THE RHEOLOGV OF BLOOD 



141 



of a, however, cannot be precisely evaluated vvitliout 

 making some rather arbitrary assumptions. 



Various empirical, or semiempirical, cqviations 

 relating relative viscosity to volume fraction in more 

 concentrated suspensions have been put forward. 

 Most of them have been used to describe the relation 

 between the relative \'iscosity of plasma and the 

 protein concentration, and the relation between the 

 relative viscosity of blood and the red cell concentra- 

 tion (hematocrit value) : none of them, however, does 

 so perfectly. Since the viscosity of i:)lood is shear- 

 dependent, it is obvious that the effect of changing 

 the hematocrit \alue must be observed in some 

 defined conditions of shear, although this has not 

 always Iseen done. The most suitable is tlie asymptotic 

 condition which is approached when the shearing 

 stress is very large, and the blood behaves nearly as 

 a Newtonian fluid (see below). 



Arrhenius ( i ) modified and extended the Einstein 

 equation, and arrived at the expression : 



the factor b must be made a function of a. If we make 

 h = i/a-i\ we get: 



log v^ = 



(9) 



where c is the "mole fraction" of the suspended 

 particles, i.e., the ratio (number of moles in disperse 

 phase)/ (number of moles in continuous phase). 

 Roughly, if the densities of the two phases are not 

 very different, we can put c = a/(i — a). Empirically, 

 however, it was found necessary to increase tiie value 

 of c by an arbitrary factor representing the volume of 

 suspending fluid which is carried along with the 

 suspended particles. This equation does not fit very 

 well when applied to the viscosity of dog defibrinated 

 blood (3). An equation of this form, however, in 

 which the hematocrit value (i.e., a) is in.serted in 

 place of c describes adequately the viscosities of 

 suspensions of red cells in acid citrate-dextrose 

 solutions (23). 



Bingham & White (8) and Hess (25) independently, 

 but using the same general conceptions, different from 

 those u.sed bv Einstein, arrixed at an expression of 

 the form : 



1/(1 - b-a) 



(10) 



where b is an arbitrary factor representing the 

 increase in the "effective" volume fraction of the 

 suspended particles. This equation may be used to 

 estimate the viscosity of plasma or serum from the 

 protein concentration, within the limits of variation 

 likely to occur in li\ing animals. If the protein con- 

 centration is expressed in grams per 100 ml, the 

 value of b is 0.06. When applied to blood, however, 



v> 



1/(1 



"} 



(1.) 



which is the expression derived by Hatschek (22) for 

 tlie flow of an emulsion so concentrated that the 

 particles are deformed into flat polyhedra. The 

 asymptotic minimum viscosity of dog defibrinated 

 blood has been found to be a constant fraction of 

 that deduced from the Hatschek expression, over a 

 wide range of hematocrit values (3, 42). If the radius 

 of the tube is 200/i or over, this fraction is approxi- 

 mately 0.6 at 37 °C; for smaller tubes, it is smaller by 

 an amount which is plotted in figure 2. 



Effect of Temperature on the Viscosity of Blood 



The relative viscosity of a suspension or solution 

 should be unaffected by temperature unless the 

 volume fraction or the shape of the suspended or 

 dissolved particles changes; its absolute viscosity, 

 therefore, should depend on temperature to the same 

 extent as does that of the suspending fluid. This is 

 true of plasma or serum, and the viscosity measured 

 at one temperature may be corrected to some other 

 temperature by reference to the tabulated values of 

 the viscosity of water (27, 29). 



The relative viscosity of blood, however, rises by 

 about 10 per cent when the temperature is reduced 

 from 37°C to about I7°C (3). This temperature 

 efi"ect is sensibly independent of the hematocrit 

 value of the blood when this is greater than about 30 

 per cent. It is probable, though not definitely estab- 

 lished, that the increase in the relative viscosity as 

 the temperature is reduced is due to a small increase 

 in the volume of each red cell, together with a change 

 of shape towards a more spherical and less disc-like 

 form. There is no reason to believe, however, that the 

 rheological properties of blood are substantially 

 altered when it is allowed to cool to room temperature. 

 These properties are, in fact, more stable at the lower 

 temperature, and many of the obsers^ations to be 

 referred to subsequently have been made at about 



20°C. 



THE NON-NEWTONIAN FLOW OF BLOOD 



Let US take a sample of blood of approximately 

 normal composition, make it flow through a tube, 

 and plot the relation between the volume rate of flow 

 and the pressure head applied across the ends of the 



