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HANDBOOK OF PHYSIOLOGY 



CIRCULATION I 



Although the empirical correlations cited above 

 may be better than nothing when one is trying to 

 salvage data, they are the best of a bad bargain. 

 The correlations were obtained in a specified set of 

 experiments, in particular only in those in which 

 recirculation was sufficiently late to permit application 

 of Hamilton's exponential extrapolation. The neces- 

 sity for their use arises in another set of conditions 

 in which recirculation is an early event. There is 

 no information as to whether or not the correlations 

 from the first set of experiments can properly be 

 applied to the second. Furthermore, none of the 

 correlations applies to experiments in which injec- 

 tion or collection, or both, is from other parts of the 

 cardiopulmonary circuit or in which other vascular 

 beds are studied. Finally, these empirical formulas 

 yield approximations only of the area of the curve. 

 They therefore can be used only for estimates of 

 flow and not of volume. 



THEORETICAL APPROACHES 



Conceivably, a complete description of the in- 

 dicator concentration-time curve or of the distribu- 

 tion of transit times, h{t), might require a thorough 

 understanding of the anatomy of the bed under 

 study, including all path lengths, the rheological 

 properties of blood, the driving pressure, the periph- 

 eral resistance, the effect of pulsatile flow in an 

 elastic system, the elastic response of the system, and 

 the neural and humoral factors which affect redistri- 

 bution of blood along various paths as a function of 

 time. Very little is known about any of these, al- 

 though some information may permit first approxi- 

 mations in certain areas. For the present, the diffi- 

 culties seem insurmountable but significant attempts 

 have been made. 



Laminar Flow Through Straight Tubes 

 and More About Catheters 



When a Newtonian fluid flows through a long 

 straight tube of uniform bore, providing the Reynolds 

 number of the system is below a certain critical 

 value, its behavior is described as laminar and it 

 obeys Poiseuille's law. Several treatments of a system 

 in which indicator is distributed through a laminar 

 flow system have been given (24, 28, 29). The follow- 

 ing is adopted from that of Sheppard et al. (28). 

 Imagine that the cylinder of liquid is divided into 

 concentric sleeves. Each sleeve has an inne"- radius 



r and an outer radius r -\- dr, 3. thickness, therefore, 

 of dr and a length L, which is the length of the total 

 system between inflow and outflow. The velocity 

 of fluid in the central core is the greatest. The velocity 

 of fluid at the wall of the container, of radius R, is 

 zero. The velocity of fluid in all other sleeves de- 

 creases from the center to the wall in a parabolic 

 fashion, that is, it decreases according to the square 

 of its distance, r, from the center. Therefore, velocity, 

 u, as a function of radius, u{r) = UmuxCi ~ r'/R') = 

 L/t, where u„^^s- is the maximum velocity in the 

 central stream. It is a basic property of parabolic 

 flow that the maximum velocity is twice the mean 

 velocity (this follows from the fact that the volume 

 of a cylinder is twice that of a contained paraboloid 

 of revolution). The mean velocity is simply the flow, 

 F, divided by the cross-sectional area of the cylinder, 

 irR-. Substituting for u„^!^^ in the equation above. 



u{r) 





(49) 



The volume of a sleeve is 7r(r -j- dr)- L — wr- L or 

 ■K-i r dr- L, and the volume of a sleeve as a fraction of 

 total cylinder volume is 2 r dr I R?. 



Now introduce a quantity, 9, of indicator into the 

 inflow of the cylinder so that it is distributed im- 

 mediately and uniformly over the entire cross-section 

 of the inflow of the tube. Then the fraction of q in 

 any sleeve, at any time, is the ratio of the volume of 

 the sleeve to the volume of the cylinder, and the 

 quantity of indicator in any sleeve is 2 9 r dr/R^. 

 This quantity of indicator will reach the outflow 

 site when it has traveled the length of the cylinder L, 

 and since it does so at velocity u(r), it \\ill all appear 

 at outflow between time t and t + dt, where 

 ; = L/u(r). The quantity of fluid leaving the system 

 between time / and t -\- dt is F dt. The concentration 

 of indicator leaving the system, as a function of time, 

 is therefore the amount leaving in each sleeve divided 

 by the quantity of fluid, or 



c(t) = 2 qr drIB? F dt (50) 



Equation 49 gives the radius r as a function of time. 

 Differentiating equation 49, 



du = - ^iFrdr/irR* = - Ldt/t- (51) 



Substituting in equation 50 from equation 51, 



c{t) = q V/iF'f' (52) 



where V, the volume of the cylinder, is i:R- L. Since 

 V/F = t, equation 52 becomes 



c{t) = q 111 F e 



(53) 



