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HANDBOOK OF PHYSIOLOGY 



CIRCULATION I 



PIG. 9. Histogram of indicator concentration versus time in 

 a recirculating system. Sudden-injection of indicator at origin. 

 First four columns represent initial circulation and define dis- 

 tribution of transit times through the system. .\s indicator re- 

 injects itself into the system each slug is distributed according 

 to the frequency pattern illustrated during the first circulation 

 Distribution of successive reinjected slugs is marked by alternate 

 white and shaded patterns. [From Zierler (38).] 



constant-injection, is thus a special case of equation 24 

 in which i(t) = o for /; < o, and i(t) = /, a constant, 

 for t ^ o. 



If a quantity of indicator is injected into a closed 

 (the primary) system (single input, single output) to 

 which it returns repetitively by way of some other 

 (the recirculating) system, we may analyze the con- 

 centration at outflow from the primary systein as 

 follows : 



There is a distribution of transit times through the 

 primary system, described by the function /i{t). There 

 is also a distribution of transit times through the 

 recirculating system (from output of primary system 

 back to input of primary system) which we will 

 describe by the function /(/). There is also a distribu- 

 tion of transit times through the system as a whole, 

 from input of the primary system, through output and 

 back to input, which we will describe bv the function 



git)- 



For the case of sudden-injection, in which a 

 quantity, q, of indicator is introduced into the system 

 at its input, the concentration of indicator at output, 

 c„(t), from equation 6 is 



c„(0 = q h{t)/F 



If the concentration of indicator is measured at the 

 input to the primary system, C,(/), there will be no 

 indicator for / < o, there will be a concentration 

 C,(o), at time / = o when indicator is introduced 



suddenly into the input, and then there will reappear 

 at the input an amount of indicator per unit time 

 described by the distribution of transit times for the 

 total system, q g{t), at concentration q g(t)/F. The 

 concentration of indicator at the input, considering 

 now only the initial condition and the first recircula- 

 tion, is 



where 



CM = qgit)/F 



qg(o)/F= C.(o) 



(25) 



The rate at which indicator enters the primary sys- 

 tem is F C.(0 = qg(t). 



The concentration of indicator at the output from 

 the primary system is expressed by the action of the 

 distribution /;(/) on the input F C,(t). Substituting 

 F Ci{t) for the input function /(/) in equation 24, 

 the concentration of indicator at output from the 

 priinary system is 



C„(0 





d {t - s) his) ds 



(26) 



Equation 26 is valid for all t and for any number 

 of repeated circulations, because every C,(0, when- 

 ever it appears, is redistributed according to /;(/). 

 Equation 25 describes C,(/) only for the first circula- 

 tion. During the second circulation, Ci{t) is distributed 

 once more in accordance with the function ^(0, 

 so that when indicator returns a second time to the 

 primary input, its concentration is 



:i(0 = f , C, 

 ■^0 3 



(( - s) g(s) ds 



(27) 



With each subsequent recirculation, C,(0 will be 

 redistributed through g{l). With each recirculation, 

 indicator particles entering the primary system for, 

 say, the nth time are dispersed more and more with 

 respect to time (fig. 9), so that as / grows large, C,(/) 

 tends to approach the average concentration in the 

 system, which of course is the mass of indicator, q, 

 divided by the total volume of the system, Vt . There- 

 fore, 



lim CM 



=r 



C, (t - s) g(s) ds = q/Vr 



(28) 



Eciuation 28 provides a basis for measurement of 

 volume of certain body fluid compartments. We are 

 not concerned with that problem in this chapter but 

 with llic use of the asymptotic behavior of C,(/), 

 to which we shall return shortlv. 



