November 2024
Volume 65, Issue 13
Open Access
Retina  |   November 2024
A High-Fidelity Computational Model for Predicting Blood Cell Trafficking and 3D Capillary Hemodynamics in Retinal Microvascular Networks
Author Affiliations & Notes
  • Saman Ebrahimi
    Mechanical and Aerospace Engineering Department, Rutgers, The State University of New Jersey, Piscataway, New Jersey, United States
  • Phillip Bedggood
    Department of Optometry and Vision Sciences, University of Melbourne, Melbourne, Australia
  • Yifu Ding
    Department of Optometry and Vision Sciences, University of Melbourne, Melbourne, Australia
  • Andrew Metha
    Department of Optometry and Vision Sciences, University of Melbourne, Melbourne, Australia
  • Prosenjit Bagchi
    Mechanical and Aerospace Engineering Department, Rutgers, The State University of New Jersey, Piscataway, New Jersey, United States
  • Correspondence: Prosenjit Bagchi, Mechanical and Aerospace Engineering Department, Rutgers, The State University of New Jersey, 98 Brett Road, Piscataway, NJ 08854, USA; [email protected]
Investigative Ophthalmology & Visual Science November 2024, Vol.65, 37. doi:https://doi.org/10.1167/iovs.65.13.37
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      Saman Ebrahimi, Phillip Bedggood, Yifu Ding, Andrew Metha, Prosenjit Bagchi; A High-Fidelity Computational Model for Predicting Blood Cell Trafficking and 3D Capillary Hemodynamics in Retinal Microvascular Networks. Invest. Ophthalmol. Vis. Sci. 2024;65(13):37. https://doi.org/10.1167/iovs.65.13.37.

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      © ARVO (1962-2015); The Authors (2016-present)

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Abstract

Purpose: To present a first principle-based, high-fidelity computational model for predicting full three-dimensional (3D) and time-resolved retinal microvascular hemodynamics taking into consideration the flow and deformation of individual blood cells.

Methods: The computational model is a 3D fluid—structure interaction model based on combined finite volume/finite element/immersed-boundary methods. Three in silico microvascular networks are built from high-resolution in vivo motion contrast images of the superficial capillary plexus in the parafoveal region of the human retina. The maximum tissue area represented in the model is approximately 500 × 500 µm2, and vessel lumen diameters ranged from 5.5 to 25 µm covering capillaries, arterioles, and venules. Blood is modeled as a suspension of individual blood cells, namely, erythrocytes (RBC), leukocytes (WBC), and platelets in plasma. An accurate and detailed biophysical modeling of each blood cell and their flow-induced deformation is considered. A physiological, pulsatile boundary condition corresponding to an average cardiac cycle of 0.9 second is used.

Results: Detailed quantitative data and analysis of 3D retinal microvascular hemodynamics are presented, and their relationship to RBC flow dynamics is illustrated. Blood velocity is shown to have temporal oscillations superimposed on the background pulsatile variation, which arise because of the way RBCs partition at vascular junctions, causing repeated clogging and unclogging of vessels. Temporal variations in RBC velocity and hematocrit are anti-correlated in a given vessel, but their time-averaged distributions are positively correlated across the network. Whole blood velocity is 65% to 85% of RBC velocity, with the discrepancy related to the formation of an RBC-free region, adjacent to the vascular endothelium and typically 0.8 to 1.8 µm thick. The 3D velocity and RBC concentration profiles are shown to be oppositely skewed with respect to each other, because of the way that RBCs “hug” the apex of each bifurcation. RBC deformation is predicted to have biphasic behavior with respect to vessel diameter, with minimal cell length for vessels approximately 7 µm in diameter. The wall shear stress (WSS) exhibits a strongly 3D distribution with local regions of high value and gradient spanning a range of 10 to 80 dyn/cm2. WSS is highest where there is faster flow, greater curvature of the vessel wall, capillary bifurcations, and at locations of RBC crowding and associated thinning of the cell-free layer.

Conclusions: This study highlights the usefulness of high-fidelity cell-resolved modeling to obtain accurate and detailed 3D, time-resolved retinal hemodynamic parameters that are not readily available through noninvasive imaging approaches. The results presented are expected to complement and enhance the interpretation of in vivo data, as well as open new avenues to study retinal hemodynamics in health and disease.

The retina is one of the most metabolically active tissues with the highest oxygen consumption per unit weight.1 The retinal microcirculation, comprising an intricate network of arterioles, capillaries, and venules, along with choriocapillaris supports the metabolic demands of photoreceptors and synaptic cells.24 Simultaneously the retinal vasculature must also be sparse enough to have minimal interference with the transmitted light.58 The retinal vascular topology is designed to meet these opposing requirements, and any disruption to the vascular topology could cause visual impairment.9,10 Dysfunction in the retinal microcirculation has been associated with diabetic retinopathy, age-related macular degeneration, glaucoma, retinal vein occlusion, and sickle cell retinopathy.1116 Alteration in the retinal microcirculation also serves as an indicator for cardiovascular and cerebral disorders.17,18 Prior studies have shown that alteration in the retinal blood flow often precedes the morphological alteration in the vasculature.1923 Therefore a detailed and improved understanding of the blood flow in the retinal vasculature is critical for understanding the pathophysiology of these diseases. 
Several experimental techniques have been developed to measure retinal blood flow.24,25 Among these, laser Doppler velocimetry and Doppler optical coherence tomography (OCT) provide label-free measurements of blood velocity but suffer from low spatial resolution, and, hence, they are not suitable for capillary vessels. Quantifying blood flow in capillaries is important to understanding disease progression because these vessels are most vulnerable to hemodynamic and morphological alteration.1922,2629 OCT angiography (OCTA) has been used to study capillary perfusion.3032 However, it does not provide quantitative measures of blood velocity. In recent years, adaptive optics methods have been developed that provide quantitative measures of single blood cell velocity and can achieve higher spatial resolution necessary for capillary vessels.33,34 These approaches have been used to provide spatial and temporal variation of blood flow in single, targeted vessels and in multiple interconnected capillary vessels.3541 It has also been used to visualize blood cells in capillaries, and to measure cross-sectional blood velocity profiles.36,39,41 Apart from these noninvasive techniques, confocal imaging with fluorescently labeled erythrocytes has also been used to quantify retinal blood flow in animal models.42 One limitation of the imaging techniques, which rely on cell tracking, is that the measured cell velocity may not represent actual blood velocity.4346 Furthermore, because RBC distribution in the microvasculature is spatially and temporarily heterogeneous, the measurement accuracy may vary from vessel to vessel.47,48 Also, measurement of the cross-sectional velocity and RBC distribution profiles remains difficult and not reported, barring reference 39, which gave two-dimensional (2D) velocity profiles in vessels \( \mathbin{\lower.3ex\hbox{$\buildrel>\over {\smash{\scriptstyle \sim}\vphantom{_x}}$}} \) 20 µm diameter. 
The blood velocity profiles in the microvascular network are generally three-dimensional (3D) and significantly different from the axisymmetric profile of Poiseuille's flow. The RBC distribution over a vessel cross-section is also nonuniform, with a higher concentration near the vessel center and a reduced concentration near the wall, where a cell-free layer of plasma (CFL) develops.4446 A knowledge of the 3D velocity profile and RBC distribution is needed not only for an accurate prediction of retinal hemodynamics but also for an accurate evaluation of critical physiological quantities, such as the wall shear stress (WSS) and CFL. The WSS plays critical roles in blood flow regulation, while the CFL tends to reduce the blood viscosity in small vessels and affects the cellular margination and transvascular exchange.4450 The aforementioned imaging techniques, however, cannot provide the full 3D blood velocity, RBC distribution, WSS, and CFL. 
Theoretical models have been used to predict flow in the retinal microcirculation. In one class of models, vessels are treated as one-dimensional (1D) conduits, and Poiseuille's pressure-flow relation is used along with empirically derived laws of blood viscosity.51 This approach has been used for retinal arteriolar tree to predict blood flow regulation and tissue oxygenation.52,53 Combined with VEGF dynamics and activation dynamics of adhesion molecules, the method was used to model the progression of diabetic capillary occlusion.54 Extension of the 1D model to include vessel elasticity and pulsatile flow was also considered.55 Image-based 3D models have also been used in which the geometric features of the retinal microvessels, such as vessel curvature and nonuniform cross-section, were retained and the full equations of the fluid motion were solved.5659 In these studies, however, blood was treated as a single phase fluid of Newtonian or non-Newtonian rheology without direct modeling of RBC flow and deformation. 
Computational models that account for RBC flow and deformation have grown in recent years.6064 One such model has been developed by our laboratory and used to predict RBC trafficking and capillary hemodynamics in physiologically realistic and 3D microvascular networks comprising of multiple vessels and vascular junctions.65,66 Here we extend this model to the retinal microcirculation focusing on the superficial plexus near the foveal avascular zone (FAZ) in the human retina. We present a quantitative analysis of the discrepancy between RBC and blood velocities and its relation to the CFL, high-frequency variations of flow and hematocrit and the underlying cellular mechanisms, and spatial heterogeneity of flow and hematocrit. We further present and analyze the 3D distributions of velocity, RBC concentration and WSS. RBC deformation is also characterized and the relationship of the RBC dynamics to temporal and spatial variations of the hemodynamic parameters is illustrated. 
Material and Methods
The computational model is a fully coupled fluid—structure interaction model based on an integration of the finite volume/finite element/immersed-boundary methods.65,66 The topological organization of the vascular network near the FAZ, and three-dimensionality of each vessel, including capillaries, arterioles, and venules, as well as of vascular junctions, are fully retained in the model. Here, capillaries are defined as vessels with diameter less than 9 µm, arterioles as having diameter larger than this and dividing into smaller vessels, and venules being the post-capillary vessels formed by vessel merging. The flow and deformation of each of approximately 1000 to 1400 blood cells, occupying a network at any instant, are modeled with high accuracy. The spatial resolution is 0.25 µm for the vascular networks, and 0.23 µm for the cells. The temporal resolution is 1 µs. Figure 1 shows visualizations from the networks simulated, and Figure 2 shows the essential components of the model that are discussed below. 
Figure 1.
 
(AC) In vivo images of the human parafoveal microvascular networks obtained from references 4 and 30 (with permission). Green box indicates the tissue area modeled. Corresponding in silico models and RBC distribution as predicted by our model at one time instant are shown in DF. Arrows indicate flow directions at inlets and outlets, and a and v indicate arterioles and venules, respectively. Animations of predicted RBC flow in the modeled in silico vasculatures are given in Supplementary Materials.
Figure 1.
 
(AC) In vivo images of the human parafoveal microvascular networks obtained from references 4 and 30 (with permission). Green box indicates the tissue area modeled. Corresponding in silico models and RBC distribution as predicted by our model at one time instant are shown in DF. Arrows indicate flow directions at inlets and outlets, and a and v indicate arterioles and venules, respectively. Animations of predicted RBC flow in the modeled in silico vasculatures are given in Supplementary Materials.
Figure 2.
 
(A) Triangular mesh on vessel surface in a small segment of a microvasculature in silico. (BD) Resting shapes and sizes of a RBC, a mononucleated WBC, and a platelet. An example mesh on the RBC surface is shown. Biophysical properties of the cells used in the model are also noted. For WBCs and platelets, shear and area dilation moduli are taken 10 times higher since they are much less deformable than RBCs. (E) Pulsatile profile specified at an inlet vessel. (FI) Different RBC flow patterns as predicted by the model are shown in a few vessels. (F) Single-file RBC flow in one capillary, (H) double file and zig-zag patterns, and (G, I) multi-file flow in larger vessels. RBCs are shown in red, platelets in yellow, and WBC in gray.
Figure 2.
 
(A) Triangular mesh on vessel surface in a small segment of a microvasculature in silico. (BD) Resting shapes and sizes of a RBC, a mononucleated WBC, and a platelet. An example mesh on the RBC surface is shown. Biophysical properties of the cells used in the model are also noted. For WBCs and platelets, shear and area dilation moduli are taken 10 times higher since they are much less deformable than RBCs. (E) Pulsatile profile specified at an inlet vessel. (FI) Different RBC flow patterns as predicted by the model are shown in a few vessels. (F) Single-file RBC flow in one capillary, (H) double file and zig-zag patterns, and (G, I) multi-file flow in larger vessels. RBCs are shown in red, platelets in yellow, and WBC in gray.
Retinal Microvasculature In Silico
Three in silico vasculatures are built from in vivo images of the microvascular networks rendered en face at the superficial plexus layer in the parafoveal region of the human retina.4,30 Each in silico model contains at least one arteriole-venule pair, with multiple capillaries generally traversing circumferentially around the FAZ. Each model also includes multiple vascular bifurcations and mergers. Maximum tissue area represented in the model is about 500 × 500 µm2, and vessel diameters range from 5.5 to 25 µm. Vessel centerlines are first traced from the in vivo images, and then a CAD software is used to add three-dimensionality to each vessel. Note that vessel diameters are not obtained from the images because the motion contrast methods usually visualize the central cellular part of a vessel. Instead, we assign the lumen diameter of the terminal capillaries in the range 5.5 to 6.5 µm and use Horton's law to estimate the average diameter of larger vessels progressing through the vascular hierarchy6770 (see Supplementary Materials for details). Also note that the fundamental equations solved in our model ensure conservation of flow and red cell flux at arteriolar and venular junctions (see Supplementary Materials for details). Vessels are considered nondeformable with circular cross-sections but may have varying diameter as they approach a junction. Also note that our models are not strictly planar; there are vessels that run over and above other vessels and the layer thickness is 35 to 48 µm (see Supplementary Materials for details). Once the geometry is built, vessel surfaces are discretized with a triangular mesh (maximum number of triangles used ∼13 million). 
Blood Cells and Plasma
Blood is modeled as a suspension of erythrocytes (RBC), leukocytes (WBC), and platelets in plasma with physiological proportion of each cell type. Accurate and comprehensive biophysical modeling of each cell is considered. RBCs are extremely deformable—a property that allows them to squeeze through narrow capillaries. A healthy human RBC is non-nucleated, made of hemoglobin enclosed by a membrane (cytoskeleton + lipid bilayer) and has an undeformed shape of a biconcave discocyte. This structural characteristic is exactly represented in our model in a continuum sense without resolving the molecular details (Fig. 2). In addition, the mechanical properties of the RBC membrane include resistance against shearing deformation, surface area dilation, and membrane bending, which are also modeled using experimentally-derived nonlinear, rheological constitutive laws.71,72 Hemoglobin is modeled as a fluid of viscosity 0.006 Pa-s. WBCs and platelets are modeled similarly but with appropriate shape and biophysical properties (Fig. 2). Blood plasma is modeled as a fluid of viscosity 0.0012 Pa-s. 
Each cell surface is meshed by 5120 triangles, and a finite element method is used to obtain the stresses generated in the membrane as a result of deformation.73 
Flow Model and Cell/Flow/Vessel Integration
Flow equations are the unsteady, 3D Navier-Stokes equations. The computation domain is a rectangular box containing an in silico vascular network. The box is discretized using a rectangular mesh comprising 160 to 600 million points. The immersed boundary method (IBM) is used to deal with the nonconformity of the vascular geometry with regard to the flow domain mesh and to apply the no-slip velocity condition on the vessel surface.65 The IBM is also used for the cell and flow coupling. Physiological pressure boundary condition, corresponding to a pulsatile flow cycle of 0.9 second for an average healthy adult is specified at the inlet and outlet vessels. Cells are introduced at the inlet boundary with an average hematocrit of ∼30%. This is less than the systematic hematocrit (∼40%–45%), in accordance with previous studies in the microcirculation.4348 RBCs are naturally distributed by the flow throughout the vasculature; as such, individual vessels can have higher hematocrit than that maintained at the boundary. Rigorous validation of model was presented in our previous publications using in vitro and in vivo data: This included validating the individual model component for RBC deformation, fluid motion, and IBM methods, as well as the integrated model for mesenteric microvascular network hemodynamics.65,66,7476 Additional numerical details are given in Supplementary Materials
Results
The outcome of the model is the full 3D, time-resolved blood velocity, pressure, and the shape, position and velocity of each blood cell, which are processed a posteriori to obtain different hemodynamic parameters of interest. Visualizations of the cell distribution at one time instant in entire networks are shown in Figure 1, and close-up views of RBC shapes in a few vessel segments are shown in Figure 2. Spatially heterogeneous RBC distribution, which is a hallmark of microvascular blood flow including the retinal microcirculation, can be observed. Highly deformed RBC shapes, characterized as parachute and slipper shapes as observed in vivo, are also predicted and shown in Figure 2. A wide range of RBC flow patterns as observed in vivo (i.e., single-file flow in the small diameter capillary vessels, zig-zag patterns in larger capillaries and multi-file flow in even larger vessels) are also predicted as seen in Figure 2
Validation Against In Vivo Data
Figure 3 compares our model predictions against in vivo data reported for human retinal capillary vessels exhibiting single file RBC flow. The pulsatile RBC velocity (VRBC) in one cardiac cycle obtained from four different vessels in our model are compared in Figure 3A against the data of Bedggood and Metha.38 For each vessel, VRBC is obtained by averaging the velocity of all RBCs moving through it. Note that velocity can differ between cells in a vessel. Also, a zone near a vascular junction is excluded in the averaging because velocity decelerates near the junction. It should be noted that the simulated capillary networks are geometrically different from those of Bedggood and Metha.38 As such the flow distribution and pulsatile profiles predicted by our model and those measured in vivo need not exactly match for all vessels. The agreement seen in the figure, although coincidental for specific vessels, tends to suggest that the predictions from our model are in the physiological range of retinal capillary blood flow. Additional quantities compared are the pulsatility index of the RBC velocity defined using the maximum and minimum as (VmaxVmin)/(Vmax + Vmin) in Figure 3B, and the distribution of RBC lineal density defined as number of cells per 50 µm in Figure 3C. Predicted pulsatility index is in the similar range of the in vivo data of Neriyanuri et al.40 and Gu et al.37 and follows the general trend with regard to RBC velocity. Predicted RBC lineal density also agrees with Gu et al.77 
Figure 3.
 
Comparison of the model predictions against in vivo data reported for human retinal microcirculation. (A) Pulsatile profile of the RBC velocity as predicted by the model (continuous curves in color) in a few capillary vessels compared against Bedggood & Metha38 (dash black curve). (B) Predicted pulsatility index and minimum RBC velocity Vmin compared against Neriyanuri et al.40 and Gu et al.37. (C) RBC lineal density predicted against Gu et al.77
Figure 3.
 
Comparison of the model predictions against in vivo data reported for human retinal microcirculation. (A) Pulsatile profile of the RBC velocity as predicted by the model (continuous curves in color) in a few capillary vessels compared against Bedggood & Metha38 (dash black curve). (B) Predicted pulsatility index and minimum RBC velocity Vmin compared against Neriyanuri et al.40 and Gu et al.37. (C) RBC lineal density predicted against Gu et al.77
RBC Velocity Versus Whole Blood Velocity
Next, we quantify the discrepancy between the RBC velocity VRBC and whole blood velocity VBL per vessel as predicted by our model. To compute VBL we average the space-and-time resolved velocity obtained from the model over the volume of each vessel, yielding a time-dependent velocity. Computation of VRBC was described earlier. As seen in Figure 4A for a randomly selected vessel, VBL is less than VRBC. We further compute time-averaged (i.e., averaged over one cardiac cycle) RBC velocity and blood velocity, denoted as \({\bar{V}_{RBC}}\) and \({\bar{V}_{BL}}\), respectively, for all vessels (Fig. 4B). As seen, RBC velocity overestimates the whole-blood velocity in all vessels, including capillaries, arterioles and venules. The ratio \({\bar{V}_{BL}}/{\bar{V}_{RBC}}\) ranges as \({\sim}0.65\)–0.85 in the vessel diameter range 5.5–18 µm and generally increases with increasing diameter (Fig. 4C). 
Figure 4.
 
Quantification of RBC velocity and whole blood velocity differences. (A) Pulsatile profiles of RBC velocity (dash) and whole blood velocity (solid) in one vessel. (B) Time-averaged RBC (\({\bar{V}_{RBC}}\), filled symbols) and whole blood (\({\bar{V}_{BL}}\), open symbols) velocity versus vessel diameter in all vessels of a simulated vascular network. (C) The ratio \({\bar{V}_{BL}}/{\bar{V}_{RBC}}\) versus vessel diameter. For B and C, red, green, and blue symbols indicate capillaries, arterioles, and venules, respectively.
Figure 4.
 
Quantification of RBC velocity and whole blood velocity differences. (A) Pulsatile profiles of RBC velocity (dash) and whole blood velocity (solid) in one vessel. (B) Time-averaged RBC (\({\bar{V}_{RBC}}\), filled symbols) and whole blood (\({\bar{V}_{BL}}\), open symbols) velocity versus vessel diameter in all vessels of a simulated vascular network. (C) The ratio \({\bar{V}_{BL}}/{\bar{V}_{RBC}}\) versus vessel diameter. For B and C, red, green, and blue symbols indicate capillaries, arterioles, and venules, respectively.
To understand why VRBC overestimates VBL, we investigate the RBC distribution within a vessel. Figure 5A shows an instantaneous RBC distribution in a vessel segment, and Figure 5B shows the time-averaged RBC concentration and blood velocity distributions over the vessel cross-section. As seen, RBCs tend to distance from the vessel wall, which results in the formation of the plasma-rich CFL. The fluid velocity is higher near the vessel center than near the wall, and RBCs flow with the local fluid velocity; as a result, VRBC is higher than VBL. To investigate if the discrepancy in RBC and whole blood velocity correlates with the CFL, we further compute the thickness of the CFL, denoted as δ, for each vessel. Details of the CFL calculation is given in the Supplementary Materials. The ratio of the CFL thickness to vessel diameter, δ/d, shows an increasing trend with decreasing vessel diameter (Fig. 5C). Note that δ/d is the more relevant physical parameter than δ, because it gives a measure of how significant the layer is relative to the vessel diameter. We then plot \({\bar{V}_{BL}}/{\bar{V}_{RBC}}\) versus δ/d in Figure 5D and find that \({\bar{V}_{BL}}/{\bar{V}_{RBC}}\) decreases with increasing δ/d. The result implies that the discrepancy between \({\bar{V}_{BL}}\) and \({\bar{V}_{RBC}}\) is related to and increases with the proportion of the lumen occupied by the CFL. 
Figure 5.
 
(A) A snapshot of RBC distribution in a vessel segment in a simulated vasculature. (B) Radial distribution of blood velocity (right axis, black curve) and RBC concentration H (left axis, red curve) in the vessel. CFL is marked in A and B by dashed lines. (C) Ratio of CFL thickness to vessel diameter versus vessel diameter for all vessels in one vasculature (R2 = 0.38, P < 0.001). Inset shows CFL versus vessel diameter with no significance dependence. (D) The ratio of blood velocity to RBC velocity in each vessel versus CFL thickness to diameter ratio (R2 = 0.18, P = 0.013). For C and D, red, green, and blue symbols represent capillaries, arterioles and venules, respectively.
Figure 5.
 
(A) A snapshot of RBC distribution in a vessel segment in a simulated vasculature. (B) Radial distribution of blood velocity (right axis, black curve) and RBC concentration H (left axis, red curve) in the vessel. CFL is marked in A and B by dashed lines. (C) Ratio of CFL thickness to vessel diameter versus vessel diameter for all vessels in one vasculature (R2 = 0.38, P < 0.001). Inset shows CFL versus vessel diameter with no significance dependence. (D) The ratio of blood velocity to RBC velocity in each vessel versus CFL thickness to diameter ratio (R2 = 0.18, P = 0.013). For C and D, red, green, and blue symbols represent capillaries, arterioles and venules, respectively.
Note that CFL becomes proportionately smaller as vessel diameter becomes larger (i.e., δ/d becomes small), and \({\bar{V}_{BL}}/{\bar{V}_{RBC}}\) approaches one. Therefore the trend seen in Figure 4C is not a linear one. Also, the CFL thickness can be different between vessels of the same caliber due to variations in hematocrit and flow rate. 
RBC Deformation
RBC deformation is computed as the length between a cell's leading and trailing edges. Figure 6 shows the cell length in each vessel obtained by averaging over all cells that flow through the vessel over the entire pulsatile period. We also compute the cell length during peak and low flow of the pulsatile period. As seen, cell length varies in time within a pulsation period and is higher during the peak of the pulsation cycle. Furthermore, a nonmonotonic trend of the cell length with respect to vessel diameter is predicted. Average lengths are higher in vessels \( \mathbin{\lower.3ex\hbox{$\buildrel<\over {\smash{\scriptstyle \sim}\vphantom{_x}}$}} \)6 µm and \( \mathbin{\lower.3ex\hbox{$\buildrel>\over {\smash{\scriptstyle \sim}\vphantom{_x}}$}} 12\) µm diameter. The minimum cell lengths are predicted in diameter range ∼7–11 µm. For vessel diameter less than ∼7 µm, cell length increases with decreasing diameter, which tends to agree with in vivo observation41,78 and vice versa in the range \( \mathbin{\lower.3ex\hbox{$\buildrel>\over {\smash{\scriptstyle \sim}\vphantom{_x}}$}} 11\) µm. This trend arises because of different flow patterns of RBCs in different diameter vessels, which are also shown in the figure for a few cases. In vessels less than ∼7 µm diameter, the geometric confinement imposed by the vessel causes high elongation of the RBCs. In vessels of diameter \( \mathbin{\lower.3ex\hbox{$\buildrel>\over {\smash{\scriptstyle \sim}\vphantom{_x}}$}} 12\) µm, the blood velocity is generally high, causing the cells to deform more. Also, cells in these vessels flow in double or multi-file pattern. The RBCs flowing near the vessel wall are subjected to higher shear rate compared to those near the vessel center, resulting in higher elongation with their long axis at an inclination to the flow direction. In the intermediate range (∼7–11 µm diameter), RBCs mostly flow in a single-file pattern along the center of the vessel. This causes less geometric confinement by the vessel wall, resulting in the reduced cell length. Heterogeneity in RBC length is also observed as vessels of similar caliber yield different values which is caused by heterogeneous distribution of blood velocity and hematocrit. Specifically, at a higher hematocrit, RBCs stack back-to-back with long axis nearly perpendicular to the flow direction, causing a reduction in cell length. 
Figure 6.
 
Predicted RBC lengths in a network as a function of vessel diameter. Green delta represents average cell length over one pulsatile period. Red squares and black circles represent cell lengths over a 0.2-second window around the peak and low flow rates, respectively. Also shown are visualizations in a few vessel segments illustrating the role of confinement, and different flow patterns on RBC deformation.
Figure 6.
 
Predicted RBC lengths in a network as a function of vessel diameter. Green delta represents average cell length over one pulsatile period. Red squares and black circles represent cell lengths over a 0.2-second window around the peak and low flow rates, respectively. Also shown are visualizations in a few vessel segments illustrating the role of confinement, and different flow patterns on RBC deformation.
Velocity and Hematocrit Oscillations
Analysis of time-dependent hemodynamics is considered next. RBC velocity from several capillaries is presented in Figure 7A, which shows significant fluctuations superimposed on a pulsatile variation, in agreement with previous in vivo observations37,38,79 (also see Figs. 3A, 4A). Similar fluctuations are also predicted in arterioles and venules, as well as for the whole blood velocity (Fig. 4A). The time-dependent hematocrit H in several vessels is presented in Figure 7B, which also shows large fluctuations over time within the pulsation period. A background pulsatile variation is not seen for the hematocrit. Of interest is the quantification of such fluctuations and their correlation. For each vessel, we quantify the velocity fluctuations by computing the mean absolute deviation (MAD) with respect to the background pulsatile profile obtained by filtering out the fluctuations using local averaging (Fig. 7A). For the hematocrit fluctuations, MAD is computed with regard to the time-averaged hematocrit. Figures 7C and 7D show that the MAD of VRBC, VBL and H scaled by their respective time-average quantities is higher in capillaries and decreases with increasing vessel diameter. Furthermore, the MAD of RBC velocity is higher than that of the whole blood velocity. Their difference increases when unscaled MAD is considered since \({\bar{V}_{BL}}\) is less than \({\bar{V}_{RBC}}\) (data not shown). 
Figure 7.
 
Quantifying time-dependent fluctuations. (A) RBC velocity versus time shown in several vessels appears as superposition of a background pulsatile profile (black dashed curve) and fluctuations. (B) Hematocrit versus time shows similar fluctuations, but no pulsatile profile. (C) MAD of RBC (VRBC, filled symbols) and whole blood (VBL, open symbols) velocity fluctuations w.r.t background pulsatile profile. (D) MAD for hematocrit (H′). In C and D, different colors indicate capillaries, arterioles, and venules as noted.
Figure 7.
 
Quantifying time-dependent fluctuations. (A) RBC velocity versus time shown in several vessels appears as superposition of a background pulsatile profile (black dashed curve) and fluctuations. (B) Hematocrit versus time shows similar fluctuations, but no pulsatile profile. (C) MAD of RBC (VRBC, filled symbols) and whole blood (VBL, open symbols) velocity fluctuations w.r.t background pulsatile profile. (D) MAD for hematocrit (H′). In C and D, different colors indicate capillaries, arterioles, and venules as noted.
Velocity and Hematocrit Oscillations and RBC Dynamics
Such fluctuations in velocity and hematocrit arise from the way RBCs interact with the vessels and partition at vascular bifurcations. To investigate this, we plot VRBC and H in four connected vessels in Figures 8A through 8D. The figure shows that for each vessel, the time-history of VRBC and H are generally anti-correlated: at any time, an increase in one quantity is accompanied by a decrease in the other. This is due to the repeated clogging and unclogging of a vessel by RBCs as illustrated by the RBC distributions at two instances in Figures 8E and 8F. This finding is similar to those reported for retinal capillary vessels in vivo.79 Also, clogging and unclogging often occurs cyclically (independent of cardiac rhythm) in two daughter branches of a bifurcation and over a time window that is shorter than the pulsation period. For example, as seen in the figure, at t = 1.05 seconds, vessel 42 receives a higher number of cells causing a spike in the hematocrit and an accompanying drop in the velocity, whereas vessel 43, which comes off the same upstream bifurcation, receives a smaller number of cells causing a drop in the hematocrit and a spike in the velocity. Subsequently, the higher velocity in vessel 43 draws more RBCs as they come out of the bifurcation, causing vessel 42 nearly empty of cells. This leads to the increased hematocrit and reduced velocity in 43, and vice versa in 42 at a later time t = 1.1 seconds. 
Figure 8.
 
RBC dynamics causes fluctuations in velocity and hematocrit. (AD) Time history of RBC velocity (left axis, black solid curve) and hematocrit (right axis, red dash curves) in a few interconnected vessels over one cardiac cycle. The vessels are shown in E and F and identified by numbers 42–44, 46. (E, F) RBC distributions at two instances (also marked in AD by colored vertical lines). Arrows give flow direction. (G) Cross-correlation between RBC velocity and blood velocity (red squares) and between RBC velocity and hematocrit (black circles) in each vessel.
Figure 8.
 
RBC dynamics causes fluctuations in velocity and hematocrit. (AD) Time history of RBC velocity (left axis, black solid curve) and hematocrit (right axis, red dash curves) in a few interconnected vessels over one cardiac cycle. The vessels are shown in E and F and identified by numbers 42–44, 46. (E, F) RBC distributions at two instances (also marked in AD by colored vertical lines). Arrows give flow direction. (G) Cross-correlation between RBC velocity and blood velocity (red squares) and between RBC velocity and hematocrit (black circles) in each vessel.
We further compute the cross-correlations between the time series of VRBC and VBL, and VRBC and H (Fig. 8G). For this, the mean pulsatile profiles were subtracted from the time-series data before calculating the cross-correlation. Although VRBC and VBL are predicted to be positively correlated, VRBC and H are negatively correlated because of the RBC dynamics described above. The cross-correlations, however, do not strongly vary with regard to vessel diameter. 
Spatial Heterogeneity
Even though the time-dependent VRBC and H are negatively correlated, their time-averaged values are positively correlated. Because VRBC and VBL are positively correlated as seen earlier, a positive correlation is also expected for time-averaged \({\bar{V}_{BL}}\) and \(\bar{H}\). This is shown in Figure 9A, where \(\bar{H}\) is seen to increase with increasing \({\bar{V}_{BL}}\). This further means that, on average, more RBCs would flow into a vessel with a higher mean velocity, and vice versa, which is consistent with prior in vivo observations of microcirculatory blood flow.4547,51 Figure 9B presents \(\bar{H}\) versus vessel diameter. The scatter of the data is indicative of spatially heterogeneous distribution of hematocrit as predicted by our model and is known to be a hallmark of microcirculatory blood flow.4548 Among all capillary vessels, \(\bar{H}\) ranges ∼0.14–0.31. Heterogeneity in \(\bar{H}\) is more in capillary vessels than in arterioles and venules. The mean of all capillary vessels \(\bar{H}\) is ∼0.22, which is significantly below the systematic hematocrit, consistent with previous studies on the microcirculation.4348 Additionally, \(\bar{H}\) is predicted to increase with increasing vessel diameter, which is also consistent with previous in vivo studies.4348 
Figure 9.
 
(A, B) Average hematocrit in each vessel plotted against the average whole blood velocity and vessel diameter, respectively. R2 = 0.31, P < 0.001 for A, and R2 = 0.15, P = 0.002 for B. Red, green, and blue symbols correspond to capillaries, arterioles, and venules, respectively. (C, D) Maps of hematocrit and whole blood velocity (mm/s), respectively. Arrows indicate vessels near the FAZ with low velocity and hematocrit.
Figure 9.
 
(A, B) Average hematocrit in each vessel plotted against the average whole blood velocity and vessel diameter, respectively. R2 = 0.31, P < 0.001 for A, and R2 = 0.15, P = 0.002 for B. Red, green, and blue symbols correspond to capillaries, arterioles, and venules, respectively. (C, D) Maps of hematocrit and whole blood velocity (mm/s), respectively. Arrows indicate vessels near the FAZ with low velocity and hematocrit.
Maps of \(\bar{H}\) and \({\bar{V}_{BL}}\) are presented in Figures 9C and 9D. Spatial heterogeneity of both quantities can also be noted here. Specifically, a few terminal capillaries bordering the FAZ are predicted to have reduced flow and hematocrit because of longer paths for the blood to travel from the feeding arteriole (shown by arrows in the figure). 
The 3D Velocity and Hematocrit Profiles
As noted previously, our model provides full 3D velocity and hematocrit distributions in every vessel. From this, we find that velocity and hematocrit distributions over a vessel cross-section are not symmetric about the vessel center; rather they are skewed (Fig. 10). Such skewed distributions are due to the presence of vascular bifurcations and vessel curvature. When RBCs flow through the daughter vessels of a bifurcation, they flow next to the side of a branch that is closer to the apex of the bifurcation (Fig. 10A). This creates a wider plasma layer along the opposite side. Consequently, the hematocrit distribution becomes skewed to the side of the vessel that is closer to the apex, whereas the velocity becomes skewed to the other side (Fig. 10B). We compute the skewness of the hematocrit and velocity distributions, SH and SV, respectively. If the distribution is skewed to the right side of a vessel looking down the flow direction, the skewness is positive, and vice versa, whereas it is zero for a symmetric distribution. We plot SH versus SV in Figure 10E immediate downstream from a bifurcation and in Figure 10F near the end of a vessel. As seen, for most vessels, SH and SV have opposite signs, meaning that hematocrit and velocity are oppositely skewed. Higher magnitudes of SH and SV are predicted immediately after the bifurcations compared to those near the vessel end. As the flow develops along a vessel, the asymmetry in velocity and hematocrit tends to reduce due to the cross-streamline movement of the RBCs toward the vessel center. This causes a reduction in the magnitudes of SH and SV and their scatter near the vessel end. 
Figure 10.
 
(A) An instantaneous image showing RBCs flow through a bifurcation. RBCs flow along the side of a daughter vessel that is closer to the apex of a bifurcation. (B) Resulting blood velocity and hematocrit distributions are skewed to opposite sides in the daughter vessels as marked 1 and 2. (C, D) The 2D profiles of velocity (black) and hematocrit (red) along the vessel diameter for 1 and 2, respectively. (E) Velocity skewness SV versus hematocrit skewness SH at the beginning and (F) end of a vessel, respectively. Red, green, and blue symbols represent capillaries, arterioles, and venules, respectively.
Figure 10.
 
(A) An instantaneous image showing RBCs flow through a bifurcation. RBCs flow along the side of a daughter vessel that is closer to the apex of a bifurcation. (B) Resulting blood velocity and hematocrit distributions are skewed to opposite sides in the daughter vessels as marked 1 and 2. (C, D) The 2D profiles of velocity (black) and hematocrit (red) along the vessel diameter for 1 and 2, respectively. (E) Velocity skewness SV versus hematocrit skewness SH at the beginning and (F) end of a vessel, respectively. Red, green, and blue symbols represent capillaries, arterioles, and venules, respectively.
WSS
Method to calculate WSS is described in the Supplementary Materials. The predicted 3D velocity distribution further allows us to compute 3D distribution of WSS as shown in Figure 11A. Our results show a strong heterogeneity in WSS across the network. The WSS distribution shows the focal nature as well as a strong spatial gradient even within a single vessel. The 1D network blood flow models and most prior in vivo studies provide one value of WSS per vessel.49,5153,55 In contrast, our model predicts that there are significant spatial variations of WSS within a vessel. We also compute WSS in the same network considering a cell-free Newtonian fluid (Fig. 11B), which does not show such strong spatial WSS gradients within a vessel. Heterogeneity of WSS across the network for the Newtonian fluid is also less. One reason why such focal distribution of WSS occurs is because RBCs do not flow in a continuous manner; rather, individual or a group of RBCs flow followed by a plasma gap.79 Also, the number of RBCs in any group varies. Generally, a higher number of RBCs tends to locally reduce the CFL and creates a higher velocity gradient leading to the higher WSS. 
Figure 11.
 
(A, B) The 3D distribution of WSS in the full (RBC included) model and using cell-free Newtonian fluid, respectively. Thick arrows indicate focal nature of WSS; circles indicate high WSS regions near the bifurcations; thin arrows indicate WSS variation in high curvature regions. (C, D) Close-up view of the WSS and its gradient at a bifurcation and RBC lingering causing such high values. (E, F) The WSS distribution in a curved vessel and the skewed velocity profile that causes such. Dashed line indicates the location for which the velocity profile is shown. (G) WSS per vessel as a function of diameter. Red and green symbols correspond to peak and low of the pulsatile cycle, and black symbols represent the average over the entire cycle. (H) Comparison of average WSS from the full (RBC included) model (black symbols) and cell-free Newtonian fluid model (blue). (IK) WSS per vessel against whole blood velocity, CFL, and hematocrit.
Figure 11.
 
(A, B) The 3D distribution of WSS in the full (RBC included) model and using cell-free Newtonian fluid, respectively. Thick arrows indicate focal nature of WSS; circles indicate high WSS regions near the bifurcations; thin arrows indicate WSS variation in high curvature regions. (C, D) Close-up view of the WSS and its gradient at a bifurcation and RBC lingering causing such high values. (E, F) The WSS distribution in a curved vessel and the skewed velocity profile that causes such. Dashed line indicates the location for which the velocity profile is shown. (G) WSS per vessel as a function of diameter. Red and green symbols correspond to peak and low of the pulsatile cycle, and black symbols represent the average over the entire cycle. (H) Comparison of average WSS from the full (RBC included) model (black symbols) and cell-free Newtonian fluid model (blue). (IK) WSS per vessel against whole blood velocity, CFL, and hematocrit.
We further find a highly elevated WSS and its gradient near the capillary bifurcations. This is attributed to the specific RBC dynamics at such locations. RBCs often are observed to “slow down” near the apex of the bifurcation, which is a flow stagnation region. A few RBCs linger behind the leading RBC that straddles around the bifurcation apex causing a partial blockage and reduction of the CFL in this region (Figs. 11C, 11D). The blood plasma then escapes through the narrow CFL, and a high-velocity gradient is formed leading to a high local WSS. The same mechanism also creates a high spatial gradient of WSS around the capillary bifurcations. 
A higher spatial gradient of WSS is also predicted in vessel regions with a high curvature. This is because at low inertia flow in capillary vessels, the velocity profiles are skewed toward the side of a vessel that has a higher curvature, unlike in large blood vessels where the trend is opposite.80,81 As a result, the WSS on this side of the vessel is higher compared to the opposite side (Figs. 11E, 11F). For the example vessel shown, the average WSS for the entire vessel is 28 dyn/cm2, whereas the WSS on the higher and lower curvature sides is 40 and 18 dyn/cm2, respectively. 
Average WSS per vessel as a function of diameter is shown in Figure 11G. Also shown are the WSS during the peak and low of the pulsatile cycle. As seen, the WSS in each vessel varies within the pulsatile cycle with higher values occurring around the peak and vice versa. The ratio of maximum to minimum WSS in any vessel within the cycle varies from about 1.3 to 2 across the network, suggesting a strong temporal heterogeneity. 
The data in Figure 11G show that WSS is a function of vessel diameter, unlike Murray's law in which WSS remains constant across a network.82 Furthermore, significantly different values of WSS are predicted for vessels of similar caliber, which indicates a further deviation from Murray's law. This variation of WSS for the same vessel caliber is due to the spatial heterogeneity of the hematocrit and blood velocity discussed earlier. Consistent with the in vivo findings, WSS in the arterioles is predicted to be higher than in venules.49 The WSS per vessel obtained from the Newtonian model is also compared against the full model in Figure 11H and found to be significantly less. Figures 11I through 11K show WSS against whole blood velocity, CFL, and hematocrit. A strong positive correlation exists between WSS and blood velocity, and a moderate inverse relation exists between WSS and CFL, whereas no direct relationship can be inferred between WSS and H
Discussion
In this article, we extended a previously developed 3D, high-fidelity computational model to predict blood cell trafficking and microvascular hemodynamics in the human retina. Contrary to 1D network blood flow models, as well as most in vivo imaging techniques, our model provides full 3D profiles of blood velocity, RBC concentration, and WSS in every vessel including the smallest diameter capillaries with single file RBC flow and larger diameter vessels with multifile flow. In recent years, 3D models of the retinal microvascular networks have been considered, but without the inclusion of RBCs and using either Newtonian or non-Newtonian viscosity laws.5659 The use of the Newtonian viscosity has limitations as blood in the microcirculation is known to have non-Newtonian rheology. Also, the non-Newtonian viscosity models (e.g., the widely used Carreau-Yasuda and power-law models) involve empirical constants, the choice of which may affect the prediction. 
We provide quantitative predictions of the discrepancy between the whole blood velocity and the RBC velocity, and predict that their ratio varies in the range 0.65 to 0.85 for vessels <25 µm in diameter. What causes the difference between the whole blood and RBC velocity is the tendency of RBCs to move away from the vessel wall resulting in the formation of the CFL? We provide quantitative data on the CFL thickness and show that the whole blood to RBC velocity ratio decreases with increasing CFL thickness to diameter ratio. The data presented here can be used as a correction to in vivo measured RBC velocity to obtain the whole blood velocity. It may be noted that in addition to vessel diameter, the CFL thickness also depends on the vessel hematocrit and flow rate. This leads to a heterogeneous distribution of the CFL thickness as predicted by our model. 
Spatial heterogeneity of hemodynamic quantities is a hallmark of the microcirculation and is predicted by our model. Although the hematocrit in capillaries is predicted to be less than the systematic value in agreement with in vivo observations,4348 more heterogeneity in the hematocrit is found in capillary vessels than in arterioles and venules. Time-averaged hematocrit is predicted to increase with increasing blood velocity, implying that more RBCs are drawn into vessels with higher flow, a result that is also consistent with the so-called RBC partitioning at vascular bifurcations as observed in vivo46 (also see Supplementary Materials). Specific to the retinal vasculature, our model predicts a few terminal capillaries bordering the FAZ to have reduced flow and hematocrit. It suggests that these vessels could potentially regress in the early phase of retinal diseases, leading to an enlargement of the FAZ.21 
Recently, adaptive optics has been used to visualize the deformed RBC shapes in the human and mouse retina for single and double file flow.36,41,78 However, the technique may not be extendable to the multifile flow because of cell crowding, which obfuscates tracking of cells. Here we provide quantitative data on the RBC length in vessels 5.5 to 25 µm diameter, covering an entire range of capillaries, arterioles and venules and different RBC flow patterns. We find a nonmonotonic trend with longer RBC shapes in small diameter capillaries and in large-diameter arterioles and venules, but shorter cells in large-diameter capillaries. We attribute this finding to different patterns of RBC flow and geometric confinement. We further find that RBC lengths within a vessel vary with time due to the flow pulsatility. These quantitative data may help improve the interpretation of RBC deformation from adaptive optics images. 
Our model also predicts temporal heterogeneity of the hemodynamic quantities. Specifically, we predict high-frequency oscillations (approximately <100 ms duration) in the hematocrit and RBC velocity, with the latter superimposed on the background pulsatile profile, similar to those reported in previous in vivo studies with human retina.37,38,79 Such oscillations in vivo may arise for multiple reasons, including the passage of a WBC a closely packed train of RBCs or RBC aggregates, or a momentary increase or drop in flow rate or hematocrit caused by some events in neighboring vessels. Here, we show that these oscillations may result from the way RBCs partition at a vascular bifurcation. On a short time scale, smaller vessels downstream from a bifurcation repeatedly clog and unclog resulting in increased hematocrit and reduced RBC velocity, and vice versa. As such, the time-dependent RBC velocity (and, hence, the whole blood velocity) and hematocrit are anti-correlated within a vessel, even though their time averages are positively correlated across vessels. We provide a quantitative analysis of these oscillations and show that they have higher amplitudes and more heterogeneity in capillaries than in arterioles and venules, implying that capillary hemodynamics are more affected by the passage of individual cells. 
Our model provides the full 3D profiles of the whole blood velocity and RBC concentration in every vessel of a network. Recently, adaptive optics scanning laser ophthalmoscope (AOSLO) techniques have been used to obtain 2D profiles (i.e., along a vessel diameter) of the velocity.39 This method requires enough number of RBCs flowing across a vessel diameter and, hence, large enough diameter (>20 µm reported in reference 39) to generate a smooth and continuous velocity profile. As the vessel diameter decreases, causing a reduction in the number of RBCs across the diameter, particularly for single- and double-file RBC flows, a continuous velocity profile may be difficult to obtain. Also, approximations are needed to extrapolate the velocity profile to the CFL. Such issues are not present in the current model which provides a continuous velocity profile for all regimes of flow and vessel types. Our predicted 3D blood velocity and RBC concentration profiles show highly asymmetric distributions due to the presence of bifurcations and tortuous vessels. Furthermore, in most vessels, they are biased to the opposite sides because of the way RBCs flow into the daughter vessels immediately after a bifurcation. 
Predicted 3D velocity profiles allow us to compute the 3D distribution of WSS and its gradient. The average WSS per vessel ranges from ∼10–80 dyn/cm2, in agreement with in vivo data.49,51,83,84 Our model yields WSS gradient values as high as ∼104 dyn/cm3. This value is much higher than the WSS gradient known to trigger endothelial cell response in vitro (∼103 dyn/cm3).85 The predicted WSS variation from one vessel to another in a network, and from one location to another within a vessel both contradict Murray's law in which WSS remains constant across a network. This deviation from Murray's law happens, in part, due to the assumptions of the law, such as constant blood viscosity, which is not the case as dictated by the Fahraeus-Lindqvist effect.86 This deviation was observed previously in the microcirculation.49,51,83,84 Predicted spatial variations in WSS are attributed to the local RBC dynamics. 
Limitations of the current study and consideration for future development may be noted. Although the model is 3D (i.e., cylindrical-shaped vessels), the input en face images are 2D and taken from the superficial layer bordering the FAZ. It may be noted that OCTA can provide 3D maps, but most images provided are planar. In real 3D vasculature, some capillaries move up and down across different layers, which is neglected in our model. Consideration of the full retinal microvascular network which consists of three primary layers with connecting vessels is currently underway.67 Furthermore, RBC trafficking in the presence of vascular abnormalities, such as microaneurysms of different shapes, occluded vessels, and highly tortuous vessels, that are hallmarks of many retinal diseases, can be modeled.16,21,22,2631 In the present study, the vessels are modeled as nondeformable conduits. Work is currently underway to model flow of blood cells in deforming vessels with the goal of modeling blood flow regulation which is critical to maintain the perfusion.4,11,87 Furthermore, our model allows provisions to alter RBC deformability88; thus it can be used to predict hemodynamic alterations in retinal diseases associated with altered RBC deformability, such as diabetic and sickle cell retinopathy.89,90 
Acknowledgments
The authors thank Computational resources at Texas Advanced Computing Center, Anvil cluster in Purdue University, and Rutgers University. 
Supported by a grant from the National Institute of Health (R01EY033003). 
Disclosure: S. Ebrahimi, None; P. Bedggood, None; Y. Ding, None; A. Metha, None; P. Bagchi, None 
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Figure 1.
 
(AC) In vivo images of the human parafoveal microvascular networks obtained from references 4 and 30 (with permission). Green box indicates the tissue area modeled. Corresponding in silico models and RBC distribution as predicted by our model at one time instant are shown in DF. Arrows indicate flow directions at inlets and outlets, and a and v indicate arterioles and venules, respectively. Animations of predicted RBC flow in the modeled in silico vasculatures are given in Supplementary Materials.
Figure 1.
 
(AC) In vivo images of the human parafoveal microvascular networks obtained from references 4 and 30 (with permission). Green box indicates the tissue area modeled. Corresponding in silico models and RBC distribution as predicted by our model at one time instant are shown in DF. Arrows indicate flow directions at inlets and outlets, and a and v indicate arterioles and venules, respectively. Animations of predicted RBC flow in the modeled in silico vasculatures are given in Supplementary Materials.
Figure 2.
 
(A) Triangular mesh on vessel surface in a small segment of a microvasculature in silico. (BD) Resting shapes and sizes of a RBC, a mononucleated WBC, and a platelet. An example mesh on the RBC surface is shown. Biophysical properties of the cells used in the model are also noted. For WBCs and platelets, shear and area dilation moduli are taken 10 times higher since they are much less deformable than RBCs. (E) Pulsatile profile specified at an inlet vessel. (FI) Different RBC flow patterns as predicted by the model are shown in a few vessels. (F) Single-file RBC flow in one capillary, (H) double file and zig-zag patterns, and (G, I) multi-file flow in larger vessels. RBCs are shown in red, platelets in yellow, and WBC in gray.
Figure 2.
 
(A) Triangular mesh on vessel surface in a small segment of a microvasculature in silico. (BD) Resting shapes and sizes of a RBC, a mononucleated WBC, and a platelet. An example mesh on the RBC surface is shown. Biophysical properties of the cells used in the model are also noted. For WBCs and platelets, shear and area dilation moduli are taken 10 times higher since they are much less deformable than RBCs. (E) Pulsatile profile specified at an inlet vessel. (FI) Different RBC flow patterns as predicted by the model are shown in a few vessels. (F) Single-file RBC flow in one capillary, (H) double file and zig-zag patterns, and (G, I) multi-file flow in larger vessels. RBCs are shown in red, platelets in yellow, and WBC in gray.
Figure 3.
 
Comparison of the model predictions against in vivo data reported for human retinal microcirculation. (A) Pulsatile profile of the RBC velocity as predicted by the model (continuous curves in color) in a few capillary vessels compared against Bedggood & Metha38 (dash black curve). (B) Predicted pulsatility index and minimum RBC velocity Vmin compared against Neriyanuri et al.40 and Gu et al.37. (C) RBC lineal density predicted against Gu et al.77
Figure 3.
 
Comparison of the model predictions against in vivo data reported for human retinal microcirculation. (A) Pulsatile profile of the RBC velocity as predicted by the model (continuous curves in color) in a few capillary vessels compared against Bedggood & Metha38 (dash black curve). (B) Predicted pulsatility index and minimum RBC velocity Vmin compared against Neriyanuri et al.40 and Gu et al.37. (C) RBC lineal density predicted against Gu et al.77
Figure 4.
 
Quantification of RBC velocity and whole blood velocity differences. (A) Pulsatile profiles of RBC velocity (dash) and whole blood velocity (solid) in one vessel. (B) Time-averaged RBC (\({\bar{V}_{RBC}}\), filled symbols) and whole blood (\({\bar{V}_{BL}}\), open symbols) velocity versus vessel diameter in all vessels of a simulated vascular network. (C) The ratio \({\bar{V}_{BL}}/{\bar{V}_{RBC}}\) versus vessel diameter. For B and C, red, green, and blue symbols indicate capillaries, arterioles, and venules, respectively.
Figure 4.
 
Quantification of RBC velocity and whole blood velocity differences. (A) Pulsatile profiles of RBC velocity (dash) and whole blood velocity (solid) in one vessel. (B) Time-averaged RBC (\({\bar{V}_{RBC}}\), filled symbols) and whole blood (\({\bar{V}_{BL}}\), open symbols) velocity versus vessel diameter in all vessels of a simulated vascular network. (C) The ratio \({\bar{V}_{BL}}/{\bar{V}_{RBC}}\) versus vessel diameter. For B and C, red, green, and blue symbols indicate capillaries, arterioles, and venules, respectively.
Figure 5.
 
(A) A snapshot of RBC distribution in a vessel segment in a simulated vasculature. (B) Radial distribution of blood velocity (right axis, black curve) and RBC concentration H (left axis, red curve) in the vessel. CFL is marked in A and B by dashed lines. (C) Ratio of CFL thickness to vessel diameter versus vessel diameter for all vessels in one vasculature (R2 = 0.38, P < 0.001). Inset shows CFL versus vessel diameter with no significance dependence. (D) The ratio of blood velocity to RBC velocity in each vessel versus CFL thickness to diameter ratio (R2 = 0.18, P = 0.013). For C and D, red, green, and blue symbols represent capillaries, arterioles and venules, respectively.
Figure 5.
 
(A) A snapshot of RBC distribution in a vessel segment in a simulated vasculature. (B) Radial distribution of blood velocity (right axis, black curve) and RBC concentration H (left axis, red curve) in the vessel. CFL is marked in A and B by dashed lines. (C) Ratio of CFL thickness to vessel diameter versus vessel diameter for all vessels in one vasculature (R2 = 0.38, P < 0.001). Inset shows CFL versus vessel diameter with no significance dependence. (D) The ratio of blood velocity to RBC velocity in each vessel versus CFL thickness to diameter ratio (R2 = 0.18, P = 0.013). For C and D, red, green, and blue symbols represent capillaries, arterioles and venules, respectively.
Figure 6.
 
Predicted RBC lengths in a network as a function of vessel diameter. Green delta represents average cell length over one pulsatile period. Red squares and black circles represent cell lengths over a 0.2-second window around the peak and low flow rates, respectively. Also shown are visualizations in a few vessel segments illustrating the role of confinement, and different flow patterns on RBC deformation.
Figure 6.
 
Predicted RBC lengths in a network as a function of vessel diameter. Green delta represents average cell length over one pulsatile period. Red squares and black circles represent cell lengths over a 0.2-second window around the peak and low flow rates, respectively. Also shown are visualizations in a few vessel segments illustrating the role of confinement, and different flow patterns on RBC deformation.
Figure 7.
 
Quantifying time-dependent fluctuations. (A) RBC velocity versus time shown in several vessels appears as superposition of a background pulsatile profile (black dashed curve) and fluctuations. (B) Hematocrit versus time shows similar fluctuations, but no pulsatile profile. (C) MAD of RBC (VRBC, filled symbols) and whole blood (VBL, open symbols) velocity fluctuations w.r.t background pulsatile profile. (D) MAD for hematocrit (H′). In C and D, different colors indicate capillaries, arterioles, and venules as noted.
Figure 7.
 
Quantifying time-dependent fluctuations. (A) RBC velocity versus time shown in several vessels appears as superposition of a background pulsatile profile (black dashed curve) and fluctuations. (B) Hematocrit versus time shows similar fluctuations, but no pulsatile profile. (C) MAD of RBC (VRBC, filled symbols) and whole blood (VBL, open symbols) velocity fluctuations w.r.t background pulsatile profile. (D) MAD for hematocrit (H′). In C and D, different colors indicate capillaries, arterioles, and venules as noted.
Figure 8.
 
RBC dynamics causes fluctuations in velocity and hematocrit. (AD) Time history of RBC velocity (left axis, black solid curve) and hematocrit (right axis, red dash curves) in a few interconnected vessels over one cardiac cycle. The vessels are shown in E and F and identified by numbers 42–44, 46. (E, F) RBC distributions at two instances (also marked in AD by colored vertical lines). Arrows give flow direction. (G) Cross-correlation between RBC velocity and blood velocity (red squares) and between RBC velocity and hematocrit (black circles) in each vessel.
Figure 8.
 
RBC dynamics causes fluctuations in velocity and hematocrit. (AD) Time history of RBC velocity (left axis, black solid curve) and hematocrit (right axis, red dash curves) in a few interconnected vessels over one cardiac cycle. The vessels are shown in E and F and identified by numbers 42–44, 46. (E, F) RBC distributions at two instances (also marked in AD by colored vertical lines). Arrows give flow direction. (G) Cross-correlation between RBC velocity and blood velocity (red squares) and between RBC velocity and hematocrit (black circles) in each vessel.
Figure 9.
 
(A, B) Average hematocrit in each vessel plotted against the average whole blood velocity and vessel diameter, respectively. R2 = 0.31, P < 0.001 for A, and R2 = 0.15, P = 0.002 for B. Red, green, and blue symbols correspond to capillaries, arterioles, and venules, respectively. (C, D) Maps of hematocrit and whole blood velocity (mm/s), respectively. Arrows indicate vessels near the FAZ with low velocity and hematocrit.
Figure 9.
 
(A, B) Average hematocrit in each vessel plotted against the average whole blood velocity and vessel diameter, respectively. R2 = 0.31, P < 0.001 for A, and R2 = 0.15, P = 0.002 for B. Red, green, and blue symbols correspond to capillaries, arterioles, and venules, respectively. (C, D) Maps of hematocrit and whole blood velocity (mm/s), respectively. Arrows indicate vessels near the FAZ with low velocity and hematocrit.
Figure 10.
 
(A) An instantaneous image showing RBCs flow through a bifurcation. RBCs flow along the side of a daughter vessel that is closer to the apex of a bifurcation. (B) Resulting blood velocity and hematocrit distributions are skewed to opposite sides in the daughter vessels as marked 1 and 2. (C, D) The 2D profiles of velocity (black) and hematocrit (red) along the vessel diameter for 1 and 2, respectively. (E) Velocity skewness SV versus hematocrit skewness SH at the beginning and (F) end of a vessel, respectively. Red, green, and blue symbols represent capillaries, arterioles, and venules, respectively.
Figure 10.
 
(A) An instantaneous image showing RBCs flow through a bifurcation. RBCs flow along the side of a daughter vessel that is closer to the apex of a bifurcation. (B) Resulting blood velocity and hematocrit distributions are skewed to opposite sides in the daughter vessels as marked 1 and 2. (C, D) The 2D profiles of velocity (black) and hematocrit (red) along the vessel diameter for 1 and 2, respectively. (E) Velocity skewness SV versus hematocrit skewness SH at the beginning and (F) end of a vessel, respectively. Red, green, and blue symbols represent capillaries, arterioles, and venules, respectively.
Figure 11.
 
(A, B) The 3D distribution of WSS in the full (RBC included) model and using cell-free Newtonian fluid, respectively. Thick arrows indicate focal nature of WSS; circles indicate high WSS regions near the bifurcations; thin arrows indicate WSS variation in high curvature regions. (C, D) Close-up view of the WSS and its gradient at a bifurcation and RBC lingering causing such high values. (E, F) The WSS distribution in a curved vessel and the skewed velocity profile that causes such. Dashed line indicates the location for which the velocity profile is shown. (G) WSS per vessel as a function of diameter. Red and green symbols correspond to peak and low of the pulsatile cycle, and black symbols represent the average over the entire cycle. (H) Comparison of average WSS from the full (RBC included) model (black symbols) and cell-free Newtonian fluid model (blue). (IK) WSS per vessel against whole blood velocity, CFL, and hematocrit.
Figure 11.
 
(A, B) The 3D distribution of WSS in the full (RBC included) model and using cell-free Newtonian fluid, respectively. Thick arrows indicate focal nature of WSS; circles indicate high WSS regions near the bifurcations; thin arrows indicate WSS variation in high curvature regions. (C, D) Close-up view of the WSS and its gradient at a bifurcation and RBC lingering causing such high values. (E, F) The WSS distribution in a curved vessel and the skewed velocity profile that causes such. Dashed line indicates the location for which the velocity profile is shown. (G) WSS per vessel as a function of diameter. Red and green symbols correspond to peak and low of the pulsatile cycle, and black symbols represent the average over the entire cycle. (H) Comparison of average WSS from the full (RBC included) model (black symbols) and cell-free Newtonian fluid model (blue). (IK) WSS per vessel against whole blood velocity, CFL, and hematocrit.
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