CIRCULATION TIMES AND THEORY OF INDICATOR-DILUTION METHODS 



595 



and a fraction (i — h) through ^i . The flow through 

 /?2 is thus a b F and, through Ri , a (i — b) F. The 

 flow through R,, is the sum of the flow through J^i and 

 P^', or (i — a b) F. The flow through R^ divides. A 

 fraction c returns to the upper channel to join the 

 flow from /?2 . The fractions a, b, and c are each 

 greater than zero and less than i . If indicator is 

 injected into the system at P/ at constant rate, /, after 

 the initial transients have disappeared, its steady 

 concentration as a function of the channel in which 

 concentration is measured, assuming mixing of fluid 

 is everywhere complete, will be: 



C (A') = 4 ; C {P.') = o; C (R,) = /' ~ ^-J^ = C (Q/) 

 ap (1 — a b) F 



C (Qi') = 



(c - b c + b) I 

 {c — a l> c -{- a b) F 



If sampling is at Q,i'. the requirement for a useful 

 measurement of flow is that C (Q,/) be very nearly 

 equal to I/F. This will be true whenever a is very 

 nearly i, that is, whenever nearly all the flow passes 

 through Pi' and nearly none through P«'. In that 

 event, the system is obviously almost that of the case 

 of single input-many outputs. If « issmall, C (Q_i') may 

 still be nearly equal to I/F if c is very nearly i, for 

 then the system is almost that of many inputs-single 

 output. Even if c is not close to i , but is large com- 

 pared to a and b, C (Q./) will be approximately 

 equal to I/F. 



However, if both a and f are small, then C (Q/) 

 will differ greatly from I/F. This is to be expected, 

 for, in such a case, only a small portion of the total 

 flow passes through the system Pi' — > Q/. 



In more complex systems, even if no single channel 

 ever receives a major portion of the total input, C 

 iQj') may be nearly equal to I/F. For example, if 

 one adds a second pair of communicating channels, 

 like Ri and R^ in figure 8, and if at every bifurcation 

 half the flow takes one path (i.e., 0.5 = a = b = c =), 

 then C (Q,/) will exceed I/F by less than 5 per cent. 



Measurement of volume in an open system is more 

 tenuous. If, in figure 8, a and b are large, that is, if 

 most of the fluid takes the path P ^> Pi' —> R2 —* 

 ()i', then t will be small and the volume measured 

 will be nearly that of the preferential channels. 

 Unless there are many communicating branches 

 through which fluid from the inflow channels can 

 intermingle luxuriously, there is little likelihood that 

 the volume measured by the equations developed so 

 far will resemble closely the true volume of the sys- 

 tem. The criterion for satisfactory intermingling is 



FIG. 8. Intercommunicating flow system through most of 

 which indicator may be distributed, even if injection is only 

 into a branch. Arrows within channels indicate direction of 

 flow. Symbols within channels indicate fraction of flow carried 

 by each channel. 



simply that, for the case of constant-injection, the 

 limiting concentration of indicator in every output 

 channel must be the same. For certain vascular beds 

 this can be and has been tested (2). 



T/ie System h Xonstationary 



The "stationarity" condition, that is, that the 

 distribution of traversal times for entering particles 

 does not change with time, is violated in real vascular 

 systems. It is certainly violated in pulsatile systems, 

 particularly those which include the heart, and it is 

 apt to be violated by vasomotor actixity in which 

 entering blood is distributed chiefly through one 

 branch and then another as peripheral resistance is 

 shifted by alterations in arteriolar diameter. In 

 pulsatile and vasoactive systems there are not only 

 changes in flow but also changes in volume with 

 time. If phasic, :iot necessarily regular, alterations in 

 distribution of traversal times occur many times 

 during the esolution of the sudden-injection indicator- 

 dilution curve, then the violation of "stationarity" 

 may not be important. The constant-injection 

 method will yield a measure of flow and volume in 

 nonstationary systems because the input-output 

 equations still hold, although there may not be a 

 plateau concentration, but a regular or irregular 

 oscillation about a plateau. 



Flow or Volume, or Both, not Constant 



The sudden-injection method cannot measure 

 either flow or volume if flow and volume change 

 during the course of the indicator-dilution curve, 

 unless the changes are rapid and phasic. The con- 

 stant-injection method cannot measure a changing 

 volume but it may measure flow. If the changes in 

 flow are slow or move from one steady flow to another. 



