588 HANDBOOK OF PHYSIOLOGY ^ CIRCULATION I 



40r- 



30 - 



20- 



5 10 15 20 25 



FIG. 2. A: concentration of indicator as function of time after 

 sudden-injection into systemic vein, sampling from systemic 

 artery. Recirculation of indicator begins at arrow and dashed 

 line is extrapolation of concentration during first circulation. 

 B: concentration of indicator as function of time during con- 

 stant injection through same bed as A, to show that/? is integral 

 of A (scale factor used on concentration axis) and that recircu- 

 lation is evident at same time in B as in A. Dashed line is ex- 

 trapolation to plateau concentration. [From Meier & Zierler 

 (i9)-] 



system at any moment, s, is the product of its concen- 

 tration at that moment, c(s), and the unknown flow, 

 F. Eventually the entire ma.ss, q, of injected indicator 

 must leave the system. If all products of c(s) and F 

 are summed, this sum must equal the amount of 

 injected indicator, or. 



r 



F I cit) lit 

 Jo 



(3) 



F, the unknown flow, can therefore be measured 

 from either equation 2 or •], using constant- or sudden- 

 injection, whichever is appropriate. Since both meth- 

 ods measure the same flow, combining equations 

 ■2 and 3 yields 



1/1 



= (/■""•■')/' 



(4) 



If the magniiudcs of q and / are chosen so that q/I = 



FIG. 3. Schema of vascular bed with injection sites P, P' and 

 sampling sites Q, Q_'. [From Meier & Zierler (19).] 



I, Cmax = jo c(l) dt- Hamilton & Remington (13) 

 first pointed out that constant-injection could be 

 considered an integral of sudden-injection and that 

 Cn.ax must therefore be the integral of the concentra- 

 tion-time curve obtained by single injection, a point 

 to which we shall return later. 



Measurement of I'olume 



Measurement of flow is unequivocal, if there is no 

 recirculation, by either sudden- or constant-injection 

 of indicator. There has been, however, some dispute 

 over the measurement of volume by indicator-dilu- 

 tion methods. We shall develop a rigorous demonstra- 

 tion of the relation between the volume of a system 

 and the concentration of indicator as a function of 

 time at the output from the system. 



First consider a "closed" flow system, that is, one 

 with a single inflow orifice, P, and a single outflow 

 orifice, Q, (fig- 3)- The system contains a volume, I', 

 of fluid which flows into and out of it at constant 

 flow, F, in units of volume per time. The internal 

 structure may be that of a vascular net, consisting of 

 many branches and interlacing of vessels, but the 

 internal structure need not be specified and the argu- 

 ment which follows is independent of any assumptions 

 concerning structure. Consider that the fluid can be 

 treated as though it were made up of many individual 

 particles. Particles of fluid entering P at the same 

 time require varying amounts of time to reach Q^, 

 the time required for any particle depending on the 

 path taken and the velocity with which the particle 

 travels. The fluid, therefore, does not have a single 

 traversal time from P to Q^ but rather a distribution of 

 traversal times. It is unnecessary to make any assump- 

 tions about the relative proportion of particles having 

 long or short traversal times. The distribution of 

 traversal times is determined solely by the experiment 

 and is not part of the theoretical structure. 



Several restrictions on the system must be made. 

 a) The distribution of traversal times for entering 

 particles does not change with time. Particles enter- 

 ing P at any time are dispersed when they leave at Q_ 



