CIRCULATION TIMES AND THEORY OF INDICATOR-DILUTION METHODS 



607 



Equation 53 must be restricted so that c{t) = o 

 for t less than the appearance time. Tlie appearance 

 time is the time required by tlie fastest particles, 

 those in the central stream with velocity Wmax = 

 2 F/tB^, and is, therefore, L/u„,^^ = VI 2 F = }^ I. 



Equation 53 also states that in parabolic flow the 

 frequency function of transit times, h(t), is I/2 f, 

 since we have defined h{t) = F c(t)/q. 



Equation 53 does not in fact resemble the con- 

 centration-time curve obtained with sudden-injection 

 of indicator into a vascular bed. Unlike the experi- 

 mental curve, which rises to a maximum and falls 

 to zero, equation 53 describes a function which is 

 monotone decreasing. Its greatest value is its initial 

 value, occurring at appearance time 3-2 ^> ^nd this 

 maximum concentration is 2 q/V, as though all the 

 indicator were suddenly mixed in half the volume 

 of the cylinder. This is true because the assumptions 

 were implicitly equivalent to stating that indicator 

 was mixed in the volume of the contained paraboloid 

 of revolution which is half the volume of the cylinder. 



A derivation similar to Sheppard's has been made 

 by Sherman et al. (29). However, instead of a.ssuming 

 that the velocity of the outermost sleeve was zero, 

 they assumed that it had some positive value, u„ . 

 The derivation, which need not be repoduced here, 

 leads to 



c{l) = q 111 P {F - ttR^ Uo) 



(54) 



which is identical with equation 53 when ;/<, = o, 

 and which applies only for I/2 ^ t < L/uo . For 

 L/uo < t and for t < t/2, c{t) = o. 



Like equation 53, equation 54 describes a monotone 

 decreasing function and, for the same reason, does 

 not resemble a real indicator-dilution curve. 



The unrealistic nature of equation 53 follows from 

 the fact that the initial injectate was assumed to 

 have been introduced at inflow into a volume tt R- dx 

 in which dx differed from zero by less than any 

 arbitrarily small number, that is, the injectate had 

 area but essentially no thickness. It was therefore 

 distributed only as a thin skin forming the paraboloid 

 of revolution. 



Several attempts have been made to produce 

 equations through parabolic flow systeins which 

 more nearly resemble real indicator-dilution curves. 

 In general, the outcome depends on the exact nature 

 of arbitrary boundary conditions which have been 

 set by letting the injection bolus occupy a finite 

 volume at inflow at zero time, for example, in the 

 several solutions offered by Rossi et al. (24). 



We will not follow their derivation but instead will 



treat the problem in a manner consistent with the 

 general development used in this chapter, based on 

 the fact that indicator concentration at outflow is 

 the integral of convolutions. Recall equation 24, 



C(t) 





I (t — s) h{s) ds 



To illustrate the use of this equation for the case of 

 parabolic flow, take the case in which indicator is 

 introduced as a square wave, that is, the rate of in- 

 jection, in units of mass/time, is constant, /, for a 

 definite time, T. We have already established by 

 equation 53 that h{t) through a parabolic flow system 

 is J/2 <', and that the appearance time is ^2 t. 



W^e can therefore set boundary conditions: 



i it - s) 



h{s) 



(55) 



(56) 



These boundaries, applied to the convolution 

 integral, lead immediately to the expressions of con- 

 centration: 



C{t) = 0,1 < \i t 



C{i) 



I ds, 



■2 S- 



< t < 



e-O 



Cit) 



P Jin 



(57) 



(58) 



/ Ti 



(59) 



iF t {t - T) 



Equations 57, 58, and 59 describe concentration 

 rising from zero to a peak (equation 58) and falling 

 from that peak toward zero (equation 59). They are 

 identical with one of the sets of equations developed 

 by Rossi et al. (24). 



If T is extended to /, equation 58 describes con- 

 centration during constant injection of indicator 

 through a parabolic flow system and is asymptotic to 

 I/F, as it must be if it is to be consistent with our 

 general treatment of constant injection. 



If T is decreased toward zero, the limits of equa- 

 tion 58 approach one another so that, for T = o. 



