6l2 



HANDBOOK OF PHYSIOLOGY 



CIRCULATION 



Thus, the notions that k = F/V and that the 

 initial concentration is q/V are incompatible with a 

 lag exponential. 



One can consider the effect of connecting mixing 

 volumes in series and in parallel. The argument is 

 the same as that advanced earlier for parabolic flow 

 systems. In parallel circuits, output concentrations 

 are simply additive. As in the case of parallel parabolic 

 flow systems (see fig. ii), the final curve will reflect 

 the distribution of mean transit times among the 

 members of the parallel system, and the downlimb 

 after sufficiently long time will assume the slope of 

 that container through which mean transit time is 

 longest, although this might not occur until quite late. 

 The equation describing concentration is the sum of 

 n exponentials, each of the form of equation 66, 

 where there are n parallel containers. 



Newman el al. (21) have considered the effect of 

 series connnections of instantaneous mixing containers 

 and proposed that the downslope of the concentration- 

 time curve at outflow is determined by that container 

 for which k is least, and, since they equate k to F/]', 

 for which I' is greatest. 



We examine this proposal by applying the con- 

 volution equation 24, C{t) = i/Fj'(,i{t — s) k(s) ds. 

 Output from the first container, of volume I'l , is, 



from equation 66, F Ci(t) = F Ca^ 



-ki ((-.i|) 



, and 



this is the input, ;(/), to the second container. The 

 subscript, i, refers to properties of the first container. 

 The distribution function, /((/), is obtained from the 

 relation, equation 6, h{t) = F c{t)/q. Substituting 

 for (it) in equation 66, the distribution function, 

 Ihii), through the second container in the series is 

 h-iit) = {F Ca„/q)e~ "' ~°' . Concentration of in- 

 dicator at outflow from the second container is there- 

 fore 



c{t) 



'if 



F Cai e-" 



<:i«- 



FCa. 



■ e-'^at^-oj' ^^ 



whereCa, = q/{Vi — oi F) and C, = 9/(^2 — a-2 F). 

 Integration yields 



F C„ r.„ /•*S ''2+*:i a I 

 q (ki — ki) 



(67) 



') 



Equation 67 states that, for two instantaneous- 

 mixing containers connected in scries, indicator 

 concentration at outflow is proportional to the 

 difference between two exponentials as a function of 

 time. The curve has an appearance time f«i -|- «•_.), 



rises to a maximum at time < = ai -|- (32 + 1/(^:2 — ^1) 

 In ki/ki , if ^2 5^ ^1 , and falls with time, ultimately at 

 a rate determined largely by the smaller of the 

 values A'l and /.o . 



Equation 67 differs from that given by Newman 

 et al. (21) (their equation 9), but can be reduced to 

 the form which they offer if ki is set to equal F/ Vi , k-^ 

 is set to equal F/V2 , and Ca^ and C,,, are made equal 

 to q over the respective values of t', and a-2 equals 

 zero, that is, they permitted the indicator to take a 

 finite time to traverse the first container but not the 

 second. Again, ki will equal F/l\ and ki will equal 

 F/V2 only if fli and a^ are zero. In the physical models 

 which they studied, appearance time was small 

 compared to mean time, so that the approximation 

 which they used gave a reasonably good agreement 

 with the observed curve. When appearance times 

 are negligible and when ki « ^2 , then the downslope 

 is approximately described by Aj = F/Vj , where 

 I'l is the larger volume, or by k-2 = F/V2 , where 

 ki ^ k^ and Vo is the larger volume. 



Newman and his colleagues have used this method 

 for estimating volume through which indicator is 

 injected following its sudden-injection into cardio- 

 pulmonary or central circulatory system. The volume 

 estimated by the slope method, I' = k/F, is less than 

 that estimated by the conventional method, V = F t. 

 Nor is it correct to say that this is so because they 

 are measuring only the largest volume in a series of 

 instantaneous-mixing chambers. Because appearance 

 time is a large fraction of mean transit time their 

 assumption that ki = F/l'i is apt to be erroneous. 

 Finally, if any parts of the cardiovascular system 

 behave like instantaneous mixing chambers, they 

 ought to be chambers of the heart, distribution 

 through the pulmonary vascular bed can hardly fit 

 the simple model of a single continuously stirred 

 chamber. 



We may ask what the indicator concentration-time 

 curve looks like when an instantaneous mixing 

 chamber is placed in series with an unspecified 

 distribution function, /((/). 



C(() = ^ /" e-'' ('-°-») his) ds 



e'" his) ds 



Obviously, although the shape of the concentration- 

 time curve cannot be predicted without knowledge of 

 /)(/), it would be surprising if after sufficiently long 

 time tlie downlimb did not resemble an exponential 



