6o8 



HANDBOOK OF PHYSIOLOGY 



CIRCULATION I 



C{t) vanishes. Thus, for T = o we are left with only 

 equation 59 which has the limit lim7-^oC(0 = 

 gt/2 F f-, where limr_o/ T = q. This result is 

 exactly equation 53, which is therefore a special 

 case of equations 58 and 59 for T = o. 



How pertinent is study of parabolic flow system 

 to the real problem? 



One obvious application is to the effect of the col- 

 lecting catheter which is usually a single tube of 

 uniform bore, though not necessarily straight. Even 

 here, however, owing to perturbations at catheter 

 input, if not also at output, and to the unusual 

 rheological properties of blood, the description is 

 not perfect. Sheppard et al. (28) used equation 53 

 and Sherman et al. (29) used equation 54 to examine 

 the influence of the collecting catheter on the con- 

 centration-time curve. Sheppard et al. (28) also 

 examined the effect of a collecting tube as a con- 

 voluter of a more complex input function, ;'(/). 

 i{t) was generated experimentally by letting indicator 

 flow through a labyrinth — a container filled with 

 glass beads — from which it appeared as the input 

 into a straight tube. 



Sherman et al. (29) proposed what they called "a 

 figure of merit" for catheter sampling systems. In 

 order to give satisfactory resolution of indicator con- 

 centration with respect to time in systems in which 

 rather abrupt changes occur, as in pulsatile systems, 

 regurgitant flow, pulsatile pump models of the 

 circulation, and so on, the smearing effect of the 

 catheter on the concentration-time curve should be 

 less than the interval between important changes in 

 indicator input to the catheter. Their meritorious 

 figure is "the volume of the catheter divided by 

 twice the flow rate." This, of course, is exactly ^0^ 

 which is, as we have seen, the appearance time 

 through a parabolic flow system. In terms of the 

 convolution equation, recalling that h{t) for a catheter 

 with nonturbulent flow is t/2 t", then as / goes to 

 zero, //(/) disappears and the input is not distorted. 

 The implication is that flow through the catheter 

 should be as large as possible and the volume of the 

 catheter as small as possible. 



Do these considerations of parabolic flow systems 

 apply to real vascular beds? Obviously, although, 

 e.xcept for transient eddies at branches, blood flow 

 through vascular beds is nonturbulent, it does not 

 obey the simple law evolved here. Nevertheless, 

 using /;(/) = t/2 t- as a first approximation, we may 

 ask what effect there is on indicator concentration 

 when indicator is made to flow through tubes in 

 parallel and in series. 



Consider parabolic flow through parallel tubes. 

 A quantity, q, of indicator is introduced at a com- 

 mon inflow at time zero and thoroughly mixed with 

 inflowing fluid. It is carried immediately to one or 

 another of parallel branches in proportion to the 

 relative flow through each branch. Thus the rate at 

 which indicator enters the /-th branch is q Fi/F dt, 

 where F, is flow through the /-th branch and F is 

 total flow through the system. The fraction of indicator 

 particles leaving the i-l\i branch per unit time be- 

 tween time t and t -\- dt\s q Ft hi(t)/F, where hi{t) = 

 ii/2 t"^. Indicator particles exiting from the /-th branch 

 immediately merge with total flow, so that the 

 contribution to indicator concentration at outflow 

 from the /-th branch is 



CM = 



o, t < }4 h 

 q F, I 



2 F'ifl 



/ g 1^ li 



where t, is mean transit time through /-th branch. 

 Contributions from all parallel branches are addi- 

 tive, so that the observed concentration at outflow, 

 summed for a system of n branches, is 



c(t) = CM + CM + ■■■ + CM) 



^F'^'iTi 



E f . '. 



(60) 



E^'. 



2 F' fl ;__ 



where I', , the volume of the /-th branch, = F, U , 

 and where C(/) = o, / < J 2 '2 , the shortest appearance 

 time in the system. Since the total volume of the 

 system F = 2Z"_i 1', , then once all branches have 

 begun to contribute to outflow, C(t) = q V/2 F^ t- = 

 q 1/2 F t-, t ^ 3'2 '" ) th^ longest appearance time in 

 the system. 



Thus, as soon as the initial transients have been 

 completed and all branches are contributing indi- 

 cator, the system of parallel tubes with parabolic 

 flow behaves as though it were a single tube with 

 parabolic flow, of volume I' and mean transit time /. 



Concentration in a parallel tube system is il- 

 lustrated in figure 11. 



Now consider parabolic flow systems in series. 

 The output from one tube is the input to a second 

 tube. Concentration at output from the second tube 

 is the integral of the convolution of the distribution 

 function through parabolic systems. 



