CIRCULATION TIMES AND THEORY OF INDICATOR-DILUTION METHODS 



609 



0.24 



0.20 



A+B + C 



008- 



0.04- 



Therefore, 



C(0 = 



9 ■ r 

 TIME 



FIG. 1 1 . Histogram of distribution of concentration of indicator as function of time through three 

 branches, ^-1, B, and C and through common output, A + B + C, of parallel branch system. Fre- 

 quency function through A, B, and C is that of parabolic flow system. Mean transit time shortest 

 through A, longest through B. Areas drawn so that '^ of total flow goes through .-1, '2 through B, 

 I4 through C. .Appearance time and time at which peak concentration occurs coincide for each of 

 the three branches, but not for the summated output, because the branches do not all have the same 

 appearance time. The ordinate for A, B, and C is not the concentration within the individual channe. 

 but is the output from the individual channel divided by the total flow through all three channelsl 



that of a system through which there was laminar 

 flow, rather tlian with that of a system in which 

 there was a single well-stirred pool. 



(61) 



a h u r I I 



^-^ / ■ - ■ ds t > 



h + t-i 



where subscripts i and 2 refer, respectively, to tube 

 entered first and second. The integral in equation 61 

 is deceptively simple. It cannot be integrated over the 

 limits set and its Laplace transform cannot be in- 

 verted readily. It can, however, be computed, and 

 it appears to assume at least a quasi-exponential 

 form. 



A system of dichotomous branching or of Y-tubes 

 is simply a combination of series and parallel tubes, 

 the solution of which is obtained by combining 

 equations 60 and 61. The downlimb of the resulting 

 concentration curve will, after a sufficiently long 

 time, assume the characteristics of the downlimb of 

 equation 61. 



Parrish et al. (22), by an analog computer technique 

 to which reference was made in a previous section, 

 obtained the admittance function of the pulmonary 

 vascular bed in dogs and found it compatible with 



Random ]\'alk and Other Prohahi/ilv Functions 



On the grounds that real vascular beds are ran- 

 domizing nets, too complicated for assessment of 

 the contribution of each of the factors tending to 

 distribute indicator, Sheppard (26-28) has in- 

 troduced and discussed critically the application of 

 several probability functions. 



One of the most interesting of these is the one- 

 dimensional random walk, although it has failed to 

 describe indicator dilution curves in several im- 

 portant details. The one-dimensional random walk 

 is constructed as follows. 



Consider a number, n, of chambers of equal size 

 and shape, connected in series by tubes of negligible 

 volume. Place indicator in the first chamber. By 

 some means or other, the exact nature of which need 

 not concern us, after a number of discrete jumps in 

 random directions, an indicator particle will find 

 its way into the second chamber. After another 

 series of jumps the indicator particle may have 



