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HANDBOOK OF PHYSIOLOGY 



CIRCULATION I 



returned to the first chamber or may have advanced 

 to the third chamber, or beyond. Now if we let fluid 

 flow through the system in the direction from the 

 first to the n-th. chamber, the probability of indicator 

 particles moving in the direction of fluid flow, rather 

 than in anv other direction, is increased, that is, 

 the random walk of indicator particles tends to be in 

 one direction or is one-dimensional. If an indicator 

 particle must take a large number of jumps, or make 

 many trials, in order to score a success, that is, in 

 order to advance to the next chamber, the prob- 

 ability of success increases uniformly with the number 

 of trials according to the Poisson distribution, and, 

 since each jump requires time, the probability that a 

 given indicator particle will escape from one chamber 

 to the next, and so on through the series to the out- 

 flow from the n-th chamber, is a function of time. 

 We can therefore consider transit time as a substitute 

 for the required number of successes. 



If now we let the volume of each chamber, dV, 

 approach zero and the number of chambers approach 

 infinity, the Poisson distribution does not apply 

 because the number of jumps required to move from 

 one chamber to the next becomes small. Sheppard 

 (27) predicts a Gaussian distribution for the number 

 of successes from one element of volume to the next. 

 The Gaussian input to the second element of volume 

 must pass through a Gaussian distribution in order 

 to escape to the third element of volume, and so on. 

 The probability of advancing from one element of 

 volume to the next is still increased by the fact that 

 this is the direction in which the mother fluid flows. 

 In this way the distribution function becomes skew 

 and the standard deviation of transit times spreads 

 out. The standard deviation of transit times is a 

 characteristic of the vascular bed, related to the 

 distribution of flow through the volume. Sheppard 

 introduces a randomizing constant, K, which is 

 \/2 X standard deviation of transit times, as a 

 measure of this characteristic distribution, and the 

 dimensionless variable, r, which is the fraction of the 

 volume displaced by the flow from time zero to 

 some time between t and t + dt. That is, r = F t/ V = 

 l/i, or T is transit time as a fraction of mean transit 

 time. (Note that this is an equivalent definition of 

 mean transit time. When t = t, the volume displaced 

 by the flow is exactly the volume of the system.) 

 The indicator-distribution model which Sheppard 

 uses is obviously related closely to that used by Ein- 

 stein (8) for study of Brownian movement, and the 

 solution, provided by Einstein in 1 905 is 



The exponent of « contains the factor (i — r), which 

 is (i — t)/l, that is, it is a measure of the difference 

 between the time required by those particles which 

 take some time between / and t -\- dt to traverse the 

 system and the mean transit time. 



Equation 62 contains two constants. A' and V, 

 the latter being implicit in the variable r = F t/V, 

 where it is assumed that F can be measured correctly 

 in the usual way from the area under the curve. 

 The job of examining the closeness with which equa- 

 tion 62 describes an indicator-dilution curve boils 

 down to empirical selection of values for A' and V 

 by curve-fitting. Despite the advantage of having 

 not one but two constants to adjust, although some 

 indicator-dilution curves could be well matched, in 

 general Sheppard had to select values for ]' which 

 were smaller than the known values and he had to 

 displace the tiine axis to an arbitrary zero. Sheppard 

 concludes that in real vascular beds the probability 

 of successes is not constant as indicator moves along 

 the system from one type of vessel to another o 

 different diameter, and so on, whereas equation 62 

 depends on the assumption of constant probability 

 of success. 



Sheppard's distribution function can of course be 

 con\'oluted upon others, a parabolic flow distribution, 

 for example, which he has done successfully for 

 certain models. 



Other known probability functions might be 

 tested. Sheppard (25) originally pointed out that 

 sudden-injection indicator-dilution curves resemble 

 the normal error function plotted on a logarithmic 

 time scale. Stow & Hetzel (36) pursued this suggestion 

 and, by analogy with the error function, ofTered the 

 following expression for concentration of indicator: 



C{t) = Cpe-'''"-'- 



(63) 



«r) = - 



K V" 



(62) 



where Cp is concentration at the peak of the observed 

 curve, r = (t — a)/(p — a), where a is appearance 

 time and p is time at which Cp occurs, and k is a 

 constant for which a value is obtained by plugging 

 in an observed value for C at any convenient /. 

 They reported reasonably good agreement between 

 observed curves (through the central circulation in 

 man) and those predicted from equation 63. The 

 downlimb of equation 63 falls more rapidly than 

 Hamilton's simple exponential, so that the area under 

 the curve extrapolated by equation 63 is less than 

 that imder the curve extrapolated by Hamilton's 

 method, and the flow estimated by the former is 

 therefore greater. 



It is difficult to accept any of these extrapolations. 

 If a method of extrapolation yields a good fit with 



