In Silico Investigation of the RBC Velocity Fluctuations in Ex Vivo Capillaries

Abstract

A properly functioning capillary microcirculation is essential for sufficient oxygen and nutrient delivery to the central nervous system. The physical mechanisms governing the transport of red blood cells (RBCs) inside the narrow and irregularly shaped capillary lumen are complex, but understanding them is essential for identifying the root causes of neurological disorders like cerebral ischemia, Alzheimer’s disease, and other neurodegenerative conditions such as concussion and cognitive dysfunction in systemic inflammatory conditions. In this work, we conducted numerical simulations of three-dimensional capillary models, which were acquired ex vivo from a mouse retina, to characterize RBC transport. We show how the spatiotemporal velocity of the RBCs deviates in realistic capillaries and equivalent cylindrical tubes, as well as how this profile is affected by hematocrit and red cell distribution width (RDW). Our results show a previously unprecedented level of RBC velocity fluctuations in capillaries that depends on the geometric features of different confinement regions and a capillary circularity index $\left(I_{cc}\right)$ that represents luminal irregularity. This velocity fluctuation is aggravated by high hematocrit conditions, without any further effect on RDW. These results can provide a better understanding of the underlying mechanisms of pathologically high capillary transit time heterogeneity that results in microcirculatory dysfunction.

1. Introduction

The importance of capillary blood flow was stated back in 1922 by Freedlander and Lenhart as follows: “It is evident that the rest of the cardiovascular system exist only to regulate the blood flow through the capillaries” [1]. Indeed, the high and dynamic energy requirements of the central nervous system (CNS) require a continuous supply of oxygen and nutrients, which must be provided by a highly efficient microcirculation. As most acute and chronic neurodegenerative conditions have a component of microcirculatory dysfunction in their pathogenesis, it is necessary to understand the physical mechanisms that govern cell transport in capillaries.

Capillary dysfunction arises when there is extremely high capillary transit time heterogeneity (CTTH) in the microcirculatory network [2]. High CTTH results in rapid oxygenated RBC shunting, leading to suboptimal tissue oxygenation, even when there is no flow-limiting condition in the large arteries that feed the capillary network [3,4]. The underlying causes of high CTTH are subject to active investigation. At any given time, cell fluxes, flow velocities, and hematocrit distributions can vary significantly between individual interconnected parenchymal capillaries. Many in vivo studies have suggested prominent fluctuations in cell flux and hematocrit level in a given capillary [5,6,7]. Observed heterogeneity, studied by utilizing ex vivo, in vitro, and in silico techniques, suggests possible contributors such as RBC lingering at the bifurcation apex, vessel plugging by leukocytes or platelets, and changes in the diameters of vessels upstream and downstream [6,8,9,10,11,12]. However, despite its importance, the detailed biomechanics of RBC flow in relation to natural luminal irregularities and its contribution to flow heterogeneity are still relatively underexplored. In silico investigations therefore present a significant opportunity to deepen this understanding.

Several numerical approaches have been developed over the years to investigate blood flow in silico. For large vessels, where the diameter is sufficient to create a cell-free layer (CFL), also known as the Fåhræus–Lindqvist effect, it is possible to determine the hematocrit and/or the bulk velocity of the blood using correlation-based models and non-Newtonian viscosity assumptions [13,14,15,16]. In correlation-based models, the diameter of the vessels and the inlet hematocrit (Ht) ratio determine the local viscosity, and the capillary networks can be modeled and solved with Hardy-Cross-like methods [17]. Similarly, a single-phase non-Newtonian viscosity assumption can be a valid and useful approach if a CFL is present [16]. In this approach, it can be assumed that the viscosity of the single-phase fluid can be assumed to be shear-thinning (such as a Casson fluid), and the viscosity of the bulk fluid will be lower near the walls, where the shear rate is high and the Fåhræus–Lindqvist effect prevails. Both of these approaches yield good results that match the experimental data, as long as the capillary diameter is large enough to create a CFL.

However, these approaches ignore the local dynamics of RBC transport, such as RBC-to-RBC and RBC-to-wall interactions, as well as the non-linear behavior of RBCs due to their hyperelastic membrane when a CFL is not present. In flow configurations where RBCs move in a single-file manner [18], such as narrow capillaries, the assumptions of these models are not valid. Hence, these are low-fidelity approaches for hemodynamics in small channels and cannot be used to understand the local dynamics of RBCs. For small and realistic capillaries, it is essential to model the RBCs explicitly, since the capillary diameters are between 5 and 10 μm, which means that a CFL is not present.

To the best of our best knowledge, the literature review does not show a direct comparison of the differences in RBC transit time between realistic capillaries that harbor luminal irregularities and their equivalent tube geometries. Detailed physical principles that affect the motion of multiple RBCs in an irregular capillary lumen, which represent the actual biological condition, and how flux dynamics are regulated by differences in capillary shape and size have not been studied in detail. Most prior in silico studies have, in fact, used idealized capillaries with regular luminal profiles and in some cases did not take into account interactions between multiple RBCs in cases of varying hematocrit levels and variations in red blood cell diameter distribution. For this task, we used mouse retina capillaries, one of the most active zones of the CNS with high metabolic demand and therefore one of the most vulnerable tissues. Understanding how RBC transport is affected by capillary geometry, variations in inlet hematocrit, and RDW can be very useful to demonstrate the possible origins of elevated capillary transit time heterogeneity and microcirculatory dysfunction, which are relatively unexplored topics in hemodynamic physiology.

2. Physical and Mathematical Description of the Problem

In a healthy person, blood consists of 55% plasma. We can reasonably assume that plasma behaves similarly to water, as only 8% of its composition comprises microscale proteins and solid particles, with the remainder being water [19]. The remaining 45% of the blood is occupied by RBCs, with less than 1% being other blood cells.

RBCs are biconcave, cytoplasm-filled blood cells that are primarily responsible for the transport of oxygen and nutrients to tissues. Although the cytoplasm viscosity is five times greater than that of plasma, for computational simplification we assume that it has the same viscosity as plasma.

The ex vivo capillary segments used in this study are third-order capillaries, meaning that they are located after the third bifurcation following the arteriole, as illustrated in Figure 1a. These specific capillary geometries were sourced from an external work (see the Acknowledgment Section), ensuring that no animals were sacrificed exclusively for this study. Although the exact method to obtain the STL files is beyond the scope of the current study, the following explanation is quoted from the experimentalist regarding geometry preparation as it was asked by the reviewer: “Immediately prior to animal termination, retinas were extracted and fixed with methanol, and capillaries were perfused with gelatin, with no other tissue processing performed. This approach ensured that the three-dimensional models, acquired with confocal microscopy, are representative of their actual in vivo conditions, a comparison previously validated [20]”. The average hydraulic diameter of these capillaries and their deviation are shown in Figure 1b. These measured values align with prior ex vivo and in vivo measurements from similarly ordered capillaries in the mouse retina [21,22].

To facilitate the comparison, equivalent straight tubes were generated using the average hydraulic diameter of the corresponding capillaries. Since our focus is solely on the transport of RBCs within single channels, both plasma and RBCs enter the channels from an inlet and exit from an outlet. The flow rates for the different capillaries were determined on the basis of their average hydraulic diameter, while the flow rates of the equivalent tubes were matched to their corresponding capillaries. Throughout these simulations, the flow is consistently assumed to be isothermal.

2.1. Fluid Modeling

It is well known that the velocity of RBCs inside the capillaries is in the range of 0.4–1.8 mm/s [23]. However, it is physiologically impossible to determine a single velocity that is valid for all capillaries. The velocity in each individual capillary is regulated by the downstream and upstream tube hematocrit due to partitioning and lingering events [8,11,24,25]. This means that the inlet velocity (and consequently the flow rate) of the capillaries is inherently a transient phenomenon. For comparison purposes, we set the maximum average inlet velocity of the largest capillary (S4) to 1.5 mm/s, based on the maximum average velocity from the experimental study by Ivanov et al. [23]. The average inlet velocities of the remaining capillaries are calculated by adapting an ‘inverse’ Hagen–Poiseuille relation, where velocity is assumed to be inversely proportional to the square of the average hydraulic diameter. This approach ensures that the flow rate is reduced when the average hydraulic diameter is lower, which is physiologically accurate.

The flow field is solved using LBM (D3Q19) [26,27] with the BGK operator [28] and the forcing term [29]. A fine lattice distance of $\Delta x=0.25$ μm is used for spatial discretization to accurately solve the flow field in narrow capillaries, and $\tau \approx 1$ prevails for all simulations.

2.2. RBC Modeling

RBCs are elastic objects, and in this study we assume that they are composed of discrete elements that form a thin elastic membrane immersed in the surrounding incompressible fluid. We use the FEM-based Zavodszky material model [30] to represent the RBCs and the immersed boundary method (IBM) [31,32] for fluid–solid coupling. The forces acting on the RBC are $F_{link}$, $F_{bend}$, $F_{area}$, and $F_{volume}$. We assume that during small deformations, all forces change linearly and are independent of each other.

Equation (1) shows the link force, which acts between surface points and is responsible for the response to the stretching and compression of the spectrin network. Here, p is a biological value, specifically the spectrin link length, taken as $7.5$ pN [33]. The relative extension is given by $dL=(L_i-L_0)/L_0$, where $L_0$ is the initial distance between the vertices.

Equation (2) shows the bending force, which acts between neighboring faces and is responsible for the bending response of the RBC membrane. This response is due to the ∼29 nm thickness of the spectrin network [34], where $d\theta =({\theta }_i-{\theta }_0)/{\theta }_0$, and ${\theta }_0$ is the initial angle between the faces.

Equation (3) shows the area force, which represents the membrane’s response to stretching and compression at the surface area of each element. Here, $dA=(A_i-A_0)/A_0$, where $A_0$ is the initial area of the triangle.

Equation (4) shows the volume force, a global force that enforces a pseudoincompressibility to the RBC. Here, $dV=(V_i-V_0)/V_0$, and $V_0$ is the initial volume of the RBC. This force ensures that the volume is conserved at a specified ${\tau }_{vol}$.

The model parameters are provided in

Table 1. RBC material model constants.

$F_{\mathit{link}}$	15	${\tau }_{\mathit{link}}$	3
$F_{bend}$	80	${\tau }_{bend}$	$\pi /6$
$F_{area}$	5	${\tau }_{area}$	$0.3$
$F_{vol}$	20	${\tau }_{vol}$	$0.01$

, yielding a Young’s modulus ($E_s$) of 27.82 μN/m, a shear modulus ($K_s$) of 10.72 μN/m, and a Poisson ratio of 0.29. These continuum mechanical properties align with the experimental measurement range of previous studies [35,36,37].

It should be noted that these model parameters are dependent on the Lagrangian resolution. The shared values are for a 0.5 μm spacing. When the spacing changes, constants are modified to yield the same macroscopic material properties (e.g., shear moduli and Poisson ratio). For example, in the main simulations, due to the refined RBC membrane mesh, $\Delta s/\Delta x\approx 0.74$, where s is the Lagrangian particle distance of the RBC. This also applies to the RDW cases, where the diameters of the smaller and larger RBCs are 5.5 and 6.1 μm, respectively.

The Föppl–von Kármán number, given in Equation (5), can be used to characterize RBC deflection. It is generally taken as ∼400 [38], although higher values [39,40,41,42] or lower values [43,44] are also commonly observed.

$$F_{link}=-{\displaystyle \frac{{\kappa }_{link}dL}{p}}[1+{\displaystyle \frac{1}{{\tau }_{link}^2-d{L}^2}}]$$

(1)

$$F_{bend}=-{\displaystyle \frac{{\kappa }_{bend}d\theta }{L_0}}[1+{\displaystyle \frac{1}{{\tau }_{bend}^2-d{\theta }^2}}]$$

(2)

$$F_{area}=-{\displaystyle \frac{{\kappa }_{area}dA}{A_0}}[1+{\displaystyle \frac{1}{{\tau }_{area}^2-d{A}^2}}]$$

(3)

$$F_{volume}=-{\displaystyle \frac{{\kappa }_{volume}dV}{V_0}}[1+{\displaystyle \frac{1}{{\tau }_{volume}^2-d{V}^2}}]$$

(4)

$$1/\Gamma =K_B/K_s{r}^2$$

(5)

Currently, this material model has been shown to be stable in the low/high shear regions [30]. To our knowledge, there is no publication that uses this model in capillaries. Therefore, this study also proves that the Zavodovski model can be used in capillary blood flow where the length scale is in the vicinity of 5 μm. In a microcirculatory network-scale study, introducing viscosity contrast may be physically more accurate, but it is not applicable to single-file motion in capillaries due to a lack of CFL. The current code also lacks adaptive mesh refinement, which is also an unexplored topic in micron-scale hemodynamics.

2.3. Boundary Conditions

For the main simulations, a fixed Zou–He density is used at the capillary outlet, a bounce-back condition at the walls, and a transient Zou–He velocity condition at the inlet. This transient condition is provided by a periodic feeding section where the RBCs are initialized and fed into the main simulation domain. In this periodic domain, an external force drives the fluid, specifically using Guo’s forcing method [29]. The outlet of this periodic feeding section intersects with the feed section of the main simulation domain, and the values from its outlet are extrapolated to the main simulation domain, as shown in Figure 2(top). The RBCs are randomly fed into the main simulation domain; they are initially positioned off the center at varying angles to ensure that no bias is introduced into the main domain. Figure 2(bottom) shows an example feed setup. Different sets of boundary conditions are used in validation studies, which are detailed in their corresponding sections.

Before initializing the RBCs, pure plasma flow is solved to obtain a converged velocity field in the capillaries. Since LBM is inherently transient, convergence is achieved by running the simulations until the plasma flow becomes steady, after which the RBCs are initialized. Due to singularity problems near the wall [45], discrete particles of the RBCs never touch the wall boundaries. This is accomplished by employing a repulsive potential between the wall boundary and the RBC, which decays with distance.

3. Validation

A series of validation studies was conducted to rigorously assess the predictive capacity of the proposed fluid–structure interaction model to capture the mechanics of RBCs under physiologically relevant conditions. These validations encompassed three canonical test cases: First, the stretching of a single RBC in a stationary fluid is investigated. Second, the shearing behavior of an RBC positioned at the center of a shear flow in which the velocity ranges from −u to u and zero at the middle is determined. Finally, we investigate the transport of a single RBC through a confined slit. The first two scenarios primarily served to verify the model’s ability to reproduce the mechanical response of the RBCs to external forces and shear stress. However, the third case, involving cell transport near bounding walls, was particularly critical, since RBCs always move in the vicinity of the capillary walls, where confinement is always high. The accurate computation of wall effects is essential for a realistic simulation of microcirculatory flows. Unless stated otherwise, in all validation simulations, the lattice spacing is 0.5 μm, while the single relaxation time is $\tau =1$.

The first validation study is conducted on the cell stretch experiment, in which a single RBC is placed in a stationary fluid. The RBC is fixed on one side and stretched by applying force on the other side. Figure 3(left) shows an example of a stretched RBC. This study was carried out experimentally by Mills et al. [46], who used silica beads to fix the RBC on one side and stretch it by applying force on the other side. Figure 3(right) presents a comparison of the result with the data from Mills et al. [46]. Our current simulation results are in agreement with the experimental data, where the axial diameter is accurately calculated, and the transverse diameter calculation is in the upper range of the measurements. However, the reason for the slightly less accurate transverse diameter in low-force regions (20–70 pN) lies in the shear modulus value. We have chosen the shear modulus in the upper range of the experimental values [35,36,37]; therefore, it could represent a slightly more rigid RBC, which explains the higher transverse diameter in the low-force regime due to the slightly less deformed RBC behavior.

The second validation study involves a single RBC in shear flow. A single RBC is placed in the center of a fully periodic domain, and Couette flow conditions are applied where the velocity is zero at the middle of the domain, and lowest and highest velocities are −u and u, respectively. The left image of Figure 4(left) shows an example simulation snapshot, while the right image of Figure 4(right) presents the comparison with the experimental study by Yao et al. [47]. Our current results show good agreement with the experimental data and consistently fall within the measurement error range. The deviation from the curve fit line of the Yao (black) shows a larger deformation, which is caused again by the selected shear modulus value. A real RBC can have a shear modulus between 5 and 10 μN/m, but we use the shear modulus that is in the vicinity of the upper range of the experimental measurements. Therefore, a slightly lower deformation index observation is an expected outcome with the current material model constants.

The final validation study examines the passage of the RBC through a 1.5 μm slit, as illustrated in Figure 5(left). This particular study has been conducted numerically by both Salehyar [48] and Ye [49]. The computational domain consists of $100\times 19\times 15.2$ μm³. A Zou–He condition is applied to the inlet and outlet, with the remaining boundary conditions set to periodic. The gap between the upper/lower walls and the slit walls is 0.2 μm. The RBC is initialized 10 μm away from the slit entrance ($L_2$), and the average inlet velocity is set to 50 μm/s. For this case specifically, the lattice size is reduced to 0.1 μm to accurately capture the flow within the slit. Consequently, the Lagrangian resolution (particle-to-particle distance) is matched to the lattice resolution, with $\Delta s/\Delta x\approx 1.42$, and the material model parameters are adjusted to maintain the same mechanical behavior.

Figure 5(right) shows the deformation of RBCs in three different initial orientations: a, b, and c. Although the resulting RBC shapes show minor differences from the reference studies, this is an expected outcome due to variations in computational methodologies. Even with the same methodology, the resulting RBC shapes can differ [48,50,51]. However, general behavior and transport times are not significantly affected by these small local shape differences; therefore, capturing the exact shape is unnecessary. It is evident that the current model can successfully calculate the RBC transit time in extremely confined geometries without any instability.

4. Results and Discussion

Each simulation runs for 1 s of physical time, revealing the spatiotemporal RBC velocity profiles in four different capillaries with varying levels of luminal irregularities and their straight tube equivalents. Three different inlet ht levels are investigated in each capillary, since ht could be a major determinant of velocity characteristics. The RDW was also investigated to represent the actual biological situation since red blood cells actually have heterogeneity in their diameters. The diameter of the standard RBCs is selected as 5.8 μm to represent the mouse RBCs, since the capillary geometers employed are acquired from a mouse (ex vivo). RBC data are collected only when the RBC moves across the original length of the capillaries; i.e., if the RBC does not move across the capillary by the end time, that data is discarded. The coefficient of variation (CoV) ($CoV=V_{STD}/V_{avg}$) [52] is also presented to reveal the velocity heterogeneity in different configurations, where STD is the standard deviation and avg is the average.

Due to the unique shape of the capillaries, the evaluation of the results using only the hydraulic diameter and the deviation is not sufficient to explain the complex data. Hence, here we define a capillary circularity index ($I_{cc}$) to quantify the irregularity of the capillaries. Equation (6) shows the definition, where $\alpha$ is the aspect ratio, $\chi =D_{RBC}/D$ is the confinement ratio, $D_{ah}$ is the average hydraulic diameter of the geometry, D is the hydraulic diameter, and n is the number of cross-section slices representing the capillary. When the cross-section is not circular due to the $\alpha$ and $\chi$ terms and when there is a diameter variation due to the final term, $I_{cc}$ will start to reduce. Here, the term $\alpha$ is responsible for capturing the torosity when used together with the final term, and the term $\chi$ is responsible for capturing the confinement. In a perfect cylinder the final term will be zero with $I_{cc}$ = 1, and any number less than unity means that the geometry is not a straight cylinder, and the velocity across the geometry will deviate more as $I_{cc}$ decreases. For example, in a converging–diverging nozzle-type confined geometry, which is widely used in in vivo RBC studies, $\alpha$ and the final term will always be one, but $\chi$ will increase in the confinement region, so $I_{cc}$ will reduce. Another example is a rectangular channel, where $\alpha$ and $\chi$ are equal to unity, but the final term will increase due to the average hydraulic diameter, so $I_{cc}$ will reduce. It is important to note that, due to the inclusion of the term $D_{avg}$, this index is intended to compare a given geometry with its ideal tubular equivalent and is not suitable for comparing different capillaries with each other.

The average hydraulic diameter, length, average input velocity $I_{cc}$, and the number of capillaries of each capillary are given in

Table 2. Flow–capillary properties.

Capillary	S1	S2	S3	S4
Average hydraulic diameter (μm)	4.89	5.22	6.77	6.9
Length (μm)	46.9	48	55	49.6
Average Inlet velocity (mm/s)	0.75	0.86	1.44	1.5
Flow rate (μm3/s)	13,992	21,365	53,266	56,156
$I_{cc}$	0.91	0.93	0.84	0.67
Capillary Number	0.07	0.08	0.13	0.14

. Here, the capillary number is calculated in equivalent tubes with the average inlet velocity.

$$I_{CC}=1-{\displaystyle \frac{1}{n}}\sum _{i=1}^n{\displaystyle \frac{1}{{\alpha }_i}}{\displaystyle \frac{1}{{\chi }_i}}{\displaystyle \frac{|D_i-D_{ah}|}{D_{ah}}}$$

(6)

Three different ht levels were used in the feeding section to achieve low, medium, and high hematocrit levels for each capillary. The orientation of the RBCs and the number of RBCs are equivalent in each hematocrit level to compare the RBC transport differences in each capillary and their equivalent tubes while feeding the RBCs almost in the same manner. Simulation visualizations are shared in the Supplementary Materials for each capillary (S1, S2, S3, S4) with three different ht.

This investigation is conducted in four stages. First, we present the data of the RBCs averaged in each 5 μm interval throughout the capillary, along with their average values and deviation. Second, temporal variations during the simulation time are incorporated into the spatial average evaluation. After revealing the spatial variations, as a third step, we individually calculate the temporal average of each RBC data and study the transport variation between the RBCs. Finally, in the fourth step, RDW is introduced in cases of high hematocrit and compared with non-RDW RBC transport in capillaries.

The hematocrit values achieved with these settings in all capillaries and their equivalent regularized tubes are shown in Figure 6a,b with respect to $I_{cc}$ and ht. We observe the lowest ht in the S4 capillary, which has the highest average hydraulic diameter and the lowest $I_{cc}$, while the highest $I_{cc}$ is observed with the S2 capillary. As a general trend, ht is almost always higher in tubes except the S2 capillary, which is the most uniform capillary. Although their confinement ratio is relatively the same, the $I_{cc}$ values of the S1 and S2 capillaries are 0.91 and 0.93, respectively, and it can clearly be observed that the deviation from the tube is higher in the S1 capillary.

Figure 6c shows the averaged velocities (in each 5 μm) of the red blood cells across the capillaries and the tubes, and

Table 3. Average RBC velocity difference (%) in each 5 μm interval for different ht levels (low, mid, and high).

Capillary	${\mathit{I}}_{\mathit{cc}}$ (%)	Ht Low	Ht Mid	Ht High
S4	69	55.5	56.88	57.27
S3	84	33.23	33.38	36.72
S2	93	24.99	25.15	26.33
S1	91	18.22	18.41	18.87

shows the difference of the average velocities between the capillaries and their corresponding tubes. The tube results show nearly uniform behavior, whereas in the capillaries the velocity of the RBC varies with respect to the position due to changing $I_{cc}$. Since an average velocity value is imposed at the inlet boundary condition and RBCs tend to stay near the center of the channels, it is expected to see a higher RBC velocity than the average inlet velocity in these channels. Table 3 shows the highest velocity difference in each capillary calculated 5 μm apart. We can clearly observe an inversely proportional relation with $I_{cc}$, and it has a minimum of 18% in the S2 capillary ($I_{cc}$ = 0.91) and 57% in the S4 capillary ($I_{cc}$ = 0.67). The velocity difference increases with an increase in ht in each capillary, but it is negligible compared to the effect of $I_{cc}$ on the velocity difference. Therefore, it can be clearly stated that the $I_{cc}$ change is more crucial than the ht change for the 8–22 ht range. In regular tubes, the velocity difference is at most 1.5% for all cases due to the constant diameter and $I_{cc}=1$. This indicates that even a small deviation from the ideal channel can cause spatial heterogeneity in RBC velocity, and for more complex capillaries, such as S4, this difference can reach 57%.

In Figure 7, the second investigation is shown. The spatiotemporal profile of the average RBC velocity inside the capillary segments (Figure 7a–d) shows, unsurprisingly, that the mean RBC velocity can vary considerably, up to 30% on average, and higher locally, due to the effect of luminal irregularities. Tube channels show a uniform distribution of RBC velocity throughout their length, while in capillaries, RBCs tend to move faster in high-confinement regions. This trend is the most obvious for S4, the capillary with the lowest $I_{cc}$. RBC velocity fluctuates more along this segment, where the increase in velocity is almost 80% near the center of the capillary. This velocity fluctuation increases as the hematocrit increases, pointing to the contribution of the RBC–RBC interactions to the RBC velocity profile in the capillaries.

The spatial CoV distribution across the capillaries is shown in Figure 7e–h. It is clear that CoV is always below 4% for tube geometries. This indicates that the RBC velocity heterogeneity in the ideal tubes is always low, regardless of the ht level. However, in capillaries, even when $I_{cc}\ge 0.91$, a CoV between 5 and 7 can be observed in some regions of the capillary. In low $I_{CC}$ cases, the CoV can reach 14% near high-confinement regions, and the CoV increases with increasing ht. It is important to point out that the highest CoV is not calculated at the highest-velocity regions, but near the highest-velocity regions. his suggests that the greatest heterogeneity does 349 not occur in areas of extreme confinement but rather as the RBC recovers its original shape 350 after leaving the highly confined region.

Mixed ANOVA statistics revealed p-values of 0.392, 0.399, 0.0121, and approximately 0 for comparisons between capillaries and tubes in different ht configurations, with respect to segment number. This finding aligns with the observed velocity profiles, which indicate that velocity fluctuations are lower and more consistent with tube measurements in the S1 and S2 capillaries, whereas in S3 and S4, velocity fluctuations are higher and deviate significantly from the results obtained from the tubes.

For the third step, the temporal average velocity of the RBCs for each simulation with respect to ht, as well as the deviation from the temporal average, is shown in Figure 8a–d. This temporal average was calculated by calculating the temporal average velocity of each individual RBC that traveled through the capillary and taking the arithmetic average of the temporal averages. The temporal average velocity of the RBCs in the S4 capillary shows the highest deviation, where it can reach a 75% difference for the highest ht and 35% for the lowest ht. Whereas in capillaries that have higher $I_{cc}$ such as S1 and S2, temporal average velocity deviations are much smaller, and for the S2 capillary there is a negligible difference between different hematocrits, as well as the capillary and the tube. The average hydraulic diameters of the S1 and S2 capillaries are 4.89 and 5.22 μm, respectively, and the confinement $\chi$ is equal to almost unity across their length. This means that the RBCs are always forced to show parachuting behavior [53], and due to the restricting channel, the temporal average of each RBC is relatively the same. However, the S3 and S4 capillaries show a great deviation, while their equivalent tube results do not show a significant deviation. When $I_{cc}$ values are compared, the S2 capillary has the highest $I_{cc}$, and the S4 capillary has the lowest $I_{cc}$. Although there is no linear correlation between $I_{cc}$ and the deviation in ht or temporal average velocity; $I_{cc}$ can be used to anticipate the highest fluctuations in the corresponding capillaries.

The CoV is shared in Figure 8e,f. It is clear that all tube results are below 5%. That means that in tubes where ($I_{cc}$ = 1), the deviation is always small, whereas even in cases with high $I_{cc}$, such as the S1 and S2 cases, the CoV varies near 15% and 10%, respectively. The S4 capillary on the other hand ($I_{cc}$ = 0.67) has a significantly higher CoV than other capillaries, even though the avarage hydraulic diameter of S3 and S4 is close. The difference in $I_{cc}$ results in a nearly two-fold variance in CoV. This reinforces the idea that realistic capillary usage can change the result significantly even with capillaries that have similar average diameters and similar inlet conditions due to their unique shapes.

One-way analysis of variance (ANOVA) and post hoc analysis revealed highly significant differences $(p\approx 0)$ in RBC velocities compared between capillaries and tubes. Specifically, for the S2 capillary, the adjusted p-value was 0.0234 when comparing low-hematocrit (Ht) velocities and 1 × 10⁻⁴ when comparing high-Ht velocities. These results indicate a significant departure from previous statistical analyses. This difference arises because the current analysis considers the temporal average velocity of individual RBCs, whereas prior analyses averaged the RBC velocities with respect to their spatial positions. This methodological distinction naturally strengthens the hypothesis that RBC velocities exhibit greater fluctuations along the capillary and demonstrate lower deviations within smaller temporal or spatial intervals.

Finally, the effect of RDW was investigated by comparing the results of the high-ht cases in the capillaries. Our hypothesis was that due to the diameter (and possibly velocity) variance between individual RBCs, the overall capillary flow would fluctuate more, especially in the case of high-hematocrit conditions, which allow more interactions between RBCs. RDW cases were utilized by feeding 43% regular RBCs and 28.5% smaller ($R=2.75$ μm) and 28.5% larger ($R=3.05$ μm) RBCs to the capillaries. Figure 9a clearly shows that in high-$I_{cc}$ capillaries (S1 and S2) RDW causes no significant change in the local RBC velocity distribution, yet in low-$I_{cc}$ capillaries (S3 and S4), an increase in spatial RBC velocity is observed near the middle of the capillaries. In the S1 and S2 capillaries, the uniformity and high confinement of the capillaries cause each RBC to show parachuting behavior regardless of their diameter variation since the small RBCs are also larger than the channel diameter, and hence the RDW loses its effect on these capillaries with high $I_{cc}$ values and low hydraulic diameters. The S3 and S4 results however ($I_{cc}$ = 0.84 and 0.67) show a slight increase in the temporal average velocities while constraining the deviation from the temporal average. The reason behind this increase and the lower deviation can be explained by the RBC–RBC interaction and confinement. In the confined regions of these capillaries, smaller RBCs can accelerate more than regular or larger RBCs, increasing the average velocity. The CoV results are shown in Figure 9d, and using RDW does not cause a significant change in the CoV regardless of the $I_{cc}$ of the capillary. The ANOVA test also shows no significance using RDW.

The main emphasis of this study is on the critical need to consider the actual three-dimensional profile of a capillary, moving beyond simplified geometries that focus solely on the overall diameter. This detailed approach is essential for accurately evaluating the velocity distribution in capillaries in both space and time. Computer simulations are crucial for revealing the velocity profiles of RBCs, which are currently inaccessible with available in vivo microscopy technologies.

It should be noted that our model, although designed to be as realistic as possible in terms of mechanical RBC features and capillary geometries, completely omits the compliance of endothelia and ignores the active or passive reactions in capillary tonus and diameter to changes in flow. In reality, as cells are flowing inside a capillary segment, the vessel wall may passively respond to allow easier passage of the cell, and this is determined by endothelial rigidity or capillary tonus. Beyond the passive adaptability of the capillary wall, endothelial mechanosensors like Piezo1 channels can sense fluctuations in RBC flow or their forces applied to the capillary lumen and trigger appropriate effector responses to actively modulate the capillary tonus and geometry [54,55]. This is mainly established by endothelial–pericyte signaling, which forms appropriate feedback mechanisms for this regulation [56]. Capillary pericytes that surround the endothelia with their contractile machinery can monitor neuronal activity and oxygenation along with the mechanical status of the capillary lumen to calculate an adaptive contractile response that can result in either the constriction or dilation of the capillary lumen, and this immediately results in a change in RBC flow and the redistribution of pressure gradients in the adjacent capillary network [57,58]. Finally, RBCs have been reported to actively respond and adapt to passages inside tight capillaries by adjusting their calcium level [59] or shape and size [60], which was not modeled in our simulations. This collective regulatory effort of RBCs themselves, endothelia, and pericytes can adjust the RBC flow fluctuations, dampening them with appropriate feedback mechanisms. Therefore, in reality, the measurements reported here that arise only from physical elements may not be observable in vivo to this extent, or in the case of excessive RBC slowing and aggregation, a reflex spontaneous dilatation of the capillary can help the resolution of the flow.

A constant number of RBCs with a predetermined velocity are fed into the capillaries throughout the duration of simulations, which is a limitation. In reality, the capillary network continuously adjusts how many RBCs enter a specific capillary segment and at what velocity, depending on the hydraulic resistance of that specific capillary segment. Furthermore, passively, the interactions and dynamics of RBC–RBC and RBC–white blood cells at the capillary junctions affect the distribution of hematocrit in individual capillary segments [8]. Such network interactions are out of the scope of this study, as we focused only on single capillary segments to investigate a fundamental question. To extend this study, at least a network of some connected realistic capillaries must be investigated to reveal the full extent of the RBC velocity and hematocrit distributions.

Finally, simulation times are limited to 1 s of physical time after examining the fluctuation on the RBC velocities. Although the RBC velocity deviation does not increase with respect to time, overextending the physical simulation time (such as 30 s) may demonstrate velocity fluctuations on a higher scale due to the random feeding of the RBCs.

5. Pathophysiological Relevance

Understanding the physical principles behind the RBC flow dynamics in individual capillaries is necessary for a better understanding of microcirculatory dysfunction in the central nervous system. Our findings indicate slight but persistent temporal fluctuations in RBC velocity, which are influenced by the level of hematocrit and capillary lumen irregularity. Although direct in vivo measurements of capillary velocity at the spatiotemporal resolution of our simulations is highly challenging, self-fluctuation of the RBC flux and velocity has been experimentally shown in brain capillaries in a recent study [61], underscoring the applicability of our findings to in vivo conditions. Such similar fluctuations in RBC flow were previously attributed to variations in cell size and rheological features, changes in capillary wall tonus, or the presence of leukocytes [62], while under high-hematocrit conditions, the underlying heterogeneity of the capillary luminal profile can be sufficient to trigger oscillations even with the uniform type of flowing RBCs. Our reported temporal fluctuations are small in absolute measurements. However, because the central nervous system contains a dense, interconnected network of capillaries, even minor velocity changes in individual vessels can build up cumulatively to affect blood flow dynamics at macroscopic levels. It has been shown in computational models that flow abnormality in a capillary can readily affect nearby interconnected capillaries; for example, a block can increase the probability of blocks in the nearby capillaries, spatially expanding the abnormality [63]. Therefore, small but repetitive fluctuations at multiple adjacent capillaries can, in theory, synchronize and amplify over time and space, leading to spontaneous oscillations in blood flow. If these fluctuations rise to a certain level, they can lead to the formation of capillary stalls, causing RBCs to come to a temporary halt in a capillary segment [64,65,66]. This is analogous to slight velocity disturbances in rapidly moving traffic dramatically escalating to a full traffic jam [67]. Although RBC stalls can occur to some extent in physiological situations, the number and frequency of capillary stalls increase prominently under acute and chronic neurological disorders like cerebral ischemia, epilepsy, Alzheimer’s disease, sepsis, and even SARS-CoV-2 infection [60,68,69,70,71,72]. The exact physical causes of capillary stalls are currently under active investigation, and our work and the future research on spontaneous capillary RBC velocity fluctuations can contribute to an understanding of these dynamic flow interruptions. It is now well established that the level of oxygen extraction in the central nervous system depends not only on the level of blood flow but also almost equally on the distribution of blood flow velocity across capillaries over space and time [73]. If this distribution is highly heterogeneous (i.e., high CTTH is present), oxygenated RBCs either shunt rapidly across low-resistance paths without delivering their oxygen to the tissue [74] or end up in very long anomalous transport routes [75], traveling in the tissue without benefits after being deprived of their oxygen content. Microcirculatory dysfunction due to excessive capillary transit time heterogeneity is found to be responsible for the pathophysiology of ischemic stroke, Alzheimer’s disease, subarachnoid hemorrhage, traumatic brain injury, and many other debilitating conditions [76,77,78,79]. Therefore, the spontaneous oscillations of RBC flow can have profound effects on tissue health and viability.

Our results point to a strong contributing effect of high capillary hematocrit in the temporal velocity fluctuations of RBCs. Capillary hematocrit is dependent on arterial or venous hematocrit levels, which are measured routinely in clinical studies. However, local capillary hematocrit is also determined by RBC–RBC interactions in bifurcations [8] and also by the contribution of slow-moving cells such as white blood cells as they concentrate RBCs behind them [80]. Although our work did not test the direct contribution of white blood cells to velocity profiles, it can help elucidate microcirculatory dysfunction triggered by WBCs in Alzheimer’s disease, stroke, or systemic inflammatory conditions [60,68,72,81] due to their secondary effects on RBC hematocrit distributions. It should also be added that high hematocrit is a risk factor for severe stroke or myocardial infarction [82,83].

Similar to elevated hematocrit and blood viscosity, RDW has been recognized as a poor prognostic factor in a variety of disorders, particularly those associated with inflammation and microcirculatory dysfunction [84]. Computer models suggested a greater interaction of blood cells with vessel walls, which affected intravascular hemodynamics [85], but these models are based on large vessels. No relationship was detected between RDW, microcirculatory alterations, and prognoses in septic patients [86]. Our results support the notion that even if RDW could have a poor prognostic effect, it could arise from a large vasculature and not from the dynamics of individual blood flow in capillaries. Of course, this needs to be further explored using appropriate experimental and clinical models.

6. Conclussions

Here, an investigation of RBC transport in four different ex vivo capillaries with explicit RBC modeling is presented. The investigation includes a realistic comparison of capillary and cylindrical tubes, the tube hematocrit effect, and the RDW effect.

The capillary circularity index is used to characterize ex vivo capillaries depending on their cross-sectional characteristics. Although $I_{cc}$ can be used to anticipate the irregularity of the capillaries, we believe that this definition is still in its early stage, and more work should be conducted on this index to develop it for broader applications.

It is evident that the use of cylindrical tubes overestimates the actual ht established in the capillary lumen, and this overestimation increases as $I_{cc}$ decreases. The fluctuation of RBC velocity in tubes is much smaller than in the capillaries, resulting in a more uniform flow but at the cost of less realistic results. In relatively smaller capillaries with high $I_{cc}$, such as S1 and S2, velocity fluctuations with respect to ht are significantly lower than in low-$I_{cc}$ capillaries due to a more uniform flow field and a high confinement ratio, where RBCs are always forced to show parachuting behavior. Using RDW does not have a significant effect on individual RBC velocity fluctuations, but spatial RBC velocity shows an increase in low-$I_{cc}$ cases, and CoV is mostly lower in RDW cases. Although velocity fluctuations on RDW cases are on a small scale, the effect of these fluctuations is observed only in a single capillary, whereas microcirculation is vast, and the observed fluctuations can cause a major difference in the downstream bifurcation events, and the hematocrit of the following capillaries, consequently the CTTH, the most important parameter for the oxygenation, can change drastically.