6o4 



HANDBOOK OF PHYSIOLOGY 



CIRCULATION I 



theless, the mean transit time at output, /„ = (5 X 

 0.125) + (6 X 0.3) + (7 X 0.3125) + (8 X 0.1875) 

 + (9 X 0.0625) + ('" X 0.0125) = 6.8, is identical 

 with that in figure loB, and tlie true mean transit 

 / = 4 — ^, = 6.3 is estimated correctly. 



Therefore, the mean transit time of fluid particles 

 through the system, / = J"< h{t) dl, which is the 

 datum desired, can always be calculated as the 

 difTerence between the observed time of indicator 

 particles, /" t C(t) dt/jo C(t) dt, and the mean time 

 of the injection process. The latter may be determined 

 experimentally, for example, by injecting into a 

 system in which /;(/) is known. However, if the 

 injection is very rapid, it may be sufficiently ac- 

 curate to consider that it is a square wave. 



What may seem to be an excessive amount of 

 space has been devoted to an explanation of the 

 fact that mean times are additive in order to make 

 the principle clear, because uses of the principle 

 recur in other practical aspects of the problem, 

 particularly with those concerning the effects of 

 collecting systems which will be considered later. 



Injection at Constant Acceleration 



There may be certain advantages in special situ- 

 ations in making the injection follow certain known 

 functions. For example, the following interesting 

 property occurs when injection is delivered at constant 

 acceleration (38). Let the rate of injection ;'(/) = o 

 for / < o and i{t) = a-t, where a is constant for 

 t ^ o. 



Then, 



f) = - / (/ - s) fl(,s) ds 

 a t C' a r' 



s Ilis) ds 



(46) 



lim C(() = - (/ - t) 

 (-.00 l" 



The concentration of indicator at inflow is a t/F. 

 Therefore, the difference between the concentration 

 at inflow and concentration at outflow, after initial 

 transients, is exactly a- t/F. Then if F is measured 

 independently, t can be measured at once and more 

 simply than by any of the other methods. 



EFFECT OF COLLECTING CATHETER 



If blood is led from the system under study through 

 a device such as a catheter, the distriliution of transit 



times through the system, h{t), is convoluted upon 

 the distribution of transit times through the catheter, 

 //„(/). If the sudden-injection technique is used and 

 injection is a delta function, then the observed con- 

 centration at outflow from the catheter is 



CU) 





/i U - s) h„{s) ds 



This is identical to the case in equation 44, in which 

 an inflow distribution is convoluted through the 

 function h{t) because the sequence in which the 

 convolution occurs makes no difference. That is 

 Jo/ it - s) Ms) ds = i'ofi {t - s) Ms) ds. There- 

 fore, all the arguments used in the previous section, 

 with reference to convolution of injection upon the 

 distribution function of the system, apply equally 

 to convolution of the distribution through the catheter 

 upon the distribution h{t). Again, the mean times are 

 additive. The true mean transit time through the 

 system (excluding the catheter) is the difference 

 between the observed mean time, jo t C{t) dt/ 

 jo C(t) dt, and the mean time through the catheter. 

 The latter is easily obtained, since it is simply the 

 volume of the catheter divided by the flow through 

 the catheter. 



As was argued in the previous section, flow through 

 the system is measured correctly by the relation 



F = q/j^ C(t) dt. 



Several investigators have examined the problem 

 of the effect of catheter sampling on the shape of 

 the indicator concentration-time curve. These studies 

 will be considered in detail in the next section. It is 

 important to note, however, that for the calculation 

 of flow we are not concerned with the shape of the 

 outflow concentration curve, only with its area, and 

 that for the calculation of volume we need to de- 

 termine only the mean time of the outflow curve 

 (which we can do by numerical calculation) and 

 the mean time of the catheter. The shape of the 

 outflow curve is immaterial. 



FORMAL EXPRESSIONS FOR THE DISTRIBUTION FUNCTION 



In all the previous sections we have not cared what 

 form the distribution function of transit times, /;(/), 

 might take. It has been simplest and completely 

 accurate to let the flow system (e.g., a vascular bed) 

 determine the function for us by delivering an in- 

 dicator at a concentration which is measurable 

 experimentally as a function of time. 



However, it is a legitimate object of scientific 

 curiosity to ask whether /((/) might be constrained 



