INTRODUCTION

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EXPERIMENTAL STUDIES of blood flow in capillaries have shown tube hematocrits much lower than systemic hematocrit (12). The observed levels of tube hematocrit are only partly accounted for by the Fahraeus effect as measured in glass tubes and by the network Fahraeus effect (15). Desjardins and Duling (4, 5) proposed that the glycocalyx, a layer of macromolecules bound or adsorbed to the endothelial surface, may retard plasma motion in a zone adjacent to capillary walls. The slowly moving plasma in such a layer contributes little to plasma flow through the vessel but forms part of the luminal volume used to estimate tube hematocrit, implying a reduction of tube hematocrit. Support for this concept came from measurements (25) of the width of the columns of red blood cells and labeled Dextran 70 in capillaries in the hamster cremaster muscle. After a light dye treatment, the widths of these columns increased 0.8-1 µm without observable increases in capillary anatomic diameters. These observations imply the existence of a layer at least 0.4-0.5 µm in width adjacent to the endothelium, which can exclude red blood cells and some macromolecules.

The presence of such a layer of retarded plasma flow would be expected to cause a substantial increase in flow resistance because it decreases the effective diameter of the vessel available for plasma and red blood cell motion, and flow resistance varies approximately as the inverse fourth power of tube diameter. Indeed, experimental studies of blood flow in microvascular networks (14, 16, 17) imply levels of flow resistance much higher than in glass tubes with comparable diameters. In recent work (18), enzymes targeted at oligosaccharide side chains of the glycocalyx were microinfused into microvascular networks of the rat mesentery. Infusion of heparinase resulted in sustained decreases in flow resistance by 14-21% in flow pathways downstream of the infusion point, implying that the glycocalyx does indeed contribute to resistance. These data may underestimate the contribution of the glycocalyx inasmuch as it may have been only partially removed by the treatment. Also, the flow pathways included a range of vessel diameters, and the data do not show how the resistance change varied with diameter.

In this study, a theoretical model is used to analyze the motion and deformation of red blood cells in a glycocalyx-lined capillary and to predict the effects of the glycocalyx on flow resistance and hematocrit. Previous theoretical studies (3, 26) have considered the motion of rigid spherical particles through cylindrical tubes lined with a porous wall layer (representing the glycocalyx). The present model includes effects of red blood cell shape and deformability.

	FORMULATION OF MODEL

Red blood cell mechanics. The red blood cell is modeled as an axisymmetric viscoelastic membrane containing an incompressible viscous fluid. Single-file flow of red blood cells is considered, and cell-to-cell interactions are neglected. Steady red blood cell motion is assumed, with constant deformation, and so only the elastic properties of the membrane have to be specified. The elastic resistance of the membrane to shear deformation is included in the model, but the elastic bending resistance of the cell membrane is neglected (23). This simplifies the computations and leads to cell shapes with a sharp cusp at the trailing edge. The concave rear of the cell is represented by part of a sphere. Previous studies (21) have shown that neglect of membrane bending resistance does not lead to substantial errors. The membrane strongly resists area changes, and so a fixed membrane area is assumed. A spherical reference shape for the membrane is assumed (23). Actual red blood cell shapes are not axisymmetric, but this has little effect on flow resistance (11).

View larger version (24K): [in this window] [in a new window]   Fig. 1.   Configuration assumed in model, showing variables used to analyze red blood cell motion. Semicircle shows reference shape of membrane. Dashed line connects corresponding points in membrane in the reference and deformed shapes. , arc length measured along cell from origin in reference shape; r0(), radial position of material element in reference shape; s(), arc length measured along cell from origin; (), angle between normal to membrane and axis; w, glycocalyx width; z, distance along axis from origin.

Plasma flow mechanics. Lubrication theory is used to describe the motion of plasma around the cell and in the glycocalyx. This theory is appropriate for analyzing viscous flows between two surfaces, when the Reynolds number is very small, and the gap between the surfaces is narrow compared with the other dimensions. Lubrication theory can yield good approximations to exact solutions even when the gap width is not uniformly narrow (23). Fluid pressure p is assumed to be uniform across the gap between the cell and the wall, including the glycocalyx. The glycocalyx is modeled as a porous matrix, with a radially varying hydraulic resistivity K(). This quantity, which gives the pressure gradient required to produce a unit mean flow velocity in the matrix, is the inverse of the hydraulic conductance or hydraulic permeability. Our notation follows that of Damiano et al. (3), who give the following equation for the axial component v_z of plasma velocity

(1)

where µ is the fluid viscosity, and the solid fraction of the matrix is assumed small (see also Ref. 6). One modification has been made, in that the glycocalyx is assumed to have a diffuse boundary, and K() varies smoothly with distance from the wall where erfc denotes the complementary error function, giving an appropriate sigmoidal variation (Fig. 2), D is the capillary diameter, w is the nominal width of the layer, and L determines the width of its diffuse boundary (see Glycocalyx stiffness).

View larger version (12K): [in this window] [in a new window]   Fig. 2.   Assumed variation of hydraulic resistivity K and radial force f in the glycocalyx, with distance from the vessel wall. p, increase in colloidal osmotic pressure within glycocalyx. K0, glycocalyx hydraulic resistivity.

Glycocalyx stiffness. The physical mechanism by which the glycocalyx resists compression is not known, but some assumptions regarding its properties are required for the model. According to observations (25), the glycocalyx is sufficiently stiff to resist penetration by flowing red blood cells. Damiano et al. (3) represented the layer as a linearly elastic solid under deformation. Pries et al. (18) hypothesized that the stiffness of the layer results from colloid osmotic forces generated by plasma proteins adsorbed to the glycocalyx. The model used here is based on this hypothesis. Suppose that adsorbed plasma proteins generate an increase _p in colloidal osmotic pressure within the glycocalyx (above that of free plasma). This additional osmotic pressure is balanced by tension in membrane-bound glycoprotein chains (24), which anchor the layer to the endothelial surface and extend radially through it. An applied mechanical force tending to compress the glycocalyx must then overcome the increment in colloidal osmotic pressure within the layer, to reduce its width. For equilibrium where t_g is the radial tension in the glycocalyx (per unit membrane area) and f is the compressive force (per unit area) applied to the layer. If it is assumed that the macromolecules forming the glycocalyx cannot support compression, then t_g must be positive. If f exceeds _p, then the layer is compressed.

Parameter values. A typical capillary diameter (D) of 6 µm and a typical mean flow velocity in the capillary () of 0.5 mm/s are assumed. The corresponding red blood cell velocity is ~1 mm/s. Effects of changes in capillary diameter and flow velocity are examined. Plasma viscosity µ = 0.01 dyn · s/cm² is assumed. A red blood cell is assumed to have a volume of 90 µm³ and an area of 135 µm², and the shear elastic modulus of its membrane is taken as = 0.006 dyn/cm (10). These properties are typical for human red blood cells. In comparisons with experimental data from rats, the observed vessel diameters are scaled up according to the cube root of the ratio of the assumed volume to the mean volume (55 µm³) of rat red blood cells (17), i.e., by a factor of 1.18.

	RESULTS

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Figure 3 shows examples of computed red blood cell shapes and flow velocity profiles. With no glycocalyx, the red blood cell almost fills the capillary lumen, with a minimum gap width of 0.41 µm. The presence of the glycocalyx leads to longer and narrower red blood cell shapes, with a minimum gap width of 1 µm. The velocity profile is sensitive to the hydraulic resistivity K₀. This may be understood in terms of the variation of  = (µ/K₀)^1/2, representing the typical distance that flow driven by an imposed velocity at the edge of the glycocalyx penetrates within the layer. When K₀ is relatively small (10⁶ dyn · s/cm⁴),  is large (1 µm) and plasma flows throughout the layer. At large K₀ (10⁹ dyn · s/cm⁴),  is reduced (0.03 µm) and flow penetrates only a short distance into the layer. The cell shape varies only slightly with K₀, except at the highest values of K₀ considered.

View larger version (56K): [in this window] [in a new window]   Fig. 3.   Predicted red blood cell shapes and velocity profiles. Axes are labeled in µm, with origin at front of red blood cell. In each case, capillary diameter (D) = 6 µm, w = 1 µm, mean flow velocity () = 0.5 mm/s, and velocity profiles (axial flow velocities as function of radial position) are shown in a cross-section through a cell and in a cell-free cross-section. A: no glycocalyx. For B-E, K0 = 106, 107, 108, and 109 dyn · s/cm4, respectively.

The effects of varying glycocalyx properties on the Fahraeus effect and on flow resistance are examined in Fig. 4. The Fahraeus effect is expressed as the ratio of tube hematocrit (H_T) to discharge hematocrit (H_D) where is the mean flow velocity and V_rbc is the velocity of the red blood cells. Presence of a glycocalyx increases the Fahraeus effect because the glycocalyx retards plasma flow within the layer. Increasing the width or resistivity accentuates this effect. Flow resistance (R) is expressed relative to the flow resistance in a vessel with the same length and diameter with no glycocalyx or red blood cells, i.e., in terms of the relative apparent viscosity. Because of the assumptions of the model, resistance varies linearly with hematocrit where R₀ is the resistance at zero hematocrit and K_T is the apparent intrinsic viscosity. The resistance at discharge hematocrit 45% is given by As expected, R₀ increases with both w and K₀, but R₄₅ shows more complicated behavior. At small values of K₀, R₄₅ decreases with increasing w, and at large values, R₄₅ decreases with increasing K₀. This behavior is discussed in the next section.

View larger version (34K): [in this window] [in a new window]   Fig. 4.   Variation of Fahraeus effect and flow resistance with glycocalyx resistivity K0 and thickness w, for D = 6 µm and  = 0.5 mm/s. In each case, w is varied from 0.6 to 1.4 µm, in steps of 0.2 µm. Results for w = 1 µm are shown by thick curves. A: Fahraeus effect, expressed as ratio of tube hematocrit to discharge hematocrit (HT/HD). B: resistance in absence of red blood cells (R0). C: apparent intrinsic viscosity (KT). D: resistance at hematocrit 45% (R45); dashed horizontal line (based on data from Ref. 17) shows estimated flow resistance in vivo for a 6-µm capillary.

According to experimental results (17), flow resistance (relative apparent viscosity) in 5-µm capillaries of the rat mesentery at discharge hematocrit 45% is ~10, much higher than expected based on in vitro measurements. Although several factors may contribute to this difference, it appears that the glycocalyx is the major contributor (22). Therefore, we consider what glycocalyx properties could lead to predicted R₄₅ = 10. Taking into account the relative size of rat red blood cells, as described earlier, we assume a corresponding capillary diameter of 6 µm. As Fig. 4 shows, a glycocalyx thickness of 0.8 µm is insufficient. However, R₄₅  10 when w = 1 µm and K₀ = 10⁸ dyn · s/cm⁴, and this case is now considered further. Other combinations of w and K₀ could lead to the same level of resistance, but w must be at least 0.8 µm.

The effects of the glycocalyx as a function of diameter are examined in Fig. 5, assuming that the glycocalyx width and other properties are independent of diameter and that mean flow velocity is held constant. With increasing diameter, the influence of the glycocalyx on flow resistance decreases. The diameter dependence of the predicted resistance R₄₅ is compared in Fig. 5D with experimental results (17), showing reasonable agreement for diameters in the range 5-7 µm for the chosen values of w and K₀.

View larger version (22K): [in this window] [in a new window]   Fig. 5.   Variation of Fahraeus effect and flow resistance with capillary diameter D for K0 = 108 dyn · s/cm4, w = 1 µm, and  = 0.5 mm/s. A: Fahraeus effect. B: resistance in absence of red blood cells. C: apparent intrinsic viscosity. D: resistance at hematocrit 45%. In vivo and in vitro curves in D are based on data from Ref. 16.

The results presented so far are for a mean flow velocity of 0.5 mm/s. Theoretical and experimental studies (23) have shown that the Fahraeus effect and resistance to blood flow in smooth-walled tubes depend on flow velocity. However, in the presence of a glycocalyx, the predicted values of H_T/H_D and R₄₅ in a 6-µm capillary are almost independent of velocity for values of  > 0.01 mm/s, as shown in Fig. 6. In a 9-µm capillary, slight decreases in H_T/H_D and R₄₅ are predicted with increasing velocity over this range.

View larger version (14K): [in this window] [in a new window]   Fig. 6.   Variation of Fahraeus effect and flow resistance with mean flow velocity for K0 = 108 dyn · s/cm4, D = 6 µm and 9 µm, and w = 1 µm. A: Fåhraeus effect. B: resistance at hematocrit 45%.

Further simulations were carried out to test the effects of the increment _p in colloidal osmotic pressure and of the parameter L, giving the width of the diffuse boundary of the layer, for the case D = 6 µm and  = 0.5 mm/s. Varying _p over the range 40-800 dyn/cm² led to changes in H_T/H_D by up to 0.03 and in R₄₅ by up to 12%. Varying L over the range 0.05-0.4 µm altered H_T/H_D by up to 0.02 and R₄₅ by up to 12%. Thus the results were insensitive to the values of these two parameters within the ranges considered.

	DISCUSSION

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The principal result of this study is that the presence of a glycocalyx of width ~1 µm can largely account for the discrepancy between in vivo and in vitro estimates of resistance to blood flow (17) in capillaries of the rat mesentery. Although other mechanisms, such as capillary irregularity and the presence of white blood cells, may also contribute to the discrepancy, their effects appear to be relatively small (18, 22), suggesting that the actual contribution of the glycocalyx to flow resistance is comparable to that illustrated in these simulations.

The glycocalyx hydraulic resistivity required to achieve this increase in resistance, ~10⁸ dyn · s/cm⁴, is lower than that of several other water-permeable biological structures mentioned earlier and can be produced by a very dilute matrix of macromolecules. For flow perpendicular to evenly spaced cylindrical fibers, an estimate of the hydraulic resistivity is (9) where r_f is the fiber radius and S is the fractional volume occupied by fibers. If r_f = 0.6 nm (8), then K₀ = 10⁸ dyn · s/cm⁴ when S = 4.1 × 10⁵, whereas r_f = 3 nm gives S = 7.0 × 10⁴. The corresponding spacings of the fibers are 166 and 200 nm, respectively. The fragility of the glycocalyx and the difficulty in directly visualizing it, evident from experimental studies of the endothelium, are easily understood if these estimates are correct. Furthermore, such a dilute structure seems unlikely to be able to resist significant compressive stresses by purely mechanical means. Consequently, the present hypothesis that glycocalyx stiffness is osmotic in origin appears more plausible than a simple elastic model. However, the results presented here are not crucially dependent on this hypothesis. The simulations show that the thickness and hydraulic resistivity of the layer largely determine its effect on flow resistance and hematocrit. Assuming a different mechanism for glycocalyx stiffness would not lead to substantially different predictions.

A glycocalyx width of 1 µm is an order of magnitude larger than estimates obtained by electron microscopy (25). However, the apparent thickness may be greatly reduced by dehydration before electron microscopy. Several other studies imply a glycocalyx width of similar magnitude to that assumed here. Observations of hematocrit reduction in capillaries (4) implied effective layer thicknesses of 0.8-1.8 µm. The visualization technique of Vink and Duling (25) implied that the layer was at least 0.4-0.5 µm wide. Assuming a uniform layer width throughout a microvascular network, Pries et al. (18) found that a layer thickness of 1.5 µm would produce the difference between flow resistance measured in rat mesenteric networks and that predicted on the basis of in vitro measurements of apparent blood viscosity. The physical and chemical mechanisms that could lead to the formation of a glycocalyx with width of order 1 µm are not known at present. This dimension is much larger than the length of individual glycoprotein molecules, but such molecules may link together to form long chains (24), possibly in combination with oligomeric carbohydrate side chains and/or plasma protein molecules.

For very low glycocalyx resistivities (K₀ < 10⁶ dyn · s/cm⁴), the predicted flow resistance R₄₅ decreases with increasing glycocalyx thickness (Fig. 4) and is less than its value with no glycocalyx, as suggested by Copley (2). In this case, the presence of the glycocalyx increases the width of the lubrication layer, while offering little resistance to plasma motion through it. However, this range of K₀ is not consistent with the experimentally observed flow resistance in vivo. At very high values of K₀, the predicted cell shape is narrower (Fig. 3), and a slight decline in flow resistance is predicted for some values of w (Fig. 4). In this case, the glycocalyx behaves like an impermeable surface with respect to fluid flow, and a lubrication layer is formed outside the glycocalyx.

The parameters of the model were chosen to give R₄₅ values close to the experimentally determined curve for diameters in the range 5-7 µm (Fig. 5). At larger diameters, the predicted R₄₅ falls below the experimental curve. The model may underestimate resistance in this range of diameters because it assumes single-file flow of red blood cells with relatively wide plasma layers, whereas multifile flow is typically observed at such diameters. Also, the layer thickness may increase with capillary diameter but is assumed constant in the model.

In the presence of a glycocalyx, the predicted dependence of the Fahraeus effect and flow resistance on velocity is quite different from that obtained when no glycocalyx is present (Fig. 6). With no glycocalyx, red blood cell shapes change significantly with velocity, and this is reflected by the H_T/H_D and R₄₅ values. At low velocities, the cells bulge outward, almost filling a 6-µm capillary, and they become more elongated and streamlined with increasing velocity (19). In the presence of the glycocalyx, the red blood cells are confined to a narrower channel, and shape changes are inhibited, so that H_T/H_D and R₄₅ are nearly constant. In a 9-µm capillary, more variation in cell shape can occur, as reflected in the predicted decreases in H_T/H_D and R₄₅ with increasing velocity. It is noteworthy that inclusion of flow-dependent resistance did not improve the agreement between predicted and observed hemodynamic parameters in the study by Pries et al. (17), consistent with this result. However, the present model does not take into account the possibility of flow-dependent effects on the glycocalyx itself, and thus it may underestimate the dependence of resistance on flow velocity. For example, Vink and Duling (25) observed that stationary red blood cells can penetrate the glycocalyx. This may reflect a relatively slow dynamic process of plasma protein exchange between the glycocalyx and free plasma, which is not considered here.

The model predicts that the presence of a glycocalyx accentuates the Fahraeus effect. In this respect, it provides a further analysis of ideas developed by Desjardins and Duling (4). However, the present model predicts H_T/H_D values of ~0.5 in 6-µm capillaries, on the basis of the values w = 1 µm and K₀ = 10⁸ dyn · s/cm⁴, whereas ratios ranging from 0.3 to 0.05 were reported (4). The conditions assumed here may underestimate the effective width of the glycocalyx present in those experiments, in which case the effect of the glycocalyx on flow resistance may have been even larger than in the rat mesentery. As shown in Fig. 4, H_T/H_D values as low as 0.25 are possible under the assumptions of this model, but values as low as 0.05 cannot be accounted for within this framework. The mechanisms responsible for such large reductions in tube hematocrit remain poorly understood.

In summary, this model shows that a very dilute, endothelium-bound macromolecular matrix ~1 µm wide, possessing a slightly increased colloidal osmotic pressure, can substantially increase flow resistance in capillaries, to a level consistent with experimental observations. Further work is clearly needed to learn the biochemical and biophysical mechanisms by which such a structure may be formed and maintained.