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HANDBOOK OF PHYSIOLOGY -^ CIRCULATION I 



expression for concentration of indicator at outflow 

 from time zero is therefore 



Whence, 



la 



f 



CAD = '-^ h(t) + I Ca (I - r) g(r) dr (35) 



F Jo 



Now let there be introduced into the outflow site a 

 quantity, qi, , of indicator so that the concentration 

 of indicator at outflow at zero time is Ch{o). When 

 this indicator reappears at the outflow site for the 

 first time it will have been distributed through the 

 entire system so that its concentration will be 

 Ci,{t) = Ci,(o) g{t) dt. This distribution will in turn 

 be redistributed in accordance with the function, g{t), 

 and the complete expression for concentration of 

 the second indicator at outflow is 



a(/) = Ctio) g(t) dt + C„U - r) g(r) dr (36) 



Equations 35 and 36 yield the measured outflow 

 concentrations of the two indicators as a function of 

 two unknown distributions, h{t), through the primary 

 system, and g(t) through the entire system. The 

 possibility of calculating /;(/) is therefore a real one. 



The integrals in equations 24, 26, 27, 29, 35, and 

 36 are of the class called convolution integrals. They 

 describe the fact that one function is made to pass 

 through or is convoluted by a second function, a 

 fact pointed out by Stephenson (30), Sheppard (27), 

 and Zierler (38). Providing the functions meet cer- 

 tain requirements they can be operated upon so as to 

 yield more tractable equations. The following are 

 sufficient conditions for the transform. 



The functions must be .sectionally continuous on a 

 finite interval. Frequency distributions of transit 

 time, in practice, are continuous. The functions must 

 be of exponential order as the variable t approaches 

 infinity. This means that if the function is h{t) the 

 product e^"' \ hit) \ is less than some constant. A, 

 for all / greater than some finite number T. That is, 

 I /;(/) I does not grow more rapidly than A e" as 

 / — > 00, where a is a constant. Since the distribution 

 function, such as h{l), always lies between zero and 

 one, it is satisfactory for transformation. 



We introduce the function y{s), the Laplace trans- 

 form of the function h{l), which we have shown to be 

 suitable for Laplace transformation, where y{s) = 

 /" e~'^ h{t) dt. The advantage of the Laplace trans- 

 formation is that the integral of the convolution of 

 two functions becomes the product of the Laplace 

 transforms of the two functions. Therefore, equation 

 35 is transformed to 



ka[s) = qay{s)/F + kjs) z{s) 



Lis) = 



q„ y(s)/F 



I - Z(s) 



(37) 



where k„{s) is the Laplace transform of C„(/) and c(-f) 

 is the Laplace transform of ^(/). 



Since o g z(s) < i, equation 37 may undergo 

 binomial expansion to yield 



kaU) = ^^"^ [. -I- zis) + zHs) + zKs) + ■■■] (38) 



r 



Equation 38 states explicitly, as pointed out by Shep- 

 pard (27), that the concentration of indicator at 

 outflow is, as one expects, the algebraic sum of all 

 convolutions of the outflow concentration through the 

 system in its first, second, third, fourth, and so on, 

 circuit. 



The Laplace transform of equation 36 is 



A-tCO = a(o) dt z(s) + kbis) z(s) 



where kb(s) is the Laplace transform of Ci,{t), and 



Ct(o) dt zis) 



(39) 



k,,is) = 



I - zis) 



= a(o) dt [zis) + zHs) + zKs) + ■■■] 



Combining equations 37 and 39, and solving for 

 y(s) with elimination of c(^), 



yis) = A 



k.is) 



Ctio)dt + ktis) 



(40) 



where A is the constant F Ci,(o) dt/qa = Ci,(o)/Ca(o). 

 The inverse transform of equation 40 is 



hit) = .-!£- 



kais) 



Ci(0) dt + ktis) 



(40 



Thus, //(/), which is tlie desired function, is given 

 as the product of a measurable constant and the 

 inverse transform of a function only of observed con- 

 centrations of indicators. Equation 41 can be solved 

 if the form of the functions is known, although at the 

 moment it is unlikely that a formal expression of the 

 concentration functions is known with sufticient 

 accuracy. However, because the concentration 

 functions are available experimentally, they can be 

 manipulated by computer techniques and a solution 

 for h{t) obtained empirically. A preliminary report 

 states that something along these lines has been done 

 by Cheesman et al. (4). 



A relatively simpler, but clo.sely related, procedure 

 has been used by Parrish et al. (22), based on an 



