INTRODUCTION

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THE INTERIOR SURFACES of blood vessels are lined with a glycocalyx, consisting of bound and adsorbed macromolecules. Electron-microscopic studies have typically revealed a layer several tens of nanometers thick. In vivo investigations of blood flow in capillaries have shown the presence of a much thicker layer, estimated to be ~0.4-1 µm thick, known as the endothelial surface layer (ESL) (10), which excludes red blood cells and impedes plasma flow. This layer is believed to be a major factor leading to the low tube hematocrits observed in capillaries (9) and levels of flow resistance in microvessels that are substantially higher than in glass tubes with corresponding diameters (11). Further evidence for the existence of such a relatively thick layer was provided by the experiments of Vink and Duling (21), who measured the widths of the columns of red blood cells and labeled dextran-70 in capillaries in the hamster cremaster muscle. The widths of these columns increased 0.8-1 µm after a light-dye treatment, without observable changes in capillaries' anatomic diameters. These results were interpreted to show that disrupting the glycocalyx (ESL) by a photodynamic process increased the space available for red blood cell motion and plasma flow.

Vink and Duling (21) also examined the effect of red blood cell velocity on the width of the zone at the endothelial surface from which red blood cells were excluded. They found that the width of the zone decreased with decreasing cell velocity below ~200 µm/s. When motion of red blood cells ceased, red blood cells expanded to fill the capillary, such that no space could be detected between the red blood cell surface and the anatomic wall of the capillary at the widest part of the red blood cell. This observation is of interest, because it suggests that the physiological effects of the ESL, such as its effects on oxygen transport, may vary with red blood cell velocity. Also it provides possible indications of the biophysical nature of the ESL.

Feng and Weinbaum (5) proposed that fluid dynamic lubrication forces generated within the ESL are responsible for the exclusion of flowing red blood cells from the ESL. Because the lubrication forces are generated by fluid flow, such forces cease when flow stops. This provides a possible explanation for the observation (21) that stationary red blood cells expand to fill capillaries. In the theoretical analysis (5), the red blood cell is treated as an object with a prescribed shape. Other previous theoretical studies considered the motion of rigid spherical particles (2, 23) or deformable red blood cells (1, 17) through cylindrical tubes lined with a porous wall layer but did not consider cases in which the red blood cell penetrates or compresses the layer substantially.

In the present study, a modified version of a previously developed theoretical model (17) is used to analyze the motion and deformation of red blood cells in capillaries lined with an ESL as a function of red blood cell velocity. The mechanical properties of the red blood cell, including the elastic resistance of the cell membrane to shear and bending deformations, are included in the model. The ESL is assumed to be compressible and permeable to water and to exert a small radial force resisting penetration by red blood cells. With this approach, quantitative relationships can be established between the biophysical properties of the ESL and the velocity-dependent deformation of red blood cells within a capillary.

	FORMULATION OF THE MODEL

TOP ABSTRACT INTRODUCTION FORMULATION OF THE MODEL RESULTS DISCUSSION REFERENCES

Red blood cell mechanics. Single-file flow of mammalian red blood cells is considered, and fluid mechanical interactions between red blood cells are neglected. Axisymmetric red cell shapes are assumed. The red blood cell is represented as a viscoelastic membrane containing an incompressible viscous fluid. Our previous model (17) included the elastic resistance of the red cell membrane to shear deformation but neglected bending resistance of the membrane. Although the bending resistance is small, its effects may be significant at very low red cell velocities, when fluid mechanical forces are very small, and so effects of bending resistance are included here. The elastic resistance of the red cell membrane to area changes is very large, so it is assumed to deform without area change. Transient changes of red cell shapes are considered, and so the viscous resistance of the membrane to shear deformation is included. Previous studies of red cell motion in capillaries with variable diameters (15) have shown that the energy dissipation in the cell interior associated with transient changes in cell shape is much less than the dissipation in the membrane, and such internal dissipation is neglected here. The hydrostatic pressure is therefore considered to be uniform within the cell, and internal shear stresses are neglected.

View larger version (24K): [in this window] [in a new window]   Fig. 1.   Configuration assumed in model. Variables used to analyze red blood cell motion are shown: , arc length measured along the cell from the origin in the reference shape; , angle; s, arc length; r, radius; z, cylindrical polar coordinate. Biconcave reference shape of membrane is also shown. Dashed line connects corresponding points in the membrane in the reference and deformed shapes. Top horizontal line, capillary wall; bottom horizontal line, capillary axis. Width of endothelial surface layer (ESL, w) in absence of red blood cells is indicated by shaded zone adjacent to capillary wall.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Plasma flow mechanics. The flow of plasma around the exterior of the cell and through the ESL is described by invoking the approximations used in lubrication theory, which results in a simpler form of the equations governing fluid flow in a narrow space between two surfaces when the Reynolds number is very small. In this theory, the fluid pressure p is assumed to be uniform across the gap between the cell and the wall, including the ESL (2, 17). The ESL is modeled as a porous matrix. The hydraulic resistivity, i.e., the pressure gradient divided by the mean flow velocity in the matrix, is denoted by K(). This assumption leads to the following equation for the axial component of plasma velocity (v_z)

(8)

where µ is the fluid viscosity and the solid fraction of the matrix is assumed to be small. The boundary conditions are as follows: v_z = 0 when  = D/2, where D is the capillary diameter, and v_z = V_rbc when  = r, where r is the cell radius and V_rbc is the cell velocity. In the previous study (17), K() was assumed to vary smoothly with distance from the wall, with a parameter L describing the width of the diffuse boundary of the layer. However, the results were found to be relatively insensitive to the value of L. For simplicity, therefore, a sharp boundary to the layer is assumed here; i.e., K() = K₀ when  > D/2  w and K() = 0 otherwise, where w is width of the ESL. Possible increases in K₀ with compression of the ESL (5) are neglected.

ESL stiffness. Two types of evidence suggest that the ESL has a finite resistance to compression. First, a structure with no resistance to compression would be flattened by the shear stress exerted by flowing blood (10). Second, the ESL in a capillary regains its width in ~1 s after being flattened by a passing white blood cell (22). This shows the existence of a force tending to restore the layer thickness after compression. The biophysical origin of this force is not known. One hypothesis (12) is that it results from colloid osmotic forces generated by plasma proteins adsorbed to the glycocalyx. According to this hypothesis, the colloid osmotic pressure within the ESL is increased by an amount _p above that of free plasma, and this additional pressure is balanced by tension in membrane-bound glycoprotein chains (19). An applied mechanical force tending to compress the ESL, such as that exerted by a red blood cell, relieves the tension in the chains and must therefore exert force against the increment in colloid osmotic pressure within the layer to reduce its width.

Parameter values. Red blood cells are assumed to have volume of 90 µm³ and area of 135 µm². The membrane is assumed to have  = 0.006 dyn/cm and µ_m = 0.001 dyn · s · cm¹ (8). B is estimated to be 1.8 × 10¹² dyn · cm (3). These properties are typical for human red blood cells. A typical capillary diameter (D) of 6 µm is assumed. In comparisons with experimental data from hamsters, the observed vessel diameters are scaled up according to the cube root of the ratio of the assumed volume to the mean volume (61 µm³) of hamster red blood cells (21), i.e., by a factor of 1.14. Plasma viscosity (µ) of 0.01 dyn · s · cm² is assumed.

4

	RESULTS

TOP ABSTRACT INTRODUCTION FORMULATION OF THE MODEL RESULTS DISCUSSION REFERENCES

Figure 2 shows computed steady-state red blood cell shapes corresponding to cell velocities of 1 µm/s-3 mm/s. At very low velocities, the cell is predicted to bulge outward, almost filling the capillary. As velocity increases, the cell shape becomes narrower and more elongated, and at a velocity between 300 µm/s and 1 mm/s the cell is clear of the ESL, with a layer of plasma between the ESL and the cell.

View larger version (86K): [in this window] [in a new window]   Fig. 2.   Predicted red blood cell shapes. Axes are labeled in µm, with the origin at the front of the cell. In each case, diameter (D) = 6 µm, thickness (w) = 0.7 µm, hydraulic resistivity (K0) = 2 × 108 dyn · s · cm4. Each panel corresponds to a different red blood cell velocity. Panels are ordered with velocities increasing from top to bottom and from left to right. Width of ESL in absence of red blood cells is indicated by shaded zones.

The predicted variation of the gap width between the cell and vessel wall is shown in Fig. 3 and compared with experimental data of Vink and Duling (21), with gap widths increased by a factor 1.14 as discussed earlier. Here, gap width is defined as the average along the length of the cell, from the point of minimum gap near the trailing edge of the cell to the point near the front of the cell where the angle between the membrane and the vessel wall equals 11°. In Fig. 3A, results are shown for the case when no ESL is present, and for an ESL with w = 0.7 µm and K₀ = 2 × 10⁸ dyn · s · cm⁴. These parameter values, which are within the expected range, were chosen to give a good fit to the experimental data (21) for velocities of 20-220 µm/s, as shown in Fig. 3A. In the absence of an ESL, the predicted gap width increases with increasing cell velocity but is always much less than the experimentally observed value.

View larger version (22K): [in this window] [in a new window]   Fig. 3.   Predicted and experimentally observed variation of gap width with red blood cell velocity. A: model results with ESL (w = 0.7 µm, K0 = 2 × 108 dyn · s · cm4) and without ESL (solid lines) and experimental data (dots and dashed line) (21). B: model results for 3 combinations of w and K0 chosen to give the same average gap width at red cell velocity = 100 µm/s (solid lines) and experimental data (dots and dashed line) (21). Ordinate scales differ in A and B.

The sensitivity of the results to the parameter values is indicated in Fig. 3B, which shows results for w = 0.6, 0.7, and 0.8 µm. In each case, K₀ was chosen to give approximately the same average gap width at a velocity of 100 µm/s as observed experimentally. Although best agreement with the experimental data is seen with w = 0.7 µm, the results for the other two cases are probably within the experimental uncertainty of the experimental results.

Theoretical predictions of gap width and rheological parameters over a wide range of velocities are summarized in Fig. 4 for the case when w = 0.7 µm and K₀ = 2 × 10⁸ dyn · s · cm⁴. Corresponding results in the absence of an ESL are shown by dashed lines. The discontinuity in slope of the results with ESL corresponds to the velocity at which the cell starts to ride above the ESL, i.e., ~300 µm/s. Figure 4B shows a strong variation in the Fåhraeus effect with cell velocity. The Fåhraeus effect is expressed as

(9)

where H_T is the tube hematocrit, H_D is the discharge hematocrit, is the mean flow velocity, and V_rbc is the velocity of the red blood cells. At very low cell velocities, the cell almost fills the capillary, and so V_rbc  . At higher velocities, the red blood cell moves with the plasma lying outside the ESL, and the plasma within the ESL is almost stationary, so V_rbc > .

View larger version (24K): [in this window] [in a new window]   Fig. 4.   Variation of gap width, Fåhraeus effect, and flow resistance with cell velocity (Vrbc). A: gap width. B: Fåhraeus effect (HT/HD, where HT and HD represent tube and discharge hematocrit). C: apparent intrinsic viscosity (KT). D: resistance at hematocrit 45% (R45). Solid and dotted curves, predictions with ESL (D = 6 µm, w = 0.7 µm, K0 = 2 × 108 dyn · s · cm4); dotted parts of curves denote velocity range in which results may be affected by the finite recovery time of the ESL after red cell passage. Dashed curves, predictions without ESL (D = 6 µm).

In this model, interactions between red blood cells are neglected, and so flow resistance varies linearly with hematocrit

(10)

where R₀ is the resistance at zero hematocrit and K_T is the apparent intrinsic viscosity. Here, R is expressed as a relative apparent viscosity, i.e., the viscosity that would lead to the same flow resistance as a Poiseuille flow in the absence of red blood cells and an ESL. With an ESL present but no red blood cells, R₀ = 2.54, while R₀ = 1 in the absence of both red blood cells and an ESL. The resistance at discharge hematocrit 45% (R₄₅) is given by

(11)

K_T and R₄₅ decrease substantially with increasing flow velocity (Fig. 4, C and D). This reflects two factors. First, at low velocities, the cell membrane is inclined to the tube axis along much of the length of the cell, and the normal force exerted by the ESL on the membrane has a component opposing the cell's motion. Second, at high velocities, the increasing width of the plasma layer between the cell and the ESL leads to a reduction in flow resistance.

In comparison with the results in the absence of an ESL (dashed curves in Fig. 4), the ESL is predicted to cause a stronger Fåhraeus effect (lower H_T/H_D) and increased flow resistance (higher R₄₅) at corresponding velocities. The increase in R₄₅ reflects the increased R₀ in the absence of red blood cells resulting from the ESL and the increased proportional effect of red blood cells on resistance (higher K_T), which together more than compensate for the reduced tube hematocrit in the presence of an ESL.

The model can also be used to investigate the transient changes in red blood cell shape when flow is suddenly started. This process is modeled by assuming that the driving pressure is suddenly increased from an extremely low value, corresponding to a cell velocity of 1 µm/s, to one corresponding to a given steady-state cell velocity. Predicted sequences of cell shapes are shown in Fig. 5 for final velocities of 0.1 and 1 mm/s. In each case, the corresponding steady driving pressure (113 and 558 dyn/cm²) is imposed over a 30-µm length of the capillary containing the cell. The predicted variation of cell shape with distance traveled is similar in both cases. After traveling ~4 µm, the cell temporarily takes on a conical shape unlike any of the steady-state shapes shown in Fig. 2. The time taken by the cell to approach its final steady-state shape depends on the eventual velocity and is ~200 ms for a final velocity of 0.1 mm/s and ~40 ms for a final velocity of 1 mm/s. However, the distance traveled when the cell has nearly attained its final shape is approximately the same in each case and is ~20 µm.

View larger version (57K): [in this window] [in a new window]   Fig. 5.   Transient changes in red blood cell shape after a step increase in driving pressure. Initial velocity is 1 µm/s. Each shape is labeled with time after pressure increase. Axis markings show distance traveled by cell after pressure increase. Width of ESL in absence of red blood cells is indicated by shaded zones. Pressure drop over a length of 30 µm containing a cell is 113 dyn/cm2 with a final velocity of 100 µm/s (A) and 558 dyn/cm2 with a final velocity of 1 mm/s (B).

	DISCUSSION

TOP ABSTRACT INTRODUCTION FORMULATION OF THE MODEL RESULTS DISCUSSION REFERENCES

The principal result of this study is that fluid dynamical pressures generated within the ESL can lead to variation of red cell shape and gap width with flow velocity in capillaries, consistent with the observations of Vink and Duling (21). This mechanism was proposed by Feng and Weinbaum (5) in a model in which the red blood cell was represented by an inclined plane riding over a planar surface coated with a porous layer. The present model, which includes the deformability of the red blood cell, predicts more complex cell shapes in which the angle between the cell membrane and the vessel wall varies significantly over the length of the cell. The theory of Feng and Weinbaum takes into account increased hydraulic resistivity of the layer when it is compressed. In the present model, the resistivity of the layer was assumed to be constant, independent of compression. Consequently, the lubrication pressures predicted by this model are less than would be predicted by the model of Feng and Weinbaum but are still sufficient to account for the observed variation in gap width. Significant variation of gap width with red blood cell velocity is predicted over a wide range of velocities, from <1 µm/s to >1 mm/s (Fig. 4). This corresponds to the range of red cell velocities observed in capillaries in vivo.

In the absence of an ESL, the fluid dynamical forces generated in the lubrication layer cause an increase in the average gap width with increasing velocity. However, the predicted gap width in that case is less than half of that observed experimentally. In contrast, a much wider gap is predicted in the presence of an ESL. This result provides further evidence supporting the existence of an ESL that exerts substantial effects on the motion of plasma and red blood cells.

The key parameters defining the properties of the ESL in this model are w, K₀, and the outward force (i.e., _p) that it exerts on red blood cells that penetrate it. According to the results shown in Fig. 3B, reasonable agreement between the model predictions and the experimental results (21) is obtained for w = 0.6-0.8 µm and K₀ = 10⁸-10⁹ dyn · s · cm⁴. These estimates are similar to those obtained by Secomb et al. (17). With the assumed scaling of radial dimensions by a factor of 1.14 from hamster to human taken into account, these results suggest that w in the experiments (21) was 0.5-0.7 µm. However, relatively small changes in the experimental data could lead to estimates of w and K₀ outside these ranges. The other key parameter, _p, is assumed to be 20 dyn/cm². In further simulations, _p was varied. When _p was decreased to 10 dyn/cm², the predicted gap width decreased slightly. When _p was increased to 40 dyn/cm², the gap width did not approach zero as cell velocity decreased. These results suggest that 20 dyn/cm² is an approximate upper bound on _p. An argument based on the ability of the ESL to withstand shear stress exerted by flowing blood (10) suggests that the force has about this magnitude.

In the model, the outward force (_p) is assumed to be independent of the degree of compression of the layer. Compression of the layer might be expected to lead to an increase in _p. However, assuming a substantial increase in _p with decreasing layer thickness leads to predicted gap widths that do not approach zero as cell velocity decreases, and such behavior is not consistent with the observations (21).

In the model, the unstressed shape of the red blood cell membrane is assumed to be a biconcave disk, the axis of symmetry of which coincides with the capillary axis. Assuming that the membrane is unstressed in a spherical shape with the same area (17) would lead to an underestimate of the outward force exerted by the red blood cell on the layer. However, observations (7) suggest that red blood cells actually enter capillaries "edge on," i.e., with the axis of the disk perpendicular to the capillary axis. A red blood cell entering a narrow capillary edge on requires less deformation than one entering "face on." Thus, for a cell entering edge on, the outward force may vary around the circumference of the capillary and be lower on average than that assumed here. In that case, the red blood cell shape is not axisymmetric, and prediction of the shape would require a fully three-dimensional analysis, which is beyond the scope of the present model. The cell shapes predicted by the present model represent axisymmetric approximations to the actual shapes taken by cells entering capillaries edge on. The model cannot predict details such as the asymmetric concavity or cleft often seen at the rear of cells traveling along capillaries.

The predictions of the model also depend on the assumed mechanical properties of the red blood cell membrane, i.e.,  and B. Further simulations were carried out in which B was varied, showing that B determines the membrane curvature in the sharply curved region at the rear of the cell but otherwise has little effect on cell shape (16). However, the model predictions are sensitive to . For a given cell shape, the outward force exerted by the red blood cell on the surrounding ESL or fluid is proportional to . At very low velocities, the compression of the ESL by the cell is determined mainly by the ratio of  to _p, while the cell shapes predicted at higher velocities depend mainly on the ratio of  to µV_rbc.

According to the model, the changes in cell width with increasing velocity are accompanied by decreasing flow resistance and increasing Fåhraeus effect (i.e., decreasing H_T/H_D). As shown in Fig. 4, C and D, the predicted flow resistance increases markedly with decreasing flow velocity, as the red blood cell increasingly enters the layer rather than flowing outside it. The Fåhraeus effect vanishes (H_T/H_D = 1) at extremely low velocities (Fig. 4B), because the cell completely fills the capillary cross section, and cell velocity necessarily equals mean flow velocity in the capillary. However, at velocities above ~10 µm/s, the Fåhraeus effect is much more marked than would be the case in the absence of the ESL (Fig. 4, dashed line).

The question arises whether the model predicts flow cessation in capillaries at positive driving pressures, as observed experimentally (14). For this to occur, the slope of a log-log plot of flow resistance vs. velocity must be 1 or more negative at low velocities (14). The slope of the graph in Fig. 4D at low velocities is approximately 0.6, and so flow cessation is not predicted. Flow cessation may occur when red blood cells encounter irregularities in capillary cross section (14). In that case, the presence of the ESL would increase the minimum driving pressure needed to sustain flow.

Several other effects, not included in the model, may influence the behavior of red blood cells flowing in a capillary with an ESL. The present model does not include effects of interactions between red blood cells. In particular, each red blood cell is assumed to encounter an unperturbed ESL, with the specified nominal thickness. At low velocities, according to the model, each passing cell significantly compresses the layer. According to Vink et al. (22), the layer takes ~1 s to recover its initial width after compression. If a following red blood cell arrives before recovery is complete, it may encounter a significantly compressed ESL, and so the gap between the cell and the capillary wall would be smaller than predicted by the present model. This could occur when the red cell flux exceeds 1/s, which corresponds to a red cell velocity of 10 µm/s, for the assumed parameter values. Conversely, at higher flow velocities, about 300 µm/s, the red blood cells are predicted to ride above the layer. Thus, for flow velocities of 10-300 µm/s, the present theory may overestimate the gap between red blood cell and vessel wall, leading to underestimates of tube hematocrit and flow resistance. Model predictions corresponding to this range are shown as dotted lines in Fig. 4 to indicate that they are subject to this uncertainty. Another effect not included in the model is the possible compression of the ESL resulting from wall shear stress at high flow rates (10). If these phenomena are significant, the actual width of the layer may be larger than deduced here (i.e., 0.5-0.7 µm) to explain the experimental results (21). Inclusion of these effects would lead to altered estimates of parameter values, including K₀, for agreement between predicted and observed velocity-dependent gap width but would not lead to qualitatively different predicted dependence of flow resistance and hematocrit on velocity.

In summary, this model shows that the exclusion of red blood cells from the ESL in flowing capillaries can be explained by hydrodynamic forces generated by plasma flow through and adjacent to the ESL, as first proposed by Feng and Weinbaum (5). Comparison of model predictions with experimental observations (21) leads to approximate bounds on key parameters characterizing the mechanical properties of the ESL. According to the model, the ESL causes a substantial increase in flow resistance at low velocities, and this may play a role in flow cessation in capillaries at low driving pressures.