CIRCULATION TIMES AND THEORY OF INDICATOR-DILUTION METHODS 



60 I 



application by Paynter (23) to a flood control problem 

 concerned with water flow through drainage basins. 

 The formal treatment rests on the Duhamel super- 

 position theorem. 



Consider a closed system in which any given input 

 is distributed with time such that, for a unit step ( = 

 brief square wave) input, the output is described by 

 the function A(t), called the indicial response or 

 admittance function. By arguments similar to those 

 we have used previously, as in development of the 

 convolution equation 24, if the input has magnitude 

 dx at time s, the output will be A (t — s) dx. Summing 

 for all such stepwise inputs to obtain the input, x{t), 

 yields the output 



y(t) = x,A{l) + f A - s) dxis) 

 Jo 



y(t) = A^x(t) + X it - s) d Ais) 



(42) 



i 



where Xo and Ao refer to initial conditions. 



The relationship between the Duhamel integral 

 and equations developed earlier in this chapter is 

 obvious. 



Equation 42 can be handled numerically Ijy ex- 

 pressing the output as a stepwise function of the 

 input and the admittance fimction. Thus 



y(t) = J2a (t + i - k)-x (k) 



(43) 



where k assumes only integer values for integer values 

 of t. For example, 



y{i) = ^(i)-A-(i) 

 >(2) = A{2)-x(i) + A{i)-x{-2) 

 y{t) = A{t) -xd) + A(t- i)-.v(2) + ■■■ 

 + A{,i)-x{t) 



Therefore, if v and x are determined experimentally, 

 ^(i), A{2), etc. can be calculated, and the admittance 

 function can be described numerically. 



Parrish et al. (22) injected proximally to the input 

 of the system under study (the pulmonary vascular 

 system) and measured both the input and the output 

 concentrations of indicator as a function of time. 

 The concentration-time curve obtained from the input 

 constituted the input to an analog computer. The 

 coefficients of the computer input were then ad- 

 justed until the computer output duplicated the 

 observed concentration of indicator at output from 

 the vascular system. A square wave was then fed 

 through the analog and the admittance function was 



registered. From this distribution function the mean 

 transit time was calculated. 



All the methods described so far for dealing with 

 systems in which recirculation occurs are complex 

 experimentally and analytically. It would be far 

 simpler if the shape of the distribution function were 

 known so that its distortion by recirculating indi- 

 cator could be obliterated by extrapolation of the 

 distribution function beneath the distortion. We shall 

 consider later some of the attempts to restrict the 

 entire distribution function to some formal expression. 

 For the time being we are concerned only with the 

 downslope of the function /;(/). 



Hamilton et al. (10) discovered that, when an 

 indicator was injected suddenly intravenously in 

 the dog and the concentration of indicator measured 

 with time in arterial blood, the downlimb of the 

 concentration sooner or later fitted an exponential 

 of the form C„e~'" until it was evidently interrupted 

 by an increase in concentration attributed to the 

 first recirculation of indicator. This means that a 

 plot of the logarithm of indicator concentration versus 

 linear time yields a reasonably straight line until 

 recirculation appears. When recirculation does ap- 

 pear, the logarithmic downslope is simply extrap- 

 olated to very small concentrations. The extrapo- 

 lated concentration-time curve is then replotted on 

 linear coordinates, and, with recirculation thus elimi- 

 nated, the area and the mean transit time of the 

 primary curve are calculated. 



This has been an extraordinarily useful artifice, 

 although there is no convincing theoretical reason 

 for the downlimb to assimic the form C^f"*'. Indeed, 

 it is a matter of experience that it by no means always 

 does so (7). 



In specific instances it may be possible to find other 

 useful approximations. For example, in the case of 

 measurement of blood flow through the forearm of 

 man by constant-injection of indicator, recirculation, 

 which must pass through the large bulk of the rest 

 of the total body vascular bed, is a late event. Conse- 

 cjuently, indicator, which accumulates in the total 

 plasma volume at rate /, is almost completely dis- 

 tributed at its equilibrium concentration when it 

 returns to the forearm. Therefore, the rate at which 

 indicator re-enters the forearm is approximately 

 F I-(t — b)/f'T, where IV is the total volume in 

 which indicator is distributed and ^ is a correction 

 for the time lag. The rate at which indicator is 

 introduced into the forearm is therefore i{t) = / + 

 a (1 — b), where a (t — b) = o for / < o and 

 a {t — b) = [t — b] a for t ^ o, where a = F //TV • 



