics that can be related to physiological or pathological conditions of porous tissues and organs. Specifically, our focus is to quantify the correlations between cell-scale haemodynamics (e.g. flow patterns, RBC partitioning, haematocrit distribution) and key metrics of the porous medium, such as porosity and disorder. Primarily, we tackle the task computationally, with the aid of microfluidic experiments in equivalent flow systems for validation.

Design of canonical disordered porous media
We consider two typical designs of planar (quasi-twodimensional) canonical disordered porous media (DPM) for our three-dimensional simulations and validating experiments: locally perturbed media (LPM) (figure 1a) representing weakdisorder systems and globally random media (GRM) (figure 1c) representing strong-disorder systems, both of which are in the form of non-overlapping uniformly sized cylinders. The design royalsocietypublishing.org/journal/rsfs Interface Focus 12: 20220037 process is inspired by the 'bundle of tubes (uniform)' model implemented by Gostick et al. [27].
To obtain LPM, we first construct an ordered porous medium (OPM) by placing cylinders of constant diameter D c (determined by porosity ϕ and domain size) on a square grid with length L and width W (figure 1a inset), which can be regarded as a generalized capillary bundle model. Three porosity values ϕ = 0.57, 0.62, 0.67 are considered to account for distinct levels of confinement that cellular blood flow experiences in the capillary bed and extravascular tissues, corresponding to a confinement ratio of χ = 1.2, 1.0, 0.9 (defined as χ = D RBC /W throat , which compares the effective RBC diameter with the size of pore-throats) in the baseline OPM geometry, respectively. The alteration of porosity is achieved by varying D c while keeping the cylinder positions unchanged. The structural disorder of the LPM is then created by introducing random perturbations to the cylinder positions (x, y) with standard deviation σ = 0.1, 0.3, 0.5, 0.7, where (x, y) are drawn from a bivariate normal distribution. For a relatively large σ (e.g. σ = 0.7), the perturbations are likely to cause overlapping cylinders and several attempts may be needed to construct a non-overlapping layout. The resulting distributions of throat widths in the LPM are Gaussian (figure 1b).
The GRM is constructed by placing a uniformly random spatial distribution of cylinders in a domain of given size for a designated porosity (ϕ = 0.66), while enforcing a minimal separation distance of 0.4D RBC between neighbouring cylinders (figure 1c). The resulting distribution of throat widths in the GRM is approximately exponential (figure 1d ). Three GRM realizations are tested in the present study.

Numerical model
The immersed-boundary lattice-Boltzmann method (IB-LBM [28,29]) simulates cellular blood flow as a suspension of deformable RBCs (see the electronic supplementary material, section S1 for more details of the model) through a shallow Hele-Shaw bed (L × W × H = 258 × 129 × 6 μm 3 ) of vertical cylinders, which is constructed through extrusion over a distance (H) of the desired OPM or DPM (LPM and GRM) designs of dimension L × W in the depth direction. Each RBC is modelled using our previous approach [30,31] as a closed hyperelastic membrane with the unstressed shape of a biconcave discoid, which has been calibrated and benchmarked against experimental measurements of healthy human RBC [32]. The morphological deformation of RBCs in confined shear flows is primarily governed by the shear and bending elasticity of the RBC membrane, which contributes to characteristic features at the cell's front and rear ends (Fig. S1 in the electronic supplementary material). To tackle close cell-cell and cell-wall interactions, a repulsion potential decaying with inverse distance between neighbouring surfaces is numerically implemented with interaction intensities comparable to the bending elasticity of the RBC membrane [31]. For a list of key simulation parameters, refer to the electronic supplementary material, table TS1.
In the simulations, an RBC-free plasma flow (Newtonian) is initialized from left to right along the channel axis (length direction) with a designated volumetric flow rate Q 0 (figure 2a). The flow is driven by imposing a parabolic velocity profile at the inlet (assuming Poiseuille flow under Q 0 ) and a reference pressure at the outlet ( p 0 ), through two cylindrical flow extensions smoothly stitched to the porous bed of rectangular crosssection (W × H). The no-slip condition is imposed at the wall and all fluid-solid interfaces. Once the plasma flow is converged, RBCs are randomly inserted from the inlet flow extension in a continuous manner with designated feeding discharge haematocrit (i.e. flow-weighted RBC volume fraction), H F = 0.1, 0.2, 0.3. RBCs reaching the outlet flow extension are removed from the system (see electronic supplementary material, movie S1 for an example simulation in the OPM-ϕ0.57 geometry). For the definition of other flow-related quantities, refer to appendix A.

Microfluidic experiments
In the microfluidic experiments, a suspension of silicone oil droplets (Sigma Aldrich, viscosity ν = 20 cSt with paraffin oil dye) in a mixture of water and glycerol (Sigma Aldrich, 80 : 20 by volume with 0.2% SDS, viscosity 0.24 Pa s at 21 C) was used as biomimetic model for blood. The droplets were generated in a flow-focusing device [33] which was connected to the porous   medium via a bent glass capillary (length 100 mm, outer diameter 1 mm, inner diameter 0.58 mm). Droplets with a diameter of 240 μm were produced by flowing the inner phase (silicone oil) at 4 μl min −1 and introducing the outer phase (water/glycerol) through a cross-junction at 5 μl min −1 immediately upstream of the flow-focusing channel constriction. Downstream of the droplet formation, another 11 μl min −1 of water-glycerol mix was added to set the droplet flow fraction to H F = 0.2. Flow control at the different inlets was achieved with a combination of a pressure controller (Elvesys) fitted with flow meters and a syringe pump (KD Scientific). Both the flow-focusing device and the porous medium were made from PDMS ( polydimethylsiloxane, Sylgard 184, Dow Corning). To obtain hydrophilic wetting behaviour, the devices were oxidized by oxygen plasma treatment (Henniker plasma HPT-100) and immediately filled with water. The porous medium corresponds to the LPM-σ0.5-ϕ0.57 geometry (i.e. disorder 0.5 and porosity 0.57) used in numerical simulations, scaled by the ratio of the droplet diameter to the equivalent diameter of simulated RBCs (D RBC = 6.68 μm). The suspension flow was imaged with a monochrome CMOS camera (PCO 1200hs) and analysed off-line using the ImageJ platform TrackMate [34].

Results and discussion
We first cross-validate the simulated RBC dynamics in the porous media with a generic model of soft particle suspension flow using microfluidic droplets, aimed at reproducing the physical behaviour of cellular blood flow. Then, we demonstrate the effect of porous medium structure and the presence of RBCs on the flow resistance. Lastly, we investigate the RBCs' contribution to pore-scale flow redistribution and their spatio-temporal dynamics in relation to structural characteristics of the porous domain.

Cross-validation of simulation and experiment
Good agreement between the simulated RBC dynamics and experimented droplet dynamics is obtained (see figure 2c-f and electronic supplementary material, movies S2 and S3 for comparison in an example geometry), including the pattern of particle trajectories (figure 2c,d), the distribution of particle fluxes (figure 2e) and the distribution of particle velocities (time-average speed for particles crossing individual porethroats, figure 2f ). Both the RBCs and droplets demonstrate evident preferential pathways or regional shunts when travelling across the porous domain, with discrete clusters of high-flux and low-flux throats. Similarly, the mean particle velocities at individual throats feature a bimodal distribution where high-velocity and low-velocity populations coexist. Some quantitative discrepancies, e.g. in the magnitude of pore-scale velocities (figure 2f ), stem from subtle differences between the two particle types [35]. First, the unstressed shape of a droplet is spherical, whereas that of the RBC is biconcave (suggesting a large excess area compared to an equivalent sphere of the same volume). Second, the intrinsic surface properties of a droplet and an RBC differ, the former of which relies on surface tension to resist deformation in shear flow, whereas the latter is primarily subject to membrane elasticity governing shear and bending stresses under low-moderate flow velocities [36] (as in the case of viscous flow through disordered porous media).
The encouraging agreement observed in the overall behaviour of particle trajectories, fluxes, velocities between the simulated RBCs and microfluidic droplets underpins the existence of potential physical determinants of the cell-scale haemodynamics in confined pore-space networks, e.g. the channel confinement ( particle size relative to throat size) and the particle deformability (deforming stress relative to restoring stress). Such physical determinants can be universal regardless of certain particle properties such as size, shape or surface composition, therefore dominating the particulate flow behaviour in canonical porous media. Further characterization and understanding of these determinants can contribute to the development of robust blood analogues [37] that may serve as model systems of real blood with high-level rheological and haemodynamic similarity. Emerging biomimetic counterparts of RBCs include droplets, vesicles and capsules given their stable properties and versatile fabrication methods [37,38], which can reproduce key features of RBCs in shear flow and are envisioned for mimicking particulate blood flow in health and disease under well-controlled conditions.

Interplay of the porous structure and RBC presence on flow resistance and blood viscosity
To assess the resistance to plasma flow (RBC-free) and suspension flow (RBC-laden, with default feeding haematocrit H F = 0.2), we compare the overall pressure drop ΔP across different porous media under an identical feeding volumetric flow rate at the inlet (Q 0 = 0.4 μl min −1 in simulations, unless otherwise specified). For suspension flow, we also examine the relative apparent viscosity η rel comparing the suspension apparent viscosity η app to that of plasma η 0 (1 × 10 −3 Pa s), which can be calculated as the ratio of pressure drops for equivalent RBC suspension and Newtonian plasma flows under same inflow (Q S = Q N = Q 0 ), i.e. η rel = η app /η 0 = ΔP S /ΔP N . The porosity is found to have a prominent effect on the plasma flow resistance (figure 3a). For instance, an increase of ϕ = 0.57 to 0.62 and 0.67 for the intermediately disordered LPM (σ = 0.5) leads to 24.1% and 39.4% reduction in the pressure drop, respectively. Similarly, the suspension also experiences lower flow resistance: 24.7% and 40.0% reduction, respectively (figure 3a). However, the relative apparent viscosity of the suspension remains approximately constant, therefore suggesting a small effect of porosity within the studied range (figure 3a). The same trends of pressure drop and relative apparent viscosity against increasing porosity are found for the baseline OPM geometry too (electronic supplementary material, figure S2).
For a given porosity of ϕ = 0.57, our simulations in LPM geometries reveal a marginal effect of incremental disorder on the resistance of plasma flow, but an evident effect on that of the suspension flow (figure 3b). The relative apparent viscosity increases substantially beyond a critical degree of disorder; η rel is 16.4% higher at σ = 0.5 compared with its value in the ordered medium (σ = 0).
We further investigate the effect of stronger disorder on the relative apparent viscosity of suspension flow with three GRM realizations of distinct topology (but identical porosity and similar throat-width distribution, see figure 1d), for which spatial disorder is introduced globally. Different haematocrit levels are examined, where the relative apparent viscosity can rise by nearly 66% as the feeding discharge haematocrit is increased from H F = 0. The significantly different effects of porosity and disorder on the relative apparent viscosity are surprising; both decreasing ϕ and increasing σ are expected to contribute to stronger pore-scale confinement that impedes the transport of RBCs [41] (see definition of the confinement ratio χ in §2.1). We believe that this difference in behaviour stems from the sensitivity of local confinement to changes in ϕ and σ. Changing ϕ uniformly modifies all throats (for ϕ ∈ [0.57, 0.67]; the mean throat width only varies in the range of W throat [ ½5:5, 7:5 mm). In contrast, a change in σ augments the possibility of severe confinement in the system by broadening the W throat distribution and giving rise to exceedingly small throats (W throat < 3 μm, see figure 1b). Note, however, that the impact of occasional severe confinement in some throats when increasing σ might be partially compensated by the simultaneously introduced larger throats, leading to an overall minor increment in the relative apparent viscosity.
Theoretically, when the size of a channel approaches that of an RBC, the cell-free layer vanishes due to confinement and the cells are in direct contact with the wall, consequently increasing the relative apparent viscosity of blood in straight tubes as postulated by the well-known Fåhraeus-Lindqvist effect [39] (figure 3c). However, depending on its severity, such confinement may also contribute to lower relative apparent viscosity by forcing the organization of RBCs into streamlined single-file form, thus excluding excessive cell-cell collisions that would otherwise increase flow resistance [15]. For the disordered geometries studied here, where a tightly interconnected network of pores and throats (rather than isolated channels) are involved, it is expected that both effects exist and compete with each other in determining the relative apparent viscosity of the suspension flow. Further, because each throat has a varying cross-section of contraction and expansion, the spatial organization of RBCs constantly re-arranges without reaching a fully developed profile, which constitutes a mechanism for higher relative apparent viscosity in the porous media than straight tubes of geometrically equivalent diameter (figure 3c). These counteracting mechanisms associated with structural alterations sometimes balance each other and the overall viscosity of the suspension remains roughly unchanged under certain circumstances (e.g. figure 3a).

RBC traffic dynamically alters the pore-scale flow distribution
For both the plasma flow and suspension flow (figure 4a), a larger degree of structural disorder (i.e. increasing σ) is found to promote higher perfusion of the pore space network as indicated by a smaller percentage of throats experiencing negligible flux (compare figure 4b with the electronic supplementary material, figure S3A). In both RBC-free and RBC-laden scenarios, the flow distributions evolve from discrete high-/low-flow clusters towards a continuous exponential-like profile as the level of medium disorder increases, similar to earlier findings on Newtonian flows through perturbed cylinder arrays [11]. However, this evolution is noticeably accelerated in the presence of RBCs, i.e. achieving an exponential-like flow distribution at lower level of disorder ( figure 4b). Given that such exponential distribution is underpinned by less extreme local flow fractions at nodal points (i.e. bifurcations) [11], the RBCs contribute to more robust perfusion of the porous domain, which is in line with a recent report on the role of RBCs in stabilizing blood flow in the capillary bed [42]. Indeed, for strongly disordered geometry where flow channellization (known to be royalsocietypublishing.org/journal/rsfs Interface Focus 12: 20220037 associated with anomalous solute transport in heterogeneous media [43]) prevails in the plasma flow, the presence of RBCs is found to break down some dominant pathways and enhance regional perfusion (see the electronic supplementary material, figure S3B).
Examination of the pore-scale dynamics reveals that the presence of RBCs introduces intermittent and/or permanent occlusion of narrow throats (figure 4a and electronic supplementary material, movie S2), featuring trapped RBCs between adjacent cylinders as observed in a recent experimental study [44]. Accompanied by such occlusion events are occasional flow reversals found in throats located in the vicinity (see figure 4a insets, 'a1' and 'a2'). These occlusions may in turn contribute to the diversion of flow (via occlusive pressure feedbacks [45]) to otherwise poorly infused regions (e.g. transversely oriented throats) and lead to more evenly distributed suspension flow in the system (see the enhanced transverse flows in figure 4c, right panel) compared with plasma-only flow (figure 4c, left panel) under identical inflow conditions. Indeed, the underlying streamline patterns in figure 4d suggest that some negligibly perfused throats in the plasma scenario become better perfused in the suspension scenario.
Furthermore, the RBCs impose a temporal signature on the pore-scale flow which is characterized by distinct patterns of flow channellization over time (electronic supplementary material, figure S4). To quantify the temporal variation, three throats of interest (figure 4e) are analysed for their instantaneous flow rates over 15 time instants. Wide distributions of flow rates are found, and the time-averaged blood flow at the throats is roughly two to four times as high as in the plasma-only scenario. It is noteworthy that despite the highly dynamic RBC traffic and flow fluctuations at the pore scale, the pressure drop across the entire porous bed is essentially steady (see figure 3 for the narrow error bar ranges).

Symmetry-breaking promotes more homogeneous RBC transport and haematocrit distribution
RBC trajectories in the porous bed provide information about the spatio-temporal dynamics of the cells (compiled with over 1000 cells, figure 5a). Figure 5b shows the corresponding distributions of RBC transit time in geometries with different medium disorder. The data reveal that both the distribution shape and median transit time are quantitatively similar in cases investigated here (ϕ = 0.67), provided that the porosity remains constant (confirmed for ϕ = 0.57 cases in the electronic supplementary material, figure S5A). In other words, the transit time distributions are insensitive to the medium disorder under our studied range of porosity magnitudes. The insensitivity is likely caused by two opposing effects of structural disorder on the motion of RBCs: increasing the tortuosity of existing pathways slows down individual cell transits, but enables more flow pathways by breaking the spatial symmetry and introducing transverse pressure gradients locally. This result suggests that, in a disordered porous medium such as the placental IVS [23], the average residence royalsocietypublishing.org/journal/rsfs Interface Focus 12: 20220037 time of RBCs (associated with oxygenation and solute transport processes) within representative tissue volumes can be independent of the inherent structural heterogeneity if the tissue porosity remains roughly the same. On the other hand, increasing the porosity from ϕ = 0.57 to ϕ = 0.67, while keeping the disorder unchanged, leads to an increase in median transit time (compare 'LPM-σ0.5-ϕ0.57' and 'LPM-σ0.5-ϕ0.67' in figure 5b). This seemingly counterintuitive effect (i.e. slower flow under relaxed confinement) is due to an overall lower interstitial flow speed at higher porosity under the imposition of a constant feeding flow rate Q 0 (rather than constant pressure drop Δp). The evidently altered pattern of RBC transit time by a moderate change in the medium porosity suggests a potential mechanism for impaired placental function in pre-eclamptic or diabetic pathologies, where the porous tissues have been found either overly sparser or denser than under normal conditions [46].
Finally, we examine the spatio-temporal RBC distribution in more detail. A visual inspection of the time-lapse RBC perfusion reveals that there are still plenty of throats devoid of RBCs for the intermediate disorder σ = 0.5 (electronic supplementary material, movie S4), but for higher degree of a locally perturbed disorder σ = 0.7 and the global disorder, the pore space network is more strongly perfused (figure 5a and electronic supplementary material, movies S5 and S6). Figure 5c shows the RBC flux distributions in the throats of the networks. These data confirm that the locally perturbed 'LPM-σ0.7' and the globally random 'GRM' have comparable RBC distributions while the less disordered 'LPM-σ0.5' features a large number of throats without RBCs.
The discharge haematocrits H d of individual throats tend to be close to the feeding discharge haematocrit H F = 0.2 as the structural disorder becomes sufficiently large in the system (e.g. σ ≥ 0.5), featuring a primary peak located around H d = H F in the distribution profile (figure 5d). This concentrated distribution around H d = 0.2 is still visible for weakly disordered geometries with σ ≤ 0.3, but outweighed by a more dominant peak at zero (see the electronic supplementary material, figure S5B). royalsocietypublishing.org/journal/rsfs Interface Focus 12: 20220037 Our findings have implications for the optimization of hollow-fibre bundles in extracorporeal membrane oxygenation (ECMO) [47] or microfluidic blood oxygenators in lung assist device (LAD) [26], which adopt designs similar to our studied porous media for oxygen delivery. In both cases, blood is perfused through a porous geometry with uniformly spaced obstacles. Multiple parameters of operation need to be considered for achieving maximal oxygen uptake (e.g. by increasing RBC residence time within the device) while maintaining minimal pressure drop and dampening the temporal fluctuations of oxygenation characteristics induced by flow pulsatility. Our results demonstrate that disorder is another handle to consider for enhancing RBC perfusion, which has not been explored in existing oxygenator designs.

Conclusion
We have investigated the micro-haemodynamics of cellular blood flow in disordered porous media as simplistic models for extravascular biological tissues, for instance, the intervillous space in the human placenta. Two types of canonical porous media with varying degrees of porosity and disorder were considered: weakly disordered porous media based on a regular square array of non-overlapping circular obstacles, and a random porous medium with intrinsically disordered arrays. We employed simulations combining the lattice-Boltzmann, finite-element and immersed-boundary methods to explore the effects of haematocrit, porosity and structural disorder on the blood rheology, RBC perfusion and haematocrit distribution in the porous media. We also conducted scaled-up experiments using microfluidic droplets to validate the numerical model, where the particle dynamics are found to be qualitatively and quantitatively similar (e.g. flux and velocity distributions). The numerical-experimental framework allows us to examine correlations between the structural characteristics and cellular haemodynamics in canonical disordered porous media, aimed at inferring the structure-function relationship in extravascular biological tissues of high-level heterogeneity and excessive irregularity that remains to be validated using more complicated realistic domains in the future.
The main findings from our study are threefold: -First, the relative apparent viscosity of cellular blood increases with the structural disorder of the porous media considered herein, but it is largely independent of the studied porosity. These counterintuitive findings are likely caused by competing effects between cell-free layer development (i.e. evolving thickness of depletion layer) and cell re-ordering (i.e. alternating single-file or two/multi-file organization) in narrow channels. -Second, the presence of RBCs dynamically alters the flow distribution in the investigated porous media. Throats that are only weakly perfused in the absence of RBCs can receive significantly higher flow due to RBCs partially blocking the flow in other, better-perfused throats. Due to the motion of the RBCs, the flow rates in the throats can fluctuate strongly. -Third, breaking the symmetry of canonical porous media by introducing moderate structural disorder into the system (without causing prohibitive channel occlusions) can promote more homogeneous distribution of RBCs as measured by cell fluxes and discharge haematocrits. While RBCs favour fast lanes through ordered or weakly disordered porous media, a stronger perturbation creates new pathways for the RBCs and lead to more homogeneous haematocrit distribution throughout.
Our reported effects of the porous structure on RBC perfusion may inform more efficient oxygenator designs and our model can be readily applied for simulation as well as optimization of the design. We envision that a multidisciplinary approach, cross-validating simulations and experiments to extract generalized constitutive relationships for RBC flow in complex geometries, will help bridge the gap between microscopic characterization and tissue/ organ-level modelling which are both necessary to reveal the relationship between structure and function of biological tissues and organs.
Data accessibility. Data and information supporting this article are provided in the electronic supplementary materials, including one PDF document, six simulation/experiment movies and a ZIP package (SimulationFiles.zip) where simulation parameter files and RBC trajectory data can be found. The source code HemeLB for the blood flow simulations in this study is available at https://github.com/ hemelb-codes/hemelb. The code version used was f85dac87900 e082a6f5fd125a2b6366c94c752e5. The electronic supplementary materials are available online at [48].