CIRCULATION TIMES AND THEORY OF INDICATOR-DILUTION METHODS 



59' 



FIG. 6. Series of 3-dimcnsional drawings to illustrate that experimental elapsed time is identical 

 with indicator transit time and that V = Ft. A: distribution of transit times through system (same 

 distribution as fig. 4). Total flow during first time unit following injection of indicator. White cubes 

 in this and subsequent drawings are untagged particles. Number on face of each cube indicates its 

 transit time. B: total flow during first three time units following injection of indicator. Fastest indi- 

 cator particles (shaded cubes) have just appeared. C: by the end of the seventh time unit all indi- 

 cator particles have pushed out of the system all untagged particles of the same transit time. The 

 total number of cubes is therefore the volume of the system. Height of each column (of cubes of the 

 same transit time) is F h(t), length is t, and width is dt. Volume of each column is therefore F t h {t)dt. 

 Sum of volumes of all columns is total volume of system, F ^_, ^ h(l) dt. [From Zierler (38).] 



centration of indicator, from equation 8, since /;(/) 

 f(/)//o" c{t) dt. 



dV = F I h{t) dt = 



F t c(i) dt 



J 



Jo 

 Summing for all such time intervals 



/ t c(t) dt 

 Jo 



/ c(t) dt 

 Jo 



c{t) dt 



V = F 



(..) 



where 



/' 



Jo 



t dt) dl I I c{l) dt 



It is important to remember that the derivation of 

 equations 9, 10, and 1 1, v^fhich describe the relation- 

 ship between volume, flow, and mean transit time, 

 requires no assumptions about the form of li{t) which 

 is determined solely experimentally by the observed 

 curve of c{i) versus time. Development of the relation 

 between mean transit time and volume is illustrated 

 in figure 6. 



There has been some misunderstanding of the 

 definition of mean transit time. Hamilton et al. (10) 

 originally and incorrectly defined the mean time as 

 that time which divided in half the area under the 

 concentration-time curve following sudden-injection. 

 This, of course, is the "median" time and differs 

 from the mean time in all but svmmetrical curves. 



Hamilton et al. (12) soon corrected the error but it 

 appeared persistently in the works of some others 

 for more than a decade. 



~t = i'o I h{t) dt is literally the mean or average 

 value of t. This formula for the mean is not familiar 

 to many biologists who are more accustomed to 

 seeing it expressed as (^"^oti)/N, where A' is the 

 total number of observations of t. The identity of the 

 two expressions for / may be illustrated by considering 

 a population of / in which the value to appears oo 

 times, the value /i appears ai times, and so on. 

 Clearly, / = (aula + fli/i + • ■ • -f a„t„)/ 

 (ao + ai + • • ■ + a„) = (aolo + aiti + ■ • ■ + a„t„)/ 

 N = to(ao/N) + t,(ai/N) + ■■■ + t„{ajN) = 

 '^j=otj(aj/N). aj/N is exactly the frequency with 

 which the time tj occurs, that is, it is the fraction of 

 the total observations which includes tj . It is there- 

 lore equivalent to h(t) dt. 



In hydraulic engineering the volume of water in 

 conduits has been measured by an indicator dilution 

 method known as Allen's method (i), although Allen 

 acknowledged his indebtedness to Stewart. Like 

 Stewart, Allen added salt to the flowing system and 

 measured the change in conductivity at some distant 

 point. However, it was intrinsic in the conduit system 

 that the appearance time was long and dispersion of 

 the saline indicator was small so that no great error 

 was introduced by using any time during the arrival 

 of salt at the sampling site. Allen, in fact, tested the 

 appearance time, the time at which peak concentra- 

 tion occurred and the mean time in the equation 

 volume = flow X time, and found that the mean time 



