The Poiseuille equation (see Equation 17-9) is based on several important assumptions (see pp. 416–418). On p. 416, we considered the first five, and here we consider the sixth: whether the viscosity of whole blood is constant in a newtonian sense (see pp. 416–418).
Whole blood has an anomalous viscosity
Water and saline solutions are homogeneous or newtonian liquids. For these, the relationship between shear stress (force needed to move one lamina faster than its neighbor) and shear rate (velocity gradient between laminae) is linear and passes through the origin (Fig. 18-7, blue line). Viscosity (see Equation 17-12) is the slope of this line. The relationship of shear stress to shear rate for a non-newtonian fluid, such as latex house paint, is nonlinear. Blood plasma (see Fig. 18-7, yellow line) and serum are nearly newtonian. However, normal whole blood is non-newtonian: it has a nonlinear shear stress–to–shear rate relationship that intersects the y-axis above the origin (see Fig. 18-7, red curve). In other words, one has to apply some threshold force (i.e., the yield shear stress) before the fluid will move at all. At lower forces, the fluid is immobile. However, at higher shear rates (such as those achieved physiologically after the velocity of blood flow has increased sufficiently), the relationship between shear stress and shear rate assumes the newtonian ideal, with a slope that corresponds to a viscosity of ~3.2 centipoise (cP).
FIGURE 18-7 Anomalous viscosity of blood. To emphasize deviations from linearity, we plot the square roots of shear stress (force/area) and shear rate (Δv/Δx).
The concept of yield shear stress can best be understood by comparing two familiar viscous solutions: honey and mayonnaise. If one estimates the effort needed to stir a pot of honey or a jar of mayonnaise with a spoon, one might assume that their apparent viscosity is the same because each solution offers about the same resistance to deformation. However, with use of a more modest deforming force, such as gravity, a striking difference between the two fluids emerges. As one removes the spoon from the pot of honey, gravity causes the honey to drip from the spoon in a continuous stream. In contrast, a glob of mayonnaise stays attached to the spoon. The velocity of the mayonnaise is zero because its yield shear stress is greater than the force of gravity. Honey is a newtonian fluid that behaves similarly at low and high forces of deformation. Mayonnaise is similar to blood in that its viscosity appears to be infinite at low forces of deformation.
Blood viscosity increases with the hematocrit and the fibrinogen plasma concentration
In practice, the effective viscosity of whole blood depends on several physiological factors: (1) fibrinogen concentration, (2) hematocrit, (3) vessel radius, (4) linear velocity, and (5) temperature. As noted previously, the viscosity of whole blood in the linear (or newtonian) region of Figure 18-7 is ~3.2 cP, assuming a typical fibrinogen concentration of 260 mg/dL, a hematocrit of 40%, and a temperature of 37°C. We will now see how each of these five factors influences the viscosity of blood.
Fibrinogen is a major protein component of human plasma (see p. 429) and is a key element in the coagulation cascade. The main reason for the non-newtonian behavior of blood is the interaction of this fibrinogen with RBCs. Thus, plasma (which contains fibrinogen but not RBCs) is newtonian, as is a suspension of washed RBCs in saline (without fibrinogen). However, the simultaneous presence of fibrinogen and blood cells causes the nonlinear behavior illustrated by the red curve in Figure 18-7. At normal hematocrits, fibrinogen and, perhaps, low-density lipoproteins (see p. 968), electrophoretically seen as β-lipoproteins, are the only plasma proteins capable of creating a yield shear stress.
At normal hematocrits, the absence of fibrinogen (in congenital afibrinogenemia) eliminates the yield shear stress altogether. N18-4 Conversely, hyperfibrinogenemia elevates yield shear stress and, in the extreme, leads to a clustering of RBCs that increases their effective density. This increased effective density, in turn, causes the RBCs to settle toward the bottom of a vertical tube—easily measured as an increased erythrocyte sedimentation rate (see p. 430). Note that fibrinogen levels tend to increase with age and with smoking.
Effect of Fibrinogen on Blood Viscosity
Contributed by Emile Boulpaep
As shown by Figure 18-7, at very low shear rates (x-axis), the viscosity of whole blood (the value of shear stress divided the shear rate for the red curve) is greater than at high shear rates because of the non-newtonian nature of whole blood. The “yield shear stress” (the intercept at the y-axis) is the minimum force (i.e., shear stress) that one needs to apply to achieve any blood flow (i.e., shearing) at all.
The yield shear stress increases with the square of the fibrinogen concentration. Fibrinogen is a large glycoprotein (molecular weight ≅ 340 kDa) that is a hexamer composed, in turn, of a dimer of three chains, α, β, γ.
All known congenital fibrinogen disorders cause fibrinogen levels to be low. Congenital afibrinogenemia is a rare autosomal recessive disorder that leads to severely impaired hemostasis. As would be expected—because of the lack of fibrinogen—this disease is associated with near-newtonian behavior of whole-blood viscosity. Congenital hypofibrinogenemia is a rare genetic condition caused by mutations in one of the three fibrinogen chain genes (α, β, γ). As a result, heterozygotes exhibit poor assembly and secretion of mature fibrinogen by the liver.
Acquired alterations in fibrinogen levels include both hypofibrinogenemia and hyperfibrinogenemia. Hypofibrinogenemia can result from liver failure. Elevated fibrinogen levels are part of the acute-phase reaction during inflammation. Higher fibrinogen levels are associated with higher risk of ischemic heart disease—although there is not necessarily a cause-and-effect relationship.
Increases in hematocrit elevate blood viscosity by two mechanisms (Fig. 18-8). One prevails at physiological hematocrits, the other at higher hematocrits. Starting from values of 30%, raising the hematocrit increases the interactions among RBCs—both directly and by proteins such as fibrinogen—and thereby increases viscosity. The reason is that forcing two RBCs closer together at higher hematocrits makes them more likely to stick to each other. At hematocrits >60%, the cells are so tightly packed that further increases lead to cell-cell interactions that increasingly deform the RBCs, thereby increasing viscosity. In patients with polycythemia, not only is viscosity high, but the yield stress can be >4-fold higher than normal (see Fig. 18-7). Obviously, the combination of a high hematocrit and a high fibrinogen level can be expected to lead to extremely high viscosities.
FIGURE 18-8 Dependence of viscosity on hematocrit.
In reasonably large vessels (radius greater than ~1 mm), blood viscosity is independent of vessel radius Fig. 18-9). However, the viscosity decreases steeply at lower radii. This Fahraeus-Lindqvist phenomenon has four major causes.
FIGURE 18-9 Dependence of viscosity on vessel radius.
Poiseuille observed not only that RBCs move faster in the center of an arteriole or venule than at the periphery but also that the concentration of RBCs is greater at the center. Indeed, very near the wall, he observed a “transparent space” occupied only by plasma (Fig. 18-10A). This axial accumulation of RBCs occurs because the plasma imparts a spin to an erythrocyte caught between two layers of plasma sliding past one another at different velocities (see Fig. 17-5B). This spin causes the cell to move toward the center of the vessel, much like a billiard ball curves when struck off center (see Fig. 18-10A, inset of upper drawing). One consequence of axial accumulation is that local viscosity is lowest in the cell-poor region near the vessel wall and greatest in the cell-enriched core. The net effect in smaller vessels is that the overall viscosity of the blood is decreased because the cell-poor plasma (low intrinsic viscosity) moves to the periphery where the shearing forces are the greatest, whereas the cell-enriched blood (high intrinsic viscosity) is left along the central axis where the shearing forces are least. A second consequence of axial accumulation is that branch vessels preferentially skim the plasma (plasma skimming) from the main stream of the parent vessel, which leads to a lower hematocrit in branch vessels. However, some anatomical sites prevent skimming by means of an arterial cushion (see Fig. 18-10A, lower drawing).
FIGURE 18-10 Flow of blood in small vessels.
The Poiseuille equation does not hold in vessels small enough to contain only a few RBCs in their cross section. Hagen's derivation of the equation assumes an infinite number of concentric laminae inside the vessel. In reality, a lamina can be no smaller than the thickness of an erythrocyte, which severely limits the number of laminae in small vessels (see Fig. 18-10B). If we were to re-derive a Poiseuille-like equation to model a small vessel, it would predict a viscosity that is lower than that predicted by the Poiseuille equation for a large vessel.
In vessels (e.g., capillaries) so small that their diameter is about the size of a single erythrocyte, we can no longer speak of the friction between concentric laminae. Instead, in small capillaries, the membrane of the RBC rolls around the cytoplasm in a movement called tank treading, similar to that of the track of a bulldozer (see Fig. 18-10C). As two treading erythrocytes shoot down a capillary, they spin the bolus of plasma trapped between them (bolus flow).
In vessels smaller than an erythrocyte, highly deformed—literally bullet-shaped—RBCs squeeze through the capillaries, so that the effective viscosity falls even farther below that of the bulk solution (see Fig. 18-10D). Under these conditions, a layer of plasma separates the erythrocytes from the capillary wall. The cells automatically “focus” themselves to the centerline of the capillary and maintain a fixed distance between two successive cells.
Velocity of Flow
As for an ideal fluid (see Fig. 17-6A), the dependence of flow on pressure is linear for blood plasma (Fig. 18-11, gold curve). However, for whole blood, the pressure-flow relationship deviates slightly from linearity at velocities close to zero, and the situation is even worse for polycythemia. These deviations at very low flows have two explanations. First, at low flows, the shear rate is also low, which causes the whole blood (like mayonnaise) to behave in a non-newtonian manner (see Fig. 18-7) and to have a high apparent viscosity. In fact, as was already seen, one must apply a threshold force to get the blood to move at all. Thus, the red and green curves in Figure 18-11 have shallower slopes at low flows than at high flows. The second reason for the nonlinear pressure-flow relationship is that the tendency for RBCs to move to the center of the stream—thereby lowering viscosity—requires a modest flow. After this axial accumulation “saturates,” however, the relationship becomes linear.
FIGURE 18-11 Pressure-flow relationships over a range of hematocrits. The slope of the linear portion of each curve is 1/resistance; resistance (or viscosity) increases from plasma to normal whole blood to polycythemic blood.
Cooling normal whole blood from 37°C to 0°C increases its viscosity ~2.5-fold. However, physiologically, this effect is negligible in humans except during intense cooling of the extremities. On the other hand, in some patients, the presence of cryoglobulins in the blood can cause an abnormal rise in viscosity even with less intense cooling of limbs. Cryoglobulins are immunoglobulins that precipitate at a temperature that is <37°C but can partially resolubilize on warming. Different cryoglobulins are associated with infections, particularly hepatitis C, as well as with several autoimmune and lymphoproliferative disorders. High blood viscosity from the precipitated cryoglobulins can lead to vessel obstruction and local thrombosis.