"locally") takes place only when the flow rate is 

 changing in time. Pressure differences between two 

 points which are caused by local acceleration are 

 proportional to the differential quotient of the flow 

 velocity, to the fluid's density, and to the distance 

 between both points. Thus, we have, according to 

 Frank (39), the following equation: 



(8) 



• p **F, v 



* ?* v . 



n 



dv 



* ?"» 



m 



where Pi and P2 = instantaneous pressures at two 

 different points; v = instantaneous flow velocity; 

 t = time; C u C 2 , and C 3 = coefficients; I = frictional 

 term; II = inertia term due to convective accelera- 

 tion (Bernoulli); III = inertia term due to local 

 acceleration. The pressure difference (Pi — Pj) is 

 recorded by a suitable differential manometer (see 

 below). In most experimental cases, the coefficients 

 are determined by practical calibration although they 

 are, under certain conditions, calculable from the 

 tube dimensions and from density and viscosity, 

 respectively, of the fluid. If, instead of the linear 

 velocity v, the rate of volume flow is used in equation 

 8, then d is equivalent to the so-called effective mass 

 M' (Frank): 



M -k 



<°L 



(9) 



where p = density of fluid, L and A = length and 

 cross-sectional area, respectively, of the fluid column 

 contained in the tube between both points and k = 

 correction factor for velocity distribution within this 

 column; k = 1.0 if the velocity profile is flat [see 

 (46, 48)]. As Ranke (107) pointed out, equation 8 

 must be regarded as an approximation, since the 

 coefficients change with the Reynolds number for the 

 flow and further terms may have to be taken into 

 account. 



The properties of most differential-pressure flow- 

 meters which respond to pulsatile flow are dependent 

 upon the three terms of equation 8, although, for 

 certain models, one or two terms may play a dominat- 

 ing role. If only mean flow is recorded, term III can 

 be ignored; nevertheless, a great effective mass 

 (coefficient C 3 ) should be avoided because it alters 

 hemodynamic conditions in the case of pulsatile 

 flow [cf McDonald (93)]. If the tube diameter is 

 large, as in great central vessels, term I usually has 

 little significance as compared to term II. 



If the pressure difference is generated mainly by 

 friction as in figure 2, the device must be constructed 

 in such a way that the resulting pressure drop will 



METHODS OF MEASURING BLOOD FLOW 1 297 

 UPO DPO 



fig. 2. Friction device. UPO, DPO = lateral openings 

 upstream and downstream from the constriction for connection 

 with differential manometer. [From Green (50).] 



not be so great as to disturb the physiological condi- 

 tions. A friction device consisting of a long, narrow 

 plastic tube inserted into a blood vessel was applied 

 by Ueno & Takenata (129) for recording the mean 

 flow; the pressure drop was measured by a rolling 

 manometer. It seems likely that it interferes with 

 normal blood flow. 



An older method may be mentioned here. In 1935, 

 Green et al. (53) tried to estimate the systolic and 

 diastolic coronary-artery flow from the pressure 

 difference between the aorta and a peripheral coro- 

 nary branch. There is, however, no simple relation- 

 ship between these magnitudes, because waves travel- 

 ing in elastic tubes are concerned. Therefore, the 

 method was abandoned [cf Gregg's criticism (54) and 

 Chapter 7, vol. I, this Handbook]. 



The principle of the Venturi meters is based on the 

 generation of convective acceleration by a variation 

 in the cross-sectional area of a tube (Venturi 1797; 

 Herschel 1887). As shown in figure 3, the fluid has to 

 move from a wider into a narrower tube segment. 

 According to the continuity law, equal quantities of 

 an incompressible fluid must pass each cross section 

 of a rigid tube during the same time interval. The 

 fluid's linear velocity is therefore augmented in the 

 narrow segment so that here the kinetic energy is 

 increased and the lateral pressure is decreased. This 

 results in a pressure difference between UPO and 

 DPO in figure 3, which is proportional to the square 

 of the average flow velocity (term II in equation 8). 

 If the tube widens again downstream from DPO to 

 the same cross-sectional area as before, the former 

 pressure is restored. The additional influence of fric- 

 tion will augment the pressure difference between 

 both points (term I); this part of the pressure drop, 

 of course, is not reversible by rewidening of the tube. 

 When the rate of volume flow is changing in time, a 

 third kind of pressure difference corresponding to 

 term III appears which should be kept minimal 

 because it distorts the records. Devisers of such flow- 

 meters often failed to take this source of error into 

 consideration. Lauber's Venturi cannula (87), for 

 instance, was criticized by Frank (42) because its 



