594 



HANDBOOK OF PHYSIOLOGY 



CIRCULATION I 



and 



dCit) 

 dt 



c{t) 



(23) 



Equations 22 and 23 emphasize tliat tlie concentra- 

 tion at outflow during constant-injection, C{t), is a 

 constant, I/q, multiplied by the concentration at 

 outflow following sudden-injection, c{t). This means 

 that the time course of the concentration curve fol- 

 lowing sudden-injection coincides with the time 

 course of the transient or rising phase of the concen- 

 tration curve during constant-injection. When no 

 indicator appears at the outflow site, both concen- 

 trations are zero. The instant some indicator appears 

 so that c{t) just assumes a non-zero value, C{t) also 

 just assumes a non-zero value, that is, appearance 

 time is the same for both. When c{t) reaches its 

 maximum value, i.e., when its first derivative is zero, 

 C{t) will simultaneously have a flex point. When 

 c{t) returns to zero, d C(t)/dt becomes zero and C{t) 

 reaches its maximum, or plateau, value, Cmax • 

 Irregularities in one curve, say c{t), will appear as 

 simultaneous irregularities in the other. For example, 

 when recirculation is apparent in c{t) it will also be 

 apparent in C{t). Manipulation of one curve, for 

 example, in an effort to exclude the effect of recircu- 

 lation, is as simple or as difficult for the other curve. 

 The choice between sudden- and constant-injection 

 techniques, therefore, lies not in the formal treat- 

 ment of the data but in the individual experiment. 



EFFECTS OF VIOLATION OF THE ASSUMPTIONS: 

 RELATION OF THE MODEL TO REAL 

 VASCULAR SYSTEMS 



The System Is not Closed 



The Stipulation, from whicli the equations were 

 derived, that the system be closed, that is, that it have 

 a single input and single output, may not often be 

 met in real vascular systems. Let us return to con- 

 sideration of the vascular net illustrated in figure 3. 

 A single artery, P, divides into a number of branches, 

 one of which is P'. There is also a confluence of venous 

 channels, one of which is ()'. Ultimately all venous 

 effluent passes through Q,- We have so far examined 

 only the case of injection at P and sampling at Q,- 



Supposing there are many inputs and but one out- 

 put. This is the case of injection at P' and sampling 

 at Q. Because all the indicator must exit at Q,, the 

 equations developed for measurement of flow are still 



valid. However, the volume measured by the closed- 

 system equations includes some portion of other input 

 channels. From the argument leading to equation 10, 

 V = Ft, it follows that one is required to find sites 

 P", P'", and so on along all input channels such that 

 the mean transit time from P" to Q,, from P'" to Q,, 

 and so on, is exactly the same as that observed from 

 P' to Q. The volume of the system defined by F t 

 therefore begins at P', P", etc. The proof is straight- 

 forward. 



Let the flow past site P' be /i , past P" be /i and 

 so on. Then there is a volume, beginning at P', Vi = 

 /i t, and there is a volume, beginning at P", V2 = 

 fi t, and so on. Summing these volumes to find the 

 total volume. 



V = Vi + V, + ■ ■ ■ + V„ = /rt + f2i + ■ ■ ■ + f^i 



Ft 



Now consider the case of a single input and many 

 outputs. In figure 3 this is illustrated by injection at 

 P and sampling at Q^'. The equations developed for 

 flow again hold and equation 10 is also valid. The 

 volume includes a portion of each output channel up 

 to that site at which the mean transit time is the same 

 as that determined for the sampling site actually 

 used. 



In the cases P —* (l, P' —> (I, and P — > Q,', the total 

 flow passes either entrance or exit sampling site or 

 both. Because indicator must in any of these cases 

 mi.x ultimately with the total flow, the total flow can 

 be determined. For constant injection, the limiting 

 concentration Cmax ^vill always be I/F, no matter 

 where the mixing of / and F occurs, and so flow can 

 be measured. This is true also for the case of injection 

 at P' and sampling at Q,' (many inputs and many 

 outputs) providing there is at least one channel inside 

 the system through which all flow must pass and in 

 which mixing occurs. This is the case in systems which 

 include the normal heart, providing it can be shown 

 that mixing of indicator and all blood is complete. 



However, it is not essential that all the fluid pass 

 through a single channel. If fluid from each input 

 channel mixes with that from every other input 

 channel before it leaves the system, it may still be 

 possible to measure flow. Consider the simple inter- 

 communicating system in figure 8. 



P, P', R, Q_', and Q, are vascular channels in which 

 the direction of flow is indicated by the arrows within 

 channels. Symbols within the channels represent the 

 fraction of flow within each channel. All the flow, 

 F, passes through P. A fraction a of F passes through 

 Pi and the remainder, (i — a) F, passes through 

 P/. Of the fraction a, a fraction h passes through R< 



