592 



HANDBOOK OF PHYSIOLOGY 



CIRCULATION 



did not yield a solution for volume which agreed best 

 with the known volume of his test system. Empirically, 

 Allen chose the time at which peak concentration 

 appeared. Since the mean time is the only time which 

 is formally correct, Allen's result must be attributed 

 to experimental error in either his independent 

 measure of flow or volume, or in his measure of time. 

 It may have been caused by dispersion of indicator 

 at the site of introduction because, as will be shown 

 later, this leads to an overestimate of mean transit 

 time. 



We turn now to consideration of measurement of 

 volume by constant-injection of indicator. Indicator 

 is introduced into the system at P (fig. 3), at constant 

 rate, /, in units of mass-time"^. Assumption e (page 

 589) again demands that indicator and fluid mix 

 completely at P. Therefore, the concentration of 

 indicator at P is I/F, where F is flow in units of 

 volume- time"'. 



We have already seen that the concentration of indi- 

 cator at the outflow, C{t), increases with time to a 

 maximum, Cmax , which equals I/F (equation 2). 

 This is so because indicator particles, entering the 

 system at concentration I/F, gradually displace all 

 indicator-free elements until every element of volume 

 within the system contains indicator at concentration 

 I/F. When the concentration at outflow finally 

 reaches I/F, the concentration of indicator through- 

 out the volume, V, must also be I/F. This means 

 that there is within the system a mass of indicator, 

 M, distributed over V at concentration I/F, or 



M/V = I/F 



(12) 



Since / is known and F is calculated from equation 2, 

 I/F = Cniax , if ^ can be determined we have a solu- 

 tion for V. M is determined as follows: 



The amount of indicator in the system at any time, 

 /, is 



M(l) = (input up to time t) — (output up to time t) 



= 1 1 - f F Cit) dt 

 Jo 



= f \I - F C{t)] dt 

 Jo 



(13) 



= F f [c,„,, - ao] dt 



Jo 



at any time, /, is 



M(l) F 



Mil) _ F r 



y ~ ^ L 



Jo 



rC„,ax - C(t)\dl 



(14) 



The limit of this concentration, as we have seen, 

 is Cniax = I/F. Therefore 



lim 



(-•00 



M(t 





Whence, 



V = 



r 



[C„ 



CO)] dt = C„, 



Cit)] dt 



(■5) 



J [Cmax — C(0] dt is the area between the line C,nax 

 and the curve C{t), that is, the area above the con- 

 centration-time curve for constant-injection up to 

 the line C„,ax extended back to zero time. 



Relationship Between Equations for 

 Sudden- and Constant-Injection 



Equation 10 stated that for the case of sudden- 

 injection, volume = flow X t. Equation 15, de- 

 veloped for the case of constant-injection, states that 



volume = flow X (l/C,„.ax) J?" [C^ax - C(t)] dt. 



Since we are dealing with the same volume and flow 

 whether indicator is injected suddenly or constantly, 

 it must be true that (i/Cn,ax) /" [Cmax — C(t)] dt 

 is also the mean transit time, but the identity is not 

 obvious. To prove it we proceed as follows. 



We introduce the function II{t), which is the 

 integral of hit), the distrilnition function. 



Hit) = f his) ds 

 Jo 



(16) 



The concentration of indicator within the svstem 



We must now relate the cumulative distribution 

 function, //(/), to the observed concentration at 

 outflow, C{t). 



When indicator is introduced at the inflow at 

 constant rate, /, the concentration of indicator at 

 outflow, Cit), is determined by / and F and by the 

 frequency function of traversal times, liit). Consider 

 the contribution to the rate at which indicator leaves 

 the .system at time t made by indicator introduced 

 into the system during the time interval occurring 

 between .v and s -\- ds time units before /. The amount 

 of indicator introduced during this time interval is 

 Ids. 



Recall that hit) was defined in equation 6 as the 

 fraction of injected indicator (for sudden-injection) 

 leaving the system per unit lime at time /. But, be- 



