146 



HANDBOOK OF PHYSIOLOGY 



CIRCULATION I 



viscosity as deduced from the pressure-to-flow ratio 

 will fall as a becomes smaller and the ratio zl a be- 

 comes larger. This effect will be superimposed on 

 that produced by the reduction in the viscosity of 

 the blood itself, according to equation 18. 



If we know the values of the pressure-to-flow 

 ratios for a given sample of blood, under the same 

 conditions of shear, in tubes of several difTerent radii, 

 we can deduce from equations 18 and 22 the values 

 of c, 5, and tj^. Thecomputation, however, is elaborate, 

 and the experimental methods are not, at present, of 

 sufficient precision for it to be possible to derive other 

 than very approximate \alues. If one neglects the 

 finite summation correction (equation 18), the analy- 

 sis is greatly simplified, although the results may be 

 subject to a systematic error. The quantities i and 6, 

 however, are both probably related to the dimensions 

 of the red cells, and are thus likely to be more or less 

 equal in magnitude; the error is not likely to be 

 serious except when the radius of the tube used is 

 extremely small (say 30^ or less). The results of 

 analyses of this kind (5, 34) indicate that in blood 

 with hematocrit values between 40 and 50 per cent 

 the width of the marginal sheath lies between iju and 

 5/i, and is probably between i/ii and 3^4; it is thus 

 about the value to be expected from the wall effect. 

 The lines drawn in figure 2 are calculated from 

 equations 18 and 22, putting r/t = 3.15 (in a tube of 

 infinite radius) and either c = 5 = i .0 ju or c = 5 = 

 2.0 p.. 



Transit Times of Cells and Plasma and 

 the Dynamic Hematocrit 



When blood flows through a tube, the existence of 

 a marginal sheath and an axial core has the effect of 

 partially separating the red cells from the plasma, so 

 that their average velocities are not the same. The 

 relative velocities of cells and plasma will be inversely 

 proportional to the relative "transit times," that is, to 

 the average times taken by the cells and the plasma, 

 respectively, to traverse the tube or system of tubes, 

 such as an organ or tissue of an animal. These times 

 may be measured experimentally by "labeling" 

 suitably the cells and the plasma. 



Further, in unit time the volume of cells emerging 

 from the tube or system of tubes must be equal to the 

 volume of cells present in each unit length of the tube, 

 multiplied by the average velocity of the cells. Now 

 the volume of cells in unit length of a tube will be 

 equal to the cross-sectional area of the tube, multi- 

 plied by the hematocrit value of the blood while it is 



flowing in the tube. This last is called the "dynamic 

 hematocrit"; it may be measured experimentally by 

 suddenly stopping the flow and estimating the 

 relati\'e volume of cells and plasma in the tube or 

 system ("suddenly" means in a time very short 

 compared with the transit time). The relation of the 

 dynamic hematocrit (ctu) to the ordinary or "bulk" 

 hematocrit (a,,) is thus defined by the equation: 



cto'Qfi — Tra--aD-Uc 



where a is the radius of the tube, and Uc is the average 

 linear velocity of the cells. An analogous equality 

 must apply to the rate of emergence of the plasma, so 

 that we have : 



(l — ao)Q,b = "0^(1 — ao)-'ip 



If the transit times of cells and plasma, respectively, 

 are tc and tp, we thus find : 



(l — ore) 



(I 



o) 



(24) 



This is a perfectly general equation relating transit 

 times to the dynamic hematocrit. No assumptions 

 have been made as to the origin of the difference 

 between the velocity of the cells and the velocity of 

 the plasma; it might even be, as an extreme example, 

 that the cells and the plasma travel in diff"erent 

 channels, of different dimensions, in parallel with one 

 another. But in so far as the difference in velocity is 

 due only to the existence of a marginal sheath and an 

 axial core, it is possible to deduce the relations 

 between the dynamic hematocrit, or the relative 

 transit times, and the fractional width of the marginal 

 sheath (i.e., the ratio z/a or the quantity 7). But we 

 must now define the hypothetical cell-free marginal 

 sheath in the second of the two ways mentioned 

 above; it must be of .such a width that the volume of 

 cells in unit length of the tube (now in the axial core 

 only) is the same as that which actually exists in the 

 same conditions of flow. This definition leads to a 

 value of the equi\alent cell-free marginal sheath 

 which is not necessarily identical with that resulting 

 from the definition used in the previous section. The 

 matter is not of great practical importance, however, 

 since we cannot measure either of the values, except 

 very approximately. If the radius of the tube is a, 

 and that of the axial core is ya, we have: 



Volume of cells in imit length 



= ira--aD 



Try a- ■ Uc^ 



