602 



HANDBOOK OF PHYSIOLOGY 



CIRCULATION I 



Substituting for /(O in equation 24, 



C(/) = '- f [I + a it - b)] his) ds 



I a c" ab 



= - Hit) + - t his) ds - — Hit) (44) 



I a 1 a b I I ■ i't - b) 

 U^^Cit) = - + y-- = -+—^^-- 



It has been verified experimentally that indicator 

 recirculating concentration in plasma to the forearm 

 increases linearly (2). Therefore, to correct for re- 

 circulation in this system it is necessary to measure 

 the recirculating concentration only sufficiently to 

 establish the slope and intercept of the recirculating 

 concentration. 



Summary of Treatment of Recirculation 



If recirculation is a late event so that, for the case of 

 sudden-injection, indicator concentration has almost 

 returned to zero before the effect of recirculation is 

 evident, or, for the case of constant-injection, indicator 

 concentration has almost reached plateau, then it is 

 far simpler to make some arbitrary extrapolation of 

 the primary concentration curve. It is obvious that 

 under these conditions no great error will be in- 

 troduced by any reasonable extrapolation. In most 

 cases the exponential extrapolation of Hamilton 

 et al. (10) is quite satisfactory. 



However, if recirculation occurs early so that the 

 downlimb of the concentration curve following 

 sudden-injection is obscured, then no correction for 

 recirculation can be made with confidence. Under 

 these circumstances it is necessary to use one of the 

 other methods which demand measurement of con- 

 centration at two sites, such as the procedure pro- 

 posed by Stephenson (30). These methods, although 

 complex, have the important advantage that they 

 make no assumptions about the form of the distribu- 

 tion function. 



EFFECT OF INJECTION WHICH IS NEITHER 

 SUDDEN NOR CONSTANT 



Sudden-Injection Which Is Sot Truly Instantaneous 



For some years it was customary to regard sudden- 

 injection of indicator as requiring no time. That is, 

 the injectate was delivered truly instantaneously 

 into the vascular s\stem where it was dissolved at 



once in an element of volume, dV, so that its con- 

 centration was q/d]'. As dV is permitted to become 

 as small as we please, the input concentration, q/dV, 

 obviously goes to infinity. Howe\er, a plot of input 

 concentration vs. time required for input must always 

 have a finite area, for if /;,(/) is the frequency distri- 

 bution of injection times, then Ci{t) = q/'dV = 

 q h,{t)iF, from equation 6, and /"<:,(<) dt = q/F, 

 which is finite. Distribution functions of this form 

 are essentially spikes and are known formally as 

 delta functions. 



Obviously, in practice it is impossible literally to 

 inject instantaneously. If the mean transit time 

 through the system is long compared to the mean 

 time of injection, it is usually sufficiently accurate to 

 ignore the fact that the injection is not really in- 

 stantaneous. However, if the mean time of the distri- 

 bution of injection is significant compared to the 

 mean transit time through the system, then equation 

 6, c{t) = q h{t)/F, does not hold and the calculation 

 of J and, therefore, volume will be erroneous. 



Let hi{t) be the distribution of injection with 

 time, where q hi(t) dt is the fraction of injectate, q, 

 introduced into the system between time / and time 

 t and dt. Then q h,{t) is the function lit) of the general 

 equation 24, and the concentration at outflow is 



Cit) 





h, it - s) his) ds 



(45) 



The calculation of flow from the area of the out- 

 flow concentration-time curve will still be correct. 

 The rate at which indicator leaves the system is 

 F C(t). The total amount of indicator which leaves 

 the system is FjoC{t) dt, which must equal the 

 amount introduced, so that the equation F = 

 q/joC{t) dt still applies. 



However the mean time, calculated from 

 jot C(t) dt/fo C(t) dt, will be larger than the true mean 

 transit time, jot h{t) dt. Advantage is now taken of an 

 important property of frequency functions. 



Given the frequency function, /i(0. which is the 

 result of the convolution of one frequency function, 

 /2(0 on another frequency function, fs(t), or 



MO 



-/■ 



Jo 



fz (t - s) Ms) ds 



then the mean lime h of the con\olution fi(t) is the 

 sum of the mean time /■. of the function /■> and the 

 mean time /j of /j. 



The important relation is illustrated in figure 10 

 ijy the use of histograms representing distribution 



